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Real-World Video
ESSENTIAL QUESTION
myhrwcom
How can you use real numbers to solve real-world problems
Real Numbers 1
Get immediate feedback and help as
you work through practice sets
Personal Math Trainer
Interactively explore key concepts to see
how math works
Animated Math
Go digital with your write-in student
edition accessible on any device
Scan with your smart phone to jump directly to the online edition
video tutor and more
Math On the Spot
MODULE
Living creatures can be classified into groups The sea otter belongs to the kingdom Animalia and class Mammalia Numbers can also be classified into groups such as rational numbers and integers
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LESSON 11
Rational and Irrational Numbers
8NS1 8NS2 8EE2
LESSON 12
Sets of Real Numbers8NS1
LESSON 13
Ordering Real Numbers8NS2
Since every rational and irrational number is a real number any real-world problem that can be modeled and solved with rational or irrational numbers can be modeled and solved with real numbers
3
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age C
redit
s copy
Danie
l Her
shm
anG
etty
Imag
es
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8_MCAAESE206984_U1MO01indd 3 220513 109 AM
3 Module 1
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
YOUAre ReadyPersonal
Math Trainer
Online Practice and Helpmyhrwcom
Complete these exercises to review skills you will need for this module
Find the Square of a NumberEXAMPLE Find the square of 2 _ 3
2 _ 3 times 2 _ 3 = 2 timesthinsp2 ____ 3 timesthinsp3
= 4 _ 9
Find the square of each number
1 7 2 21 3 -3 4 4 _ 5
5 27 6 thinsp- 1 _ 4 7 thinsp-57 8 1 2 _ 5
ExponentsEXAMPLE 5 3 = 5 times 5 times 5
thinsp = 25 times 5 thinsp = 125
Simplify each exponential expression
9 9 2 10 2 4 11 ( 1 _ 3 ) 2 12 (-7) 2
13 4 3 14 (-1) 5 15 45 2 16 10 5
Write a Mixed Number as an Improper FractionEXAMPLE 2 2 _ 5 = 2 + 2 _ 5
thinsp = 10 __ 5 + 2 _ 5
thinsp = 12 __ 5
Write each mixed number as an improper fraction
17 3 1 _ 3 18 1 5 _ 8 19 2 3 _ 7 20 5 5 _ 6
Write the mixed number as a sum of a whole number and a fractionWrite the whole number as an equivalent fraction with the same denominator as the fraction in the mixed numberAdd the numerators
Use the base 5 as a factor 3 timesMultiply from left to right
Multiply the number by itself
Simplify
49 441 9
729
81
64 -1
16
2025 100000
49
16 __ 25
1 _ 9
10 __ 3 13 __ 8 17 __ 7 35 __ 6
1 __ 16 1 24 __ 25 or 1963249
Unit 14
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8_MCAAESE206984_U1MO01indd 4 230513 448 PM
Math TrainerOnline Assessment
and Intervention
Personal
myhrwcom
1
2
3 Response toIntervention
Professional Development
PROFESSIONAL DEVELOPMENT VIDEO
Are You ReadyAssess ReadinessUse the assessment on this page to determine if students need intensive or strategic intervention for the modulersquos prerequisite skills
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myhrwcom
Interactive WhiteboardsEngage students with interactive whiteboard-ready lessons and activities
Personal Math Trainer Online Assessment and InterventionAssign automatically graded homework quizzes tests and intervention activitiesPrepare your students with updated practice tests aligned with Common Core
Online Teacher EditionAccess a full suite of teaching resources onlinemdashplan present and manage classes and assignments
ePlannerEasily plan your classes and access all your resources online
Interactive Answers and SolutionsCustomize answer keys to print or display in the classroom Choose to include answers only or full solutions to all lesson exercises
Intervention Enrichment
Access Are You Ready assessment online and receive instant scoring feedback and customized intervention or enrichment
Online and Print Resources
Skills Intervention worksheets
bull Skill 11 Find the Square of a Number
bull Skill 12 Exponents
bull Skill 22 Write a Mixed Number as an Improper Fraction
Differentiated Instruction
bull Challenge worksheets PRE-AP
Extend the Math PRE-AP Lesson Activities in TE
Real-World Video Viewing GuideAfter students have watched the video discuss the following bull What are some different ways that biologists classify animals bull What are some classifications of numbers mentioned in the video natural numbers integers rational numbers
Author Juli Dixon models successful teaching practices as she explores the concept of real numbers in an actual eighth-grade classroom
Real Numbers 4
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
Reading Start-Up
Active ReadingLayered Book Before beginning the lessons in this module create a layered book to help you learn the concepts in this module Label the flaps ldquoRational Numbersrdquo ldquoIrrational Numbersrdquo ldquoSquare Rootsrdquo and ldquoReal Numbersrdquo As you study each lesson write important ideas such as vocabulary models and sample problems under the appropriate flap
VocabularyReview Words integers (enteros) negative numbers
(nuacutemeros negativos)positive numbers
(nuacutemeros positivos)whole number (nuacutemero
entero)
Preview Words cube root (raiz cuacutebica) irrational numbers (nuacutemero
irracional) perfect cube (cubo
perfecto) perfect square (cuadrado
perfecto) principal square root (raiacutez
cuadrada principal) rational number (nuacutemero
racional) real numbers (nuacutemero real) repeating decimal (decimal
perioacutedico) square root (raiacutez cuadrada) terminating decimal
(decimal finito)
Visualize VocabularyUse the words to complete the graphic You can put more than one word in each section of the triangle
Understand VocabularyComplete the sentences using the preview words
1 One of the two equal factors of a number is a
2 A has integers as its square roots
3 The is the nonnegative square root of a number
Integers
0 83 308
1 45 192
-21 -78 -93
square root
perfect square
principal square root
whole numbers
negative numbers
positive numberswhole numbers
5Module 1
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8_MCAAESE206984_U1MO01indd 5 180513 1045 AM
Reading Start-Up Have students complete the activities on this page by working alone or with others
Strategies for English LearnersEach lesson in the TE contains specific strategies to help English Learners of all levels succeedEmerging Students at this level typically progress very quickly learning to use English for immediate needs as well as beginning to understand and use academic vocabulary and other features of academic language Expanding Students at this level are challenged to increase their English skills in more contexts and learn a greater variety of vocabulary and linguistic structures applying their growing language skills in more sophisticated ways appropriate to their age and grade level Bridging Students at this level continue to learn and apply a range of high-level English language skills in a wide variety of contexts includ-ing comprehension and production of highly technical texts
Active ReadingIntegrating Language ArtsStudents can use these reading and note-taking strategies to help them organize and understand new concepts and vocabulary
Additional ResourcesDifferentiated Instruction
bull Reading Strategies EL
EL
After
Students will connect that bull the rational numbers are those with decimal expansions that terminate in 0s or eventually repeat
bull non-rational numbers are called irrational numbers
In this moduleStudents will learn how to bull express a rational number as a decimal bull approximate the value of an irrational number bull describe the relationship between sets of real numbers bull order a set of real numbers arising from mathematical and real-world contexts
Before
Students understand bull write rational numbers as decimals bull describe relationships between sets and subsets of rational numbers
bull compare rational numbers
Tracking Your Learning Progression
Focus | Coherence | Rigor
5 Module 1
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
myhrwcom
What It Means to YouYou will learn to estimate the values of irrational numbers
What It Means to YouYou will recognize a number as rational or irrational by looking at its fraction or decimal form
Estimate the value of radic_
8
8 is between the perfect squares 4 and 9So radic
_ 8 is between radic
_ 4 and radic
_ 9
radic_
8 is between 2 and 3
8 is closer to 9 so radic_
8 is closer to 3 28 2 = 784 29 2 = 841 radic
_ 8 is between 28 and 29
A good estimate for radic_
8 is 285
Classify each number as rational or irrational
0 _
3 = 1 _ 3 025 = 1 _ 4
These numbers are rational because they can be written as ratios of integers or as repeating or terminating decimals
π asymp 3141592654hellip radic_ 5 asymp 2236067977hellip
These numbers are irrational because they cannot be written as ratios of integers or as repeating or terminating decimals
Understanding the standards and the vocabulary terms in the standards will help you know exactly what you are expected to learn in this module
Real NumbersGETTING READY FOR
Know that numbers that are not rational are called irrational Understand informally that every number has a decimal expansion for rational numbers show that the decimal expansion repeats eventually and convert a decimal expansion which repeats eventually into a rational number
Key Vocabularyrational number (nuacutemero
racional) A number that can be expressed as a ratio of two integers
irrational number (nuacutemero irracional)A number that cannot be expressed as a ratio of two integers or as a repeating or terminating decimal
Use rational approximations of irrational numbers to compare the size of irrational numbers locate them approximately on a number line diagram and estimate the value of expressions (eg π2)
EXAMPLE 8NS1
EXAMPLE 8NS2
8NS2
8NS1
Visit myhrwcom to see all CA Common Core Standards explained
8 is not a perfect square Find the two perfect squares closest to 8
Unit 16
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Miff
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8_MCABESE206984_U1MO01indd 6 102913 1123 PM
GETTING READY FOR
Real NumbersUse the examples on the page to help students know exactly what they are expected to learn in this module
myhrwcom
California Common Core Standards Lesson 11
Lesson 12
Lesson 13
8NS1 Know that numbers that are not rational are called irrational Understand informally that every number has a decimal expansion for rational numbers show that the decimal expansion repeats eventually and convert a decimal expansion which repeats eventually into a rational number
8NS2 Use rational approximations of irrational numbers to compare the size of irrational numbers locate them approximately on a number line diagram and estimate the value of expressions (eg π2)
8EE2 Use square root and cube root symbols to represent solutions to equations of the form x 2 = p and x 3 = p where p is a positive rational number Evaluate square roots of small perfect squares and cube roots of small perfect cubes Know that radic
_ 2 is irrational
Go online to see a complete unpacking of the CA Common Core Standards
CA Common Core Standards
Content Areas
The Number Systemmdash8NS
Cluster Know that there are numbers that are not rational and approximate them by rational numbers
Real Numbers 6
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=B
Lesson Support Content Objective Students will learn to rewrite rational numbers and decimals take square roots and
cube roots and approximate irrational numbers
Language Objective Students will show and explain how to rewrite rational numbers and decimals take square roots and cube roots and approximate irrational numbers
LESSON 11 Rational and Irrational Numbers
Building BackgroundEliciting Prior Knowledge Have students work with a partner to review the relationship between fractions and decimals Ask students to provide an example of writing a fraction or mixed number as a decimal and vice versa Discuss how students chose and wrote their examples
Learning ProgressionsIn this lesson students work with positive rational and irrational numbers They make connections among the real numbers by converting fractions and decimals and approximating irrational numbers Important understandings for students include the following
bull Understand that every number has a decimal expansion bull Convert a repeating decimal to a rational number bull Evaluate square roots of perfect squares and cube roots of perfect cubes
bull Estimate an irrational number
Work with the real number system will continue in this unit as students extend the positive rational and irrational numbers to include negative numbers and compare and order real numbers
Cluster ConnectionsThis lesson provides an excellent opportunity to connect ideas in this cluster Know that there are numbers that are not rational and approximate them by rational numbers Tell students ldquoA square garden has an area of 20 square feetrdquo
Have students explain why the side length cannot be rational Then have them approximate the length of each side of the garden to the nearest tenth and hundredth Sample answer The length is the solution to s 2 = 20 radic
_ 20 which is not a rational
number 45 ft 447 ft The length is between 4 and 5 feet 20 is closer to 45 2 than to 44 2 or 46 2 It is also closer to 447 2 than to 446 2 or 448 2
3 _ 4
= 075 1 2 _ 3
= 1 _
6
7 _ 10
= 07 45 = 4 1 _ 2
20 ft 2
California Common Core Standards
8NS1 Know that numbers that are not rational are called irrational Understand informally that every number has a decimal expansion for rational numbers show that the decimal expansion repeats eventually and convert a decimal expansion which repeats eventually into a rational number
8NS2 Use rational approximations of irrational numbers to compare the size of irrational numbers locate them approximately on a number line diagram and estimate the value of expressions (eg π 2 )
8EE2 Use square root and cube root symbols to represent solutions to equations of the form x 2 = p and x 3 = p where p is a positive rational number Evaluate square roots of small perfect squares and cube roots of small perfect cubes Know that radic
_ 2 is irrational
MP6 Attend to precision
7A
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=B
Math Talk
Language Support EL
PROFESSIONAL DEVELOPMENT
Linguistic Support EL
AcademicContent Vocabulary
square ndash In this lesson the word square has multiple meanings which can cause confusion For example to square as in to take the square root of a number is a verb It is different from the nouns square or square of a number The text also refers to perfect square and principal square root of a number and the square root symbol is used These different usages of square as a mathematical term need to be clarified Sentence frames can be used to help define the meaning
To square a number means to _______The perfect square of a number means _______
Background Knowledge
suffixes ndash When added to a root word the suffix -th is used in math to indicate one of a specified number of parts such as tenth hundredth or thousandth Remind students that the suffix -th also indicates place value Note that Spanish Vietnamese Mandarin and other languages do not have the ending th sound so teachers need to enunciate carefully
cognates ndash The words terminating and terminal used in this lesson are cognates in Spanish terminar meaning ldquoto endrdquo or ldquoto finishrdquo A Spanish cognate for approximate is aproximar
Leveled Strategies for English Learners
Emerging Use cards with root words ten hundred and thousand and a card with the -th suffix Have students place them together to show place value Then complete a sentence Use the same procedure to identify decimals
Expanding Support students at this level of English proficiency by providing sentence frames for them to use to describe their mathematical reasoning
To write the fraction _______ as a decimal I _______
Bridging Have students identify different meanings of the term square by matching examples of math problems with a written out sentence frame that defines the usage of the term square to square a number perfect square square root Use this procedure also with the term cube
Be sure to clarify the different uses of the term square when referring to square roots perfect squares and so on
EL
California ELD Standards
Emerging 2I12b Selecting language resources ndash Use knowledge of morphology to appropriately select affixes in basic ways
Expanding 2I12b Selecting language resources ndash Use knowledge of morphology to appropriately select affixes in a growing number of ways to manipulate language
Bridging 2I12b Selecting language resources ndash Use knowledge of morphology to appropriately select affixes in a variety of ways to manipulate language
Rational and Irrational Numbers 7B
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
11L E S S O N
Rational and Irrational Numbers
EngageESSENTIAL QUESTION
How do you rewrite rational numbers and decimals take square roots and cube roots and approximate irrational numbers To express as a decimal divide the numerator by the denominator To take a square root or cube root of a number find the number that when squared or cubed equals the original number To approximate an irrational number estimate a number between two consecutive perfect squares
Motivate the LessonAsk Which type of rational number do you see more often fractions or decimals Which do you prefer to use Why
ExploreHave students write examples of ratios and then share with the class the various notations for ratios that they used (for example 25 2 to 5 2 __ 5 ) Point out the connection between the word ratio and the meaning of rational number See also Explore Activity in student text
ExplainEXAMPLE 1
Questioning Strategies Mathematical Practices bull How does the denominator of a fraction in simplest form tell whether the decimal equivalent of the fraction is a terminating decimal The decimal will terminate if the denominator is an even number a multiple of 5 or a multiple of 10
Avoid Common ErrorsTo avoid interpreting 1 __ 4 as 4 divided by 1 tell students to start at the top of the fraction and read the bar as ldquodivided byrdquo
YOUR TURNTalk About ItCheck for Understanding
Ask Can an improper fraction be written as a decimal Give an example to support your answer Yes 5 __ 4 = 125
EXAMPLE 2Questioning Strategies Mathematical Practices bull How can you use place value to write a terminating decimal as a fraction with a power of ten in the denominator Start by identifying the place value of the decimals last digit and then use the corresponding power of 10 as the denominator of the fraction
bull How can you tell if a decimal can be written as a rational number If the decimal is a terminating or repeating decimal then it can be written as a rational number
Interactive Whiteboard Interactive example available online
ADDITIONAL EXAMPLE 1Write each fraction as a decimal
A 2 _ 5
04 B 5 _ 9
0 _
5
myhrwcom
Interactive Whiteboard Interactive example available online
ADDITIONAL EXAMPLE 2Write each decimal as a fraction in simplest form
A 0355 71 ___ 200
B 0 _
43 43 __ 99
myhrwcom
CA Common CoreStandards
The student is expected to
The Number Systemmdash8NS1
Know that numbers that are not rational are called irrational Understand informally that every number has a decimal expansion for rational numbers show that the decimal expansion repeats eventually and convert a decimal expansion which repeats eventually into a rational number
The Number Systemmdash8NS2
Use rational approximations of irrational numbers to compare the size of irrational numbers locate them approximately on a number line diagram and estimate the value of expressions (eg π 2 )
Expressions and Equationsmdash8EE2
Use square root and cube root symbols to represent solutions to equations of the form x 2 = p and x 3 = p where p is a positive rational number Evaluate square roots of small perfect squares and cube roots of small perfect cubes Know that radic
_ 2 is irrational
Mathematical Practices
MP6 Precision
The student is expected to
the value of expressions (eg
7 Lesson 11
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
My Notes
Math On the Spotmyhrwcom
Math TrainerOnline Practice
and Help
Personal
myhrwcom
Expressing Decimals as Rational NumbersYou can express terminating and repeating decimals as rational numbers
Write each decimal as a fraction in simplest form
0825
The decimal 0825 means ldquo825 thousandthsrdquo Write this as a fraction
825 ____ 1000
Then simplify the fraction
825 divide 25 ________ 1000 divide 25 = 33 __ 40
0825 = 33 __ 40
0 _
37
Let x = 0 _
37 The number 0 _
37 has 2 repeating digits so multiply each side of the equation x = 0
_ 37 by 10 2 or 100
x = 0 _
37
(100)x = 100(0 _
37 )
100x = 37 _
37
Because x = 0 _
37 you can subtract x from one side and 0 _
37 from the other
100x = 37 _
37
minusx minus0 _
37
99x = 37
Now solve the equation for x Simplify if necessary
99x ___ 99 = 37 __ 99
x = 37 __ 99
EXAMPLE 2
A
B
Write each fraction as a decimal
YOUR TURN
1 5 __ 11 2 1 _ 8 3 2 1 _ 3
8NS1
To write ldquo825 thousandthsrdquo put 825 over 1000
Divide both the numerator and the denominator by 25
100 times 0 _
37 is 37 _
37
37 _
37 minus 0 _
37 is 37
Divide both sides of the equation by 99
0 _
45 0125 2 _
3
Unit 18
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8_MCAAESE206984_U1M01L1indd 8 120413 838 PM
My Notes
Math On the Spot
myhrwcom
= 033333333333331mdash3
ESSENTIAL QUESTION
Expressing Rational Numbers as DecimalsA rational number is any number that can be written as a ratio in the form a _ b where a and b are integers and b is not 0 Examples of rational numbers are 6 and 05
6 can be written as 6 _ 1 05 can be written as 1 _ 2
Every rational number can be written as a terminating decimal or a repeating decimal A terminating decimal such as 05 has a finite number of digits A repeating decimal has a block of one or more digits that repeat indefinitely
Write each fraction as a decimal
1 _ 4
1 _ 4 = 025
1 _ 3
1 _ 3 = 0 _
3
EXAMPLEXAMPLE 1
A
B
0333 3 ⟌ ⎯ 1000 minus9 10 minus9 10 minus9 1
025 4 ⟌ ⎯ 100 -8 20 -20
0
L E S S O N
11Rational and Irrational Numbers
How do you rewrite rational numbers and decimals take square roots and cube roots and approximate irrational numbers
8NS1
Know that numbers that are not rational are called irrational Understand informally that every number has a decimal expansion for rational numbers show that the decimal expansion repeats eventually and convert a decimal expansion which repeats eventually into a relation number Also 8NS2 8EE2
8NS1
Remember that the fraction bar means ldquodivided byrdquo Divide the numerator by the denominator
Divide until the remainder is zero adding zeros after the decimal point in the dividend as needed
Divide until the remainder is zero or until the digits in the quotient begin to repeat
Add zeros after the decimal point in the dividend as needed
When a decimal has one or more digits that repeat indefinitely write the decimal with a bar over the repeating digit(s)
7Lesson 11
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Miff
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8_MCABESE206984_U1M01L1indd 7 11113 128 AM
PROFESSIONAL DEVELOPMENT
Math BackgroundSome decimals may have a pattern but still not be a repeating decimal that is rational For example in 312112111211112hellip you can predict the next digit and describe the pattern (There is one more 1 each time before the 2) However this is not a terminating decimal nor is it a repeating decimal and it is therefore NOT a rational number
Integrate Mathematical Practices MP6
This lesson provides an opportunity to address this Mathematical Practices standard It calls for students to attend to precision Students learn to express rational numbers accurately and precisely in both fractional and decimal forms and learn to translate from one form to the other They also learn how to precisely represent and communicate ideas about irrational numbers square roots and cube roots
Rational and Irrational Numbers 8
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=B
Focus on Technology Mathematical PracticesPoint out the importance of entering a repeating decimal correctly when using a graphing calculator to convert the decimal to a fraction The decimal 0
_ 59 must be entered as
0595959595959 not 059
YOUR TURNFocus on Math ConnectionsMake sure students understand that the place value of the last digit in Exercises 4 and 6 determines the denominator of the corresponding fraction or mixed number So for Exercise 4 the place value hundredths gives a denominator of 100 and for Exercise 6 the place value tenths gives a denominator of 10
EXAMPLE 3Questioning Strategies Mathematical Practices bull How can a solution of an equation of the form x 2 = p be negative if p is a positive number Since the square of a negative number is positive a negative number is also a solution of x 2 equals a positive number
bull When is a solution of an equation of the form x 3 = p larger than p The solution is larger than p if p is a number between 0 and 1
Focus on Math Connections Make sure students understand the difference in finding radic
_ 121 and solving x 2 = 121 The
symbol radic_
indicates the positive or principal square root only while the equation x 2 = 121 has two roots the principal square root and its opposite
YOUR TURNAvoid Common ErrorsTo avoid sign errors in Exercise 9 make sure that students understand that the cube of a negative number is not a positive number Therefore -8 is not a solution of x 3 = 512
Talk About ItCheck for Understanding
Ask Kris predicts that there are two real solutions for Exercises 7 and 8 and that there are three real solutions for Exercises 9 and 10 Is his prediction correct
Explain His prediction is correct for Exercises 7 and 8 because there are two numbers whose squares are the same positive number given in the exercises His prediction is not correct for Exercises 9 and 10 however because there is only one real number whose cube is the same positive number given in the exercises
EXPLORE ACTIVITYQuestioning Strategies Mathematical Practices bull Compare the values for 13 2 and 13 2 The digits are the same but 13 2 has two decimal places (169) while 13 2 has none (169)
bull How do you know whether radic_
2 will be closer to 1 or closer to 2 It will be closer to 1 because 2 is between the perfect squares of 1 and 4 but closer to 1 than it is to 4
Connect Vocabulary EL
Explain to students that the word irrational when used as an ordinary word in English means without logic or reason In mathematics when we say that a number is irrational it means only that the number cannot be written as the quotient of two integers
Engage with the WhiteboardHave students extend the number line in both directions and label the locations of the whole numbers 1 and 2 These are the roots of the consecutive perfect squares
1 and 4 used to estimate radic_
7
Interactive Whiteboard Interactive example available online
ADDITIONAL EXAMPLE 3Solve each equation for x
A x 2 = 324 18 -18
B x 2 = 25 ___ 144 5 __ 12 - 5 __ 12
C 343 = x 3 7
D x 3 = 125 ___ 512 5 __ 8
myhrwcom
9 Lesson 11
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
Math TrainerOnline Practice
and Help
Personal
myhrwcom
EXPLORE ACTIVITY
lt 2 lt
radic_
lt radic
_ 2 lt
radic_
lt radic
_ 2 lt
The solution is 9
The solution is 2 _ 5
C
D
729 = x 3
3 radic_ 729 = 3 radic
_ x 3
3 radic_ 729 = x
9 = x
x 3 = 8 ___ 125
3 radic_
x 3 =thinsp 3 radic_ 8 ___ 125
x =thinsp 3 radic_ 8 ___ 125
x = 2 _ 5
Solve each equation for x
YOUR TURN
7 x 2 = 196 8 x 2 = 9 ___ 256
9 x 3 = 512 10 x 3 = 64 ___ 343
Estimating Irrational NumbersIrrational numbers are numbers that are not rational In other words they cannot be written in the form a _ b where a and b are integers and b is not 0 Square roots of perfect squares are rational numbers Square roots of numbers that are not perfect squares are irrational Some equations like those in Example 3 involve square roots of numbers that are not perfect squares
x 2 = 2 x = plusmn radic_
2
Estimate the value of radic_
2
Find two consecutive perfect squares that 2 is between Complete the inequality by writing these perfect squares in the boxes
Now take the square root of each number
Simplify the square roots of perfect squares
radic_
2 is between and
A
B
C
8NS2 8EE2
Solve for x by taking the cube root of both sides
Solve for x by taking the cube root of both sides
Apply the definition of cube root
Think What number cubed equals 729
Apply the definition of cube root
Think What number cubed equals 8 ____ 125
radic_
2 is irrational
x = plusmn14 x = plusmn 3 __ 16
x = 8 x = 4 _ 7
1 2
1 4
1 4
1 2
Unit 110
copy H
ough
ton
Miff
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arco
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ublis
hing
Com
pany
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
8_MCAAESE206984_U1M01L1indd 10 41613 1211 AM
Math TrainerOnline Practice
and Help
Personal
myhrwcom
Math On the Spot
myhrwcom
Write each decimal as a fraction in simplest form
YOUR TURN
Finding Square Roots and Cube RootsThe square root of a positive number p is x if x 2 = p There are two square roots for every positive number For example the square roots of 36 are 6 and minus6 because 6 2 = 36 and (minus6) 2 = 36 The square roots of 1 __ 25 are 1 _ 5 and minus 1 _ 5 You can write the square roots of 1 __ 25 as plusmn 1 _ 5 The symbol radic
_ 5 indicates the positive
or principal square root
A number that is a perfect square has square roots that are integers The number 81 is a perfect square because its square roots are 9 and minus9
The cube root of a positive number p is x if x 3 = p There is one cube root for every positive number For example the cube root of 8 is 2 because 2 3 = 8 The cube root of 1 __ 27 is 1 _ 3 because ( 1 _ 3 )
3
= 1 __ 27 The symbol 3 radic_ 1 indicates the
cube root
A number that is a perfect cube has a cube root that is an integer The number 125 is a perfect cube because its cube root is 5
Solve each equation for x
The solutions are 11 and minus11
The solutions are 4 __ 13 and minus 4 __ 13
EXAMPLEXAMPLE 3
A x 2 = 121
x 2 = 121
x = plusmn radic_
121
x = plusmn11
B x 2 = 16 ___ 169
x 2 = 16 ___ 169
x = plusmn radic_
16 ___ 169
x = plusmn 4 __ 13
4 012 5 0 _
57 6 14
Can you square an integer and get a negative number
What does this indicate about whether negative
numbers have square roots
Math TalkMathematical Practices
8EE2
Solve for x by taking the square root of both sides
Apply the definition of square root
Think What numbers squared equal 121
Solve for x by taking the square root of both sides
Apply the definition of square root
Think What numbers squared equal 16 ____ 169
3 __ 25 19 __ 33 1 2 _ 5
No the square of a positive integer is positive the square of a negative integer is positive and the square of 0 is 0 So negative numbers do not have (real) square roots
9Lesson 11
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ough
ton
Miff
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hing
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pany
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
8_MCAAESE206984_U1M01L1indd 9 41913 240 PM
Critical ThinkingIn the Explore Activity students estimated the location of radic
_ 2 on a number line Ask students
whether they think that it is possible to locate more precisely the point that represents radic
_ 2 In
other words can you graph irrational numbers exactly on a number line along with rational numbers Students should understand that radic
_ 2
is a real number and all real numbers can be located on a real number line A more precise estimate will allow more precise placement on a number line
The Modeling note tells one way to do this
ModelingHave students use a ruler to represent a number line with a unit that is one inch long Have them draw a square with a side of one inch and draw the diagonal to make two isosceles triangles Lead students to understand that the length of the diagonal (or hypotenuse) is radic
_ 2
Have them copy the length of their diagonal onto their ruler or number line starting at zero The end point of the diagonal represents the exact point for the irrational number radic
_ 2 on a
number line
Additional ResourcesDifferentiated Instruction includes bull Reading Strategies bull Success for English Learners EL
bull Reteach bull Challenge PRE-AP
DIFFERENTIATE INSTRUCTION
Rational and Irrational Numbers 10
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
ElaborateTalk About ItSummarize the Lesson
Ask If someone claims that a certain number is irrational but you know it is actually rational how could you prove to that person that the number is rational
You could find a fraction equal to the number such that the number is the ratio of two integers with the denominator not equal to zero
GUIDED PRACTICEEngage with the Whiteboard
Have students plot each number in Exercises 16ndash18 on a number line Students should label each point with the irrational number written as a radical and as a
decimal
Avoid Common ErrorsExercises 1ndash6 To avoid reversing the order of the dividend and divisor tell students to start at the top of the fraction and read the bar as ldquodivided byrdquo
Focus on TechnologyHave students use a calculator to investigate the decimal equivalents of such fractions as 1 __ 9 2 __ 9 8 __ 9 and 1 __ 11 2 __ 11 10
__ 11 Ask them to describe the patterns they find as a result of these investigations
11 Lesson 11
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
Guided Practice
7 0675 8 56 9 044
10 0 _
4
10x =
x =
11 0 _
26
100x =
x =
12 0 _
325
1000x =
x =
Solve each equation for x (Example 3 and Explore Activity)
- x
-
_______________
x =
- x
-
___________________
x =
- x
-
_______________________
x =
Write each fraction or mixed number as a decimal (Example 1)
1 2 _ 5 2 8 _ 9 3 3 3 _ 4
4 7 __ 10 5 2 3 _ 8 6 5 _ 6
Write each decimal as a fraction or mixed number in simplest form (Example 2)
13 x 2 = 17 14 x 2 = 25 ___ 289 15 x 3 = 216
Approximate each irrational number to one decimal place without a calculator
x = plusmn radic__
asymp plusmn x = 3
radic__
=
(Explore Activity)
16 radic_
5 asymp
17 radic_
3 asymp
18 radic_
10 asymp
19 What is the difference between rational and irrational numbers
CHECK-INESSENTIAL QUESTION
x = plusmn radic__
__________ = plusmn _____
4 _
4
0 _
4
4 99
6216
269
41 25 5
17289
17
22 17 32
04
07
27__40
26 __ 99 325 ___ 999 4 _ 9
11__255 3_5
0 _
8
2375
375
08 _
3
26 _
26
0 _
26
325 _
325
0 _
325
999 325
Rational numbers can be written in the form a __ b where
a and b are integers and b ne 0 Irrational numbers cannot
be written in this form
Unit 112
copy H
ough
ton
Miff
lin H
arco
urt P
ublis
hing
Com
pany
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
8_MCAAESE206984_U1M01L1indd 12 41613 1211 AM
11 12 13 14 15
radic2 asymp 14
141 142 143 144 145
radic2 asymp 141
0 1 2 3 4
radic2 asymp 15
Estimate that radic_
2 asymp 15
To find a better estimate first choose some numbers between 1 and 2 and square them For example choose 13 14 and 15
1 3 2 = 1 4 2 = 1 5 2 =
Is radic_
2 between 13 and 14 How do you know
Is radic_
2 between 14 and 15 How do you know
2 is closer to than to so radic_
2 asymp
Locate and label this value on the number line
Reflect 11 How could you find an even better estimate of radic
_ 2
12 Find a better estimate of radic_
2
1 41 2 = 1 42 2 = 1 43 2 =
2 is closer to than to so radic_
2 asymp
Draw a number line and locate and label your estimate
13 Solve x 2 = 7 Write your answer as a radical expression Then estimate to one decimal place
D
E
F
No 2 is not between 169 and 196
Yes 2 is between 196 and 225
196
19881
19881
225
20164
20164
14
141
20449
169 196 225
Test the squares of numbers between 14 and 15
x = plusmn radic_
7 x asymp plusmn26
11Lesson 11
copy H
ough
ton
Miff
lin H
arco
urt P
ublis
hing
Com
pany
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
8_MCAAESE206984_U1M01L1indd 11 41613 1211 AM
Rational and Irrational Numbers 12
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
Math TrainerOnline Assessment
and Intervention
Personal
Online homework assignment available
myhrwcom
EvaluateGUIDED AND INDEPENDENT PRACTICE
Concepts amp Skills Practice
Example 1Expressing Rational Numbers as Decimals
Exercises 1ndash6 20ndash21 24ndash25
Example 2Expressing Decimals as Rational Numbers
Exercises 7ndash12 22ndash23 26ndash27
Example 3Finding Square Roots and Cube Roots
Exercises 13ndash15 28 30ndash31 35
Explore ActivityEstimating Irrational Numbers
Exercises 13 16ndash18 29 32ndash34
Additional ResourcesDifferentiated Instruction includes bull Leveled Practice worksheets
Lesson Quiz available online
11 LESSON QUIZ
1 Write as a decimal 2 5 __ 8 1 7 __ 12
2 Write as a fraction 034 1 _
24
3 Solve x 2 = 9 __ 49 for x
4 Solve x 3 = 216 for x
5 Estimate the value of radic_
13 to one decimal place without using a calculator
myhrwcom
Focus | Coherence | Rigor
Exercise Depth of Knowledge (DOK) Mathematical Practices
20ndash27 2 SkillsConcepts MP4 Modeling
28 3 Strategic Thinking MP4 Modeling
29ndash32 2 SkillsConcepts MP6 Precision
33 3 Strategic Thinking MP7 Using Structure
34 2 SkillsConcepts MP3 Logic
35 2 SkillsConcepts MP4 Modeling
36 3 Strategic Thinking MP3 Logic
37 3 Strategic Thinking MP7 Using Structure
38 3 Strategic Thinking MP2 Reasoning
8NS1 8NS2 8EE2
8NS1 8NS2 8EE2
Answers1 2625 158
_ 3
2 17 __ 50 1 8 __ 33
3 x = plusmn 3 __ 7
4 x = 6
5 36
13 Lesson 11
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
Work Area
33 Analyze Relationships To find radic_
15 Beau found 3 2 = 9 and 4 2 = 16 He said that since 15 is between 9 and 16 radic
_ 15 must be between 3 and 4 He
thinks a good estimate for radic_
15 is 3 + 4 ____ 2 = 35 Is Beaursquos estimate high low
or correct Explain
34 Justify Reasoning What is a good estimate for the solution to the equation x 3 = 95 How did you come up with your estimate
35 The volume of a sphere is 36π f t 3 What is the radius of the sphere Use the formula V = 4 _ 3 π r 3 to find your answer
36 Draw Conclusions Can you find the cube root of a negative number If so is it positive or negative Explain your reasoning
37 Make a Conjecture Evaluate and compare the following expressions
radic_
4 __ 25 and radic
_ 4 ____
radic_
25 radic
_
16 __ 81 and radic_
16 ____
radic_
81 radic
_
36 __ 49 and radic_
36 ____
radic_
49
Use your results to make a conjecture about a division rule for square roots Since division is multiplication by the reciprocal make a conjecture about a multiplication rule for square roots
38 Persevere in Problem Solving The difference between the solutions to the equation x 2 = a is 30 What is a Show that your answer is correct
FOCUS ON HIGHER ORDER THINKING
His estimate is low because 15 is very close to 16
so radic_
15 is very close to radic_
16 or 4 A better estimate
would be 38 or 39
Sample answer about 45 4 3 = 64 and 5 3 = 125
Because 95 is about halfway between 64 and 125 try 45
45 3 = 91125 which is a good estimate
3 feet
Yes the cube root of a negative number is negative
because a negative number cubed is always negative
and a nonnegative number cubed is always nonnegative
radic_
4 __ 25 = 2 _ 5 = radic
_ 4 ____
radic_
25 radic
_
16 __ 81 = 4 _ 9 = radic_
16 ____
radic_
81 radic
_
36 __ 49 = 6 _ 7 = radic_
36 ____
radic_
49
225 the solutions to x 2 = a are x = plusmn15 and
radic_
a ___
radic_
b = radic
_ a __
b radic
_ a radic
_ b = radic
_ a b
15 - (-15) = 30
Unit 114
copy H
ough
ton
Miff
lin H
arco
urt P
ublis
hing
Com
pany
bull copy
Ilen
e Mac
Dona
ldA
lamy I
mag
es
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=B
8_MCABESE206984_U1M01L1indd 14 102913 1142 PM
Personal Math Trainer
Online Practice and Helpmyhrwcom
Name Class Date
Independent Practice11
20 A 7 __ 16 -inch-long bolt is used in a machine What is this length written as a decimal
21 The weight of an object on the moon is 1 _ 6 its weight on Earth Write 1 _ 6 as a decimal
22 The distance to the nearest gas station is 2 4 _ 5 kilometers What is this distance written as a decimal
23 A baseball pitcher has pitched 98 2 _ 3 innings What is the number of innings written as a decimal
24 A heartbeat takes 08 second How many seconds is this written as a fraction
25 There are 262 miles in a marathon Write the number of miles using a fraction
26 The average score on a biology test was 72
_ 1 Write the average score using a
fraction
27 The metal in a penny is worth about 0505 cent How many cents is this written as a fraction
28 Multistep An artist wants to frame a square painting with an area of 400 square inches She wants to know the length of the wood trim that is needed to go around the painting
a If x is the length of one side of the painting what equation can you set up to find the length of a side How many solutions does the equation have
b Do all of the solutions that you found make sense in the context of the problem Explain
c What is the length of the wood trim needed to go around the painting
Solve each equation for x Write your answers as radical expressions Then estimate to one decimal place if necessary
29 x 2 = 14 30 x 3 = 1331
31 x 2 = 144 32 x 2 = 29
8NS1 8NS2 8EE2
04375 in 01 _6
28 km 98 _6 innings
x 2 = 400 x = plusmnthinsp20 the equation has 2 solutions
x = 20 makes sense but x = -20 doesnrsquot because a
painting cannot have a side length of -20 inches
4 times 20 = 80 inches
x = plusmn radic_
14 asymp plusmn37
x = plusmn radic_
144 = plusmn12 x = plusmn radic_
29 asymp plusmn54
x = 3 radic_ 1331 = 11
4_5 second 26 1_5 mi
72 1 _ 9 101 ___ 200 cent
13Lesson 11
copy H
ough
ton
Miff
lin H
arco
urt P
ublis
hing
Com
pany
bull copy
Phot
odisc
Get
ty Im
ages
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
8_MCAAESE206984_U1M01L1indd 13 41613 1211 AM
myhrwcomActivity available onlineEXTEND THE MATH PRE-AP
Activity Write radic_
09 on the board and invite students to conjecture what the value might be Have them check their conjectures by squaring Invite them to suggest ways to estimate radic
_ 09 As a hint point out that 09 is close to 10 and so they might
use that to help guide their estimates Lead them to see that since 092 is 081 and 102 is 1 the value of radic
_ 09 is greater than 09 and less than 10 Try squaring 095 to get
09025 A good estimate for radic_
09 is 095
Rational and Irrational Numbers 14
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=B
Integers
Rational Numbers IrrationalNumbers
Real Numbers
WholeNumbers
-3-4-5 -2-1 1 2 3 50 4
23
34-4 -π -1 25
radic2
Lesson Support Content Objective Students will learn to describe relationships between sets of numbers
Language Objective Students will explain how to describe relationships between sets of real numbers
LESSON 12 Sets of Real Numbers
Building BackgroundEliciting Prior Knowledge Have students draw a number line from -5 to 5 Ask them to plot points on the number line to approximate the location of rational and irrational numbers such as -1 3 __ 4 25 -4 2 __ 3 radic
_ 2 and -π
Learning ProgressionsIn this lesson students clarify their understanding of the real number system They characterize sets and subsets of the real numbers They also identify sets for real-world situations Important understandings for students include the following
bull Identify all of the possible subsets of the real numbers for a given number
bull Decide whether a statement about a subset of the real numbers is true or false
bull Identify the set of numbers that best describes a real-world situation
Understanding the relationships among the sets of numbers that make up the real numbers is essential as students are introduced to different forms of numbers throughout the school year This lesson provides a foundation for the comparing and ordering of real numbers in the next lesson
Cluster ConnectionsThis lesson provides an excellent opportunity to connect ideas in this cluster Know that there are numbers that are not rational and approximate them by rational numbers Have students copy this diagram which relates the sets of real numbers
Ask students to complete the diagram by writing three examples for each set of numbers Have students share examples and explain how they knew each number they selected belonged in the appropriate set Answers may vary Check studentsrsquo work
Focus | Coherence | Rigor
California Common Core Standards
8NS1 Know that numbers that are not rational are called irrational Understand informally that every number has a decimal expansion for rational numbers show that the decimal expansion repeats eventually and convert a decimal expansion which repeats eventually into a rational number
MP7 Look for and make use of structure
15A
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
Math Talk
Language Support EL
PROFESSIONAL DEVELOPMENT
Linguistic Support EL
AcademicContent Vocabulary
Venn diagrams ndash Students need descriptive language to describe the categories that the different areas and colors of a Venn diagram represent the concept of a set and how sets are distinct or can overlap Use sentence frames such as
The big oval represents __________The darklight blue color in the middle of the
big ovals represents __________These sets overlap because __________
In this way students have the language and structure to identify the criteria that distinguish a set and to explain the abstract representation Also point out the use of the prefix sub- meaning ldquounderrdquo in the term subset
Rules and Patterns
Abbreviations ndash In this lesson the abbreviation mph is used Be sure to point out that mph stands for miles per hour and is used to give units in a rate of speed Students may also have seen mpg (miles per gallon) which gives the units in a rate of fuel efficiency
Borrowed Words ndash Terminology used in baseball such as inning and pitcher may require some explanation Spanish as well as some other languages have borrowed these terms from English so some students may be familiar with these words already Despite this whenever a word is critical to students understanding the word problem it is best to explain the meaning
Leveled Strategies for English Learners
Emerging Allow students to indicate true or false orally in Guided Practice Exercises 9 and 10
Expanding Have students use sentence frames to describe the meaning of regions and colors used in a Venn diagram Then give them similar sentence frames orally and have them draw and shade a Venn diagram based on the oral prompts
Bridging Have students work in groups to draw a Venn diagram to represent sets based on real-world examples in the lesson
To help students answer the question posed in Math Talk provide a sentence frame for their answer
The numbers between 31 and 39 on a number line are __________ because __________
EL
California ELD Standards
Emerging 2II5 Modifying to add details ndash Expand sentences with simple adverbials to provide details about a familiar activity or process
Expanding 2II5 Modifying to add details ndash Expand sentences with adverbials to provide details about a familiar or new activity or process
Bridging 2II5 Modifying to add details ndash Expand sentences with increasingly complex adverbials to provide details about a variety of familiar and new activities and processes
Sets of Real Numbers 15B
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
12L E S S O N
Sets of Real Numbers
EngageESSENTIAL QUESTION
How can you describe relationships between sets of real numbers Sample answer Describe them as two different sets or one set as being a subset of another
Motivate the LessonAsk How many different types of tigers can you name How does the set of Bengal tigers relate to the set of tigers
ExplorePoint to different locations in the Animals diagram and ask for examples for that classification Do the same for the Real Numbers diagram Students should understand that everything within a region is part of the set for example both -3 and 2 are integers
ExplainEXAMPLE 1
Questioning Strategies Mathematical Practices bull In A why is 5 not a perfect square It does not have rational numbers as its square roots
bull Can the number in B be written as a fraction Why or why not Yes it is a terminating decimal so it is a rational number
Engage with the WhiteboardHave students place the numbers in Example 1 and Additional Example 1 in the Venn diagram for numbers
YOUR TURNAvoid Common ErrorsBe sure that students read Exercise 2 carefully before answering The number given in the problem 10 is the area not the side length
EXAMPLE 2Questioning Strategies Mathematical Practices bull What two major sets are the real numbers composed of rational and irrational numbers
bull What is the location of the set of whole numbers in the Venn diagram in relation to the set of rational numbers Explain Inside it whole numbers are rational numbers
Focus on Reasoning Mathematical PracticesRemind students that it takes only one counterexample to show that a statement is false
Interactive Whiteboard Interactive example available online
ADDITIONAL EXAMPLE 1Write all names that apply to each number
A -10integer rational real
B 12 _ 3
whole integer rational real
myhrwcom
Interactive Whiteboard Interactive example available online
ADDITIONAL EXAMPLE 2Tell whether the given statement is true or false Explain your choice
No integers are whole numbers
False every whole number is also an integer
myhrwcom
Animated MathClassifying Numbers
Students build fluency in classifying numbers in this engaging fast-paced game
myhrwcom
CA Common CoreStandards
The student is expected to
The Number Systemmdash8NS1
Know that numbers that are not rational are called irrational Understand informally that every number has a decimal expansion for rational numbers show that the decimal expansion repeats eventually and convert a decimal expansion which repeats eventually into a rational numberMathematical Practices
MP7 Using Structure
The student is expected to
15 Lesson 12
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=B
Math TrainerOnline Practice
and Help
Personal
myhrwcom
Math TrainerOnline Practice
and Help
Personal
myhrwcom
Math On the Spotmyhrwcom
Understanding Sets and Subsets of Real NumbersBy understanding which sets are subsets of types of numbers you can verify whether statements about the relationships between sets are true or false
Tell whether the given statement is true or false Explain your choice
All irrational numbers are real numbers
True Every irrational number is included in the set of real numbers The irrational numbers are a subset of the real numbers
No rational numbers are whole numbers
False A whole number can be written as a fraction with a denominator of 1 so every whole number is included in the set of rational numbers The whole numbers are a subset of the rational numbers
EXAMPLE 2
A
B
Write all names that apply to each number
1 A baseball pitcher has pitched 12 2 _ 3 innings
2 The length of the side of a square that has an
area of 10 square yards
YOUR TURN
Tell whether the given statement is true or false Explain your choice
3 All rational numbers are integers
4 Some irrational numbers are integers
YOUR TURN
Give an example of a rational number that is a
whole number Show that the number is both whole
and rational
Math TalkMathematical Practices
Give an example of a
8NS1
False Every integer is a rational number but not every
False Real numbers are either rational or irrational numbers
Integers are rational numbers so no integers are irrational numbers
rational real
irrational real
Sample answer 8 8 = 8_
1
and -thinsp 5 _ 2 are not integers
rational number is an integer Rational numbers such as 3 _ 5
Unit 116
copy H
ough
ton
Miff
lin H
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hing
Com
pany
bull Im
age C
redi
ts D
igita
l Im
age c
opyr
ight
copy20
04 Ey
ewire
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
8_MCAAESE206984_U1M01L2indd 16 41613 136 AM
Math On the Spot
myhrwcom
Vertebrates
Birds
Passerines
Animals
Integers
Rational Numbers IrrationalNumbers
Real Numbers
WholeNumbers
1
45
3
0
274
67
radic4
-
-3
-2
-1
03
radic2
radic17
radic11-
π
Animated Math
myhrwcom
Classifying Real NumbersBiologists classify animals based on shared characteristics A cardinal is an animal a vertebrate a bird and a passerine
You already know that the set of rational numbers consists of whole numbers integers and fractions The set of real numbers consists of the set of rational numbers and the set of irrational numbers
Write all names that apply to each number
radic_
5 irrational real
ndash1784rational real
whole integer rational real
EXAMPLEXAMPLE 1
A
B
C radic_ 81 ____ 9
L E S S O N
12Sets of Real Numbers
ESSENTIAL QUESTIONHow can you describe relationships between sets of real numbers
Passerines such as the cardinal are also called ldquoperching birdsrdquo
What types of numbers are between 31 and 39 on a
number line
Math TalkMathematical Practices
What types of numbers are
8NS1
8NS1
Know that numbers that are not rational are called irrational Understand informally that every number has a decimal expansion for rational numbers show that the decimal expansion repeats eventually and convert a decimal expansion which repeats eventually into a relation number
ndash1784 is a terminating decimal
5 is a whole number that is not a perfect square
radic_
81 _____ 9 = 9 __ 9 = 1 rational irrational real
15Lesson 12
copy H
ough
ton
Miff
lin H
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ublis
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Com
pany
bull Im
age C
redi
ts copy
Wiki
med
ia Co
mm
ons
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=B
8_MCABESE206984_U1M01L2indd 15 061113 1144 AM
PROFESSIONAL DEVELOPMENT
Math BackgroundThe relationships between sets of numbers extend to include complex numbers A complex number can be written as a sum of a real number a and an imaginary number bi
a + bi
An imaginary number is a special number that when squared gives a negative value When you square a real number you get a nonnegative number When you square an imaginary number you get a negative value The imaginary unit is i
i = radic_
-1
Integrate Mathematical Practices MP7
This lesson provides an opportunity to address this Mathematical Practices standard It calls for students to discern structure to connect and communicate mathematical ideas
Students use a Venn diagram to structure relationships between sets of numbers They connect and communicate mathematical ideas when they make logical statements about the sets and describe which set best describes numbers applied to real-life situations
Sets of Real Numbers 16
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=B
YOUR TURNAvoid Common ErrorsStudents may see the word ldquoAllldquo or rdquoNordquo in Exercises 3 and 4 and immediately assume that any absolute statements like these are false Remind them that there are true statements that begin with these words and encourage them to provide examples
EXAMPLE 3Questioning Strategies Mathematical Practices bull In A how does the phrase ldquonumber of rdquo give you a clue about the number classification It indicates a counting number
bull What is the relationship between the circumference of a circle and the diameter The circumference is diameter times π
Focus on Critical Thinking Mathematical PracticesIn B suppose the diameters in inches were 25
__ π 28 __ π
31 __ π and so on What set of numbers would
best describe the circumferences Explain Whole numbers the circumferences would be the whole numbers 25 28 31 and so on
YOUR TURNFocus on Critical Thinking Mathematical PracticesHave students compare and contrast the classification of numbers in the answers in Exercises 5 and 6
ElaborateTalk About ItSummarize the Lesson
Ask What are some ways that number sets can be related Sets may be subsets of other sets or they may be separate from other sets
GUIDED PRACTICEEngage with the Whiteboard
Have students place the numbers in Exercises 1ndashthinsp8 in the Venn diagram for numbers at the beginning of the lesson
Integrating Language Arts EL
Encourage English learners to ask for clarification on any terms or phrases that they do not understand
Avoid Common ErrorsExercise 7 Remind students that a repeating decimal is a rational numberExercises 9ndash10 Remind students that it only takes one counterexample to show that a statement is false
Interactive Whiteboard Interactive example available online
ADDITIONAL EXAMPLE 3Identify the set of numbers that best describes the situation Explain your choice
A the amount of time that has passed since midnight
The set of real numbers time is continuous so the amount of time can be rational or irrational
B the number of tickets sold to a basketball game
The set of whole numbers the number of tickets sold may be 0 or a counting number
myhrwcom
17 Lesson 12
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
1IN
116 inch
Guided Practice
Write all names that apply to each number (Example 1)
1 7 _ 8 2 radic_
36
3 radic_
24 4 075
5 0 6 - radic_ 100
7 5 _
45 8 - 18 __ 6
Tell whether the given statement is true or false Explain your choice (Example 2)
9 All whole numbers are rational numbers
10 No irrational numbers are whole numbers
Identify the set of numbers that best describes each situation Explain your choice (Example 3)
11 the change in the value of an account when given to the nearest dollar
12 the markings on a standard ruler
13 What are some ways to describe the relationships between sets of numbers
CHECK-INESSENTIAL QUESTION
rational real
rational real
True Whole numbers are rational numbers
Rational numbers the ruler is marked every 1 __ 16 th inch
Sample answer Describe one set as being a subset of
another or show their relationships in a Venn diagram
Integers the change can be a whole dollar amount
and can be positive negative or zero
True Whole numbers are a subset of the set of rational numbers
and can be written as a ratio of the whole number to 1
irrational real
whole integer rational real
whole integer rational real
rational real
integer rational real
integer rational real
Unit 118
copy H
ough
ton
Miff
lin H
arco
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ublis
hing
Com
pany
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
8_MCAAESE206984_U1M01L2indd 18 41613 136 AM
My Notes
Math TrainerOnline Practice
and Help
Personal
myhrwcom
Math On the Spot
myhrwcom
Identifying Sets for Real-World SituationsReal numbers can be used to represent real-world quantities Highways have posted speed limit signs that are represented by natural numbers such as 55 mph Integers appear on thermometers Rational numbers are used in many daily activities including cooking For example ingredients in a recipe are often given in fractional amounts such as 2 _ 3 cup flour
Identify the set of numbers that best describes each situation Explain your choice
the number of people wearing glasses in a room
The set of whole numbers best describes the situation The number of people wearing glasses may be 0 or a counting number
the circumference of a flying disk has a diameter of 8 9 10 11 or 14 inches
The set of irrational numbers best describes the situation Each circumference would be a product of π and the diameter and any multiple of π is irrational
EXAMPLEXAMPLE 3
A
B
Identify the set of numbers that best describes the situation Explain your choice
5 the amount of water in a glass as it evaporates
6 the weight of a person in pounds
YOUR TURN
8NS1
Rational numbers a personrsquos weight can be a decimal
such as 835 pounds
Real numbers the amount can be any number greater
than 0
17Lesson 12
copy H
ough
ton
Miff
lin H
arco
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ublis
hing
Com
pany
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
8_MCAAESE206984_U1M01L2indd 17 41613 520 AM
Graphic OrganizersGive students a list of numbers (including terminating and repeating decimals fractions integers and rational and irrational square roots) and a graphic organizer as shown below
Real Numbers
Rational numbers Irrational numbers
Integer numbers
Whole numbers
Ask students to write each number in the list in the correct section of the organizer
Number SensePoint out to students that knowing the types of numbers to expect in different situations can alert them to incorrect math as well as to impossible situations For example 135 shots made in basketballs is not possible but an average number of shots can equal 135
Additional ResourcesDifferentiated Instruction includes bull Reading Strategies bull Success for English Learners EL
bull Reteach bull Challenge PRE-AP
DIFFERENTIATE INSTRUCTION
Sets of Real Numbers 18
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
Math TrainerOnline Assessment
and Intervention
Personal
Online homework assignment available
myhrwcom
Lesson Quiz available online
12 LESSON QUIZ
1 Write all the names that apply to the number
2 Tell whether the given statement is true or false Explain your choice All numbers between 1 and 2 are rational numbers
3 Identify the set of numbers that best describes the situation Explain your choiceThe choices on a survey question change the total points for the survey by -2 -1 0 1 or 2 points
-1 _
5
myhrwcom
Additional ResourcesDifferentiated Instruction includes bull Leveled Practice worksheets
EvaluateGUIDED AND INDEPENDENT PRACTICE
Concepts amp Skills Practice
Example 1Classifying Real Numbers
Exercises 1ndash8 14ndash19 22ndash24
Example 2Understanding Sets and Subsets of Real Numbers
Exercises 9ndash10
Example 3Identifying Sets for Real-World Situations
Exercises 11ndash12 20ndash21 25
Focus | Coherence | Rigor
Exercise Depth of Knowledge (DOK) Mathematical Practices
14ndash19 2 SkillsConcepts MP7 Using Structure
20ndash21 2 SkillsConcepts MP6 Precision
22ndash23 2 SkillsConcepts MP3 Logic
24 1 Recall of Information MP7 Using Structure
25 2 SkillsConcepts MP2 Reasoning
26ndash27 3 Strategic Thinking MP3 Logic
28 3 Strategic Thinking MP8 Patterns
29 3 Strategic Thinking MP3 Logic
8NS1
8NS1
Exercise 29 combines concepts from the California Common Core cluster ldquoKnow that there are numbers that are not rational and approximate them by rational numbersrdquo
Answers1 rational real
2 False radic_
2 is an example of an irrational number between 1 and 2
3 Integers each number is an integer but only three are whole numbers
19 Lesson 12
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
Work Area
π mi23 Critique Reasoning The circumference of a circular region is shown
What type of number best describes the diameter of the circle Explain
your answer
24 Critical Thinking A number is not an integer What type of number can it be
25 A grocery store has a shelf with half-gallon containers of milk What type of number best represents the total number of gallons
26 Explain the Error Katie said ldquoNegative numbers are integersrdquo What was her error
27 Justify Reasoning Can you ever use a calculator to determine if a number is rational or irrational Explain
28 Draw Conclusions The decimal 0 _
3 represents 1 _ 3 What type of number best describes 0
_ 9 which is 3 middot 0
_ 3 Explain
29 Communicate Mathematical Ideas Irrational numbers can never be precisely represented in decimal form Why is this
FOCUS ON HIGHER ORDER THINKING
It can be a rational number that is not an integer or an irrational number
rational number
The set of negative numbers also includes non-integer
rational numbers and irrational numbers
Sample answer If the calculator shows a decimal that
terminates in fewer digits than what the calculator screen
allows then you can tell that the number is rational If not
you cannot tell from the calculator display whether the
number terminates because you see a limited number
of digits It may be a repeating decimal (rational) or
non-terminating non-repeating decimal (irrational)
Whole 3 middot 0 _
3 represents 3 middot 1 _ 3 = 1 so 0 _
9 is exactly 1
Sample answer In decimal form irrational numbers never
terminate and never repeat Therefore no matter how
many decimal places you include the number will never
be precisely represented There are always more digits
Whole the diameter is π _ π = 1 mile
Unit 120
copy H
ough
ton
Miff
lin H
arco
urt P
ublis
hing
Com
pany
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
8_MCAAESE206984_U1M01L2indd 20 120413 909 PM
Integers
Rational Numbers Irrational Numbers
Real Numbers
Whole Numbers
257
radic16
166
radic9
128 radic50
Personal Math Trainer
Online Practice and Helpmyhrwcom
Name Class Date
Independent Practice
Identify the set of numbers that best describes each situation Explain your choice
20 the height of an airplane as it descends to an airport runway
21 the score with respect to par of several golfers 2 ndash 3 5 0 ndash 1
22 Critique Reasoning Ronald states that the number 1 __ 11 is not rational because when converted into a decimal it does not terminate Nathaniel says it is rational because it is a fraction Which boy is correct Explain
12
14 - radic_
9 15 257
16 radic_
50 17 8 1 _ 2
18 166 19 radic_
16
Write all names that apply to each number Then place the numbers in the correct location on the Venn diagram
8NS1
Real numbers the height can be any number greater than zero
integer rational real whole integer rational real
whole integer rational real
irrational real
rational real
rational real
Integers the scores are counting numbers their
opposites and zero
Nathaniel is correct A rational number is a number that can be written as a fraction and 1 __ 11 is a fraction
19Lesson 12
copy H
ough
ton
Miff
lin H
arco
urt P
ublis
hing
Com
pany
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
8_MCAAESE206984_U1M01L2indd 19 41613 136 AM
myhrwcomActivity available onlineEXTEND THE MATH PRE-AP
Activity Have students consider the concept of restricted domain for the sets of numbers that describe situations For example the number of sisters a person has can best be described by whole numbers but no one has ever had 1500 sisters An area code is an integer or whole number between 200 and 999
Have students use a source such as the Guinness Book of World Records and give examples of sets of numbers that describe situations where the domain is restricted Ask whether the restriction may be changed in the future
Sets of Real Numbers 20
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
-3-4-5 -2-1 1 2 3 50 4
12-4 -radic5
Lesson Support Content Objective Students will learn to order a set of real numbers
Language Objective Students will show and describe how to order a set of real numbers
LESSON 13 Ordering Real Numbers
Building BackgroundEliciting Prior Knowledge Have students draw a number line to compare a rational number and an irrational number such as - radic
_ 5 and -4 1 __ 2 Ask them to explain how
they approximated the irrational number on the number line Then have them identify the greater and the lesser real number Repeat with several other pairs of real numbers in different forms
Learning ProgressionsIn this lesson students order a set of real numbers They use rational approximations to compare the sizes of irrational numbers They also order numbers for real-world situations Important understandings for students include the following
bull Compare irrational numbers bull Estimate the value of expressions with irrational numbers bull Order a set of real numbers bull Order real numbers in a real-world context
Work with real numbers continues throughout Grade 8 and into high school This lesson provides students with a foundation for understanding the relative sizes of numbers in different forms in the real number system
Cluster ConnectionsThis lesson provides an excellent opportunity to connect ideas in this cluster Know that there are numbers that are not rational and approximate them by rational numbers Tell students that there is a special number called the golden ratio with applications in mathematics geometry art and architecture The golden ratio is called phi and is represented by the Greek letter ϕ It includes an irrational number in its definition
Have students explain why the golden ratio is irrational Ask them to find the two whole numbers the golden ratio lies between Then challenge them to approximate the golden ratio to the nearest tenth It is irrational because it includes an irrational number in its definition It lies between 1 and 2 To the nearest tenth ϕ = 16
ϕ = 1 + radic_
5 _ 2
Focus | Coherence | Rigor
California Common Core Standards
8NS2 Use rational approximations of irrational numbers to compare the size of irrational numbers locate them approximately on a number line diagram and estimate the value of expressions (eg π 2 )
MP4 Model with mathematics
21A
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
Math Talk
Language Support EL
PROFESSIONAL DEVELOPMENT
Linguistic Support EL
AcademicContent Vocabulary
Post a chart like this to remind students of the regular comparative forms of adjectives that use the -er and -est suffixes Add to the chart for terms that appear in examples and exercises in each lesson Include any irregular verb forms
Background Knowledge
Go On ndash the title of the module review or quiz is Ready to Go On This title uses an idiomatic expression In this context to go on means ldquoto move aheadrdquo or ldquoto proceedrdquo It is different from the use of go on that means having enough facts to use meaningfully as in having enough to go on Also the intonation used in pronouncing an expression can give it different meanings For example when the speaker emphasizes the word on he or she might be expressing disbelief as in ldquoGo ON Yoursquore kidding rightrdquo Discuss with students other ways that the phrase go on may be used
Leveled Strategies for English Learners
Emerging Label points on a number line with the terms used in ordering greater greatest less lesser least Use sentence frames to insert the correct terms
Expanding Have students give two or three complete sentences to compare the placement of numbers on a number line using the correct forms of the comparative and superlative adjectives
Bridging Have students work in pairs with one student giving directions to the other in complete sentences to order numbers on a number line
To help students answer the question posed in Math Talk make sure that students have a command of the forms for making comparisons and the superlative and the concept of opposite order so that the focus is on the math concept instead of the language skills needed to describe and explain order
EL
Adjective Comparative Superlative
Far Farther Farthest
Large Larger Largest
Great Greater Greatest
Some Less Least
Some More Most
California ELD Standards
Emerging 2I8 Analyzing language choices ndash Explain how phrasing or different common words with similar meanings produce different effects on the audience
Expanding 2I8 Analyzing language choices ndash Explain how phrasing or different words with similar meanings or figurative language produce shades of meaning and different effects on the audience
Bridging 2I8 Analyzing language choices ndash Explain how phrasing or different words with similar meanings or figurative language produce shades of meaning nuances and different effects on the audience
Ordering Real Numbers 21B
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
13L E S S O N
Ordering Real Numbers
Interactive Whiteboard Interactive example available online
ADDITIONAL EXAMPLE 1Compare Write lt gt or =
A radic_
8 - 2 4 - radic_
8 lt
B radic_
20 + 1 3 + radic_
2 gt
EngageESSENTIAL QUESTION
How do you order a set of real numbers Sample answer Find their approximate decimal values and order them
Motivate the LessonAsk What kind of numbers are you comparing when you compare the price of gasoline at two different gas stations
ExploreGive students two rational numbers and ask them to name a number between them Repeat a few times and then give them two irrational numbers and ask them to name a number between them
ExplainEXAMPLE 1
Questioning Strategies Mathematical Practices bull Which is greater the difference between 5 and 3 or the difference between radic
_ 5 and radic
_ 3
The difference between 5 and 3 is 2 the difference between radic_
5 and radic_
3 is approximately 1 So the difference between 5 and 3 is greater
Avoid Common ErrorsCaution students to read the problem carefully and think about what the radical sign means so that they do not misread the problem and answer that the two sides are equal
YOUR TURNFocus on TechnologyCalculators should not be used at this point because developing number sense is the goal
EXAMPLE 2Questioning Strategies Mathematical Practices bull How do you determine whether radic
_ 22 is less than or greater than 45 The square of 45 is
2025 which is less than 22 so the square root of 22 must be greater than 45
Engage with the WhiteboardHave students graph and label various real numbers between 42 and 44 and between 47 and 5
YOUR TURNFocus on Modeling Mathematical PracticesHave students label the integers on the number line with their equivalent square root For example 1 2 and 3 on the number line would be labeled radic
_ 1 radic
_ 4 and radic
_ 9
Interactive Whiteboard Interactive example available online
ADDITIONAL EXAMPLE 2Order 3π radic
_ 10 and 325 from greatest
to least
3π 325 radic_
10
myhrwcom
myhrwcom
CA Common CoreStandards
The student is expected to
The Number Systemmdash8NS2
Use rational approximations of irrational numbers to compare the size of irrational numbers locate them approximately on a number line diagram and estimate the value of expressions (eg π 2 )
Mathematical Practices
MP4 Modeling
The student is expected to
21 Lesson 13
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
Math TrainerOnline Practice
and Help
Personal
myhrwcom
Math On the Spotmyhrwcom
0 05 1 15 2 25 3 35 4
radic5radic3
π2
8 85 9 95 10 105 11 115 12
radic75
4 42 44 46 48 5
radic224 12π + 1
Ordering Real Numbers You can compare and order real numbers and list them from least to greatest
Order radic_
22 π + 1 and 4 1 _ 2 from least to greatest
First approximate radic_
22
radic_
22 is between 4 and 5 Since you donrsquot know where it falls between 4 and 5 you need to find a better estimate for radic
_ 22 so
you can compare it to 4 1 _ 2
Since 22 is closer to 25 than 16 use squares of numbers between 45 and 5 to find a better estimate of radic
_ 22
45 2 = 2025 46 2 = 2116 47 2 = 2209 48 2 = 2304
Since 47 2 = 2209 an approximate value for radic_
22 is 47
An approximate value of π is 314 So an approximate value of π +1 is 414
Plot radic_
22 π + 1 and 4 1 _ 2 on a number line
Read the numbers from left to right to place them in order from least to greatest
From least to greatest the numbers are π + 1 4 1 _ 2 and radic_
22
EXAMPLE 2
STEP 1
STEP 2
Order the numbers from least to greatest Then graph them on the number line
YOUR TURN
5 radic_
5 25 radic_
3
6 π 2 10 radic_
75
If real numbers a b and c are in order from least to greatest what is the order
of their opposites from least to greatest
Explain
Math TalkMathematical Practices
8NS2
radic_
3 radic_
5 25
radic_
75 π2 10
Math Talk answer -c -b -a -c is farthest to the left on a number line -b is in the middle and -a is farthest to the right
Unit 122
copy H
ough
ton
Miff
lin H
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ublis
hing
Com
pany
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
8_MCAAESE206984_U1M01L3indd 22 41613 447 AM
My Notes
Math TrainerOnline Practice
and Help
Personal
myhrwcom
Math On the Spot
myhrwcom
Comparing Irrational NumbersBetween any two real numbers is another real number To compare and order real numbers you can approximate irrational numbers as decimals
Compare radic_
3 + 5 3 + radic_
5 Write lt gt or =
First approximate radic_
3
radic_
3 is between 1 and 2
Next approximate radic_
5
radic_
5 is between 2 and 3
Then use your approximations to simplify the expressions
radic_
3 + 5 is between 6 and 7
3 + radic_
5 is between 5 and 6
So radic_
3 + 5 gt 3 + radic_
5
Reflect1 If 7 + radic
_ 5 is equal to radic
_ 5 plus a number what do you know about the
number Why
2 What are the closest two integers that radic_
300 is between
EXAMPLEXAMPLE 1
STEP 1
STEP 2
Compare Write lt gt or =
YOUR TURN
3 radic_
2 + 4 2 + radic_
4 4 radic_
12 + 6 12 + radic_
6
L E S S O N
13 Ordering Real Numbers
ESSENTIAL QUESTIONHow do you order a set of real numbers
8NS2
Use rational approximations of irrational numbers to compare the size of irrational numbers locate them approximately on a number line diagram and estimate the value of expressions (eg π 2 )
8NS2
Use perfect squares to estimate square roots
1 2 = 1 2 2 = 4 3 2 = 9
The number is 7 both expressions must equal 7 + radic_
5
17 and 18
gt lt
21Lesson 13
copy H
ough
ton
Miff
lin H
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ublis
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Com
pany
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
8_MCAAESE206984_U1M01L3indd 21 41913 246 PM
PROFESSIONAL DEVELOPMENT
Math BackgroundIn this lesson students estimate irrational numbers in the form of square roots of nonper-fect squares by finding two perfect squares between which the number falls A more precise method involves repeated division For example to find radic
_ 28 find a whole number whose perfect
square is close to 28 such as 5 Divide 28 by that number 28 divide 5 = 56 Find the average of the quotient and divisor 5 + 56
_____ 2 = 53 Continue dividing 28 by each result and averaging until you get the desired accuracy
Integrate Mathematical Practices MP4
This lesson provides an opportunity to address this Mathematical Practices standard It calls for students to model relationships using multiple representations including diagrams graphs and language as appropriate Students use multiple representations when they use number lines to estimate the locations of and order rational and irrational numbers given as symbols
Ordering Real Numbers 22
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
Interactive Whiteboard Interactive example available online
ADDITIONAL EXAMPLE 3The diameter of a meteorite in millimeters is calculated by four different methods Order the results from least to greatest
Joe radic_
18 mm Lisa 13 __ 3 mm
Pablo 46 mm Julien 4π __ 3 mm
Julien 4π __ 3 mm Lisa 13 __ 3 mm
Joe radic_
18 mm Pablo 46 mm
EXAMPLE 3Questioning Strategies Mathematical Practices bull How can you verify that radic
_ 28 is between 52 and 53 5 2 2 = 2704 and 5 3 2 = 2809
bull Explain how to determine which number is greater 5 _
5 or 55 When the repeating decimal is rounded to the nearest tenth or hundredth you can see that it is greater
Connect to Daily LifeDiscuss how measuring across a canyon might involve different methods than measuring along a road Explain that measurements like these are often done using calculations that approximate the distance
YOUR TURNFocus on Critical Thinking Mathematical PracticesDiscuss with students which number is greater 3
_ 45 or 3450 3
_ 45 or 3455 and why Explain
that 3 _
45 can be written out as 34545hellipMake sure they understand that 3 _
45 is greater than 345 but less than 3455
ElaborateTalk About ItSummarize the Lesson
Ask How can you order two numbers in different forms whose decimal approxi-mations appear to be equal Approximate one or both numbers to an additional
number of decimal places
GUIDED PRACTICEEngage with the Whiteboard
Have students place and label additional points on the number line in Exercise 9 Allow the points to be in any format other than decimal
Avoid Common ErrorsExercises 3ndash4 Caution students to read the problem carefully so that they do not misread the problem as the same numbers combined by addition on each side of the circleExercise 10 Remind students that the calculations have units
myhrwcom
23 Lesson 13
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
0 05 1 15 2 25 3 35 4 45 5 55 6 65 7
2πradic3
Compare Write lt gt or = (Example 1)
1 radic_
3 + 2 radic_
3 + 3 2 radic_
8 + 17 radic_
11 + 15
3 radic_
6 + 5 6 + radic_
5 4 radic_
9 + 3 9 + radic_
3
5 radic_
17 - 3 -2 + radic_
5 6 12 - radic_
2 14 - radic_
8
7 radic_
7 + 2 radic_
10 - 1 8 radic_
17 + 3 3 + radic_
11
9 Order radic_
3 2π and 15 from least to greatest Then graph them on the number line (Example 2)
radic_
3 is between and so radic_
3 asymp
π asymp 314 so 2π asymp
From least to greatest the numbers are
10 Four people have found the perimeter of a forest using different methods Their results are given in the table Order their calculations from greatest to least (Example 3)
11 Explain how to order a set of real numbers
CHECK-INESSENTIAL QUESTION
Forest Perimeter (km)
Leon Mika Jason Ashley
radic_
17 - 2 1 +thinsp π __ 2 12 ___ 5 25
Guided Practice
17
15
1 + π _ 2 km 25 km 12 __ 5 km radic_
17 - 2 km
2π radic
_ 3
18 175
628
Sample answer Convert each number to a decimal
equivalent using estimation to find equivalents for
irrational numbers Graph each number on a number line
Read the numbers from left to right for least to greatest
Read the numbers from right to left for greatest to least
lt gt
lt lt
ltgt
gt gt
24 Unit 1
copy H
ough
ton
Miff
lin H
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urt P
ublis
hing
Com
pany
bull Im
age C
redi
ts copy
Elena
Eliss
eeva
Alam
y Im
ages
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
8_MCAAESE206984_U1M01L3indd 24 41613 448 AM
My Notes
5 52 54 56 58 6
radic28 5 12
23455
Math TrainerOnline Practice
and Help
Personal
myhrwcom
Math On the Spot
myhrwcom
Ordering Real Numbers in a Real-World Context Calculations and estimations in the real world may differ It can be important to know not only which are the most accurate but which give the greatest or least values depending upon the context
Four people have found the distance in kilometers across a canyon using different methods Their results are given in the table Order the distances from greatest to least
Distance Across Quarry Canyon (km)
Juana Lee Ann Ryne Jackson
radic_
28 23 __ 4 5 _
5 5 1 _ 2
Write each value as a decimal
radic_
28 is between 52 and 53 Since 53 2 = 2809 an approximate value for radic
_ 28 is 53
23 __ 4 = 575
5 _
5 is 5555hellip so 5 _
5 to the nearest hundredth is 556
5 1 _ 2 = 55
Plot radic_
28 23 __ 4 5 _
5 and 5 1 _ 2 on a number line
From greatest to least the distances are
23 __ 4 km 5 _
5 km 5 1 _ 2 km radic_
28 km
EXAMPLEXAMPLE 3
STEP 1
STEP 2
7 Four people have found the distance in miles across a crater using different methods Their results are given below
Jonathan 10 __ 3 Elaine 3 _
45 Joseacute 3 1 _ 2 Lashonda radic_
10
Order the distances from greatest to least
YOUR TURN
8NS2
3 1 _ 2 mi 3 _
45 mi 10 __ 3 mi radic_
10 mi
23Lesson 13
copy H
ough
ton
Miff
lin H
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Com
pany
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8_MCAAESE206984_U1M01L3indd 23 41613 447 AM
ModelingPlace papers around the room with the numbers from 1 to 5 one per sheet Give each student a card showing a number between 1 and 5 in different forms Have students place his or her card between the correct integers and decide where the number goes in relation to any numbers already placed
Multiple RepresentationsGive students a vertical number line which some students might find easier to use than a horizontal one Have them decide whether to place points for rational and irrational numbers above or below existing points
Additional ResourcesDifferentiated Instruction includes bull Reading Strategies bull Success for English Learners EL
bull Reteach bull Challenge PRE-AP
DIFFERENTIATE INSTRUCTION
Ordering Real Numbers 24
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
Math TrainerOnline Assessment
and Intervention
Personal
Online homework assignment available
myhrwcom
myhrwcom
Lesson Quiz available online
13 LESSON QUIZ
1 Compare Write lt gt or =
radic_
95 - 5 radic_
62 - 2
2 Order 105 radic_
105 and 3π + 1 from greatest to least
3 A length in centimeters is calculated differently by four different people Order their calculations from least to greatest
KD 11 __ 2 cm Silvio 5 __ 3 π cm
Paula 5 _
4 cm Luis radic_
33 cm
EvaluateGUIDED AND INDEPENDENT PRACTICE
Concepts amp Skills Practice
Example 1Comparing Irrational Numbers
Exercises 1ndash8
Example 2Ordering Real Numbers
Exercises 9 12ndash15 18ndash21
Example 3Ordering Real Numbers in a Real-World Context
Exercises 10 16ndash17
Additional ResourcesDifferentiated Instruction includes bull Leveled Practice worksheets
Focus | Coherence | Rigor
Exercise Depth of Knowledge (DOK) Mathematical Practices
12ndash15 1 Recall of Information MP5 Using Tools
16 2 SkillsConcepts MP2 Reasoning
17 2 SkillsConcepts MP6 Precision
18ndash21 2 SkillsConcepts MP2 Reasoning
22 3 Strategic Thinking MP4 Modeling
23ndash24 3 Strategic Thinking MP3 Logic
8NS2
8NS2
Answers1 radic
_ 95 - 5 lt radic
_ 62 - 2
2 radic_
105 3π + 1 105
3 Silvio 5 __ 3 π cm Paula 5 _
4 cm
KD 11
__ 2 cm Luis radic_
33 cm
25 Lesson 13
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
Work Area
3140 3141 3142 3143
314 π227
20 A teacher asks his students to write the numbers shown in order from least to greatest Paul thinks the numbers are already in order Sandra thinks the order should be reversed Who is right
21 Math History There is a famous irrational number called Eulerrsquos number symbolized with an e Like π its decimal form never ends or repeats The first few digits of e are 27182818284
a Between which two square roots of integers could you find this number
b Between which two square roots of integers can you find π
22 Analyze Relationships There are several approximations used for π including 314 and 22 __ 7 π is approximately 314159265358979
a Label π and the two approximations on the number line
b Which of the two approximations is a better estimate for π Explain
c Find a whole number x so that the ratio x ___ 113 is a better estimate for π
than the two given approximations
23 Communicate Mathematical Ideas If a set of six numbers that include both rational and irrational numbers is graphed on a number line what is the fewest number of distinct points that need to be graphed Explain
24 Critique Reasoning Jill says that 12 _
6 is less than 1263 Explain her error
FOCUS ON HIGHER ORDER THINKING
radic_
115 115 ___ 11 and 105624
between radic_
7 asymp 265 and radic_
8 asymp 283
between radic_
9 = 3 and radic_
10 asymp 316
22 __ 7 it is closer to π on the number line
She did not consider the repeating digit 1266
2 rational numbers can have the same location and
irrational numbers can have the same location but they
cannot share a location
355
Neither student is correct The answer
should be 115 ___ 11 105624 radic_
115
Unit 126
copy H
ough
ton M
ifflin
Har
cour
t Pub
lishin
g Com
pany
Imag
e Cre
dits
copy3D
Stoc
kiSt
ockP
hoto
com
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
8_MCAAESE206984_U1M01L3indd 26 210513 801 AM
Personal Math Trainer
Online Practice and Helpmyhrwcom
Name Class Date
Independent Practice
16 Your sister is considering two different shapes for her garden One is a square with side lengths of 35 meters and the other is a circle with a diameter of 4 meters
a Find the area of the square
b Find the area of the circle
c Compare your answers from parts a and b Which garden would give your sister the most space to plant
17 Winnie measured the length of her fatherrsquos ranch four times and got four different distances Her measurements are shown in the table
a To estimate the actual length Winnie first approximated each distance to the nearest hundredth Then she averaged the four numbers Using a calculator find Winniersquos estimate
b Winniersquos father estimated the distance across his ranch to be radic_
56 km How does this distance compare to Winniersquos estimate
Give an example of each type of number
18 a real number between radic_
13 and radic_
14
19 an irrational number between 5 and 7
Order the numbers from least to greatest
12 radic_
7 2 radic_
8 ___ 2 13 radic_
10 π 35
14 radic_
220 -10 radic_
100 115 15 radic_
8 -375 3 9 _ 4
Distance Across Fatherrsquos Ranch (km)
1 2 3 4
radic_
60 58 __ 8 7 _
3 7 3 _ 5
138NS2
radic_
8 ___ 2 2 radic_
7
-10 radic_
100 115 radic_
220
radic_
60 asymp 775 58 __ 8 = 725 7 _
3 asymp 733 7 3 _ 5 = 760 so the average
π radic_
10 35
-375 9 _ 4 radic_
8 3
is 74825 km
1225 m2
4π m2 or approximately 126 m2
They are nearly identical radic_
56 is approximately 74833hellip
The circle would give her more space to plant because it has a
larger area
Sample answer 37
Sample answer radic_
31
25Lesson 13
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ough
ton
Miff
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arco
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ublis
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Com
pany
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
8_MCAAESE206984_U1M01L3indd 25 41613 448 AM
Activity available online myhrwcomEXTEND THE MATH PRE-AP
Activity Have students investigate whether there are infinitely many numbers between two numbers by giving examples for each of the following
bull Between any two rational numbers there is at least one other rational number Sample answer 45 is between 41 and 48
bull Between any two irrational numbers there is at least one rational number Sample answer 45 is between radic
_ 11 and radic
_ 29
bull Between any two rational numbers there is at least one irrational number Sample answer radic
_ 11 is between 31 and 36
bull Between any two irrational numbers there is at least one irrational number Sample answer radic
_ 17 is between radic
_ 11 and radic
_ 29
Ordering Real Numbers 26
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
ReadyMath Trainer
Online Practiceand Help
Personal
myhrwcom
Module Quiz
11ensp RationalenspandenspIrrationalenspNumbersWrite each fraction as a decimal or each decimal as a fraction
1 7__20 2 1___
27 3 17_8
Solve each equation for x
4 x2=81 5 x3=343 6 x2= 1___100
7 Asquarepatiohasanareaof200squarefeetHowlongiseachside
ofthepatiotothenearesttenth
12ensp SetsenspofenspRealenspNumbersWrite all names that apply to each number
8 121____radic
____121
9 π__2
10 TellwhetherthestatementldquoAllintegersarerationalnumbersrdquoistrueorfalseExplainyourchoice
13ensp OrderingenspRealenspNumbersCompare Write lt gt or =
11 radic__
8+3 8+radic__
3 12 radic__
5+11emsp emsp emsp 5+radic___
11
Order the numbers from least to greatest
13 radic___
99π29__
8 14 radic___
1__251_40__
2
15 Howarerealnumbersusedtodescribereal-worldsituations
ESSENTIAL QUESTION
035
9-9
141ft
7 1__10- 1__10
14__11 1875
wholeintegerrationalreal
Trueintegerscanbewrittenasthequotientoftwointegers
SampleanswerRealnumberssuchastherational
π29__
8radic___
99
irrationalreal
lt gt
number1_4candescribeamountsusedincooking
radic___
1__250__
21_4
27Module1
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ough
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pany
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8_MCAAESE206984_U1M01RTindd 27 41513 1113 PM
Math TrainerOnline Assessment
and Intervention
Personal
myhrwcom
1
2
3 Response toIntervention
Intervention Enrichment
Access Ready to Go On assessment online and receive instant scoring feedback and customized intervention or enrichment
Online and Print Resources
Differentiated Instruction
bull Reteach worksheets
bull Reading Strategies EL
bull Success for English Learners EL
Differentiated Instruction
bull Challenge worksheets PRE-AP
Extend the Math PRE-AP
Lesson Activities in TE
Additional ResourcesAssessment Resources includes bull Leveled Module Quizzes
Ready to Go OnAssess MasteryUse the assessment on this page to determine if students have mastered the concepts and standards covered in this module
California Common Core Standards
Lesson Exercises Common Core Standards
11 1ndash7 8NS1 8NS2 8EE2
12 8ndash10 8NS1
13 11ndash14 8NS2
27 Unit 1 Module 1
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
Personal Math Trainer
Online Practice and HelpmyhrwcomAssessment Readiness
Module 1 MIXed ReVIeW
1 Look at each number Is the number between 2π and radic___
52
Select Yes or No for expressions AndashC
A 6 2 _ 3 Yes No
B 5π __ 2 Yes No
C 3 radic__
5 Yes No
2 Consider the number - 11 __ 15
Choose True or False for each statement
A The number is rational True False
B The number can be written as True Falsea repeating decimal
C The number is less than ndash08 True False
3 The volume of a cube is given by V = x3 where x is the length of an edge of the cube A cube-shaped end table has a volume of 3 3 _ 8 cubic feet What is the length of an edge of the end table Explain how you solved this problem
4 A student says that radic___
83 is greater than 29 __ 3 Is the student correct Justify your
reasoning
1 1 _ 2 ft Sample answer The equation x3 = 3 3 _ 8 can be used
to find the edge length in feet To solve the equation
write the mixed number as a fraction greater than 1
x3 = 27 __ 8 Then take the cube root of both sides x = 3 _ 2 = 1 1 _ 2
No Sample answer radic___
83 asymp 91 and 29 __ 3 = 9
__ 6
Because 91 lt 9 __
6 radic___
83 lt 29 __ 3
28 Unit 1
copy H
ough
ton
Miff
lin H
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urt P
ublis
hing
Com
pany
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=A
8_MCAAESE206984_U1M01RTindd 28 240413 946 AM
Personal Math Trainer
Online Assessment and
Interventionmyhrwcom
Scoring GuideItem 3 Award the student 1 point for finding the edge length of the cube and 1 point for correctly explaining how to use a cube root to solve the problem
Item 4 Award the student 1 point for determining that the student is incorrect and 1 point for correctly justifying the reasoning for this conclusion
Additional ResourcesTo assign this assessment online login to your Assignment Manager at myhrwcom
Assessment Readiness
California Common Core Standards
Items Grade 8 Standards Mathematical Practices
1 8NS2 MP7
2 7NS2b 7NS2d 8NS1 MP7
3 8EE2 MP1 MP4
4 8NS1 8NS2 MP3
Item integrates mixed review concepts from previous modules or a previous course
Item 4 combines concepts from the California Common Core cluster ldquoKnow that there are numbers that are not rational and approximate them by rational numbersrdquo
Real Numbers 28
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
YOUAre ReadyPersonal
Math Trainer
Online Practice and Helpmyhrwcom
Complete these exercises to review skills you will need for this module
Find the Square of a NumberEXAMPLE Find the square of 2 _ 3
2 _ 3 times 2 _ 3 = 2 timesthinsp2 ____ 3 timesthinsp3
= 4 _ 9
Find the square of each number
1 7 2 21 3 -3 4 4 _ 5
5 27 6 thinsp- 1 _ 4 7 thinsp-57 8 1 2 _ 5
ExponentsEXAMPLE 5 3 = 5 times 5 times 5
thinsp = 25 times 5 thinsp = 125
Simplify each exponential expression
9 9 2 10 2 4 11 ( 1 _ 3 ) 2 12 (-7) 2
13 4 3 14 (-1) 5 15 45 2 16 10 5
Write a Mixed Number as an Improper FractionEXAMPLE 2 2 _ 5 = 2 + 2 _ 5
thinsp = 10 __ 5 + 2 _ 5
thinsp = 12 __ 5
Write each mixed number as an improper fraction
17 3 1 _ 3 18 1 5 _ 8 19 2 3 _ 7 20 5 5 _ 6
Write the mixed number as a sum of a whole number and a fractionWrite the whole number as an equivalent fraction with the same denominator as the fraction in the mixed numberAdd the numerators
Use the base 5 as a factor 3 timesMultiply from left to right
Multiply the number by itself
Simplify
49 441 9
729
81
64 -1
16
2025 100000
49
16 __ 25
1 _ 9
10 __ 3 13 __ 8 17 __ 7 35 __ 6
1 __ 16 1 24 __ 25 or 1963249
Unit 14
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t Pub
lishin
g Com
pany
DO NOT EDIT--Changes must be made through File infoCorrectionKey=A
8_MCAAESE206984_U1MO01indd 4 230513 448 PM
Math TrainerOnline Assessment
and Intervention
Personal
myhrwcom
1
2
3 Response toIntervention
Professional Development
PROFESSIONAL DEVELOPMENT VIDEO
Are You ReadyAssess ReadinessUse the assessment on this page to determine if students need intensive or strategic intervention for the modulersquos prerequisite skills
myhrwcom
myhrwcom
Interactive WhiteboardsEngage students with interactive whiteboard-ready lessons and activities
Personal Math Trainer Online Assessment and InterventionAssign automatically graded homework quizzes tests and intervention activitiesPrepare your students with updated practice tests aligned with Common Core
Online Teacher EditionAccess a full suite of teaching resources onlinemdashplan present and manage classes and assignments
ePlannerEasily plan your classes and access all your resources online
Interactive Answers and SolutionsCustomize answer keys to print or display in the classroom Choose to include answers only or full solutions to all lesson exercises
Intervention Enrichment
Access Are You Ready assessment online and receive instant scoring feedback and customized intervention or enrichment
Online and Print Resources
Skills Intervention worksheets
bull Skill 11 Find the Square of a Number
bull Skill 12 Exponents
bull Skill 22 Write a Mixed Number as an Improper Fraction
Differentiated Instruction
bull Challenge worksheets PRE-AP
Extend the Math PRE-AP Lesson Activities in TE
Real-World Video Viewing GuideAfter students have watched the video discuss the following bull What are some different ways that biologists classify animals bull What are some classifications of numbers mentioned in the video natural numbers integers rational numbers
Author Juli Dixon models successful teaching practices as she explores the concept of real numbers in an actual eighth-grade classroom
Real Numbers 4
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
Reading Start-Up
Active ReadingLayered Book Before beginning the lessons in this module create a layered book to help you learn the concepts in this module Label the flaps ldquoRational Numbersrdquo ldquoIrrational Numbersrdquo ldquoSquare Rootsrdquo and ldquoReal Numbersrdquo As you study each lesson write important ideas such as vocabulary models and sample problems under the appropriate flap
VocabularyReview Words integers (enteros) negative numbers
(nuacutemeros negativos)positive numbers
(nuacutemeros positivos)whole number (nuacutemero
entero)
Preview Words cube root (raiz cuacutebica) irrational numbers (nuacutemero
irracional) perfect cube (cubo
perfecto) perfect square (cuadrado
perfecto) principal square root (raiacutez
cuadrada principal) rational number (nuacutemero
racional) real numbers (nuacutemero real) repeating decimal (decimal
perioacutedico) square root (raiacutez cuadrada) terminating decimal
(decimal finito)
Visualize VocabularyUse the words to complete the graphic You can put more than one word in each section of the triangle
Understand VocabularyComplete the sentences using the preview words
1 One of the two equal factors of a number is a
2 A has integers as its square roots
3 The is the nonnegative square root of a number
Integers
0 83 308
1 45 192
-21 -78 -93
square root
perfect square
principal square root
whole numbers
negative numbers
positive numberswhole numbers
5Module 1
copy H
ough
ton M
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cour
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lishin
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pany
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8_MCAAESE206984_U1MO01indd 5 180513 1045 AM
Reading Start-Up Have students complete the activities on this page by working alone or with others
Strategies for English LearnersEach lesson in the TE contains specific strategies to help English Learners of all levels succeedEmerging Students at this level typically progress very quickly learning to use English for immediate needs as well as beginning to understand and use academic vocabulary and other features of academic language Expanding Students at this level are challenged to increase their English skills in more contexts and learn a greater variety of vocabulary and linguistic structures applying their growing language skills in more sophisticated ways appropriate to their age and grade level Bridging Students at this level continue to learn and apply a range of high-level English language skills in a wide variety of contexts includ-ing comprehension and production of highly technical texts
Active ReadingIntegrating Language ArtsStudents can use these reading and note-taking strategies to help them organize and understand new concepts and vocabulary
Additional ResourcesDifferentiated Instruction
bull Reading Strategies EL
EL
After
Students will connect that bull the rational numbers are those with decimal expansions that terminate in 0s or eventually repeat
bull non-rational numbers are called irrational numbers
In this moduleStudents will learn how to bull express a rational number as a decimal bull approximate the value of an irrational number bull describe the relationship between sets of real numbers bull order a set of real numbers arising from mathematical and real-world contexts
Before
Students understand bull write rational numbers as decimals bull describe relationships between sets and subsets of rational numbers
bull compare rational numbers
Tracking Your Learning Progression
Focus | Coherence | Rigor
5 Module 1
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
myhrwcom
What It Means to YouYou will learn to estimate the values of irrational numbers
What It Means to YouYou will recognize a number as rational or irrational by looking at its fraction or decimal form
Estimate the value of radic_
8
8 is between the perfect squares 4 and 9So radic
_ 8 is between radic
_ 4 and radic
_ 9
radic_
8 is between 2 and 3
8 is closer to 9 so radic_
8 is closer to 3 28 2 = 784 29 2 = 841 radic
_ 8 is between 28 and 29
A good estimate for radic_
8 is 285
Classify each number as rational or irrational
0 _
3 = 1 _ 3 025 = 1 _ 4
These numbers are rational because they can be written as ratios of integers or as repeating or terminating decimals
π asymp 3141592654hellip radic_ 5 asymp 2236067977hellip
These numbers are irrational because they cannot be written as ratios of integers or as repeating or terminating decimals
Understanding the standards and the vocabulary terms in the standards will help you know exactly what you are expected to learn in this module
Real NumbersGETTING READY FOR
Know that numbers that are not rational are called irrational Understand informally that every number has a decimal expansion for rational numbers show that the decimal expansion repeats eventually and convert a decimal expansion which repeats eventually into a rational number
Key Vocabularyrational number (nuacutemero
racional) A number that can be expressed as a ratio of two integers
irrational number (nuacutemero irracional)A number that cannot be expressed as a ratio of two integers or as a repeating or terminating decimal
Use rational approximations of irrational numbers to compare the size of irrational numbers locate them approximately on a number line diagram and estimate the value of expressions (eg π2)
EXAMPLE 8NS1
EXAMPLE 8NS2
8NS2
8NS1
Visit myhrwcom to see all CA Common Core Standards explained
8 is not a perfect square Find the two perfect squares closest to 8
Unit 16
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Miff
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8_MCABESE206984_U1MO01indd 6 102913 1123 PM
GETTING READY FOR
Real NumbersUse the examples on the page to help students know exactly what they are expected to learn in this module
myhrwcom
California Common Core Standards Lesson 11
Lesson 12
Lesson 13
8NS1 Know that numbers that are not rational are called irrational Understand informally that every number has a decimal expansion for rational numbers show that the decimal expansion repeats eventually and convert a decimal expansion which repeats eventually into a rational number
8NS2 Use rational approximations of irrational numbers to compare the size of irrational numbers locate them approximately on a number line diagram and estimate the value of expressions (eg π2)
8EE2 Use square root and cube root symbols to represent solutions to equations of the form x 2 = p and x 3 = p where p is a positive rational number Evaluate square roots of small perfect squares and cube roots of small perfect cubes Know that radic
_ 2 is irrational
Go online to see a complete unpacking of the CA Common Core Standards
CA Common Core Standards
Content Areas
The Number Systemmdash8NS
Cluster Know that there are numbers that are not rational and approximate them by rational numbers
Real Numbers 6
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=B
Lesson Support Content Objective Students will learn to rewrite rational numbers and decimals take square roots and
cube roots and approximate irrational numbers
Language Objective Students will show and explain how to rewrite rational numbers and decimals take square roots and cube roots and approximate irrational numbers
LESSON 11 Rational and Irrational Numbers
Building BackgroundEliciting Prior Knowledge Have students work with a partner to review the relationship between fractions and decimals Ask students to provide an example of writing a fraction or mixed number as a decimal and vice versa Discuss how students chose and wrote their examples
Learning ProgressionsIn this lesson students work with positive rational and irrational numbers They make connections among the real numbers by converting fractions and decimals and approximating irrational numbers Important understandings for students include the following
bull Understand that every number has a decimal expansion bull Convert a repeating decimal to a rational number bull Evaluate square roots of perfect squares and cube roots of perfect cubes
bull Estimate an irrational number
Work with the real number system will continue in this unit as students extend the positive rational and irrational numbers to include negative numbers and compare and order real numbers
Cluster ConnectionsThis lesson provides an excellent opportunity to connect ideas in this cluster Know that there are numbers that are not rational and approximate them by rational numbers Tell students ldquoA square garden has an area of 20 square feetrdquo
Have students explain why the side length cannot be rational Then have them approximate the length of each side of the garden to the nearest tenth and hundredth Sample answer The length is the solution to s 2 = 20 radic
_ 20 which is not a rational
number 45 ft 447 ft The length is between 4 and 5 feet 20 is closer to 45 2 than to 44 2 or 46 2 It is also closer to 447 2 than to 446 2 or 448 2
3 _ 4
= 075 1 2 _ 3
= 1 _
6
7 _ 10
= 07 45 = 4 1 _ 2
20 ft 2
California Common Core Standards
8NS1 Know that numbers that are not rational are called irrational Understand informally that every number has a decimal expansion for rational numbers show that the decimal expansion repeats eventually and convert a decimal expansion which repeats eventually into a rational number
8NS2 Use rational approximations of irrational numbers to compare the size of irrational numbers locate them approximately on a number line diagram and estimate the value of expressions (eg π 2 )
8EE2 Use square root and cube root symbols to represent solutions to equations of the form x 2 = p and x 3 = p where p is a positive rational number Evaluate square roots of small perfect squares and cube roots of small perfect cubes Know that radic
_ 2 is irrational
MP6 Attend to precision
7A
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=B
Math Talk
Language Support EL
PROFESSIONAL DEVELOPMENT
Linguistic Support EL
AcademicContent Vocabulary
square ndash In this lesson the word square has multiple meanings which can cause confusion For example to square as in to take the square root of a number is a verb It is different from the nouns square or square of a number The text also refers to perfect square and principal square root of a number and the square root symbol is used These different usages of square as a mathematical term need to be clarified Sentence frames can be used to help define the meaning
To square a number means to _______The perfect square of a number means _______
Background Knowledge
suffixes ndash When added to a root word the suffix -th is used in math to indicate one of a specified number of parts such as tenth hundredth or thousandth Remind students that the suffix -th also indicates place value Note that Spanish Vietnamese Mandarin and other languages do not have the ending th sound so teachers need to enunciate carefully
cognates ndash The words terminating and terminal used in this lesson are cognates in Spanish terminar meaning ldquoto endrdquo or ldquoto finishrdquo A Spanish cognate for approximate is aproximar
Leveled Strategies for English Learners
Emerging Use cards with root words ten hundred and thousand and a card with the -th suffix Have students place them together to show place value Then complete a sentence Use the same procedure to identify decimals
Expanding Support students at this level of English proficiency by providing sentence frames for them to use to describe their mathematical reasoning
To write the fraction _______ as a decimal I _______
Bridging Have students identify different meanings of the term square by matching examples of math problems with a written out sentence frame that defines the usage of the term square to square a number perfect square square root Use this procedure also with the term cube
Be sure to clarify the different uses of the term square when referring to square roots perfect squares and so on
EL
California ELD Standards
Emerging 2I12b Selecting language resources ndash Use knowledge of morphology to appropriately select affixes in basic ways
Expanding 2I12b Selecting language resources ndash Use knowledge of morphology to appropriately select affixes in a growing number of ways to manipulate language
Bridging 2I12b Selecting language resources ndash Use knowledge of morphology to appropriately select affixes in a variety of ways to manipulate language
Rational and Irrational Numbers 7B
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
11L E S S O N
Rational and Irrational Numbers
EngageESSENTIAL QUESTION
How do you rewrite rational numbers and decimals take square roots and cube roots and approximate irrational numbers To express as a decimal divide the numerator by the denominator To take a square root or cube root of a number find the number that when squared or cubed equals the original number To approximate an irrational number estimate a number between two consecutive perfect squares
Motivate the LessonAsk Which type of rational number do you see more often fractions or decimals Which do you prefer to use Why
ExploreHave students write examples of ratios and then share with the class the various notations for ratios that they used (for example 25 2 to 5 2 __ 5 ) Point out the connection between the word ratio and the meaning of rational number See also Explore Activity in student text
ExplainEXAMPLE 1
Questioning Strategies Mathematical Practices bull How does the denominator of a fraction in simplest form tell whether the decimal equivalent of the fraction is a terminating decimal The decimal will terminate if the denominator is an even number a multiple of 5 or a multiple of 10
Avoid Common ErrorsTo avoid interpreting 1 __ 4 as 4 divided by 1 tell students to start at the top of the fraction and read the bar as ldquodivided byrdquo
YOUR TURNTalk About ItCheck for Understanding
Ask Can an improper fraction be written as a decimal Give an example to support your answer Yes 5 __ 4 = 125
EXAMPLE 2Questioning Strategies Mathematical Practices bull How can you use place value to write a terminating decimal as a fraction with a power of ten in the denominator Start by identifying the place value of the decimals last digit and then use the corresponding power of 10 as the denominator of the fraction
bull How can you tell if a decimal can be written as a rational number If the decimal is a terminating or repeating decimal then it can be written as a rational number
Interactive Whiteboard Interactive example available online
ADDITIONAL EXAMPLE 1Write each fraction as a decimal
A 2 _ 5
04 B 5 _ 9
0 _
5
myhrwcom
Interactive Whiteboard Interactive example available online
ADDITIONAL EXAMPLE 2Write each decimal as a fraction in simplest form
A 0355 71 ___ 200
B 0 _
43 43 __ 99
myhrwcom
CA Common CoreStandards
The student is expected to
The Number Systemmdash8NS1
Know that numbers that are not rational are called irrational Understand informally that every number has a decimal expansion for rational numbers show that the decimal expansion repeats eventually and convert a decimal expansion which repeats eventually into a rational number
The Number Systemmdash8NS2
Use rational approximations of irrational numbers to compare the size of irrational numbers locate them approximately on a number line diagram and estimate the value of expressions (eg π 2 )
Expressions and Equationsmdash8EE2
Use square root and cube root symbols to represent solutions to equations of the form x 2 = p and x 3 = p where p is a positive rational number Evaluate square roots of small perfect squares and cube roots of small perfect cubes Know that radic
_ 2 is irrational
Mathematical Practices
MP6 Precision
The student is expected to
the value of expressions (eg
7 Lesson 11
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
My Notes
Math On the Spotmyhrwcom
Math TrainerOnline Practice
and Help
Personal
myhrwcom
Expressing Decimals as Rational NumbersYou can express terminating and repeating decimals as rational numbers
Write each decimal as a fraction in simplest form
0825
The decimal 0825 means ldquo825 thousandthsrdquo Write this as a fraction
825 ____ 1000
Then simplify the fraction
825 divide 25 ________ 1000 divide 25 = 33 __ 40
0825 = 33 __ 40
0 _
37
Let x = 0 _
37 The number 0 _
37 has 2 repeating digits so multiply each side of the equation x = 0
_ 37 by 10 2 or 100
x = 0 _
37
(100)x = 100(0 _
37 )
100x = 37 _
37
Because x = 0 _
37 you can subtract x from one side and 0 _
37 from the other
100x = 37 _
37
minusx minus0 _
37
99x = 37
Now solve the equation for x Simplify if necessary
99x ___ 99 = 37 __ 99
x = 37 __ 99
EXAMPLE 2
A
B
Write each fraction as a decimal
YOUR TURN
1 5 __ 11 2 1 _ 8 3 2 1 _ 3
8NS1
To write ldquo825 thousandthsrdquo put 825 over 1000
Divide both the numerator and the denominator by 25
100 times 0 _
37 is 37 _
37
37 _
37 minus 0 _
37 is 37
Divide both sides of the equation by 99
0 _
45 0125 2 _
3
Unit 18
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Miff
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pany
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8_MCAAESE206984_U1M01L1indd 8 120413 838 PM
My Notes
Math On the Spot
myhrwcom
= 033333333333331mdash3
ESSENTIAL QUESTION
Expressing Rational Numbers as DecimalsA rational number is any number that can be written as a ratio in the form a _ b where a and b are integers and b is not 0 Examples of rational numbers are 6 and 05
6 can be written as 6 _ 1 05 can be written as 1 _ 2
Every rational number can be written as a terminating decimal or a repeating decimal A terminating decimal such as 05 has a finite number of digits A repeating decimal has a block of one or more digits that repeat indefinitely
Write each fraction as a decimal
1 _ 4
1 _ 4 = 025
1 _ 3
1 _ 3 = 0 _
3
EXAMPLEXAMPLE 1
A
B
0333 3 ⟌ ⎯ 1000 minus9 10 minus9 10 minus9 1
025 4 ⟌ ⎯ 100 -8 20 -20
0
L E S S O N
11Rational and Irrational Numbers
How do you rewrite rational numbers and decimals take square roots and cube roots and approximate irrational numbers
8NS1
Know that numbers that are not rational are called irrational Understand informally that every number has a decimal expansion for rational numbers show that the decimal expansion repeats eventually and convert a decimal expansion which repeats eventually into a relation number Also 8NS2 8EE2
8NS1
Remember that the fraction bar means ldquodivided byrdquo Divide the numerator by the denominator
Divide until the remainder is zero adding zeros after the decimal point in the dividend as needed
Divide until the remainder is zero or until the digits in the quotient begin to repeat
Add zeros after the decimal point in the dividend as needed
When a decimal has one or more digits that repeat indefinitely write the decimal with a bar over the repeating digit(s)
7Lesson 11
copy H
ough
ton
Miff
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pany
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=B
8_MCABESE206984_U1M01L1indd 7 11113 128 AM
PROFESSIONAL DEVELOPMENT
Math BackgroundSome decimals may have a pattern but still not be a repeating decimal that is rational For example in 312112111211112hellip you can predict the next digit and describe the pattern (There is one more 1 each time before the 2) However this is not a terminating decimal nor is it a repeating decimal and it is therefore NOT a rational number
Integrate Mathematical Practices MP6
This lesson provides an opportunity to address this Mathematical Practices standard It calls for students to attend to precision Students learn to express rational numbers accurately and precisely in both fractional and decimal forms and learn to translate from one form to the other They also learn how to precisely represent and communicate ideas about irrational numbers square roots and cube roots
Rational and Irrational Numbers 8
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=B
Focus on Technology Mathematical PracticesPoint out the importance of entering a repeating decimal correctly when using a graphing calculator to convert the decimal to a fraction The decimal 0
_ 59 must be entered as
0595959595959 not 059
YOUR TURNFocus on Math ConnectionsMake sure students understand that the place value of the last digit in Exercises 4 and 6 determines the denominator of the corresponding fraction or mixed number So for Exercise 4 the place value hundredths gives a denominator of 100 and for Exercise 6 the place value tenths gives a denominator of 10
EXAMPLE 3Questioning Strategies Mathematical Practices bull How can a solution of an equation of the form x 2 = p be negative if p is a positive number Since the square of a negative number is positive a negative number is also a solution of x 2 equals a positive number
bull When is a solution of an equation of the form x 3 = p larger than p The solution is larger than p if p is a number between 0 and 1
Focus on Math Connections Make sure students understand the difference in finding radic
_ 121 and solving x 2 = 121 The
symbol radic_
indicates the positive or principal square root only while the equation x 2 = 121 has two roots the principal square root and its opposite
YOUR TURNAvoid Common ErrorsTo avoid sign errors in Exercise 9 make sure that students understand that the cube of a negative number is not a positive number Therefore -8 is not a solution of x 3 = 512
Talk About ItCheck for Understanding
Ask Kris predicts that there are two real solutions for Exercises 7 and 8 and that there are three real solutions for Exercises 9 and 10 Is his prediction correct
Explain His prediction is correct for Exercises 7 and 8 because there are two numbers whose squares are the same positive number given in the exercises His prediction is not correct for Exercises 9 and 10 however because there is only one real number whose cube is the same positive number given in the exercises
EXPLORE ACTIVITYQuestioning Strategies Mathematical Practices bull Compare the values for 13 2 and 13 2 The digits are the same but 13 2 has two decimal places (169) while 13 2 has none (169)
bull How do you know whether radic_
2 will be closer to 1 or closer to 2 It will be closer to 1 because 2 is between the perfect squares of 1 and 4 but closer to 1 than it is to 4
Connect Vocabulary EL
Explain to students that the word irrational when used as an ordinary word in English means without logic or reason In mathematics when we say that a number is irrational it means only that the number cannot be written as the quotient of two integers
Engage with the WhiteboardHave students extend the number line in both directions and label the locations of the whole numbers 1 and 2 These are the roots of the consecutive perfect squares
1 and 4 used to estimate radic_
7
Interactive Whiteboard Interactive example available online
ADDITIONAL EXAMPLE 3Solve each equation for x
A x 2 = 324 18 -18
B x 2 = 25 ___ 144 5 __ 12 - 5 __ 12
C 343 = x 3 7
D x 3 = 125 ___ 512 5 __ 8
myhrwcom
9 Lesson 11
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
Math TrainerOnline Practice
and Help
Personal
myhrwcom
EXPLORE ACTIVITY
lt 2 lt
radic_
lt radic
_ 2 lt
radic_
lt radic
_ 2 lt
The solution is 9
The solution is 2 _ 5
C
D
729 = x 3
3 radic_ 729 = 3 radic
_ x 3
3 radic_ 729 = x
9 = x
x 3 = 8 ___ 125
3 radic_
x 3 =thinsp 3 radic_ 8 ___ 125
x =thinsp 3 radic_ 8 ___ 125
x = 2 _ 5
Solve each equation for x
YOUR TURN
7 x 2 = 196 8 x 2 = 9 ___ 256
9 x 3 = 512 10 x 3 = 64 ___ 343
Estimating Irrational NumbersIrrational numbers are numbers that are not rational In other words they cannot be written in the form a _ b where a and b are integers and b is not 0 Square roots of perfect squares are rational numbers Square roots of numbers that are not perfect squares are irrational Some equations like those in Example 3 involve square roots of numbers that are not perfect squares
x 2 = 2 x = plusmn radic_
2
Estimate the value of radic_
2
Find two consecutive perfect squares that 2 is between Complete the inequality by writing these perfect squares in the boxes
Now take the square root of each number
Simplify the square roots of perfect squares
radic_
2 is between and
A
B
C
8NS2 8EE2
Solve for x by taking the cube root of both sides
Solve for x by taking the cube root of both sides
Apply the definition of cube root
Think What number cubed equals 729
Apply the definition of cube root
Think What number cubed equals 8 ____ 125
radic_
2 is irrational
x = plusmn14 x = plusmn 3 __ 16
x = 8 x = 4 _ 7
1 2
1 4
1 4
1 2
Unit 110
copy H
ough
ton
Miff
lin H
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urt P
ublis
hing
Com
pany
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
8_MCAAESE206984_U1M01L1indd 10 41613 1211 AM
Math TrainerOnline Practice
and Help
Personal
myhrwcom
Math On the Spot
myhrwcom
Write each decimal as a fraction in simplest form
YOUR TURN
Finding Square Roots and Cube RootsThe square root of a positive number p is x if x 2 = p There are two square roots for every positive number For example the square roots of 36 are 6 and minus6 because 6 2 = 36 and (minus6) 2 = 36 The square roots of 1 __ 25 are 1 _ 5 and minus 1 _ 5 You can write the square roots of 1 __ 25 as plusmn 1 _ 5 The symbol radic
_ 5 indicates the positive
or principal square root
A number that is a perfect square has square roots that are integers The number 81 is a perfect square because its square roots are 9 and minus9
The cube root of a positive number p is x if x 3 = p There is one cube root for every positive number For example the cube root of 8 is 2 because 2 3 = 8 The cube root of 1 __ 27 is 1 _ 3 because ( 1 _ 3 )
3
= 1 __ 27 The symbol 3 radic_ 1 indicates the
cube root
A number that is a perfect cube has a cube root that is an integer The number 125 is a perfect cube because its cube root is 5
Solve each equation for x
The solutions are 11 and minus11
The solutions are 4 __ 13 and minus 4 __ 13
EXAMPLEXAMPLE 3
A x 2 = 121
x 2 = 121
x = plusmn radic_
121
x = plusmn11
B x 2 = 16 ___ 169
x 2 = 16 ___ 169
x = plusmn radic_
16 ___ 169
x = plusmn 4 __ 13
4 012 5 0 _
57 6 14
Can you square an integer and get a negative number
What does this indicate about whether negative
numbers have square roots
Math TalkMathematical Practices
8EE2
Solve for x by taking the square root of both sides
Apply the definition of square root
Think What numbers squared equal 121
Solve for x by taking the square root of both sides
Apply the definition of square root
Think What numbers squared equal 16 ____ 169
3 __ 25 19 __ 33 1 2 _ 5
No the square of a positive integer is positive the square of a negative integer is positive and the square of 0 is 0 So negative numbers do not have (real) square roots
9Lesson 11
copy H
ough
ton
Miff
lin H
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urt P
ublis
hing
Com
pany
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
8_MCAAESE206984_U1M01L1indd 9 41913 240 PM
Critical ThinkingIn the Explore Activity students estimated the location of radic
_ 2 on a number line Ask students
whether they think that it is possible to locate more precisely the point that represents radic
_ 2 In
other words can you graph irrational numbers exactly on a number line along with rational numbers Students should understand that radic
_ 2
is a real number and all real numbers can be located on a real number line A more precise estimate will allow more precise placement on a number line
The Modeling note tells one way to do this
ModelingHave students use a ruler to represent a number line with a unit that is one inch long Have them draw a square with a side of one inch and draw the diagonal to make two isosceles triangles Lead students to understand that the length of the diagonal (or hypotenuse) is radic
_ 2
Have them copy the length of their diagonal onto their ruler or number line starting at zero The end point of the diagonal represents the exact point for the irrational number radic
_ 2 on a
number line
Additional ResourcesDifferentiated Instruction includes bull Reading Strategies bull Success for English Learners EL
bull Reteach bull Challenge PRE-AP
DIFFERENTIATE INSTRUCTION
Rational and Irrational Numbers 10
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
ElaborateTalk About ItSummarize the Lesson
Ask If someone claims that a certain number is irrational but you know it is actually rational how could you prove to that person that the number is rational
You could find a fraction equal to the number such that the number is the ratio of two integers with the denominator not equal to zero
GUIDED PRACTICEEngage with the Whiteboard
Have students plot each number in Exercises 16ndash18 on a number line Students should label each point with the irrational number written as a radical and as a
decimal
Avoid Common ErrorsExercises 1ndash6 To avoid reversing the order of the dividend and divisor tell students to start at the top of the fraction and read the bar as ldquodivided byrdquo
Focus on TechnologyHave students use a calculator to investigate the decimal equivalents of such fractions as 1 __ 9 2 __ 9 8 __ 9 and 1 __ 11 2 __ 11 10
__ 11 Ask them to describe the patterns they find as a result of these investigations
11 Lesson 11
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
Guided Practice
7 0675 8 56 9 044
10 0 _
4
10x =
x =
11 0 _
26
100x =
x =
12 0 _
325
1000x =
x =
Solve each equation for x (Example 3 and Explore Activity)
- x
-
_______________
x =
- x
-
___________________
x =
- x
-
_______________________
x =
Write each fraction or mixed number as a decimal (Example 1)
1 2 _ 5 2 8 _ 9 3 3 3 _ 4
4 7 __ 10 5 2 3 _ 8 6 5 _ 6
Write each decimal as a fraction or mixed number in simplest form (Example 2)
13 x 2 = 17 14 x 2 = 25 ___ 289 15 x 3 = 216
Approximate each irrational number to one decimal place without a calculator
x = plusmn radic__
asymp plusmn x = 3
radic__
=
(Explore Activity)
16 radic_
5 asymp
17 radic_
3 asymp
18 radic_
10 asymp
19 What is the difference between rational and irrational numbers
CHECK-INESSENTIAL QUESTION
x = plusmn radic__
__________ = plusmn _____
4 _
4
0 _
4
4 99
6216
269
41 25 5
17289
17
22 17 32
04
07
27__40
26 __ 99 325 ___ 999 4 _ 9
11__255 3_5
0 _
8
2375
375
08 _
3
26 _
26
0 _
26
325 _
325
0 _
325
999 325
Rational numbers can be written in the form a __ b where
a and b are integers and b ne 0 Irrational numbers cannot
be written in this form
Unit 112
copy H
ough
ton
Miff
lin H
arco
urt P
ublis
hing
Com
pany
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
8_MCAAESE206984_U1M01L1indd 12 41613 1211 AM
11 12 13 14 15
radic2 asymp 14
141 142 143 144 145
radic2 asymp 141
0 1 2 3 4
radic2 asymp 15
Estimate that radic_
2 asymp 15
To find a better estimate first choose some numbers between 1 and 2 and square them For example choose 13 14 and 15
1 3 2 = 1 4 2 = 1 5 2 =
Is radic_
2 between 13 and 14 How do you know
Is radic_
2 between 14 and 15 How do you know
2 is closer to than to so radic_
2 asymp
Locate and label this value on the number line
Reflect 11 How could you find an even better estimate of radic
_ 2
12 Find a better estimate of radic_
2
1 41 2 = 1 42 2 = 1 43 2 =
2 is closer to than to so radic_
2 asymp
Draw a number line and locate and label your estimate
13 Solve x 2 = 7 Write your answer as a radical expression Then estimate to one decimal place
D
E
F
No 2 is not between 169 and 196
Yes 2 is between 196 and 225
196
19881
19881
225
20164
20164
14
141
20449
169 196 225
Test the squares of numbers between 14 and 15
x = plusmn radic_
7 x asymp plusmn26
11Lesson 11
copy H
ough
ton
Miff
lin H
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urt P
ublis
hing
Com
pany
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
8_MCAAESE206984_U1M01L1indd 11 41613 1211 AM
Rational and Irrational Numbers 12
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
Math TrainerOnline Assessment
and Intervention
Personal
Online homework assignment available
myhrwcom
EvaluateGUIDED AND INDEPENDENT PRACTICE
Concepts amp Skills Practice
Example 1Expressing Rational Numbers as Decimals
Exercises 1ndash6 20ndash21 24ndash25
Example 2Expressing Decimals as Rational Numbers
Exercises 7ndash12 22ndash23 26ndash27
Example 3Finding Square Roots and Cube Roots
Exercises 13ndash15 28 30ndash31 35
Explore ActivityEstimating Irrational Numbers
Exercises 13 16ndash18 29 32ndash34
Additional ResourcesDifferentiated Instruction includes bull Leveled Practice worksheets
Lesson Quiz available online
11 LESSON QUIZ
1 Write as a decimal 2 5 __ 8 1 7 __ 12
2 Write as a fraction 034 1 _
24
3 Solve x 2 = 9 __ 49 for x
4 Solve x 3 = 216 for x
5 Estimate the value of radic_
13 to one decimal place without using a calculator
myhrwcom
Focus | Coherence | Rigor
Exercise Depth of Knowledge (DOK) Mathematical Practices
20ndash27 2 SkillsConcepts MP4 Modeling
28 3 Strategic Thinking MP4 Modeling
29ndash32 2 SkillsConcepts MP6 Precision
33 3 Strategic Thinking MP7 Using Structure
34 2 SkillsConcepts MP3 Logic
35 2 SkillsConcepts MP4 Modeling
36 3 Strategic Thinking MP3 Logic
37 3 Strategic Thinking MP7 Using Structure
38 3 Strategic Thinking MP2 Reasoning
8NS1 8NS2 8EE2
8NS1 8NS2 8EE2
Answers1 2625 158
_ 3
2 17 __ 50 1 8 __ 33
3 x = plusmn 3 __ 7
4 x = 6
5 36
13 Lesson 11
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
Work Area
33 Analyze Relationships To find radic_
15 Beau found 3 2 = 9 and 4 2 = 16 He said that since 15 is between 9 and 16 radic
_ 15 must be between 3 and 4 He
thinks a good estimate for radic_
15 is 3 + 4 ____ 2 = 35 Is Beaursquos estimate high low
or correct Explain
34 Justify Reasoning What is a good estimate for the solution to the equation x 3 = 95 How did you come up with your estimate
35 The volume of a sphere is 36π f t 3 What is the radius of the sphere Use the formula V = 4 _ 3 π r 3 to find your answer
36 Draw Conclusions Can you find the cube root of a negative number If so is it positive or negative Explain your reasoning
37 Make a Conjecture Evaluate and compare the following expressions
radic_
4 __ 25 and radic
_ 4 ____
radic_
25 radic
_
16 __ 81 and radic_
16 ____
radic_
81 radic
_
36 __ 49 and radic_
36 ____
radic_
49
Use your results to make a conjecture about a division rule for square roots Since division is multiplication by the reciprocal make a conjecture about a multiplication rule for square roots
38 Persevere in Problem Solving The difference between the solutions to the equation x 2 = a is 30 What is a Show that your answer is correct
FOCUS ON HIGHER ORDER THINKING
His estimate is low because 15 is very close to 16
so radic_
15 is very close to radic_
16 or 4 A better estimate
would be 38 or 39
Sample answer about 45 4 3 = 64 and 5 3 = 125
Because 95 is about halfway between 64 and 125 try 45
45 3 = 91125 which is a good estimate
3 feet
Yes the cube root of a negative number is negative
because a negative number cubed is always negative
and a nonnegative number cubed is always nonnegative
radic_
4 __ 25 = 2 _ 5 = radic
_ 4 ____
radic_
25 radic
_
16 __ 81 = 4 _ 9 = radic_
16 ____
radic_
81 radic
_
36 __ 49 = 6 _ 7 = radic_
36 ____
radic_
49
225 the solutions to x 2 = a are x = plusmn15 and
radic_
a ___
radic_
b = radic
_ a __
b radic
_ a radic
_ b = radic
_ a b
15 - (-15) = 30
Unit 114
copy H
ough
ton
Miff
lin H
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ublis
hing
Com
pany
bull copy
Ilen
e Mac
Dona
ldA
lamy I
mag
es
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=B
8_MCABESE206984_U1M01L1indd 14 102913 1142 PM
Personal Math Trainer
Online Practice and Helpmyhrwcom
Name Class Date
Independent Practice11
20 A 7 __ 16 -inch-long bolt is used in a machine What is this length written as a decimal
21 The weight of an object on the moon is 1 _ 6 its weight on Earth Write 1 _ 6 as a decimal
22 The distance to the nearest gas station is 2 4 _ 5 kilometers What is this distance written as a decimal
23 A baseball pitcher has pitched 98 2 _ 3 innings What is the number of innings written as a decimal
24 A heartbeat takes 08 second How many seconds is this written as a fraction
25 There are 262 miles in a marathon Write the number of miles using a fraction
26 The average score on a biology test was 72
_ 1 Write the average score using a
fraction
27 The metal in a penny is worth about 0505 cent How many cents is this written as a fraction
28 Multistep An artist wants to frame a square painting with an area of 400 square inches She wants to know the length of the wood trim that is needed to go around the painting
a If x is the length of one side of the painting what equation can you set up to find the length of a side How many solutions does the equation have
b Do all of the solutions that you found make sense in the context of the problem Explain
c What is the length of the wood trim needed to go around the painting
Solve each equation for x Write your answers as radical expressions Then estimate to one decimal place if necessary
29 x 2 = 14 30 x 3 = 1331
31 x 2 = 144 32 x 2 = 29
8NS1 8NS2 8EE2
04375 in 01 _6
28 km 98 _6 innings
x 2 = 400 x = plusmnthinsp20 the equation has 2 solutions
x = 20 makes sense but x = -20 doesnrsquot because a
painting cannot have a side length of -20 inches
4 times 20 = 80 inches
x = plusmn radic_
14 asymp plusmn37
x = plusmn radic_
144 = plusmn12 x = plusmn radic_
29 asymp plusmn54
x = 3 radic_ 1331 = 11
4_5 second 26 1_5 mi
72 1 _ 9 101 ___ 200 cent
13Lesson 11
copy H
ough
ton
Miff
lin H
arco
urt P
ublis
hing
Com
pany
bull copy
Phot
odisc
Get
ty Im
ages
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
8_MCAAESE206984_U1M01L1indd 13 41613 1211 AM
myhrwcomActivity available onlineEXTEND THE MATH PRE-AP
Activity Write radic_
09 on the board and invite students to conjecture what the value might be Have them check their conjectures by squaring Invite them to suggest ways to estimate radic
_ 09 As a hint point out that 09 is close to 10 and so they might
use that to help guide their estimates Lead them to see that since 092 is 081 and 102 is 1 the value of radic
_ 09 is greater than 09 and less than 10 Try squaring 095 to get
09025 A good estimate for radic_
09 is 095
Rational and Irrational Numbers 14
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=B
Integers
Rational Numbers IrrationalNumbers
Real Numbers
WholeNumbers
-3-4-5 -2-1 1 2 3 50 4
23
34-4 -π -1 25
radic2
Lesson Support Content Objective Students will learn to describe relationships between sets of numbers
Language Objective Students will explain how to describe relationships between sets of real numbers
LESSON 12 Sets of Real Numbers
Building BackgroundEliciting Prior Knowledge Have students draw a number line from -5 to 5 Ask them to plot points on the number line to approximate the location of rational and irrational numbers such as -1 3 __ 4 25 -4 2 __ 3 radic
_ 2 and -π
Learning ProgressionsIn this lesson students clarify their understanding of the real number system They characterize sets and subsets of the real numbers They also identify sets for real-world situations Important understandings for students include the following
bull Identify all of the possible subsets of the real numbers for a given number
bull Decide whether a statement about a subset of the real numbers is true or false
bull Identify the set of numbers that best describes a real-world situation
Understanding the relationships among the sets of numbers that make up the real numbers is essential as students are introduced to different forms of numbers throughout the school year This lesson provides a foundation for the comparing and ordering of real numbers in the next lesson
Cluster ConnectionsThis lesson provides an excellent opportunity to connect ideas in this cluster Know that there are numbers that are not rational and approximate them by rational numbers Have students copy this diagram which relates the sets of real numbers
Ask students to complete the diagram by writing three examples for each set of numbers Have students share examples and explain how they knew each number they selected belonged in the appropriate set Answers may vary Check studentsrsquo work
Focus | Coherence | Rigor
California Common Core Standards
8NS1 Know that numbers that are not rational are called irrational Understand informally that every number has a decimal expansion for rational numbers show that the decimal expansion repeats eventually and convert a decimal expansion which repeats eventually into a rational number
MP7 Look for and make use of structure
15A
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
Math Talk
Language Support EL
PROFESSIONAL DEVELOPMENT
Linguistic Support EL
AcademicContent Vocabulary
Venn diagrams ndash Students need descriptive language to describe the categories that the different areas and colors of a Venn diagram represent the concept of a set and how sets are distinct or can overlap Use sentence frames such as
The big oval represents __________The darklight blue color in the middle of the
big ovals represents __________These sets overlap because __________
In this way students have the language and structure to identify the criteria that distinguish a set and to explain the abstract representation Also point out the use of the prefix sub- meaning ldquounderrdquo in the term subset
Rules and Patterns
Abbreviations ndash In this lesson the abbreviation mph is used Be sure to point out that mph stands for miles per hour and is used to give units in a rate of speed Students may also have seen mpg (miles per gallon) which gives the units in a rate of fuel efficiency
Borrowed Words ndash Terminology used in baseball such as inning and pitcher may require some explanation Spanish as well as some other languages have borrowed these terms from English so some students may be familiar with these words already Despite this whenever a word is critical to students understanding the word problem it is best to explain the meaning
Leveled Strategies for English Learners
Emerging Allow students to indicate true or false orally in Guided Practice Exercises 9 and 10
Expanding Have students use sentence frames to describe the meaning of regions and colors used in a Venn diagram Then give them similar sentence frames orally and have them draw and shade a Venn diagram based on the oral prompts
Bridging Have students work in groups to draw a Venn diagram to represent sets based on real-world examples in the lesson
To help students answer the question posed in Math Talk provide a sentence frame for their answer
The numbers between 31 and 39 on a number line are __________ because __________
EL
California ELD Standards
Emerging 2II5 Modifying to add details ndash Expand sentences with simple adverbials to provide details about a familiar activity or process
Expanding 2II5 Modifying to add details ndash Expand sentences with adverbials to provide details about a familiar or new activity or process
Bridging 2II5 Modifying to add details ndash Expand sentences with increasingly complex adverbials to provide details about a variety of familiar and new activities and processes
Sets of Real Numbers 15B
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
12L E S S O N
Sets of Real Numbers
EngageESSENTIAL QUESTION
How can you describe relationships between sets of real numbers Sample answer Describe them as two different sets or one set as being a subset of another
Motivate the LessonAsk How many different types of tigers can you name How does the set of Bengal tigers relate to the set of tigers
ExplorePoint to different locations in the Animals diagram and ask for examples for that classification Do the same for the Real Numbers diagram Students should understand that everything within a region is part of the set for example both -3 and 2 are integers
ExplainEXAMPLE 1
Questioning Strategies Mathematical Practices bull In A why is 5 not a perfect square It does not have rational numbers as its square roots
bull Can the number in B be written as a fraction Why or why not Yes it is a terminating decimal so it is a rational number
Engage with the WhiteboardHave students place the numbers in Example 1 and Additional Example 1 in the Venn diagram for numbers
YOUR TURNAvoid Common ErrorsBe sure that students read Exercise 2 carefully before answering The number given in the problem 10 is the area not the side length
EXAMPLE 2Questioning Strategies Mathematical Practices bull What two major sets are the real numbers composed of rational and irrational numbers
bull What is the location of the set of whole numbers in the Venn diagram in relation to the set of rational numbers Explain Inside it whole numbers are rational numbers
Focus on Reasoning Mathematical PracticesRemind students that it takes only one counterexample to show that a statement is false
Interactive Whiteboard Interactive example available online
ADDITIONAL EXAMPLE 1Write all names that apply to each number
A -10integer rational real
B 12 _ 3
whole integer rational real
myhrwcom
Interactive Whiteboard Interactive example available online
ADDITIONAL EXAMPLE 2Tell whether the given statement is true or false Explain your choice
No integers are whole numbers
False every whole number is also an integer
myhrwcom
Animated MathClassifying Numbers
Students build fluency in classifying numbers in this engaging fast-paced game
myhrwcom
CA Common CoreStandards
The student is expected to
The Number Systemmdash8NS1
Know that numbers that are not rational are called irrational Understand informally that every number has a decimal expansion for rational numbers show that the decimal expansion repeats eventually and convert a decimal expansion which repeats eventually into a rational numberMathematical Practices
MP7 Using Structure
The student is expected to
15 Lesson 12
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=B
Math TrainerOnline Practice
and Help
Personal
myhrwcom
Math TrainerOnline Practice
and Help
Personal
myhrwcom
Math On the Spotmyhrwcom
Understanding Sets and Subsets of Real NumbersBy understanding which sets are subsets of types of numbers you can verify whether statements about the relationships between sets are true or false
Tell whether the given statement is true or false Explain your choice
All irrational numbers are real numbers
True Every irrational number is included in the set of real numbers The irrational numbers are a subset of the real numbers
No rational numbers are whole numbers
False A whole number can be written as a fraction with a denominator of 1 so every whole number is included in the set of rational numbers The whole numbers are a subset of the rational numbers
EXAMPLE 2
A
B
Write all names that apply to each number
1 A baseball pitcher has pitched 12 2 _ 3 innings
2 The length of the side of a square that has an
area of 10 square yards
YOUR TURN
Tell whether the given statement is true or false Explain your choice
3 All rational numbers are integers
4 Some irrational numbers are integers
YOUR TURN
Give an example of a rational number that is a
whole number Show that the number is both whole
and rational
Math TalkMathematical Practices
Give an example of a
8NS1
False Every integer is a rational number but not every
False Real numbers are either rational or irrational numbers
Integers are rational numbers so no integers are irrational numbers
rational real
irrational real
Sample answer 8 8 = 8_
1
and -thinsp 5 _ 2 are not integers
rational number is an integer Rational numbers such as 3 _ 5
Unit 116
copy H
ough
ton
Miff
lin H
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ublis
hing
Com
pany
bull Im
age C
redi
ts D
igita
l Im
age c
opyr
ight
copy20
04 Ey
ewire
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
8_MCAAESE206984_U1M01L2indd 16 41613 136 AM
Math On the Spot
myhrwcom
Vertebrates
Birds
Passerines
Animals
Integers
Rational Numbers IrrationalNumbers
Real Numbers
WholeNumbers
1
45
3
0
274
67
radic4
-
-3
-2
-1
03
radic2
radic17
radic11-
π
Animated Math
myhrwcom
Classifying Real NumbersBiologists classify animals based on shared characteristics A cardinal is an animal a vertebrate a bird and a passerine
You already know that the set of rational numbers consists of whole numbers integers and fractions The set of real numbers consists of the set of rational numbers and the set of irrational numbers
Write all names that apply to each number
radic_
5 irrational real
ndash1784rational real
whole integer rational real
EXAMPLEXAMPLE 1
A
B
C radic_ 81 ____ 9
L E S S O N
12Sets of Real Numbers
ESSENTIAL QUESTIONHow can you describe relationships between sets of real numbers
Passerines such as the cardinal are also called ldquoperching birdsrdquo
What types of numbers are between 31 and 39 on a
number line
Math TalkMathematical Practices
What types of numbers are
8NS1
8NS1
Know that numbers that are not rational are called irrational Understand informally that every number has a decimal expansion for rational numbers show that the decimal expansion repeats eventually and convert a decimal expansion which repeats eventually into a relation number
ndash1784 is a terminating decimal
5 is a whole number that is not a perfect square
radic_
81 _____ 9 = 9 __ 9 = 1 rational irrational real
15Lesson 12
copy H
ough
ton
Miff
lin H
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ublis
hing
Com
pany
bull Im
age C
redi
ts copy
Wiki
med
ia Co
mm
ons
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=B
8_MCABESE206984_U1M01L2indd 15 061113 1144 AM
PROFESSIONAL DEVELOPMENT
Math BackgroundThe relationships between sets of numbers extend to include complex numbers A complex number can be written as a sum of a real number a and an imaginary number bi
a + bi
An imaginary number is a special number that when squared gives a negative value When you square a real number you get a nonnegative number When you square an imaginary number you get a negative value The imaginary unit is i
i = radic_
-1
Integrate Mathematical Practices MP7
This lesson provides an opportunity to address this Mathematical Practices standard It calls for students to discern structure to connect and communicate mathematical ideas
Students use a Venn diagram to structure relationships between sets of numbers They connect and communicate mathematical ideas when they make logical statements about the sets and describe which set best describes numbers applied to real-life situations
Sets of Real Numbers 16
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=B
YOUR TURNAvoid Common ErrorsStudents may see the word ldquoAllldquo or rdquoNordquo in Exercises 3 and 4 and immediately assume that any absolute statements like these are false Remind them that there are true statements that begin with these words and encourage them to provide examples
EXAMPLE 3Questioning Strategies Mathematical Practices bull In A how does the phrase ldquonumber of rdquo give you a clue about the number classification It indicates a counting number
bull What is the relationship between the circumference of a circle and the diameter The circumference is diameter times π
Focus on Critical Thinking Mathematical PracticesIn B suppose the diameters in inches were 25
__ π 28 __ π
31 __ π and so on What set of numbers would
best describe the circumferences Explain Whole numbers the circumferences would be the whole numbers 25 28 31 and so on
YOUR TURNFocus on Critical Thinking Mathematical PracticesHave students compare and contrast the classification of numbers in the answers in Exercises 5 and 6
ElaborateTalk About ItSummarize the Lesson
Ask What are some ways that number sets can be related Sets may be subsets of other sets or they may be separate from other sets
GUIDED PRACTICEEngage with the Whiteboard
Have students place the numbers in Exercises 1ndashthinsp8 in the Venn diagram for numbers at the beginning of the lesson
Integrating Language Arts EL
Encourage English learners to ask for clarification on any terms or phrases that they do not understand
Avoid Common ErrorsExercise 7 Remind students that a repeating decimal is a rational numberExercises 9ndash10 Remind students that it only takes one counterexample to show that a statement is false
Interactive Whiteboard Interactive example available online
ADDITIONAL EXAMPLE 3Identify the set of numbers that best describes the situation Explain your choice
A the amount of time that has passed since midnight
The set of real numbers time is continuous so the amount of time can be rational or irrational
B the number of tickets sold to a basketball game
The set of whole numbers the number of tickets sold may be 0 or a counting number
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17 Lesson 12
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
1IN
116 inch
Guided Practice
Write all names that apply to each number (Example 1)
1 7 _ 8 2 radic_
36
3 radic_
24 4 075
5 0 6 - radic_ 100
7 5 _
45 8 - 18 __ 6
Tell whether the given statement is true or false Explain your choice (Example 2)
9 All whole numbers are rational numbers
10 No irrational numbers are whole numbers
Identify the set of numbers that best describes each situation Explain your choice (Example 3)
11 the change in the value of an account when given to the nearest dollar
12 the markings on a standard ruler
13 What are some ways to describe the relationships between sets of numbers
CHECK-INESSENTIAL QUESTION
rational real
rational real
True Whole numbers are rational numbers
Rational numbers the ruler is marked every 1 __ 16 th inch
Sample answer Describe one set as being a subset of
another or show their relationships in a Venn diagram
Integers the change can be a whole dollar amount
and can be positive negative or zero
True Whole numbers are a subset of the set of rational numbers
and can be written as a ratio of the whole number to 1
irrational real
whole integer rational real
whole integer rational real
rational real
integer rational real
integer rational real
Unit 118
copy H
ough
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Miff
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ublis
hing
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pany
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
8_MCAAESE206984_U1M01L2indd 18 41613 136 AM
My Notes
Math TrainerOnline Practice
and Help
Personal
myhrwcom
Math On the Spot
myhrwcom
Identifying Sets for Real-World SituationsReal numbers can be used to represent real-world quantities Highways have posted speed limit signs that are represented by natural numbers such as 55 mph Integers appear on thermometers Rational numbers are used in many daily activities including cooking For example ingredients in a recipe are often given in fractional amounts such as 2 _ 3 cup flour
Identify the set of numbers that best describes each situation Explain your choice
the number of people wearing glasses in a room
The set of whole numbers best describes the situation The number of people wearing glasses may be 0 or a counting number
the circumference of a flying disk has a diameter of 8 9 10 11 or 14 inches
The set of irrational numbers best describes the situation Each circumference would be a product of π and the diameter and any multiple of π is irrational
EXAMPLEXAMPLE 3
A
B
Identify the set of numbers that best describes the situation Explain your choice
5 the amount of water in a glass as it evaporates
6 the weight of a person in pounds
YOUR TURN
8NS1
Rational numbers a personrsquos weight can be a decimal
such as 835 pounds
Real numbers the amount can be any number greater
than 0
17Lesson 12
copy H
ough
ton
Miff
lin H
arco
urt P
ublis
hing
Com
pany
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
8_MCAAESE206984_U1M01L2indd 17 41613 520 AM
Graphic OrganizersGive students a list of numbers (including terminating and repeating decimals fractions integers and rational and irrational square roots) and a graphic organizer as shown below
Real Numbers
Rational numbers Irrational numbers
Integer numbers
Whole numbers
Ask students to write each number in the list in the correct section of the organizer
Number SensePoint out to students that knowing the types of numbers to expect in different situations can alert them to incorrect math as well as to impossible situations For example 135 shots made in basketballs is not possible but an average number of shots can equal 135
Additional ResourcesDifferentiated Instruction includes bull Reading Strategies bull Success for English Learners EL
bull Reteach bull Challenge PRE-AP
DIFFERENTIATE INSTRUCTION
Sets of Real Numbers 18
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
Math TrainerOnline Assessment
and Intervention
Personal
Online homework assignment available
myhrwcom
Lesson Quiz available online
12 LESSON QUIZ
1 Write all the names that apply to the number
2 Tell whether the given statement is true or false Explain your choice All numbers between 1 and 2 are rational numbers
3 Identify the set of numbers that best describes the situation Explain your choiceThe choices on a survey question change the total points for the survey by -2 -1 0 1 or 2 points
-1 _
5
myhrwcom
Additional ResourcesDifferentiated Instruction includes bull Leveled Practice worksheets
EvaluateGUIDED AND INDEPENDENT PRACTICE
Concepts amp Skills Practice
Example 1Classifying Real Numbers
Exercises 1ndash8 14ndash19 22ndash24
Example 2Understanding Sets and Subsets of Real Numbers
Exercises 9ndash10
Example 3Identifying Sets for Real-World Situations
Exercises 11ndash12 20ndash21 25
Focus | Coherence | Rigor
Exercise Depth of Knowledge (DOK) Mathematical Practices
14ndash19 2 SkillsConcepts MP7 Using Structure
20ndash21 2 SkillsConcepts MP6 Precision
22ndash23 2 SkillsConcepts MP3 Logic
24 1 Recall of Information MP7 Using Structure
25 2 SkillsConcepts MP2 Reasoning
26ndash27 3 Strategic Thinking MP3 Logic
28 3 Strategic Thinking MP8 Patterns
29 3 Strategic Thinking MP3 Logic
8NS1
8NS1
Exercise 29 combines concepts from the California Common Core cluster ldquoKnow that there are numbers that are not rational and approximate them by rational numbersrdquo
Answers1 rational real
2 False radic_
2 is an example of an irrational number between 1 and 2
3 Integers each number is an integer but only three are whole numbers
19 Lesson 12
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
Work Area
π mi23 Critique Reasoning The circumference of a circular region is shown
What type of number best describes the diameter of the circle Explain
your answer
24 Critical Thinking A number is not an integer What type of number can it be
25 A grocery store has a shelf with half-gallon containers of milk What type of number best represents the total number of gallons
26 Explain the Error Katie said ldquoNegative numbers are integersrdquo What was her error
27 Justify Reasoning Can you ever use a calculator to determine if a number is rational or irrational Explain
28 Draw Conclusions The decimal 0 _
3 represents 1 _ 3 What type of number best describes 0
_ 9 which is 3 middot 0
_ 3 Explain
29 Communicate Mathematical Ideas Irrational numbers can never be precisely represented in decimal form Why is this
FOCUS ON HIGHER ORDER THINKING
It can be a rational number that is not an integer or an irrational number
rational number
The set of negative numbers also includes non-integer
rational numbers and irrational numbers
Sample answer If the calculator shows a decimal that
terminates in fewer digits than what the calculator screen
allows then you can tell that the number is rational If not
you cannot tell from the calculator display whether the
number terminates because you see a limited number
of digits It may be a repeating decimal (rational) or
non-terminating non-repeating decimal (irrational)
Whole 3 middot 0 _
3 represents 3 middot 1 _ 3 = 1 so 0 _
9 is exactly 1
Sample answer In decimal form irrational numbers never
terminate and never repeat Therefore no matter how
many decimal places you include the number will never
be precisely represented There are always more digits
Whole the diameter is π _ π = 1 mile
Unit 120
copy H
ough
ton
Miff
lin H
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ublis
hing
Com
pany
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
8_MCAAESE206984_U1M01L2indd 20 120413 909 PM
Integers
Rational Numbers Irrational Numbers
Real Numbers
Whole Numbers
257
radic16
166
radic9
128 radic50
Personal Math Trainer
Online Practice and Helpmyhrwcom
Name Class Date
Independent Practice
Identify the set of numbers that best describes each situation Explain your choice
20 the height of an airplane as it descends to an airport runway
21 the score with respect to par of several golfers 2 ndash 3 5 0 ndash 1
22 Critique Reasoning Ronald states that the number 1 __ 11 is not rational because when converted into a decimal it does not terminate Nathaniel says it is rational because it is a fraction Which boy is correct Explain
12
14 - radic_
9 15 257
16 radic_
50 17 8 1 _ 2
18 166 19 radic_
16
Write all names that apply to each number Then place the numbers in the correct location on the Venn diagram
8NS1
Real numbers the height can be any number greater than zero
integer rational real whole integer rational real
whole integer rational real
irrational real
rational real
rational real
Integers the scores are counting numbers their
opposites and zero
Nathaniel is correct A rational number is a number that can be written as a fraction and 1 __ 11 is a fraction
19Lesson 12
copy H
ough
ton
Miff
lin H
arco
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ublis
hing
Com
pany
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
8_MCAAESE206984_U1M01L2indd 19 41613 136 AM
myhrwcomActivity available onlineEXTEND THE MATH PRE-AP
Activity Have students consider the concept of restricted domain for the sets of numbers that describe situations For example the number of sisters a person has can best be described by whole numbers but no one has ever had 1500 sisters An area code is an integer or whole number between 200 and 999
Have students use a source such as the Guinness Book of World Records and give examples of sets of numbers that describe situations where the domain is restricted Ask whether the restriction may be changed in the future
Sets of Real Numbers 20
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
-3-4-5 -2-1 1 2 3 50 4
12-4 -radic5
Lesson Support Content Objective Students will learn to order a set of real numbers
Language Objective Students will show and describe how to order a set of real numbers
LESSON 13 Ordering Real Numbers
Building BackgroundEliciting Prior Knowledge Have students draw a number line to compare a rational number and an irrational number such as - radic
_ 5 and -4 1 __ 2 Ask them to explain how
they approximated the irrational number on the number line Then have them identify the greater and the lesser real number Repeat with several other pairs of real numbers in different forms
Learning ProgressionsIn this lesson students order a set of real numbers They use rational approximations to compare the sizes of irrational numbers They also order numbers for real-world situations Important understandings for students include the following
bull Compare irrational numbers bull Estimate the value of expressions with irrational numbers bull Order a set of real numbers bull Order real numbers in a real-world context
Work with real numbers continues throughout Grade 8 and into high school This lesson provides students with a foundation for understanding the relative sizes of numbers in different forms in the real number system
Cluster ConnectionsThis lesson provides an excellent opportunity to connect ideas in this cluster Know that there are numbers that are not rational and approximate them by rational numbers Tell students that there is a special number called the golden ratio with applications in mathematics geometry art and architecture The golden ratio is called phi and is represented by the Greek letter ϕ It includes an irrational number in its definition
Have students explain why the golden ratio is irrational Ask them to find the two whole numbers the golden ratio lies between Then challenge them to approximate the golden ratio to the nearest tenth It is irrational because it includes an irrational number in its definition It lies between 1 and 2 To the nearest tenth ϕ = 16
ϕ = 1 + radic_
5 _ 2
Focus | Coherence | Rigor
California Common Core Standards
8NS2 Use rational approximations of irrational numbers to compare the size of irrational numbers locate them approximately on a number line diagram and estimate the value of expressions (eg π 2 )
MP4 Model with mathematics
21A
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
Math Talk
Language Support EL
PROFESSIONAL DEVELOPMENT
Linguistic Support EL
AcademicContent Vocabulary
Post a chart like this to remind students of the regular comparative forms of adjectives that use the -er and -est suffixes Add to the chart for terms that appear in examples and exercises in each lesson Include any irregular verb forms
Background Knowledge
Go On ndash the title of the module review or quiz is Ready to Go On This title uses an idiomatic expression In this context to go on means ldquoto move aheadrdquo or ldquoto proceedrdquo It is different from the use of go on that means having enough facts to use meaningfully as in having enough to go on Also the intonation used in pronouncing an expression can give it different meanings For example when the speaker emphasizes the word on he or she might be expressing disbelief as in ldquoGo ON Yoursquore kidding rightrdquo Discuss with students other ways that the phrase go on may be used
Leveled Strategies for English Learners
Emerging Label points on a number line with the terms used in ordering greater greatest less lesser least Use sentence frames to insert the correct terms
Expanding Have students give two or three complete sentences to compare the placement of numbers on a number line using the correct forms of the comparative and superlative adjectives
Bridging Have students work in pairs with one student giving directions to the other in complete sentences to order numbers on a number line
To help students answer the question posed in Math Talk make sure that students have a command of the forms for making comparisons and the superlative and the concept of opposite order so that the focus is on the math concept instead of the language skills needed to describe and explain order
EL
Adjective Comparative Superlative
Far Farther Farthest
Large Larger Largest
Great Greater Greatest
Some Less Least
Some More Most
California ELD Standards
Emerging 2I8 Analyzing language choices ndash Explain how phrasing or different common words with similar meanings produce different effects on the audience
Expanding 2I8 Analyzing language choices ndash Explain how phrasing or different words with similar meanings or figurative language produce shades of meaning and different effects on the audience
Bridging 2I8 Analyzing language choices ndash Explain how phrasing or different words with similar meanings or figurative language produce shades of meaning nuances and different effects on the audience
Ordering Real Numbers 21B
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
13L E S S O N
Ordering Real Numbers
Interactive Whiteboard Interactive example available online
ADDITIONAL EXAMPLE 1Compare Write lt gt or =
A radic_
8 - 2 4 - radic_
8 lt
B radic_
20 + 1 3 + radic_
2 gt
EngageESSENTIAL QUESTION
How do you order a set of real numbers Sample answer Find their approximate decimal values and order them
Motivate the LessonAsk What kind of numbers are you comparing when you compare the price of gasoline at two different gas stations
ExploreGive students two rational numbers and ask them to name a number between them Repeat a few times and then give them two irrational numbers and ask them to name a number between them
ExplainEXAMPLE 1
Questioning Strategies Mathematical Practices bull Which is greater the difference between 5 and 3 or the difference between radic
_ 5 and radic
_ 3
The difference between 5 and 3 is 2 the difference between radic_
5 and radic_
3 is approximately 1 So the difference between 5 and 3 is greater
Avoid Common ErrorsCaution students to read the problem carefully and think about what the radical sign means so that they do not misread the problem and answer that the two sides are equal
YOUR TURNFocus on TechnologyCalculators should not be used at this point because developing number sense is the goal
EXAMPLE 2Questioning Strategies Mathematical Practices bull How do you determine whether radic
_ 22 is less than or greater than 45 The square of 45 is
2025 which is less than 22 so the square root of 22 must be greater than 45
Engage with the WhiteboardHave students graph and label various real numbers between 42 and 44 and between 47 and 5
YOUR TURNFocus on Modeling Mathematical PracticesHave students label the integers on the number line with their equivalent square root For example 1 2 and 3 on the number line would be labeled radic
_ 1 radic
_ 4 and radic
_ 9
Interactive Whiteboard Interactive example available online
ADDITIONAL EXAMPLE 2Order 3π radic
_ 10 and 325 from greatest
to least
3π 325 radic_
10
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CA Common CoreStandards
The student is expected to
The Number Systemmdash8NS2
Use rational approximations of irrational numbers to compare the size of irrational numbers locate them approximately on a number line diagram and estimate the value of expressions (eg π 2 )
Mathematical Practices
MP4 Modeling
The student is expected to
21 Lesson 13
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
Math TrainerOnline Practice
and Help
Personal
myhrwcom
Math On the Spotmyhrwcom
0 05 1 15 2 25 3 35 4
radic5radic3
π2
8 85 9 95 10 105 11 115 12
radic75
4 42 44 46 48 5
radic224 12π + 1
Ordering Real Numbers You can compare and order real numbers and list them from least to greatest
Order radic_
22 π + 1 and 4 1 _ 2 from least to greatest
First approximate radic_
22
radic_
22 is between 4 and 5 Since you donrsquot know where it falls between 4 and 5 you need to find a better estimate for radic
_ 22 so
you can compare it to 4 1 _ 2
Since 22 is closer to 25 than 16 use squares of numbers between 45 and 5 to find a better estimate of radic
_ 22
45 2 = 2025 46 2 = 2116 47 2 = 2209 48 2 = 2304
Since 47 2 = 2209 an approximate value for radic_
22 is 47
An approximate value of π is 314 So an approximate value of π +1 is 414
Plot radic_
22 π + 1 and 4 1 _ 2 on a number line
Read the numbers from left to right to place them in order from least to greatest
From least to greatest the numbers are π + 1 4 1 _ 2 and radic_
22
EXAMPLE 2
STEP 1
STEP 2
Order the numbers from least to greatest Then graph them on the number line
YOUR TURN
5 radic_
5 25 radic_
3
6 π 2 10 radic_
75
If real numbers a b and c are in order from least to greatest what is the order
of their opposites from least to greatest
Explain
Math TalkMathematical Practices
8NS2
radic_
3 radic_
5 25
radic_
75 π2 10
Math Talk answer -c -b -a -c is farthest to the left on a number line -b is in the middle and -a is farthest to the right
Unit 122
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ough
ton
Miff
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pany
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
8_MCAAESE206984_U1M01L3indd 22 41613 447 AM
My Notes
Math TrainerOnline Practice
and Help
Personal
myhrwcom
Math On the Spot
myhrwcom
Comparing Irrational NumbersBetween any two real numbers is another real number To compare and order real numbers you can approximate irrational numbers as decimals
Compare radic_
3 + 5 3 + radic_
5 Write lt gt or =
First approximate radic_
3
radic_
3 is between 1 and 2
Next approximate radic_
5
radic_
5 is between 2 and 3
Then use your approximations to simplify the expressions
radic_
3 + 5 is between 6 and 7
3 + radic_
5 is between 5 and 6
So radic_
3 + 5 gt 3 + radic_
5
Reflect1 If 7 + radic
_ 5 is equal to radic
_ 5 plus a number what do you know about the
number Why
2 What are the closest two integers that radic_
300 is between
EXAMPLEXAMPLE 1
STEP 1
STEP 2
Compare Write lt gt or =
YOUR TURN
3 radic_
2 + 4 2 + radic_
4 4 radic_
12 + 6 12 + radic_
6
L E S S O N
13 Ordering Real Numbers
ESSENTIAL QUESTIONHow do you order a set of real numbers
8NS2
Use rational approximations of irrational numbers to compare the size of irrational numbers locate them approximately on a number line diagram and estimate the value of expressions (eg π 2 )
8NS2
Use perfect squares to estimate square roots
1 2 = 1 2 2 = 4 3 2 = 9
The number is 7 both expressions must equal 7 + radic_
5
17 and 18
gt lt
21Lesson 13
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ough
ton
Miff
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Com
pany
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
8_MCAAESE206984_U1M01L3indd 21 41913 246 PM
PROFESSIONAL DEVELOPMENT
Math BackgroundIn this lesson students estimate irrational numbers in the form of square roots of nonper-fect squares by finding two perfect squares between which the number falls A more precise method involves repeated division For example to find radic
_ 28 find a whole number whose perfect
square is close to 28 such as 5 Divide 28 by that number 28 divide 5 = 56 Find the average of the quotient and divisor 5 + 56
_____ 2 = 53 Continue dividing 28 by each result and averaging until you get the desired accuracy
Integrate Mathematical Practices MP4
This lesson provides an opportunity to address this Mathematical Practices standard It calls for students to model relationships using multiple representations including diagrams graphs and language as appropriate Students use multiple representations when they use number lines to estimate the locations of and order rational and irrational numbers given as symbols
Ordering Real Numbers 22
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
Interactive Whiteboard Interactive example available online
ADDITIONAL EXAMPLE 3The diameter of a meteorite in millimeters is calculated by four different methods Order the results from least to greatest
Joe radic_
18 mm Lisa 13 __ 3 mm
Pablo 46 mm Julien 4π __ 3 mm
Julien 4π __ 3 mm Lisa 13 __ 3 mm
Joe radic_
18 mm Pablo 46 mm
EXAMPLE 3Questioning Strategies Mathematical Practices bull How can you verify that radic
_ 28 is between 52 and 53 5 2 2 = 2704 and 5 3 2 = 2809
bull Explain how to determine which number is greater 5 _
5 or 55 When the repeating decimal is rounded to the nearest tenth or hundredth you can see that it is greater
Connect to Daily LifeDiscuss how measuring across a canyon might involve different methods than measuring along a road Explain that measurements like these are often done using calculations that approximate the distance
YOUR TURNFocus on Critical Thinking Mathematical PracticesDiscuss with students which number is greater 3
_ 45 or 3450 3
_ 45 or 3455 and why Explain
that 3 _
45 can be written out as 34545hellipMake sure they understand that 3 _
45 is greater than 345 but less than 3455
ElaborateTalk About ItSummarize the Lesson
Ask How can you order two numbers in different forms whose decimal approxi-mations appear to be equal Approximate one or both numbers to an additional
number of decimal places
GUIDED PRACTICEEngage with the Whiteboard
Have students place and label additional points on the number line in Exercise 9 Allow the points to be in any format other than decimal
Avoid Common ErrorsExercises 3ndash4 Caution students to read the problem carefully so that they do not misread the problem as the same numbers combined by addition on each side of the circleExercise 10 Remind students that the calculations have units
myhrwcom
23 Lesson 13
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
0 05 1 15 2 25 3 35 4 45 5 55 6 65 7
2πradic3
Compare Write lt gt or = (Example 1)
1 radic_
3 + 2 radic_
3 + 3 2 radic_
8 + 17 radic_
11 + 15
3 radic_
6 + 5 6 + radic_
5 4 radic_
9 + 3 9 + radic_
3
5 radic_
17 - 3 -2 + radic_
5 6 12 - radic_
2 14 - radic_
8
7 radic_
7 + 2 radic_
10 - 1 8 radic_
17 + 3 3 + radic_
11
9 Order radic_
3 2π and 15 from least to greatest Then graph them on the number line (Example 2)
radic_
3 is between and so radic_
3 asymp
π asymp 314 so 2π asymp
From least to greatest the numbers are
10 Four people have found the perimeter of a forest using different methods Their results are given in the table Order their calculations from greatest to least (Example 3)
11 Explain how to order a set of real numbers
CHECK-INESSENTIAL QUESTION
Forest Perimeter (km)
Leon Mika Jason Ashley
radic_
17 - 2 1 +thinsp π __ 2 12 ___ 5 25
Guided Practice
17
15
1 + π _ 2 km 25 km 12 __ 5 km radic_
17 - 2 km
2π radic
_ 3
18 175
628
Sample answer Convert each number to a decimal
equivalent using estimation to find equivalents for
irrational numbers Graph each number on a number line
Read the numbers from left to right for least to greatest
Read the numbers from right to left for greatest to least
lt gt
lt lt
ltgt
gt gt
24 Unit 1
copy H
ough
ton
Miff
lin H
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ublis
hing
Com
pany
bull Im
age C
redi
ts copy
Elena
Eliss
eeva
Alam
y Im
ages
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
8_MCAAESE206984_U1M01L3indd 24 41613 448 AM
My Notes
5 52 54 56 58 6
radic28 5 12
23455
Math TrainerOnline Practice
and Help
Personal
myhrwcom
Math On the Spot
myhrwcom
Ordering Real Numbers in a Real-World Context Calculations and estimations in the real world may differ It can be important to know not only which are the most accurate but which give the greatest or least values depending upon the context
Four people have found the distance in kilometers across a canyon using different methods Their results are given in the table Order the distances from greatest to least
Distance Across Quarry Canyon (km)
Juana Lee Ann Ryne Jackson
radic_
28 23 __ 4 5 _
5 5 1 _ 2
Write each value as a decimal
radic_
28 is between 52 and 53 Since 53 2 = 2809 an approximate value for radic
_ 28 is 53
23 __ 4 = 575
5 _
5 is 5555hellip so 5 _
5 to the nearest hundredth is 556
5 1 _ 2 = 55
Plot radic_
28 23 __ 4 5 _
5 and 5 1 _ 2 on a number line
From greatest to least the distances are
23 __ 4 km 5 _
5 km 5 1 _ 2 km radic_
28 km
EXAMPLEXAMPLE 3
STEP 1
STEP 2
7 Four people have found the distance in miles across a crater using different methods Their results are given below
Jonathan 10 __ 3 Elaine 3 _
45 Joseacute 3 1 _ 2 Lashonda radic_
10
Order the distances from greatest to least
YOUR TURN
8NS2
3 1 _ 2 mi 3 _
45 mi 10 __ 3 mi radic_
10 mi
23Lesson 13
copy H
ough
ton
Miff
lin H
arco
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ublis
hing
Com
pany
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
8_MCAAESE206984_U1M01L3indd 23 41613 447 AM
ModelingPlace papers around the room with the numbers from 1 to 5 one per sheet Give each student a card showing a number between 1 and 5 in different forms Have students place his or her card between the correct integers and decide where the number goes in relation to any numbers already placed
Multiple RepresentationsGive students a vertical number line which some students might find easier to use than a horizontal one Have them decide whether to place points for rational and irrational numbers above or below existing points
Additional ResourcesDifferentiated Instruction includes bull Reading Strategies bull Success for English Learners EL
bull Reteach bull Challenge PRE-AP
DIFFERENTIATE INSTRUCTION
Ordering Real Numbers 24
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
Math TrainerOnline Assessment
and Intervention
Personal
Online homework assignment available
myhrwcom
myhrwcom
Lesson Quiz available online
13 LESSON QUIZ
1 Compare Write lt gt or =
radic_
95 - 5 radic_
62 - 2
2 Order 105 radic_
105 and 3π + 1 from greatest to least
3 A length in centimeters is calculated differently by four different people Order their calculations from least to greatest
KD 11 __ 2 cm Silvio 5 __ 3 π cm
Paula 5 _
4 cm Luis radic_
33 cm
EvaluateGUIDED AND INDEPENDENT PRACTICE
Concepts amp Skills Practice
Example 1Comparing Irrational Numbers
Exercises 1ndash8
Example 2Ordering Real Numbers
Exercises 9 12ndash15 18ndash21
Example 3Ordering Real Numbers in a Real-World Context
Exercises 10 16ndash17
Additional ResourcesDifferentiated Instruction includes bull Leveled Practice worksheets
Focus | Coherence | Rigor
Exercise Depth of Knowledge (DOK) Mathematical Practices
12ndash15 1 Recall of Information MP5 Using Tools
16 2 SkillsConcepts MP2 Reasoning
17 2 SkillsConcepts MP6 Precision
18ndash21 2 SkillsConcepts MP2 Reasoning
22 3 Strategic Thinking MP4 Modeling
23ndash24 3 Strategic Thinking MP3 Logic
8NS2
8NS2
Answers1 radic
_ 95 - 5 lt radic
_ 62 - 2
2 radic_
105 3π + 1 105
3 Silvio 5 __ 3 π cm Paula 5 _
4 cm
KD 11
__ 2 cm Luis radic_
33 cm
25 Lesson 13
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
Work Area
3140 3141 3142 3143
314 π227
20 A teacher asks his students to write the numbers shown in order from least to greatest Paul thinks the numbers are already in order Sandra thinks the order should be reversed Who is right
21 Math History There is a famous irrational number called Eulerrsquos number symbolized with an e Like π its decimal form never ends or repeats The first few digits of e are 27182818284
a Between which two square roots of integers could you find this number
b Between which two square roots of integers can you find π
22 Analyze Relationships There are several approximations used for π including 314 and 22 __ 7 π is approximately 314159265358979
a Label π and the two approximations on the number line
b Which of the two approximations is a better estimate for π Explain
c Find a whole number x so that the ratio x ___ 113 is a better estimate for π
than the two given approximations
23 Communicate Mathematical Ideas If a set of six numbers that include both rational and irrational numbers is graphed on a number line what is the fewest number of distinct points that need to be graphed Explain
24 Critique Reasoning Jill says that 12 _
6 is less than 1263 Explain her error
FOCUS ON HIGHER ORDER THINKING
radic_
115 115 ___ 11 and 105624
between radic_
7 asymp 265 and radic_
8 asymp 283
between radic_
9 = 3 and radic_
10 asymp 316
22 __ 7 it is closer to π on the number line
She did not consider the repeating digit 1266
2 rational numbers can have the same location and
irrational numbers can have the same location but they
cannot share a location
355
Neither student is correct The answer
should be 115 ___ 11 105624 radic_
115
Unit 126
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lishin
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pany
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e Cre
dits
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Stoc
kiSt
ockP
hoto
com
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
8_MCAAESE206984_U1M01L3indd 26 210513 801 AM
Personal Math Trainer
Online Practice and Helpmyhrwcom
Name Class Date
Independent Practice
16 Your sister is considering two different shapes for her garden One is a square with side lengths of 35 meters and the other is a circle with a diameter of 4 meters
a Find the area of the square
b Find the area of the circle
c Compare your answers from parts a and b Which garden would give your sister the most space to plant
17 Winnie measured the length of her fatherrsquos ranch four times and got four different distances Her measurements are shown in the table
a To estimate the actual length Winnie first approximated each distance to the nearest hundredth Then she averaged the four numbers Using a calculator find Winniersquos estimate
b Winniersquos father estimated the distance across his ranch to be radic_
56 km How does this distance compare to Winniersquos estimate
Give an example of each type of number
18 a real number between radic_
13 and radic_
14
19 an irrational number between 5 and 7
Order the numbers from least to greatest
12 radic_
7 2 radic_
8 ___ 2 13 radic_
10 π 35
14 radic_
220 -10 radic_
100 115 15 radic_
8 -375 3 9 _ 4
Distance Across Fatherrsquos Ranch (km)
1 2 3 4
radic_
60 58 __ 8 7 _
3 7 3 _ 5
138NS2
radic_
8 ___ 2 2 radic_
7
-10 radic_
100 115 radic_
220
radic_
60 asymp 775 58 __ 8 = 725 7 _
3 asymp 733 7 3 _ 5 = 760 so the average
π radic_
10 35
-375 9 _ 4 radic_
8 3
is 74825 km
1225 m2
4π m2 or approximately 126 m2
They are nearly identical radic_
56 is approximately 74833hellip
The circle would give her more space to plant because it has a
larger area
Sample answer 37
Sample answer radic_
31
25Lesson 13
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8_MCAAESE206984_U1M01L3indd 25 41613 448 AM
Activity available online myhrwcomEXTEND THE MATH PRE-AP
Activity Have students investigate whether there are infinitely many numbers between two numbers by giving examples for each of the following
bull Between any two rational numbers there is at least one other rational number Sample answer 45 is between 41 and 48
bull Between any two irrational numbers there is at least one rational number Sample answer 45 is between radic
_ 11 and radic
_ 29
bull Between any two rational numbers there is at least one irrational number Sample answer radic
_ 11 is between 31 and 36
bull Between any two irrational numbers there is at least one irrational number Sample answer radic
_ 17 is between radic
_ 11 and radic
_ 29
Ordering Real Numbers 26
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
ReadyMath Trainer
Online Practiceand Help
Personal
myhrwcom
Module Quiz
11ensp RationalenspandenspIrrationalenspNumbersWrite each fraction as a decimal or each decimal as a fraction
1 7__20 2 1___
27 3 17_8
Solve each equation for x
4 x2=81 5 x3=343 6 x2= 1___100
7 Asquarepatiohasanareaof200squarefeetHowlongiseachside
ofthepatiotothenearesttenth
12ensp SetsenspofenspRealenspNumbersWrite all names that apply to each number
8 121____radic
____121
9 π__2
10 TellwhetherthestatementldquoAllintegersarerationalnumbersrdquoistrueorfalseExplainyourchoice
13ensp OrderingenspRealenspNumbersCompare Write lt gt or =
11 radic__
8+3 8+radic__
3 12 radic__
5+11emsp emsp emsp 5+radic___
11
Order the numbers from least to greatest
13 radic___
99π29__
8 14 radic___
1__251_40__
2
15 Howarerealnumbersusedtodescribereal-worldsituations
ESSENTIAL QUESTION
035
9-9
141ft
7 1__10- 1__10
14__11 1875
wholeintegerrationalreal
Trueintegerscanbewrittenasthequotientoftwointegers
SampleanswerRealnumberssuchastherational
π29__
8radic___
99
irrationalreal
lt gt
number1_4candescribeamountsusedincooking
radic___
1__250__
21_4
27Module1
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ough
ton
Miff
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hing
Com
pany
DONOTEDIT--ChangesmustbemadethroughldquoFileinfordquoCorrectionKey=A
8_MCAAESE206984_U1M01RTindd 27 41513 1113 PM
Math TrainerOnline Assessment
and Intervention
Personal
myhrwcom
1
2
3 Response toIntervention
Intervention Enrichment
Access Ready to Go On assessment online and receive instant scoring feedback and customized intervention or enrichment
Online and Print Resources
Differentiated Instruction
bull Reteach worksheets
bull Reading Strategies EL
bull Success for English Learners EL
Differentiated Instruction
bull Challenge worksheets PRE-AP
Extend the Math PRE-AP
Lesson Activities in TE
Additional ResourcesAssessment Resources includes bull Leveled Module Quizzes
Ready to Go OnAssess MasteryUse the assessment on this page to determine if students have mastered the concepts and standards covered in this module
California Common Core Standards
Lesson Exercises Common Core Standards
11 1ndash7 8NS1 8NS2 8EE2
12 8ndash10 8NS1
13 11ndash14 8NS2
27 Unit 1 Module 1
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
Personal Math Trainer
Online Practice and HelpmyhrwcomAssessment Readiness
Module 1 MIXed ReVIeW
1 Look at each number Is the number between 2π and radic___
52
Select Yes or No for expressions AndashC
A 6 2 _ 3 Yes No
B 5π __ 2 Yes No
C 3 radic__
5 Yes No
2 Consider the number - 11 __ 15
Choose True or False for each statement
A The number is rational True False
B The number can be written as True Falsea repeating decimal
C The number is less than ndash08 True False
3 The volume of a cube is given by V = x3 where x is the length of an edge of the cube A cube-shaped end table has a volume of 3 3 _ 8 cubic feet What is the length of an edge of the end table Explain how you solved this problem
4 A student says that radic___
83 is greater than 29 __ 3 Is the student correct Justify your
reasoning
1 1 _ 2 ft Sample answer The equation x3 = 3 3 _ 8 can be used
to find the edge length in feet To solve the equation
write the mixed number as a fraction greater than 1
x3 = 27 __ 8 Then take the cube root of both sides x = 3 _ 2 = 1 1 _ 2
No Sample answer radic___
83 asymp 91 and 29 __ 3 = 9
__ 6
Because 91 lt 9 __
6 radic___
83 lt 29 __ 3
28 Unit 1
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ough
ton
Miff
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pany
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8_MCAAESE206984_U1M01RTindd 28 240413 946 AM
Personal Math Trainer
Online Assessment and
Interventionmyhrwcom
Scoring GuideItem 3 Award the student 1 point for finding the edge length of the cube and 1 point for correctly explaining how to use a cube root to solve the problem
Item 4 Award the student 1 point for determining that the student is incorrect and 1 point for correctly justifying the reasoning for this conclusion
Additional ResourcesTo assign this assessment online login to your Assignment Manager at myhrwcom
Assessment Readiness
California Common Core Standards
Items Grade 8 Standards Mathematical Practices
1 8NS2 MP7
2 7NS2b 7NS2d 8NS1 MP7
3 8EE2 MP1 MP4
4 8NS1 8NS2 MP3
Item integrates mixed review concepts from previous modules or a previous course
Item 4 combines concepts from the California Common Core cluster ldquoKnow that there are numbers that are not rational and approximate them by rational numbersrdquo
Real Numbers 28
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
Reading Start-Up
Active ReadingLayered Book Before beginning the lessons in this module create a layered book to help you learn the concepts in this module Label the flaps ldquoRational Numbersrdquo ldquoIrrational Numbersrdquo ldquoSquare Rootsrdquo and ldquoReal Numbersrdquo As you study each lesson write important ideas such as vocabulary models and sample problems under the appropriate flap
VocabularyReview Words integers (enteros) negative numbers
(nuacutemeros negativos)positive numbers
(nuacutemeros positivos)whole number (nuacutemero
entero)
Preview Words cube root (raiz cuacutebica) irrational numbers (nuacutemero
irracional) perfect cube (cubo
perfecto) perfect square (cuadrado
perfecto) principal square root (raiacutez
cuadrada principal) rational number (nuacutemero
racional) real numbers (nuacutemero real) repeating decimal (decimal
perioacutedico) square root (raiacutez cuadrada) terminating decimal
(decimal finito)
Visualize VocabularyUse the words to complete the graphic You can put more than one word in each section of the triangle
Understand VocabularyComplete the sentences using the preview words
1 One of the two equal factors of a number is a
2 A has integers as its square roots
3 The is the nonnegative square root of a number
Integers
0 83 308
1 45 192
-21 -78 -93
square root
perfect square
principal square root
whole numbers
negative numbers
positive numberswhole numbers
5Module 1
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pany
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8_MCAAESE206984_U1MO01indd 5 180513 1045 AM
Reading Start-Up Have students complete the activities on this page by working alone or with others
Strategies for English LearnersEach lesson in the TE contains specific strategies to help English Learners of all levels succeedEmerging Students at this level typically progress very quickly learning to use English for immediate needs as well as beginning to understand and use academic vocabulary and other features of academic language Expanding Students at this level are challenged to increase their English skills in more contexts and learn a greater variety of vocabulary and linguistic structures applying their growing language skills in more sophisticated ways appropriate to their age and grade level Bridging Students at this level continue to learn and apply a range of high-level English language skills in a wide variety of contexts includ-ing comprehension and production of highly technical texts
Active ReadingIntegrating Language ArtsStudents can use these reading and note-taking strategies to help them organize and understand new concepts and vocabulary
Additional ResourcesDifferentiated Instruction
bull Reading Strategies EL
EL
After
Students will connect that bull the rational numbers are those with decimal expansions that terminate in 0s or eventually repeat
bull non-rational numbers are called irrational numbers
In this moduleStudents will learn how to bull express a rational number as a decimal bull approximate the value of an irrational number bull describe the relationship between sets of real numbers bull order a set of real numbers arising from mathematical and real-world contexts
Before
Students understand bull write rational numbers as decimals bull describe relationships between sets and subsets of rational numbers
bull compare rational numbers
Tracking Your Learning Progression
Focus | Coherence | Rigor
5 Module 1
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
myhrwcom
What It Means to YouYou will learn to estimate the values of irrational numbers
What It Means to YouYou will recognize a number as rational or irrational by looking at its fraction or decimal form
Estimate the value of radic_
8
8 is between the perfect squares 4 and 9So radic
_ 8 is between radic
_ 4 and radic
_ 9
radic_
8 is between 2 and 3
8 is closer to 9 so radic_
8 is closer to 3 28 2 = 784 29 2 = 841 radic
_ 8 is between 28 and 29
A good estimate for radic_
8 is 285
Classify each number as rational or irrational
0 _
3 = 1 _ 3 025 = 1 _ 4
These numbers are rational because they can be written as ratios of integers or as repeating or terminating decimals
π asymp 3141592654hellip radic_ 5 asymp 2236067977hellip
These numbers are irrational because they cannot be written as ratios of integers or as repeating or terminating decimals
Understanding the standards and the vocabulary terms in the standards will help you know exactly what you are expected to learn in this module
Real NumbersGETTING READY FOR
Know that numbers that are not rational are called irrational Understand informally that every number has a decimal expansion for rational numbers show that the decimal expansion repeats eventually and convert a decimal expansion which repeats eventually into a rational number
Key Vocabularyrational number (nuacutemero
racional) A number that can be expressed as a ratio of two integers
irrational number (nuacutemero irracional)A number that cannot be expressed as a ratio of two integers or as a repeating or terminating decimal
Use rational approximations of irrational numbers to compare the size of irrational numbers locate them approximately on a number line diagram and estimate the value of expressions (eg π2)
EXAMPLE 8NS1
EXAMPLE 8NS2
8NS2
8NS1
Visit myhrwcom to see all CA Common Core Standards explained
8 is not a perfect square Find the two perfect squares closest to 8
Unit 16
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8_MCABESE206984_U1MO01indd 6 102913 1123 PM
GETTING READY FOR
Real NumbersUse the examples on the page to help students know exactly what they are expected to learn in this module
myhrwcom
California Common Core Standards Lesson 11
Lesson 12
Lesson 13
8NS1 Know that numbers that are not rational are called irrational Understand informally that every number has a decimal expansion for rational numbers show that the decimal expansion repeats eventually and convert a decimal expansion which repeats eventually into a rational number
8NS2 Use rational approximations of irrational numbers to compare the size of irrational numbers locate them approximately on a number line diagram and estimate the value of expressions (eg π2)
8EE2 Use square root and cube root symbols to represent solutions to equations of the form x 2 = p and x 3 = p where p is a positive rational number Evaluate square roots of small perfect squares and cube roots of small perfect cubes Know that radic
_ 2 is irrational
Go online to see a complete unpacking of the CA Common Core Standards
CA Common Core Standards
Content Areas
The Number Systemmdash8NS
Cluster Know that there are numbers that are not rational and approximate them by rational numbers
Real Numbers 6
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=B
Lesson Support Content Objective Students will learn to rewrite rational numbers and decimals take square roots and
cube roots and approximate irrational numbers
Language Objective Students will show and explain how to rewrite rational numbers and decimals take square roots and cube roots and approximate irrational numbers
LESSON 11 Rational and Irrational Numbers
Building BackgroundEliciting Prior Knowledge Have students work with a partner to review the relationship between fractions and decimals Ask students to provide an example of writing a fraction or mixed number as a decimal and vice versa Discuss how students chose and wrote their examples
Learning ProgressionsIn this lesson students work with positive rational and irrational numbers They make connections among the real numbers by converting fractions and decimals and approximating irrational numbers Important understandings for students include the following
bull Understand that every number has a decimal expansion bull Convert a repeating decimal to a rational number bull Evaluate square roots of perfect squares and cube roots of perfect cubes
bull Estimate an irrational number
Work with the real number system will continue in this unit as students extend the positive rational and irrational numbers to include negative numbers and compare and order real numbers
Cluster ConnectionsThis lesson provides an excellent opportunity to connect ideas in this cluster Know that there are numbers that are not rational and approximate them by rational numbers Tell students ldquoA square garden has an area of 20 square feetrdquo
Have students explain why the side length cannot be rational Then have them approximate the length of each side of the garden to the nearest tenth and hundredth Sample answer The length is the solution to s 2 = 20 radic
_ 20 which is not a rational
number 45 ft 447 ft The length is between 4 and 5 feet 20 is closer to 45 2 than to 44 2 or 46 2 It is also closer to 447 2 than to 446 2 or 448 2
3 _ 4
= 075 1 2 _ 3
= 1 _
6
7 _ 10
= 07 45 = 4 1 _ 2
20 ft 2
California Common Core Standards
8NS1 Know that numbers that are not rational are called irrational Understand informally that every number has a decimal expansion for rational numbers show that the decimal expansion repeats eventually and convert a decimal expansion which repeats eventually into a rational number
8NS2 Use rational approximations of irrational numbers to compare the size of irrational numbers locate them approximately on a number line diagram and estimate the value of expressions (eg π 2 )
8EE2 Use square root and cube root symbols to represent solutions to equations of the form x 2 = p and x 3 = p where p is a positive rational number Evaluate square roots of small perfect squares and cube roots of small perfect cubes Know that radic
_ 2 is irrational
MP6 Attend to precision
7A
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=B
Math Talk
Language Support EL
PROFESSIONAL DEVELOPMENT
Linguistic Support EL
AcademicContent Vocabulary
square ndash In this lesson the word square has multiple meanings which can cause confusion For example to square as in to take the square root of a number is a verb It is different from the nouns square or square of a number The text also refers to perfect square and principal square root of a number and the square root symbol is used These different usages of square as a mathematical term need to be clarified Sentence frames can be used to help define the meaning
To square a number means to _______The perfect square of a number means _______
Background Knowledge
suffixes ndash When added to a root word the suffix -th is used in math to indicate one of a specified number of parts such as tenth hundredth or thousandth Remind students that the suffix -th also indicates place value Note that Spanish Vietnamese Mandarin and other languages do not have the ending th sound so teachers need to enunciate carefully
cognates ndash The words terminating and terminal used in this lesson are cognates in Spanish terminar meaning ldquoto endrdquo or ldquoto finishrdquo A Spanish cognate for approximate is aproximar
Leveled Strategies for English Learners
Emerging Use cards with root words ten hundred and thousand and a card with the -th suffix Have students place them together to show place value Then complete a sentence Use the same procedure to identify decimals
Expanding Support students at this level of English proficiency by providing sentence frames for them to use to describe their mathematical reasoning
To write the fraction _______ as a decimal I _______
Bridging Have students identify different meanings of the term square by matching examples of math problems with a written out sentence frame that defines the usage of the term square to square a number perfect square square root Use this procedure also with the term cube
Be sure to clarify the different uses of the term square when referring to square roots perfect squares and so on
EL
California ELD Standards
Emerging 2I12b Selecting language resources ndash Use knowledge of morphology to appropriately select affixes in basic ways
Expanding 2I12b Selecting language resources ndash Use knowledge of morphology to appropriately select affixes in a growing number of ways to manipulate language
Bridging 2I12b Selecting language resources ndash Use knowledge of morphology to appropriately select affixes in a variety of ways to manipulate language
Rational and Irrational Numbers 7B
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
11L E S S O N
Rational and Irrational Numbers
EngageESSENTIAL QUESTION
How do you rewrite rational numbers and decimals take square roots and cube roots and approximate irrational numbers To express as a decimal divide the numerator by the denominator To take a square root or cube root of a number find the number that when squared or cubed equals the original number To approximate an irrational number estimate a number between two consecutive perfect squares
Motivate the LessonAsk Which type of rational number do you see more often fractions or decimals Which do you prefer to use Why
ExploreHave students write examples of ratios and then share with the class the various notations for ratios that they used (for example 25 2 to 5 2 __ 5 ) Point out the connection between the word ratio and the meaning of rational number See also Explore Activity in student text
ExplainEXAMPLE 1
Questioning Strategies Mathematical Practices bull How does the denominator of a fraction in simplest form tell whether the decimal equivalent of the fraction is a terminating decimal The decimal will terminate if the denominator is an even number a multiple of 5 or a multiple of 10
Avoid Common ErrorsTo avoid interpreting 1 __ 4 as 4 divided by 1 tell students to start at the top of the fraction and read the bar as ldquodivided byrdquo
YOUR TURNTalk About ItCheck for Understanding
Ask Can an improper fraction be written as a decimal Give an example to support your answer Yes 5 __ 4 = 125
EXAMPLE 2Questioning Strategies Mathematical Practices bull How can you use place value to write a terminating decimal as a fraction with a power of ten in the denominator Start by identifying the place value of the decimals last digit and then use the corresponding power of 10 as the denominator of the fraction
bull How can you tell if a decimal can be written as a rational number If the decimal is a terminating or repeating decimal then it can be written as a rational number
Interactive Whiteboard Interactive example available online
ADDITIONAL EXAMPLE 1Write each fraction as a decimal
A 2 _ 5
04 B 5 _ 9
0 _
5
myhrwcom
Interactive Whiteboard Interactive example available online
ADDITIONAL EXAMPLE 2Write each decimal as a fraction in simplest form
A 0355 71 ___ 200
B 0 _
43 43 __ 99
myhrwcom
CA Common CoreStandards
The student is expected to
The Number Systemmdash8NS1
Know that numbers that are not rational are called irrational Understand informally that every number has a decimal expansion for rational numbers show that the decimal expansion repeats eventually and convert a decimal expansion which repeats eventually into a rational number
The Number Systemmdash8NS2
Use rational approximations of irrational numbers to compare the size of irrational numbers locate them approximately on a number line diagram and estimate the value of expressions (eg π 2 )
Expressions and Equationsmdash8EE2
Use square root and cube root symbols to represent solutions to equations of the form x 2 = p and x 3 = p where p is a positive rational number Evaluate square roots of small perfect squares and cube roots of small perfect cubes Know that radic
_ 2 is irrational
Mathematical Practices
MP6 Precision
The student is expected to
the value of expressions (eg
7 Lesson 11
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
My Notes
Math On the Spotmyhrwcom
Math TrainerOnline Practice
and Help
Personal
myhrwcom
Expressing Decimals as Rational NumbersYou can express terminating and repeating decimals as rational numbers
Write each decimal as a fraction in simplest form
0825
The decimal 0825 means ldquo825 thousandthsrdquo Write this as a fraction
825 ____ 1000
Then simplify the fraction
825 divide 25 ________ 1000 divide 25 = 33 __ 40
0825 = 33 __ 40
0 _
37
Let x = 0 _
37 The number 0 _
37 has 2 repeating digits so multiply each side of the equation x = 0
_ 37 by 10 2 or 100
x = 0 _
37
(100)x = 100(0 _
37 )
100x = 37 _
37
Because x = 0 _
37 you can subtract x from one side and 0 _
37 from the other
100x = 37 _
37
minusx minus0 _
37
99x = 37
Now solve the equation for x Simplify if necessary
99x ___ 99 = 37 __ 99
x = 37 __ 99
EXAMPLE 2
A
B
Write each fraction as a decimal
YOUR TURN
1 5 __ 11 2 1 _ 8 3 2 1 _ 3
8NS1
To write ldquo825 thousandthsrdquo put 825 over 1000
Divide both the numerator and the denominator by 25
100 times 0 _
37 is 37 _
37
37 _
37 minus 0 _
37 is 37
Divide both sides of the equation by 99
0 _
45 0125 2 _
3
Unit 18
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Miff
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pany
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
8_MCAAESE206984_U1M01L1indd 8 120413 838 PM
My Notes
Math On the Spot
myhrwcom
= 033333333333331mdash3
ESSENTIAL QUESTION
Expressing Rational Numbers as DecimalsA rational number is any number that can be written as a ratio in the form a _ b where a and b are integers and b is not 0 Examples of rational numbers are 6 and 05
6 can be written as 6 _ 1 05 can be written as 1 _ 2
Every rational number can be written as a terminating decimal or a repeating decimal A terminating decimal such as 05 has a finite number of digits A repeating decimal has a block of one or more digits that repeat indefinitely
Write each fraction as a decimal
1 _ 4
1 _ 4 = 025
1 _ 3
1 _ 3 = 0 _
3
EXAMPLEXAMPLE 1
A
B
0333 3 ⟌ ⎯ 1000 minus9 10 minus9 10 minus9 1
025 4 ⟌ ⎯ 100 -8 20 -20
0
L E S S O N
11Rational and Irrational Numbers
How do you rewrite rational numbers and decimals take square roots and cube roots and approximate irrational numbers
8NS1
Know that numbers that are not rational are called irrational Understand informally that every number has a decimal expansion for rational numbers show that the decimal expansion repeats eventually and convert a decimal expansion which repeats eventually into a relation number Also 8NS2 8EE2
8NS1
Remember that the fraction bar means ldquodivided byrdquo Divide the numerator by the denominator
Divide until the remainder is zero adding zeros after the decimal point in the dividend as needed
Divide until the remainder is zero or until the digits in the quotient begin to repeat
Add zeros after the decimal point in the dividend as needed
When a decimal has one or more digits that repeat indefinitely write the decimal with a bar over the repeating digit(s)
7Lesson 11
copy H
ough
ton
Miff
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pany
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=B
8_MCABESE206984_U1M01L1indd 7 11113 128 AM
PROFESSIONAL DEVELOPMENT
Math BackgroundSome decimals may have a pattern but still not be a repeating decimal that is rational For example in 312112111211112hellip you can predict the next digit and describe the pattern (There is one more 1 each time before the 2) However this is not a terminating decimal nor is it a repeating decimal and it is therefore NOT a rational number
Integrate Mathematical Practices MP6
This lesson provides an opportunity to address this Mathematical Practices standard It calls for students to attend to precision Students learn to express rational numbers accurately and precisely in both fractional and decimal forms and learn to translate from one form to the other They also learn how to precisely represent and communicate ideas about irrational numbers square roots and cube roots
Rational and Irrational Numbers 8
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=B
Focus on Technology Mathematical PracticesPoint out the importance of entering a repeating decimal correctly when using a graphing calculator to convert the decimal to a fraction The decimal 0
_ 59 must be entered as
0595959595959 not 059
YOUR TURNFocus on Math ConnectionsMake sure students understand that the place value of the last digit in Exercises 4 and 6 determines the denominator of the corresponding fraction or mixed number So for Exercise 4 the place value hundredths gives a denominator of 100 and for Exercise 6 the place value tenths gives a denominator of 10
EXAMPLE 3Questioning Strategies Mathematical Practices bull How can a solution of an equation of the form x 2 = p be negative if p is a positive number Since the square of a negative number is positive a negative number is also a solution of x 2 equals a positive number
bull When is a solution of an equation of the form x 3 = p larger than p The solution is larger than p if p is a number between 0 and 1
Focus on Math Connections Make sure students understand the difference in finding radic
_ 121 and solving x 2 = 121 The
symbol radic_
indicates the positive or principal square root only while the equation x 2 = 121 has two roots the principal square root and its opposite
YOUR TURNAvoid Common ErrorsTo avoid sign errors in Exercise 9 make sure that students understand that the cube of a negative number is not a positive number Therefore -8 is not a solution of x 3 = 512
Talk About ItCheck for Understanding
Ask Kris predicts that there are two real solutions for Exercises 7 and 8 and that there are three real solutions for Exercises 9 and 10 Is his prediction correct
Explain His prediction is correct for Exercises 7 and 8 because there are two numbers whose squares are the same positive number given in the exercises His prediction is not correct for Exercises 9 and 10 however because there is only one real number whose cube is the same positive number given in the exercises
EXPLORE ACTIVITYQuestioning Strategies Mathematical Practices bull Compare the values for 13 2 and 13 2 The digits are the same but 13 2 has two decimal places (169) while 13 2 has none (169)
bull How do you know whether radic_
2 will be closer to 1 or closer to 2 It will be closer to 1 because 2 is between the perfect squares of 1 and 4 but closer to 1 than it is to 4
Connect Vocabulary EL
Explain to students that the word irrational when used as an ordinary word in English means without logic or reason In mathematics when we say that a number is irrational it means only that the number cannot be written as the quotient of two integers
Engage with the WhiteboardHave students extend the number line in both directions and label the locations of the whole numbers 1 and 2 These are the roots of the consecutive perfect squares
1 and 4 used to estimate radic_
7
Interactive Whiteboard Interactive example available online
ADDITIONAL EXAMPLE 3Solve each equation for x
A x 2 = 324 18 -18
B x 2 = 25 ___ 144 5 __ 12 - 5 __ 12
C 343 = x 3 7
D x 3 = 125 ___ 512 5 __ 8
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9 Lesson 11
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
Math TrainerOnline Practice
and Help
Personal
myhrwcom
EXPLORE ACTIVITY
lt 2 lt
radic_
lt radic
_ 2 lt
radic_
lt radic
_ 2 lt
The solution is 9
The solution is 2 _ 5
C
D
729 = x 3
3 radic_ 729 = 3 radic
_ x 3
3 radic_ 729 = x
9 = x
x 3 = 8 ___ 125
3 radic_
x 3 =thinsp 3 radic_ 8 ___ 125
x =thinsp 3 radic_ 8 ___ 125
x = 2 _ 5
Solve each equation for x
YOUR TURN
7 x 2 = 196 8 x 2 = 9 ___ 256
9 x 3 = 512 10 x 3 = 64 ___ 343
Estimating Irrational NumbersIrrational numbers are numbers that are not rational In other words they cannot be written in the form a _ b where a and b are integers and b is not 0 Square roots of perfect squares are rational numbers Square roots of numbers that are not perfect squares are irrational Some equations like those in Example 3 involve square roots of numbers that are not perfect squares
x 2 = 2 x = plusmn radic_
2
Estimate the value of radic_
2
Find two consecutive perfect squares that 2 is between Complete the inequality by writing these perfect squares in the boxes
Now take the square root of each number
Simplify the square roots of perfect squares
radic_
2 is between and
A
B
C
8NS2 8EE2
Solve for x by taking the cube root of both sides
Solve for x by taking the cube root of both sides
Apply the definition of cube root
Think What number cubed equals 729
Apply the definition of cube root
Think What number cubed equals 8 ____ 125
radic_
2 is irrational
x = plusmn14 x = plusmn 3 __ 16
x = 8 x = 4 _ 7
1 2
1 4
1 4
1 2
Unit 110
copy H
ough
ton
Miff
lin H
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urt P
ublis
hing
Com
pany
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
8_MCAAESE206984_U1M01L1indd 10 41613 1211 AM
Math TrainerOnline Practice
and Help
Personal
myhrwcom
Math On the Spot
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Write each decimal as a fraction in simplest form
YOUR TURN
Finding Square Roots and Cube RootsThe square root of a positive number p is x if x 2 = p There are two square roots for every positive number For example the square roots of 36 are 6 and minus6 because 6 2 = 36 and (minus6) 2 = 36 The square roots of 1 __ 25 are 1 _ 5 and minus 1 _ 5 You can write the square roots of 1 __ 25 as plusmn 1 _ 5 The symbol radic
_ 5 indicates the positive
or principal square root
A number that is a perfect square has square roots that are integers The number 81 is a perfect square because its square roots are 9 and minus9
The cube root of a positive number p is x if x 3 = p There is one cube root for every positive number For example the cube root of 8 is 2 because 2 3 = 8 The cube root of 1 __ 27 is 1 _ 3 because ( 1 _ 3 )
3
= 1 __ 27 The symbol 3 radic_ 1 indicates the
cube root
A number that is a perfect cube has a cube root that is an integer The number 125 is a perfect cube because its cube root is 5
Solve each equation for x
The solutions are 11 and minus11
The solutions are 4 __ 13 and minus 4 __ 13
EXAMPLEXAMPLE 3
A x 2 = 121
x 2 = 121
x = plusmn radic_
121
x = plusmn11
B x 2 = 16 ___ 169
x 2 = 16 ___ 169
x = plusmn radic_
16 ___ 169
x = plusmn 4 __ 13
4 012 5 0 _
57 6 14
Can you square an integer and get a negative number
What does this indicate about whether negative
numbers have square roots
Math TalkMathematical Practices
8EE2
Solve for x by taking the square root of both sides
Apply the definition of square root
Think What numbers squared equal 121
Solve for x by taking the square root of both sides
Apply the definition of square root
Think What numbers squared equal 16 ____ 169
3 __ 25 19 __ 33 1 2 _ 5
No the square of a positive integer is positive the square of a negative integer is positive and the square of 0 is 0 So negative numbers do not have (real) square roots
9Lesson 11
copy H
ough
ton
Miff
lin H
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ublis
hing
Com
pany
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
8_MCAAESE206984_U1M01L1indd 9 41913 240 PM
Critical ThinkingIn the Explore Activity students estimated the location of radic
_ 2 on a number line Ask students
whether they think that it is possible to locate more precisely the point that represents radic
_ 2 In
other words can you graph irrational numbers exactly on a number line along with rational numbers Students should understand that radic
_ 2
is a real number and all real numbers can be located on a real number line A more precise estimate will allow more precise placement on a number line
The Modeling note tells one way to do this
ModelingHave students use a ruler to represent a number line with a unit that is one inch long Have them draw a square with a side of one inch and draw the diagonal to make two isosceles triangles Lead students to understand that the length of the diagonal (or hypotenuse) is radic
_ 2
Have them copy the length of their diagonal onto their ruler or number line starting at zero The end point of the diagonal represents the exact point for the irrational number radic
_ 2 on a
number line
Additional ResourcesDifferentiated Instruction includes bull Reading Strategies bull Success for English Learners EL
bull Reteach bull Challenge PRE-AP
DIFFERENTIATE INSTRUCTION
Rational and Irrational Numbers 10
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
ElaborateTalk About ItSummarize the Lesson
Ask If someone claims that a certain number is irrational but you know it is actually rational how could you prove to that person that the number is rational
You could find a fraction equal to the number such that the number is the ratio of two integers with the denominator not equal to zero
GUIDED PRACTICEEngage with the Whiteboard
Have students plot each number in Exercises 16ndash18 on a number line Students should label each point with the irrational number written as a radical and as a
decimal
Avoid Common ErrorsExercises 1ndash6 To avoid reversing the order of the dividend and divisor tell students to start at the top of the fraction and read the bar as ldquodivided byrdquo
Focus on TechnologyHave students use a calculator to investigate the decimal equivalents of such fractions as 1 __ 9 2 __ 9 8 __ 9 and 1 __ 11 2 __ 11 10
__ 11 Ask them to describe the patterns they find as a result of these investigations
11 Lesson 11
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
Guided Practice
7 0675 8 56 9 044
10 0 _
4
10x =
x =
11 0 _
26
100x =
x =
12 0 _
325
1000x =
x =
Solve each equation for x (Example 3 and Explore Activity)
- x
-
_______________
x =
- x
-
___________________
x =
- x
-
_______________________
x =
Write each fraction or mixed number as a decimal (Example 1)
1 2 _ 5 2 8 _ 9 3 3 3 _ 4
4 7 __ 10 5 2 3 _ 8 6 5 _ 6
Write each decimal as a fraction or mixed number in simplest form (Example 2)
13 x 2 = 17 14 x 2 = 25 ___ 289 15 x 3 = 216
Approximate each irrational number to one decimal place without a calculator
x = plusmn radic__
asymp plusmn x = 3
radic__
=
(Explore Activity)
16 radic_
5 asymp
17 radic_
3 asymp
18 radic_
10 asymp
19 What is the difference between rational and irrational numbers
CHECK-INESSENTIAL QUESTION
x = plusmn radic__
__________ = plusmn _____
4 _
4
0 _
4
4 99
6216
269
41 25 5
17289
17
22 17 32
04
07
27__40
26 __ 99 325 ___ 999 4 _ 9
11__255 3_5
0 _
8
2375
375
08 _
3
26 _
26
0 _
26
325 _
325
0 _
325
999 325
Rational numbers can be written in the form a __ b where
a and b are integers and b ne 0 Irrational numbers cannot
be written in this form
Unit 112
copy H
ough
ton
Miff
lin H
arco
urt P
ublis
hing
Com
pany
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
8_MCAAESE206984_U1M01L1indd 12 41613 1211 AM
11 12 13 14 15
radic2 asymp 14
141 142 143 144 145
radic2 asymp 141
0 1 2 3 4
radic2 asymp 15
Estimate that radic_
2 asymp 15
To find a better estimate first choose some numbers between 1 and 2 and square them For example choose 13 14 and 15
1 3 2 = 1 4 2 = 1 5 2 =
Is radic_
2 between 13 and 14 How do you know
Is radic_
2 between 14 and 15 How do you know
2 is closer to than to so radic_
2 asymp
Locate and label this value on the number line
Reflect 11 How could you find an even better estimate of radic
_ 2
12 Find a better estimate of radic_
2
1 41 2 = 1 42 2 = 1 43 2 =
2 is closer to than to so radic_
2 asymp
Draw a number line and locate and label your estimate
13 Solve x 2 = 7 Write your answer as a radical expression Then estimate to one decimal place
D
E
F
No 2 is not between 169 and 196
Yes 2 is between 196 and 225
196
19881
19881
225
20164
20164
14
141
20449
169 196 225
Test the squares of numbers between 14 and 15
x = plusmn radic_
7 x asymp plusmn26
11Lesson 11
copy H
ough
ton
Miff
lin H
arco
urt P
ublis
hing
Com
pany
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
8_MCAAESE206984_U1M01L1indd 11 41613 1211 AM
Rational and Irrational Numbers 12
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
Math TrainerOnline Assessment
and Intervention
Personal
Online homework assignment available
myhrwcom
EvaluateGUIDED AND INDEPENDENT PRACTICE
Concepts amp Skills Practice
Example 1Expressing Rational Numbers as Decimals
Exercises 1ndash6 20ndash21 24ndash25
Example 2Expressing Decimals as Rational Numbers
Exercises 7ndash12 22ndash23 26ndash27
Example 3Finding Square Roots and Cube Roots
Exercises 13ndash15 28 30ndash31 35
Explore ActivityEstimating Irrational Numbers
Exercises 13 16ndash18 29 32ndash34
Additional ResourcesDifferentiated Instruction includes bull Leveled Practice worksheets
Lesson Quiz available online
11 LESSON QUIZ
1 Write as a decimal 2 5 __ 8 1 7 __ 12
2 Write as a fraction 034 1 _
24
3 Solve x 2 = 9 __ 49 for x
4 Solve x 3 = 216 for x
5 Estimate the value of radic_
13 to one decimal place without using a calculator
myhrwcom
Focus | Coherence | Rigor
Exercise Depth of Knowledge (DOK) Mathematical Practices
20ndash27 2 SkillsConcepts MP4 Modeling
28 3 Strategic Thinking MP4 Modeling
29ndash32 2 SkillsConcepts MP6 Precision
33 3 Strategic Thinking MP7 Using Structure
34 2 SkillsConcepts MP3 Logic
35 2 SkillsConcepts MP4 Modeling
36 3 Strategic Thinking MP3 Logic
37 3 Strategic Thinking MP7 Using Structure
38 3 Strategic Thinking MP2 Reasoning
8NS1 8NS2 8EE2
8NS1 8NS2 8EE2
Answers1 2625 158
_ 3
2 17 __ 50 1 8 __ 33
3 x = plusmn 3 __ 7
4 x = 6
5 36
13 Lesson 11
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
Work Area
33 Analyze Relationships To find radic_
15 Beau found 3 2 = 9 and 4 2 = 16 He said that since 15 is between 9 and 16 radic
_ 15 must be between 3 and 4 He
thinks a good estimate for radic_
15 is 3 + 4 ____ 2 = 35 Is Beaursquos estimate high low
or correct Explain
34 Justify Reasoning What is a good estimate for the solution to the equation x 3 = 95 How did you come up with your estimate
35 The volume of a sphere is 36π f t 3 What is the radius of the sphere Use the formula V = 4 _ 3 π r 3 to find your answer
36 Draw Conclusions Can you find the cube root of a negative number If so is it positive or negative Explain your reasoning
37 Make a Conjecture Evaluate and compare the following expressions
radic_
4 __ 25 and radic
_ 4 ____
radic_
25 radic
_
16 __ 81 and radic_
16 ____
radic_
81 radic
_
36 __ 49 and radic_
36 ____
radic_
49
Use your results to make a conjecture about a division rule for square roots Since division is multiplication by the reciprocal make a conjecture about a multiplication rule for square roots
38 Persevere in Problem Solving The difference between the solutions to the equation x 2 = a is 30 What is a Show that your answer is correct
FOCUS ON HIGHER ORDER THINKING
His estimate is low because 15 is very close to 16
so radic_
15 is very close to radic_
16 or 4 A better estimate
would be 38 or 39
Sample answer about 45 4 3 = 64 and 5 3 = 125
Because 95 is about halfway between 64 and 125 try 45
45 3 = 91125 which is a good estimate
3 feet
Yes the cube root of a negative number is negative
because a negative number cubed is always negative
and a nonnegative number cubed is always nonnegative
radic_
4 __ 25 = 2 _ 5 = radic
_ 4 ____
radic_
25 radic
_
16 __ 81 = 4 _ 9 = radic_
16 ____
radic_
81 radic
_
36 __ 49 = 6 _ 7 = radic_
36 ____
radic_
49
225 the solutions to x 2 = a are x = plusmn15 and
radic_
a ___
radic_
b = radic
_ a __
b radic
_ a radic
_ b = radic
_ a b
15 - (-15) = 30
Unit 114
copy H
ough
ton
Miff
lin H
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ublis
hing
Com
pany
bull copy
Ilen
e Mac
Dona
ldA
lamy I
mag
es
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=B
8_MCABESE206984_U1M01L1indd 14 102913 1142 PM
Personal Math Trainer
Online Practice and Helpmyhrwcom
Name Class Date
Independent Practice11
20 A 7 __ 16 -inch-long bolt is used in a machine What is this length written as a decimal
21 The weight of an object on the moon is 1 _ 6 its weight on Earth Write 1 _ 6 as a decimal
22 The distance to the nearest gas station is 2 4 _ 5 kilometers What is this distance written as a decimal
23 A baseball pitcher has pitched 98 2 _ 3 innings What is the number of innings written as a decimal
24 A heartbeat takes 08 second How many seconds is this written as a fraction
25 There are 262 miles in a marathon Write the number of miles using a fraction
26 The average score on a biology test was 72
_ 1 Write the average score using a
fraction
27 The metal in a penny is worth about 0505 cent How many cents is this written as a fraction
28 Multistep An artist wants to frame a square painting with an area of 400 square inches She wants to know the length of the wood trim that is needed to go around the painting
a If x is the length of one side of the painting what equation can you set up to find the length of a side How many solutions does the equation have
b Do all of the solutions that you found make sense in the context of the problem Explain
c What is the length of the wood trim needed to go around the painting
Solve each equation for x Write your answers as radical expressions Then estimate to one decimal place if necessary
29 x 2 = 14 30 x 3 = 1331
31 x 2 = 144 32 x 2 = 29
8NS1 8NS2 8EE2
04375 in 01 _6
28 km 98 _6 innings
x 2 = 400 x = plusmnthinsp20 the equation has 2 solutions
x = 20 makes sense but x = -20 doesnrsquot because a
painting cannot have a side length of -20 inches
4 times 20 = 80 inches
x = plusmn radic_
14 asymp plusmn37
x = plusmn radic_
144 = plusmn12 x = plusmn radic_
29 asymp plusmn54
x = 3 radic_ 1331 = 11
4_5 second 26 1_5 mi
72 1 _ 9 101 ___ 200 cent
13Lesson 11
copy H
ough
ton
Miff
lin H
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ublis
hing
Com
pany
bull copy
Phot
odisc
Get
ty Im
ages
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
8_MCAAESE206984_U1M01L1indd 13 41613 1211 AM
myhrwcomActivity available onlineEXTEND THE MATH PRE-AP
Activity Write radic_
09 on the board and invite students to conjecture what the value might be Have them check their conjectures by squaring Invite them to suggest ways to estimate radic
_ 09 As a hint point out that 09 is close to 10 and so they might
use that to help guide their estimates Lead them to see that since 092 is 081 and 102 is 1 the value of radic
_ 09 is greater than 09 and less than 10 Try squaring 095 to get
09025 A good estimate for radic_
09 is 095
Rational and Irrational Numbers 14
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=B
Integers
Rational Numbers IrrationalNumbers
Real Numbers
WholeNumbers
-3-4-5 -2-1 1 2 3 50 4
23
34-4 -π -1 25
radic2
Lesson Support Content Objective Students will learn to describe relationships between sets of numbers
Language Objective Students will explain how to describe relationships between sets of real numbers
LESSON 12 Sets of Real Numbers
Building BackgroundEliciting Prior Knowledge Have students draw a number line from -5 to 5 Ask them to plot points on the number line to approximate the location of rational and irrational numbers such as -1 3 __ 4 25 -4 2 __ 3 radic
_ 2 and -π
Learning ProgressionsIn this lesson students clarify their understanding of the real number system They characterize sets and subsets of the real numbers They also identify sets for real-world situations Important understandings for students include the following
bull Identify all of the possible subsets of the real numbers for a given number
bull Decide whether a statement about a subset of the real numbers is true or false
bull Identify the set of numbers that best describes a real-world situation
Understanding the relationships among the sets of numbers that make up the real numbers is essential as students are introduced to different forms of numbers throughout the school year This lesson provides a foundation for the comparing and ordering of real numbers in the next lesson
Cluster ConnectionsThis lesson provides an excellent opportunity to connect ideas in this cluster Know that there are numbers that are not rational and approximate them by rational numbers Have students copy this diagram which relates the sets of real numbers
Ask students to complete the diagram by writing three examples for each set of numbers Have students share examples and explain how they knew each number they selected belonged in the appropriate set Answers may vary Check studentsrsquo work
Focus | Coherence | Rigor
California Common Core Standards
8NS1 Know that numbers that are not rational are called irrational Understand informally that every number has a decimal expansion for rational numbers show that the decimal expansion repeats eventually and convert a decimal expansion which repeats eventually into a rational number
MP7 Look for and make use of structure
15A
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
Math Talk
Language Support EL
PROFESSIONAL DEVELOPMENT
Linguistic Support EL
AcademicContent Vocabulary
Venn diagrams ndash Students need descriptive language to describe the categories that the different areas and colors of a Venn diagram represent the concept of a set and how sets are distinct or can overlap Use sentence frames such as
The big oval represents __________The darklight blue color in the middle of the
big ovals represents __________These sets overlap because __________
In this way students have the language and structure to identify the criteria that distinguish a set and to explain the abstract representation Also point out the use of the prefix sub- meaning ldquounderrdquo in the term subset
Rules and Patterns
Abbreviations ndash In this lesson the abbreviation mph is used Be sure to point out that mph stands for miles per hour and is used to give units in a rate of speed Students may also have seen mpg (miles per gallon) which gives the units in a rate of fuel efficiency
Borrowed Words ndash Terminology used in baseball such as inning and pitcher may require some explanation Spanish as well as some other languages have borrowed these terms from English so some students may be familiar with these words already Despite this whenever a word is critical to students understanding the word problem it is best to explain the meaning
Leveled Strategies for English Learners
Emerging Allow students to indicate true or false orally in Guided Practice Exercises 9 and 10
Expanding Have students use sentence frames to describe the meaning of regions and colors used in a Venn diagram Then give them similar sentence frames orally and have them draw and shade a Venn diagram based on the oral prompts
Bridging Have students work in groups to draw a Venn diagram to represent sets based on real-world examples in the lesson
To help students answer the question posed in Math Talk provide a sentence frame for their answer
The numbers between 31 and 39 on a number line are __________ because __________
EL
California ELD Standards
Emerging 2II5 Modifying to add details ndash Expand sentences with simple adverbials to provide details about a familiar activity or process
Expanding 2II5 Modifying to add details ndash Expand sentences with adverbials to provide details about a familiar or new activity or process
Bridging 2II5 Modifying to add details ndash Expand sentences with increasingly complex adverbials to provide details about a variety of familiar and new activities and processes
Sets of Real Numbers 15B
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
12L E S S O N
Sets of Real Numbers
EngageESSENTIAL QUESTION
How can you describe relationships between sets of real numbers Sample answer Describe them as two different sets or one set as being a subset of another
Motivate the LessonAsk How many different types of tigers can you name How does the set of Bengal tigers relate to the set of tigers
ExplorePoint to different locations in the Animals diagram and ask for examples for that classification Do the same for the Real Numbers diagram Students should understand that everything within a region is part of the set for example both -3 and 2 are integers
ExplainEXAMPLE 1
Questioning Strategies Mathematical Practices bull In A why is 5 not a perfect square It does not have rational numbers as its square roots
bull Can the number in B be written as a fraction Why or why not Yes it is a terminating decimal so it is a rational number
Engage with the WhiteboardHave students place the numbers in Example 1 and Additional Example 1 in the Venn diagram for numbers
YOUR TURNAvoid Common ErrorsBe sure that students read Exercise 2 carefully before answering The number given in the problem 10 is the area not the side length
EXAMPLE 2Questioning Strategies Mathematical Practices bull What two major sets are the real numbers composed of rational and irrational numbers
bull What is the location of the set of whole numbers in the Venn diagram in relation to the set of rational numbers Explain Inside it whole numbers are rational numbers
Focus on Reasoning Mathematical PracticesRemind students that it takes only one counterexample to show that a statement is false
Interactive Whiteboard Interactive example available online
ADDITIONAL EXAMPLE 1Write all names that apply to each number
A -10integer rational real
B 12 _ 3
whole integer rational real
myhrwcom
Interactive Whiteboard Interactive example available online
ADDITIONAL EXAMPLE 2Tell whether the given statement is true or false Explain your choice
No integers are whole numbers
False every whole number is also an integer
myhrwcom
Animated MathClassifying Numbers
Students build fluency in classifying numbers in this engaging fast-paced game
myhrwcom
CA Common CoreStandards
The student is expected to
The Number Systemmdash8NS1
Know that numbers that are not rational are called irrational Understand informally that every number has a decimal expansion for rational numbers show that the decimal expansion repeats eventually and convert a decimal expansion which repeats eventually into a rational numberMathematical Practices
MP7 Using Structure
The student is expected to
15 Lesson 12
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=B
Math TrainerOnline Practice
and Help
Personal
myhrwcom
Math TrainerOnline Practice
and Help
Personal
myhrwcom
Math On the Spotmyhrwcom
Understanding Sets and Subsets of Real NumbersBy understanding which sets are subsets of types of numbers you can verify whether statements about the relationships between sets are true or false
Tell whether the given statement is true or false Explain your choice
All irrational numbers are real numbers
True Every irrational number is included in the set of real numbers The irrational numbers are a subset of the real numbers
No rational numbers are whole numbers
False A whole number can be written as a fraction with a denominator of 1 so every whole number is included in the set of rational numbers The whole numbers are a subset of the rational numbers
EXAMPLE 2
A
B
Write all names that apply to each number
1 A baseball pitcher has pitched 12 2 _ 3 innings
2 The length of the side of a square that has an
area of 10 square yards
YOUR TURN
Tell whether the given statement is true or false Explain your choice
3 All rational numbers are integers
4 Some irrational numbers are integers
YOUR TURN
Give an example of a rational number that is a
whole number Show that the number is both whole
and rational
Math TalkMathematical Practices
Give an example of a
8NS1
False Every integer is a rational number but not every
False Real numbers are either rational or irrational numbers
Integers are rational numbers so no integers are irrational numbers
rational real
irrational real
Sample answer 8 8 = 8_
1
and -thinsp 5 _ 2 are not integers
rational number is an integer Rational numbers such as 3 _ 5
Unit 116
copy H
ough
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Miff
lin H
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ublis
hing
Com
pany
bull Im
age C
redi
ts D
igita
l Im
age c
opyr
ight
copy20
04 Ey
ewire
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
8_MCAAESE206984_U1M01L2indd 16 41613 136 AM
Math On the Spot
myhrwcom
Vertebrates
Birds
Passerines
Animals
Integers
Rational Numbers IrrationalNumbers
Real Numbers
WholeNumbers
1
45
3
0
274
67
radic4
-
-3
-2
-1
03
radic2
radic17
radic11-
π
Animated Math
myhrwcom
Classifying Real NumbersBiologists classify animals based on shared characteristics A cardinal is an animal a vertebrate a bird and a passerine
You already know that the set of rational numbers consists of whole numbers integers and fractions The set of real numbers consists of the set of rational numbers and the set of irrational numbers
Write all names that apply to each number
radic_
5 irrational real
ndash1784rational real
whole integer rational real
EXAMPLEXAMPLE 1
A
B
C radic_ 81 ____ 9
L E S S O N
12Sets of Real Numbers
ESSENTIAL QUESTIONHow can you describe relationships between sets of real numbers
Passerines such as the cardinal are also called ldquoperching birdsrdquo
What types of numbers are between 31 and 39 on a
number line
Math TalkMathematical Practices
What types of numbers are
8NS1
8NS1
Know that numbers that are not rational are called irrational Understand informally that every number has a decimal expansion for rational numbers show that the decimal expansion repeats eventually and convert a decimal expansion which repeats eventually into a relation number
ndash1784 is a terminating decimal
5 is a whole number that is not a perfect square
radic_
81 _____ 9 = 9 __ 9 = 1 rational irrational real
15Lesson 12
copy H
ough
ton
Miff
lin H
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ublis
hing
Com
pany
bull Im
age C
redi
ts copy
Wiki
med
ia Co
mm
ons
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=B
8_MCABESE206984_U1M01L2indd 15 061113 1144 AM
PROFESSIONAL DEVELOPMENT
Math BackgroundThe relationships between sets of numbers extend to include complex numbers A complex number can be written as a sum of a real number a and an imaginary number bi
a + bi
An imaginary number is a special number that when squared gives a negative value When you square a real number you get a nonnegative number When you square an imaginary number you get a negative value The imaginary unit is i
i = radic_
-1
Integrate Mathematical Practices MP7
This lesson provides an opportunity to address this Mathematical Practices standard It calls for students to discern structure to connect and communicate mathematical ideas
Students use a Venn diagram to structure relationships between sets of numbers They connect and communicate mathematical ideas when they make logical statements about the sets and describe which set best describes numbers applied to real-life situations
Sets of Real Numbers 16
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=B
YOUR TURNAvoid Common ErrorsStudents may see the word ldquoAllldquo or rdquoNordquo in Exercises 3 and 4 and immediately assume that any absolute statements like these are false Remind them that there are true statements that begin with these words and encourage them to provide examples
EXAMPLE 3Questioning Strategies Mathematical Practices bull In A how does the phrase ldquonumber of rdquo give you a clue about the number classification It indicates a counting number
bull What is the relationship between the circumference of a circle and the diameter The circumference is diameter times π
Focus on Critical Thinking Mathematical PracticesIn B suppose the diameters in inches were 25
__ π 28 __ π
31 __ π and so on What set of numbers would
best describe the circumferences Explain Whole numbers the circumferences would be the whole numbers 25 28 31 and so on
YOUR TURNFocus on Critical Thinking Mathematical PracticesHave students compare and contrast the classification of numbers in the answers in Exercises 5 and 6
ElaborateTalk About ItSummarize the Lesson
Ask What are some ways that number sets can be related Sets may be subsets of other sets or they may be separate from other sets
GUIDED PRACTICEEngage with the Whiteboard
Have students place the numbers in Exercises 1ndashthinsp8 in the Venn diagram for numbers at the beginning of the lesson
Integrating Language Arts EL
Encourage English learners to ask for clarification on any terms or phrases that they do not understand
Avoid Common ErrorsExercise 7 Remind students that a repeating decimal is a rational numberExercises 9ndash10 Remind students that it only takes one counterexample to show that a statement is false
Interactive Whiteboard Interactive example available online
ADDITIONAL EXAMPLE 3Identify the set of numbers that best describes the situation Explain your choice
A the amount of time that has passed since midnight
The set of real numbers time is continuous so the amount of time can be rational or irrational
B the number of tickets sold to a basketball game
The set of whole numbers the number of tickets sold may be 0 or a counting number
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17 Lesson 12
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
1IN
116 inch
Guided Practice
Write all names that apply to each number (Example 1)
1 7 _ 8 2 radic_
36
3 radic_
24 4 075
5 0 6 - radic_ 100
7 5 _
45 8 - 18 __ 6
Tell whether the given statement is true or false Explain your choice (Example 2)
9 All whole numbers are rational numbers
10 No irrational numbers are whole numbers
Identify the set of numbers that best describes each situation Explain your choice (Example 3)
11 the change in the value of an account when given to the nearest dollar
12 the markings on a standard ruler
13 What are some ways to describe the relationships between sets of numbers
CHECK-INESSENTIAL QUESTION
rational real
rational real
True Whole numbers are rational numbers
Rational numbers the ruler is marked every 1 __ 16 th inch
Sample answer Describe one set as being a subset of
another or show their relationships in a Venn diagram
Integers the change can be a whole dollar amount
and can be positive negative or zero
True Whole numbers are a subset of the set of rational numbers
and can be written as a ratio of the whole number to 1
irrational real
whole integer rational real
whole integer rational real
rational real
integer rational real
integer rational real
Unit 118
copy H
ough
ton
Miff
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ublis
hing
Com
pany
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
8_MCAAESE206984_U1M01L2indd 18 41613 136 AM
My Notes
Math TrainerOnline Practice
and Help
Personal
myhrwcom
Math On the Spot
myhrwcom
Identifying Sets for Real-World SituationsReal numbers can be used to represent real-world quantities Highways have posted speed limit signs that are represented by natural numbers such as 55 mph Integers appear on thermometers Rational numbers are used in many daily activities including cooking For example ingredients in a recipe are often given in fractional amounts such as 2 _ 3 cup flour
Identify the set of numbers that best describes each situation Explain your choice
the number of people wearing glasses in a room
The set of whole numbers best describes the situation The number of people wearing glasses may be 0 or a counting number
the circumference of a flying disk has a diameter of 8 9 10 11 or 14 inches
The set of irrational numbers best describes the situation Each circumference would be a product of π and the diameter and any multiple of π is irrational
EXAMPLEXAMPLE 3
A
B
Identify the set of numbers that best describes the situation Explain your choice
5 the amount of water in a glass as it evaporates
6 the weight of a person in pounds
YOUR TURN
8NS1
Rational numbers a personrsquos weight can be a decimal
such as 835 pounds
Real numbers the amount can be any number greater
than 0
17Lesson 12
copy H
ough
ton
Miff
lin H
arco
urt P
ublis
hing
Com
pany
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
8_MCAAESE206984_U1M01L2indd 17 41613 520 AM
Graphic OrganizersGive students a list of numbers (including terminating and repeating decimals fractions integers and rational and irrational square roots) and a graphic organizer as shown below
Real Numbers
Rational numbers Irrational numbers
Integer numbers
Whole numbers
Ask students to write each number in the list in the correct section of the organizer
Number SensePoint out to students that knowing the types of numbers to expect in different situations can alert them to incorrect math as well as to impossible situations For example 135 shots made in basketballs is not possible but an average number of shots can equal 135
Additional ResourcesDifferentiated Instruction includes bull Reading Strategies bull Success for English Learners EL
bull Reteach bull Challenge PRE-AP
DIFFERENTIATE INSTRUCTION
Sets of Real Numbers 18
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
Math TrainerOnline Assessment
and Intervention
Personal
Online homework assignment available
myhrwcom
Lesson Quiz available online
12 LESSON QUIZ
1 Write all the names that apply to the number
2 Tell whether the given statement is true or false Explain your choice All numbers between 1 and 2 are rational numbers
3 Identify the set of numbers that best describes the situation Explain your choiceThe choices on a survey question change the total points for the survey by -2 -1 0 1 or 2 points
-1 _
5
myhrwcom
Additional ResourcesDifferentiated Instruction includes bull Leveled Practice worksheets
EvaluateGUIDED AND INDEPENDENT PRACTICE
Concepts amp Skills Practice
Example 1Classifying Real Numbers
Exercises 1ndash8 14ndash19 22ndash24
Example 2Understanding Sets and Subsets of Real Numbers
Exercises 9ndash10
Example 3Identifying Sets for Real-World Situations
Exercises 11ndash12 20ndash21 25
Focus | Coherence | Rigor
Exercise Depth of Knowledge (DOK) Mathematical Practices
14ndash19 2 SkillsConcepts MP7 Using Structure
20ndash21 2 SkillsConcepts MP6 Precision
22ndash23 2 SkillsConcepts MP3 Logic
24 1 Recall of Information MP7 Using Structure
25 2 SkillsConcepts MP2 Reasoning
26ndash27 3 Strategic Thinking MP3 Logic
28 3 Strategic Thinking MP8 Patterns
29 3 Strategic Thinking MP3 Logic
8NS1
8NS1
Exercise 29 combines concepts from the California Common Core cluster ldquoKnow that there are numbers that are not rational and approximate them by rational numbersrdquo
Answers1 rational real
2 False radic_
2 is an example of an irrational number between 1 and 2
3 Integers each number is an integer but only three are whole numbers
19 Lesson 12
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
Work Area
π mi23 Critique Reasoning The circumference of a circular region is shown
What type of number best describes the diameter of the circle Explain
your answer
24 Critical Thinking A number is not an integer What type of number can it be
25 A grocery store has a shelf with half-gallon containers of milk What type of number best represents the total number of gallons
26 Explain the Error Katie said ldquoNegative numbers are integersrdquo What was her error
27 Justify Reasoning Can you ever use a calculator to determine if a number is rational or irrational Explain
28 Draw Conclusions The decimal 0 _
3 represents 1 _ 3 What type of number best describes 0
_ 9 which is 3 middot 0
_ 3 Explain
29 Communicate Mathematical Ideas Irrational numbers can never be precisely represented in decimal form Why is this
FOCUS ON HIGHER ORDER THINKING
It can be a rational number that is not an integer or an irrational number
rational number
The set of negative numbers also includes non-integer
rational numbers and irrational numbers
Sample answer If the calculator shows a decimal that
terminates in fewer digits than what the calculator screen
allows then you can tell that the number is rational If not
you cannot tell from the calculator display whether the
number terminates because you see a limited number
of digits It may be a repeating decimal (rational) or
non-terminating non-repeating decimal (irrational)
Whole 3 middot 0 _
3 represents 3 middot 1 _ 3 = 1 so 0 _
9 is exactly 1
Sample answer In decimal form irrational numbers never
terminate and never repeat Therefore no matter how
many decimal places you include the number will never
be precisely represented There are always more digits
Whole the diameter is π _ π = 1 mile
Unit 120
copy H
ough
ton
Miff
lin H
arco
urt P
ublis
hing
Com
pany
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
8_MCAAESE206984_U1M01L2indd 20 120413 909 PM
Integers
Rational Numbers Irrational Numbers
Real Numbers
Whole Numbers
257
radic16
166
radic9
128 radic50
Personal Math Trainer
Online Practice and Helpmyhrwcom
Name Class Date
Independent Practice
Identify the set of numbers that best describes each situation Explain your choice
20 the height of an airplane as it descends to an airport runway
21 the score with respect to par of several golfers 2 ndash 3 5 0 ndash 1
22 Critique Reasoning Ronald states that the number 1 __ 11 is not rational because when converted into a decimal it does not terminate Nathaniel says it is rational because it is a fraction Which boy is correct Explain
12
14 - radic_
9 15 257
16 radic_
50 17 8 1 _ 2
18 166 19 radic_
16
Write all names that apply to each number Then place the numbers in the correct location on the Venn diagram
8NS1
Real numbers the height can be any number greater than zero
integer rational real whole integer rational real
whole integer rational real
irrational real
rational real
rational real
Integers the scores are counting numbers their
opposites and zero
Nathaniel is correct A rational number is a number that can be written as a fraction and 1 __ 11 is a fraction
19Lesson 12
copy H
ough
ton
Miff
lin H
arco
urt P
ublis
hing
Com
pany
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
8_MCAAESE206984_U1M01L2indd 19 41613 136 AM
myhrwcomActivity available onlineEXTEND THE MATH PRE-AP
Activity Have students consider the concept of restricted domain for the sets of numbers that describe situations For example the number of sisters a person has can best be described by whole numbers but no one has ever had 1500 sisters An area code is an integer or whole number between 200 and 999
Have students use a source such as the Guinness Book of World Records and give examples of sets of numbers that describe situations where the domain is restricted Ask whether the restriction may be changed in the future
Sets of Real Numbers 20
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
-3-4-5 -2-1 1 2 3 50 4
12-4 -radic5
Lesson Support Content Objective Students will learn to order a set of real numbers
Language Objective Students will show and describe how to order a set of real numbers
LESSON 13 Ordering Real Numbers
Building BackgroundEliciting Prior Knowledge Have students draw a number line to compare a rational number and an irrational number such as - radic
_ 5 and -4 1 __ 2 Ask them to explain how
they approximated the irrational number on the number line Then have them identify the greater and the lesser real number Repeat with several other pairs of real numbers in different forms
Learning ProgressionsIn this lesson students order a set of real numbers They use rational approximations to compare the sizes of irrational numbers They also order numbers for real-world situations Important understandings for students include the following
bull Compare irrational numbers bull Estimate the value of expressions with irrational numbers bull Order a set of real numbers bull Order real numbers in a real-world context
Work with real numbers continues throughout Grade 8 and into high school This lesson provides students with a foundation for understanding the relative sizes of numbers in different forms in the real number system
Cluster ConnectionsThis lesson provides an excellent opportunity to connect ideas in this cluster Know that there are numbers that are not rational and approximate them by rational numbers Tell students that there is a special number called the golden ratio with applications in mathematics geometry art and architecture The golden ratio is called phi and is represented by the Greek letter ϕ It includes an irrational number in its definition
Have students explain why the golden ratio is irrational Ask them to find the two whole numbers the golden ratio lies between Then challenge them to approximate the golden ratio to the nearest tenth It is irrational because it includes an irrational number in its definition It lies between 1 and 2 To the nearest tenth ϕ = 16
ϕ = 1 + radic_
5 _ 2
Focus | Coherence | Rigor
California Common Core Standards
8NS2 Use rational approximations of irrational numbers to compare the size of irrational numbers locate them approximately on a number line diagram and estimate the value of expressions (eg π 2 )
MP4 Model with mathematics
21A
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
Math Talk
Language Support EL
PROFESSIONAL DEVELOPMENT
Linguistic Support EL
AcademicContent Vocabulary
Post a chart like this to remind students of the regular comparative forms of adjectives that use the -er and -est suffixes Add to the chart for terms that appear in examples and exercises in each lesson Include any irregular verb forms
Background Knowledge
Go On ndash the title of the module review or quiz is Ready to Go On This title uses an idiomatic expression In this context to go on means ldquoto move aheadrdquo or ldquoto proceedrdquo It is different from the use of go on that means having enough facts to use meaningfully as in having enough to go on Also the intonation used in pronouncing an expression can give it different meanings For example when the speaker emphasizes the word on he or she might be expressing disbelief as in ldquoGo ON Yoursquore kidding rightrdquo Discuss with students other ways that the phrase go on may be used
Leveled Strategies for English Learners
Emerging Label points on a number line with the terms used in ordering greater greatest less lesser least Use sentence frames to insert the correct terms
Expanding Have students give two or three complete sentences to compare the placement of numbers on a number line using the correct forms of the comparative and superlative adjectives
Bridging Have students work in pairs with one student giving directions to the other in complete sentences to order numbers on a number line
To help students answer the question posed in Math Talk make sure that students have a command of the forms for making comparisons and the superlative and the concept of opposite order so that the focus is on the math concept instead of the language skills needed to describe and explain order
EL
Adjective Comparative Superlative
Far Farther Farthest
Large Larger Largest
Great Greater Greatest
Some Less Least
Some More Most
California ELD Standards
Emerging 2I8 Analyzing language choices ndash Explain how phrasing or different common words with similar meanings produce different effects on the audience
Expanding 2I8 Analyzing language choices ndash Explain how phrasing or different words with similar meanings or figurative language produce shades of meaning and different effects on the audience
Bridging 2I8 Analyzing language choices ndash Explain how phrasing or different words with similar meanings or figurative language produce shades of meaning nuances and different effects on the audience
Ordering Real Numbers 21B
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
13L E S S O N
Ordering Real Numbers
Interactive Whiteboard Interactive example available online
ADDITIONAL EXAMPLE 1Compare Write lt gt or =
A radic_
8 - 2 4 - radic_
8 lt
B radic_
20 + 1 3 + radic_
2 gt
EngageESSENTIAL QUESTION
How do you order a set of real numbers Sample answer Find their approximate decimal values and order them
Motivate the LessonAsk What kind of numbers are you comparing when you compare the price of gasoline at two different gas stations
ExploreGive students two rational numbers and ask them to name a number between them Repeat a few times and then give them two irrational numbers and ask them to name a number between them
ExplainEXAMPLE 1
Questioning Strategies Mathematical Practices bull Which is greater the difference between 5 and 3 or the difference between radic
_ 5 and radic
_ 3
The difference between 5 and 3 is 2 the difference between radic_
5 and radic_
3 is approximately 1 So the difference between 5 and 3 is greater
Avoid Common ErrorsCaution students to read the problem carefully and think about what the radical sign means so that they do not misread the problem and answer that the two sides are equal
YOUR TURNFocus on TechnologyCalculators should not be used at this point because developing number sense is the goal
EXAMPLE 2Questioning Strategies Mathematical Practices bull How do you determine whether radic
_ 22 is less than or greater than 45 The square of 45 is
2025 which is less than 22 so the square root of 22 must be greater than 45
Engage with the WhiteboardHave students graph and label various real numbers between 42 and 44 and between 47 and 5
YOUR TURNFocus on Modeling Mathematical PracticesHave students label the integers on the number line with their equivalent square root For example 1 2 and 3 on the number line would be labeled radic
_ 1 radic
_ 4 and radic
_ 9
Interactive Whiteboard Interactive example available online
ADDITIONAL EXAMPLE 2Order 3π radic
_ 10 and 325 from greatest
to least
3π 325 radic_
10
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CA Common CoreStandards
The student is expected to
The Number Systemmdash8NS2
Use rational approximations of irrational numbers to compare the size of irrational numbers locate them approximately on a number line diagram and estimate the value of expressions (eg π 2 )
Mathematical Practices
MP4 Modeling
The student is expected to
21 Lesson 13
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
Math TrainerOnline Practice
and Help
Personal
myhrwcom
Math On the Spotmyhrwcom
0 05 1 15 2 25 3 35 4
radic5radic3
π2
8 85 9 95 10 105 11 115 12
radic75
4 42 44 46 48 5
radic224 12π + 1
Ordering Real Numbers You can compare and order real numbers and list them from least to greatest
Order radic_
22 π + 1 and 4 1 _ 2 from least to greatest
First approximate radic_
22
radic_
22 is between 4 and 5 Since you donrsquot know where it falls between 4 and 5 you need to find a better estimate for radic
_ 22 so
you can compare it to 4 1 _ 2
Since 22 is closer to 25 than 16 use squares of numbers between 45 and 5 to find a better estimate of radic
_ 22
45 2 = 2025 46 2 = 2116 47 2 = 2209 48 2 = 2304
Since 47 2 = 2209 an approximate value for radic_
22 is 47
An approximate value of π is 314 So an approximate value of π +1 is 414
Plot radic_
22 π + 1 and 4 1 _ 2 on a number line
Read the numbers from left to right to place them in order from least to greatest
From least to greatest the numbers are π + 1 4 1 _ 2 and radic_
22
EXAMPLE 2
STEP 1
STEP 2
Order the numbers from least to greatest Then graph them on the number line
YOUR TURN
5 radic_
5 25 radic_
3
6 π 2 10 radic_
75
If real numbers a b and c are in order from least to greatest what is the order
of their opposites from least to greatest
Explain
Math TalkMathematical Practices
8NS2
radic_
3 radic_
5 25
radic_
75 π2 10
Math Talk answer -c -b -a -c is farthest to the left on a number line -b is in the middle and -a is farthest to the right
Unit 122
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ough
ton
Miff
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pany
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
8_MCAAESE206984_U1M01L3indd 22 41613 447 AM
My Notes
Math TrainerOnline Practice
and Help
Personal
myhrwcom
Math On the Spot
myhrwcom
Comparing Irrational NumbersBetween any two real numbers is another real number To compare and order real numbers you can approximate irrational numbers as decimals
Compare radic_
3 + 5 3 + radic_
5 Write lt gt or =
First approximate radic_
3
radic_
3 is between 1 and 2
Next approximate radic_
5
radic_
5 is between 2 and 3
Then use your approximations to simplify the expressions
radic_
3 + 5 is between 6 and 7
3 + radic_
5 is between 5 and 6
So radic_
3 + 5 gt 3 + radic_
5
Reflect1 If 7 + radic
_ 5 is equal to radic
_ 5 plus a number what do you know about the
number Why
2 What are the closest two integers that radic_
300 is between
EXAMPLEXAMPLE 1
STEP 1
STEP 2
Compare Write lt gt or =
YOUR TURN
3 radic_
2 + 4 2 + radic_
4 4 radic_
12 + 6 12 + radic_
6
L E S S O N
13 Ordering Real Numbers
ESSENTIAL QUESTIONHow do you order a set of real numbers
8NS2
Use rational approximations of irrational numbers to compare the size of irrational numbers locate them approximately on a number line diagram and estimate the value of expressions (eg π 2 )
8NS2
Use perfect squares to estimate square roots
1 2 = 1 2 2 = 4 3 2 = 9
The number is 7 both expressions must equal 7 + radic_
5
17 and 18
gt lt
21Lesson 13
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ough
ton
Miff
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Com
pany
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
8_MCAAESE206984_U1M01L3indd 21 41913 246 PM
PROFESSIONAL DEVELOPMENT
Math BackgroundIn this lesson students estimate irrational numbers in the form of square roots of nonper-fect squares by finding two perfect squares between which the number falls A more precise method involves repeated division For example to find radic
_ 28 find a whole number whose perfect
square is close to 28 such as 5 Divide 28 by that number 28 divide 5 = 56 Find the average of the quotient and divisor 5 + 56
_____ 2 = 53 Continue dividing 28 by each result and averaging until you get the desired accuracy
Integrate Mathematical Practices MP4
This lesson provides an opportunity to address this Mathematical Practices standard It calls for students to model relationships using multiple representations including diagrams graphs and language as appropriate Students use multiple representations when they use number lines to estimate the locations of and order rational and irrational numbers given as symbols
Ordering Real Numbers 22
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
Interactive Whiteboard Interactive example available online
ADDITIONAL EXAMPLE 3The diameter of a meteorite in millimeters is calculated by four different methods Order the results from least to greatest
Joe radic_
18 mm Lisa 13 __ 3 mm
Pablo 46 mm Julien 4π __ 3 mm
Julien 4π __ 3 mm Lisa 13 __ 3 mm
Joe radic_
18 mm Pablo 46 mm
EXAMPLE 3Questioning Strategies Mathematical Practices bull How can you verify that radic
_ 28 is between 52 and 53 5 2 2 = 2704 and 5 3 2 = 2809
bull Explain how to determine which number is greater 5 _
5 or 55 When the repeating decimal is rounded to the nearest tenth or hundredth you can see that it is greater
Connect to Daily LifeDiscuss how measuring across a canyon might involve different methods than measuring along a road Explain that measurements like these are often done using calculations that approximate the distance
YOUR TURNFocus on Critical Thinking Mathematical PracticesDiscuss with students which number is greater 3
_ 45 or 3450 3
_ 45 or 3455 and why Explain
that 3 _
45 can be written out as 34545hellipMake sure they understand that 3 _
45 is greater than 345 but less than 3455
ElaborateTalk About ItSummarize the Lesson
Ask How can you order two numbers in different forms whose decimal approxi-mations appear to be equal Approximate one or both numbers to an additional
number of decimal places
GUIDED PRACTICEEngage with the Whiteboard
Have students place and label additional points on the number line in Exercise 9 Allow the points to be in any format other than decimal
Avoid Common ErrorsExercises 3ndash4 Caution students to read the problem carefully so that they do not misread the problem as the same numbers combined by addition on each side of the circleExercise 10 Remind students that the calculations have units
myhrwcom
23 Lesson 13
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
0 05 1 15 2 25 3 35 4 45 5 55 6 65 7
2πradic3
Compare Write lt gt or = (Example 1)
1 radic_
3 + 2 radic_
3 + 3 2 radic_
8 + 17 radic_
11 + 15
3 radic_
6 + 5 6 + radic_
5 4 radic_
9 + 3 9 + radic_
3
5 radic_
17 - 3 -2 + radic_
5 6 12 - radic_
2 14 - radic_
8
7 radic_
7 + 2 radic_
10 - 1 8 radic_
17 + 3 3 + radic_
11
9 Order radic_
3 2π and 15 from least to greatest Then graph them on the number line (Example 2)
radic_
3 is between and so radic_
3 asymp
π asymp 314 so 2π asymp
From least to greatest the numbers are
10 Four people have found the perimeter of a forest using different methods Their results are given in the table Order their calculations from greatest to least (Example 3)
11 Explain how to order a set of real numbers
CHECK-INESSENTIAL QUESTION
Forest Perimeter (km)
Leon Mika Jason Ashley
radic_
17 - 2 1 +thinsp π __ 2 12 ___ 5 25
Guided Practice
17
15
1 + π _ 2 km 25 km 12 __ 5 km radic_
17 - 2 km
2π radic
_ 3
18 175
628
Sample answer Convert each number to a decimal
equivalent using estimation to find equivalents for
irrational numbers Graph each number on a number line
Read the numbers from left to right for least to greatest
Read the numbers from right to left for greatest to least
lt gt
lt lt
ltgt
gt gt
24 Unit 1
copy H
ough
ton
Miff
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ublis
hing
Com
pany
bull Im
age C
redi
ts copy
Elena
Eliss
eeva
Alam
y Im
ages
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
8_MCAAESE206984_U1M01L3indd 24 41613 448 AM
My Notes
5 52 54 56 58 6
radic28 5 12
23455
Math TrainerOnline Practice
and Help
Personal
myhrwcom
Math On the Spot
myhrwcom
Ordering Real Numbers in a Real-World Context Calculations and estimations in the real world may differ It can be important to know not only which are the most accurate but which give the greatest or least values depending upon the context
Four people have found the distance in kilometers across a canyon using different methods Their results are given in the table Order the distances from greatest to least
Distance Across Quarry Canyon (km)
Juana Lee Ann Ryne Jackson
radic_
28 23 __ 4 5 _
5 5 1 _ 2
Write each value as a decimal
radic_
28 is between 52 and 53 Since 53 2 = 2809 an approximate value for radic
_ 28 is 53
23 __ 4 = 575
5 _
5 is 5555hellip so 5 _
5 to the nearest hundredth is 556
5 1 _ 2 = 55
Plot radic_
28 23 __ 4 5 _
5 and 5 1 _ 2 on a number line
From greatest to least the distances are
23 __ 4 km 5 _
5 km 5 1 _ 2 km radic_
28 km
EXAMPLEXAMPLE 3
STEP 1
STEP 2
7 Four people have found the distance in miles across a crater using different methods Their results are given below
Jonathan 10 __ 3 Elaine 3 _
45 Joseacute 3 1 _ 2 Lashonda radic_
10
Order the distances from greatest to least
YOUR TURN
8NS2
3 1 _ 2 mi 3 _
45 mi 10 __ 3 mi radic_
10 mi
23Lesson 13
copy H
ough
ton
Miff
lin H
arco
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ublis
hing
Com
pany
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
8_MCAAESE206984_U1M01L3indd 23 41613 447 AM
ModelingPlace papers around the room with the numbers from 1 to 5 one per sheet Give each student a card showing a number between 1 and 5 in different forms Have students place his or her card between the correct integers and decide where the number goes in relation to any numbers already placed
Multiple RepresentationsGive students a vertical number line which some students might find easier to use than a horizontal one Have them decide whether to place points for rational and irrational numbers above or below existing points
Additional ResourcesDifferentiated Instruction includes bull Reading Strategies bull Success for English Learners EL
bull Reteach bull Challenge PRE-AP
DIFFERENTIATE INSTRUCTION
Ordering Real Numbers 24
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
Math TrainerOnline Assessment
and Intervention
Personal
Online homework assignment available
myhrwcom
myhrwcom
Lesson Quiz available online
13 LESSON QUIZ
1 Compare Write lt gt or =
radic_
95 - 5 radic_
62 - 2
2 Order 105 radic_
105 and 3π + 1 from greatest to least
3 A length in centimeters is calculated differently by four different people Order their calculations from least to greatest
KD 11 __ 2 cm Silvio 5 __ 3 π cm
Paula 5 _
4 cm Luis radic_
33 cm
EvaluateGUIDED AND INDEPENDENT PRACTICE
Concepts amp Skills Practice
Example 1Comparing Irrational Numbers
Exercises 1ndash8
Example 2Ordering Real Numbers
Exercises 9 12ndash15 18ndash21
Example 3Ordering Real Numbers in a Real-World Context
Exercises 10 16ndash17
Additional ResourcesDifferentiated Instruction includes bull Leveled Practice worksheets
Focus | Coherence | Rigor
Exercise Depth of Knowledge (DOK) Mathematical Practices
12ndash15 1 Recall of Information MP5 Using Tools
16 2 SkillsConcepts MP2 Reasoning
17 2 SkillsConcepts MP6 Precision
18ndash21 2 SkillsConcepts MP2 Reasoning
22 3 Strategic Thinking MP4 Modeling
23ndash24 3 Strategic Thinking MP3 Logic
8NS2
8NS2
Answers1 radic
_ 95 - 5 lt radic
_ 62 - 2
2 radic_
105 3π + 1 105
3 Silvio 5 __ 3 π cm Paula 5 _
4 cm
KD 11
__ 2 cm Luis radic_
33 cm
25 Lesson 13
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
Work Area
3140 3141 3142 3143
314 π227
20 A teacher asks his students to write the numbers shown in order from least to greatest Paul thinks the numbers are already in order Sandra thinks the order should be reversed Who is right
21 Math History There is a famous irrational number called Eulerrsquos number symbolized with an e Like π its decimal form never ends or repeats The first few digits of e are 27182818284
a Between which two square roots of integers could you find this number
b Between which two square roots of integers can you find π
22 Analyze Relationships There are several approximations used for π including 314 and 22 __ 7 π is approximately 314159265358979
a Label π and the two approximations on the number line
b Which of the two approximations is a better estimate for π Explain
c Find a whole number x so that the ratio x ___ 113 is a better estimate for π
than the two given approximations
23 Communicate Mathematical Ideas If a set of six numbers that include both rational and irrational numbers is graphed on a number line what is the fewest number of distinct points that need to be graphed Explain
24 Critique Reasoning Jill says that 12 _
6 is less than 1263 Explain her error
FOCUS ON HIGHER ORDER THINKING
radic_
115 115 ___ 11 and 105624
between radic_
7 asymp 265 and radic_
8 asymp 283
between radic_
9 = 3 and radic_
10 asymp 316
22 __ 7 it is closer to π on the number line
She did not consider the repeating digit 1266
2 rational numbers can have the same location and
irrational numbers can have the same location but they
cannot share a location
355
Neither student is correct The answer
should be 115 ___ 11 105624 radic_
115
Unit 126
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ifflin
Har
cour
t Pub
lishin
g Com
pany
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e Cre
dits
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Stoc
kiSt
ockP
hoto
com
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8_MCAAESE206984_U1M01L3indd 26 210513 801 AM
Personal Math Trainer
Online Practice and Helpmyhrwcom
Name Class Date
Independent Practice
16 Your sister is considering two different shapes for her garden One is a square with side lengths of 35 meters and the other is a circle with a diameter of 4 meters
a Find the area of the square
b Find the area of the circle
c Compare your answers from parts a and b Which garden would give your sister the most space to plant
17 Winnie measured the length of her fatherrsquos ranch four times and got four different distances Her measurements are shown in the table
a To estimate the actual length Winnie first approximated each distance to the nearest hundredth Then she averaged the four numbers Using a calculator find Winniersquos estimate
b Winniersquos father estimated the distance across his ranch to be radic_
56 km How does this distance compare to Winniersquos estimate
Give an example of each type of number
18 a real number between radic_
13 and radic_
14
19 an irrational number between 5 and 7
Order the numbers from least to greatest
12 radic_
7 2 radic_
8 ___ 2 13 radic_
10 π 35
14 radic_
220 -10 radic_
100 115 15 radic_
8 -375 3 9 _ 4
Distance Across Fatherrsquos Ranch (km)
1 2 3 4
radic_
60 58 __ 8 7 _
3 7 3 _ 5
138NS2
radic_
8 ___ 2 2 radic_
7
-10 radic_
100 115 radic_
220
radic_
60 asymp 775 58 __ 8 = 725 7 _
3 asymp 733 7 3 _ 5 = 760 so the average
π radic_
10 35
-375 9 _ 4 radic_
8 3
is 74825 km
1225 m2
4π m2 or approximately 126 m2
They are nearly identical radic_
56 is approximately 74833hellip
The circle would give her more space to plant because it has a
larger area
Sample answer 37
Sample answer radic_
31
25Lesson 13
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Miff
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8_MCAAESE206984_U1M01L3indd 25 41613 448 AM
Activity available online myhrwcomEXTEND THE MATH PRE-AP
Activity Have students investigate whether there are infinitely many numbers between two numbers by giving examples for each of the following
bull Between any two rational numbers there is at least one other rational number Sample answer 45 is between 41 and 48
bull Between any two irrational numbers there is at least one rational number Sample answer 45 is between radic
_ 11 and radic
_ 29
bull Between any two rational numbers there is at least one irrational number Sample answer radic
_ 11 is between 31 and 36
bull Between any two irrational numbers there is at least one irrational number Sample answer radic
_ 17 is between radic
_ 11 and radic
_ 29
Ordering Real Numbers 26
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
ReadyMath Trainer
Online Practiceand Help
Personal
myhrwcom
Module Quiz
11ensp RationalenspandenspIrrationalenspNumbersWrite each fraction as a decimal or each decimal as a fraction
1 7__20 2 1___
27 3 17_8
Solve each equation for x
4 x2=81 5 x3=343 6 x2= 1___100
7 Asquarepatiohasanareaof200squarefeetHowlongiseachside
ofthepatiotothenearesttenth
12ensp SetsenspofenspRealenspNumbersWrite all names that apply to each number
8 121____radic
____121
9 π__2
10 TellwhetherthestatementldquoAllintegersarerationalnumbersrdquoistrueorfalseExplainyourchoice
13ensp OrderingenspRealenspNumbersCompare Write lt gt or =
11 radic__
8+3 8+radic__
3 12 radic__
5+11emsp emsp emsp 5+radic___
11
Order the numbers from least to greatest
13 radic___
99π29__
8 14 radic___
1__251_40__
2
15 Howarerealnumbersusedtodescribereal-worldsituations
ESSENTIAL QUESTION
035
9-9
141ft
7 1__10- 1__10
14__11 1875
wholeintegerrationalreal
Trueintegerscanbewrittenasthequotientoftwointegers
SampleanswerRealnumberssuchastherational
π29__
8radic___
99
irrationalreal
lt gt
number1_4candescribeamountsusedincooking
radic___
1__250__
21_4
27Module1
copy H
ough
ton
Miff
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ublis
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Com
pany
DONOTEDIT--ChangesmustbemadethroughldquoFileinfordquoCorrectionKey=A
8_MCAAESE206984_U1M01RTindd 27 41513 1113 PM
Math TrainerOnline Assessment
and Intervention
Personal
myhrwcom
1
2
3 Response toIntervention
Intervention Enrichment
Access Ready to Go On assessment online and receive instant scoring feedback and customized intervention or enrichment
Online and Print Resources
Differentiated Instruction
bull Reteach worksheets
bull Reading Strategies EL
bull Success for English Learners EL
Differentiated Instruction
bull Challenge worksheets PRE-AP
Extend the Math PRE-AP
Lesson Activities in TE
Additional ResourcesAssessment Resources includes bull Leveled Module Quizzes
Ready to Go OnAssess MasteryUse the assessment on this page to determine if students have mastered the concepts and standards covered in this module
California Common Core Standards
Lesson Exercises Common Core Standards
11 1ndash7 8NS1 8NS2 8EE2
12 8ndash10 8NS1
13 11ndash14 8NS2
27 Unit 1 Module 1
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
Personal Math Trainer
Online Practice and HelpmyhrwcomAssessment Readiness
Module 1 MIXed ReVIeW
1 Look at each number Is the number between 2π and radic___
52
Select Yes or No for expressions AndashC
A 6 2 _ 3 Yes No
B 5π __ 2 Yes No
C 3 radic__
5 Yes No
2 Consider the number - 11 __ 15
Choose True or False for each statement
A The number is rational True False
B The number can be written as True Falsea repeating decimal
C The number is less than ndash08 True False
3 The volume of a cube is given by V = x3 where x is the length of an edge of the cube A cube-shaped end table has a volume of 3 3 _ 8 cubic feet What is the length of an edge of the end table Explain how you solved this problem
4 A student says that radic___
83 is greater than 29 __ 3 Is the student correct Justify your
reasoning
1 1 _ 2 ft Sample answer The equation x3 = 3 3 _ 8 can be used
to find the edge length in feet To solve the equation
write the mixed number as a fraction greater than 1
x3 = 27 __ 8 Then take the cube root of both sides x = 3 _ 2 = 1 1 _ 2
No Sample answer radic___
83 asymp 91 and 29 __ 3 = 9
__ 6
Because 91 lt 9 __
6 radic___
83 lt 29 __ 3
28 Unit 1
copy H
ough
ton
Miff
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arco
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ublis
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Com
pany
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=A
8_MCAAESE206984_U1M01RTindd 28 240413 946 AM
Personal Math Trainer
Online Assessment and
Interventionmyhrwcom
Scoring GuideItem 3 Award the student 1 point for finding the edge length of the cube and 1 point for correctly explaining how to use a cube root to solve the problem
Item 4 Award the student 1 point for determining that the student is incorrect and 1 point for correctly justifying the reasoning for this conclusion
Additional ResourcesTo assign this assessment online login to your Assignment Manager at myhrwcom
Assessment Readiness
California Common Core Standards
Items Grade 8 Standards Mathematical Practices
1 8NS2 MP7
2 7NS2b 7NS2d 8NS1 MP7
3 8EE2 MP1 MP4
4 8NS1 8NS2 MP3
Item integrates mixed review concepts from previous modules or a previous course
Item 4 combines concepts from the California Common Core cluster ldquoKnow that there are numbers that are not rational and approximate them by rational numbersrdquo
Real Numbers 28
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
myhrwcom
What It Means to YouYou will learn to estimate the values of irrational numbers
What It Means to YouYou will recognize a number as rational or irrational by looking at its fraction or decimal form
Estimate the value of radic_
8
8 is between the perfect squares 4 and 9So radic
_ 8 is between radic
_ 4 and radic
_ 9
radic_
8 is between 2 and 3
8 is closer to 9 so radic_
8 is closer to 3 28 2 = 784 29 2 = 841 radic
_ 8 is between 28 and 29
A good estimate for radic_
8 is 285
Classify each number as rational or irrational
0 _
3 = 1 _ 3 025 = 1 _ 4
These numbers are rational because they can be written as ratios of integers or as repeating or terminating decimals
π asymp 3141592654hellip radic_ 5 asymp 2236067977hellip
These numbers are irrational because they cannot be written as ratios of integers or as repeating or terminating decimals
Understanding the standards and the vocabulary terms in the standards will help you know exactly what you are expected to learn in this module
Real NumbersGETTING READY FOR
Know that numbers that are not rational are called irrational Understand informally that every number has a decimal expansion for rational numbers show that the decimal expansion repeats eventually and convert a decimal expansion which repeats eventually into a rational number
Key Vocabularyrational number (nuacutemero
racional) A number that can be expressed as a ratio of two integers
irrational number (nuacutemero irracional)A number that cannot be expressed as a ratio of two integers or as a repeating or terminating decimal
Use rational approximations of irrational numbers to compare the size of irrational numbers locate them approximately on a number line diagram and estimate the value of expressions (eg π2)
EXAMPLE 8NS1
EXAMPLE 8NS2
8NS2
8NS1
Visit myhrwcom to see all CA Common Core Standards explained
8 is not a perfect square Find the two perfect squares closest to 8
Unit 16
copy H
ough
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Miff
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8_MCABESE206984_U1MO01indd 6 102913 1123 PM
GETTING READY FOR
Real NumbersUse the examples on the page to help students know exactly what they are expected to learn in this module
myhrwcom
California Common Core Standards Lesson 11
Lesson 12
Lesson 13
8NS1 Know that numbers that are not rational are called irrational Understand informally that every number has a decimal expansion for rational numbers show that the decimal expansion repeats eventually and convert a decimal expansion which repeats eventually into a rational number
8NS2 Use rational approximations of irrational numbers to compare the size of irrational numbers locate them approximately on a number line diagram and estimate the value of expressions (eg π2)
8EE2 Use square root and cube root symbols to represent solutions to equations of the form x 2 = p and x 3 = p where p is a positive rational number Evaluate square roots of small perfect squares and cube roots of small perfect cubes Know that radic
_ 2 is irrational
Go online to see a complete unpacking of the CA Common Core Standards
CA Common Core Standards
Content Areas
The Number Systemmdash8NS
Cluster Know that there are numbers that are not rational and approximate them by rational numbers
Real Numbers 6
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=B
Lesson Support Content Objective Students will learn to rewrite rational numbers and decimals take square roots and
cube roots and approximate irrational numbers
Language Objective Students will show and explain how to rewrite rational numbers and decimals take square roots and cube roots and approximate irrational numbers
LESSON 11 Rational and Irrational Numbers
Building BackgroundEliciting Prior Knowledge Have students work with a partner to review the relationship between fractions and decimals Ask students to provide an example of writing a fraction or mixed number as a decimal and vice versa Discuss how students chose and wrote their examples
Learning ProgressionsIn this lesson students work with positive rational and irrational numbers They make connections among the real numbers by converting fractions and decimals and approximating irrational numbers Important understandings for students include the following
bull Understand that every number has a decimal expansion bull Convert a repeating decimal to a rational number bull Evaluate square roots of perfect squares and cube roots of perfect cubes
bull Estimate an irrational number
Work with the real number system will continue in this unit as students extend the positive rational and irrational numbers to include negative numbers and compare and order real numbers
Cluster ConnectionsThis lesson provides an excellent opportunity to connect ideas in this cluster Know that there are numbers that are not rational and approximate them by rational numbers Tell students ldquoA square garden has an area of 20 square feetrdquo
Have students explain why the side length cannot be rational Then have them approximate the length of each side of the garden to the nearest tenth and hundredth Sample answer The length is the solution to s 2 = 20 radic
_ 20 which is not a rational
number 45 ft 447 ft The length is between 4 and 5 feet 20 is closer to 45 2 than to 44 2 or 46 2 It is also closer to 447 2 than to 446 2 or 448 2
3 _ 4
= 075 1 2 _ 3
= 1 _
6
7 _ 10
= 07 45 = 4 1 _ 2
20 ft 2
California Common Core Standards
8NS1 Know that numbers that are not rational are called irrational Understand informally that every number has a decimal expansion for rational numbers show that the decimal expansion repeats eventually and convert a decimal expansion which repeats eventually into a rational number
8NS2 Use rational approximations of irrational numbers to compare the size of irrational numbers locate them approximately on a number line diagram and estimate the value of expressions (eg π 2 )
8EE2 Use square root and cube root symbols to represent solutions to equations of the form x 2 = p and x 3 = p where p is a positive rational number Evaluate square roots of small perfect squares and cube roots of small perfect cubes Know that radic
_ 2 is irrational
MP6 Attend to precision
7A
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=B
Math Talk
Language Support EL
PROFESSIONAL DEVELOPMENT
Linguistic Support EL
AcademicContent Vocabulary
square ndash In this lesson the word square has multiple meanings which can cause confusion For example to square as in to take the square root of a number is a verb It is different from the nouns square or square of a number The text also refers to perfect square and principal square root of a number and the square root symbol is used These different usages of square as a mathematical term need to be clarified Sentence frames can be used to help define the meaning
To square a number means to _______The perfect square of a number means _______
Background Knowledge
suffixes ndash When added to a root word the suffix -th is used in math to indicate one of a specified number of parts such as tenth hundredth or thousandth Remind students that the suffix -th also indicates place value Note that Spanish Vietnamese Mandarin and other languages do not have the ending th sound so teachers need to enunciate carefully
cognates ndash The words terminating and terminal used in this lesson are cognates in Spanish terminar meaning ldquoto endrdquo or ldquoto finishrdquo A Spanish cognate for approximate is aproximar
Leveled Strategies for English Learners
Emerging Use cards with root words ten hundred and thousand and a card with the -th suffix Have students place them together to show place value Then complete a sentence Use the same procedure to identify decimals
Expanding Support students at this level of English proficiency by providing sentence frames for them to use to describe their mathematical reasoning
To write the fraction _______ as a decimal I _______
Bridging Have students identify different meanings of the term square by matching examples of math problems with a written out sentence frame that defines the usage of the term square to square a number perfect square square root Use this procedure also with the term cube
Be sure to clarify the different uses of the term square when referring to square roots perfect squares and so on
EL
California ELD Standards
Emerging 2I12b Selecting language resources ndash Use knowledge of morphology to appropriately select affixes in basic ways
Expanding 2I12b Selecting language resources ndash Use knowledge of morphology to appropriately select affixes in a growing number of ways to manipulate language
Bridging 2I12b Selecting language resources ndash Use knowledge of morphology to appropriately select affixes in a variety of ways to manipulate language
Rational and Irrational Numbers 7B
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
11L E S S O N
Rational and Irrational Numbers
EngageESSENTIAL QUESTION
How do you rewrite rational numbers and decimals take square roots and cube roots and approximate irrational numbers To express as a decimal divide the numerator by the denominator To take a square root or cube root of a number find the number that when squared or cubed equals the original number To approximate an irrational number estimate a number between two consecutive perfect squares
Motivate the LessonAsk Which type of rational number do you see more often fractions or decimals Which do you prefer to use Why
ExploreHave students write examples of ratios and then share with the class the various notations for ratios that they used (for example 25 2 to 5 2 __ 5 ) Point out the connection between the word ratio and the meaning of rational number See also Explore Activity in student text
ExplainEXAMPLE 1
Questioning Strategies Mathematical Practices bull How does the denominator of a fraction in simplest form tell whether the decimal equivalent of the fraction is a terminating decimal The decimal will terminate if the denominator is an even number a multiple of 5 or a multiple of 10
Avoid Common ErrorsTo avoid interpreting 1 __ 4 as 4 divided by 1 tell students to start at the top of the fraction and read the bar as ldquodivided byrdquo
YOUR TURNTalk About ItCheck for Understanding
Ask Can an improper fraction be written as a decimal Give an example to support your answer Yes 5 __ 4 = 125
EXAMPLE 2Questioning Strategies Mathematical Practices bull How can you use place value to write a terminating decimal as a fraction with a power of ten in the denominator Start by identifying the place value of the decimals last digit and then use the corresponding power of 10 as the denominator of the fraction
bull How can you tell if a decimal can be written as a rational number If the decimal is a terminating or repeating decimal then it can be written as a rational number
Interactive Whiteboard Interactive example available online
ADDITIONAL EXAMPLE 1Write each fraction as a decimal
A 2 _ 5
04 B 5 _ 9
0 _
5
myhrwcom
Interactive Whiteboard Interactive example available online
ADDITIONAL EXAMPLE 2Write each decimal as a fraction in simplest form
A 0355 71 ___ 200
B 0 _
43 43 __ 99
myhrwcom
CA Common CoreStandards
The student is expected to
The Number Systemmdash8NS1
Know that numbers that are not rational are called irrational Understand informally that every number has a decimal expansion for rational numbers show that the decimal expansion repeats eventually and convert a decimal expansion which repeats eventually into a rational number
The Number Systemmdash8NS2
Use rational approximations of irrational numbers to compare the size of irrational numbers locate them approximately on a number line diagram and estimate the value of expressions (eg π 2 )
Expressions and Equationsmdash8EE2
Use square root and cube root symbols to represent solutions to equations of the form x 2 = p and x 3 = p where p is a positive rational number Evaluate square roots of small perfect squares and cube roots of small perfect cubes Know that radic
_ 2 is irrational
Mathematical Practices
MP6 Precision
The student is expected to
the value of expressions (eg
7 Lesson 11
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
My Notes
Math On the Spotmyhrwcom
Math TrainerOnline Practice
and Help
Personal
myhrwcom
Expressing Decimals as Rational NumbersYou can express terminating and repeating decimals as rational numbers
Write each decimal as a fraction in simplest form
0825
The decimal 0825 means ldquo825 thousandthsrdquo Write this as a fraction
825 ____ 1000
Then simplify the fraction
825 divide 25 ________ 1000 divide 25 = 33 __ 40
0825 = 33 __ 40
0 _
37
Let x = 0 _
37 The number 0 _
37 has 2 repeating digits so multiply each side of the equation x = 0
_ 37 by 10 2 or 100
x = 0 _
37
(100)x = 100(0 _
37 )
100x = 37 _
37
Because x = 0 _
37 you can subtract x from one side and 0 _
37 from the other
100x = 37 _
37
minusx minus0 _
37
99x = 37
Now solve the equation for x Simplify if necessary
99x ___ 99 = 37 __ 99
x = 37 __ 99
EXAMPLE 2
A
B
Write each fraction as a decimal
YOUR TURN
1 5 __ 11 2 1 _ 8 3 2 1 _ 3
8NS1
To write ldquo825 thousandthsrdquo put 825 over 1000
Divide both the numerator and the denominator by 25
100 times 0 _
37 is 37 _
37
37 _
37 minus 0 _
37 is 37
Divide both sides of the equation by 99
0 _
45 0125 2 _
3
Unit 18
copy H
ough
ton
Miff
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pany
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8_MCAAESE206984_U1M01L1indd 8 120413 838 PM
My Notes
Math On the Spot
myhrwcom
= 033333333333331mdash3
ESSENTIAL QUESTION
Expressing Rational Numbers as DecimalsA rational number is any number that can be written as a ratio in the form a _ b where a and b are integers and b is not 0 Examples of rational numbers are 6 and 05
6 can be written as 6 _ 1 05 can be written as 1 _ 2
Every rational number can be written as a terminating decimal or a repeating decimal A terminating decimal such as 05 has a finite number of digits A repeating decimal has a block of one or more digits that repeat indefinitely
Write each fraction as a decimal
1 _ 4
1 _ 4 = 025
1 _ 3
1 _ 3 = 0 _
3
EXAMPLEXAMPLE 1
A
B
0333 3 ⟌ ⎯ 1000 minus9 10 minus9 10 minus9 1
025 4 ⟌ ⎯ 100 -8 20 -20
0
L E S S O N
11Rational and Irrational Numbers
How do you rewrite rational numbers and decimals take square roots and cube roots and approximate irrational numbers
8NS1
Know that numbers that are not rational are called irrational Understand informally that every number has a decimal expansion for rational numbers show that the decimal expansion repeats eventually and convert a decimal expansion which repeats eventually into a relation number Also 8NS2 8EE2
8NS1
Remember that the fraction bar means ldquodivided byrdquo Divide the numerator by the denominator
Divide until the remainder is zero adding zeros after the decimal point in the dividend as needed
Divide until the remainder is zero or until the digits in the quotient begin to repeat
Add zeros after the decimal point in the dividend as needed
When a decimal has one or more digits that repeat indefinitely write the decimal with a bar over the repeating digit(s)
7Lesson 11
copy H
ough
ton
Miff
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pany
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8_MCABESE206984_U1M01L1indd 7 11113 128 AM
PROFESSIONAL DEVELOPMENT
Math BackgroundSome decimals may have a pattern but still not be a repeating decimal that is rational For example in 312112111211112hellip you can predict the next digit and describe the pattern (There is one more 1 each time before the 2) However this is not a terminating decimal nor is it a repeating decimal and it is therefore NOT a rational number
Integrate Mathematical Practices MP6
This lesson provides an opportunity to address this Mathematical Practices standard It calls for students to attend to precision Students learn to express rational numbers accurately and precisely in both fractional and decimal forms and learn to translate from one form to the other They also learn how to precisely represent and communicate ideas about irrational numbers square roots and cube roots
Rational and Irrational Numbers 8
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=B
Focus on Technology Mathematical PracticesPoint out the importance of entering a repeating decimal correctly when using a graphing calculator to convert the decimal to a fraction The decimal 0
_ 59 must be entered as
0595959595959 not 059
YOUR TURNFocus on Math ConnectionsMake sure students understand that the place value of the last digit in Exercises 4 and 6 determines the denominator of the corresponding fraction or mixed number So for Exercise 4 the place value hundredths gives a denominator of 100 and for Exercise 6 the place value tenths gives a denominator of 10
EXAMPLE 3Questioning Strategies Mathematical Practices bull How can a solution of an equation of the form x 2 = p be negative if p is a positive number Since the square of a negative number is positive a negative number is also a solution of x 2 equals a positive number
bull When is a solution of an equation of the form x 3 = p larger than p The solution is larger than p if p is a number between 0 and 1
Focus on Math Connections Make sure students understand the difference in finding radic
_ 121 and solving x 2 = 121 The
symbol radic_
indicates the positive or principal square root only while the equation x 2 = 121 has two roots the principal square root and its opposite
YOUR TURNAvoid Common ErrorsTo avoid sign errors in Exercise 9 make sure that students understand that the cube of a negative number is not a positive number Therefore -8 is not a solution of x 3 = 512
Talk About ItCheck for Understanding
Ask Kris predicts that there are two real solutions for Exercises 7 and 8 and that there are three real solutions for Exercises 9 and 10 Is his prediction correct
Explain His prediction is correct for Exercises 7 and 8 because there are two numbers whose squares are the same positive number given in the exercises His prediction is not correct for Exercises 9 and 10 however because there is only one real number whose cube is the same positive number given in the exercises
EXPLORE ACTIVITYQuestioning Strategies Mathematical Practices bull Compare the values for 13 2 and 13 2 The digits are the same but 13 2 has two decimal places (169) while 13 2 has none (169)
bull How do you know whether radic_
2 will be closer to 1 or closer to 2 It will be closer to 1 because 2 is between the perfect squares of 1 and 4 but closer to 1 than it is to 4
Connect Vocabulary EL
Explain to students that the word irrational when used as an ordinary word in English means without logic or reason In mathematics when we say that a number is irrational it means only that the number cannot be written as the quotient of two integers
Engage with the WhiteboardHave students extend the number line in both directions and label the locations of the whole numbers 1 and 2 These are the roots of the consecutive perfect squares
1 and 4 used to estimate radic_
7
Interactive Whiteboard Interactive example available online
ADDITIONAL EXAMPLE 3Solve each equation for x
A x 2 = 324 18 -18
B x 2 = 25 ___ 144 5 __ 12 - 5 __ 12
C 343 = x 3 7
D x 3 = 125 ___ 512 5 __ 8
myhrwcom
9 Lesson 11
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
Math TrainerOnline Practice
and Help
Personal
myhrwcom
EXPLORE ACTIVITY
lt 2 lt
radic_
lt radic
_ 2 lt
radic_
lt radic
_ 2 lt
The solution is 9
The solution is 2 _ 5
C
D
729 = x 3
3 radic_ 729 = 3 radic
_ x 3
3 radic_ 729 = x
9 = x
x 3 = 8 ___ 125
3 radic_
x 3 =thinsp 3 radic_ 8 ___ 125
x =thinsp 3 radic_ 8 ___ 125
x = 2 _ 5
Solve each equation for x
YOUR TURN
7 x 2 = 196 8 x 2 = 9 ___ 256
9 x 3 = 512 10 x 3 = 64 ___ 343
Estimating Irrational NumbersIrrational numbers are numbers that are not rational In other words they cannot be written in the form a _ b where a and b are integers and b is not 0 Square roots of perfect squares are rational numbers Square roots of numbers that are not perfect squares are irrational Some equations like those in Example 3 involve square roots of numbers that are not perfect squares
x 2 = 2 x = plusmn radic_
2
Estimate the value of radic_
2
Find two consecutive perfect squares that 2 is between Complete the inequality by writing these perfect squares in the boxes
Now take the square root of each number
Simplify the square roots of perfect squares
radic_
2 is between and
A
B
C
8NS2 8EE2
Solve for x by taking the cube root of both sides
Solve for x by taking the cube root of both sides
Apply the definition of cube root
Think What number cubed equals 729
Apply the definition of cube root
Think What number cubed equals 8 ____ 125
radic_
2 is irrational
x = plusmn14 x = plusmn 3 __ 16
x = 8 x = 4 _ 7
1 2
1 4
1 4
1 2
Unit 110
copy H
ough
ton
Miff
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ublis
hing
Com
pany
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
8_MCAAESE206984_U1M01L1indd 10 41613 1211 AM
Math TrainerOnline Practice
and Help
Personal
myhrwcom
Math On the Spot
myhrwcom
Write each decimal as a fraction in simplest form
YOUR TURN
Finding Square Roots and Cube RootsThe square root of a positive number p is x if x 2 = p There are two square roots for every positive number For example the square roots of 36 are 6 and minus6 because 6 2 = 36 and (minus6) 2 = 36 The square roots of 1 __ 25 are 1 _ 5 and minus 1 _ 5 You can write the square roots of 1 __ 25 as plusmn 1 _ 5 The symbol radic
_ 5 indicates the positive
or principal square root
A number that is a perfect square has square roots that are integers The number 81 is a perfect square because its square roots are 9 and minus9
The cube root of a positive number p is x if x 3 = p There is one cube root for every positive number For example the cube root of 8 is 2 because 2 3 = 8 The cube root of 1 __ 27 is 1 _ 3 because ( 1 _ 3 )
3
= 1 __ 27 The symbol 3 radic_ 1 indicates the
cube root
A number that is a perfect cube has a cube root that is an integer The number 125 is a perfect cube because its cube root is 5
Solve each equation for x
The solutions are 11 and minus11
The solutions are 4 __ 13 and minus 4 __ 13
EXAMPLEXAMPLE 3
A x 2 = 121
x 2 = 121
x = plusmn radic_
121
x = plusmn11
B x 2 = 16 ___ 169
x 2 = 16 ___ 169
x = plusmn radic_
16 ___ 169
x = plusmn 4 __ 13
4 012 5 0 _
57 6 14
Can you square an integer and get a negative number
What does this indicate about whether negative
numbers have square roots
Math TalkMathematical Practices
8EE2
Solve for x by taking the square root of both sides
Apply the definition of square root
Think What numbers squared equal 121
Solve for x by taking the square root of both sides
Apply the definition of square root
Think What numbers squared equal 16 ____ 169
3 __ 25 19 __ 33 1 2 _ 5
No the square of a positive integer is positive the square of a negative integer is positive and the square of 0 is 0 So negative numbers do not have (real) square roots
9Lesson 11
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ough
ton
Miff
lin H
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Com
pany
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
8_MCAAESE206984_U1M01L1indd 9 41913 240 PM
Critical ThinkingIn the Explore Activity students estimated the location of radic
_ 2 on a number line Ask students
whether they think that it is possible to locate more precisely the point that represents radic
_ 2 In
other words can you graph irrational numbers exactly on a number line along with rational numbers Students should understand that radic
_ 2
is a real number and all real numbers can be located on a real number line A more precise estimate will allow more precise placement on a number line
The Modeling note tells one way to do this
ModelingHave students use a ruler to represent a number line with a unit that is one inch long Have them draw a square with a side of one inch and draw the diagonal to make two isosceles triangles Lead students to understand that the length of the diagonal (or hypotenuse) is radic
_ 2
Have them copy the length of their diagonal onto their ruler or number line starting at zero The end point of the diagonal represents the exact point for the irrational number radic
_ 2 on a
number line
Additional ResourcesDifferentiated Instruction includes bull Reading Strategies bull Success for English Learners EL
bull Reteach bull Challenge PRE-AP
DIFFERENTIATE INSTRUCTION
Rational and Irrational Numbers 10
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
ElaborateTalk About ItSummarize the Lesson
Ask If someone claims that a certain number is irrational but you know it is actually rational how could you prove to that person that the number is rational
You could find a fraction equal to the number such that the number is the ratio of two integers with the denominator not equal to zero
GUIDED PRACTICEEngage with the Whiteboard
Have students plot each number in Exercises 16ndash18 on a number line Students should label each point with the irrational number written as a radical and as a
decimal
Avoid Common ErrorsExercises 1ndash6 To avoid reversing the order of the dividend and divisor tell students to start at the top of the fraction and read the bar as ldquodivided byrdquo
Focus on TechnologyHave students use a calculator to investigate the decimal equivalents of such fractions as 1 __ 9 2 __ 9 8 __ 9 and 1 __ 11 2 __ 11 10
__ 11 Ask them to describe the patterns they find as a result of these investigations
11 Lesson 11
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
Guided Practice
7 0675 8 56 9 044
10 0 _
4
10x =
x =
11 0 _
26
100x =
x =
12 0 _
325
1000x =
x =
Solve each equation for x (Example 3 and Explore Activity)
- x
-
_______________
x =
- x
-
___________________
x =
- x
-
_______________________
x =
Write each fraction or mixed number as a decimal (Example 1)
1 2 _ 5 2 8 _ 9 3 3 3 _ 4
4 7 __ 10 5 2 3 _ 8 6 5 _ 6
Write each decimal as a fraction or mixed number in simplest form (Example 2)
13 x 2 = 17 14 x 2 = 25 ___ 289 15 x 3 = 216
Approximate each irrational number to one decimal place without a calculator
x = plusmn radic__
asymp plusmn x = 3
radic__
=
(Explore Activity)
16 radic_
5 asymp
17 radic_
3 asymp
18 radic_
10 asymp
19 What is the difference between rational and irrational numbers
CHECK-INESSENTIAL QUESTION
x = plusmn radic__
__________ = plusmn _____
4 _
4
0 _
4
4 99
6216
269
41 25 5
17289
17
22 17 32
04
07
27__40
26 __ 99 325 ___ 999 4 _ 9
11__255 3_5
0 _
8
2375
375
08 _
3
26 _
26
0 _
26
325 _
325
0 _
325
999 325
Rational numbers can be written in the form a __ b where
a and b are integers and b ne 0 Irrational numbers cannot
be written in this form
Unit 112
copy H
ough
ton
Miff
lin H
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urt P
ublis
hing
Com
pany
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
8_MCAAESE206984_U1M01L1indd 12 41613 1211 AM
11 12 13 14 15
radic2 asymp 14
141 142 143 144 145
radic2 asymp 141
0 1 2 3 4
radic2 asymp 15
Estimate that radic_
2 asymp 15
To find a better estimate first choose some numbers between 1 and 2 and square them For example choose 13 14 and 15
1 3 2 = 1 4 2 = 1 5 2 =
Is radic_
2 between 13 and 14 How do you know
Is radic_
2 between 14 and 15 How do you know
2 is closer to than to so radic_
2 asymp
Locate and label this value on the number line
Reflect 11 How could you find an even better estimate of radic
_ 2
12 Find a better estimate of radic_
2
1 41 2 = 1 42 2 = 1 43 2 =
2 is closer to than to so radic_
2 asymp
Draw a number line and locate and label your estimate
13 Solve x 2 = 7 Write your answer as a radical expression Then estimate to one decimal place
D
E
F
No 2 is not between 169 and 196
Yes 2 is between 196 and 225
196
19881
19881
225
20164
20164
14
141
20449
169 196 225
Test the squares of numbers between 14 and 15
x = plusmn radic_
7 x asymp plusmn26
11Lesson 11
copy H
ough
ton
Miff
lin H
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urt P
ublis
hing
Com
pany
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
8_MCAAESE206984_U1M01L1indd 11 41613 1211 AM
Rational and Irrational Numbers 12
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
Math TrainerOnline Assessment
and Intervention
Personal
Online homework assignment available
myhrwcom
EvaluateGUIDED AND INDEPENDENT PRACTICE
Concepts amp Skills Practice
Example 1Expressing Rational Numbers as Decimals
Exercises 1ndash6 20ndash21 24ndash25
Example 2Expressing Decimals as Rational Numbers
Exercises 7ndash12 22ndash23 26ndash27
Example 3Finding Square Roots and Cube Roots
Exercises 13ndash15 28 30ndash31 35
Explore ActivityEstimating Irrational Numbers
Exercises 13 16ndash18 29 32ndash34
Additional ResourcesDifferentiated Instruction includes bull Leveled Practice worksheets
Lesson Quiz available online
11 LESSON QUIZ
1 Write as a decimal 2 5 __ 8 1 7 __ 12
2 Write as a fraction 034 1 _
24
3 Solve x 2 = 9 __ 49 for x
4 Solve x 3 = 216 for x
5 Estimate the value of radic_
13 to one decimal place without using a calculator
myhrwcom
Focus | Coherence | Rigor
Exercise Depth of Knowledge (DOK) Mathematical Practices
20ndash27 2 SkillsConcepts MP4 Modeling
28 3 Strategic Thinking MP4 Modeling
29ndash32 2 SkillsConcepts MP6 Precision
33 3 Strategic Thinking MP7 Using Structure
34 2 SkillsConcepts MP3 Logic
35 2 SkillsConcepts MP4 Modeling
36 3 Strategic Thinking MP3 Logic
37 3 Strategic Thinking MP7 Using Structure
38 3 Strategic Thinking MP2 Reasoning
8NS1 8NS2 8EE2
8NS1 8NS2 8EE2
Answers1 2625 158
_ 3
2 17 __ 50 1 8 __ 33
3 x = plusmn 3 __ 7
4 x = 6
5 36
13 Lesson 11
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
Work Area
33 Analyze Relationships To find radic_
15 Beau found 3 2 = 9 and 4 2 = 16 He said that since 15 is between 9 and 16 radic
_ 15 must be between 3 and 4 He
thinks a good estimate for radic_
15 is 3 + 4 ____ 2 = 35 Is Beaursquos estimate high low
or correct Explain
34 Justify Reasoning What is a good estimate for the solution to the equation x 3 = 95 How did you come up with your estimate
35 The volume of a sphere is 36π f t 3 What is the radius of the sphere Use the formula V = 4 _ 3 π r 3 to find your answer
36 Draw Conclusions Can you find the cube root of a negative number If so is it positive or negative Explain your reasoning
37 Make a Conjecture Evaluate and compare the following expressions
radic_
4 __ 25 and radic
_ 4 ____
radic_
25 radic
_
16 __ 81 and radic_
16 ____
radic_
81 radic
_
36 __ 49 and radic_
36 ____
radic_
49
Use your results to make a conjecture about a division rule for square roots Since division is multiplication by the reciprocal make a conjecture about a multiplication rule for square roots
38 Persevere in Problem Solving The difference between the solutions to the equation x 2 = a is 30 What is a Show that your answer is correct
FOCUS ON HIGHER ORDER THINKING
His estimate is low because 15 is very close to 16
so radic_
15 is very close to radic_
16 or 4 A better estimate
would be 38 or 39
Sample answer about 45 4 3 = 64 and 5 3 = 125
Because 95 is about halfway between 64 and 125 try 45
45 3 = 91125 which is a good estimate
3 feet
Yes the cube root of a negative number is negative
because a negative number cubed is always negative
and a nonnegative number cubed is always nonnegative
radic_
4 __ 25 = 2 _ 5 = radic
_ 4 ____
radic_
25 radic
_
16 __ 81 = 4 _ 9 = radic_
16 ____
radic_
81 radic
_
36 __ 49 = 6 _ 7 = radic_
36 ____
radic_
49
225 the solutions to x 2 = a are x = plusmn15 and
radic_
a ___
radic_
b = radic
_ a __
b radic
_ a radic
_ b = radic
_ a b
15 - (-15) = 30
Unit 114
copy H
ough
ton
Miff
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ublis
hing
Com
pany
bull copy
Ilen
e Mac
Dona
ldA
lamy I
mag
es
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=B
8_MCABESE206984_U1M01L1indd 14 102913 1142 PM
Personal Math Trainer
Online Practice and Helpmyhrwcom
Name Class Date
Independent Practice11
20 A 7 __ 16 -inch-long bolt is used in a machine What is this length written as a decimal
21 The weight of an object on the moon is 1 _ 6 its weight on Earth Write 1 _ 6 as a decimal
22 The distance to the nearest gas station is 2 4 _ 5 kilometers What is this distance written as a decimal
23 A baseball pitcher has pitched 98 2 _ 3 innings What is the number of innings written as a decimal
24 A heartbeat takes 08 second How many seconds is this written as a fraction
25 There are 262 miles in a marathon Write the number of miles using a fraction
26 The average score on a biology test was 72
_ 1 Write the average score using a
fraction
27 The metal in a penny is worth about 0505 cent How many cents is this written as a fraction
28 Multistep An artist wants to frame a square painting with an area of 400 square inches She wants to know the length of the wood trim that is needed to go around the painting
a If x is the length of one side of the painting what equation can you set up to find the length of a side How many solutions does the equation have
b Do all of the solutions that you found make sense in the context of the problem Explain
c What is the length of the wood trim needed to go around the painting
Solve each equation for x Write your answers as radical expressions Then estimate to one decimal place if necessary
29 x 2 = 14 30 x 3 = 1331
31 x 2 = 144 32 x 2 = 29
8NS1 8NS2 8EE2
04375 in 01 _6
28 km 98 _6 innings
x 2 = 400 x = plusmnthinsp20 the equation has 2 solutions
x = 20 makes sense but x = -20 doesnrsquot because a
painting cannot have a side length of -20 inches
4 times 20 = 80 inches
x = plusmn radic_
14 asymp plusmn37
x = plusmn radic_
144 = plusmn12 x = plusmn radic_
29 asymp plusmn54
x = 3 radic_ 1331 = 11
4_5 second 26 1_5 mi
72 1 _ 9 101 ___ 200 cent
13Lesson 11
copy H
ough
ton
Miff
lin H
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ublis
hing
Com
pany
bull copy
Phot
odisc
Get
ty Im
ages
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
8_MCAAESE206984_U1M01L1indd 13 41613 1211 AM
myhrwcomActivity available onlineEXTEND THE MATH PRE-AP
Activity Write radic_
09 on the board and invite students to conjecture what the value might be Have them check their conjectures by squaring Invite them to suggest ways to estimate radic
_ 09 As a hint point out that 09 is close to 10 and so they might
use that to help guide their estimates Lead them to see that since 092 is 081 and 102 is 1 the value of radic
_ 09 is greater than 09 and less than 10 Try squaring 095 to get
09025 A good estimate for radic_
09 is 095
Rational and Irrational Numbers 14
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=B
Integers
Rational Numbers IrrationalNumbers
Real Numbers
WholeNumbers
-3-4-5 -2-1 1 2 3 50 4
23
34-4 -π -1 25
radic2
Lesson Support Content Objective Students will learn to describe relationships between sets of numbers
Language Objective Students will explain how to describe relationships between sets of real numbers
LESSON 12 Sets of Real Numbers
Building BackgroundEliciting Prior Knowledge Have students draw a number line from -5 to 5 Ask them to plot points on the number line to approximate the location of rational and irrational numbers such as -1 3 __ 4 25 -4 2 __ 3 radic
_ 2 and -π
Learning ProgressionsIn this lesson students clarify their understanding of the real number system They characterize sets and subsets of the real numbers They also identify sets for real-world situations Important understandings for students include the following
bull Identify all of the possible subsets of the real numbers for a given number
bull Decide whether a statement about a subset of the real numbers is true or false
bull Identify the set of numbers that best describes a real-world situation
Understanding the relationships among the sets of numbers that make up the real numbers is essential as students are introduced to different forms of numbers throughout the school year This lesson provides a foundation for the comparing and ordering of real numbers in the next lesson
Cluster ConnectionsThis lesson provides an excellent opportunity to connect ideas in this cluster Know that there are numbers that are not rational and approximate them by rational numbers Have students copy this diagram which relates the sets of real numbers
Ask students to complete the diagram by writing three examples for each set of numbers Have students share examples and explain how they knew each number they selected belonged in the appropriate set Answers may vary Check studentsrsquo work
Focus | Coherence | Rigor
California Common Core Standards
8NS1 Know that numbers that are not rational are called irrational Understand informally that every number has a decimal expansion for rational numbers show that the decimal expansion repeats eventually and convert a decimal expansion which repeats eventually into a rational number
MP7 Look for and make use of structure
15A
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
Math Talk
Language Support EL
PROFESSIONAL DEVELOPMENT
Linguistic Support EL
AcademicContent Vocabulary
Venn diagrams ndash Students need descriptive language to describe the categories that the different areas and colors of a Venn diagram represent the concept of a set and how sets are distinct or can overlap Use sentence frames such as
The big oval represents __________The darklight blue color in the middle of the
big ovals represents __________These sets overlap because __________
In this way students have the language and structure to identify the criteria that distinguish a set and to explain the abstract representation Also point out the use of the prefix sub- meaning ldquounderrdquo in the term subset
Rules and Patterns
Abbreviations ndash In this lesson the abbreviation mph is used Be sure to point out that mph stands for miles per hour and is used to give units in a rate of speed Students may also have seen mpg (miles per gallon) which gives the units in a rate of fuel efficiency
Borrowed Words ndash Terminology used in baseball such as inning and pitcher may require some explanation Spanish as well as some other languages have borrowed these terms from English so some students may be familiar with these words already Despite this whenever a word is critical to students understanding the word problem it is best to explain the meaning
Leveled Strategies for English Learners
Emerging Allow students to indicate true or false orally in Guided Practice Exercises 9 and 10
Expanding Have students use sentence frames to describe the meaning of regions and colors used in a Venn diagram Then give them similar sentence frames orally and have them draw and shade a Venn diagram based on the oral prompts
Bridging Have students work in groups to draw a Venn diagram to represent sets based on real-world examples in the lesson
To help students answer the question posed in Math Talk provide a sentence frame for their answer
The numbers between 31 and 39 on a number line are __________ because __________
EL
California ELD Standards
Emerging 2II5 Modifying to add details ndash Expand sentences with simple adverbials to provide details about a familiar activity or process
Expanding 2II5 Modifying to add details ndash Expand sentences with adverbials to provide details about a familiar or new activity or process
Bridging 2II5 Modifying to add details ndash Expand sentences with increasingly complex adverbials to provide details about a variety of familiar and new activities and processes
Sets of Real Numbers 15B
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
12L E S S O N
Sets of Real Numbers
EngageESSENTIAL QUESTION
How can you describe relationships between sets of real numbers Sample answer Describe them as two different sets or one set as being a subset of another
Motivate the LessonAsk How many different types of tigers can you name How does the set of Bengal tigers relate to the set of tigers
ExplorePoint to different locations in the Animals diagram and ask for examples for that classification Do the same for the Real Numbers diagram Students should understand that everything within a region is part of the set for example both -3 and 2 are integers
ExplainEXAMPLE 1
Questioning Strategies Mathematical Practices bull In A why is 5 not a perfect square It does not have rational numbers as its square roots
bull Can the number in B be written as a fraction Why or why not Yes it is a terminating decimal so it is a rational number
Engage with the WhiteboardHave students place the numbers in Example 1 and Additional Example 1 in the Venn diagram for numbers
YOUR TURNAvoid Common ErrorsBe sure that students read Exercise 2 carefully before answering The number given in the problem 10 is the area not the side length
EXAMPLE 2Questioning Strategies Mathematical Practices bull What two major sets are the real numbers composed of rational and irrational numbers
bull What is the location of the set of whole numbers in the Venn diagram in relation to the set of rational numbers Explain Inside it whole numbers are rational numbers
Focus on Reasoning Mathematical PracticesRemind students that it takes only one counterexample to show that a statement is false
Interactive Whiteboard Interactive example available online
ADDITIONAL EXAMPLE 1Write all names that apply to each number
A -10integer rational real
B 12 _ 3
whole integer rational real
myhrwcom
Interactive Whiteboard Interactive example available online
ADDITIONAL EXAMPLE 2Tell whether the given statement is true or false Explain your choice
No integers are whole numbers
False every whole number is also an integer
myhrwcom
Animated MathClassifying Numbers
Students build fluency in classifying numbers in this engaging fast-paced game
myhrwcom
CA Common CoreStandards
The student is expected to
The Number Systemmdash8NS1
Know that numbers that are not rational are called irrational Understand informally that every number has a decimal expansion for rational numbers show that the decimal expansion repeats eventually and convert a decimal expansion which repeats eventually into a rational numberMathematical Practices
MP7 Using Structure
The student is expected to
15 Lesson 12
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=B
Math TrainerOnline Practice
and Help
Personal
myhrwcom
Math TrainerOnline Practice
and Help
Personal
myhrwcom
Math On the Spotmyhrwcom
Understanding Sets and Subsets of Real NumbersBy understanding which sets are subsets of types of numbers you can verify whether statements about the relationships between sets are true or false
Tell whether the given statement is true or false Explain your choice
All irrational numbers are real numbers
True Every irrational number is included in the set of real numbers The irrational numbers are a subset of the real numbers
No rational numbers are whole numbers
False A whole number can be written as a fraction with a denominator of 1 so every whole number is included in the set of rational numbers The whole numbers are a subset of the rational numbers
EXAMPLE 2
A
B
Write all names that apply to each number
1 A baseball pitcher has pitched 12 2 _ 3 innings
2 The length of the side of a square that has an
area of 10 square yards
YOUR TURN
Tell whether the given statement is true or false Explain your choice
3 All rational numbers are integers
4 Some irrational numbers are integers
YOUR TURN
Give an example of a rational number that is a
whole number Show that the number is both whole
and rational
Math TalkMathematical Practices
Give an example of a
8NS1
False Every integer is a rational number but not every
False Real numbers are either rational or irrational numbers
Integers are rational numbers so no integers are irrational numbers
rational real
irrational real
Sample answer 8 8 = 8_
1
and -thinsp 5 _ 2 are not integers
rational number is an integer Rational numbers such as 3 _ 5
Unit 116
copy H
ough
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Miff
lin H
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ublis
hing
Com
pany
bull Im
age C
redi
ts D
igita
l Im
age c
opyr
ight
copy20
04 Ey
ewire
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
8_MCAAESE206984_U1M01L2indd 16 41613 136 AM
Math On the Spot
myhrwcom
Vertebrates
Birds
Passerines
Animals
Integers
Rational Numbers IrrationalNumbers
Real Numbers
WholeNumbers
1
45
3
0
274
67
radic4
-
-3
-2
-1
03
radic2
radic17
radic11-
π
Animated Math
myhrwcom
Classifying Real NumbersBiologists classify animals based on shared characteristics A cardinal is an animal a vertebrate a bird and a passerine
You already know that the set of rational numbers consists of whole numbers integers and fractions The set of real numbers consists of the set of rational numbers and the set of irrational numbers
Write all names that apply to each number
radic_
5 irrational real
ndash1784rational real
whole integer rational real
EXAMPLEXAMPLE 1
A
B
C radic_ 81 ____ 9
L E S S O N
12Sets of Real Numbers
ESSENTIAL QUESTIONHow can you describe relationships between sets of real numbers
Passerines such as the cardinal are also called ldquoperching birdsrdquo
What types of numbers are between 31 and 39 on a
number line
Math TalkMathematical Practices
What types of numbers are
8NS1
8NS1
Know that numbers that are not rational are called irrational Understand informally that every number has a decimal expansion for rational numbers show that the decimal expansion repeats eventually and convert a decimal expansion which repeats eventually into a relation number
ndash1784 is a terminating decimal
5 is a whole number that is not a perfect square
radic_
81 _____ 9 = 9 __ 9 = 1 rational irrational real
15Lesson 12
copy H
ough
ton
Miff
lin H
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ublis
hing
Com
pany
bull Im
age C
redi
ts copy
Wiki
med
ia Co
mm
ons
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=B
8_MCABESE206984_U1M01L2indd 15 061113 1144 AM
PROFESSIONAL DEVELOPMENT
Math BackgroundThe relationships between sets of numbers extend to include complex numbers A complex number can be written as a sum of a real number a and an imaginary number bi
a + bi
An imaginary number is a special number that when squared gives a negative value When you square a real number you get a nonnegative number When you square an imaginary number you get a negative value The imaginary unit is i
i = radic_
-1
Integrate Mathematical Practices MP7
This lesson provides an opportunity to address this Mathematical Practices standard It calls for students to discern structure to connect and communicate mathematical ideas
Students use a Venn diagram to structure relationships between sets of numbers They connect and communicate mathematical ideas when they make logical statements about the sets and describe which set best describes numbers applied to real-life situations
Sets of Real Numbers 16
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=B
YOUR TURNAvoid Common ErrorsStudents may see the word ldquoAllldquo or rdquoNordquo in Exercises 3 and 4 and immediately assume that any absolute statements like these are false Remind them that there are true statements that begin with these words and encourage them to provide examples
EXAMPLE 3Questioning Strategies Mathematical Practices bull In A how does the phrase ldquonumber of rdquo give you a clue about the number classification It indicates a counting number
bull What is the relationship between the circumference of a circle and the diameter The circumference is diameter times π
Focus on Critical Thinking Mathematical PracticesIn B suppose the diameters in inches were 25
__ π 28 __ π
31 __ π and so on What set of numbers would
best describe the circumferences Explain Whole numbers the circumferences would be the whole numbers 25 28 31 and so on
YOUR TURNFocus on Critical Thinking Mathematical PracticesHave students compare and contrast the classification of numbers in the answers in Exercises 5 and 6
ElaborateTalk About ItSummarize the Lesson
Ask What are some ways that number sets can be related Sets may be subsets of other sets or they may be separate from other sets
GUIDED PRACTICEEngage with the Whiteboard
Have students place the numbers in Exercises 1ndashthinsp8 in the Venn diagram for numbers at the beginning of the lesson
Integrating Language Arts EL
Encourage English learners to ask for clarification on any terms or phrases that they do not understand
Avoid Common ErrorsExercise 7 Remind students that a repeating decimal is a rational numberExercises 9ndash10 Remind students that it only takes one counterexample to show that a statement is false
Interactive Whiteboard Interactive example available online
ADDITIONAL EXAMPLE 3Identify the set of numbers that best describes the situation Explain your choice
A the amount of time that has passed since midnight
The set of real numbers time is continuous so the amount of time can be rational or irrational
B the number of tickets sold to a basketball game
The set of whole numbers the number of tickets sold may be 0 or a counting number
myhrwcom
17 Lesson 12
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
1IN
116 inch
Guided Practice
Write all names that apply to each number (Example 1)
1 7 _ 8 2 radic_
36
3 radic_
24 4 075
5 0 6 - radic_ 100
7 5 _
45 8 - 18 __ 6
Tell whether the given statement is true or false Explain your choice (Example 2)
9 All whole numbers are rational numbers
10 No irrational numbers are whole numbers
Identify the set of numbers that best describes each situation Explain your choice (Example 3)
11 the change in the value of an account when given to the nearest dollar
12 the markings on a standard ruler
13 What are some ways to describe the relationships between sets of numbers
CHECK-INESSENTIAL QUESTION
rational real
rational real
True Whole numbers are rational numbers
Rational numbers the ruler is marked every 1 __ 16 th inch
Sample answer Describe one set as being a subset of
another or show their relationships in a Venn diagram
Integers the change can be a whole dollar amount
and can be positive negative or zero
True Whole numbers are a subset of the set of rational numbers
and can be written as a ratio of the whole number to 1
irrational real
whole integer rational real
whole integer rational real
rational real
integer rational real
integer rational real
Unit 118
copy H
ough
ton
Miff
lin H
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ublis
hing
Com
pany
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
8_MCAAESE206984_U1M01L2indd 18 41613 136 AM
My Notes
Math TrainerOnline Practice
and Help
Personal
myhrwcom
Math On the Spot
myhrwcom
Identifying Sets for Real-World SituationsReal numbers can be used to represent real-world quantities Highways have posted speed limit signs that are represented by natural numbers such as 55 mph Integers appear on thermometers Rational numbers are used in many daily activities including cooking For example ingredients in a recipe are often given in fractional amounts such as 2 _ 3 cup flour
Identify the set of numbers that best describes each situation Explain your choice
the number of people wearing glasses in a room
The set of whole numbers best describes the situation The number of people wearing glasses may be 0 or a counting number
the circumference of a flying disk has a diameter of 8 9 10 11 or 14 inches
The set of irrational numbers best describes the situation Each circumference would be a product of π and the diameter and any multiple of π is irrational
EXAMPLEXAMPLE 3
A
B
Identify the set of numbers that best describes the situation Explain your choice
5 the amount of water in a glass as it evaporates
6 the weight of a person in pounds
YOUR TURN
8NS1
Rational numbers a personrsquos weight can be a decimal
such as 835 pounds
Real numbers the amount can be any number greater
than 0
17Lesson 12
copy H
ough
ton
Miff
lin H
arco
urt P
ublis
hing
Com
pany
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
8_MCAAESE206984_U1M01L2indd 17 41613 520 AM
Graphic OrganizersGive students a list of numbers (including terminating and repeating decimals fractions integers and rational and irrational square roots) and a graphic organizer as shown below
Real Numbers
Rational numbers Irrational numbers
Integer numbers
Whole numbers
Ask students to write each number in the list in the correct section of the organizer
Number SensePoint out to students that knowing the types of numbers to expect in different situations can alert them to incorrect math as well as to impossible situations For example 135 shots made in basketballs is not possible but an average number of shots can equal 135
Additional ResourcesDifferentiated Instruction includes bull Reading Strategies bull Success for English Learners EL
bull Reteach bull Challenge PRE-AP
DIFFERENTIATE INSTRUCTION
Sets of Real Numbers 18
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
Math TrainerOnline Assessment
and Intervention
Personal
Online homework assignment available
myhrwcom
Lesson Quiz available online
12 LESSON QUIZ
1 Write all the names that apply to the number
2 Tell whether the given statement is true or false Explain your choice All numbers between 1 and 2 are rational numbers
3 Identify the set of numbers that best describes the situation Explain your choiceThe choices on a survey question change the total points for the survey by -2 -1 0 1 or 2 points
-1 _
5
myhrwcom
Additional ResourcesDifferentiated Instruction includes bull Leveled Practice worksheets
EvaluateGUIDED AND INDEPENDENT PRACTICE
Concepts amp Skills Practice
Example 1Classifying Real Numbers
Exercises 1ndash8 14ndash19 22ndash24
Example 2Understanding Sets and Subsets of Real Numbers
Exercises 9ndash10
Example 3Identifying Sets for Real-World Situations
Exercises 11ndash12 20ndash21 25
Focus | Coherence | Rigor
Exercise Depth of Knowledge (DOK) Mathematical Practices
14ndash19 2 SkillsConcepts MP7 Using Structure
20ndash21 2 SkillsConcepts MP6 Precision
22ndash23 2 SkillsConcepts MP3 Logic
24 1 Recall of Information MP7 Using Structure
25 2 SkillsConcepts MP2 Reasoning
26ndash27 3 Strategic Thinking MP3 Logic
28 3 Strategic Thinking MP8 Patterns
29 3 Strategic Thinking MP3 Logic
8NS1
8NS1
Exercise 29 combines concepts from the California Common Core cluster ldquoKnow that there are numbers that are not rational and approximate them by rational numbersrdquo
Answers1 rational real
2 False radic_
2 is an example of an irrational number between 1 and 2
3 Integers each number is an integer but only three are whole numbers
19 Lesson 12
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
Work Area
π mi23 Critique Reasoning The circumference of a circular region is shown
What type of number best describes the diameter of the circle Explain
your answer
24 Critical Thinking A number is not an integer What type of number can it be
25 A grocery store has a shelf with half-gallon containers of milk What type of number best represents the total number of gallons
26 Explain the Error Katie said ldquoNegative numbers are integersrdquo What was her error
27 Justify Reasoning Can you ever use a calculator to determine if a number is rational or irrational Explain
28 Draw Conclusions The decimal 0 _
3 represents 1 _ 3 What type of number best describes 0
_ 9 which is 3 middot 0
_ 3 Explain
29 Communicate Mathematical Ideas Irrational numbers can never be precisely represented in decimal form Why is this
FOCUS ON HIGHER ORDER THINKING
It can be a rational number that is not an integer or an irrational number
rational number
The set of negative numbers also includes non-integer
rational numbers and irrational numbers
Sample answer If the calculator shows a decimal that
terminates in fewer digits than what the calculator screen
allows then you can tell that the number is rational If not
you cannot tell from the calculator display whether the
number terminates because you see a limited number
of digits It may be a repeating decimal (rational) or
non-terminating non-repeating decimal (irrational)
Whole 3 middot 0 _
3 represents 3 middot 1 _ 3 = 1 so 0 _
9 is exactly 1
Sample answer In decimal form irrational numbers never
terminate and never repeat Therefore no matter how
many decimal places you include the number will never
be precisely represented There are always more digits
Whole the diameter is π _ π = 1 mile
Unit 120
copy H
ough
ton
Miff
lin H
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ublis
hing
Com
pany
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
8_MCAAESE206984_U1M01L2indd 20 120413 909 PM
Integers
Rational Numbers Irrational Numbers
Real Numbers
Whole Numbers
257
radic16
166
radic9
128 radic50
Personal Math Trainer
Online Practice and Helpmyhrwcom
Name Class Date
Independent Practice
Identify the set of numbers that best describes each situation Explain your choice
20 the height of an airplane as it descends to an airport runway
21 the score with respect to par of several golfers 2 ndash 3 5 0 ndash 1
22 Critique Reasoning Ronald states that the number 1 __ 11 is not rational because when converted into a decimal it does not terminate Nathaniel says it is rational because it is a fraction Which boy is correct Explain
12
14 - radic_
9 15 257
16 radic_
50 17 8 1 _ 2
18 166 19 radic_
16
Write all names that apply to each number Then place the numbers in the correct location on the Venn diagram
8NS1
Real numbers the height can be any number greater than zero
integer rational real whole integer rational real
whole integer rational real
irrational real
rational real
rational real
Integers the scores are counting numbers their
opposites and zero
Nathaniel is correct A rational number is a number that can be written as a fraction and 1 __ 11 is a fraction
19Lesson 12
copy H
ough
ton
Miff
lin H
arco
urt P
ublis
hing
Com
pany
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
8_MCAAESE206984_U1M01L2indd 19 41613 136 AM
myhrwcomActivity available onlineEXTEND THE MATH PRE-AP
Activity Have students consider the concept of restricted domain for the sets of numbers that describe situations For example the number of sisters a person has can best be described by whole numbers but no one has ever had 1500 sisters An area code is an integer or whole number between 200 and 999
Have students use a source such as the Guinness Book of World Records and give examples of sets of numbers that describe situations where the domain is restricted Ask whether the restriction may be changed in the future
Sets of Real Numbers 20
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
-3-4-5 -2-1 1 2 3 50 4
12-4 -radic5
Lesson Support Content Objective Students will learn to order a set of real numbers
Language Objective Students will show and describe how to order a set of real numbers
LESSON 13 Ordering Real Numbers
Building BackgroundEliciting Prior Knowledge Have students draw a number line to compare a rational number and an irrational number such as - radic
_ 5 and -4 1 __ 2 Ask them to explain how
they approximated the irrational number on the number line Then have them identify the greater and the lesser real number Repeat with several other pairs of real numbers in different forms
Learning ProgressionsIn this lesson students order a set of real numbers They use rational approximations to compare the sizes of irrational numbers They also order numbers for real-world situations Important understandings for students include the following
bull Compare irrational numbers bull Estimate the value of expressions with irrational numbers bull Order a set of real numbers bull Order real numbers in a real-world context
Work with real numbers continues throughout Grade 8 and into high school This lesson provides students with a foundation for understanding the relative sizes of numbers in different forms in the real number system
Cluster ConnectionsThis lesson provides an excellent opportunity to connect ideas in this cluster Know that there are numbers that are not rational and approximate them by rational numbers Tell students that there is a special number called the golden ratio with applications in mathematics geometry art and architecture The golden ratio is called phi and is represented by the Greek letter ϕ It includes an irrational number in its definition
Have students explain why the golden ratio is irrational Ask them to find the two whole numbers the golden ratio lies between Then challenge them to approximate the golden ratio to the nearest tenth It is irrational because it includes an irrational number in its definition It lies between 1 and 2 To the nearest tenth ϕ = 16
ϕ = 1 + radic_
5 _ 2
Focus | Coherence | Rigor
California Common Core Standards
8NS2 Use rational approximations of irrational numbers to compare the size of irrational numbers locate them approximately on a number line diagram and estimate the value of expressions (eg π 2 )
MP4 Model with mathematics
21A
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
Math Talk
Language Support EL
PROFESSIONAL DEVELOPMENT
Linguistic Support EL
AcademicContent Vocabulary
Post a chart like this to remind students of the regular comparative forms of adjectives that use the -er and -est suffixes Add to the chart for terms that appear in examples and exercises in each lesson Include any irregular verb forms
Background Knowledge
Go On ndash the title of the module review or quiz is Ready to Go On This title uses an idiomatic expression In this context to go on means ldquoto move aheadrdquo or ldquoto proceedrdquo It is different from the use of go on that means having enough facts to use meaningfully as in having enough to go on Also the intonation used in pronouncing an expression can give it different meanings For example when the speaker emphasizes the word on he or she might be expressing disbelief as in ldquoGo ON Yoursquore kidding rightrdquo Discuss with students other ways that the phrase go on may be used
Leveled Strategies for English Learners
Emerging Label points on a number line with the terms used in ordering greater greatest less lesser least Use sentence frames to insert the correct terms
Expanding Have students give two or three complete sentences to compare the placement of numbers on a number line using the correct forms of the comparative and superlative adjectives
Bridging Have students work in pairs with one student giving directions to the other in complete sentences to order numbers on a number line
To help students answer the question posed in Math Talk make sure that students have a command of the forms for making comparisons and the superlative and the concept of opposite order so that the focus is on the math concept instead of the language skills needed to describe and explain order
EL
Adjective Comparative Superlative
Far Farther Farthest
Large Larger Largest
Great Greater Greatest
Some Less Least
Some More Most
California ELD Standards
Emerging 2I8 Analyzing language choices ndash Explain how phrasing or different common words with similar meanings produce different effects on the audience
Expanding 2I8 Analyzing language choices ndash Explain how phrasing or different words with similar meanings or figurative language produce shades of meaning and different effects on the audience
Bridging 2I8 Analyzing language choices ndash Explain how phrasing or different words with similar meanings or figurative language produce shades of meaning nuances and different effects on the audience
Ordering Real Numbers 21B
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
13L E S S O N
Ordering Real Numbers
Interactive Whiteboard Interactive example available online
ADDITIONAL EXAMPLE 1Compare Write lt gt or =
A radic_
8 - 2 4 - radic_
8 lt
B radic_
20 + 1 3 + radic_
2 gt
EngageESSENTIAL QUESTION
How do you order a set of real numbers Sample answer Find their approximate decimal values and order them
Motivate the LessonAsk What kind of numbers are you comparing when you compare the price of gasoline at two different gas stations
ExploreGive students two rational numbers and ask them to name a number between them Repeat a few times and then give them two irrational numbers and ask them to name a number between them
ExplainEXAMPLE 1
Questioning Strategies Mathematical Practices bull Which is greater the difference between 5 and 3 or the difference between radic
_ 5 and radic
_ 3
The difference between 5 and 3 is 2 the difference between radic_
5 and radic_
3 is approximately 1 So the difference between 5 and 3 is greater
Avoid Common ErrorsCaution students to read the problem carefully and think about what the radical sign means so that they do not misread the problem and answer that the two sides are equal
YOUR TURNFocus on TechnologyCalculators should not be used at this point because developing number sense is the goal
EXAMPLE 2Questioning Strategies Mathematical Practices bull How do you determine whether radic
_ 22 is less than or greater than 45 The square of 45 is
2025 which is less than 22 so the square root of 22 must be greater than 45
Engage with the WhiteboardHave students graph and label various real numbers between 42 and 44 and between 47 and 5
YOUR TURNFocus on Modeling Mathematical PracticesHave students label the integers on the number line with their equivalent square root For example 1 2 and 3 on the number line would be labeled radic
_ 1 radic
_ 4 and radic
_ 9
Interactive Whiteboard Interactive example available online
ADDITIONAL EXAMPLE 2Order 3π radic
_ 10 and 325 from greatest
to least
3π 325 radic_
10
myhrwcom
myhrwcom
CA Common CoreStandards
The student is expected to
The Number Systemmdash8NS2
Use rational approximations of irrational numbers to compare the size of irrational numbers locate them approximately on a number line diagram and estimate the value of expressions (eg π 2 )
Mathematical Practices
MP4 Modeling
The student is expected to
21 Lesson 13
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
Math TrainerOnline Practice
and Help
Personal
myhrwcom
Math On the Spotmyhrwcom
0 05 1 15 2 25 3 35 4
radic5radic3
π2
8 85 9 95 10 105 11 115 12
radic75
4 42 44 46 48 5
radic224 12π + 1
Ordering Real Numbers You can compare and order real numbers and list them from least to greatest
Order radic_
22 π + 1 and 4 1 _ 2 from least to greatest
First approximate radic_
22
radic_
22 is between 4 and 5 Since you donrsquot know where it falls between 4 and 5 you need to find a better estimate for radic
_ 22 so
you can compare it to 4 1 _ 2
Since 22 is closer to 25 than 16 use squares of numbers between 45 and 5 to find a better estimate of radic
_ 22
45 2 = 2025 46 2 = 2116 47 2 = 2209 48 2 = 2304
Since 47 2 = 2209 an approximate value for radic_
22 is 47
An approximate value of π is 314 So an approximate value of π +1 is 414
Plot radic_
22 π + 1 and 4 1 _ 2 on a number line
Read the numbers from left to right to place them in order from least to greatest
From least to greatest the numbers are π + 1 4 1 _ 2 and radic_
22
EXAMPLE 2
STEP 1
STEP 2
Order the numbers from least to greatest Then graph them on the number line
YOUR TURN
5 radic_
5 25 radic_
3
6 π 2 10 radic_
75
If real numbers a b and c are in order from least to greatest what is the order
of their opposites from least to greatest
Explain
Math TalkMathematical Practices
8NS2
radic_
3 radic_
5 25
radic_
75 π2 10
Math Talk answer -c -b -a -c is farthest to the left on a number line -b is in the middle and -a is farthest to the right
Unit 122
copy H
ough
ton
Miff
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Com
pany
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
8_MCAAESE206984_U1M01L3indd 22 41613 447 AM
My Notes
Math TrainerOnline Practice
and Help
Personal
myhrwcom
Math On the Spot
myhrwcom
Comparing Irrational NumbersBetween any two real numbers is another real number To compare and order real numbers you can approximate irrational numbers as decimals
Compare radic_
3 + 5 3 + radic_
5 Write lt gt or =
First approximate radic_
3
radic_
3 is between 1 and 2
Next approximate radic_
5
radic_
5 is between 2 and 3
Then use your approximations to simplify the expressions
radic_
3 + 5 is between 6 and 7
3 + radic_
5 is between 5 and 6
So radic_
3 + 5 gt 3 + radic_
5
Reflect1 If 7 + radic
_ 5 is equal to radic
_ 5 plus a number what do you know about the
number Why
2 What are the closest two integers that radic_
300 is between
EXAMPLEXAMPLE 1
STEP 1
STEP 2
Compare Write lt gt or =
YOUR TURN
3 radic_
2 + 4 2 + radic_
4 4 radic_
12 + 6 12 + radic_
6
L E S S O N
13 Ordering Real Numbers
ESSENTIAL QUESTIONHow do you order a set of real numbers
8NS2
Use rational approximations of irrational numbers to compare the size of irrational numbers locate them approximately on a number line diagram and estimate the value of expressions (eg π 2 )
8NS2
Use perfect squares to estimate square roots
1 2 = 1 2 2 = 4 3 2 = 9
The number is 7 both expressions must equal 7 + radic_
5
17 and 18
gt lt
21Lesson 13
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ough
ton
Miff
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Com
pany
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
8_MCAAESE206984_U1M01L3indd 21 41913 246 PM
PROFESSIONAL DEVELOPMENT
Math BackgroundIn this lesson students estimate irrational numbers in the form of square roots of nonper-fect squares by finding two perfect squares between which the number falls A more precise method involves repeated division For example to find radic
_ 28 find a whole number whose perfect
square is close to 28 such as 5 Divide 28 by that number 28 divide 5 = 56 Find the average of the quotient and divisor 5 + 56
_____ 2 = 53 Continue dividing 28 by each result and averaging until you get the desired accuracy
Integrate Mathematical Practices MP4
This lesson provides an opportunity to address this Mathematical Practices standard It calls for students to model relationships using multiple representations including diagrams graphs and language as appropriate Students use multiple representations when they use number lines to estimate the locations of and order rational and irrational numbers given as symbols
Ordering Real Numbers 22
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
Interactive Whiteboard Interactive example available online
ADDITIONAL EXAMPLE 3The diameter of a meteorite in millimeters is calculated by four different methods Order the results from least to greatest
Joe radic_
18 mm Lisa 13 __ 3 mm
Pablo 46 mm Julien 4π __ 3 mm
Julien 4π __ 3 mm Lisa 13 __ 3 mm
Joe radic_
18 mm Pablo 46 mm
EXAMPLE 3Questioning Strategies Mathematical Practices bull How can you verify that radic
_ 28 is between 52 and 53 5 2 2 = 2704 and 5 3 2 = 2809
bull Explain how to determine which number is greater 5 _
5 or 55 When the repeating decimal is rounded to the nearest tenth or hundredth you can see that it is greater
Connect to Daily LifeDiscuss how measuring across a canyon might involve different methods than measuring along a road Explain that measurements like these are often done using calculations that approximate the distance
YOUR TURNFocus on Critical Thinking Mathematical PracticesDiscuss with students which number is greater 3
_ 45 or 3450 3
_ 45 or 3455 and why Explain
that 3 _
45 can be written out as 34545hellipMake sure they understand that 3 _
45 is greater than 345 but less than 3455
ElaborateTalk About ItSummarize the Lesson
Ask How can you order two numbers in different forms whose decimal approxi-mations appear to be equal Approximate one or both numbers to an additional
number of decimal places
GUIDED PRACTICEEngage with the Whiteboard
Have students place and label additional points on the number line in Exercise 9 Allow the points to be in any format other than decimal
Avoid Common ErrorsExercises 3ndash4 Caution students to read the problem carefully so that they do not misread the problem as the same numbers combined by addition on each side of the circleExercise 10 Remind students that the calculations have units
myhrwcom
23 Lesson 13
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
0 05 1 15 2 25 3 35 4 45 5 55 6 65 7
2πradic3
Compare Write lt gt or = (Example 1)
1 radic_
3 + 2 radic_
3 + 3 2 radic_
8 + 17 radic_
11 + 15
3 radic_
6 + 5 6 + radic_
5 4 radic_
9 + 3 9 + radic_
3
5 radic_
17 - 3 -2 + radic_
5 6 12 - radic_
2 14 - radic_
8
7 radic_
7 + 2 radic_
10 - 1 8 radic_
17 + 3 3 + radic_
11
9 Order radic_
3 2π and 15 from least to greatest Then graph them on the number line (Example 2)
radic_
3 is between and so radic_
3 asymp
π asymp 314 so 2π asymp
From least to greatest the numbers are
10 Four people have found the perimeter of a forest using different methods Their results are given in the table Order their calculations from greatest to least (Example 3)
11 Explain how to order a set of real numbers
CHECK-INESSENTIAL QUESTION
Forest Perimeter (km)
Leon Mika Jason Ashley
radic_
17 - 2 1 +thinsp π __ 2 12 ___ 5 25
Guided Practice
17
15
1 + π _ 2 km 25 km 12 __ 5 km radic_
17 - 2 km
2π radic
_ 3
18 175
628
Sample answer Convert each number to a decimal
equivalent using estimation to find equivalents for
irrational numbers Graph each number on a number line
Read the numbers from left to right for least to greatest
Read the numbers from right to left for greatest to least
lt gt
lt lt
ltgt
gt gt
24 Unit 1
copy H
ough
ton
Miff
lin H
arco
urt P
ublis
hing
Com
pany
bull Im
age C
redi
ts copy
Elena
Eliss
eeva
Alam
y Im
ages
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
8_MCAAESE206984_U1M01L3indd 24 41613 448 AM
My Notes
5 52 54 56 58 6
radic28 5 12
23455
Math TrainerOnline Practice
and Help
Personal
myhrwcom
Math On the Spot
myhrwcom
Ordering Real Numbers in a Real-World Context Calculations and estimations in the real world may differ It can be important to know not only which are the most accurate but which give the greatest or least values depending upon the context
Four people have found the distance in kilometers across a canyon using different methods Their results are given in the table Order the distances from greatest to least
Distance Across Quarry Canyon (km)
Juana Lee Ann Ryne Jackson
radic_
28 23 __ 4 5 _
5 5 1 _ 2
Write each value as a decimal
radic_
28 is between 52 and 53 Since 53 2 = 2809 an approximate value for radic
_ 28 is 53
23 __ 4 = 575
5 _
5 is 5555hellip so 5 _
5 to the nearest hundredth is 556
5 1 _ 2 = 55
Plot radic_
28 23 __ 4 5 _
5 and 5 1 _ 2 on a number line
From greatest to least the distances are
23 __ 4 km 5 _
5 km 5 1 _ 2 km radic_
28 km
EXAMPLEXAMPLE 3
STEP 1
STEP 2
7 Four people have found the distance in miles across a crater using different methods Their results are given below
Jonathan 10 __ 3 Elaine 3 _
45 Joseacute 3 1 _ 2 Lashonda radic_
10
Order the distances from greatest to least
YOUR TURN
8NS2
3 1 _ 2 mi 3 _
45 mi 10 __ 3 mi radic_
10 mi
23Lesson 13
copy H
ough
ton
Miff
lin H
arco
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ublis
hing
Com
pany
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
8_MCAAESE206984_U1M01L3indd 23 41613 447 AM
ModelingPlace papers around the room with the numbers from 1 to 5 one per sheet Give each student a card showing a number between 1 and 5 in different forms Have students place his or her card between the correct integers and decide where the number goes in relation to any numbers already placed
Multiple RepresentationsGive students a vertical number line which some students might find easier to use than a horizontal one Have them decide whether to place points for rational and irrational numbers above or below existing points
Additional ResourcesDifferentiated Instruction includes bull Reading Strategies bull Success for English Learners EL
bull Reteach bull Challenge PRE-AP
DIFFERENTIATE INSTRUCTION
Ordering Real Numbers 24
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
Math TrainerOnline Assessment
and Intervention
Personal
Online homework assignment available
myhrwcom
myhrwcom
Lesson Quiz available online
13 LESSON QUIZ
1 Compare Write lt gt or =
radic_
95 - 5 radic_
62 - 2
2 Order 105 radic_
105 and 3π + 1 from greatest to least
3 A length in centimeters is calculated differently by four different people Order their calculations from least to greatest
KD 11 __ 2 cm Silvio 5 __ 3 π cm
Paula 5 _
4 cm Luis radic_
33 cm
EvaluateGUIDED AND INDEPENDENT PRACTICE
Concepts amp Skills Practice
Example 1Comparing Irrational Numbers
Exercises 1ndash8
Example 2Ordering Real Numbers
Exercises 9 12ndash15 18ndash21
Example 3Ordering Real Numbers in a Real-World Context
Exercises 10 16ndash17
Additional ResourcesDifferentiated Instruction includes bull Leveled Practice worksheets
Focus | Coherence | Rigor
Exercise Depth of Knowledge (DOK) Mathematical Practices
12ndash15 1 Recall of Information MP5 Using Tools
16 2 SkillsConcepts MP2 Reasoning
17 2 SkillsConcepts MP6 Precision
18ndash21 2 SkillsConcepts MP2 Reasoning
22 3 Strategic Thinking MP4 Modeling
23ndash24 3 Strategic Thinking MP3 Logic
8NS2
8NS2
Answers1 radic
_ 95 - 5 lt radic
_ 62 - 2
2 radic_
105 3π + 1 105
3 Silvio 5 __ 3 π cm Paula 5 _
4 cm
KD 11
__ 2 cm Luis radic_
33 cm
25 Lesson 13
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
Work Area
3140 3141 3142 3143
314 π227
20 A teacher asks his students to write the numbers shown in order from least to greatest Paul thinks the numbers are already in order Sandra thinks the order should be reversed Who is right
21 Math History There is a famous irrational number called Eulerrsquos number symbolized with an e Like π its decimal form never ends or repeats The first few digits of e are 27182818284
a Between which two square roots of integers could you find this number
b Between which two square roots of integers can you find π
22 Analyze Relationships There are several approximations used for π including 314 and 22 __ 7 π is approximately 314159265358979
a Label π and the two approximations on the number line
b Which of the two approximations is a better estimate for π Explain
c Find a whole number x so that the ratio x ___ 113 is a better estimate for π
than the two given approximations
23 Communicate Mathematical Ideas If a set of six numbers that include both rational and irrational numbers is graphed on a number line what is the fewest number of distinct points that need to be graphed Explain
24 Critique Reasoning Jill says that 12 _
6 is less than 1263 Explain her error
FOCUS ON HIGHER ORDER THINKING
radic_
115 115 ___ 11 and 105624
between radic_
7 asymp 265 and radic_
8 asymp 283
between radic_
9 = 3 and radic_
10 asymp 316
22 __ 7 it is closer to π on the number line
She did not consider the repeating digit 1266
2 rational numbers can have the same location and
irrational numbers can have the same location but they
cannot share a location
355
Neither student is correct The answer
should be 115 ___ 11 105624 radic_
115
Unit 126
copy H
ough
ton M
ifflin
Har
cour
t Pub
lishin
g Com
pany
Imag
e Cre
dits
copy3D
Stoc
kiSt
ockP
hoto
com
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
8_MCAAESE206984_U1M01L3indd 26 210513 801 AM
Personal Math Trainer
Online Practice and Helpmyhrwcom
Name Class Date
Independent Practice
16 Your sister is considering two different shapes for her garden One is a square with side lengths of 35 meters and the other is a circle with a diameter of 4 meters
a Find the area of the square
b Find the area of the circle
c Compare your answers from parts a and b Which garden would give your sister the most space to plant
17 Winnie measured the length of her fatherrsquos ranch four times and got four different distances Her measurements are shown in the table
a To estimate the actual length Winnie first approximated each distance to the nearest hundredth Then she averaged the four numbers Using a calculator find Winniersquos estimate
b Winniersquos father estimated the distance across his ranch to be radic_
56 km How does this distance compare to Winniersquos estimate
Give an example of each type of number
18 a real number between radic_
13 and radic_
14
19 an irrational number between 5 and 7
Order the numbers from least to greatest
12 radic_
7 2 radic_
8 ___ 2 13 radic_
10 π 35
14 radic_
220 -10 radic_
100 115 15 radic_
8 -375 3 9 _ 4
Distance Across Fatherrsquos Ranch (km)
1 2 3 4
radic_
60 58 __ 8 7 _
3 7 3 _ 5
138NS2
radic_
8 ___ 2 2 radic_
7
-10 radic_
100 115 radic_
220
radic_
60 asymp 775 58 __ 8 = 725 7 _
3 asymp 733 7 3 _ 5 = 760 so the average
π radic_
10 35
-375 9 _ 4 radic_
8 3
is 74825 km
1225 m2
4π m2 or approximately 126 m2
They are nearly identical radic_
56 is approximately 74833hellip
The circle would give her more space to plant because it has a
larger area
Sample answer 37
Sample answer radic_
31
25Lesson 13
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pany
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8_MCAAESE206984_U1M01L3indd 25 41613 448 AM
Activity available online myhrwcomEXTEND THE MATH PRE-AP
Activity Have students investigate whether there are infinitely many numbers between two numbers by giving examples for each of the following
bull Between any two rational numbers there is at least one other rational number Sample answer 45 is between 41 and 48
bull Between any two irrational numbers there is at least one rational number Sample answer 45 is between radic
_ 11 and radic
_ 29
bull Between any two rational numbers there is at least one irrational number Sample answer radic
_ 11 is between 31 and 36
bull Between any two irrational numbers there is at least one irrational number Sample answer radic
_ 17 is between radic
_ 11 and radic
_ 29
Ordering Real Numbers 26
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
ReadyMath Trainer
Online Practiceand Help
Personal
myhrwcom
Module Quiz
11ensp RationalenspandenspIrrationalenspNumbersWrite each fraction as a decimal or each decimal as a fraction
1 7__20 2 1___
27 3 17_8
Solve each equation for x
4 x2=81 5 x3=343 6 x2= 1___100
7 Asquarepatiohasanareaof200squarefeetHowlongiseachside
ofthepatiotothenearesttenth
12ensp SetsenspofenspRealenspNumbersWrite all names that apply to each number
8 121____radic
____121
9 π__2
10 TellwhetherthestatementldquoAllintegersarerationalnumbersrdquoistrueorfalseExplainyourchoice
13ensp OrderingenspRealenspNumbersCompare Write lt gt or =
11 radic__
8+3 8+radic__
3 12 radic__
5+11emsp emsp emsp 5+radic___
11
Order the numbers from least to greatest
13 radic___
99π29__
8 14 radic___
1__251_40__
2
15 Howarerealnumbersusedtodescribereal-worldsituations
ESSENTIAL QUESTION
035
9-9
141ft
7 1__10- 1__10
14__11 1875
wholeintegerrationalreal
Trueintegerscanbewrittenasthequotientoftwointegers
SampleanswerRealnumberssuchastherational
π29__
8radic___
99
irrationalreal
lt gt
number1_4candescribeamountsusedincooking
radic___
1__250__
21_4
27Module1
copy H
ough
ton
Miff
lin H
arco
urt P
ublis
hing
Com
pany
DONOTEDIT--ChangesmustbemadethroughldquoFileinfordquoCorrectionKey=A
8_MCAAESE206984_U1M01RTindd 27 41513 1113 PM
Math TrainerOnline Assessment
and Intervention
Personal
myhrwcom
1
2
3 Response toIntervention
Intervention Enrichment
Access Ready to Go On assessment online and receive instant scoring feedback and customized intervention or enrichment
Online and Print Resources
Differentiated Instruction
bull Reteach worksheets
bull Reading Strategies EL
bull Success for English Learners EL
Differentiated Instruction
bull Challenge worksheets PRE-AP
Extend the Math PRE-AP
Lesson Activities in TE
Additional ResourcesAssessment Resources includes bull Leveled Module Quizzes
Ready to Go OnAssess MasteryUse the assessment on this page to determine if students have mastered the concepts and standards covered in this module
California Common Core Standards
Lesson Exercises Common Core Standards
11 1ndash7 8NS1 8NS2 8EE2
12 8ndash10 8NS1
13 11ndash14 8NS2
27 Unit 1 Module 1
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
Personal Math Trainer
Online Practice and HelpmyhrwcomAssessment Readiness
Module 1 MIXed ReVIeW
1 Look at each number Is the number between 2π and radic___
52
Select Yes or No for expressions AndashC
A 6 2 _ 3 Yes No
B 5π __ 2 Yes No
C 3 radic__
5 Yes No
2 Consider the number - 11 __ 15
Choose True or False for each statement
A The number is rational True False
B The number can be written as True Falsea repeating decimal
C The number is less than ndash08 True False
3 The volume of a cube is given by V = x3 where x is the length of an edge of the cube A cube-shaped end table has a volume of 3 3 _ 8 cubic feet What is the length of an edge of the end table Explain how you solved this problem
4 A student says that radic___
83 is greater than 29 __ 3 Is the student correct Justify your
reasoning
1 1 _ 2 ft Sample answer The equation x3 = 3 3 _ 8 can be used
to find the edge length in feet To solve the equation
write the mixed number as a fraction greater than 1
x3 = 27 __ 8 Then take the cube root of both sides x = 3 _ 2 = 1 1 _ 2
No Sample answer radic___
83 asymp 91 and 29 __ 3 = 9
__ 6
Because 91 lt 9 __
6 radic___
83 lt 29 __ 3
28 Unit 1
copy H
ough
ton
Miff
lin H
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pany
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=A
8_MCAAESE206984_U1M01RTindd 28 240413 946 AM
Personal Math Trainer
Online Assessment and
Interventionmyhrwcom
Scoring GuideItem 3 Award the student 1 point for finding the edge length of the cube and 1 point for correctly explaining how to use a cube root to solve the problem
Item 4 Award the student 1 point for determining that the student is incorrect and 1 point for correctly justifying the reasoning for this conclusion
Additional ResourcesTo assign this assessment online login to your Assignment Manager at myhrwcom
Assessment Readiness
California Common Core Standards
Items Grade 8 Standards Mathematical Practices
1 8NS2 MP7
2 7NS2b 7NS2d 8NS1 MP7
3 8EE2 MP1 MP4
4 8NS1 8NS2 MP3
Item integrates mixed review concepts from previous modules or a previous course
Item 4 combines concepts from the California Common Core cluster ldquoKnow that there are numbers that are not rational and approximate them by rational numbersrdquo
Real Numbers 28
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
Lesson Support Content Objective Students will learn to rewrite rational numbers and decimals take square roots and
cube roots and approximate irrational numbers
Language Objective Students will show and explain how to rewrite rational numbers and decimals take square roots and cube roots and approximate irrational numbers
LESSON 11 Rational and Irrational Numbers
Building BackgroundEliciting Prior Knowledge Have students work with a partner to review the relationship between fractions and decimals Ask students to provide an example of writing a fraction or mixed number as a decimal and vice versa Discuss how students chose and wrote their examples
Learning ProgressionsIn this lesson students work with positive rational and irrational numbers They make connections among the real numbers by converting fractions and decimals and approximating irrational numbers Important understandings for students include the following
bull Understand that every number has a decimal expansion bull Convert a repeating decimal to a rational number bull Evaluate square roots of perfect squares and cube roots of perfect cubes
bull Estimate an irrational number
Work with the real number system will continue in this unit as students extend the positive rational and irrational numbers to include negative numbers and compare and order real numbers
Cluster ConnectionsThis lesson provides an excellent opportunity to connect ideas in this cluster Know that there are numbers that are not rational and approximate them by rational numbers Tell students ldquoA square garden has an area of 20 square feetrdquo
Have students explain why the side length cannot be rational Then have them approximate the length of each side of the garden to the nearest tenth and hundredth Sample answer The length is the solution to s 2 = 20 radic
_ 20 which is not a rational
number 45 ft 447 ft The length is between 4 and 5 feet 20 is closer to 45 2 than to 44 2 or 46 2 It is also closer to 447 2 than to 446 2 or 448 2
3 _ 4
= 075 1 2 _ 3
= 1 _
6
7 _ 10
= 07 45 = 4 1 _ 2
20 ft 2
California Common Core Standards
8NS1 Know that numbers that are not rational are called irrational Understand informally that every number has a decimal expansion for rational numbers show that the decimal expansion repeats eventually and convert a decimal expansion which repeats eventually into a rational number
8NS2 Use rational approximations of irrational numbers to compare the size of irrational numbers locate them approximately on a number line diagram and estimate the value of expressions (eg π 2 )
8EE2 Use square root and cube root symbols to represent solutions to equations of the form x 2 = p and x 3 = p where p is a positive rational number Evaluate square roots of small perfect squares and cube roots of small perfect cubes Know that radic
_ 2 is irrational
MP6 Attend to precision
7A
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=B
Math Talk
Language Support EL
PROFESSIONAL DEVELOPMENT
Linguistic Support EL
AcademicContent Vocabulary
square ndash In this lesson the word square has multiple meanings which can cause confusion For example to square as in to take the square root of a number is a verb It is different from the nouns square or square of a number The text also refers to perfect square and principal square root of a number and the square root symbol is used These different usages of square as a mathematical term need to be clarified Sentence frames can be used to help define the meaning
To square a number means to _______The perfect square of a number means _______
Background Knowledge
suffixes ndash When added to a root word the suffix -th is used in math to indicate one of a specified number of parts such as tenth hundredth or thousandth Remind students that the suffix -th also indicates place value Note that Spanish Vietnamese Mandarin and other languages do not have the ending th sound so teachers need to enunciate carefully
cognates ndash The words terminating and terminal used in this lesson are cognates in Spanish terminar meaning ldquoto endrdquo or ldquoto finishrdquo A Spanish cognate for approximate is aproximar
Leveled Strategies for English Learners
Emerging Use cards with root words ten hundred and thousand and a card with the -th suffix Have students place them together to show place value Then complete a sentence Use the same procedure to identify decimals
Expanding Support students at this level of English proficiency by providing sentence frames for them to use to describe their mathematical reasoning
To write the fraction _______ as a decimal I _______
Bridging Have students identify different meanings of the term square by matching examples of math problems with a written out sentence frame that defines the usage of the term square to square a number perfect square square root Use this procedure also with the term cube
Be sure to clarify the different uses of the term square when referring to square roots perfect squares and so on
EL
California ELD Standards
Emerging 2I12b Selecting language resources ndash Use knowledge of morphology to appropriately select affixes in basic ways
Expanding 2I12b Selecting language resources ndash Use knowledge of morphology to appropriately select affixes in a growing number of ways to manipulate language
Bridging 2I12b Selecting language resources ndash Use knowledge of morphology to appropriately select affixes in a variety of ways to manipulate language
Rational and Irrational Numbers 7B
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
11L E S S O N
Rational and Irrational Numbers
EngageESSENTIAL QUESTION
How do you rewrite rational numbers and decimals take square roots and cube roots and approximate irrational numbers To express as a decimal divide the numerator by the denominator To take a square root or cube root of a number find the number that when squared or cubed equals the original number To approximate an irrational number estimate a number between two consecutive perfect squares
Motivate the LessonAsk Which type of rational number do you see more often fractions or decimals Which do you prefer to use Why
ExploreHave students write examples of ratios and then share with the class the various notations for ratios that they used (for example 25 2 to 5 2 __ 5 ) Point out the connection between the word ratio and the meaning of rational number See also Explore Activity in student text
ExplainEXAMPLE 1
Questioning Strategies Mathematical Practices bull How does the denominator of a fraction in simplest form tell whether the decimal equivalent of the fraction is a terminating decimal The decimal will terminate if the denominator is an even number a multiple of 5 or a multiple of 10
Avoid Common ErrorsTo avoid interpreting 1 __ 4 as 4 divided by 1 tell students to start at the top of the fraction and read the bar as ldquodivided byrdquo
YOUR TURNTalk About ItCheck for Understanding
Ask Can an improper fraction be written as a decimal Give an example to support your answer Yes 5 __ 4 = 125
EXAMPLE 2Questioning Strategies Mathematical Practices bull How can you use place value to write a terminating decimal as a fraction with a power of ten in the denominator Start by identifying the place value of the decimals last digit and then use the corresponding power of 10 as the denominator of the fraction
bull How can you tell if a decimal can be written as a rational number If the decimal is a terminating or repeating decimal then it can be written as a rational number
Interactive Whiteboard Interactive example available online
ADDITIONAL EXAMPLE 1Write each fraction as a decimal
A 2 _ 5
04 B 5 _ 9
0 _
5
myhrwcom
Interactive Whiteboard Interactive example available online
ADDITIONAL EXAMPLE 2Write each decimal as a fraction in simplest form
A 0355 71 ___ 200
B 0 _
43 43 __ 99
myhrwcom
CA Common CoreStandards
The student is expected to
The Number Systemmdash8NS1
Know that numbers that are not rational are called irrational Understand informally that every number has a decimal expansion for rational numbers show that the decimal expansion repeats eventually and convert a decimal expansion which repeats eventually into a rational number
The Number Systemmdash8NS2
Use rational approximations of irrational numbers to compare the size of irrational numbers locate them approximately on a number line diagram and estimate the value of expressions (eg π 2 )
Expressions and Equationsmdash8EE2
Use square root and cube root symbols to represent solutions to equations of the form x 2 = p and x 3 = p where p is a positive rational number Evaluate square roots of small perfect squares and cube roots of small perfect cubes Know that radic
_ 2 is irrational
Mathematical Practices
MP6 Precision
The student is expected to
the value of expressions (eg
7 Lesson 11
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
My Notes
Math On the Spotmyhrwcom
Math TrainerOnline Practice
and Help
Personal
myhrwcom
Expressing Decimals as Rational NumbersYou can express terminating and repeating decimals as rational numbers
Write each decimal as a fraction in simplest form
0825
The decimal 0825 means ldquo825 thousandthsrdquo Write this as a fraction
825 ____ 1000
Then simplify the fraction
825 divide 25 ________ 1000 divide 25 = 33 __ 40
0825 = 33 __ 40
0 _
37
Let x = 0 _
37 The number 0 _
37 has 2 repeating digits so multiply each side of the equation x = 0
_ 37 by 10 2 or 100
x = 0 _
37
(100)x = 100(0 _
37 )
100x = 37 _
37
Because x = 0 _
37 you can subtract x from one side and 0 _
37 from the other
100x = 37 _
37
minusx minus0 _
37
99x = 37
Now solve the equation for x Simplify if necessary
99x ___ 99 = 37 __ 99
x = 37 __ 99
EXAMPLE 2
A
B
Write each fraction as a decimal
YOUR TURN
1 5 __ 11 2 1 _ 8 3 2 1 _ 3
8NS1
To write ldquo825 thousandthsrdquo put 825 over 1000
Divide both the numerator and the denominator by 25
100 times 0 _
37 is 37 _
37
37 _
37 minus 0 _
37 is 37
Divide both sides of the equation by 99
0 _
45 0125 2 _
3
Unit 18
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ton
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pany
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8_MCAAESE206984_U1M01L1indd 8 120413 838 PM
My Notes
Math On the Spot
myhrwcom
= 033333333333331mdash3
ESSENTIAL QUESTION
Expressing Rational Numbers as DecimalsA rational number is any number that can be written as a ratio in the form a _ b where a and b are integers and b is not 0 Examples of rational numbers are 6 and 05
6 can be written as 6 _ 1 05 can be written as 1 _ 2
Every rational number can be written as a terminating decimal or a repeating decimal A terminating decimal such as 05 has a finite number of digits A repeating decimal has a block of one or more digits that repeat indefinitely
Write each fraction as a decimal
1 _ 4
1 _ 4 = 025
1 _ 3
1 _ 3 = 0 _
3
EXAMPLEXAMPLE 1
A
B
0333 3 ⟌ ⎯ 1000 minus9 10 minus9 10 minus9 1
025 4 ⟌ ⎯ 100 -8 20 -20
0
L E S S O N
11Rational and Irrational Numbers
How do you rewrite rational numbers and decimals take square roots and cube roots and approximate irrational numbers
8NS1
Know that numbers that are not rational are called irrational Understand informally that every number has a decimal expansion for rational numbers show that the decimal expansion repeats eventually and convert a decimal expansion which repeats eventually into a relation number Also 8NS2 8EE2
8NS1
Remember that the fraction bar means ldquodivided byrdquo Divide the numerator by the denominator
Divide until the remainder is zero adding zeros after the decimal point in the dividend as needed
Divide until the remainder is zero or until the digits in the quotient begin to repeat
Add zeros after the decimal point in the dividend as needed
When a decimal has one or more digits that repeat indefinitely write the decimal with a bar over the repeating digit(s)
7Lesson 11
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ough
ton
Miff
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pany
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8_MCABESE206984_U1M01L1indd 7 11113 128 AM
PROFESSIONAL DEVELOPMENT
Math BackgroundSome decimals may have a pattern but still not be a repeating decimal that is rational For example in 312112111211112hellip you can predict the next digit and describe the pattern (There is one more 1 each time before the 2) However this is not a terminating decimal nor is it a repeating decimal and it is therefore NOT a rational number
Integrate Mathematical Practices MP6
This lesson provides an opportunity to address this Mathematical Practices standard It calls for students to attend to precision Students learn to express rational numbers accurately and precisely in both fractional and decimal forms and learn to translate from one form to the other They also learn how to precisely represent and communicate ideas about irrational numbers square roots and cube roots
Rational and Irrational Numbers 8
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=B
Focus on Technology Mathematical PracticesPoint out the importance of entering a repeating decimal correctly when using a graphing calculator to convert the decimal to a fraction The decimal 0
_ 59 must be entered as
0595959595959 not 059
YOUR TURNFocus on Math ConnectionsMake sure students understand that the place value of the last digit in Exercises 4 and 6 determines the denominator of the corresponding fraction or mixed number So for Exercise 4 the place value hundredths gives a denominator of 100 and for Exercise 6 the place value tenths gives a denominator of 10
EXAMPLE 3Questioning Strategies Mathematical Practices bull How can a solution of an equation of the form x 2 = p be negative if p is a positive number Since the square of a negative number is positive a negative number is also a solution of x 2 equals a positive number
bull When is a solution of an equation of the form x 3 = p larger than p The solution is larger than p if p is a number between 0 and 1
Focus on Math Connections Make sure students understand the difference in finding radic
_ 121 and solving x 2 = 121 The
symbol radic_
indicates the positive or principal square root only while the equation x 2 = 121 has two roots the principal square root and its opposite
YOUR TURNAvoid Common ErrorsTo avoid sign errors in Exercise 9 make sure that students understand that the cube of a negative number is not a positive number Therefore -8 is not a solution of x 3 = 512
Talk About ItCheck for Understanding
Ask Kris predicts that there are two real solutions for Exercises 7 and 8 and that there are three real solutions for Exercises 9 and 10 Is his prediction correct
Explain His prediction is correct for Exercises 7 and 8 because there are two numbers whose squares are the same positive number given in the exercises His prediction is not correct for Exercises 9 and 10 however because there is only one real number whose cube is the same positive number given in the exercises
EXPLORE ACTIVITYQuestioning Strategies Mathematical Practices bull Compare the values for 13 2 and 13 2 The digits are the same but 13 2 has two decimal places (169) while 13 2 has none (169)
bull How do you know whether radic_
2 will be closer to 1 or closer to 2 It will be closer to 1 because 2 is between the perfect squares of 1 and 4 but closer to 1 than it is to 4
Connect Vocabulary EL
Explain to students that the word irrational when used as an ordinary word in English means without logic or reason In mathematics when we say that a number is irrational it means only that the number cannot be written as the quotient of two integers
Engage with the WhiteboardHave students extend the number line in both directions and label the locations of the whole numbers 1 and 2 These are the roots of the consecutive perfect squares
1 and 4 used to estimate radic_
7
Interactive Whiteboard Interactive example available online
ADDITIONAL EXAMPLE 3Solve each equation for x
A x 2 = 324 18 -18
B x 2 = 25 ___ 144 5 __ 12 - 5 __ 12
C 343 = x 3 7
D x 3 = 125 ___ 512 5 __ 8
myhrwcom
9 Lesson 11
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
Math TrainerOnline Practice
and Help
Personal
myhrwcom
EXPLORE ACTIVITY
lt 2 lt
radic_
lt radic
_ 2 lt
radic_
lt radic
_ 2 lt
The solution is 9
The solution is 2 _ 5
C
D
729 = x 3
3 radic_ 729 = 3 radic
_ x 3
3 radic_ 729 = x
9 = x
x 3 = 8 ___ 125
3 radic_
x 3 =thinsp 3 radic_ 8 ___ 125
x =thinsp 3 radic_ 8 ___ 125
x = 2 _ 5
Solve each equation for x
YOUR TURN
7 x 2 = 196 8 x 2 = 9 ___ 256
9 x 3 = 512 10 x 3 = 64 ___ 343
Estimating Irrational NumbersIrrational numbers are numbers that are not rational In other words they cannot be written in the form a _ b where a and b are integers and b is not 0 Square roots of perfect squares are rational numbers Square roots of numbers that are not perfect squares are irrational Some equations like those in Example 3 involve square roots of numbers that are not perfect squares
x 2 = 2 x = plusmn radic_
2
Estimate the value of radic_
2
Find two consecutive perfect squares that 2 is between Complete the inequality by writing these perfect squares in the boxes
Now take the square root of each number
Simplify the square roots of perfect squares
radic_
2 is between and
A
B
C
8NS2 8EE2
Solve for x by taking the cube root of both sides
Solve for x by taking the cube root of both sides
Apply the definition of cube root
Think What number cubed equals 729
Apply the definition of cube root
Think What number cubed equals 8 ____ 125
radic_
2 is irrational
x = plusmn14 x = plusmn 3 __ 16
x = 8 x = 4 _ 7
1 2
1 4
1 4
1 2
Unit 110
copy H
ough
ton
Miff
lin H
arco
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ublis
hing
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pany
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
8_MCAAESE206984_U1M01L1indd 10 41613 1211 AM
Math TrainerOnline Practice
and Help
Personal
myhrwcom
Math On the Spot
myhrwcom
Write each decimal as a fraction in simplest form
YOUR TURN
Finding Square Roots and Cube RootsThe square root of a positive number p is x if x 2 = p There are two square roots for every positive number For example the square roots of 36 are 6 and minus6 because 6 2 = 36 and (minus6) 2 = 36 The square roots of 1 __ 25 are 1 _ 5 and minus 1 _ 5 You can write the square roots of 1 __ 25 as plusmn 1 _ 5 The symbol radic
_ 5 indicates the positive
or principal square root
A number that is a perfect square has square roots that are integers The number 81 is a perfect square because its square roots are 9 and minus9
The cube root of a positive number p is x if x 3 = p There is one cube root for every positive number For example the cube root of 8 is 2 because 2 3 = 8 The cube root of 1 __ 27 is 1 _ 3 because ( 1 _ 3 )
3
= 1 __ 27 The symbol 3 radic_ 1 indicates the
cube root
A number that is a perfect cube has a cube root that is an integer The number 125 is a perfect cube because its cube root is 5
Solve each equation for x
The solutions are 11 and minus11
The solutions are 4 __ 13 and minus 4 __ 13
EXAMPLEXAMPLE 3
A x 2 = 121
x 2 = 121
x = plusmn radic_
121
x = plusmn11
B x 2 = 16 ___ 169
x 2 = 16 ___ 169
x = plusmn radic_
16 ___ 169
x = plusmn 4 __ 13
4 012 5 0 _
57 6 14
Can you square an integer and get a negative number
What does this indicate about whether negative
numbers have square roots
Math TalkMathematical Practices
8EE2
Solve for x by taking the square root of both sides
Apply the definition of square root
Think What numbers squared equal 121
Solve for x by taking the square root of both sides
Apply the definition of square root
Think What numbers squared equal 16 ____ 169
3 __ 25 19 __ 33 1 2 _ 5
No the square of a positive integer is positive the square of a negative integer is positive and the square of 0 is 0 So negative numbers do not have (real) square roots
9Lesson 11
copy H
ough
ton
Miff
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pany
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
8_MCAAESE206984_U1M01L1indd 9 41913 240 PM
Critical ThinkingIn the Explore Activity students estimated the location of radic
_ 2 on a number line Ask students
whether they think that it is possible to locate more precisely the point that represents radic
_ 2 In
other words can you graph irrational numbers exactly on a number line along with rational numbers Students should understand that radic
_ 2
is a real number and all real numbers can be located on a real number line A more precise estimate will allow more precise placement on a number line
The Modeling note tells one way to do this
ModelingHave students use a ruler to represent a number line with a unit that is one inch long Have them draw a square with a side of one inch and draw the diagonal to make two isosceles triangles Lead students to understand that the length of the diagonal (or hypotenuse) is radic
_ 2
Have them copy the length of their diagonal onto their ruler or number line starting at zero The end point of the diagonal represents the exact point for the irrational number radic
_ 2 on a
number line
Additional ResourcesDifferentiated Instruction includes bull Reading Strategies bull Success for English Learners EL
bull Reteach bull Challenge PRE-AP
DIFFERENTIATE INSTRUCTION
Rational and Irrational Numbers 10
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
ElaborateTalk About ItSummarize the Lesson
Ask If someone claims that a certain number is irrational but you know it is actually rational how could you prove to that person that the number is rational
You could find a fraction equal to the number such that the number is the ratio of two integers with the denominator not equal to zero
GUIDED PRACTICEEngage with the Whiteboard
Have students plot each number in Exercises 16ndash18 on a number line Students should label each point with the irrational number written as a radical and as a
decimal
Avoid Common ErrorsExercises 1ndash6 To avoid reversing the order of the dividend and divisor tell students to start at the top of the fraction and read the bar as ldquodivided byrdquo
Focus on TechnologyHave students use a calculator to investigate the decimal equivalents of such fractions as 1 __ 9 2 __ 9 8 __ 9 and 1 __ 11 2 __ 11 10
__ 11 Ask them to describe the patterns they find as a result of these investigations
11 Lesson 11
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
Guided Practice
7 0675 8 56 9 044
10 0 _
4
10x =
x =
11 0 _
26
100x =
x =
12 0 _
325
1000x =
x =
Solve each equation for x (Example 3 and Explore Activity)
- x
-
_______________
x =
- x
-
___________________
x =
- x
-
_______________________
x =
Write each fraction or mixed number as a decimal (Example 1)
1 2 _ 5 2 8 _ 9 3 3 3 _ 4
4 7 __ 10 5 2 3 _ 8 6 5 _ 6
Write each decimal as a fraction or mixed number in simplest form (Example 2)
13 x 2 = 17 14 x 2 = 25 ___ 289 15 x 3 = 216
Approximate each irrational number to one decimal place without a calculator
x = plusmn radic__
asymp plusmn x = 3
radic__
=
(Explore Activity)
16 radic_
5 asymp
17 radic_
3 asymp
18 radic_
10 asymp
19 What is the difference between rational and irrational numbers
CHECK-INESSENTIAL QUESTION
x = plusmn radic__
__________ = plusmn _____
4 _
4
0 _
4
4 99
6216
269
41 25 5
17289
17
22 17 32
04
07
27__40
26 __ 99 325 ___ 999 4 _ 9
11__255 3_5
0 _
8
2375
375
08 _
3
26 _
26
0 _
26
325 _
325
0 _
325
999 325
Rational numbers can be written in the form a __ b where
a and b are integers and b ne 0 Irrational numbers cannot
be written in this form
Unit 112
copy H
ough
ton
Miff
lin H
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urt P
ublis
hing
Com
pany
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
8_MCAAESE206984_U1M01L1indd 12 41613 1211 AM
11 12 13 14 15
radic2 asymp 14
141 142 143 144 145
radic2 asymp 141
0 1 2 3 4
radic2 asymp 15
Estimate that radic_
2 asymp 15
To find a better estimate first choose some numbers between 1 and 2 and square them For example choose 13 14 and 15
1 3 2 = 1 4 2 = 1 5 2 =
Is radic_
2 between 13 and 14 How do you know
Is radic_
2 between 14 and 15 How do you know
2 is closer to than to so radic_
2 asymp
Locate and label this value on the number line
Reflect 11 How could you find an even better estimate of radic
_ 2
12 Find a better estimate of radic_
2
1 41 2 = 1 42 2 = 1 43 2 =
2 is closer to than to so radic_
2 asymp
Draw a number line and locate and label your estimate
13 Solve x 2 = 7 Write your answer as a radical expression Then estimate to one decimal place
D
E
F
No 2 is not between 169 and 196
Yes 2 is between 196 and 225
196
19881
19881
225
20164
20164
14
141
20449
169 196 225
Test the squares of numbers between 14 and 15
x = plusmn radic_
7 x asymp plusmn26
11Lesson 11
copy H
ough
ton
Miff
lin H
arco
urt P
ublis
hing
Com
pany
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
8_MCAAESE206984_U1M01L1indd 11 41613 1211 AM
Rational and Irrational Numbers 12
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
Math TrainerOnline Assessment
and Intervention
Personal
Online homework assignment available
myhrwcom
EvaluateGUIDED AND INDEPENDENT PRACTICE
Concepts amp Skills Practice
Example 1Expressing Rational Numbers as Decimals
Exercises 1ndash6 20ndash21 24ndash25
Example 2Expressing Decimals as Rational Numbers
Exercises 7ndash12 22ndash23 26ndash27
Example 3Finding Square Roots and Cube Roots
Exercises 13ndash15 28 30ndash31 35
Explore ActivityEstimating Irrational Numbers
Exercises 13 16ndash18 29 32ndash34
Additional ResourcesDifferentiated Instruction includes bull Leveled Practice worksheets
Lesson Quiz available online
11 LESSON QUIZ
1 Write as a decimal 2 5 __ 8 1 7 __ 12
2 Write as a fraction 034 1 _
24
3 Solve x 2 = 9 __ 49 for x
4 Solve x 3 = 216 for x
5 Estimate the value of radic_
13 to one decimal place without using a calculator
myhrwcom
Focus | Coherence | Rigor
Exercise Depth of Knowledge (DOK) Mathematical Practices
20ndash27 2 SkillsConcepts MP4 Modeling
28 3 Strategic Thinking MP4 Modeling
29ndash32 2 SkillsConcepts MP6 Precision
33 3 Strategic Thinking MP7 Using Structure
34 2 SkillsConcepts MP3 Logic
35 2 SkillsConcepts MP4 Modeling
36 3 Strategic Thinking MP3 Logic
37 3 Strategic Thinking MP7 Using Structure
38 3 Strategic Thinking MP2 Reasoning
8NS1 8NS2 8EE2
8NS1 8NS2 8EE2
Answers1 2625 158
_ 3
2 17 __ 50 1 8 __ 33
3 x = plusmn 3 __ 7
4 x = 6
5 36
13 Lesson 11
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
Work Area
33 Analyze Relationships To find radic_
15 Beau found 3 2 = 9 and 4 2 = 16 He said that since 15 is between 9 and 16 radic
_ 15 must be between 3 and 4 He
thinks a good estimate for radic_
15 is 3 + 4 ____ 2 = 35 Is Beaursquos estimate high low
or correct Explain
34 Justify Reasoning What is a good estimate for the solution to the equation x 3 = 95 How did you come up with your estimate
35 The volume of a sphere is 36π f t 3 What is the radius of the sphere Use the formula V = 4 _ 3 π r 3 to find your answer
36 Draw Conclusions Can you find the cube root of a negative number If so is it positive or negative Explain your reasoning
37 Make a Conjecture Evaluate and compare the following expressions
radic_
4 __ 25 and radic
_ 4 ____
radic_
25 radic
_
16 __ 81 and radic_
16 ____
radic_
81 radic
_
36 __ 49 and radic_
36 ____
radic_
49
Use your results to make a conjecture about a division rule for square roots Since division is multiplication by the reciprocal make a conjecture about a multiplication rule for square roots
38 Persevere in Problem Solving The difference between the solutions to the equation x 2 = a is 30 What is a Show that your answer is correct
FOCUS ON HIGHER ORDER THINKING
His estimate is low because 15 is very close to 16
so radic_
15 is very close to radic_
16 or 4 A better estimate
would be 38 or 39
Sample answer about 45 4 3 = 64 and 5 3 = 125
Because 95 is about halfway between 64 and 125 try 45
45 3 = 91125 which is a good estimate
3 feet
Yes the cube root of a negative number is negative
because a negative number cubed is always negative
and a nonnegative number cubed is always nonnegative
radic_
4 __ 25 = 2 _ 5 = radic
_ 4 ____
radic_
25 radic
_
16 __ 81 = 4 _ 9 = radic_
16 ____
radic_
81 radic
_
36 __ 49 = 6 _ 7 = radic_
36 ____
radic_
49
225 the solutions to x 2 = a are x = plusmn15 and
radic_
a ___
radic_
b = radic
_ a __
b radic
_ a radic
_ b = radic
_ a b
15 - (-15) = 30
Unit 114
copy H
ough
ton
Miff
lin H
arco
urt P
ublis
hing
Com
pany
bull copy
Ilen
e Mac
Dona
ldA
lamy I
mag
es
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=B
8_MCABESE206984_U1M01L1indd 14 102913 1142 PM
Personal Math Trainer
Online Practice and Helpmyhrwcom
Name Class Date
Independent Practice11
20 A 7 __ 16 -inch-long bolt is used in a machine What is this length written as a decimal
21 The weight of an object on the moon is 1 _ 6 its weight on Earth Write 1 _ 6 as a decimal
22 The distance to the nearest gas station is 2 4 _ 5 kilometers What is this distance written as a decimal
23 A baseball pitcher has pitched 98 2 _ 3 innings What is the number of innings written as a decimal
24 A heartbeat takes 08 second How many seconds is this written as a fraction
25 There are 262 miles in a marathon Write the number of miles using a fraction
26 The average score on a biology test was 72
_ 1 Write the average score using a
fraction
27 The metal in a penny is worth about 0505 cent How many cents is this written as a fraction
28 Multistep An artist wants to frame a square painting with an area of 400 square inches She wants to know the length of the wood trim that is needed to go around the painting
a If x is the length of one side of the painting what equation can you set up to find the length of a side How many solutions does the equation have
b Do all of the solutions that you found make sense in the context of the problem Explain
c What is the length of the wood trim needed to go around the painting
Solve each equation for x Write your answers as radical expressions Then estimate to one decimal place if necessary
29 x 2 = 14 30 x 3 = 1331
31 x 2 = 144 32 x 2 = 29
8NS1 8NS2 8EE2
04375 in 01 _6
28 km 98 _6 innings
x 2 = 400 x = plusmnthinsp20 the equation has 2 solutions
x = 20 makes sense but x = -20 doesnrsquot because a
painting cannot have a side length of -20 inches
4 times 20 = 80 inches
x = plusmn radic_
14 asymp plusmn37
x = plusmn radic_
144 = plusmn12 x = plusmn radic_
29 asymp plusmn54
x = 3 radic_ 1331 = 11
4_5 second 26 1_5 mi
72 1 _ 9 101 ___ 200 cent
13Lesson 11
copy H
ough
ton
Miff
lin H
arco
urt P
ublis
hing
Com
pany
bull copy
Phot
odisc
Get
ty Im
ages
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
8_MCAAESE206984_U1M01L1indd 13 41613 1211 AM
myhrwcomActivity available onlineEXTEND THE MATH PRE-AP
Activity Write radic_
09 on the board and invite students to conjecture what the value might be Have them check their conjectures by squaring Invite them to suggest ways to estimate radic
_ 09 As a hint point out that 09 is close to 10 and so they might
use that to help guide their estimates Lead them to see that since 092 is 081 and 102 is 1 the value of radic
_ 09 is greater than 09 and less than 10 Try squaring 095 to get
09025 A good estimate for radic_
09 is 095
Rational and Irrational Numbers 14
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=B
Integers
Rational Numbers IrrationalNumbers
Real Numbers
WholeNumbers
-3-4-5 -2-1 1 2 3 50 4
23
34-4 -π -1 25
radic2
Lesson Support Content Objective Students will learn to describe relationships between sets of numbers
Language Objective Students will explain how to describe relationships between sets of real numbers
LESSON 12 Sets of Real Numbers
Building BackgroundEliciting Prior Knowledge Have students draw a number line from -5 to 5 Ask them to plot points on the number line to approximate the location of rational and irrational numbers such as -1 3 __ 4 25 -4 2 __ 3 radic
_ 2 and -π
Learning ProgressionsIn this lesson students clarify their understanding of the real number system They characterize sets and subsets of the real numbers They also identify sets for real-world situations Important understandings for students include the following
bull Identify all of the possible subsets of the real numbers for a given number
bull Decide whether a statement about a subset of the real numbers is true or false
bull Identify the set of numbers that best describes a real-world situation
Understanding the relationships among the sets of numbers that make up the real numbers is essential as students are introduced to different forms of numbers throughout the school year This lesson provides a foundation for the comparing and ordering of real numbers in the next lesson
Cluster ConnectionsThis lesson provides an excellent opportunity to connect ideas in this cluster Know that there are numbers that are not rational and approximate them by rational numbers Have students copy this diagram which relates the sets of real numbers
Ask students to complete the diagram by writing three examples for each set of numbers Have students share examples and explain how they knew each number they selected belonged in the appropriate set Answers may vary Check studentsrsquo work
Focus | Coherence | Rigor
California Common Core Standards
8NS1 Know that numbers that are not rational are called irrational Understand informally that every number has a decimal expansion for rational numbers show that the decimal expansion repeats eventually and convert a decimal expansion which repeats eventually into a rational number
MP7 Look for and make use of structure
15A
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
Math Talk
Language Support EL
PROFESSIONAL DEVELOPMENT
Linguistic Support EL
AcademicContent Vocabulary
Venn diagrams ndash Students need descriptive language to describe the categories that the different areas and colors of a Venn diagram represent the concept of a set and how sets are distinct or can overlap Use sentence frames such as
The big oval represents __________The darklight blue color in the middle of the
big ovals represents __________These sets overlap because __________
In this way students have the language and structure to identify the criteria that distinguish a set and to explain the abstract representation Also point out the use of the prefix sub- meaning ldquounderrdquo in the term subset
Rules and Patterns
Abbreviations ndash In this lesson the abbreviation mph is used Be sure to point out that mph stands for miles per hour and is used to give units in a rate of speed Students may also have seen mpg (miles per gallon) which gives the units in a rate of fuel efficiency
Borrowed Words ndash Terminology used in baseball such as inning and pitcher may require some explanation Spanish as well as some other languages have borrowed these terms from English so some students may be familiar with these words already Despite this whenever a word is critical to students understanding the word problem it is best to explain the meaning
Leveled Strategies for English Learners
Emerging Allow students to indicate true or false orally in Guided Practice Exercises 9 and 10
Expanding Have students use sentence frames to describe the meaning of regions and colors used in a Venn diagram Then give them similar sentence frames orally and have them draw and shade a Venn diagram based on the oral prompts
Bridging Have students work in groups to draw a Venn diagram to represent sets based on real-world examples in the lesson
To help students answer the question posed in Math Talk provide a sentence frame for their answer
The numbers between 31 and 39 on a number line are __________ because __________
EL
California ELD Standards
Emerging 2II5 Modifying to add details ndash Expand sentences with simple adverbials to provide details about a familiar activity or process
Expanding 2II5 Modifying to add details ndash Expand sentences with adverbials to provide details about a familiar or new activity or process
Bridging 2II5 Modifying to add details ndash Expand sentences with increasingly complex adverbials to provide details about a variety of familiar and new activities and processes
Sets of Real Numbers 15B
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
12L E S S O N
Sets of Real Numbers
EngageESSENTIAL QUESTION
How can you describe relationships between sets of real numbers Sample answer Describe them as two different sets or one set as being a subset of another
Motivate the LessonAsk How many different types of tigers can you name How does the set of Bengal tigers relate to the set of tigers
ExplorePoint to different locations in the Animals diagram and ask for examples for that classification Do the same for the Real Numbers diagram Students should understand that everything within a region is part of the set for example both -3 and 2 are integers
ExplainEXAMPLE 1
Questioning Strategies Mathematical Practices bull In A why is 5 not a perfect square It does not have rational numbers as its square roots
bull Can the number in B be written as a fraction Why or why not Yes it is a terminating decimal so it is a rational number
Engage with the WhiteboardHave students place the numbers in Example 1 and Additional Example 1 in the Venn diagram for numbers
YOUR TURNAvoid Common ErrorsBe sure that students read Exercise 2 carefully before answering The number given in the problem 10 is the area not the side length
EXAMPLE 2Questioning Strategies Mathematical Practices bull What two major sets are the real numbers composed of rational and irrational numbers
bull What is the location of the set of whole numbers in the Venn diagram in relation to the set of rational numbers Explain Inside it whole numbers are rational numbers
Focus on Reasoning Mathematical PracticesRemind students that it takes only one counterexample to show that a statement is false
Interactive Whiteboard Interactive example available online
ADDITIONAL EXAMPLE 1Write all names that apply to each number
A -10integer rational real
B 12 _ 3
whole integer rational real
myhrwcom
Interactive Whiteboard Interactive example available online
ADDITIONAL EXAMPLE 2Tell whether the given statement is true or false Explain your choice
No integers are whole numbers
False every whole number is also an integer
myhrwcom
Animated MathClassifying Numbers
Students build fluency in classifying numbers in this engaging fast-paced game
myhrwcom
CA Common CoreStandards
The student is expected to
The Number Systemmdash8NS1
Know that numbers that are not rational are called irrational Understand informally that every number has a decimal expansion for rational numbers show that the decimal expansion repeats eventually and convert a decimal expansion which repeats eventually into a rational numberMathematical Practices
MP7 Using Structure
The student is expected to
15 Lesson 12
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=B
Math TrainerOnline Practice
and Help
Personal
myhrwcom
Math TrainerOnline Practice
and Help
Personal
myhrwcom
Math On the Spotmyhrwcom
Understanding Sets and Subsets of Real NumbersBy understanding which sets are subsets of types of numbers you can verify whether statements about the relationships between sets are true or false
Tell whether the given statement is true or false Explain your choice
All irrational numbers are real numbers
True Every irrational number is included in the set of real numbers The irrational numbers are a subset of the real numbers
No rational numbers are whole numbers
False A whole number can be written as a fraction with a denominator of 1 so every whole number is included in the set of rational numbers The whole numbers are a subset of the rational numbers
EXAMPLE 2
A
B
Write all names that apply to each number
1 A baseball pitcher has pitched 12 2 _ 3 innings
2 The length of the side of a square that has an
area of 10 square yards
YOUR TURN
Tell whether the given statement is true or false Explain your choice
3 All rational numbers are integers
4 Some irrational numbers are integers
YOUR TURN
Give an example of a rational number that is a
whole number Show that the number is both whole
and rational
Math TalkMathematical Practices
Give an example of a
8NS1
False Every integer is a rational number but not every
False Real numbers are either rational or irrational numbers
Integers are rational numbers so no integers are irrational numbers
rational real
irrational real
Sample answer 8 8 = 8_
1
and -thinsp 5 _ 2 are not integers
rational number is an integer Rational numbers such as 3 _ 5
Unit 116
copy H
ough
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Miff
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Com
pany
bull Im
age C
redi
ts D
igita
l Im
age c
opyr
ight
copy20
04 Ey
ewire
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
8_MCAAESE206984_U1M01L2indd 16 41613 136 AM
Math On the Spot
myhrwcom
Vertebrates
Birds
Passerines
Animals
Integers
Rational Numbers IrrationalNumbers
Real Numbers
WholeNumbers
1
45
3
0
274
67
radic4
-
-3
-2
-1
03
radic2
radic17
radic11-
π
Animated Math
myhrwcom
Classifying Real NumbersBiologists classify animals based on shared characteristics A cardinal is an animal a vertebrate a bird and a passerine
You already know that the set of rational numbers consists of whole numbers integers and fractions The set of real numbers consists of the set of rational numbers and the set of irrational numbers
Write all names that apply to each number
radic_
5 irrational real
ndash1784rational real
whole integer rational real
EXAMPLEXAMPLE 1
A
B
C radic_ 81 ____ 9
L E S S O N
12Sets of Real Numbers
ESSENTIAL QUESTIONHow can you describe relationships between sets of real numbers
Passerines such as the cardinal are also called ldquoperching birdsrdquo
What types of numbers are between 31 and 39 on a
number line
Math TalkMathematical Practices
What types of numbers are
8NS1
8NS1
Know that numbers that are not rational are called irrational Understand informally that every number has a decimal expansion for rational numbers show that the decimal expansion repeats eventually and convert a decimal expansion which repeats eventually into a relation number
ndash1784 is a terminating decimal
5 is a whole number that is not a perfect square
radic_
81 _____ 9 = 9 __ 9 = 1 rational irrational real
15Lesson 12
copy H
ough
ton
Miff
lin H
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Com
pany
bull Im
age C
redi
ts copy
Wiki
med
ia Co
mm
ons
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=B
8_MCABESE206984_U1M01L2indd 15 061113 1144 AM
PROFESSIONAL DEVELOPMENT
Math BackgroundThe relationships between sets of numbers extend to include complex numbers A complex number can be written as a sum of a real number a and an imaginary number bi
a + bi
An imaginary number is a special number that when squared gives a negative value When you square a real number you get a nonnegative number When you square an imaginary number you get a negative value The imaginary unit is i
i = radic_
-1
Integrate Mathematical Practices MP7
This lesson provides an opportunity to address this Mathematical Practices standard It calls for students to discern structure to connect and communicate mathematical ideas
Students use a Venn diagram to structure relationships between sets of numbers They connect and communicate mathematical ideas when they make logical statements about the sets and describe which set best describes numbers applied to real-life situations
Sets of Real Numbers 16
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=B
YOUR TURNAvoid Common ErrorsStudents may see the word ldquoAllldquo or rdquoNordquo in Exercises 3 and 4 and immediately assume that any absolute statements like these are false Remind them that there are true statements that begin with these words and encourage them to provide examples
EXAMPLE 3Questioning Strategies Mathematical Practices bull In A how does the phrase ldquonumber of rdquo give you a clue about the number classification It indicates a counting number
bull What is the relationship between the circumference of a circle and the diameter The circumference is diameter times π
Focus on Critical Thinking Mathematical PracticesIn B suppose the diameters in inches were 25
__ π 28 __ π
31 __ π and so on What set of numbers would
best describe the circumferences Explain Whole numbers the circumferences would be the whole numbers 25 28 31 and so on
YOUR TURNFocus on Critical Thinking Mathematical PracticesHave students compare and contrast the classification of numbers in the answers in Exercises 5 and 6
ElaborateTalk About ItSummarize the Lesson
Ask What are some ways that number sets can be related Sets may be subsets of other sets or they may be separate from other sets
GUIDED PRACTICEEngage with the Whiteboard
Have students place the numbers in Exercises 1ndashthinsp8 in the Venn diagram for numbers at the beginning of the lesson
Integrating Language Arts EL
Encourage English learners to ask for clarification on any terms or phrases that they do not understand
Avoid Common ErrorsExercise 7 Remind students that a repeating decimal is a rational numberExercises 9ndash10 Remind students that it only takes one counterexample to show that a statement is false
Interactive Whiteboard Interactive example available online
ADDITIONAL EXAMPLE 3Identify the set of numbers that best describes the situation Explain your choice
A the amount of time that has passed since midnight
The set of real numbers time is continuous so the amount of time can be rational or irrational
B the number of tickets sold to a basketball game
The set of whole numbers the number of tickets sold may be 0 or a counting number
myhrwcom
17 Lesson 12
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
1IN
116 inch
Guided Practice
Write all names that apply to each number (Example 1)
1 7 _ 8 2 radic_
36
3 radic_
24 4 075
5 0 6 - radic_ 100
7 5 _
45 8 - 18 __ 6
Tell whether the given statement is true or false Explain your choice (Example 2)
9 All whole numbers are rational numbers
10 No irrational numbers are whole numbers
Identify the set of numbers that best describes each situation Explain your choice (Example 3)
11 the change in the value of an account when given to the nearest dollar
12 the markings on a standard ruler
13 What are some ways to describe the relationships between sets of numbers
CHECK-INESSENTIAL QUESTION
rational real
rational real
True Whole numbers are rational numbers
Rational numbers the ruler is marked every 1 __ 16 th inch
Sample answer Describe one set as being a subset of
another or show their relationships in a Venn diagram
Integers the change can be a whole dollar amount
and can be positive negative or zero
True Whole numbers are a subset of the set of rational numbers
and can be written as a ratio of the whole number to 1
irrational real
whole integer rational real
whole integer rational real
rational real
integer rational real
integer rational real
Unit 118
copy H
ough
ton
Miff
lin H
arco
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ublis
hing
Com
pany
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
8_MCAAESE206984_U1M01L2indd 18 41613 136 AM
My Notes
Math TrainerOnline Practice
and Help
Personal
myhrwcom
Math On the Spot
myhrwcom
Identifying Sets for Real-World SituationsReal numbers can be used to represent real-world quantities Highways have posted speed limit signs that are represented by natural numbers such as 55 mph Integers appear on thermometers Rational numbers are used in many daily activities including cooking For example ingredients in a recipe are often given in fractional amounts such as 2 _ 3 cup flour
Identify the set of numbers that best describes each situation Explain your choice
the number of people wearing glasses in a room
The set of whole numbers best describes the situation The number of people wearing glasses may be 0 or a counting number
the circumference of a flying disk has a diameter of 8 9 10 11 or 14 inches
The set of irrational numbers best describes the situation Each circumference would be a product of π and the diameter and any multiple of π is irrational
EXAMPLEXAMPLE 3
A
B
Identify the set of numbers that best describes the situation Explain your choice
5 the amount of water in a glass as it evaporates
6 the weight of a person in pounds
YOUR TURN
8NS1
Rational numbers a personrsquos weight can be a decimal
such as 835 pounds
Real numbers the amount can be any number greater
than 0
17Lesson 12
copy H
ough
ton
Miff
lin H
arco
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ublis
hing
Com
pany
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
8_MCAAESE206984_U1M01L2indd 17 41613 520 AM
Graphic OrganizersGive students a list of numbers (including terminating and repeating decimals fractions integers and rational and irrational square roots) and a graphic organizer as shown below
Real Numbers
Rational numbers Irrational numbers
Integer numbers
Whole numbers
Ask students to write each number in the list in the correct section of the organizer
Number SensePoint out to students that knowing the types of numbers to expect in different situations can alert them to incorrect math as well as to impossible situations For example 135 shots made in basketballs is not possible but an average number of shots can equal 135
Additional ResourcesDifferentiated Instruction includes bull Reading Strategies bull Success for English Learners EL
bull Reteach bull Challenge PRE-AP
DIFFERENTIATE INSTRUCTION
Sets of Real Numbers 18
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
Math TrainerOnline Assessment
and Intervention
Personal
Online homework assignment available
myhrwcom
Lesson Quiz available online
12 LESSON QUIZ
1 Write all the names that apply to the number
2 Tell whether the given statement is true or false Explain your choice All numbers between 1 and 2 are rational numbers
3 Identify the set of numbers that best describes the situation Explain your choiceThe choices on a survey question change the total points for the survey by -2 -1 0 1 or 2 points
-1 _
5
myhrwcom
Additional ResourcesDifferentiated Instruction includes bull Leveled Practice worksheets
EvaluateGUIDED AND INDEPENDENT PRACTICE
Concepts amp Skills Practice
Example 1Classifying Real Numbers
Exercises 1ndash8 14ndash19 22ndash24
Example 2Understanding Sets and Subsets of Real Numbers
Exercises 9ndash10
Example 3Identifying Sets for Real-World Situations
Exercises 11ndash12 20ndash21 25
Focus | Coherence | Rigor
Exercise Depth of Knowledge (DOK) Mathematical Practices
14ndash19 2 SkillsConcepts MP7 Using Structure
20ndash21 2 SkillsConcepts MP6 Precision
22ndash23 2 SkillsConcepts MP3 Logic
24 1 Recall of Information MP7 Using Structure
25 2 SkillsConcepts MP2 Reasoning
26ndash27 3 Strategic Thinking MP3 Logic
28 3 Strategic Thinking MP8 Patterns
29 3 Strategic Thinking MP3 Logic
8NS1
8NS1
Exercise 29 combines concepts from the California Common Core cluster ldquoKnow that there are numbers that are not rational and approximate them by rational numbersrdquo
Answers1 rational real
2 False radic_
2 is an example of an irrational number between 1 and 2
3 Integers each number is an integer but only three are whole numbers
19 Lesson 12
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
Work Area
π mi23 Critique Reasoning The circumference of a circular region is shown
What type of number best describes the diameter of the circle Explain
your answer
24 Critical Thinking A number is not an integer What type of number can it be
25 A grocery store has a shelf with half-gallon containers of milk What type of number best represents the total number of gallons
26 Explain the Error Katie said ldquoNegative numbers are integersrdquo What was her error
27 Justify Reasoning Can you ever use a calculator to determine if a number is rational or irrational Explain
28 Draw Conclusions The decimal 0 _
3 represents 1 _ 3 What type of number best describes 0
_ 9 which is 3 middot 0
_ 3 Explain
29 Communicate Mathematical Ideas Irrational numbers can never be precisely represented in decimal form Why is this
FOCUS ON HIGHER ORDER THINKING
It can be a rational number that is not an integer or an irrational number
rational number
The set of negative numbers also includes non-integer
rational numbers and irrational numbers
Sample answer If the calculator shows a decimal that
terminates in fewer digits than what the calculator screen
allows then you can tell that the number is rational If not
you cannot tell from the calculator display whether the
number terminates because you see a limited number
of digits It may be a repeating decimal (rational) or
non-terminating non-repeating decimal (irrational)
Whole 3 middot 0 _
3 represents 3 middot 1 _ 3 = 1 so 0 _
9 is exactly 1
Sample answer In decimal form irrational numbers never
terminate and never repeat Therefore no matter how
many decimal places you include the number will never
be precisely represented There are always more digits
Whole the diameter is π _ π = 1 mile
Unit 120
copy H
ough
ton
Miff
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ublis
hing
Com
pany
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
8_MCAAESE206984_U1M01L2indd 20 120413 909 PM
Integers
Rational Numbers Irrational Numbers
Real Numbers
Whole Numbers
257
radic16
166
radic9
128 radic50
Personal Math Trainer
Online Practice and Helpmyhrwcom
Name Class Date
Independent Practice
Identify the set of numbers that best describes each situation Explain your choice
20 the height of an airplane as it descends to an airport runway
21 the score with respect to par of several golfers 2 ndash 3 5 0 ndash 1
22 Critique Reasoning Ronald states that the number 1 __ 11 is not rational because when converted into a decimal it does not terminate Nathaniel says it is rational because it is a fraction Which boy is correct Explain
12
14 - radic_
9 15 257
16 radic_
50 17 8 1 _ 2
18 166 19 radic_
16
Write all names that apply to each number Then place the numbers in the correct location on the Venn diagram
8NS1
Real numbers the height can be any number greater than zero
integer rational real whole integer rational real
whole integer rational real
irrational real
rational real
rational real
Integers the scores are counting numbers their
opposites and zero
Nathaniel is correct A rational number is a number that can be written as a fraction and 1 __ 11 is a fraction
19Lesson 12
copy H
ough
ton
Miff
lin H
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urt P
ublis
hing
Com
pany
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
8_MCAAESE206984_U1M01L2indd 19 41613 136 AM
myhrwcomActivity available onlineEXTEND THE MATH PRE-AP
Activity Have students consider the concept of restricted domain for the sets of numbers that describe situations For example the number of sisters a person has can best be described by whole numbers but no one has ever had 1500 sisters An area code is an integer or whole number between 200 and 999
Have students use a source such as the Guinness Book of World Records and give examples of sets of numbers that describe situations where the domain is restricted Ask whether the restriction may be changed in the future
Sets of Real Numbers 20
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
-3-4-5 -2-1 1 2 3 50 4
12-4 -radic5
Lesson Support Content Objective Students will learn to order a set of real numbers
Language Objective Students will show and describe how to order a set of real numbers
LESSON 13 Ordering Real Numbers
Building BackgroundEliciting Prior Knowledge Have students draw a number line to compare a rational number and an irrational number such as - radic
_ 5 and -4 1 __ 2 Ask them to explain how
they approximated the irrational number on the number line Then have them identify the greater and the lesser real number Repeat with several other pairs of real numbers in different forms
Learning ProgressionsIn this lesson students order a set of real numbers They use rational approximations to compare the sizes of irrational numbers They also order numbers for real-world situations Important understandings for students include the following
bull Compare irrational numbers bull Estimate the value of expressions with irrational numbers bull Order a set of real numbers bull Order real numbers in a real-world context
Work with real numbers continues throughout Grade 8 and into high school This lesson provides students with a foundation for understanding the relative sizes of numbers in different forms in the real number system
Cluster ConnectionsThis lesson provides an excellent opportunity to connect ideas in this cluster Know that there are numbers that are not rational and approximate them by rational numbers Tell students that there is a special number called the golden ratio with applications in mathematics geometry art and architecture The golden ratio is called phi and is represented by the Greek letter ϕ It includes an irrational number in its definition
Have students explain why the golden ratio is irrational Ask them to find the two whole numbers the golden ratio lies between Then challenge them to approximate the golden ratio to the nearest tenth It is irrational because it includes an irrational number in its definition It lies between 1 and 2 To the nearest tenth ϕ = 16
ϕ = 1 + radic_
5 _ 2
Focus | Coherence | Rigor
California Common Core Standards
8NS2 Use rational approximations of irrational numbers to compare the size of irrational numbers locate them approximately on a number line diagram and estimate the value of expressions (eg π 2 )
MP4 Model with mathematics
21A
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
Math Talk
Language Support EL
PROFESSIONAL DEVELOPMENT
Linguistic Support EL
AcademicContent Vocabulary
Post a chart like this to remind students of the regular comparative forms of adjectives that use the -er and -est suffixes Add to the chart for terms that appear in examples and exercises in each lesson Include any irregular verb forms
Background Knowledge
Go On ndash the title of the module review or quiz is Ready to Go On This title uses an idiomatic expression In this context to go on means ldquoto move aheadrdquo or ldquoto proceedrdquo It is different from the use of go on that means having enough facts to use meaningfully as in having enough to go on Also the intonation used in pronouncing an expression can give it different meanings For example when the speaker emphasizes the word on he or she might be expressing disbelief as in ldquoGo ON Yoursquore kidding rightrdquo Discuss with students other ways that the phrase go on may be used
Leveled Strategies for English Learners
Emerging Label points on a number line with the terms used in ordering greater greatest less lesser least Use sentence frames to insert the correct terms
Expanding Have students give two or three complete sentences to compare the placement of numbers on a number line using the correct forms of the comparative and superlative adjectives
Bridging Have students work in pairs with one student giving directions to the other in complete sentences to order numbers on a number line
To help students answer the question posed in Math Talk make sure that students have a command of the forms for making comparisons and the superlative and the concept of opposite order so that the focus is on the math concept instead of the language skills needed to describe and explain order
EL
Adjective Comparative Superlative
Far Farther Farthest
Large Larger Largest
Great Greater Greatest
Some Less Least
Some More Most
California ELD Standards
Emerging 2I8 Analyzing language choices ndash Explain how phrasing or different common words with similar meanings produce different effects on the audience
Expanding 2I8 Analyzing language choices ndash Explain how phrasing or different words with similar meanings or figurative language produce shades of meaning and different effects on the audience
Bridging 2I8 Analyzing language choices ndash Explain how phrasing or different words with similar meanings or figurative language produce shades of meaning nuances and different effects on the audience
Ordering Real Numbers 21B
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
13L E S S O N
Ordering Real Numbers
Interactive Whiteboard Interactive example available online
ADDITIONAL EXAMPLE 1Compare Write lt gt or =
A radic_
8 - 2 4 - radic_
8 lt
B radic_
20 + 1 3 + radic_
2 gt
EngageESSENTIAL QUESTION
How do you order a set of real numbers Sample answer Find their approximate decimal values and order them
Motivate the LessonAsk What kind of numbers are you comparing when you compare the price of gasoline at two different gas stations
ExploreGive students two rational numbers and ask them to name a number between them Repeat a few times and then give them two irrational numbers and ask them to name a number between them
ExplainEXAMPLE 1
Questioning Strategies Mathematical Practices bull Which is greater the difference between 5 and 3 or the difference between radic
_ 5 and radic
_ 3
The difference between 5 and 3 is 2 the difference between radic_
5 and radic_
3 is approximately 1 So the difference between 5 and 3 is greater
Avoid Common ErrorsCaution students to read the problem carefully and think about what the radical sign means so that they do not misread the problem and answer that the two sides are equal
YOUR TURNFocus on TechnologyCalculators should not be used at this point because developing number sense is the goal
EXAMPLE 2Questioning Strategies Mathematical Practices bull How do you determine whether radic
_ 22 is less than or greater than 45 The square of 45 is
2025 which is less than 22 so the square root of 22 must be greater than 45
Engage with the WhiteboardHave students graph and label various real numbers between 42 and 44 and between 47 and 5
YOUR TURNFocus on Modeling Mathematical PracticesHave students label the integers on the number line with their equivalent square root For example 1 2 and 3 on the number line would be labeled radic
_ 1 radic
_ 4 and radic
_ 9
Interactive Whiteboard Interactive example available online
ADDITIONAL EXAMPLE 2Order 3π radic
_ 10 and 325 from greatest
to least
3π 325 radic_
10
myhrwcom
myhrwcom
CA Common CoreStandards
The student is expected to
The Number Systemmdash8NS2
Use rational approximations of irrational numbers to compare the size of irrational numbers locate them approximately on a number line diagram and estimate the value of expressions (eg π 2 )
Mathematical Practices
MP4 Modeling
The student is expected to
21 Lesson 13
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
Math TrainerOnline Practice
and Help
Personal
myhrwcom
Math On the Spotmyhrwcom
0 05 1 15 2 25 3 35 4
radic5radic3
π2
8 85 9 95 10 105 11 115 12
radic75
4 42 44 46 48 5
radic224 12π + 1
Ordering Real Numbers You can compare and order real numbers and list them from least to greatest
Order radic_
22 π + 1 and 4 1 _ 2 from least to greatest
First approximate radic_
22
radic_
22 is between 4 and 5 Since you donrsquot know where it falls between 4 and 5 you need to find a better estimate for radic
_ 22 so
you can compare it to 4 1 _ 2
Since 22 is closer to 25 than 16 use squares of numbers between 45 and 5 to find a better estimate of radic
_ 22
45 2 = 2025 46 2 = 2116 47 2 = 2209 48 2 = 2304
Since 47 2 = 2209 an approximate value for radic_
22 is 47
An approximate value of π is 314 So an approximate value of π +1 is 414
Plot radic_
22 π + 1 and 4 1 _ 2 on a number line
Read the numbers from left to right to place them in order from least to greatest
From least to greatest the numbers are π + 1 4 1 _ 2 and radic_
22
EXAMPLE 2
STEP 1
STEP 2
Order the numbers from least to greatest Then graph them on the number line
YOUR TURN
5 radic_
5 25 radic_
3
6 π 2 10 radic_
75
If real numbers a b and c are in order from least to greatest what is the order
of their opposites from least to greatest
Explain
Math TalkMathematical Practices
8NS2
radic_
3 radic_
5 25
radic_
75 π2 10
Math Talk answer -c -b -a -c is farthest to the left on a number line -b is in the middle and -a is farthest to the right
Unit 122
copy H
ough
ton
Miff
lin H
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hing
Com
pany
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
8_MCAAESE206984_U1M01L3indd 22 41613 447 AM
My Notes
Math TrainerOnline Practice
and Help
Personal
myhrwcom
Math On the Spot
myhrwcom
Comparing Irrational NumbersBetween any two real numbers is another real number To compare and order real numbers you can approximate irrational numbers as decimals
Compare radic_
3 + 5 3 + radic_
5 Write lt gt or =
First approximate radic_
3
radic_
3 is between 1 and 2
Next approximate radic_
5
radic_
5 is between 2 and 3
Then use your approximations to simplify the expressions
radic_
3 + 5 is between 6 and 7
3 + radic_
5 is between 5 and 6
So radic_
3 + 5 gt 3 + radic_
5
Reflect1 If 7 + radic
_ 5 is equal to radic
_ 5 plus a number what do you know about the
number Why
2 What are the closest two integers that radic_
300 is between
EXAMPLEXAMPLE 1
STEP 1
STEP 2
Compare Write lt gt or =
YOUR TURN
3 radic_
2 + 4 2 + radic_
4 4 radic_
12 + 6 12 + radic_
6
L E S S O N
13 Ordering Real Numbers
ESSENTIAL QUESTIONHow do you order a set of real numbers
8NS2
Use rational approximations of irrational numbers to compare the size of irrational numbers locate them approximately on a number line diagram and estimate the value of expressions (eg π 2 )
8NS2
Use perfect squares to estimate square roots
1 2 = 1 2 2 = 4 3 2 = 9
The number is 7 both expressions must equal 7 + radic_
5
17 and 18
gt lt
21Lesson 13
copy H
ough
ton
Miff
lin H
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ublis
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pany
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
8_MCAAESE206984_U1M01L3indd 21 41913 246 PM
PROFESSIONAL DEVELOPMENT
Math BackgroundIn this lesson students estimate irrational numbers in the form of square roots of nonper-fect squares by finding two perfect squares between which the number falls A more precise method involves repeated division For example to find radic
_ 28 find a whole number whose perfect
square is close to 28 such as 5 Divide 28 by that number 28 divide 5 = 56 Find the average of the quotient and divisor 5 + 56
_____ 2 = 53 Continue dividing 28 by each result and averaging until you get the desired accuracy
Integrate Mathematical Practices MP4
This lesson provides an opportunity to address this Mathematical Practices standard It calls for students to model relationships using multiple representations including diagrams graphs and language as appropriate Students use multiple representations when they use number lines to estimate the locations of and order rational and irrational numbers given as symbols
Ordering Real Numbers 22
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
Interactive Whiteboard Interactive example available online
ADDITIONAL EXAMPLE 3The diameter of a meteorite in millimeters is calculated by four different methods Order the results from least to greatest
Joe radic_
18 mm Lisa 13 __ 3 mm
Pablo 46 mm Julien 4π __ 3 mm
Julien 4π __ 3 mm Lisa 13 __ 3 mm
Joe radic_
18 mm Pablo 46 mm
EXAMPLE 3Questioning Strategies Mathematical Practices bull How can you verify that radic
_ 28 is between 52 and 53 5 2 2 = 2704 and 5 3 2 = 2809
bull Explain how to determine which number is greater 5 _
5 or 55 When the repeating decimal is rounded to the nearest tenth or hundredth you can see that it is greater
Connect to Daily LifeDiscuss how measuring across a canyon might involve different methods than measuring along a road Explain that measurements like these are often done using calculations that approximate the distance
YOUR TURNFocus on Critical Thinking Mathematical PracticesDiscuss with students which number is greater 3
_ 45 or 3450 3
_ 45 or 3455 and why Explain
that 3 _
45 can be written out as 34545hellipMake sure they understand that 3 _
45 is greater than 345 but less than 3455
ElaborateTalk About ItSummarize the Lesson
Ask How can you order two numbers in different forms whose decimal approxi-mations appear to be equal Approximate one or both numbers to an additional
number of decimal places
GUIDED PRACTICEEngage with the Whiteboard
Have students place and label additional points on the number line in Exercise 9 Allow the points to be in any format other than decimal
Avoid Common ErrorsExercises 3ndash4 Caution students to read the problem carefully so that they do not misread the problem as the same numbers combined by addition on each side of the circleExercise 10 Remind students that the calculations have units
myhrwcom
23 Lesson 13
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
0 05 1 15 2 25 3 35 4 45 5 55 6 65 7
2πradic3
Compare Write lt gt or = (Example 1)
1 radic_
3 + 2 radic_
3 + 3 2 radic_
8 + 17 radic_
11 + 15
3 radic_
6 + 5 6 + radic_
5 4 radic_
9 + 3 9 + radic_
3
5 radic_
17 - 3 -2 + radic_
5 6 12 - radic_
2 14 - radic_
8
7 radic_
7 + 2 radic_
10 - 1 8 radic_
17 + 3 3 + radic_
11
9 Order radic_
3 2π and 15 from least to greatest Then graph them on the number line (Example 2)
radic_
3 is between and so radic_
3 asymp
π asymp 314 so 2π asymp
From least to greatest the numbers are
10 Four people have found the perimeter of a forest using different methods Their results are given in the table Order their calculations from greatest to least (Example 3)
11 Explain how to order a set of real numbers
CHECK-INESSENTIAL QUESTION
Forest Perimeter (km)
Leon Mika Jason Ashley
radic_
17 - 2 1 +thinsp π __ 2 12 ___ 5 25
Guided Practice
17
15
1 + π _ 2 km 25 km 12 __ 5 km radic_
17 - 2 km
2π radic
_ 3
18 175
628
Sample answer Convert each number to a decimal
equivalent using estimation to find equivalents for
irrational numbers Graph each number on a number line
Read the numbers from left to right for least to greatest
Read the numbers from right to left for greatest to least
lt gt
lt lt
ltgt
gt gt
24 Unit 1
copy H
ough
ton
Miff
lin H
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ublis
hing
Com
pany
bull Im
age C
redi
ts copy
Elena
Eliss
eeva
Alam
y Im
ages
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
8_MCAAESE206984_U1M01L3indd 24 41613 448 AM
My Notes
5 52 54 56 58 6
radic28 5 12
23455
Math TrainerOnline Practice
and Help
Personal
myhrwcom
Math On the Spot
myhrwcom
Ordering Real Numbers in a Real-World Context Calculations and estimations in the real world may differ It can be important to know not only which are the most accurate but which give the greatest or least values depending upon the context
Four people have found the distance in kilometers across a canyon using different methods Their results are given in the table Order the distances from greatest to least
Distance Across Quarry Canyon (km)
Juana Lee Ann Ryne Jackson
radic_
28 23 __ 4 5 _
5 5 1 _ 2
Write each value as a decimal
radic_
28 is between 52 and 53 Since 53 2 = 2809 an approximate value for radic
_ 28 is 53
23 __ 4 = 575
5 _
5 is 5555hellip so 5 _
5 to the nearest hundredth is 556
5 1 _ 2 = 55
Plot radic_
28 23 __ 4 5 _
5 and 5 1 _ 2 on a number line
From greatest to least the distances are
23 __ 4 km 5 _
5 km 5 1 _ 2 km radic_
28 km
EXAMPLEXAMPLE 3
STEP 1
STEP 2
7 Four people have found the distance in miles across a crater using different methods Their results are given below
Jonathan 10 __ 3 Elaine 3 _
45 Joseacute 3 1 _ 2 Lashonda radic_
10
Order the distances from greatest to least
YOUR TURN
8NS2
3 1 _ 2 mi 3 _
45 mi 10 __ 3 mi radic_
10 mi
23Lesson 13
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ough
ton
Miff
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pany
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8_MCAAESE206984_U1M01L3indd 23 41613 447 AM
ModelingPlace papers around the room with the numbers from 1 to 5 one per sheet Give each student a card showing a number between 1 and 5 in different forms Have students place his or her card between the correct integers and decide where the number goes in relation to any numbers already placed
Multiple RepresentationsGive students a vertical number line which some students might find easier to use than a horizontal one Have them decide whether to place points for rational and irrational numbers above or below existing points
Additional ResourcesDifferentiated Instruction includes bull Reading Strategies bull Success for English Learners EL
bull Reteach bull Challenge PRE-AP
DIFFERENTIATE INSTRUCTION
Ordering Real Numbers 24
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
Math TrainerOnline Assessment
and Intervention
Personal
Online homework assignment available
myhrwcom
myhrwcom
Lesson Quiz available online
13 LESSON QUIZ
1 Compare Write lt gt or =
radic_
95 - 5 radic_
62 - 2
2 Order 105 radic_
105 and 3π + 1 from greatest to least
3 A length in centimeters is calculated differently by four different people Order their calculations from least to greatest
KD 11 __ 2 cm Silvio 5 __ 3 π cm
Paula 5 _
4 cm Luis radic_
33 cm
EvaluateGUIDED AND INDEPENDENT PRACTICE
Concepts amp Skills Practice
Example 1Comparing Irrational Numbers
Exercises 1ndash8
Example 2Ordering Real Numbers
Exercises 9 12ndash15 18ndash21
Example 3Ordering Real Numbers in a Real-World Context
Exercises 10 16ndash17
Additional ResourcesDifferentiated Instruction includes bull Leveled Practice worksheets
Focus | Coherence | Rigor
Exercise Depth of Knowledge (DOK) Mathematical Practices
12ndash15 1 Recall of Information MP5 Using Tools
16 2 SkillsConcepts MP2 Reasoning
17 2 SkillsConcepts MP6 Precision
18ndash21 2 SkillsConcepts MP2 Reasoning
22 3 Strategic Thinking MP4 Modeling
23ndash24 3 Strategic Thinking MP3 Logic
8NS2
8NS2
Answers1 radic
_ 95 - 5 lt radic
_ 62 - 2
2 radic_
105 3π + 1 105
3 Silvio 5 __ 3 π cm Paula 5 _
4 cm
KD 11
__ 2 cm Luis radic_
33 cm
25 Lesson 13
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
Work Area
3140 3141 3142 3143
314 π227
20 A teacher asks his students to write the numbers shown in order from least to greatest Paul thinks the numbers are already in order Sandra thinks the order should be reversed Who is right
21 Math History There is a famous irrational number called Eulerrsquos number symbolized with an e Like π its decimal form never ends or repeats The first few digits of e are 27182818284
a Between which two square roots of integers could you find this number
b Between which two square roots of integers can you find π
22 Analyze Relationships There are several approximations used for π including 314 and 22 __ 7 π is approximately 314159265358979
a Label π and the two approximations on the number line
b Which of the two approximations is a better estimate for π Explain
c Find a whole number x so that the ratio x ___ 113 is a better estimate for π
than the two given approximations
23 Communicate Mathematical Ideas If a set of six numbers that include both rational and irrational numbers is graphed on a number line what is the fewest number of distinct points that need to be graphed Explain
24 Critique Reasoning Jill says that 12 _
6 is less than 1263 Explain her error
FOCUS ON HIGHER ORDER THINKING
radic_
115 115 ___ 11 and 105624
between radic_
7 asymp 265 and radic_
8 asymp 283
between radic_
9 = 3 and radic_
10 asymp 316
22 __ 7 it is closer to π on the number line
She did not consider the repeating digit 1266
2 rational numbers can have the same location and
irrational numbers can have the same location but they
cannot share a location
355
Neither student is correct The answer
should be 115 ___ 11 105624 radic_
115
Unit 126
copy H
ough
ton M
ifflin
Har
cour
t Pub
lishin
g Com
pany
Imag
e Cre
dits
copy3D
Stoc
kiSt
ockP
hoto
com
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
8_MCAAESE206984_U1M01L3indd 26 210513 801 AM
Personal Math Trainer
Online Practice and Helpmyhrwcom
Name Class Date
Independent Practice
16 Your sister is considering two different shapes for her garden One is a square with side lengths of 35 meters and the other is a circle with a diameter of 4 meters
a Find the area of the square
b Find the area of the circle
c Compare your answers from parts a and b Which garden would give your sister the most space to plant
17 Winnie measured the length of her fatherrsquos ranch four times and got four different distances Her measurements are shown in the table
a To estimate the actual length Winnie first approximated each distance to the nearest hundredth Then she averaged the four numbers Using a calculator find Winniersquos estimate
b Winniersquos father estimated the distance across his ranch to be radic_
56 km How does this distance compare to Winniersquos estimate
Give an example of each type of number
18 a real number between radic_
13 and radic_
14
19 an irrational number between 5 and 7
Order the numbers from least to greatest
12 radic_
7 2 radic_
8 ___ 2 13 radic_
10 π 35
14 radic_
220 -10 radic_
100 115 15 radic_
8 -375 3 9 _ 4
Distance Across Fatherrsquos Ranch (km)
1 2 3 4
radic_
60 58 __ 8 7 _
3 7 3 _ 5
138NS2
radic_
8 ___ 2 2 radic_
7
-10 radic_
100 115 radic_
220
radic_
60 asymp 775 58 __ 8 = 725 7 _
3 asymp 733 7 3 _ 5 = 760 so the average
π radic_
10 35
-375 9 _ 4 radic_
8 3
is 74825 km
1225 m2
4π m2 or approximately 126 m2
They are nearly identical radic_
56 is approximately 74833hellip
The circle would give her more space to plant because it has a
larger area
Sample answer 37
Sample answer radic_
31
25Lesson 13
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ough
ton
Miff
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hing
Com
pany
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
8_MCAAESE206984_U1M01L3indd 25 41613 448 AM
Activity available online myhrwcomEXTEND THE MATH PRE-AP
Activity Have students investigate whether there are infinitely many numbers between two numbers by giving examples for each of the following
bull Between any two rational numbers there is at least one other rational number Sample answer 45 is between 41 and 48
bull Between any two irrational numbers there is at least one rational number Sample answer 45 is between radic
_ 11 and radic
_ 29
bull Between any two rational numbers there is at least one irrational number Sample answer radic
_ 11 is between 31 and 36
bull Between any two irrational numbers there is at least one irrational number Sample answer radic
_ 17 is between radic
_ 11 and radic
_ 29
Ordering Real Numbers 26
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
ReadyMath Trainer
Online Practiceand Help
Personal
myhrwcom
Module Quiz
11ensp RationalenspandenspIrrationalenspNumbersWrite each fraction as a decimal or each decimal as a fraction
1 7__20 2 1___
27 3 17_8
Solve each equation for x
4 x2=81 5 x3=343 6 x2= 1___100
7 Asquarepatiohasanareaof200squarefeetHowlongiseachside
ofthepatiotothenearesttenth
12ensp SetsenspofenspRealenspNumbersWrite all names that apply to each number
8 121____radic
____121
9 π__2
10 TellwhetherthestatementldquoAllintegersarerationalnumbersrdquoistrueorfalseExplainyourchoice
13ensp OrderingenspRealenspNumbersCompare Write lt gt or =
11 radic__
8+3 8+radic__
3 12 radic__
5+11emsp emsp emsp 5+radic___
11
Order the numbers from least to greatest
13 radic___
99π29__
8 14 radic___
1__251_40__
2
15 Howarerealnumbersusedtodescribereal-worldsituations
ESSENTIAL QUESTION
035
9-9
141ft
7 1__10- 1__10
14__11 1875
wholeintegerrationalreal
Trueintegerscanbewrittenasthequotientoftwointegers
SampleanswerRealnumberssuchastherational
π29__
8radic___
99
irrationalreal
lt gt
number1_4candescribeamountsusedincooking
radic___
1__250__
21_4
27Module1
copy H
ough
ton
Miff
lin H
arco
urt P
ublis
hing
Com
pany
DONOTEDIT--ChangesmustbemadethroughldquoFileinfordquoCorrectionKey=A
8_MCAAESE206984_U1M01RTindd 27 41513 1113 PM
Math TrainerOnline Assessment
and Intervention
Personal
myhrwcom
1
2
3 Response toIntervention
Intervention Enrichment
Access Ready to Go On assessment online and receive instant scoring feedback and customized intervention or enrichment
Online and Print Resources
Differentiated Instruction
bull Reteach worksheets
bull Reading Strategies EL
bull Success for English Learners EL
Differentiated Instruction
bull Challenge worksheets PRE-AP
Extend the Math PRE-AP
Lesson Activities in TE
Additional ResourcesAssessment Resources includes bull Leveled Module Quizzes
Ready to Go OnAssess MasteryUse the assessment on this page to determine if students have mastered the concepts and standards covered in this module
California Common Core Standards
Lesson Exercises Common Core Standards
11 1ndash7 8NS1 8NS2 8EE2
12 8ndash10 8NS1
13 11ndash14 8NS2
27 Unit 1 Module 1
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
Personal Math Trainer
Online Practice and HelpmyhrwcomAssessment Readiness
Module 1 MIXed ReVIeW
1 Look at each number Is the number between 2π and radic___
52
Select Yes or No for expressions AndashC
A 6 2 _ 3 Yes No
B 5π __ 2 Yes No
C 3 radic__
5 Yes No
2 Consider the number - 11 __ 15
Choose True or False for each statement
A The number is rational True False
B The number can be written as True Falsea repeating decimal
C The number is less than ndash08 True False
3 The volume of a cube is given by V = x3 where x is the length of an edge of the cube A cube-shaped end table has a volume of 3 3 _ 8 cubic feet What is the length of an edge of the end table Explain how you solved this problem
4 A student says that radic___
83 is greater than 29 __ 3 Is the student correct Justify your
reasoning
1 1 _ 2 ft Sample answer The equation x3 = 3 3 _ 8 can be used
to find the edge length in feet To solve the equation
write the mixed number as a fraction greater than 1
x3 = 27 __ 8 Then take the cube root of both sides x = 3 _ 2 = 1 1 _ 2
No Sample answer radic___
83 asymp 91 and 29 __ 3 = 9
__ 6
Because 91 lt 9 __
6 radic___
83 lt 29 __ 3
28 Unit 1
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pany
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8_MCAAESE206984_U1M01RTindd 28 240413 946 AM
Personal Math Trainer
Online Assessment and
Interventionmyhrwcom
Scoring GuideItem 3 Award the student 1 point for finding the edge length of the cube and 1 point for correctly explaining how to use a cube root to solve the problem
Item 4 Award the student 1 point for determining that the student is incorrect and 1 point for correctly justifying the reasoning for this conclusion
Additional ResourcesTo assign this assessment online login to your Assignment Manager at myhrwcom
Assessment Readiness
California Common Core Standards
Items Grade 8 Standards Mathematical Practices
1 8NS2 MP7
2 7NS2b 7NS2d 8NS1 MP7
3 8EE2 MP1 MP4
4 8NS1 8NS2 MP3
Item integrates mixed review concepts from previous modules or a previous course
Item 4 combines concepts from the California Common Core cluster ldquoKnow that there are numbers that are not rational and approximate them by rational numbersrdquo
Real Numbers 28
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
Math Talk
Language Support EL
PROFESSIONAL DEVELOPMENT
Linguistic Support EL
AcademicContent Vocabulary
square ndash In this lesson the word square has multiple meanings which can cause confusion For example to square as in to take the square root of a number is a verb It is different from the nouns square or square of a number The text also refers to perfect square and principal square root of a number and the square root symbol is used These different usages of square as a mathematical term need to be clarified Sentence frames can be used to help define the meaning
To square a number means to _______The perfect square of a number means _______
Background Knowledge
suffixes ndash When added to a root word the suffix -th is used in math to indicate one of a specified number of parts such as tenth hundredth or thousandth Remind students that the suffix -th also indicates place value Note that Spanish Vietnamese Mandarin and other languages do not have the ending th sound so teachers need to enunciate carefully
cognates ndash The words terminating and terminal used in this lesson are cognates in Spanish terminar meaning ldquoto endrdquo or ldquoto finishrdquo A Spanish cognate for approximate is aproximar
Leveled Strategies for English Learners
Emerging Use cards with root words ten hundred and thousand and a card with the -th suffix Have students place them together to show place value Then complete a sentence Use the same procedure to identify decimals
Expanding Support students at this level of English proficiency by providing sentence frames for them to use to describe their mathematical reasoning
To write the fraction _______ as a decimal I _______
Bridging Have students identify different meanings of the term square by matching examples of math problems with a written out sentence frame that defines the usage of the term square to square a number perfect square square root Use this procedure also with the term cube
Be sure to clarify the different uses of the term square when referring to square roots perfect squares and so on
EL
California ELD Standards
Emerging 2I12b Selecting language resources ndash Use knowledge of morphology to appropriately select affixes in basic ways
Expanding 2I12b Selecting language resources ndash Use knowledge of morphology to appropriately select affixes in a growing number of ways to manipulate language
Bridging 2I12b Selecting language resources ndash Use knowledge of morphology to appropriately select affixes in a variety of ways to manipulate language
Rational and Irrational Numbers 7B
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
11L E S S O N
Rational and Irrational Numbers
EngageESSENTIAL QUESTION
How do you rewrite rational numbers and decimals take square roots and cube roots and approximate irrational numbers To express as a decimal divide the numerator by the denominator To take a square root or cube root of a number find the number that when squared or cubed equals the original number To approximate an irrational number estimate a number between two consecutive perfect squares
Motivate the LessonAsk Which type of rational number do you see more often fractions or decimals Which do you prefer to use Why
ExploreHave students write examples of ratios and then share with the class the various notations for ratios that they used (for example 25 2 to 5 2 __ 5 ) Point out the connection between the word ratio and the meaning of rational number See also Explore Activity in student text
ExplainEXAMPLE 1
Questioning Strategies Mathematical Practices bull How does the denominator of a fraction in simplest form tell whether the decimal equivalent of the fraction is a terminating decimal The decimal will terminate if the denominator is an even number a multiple of 5 or a multiple of 10
Avoid Common ErrorsTo avoid interpreting 1 __ 4 as 4 divided by 1 tell students to start at the top of the fraction and read the bar as ldquodivided byrdquo
YOUR TURNTalk About ItCheck for Understanding
Ask Can an improper fraction be written as a decimal Give an example to support your answer Yes 5 __ 4 = 125
EXAMPLE 2Questioning Strategies Mathematical Practices bull How can you use place value to write a terminating decimal as a fraction with a power of ten in the denominator Start by identifying the place value of the decimals last digit and then use the corresponding power of 10 as the denominator of the fraction
bull How can you tell if a decimal can be written as a rational number If the decimal is a terminating or repeating decimal then it can be written as a rational number
Interactive Whiteboard Interactive example available online
ADDITIONAL EXAMPLE 1Write each fraction as a decimal
A 2 _ 5
04 B 5 _ 9
0 _
5
myhrwcom
Interactive Whiteboard Interactive example available online
ADDITIONAL EXAMPLE 2Write each decimal as a fraction in simplest form
A 0355 71 ___ 200
B 0 _
43 43 __ 99
myhrwcom
CA Common CoreStandards
The student is expected to
The Number Systemmdash8NS1
Know that numbers that are not rational are called irrational Understand informally that every number has a decimal expansion for rational numbers show that the decimal expansion repeats eventually and convert a decimal expansion which repeats eventually into a rational number
The Number Systemmdash8NS2
Use rational approximations of irrational numbers to compare the size of irrational numbers locate them approximately on a number line diagram and estimate the value of expressions (eg π 2 )
Expressions and Equationsmdash8EE2
Use square root and cube root symbols to represent solutions to equations of the form x 2 = p and x 3 = p where p is a positive rational number Evaluate square roots of small perfect squares and cube roots of small perfect cubes Know that radic
_ 2 is irrational
Mathematical Practices
MP6 Precision
The student is expected to
the value of expressions (eg
7 Lesson 11
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
My Notes
Math On the Spotmyhrwcom
Math TrainerOnline Practice
and Help
Personal
myhrwcom
Expressing Decimals as Rational NumbersYou can express terminating and repeating decimals as rational numbers
Write each decimal as a fraction in simplest form
0825
The decimal 0825 means ldquo825 thousandthsrdquo Write this as a fraction
825 ____ 1000
Then simplify the fraction
825 divide 25 ________ 1000 divide 25 = 33 __ 40
0825 = 33 __ 40
0 _
37
Let x = 0 _
37 The number 0 _
37 has 2 repeating digits so multiply each side of the equation x = 0
_ 37 by 10 2 or 100
x = 0 _
37
(100)x = 100(0 _
37 )
100x = 37 _
37
Because x = 0 _
37 you can subtract x from one side and 0 _
37 from the other
100x = 37 _
37
minusx minus0 _
37
99x = 37
Now solve the equation for x Simplify if necessary
99x ___ 99 = 37 __ 99
x = 37 __ 99
EXAMPLE 2
A
B
Write each fraction as a decimal
YOUR TURN
1 5 __ 11 2 1 _ 8 3 2 1 _ 3
8NS1
To write ldquo825 thousandthsrdquo put 825 over 1000
Divide both the numerator and the denominator by 25
100 times 0 _
37 is 37 _
37
37 _
37 minus 0 _
37 is 37
Divide both sides of the equation by 99
0 _
45 0125 2 _
3
Unit 18
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ough
ton
Miff
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hing
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pany
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8_MCAAESE206984_U1M01L1indd 8 120413 838 PM
My Notes
Math On the Spot
myhrwcom
= 033333333333331mdash3
ESSENTIAL QUESTION
Expressing Rational Numbers as DecimalsA rational number is any number that can be written as a ratio in the form a _ b where a and b are integers and b is not 0 Examples of rational numbers are 6 and 05
6 can be written as 6 _ 1 05 can be written as 1 _ 2
Every rational number can be written as a terminating decimal or a repeating decimal A terminating decimal such as 05 has a finite number of digits A repeating decimal has a block of one or more digits that repeat indefinitely
Write each fraction as a decimal
1 _ 4
1 _ 4 = 025
1 _ 3
1 _ 3 = 0 _
3
EXAMPLEXAMPLE 1
A
B
0333 3 ⟌ ⎯ 1000 minus9 10 minus9 10 minus9 1
025 4 ⟌ ⎯ 100 -8 20 -20
0
L E S S O N
11Rational and Irrational Numbers
How do you rewrite rational numbers and decimals take square roots and cube roots and approximate irrational numbers
8NS1
Know that numbers that are not rational are called irrational Understand informally that every number has a decimal expansion for rational numbers show that the decimal expansion repeats eventually and convert a decimal expansion which repeats eventually into a relation number Also 8NS2 8EE2
8NS1
Remember that the fraction bar means ldquodivided byrdquo Divide the numerator by the denominator
Divide until the remainder is zero adding zeros after the decimal point in the dividend as needed
Divide until the remainder is zero or until the digits in the quotient begin to repeat
Add zeros after the decimal point in the dividend as needed
When a decimal has one or more digits that repeat indefinitely write the decimal with a bar over the repeating digit(s)
7Lesson 11
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ough
ton
Miff
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pany
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8_MCABESE206984_U1M01L1indd 7 11113 128 AM
PROFESSIONAL DEVELOPMENT
Math BackgroundSome decimals may have a pattern but still not be a repeating decimal that is rational For example in 312112111211112hellip you can predict the next digit and describe the pattern (There is one more 1 each time before the 2) However this is not a terminating decimal nor is it a repeating decimal and it is therefore NOT a rational number
Integrate Mathematical Practices MP6
This lesson provides an opportunity to address this Mathematical Practices standard It calls for students to attend to precision Students learn to express rational numbers accurately and precisely in both fractional and decimal forms and learn to translate from one form to the other They also learn how to precisely represent and communicate ideas about irrational numbers square roots and cube roots
Rational and Irrational Numbers 8
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=B
Focus on Technology Mathematical PracticesPoint out the importance of entering a repeating decimal correctly when using a graphing calculator to convert the decimal to a fraction The decimal 0
_ 59 must be entered as
0595959595959 not 059
YOUR TURNFocus on Math ConnectionsMake sure students understand that the place value of the last digit in Exercises 4 and 6 determines the denominator of the corresponding fraction or mixed number So for Exercise 4 the place value hundredths gives a denominator of 100 and for Exercise 6 the place value tenths gives a denominator of 10
EXAMPLE 3Questioning Strategies Mathematical Practices bull How can a solution of an equation of the form x 2 = p be negative if p is a positive number Since the square of a negative number is positive a negative number is also a solution of x 2 equals a positive number
bull When is a solution of an equation of the form x 3 = p larger than p The solution is larger than p if p is a number between 0 and 1
Focus on Math Connections Make sure students understand the difference in finding radic
_ 121 and solving x 2 = 121 The
symbol radic_
indicates the positive or principal square root only while the equation x 2 = 121 has two roots the principal square root and its opposite
YOUR TURNAvoid Common ErrorsTo avoid sign errors in Exercise 9 make sure that students understand that the cube of a negative number is not a positive number Therefore -8 is not a solution of x 3 = 512
Talk About ItCheck for Understanding
Ask Kris predicts that there are two real solutions for Exercises 7 and 8 and that there are three real solutions for Exercises 9 and 10 Is his prediction correct
Explain His prediction is correct for Exercises 7 and 8 because there are two numbers whose squares are the same positive number given in the exercises His prediction is not correct for Exercises 9 and 10 however because there is only one real number whose cube is the same positive number given in the exercises
EXPLORE ACTIVITYQuestioning Strategies Mathematical Practices bull Compare the values for 13 2 and 13 2 The digits are the same but 13 2 has two decimal places (169) while 13 2 has none (169)
bull How do you know whether radic_
2 will be closer to 1 or closer to 2 It will be closer to 1 because 2 is between the perfect squares of 1 and 4 but closer to 1 than it is to 4
Connect Vocabulary EL
Explain to students that the word irrational when used as an ordinary word in English means without logic or reason In mathematics when we say that a number is irrational it means only that the number cannot be written as the quotient of two integers
Engage with the WhiteboardHave students extend the number line in both directions and label the locations of the whole numbers 1 and 2 These are the roots of the consecutive perfect squares
1 and 4 used to estimate radic_
7
Interactive Whiteboard Interactive example available online
ADDITIONAL EXAMPLE 3Solve each equation for x
A x 2 = 324 18 -18
B x 2 = 25 ___ 144 5 __ 12 - 5 __ 12
C 343 = x 3 7
D x 3 = 125 ___ 512 5 __ 8
myhrwcom
9 Lesson 11
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
Math TrainerOnline Practice
and Help
Personal
myhrwcom
EXPLORE ACTIVITY
lt 2 lt
radic_
lt radic
_ 2 lt
radic_
lt radic
_ 2 lt
The solution is 9
The solution is 2 _ 5
C
D
729 = x 3
3 radic_ 729 = 3 radic
_ x 3
3 radic_ 729 = x
9 = x
x 3 = 8 ___ 125
3 radic_
x 3 =thinsp 3 radic_ 8 ___ 125
x =thinsp 3 radic_ 8 ___ 125
x = 2 _ 5
Solve each equation for x
YOUR TURN
7 x 2 = 196 8 x 2 = 9 ___ 256
9 x 3 = 512 10 x 3 = 64 ___ 343
Estimating Irrational NumbersIrrational numbers are numbers that are not rational In other words they cannot be written in the form a _ b where a and b are integers and b is not 0 Square roots of perfect squares are rational numbers Square roots of numbers that are not perfect squares are irrational Some equations like those in Example 3 involve square roots of numbers that are not perfect squares
x 2 = 2 x = plusmn radic_
2
Estimate the value of radic_
2
Find two consecutive perfect squares that 2 is between Complete the inequality by writing these perfect squares in the boxes
Now take the square root of each number
Simplify the square roots of perfect squares
radic_
2 is between and
A
B
C
8NS2 8EE2
Solve for x by taking the cube root of both sides
Solve for x by taking the cube root of both sides
Apply the definition of cube root
Think What number cubed equals 729
Apply the definition of cube root
Think What number cubed equals 8 ____ 125
radic_
2 is irrational
x = plusmn14 x = plusmn 3 __ 16
x = 8 x = 4 _ 7
1 2
1 4
1 4
1 2
Unit 110
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Miff
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Com
pany
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
8_MCAAESE206984_U1M01L1indd 10 41613 1211 AM
Math TrainerOnline Practice
and Help
Personal
myhrwcom
Math On the Spot
myhrwcom
Write each decimal as a fraction in simplest form
YOUR TURN
Finding Square Roots and Cube RootsThe square root of a positive number p is x if x 2 = p There are two square roots for every positive number For example the square roots of 36 are 6 and minus6 because 6 2 = 36 and (minus6) 2 = 36 The square roots of 1 __ 25 are 1 _ 5 and minus 1 _ 5 You can write the square roots of 1 __ 25 as plusmn 1 _ 5 The symbol radic
_ 5 indicates the positive
or principal square root
A number that is a perfect square has square roots that are integers The number 81 is a perfect square because its square roots are 9 and minus9
The cube root of a positive number p is x if x 3 = p There is one cube root for every positive number For example the cube root of 8 is 2 because 2 3 = 8 The cube root of 1 __ 27 is 1 _ 3 because ( 1 _ 3 )
3
= 1 __ 27 The symbol 3 radic_ 1 indicates the
cube root
A number that is a perfect cube has a cube root that is an integer The number 125 is a perfect cube because its cube root is 5
Solve each equation for x
The solutions are 11 and minus11
The solutions are 4 __ 13 and minus 4 __ 13
EXAMPLEXAMPLE 3
A x 2 = 121
x 2 = 121
x = plusmn radic_
121
x = plusmn11
B x 2 = 16 ___ 169
x 2 = 16 ___ 169
x = plusmn radic_
16 ___ 169
x = plusmn 4 __ 13
4 012 5 0 _
57 6 14
Can you square an integer and get a negative number
What does this indicate about whether negative
numbers have square roots
Math TalkMathematical Practices
8EE2
Solve for x by taking the square root of both sides
Apply the definition of square root
Think What numbers squared equal 121
Solve for x by taking the square root of both sides
Apply the definition of square root
Think What numbers squared equal 16 ____ 169
3 __ 25 19 __ 33 1 2 _ 5
No the square of a positive integer is positive the square of a negative integer is positive and the square of 0 is 0 So negative numbers do not have (real) square roots
9Lesson 11
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pany
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
8_MCAAESE206984_U1M01L1indd 9 41913 240 PM
Critical ThinkingIn the Explore Activity students estimated the location of radic
_ 2 on a number line Ask students
whether they think that it is possible to locate more precisely the point that represents radic
_ 2 In
other words can you graph irrational numbers exactly on a number line along with rational numbers Students should understand that radic
_ 2
is a real number and all real numbers can be located on a real number line A more precise estimate will allow more precise placement on a number line
The Modeling note tells one way to do this
ModelingHave students use a ruler to represent a number line with a unit that is one inch long Have them draw a square with a side of one inch and draw the diagonal to make two isosceles triangles Lead students to understand that the length of the diagonal (or hypotenuse) is radic
_ 2
Have them copy the length of their diagonal onto their ruler or number line starting at zero The end point of the diagonal represents the exact point for the irrational number radic
_ 2 on a
number line
Additional ResourcesDifferentiated Instruction includes bull Reading Strategies bull Success for English Learners EL
bull Reteach bull Challenge PRE-AP
DIFFERENTIATE INSTRUCTION
Rational and Irrational Numbers 10
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
ElaborateTalk About ItSummarize the Lesson
Ask If someone claims that a certain number is irrational but you know it is actually rational how could you prove to that person that the number is rational
You could find a fraction equal to the number such that the number is the ratio of two integers with the denominator not equal to zero
GUIDED PRACTICEEngage with the Whiteboard
Have students plot each number in Exercises 16ndash18 on a number line Students should label each point with the irrational number written as a radical and as a
decimal
Avoid Common ErrorsExercises 1ndash6 To avoid reversing the order of the dividend and divisor tell students to start at the top of the fraction and read the bar as ldquodivided byrdquo
Focus on TechnologyHave students use a calculator to investigate the decimal equivalents of such fractions as 1 __ 9 2 __ 9 8 __ 9 and 1 __ 11 2 __ 11 10
__ 11 Ask them to describe the patterns they find as a result of these investigations
11 Lesson 11
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
Guided Practice
7 0675 8 56 9 044
10 0 _
4
10x =
x =
11 0 _
26
100x =
x =
12 0 _
325
1000x =
x =
Solve each equation for x (Example 3 and Explore Activity)
- x
-
_______________
x =
- x
-
___________________
x =
- x
-
_______________________
x =
Write each fraction or mixed number as a decimal (Example 1)
1 2 _ 5 2 8 _ 9 3 3 3 _ 4
4 7 __ 10 5 2 3 _ 8 6 5 _ 6
Write each decimal as a fraction or mixed number in simplest form (Example 2)
13 x 2 = 17 14 x 2 = 25 ___ 289 15 x 3 = 216
Approximate each irrational number to one decimal place without a calculator
x = plusmn radic__
asymp plusmn x = 3
radic__
=
(Explore Activity)
16 radic_
5 asymp
17 radic_
3 asymp
18 radic_
10 asymp
19 What is the difference between rational and irrational numbers
CHECK-INESSENTIAL QUESTION
x = plusmn radic__
__________ = plusmn _____
4 _
4
0 _
4
4 99
6216
269
41 25 5
17289
17
22 17 32
04
07
27__40
26 __ 99 325 ___ 999 4 _ 9
11__255 3_5
0 _
8
2375
375
08 _
3
26 _
26
0 _
26
325 _
325
0 _
325
999 325
Rational numbers can be written in the form a __ b where
a and b are integers and b ne 0 Irrational numbers cannot
be written in this form
Unit 112
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pany
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
8_MCAAESE206984_U1M01L1indd 12 41613 1211 AM
11 12 13 14 15
radic2 asymp 14
141 142 143 144 145
radic2 asymp 141
0 1 2 3 4
radic2 asymp 15
Estimate that radic_
2 asymp 15
To find a better estimate first choose some numbers between 1 and 2 and square them For example choose 13 14 and 15
1 3 2 = 1 4 2 = 1 5 2 =
Is radic_
2 between 13 and 14 How do you know
Is radic_
2 between 14 and 15 How do you know
2 is closer to than to so radic_
2 asymp
Locate and label this value on the number line
Reflect 11 How could you find an even better estimate of radic
_ 2
12 Find a better estimate of radic_
2
1 41 2 = 1 42 2 = 1 43 2 =
2 is closer to than to so radic_
2 asymp
Draw a number line and locate and label your estimate
13 Solve x 2 = 7 Write your answer as a radical expression Then estimate to one decimal place
D
E
F
No 2 is not between 169 and 196
Yes 2 is between 196 and 225
196
19881
19881
225
20164
20164
14
141
20449
169 196 225
Test the squares of numbers between 14 and 15
x = plusmn radic_
7 x asymp plusmn26
11Lesson 11
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ough
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Miff
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Com
pany
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
8_MCAAESE206984_U1M01L1indd 11 41613 1211 AM
Rational and Irrational Numbers 12
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
Math TrainerOnline Assessment
and Intervention
Personal
Online homework assignment available
myhrwcom
EvaluateGUIDED AND INDEPENDENT PRACTICE
Concepts amp Skills Practice
Example 1Expressing Rational Numbers as Decimals
Exercises 1ndash6 20ndash21 24ndash25
Example 2Expressing Decimals as Rational Numbers
Exercises 7ndash12 22ndash23 26ndash27
Example 3Finding Square Roots and Cube Roots
Exercises 13ndash15 28 30ndash31 35
Explore ActivityEstimating Irrational Numbers
Exercises 13 16ndash18 29 32ndash34
Additional ResourcesDifferentiated Instruction includes bull Leveled Practice worksheets
Lesson Quiz available online
11 LESSON QUIZ
1 Write as a decimal 2 5 __ 8 1 7 __ 12
2 Write as a fraction 034 1 _
24
3 Solve x 2 = 9 __ 49 for x
4 Solve x 3 = 216 for x
5 Estimate the value of radic_
13 to one decimal place without using a calculator
myhrwcom
Focus | Coherence | Rigor
Exercise Depth of Knowledge (DOK) Mathematical Practices
20ndash27 2 SkillsConcepts MP4 Modeling
28 3 Strategic Thinking MP4 Modeling
29ndash32 2 SkillsConcepts MP6 Precision
33 3 Strategic Thinking MP7 Using Structure
34 2 SkillsConcepts MP3 Logic
35 2 SkillsConcepts MP4 Modeling
36 3 Strategic Thinking MP3 Logic
37 3 Strategic Thinking MP7 Using Structure
38 3 Strategic Thinking MP2 Reasoning
8NS1 8NS2 8EE2
8NS1 8NS2 8EE2
Answers1 2625 158
_ 3
2 17 __ 50 1 8 __ 33
3 x = plusmn 3 __ 7
4 x = 6
5 36
13 Lesson 11
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
Work Area
33 Analyze Relationships To find radic_
15 Beau found 3 2 = 9 and 4 2 = 16 He said that since 15 is between 9 and 16 radic
_ 15 must be between 3 and 4 He
thinks a good estimate for radic_
15 is 3 + 4 ____ 2 = 35 Is Beaursquos estimate high low
or correct Explain
34 Justify Reasoning What is a good estimate for the solution to the equation x 3 = 95 How did you come up with your estimate
35 The volume of a sphere is 36π f t 3 What is the radius of the sphere Use the formula V = 4 _ 3 π r 3 to find your answer
36 Draw Conclusions Can you find the cube root of a negative number If so is it positive or negative Explain your reasoning
37 Make a Conjecture Evaluate and compare the following expressions
radic_
4 __ 25 and radic
_ 4 ____
radic_
25 radic
_
16 __ 81 and radic_
16 ____
radic_
81 radic
_
36 __ 49 and radic_
36 ____
radic_
49
Use your results to make a conjecture about a division rule for square roots Since division is multiplication by the reciprocal make a conjecture about a multiplication rule for square roots
38 Persevere in Problem Solving The difference between the solutions to the equation x 2 = a is 30 What is a Show that your answer is correct
FOCUS ON HIGHER ORDER THINKING
His estimate is low because 15 is very close to 16
so radic_
15 is very close to radic_
16 or 4 A better estimate
would be 38 or 39
Sample answer about 45 4 3 = 64 and 5 3 = 125
Because 95 is about halfway between 64 and 125 try 45
45 3 = 91125 which is a good estimate
3 feet
Yes the cube root of a negative number is negative
because a negative number cubed is always negative
and a nonnegative number cubed is always nonnegative
radic_
4 __ 25 = 2 _ 5 = radic
_ 4 ____
radic_
25 radic
_
16 __ 81 = 4 _ 9 = radic_
16 ____
radic_
81 radic
_
36 __ 49 = 6 _ 7 = radic_
36 ____
radic_
49
225 the solutions to x 2 = a are x = plusmn15 and
radic_
a ___
radic_
b = radic
_ a __
b radic
_ a radic
_ b = radic
_ a b
15 - (-15) = 30
Unit 114
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ough
ton
Miff
lin H
arco
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ublis
hing
Com
pany
bull copy
Ilen
e Mac
Dona
ldA
lamy I
mag
es
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=B
8_MCABESE206984_U1M01L1indd 14 102913 1142 PM
Personal Math Trainer
Online Practice and Helpmyhrwcom
Name Class Date
Independent Practice11
20 A 7 __ 16 -inch-long bolt is used in a machine What is this length written as a decimal
21 The weight of an object on the moon is 1 _ 6 its weight on Earth Write 1 _ 6 as a decimal
22 The distance to the nearest gas station is 2 4 _ 5 kilometers What is this distance written as a decimal
23 A baseball pitcher has pitched 98 2 _ 3 innings What is the number of innings written as a decimal
24 A heartbeat takes 08 second How many seconds is this written as a fraction
25 There are 262 miles in a marathon Write the number of miles using a fraction
26 The average score on a biology test was 72
_ 1 Write the average score using a
fraction
27 The metal in a penny is worth about 0505 cent How many cents is this written as a fraction
28 Multistep An artist wants to frame a square painting with an area of 400 square inches She wants to know the length of the wood trim that is needed to go around the painting
a If x is the length of one side of the painting what equation can you set up to find the length of a side How many solutions does the equation have
b Do all of the solutions that you found make sense in the context of the problem Explain
c What is the length of the wood trim needed to go around the painting
Solve each equation for x Write your answers as radical expressions Then estimate to one decimal place if necessary
29 x 2 = 14 30 x 3 = 1331
31 x 2 = 144 32 x 2 = 29
8NS1 8NS2 8EE2
04375 in 01 _6
28 km 98 _6 innings
x 2 = 400 x = plusmnthinsp20 the equation has 2 solutions
x = 20 makes sense but x = -20 doesnrsquot because a
painting cannot have a side length of -20 inches
4 times 20 = 80 inches
x = plusmn radic_
14 asymp plusmn37
x = plusmn radic_
144 = plusmn12 x = plusmn radic_
29 asymp plusmn54
x = 3 radic_ 1331 = 11
4_5 second 26 1_5 mi
72 1 _ 9 101 ___ 200 cent
13Lesson 11
copy H
ough
ton
Miff
lin H
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urt P
ublis
hing
Com
pany
bull copy
Phot
odisc
Get
ty Im
ages
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8_MCAAESE206984_U1M01L1indd 13 41613 1211 AM
myhrwcomActivity available onlineEXTEND THE MATH PRE-AP
Activity Write radic_
09 on the board and invite students to conjecture what the value might be Have them check their conjectures by squaring Invite them to suggest ways to estimate radic
_ 09 As a hint point out that 09 is close to 10 and so they might
use that to help guide their estimates Lead them to see that since 092 is 081 and 102 is 1 the value of radic
_ 09 is greater than 09 and less than 10 Try squaring 095 to get
09025 A good estimate for radic_
09 is 095
Rational and Irrational Numbers 14
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=B
Integers
Rational Numbers IrrationalNumbers
Real Numbers
WholeNumbers
-3-4-5 -2-1 1 2 3 50 4
23
34-4 -π -1 25
radic2
Lesson Support Content Objective Students will learn to describe relationships between sets of numbers
Language Objective Students will explain how to describe relationships between sets of real numbers
LESSON 12 Sets of Real Numbers
Building BackgroundEliciting Prior Knowledge Have students draw a number line from -5 to 5 Ask them to plot points on the number line to approximate the location of rational and irrational numbers such as -1 3 __ 4 25 -4 2 __ 3 radic
_ 2 and -π
Learning ProgressionsIn this lesson students clarify their understanding of the real number system They characterize sets and subsets of the real numbers They also identify sets for real-world situations Important understandings for students include the following
bull Identify all of the possible subsets of the real numbers for a given number
bull Decide whether a statement about a subset of the real numbers is true or false
bull Identify the set of numbers that best describes a real-world situation
Understanding the relationships among the sets of numbers that make up the real numbers is essential as students are introduced to different forms of numbers throughout the school year This lesson provides a foundation for the comparing and ordering of real numbers in the next lesson
Cluster ConnectionsThis lesson provides an excellent opportunity to connect ideas in this cluster Know that there are numbers that are not rational and approximate them by rational numbers Have students copy this diagram which relates the sets of real numbers
Ask students to complete the diagram by writing three examples for each set of numbers Have students share examples and explain how they knew each number they selected belonged in the appropriate set Answers may vary Check studentsrsquo work
Focus | Coherence | Rigor
California Common Core Standards
8NS1 Know that numbers that are not rational are called irrational Understand informally that every number has a decimal expansion for rational numbers show that the decimal expansion repeats eventually and convert a decimal expansion which repeats eventually into a rational number
MP7 Look for and make use of structure
15A
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
Math Talk
Language Support EL
PROFESSIONAL DEVELOPMENT
Linguistic Support EL
AcademicContent Vocabulary
Venn diagrams ndash Students need descriptive language to describe the categories that the different areas and colors of a Venn diagram represent the concept of a set and how sets are distinct or can overlap Use sentence frames such as
The big oval represents __________The darklight blue color in the middle of the
big ovals represents __________These sets overlap because __________
In this way students have the language and structure to identify the criteria that distinguish a set and to explain the abstract representation Also point out the use of the prefix sub- meaning ldquounderrdquo in the term subset
Rules and Patterns
Abbreviations ndash In this lesson the abbreviation mph is used Be sure to point out that mph stands for miles per hour and is used to give units in a rate of speed Students may also have seen mpg (miles per gallon) which gives the units in a rate of fuel efficiency
Borrowed Words ndash Terminology used in baseball such as inning and pitcher may require some explanation Spanish as well as some other languages have borrowed these terms from English so some students may be familiar with these words already Despite this whenever a word is critical to students understanding the word problem it is best to explain the meaning
Leveled Strategies for English Learners
Emerging Allow students to indicate true or false orally in Guided Practice Exercises 9 and 10
Expanding Have students use sentence frames to describe the meaning of regions and colors used in a Venn diagram Then give them similar sentence frames orally and have them draw and shade a Venn diagram based on the oral prompts
Bridging Have students work in groups to draw a Venn diagram to represent sets based on real-world examples in the lesson
To help students answer the question posed in Math Talk provide a sentence frame for their answer
The numbers between 31 and 39 on a number line are __________ because __________
EL
California ELD Standards
Emerging 2II5 Modifying to add details ndash Expand sentences with simple adverbials to provide details about a familiar activity or process
Expanding 2II5 Modifying to add details ndash Expand sentences with adverbials to provide details about a familiar or new activity or process
Bridging 2II5 Modifying to add details ndash Expand sentences with increasingly complex adverbials to provide details about a variety of familiar and new activities and processes
Sets of Real Numbers 15B
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
12L E S S O N
Sets of Real Numbers
EngageESSENTIAL QUESTION
How can you describe relationships between sets of real numbers Sample answer Describe them as two different sets or one set as being a subset of another
Motivate the LessonAsk How many different types of tigers can you name How does the set of Bengal tigers relate to the set of tigers
ExplorePoint to different locations in the Animals diagram and ask for examples for that classification Do the same for the Real Numbers diagram Students should understand that everything within a region is part of the set for example both -3 and 2 are integers
ExplainEXAMPLE 1
Questioning Strategies Mathematical Practices bull In A why is 5 not a perfect square It does not have rational numbers as its square roots
bull Can the number in B be written as a fraction Why or why not Yes it is a terminating decimal so it is a rational number
Engage with the WhiteboardHave students place the numbers in Example 1 and Additional Example 1 in the Venn diagram for numbers
YOUR TURNAvoid Common ErrorsBe sure that students read Exercise 2 carefully before answering The number given in the problem 10 is the area not the side length
EXAMPLE 2Questioning Strategies Mathematical Practices bull What two major sets are the real numbers composed of rational and irrational numbers
bull What is the location of the set of whole numbers in the Venn diagram in relation to the set of rational numbers Explain Inside it whole numbers are rational numbers
Focus on Reasoning Mathematical PracticesRemind students that it takes only one counterexample to show that a statement is false
Interactive Whiteboard Interactive example available online
ADDITIONAL EXAMPLE 1Write all names that apply to each number
A -10integer rational real
B 12 _ 3
whole integer rational real
myhrwcom
Interactive Whiteboard Interactive example available online
ADDITIONAL EXAMPLE 2Tell whether the given statement is true or false Explain your choice
No integers are whole numbers
False every whole number is also an integer
myhrwcom
Animated MathClassifying Numbers
Students build fluency in classifying numbers in this engaging fast-paced game
myhrwcom
CA Common CoreStandards
The student is expected to
The Number Systemmdash8NS1
Know that numbers that are not rational are called irrational Understand informally that every number has a decimal expansion for rational numbers show that the decimal expansion repeats eventually and convert a decimal expansion which repeats eventually into a rational numberMathematical Practices
MP7 Using Structure
The student is expected to
15 Lesson 12
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=B
Math TrainerOnline Practice
and Help
Personal
myhrwcom
Math TrainerOnline Practice
and Help
Personal
myhrwcom
Math On the Spotmyhrwcom
Understanding Sets and Subsets of Real NumbersBy understanding which sets are subsets of types of numbers you can verify whether statements about the relationships between sets are true or false
Tell whether the given statement is true or false Explain your choice
All irrational numbers are real numbers
True Every irrational number is included in the set of real numbers The irrational numbers are a subset of the real numbers
No rational numbers are whole numbers
False A whole number can be written as a fraction with a denominator of 1 so every whole number is included in the set of rational numbers The whole numbers are a subset of the rational numbers
EXAMPLE 2
A
B
Write all names that apply to each number
1 A baseball pitcher has pitched 12 2 _ 3 innings
2 The length of the side of a square that has an
area of 10 square yards
YOUR TURN
Tell whether the given statement is true or false Explain your choice
3 All rational numbers are integers
4 Some irrational numbers are integers
YOUR TURN
Give an example of a rational number that is a
whole number Show that the number is both whole
and rational
Math TalkMathematical Practices
Give an example of a
8NS1
False Every integer is a rational number but not every
False Real numbers are either rational or irrational numbers
Integers are rational numbers so no integers are irrational numbers
rational real
irrational real
Sample answer 8 8 = 8_
1
and -thinsp 5 _ 2 are not integers
rational number is an integer Rational numbers such as 3 _ 5
Unit 116
copy H
ough
ton
Miff
lin H
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ublis
hing
Com
pany
bull Im
age C
redi
ts D
igita
l Im
age c
opyr
ight
copy20
04 Ey
ewire
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8_MCAAESE206984_U1M01L2indd 16 41613 136 AM
Math On the Spot
myhrwcom
Vertebrates
Birds
Passerines
Animals
Integers
Rational Numbers IrrationalNumbers
Real Numbers
WholeNumbers
1
45
3
0
274
67
radic4
-
-3
-2
-1
03
radic2
radic17
radic11-
π
Animated Math
myhrwcom
Classifying Real NumbersBiologists classify animals based on shared characteristics A cardinal is an animal a vertebrate a bird and a passerine
You already know that the set of rational numbers consists of whole numbers integers and fractions The set of real numbers consists of the set of rational numbers and the set of irrational numbers
Write all names that apply to each number
radic_
5 irrational real
ndash1784rational real
whole integer rational real
EXAMPLEXAMPLE 1
A
B
C radic_ 81 ____ 9
L E S S O N
12Sets of Real Numbers
ESSENTIAL QUESTIONHow can you describe relationships between sets of real numbers
Passerines such as the cardinal are also called ldquoperching birdsrdquo
What types of numbers are between 31 and 39 on a
number line
Math TalkMathematical Practices
What types of numbers are
8NS1
8NS1
Know that numbers that are not rational are called irrational Understand informally that every number has a decimal expansion for rational numbers show that the decimal expansion repeats eventually and convert a decimal expansion which repeats eventually into a relation number
ndash1784 is a terminating decimal
5 is a whole number that is not a perfect square
radic_
81 _____ 9 = 9 __ 9 = 1 rational irrational real
15Lesson 12
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ough
ton
Miff
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Com
pany
bull Im
age C
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ts copy
Wiki
med
ia Co
mm
ons
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8_MCABESE206984_U1M01L2indd 15 061113 1144 AM
PROFESSIONAL DEVELOPMENT
Math BackgroundThe relationships between sets of numbers extend to include complex numbers A complex number can be written as a sum of a real number a and an imaginary number bi
a + bi
An imaginary number is a special number that when squared gives a negative value When you square a real number you get a nonnegative number When you square an imaginary number you get a negative value The imaginary unit is i
i = radic_
-1
Integrate Mathematical Practices MP7
This lesson provides an opportunity to address this Mathematical Practices standard It calls for students to discern structure to connect and communicate mathematical ideas
Students use a Venn diagram to structure relationships between sets of numbers They connect and communicate mathematical ideas when they make logical statements about the sets and describe which set best describes numbers applied to real-life situations
Sets of Real Numbers 16
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=B
YOUR TURNAvoid Common ErrorsStudents may see the word ldquoAllldquo or rdquoNordquo in Exercises 3 and 4 and immediately assume that any absolute statements like these are false Remind them that there are true statements that begin with these words and encourage them to provide examples
EXAMPLE 3Questioning Strategies Mathematical Practices bull In A how does the phrase ldquonumber of rdquo give you a clue about the number classification It indicates a counting number
bull What is the relationship between the circumference of a circle and the diameter The circumference is diameter times π
Focus on Critical Thinking Mathematical PracticesIn B suppose the diameters in inches were 25
__ π 28 __ π
31 __ π and so on What set of numbers would
best describe the circumferences Explain Whole numbers the circumferences would be the whole numbers 25 28 31 and so on
YOUR TURNFocus on Critical Thinking Mathematical PracticesHave students compare and contrast the classification of numbers in the answers in Exercises 5 and 6
ElaborateTalk About ItSummarize the Lesson
Ask What are some ways that number sets can be related Sets may be subsets of other sets or they may be separate from other sets
GUIDED PRACTICEEngage with the Whiteboard
Have students place the numbers in Exercises 1ndashthinsp8 in the Venn diagram for numbers at the beginning of the lesson
Integrating Language Arts EL
Encourage English learners to ask for clarification on any terms or phrases that they do not understand
Avoid Common ErrorsExercise 7 Remind students that a repeating decimal is a rational numberExercises 9ndash10 Remind students that it only takes one counterexample to show that a statement is false
Interactive Whiteboard Interactive example available online
ADDITIONAL EXAMPLE 3Identify the set of numbers that best describes the situation Explain your choice
A the amount of time that has passed since midnight
The set of real numbers time is continuous so the amount of time can be rational or irrational
B the number of tickets sold to a basketball game
The set of whole numbers the number of tickets sold may be 0 or a counting number
myhrwcom
17 Lesson 12
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
1IN
116 inch
Guided Practice
Write all names that apply to each number (Example 1)
1 7 _ 8 2 radic_
36
3 radic_
24 4 075
5 0 6 - radic_ 100
7 5 _
45 8 - 18 __ 6
Tell whether the given statement is true or false Explain your choice (Example 2)
9 All whole numbers are rational numbers
10 No irrational numbers are whole numbers
Identify the set of numbers that best describes each situation Explain your choice (Example 3)
11 the change in the value of an account when given to the nearest dollar
12 the markings on a standard ruler
13 What are some ways to describe the relationships between sets of numbers
CHECK-INESSENTIAL QUESTION
rational real
rational real
True Whole numbers are rational numbers
Rational numbers the ruler is marked every 1 __ 16 th inch
Sample answer Describe one set as being a subset of
another or show their relationships in a Venn diagram
Integers the change can be a whole dollar amount
and can be positive negative or zero
True Whole numbers are a subset of the set of rational numbers
and can be written as a ratio of the whole number to 1
irrational real
whole integer rational real
whole integer rational real
rational real
integer rational real
integer rational real
Unit 118
copy H
ough
ton
Miff
lin H
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Com
pany
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8_MCAAESE206984_U1M01L2indd 18 41613 136 AM
My Notes
Math TrainerOnline Practice
and Help
Personal
myhrwcom
Math On the Spot
myhrwcom
Identifying Sets for Real-World SituationsReal numbers can be used to represent real-world quantities Highways have posted speed limit signs that are represented by natural numbers such as 55 mph Integers appear on thermometers Rational numbers are used in many daily activities including cooking For example ingredients in a recipe are often given in fractional amounts such as 2 _ 3 cup flour
Identify the set of numbers that best describes each situation Explain your choice
the number of people wearing glasses in a room
The set of whole numbers best describes the situation The number of people wearing glasses may be 0 or a counting number
the circumference of a flying disk has a diameter of 8 9 10 11 or 14 inches
The set of irrational numbers best describes the situation Each circumference would be a product of π and the diameter and any multiple of π is irrational
EXAMPLEXAMPLE 3
A
B
Identify the set of numbers that best describes the situation Explain your choice
5 the amount of water in a glass as it evaporates
6 the weight of a person in pounds
YOUR TURN
8NS1
Rational numbers a personrsquos weight can be a decimal
such as 835 pounds
Real numbers the amount can be any number greater
than 0
17Lesson 12
copy H
ough
ton
Miff
lin H
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Com
pany
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
8_MCAAESE206984_U1M01L2indd 17 41613 520 AM
Graphic OrganizersGive students a list of numbers (including terminating and repeating decimals fractions integers and rational and irrational square roots) and a graphic organizer as shown below
Real Numbers
Rational numbers Irrational numbers
Integer numbers
Whole numbers
Ask students to write each number in the list in the correct section of the organizer
Number SensePoint out to students that knowing the types of numbers to expect in different situations can alert them to incorrect math as well as to impossible situations For example 135 shots made in basketballs is not possible but an average number of shots can equal 135
Additional ResourcesDifferentiated Instruction includes bull Reading Strategies bull Success for English Learners EL
bull Reteach bull Challenge PRE-AP
DIFFERENTIATE INSTRUCTION
Sets of Real Numbers 18
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
Math TrainerOnline Assessment
and Intervention
Personal
Online homework assignment available
myhrwcom
Lesson Quiz available online
12 LESSON QUIZ
1 Write all the names that apply to the number
2 Tell whether the given statement is true or false Explain your choice All numbers between 1 and 2 are rational numbers
3 Identify the set of numbers that best describes the situation Explain your choiceThe choices on a survey question change the total points for the survey by -2 -1 0 1 or 2 points
-1 _
5
myhrwcom
Additional ResourcesDifferentiated Instruction includes bull Leveled Practice worksheets
EvaluateGUIDED AND INDEPENDENT PRACTICE
Concepts amp Skills Practice
Example 1Classifying Real Numbers
Exercises 1ndash8 14ndash19 22ndash24
Example 2Understanding Sets and Subsets of Real Numbers
Exercises 9ndash10
Example 3Identifying Sets for Real-World Situations
Exercises 11ndash12 20ndash21 25
Focus | Coherence | Rigor
Exercise Depth of Knowledge (DOK) Mathematical Practices
14ndash19 2 SkillsConcepts MP7 Using Structure
20ndash21 2 SkillsConcepts MP6 Precision
22ndash23 2 SkillsConcepts MP3 Logic
24 1 Recall of Information MP7 Using Structure
25 2 SkillsConcepts MP2 Reasoning
26ndash27 3 Strategic Thinking MP3 Logic
28 3 Strategic Thinking MP8 Patterns
29 3 Strategic Thinking MP3 Logic
8NS1
8NS1
Exercise 29 combines concepts from the California Common Core cluster ldquoKnow that there are numbers that are not rational and approximate them by rational numbersrdquo
Answers1 rational real
2 False radic_
2 is an example of an irrational number between 1 and 2
3 Integers each number is an integer but only three are whole numbers
19 Lesson 12
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
Work Area
π mi23 Critique Reasoning The circumference of a circular region is shown
What type of number best describes the diameter of the circle Explain
your answer
24 Critical Thinking A number is not an integer What type of number can it be
25 A grocery store has a shelf with half-gallon containers of milk What type of number best represents the total number of gallons
26 Explain the Error Katie said ldquoNegative numbers are integersrdquo What was her error
27 Justify Reasoning Can you ever use a calculator to determine if a number is rational or irrational Explain
28 Draw Conclusions The decimal 0 _
3 represents 1 _ 3 What type of number best describes 0
_ 9 which is 3 middot 0
_ 3 Explain
29 Communicate Mathematical Ideas Irrational numbers can never be precisely represented in decimal form Why is this
FOCUS ON HIGHER ORDER THINKING
It can be a rational number that is not an integer or an irrational number
rational number
The set of negative numbers also includes non-integer
rational numbers and irrational numbers
Sample answer If the calculator shows a decimal that
terminates in fewer digits than what the calculator screen
allows then you can tell that the number is rational If not
you cannot tell from the calculator display whether the
number terminates because you see a limited number
of digits It may be a repeating decimal (rational) or
non-terminating non-repeating decimal (irrational)
Whole 3 middot 0 _
3 represents 3 middot 1 _ 3 = 1 so 0 _
9 is exactly 1
Sample answer In decimal form irrational numbers never
terminate and never repeat Therefore no matter how
many decimal places you include the number will never
be precisely represented There are always more digits
Whole the diameter is π _ π = 1 mile
Unit 120
copy H
ough
ton
Miff
lin H
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hing
Com
pany
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
8_MCAAESE206984_U1M01L2indd 20 120413 909 PM
Integers
Rational Numbers Irrational Numbers
Real Numbers
Whole Numbers
257
radic16
166
radic9
128 radic50
Personal Math Trainer
Online Practice and Helpmyhrwcom
Name Class Date
Independent Practice
Identify the set of numbers that best describes each situation Explain your choice
20 the height of an airplane as it descends to an airport runway
21 the score with respect to par of several golfers 2 ndash 3 5 0 ndash 1
22 Critique Reasoning Ronald states that the number 1 __ 11 is not rational because when converted into a decimal it does not terminate Nathaniel says it is rational because it is a fraction Which boy is correct Explain
12
14 - radic_
9 15 257
16 radic_
50 17 8 1 _ 2
18 166 19 radic_
16
Write all names that apply to each number Then place the numbers in the correct location on the Venn diagram
8NS1
Real numbers the height can be any number greater than zero
integer rational real whole integer rational real
whole integer rational real
irrational real
rational real
rational real
Integers the scores are counting numbers their
opposites and zero
Nathaniel is correct A rational number is a number that can be written as a fraction and 1 __ 11 is a fraction
19Lesson 12
copy H
ough
ton
Miff
lin H
arco
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ublis
hing
Com
pany
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
8_MCAAESE206984_U1M01L2indd 19 41613 136 AM
myhrwcomActivity available onlineEXTEND THE MATH PRE-AP
Activity Have students consider the concept of restricted domain for the sets of numbers that describe situations For example the number of sisters a person has can best be described by whole numbers but no one has ever had 1500 sisters An area code is an integer or whole number between 200 and 999
Have students use a source such as the Guinness Book of World Records and give examples of sets of numbers that describe situations where the domain is restricted Ask whether the restriction may be changed in the future
Sets of Real Numbers 20
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
-3-4-5 -2-1 1 2 3 50 4
12-4 -radic5
Lesson Support Content Objective Students will learn to order a set of real numbers
Language Objective Students will show and describe how to order a set of real numbers
LESSON 13 Ordering Real Numbers
Building BackgroundEliciting Prior Knowledge Have students draw a number line to compare a rational number and an irrational number such as - radic
_ 5 and -4 1 __ 2 Ask them to explain how
they approximated the irrational number on the number line Then have them identify the greater and the lesser real number Repeat with several other pairs of real numbers in different forms
Learning ProgressionsIn this lesson students order a set of real numbers They use rational approximations to compare the sizes of irrational numbers They also order numbers for real-world situations Important understandings for students include the following
bull Compare irrational numbers bull Estimate the value of expressions with irrational numbers bull Order a set of real numbers bull Order real numbers in a real-world context
Work with real numbers continues throughout Grade 8 and into high school This lesson provides students with a foundation for understanding the relative sizes of numbers in different forms in the real number system
Cluster ConnectionsThis lesson provides an excellent opportunity to connect ideas in this cluster Know that there are numbers that are not rational and approximate them by rational numbers Tell students that there is a special number called the golden ratio with applications in mathematics geometry art and architecture The golden ratio is called phi and is represented by the Greek letter ϕ It includes an irrational number in its definition
Have students explain why the golden ratio is irrational Ask them to find the two whole numbers the golden ratio lies between Then challenge them to approximate the golden ratio to the nearest tenth It is irrational because it includes an irrational number in its definition It lies between 1 and 2 To the nearest tenth ϕ = 16
ϕ = 1 + radic_
5 _ 2
Focus | Coherence | Rigor
California Common Core Standards
8NS2 Use rational approximations of irrational numbers to compare the size of irrational numbers locate them approximately on a number line diagram and estimate the value of expressions (eg π 2 )
MP4 Model with mathematics
21A
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
Math Talk
Language Support EL
PROFESSIONAL DEVELOPMENT
Linguistic Support EL
AcademicContent Vocabulary
Post a chart like this to remind students of the regular comparative forms of adjectives that use the -er and -est suffixes Add to the chart for terms that appear in examples and exercises in each lesson Include any irregular verb forms
Background Knowledge
Go On ndash the title of the module review or quiz is Ready to Go On This title uses an idiomatic expression In this context to go on means ldquoto move aheadrdquo or ldquoto proceedrdquo It is different from the use of go on that means having enough facts to use meaningfully as in having enough to go on Also the intonation used in pronouncing an expression can give it different meanings For example when the speaker emphasizes the word on he or she might be expressing disbelief as in ldquoGo ON Yoursquore kidding rightrdquo Discuss with students other ways that the phrase go on may be used
Leveled Strategies for English Learners
Emerging Label points on a number line with the terms used in ordering greater greatest less lesser least Use sentence frames to insert the correct terms
Expanding Have students give two or three complete sentences to compare the placement of numbers on a number line using the correct forms of the comparative and superlative adjectives
Bridging Have students work in pairs with one student giving directions to the other in complete sentences to order numbers on a number line
To help students answer the question posed in Math Talk make sure that students have a command of the forms for making comparisons and the superlative and the concept of opposite order so that the focus is on the math concept instead of the language skills needed to describe and explain order
EL
Adjective Comparative Superlative
Far Farther Farthest
Large Larger Largest
Great Greater Greatest
Some Less Least
Some More Most
California ELD Standards
Emerging 2I8 Analyzing language choices ndash Explain how phrasing or different common words with similar meanings produce different effects on the audience
Expanding 2I8 Analyzing language choices ndash Explain how phrasing or different words with similar meanings or figurative language produce shades of meaning and different effects on the audience
Bridging 2I8 Analyzing language choices ndash Explain how phrasing or different words with similar meanings or figurative language produce shades of meaning nuances and different effects on the audience
Ordering Real Numbers 21B
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
13L E S S O N
Ordering Real Numbers
Interactive Whiteboard Interactive example available online
ADDITIONAL EXAMPLE 1Compare Write lt gt or =
A radic_
8 - 2 4 - radic_
8 lt
B radic_
20 + 1 3 + radic_
2 gt
EngageESSENTIAL QUESTION
How do you order a set of real numbers Sample answer Find their approximate decimal values and order them
Motivate the LessonAsk What kind of numbers are you comparing when you compare the price of gasoline at two different gas stations
ExploreGive students two rational numbers and ask them to name a number between them Repeat a few times and then give them two irrational numbers and ask them to name a number between them
ExplainEXAMPLE 1
Questioning Strategies Mathematical Practices bull Which is greater the difference between 5 and 3 or the difference between radic
_ 5 and radic
_ 3
The difference between 5 and 3 is 2 the difference between radic_
5 and radic_
3 is approximately 1 So the difference between 5 and 3 is greater
Avoid Common ErrorsCaution students to read the problem carefully and think about what the radical sign means so that they do not misread the problem and answer that the two sides are equal
YOUR TURNFocus on TechnologyCalculators should not be used at this point because developing number sense is the goal
EXAMPLE 2Questioning Strategies Mathematical Practices bull How do you determine whether radic
_ 22 is less than or greater than 45 The square of 45 is
2025 which is less than 22 so the square root of 22 must be greater than 45
Engage with the WhiteboardHave students graph and label various real numbers between 42 and 44 and between 47 and 5
YOUR TURNFocus on Modeling Mathematical PracticesHave students label the integers on the number line with their equivalent square root For example 1 2 and 3 on the number line would be labeled radic
_ 1 radic
_ 4 and radic
_ 9
Interactive Whiteboard Interactive example available online
ADDITIONAL EXAMPLE 2Order 3π radic
_ 10 and 325 from greatest
to least
3π 325 radic_
10
myhrwcom
myhrwcom
CA Common CoreStandards
The student is expected to
The Number Systemmdash8NS2
Use rational approximations of irrational numbers to compare the size of irrational numbers locate them approximately on a number line diagram and estimate the value of expressions (eg π 2 )
Mathematical Practices
MP4 Modeling
The student is expected to
21 Lesson 13
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
Math TrainerOnline Practice
and Help
Personal
myhrwcom
Math On the Spotmyhrwcom
0 05 1 15 2 25 3 35 4
radic5radic3
π2
8 85 9 95 10 105 11 115 12
radic75
4 42 44 46 48 5
radic224 12π + 1
Ordering Real Numbers You can compare and order real numbers and list them from least to greatest
Order radic_
22 π + 1 and 4 1 _ 2 from least to greatest
First approximate radic_
22
radic_
22 is between 4 and 5 Since you donrsquot know where it falls between 4 and 5 you need to find a better estimate for radic
_ 22 so
you can compare it to 4 1 _ 2
Since 22 is closer to 25 than 16 use squares of numbers between 45 and 5 to find a better estimate of radic
_ 22
45 2 = 2025 46 2 = 2116 47 2 = 2209 48 2 = 2304
Since 47 2 = 2209 an approximate value for radic_
22 is 47
An approximate value of π is 314 So an approximate value of π +1 is 414
Plot radic_
22 π + 1 and 4 1 _ 2 on a number line
Read the numbers from left to right to place them in order from least to greatest
From least to greatest the numbers are π + 1 4 1 _ 2 and radic_
22
EXAMPLE 2
STEP 1
STEP 2
Order the numbers from least to greatest Then graph them on the number line
YOUR TURN
5 radic_
5 25 radic_
3
6 π 2 10 radic_
75
If real numbers a b and c are in order from least to greatest what is the order
of their opposites from least to greatest
Explain
Math TalkMathematical Practices
8NS2
radic_
3 radic_
5 25
radic_
75 π2 10
Math Talk answer -c -b -a -c is farthest to the left on a number line -b is in the middle and -a is farthest to the right
Unit 122
copy H
ough
ton
Miff
lin H
arco
urt P
ublis
hing
Com
pany
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
8_MCAAESE206984_U1M01L3indd 22 41613 447 AM
My Notes
Math TrainerOnline Practice
and Help
Personal
myhrwcom
Math On the Spot
myhrwcom
Comparing Irrational NumbersBetween any two real numbers is another real number To compare and order real numbers you can approximate irrational numbers as decimals
Compare radic_
3 + 5 3 + radic_
5 Write lt gt or =
First approximate radic_
3
radic_
3 is between 1 and 2
Next approximate radic_
5
radic_
5 is between 2 and 3
Then use your approximations to simplify the expressions
radic_
3 + 5 is between 6 and 7
3 + radic_
5 is between 5 and 6
So radic_
3 + 5 gt 3 + radic_
5
Reflect1 If 7 + radic
_ 5 is equal to radic
_ 5 plus a number what do you know about the
number Why
2 What are the closest two integers that radic_
300 is between
EXAMPLEXAMPLE 1
STEP 1
STEP 2
Compare Write lt gt or =
YOUR TURN
3 radic_
2 + 4 2 + radic_
4 4 radic_
12 + 6 12 + radic_
6
L E S S O N
13 Ordering Real Numbers
ESSENTIAL QUESTIONHow do you order a set of real numbers
8NS2
Use rational approximations of irrational numbers to compare the size of irrational numbers locate them approximately on a number line diagram and estimate the value of expressions (eg π 2 )
8NS2
Use perfect squares to estimate square roots
1 2 = 1 2 2 = 4 3 2 = 9
The number is 7 both expressions must equal 7 + radic_
5
17 and 18
gt lt
21Lesson 13
copy H
ough
ton
Miff
lin H
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ublis
hing
Com
pany
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
8_MCAAESE206984_U1M01L3indd 21 41913 246 PM
PROFESSIONAL DEVELOPMENT
Math BackgroundIn this lesson students estimate irrational numbers in the form of square roots of nonper-fect squares by finding two perfect squares between which the number falls A more precise method involves repeated division For example to find radic
_ 28 find a whole number whose perfect
square is close to 28 such as 5 Divide 28 by that number 28 divide 5 = 56 Find the average of the quotient and divisor 5 + 56
_____ 2 = 53 Continue dividing 28 by each result and averaging until you get the desired accuracy
Integrate Mathematical Practices MP4
This lesson provides an opportunity to address this Mathematical Practices standard It calls for students to model relationships using multiple representations including diagrams graphs and language as appropriate Students use multiple representations when they use number lines to estimate the locations of and order rational and irrational numbers given as symbols
Ordering Real Numbers 22
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
Interactive Whiteboard Interactive example available online
ADDITIONAL EXAMPLE 3The diameter of a meteorite in millimeters is calculated by four different methods Order the results from least to greatest
Joe radic_
18 mm Lisa 13 __ 3 mm
Pablo 46 mm Julien 4π __ 3 mm
Julien 4π __ 3 mm Lisa 13 __ 3 mm
Joe radic_
18 mm Pablo 46 mm
EXAMPLE 3Questioning Strategies Mathematical Practices bull How can you verify that radic
_ 28 is between 52 and 53 5 2 2 = 2704 and 5 3 2 = 2809
bull Explain how to determine which number is greater 5 _
5 or 55 When the repeating decimal is rounded to the nearest tenth or hundredth you can see that it is greater
Connect to Daily LifeDiscuss how measuring across a canyon might involve different methods than measuring along a road Explain that measurements like these are often done using calculations that approximate the distance
YOUR TURNFocus on Critical Thinking Mathematical PracticesDiscuss with students which number is greater 3
_ 45 or 3450 3
_ 45 or 3455 and why Explain
that 3 _
45 can be written out as 34545hellipMake sure they understand that 3 _
45 is greater than 345 but less than 3455
ElaborateTalk About ItSummarize the Lesson
Ask How can you order two numbers in different forms whose decimal approxi-mations appear to be equal Approximate one or both numbers to an additional
number of decimal places
GUIDED PRACTICEEngage with the Whiteboard
Have students place and label additional points on the number line in Exercise 9 Allow the points to be in any format other than decimal
Avoid Common ErrorsExercises 3ndash4 Caution students to read the problem carefully so that they do not misread the problem as the same numbers combined by addition on each side of the circleExercise 10 Remind students that the calculations have units
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23 Lesson 13
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
0 05 1 15 2 25 3 35 4 45 5 55 6 65 7
2πradic3
Compare Write lt gt or = (Example 1)
1 radic_
3 + 2 radic_
3 + 3 2 radic_
8 + 17 radic_
11 + 15
3 radic_
6 + 5 6 + radic_
5 4 radic_
9 + 3 9 + radic_
3
5 radic_
17 - 3 -2 + radic_
5 6 12 - radic_
2 14 - radic_
8
7 radic_
7 + 2 radic_
10 - 1 8 radic_
17 + 3 3 + radic_
11
9 Order radic_
3 2π and 15 from least to greatest Then graph them on the number line (Example 2)
radic_
3 is between and so radic_
3 asymp
π asymp 314 so 2π asymp
From least to greatest the numbers are
10 Four people have found the perimeter of a forest using different methods Their results are given in the table Order their calculations from greatest to least (Example 3)
11 Explain how to order a set of real numbers
CHECK-INESSENTIAL QUESTION
Forest Perimeter (km)
Leon Mika Jason Ashley
radic_
17 - 2 1 +thinsp π __ 2 12 ___ 5 25
Guided Practice
17
15
1 + π _ 2 km 25 km 12 __ 5 km radic_
17 - 2 km
2π radic
_ 3
18 175
628
Sample answer Convert each number to a decimal
equivalent using estimation to find equivalents for
irrational numbers Graph each number on a number line
Read the numbers from left to right for least to greatest
Read the numbers from right to left for greatest to least
lt gt
lt lt
ltgt
gt gt
24 Unit 1
copy H
ough
ton
Miff
lin H
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urt P
ublis
hing
Com
pany
bull Im
age C
redi
ts copy
Elena
Eliss
eeva
Alam
y Im
ages
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
8_MCAAESE206984_U1M01L3indd 24 41613 448 AM
My Notes
5 52 54 56 58 6
radic28 5 12
23455
Math TrainerOnline Practice
and Help
Personal
myhrwcom
Math On the Spot
myhrwcom
Ordering Real Numbers in a Real-World Context Calculations and estimations in the real world may differ It can be important to know not only which are the most accurate but which give the greatest or least values depending upon the context
Four people have found the distance in kilometers across a canyon using different methods Their results are given in the table Order the distances from greatest to least
Distance Across Quarry Canyon (km)
Juana Lee Ann Ryne Jackson
radic_
28 23 __ 4 5 _
5 5 1 _ 2
Write each value as a decimal
radic_
28 is between 52 and 53 Since 53 2 = 2809 an approximate value for radic
_ 28 is 53
23 __ 4 = 575
5 _
5 is 5555hellip so 5 _
5 to the nearest hundredth is 556
5 1 _ 2 = 55
Plot radic_
28 23 __ 4 5 _
5 and 5 1 _ 2 on a number line
From greatest to least the distances are
23 __ 4 km 5 _
5 km 5 1 _ 2 km radic_
28 km
EXAMPLEXAMPLE 3
STEP 1
STEP 2
7 Four people have found the distance in miles across a crater using different methods Their results are given below
Jonathan 10 __ 3 Elaine 3 _
45 Joseacute 3 1 _ 2 Lashonda radic_
10
Order the distances from greatest to least
YOUR TURN
8NS2
3 1 _ 2 mi 3 _
45 mi 10 __ 3 mi radic_
10 mi
23Lesson 13
copy H
ough
ton
Miff
lin H
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ublis
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Com
pany
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
8_MCAAESE206984_U1M01L3indd 23 41613 447 AM
ModelingPlace papers around the room with the numbers from 1 to 5 one per sheet Give each student a card showing a number between 1 and 5 in different forms Have students place his or her card between the correct integers and decide where the number goes in relation to any numbers already placed
Multiple RepresentationsGive students a vertical number line which some students might find easier to use than a horizontal one Have them decide whether to place points for rational and irrational numbers above or below existing points
Additional ResourcesDifferentiated Instruction includes bull Reading Strategies bull Success for English Learners EL
bull Reteach bull Challenge PRE-AP
DIFFERENTIATE INSTRUCTION
Ordering Real Numbers 24
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
Math TrainerOnline Assessment
and Intervention
Personal
Online homework assignment available
myhrwcom
myhrwcom
Lesson Quiz available online
13 LESSON QUIZ
1 Compare Write lt gt or =
radic_
95 - 5 radic_
62 - 2
2 Order 105 radic_
105 and 3π + 1 from greatest to least
3 A length in centimeters is calculated differently by four different people Order their calculations from least to greatest
KD 11 __ 2 cm Silvio 5 __ 3 π cm
Paula 5 _
4 cm Luis radic_
33 cm
EvaluateGUIDED AND INDEPENDENT PRACTICE
Concepts amp Skills Practice
Example 1Comparing Irrational Numbers
Exercises 1ndash8
Example 2Ordering Real Numbers
Exercises 9 12ndash15 18ndash21
Example 3Ordering Real Numbers in a Real-World Context
Exercises 10 16ndash17
Additional ResourcesDifferentiated Instruction includes bull Leveled Practice worksheets
Focus | Coherence | Rigor
Exercise Depth of Knowledge (DOK) Mathematical Practices
12ndash15 1 Recall of Information MP5 Using Tools
16 2 SkillsConcepts MP2 Reasoning
17 2 SkillsConcepts MP6 Precision
18ndash21 2 SkillsConcepts MP2 Reasoning
22 3 Strategic Thinking MP4 Modeling
23ndash24 3 Strategic Thinking MP3 Logic
8NS2
8NS2
Answers1 radic
_ 95 - 5 lt radic
_ 62 - 2
2 radic_
105 3π + 1 105
3 Silvio 5 __ 3 π cm Paula 5 _
4 cm
KD 11
__ 2 cm Luis radic_
33 cm
25 Lesson 13
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
Work Area
3140 3141 3142 3143
314 π227
20 A teacher asks his students to write the numbers shown in order from least to greatest Paul thinks the numbers are already in order Sandra thinks the order should be reversed Who is right
21 Math History There is a famous irrational number called Eulerrsquos number symbolized with an e Like π its decimal form never ends or repeats The first few digits of e are 27182818284
a Between which two square roots of integers could you find this number
b Between which two square roots of integers can you find π
22 Analyze Relationships There are several approximations used for π including 314 and 22 __ 7 π is approximately 314159265358979
a Label π and the two approximations on the number line
b Which of the two approximations is a better estimate for π Explain
c Find a whole number x so that the ratio x ___ 113 is a better estimate for π
than the two given approximations
23 Communicate Mathematical Ideas If a set of six numbers that include both rational and irrational numbers is graphed on a number line what is the fewest number of distinct points that need to be graphed Explain
24 Critique Reasoning Jill says that 12 _
6 is less than 1263 Explain her error
FOCUS ON HIGHER ORDER THINKING
radic_
115 115 ___ 11 and 105624
between radic_
7 asymp 265 and radic_
8 asymp 283
between radic_
9 = 3 and radic_
10 asymp 316
22 __ 7 it is closer to π on the number line
She did not consider the repeating digit 1266
2 rational numbers can have the same location and
irrational numbers can have the same location but they
cannot share a location
355
Neither student is correct The answer
should be 115 ___ 11 105624 radic_
115
Unit 126
copy H
ough
ton M
ifflin
Har
cour
t Pub
lishin
g Com
pany
Imag
e Cre
dits
copy3D
Stoc
kiSt
ockP
hoto
com
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
8_MCAAESE206984_U1M01L3indd 26 210513 801 AM
Personal Math Trainer
Online Practice and Helpmyhrwcom
Name Class Date
Independent Practice
16 Your sister is considering two different shapes for her garden One is a square with side lengths of 35 meters and the other is a circle with a diameter of 4 meters
a Find the area of the square
b Find the area of the circle
c Compare your answers from parts a and b Which garden would give your sister the most space to plant
17 Winnie measured the length of her fatherrsquos ranch four times and got four different distances Her measurements are shown in the table
a To estimate the actual length Winnie first approximated each distance to the nearest hundredth Then she averaged the four numbers Using a calculator find Winniersquos estimate
b Winniersquos father estimated the distance across his ranch to be radic_
56 km How does this distance compare to Winniersquos estimate
Give an example of each type of number
18 a real number between radic_
13 and radic_
14
19 an irrational number between 5 and 7
Order the numbers from least to greatest
12 radic_
7 2 radic_
8 ___ 2 13 radic_
10 π 35
14 radic_
220 -10 radic_
100 115 15 radic_
8 -375 3 9 _ 4
Distance Across Fatherrsquos Ranch (km)
1 2 3 4
radic_
60 58 __ 8 7 _
3 7 3 _ 5
138NS2
radic_
8 ___ 2 2 radic_
7
-10 radic_
100 115 radic_
220
radic_
60 asymp 775 58 __ 8 = 725 7 _
3 asymp 733 7 3 _ 5 = 760 so the average
π radic_
10 35
-375 9 _ 4 radic_
8 3
is 74825 km
1225 m2
4π m2 or approximately 126 m2
They are nearly identical radic_
56 is approximately 74833hellip
The circle would give her more space to plant because it has a
larger area
Sample answer 37
Sample answer radic_
31
25Lesson 13
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ough
ton
Miff
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pany
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8_MCAAESE206984_U1M01L3indd 25 41613 448 AM
Activity available online myhrwcomEXTEND THE MATH PRE-AP
Activity Have students investigate whether there are infinitely many numbers between two numbers by giving examples for each of the following
bull Between any two rational numbers there is at least one other rational number Sample answer 45 is between 41 and 48
bull Between any two irrational numbers there is at least one rational number Sample answer 45 is between radic
_ 11 and radic
_ 29
bull Between any two rational numbers there is at least one irrational number Sample answer radic
_ 11 is between 31 and 36
bull Between any two irrational numbers there is at least one irrational number Sample answer radic
_ 17 is between radic
_ 11 and radic
_ 29
Ordering Real Numbers 26
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
ReadyMath Trainer
Online Practiceand Help
Personal
myhrwcom
Module Quiz
11ensp RationalenspandenspIrrationalenspNumbersWrite each fraction as a decimal or each decimal as a fraction
1 7__20 2 1___
27 3 17_8
Solve each equation for x
4 x2=81 5 x3=343 6 x2= 1___100
7 Asquarepatiohasanareaof200squarefeetHowlongiseachside
ofthepatiotothenearesttenth
12ensp SetsenspofenspRealenspNumbersWrite all names that apply to each number
8 121____radic
____121
9 π__2
10 TellwhetherthestatementldquoAllintegersarerationalnumbersrdquoistrueorfalseExplainyourchoice
13ensp OrderingenspRealenspNumbersCompare Write lt gt or =
11 radic__
8+3 8+radic__
3 12 radic__
5+11emsp emsp emsp 5+radic___
11
Order the numbers from least to greatest
13 radic___
99π29__
8 14 radic___
1__251_40__
2
15 Howarerealnumbersusedtodescribereal-worldsituations
ESSENTIAL QUESTION
035
9-9
141ft
7 1__10- 1__10
14__11 1875
wholeintegerrationalreal
Trueintegerscanbewrittenasthequotientoftwointegers
SampleanswerRealnumberssuchastherational
π29__
8radic___
99
irrationalreal
lt gt
number1_4candescribeamountsusedincooking
radic___
1__250__
21_4
27Module1
copy H
ough
ton
Miff
lin H
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urt P
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hing
Com
pany
DONOTEDIT--ChangesmustbemadethroughldquoFileinfordquoCorrectionKey=A
8_MCAAESE206984_U1M01RTindd 27 41513 1113 PM
Math TrainerOnline Assessment
and Intervention
Personal
myhrwcom
1
2
3 Response toIntervention
Intervention Enrichment
Access Ready to Go On assessment online and receive instant scoring feedback and customized intervention or enrichment
Online and Print Resources
Differentiated Instruction
bull Reteach worksheets
bull Reading Strategies EL
bull Success for English Learners EL
Differentiated Instruction
bull Challenge worksheets PRE-AP
Extend the Math PRE-AP
Lesson Activities in TE
Additional ResourcesAssessment Resources includes bull Leveled Module Quizzes
Ready to Go OnAssess MasteryUse the assessment on this page to determine if students have mastered the concepts and standards covered in this module
California Common Core Standards
Lesson Exercises Common Core Standards
11 1ndash7 8NS1 8NS2 8EE2
12 8ndash10 8NS1
13 11ndash14 8NS2
27 Unit 1 Module 1
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
Personal Math Trainer
Online Practice and HelpmyhrwcomAssessment Readiness
Module 1 MIXed ReVIeW
1 Look at each number Is the number between 2π and radic___
52
Select Yes or No for expressions AndashC
A 6 2 _ 3 Yes No
B 5π __ 2 Yes No
C 3 radic__
5 Yes No
2 Consider the number - 11 __ 15
Choose True or False for each statement
A The number is rational True False
B The number can be written as True Falsea repeating decimal
C The number is less than ndash08 True False
3 The volume of a cube is given by V = x3 where x is the length of an edge of the cube A cube-shaped end table has a volume of 3 3 _ 8 cubic feet What is the length of an edge of the end table Explain how you solved this problem
4 A student says that radic___
83 is greater than 29 __ 3 Is the student correct Justify your
reasoning
1 1 _ 2 ft Sample answer The equation x3 = 3 3 _ 8 can be used
to find the edge length in feet To solve the equation
write the mixed number as a fraction greater than 1
x3 = 27 __ 8 Then take the cube root of both sides x = 3 _ 2 = 1 1 _ 2
No Sample answer radic___
83 asymp 91 and 29 __ 3 = 9
__ 6
Because 91 lt 9 __
6 radic___
83 lt 29 __ 3
28 Unit 1
copy H
ough
ton
Miff
lin H
arco
urt P
ublis
hing
Com
pany
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=A
8_MCAAESE206984_U1M01RTindd 28 240413 946 AM
Personal Math Trainer
Online Assessment and
Interventionmyhrwcom
Scoring GuideItem 3 Award the student 1 point for finding the edge length of the cube and 1 point for correctly explaining how to use a cube root to solve the problem
Item 4 Award the student 1 point for determining that the student is incorrect and 1 point for correctly justifying the reasoning for this conclusion
Additional ResourcesTo assign this assessment online login to your Assignment Manager at myhrwcom
Assessment Readiness
California Common Core Standards
Items Grade 8 Standards Mathematical Practices
1 8NS2 MP7
2 7NS2b 7NS2d 8NS1 MP7
3 8EE2 MP1 MP4
4 8NS1 8NS2 MP3
Item integrates mixed review concepts from previous modules or a previous course
Item 4 combines concepts from the California Common Core cluster ldquoKnow that there are numbers that are not rational and approximate them by rational numbersrdquo
Real Numbers 28
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
11L E S S O N
Rational and Irrational Numbers
EngageESSENTIAL QUESTION
How do you rewrite rational numbers and decimals take square roots and cube roots and approximate irrational numbers To express as a decimal divide the numerator by the denominator To take a square root or cube root of a number find the number that when squared or cubed equals the original number To approximate an irrational number estimate a number between two consecutive perfect squares
Motivate the LessonAsk Which type of rational number do you see more often fractions or decimals Which do you prefer to use Why
ExploreHave students write examples of ratios and then share with the class the various notations for ratios that they used (for example 25 2 to 5 2 __ 5 ) Point out the connection between the word ratio and the meaning of rational number See also Explore Activity in student text
ExplainEXAMPLE 1
Questioning Strategies Mathematical Practices bull How does the denominator of a fraction in simplest form tell whether the decimal equivalent of the fraction is a terminating decimal The decimal will terminate if the denominator is an even number a multiple of 5 or a multiple of 10
Avoid Common ErrorsTo avoid interpreting 1 __ 4 as 4 divided by 1 tell students to start at the top of the fraction and read the bar as ldquodivided byrdquo
YOUR TURNTalk About ItCheck for Understanding
Ask Can an improper fraction be written as a decimal Give an example to support your answer Yes 5 __ 4 = 125
EXAMPLE 2Questioning Strategies Mathematical Practices bull How can you use place value to write a terminating decimal as a fraction with a power of ten in the denominator Start by identifying the place value of the decimals last digit and then use the corresponding power of 10 as the denominator of the fraction
bull How can you tell if a decimal can be written as a rational number If the decimal is a terminating or repeating decimal then it can be written as a rational number
Interactive Whiteboard Interactive example available online
ADDITIONAL EXAMPLE 1Write each fraction as a decimal
A 2 _ 5
04 B 5 _ 9
0 _
5
myhrwcom
Interactive Whiteboard Interactive example available online
ADDITIONAL EXAMPLE 2Write each decimal as a fraction in simplest form
A 0355 71 ___ 200
B 0 _
43 43 __ 99
myhrwcom
CA Common CoreStandards
The student is expected to
The Number Systemmdash8NS1
Know that numbers that are not rational are called irrational Understand informally that every number has a decimal expansion for rational numbers show that the decimal expansion repeats eventually and convert a decimal expansion which repeats eventually into a rational number
The Number Systemmdash8NS2
Use rational approximations of irrational numbers to compare the size of irrational numbers locate them approximately on a number line diagram and estimate the value of expressions (eg π 2 )
Expressions and Equationsmdash8EE2
Use square root and cube root symbols to represent solutions to equations of the form x 2 = p and x 3 = p where p is a positive rational number Evaluate square roots of small perfect squares and cube roots of small perfect cubes Know that radic
_ 2 is irrational
Mathematical Practices
MP6 Precision
The student is expected to
the value of expressions (eg
7 Lesson 11
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
My Notes
Math On the Spotmyhrwcom
Math TrainerOnline Practice
and Help
Personal
myhrwcom
Expressing Decimals as Rational NumbersYou can express terminating and repeating decimals as rational numbers
Write each decimal as a fraction in simplest form
0825
The decimal 0825 means ldquo825 thousandthsrdquo Write this as a fraction
825 ____ 1000
Then simplify the fraction
825 divide 25 ________ 1000 divide 25 = 33 __ 40
0825 = 33 __ 40
0 _
37
Let x = 0 _
37 The number 0 _
37 has 2 repeating digits so multiply each side of the equation x = 0
_ 37 by 10 2 or 100
x = 0 _
37
(100)x = 100(0 _
37 )
100x = 37 _
37
Because x = 0 _
37 you can subtract x from one side and 0 _
37 from the other
100x = 37 _
37
minusx minus0 _
37
99x = 37
Now solve the equation for x Simplify if necessary
99x ___ 99 = 37 __ 99
x = 37 __ 99
EXAMPLE 2
A
B
Write each fraction as a decimal
YOUR TURN
1 5 __ 11 2 1 _ 8 3 2 1 _ 3
8NS1
To write ldquo825 thousandthsrdquo put 825 over 1000
Divide both the numerator and the denominator by 25
100 times 0 _
37 is 37 _
37
37 _
37 minus 0 _
37 is 37
Divide both sides of the equation by 99
0 _
45 0125 2 _
3
Unit 18
copy H
ough
ton
Miff
lin H
arco
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ublis
hing
Com
pany
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
8_MCAAESE206984_U1M01L1indd 8 120413 838 PM
My Notes
Math On the Spot
myhrwcom
= 033333333333331mdash3
ESSENTIAL QUESTION
Expressing Rational Numbers as DecimalsA rational number is any number that can be written as a ratio in the form a _ b where a and b are integers and b is not 0 Examples of rational numbers are 6 and 05
6 can be written as 6 _ 1 05 can be written as 1 _ 2
Every rational number can be written as a terminating decimal or a repeating decimal A terminating decimal such as 05 has a finite number of digits A repeating decimal has a block of one or more digits that repeat indefinitely
Write each fraction as a decimal
1 _ 4
1 _ 4 = 025
1 _ 3
1 _ 3 = 0 _
3
EXAMPLEXAMPLE 1
A
B
0333 3 ⟌ ⎯ 1000 minus9 10 minus9 10 minus9 1
025 4 ⟌ ⎯ 100 -8 20 -20
0
L E S S O N
11Rational and Irrational Numbers
How do you rewrite rational numbers and decimals take square roots and cube roots and approximate irrational numbers
8NS1
Know that numbers that are not rational are called irrational Understand informally that every number has a decimal expansion for rational numbers show that the decimal expansion repeats eventually and convert a decimal expansion which repeats eventually into a relation number Also 8NS2 8EE2
8NS1
Remember that the fraction bar means ldquodivided byrdquo Divide the numerator by the denominator
Divide until the remainder is zero adding zeros after the decimal point in the dividend as needed
Divide until the remainder is zero or until the digits in the quotient begin to repeat
Add zeros after the decimal point in the dividend as needed
When a decimal has one or more digits that repeat indefinitely write the decimal with a bar over the repeating digit(s)
7Lesson 11
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ough
ton
Miff
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hing
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pany
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=B
8_MCABESE206984_U1M01L1indd 7 11113 128 AM
PROFESSIONAL DEVELOPMENT
Math BackgroundSome decimals may have a pattern but still not be a repeating decimal that is rational For example in 312112111211112hellip you can predict the next digit and describe the pattern (There is one more 1 each time before the 2) However this is not a terminating decimal nor is it a repeating decimal and it is therefore NOT a rational number
Integrate Mathematical Practices MP6
This lesson provides an opportunity to address this Mathematical Practices standard It calls for students to attend to precision Students learn to express rational numbers accurately and precisely in both fractional and decimal forms and learn to translate from one form to the other They also learn how to precisely represent and communicate ideas about irrational numbers square roots and cube roots
Rational and Irrational Numbers 8
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=B
Focus on Technology Mathematical PracticesPoint out the importance of entering a repeating decimal correctly when using a graphing calculator to convert the decimal to a fraction The decimal 0
_ 59 must be entered as
0595959595959 not 059
YOUR TURNFocus on Math ConnectionsMake sure students understand that the place value of the last digit in Exercises 4 and 6 determines the denominator of the corresponding fraction or mixed number So for Exercise 4 the place value hundredths gives a denominator of 100 and for Exercise 6 the place value tenths gives a denominator of 10
EXAMPLE 3Questioning Strategies Mathematical Practices bull How can a solution of an equation of the form x 2 = p be negative if p is a positive number Since the square of a negative number is positive a negative number is also a solution of x 2 equals a positive number
bull When is a solution of an equation of the form x 3 = p larger than p The solution is larger than p if p is a number between 0 and 1
Focus on Math Connections Make sure students understand the difference in finding radic
_ 121 and solving x 2 = 121 The
symbol radic_
indicates the positive or principal square root only while the equation x 2 = 121 has two roots the principal square root and its opposite
YOUR TURNAvoid Common ErrorsTo avoid sign errors in Exercise 9 make sure that students understand that the cube of a negative number is not a positive number Therefore -8 is not a solution of x 3 = 512
Talk About ItCheck for Understanding
Ask Kris predicts that there are two real solutions for Exercises 7 and 8 and that there are three real solutions for Exercises 9 and 10 Is his prediction correct
Explain His prediction is correct for Exercises 7 and 8 because there are two numbers whose squares are the same positive number given in the exercises His prediction is not correct for Exercises 9 and 10 however because there is only one real number whose cube is the same positive number given in the exercises
EXPLORE ACTIVITYQuestioning Strategies Mathematical Practices bull Compare the values for 13 2 and 13 2 The digits are the same but 13 2 has two decimal places (169) while 13 2 has none (169)
bull How do you know whether radic_
2 will be closer to 1 or closer to 2 It will be closer to 1 because 2 is between the perfect squares of 1 and 4 but closer to 1 than it is to 4
Connect Vocabulary EL
Explain to students that the word irrational when used as an ordinary word in English means without logic or reason In mathematics when we say that a number is irrational it means only that the number cannot be written as the quotient of two integers
Engage with the WhiteboardHave students extend the number line in both directions and label the locations of the whole numbers 1 and 2 These are the roots of the consecutive perfect squares
1 and 4 used to estimate radic_
7
Interactive Whiteboard Interactive example available online
ADDITIONAL EXAMPLE 3Solve each equation for x
A x 2 = 324 18 -18
B x 2 = 25 ___ 144 5 __ 12 - 5 __ 12
C 343 = x 3 7
D x 3 = 125 ___ 512 5 __ 8
myhrwcom
9 Lesson 11
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
Math TrainerOnline Practice
and Help
Personal
myhrwcom
EXPLORE ACTIVITY
lt 2 lt
radic_
lt radic
_ 2 lt
radic_
lt radic
_ 2 lt
The solution is 9
The solution is 2 _ 5
C
D
729 = x 3
3 radic_ 729 = 3 radic
_ x 3
3 radic_ 729 = x
9 = x
x 3 = 8 ___ 125
3 radic_
x 3 =thinsp 3 radic_ 8 ___ 125
x =thinsp 3 radic_ 8 ___ 125
x = 2 _ 5
Solve each equation for x
YOUR TURN
7 x 2 = 196 8 x 2 = 9 ___ 256
9 x 3 = 512 10 x 3 = 64 ___ 343
Estimating Irrational NumbersIrrational numbers are numbers that are not rational In other words they cannot be written in the form a _ b where a and b are integers and b is not 0 Square roots of perfect squares are rational numbers Square roots of numbers that are not perfect squares are irrational Some equations like those in Example 3 involve square roots of numbers that are not perfect squares
x 2 = 2 x = plusmn radic_
2
Estimate the value of radic_
2
Find two consecutive perfect squares that 2 is between Complete the inequality by writing these perfect squares in the boxes
Now take the square root of each number
Simplify the square roots of perfect squares
radic_
2 is between and
A
B
C
8NS2 8EE2
Solve for x by taking the cube root of both sides
Solve for x by taking the cube root of both sides
Apply the definition of cube root
Think What number cubed equals 729
Apply the definition of cube root
Think What number cubed equals 8 ____ 125
radic_
2 is irrational
x = plusmn14 x = plusmn 3 __ 16
x = 8 x = 4 _ 7
1 2
1 4
1 4
1 2
Unit 110
copy H
ough
ton
Miff
lin H
arco
urt P
ublis
hing
Com
pany
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
8_MCAAESE206984_U1M01L1indd 10 41613 1211 AM
Math TrainerOnline Practice
and Help
Personal
myhrwcom
Math On the Spot
myhrwcom
Write each decimal as a fraction in simplest form
YOUR TURN
Finding Square Roots and Cube RootsThe square root of a positive number p is x if x 2 = p There are two square roots for every positive number For example the square roots of 36 are 6 and minus6 because 6 2 = 36 and (minus6) 2 = 36 The square roots of 1 __ 25 are 1 _ 5 and minus 1 _ 5 You can write the square roots of 1 __ 25 as plusmn 1 _ 5 The symbol radic
_ 5 indicates the positive
or principal square root
A number that is a perfect square has square roots that are integers The number 81 is a perfect square because its square roots are 9 and minus9
The cube root of a positive number p is x if x 3 = p There is one cube root for every positive number For example the cube root of 8 is 2 because 2 3 = 8 The cube root of 1 __ 27 is 1 _ 3 because ( 1 _ 3 )
3
= 1 __ 27 The symbol 3 radic_ 1 indicates the
cube root
A number that is a perfect cube has a cube root that is an integer The number 125 is a perfect cube because its cube root is 5
Solve each equation for x
The solutions are 11 and minus11
The solutions are 4 __ 13 and minus 4 __ 13
EXAMPLEXAMPLE 3
A x 2 = 121
x 2 = 121
x = plusmn radic_
121
x = plusmn11
B x 2 = 16 ___ 169
x 2 = 16 ___ 169
x = plusmn radic_
16 ___ 169
x = plusmn 4 __ 13
4 012 5 0 _
57 6 14
Can you square an integer and get a negative number
What does this indicate about whether negative
numbers have square roots
Math TalkMathematical Practices
8EE2
Solve for x by taking the square root of both sides
Apply the definition of square root
Think What numbers squared equal 121
Solve for x by taking the square root of both sides
Apply the definition of square root
Think What numbers squared equal 16 ____ 169
3 __ 25 19 __ 33 1 2 _ 5
No the square of a positive integer is positive the square of a negative integer is positive and the square of 0 is 0 So negative numbers do not have (real) square roots
9Lesson 11
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ough
ton
Miff
lin H
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ublis
hing
Com
pany
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
8_MCAAESE206984_U1M01L1indd 9 41913 240 PM
Critical ThinkingIn the Explore Activity students estimated the location of radic
_ 2 on a number line Ask students
whether they think that it is possible to locate more precisely the point that represents radic
_ 2 In
other words can you graph irrational numbers exactly on a number line along with rational numbers Students should understand that radic
_ 2
is a real number and all real numbers can be located on a real number line A more precise estimate will allow more precise placement on a number line
The Modeling note tells one way to do this
ModelingHave students use a ruler to represent a number line with a unit that is one inch long Have them draw a square with a side of one inch and draw the diagonal to make two isosceles triangles Lead students to understand that the length of the diagonal (or hypotenuse) is radic
_ 2
Have them copy the length of their diagonal onto their ruler or number line starting at zero The end point of the diagonal represents the exact point for the irrational number radic
_ 2 on a
number line
Additional ResourcesDifferentiated Instruction includes bull Reading Strategies bull Success for English Learners EL
bull Reteach bull Challenge PRE-AP
DIFFERENTIATE INSTRUCTION
Rational and Irrational Numbers 10
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
ElaborateTalk About ItSummarize the Lesson
Ask If someone claims that a certain number is irrational but you know it is actually rational how could you prove to that person that the number is rational
You could find a fraction equal to the number such that the number is the ratio of two integers with the denominator not equal to zero
GUIDED PRACTICEEngage with the Whiteboard
Have students plot each number in Exercises 16ndash18 on a number line Students should label each point with the irrational number written as a radical and as a
decimal
Avoid Common ErrorsExercises 1ndash6 To avoid reversing the order of the dividend and divisor tell students to start at the top of the fraction and read the bar as ldquodivided byrdquo
Focus on TechnologyHave students use a calculator to investigate the decimal equivalents of such fractions as 1 __ 9 2 __ 9 8 __ 9 and 1 __ 11 2 __ 11 10
__ 11 Ask them to describe the patterns they find as a result of these investigations
11 Lesson 11
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
Guided Practice
7 0675 8 56 9 044
10 0 _
4
10x =
x =
11 0 _
26
100x =
x =
12 0 _
325
1000x =
x =
Solve each equation for x (Example 3 and Explore Activity)
- x
-
_______________
x =
- x
-
___________________
x =
- x
-
_______________________
x =
Write each fraction or mixed number as a decimal (Example 1)
1 2 _ 5 2 8 _ 9 3 3 3 _ 4
4 7 __ 10 5 2 3 _ 8 6 5 _ 6
Write each decimal as a fraction or mixed number in simplest form (Example 2)
13 x 2 = 17 14 x 2 = 25 ___ 289 15 x 3 = 216
Approximate each irrational number to one decimal place without a calculator
x = plusmn radic__
asymp plusmn x = 3
radic__
=
(Explore Activity)
16 radic_
5 asymp
17 radic_
3 asymp
18 radic_
10 asymp
19 What is the difference between rational and irrational numbers
CHECK-INESSENTIAL QUESTION
x = plusmn radic__
__________ = plusmn _____
4 _
4
0 _
4
4 99
6216
269
41 25 5
17289
17
22 17 32
04
07
27__40
26 __ 99 325 ___ 999 4 _ 9
11__255 3_5
0 _
8
2375
375
08 _
3
26 _
26
0 _
26
325 _
325
0 _
325
999 325
Rational numbers can be written in the form a __ b where
a and b are integers and b ne 0 Irrational numbers cannot
be written in this form
Unit 112
copy H
ough
ton
Miff
lin H
arco
urt P
ublis
hing
Com
pany
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
8_MCAAESE206984_U1M01L1indd 12 41613 1211 AM
11 12 13 14 15
radic2 asymp 14
141 142 143 144 145
radic2 asymp 141
0 1 2 3 4
radic2 asymp 15
Estimate that radic_
2 asymp 15
To find a better estimate first choose some numbers between 1 and 2 and square them For example choose 13 14 and 15
1 3 2 = 1 4 2 = 1 5 2 =
Is radic_
2 between 13 and 14 How do you know
Is radic_
2 between 14 and 15 How do you know
2 is closer to than to so radic_
2 asymp
Locate and label this value on the number line
Reflect 11 How could you find an even better estimate of radic
_ 2
12 Find a better estimate of radic_
2
1 41 2 = 1 42 2 = 1 43 2 =
2 is closer to than to so radic_
2 asymp
Draw a number line and locate and label your estimate
13 Solve x 2 = 7 Write your answer as a radical expression Then estimate to one decimal place
D
E
F
No 2 is not between 169 and 196
Yes 2 is between 196 and 225
196
19881
19881
225
20164
20164
14
141
20449
169 196 225
Test the squares of numbers between 14 and 15
x = plusmn radic_
7 x asymp plusmn26
11Lesson 11
copy H
ough
ton
Miff
lin H
arco
urt P
ublis
hing
Com
pany
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
8_MCAAESE206984_U1M01L1indd 11 41613 1211 AM
Rational and Irrational Numbers 12
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
Math TrainerOnline Assessment
and Intervention
Personal
Online homework assignment available
myhrwcom
EvaluateGUIDED AND INDEPENDENT PRACTICE
Concepts amp Skills Practice
Example 1Expressing Rational Numbers as Decimals
Exercises 1ndash6 20ndash21 24ndash25
Example 2Expressing Decimals as Rational Numbers
Exercises 7ndash12 22ndash23 26ndash27
Example 3Finding Square Roots and Cube Roots
Exercises 13ndash15 28 30ndash31 35
Explore ActivityEstimating Irrational Numbers
Exercises 13 16ndash18 29 32ndash34
Additional ResourcesDifferentiated Instruction includes bull Leveled Practice worksheets
Lesson Quiz available online
11 LESSON QUIZ
1 Write as a decimal 2 5 __ 8 1 7 __ 12
2 Write as a fraction 034 1 _
24
3 Solve x 2 = 9 __ 49 for x
4 Solve x 3 = 216 for x
5 Estimate the value of radic_
13 to one decimal place without using a calculator
myhrwcom
Focus | Coherence | Rigor
Exercise Depth of Knowledge (DOK) Mathematical Practices
20ndash27 2 SkillsConcepts MP4 Modeling
28 3 Strategic Thinking MP4 Modeling
29ndash32 2 SkillsConcepts MP6 Precision
33 3 Strategic Thinking MP7 Using Structure
34 2 SkillsConcepts MP3 Logic
35 2 SkillsConcepts MP4 Modeling
36 3 Strategic Thinking MP3 Logic
37 3 Strategic Thinking MP7 Using Structure
38 3 Strategic Thinking MP2 Reasoning
8NS1 8NS2 8EE2
8NS1 8NS2 8EE2
Answers1 2625 158
_ 3
2 17 __ 50 1 8 __ 33
3 x = plusmn 3 __ 7
4 x = 6
5 36
13 Lesson 11
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
Work Area
33 Analyze Relationships To find radic_
15 Beau found 3 2 = 9 and 4 2 = 16 He said that since 15 is between 9 and 16 radic
_ 15 must be between 3 and 4 He
thinks a good estimate for radic_
15 is 3 + 4 ____ 2 = 35 Is Beaursquos estimate high low
or correct Explain
34 Justify Reasoning What is a good estimate for the solution to the equation x 3 = 95 How did you come up with your estimate
35 The volume of a sphere is 36π f t 3 What is the radius of the sphere Use the formula V = 4 _ 3 π r 3 to find your answer
36 Draw Conclusions Can you find the cube root of a negative number If so is it positive or negative Explain your reasoning
37 Make a Conjecture Evaluate and compare the following expressions
radic_
4 __ 25 and radic
_ 4 ____
radic_
25 radic
_
16 __ 81 and radic_
16 ____
radic_
81 radic
_
36 __ 49 and radic_
36 ____
radic_
49
Use your results to make a conjecture about a division rule for square roots Since division is multiplication by the reciprocal make a conjecture about a multiplication rule for square roots
38 Persevere in Problem Solving The difference between the solutions to the equation x 2 = a is 30 What is a Show that your answer is correct
FOCUS ON HIGHER ORDER THINKING
His estimate is low because 15 is very close to 16
so radic_
15 is very close to radic_
16 or 4 A better estimate
would be 38 or 39
Sample answer about 45 4 3 = 64 and 5 3 = 125
Because 95 is about halfway between 64 and 125 try 45
45 3 = 91125 which is a good estimate
3 feet
Yes the cube root of a negative number is negative
because a negative number cubed is always negative
and a nonnegative number cubed is always nonnegative
radic_
4 __ 25 = 2 _ 5 = radic
_ 4 ____
radic_
25 radic
_
16 __ 81 = 4 _ 9 = radic_
16 ____
radic_
81 radic
_
36 __ 49 = 6 _ 7 = radic_
36 ____
radic_
49
225 the solutions to x 2 = a are x = plusmn15 and
radic_
a ___
radic_
b = radic
_ a __
b radic
_ a radic
_ b = radic
_ a b
15 - (-15) = 30
Unit 114
copy H
ough
ton
Miff
lin H
arco
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ublis
hing
Com
pany
bull copy
Ilen
e Mac
Dona
ldA
lamy I
mag
es
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=B
8_MCABESE206984_U1M01L1indd 14 102913 1142 PM
Personal Math Trainer
Online Practice and Helpmyhrwcom
Name Class Date
Independent Practice11
20 A 7 __ 16 -inch-long bolt is used in a machine What is this length written as a decimal
21 The weight of an object on the moon is 1 _ 6 its weight on Earth Write 1 _ 6 as a decimal
22 The distance to the nearest gas station is 2 4 _ 5 kilometers What is this distance written as a decimal
23 A baseball pitcher has pitched 98 2 _ 3 innings What is the number of innings written as a decimal
24 A heartbeat takes 08 second How many seconds is this written as a fraction
25 There are 262 miles in a marathon Write the number of miles using a fraction
26 The average score on a biology test was 72
_ 1 Write the average score using a
fraction
27 The metal in a penny is worth about 0505 cent How many cents is this written as a fraction
28 Multistep An artist wants to frame a square painting with an area of 400 square inches She wants to know the length of the wood trim that is needed to go around the painting
a If x is the length of one side of the painting what equation can you set up to find the length of a side How many solutions does the equation have
b Do all of the solutions that you found make sense in the context of the problem Explain
c What is the length of the wood trim needed to go around the painting
Solve each equation for x Write your answers as radical expressions Then estimate to one decimal place if necessary
29 x 2 = 14 30 x 3 = 1331
31 x 2 = 144 32 x 2 = 29
8NS1 8NS2 8EE2
04375 in 01 _6
28 km 98 _6 innings
x 2 = 400 x = plusmnthinsp20 the equation has 2 solutions
x = 20 makes sense but x = -20 doesnrsquot because a
painting cannot have a side length of -20 inches
4 times 20 = 80 inches
x = plusmn radic_
14 asymp plusmn37
x = plusmn radic_
144 = plusmn12 x = plusmn radic_
29 asymp plusmn54
x = 3 radic_ 1331 = 11
4_5 second 26 1_5 mi
72 1 _ 9 101 ___ 200 cent
13Lesson 11
copy H
ough
ton
Miff
lin H
arco
urt P
ublis
hing
Com
pany
bull copy
Phot
odisc
Get
ty Im
ages
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
8_MCAAESE206984_U1M01L1indd 13 41613 1211 AM
myhrwcomActivity available onlineEXTEND THE MATH PRE-AP
Activity Write radic_
09 on the board and invite students to conjecture what the value might be Have them check their conjectures by squaring Invite them to suggest ways to estimate radic
_ 09 As a hint point out that 09 is close to 10 and so they might
use that to help guide their estimates Lead them to see that since 092 is 081 and 102 is 1 the value of radic
_ 09 is greater than 09 and less than 10 Try squaring 095 to get
09025 A good estimate for radic_
09 is 095
Rational and Irrational Numbers 14
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=B
Integers
Rational Numbers IrrationalNumbers
Real Numbers
WholeNumbers
-3-4-5 -2-1 1 2 3 50 4
23
34-4 -π -1 25
radic2
Lesson Support Content Objective Students will learn to describe relationships between sets of numbers
Language Objective Students will explain how to describe relationships between sets of real numbers
LESSON 12 Sets of Real Numbers
Building BackgroundEliciting Prior Knowledge Have students draw a number line from -5 to 5 Ask them to plot points on the number line to approximate the location of rational and irrational numbers such as -1 3 __ 4 25 -4 2 __ 3 radic
_ 2 and -π
Learning ProgressionsIn this lesson students clarify their understanding of the real number system They characterize sets and subsets of the real numbers They also identify sets for real-world situations Important understandings for students include the following
bull Identify all of the possible subsets of the real numbers for a given number
bull Decide whether a statement about a subset of the real numbers is true or false
bull Identify the set of numbers that best describes a real-world situation
Understanding the relationships among the sets of numbers that make up the real numbers is essential as students are introduced to different forms of numbers throughout the school year This lesson provides a foundation for the comparing and ordering of real numbers in the next lesson
Cluster ConnectionsThis lesson provides an excellent opportunity to connect ideas in this cluster Know that there are numbers that are not rational and approximate them by rational numbers Have students copy this diagram which relates the sets of real numbers
Ask students to complete the diagram by writing three examples for each set of numbers Have students share examples and explain how they knew each number they selected belonged in the appropriate set Answers may vary Check studentsrsquo work
Focus | Coherence | Rigor
California Common Core Standards
8NS1 Know that numbers that are not rational are called irrational Understand informally that every number has a decimal expansion for rational numbers show that the decimal expansion repeats eventually and convert a decimal expansion which repeats eventually into a rational number
MP7 Look for and make use of structure
15A
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
Math Talk
Language Support EL
PROFESSIONAL DEVELOPMENT
Linguistic Support EL
AcademicContent Vocabulary
Venn diagrams ndash Students need descriptive language to describe the categories that the different areas and colors of a Venn diagram represent the concept of a set and how sets are distinct or can overlap Use sentence frames such as
The big oval represents __________The darklight blue color in the middle of the
big ovals represents __________These sets overlap because __________
In this way students have the language and structure to identify the criteria that distinguish a set and to explain the abstract representation Also point out the use of the prefix sub- meaning ldquounderrdquo in the term subset
Rules and Patterns
Abbreviations ndash In this lesson the abbreviation mph is used Be sure to point out that mph stands for miles per hour and is used to give units in a rate of speed Students may also have seen mpg (miles per gallon) which gives the units in a rate of fuel efficiency
Borrowed Words ndash Terminology used in baseball such as inning and pitcher may require some explanation Spanish as well as some other languages have borrowed these terms from English so some students may be familiar with these words already Despite this whenever a word is critical to students understanding the word problem it is best to explain the meaning
Leveled Strategies for English Learners
Emerging Allow students to indicate true or false orally in Guided Practice Exercises 9 and 10
Expanding Have students use sentence frames to describe the meaning of regions and colors used in a Venn diagram Then give them similar sentence frames orally and have them draw and shade a Venn diagram based on the oral prompts
Bridging Have students work in groups to draw a Venn diagram to represent sets based on real-world examples in the lesson
To help students answer the question posed in Math Talk provide a sentence frame for their answer
The numbers between 31 and 39 on a number line are __________ because __________
EL
California ELD Standards
Emerging 2II5 Modifying to add details ndash Expand sentences with simple adverbials to provide details about a familiar activity or process
Expanding 2II5 Modifying to add details ndash Expand sentences with adverbials to provide details about a familiar or new activity or process
Bridging 2II5 Modifying to add details ndash Expand sentences with increasingly complex adverbials to provide details about a variety of familiar and new activities and processes
Sets of Real Numbers 15B
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
12L E S S O N
Sets of Real Numbers
EngageESSENTIAL QUESTION
How can you describe relationships between sets of real numbers Sample answer Describe them as two different sets or one set as being a subset of another
Motivate the LessonAsk How many different types of tigers can you name How does the set of Bengal tigers relate to the set of tigers
ExplorePoint to different locations in the Animals diagram and ask for examples for that classification Do the same for the Real Numbers diagram Students should understand that everything within a region is part of the set for example both -3 and 2 are integers
ExplainEXAMPLE 1
Questioning Strategies Mathematical Practices bull In A why is 5 not a perfect square It does not have rational numbers as its square roots
bull Can the number in B be written as a fraction Why or why not Yes it is a terminating decimal so it is a rational number
Engage with the WhiteboardHave students place the numbers in Example 1 and Additional Example 1 in the Venn diagram for numbers
YOUR TURNAvoid Common ErrorsBe sure that students read Exercise 2 carefully before answering The number given in the problem 10 is the area not the side length
EXAMPLE 2Questioning Strategies Mathematical Practices bull What two major sets are the real numbers composed of rational and irrational numbers
bull What is the location of the set of whole numbers in the Venn diagram in relation to the set of rational numbers Explain Inside it whole numbers are rational numbers
Focus on Reasoning Mathematical PracticesRemind students that it takes only one counterexample to show that a statement is false
Interactive Whiteboard Interactive example available online
ADDITIONAL EXAMPLE 1Write all names that apply to each number
A -10integer rational real
B 12 _ 3
whole integer rational real
myhrwcom
Interactive Whiteboard Interactive example available online
ADDITIONAL EXAMPLE 2Tell whether the given statement is true or false Explain your choice
No integers are whole numbers
False every whole number is also an integer
myhrwcom
Animated MathClassifying Numbers
Students build fluency in classifying numbers in this engaging fast-paced game
myhrwcom
CA Common CoreStandards
The student is expected to
The Number Systemmdash8NS1
Know that numbers that are not rational are called irrational Understand informally that every number has a decimal expansion for rational numbers show that the decimal expansion repeats eventually and convert a decimal expansion which repeats eventually into a rational numberMathematical Practices
MP7 Using Structure
The student is expected to
15 Lesson 12
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=B
Math TrainerOnline Practice
and Help
Personal
myhrwcom
Math TrainerOnline Practice
and Help
Personal
myhrwcom
Math On the Spotmyhrwcom
Understanding Sets and Subsets of Real NumbersBy understanding which sets are subsets of types of numbers you can verify whether statements about the relationships between sets are true or false
Tell whether the given statement is true or false Explain your choice
All irrational numbers are real numbers
True Every irrational number is included in the set of real numbers The irrational numbers are a subset of the real numbers
No rational numbers are whole numbers
False A whole number can be written as a fraction with a denominator of 1 so every whole number is included in the set of rational numbers The whole numbers are a subset of the rational numbers
EXAMPLE 2
A
B
Write all names that apply to each number
1 A baseball pitcher has pitched 12 2 _ 3 innings
2 The length of the side of a square that has an
area of 10 square yards
YOUR TURN
Tell whether the given statement is true or false Explain your choice
3 All rational numbers are integers
4 Some irrational numbers are integers
YOUR TURN
Give an example of a rational number that is a
whole number Show that the number is both whole
and rational
Math TalkMathematical Practices
Give an example of a
8NS1
False Every integer is a rational number but not every
False Real numbers are either rational or irrational numbers
Integers are rational numbers so no integers are irrational numbers
rational real
irrational real
Sample answer 8 8 = 8_
1
and -thinsp 5 _ 2 are not integers
rational number is an integer Rational numbers such as 3 _ 5
Unit 116
copy H
ough
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ublis
hing
Com
pany
bull Im
age C
redi
ts D
igita
l Im
age c
opyr
ight
copy20
04 Ey
ewire
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
8_MCAAESE206984_U1M01L2indd 16 41613 136 AM
Math On the Spot
myhrwcom
Vertebrates
Birds
Passerines
Animals
Integers
Rational Numbers IrrationalNumbers
Real Numbers
WholeNumbers
1
45
3
0
274
67
radic4
-
-3
-2
-1
03
radic2
radic17
radic11-
π
Animated Math
myhrwcom
Classifying Real NumbersBiologists classify animals based on shared characteristics A cardinal is an animal a vertebrate a bird and a passerine
You already know that the set of rational numbers consists of whole numbers integers and fractions The set of real numbers consists of the set of rational numbers and the set of irrational numbers
Write all names that apply to each number
radic_
5 irrational real
ndash1784rational real
whole integer rational real
EXAMPLEXAMPLE 1
A
B
C radic_ 81 ____ 9
L E S S O N
12Sets of Real Numbers
ESSENTIAL QUESTIONHow can you describe relationships between sets of real numbers
Passerines such as the cardinal are also called ldquoperching birdsrdquo
What types of numbers are between 31 and 39 on a
number line
Math TalkMathematical Practices
What types of numbers are
8NS1
8NS1
Know that numbers that are not rational are called irrational Understand informally that every number has a decimal expansion for rational numbers show that the decimal expansion repeats eventually and convert a decimal expansion which repeats eventually into a relation number
ndash1784 is a terminating decimal
5 is a whole number that is not a perfect square
radic_
81 _____ 9 = 9 __ 9 = 1 rational irrational real
15Lesson 12
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ough
ton
Miff
lin H
arco
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ublis
hing
Com
pany
bull Im
age C
redi
ts copy
Wiki
med
ia Co
mm
ons
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=B
8_MCABESE206984_U1M01L2indd 15 061113 1144 AM
PROFESSIONAL DEVELOPMENT
Math BackgroundThe relationships between sets of numbers extend to include complex numbers A complex number can be written as a sum of a real number a and an imaginary number bi
a + bi
An imaginary number is a special number that when squared gives a negative value When you square a real number you get a nonnegative number When you square an imaginary number you get a negative value The imaginary unit is i
i = radic_
-1
Integrate Mathematical Practices MP7
This lesson provides an opportunity to address this Mathematical Practices standard It calls for students to discern structure to connect and communicate mathematical ideas
Students use a Venn diagram to structure relationships between sets of numbers They connect and communicate mathematical ideas when they make logical statements about the sets and describe which set best describes numbers applied to real-life situations
Sets of Real Numbers 16
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=B
YOUR TURNAvoid Common ErrorsStudents may see the word ldquoAllldquo or rdquoNordquo in Exercises 3 and 4 and immediately assume that any absolute statements like these are false Remind them that there are true statements that begin with these words and encourage them to provide examples
EXAMPLE 3Questioning Strategies Mathematical Practices bull In A how does the phrase ldquonumber of rdquo give you a clue about the number classification It indicates a counting number
bull What is the relationship between the circumference of a circle and the diameter The circumference is diameter times π
Focus on Critical Thinking Mathematical PracticesIn B suppose the diameters in inches were 25
__ π 28 __ π
31 __ π and so on What set of numbers would
best describe the circumferences Explain Whole numbers the circumferences would be the whole numbers 25 28 31 and so on
YOUR TURNFocus on Critical Thinking Mathematical PracticesHave students compare and contrast the classification of numbers in the answers in Exercises 5 and 6
ElaborateTalk About ItSummarize the Lesson
Ask What are some ways that number sets can be related Sets may be subsets of other sets or they may be separate from other sets
GUIDED PRACTICEEngage with the Whiteboard
Have students place the numbers in Exercises 1ndashthinsp8 in the Venn diagram for numbers at the beginning of the lesson
Integrating Language Arts EL
Encourage English learners to ask for clarification on any terms or phrases that they do not understand
Avoid Common ErrorsExercise 7 Remind students that a repeating decimal is a rational numberExercises 9ndash10 Remind students that it only takes one counterexample to show that a statement is false
Interactive Whiteboard Interactive example available online
ADDITIONAL EXAMPLE 3Identify the set of numbers that best describes the situation Explain your choice
A the amount of time that has passed since midnight
The set of real numbers time is continuous so the amount of time can be rational or irrational
B the number of tickets sold to a basketball game
The set of whole numbers the number of tickets sold may be 0 or a counting number
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17 Lesson 12
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
1IN
116 inch
Guided Practice
Write all names that apply to each number (Example 1)
1 7 _ 8 2 radic_
36
3 radic_
24 4 075
5 0 6 - radic_ 100
7 5 _
45 8 - 18 __ 6
Tell whether the given statement is true or false Explain your choice (Example 2)
9 All whole numbers are rational numbers
10 No irrational numbers are whole numbers
Identify the set of numbers that best describes each situation Explain your choice (Example 3)
11 the change in the value of an account when given to the nearest dollar
12 the markings on a standard ruler
13 What are some ways to describe the relationships between sets of numbers
CHECK-INESSENTIAL QUESTION
rational real
rational real
True Whole numbers are rational numbers
Rational numbers the ruler is marked every 1 __ 16 th inch
Sample answer Describe one set as being a subset of
another or show their relationships in a Venn diagram
Integers the change can be a whole dollar amount
and can be positive negative or zero
True Whole numbers are a subset of the set of rational numbers
and can be written as a ratio of the whole number to 1
irrational real
whole integer rational real
whole integer rational real
rational real
integer rational real
integer rational real
Unit 118
copy H
ough
ton
Miff
lin H
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ublis
hing
Com
pany
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
8_MCAAESE206984_U1M01L2indd 18 41613 136 AM
My Notes
Math TrainerOnline Practice
and Help
Personal
myhrwcom
Math On the Spot
myhrwcom
Identifying Sets for Real-World SituationsReal numbers can be used to represent real-world quantities Highways have posted speed limit signs that are represented by natural numbers such as 55 mph Integers appear on thermometers Rational numbers are used in many daily activities including cooking For example ingredients in a recipe are often given in fractional amounts such as 2 _ 3 cup flour
Identify the set of numbers that best describes each situation Explain your choice
the number of people wearing glasses in a room
The set of whole numbers best describes the situation The number of people wearing glasses may be 0 or a counting number
the circumference of a flying disk has a diameter of 8 9 10 11 or 14 inches
The set of irrational numbers best describes the situation Each circumference would be a product of π and the diameter and any multiple of π is irrational
EXAMPLEXAMPLE 3
A
B
Identify the set of numbers that best describes the situation Explain your choice
5 the amount of water in a glass as it evaporates
6 the weight of a person in pounds
YOUR TURN
8NS1
Rational numbers a personrsquos weight can be a decimal
such as 835 pounds
Real numbers the amount can be any number greater
than 0
17Lesson 12
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ough
ton
Miff
lin H
arco
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ublis
hing
Com
pany
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
8_MCAAESE206984_U1M01L2indd 17 41613 520 AM
Graphic OrganizersGive students a list of numbers (including terminating and repeating decimals fractions integers and rational and irrational square roots) and a graphic organizer as shown below
Real Numbers
Rational numbers Irrational numbers
Integer numbers
Whole numbers
Ask students to write each number in the list in the correct section of the organizer
Number SensePoint out to students that knowing the types of numbers to expect in different situations can alert them to incorrect math as well as to impossible situations For example 135 shots made in basketballs is not possible but an average number of shots can equal 135
Additional ResourcesDifferentiated Instruction includes bull Reading Strategies bull Success for English Learners EL
bull Reteach bull Challenge PRE-AP
DIFFERENTIATE INSTRUCTION
Sets of Real Numbers 18
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
Math TrainerOnline Assessment
and Intervention
Personal
Online homework assignment available
myhrwcom
Lesson Quiz available online
12 LESSON QUIZ
1 Write all the names that apply to the number
2 Tell whether the given statement is true or false Explain your choice All numbers between 1 and 2 are rational numbers
3 Identify the set of numbers that best describes the situation Explain your choiceThe choices on a survey question change the total points for the survey by -2 -1 0 1 or 2 points
-1 _
5
myhrwcom
Additional ResourcesDifferentiated Instruction includes bull Leveled Practice worksheets
EvaluateGUIDED AND INDEPENDENT PRACTICE
Concepts amp Skills Practice
Example 1Classifying Real Numbers
Exercises 1ndash8 14ndash19 22ndash24
Example 2Understanding Sets and Subsets of Real Numbers
Exercises 9ndash10
Example 3Identifying Sets for Real-World Situations
Exercises 11ndash12 20ndash21 25
Focus | Coherence | Rigor
Exercise Depth of Knowledge (DOK) Mathematical Practices
14ndash19 2 SkillsConcepts MP7 Using Structure
20ndash21 2 SkillsConcepts MP6 Precision
22ndash23 2 SkillsConcepts MP3 Logic
24 1 Recall of Information MP7 Using Structure
25 2 SkillsConcepts MP2 Reasoning
26ndash27 3 Strategic Thinking MP3 Logic
28 3 Strategic Thinking MP8 Patterns
29 3 Strategic Thinking MP3 Logic
8NS1
8NS1
Exercise 29 combines concepts from the California Common Core cluster ldquoKnow that there are numbers that are not rational and approximate them by rational numbersrdquo
Answers1 rational real
2 False radic_
2 is an example of an irrational number between 1 and 2
3 Integers each number is an integer but only three are whole numbers
19 Lesson 12
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
Work Area
π mi23 Critique Reasoning The circumference of a circular region is shown
What type of number best describes the diameter of the circle Explain
your answer
24 Critical Thinking A number is not an integer What type of number can it be
25 A grocery store has a shelf with half-gallon containers of milk What type of number best represents the total number of gallons
26 Explain the Error Katie said ldquoNegative numbers are integersrdquo What was her error
27 Justify Reasoning Can you ever use a calculator to determine if a number is rational or irrational Explain
28 Draw Conclusions The decimal 0 _
3 represents 1 _ 3 What type of number best describes 0
_ 9 which is 3 middot 0
_ 3 Explain
29 Communicate Mathematical Ideas Irrational numbers can never be precisely represented in decimal form Why is this
FOCUS ON HIGHER ORDER THINKING
It can be a rational number that is not an integer or an irrational number
rational number
The set of negative numbers also includes non-integer
rational numbers and irrational numbers
Sample answer If the calculator shows a decimal that
terminates in fewer digits than what the calculator screen
allows then you can tell that the number is rational If not
you cannot tell from the calculator display whether the
number terminates because you see a limited number
of digits It may be a repeating decimal (rational) or
non-terminating non-repeating decimal (irrational)
Whole 3 middot 0 _
3 represents 3 middot 1 _ 3 = 1 so 0 _
9 is exactly 1
Sample answer In decimal form irrational numbers never
terminate and never repeat Therefore no matter how
many decimal places you include the number will never
be precisely represented There are always more digits
Whole the diameter is π _ π = 1 mile
Unit 120
copy H
ough
ton
Miff
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ublis
hing
Com
pany
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
8_MCAAESE206984_U1M01L2indd 20 120413 909 PM
Integers
Rational Numbers Irrational Numbers
Real Numbers
Whole Numbers
257
radic16
166
radic9
128 radic50
Personal Math Trainer
Online Practice and Helpmyhrwcom
Name Class Date
Independent Practice
Identify the set of numbers that best describes each situation Explain your choice
20 the height of an airplane as it descends to an airport runway
21 the score with respect to par of several golfers 2 ndash 3 5 0 ndash 1
22 Critique Reasoning Ronald states that the number 1 __ 11 is not rational because when converted into a decimal it does not terminate Nathaniel says it is rational because it is a fraction Which boy is correct Explain
12
14 - radic_
9 15 257
16 radic_
50 17 8 1 _ 2
18 166 19 radic_
16
Write all names that apply to each number Then place the numbers in the correct location on the Venn diagram
8NS1
Real numbers the height can be any number greater than zero
integer rational real whole integer rational real
whole integer rational real
irrational real
rational real
rational real
Integers the scores are counting numbers their
opposites and zero
Nathaniel is correct A rational number is a number that can be written as a fraction and 1 __ 11 is a fraction
19Lesson 12
copy H
ough
ton
Miff
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Com
pany
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8_MCAAESE206984_U1M01L2indd 19 41613 136 AM
myhrwcomActivity available onlineEXTEND THE MATH PRE-AP
Activity Have students consider the concept of restricted domain for the sets of numbers that describe situations For example the number of sisters a person has can best be described by whole numbers but no one has ever had 1500 sisters An area code is an integer or whole number between 200 and 999
Have students use a source such as the Guinness Book of World Records and give examples of sets of numbers that describe situations where the domain is restricted Ask whether the restriction may be changed in the future
Sets of Real Numbers 20
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
-3-4-5 -2-1 1 2 3 50 4
12-4 -radic5
Lesson Support Content Objective Students will learn to order a set of real numbers
Language Objective Students will show and describe how to order a set of real numbers
LESSON 13 Ordering Real Numbers
Building BackgroundEliciting Prior Knowledge Have students draw a number line to compare a rational number and an irrational number such as - radic
_ 5 and -4 1 __ 2 Ask them to explain how
they approximated the irrational number on the number line Then have them identify the greater and the lesser real number Repeat with several other pairs of real numbers in different forms
Learning ProgressionsIn this lesson students order a set of real numbers They use rational approximations to compare the sizes of irrational numbers They also order numbers for real-world situations Important understandings for students include the following
bull Compare irrational numbers bull Estimate the value of expressions with irrational numbers bull Order a set of real numbers bull Order real numbers in a real-world context
Work with real numbers continues throughout Grade 8 and into high school This lesson provides students with a foundation for understanding the relative sizes of numbers in different forms in the real number system
Cluster ConnectionsThis lesson provides an excellent opportunity to connect ideas in this cluster Know that there are numbers that are not rational and approximate them by rational numbers Tell students that there is a special number called the golden ratio with applications in mathematics geometry art and architecture The golden ratio is called phi and is represented by the Greek letter ϕ It includes an irrational number in its definition
Have students explain why the golden ratio is irrational Ask them to find the two whole numbers the golden ratio lies between Then challenge them to approximate the golden ratio to the nearest tenth It is irrational because it includes an irrational number in its definition It lies between 1 and 2 To the nearest tenth ϕ = 16
ϕ = 1 + radic_
5 _ 2
Focus | Coherence | Rigor
California Common Core Standards
8NS2 Use rational approximations of irrational numbers to compare the size of irrational numbers locate them approximately on a number line diagram and estimate the value of expressions (eg π 2 )
MP4 Model with mathematics
21A
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
Math Talk
Language Support EL
PROFESSIONAL DEVELOPMENT
Linguistic Support EL
AcademicContent Vocabulary
Post a chart like this to remind students of the regular comparative forms of adjectives that use the -er and -est suffixes Add to the chart for terms that appear in examples and exercises in each lesson Include any irregular verb forms
Background Knowledge
Go On ndash the title of the module review or quiz is Ready to Go On This title uses an idiomatic expression In this context to go on means ldquoto move aheadrdquo or ldquoto proceedrdquo It is different from the use of go on that means having enough facts to use meaningfully as in having enough to go on Also the intonation used in pronouncing an expression can give it different meanings For example when the speaker emphasizes the word on he or she might be expressing disbelief as in ldquoGo ON Yoursquore kidding rightrdquo Discuss with students other ways that the phrase go on may be used
Leveled Strategies for English Learners
Emerging Label points on a number line with the terms used in ordering greater greatest less lesser least Use sentence frames to insert the correct terms
Expanding Have students give two or three complete sentences to compare the placement of numbers on a number line using the correct forms of the comparative and superlative adjectives
Bridging Have students work in pairs with one student giving directions to the other in complete sentences to order numbers on a number line
To help students answer the question posed in Math Talk make sure that students have a command of the forms for making comparisons and the superlative and the concept of opposite order so that the focus is on the math concept instead of the language skills needed to describe and explain order
EL
Adjective Comparative Superlative
Far Farther Farthest
Large Larger Largest
Great Greater Greatest
Some Less Least
Some More Most
California ELD Standards
Emerging 2I8 Analyzing language choices ndash Explain how phrasing or different common words with similar meanings produce different effects on the audience
Expanding 2I8 Analyzing language choices ndash Explain how phrasing or different words with similar meanings or figurative language produce shades of meaning and different effects on the audience
Bridging 2I8 Analyzing language choices ndash Explain how phrasing or different words with similar meanings or figurative language produce shades of meaning nuances and different effects on the audience
Ordering Real Numbers 21B
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
13L E S S O N
Ordering Real Numbers
Interactive Whiteboard Interactive example available online
ADDITIONAL EXAMPLE 1Compare Write lt gt or =
A radic_
8 - 2 4 - radic_
8 lt
B radic_
20 + 1 3 + radic_
2 gt
EngageESSENTIAL QUESTION
How do you order a set of real numbers Sample answer Find their approximate decimal values and order them
Motivate the LessonAsk What kind of numbers are you comparing when you compare the price of gasoline at two different gas stations
ExploreGive students two rational numbers and ask them to name a number between them Repeat a few times and then give them two irrational numbers and ask them to name a number between them
ExplainEXAMPLE 1
Questioning Strategies Mathematical Practices bull Which is greater the difference between 5 and 3 or the difference between radic
_ 5 and radic
_ 3
The difference between 5 and 3 is 2 the difference between radic_
5 and radic_
3 is approximately 1 So the difference between 5 and 3 is greater
Avoid Common ErrorsCaution students to read the problem carefully and think about what the radical sign means so that they do not misread the problem and answer that the two sides are equal
YOUR TURNFocus on TechnologyCalculators should not be used at this point because developing number sense is the goal
EXAMPLE 2Questioning Strategies Mathematical Practices bull How do you determine whether radic
_ 22 is less than or greater than 45 The square of 45 is
2025 which is less than 22 so the square root of 22 must be greater than 45
Engage with the WhiteboardHave students graph and label various real numbers between 42 and 44 and between 47 and 5
YOUR TURNFocus on Modeling Mathematical PracticesHave students label the integers on the number line with their equivalent square root For example 1 2 and 3 on the number line would be labeled radic
_ 1 radic
_ 4 and radic
_ 9
Interactive Whiteboard Interactive example available online
ADDITIONAL EXAMPLE 2Order 3π radic
_ 10 and 325 from greatest
to least
3π 325 radic_
10
myhrwcom
myhrwcom
CA Common CoreStandards
The student is expected to
The Number Systemmdash8NS2
Use rational approximations of irrational numbers to compare the size of irrational numbers locate them approximately on a number line diagram and estimate the value of expressions (eg π 2 )
Mathematical Practices
MP4 Modeling
The student is expected to
21 Lesson 13
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
Math TrainerOnline Practice
and Help
Personal
myhrwcom
Math On the Spotmyhrwcom
0 05 1 15 2 25 3 35 4
radic5radic3
π2
8 85 9 95 10 105 11 115 12
radic75
4 42 44 46 48 5
radic224 12π + 1
Ordering Real Numbers You can compare and order real numbers and list them from least to greatest
Order radic_
22 π + 1 and 4 1 _ 2 from least to greatest
First approximate radic_
22
radic_
22 is between 4 and 5 Since you donrsquot know where it falls between 4 and 5 you need to find a better estimate for radic
_ 22 so
you can compare it to 4 1 _ 2
Since 22 is closer to 25 than 16 use squares of numbers between 45 and 5 to find a better estimate of radic
_ 22
45 2 = 2025 46 2 = 2116 47 2 = 2209 48 2 = 2304
Since 47 2 = 2209 an approximate value for radic_
22 is 47
An approximate value of π is 314 So an approximate value of π +1 is 414
Plot radic_
22 π + 1 and 4 1 _ 2 on a number line
Read the numbers from left to right to place them in order from least to greatest
From least to greatest the numbers are π + 1 4 1 _ 2 and radic_
22
EXAMPLE 2
STEP 1
STEP 2
Order the numbers from least to greatest Then graph them on the number line
YOUR TURN
5 radic_
5 25 radic_
3
6 π 2 10 radic_
75
If real numbers a b and c are in order from least to greatest what is the order
of their opposites from least to greatest
Explain
Math TalkMathematical Practices
8NS2
radic_
3 radic_
5 25
radic_
75 π2 10
Math Talk answer -c -b -a -c is farthest to the left on a number line -b is in the middle and -a is farthest to the right
Unit 122
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ough
ton
Miff
lin H
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hing
Com
pany
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
8_MCAAESE206984_U1M01L3indd 22 41613 447 AM
My Notes
Math TrainerOnline Practice
and Help
Personal
myhrwcom
Math On the Spot
myhrwcom
Comparing Irrational NumbersBetween any two real numbers is another real number To compare and order real numbers you can approximate irrational numbers as decimals
Compare radic_
3 + 5 3 + radic_
5 Write lt gt or =
First approximate radic_
3
radic_
3 is between 1 and 2
Next approximate radic_
5
radic_
5 is between 2 and 3
Then use your approximations to simplify the expressions
radic_
3 + 5 is between 6 and 7
3 + radic_
5 is between 5 and 6
So radic_
3 + 5 gt 3 + radic_
5
Reflect1 If 7 + radic
_ 5 is equal to radic
_ 5 plus a number what do you know about the
number Why
2 What are the closest two integers that radic_
300 is between
EXAMPLEXAMPLE 1
STEP 1
STEP 2
Compare Write lt gt or =
YOUR TURN
3 radic_
2 + 4 2 + radic_
4 4 radic_
12 + 6 12 + radic_
6
L E S S O N
13 Ordering Real Numbers
ESSENTIAL QUESTIONHow do you order a set of real numbers
8NS2
Use rational approximations of irrational numbers to compare the size of irrational numbers locate them approximately on a number line diagram and estimate the value of expressions (eg π 2 )
8NS2
Use perfect squares to estimate square roots
1 2 = 1 2 2 = 4 3 2 = 9
The number is 7 both expressions must equal 7 + radic_
5
17 and 18
gt lt
21Lesson 13
copy H
ough
ton
Miff
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arco
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ublis
hing
Com
pany
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
8_MCAAESE206984_U1M01L3indd 21 41913 246 PM
PROFESSIONAL DEVELOPMENT
Math BackgroundIn this lesson students estimate irrational numbers in the form of square roots of nonper-fect squares by finding two perfect squares between which the number falls A more precise method involves repeated division For example to find radic
_ 28 find a whole number whose perfect
square is close to 28 such as 5 Divide 28 by that number 28 divide 5 = 56 Find the average of the quotient and divisor 5 + 56
_____ 2 = 53 Continue dividing 28 by each result and averaging until you get the desired accuracy
Integrate Mathematical Practices MP4
This lesson provides an opportunity to address this Mathematical Practices standard It calls for students to model relationships using multiple representations including diagrams graphs and language as appropriate Students use multiple representations when they use number lines to estimate the locations of and order rational and irrational numbers given as symbols
Ordering Real Numbers 22
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
Interactive Whiteboard Interactive example available online
ADDITIONAL EXAMPLE 3The diameter of a meteorite in millimeters is calculated by four different methods Order the results from least to greatest
Joe radic_
18 mm Lisa 13 __ 3 mm
Pablo 46 mm Julien 4π __ 3 mm
Julien 4π __ 3 mm Lisa 13 __ 3 mm
Joe radic_
18 mm Pablo 46 mm
EXAMPLE 3Questioning Strategies Mathematical Practices bull How can you verify that radic
_ 28 is between 52 and 53 5 2 2 = 2704 and 5 3 2 = 2809
bull Explain how to determine which number is greater 5 _
5 or 55 When the repeating decimal is rounded to the nearest tenth or hundredth you can see that it is greater
Connect to Daily LifeDiscuss how measuring across a canyon might involve different methods than measuring along a road Explain that measurements like these are often done using calculations that approximate the distance
YOUR TURNFocus on Critical Thinking Mathematical PracticesDiscuss with students which number is greater 3
_ 45 or 3450 3
_ 45 or 3455 and why Explain
that 3 _
45 can be written out as 34545hellipMake sure they understand that 3 _
45 is greater than 345 but less than 3455
ElaborateTalk About ItSummarize the Lesson
Ask How can you order two numbers in different forms whose decimal approxi-mations appear to be equal Approximate one or both numbers to an additional
number of decimal places
GUIDED PRACTICEEngage with the Whiteboard
Have students place and label additional points on the number line in Exercise 9 Allow the points to be in any format other than decimal
Avoid Common ErrorsExercises 3ndash4 Caution students to read the problem carefully so that they do not misread the problem as the same numbers combined by addition on each side of the circleExercise 10 Remind students that the calculations have units
myhrwcom
23 Lesson 13
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
0 05 1 15 2 25 3 35 4 45 5 55 6 65 7
2πradic3
Compare Write lt gt or = (Example 1)
1 radic_
3 + 2 radic_
3 + 3 2 radic_
8 + 17 radic_
11 + 15
3 radic_
6 + 5 6 + radic_
5 4 radic_
9 + 3 9 + radic_
3
5 radic_
17 - 3 -2 + radic_
5 6 12 - radic_
2 14 - radic_
8
7 radic_
7 + 2 radic_
10 - 1 8 radic_
17 + 3 3 + radic_
11
9 Order radic_
3 2π and 15 from least to greatest Then graph them on the number line (Example 2)
radic_
3 is between and so radic_
3 asymp
π asymp 314 so 2π asymp
From least to greatest the numbers are
10 Four people have found the perimeter of a forest using different methods Their results are given in the table Order their calculations from greatest to least (Example 3)
11 Explain how to order a set of real numbers
CHECK-INESSENTIAL QUESTION
Forest Perimeter (km)
Leon Mika Jason Ashley
radic_
17 - 2 1 +thinsp π __ 2 12 ___ 5 25
Guided Practice
17
15
1 + π _ 2 km 25 km 12 __ 5 km radic_
17 - 2 km
2π radic
_ 3
18 175
628
Sample answer Convert each number to a decimal
equivalent using estimation to find equivalents for
irrational numbers Graph each number on a number line
Read the numbers from left to right for least to greatest
Read the numbers from right to left for greatest to least
lt gt
lt lt
ltgt
gt gt
24 Unit 1
copy H
ough
ton
Miff
lin H
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ublis
hing
Com
pany
bull Im
age C
redi
ts copy
Elena
Eliss
eeva
Alam
y Im
ages
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
8_MCAAESE206984_U1M01L3indd 24 41613 448 AM
My Notes
5 52 54 56 58 6
radic28 5 12
23455
Math TrainerOnline Practice
and Help
Personal
myhrwcom
Math On the Spot
myhrwcom
Ordering Real Numbers in a Real-World Context Calculations and estimations in the real world may differ It can be important to know not only which are the most accurate but which give the greatest or least values depending upon the context
Four people have found the distance in kilometers across a canyon using different methods Their results are given in the table Order the distances from greatest to least
Distance Across Quarry Canyon (km)
Juana Lee Ann Ryne Jackson
radic_
28 23 __ 4 5 _
5 5 1 _ 2
Write each value as a decimal
radic_
28 is between 52 and 53 Since 53 2 = 2809 an approximate value for radic
_ 28 is 53
23 __ 4 = 575
5 _
5 is 5555hellip so 5 _
5 to the nearest hundredth is 556
5 1 _ 2 = 55
Plot radic_
28 23 __ 4 5 _
5 and 5 1 _ 2 on a number line
From greatest to least the distances are
23 __ 4 km 5 _
5 km 5 1 _ 2 km radic_
28 km
EXAMPLEXAMPLE 3
STEP 1
STEP 2
7 Four people have found the distance in miles across a crater using different methods Their results are given below
Jonathan 10 __ 3 Elaine 3 _
45 Joseacute 3 1 _ 2 Lashonda radic_
10
Order the distances from greatest to least
YOUR TURN
8NS2
3 1 _ 2 mi 3 _
45 mi 10 __ 3 mi radic_
10 mi
23Lesson 13
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Miff
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pany
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8_MCAAESE206984_U1M01L3indd 23 41613 447 AM
ModelingPlace papers around the room with the numbers from 1 to 5 one per sheet Give each student a card showing a number between 1 and 5 in different forms Have students place his or her card between the correct integers and decide where the number goes in relation to any numbers already placed
Multiple RepresentationsGive students a vertical number line which some students might find easier to use than a horizontal one Have them decide whether to place points for rational and irrational numbers above or below existing points
Additional ResourcesDifferentiated Instruction includes bull Reading Strategies bull Success for English Learners EL
bull Reteach bull Challenge PRE-AP
DIFFERENTIATE INSTRUCTION
Ordering Real Numbers 24
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
Math TrainerOnline Assessment
and Intervention
Personal
Online homework assignment available
myhrwcom
myhrwcom
Lesson Quiz available online
13 LESSON QUIZ
1 Compare Write lt gt or =
radic_
95 - 5 radic_
62 - 2
2 Order 105 radic_
105 and 3π + 1 from greatest to least
3 A length in centimeters is calculated differently by four different people Order their calculations from least to greatest
KD 11 __ 2 cm Silvio 5 __ 3 π cm
Paula 5 _
4 cm Luis radic_
33 cm
EvaluateGUIDED AND INDEPENDENT PRACTICE
Concepts amp Skills Practice
Example 1Comparing Irrational Numbers
Exercises 1ndash8
Example 2Ordering Real Numbers
Exercises 9 12ndash15 18ndash21
Example 3Ordering Real Numbers in a Real-World Context
Exercises 10 16ndash17
Additional ResourcesDifferentiated Instruction includes bull Leveled Practice worksheets
Focus | Coherence | Rigor
Exercise Depth of Knowledge (DOK) Mathematical Practices
12ndash15 1 Recall of Information MP5 Using Tools
16 2 SkillsConcepts MP2 Reasoning
17 2 SkillsConcepts MP6 Precision
18ndash21 2 SkillsConcepts MP2 Reasoning
22 3 Strategic Thinking MP4 Modeling
23ndash24 3 Strategic Thinking MP3 Logic
8NS2
8NS2
Answers1 radic
_ 95 - 5 lt radic
_ 62 - 2
2 radic_
105 3π + 1 105
3 Silvio 5 __ 3 π cm Paula 5 _
4 cm
KD 11
__ 2 cm Luis radic_
33 cm
25 Lesson 13
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
Work Area
3140 3141 3142 3143
314 π227
20 A teacher asks his students to write the numbers shown in order from least to greatest Paul thinks the numbers are already in order Sandra thinks the order should be reversed Who is right
21 Math History There is a famous irrational number called Eulerrsquos number symbolized with an e Like π its decimal form never ends or repeats The first few digits of e are 27182818284
a Between which two square roots of integers could you find this number
b Between which two square roots of integers can you find π
22 Analyze Relationships There are several approximations used for π including 314 and 22 __ 7 π is approximately 314159265358979
a Label π and the two approximations on the number line
b Which of the two approximations is a better estimate for π Explain
c Find a whole number x so that the ratio x ___ 113 is a better estimate for π
than the two given approximations
23 Communicate Mathematical Ideas If a set of six numbers that include both rational and irrational numbers is graphed on a number line what is the fewest number of distinct points that need to be graphed Explain
24 Critique Reasoning Jill says that 12 _
6 is less than 1263 Explain her error
FOCUS ON HIGHER ORDER THINKING
radic_
115 115 ___ 11 and 105624
between radic_
7 asymp 265 and radic_
8 asymp 283
between radic_
9 = 3 and radic_
10 asymp 316
22 __ 7 it is closer to π on the number line
She did not consider the repeating digit 1266
2 rational numbers can have the same location and
irrational numbers can have the same location but they
cannot share a location
355
Neither student is correct The answer
should be 115 ___ 11 105624 radic_
115
Unit 126
copy H
ough
ton M
ifflin
Har
cour
t Pub
lishin
g Com
pany
Imag
e Cre
dits
copy3D
Stoc
kiSt
ockP
hoto
com
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
8_MCAAESE206984_U1M01L3indd 26 210513 801 AM
Personal Math Trainer
Online Practice and Helpmyhrwcom
Name Class Date
Independent Practice
16 Your sister is considering two different shapes for her garden One is a square with side lengths of 35 meters and the other is a circle with a diameter of 4 meters
a Find the area of the square
b Find the area of the circle
c Compare your answers from parts a and b Which garden would give your sister the most space to plant
17 Winnie measured the length of her fatherrsquos ranch four times and got four different distances Her measurements are shown in the table
a To estimate the actual length Winnie first approximated each distance to the nearest hundredth Then she averaged the four numbers Using a calculator find Winniersquos estimate
b Winniersquos father estimated the distance across his ranch to be radic_
56 km How does this distance compare to Winniersquos estimate
Give an example of each type of number
18 a real number between radic_
13 and radic_
14
19 an irrational number between 5 and 7
Order the numbers from least to greatest
12 radic_
7 2 radic_
8 ___ 2 13 radic_
10 π 35
14 radic_
220 -10 radic_
100 115 15 radic_
8 -375 3 9 _ 4
Distance Across Fatherrsquos Ranch (km)
1 2 3 4
radic_
60 58 __ 8 7 _
3 7 3 _ 5
138NS2
radic_
8 ___ 2 2 radic_
7
-10 radic_
100 115 radic_
220
radic_
60 asymp 775 58 __ 8 = 725 7 _
3 asymp 733 7 3 _ 5 = 760 so the average
π radic_
10 35
-375 9 _ 4 radic_
8 3
is 74825 km
1225 m2
4π m2 or approximately 126 m2
They are nearly identical radic_
56 is approximately 74833hellip
The circle would give her more space to plant because it has a
larger area
Sample answer 37
Sample answer radic_
31
25Lesson 13
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ough
ton
Miff
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pany
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8_MCAAESE206984_U1M01L3indd 25 41613 448 AM
Activity available online myhrwcomEXTEND THE MATH PRE-AP
Activity Have students investigate whether there are infinitely many numbers between two numbers by giving examples for each of the following
bull Between any two rational numbers there is at least one other rational number Sample answer 45 is between 41 and 48
bull Between any two irrational numbers there is at least one rational number Sample answer 45 is between radic
_ 11 and radic
_ 29
bull Between any two rational numbers there is at least one irrational number Sample answer radic
_ 11 is between 31 and 36
bull Between any two irrational numbers there is at least one irrational number Sample answer radic
_ 17 is between radic
_ 11 and radic
_ 29
Ordering Real Numbers 26
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
ReadyMath Trainer
Online Practiceand Help
Personal
myhrwcom
Module Quiz
11ensp RationalenspandenspIrrationalenspNumbersWrite each fraction as a decimal or each decimal as a fraction
1 7__20 2 1___
27 3 17_8
Solve each equation for x
4 x2=81 5 x3=343 6 x2= 1___100
7 Asquarepatiohasanareaof200squarefeetHowlongiseachside
ofthepatiotothenearesttenth
12ensp SetsenspofenspRealenspNumbersWrite all names that apply to each number
8 121____radic
____121
9 π__2
10 TellwhetherthestatementldquoAllintegersarerationalnumbersrdquoistrueorfalseExplainyourchoice
13ensp OrderingenspRealenspNumbersCompare Write lt gt or =
11 radic__
8+3 8+radic__
3 12 radic__
5+11emsp emsp emsp 5+radic___
11
Order the numbers from least to greatest
13 radic___
99π29__
8 14 radic___
1__251_40__
2
15 Howarerealnumbersusedtodescribereal-worldsituations
ESSENTIAL QUESTION
035
9-9
141ft
7 1__10- 1__10
14__11 1875
wholeintegerrationalreal
Trueintegerscanbewrittenasthequotientoftwointegers
SampleanswerRealnumberssuchastherational
π29__
8radic___
99
irrationalreal
lt gt
number1_4candescribeamountsusedincooking
radic___
1__250__
21_4
27Module1
copy H
ough
ton
Miff
lin H
arco
urt P
ublis
hing
Com
pany
DONOTEDIT--ChangesmustbemadethroughldquoFileinfordquoCorrectionKey=A
8_MCAAESE206984_U1M01RTindd 27 41513 1113 PM
Math TrainerOnline Assessment
and Intervention
Personal
myhrwcom
1
2
3 Response toIntervention
Intervention Enrichment
Access Ready to Go On assessment online and receive instant scoring feedback and customized intervention or enrichment
Online and Print Resources
Differentiated Instruction
bull Reteach worksheets
bull Reading Strategies EL
bull Success for English Learners EL
Differentiated Instruction
bull Challenge worksheets PRE-AP
Extend the Math PRE-AP
Lesson Activities in TE
Additional ResourcesAssessment Resources includes bull Leveled Module Quizzes
Ready to Go OnAssess MasteryUse the assessment on this page to determine if students have mastered the concepts and standards covered in this module
California Common Core Standards
Lesson Exercises Common Core Standards
11 1ndash7 8NS1 8NS2 8EE2
12 8ndash10 8NS1
13 11ndash14 8NS2
27 Unit 1 Module 1
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
Personal Math Trainer
Online Practice and HelpmyhrwcomAssessment Readiness
Module 1 MIXed ReVIeW
1 Look at each number Is the number between 2π and radic___
52
Select Yes or No for expressions AndashC
A 6 2 _ 3 Yes No
B 5π __ 2 Yes No
C 3 radic__
5 Yes No
2 Consider the number - 11 __ 15
Choose True or False for each statement
A The number is rational True False
B The number can be written as True Falsea repeating decimal
C The number is less than ndash08 True False
3 The volume of a cube is given by V = x3 where x is the length of an edge of the cube A cube-shaped end table has a volume of 3 3 _ 8 cubic feet What is the length of an edge of the end table Explain how you solved this problem
4 A student says that radic___
83 is greater than 29 __ 3 Is the student correct Justify your
reasoning
1 1 _ 2 ft Sample answer The equation x3 = 3 3 _ 8 can be used
to find the edge length in feet To solve the equation
write the mixed number as a fraction greater than 1
x3 = 27 __ 8 Then take the cube root of both sides x = 3 _ 2 = 1 1 _ 2
No Sample answer radic___
83 asymp 91 and 29 __ 3 = 9
__ 6
Because 91 lt 9 __
6 radic___
83 lt 29 __ 3
28 Unit 1
copy H
ough
ton
Miff
lin H
arco
urt P
ublis
hing
Com
pany
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=A
8_MCAAESE206984_U1M01RTindd 28 240413 946 AM
Personal Math Trainer
Online Assessment and
Interventionmyhrwcom
Scoring GuideItem 3 Award the student 1 point for finding the edge length of the cube and 1 point for correctly explaining how to use a cube root to solve the problem
Item 4 Award the student 1 point for determining that the student is incorrect and 1 point for correctly justifying the reasoning for this conclusion
Additional ResourcesTo assign this assessment online login to your Assignment Manager at myhrwcom
Assessment Readiness
California Common Core Standards
Items Grade 8 Standards Mathematical Practices
1 8NS2 MP7
2 7NS2b 7NS2d 8NS1 MP7
3 8EE2 MP1 MP4
4 8NS1 8NS2 MP3
Item integrates mixed review concepts from previous modules or a previous course
Item 4 combines concepts from the California Common Core cluster ldquoKnow that there are numbers that are not rational and approximate them by rational numbersrdquo
Real Numbers 28
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
My Notes
Math On the Spotmyhrwcom
Math TrainerOnline Practice
and Help
Personal
myhrwcom
Expressing Decimals as Rational NumbersYou can express terminating and repeating decimals as rational numbers
Write each decimal as a fraction in simplest form
0825
The decimal 0825 means ldquo825 thousandthsrdquo Write this as a fraction
825 ____ 1000
Then simplify the fraction
825 divide 25 ________ 1000 divide 25 = 33 __ 40
0825 = 33 __ 40
0 _
37
Let x = 0 _
37 The number 0 _
37 has 2 repeating digits so multiply each side of the equation x = 0
_ 37 by 10 2 or 100
x = 0 _
37
(100)x = 100(0 _
37 )
100x = 37 _
37
Because x = 0 _
37 you can subtract x from one side and 0 _
37 from the other
100x = 37 _
37
minusx minus0 _
37
99x = 37
Now solve the equation for x Simplify if necessary
99x ___ 99 = 37 __ 99
x = 37 __ 99
EXAMPLE 2
A
B
Write each fraction as a decimal
YOUR TURN
1 5 __ 11 2 1 _ 8 3 2 1 _ 3
8NS1
To write ldquo825 thousandthsrdquo put 825 over 1000
Divide both the numerator and the denominator by 25
100 times 0 _
37 is 37 _
37
37 _
37 minus 0 _
37 is 37
Divide both sides of the equation by 99
0 _
45 0125 2 _
3
Unit 18
copy H
ough
ton
Miff
lin H
arco
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ublis
hing
Com
pany
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
8_MCAAESE206984_U1M01L1indd 8 120413 838 PM
My Notes
Math On the Spot
myhrwcom
= 033333333333331mdash3
ESSENTIAL QUESTION
Expressing Rational Numbers as DecimalsA rational number is any number that can be written as a ratio in the form a _ b where a and b are integers and b is not 0 Examples of rational numbers are 6 and 05
6 can be written as 6 _ 1 05 can be written as 1 _ 2
Every rational number can be written as a terminating decimal or a repeating decimal A terminating decimal such as 05 has a finite number of digits A repeating decimal has a block of one or more digits that repeat indefinitely
Write each fraction as a decimal
1 _ 4
1 _ 4 = 025
1 _ 3
1 _ 3 = 0 _
3
EXAMPLEXAMPLE 1
A
B
0333 3 ⟌ ⎯ 1000 minus9 10 minus9 10 minus9 1
025 4 ⟌ ⎯ 100 -8 20 -20
0
L E S S O N
11Rational and Irrational Numbers
How do you rewrite rational numbers and decimals take square roots and cube roots and approximate irrational numbers
8NS1
Know that numbers that are not rational are called irrational Understand informally that every number has a decimal expansion for rational numbers show that the decimal expansion repeats eventually and convert a decimal expansion which repeats eventually into a relation number Also 8NS2 8EE2
8NS1
Remember that the fraction bar means ldquodivided byrdquo Divide the numerator by the denominator
Divide until the remainder is zero adding zeros after the decimal point in the dividend as needed
Divide until the remainder is zero or until the digits in the quotient begin to repeat
Add zeros after the decimal point in the dividend as needed
When a decimal has one or more digits that repeat indefinitely write the decimal with a bar over the repeating digit(s)
7Lesson 11
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ough
ton
Miff
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pany
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=B
8_MCABESE206984_U1M01L1indd 7 11113 128 AM
PROFESSIONAL DEVELOPMENT
Math BackgroundSome decimals may have a pattern but still not be a repeating decimal that is rational For example in 312112111211112hellip you can predict the next digit and describe the pattern (There is one more 1 each time before the 2) However this is not a terminating decimal nor is it a repeating decimal and it is therefore NOT a rational number
Integrate Mathematical Practices MP6
This lesson provides an opportunity to address this Mathematical Practices standard It calls for students to attend to precision Students learn to express rational numbers accurately and precisely in both fractional and decimal forms and learn to translate from one form to the other They also learn how to precisely represent and communicate ideas about irrational numbers square roots and cube roots
Rational and Irrational Numbers 8
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=B
Focus on Technology Mathematical PracticesPoint out the importance of entering a repeating decimal correctly when using a graphing calculator to convert the decimal to a fraction The decimal 0
_ 59 must be entered as
0595959595959 not 059
YOUR TURNFocus on Math ConnectionsMake sure students understand that the place value of the last digit in Exercises 4 and 6 determines the denominator of the corresponding fraction or mixed number So for Exercise 4 the place value hundredths gives a denominator of 100 and for Exercise 6 the place value tenths gives a denominator of 10
EXAMPLE 3Questioning Strategies Mathematical Practices bull How can a solution of an equation of the form x 2 = p be negative if p is a positive number Since the square of a negative number is positive a negative number is also a solution of x 2 equals a positive number
bull When is a solution of an equation of the form x 3 = p larger than p The solution is larger than p if p is a number between 0 and 1
Focus on Math Connections Make sure students understand the difference in finding radic
_ 121 and solving x 2 = 121 The
symbol radic_
indicates the positive or principal square root only while the equation x 2 = 121 has two roots the principal square root and its opposite
YOUR TURNAvoid Common ErrorsTo avoid sign errors in Exercise 9 make sure that students understand that the cube of a negative number is not a positive number Therefore -8 is not a solution of x 3 = 512
Talk About ItCheck for Understanding
Ask Kris predicts that there are two real solutions for Exercises 7 and 8 and that there are three real solutions for Exercises 9 and 10 Is his prediction correct
Explain His prediction is correct for Exercises 7 and 8 because there are two numbers whose squares are the same positive number given in the exercises His prediction is not correct for Exercises 9 and 10 however because there is only one real number whose cube is the same positive number given in the exercises
EXPLORE ACTIVITYQuestioning Strategies Mathematical Practices bull Compare the values for 13 2 and 13 2 The digits are the same but 13 2 has two decimal places (169) while 13 2 has none (169)
bull How do you know whether radic_
2 will be closer to 1 or closer to 2 It will be closer to 1 because 2 is between the perfect squares of 1 and 4 but closer to 1 than it is to 4
Connect Vocabulary EL
Explain to students that the word irrational when used as an ordinary word in English means without logic or reason In mathematics when we say that a number is irrational it means only that the number cannot be written as the quotient of two integers
Engage with the WhiteboardHave students extend the number line in both directions and label the locations of the whole numbers 1 and 2 These are the roots of the consecutive perfect squares
1 and 4 used to estimate radic_
7
Interactive Whiteboard Interactive example available online
ADDITIONAL EXAMPLE 3Solve each equation for x
A x 2 = 324 18 -18
B x 2 = 25 ___ 144 5 __ 12 - 5 __ 12
C 343 = x 3 7
D x 3 = 125 ___ 512 5 __ 8
myhrwcom
9 Lesson 11
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
Math TrainerOnline Practice
and Help
Personal
myhrwcom
EXPLORE ACTIVITY
lt 2 lt
radic_
lt radic
_ 2 lt
radic_
lt radic
_ 2 lt
The solution is 9
The solution is 2 _ 5
C
D
729 = x 3
3 radic_ 729 = 3 radic
_ x 3
3 radic_ 729 = x
9 = x
x 3 = 8 ___ 125
3 radic_
x 3 =thinsp 3 radic_ 8 ___ 125
x =thinsp 3 radic_ 8 ___ 125
x = 2 _ 5
Solve each equation for x
YOUR TURN
7 x 2 = 196 8 x 2 = 9 ___ 256
9 x 3 = 512 10 x 3 = 64 ___ 343
Estimating Irrational NumbersIrrational numbers are numbers that are not rational In other words they cannot be written in the form a _ b where a and b are integers and b is not 0 Square roots of perfect squares are rational numbers Square roots of numbers that are not perfect squares are irrational Some equations like those in Example 3 involve square roots of numbers that are not perfect squares
x 2 = 2 x = plusmn radic_
2
Estimate the value of radic_
2
Find two consecutive perfect squares that 2 is between Complete the inequality by writing these perfect squares in the boxes
Now take the square root of each number
Simplify the square roots of perfect squares
radic_
2 is between and
A
B
C
8NS2 8EE2
Solve for x by taking the cube root of both sides
Solve for x by taking the cube root of both sides
Apply the definition of cube root
Think What number cubed equals 729
Apply the definition of cube root
Think What number cubed equals 8 ____ 125
radic_
2 is irrational
x = plusmn14 x = plusmn 3 __ 16
x = 8 x = 4 _ 7
1 2
1 4
1 4
1 2
Unit 110
copy H
ough
ton
Miff
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pany
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
8_MCAAESE206984_U1M01L1indd 10 41613 1211 AM
Math TrainerOnline Practice
and Help
Personal
myhrwcom
Math On the Spot
myhrwcom
Write each decimal as a fraction in simplest form
YOUR TURN
Finding Square Roots and Cube RootsThe square root of a positive number p is x if x 2 = p There are two square roots for every positive number For example the square roots of 36 are 6 and minus6 because 6 2 = 36 and (minus6) 2 = 36 The square roots of 1 __ 25 are 1 _ 5 and minus 1 _ 5 You can write the square roots of 1 __ 25 as plusmn 1 _ 5 The symbol radic
_ 5 indicates the positive
or principal square root
A number that is a perfect square has square roots that are integers The number 81 is a perfect square because its square roots are 9 and minus9
The cube root of a positive number p is x if x 3 = p There is one cube root for every positive number For example the cube root of 8 is 2 because 2 3 = 8 The cube root of 1 __ 27 is 1 _ 3 because ( 1 _ 3 )
3
= 1 __ 27 The symbol 3 radic_ 1 indicates the
cube root
A number that is a perfect cube has a cube root that is an integer The number 125 is a perfect cube because its cube root is 5
Solve each equation for x
The solutions are 11 and minus11
The solutions are 4 __ 13 and minus 4 __ 13
EXAMPLEXAMPLE 3
A x 2 = 121
x 2 = 121
x = plusmn radic_
121
x = plusmn11
B x 2 = 16 ___ 169
x 2 = 16 ___ 169
x = plusmn radic_
16 ___ 169
x = plusmn 4 __ 13
4 012 5 0 _
57 6 14
Can you square an integer and get a negative number
What does this indicate about whether negative
numbers have square roots
Math TalkMathematical Practices
8EE2
Solve for x by taking the square root of both sides
Apply the definition of square root
Think What numbers squared equal 121
Solve for x by taking the square root of both sides
Apply the definition of square root
Think What numbers squared equal 16 ____ 169
3 __ 25 19 __ 33 1 2 _ 5
No the square of a positive integer is positive the square of a negative integer is positive and the square of 0 is 0 So negative numbers do not have (real) square roots
9Lesson 11
copy H
ough
ton
Miff
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pany
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
8_MCAAESE206984_U1M01L1indd 9 41913 240 PM
Critical ThinkingIn the Explore Activity students estimated the location of radic
_ 2 on a number line Ask students
whether they think that it is possible to locate more precisely the point that represents radic
_ 2 In
other words can you graph irrational numbers exactly on a number line along with rational numbers Students should understand that radic
_ 2
is a real number and all real numbers can be located on a real number line A more precise estimate will allow more precise placement on a number line
The Modeling note tells one way to do this
ModelingHave students use a ruler to represent a number line with a unit that is one inch long Have them draw a square with a side of one inch and draw the diagonal to make two isosceles triangles Lead students to understand that the length of the diagonal (or hypotenuse) is radic
_ 2
Have them copy the length of their diagonal onto their ruler or number line starting at zero The end point of the diagonal represents the exact point for the irrational number radic
_ 2 on a
number line
Additional ResourcesDifferentiated Instruction includes bull Reading Strategies bull Success for English Learners EL
bull Reteach bull Challenge PRE-AP
DIFFERENTIATE INSTRUCTION
Rational and Irrational Numbers 10
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
ElaborateTalk About ItSummarize the Lesson
Ask If someone claims that a certain number is irrational but you know it is actually rational how could you prove to that person that the number is rational
You could find a fraction equal to the number such that the number is the ratio of two integers with the denominator not equal to zero
GUIDED PRACTICEEngage with the Whiteboard
Have students plot each number in Exercises 16ndash18 on a number line Students should label each point with the irrational number written as a radical and as a
decimal
Avoid Common ErrorsExercises 1ndash6 To avoid reversing the order of the dividend and divisor tell students to start at the top of the fraction and read the bar as ldquodivided byrdquo
Focus on TechnologyHave students use a calculator to investigate the decimal equivalents of such fractions as 1 __ 9 2 __ 9 8 __ 9 and 1 __ 11 2 __ 11 10
__ 11 Ask them to describe the patterns they find as a result of these investigations
11 Lesson 11
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
Guided Practice
7 0675 8 56 9 044
10 0 _
4
10x =
x =
11 0 _
26
100x =
x =
12 0 _
325
1000x =
x =
Solve each equation for x (Example 3 and Explore Activity)
- x
-
_______________
x =
- x
-
___________________
x =
- x
-
_______________________
x =
Write each fraction or mixed number as a decimal (Example 1)
1 2 _ 5 2 8 _ 9 3 3 3 _ 4
4 7 __ 10 5 2 3 _ 8 6 5 _ 6
Write each decimal as a fraction or mixed number in simplest form (Example 2)
13 x 2 = 17 14 x 2 = 25 ___ 289 15 x 3 = 216
Approximate each irrational number to one decimal place without a calculator
x = plusmn radic__
asymp plusmn x = 3
radic__
=
(Explore Activity)
16 radic_
5 asymp
17 radic_
3 asymp
18 radic_
10 asymp
19 What is the difference between rational and irrational numbers
CHECK-INESSENTIAL QUESTION
x = plusmn radic__
__________ = plusmn _____
4 _
4
0 _
4
4 99
6216
269
41 25 5
17289
17
22 17 32
04
07
27__40
26 __ 99 325 ___ 999 4 _ 9
11__255 3_5
0 _
8
2375
375
08 _
3
26 _
26
0 _
26
325 _
325
0 _
325
999 325
Rational numbers can be written in the form a __ b where
a and b are integers and b ne 0 Irrational numbers cannot
be written in this form
Unit 112
copy H
ough
ton
Miff
lin H
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urt P
ublis
hing
Com
pany
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
8_MCAAESE206984_U1M01L1indd 12 41613 1211 AM
11 12 13 14 15
radic2 asymp 14
141 142 143 144 145
radic2 asymp 141
0 1 2 3 4
radic2 asymp 15
Estimate that radic_
2 asymp 15
To find a better estimate first choose some numbers between 1 and 2 and square them For example choose 13 14 and 15
1 3 2 = 1 4 2 = 1 5 2 =
Is radic_
2 between 13 and 14 How do you know
Is radic_
2 between 14 and 15 How do you know
2 is closer to than to so radic_
2 asymp
Locate and label this value on the number line
Reflect 11 How could you find an even better estimate of radic
_ 2
12 Find a better estimate of radic_
2
1 41 2 = 1 42 2 = 1 43 2 =
2 is closer to than to so radic_
2 asymp
Draw a number line and locate and label your estimate
13 Solve x 2 = 7 Write your answer as a radical expression Then estimate to one decimal place
D
E
F
No 2 is not between 169 and 196
Yes 2 is between 196 and 225
196
19881
19881
225
20164
20164
14
141
20449
169 196 225
Test the squares of numbers between 14 and 15
x = plusmn radic_
7 x asymp plusmn26
11Lesson 11
copy H
ough
ton
Miff
lin H
arco
urt P
ublis
hing
Com
pany
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
8_MCAAESE206984_U1M01L1indd 11 41613 1211 AM
Rational and Irrational Numbers 12
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
Math TrainerOnline Assessment
and Intervention
Personal
Online homework assignment available
myhrwcom
EvaluateGUIDED AND INDEPENDENT PRACTICE
Concepts amp Skills Practice
Example 1Expressing Rational Numbers as Decimals
Exercises 1ndash6 20ndash21 24ndash25
Example 2Expressing Decimals as Rational Numbers
Exercises 7ndash12 22ndash23 26ndash27
Example 3Finding Square Roots and Cube Roots
Exercises 13ndash15 28 30ndash31 35
Explore ActivityEstimating Irrational Numbers
Exercises 13 16ndash18 29 32ndash34
Additional ResourcesDifferentiated Instruction includes bull Leveled Practice worksheets
Lesson Quiz available online
11 LESSON QUIZ
1 Write as a decimal 2 5 __ 8 1 7 __ 12
2 Write as a fraction 034 1 _
24
3 Solve x 2 = 9 __ 49 for x
4 Solve x 3 = 216 for x
5 Estimate the value of radic_
13 to one decimal place without using a calculator
myhrwcom
Focus | Coherence | Rigor
Exercise Depth of Knowledge (DOK) Mathematical Practices
20ndash27 2 SkillsConcepts MP4 Modeling
28 3 Strategic Thinking MP4 Modeling
29ndash32 2 SkillsConcepts MP6 Precision
33 3 Strategic Thinking MP7 Using Structure
34 2 SkillsConcepts MP3 Logic
35 2 SkillsConcepts MP4 Modeling
36 3 Strategic Thinking MP3 Logic
37 3 Strategic Thinking MP7 Using Structure
38 3 Strategic Thinking MP2 Reasoning
8NS1 8NS2 8EE2
8NS1 8NS2 8EE2
Answers1 2625 158
_ 3
2 17 __ 50 1 8 __ 33
3 x = plusmn 3 __ 7
4 x = 6
5 36
13 Lesson 11
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
Work Area
33 Analyze Relationships To find radic_
15 Beau found 3 2 = 9 and 4 2 = 16 He said that since 15 is between 9 and 16 radic
_ 15 must be between 3 and 4 He
thinks a good estimate for radic_
15 is 3 + 4 ____ 2 = 35 Is Beaursquos estimate high low
or correct Explain
34 Justify Reasoning What is a good estimate for the solution to the equation x 3 = 95 How did you come up with your estimate
35 The volume of a sphere is 36π f t 3 What is the radius of the sphere Use the formula V = 4 _ 3 π r 3 to find your answer
36 Draw Conclusions Can you find the cube root of a negative number If so is it positive or negative Explain your reasoning
37 Make a Conjecture Evaluate and compare the following expressions
radic_
4 __ 25 and radic
_ 4 ____
radic_
25 radic
_
16 __ 81 and radic_
16 ____
radic_
81 radic
_
36 __ 49 and radic_
36 ____
radic_
49
Use your results to make a conjecture about a division rule for square roots Since division is multiplication by the reciprocal make a conjecture about a multiplication rule for square roots
38 Persevere in Problem Solving The difference between the solutions to the equation x 2 = a is 30 What is a Show that your answer is correct
FOCUS ON HIGHER ORDER THINKING
His estimate is low because 15 is very close to 16
so radic_
15 is very close to radic_
16 or 4 A better estimate
would be 38 or 39
Sample answer about 45 4 3 = 64 and 5 3 = 125
Because 95 is about halfway between 64 and 125 try 45
45 3 = 91125 which is a good estimate
3 feet
Yes the cube root of a negative number is negative
because a negative number cubed is always negative
and a nonnegative number cubed is always nonnegative
radic_
4 __ 25 = 2 _ 5 = radic
_ 4 ____
radic_
25 radic
_
16 __ 81 = 4 _ 9 = radic_
16 ____
radic_
81 radic
_
36 __ 49 = 6 _ 7 = radic_
36 ____
radic_
49
225 the solutions to x 2 = a are x = plusmn15 and
radic_
a ___
radic_
b = radic
_ a __
b radic
_ a radic
_ b = radic
_ a b
15 - (-15) = 30
Unit 114
copy H
ough
ton
Miff
lin H
arco
urt P
ublis
hing
Com
pany
bull copy
Ilen
e Mac
Dona
ldA
lamy I
mag
es
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=B
8_MCABESE206984_U1M01L1indd 14 102913 1142 PM
Personal Math Trainer
Online Practice and Helpmyhrwcom
Name Class Date
Independent Practice11
20 A 7 __ 16 -inch-long bolt is used in a machine What is this length written as a decimal
21 The weight of an object on the moon is 1 _ 6 its weight on Earth Write 1 _ 6 as a decimal
22 The distance to the nearest gas station is 2 4 _ 5 kilometers What is this distance written as a decimal
23 A baseball pitcher has pitched 98 2 _ 3 innings What is the number of innings written as a decimal
24 A heartbeat takes 08 second How many seconds is this written as a fraction
25 There are 262 miles in a marathon Write the number of miles using a fraction
26 The average score on a biology test was 72
_ 1 Write the average score using a
fraction
27 The metal in a penny is worth about 0505 cent How many cents is this written as a fraction
28 Multistep An artist wants to frame a square painting with an area of 400 square inches She wants to know the length of the wood trim that is needed to go around the painting
a If x is the length of one side of the painting what equation can you set up to find the length of a side How many solutions does the equation have
b Do all of the solutions that you found make sense in the context of the problem Explain
c What is the length of the wood trim needed to go around the painting
Solve each equation for x Write your answers as radical expressions Then estimate to one decimal place if necessary
29 x 2 = 14 30 x 3 = 1331
31 x 2 = 144 32 x 2 = 29
8NS1 8NS2 8EE2
04375 in 01 _6
28 km 98 _6 innings
x 2 = 400 x = plusmnthinsp20 the equation has 2 solutions
x = 20 makes sense but x = -20 doesnrsquot because a
painting cannot have a side length of -20 inches
4 times 20 = 80 inches
x = plusmn radic_
14 asymp plusmn37
x = plusmn radic_
144 = plusmn12 x = plusmn radic_
29 asymp plusmn54
x = 3 radic_ 1331 = 11
4_5 second 26 1_5 mi
72 1 _ 9 101 ___ 200 cent
13Lesson 11
copy H
ough
ton
Miff
lin H
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urt P
ublis
hing
Com
pany
bull copy
Phot
odisc
Get
ty Im
ages
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
8_MCAAESE206984_U1M01L1indd 13 41613 1211 AM
myhrwcomActivity available onlineEXTEND THE MATH PRE-AP
Activity Write radic_
09 on the board and invite students to conjecture what the value might be Have them check their conjectures by squaring Invite them to suggest ways to estimate radic
_ 09 As a hint point out that 09 is close to 10 and so they might
use that to help guide their estimates Lead them to see that since 092 is 081 and 102 is 1 the value of radic
_ 09 is greater than 09 and less than 10 Try squaring 095 to get
09025 A good estimate for radic_
09 is 095
Rational and Irrational Numbers 14
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=B
Integers
Rational Numbers IrrationalNumbers
Real Numbers
WholeNumbers
-3-4-5 -2-1 1 2 3 50 4
23
34-4 -π -1 25
radic2
Lesson Support Content Objective Students will learn to describe relationships between sets of numbers
Language Objective Students will explain how to describe relationships between sets of real numbers
LESSON 12 Sets of Real Numbers
Building BackgroundEliciting Prior Knowledge Have students draw a number line from -5 to 5 Ask them to plot points on the number line to approximate the location of rational and irrational numbers such as -1 3 __ 4 25 -4 2 __ 3 radic
_ 2 and -π
Learning ProgressionsIn this lesson students clarify their understanding of the real number system They characterize sets and subsets of the real numbers They also identify sets for real-world situations Important understandings for students include the following
bull Identify all of the possible subsets of the real numbers for a given number
bull Decide whether a statement about a subset of the real numbers is true or false
bull Identify the set of numbers that best describes a real-world situation
Understanding the relationships among the sets of numbers that make up the real numbers is essential as students are introduced to different forms of numbers throughout the school year This lesson provides a foundation for the comparing and ordering of real numbers in the next lesson
Cluster ConnectionsThis lesson provides an excellent opportunity to connect ideas in this cluster Know that there are numbers that are not rational and approximate them by rational numbers Have students copy this diagram which relates the sets of real numbers
Ask students to complete the diagram by writing three examples for each set of numbers Have students share examples and explain how they knew each number they selected belonged in the appropriate set Answers may vary Check studentsrsquo work
Focus | Coherence | Rigor
California Common Core Standards
8NS1 Know that numbers that are not rational are called irrational Understand informally that every number has a decimal expansion for rational numbers show that the decimal expansion repeats eventually and convert a decimal expansion which repeats eventually into a rational number
MP7 Look for and make use of structure
15A
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
Math Talk
Language Support EL
PROFESSIONAL DEVELOPMENT
Linguistic Support EL
AcademicContent Vocabulary
Venn diagrams ndash Students need descriptive language to describe the categories that the different areas and colors of a Venn diagram represent the concept of a set and how sets are distinct or can overlap Use sentence frames such as
The big oval represents __________The darklight blue color in the middle of the
big ovals represents __________These sets overlap because __________
In this way students have the language and structure to identify the criteria that distinguish a set and to explain the abstract representation Also point out the use of the prefix sub- meaning ldquounderrdquo in the term subset
Rules and Patterns
Abbreviations ndash In this lesson the abbreviation mph is used Be sure to point out that mph stands for miles per hour and is used to give units in a rate of speed Students may also have seen mpg (miles per gallon) which gives the units in a rate of fuel efficiency
Borrowed Words ndash Terminology used in baseball such as inning and pitcher may require some explanation Spanish as well as some other languages have borrowed these terms from English so some students may be familiar with these words already Despite this whenever a word is critical to students understanding the word problem it is best to explain the meaning
Leveled Strategies for English Learners
Emerging Allow students to indicate true or false orally in Guided Practice Exercises 9 and 10
Expanding Have students use sentence frames to describe the meaning of regions and colors used in a Venn diagram Then give them similar sentence frames orally and have them draw and shade a Venn diagram based on the oral prompts
Bridging Have students work in groups to draw a Venn diagram to represent sets based on real-world examples in the lesson
To help students answer the question posed in Math Talk provide a sentence frame for their answer
The numbers between 31 and 39 on a number line are __________ because __________
EL
California ELD Standards
Emerging 2II5 Modifying to add details ndash Expand sentences with simple adverbials to provide details about a familiar activity or process
Expanding 2II5 Modifying to add details ndash Expand sentences with adverbials to provide details about a familiar or new activity or process
Bridging 2II5 Modifying to add details ndash Expand sentences with increasingly complex adverbials to provide details about a variety of familiar and new activities and processes
Sets of Real Numbers 15B
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
12L E S S O N
Sets of Real Numbers
EngageESSENTIAL QUESTION
How can you describe relationships between sets of real numbers Sample answer Describe them as two different sets or one set as being a subset of another
Motivate the LessonAsk How many different types of tigers can you name How does the set of Bengal tigers relate to the set of tigers
ExplorePoint to different locations in the Animals diagram and ask for examples for that classification Do the same for the Real Numbers diagram Students should understand that everything within a region is part of the set for example both -3 and 2 are integers
ExplainEXAMPLE 1
Questioning Strategies Mathematical Practices bull In A why is 5 not a perfect square It does not have rational numbers as its square roots
bull Can the number in B be written as a fraction Why or why not Yes it is a terminating decimal so it is a rational number
Engage with the WhiteboardHave students place the numbers in Example 1 and Additional Example 1 in the Venn diagram for numbers
YOUR TURNAvoid Common ErrorsBe sure that students read Exercise 2 carefully before answering The number given in the problem 10 is the area not the side length
EXAMPLE 2Questioning Strategies Mathematical Practices bull What two major sets are the real numbers composed of rational and irrational numbers
bull What is the location of the set of whole numbers in the Venn diagram in relation to the set of rational numbers Explain Inside it whole numbers are rational numbers
Focus on Reasoning Mathematical PracticesRemind students that it takes only one counterexample to show that a statement is false
Interactive Whiteboard Interactive example available online
ADDITIONAL EXAMPLE 1Write all names that apply to each number
A -10integer rational real
B 12 _ 3
whole integer rational real
myhrwcom
Interactive Whiteboard Interactive example available online
ADDITIONAL EXAMPLE 2Tell whether the given statement is true or false Explain your choice
No integers are whole numbers
False every whole number is also an integer
myhrwcom
Animated MathClassifying Numbers
Students build fluency in classifying numbers in this engaging fast-paced game
myhrwcom
CA Common CoreStandards
The student is expected to
The Number Systemmdash8NS1
Know that numbers that are not rational are called irrational Understand informally that every number has a decimal expansion for rational numbers show that the decimal expansion repeats eventually and convert a decimal expansion which repeats eventually into a rational numberMathematical Practices
MP7 Using Structure
The student is expected to
15 Lesson 12
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=B
Math TrainerOnline Practice
and Help
Personal
myhrwcom
Math TrainerOnline Practice
and Help
Personal
myhrwcom
Math On the Spotmyhrwcom
Understanding Sets and Subsets of Real NumbersBy understanding which sets are subsets of types of numbers you can verify whether statements about the relationships between sets are true or false
Tell whether the given statement is true or false Explain your choice
All irrational numbers are real numbers
True Every irrational number is included in the set of real numbers The irrational numbers are a subset of the real numbers
No rational numbers are whole numbers
False A whole number can be written as a fraction with a denominator of 1 so every whole number is included in the set of rational numbers The whole numbers are a subset of the rational numbers
EXAMPLE 2
A
B
Write all names that apply to each number
1 A baseball pitcher has pitched 12 2 _ 3 innings
2 The length of the side of a square that has an
area of 10 square yards
YOUR TURN
Tell whether the given statement is true or false Explain your choice
3 All rational numbers are integers
4 Some irrational numbers are integers
YOUR TURN
Give an example of a rational number that is a
whole number Show that the number is both whole
and rational
Math TalkMathematical Practices
Give an example of a
8NS1
False Every integer is a rational number but not every
False Real numbers are either rational or irrational numbers
Integers are rational numbers so no integers are irrational numbers
rational real
irrational real
Sample answer 8 8 = 8_
1
and -thinsp 5 _ 2 are not integers
rational number is an integer Rational numbers such as 3 _ 5
Unit 116
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ough
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Miff
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Com
pany
bull Im
age C
redi
ts D
igita
l Im
age c
opyr
ight
copy20
04 Ey
ewire
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
8_MCAAESE206984_U1M01L2indd 16 41613 136 AM
Math On the Spot
myhrwcom
Vertebrates
Birds
Passerines
Animals
Integers
Rational Numbers IrrationalNumbers
Real Numbers
WholeNumbers
1
45
3
0
274
67
radic4
-
-3
-2
-1
03
radic2
radic17
radic11-
π
Animated Math
myhrwcom
Classifying Real NumbersBiologists classify animals based on shared characteristics A cardinal is an animal a vertebrate a bird and a passerine
You already know that the set of rational numbers consists of whole numbers integers and fractions The set of real numbers consists of the set of rational numbers and the set of irrational numbers
Write all names that apply to each number
radic_
5 irrational real
ndash1784rational real
whole integer rational real
EXAMPLEXAMPLE 1
A
B
C radic_ 81 ____ 9
L E S S O N
12Sets of Real Numbers
ESSENTIAL QUESTIONHow can you describe relationships between sets of real numbers
Passerines such as the cardinal are also called ldquoperching birdsrdquo
What types of numbers are between 31 and 39 on a
number line
Math TalkMathematical Practices
What types of numbers are
8NS1
8NS1
Know that numbers that are not rational are called irrational Understand informally that every number has a decimal expansion for rational numbers show that the decimal expansion repeats eventually and convert a decimal expansion which repeats eventually into a relation number
ndash1784 is a terminating decimal
5 is a whole number that is not a perfect square
radic_
81 _____ 9 = 9 __ 9 = 1 rational irrational real
15Lesson 12
copy H
ough
ton
Miff
lin H
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Com
pany
bull Im
age C
redi
ts copy
Wiki
med
ia Co
mm
ons
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=B
8_MCABESE206984_U1M01L2indd 15 061113 1144 AM
PROFESSIONAL DEVELOPMENT
Math BackgroundThe relationships between sets of numbers extend to include complex numbers A complex number can be written as a sum of a real number a and an imaginary number bi
a + bi
An imaginary number is a special number that when squared gives a negative value When you square a real number you get a nonnegative number When you square an imaginary number you get a negative value The imaginary unit is i
i = radic_
-1
Integrate Mathematical Practices MP7
This lesson provides an opportunity to address this Mathematical Practices standard It calls for students to discern structure to connect and communicate mathematical ideas
Students use a Venn diagram to structure relationships between sets of numbers They connect and communicate mathematical ideas when they make logical statements about the sets and describe which set best describes numbers applied to real-life situations
Sets of Real Numbers 16
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=B
YOUR TURNAvoid Common ErrorsStudents may see the word ldquoAllldquo or rdquoNordquo in Exercises 3 and 4 and immediately assume that any absolute statements like these are false Remind them that there are true statements that begin with these words and encourage them to provide examples
EXAMPLE 3Questioning Strategies Mathematical Practices bull In A how does the phrase ldquonumber of rdquo give you a clue about the number classification It indicates a counting number
bull What is the relationship between the circumference of a circle and the diameter The circumference is diameter times π
Focus on Critical Thinking Mathematical PracticesIn B suppose the diameters in inches were 25
__ π 28 __ π
31 __ π and so on What set of numbers would
best describe the circumferences Explain Whole numbers the circumferences would be the whole numbers 25 28 31 and so on
YOUR TURNFocus on Critical Thinking Mathematical PracticesHave students compare and contrast the classification of numbers in the answers in Exercises 5 and 6
ElaborateTalk About ItSummarize the Lesson
Ask What are some ways that number sets can be related Sets may be subsets of other sets or they may be separate from other sets
GUIDED PRACTICEEngage with the Whiteboard
Have students place the numbers in Exercises 1ndashthinsp8 in the Venn diagram for numbers at the beginning of the lesson
Integrating Language Arts EL
Encourage English learners to ask for clarification on any terms or phrases that they do not understand
Avoid Common ErrorsExercise 7 Remind students that a repeating decimal is a rational numberExercises 9ndash10 Remind students that it only takes one counterexample to show that a statement is false
Interactive Whiteboard Interactive example available online
ADDITIONAL EXAMPLE 3Identify the set of numbers that best describes the situation Explain your choice
A the amount of time that has passed since midnight
The set of real numbers time is continuous so the amount of time can be rational or irrational
B the number of tickets sold to a basketball game
The set of whole numbers the number of tickets sold may be 0 or a counting number
myhrwcom
17 Lesson 12
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
1IN
116 inch
Guided Practice
Write all names that apply to each number (Example 1)
1 7 _ 8 2 radic_
36
3 radic_
24 4 075
5 0 6 - radic_ 100
7 5 _
45 8 - 18 __ 6
Tell whether the given statement is true or false Explain your choice (Example 2)
9 All whole numbers are rational numbers
10 No irrational numbers are whole numbers
Identify the set of numbers that best describes each situation Explain your choice (Example 3)
11 the change in the value of an account when given to the nearest dollar
12 the markings on a standard ruler
13 What are some ways to describe the relationships between sets of numbers
CHECK-INESSENTIAL QUESTION
rational real
rational real
True Whole numbers are rational numbers
Rational numbers the ruler is marked every 1 __ 16 th inch
Sample answer Describe one set as being a subset of
another or show their relationships in a Venn diagram
Integers the change can be a whole dollar amount
and can be positive negative or zero
True Whole numbers are a subset of the set of rational numbers
and can be written as a ratio of the whole number to 1
irrational real
whole integer rational real
whole integer rational real
rational real
integer rational real
integer rational real
Unit 118
copy H
ough
ton
Miff
lin H
arco
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ublis
hing
Com
pany
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
8_MCAAESE206984_U1M01L2indd 18 41613 136 AM
My Notes
Math TrainerOnline Practice
and Help
Personal
myhrwcom
Math On the Spot
myhrwcom
Identifying Sets for Real-World SituationsReal numbers can be used to represent real-world quantities Highways have posted speed limit signs that are represented by natural numbers such as 55 mph Integers appear on thermometers Rational numbers are used in many daily activities including cooking For example ingredients in a recipe are often given in fractional amounts such as 2 _ 3 cup flour
Identify the set of numbers that best describes each situation Explain your choice
the number of people wearing glasses in a room
The set of whole numbers best describes the situation The number of people wearing glasses may be 0 or a counting number
the circumference of a flying disk has a diameter of 8 9 10 11 or 14 inches
The set of irrational numbers best describes the situation Each circumference would be a product of π and the diameter and any multiple of π is irrational
EXAMPLEXAMPLE 3
A
B
Identify the set of numbers that best describes the situation Explain your choice
5 the amount of water in a glass as it evaporates
6 the weight of a person in pounds
YOUR TURN
8NS1
Rational numbers a personrsquos weight can be a decimal
such as 835 pounds
Real numbers the amount can be any number greater
than 0
17Lesson 12
copy H
ough
ton
Miff
lin H
arco
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ublis
hing
Com
pany
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
8_MCAAESE206984_U1M01L2indd 17 41613 520 AM
Graphic OrganizersGive students a list of numbers (including terminating and repeating decimals fractions integers and rational and irrational square roots) and a graphic organizer as shown below
Real Numbers
Rational numbers Irrational numbers
Integer numbers
Whole numbers
Ask students to write each number in the list in the correct section of the organizer
Number SensePoint out to students that knowing the types of numbers to expect in different situations can alert them to incorrect math as well as to impossible situations For example 135 shots made in basketballs is not possible but an average number of shots can equal 135
Additional ResourcesDifferentiated Instruction includes bull Reading Strategies bull Success for English Learners EL
bull Reteach bull Challenge PRE-AP
DIFFERENTIATE INSTRUCTION
Sets of Real Numbers 18
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
Math TrainerOnline Assessment
and Intervention
Personal
Online homework assignment available
myhrwcom
Lesson Quiz available online
12 LESSON QUIZ
1 Write all the names that apply to the number
2 Tell whether the given statement is true or false Explain your choice All numbers between 1 and 2 are rational numbers
3 Identify the set of numbers that best describes the situation Explain your choiceThe choices on a survey question change the total points for the survey by -2 -1 0 1 or 2 points
-1 _
5
myhrwcom
Additional ResourcesDifferentiated Instruction includes bull Leveled Practice worksheets
EvaluateGUIDED AND INDEPENDENT PRACTICE
Concepts amp Skills Practice
Example 1Classifying Real Numbers
Exercises 1ndash8 14ndash19 22ndash24
Example 2Understanding Sets and Subsets of Real Numbers
Exercises 9ndash10
Example 3Identifying Sets for Real-World Situations
Exercises 11ndash12 20ndash21 25
Focus | Coherence | Rigor
Exercise Depth of Knowledge (DOK) Mathematical Practices
14ndash19 2 SkillsConcepts MP7 Using Structure
20ndash21 2 SkillsConcepts MP6 Precision
22ndash23 2 SkillsConcepts MP3 Logic
24 1 Recall of Information MP7 Using Structure
25 2 SkillsConcepts MP2 Reasoning
26ndash27 3 Strategic Thinking MP3 Logic
28 3 Strategic Thinking MP8 Patterns
29 3 Strategic Thinking MP3 Logic
8NS1
8NS1
Exercise 29 combines concepts from the California Common Core cluster ldquoKnow that there are numbers that are not rational and approximate them by rational numbersrdquo
Answers1 rational real
2 False radic_
2 is an example of an irrational number between 1 and 2
3 Integers each number is an integer but only three are whole numbers
19 Lesson 12
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
Work Area
π mi23 Critique Reasoning The circumference of a circular region is shown
What type of number best describes the diameter of the circle Explain
your answer
24 Critical Thinking A number is not an integer What type of number can it be
25 A grocery store has a shelf with half-gallon containers of milk What type of number best represents the total number of gallons
26 Explain the Error Katie said ldquoNegative numbers are integersrdquo What was her error
27 Justify Reasoning Can you ever use a calculator to determine if a number is rational or irrational Explain
28 Draw Conclusions The decimal 0 _
3 represents 1 _ 3 What type of number best describes 0
_ 9 which is 3 middot 0
_ 3 Explain
29 Communicate Mathematical Ideas Irrational numbers can never be precisely represented in decimal form Why is this
FOCUS ON HIGHER ORDER THINKING
It can be a rational number that is not an integer or an irrational number
rational number
The set of negative numbers also includes non-integer
rational numbers and irrational numbers
Sample answer If the calculator shows a decimal that
terminates in fewer digits than what the calculator screen
allows then you can tell that the number is rational If not
you cannot tell from the calculator display whether the
number terminates because you see a limited number
of digits It may be a repeating decimal (rational) or
non-terminating non-repeating decimal (irrational)
Whole 3 middot 0 _
3 represents 3 middot 1 _ 3 = 1 so 0 _
9 is exactly 1
Sample answer In decimal form irrational numbers never
terminate and never repeat Therefore no matter how
many decimal places you include the number will never
be precisely represented There are always more digits
Whole the diameter is π _ π = 1 mile
Unit 120
copy H
ough
ton
Miff
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ublis
hing
Com
pany
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
8_MCAAESE206984_U1M01L2indd 20 120413 909 PM
Integers
Rational Numbers Irrational Numbers
Real Numbers
Whole Numbers
257
radic16
166
radic9
128 radic50
Personal Math Trainer
Online Practice and Helpmyhrwcom
Name Class Date
Independent Practice
Identify the set of numbers that best describes each situation Explain your choice
20 the height of an airplane as it descends to an airport runway
21 the score with respect to par of several golfers 2 ndash 3 5 0 ndash 1
22 Critique Reasoning Ronald states that the number 1 __ 11 is not rational because when converted into a decimal it does not terminate Nathaniel says it is rational because it is a fraction Which boy is correct Explain
12
14 - radic_
9 15 257
16 radic_
50 17 8 1 _ 2
18 166 19 radic_
16
Write all names that apply to each number Then place the numbers in the correct location on the Venn diagram
8NS1
Real numbers the height can be any number greater than zero
integer rational real whole integer rational real
whole integer rational real
irrational real
rational real
rational real
Integers the scores are counting numbers their
opposites and zero
Nathaniel is correct A rational number is a number that can be written as a fraction and 1 __ 11 is a fraction
19Lesson 12
copy H
ough
ton
Miff
lin H
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urt P
ublis
hing
Com
pany
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
8_MCAAESE206984_U1M01L2indd 19 41613 136 AM
myhrwcomActivity available onlineEXTEND THE MATH PRE-AP
Activity Have students consider the concept of restricted domain for the sets of numbers that describe situations For example the number of sisters a person has can best be described by whole numbers but no one has ever had 1500 sisters An area code is an integer or whole number between 200 and 999
Have students use a source such as the Guinness Book of World Records and give examples of sets of numbers that describe situations where the domain is restricted Ask whether the restriction may be changed in the future
Sets of Real Numbers 20
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
-3-4-5 -2-1 1 2 3 50 4
12-4 -radic5
Lesson Support Content Objective Students will learn to order a set of real numbers
Language Objective Students will show and describe how to order a set of real numbers
LESSON 13 Ordering Real Numbers
Building BackgroundEliciting Prior Knowledge Have students draw a number line to compare a rational number and an irrational number such as - radic
_ 5 and -4 1 __ 2 Ask them to explain how
they approximated the irrational number on the number line Then have them identify the greater and the lesser real number Repeat with several other pairs of real numbers in different forms
Learning ProgressionsIn this lesson students order a set of real numbers They use rational approximations to compare the sizes of irrational numbers They also order numbers for real-world situations Important understandings for students include the following
bull Compare irrational numbers bull Estimate the value of expressions with irrational numbers bull Order a set of real numbers bull Order real numbers in a real-world context
Work with real numbers continues throughout Grade 8 and into high school This lesson provides students with a foundation for understanding the relative sizes of numbers in different forms in the real number system
Cluster ConnectionsThis lesson provides an excellent opportunity to connect ideas in this cluster Know that there are numbers that are not rational and approximate them by rational numbers Tell students that there is a special number called the golden ratio with applications in mathematics geometry art and architecture The golden ratio is called phi and is represented by the Greek letter ϕ It includes an irrational number in its definition
Have students explain why the golden ratio is irrational Ask them to find the two whole numbers the golden ratio lies between Then challenge them to approximate the golden ratio to the nearest tenth It is irrational because it includes an irrational number in its definition It lies between 1 and 2 To the nearest tenth ϕ = 16
ϕ = 1 + radic_
5 _ 2
Focus | Coherence | Rigor
California Common Core Standards
8NS2 Use rational approximations of irrational numbers to compare the size of irrational numbers locate them approximately on a number line diagram and estimate the value of expressions (eg π 2 )
MP4 Model with mathematics
21A
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
Math Talk
Language Support EL
PROFESSIONAL DEVELOPMENT
Linguistic Support EL
AcademicContent Vocabulary
Post a chart like this to remind students of the regular comparative forms of adjectives that use the -er and -est suffixes Add to the chart for terms that appear in examples and exercises in each lesson Include any irregular verb forms
Background Knowledge
Go On ndash the title of the module review or quiz is Ready to Go On This title uses an idiomatic expression In this context to go on means ldquoto move aheadrdquo or ldquoto proceedrdquo It is different from the use of go on that means having enough facts to use meaningfully as in having enough to go on Also the intonation used in pronouncing an expression can give it different meanings For example when the speaker emphasizes the word on he or she might be expressing disbelief as in ldquoGo ON Yoursquore kidding rightrdquo Discuss with students other ways that the phrase go on may be used
Leveled Strategies for English Learners
Emerging Label points on a number line with the terms used in ordering greater greatest less lesser least Use sentence frames to insert the correct terms
Expanding Have students give two or three complete sentences to compare the placement of numbers on a number line using the correct forms of the comparative and superlative adjectives
Bridging Have students work in pairs with one student giving directions to the other in complete sentences to order numbers on a number line
To help students answer the question posed in Math Talk make sure that students have a command of the forms for making comparisons and the superlative and the concept of opposite order so that the focus is on the math concept instead of the language skills needed to describe and explain order
EL
Adjective Comparative Superlative
Far Farther Farthest
Large Larger Largest
Great Greater Greatest
Some Less Least
Some More Most
California ELD Standards
Emerging 2I8 Analyzing language choices ndash Explain how phrasing or different common words with similar meanings produce different effects on the audience
Expanding 2I8 Analyzing language choices ndash Explain how phrasing or different words with similar meanings or figurative language produce shades of meaning and different effects on the audience
Bridging 2I8 Analyzing language choices ndash Explain how phrasing or different words with similar meanings or figurative language produce shades of meaning nuances and different effects on the audience
Ordering Real Numbers 21B
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
13L E S S O N
Ordering Real Numbers
Interactive Whiteboard Interactive example available online
ADDITIONAL EXAMPLE 1Compare Write lt gt or =
A radic_
8 - 2 4 - radic_
8 lt
B radic_
20 + 1 3 + radic_
2 gt
EngageESSENTIAL QUESTION
How do you order a set of real numbers Sample answer Find their approximate decimal values and order them
Motivate the LessonAsk What kind of numbers are you comparing when you compare the price of gasoline at two different gas stations
ExploreGive students two rational numbers and ask them to name a number between them Repeat a few times and then give them two irrational numbers and ask them to name a number between them
ExplainEXAMPLE 1
Questioning Strategies Mathematical Practices bull Which is greater the difference between 5 and 3 or the difference between radic
_ 5 and radic
_ 3
The difference between 5 and 3 is 2 the difference between radic_
5 and radic_
3 is approximately 1 So the difference between 5 and 3 is greater
Avoid Common ErrorsCaution students to read the problem carefully and think about what the radical sign means so that they do not misread the problem and answer that the two sides are equal
YOUR TURNFocus on TechnologyCalculators should not be used at this point because developing number sense is the goal
EXAMPLE 2Questioning Strategies Mathematical Practices bull How do you determine whether radic
_ 22 is less than or greater than 45 The square of 45 is
2025 which is less than 22 so the square root of 22 must be greater than 45
Engage with the WhiteboardHave students graph and label various real numbers between 42 and 44 and between 47 and 5
YOUR TURNFocus on Modeling Mathematical PracticesHave students label the integers on the number line with their equivalent square root For example 1 2 and 3 on the number line would be labeled radic
_ 1 radic
_ 4 and radic
_ 9
Interactive Whiteboard Interactive example available online
ADDITIONAL EXAMPLE 2Order 3π radic
_ 10 and 325 from greatest
to least
3π 325 radic_
10
myhrwcom
myhrwcom
CA Common CoreStandards
The student is expected to
The Number Systemmdash8NS2
Use rational approximations of irrational numbers to compare the size of irrational numbers locate them approximately on a number line diagram and estimate the value of expressions (eg π 2 )
Mathematical Practices
MP4 Modeling
The student is expected to
21 Lesson 13
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
Math TrainerOnline Practice
and Help
Personal
myhrwcom
Math On the Spotmyhrwcom
0 05 1 15 2 25 3 35 4
radic5radic3
π2
8 85 9 95 10 105 11 115 12
radic75
4 42 44 46 48 5
radic224 12π + 1
Ordering Real Numbers You can compare and order real numbers and list them from least to greatest
Order radic_
22 π + 1 and 4 1 _ 2 from least to greatest
First approximate radic_
22
radic_
22 is between 4 and 5 Since you donrsquot know where it falls between 4 and 5 you need to find a better estimate for radic
_ 22 so
you can compare it to 4 1 _ 2
Since 22 is closer to 25 than 16 use squares of numbers between 45 and 5 to find a better estimate of radic
_ 22
45 2 = 2025 46 2 = 2116 47 2 = 2209 48 2 = 2304
Since 47 2 = 2209 an approximate value for radic_
22 is 47
An approximate value of π is 314 So an approximate value of π +1 is 414
Plot radic_
22 π + 1 and 4 1 _ 2 on a number line
Read the numbers from left to right to place them in order from least to greatest
From least to greatest the numbers are π + 1 4 1 _ 2 and radic_
22
EXAMPLE 2
STEP 1
STEP 2
Order the numbers from least to greatest Then graph them on the number line
YOUR TURN
5 radic_
5 25 radic_
3
6 π 2 10 radic_
75
If real numbers a b and c are in order from least to greatest what is the order
of their opposites from least to greatest
Explain
Math TalkMathematical Practices
8NS2
radic_
3 radic_
5 25
radic_
75 π2 10
Math Talk answer -c -b -a -c is farthest to the left on a number line -b is in the middle and -a is farthest to the right
Unit 122
copy H
ough
ton
Miff
lin H
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hing
Com
pany
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
8_MCAAESE206984_U1M01L3indd 22 41613 447 AM
My Notes
Math TrainerOnline Practice
and Help
Personal
myhrwcom
Math On the Spot
myhrwcom
Comparing Irrational NumbersBetween any two real numbers is another real number To compare and order real numbers you can approximate irrational numbers as decimals
Compare radic_
3 + 5 3 + radic_
5 Write lt gt or =
First approximate radic_
3
radic_
3 is between 1 and 2
Next approximate radic_
5
radic_
5 is between 2 and 3
Then use your approximations to simplify the expressions
radic_
3 + 5 is between 6 and 7
3 + radic_
5 is between 5 and 6
So radic_
3 + 5 gt 3 + radic_
5
Reflect1 If 7 + radic
_ 5 is equal to radic
_ 5 plus a number what do you know about the
number Why
2 What are the closest two integers that radic_
300 is between
EXAMPLEXAMPLE 1
STEP 1
STEP 2
Compare Write lt gt or =
YOUR TURN
3 radic_
2 + 4 2 + radic_
4 4 radic_
12 + 6 12 + radic_
6
L E S S O N
13 Ordering Real Numbers
ESSENTIAL QUESTIONHow do you order a set of real numbers
8NS2
Use rational approximations of irrational numbers to compare the size of irrational numbers locate them approximately on a number line diagram and estimate the value of expressions (eg π 2 )
8NS2
Use perfect squares to estimate square roots
1 2 = 1 2 2 = 4 3 2 = 9
The number is 7 both expressions must equal 7 + radic_
5
17 and 18
gt lt
21Lesson 13
copy H
ough
ton
Miff
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pany
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
8_MCAAESE206984_U1M01L3indd 21 41913 246 PM
PROFESSIONAL DEVELOPMENT
Math BackgroundIn this lesson students estimate irrational numbers in the form of square roots of nonper-fect squares by finding two perfect squares between which the number falls A more precise method involves repeated division For example to find radic
_ 28 find a whole number whose perfect
square is close to 28 such as 5 Divide 28 by that number 28 divide 5 = 56 Find the average of the quotient and divisor 5 + 56
_____ 2 = 53 Continue dividing 28 by each result and averaging until you get the desired accuracy
Integrate Mathematical Practices MP4
This lesson provides an opportunity to address this Mathematical Practices standard It calls for students to model relationships using multiple representations including diagrams graphs and language as appropriate Students use multiple representations when they use number lines to estimate the locations of and order rational and irrational numbers given as symbols
Ordering Real Numbers 22
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
Interactive Whiteboard Interactive example available online
ADDITIONAL EXAMPLE 3The diameter of a meteorite in millimeters is calculated by four different methods Order the results from least to greatest
Joe radic_
18 mm Lisa 13 __ 3 mm
Pablo 46 mm Julien 4π __ 3 mm
Julien 4π __ 3 mm Lisa 13 __ 3 mm
Joe radic_
18 mm Pablo 46 mm
EXAMPLE 3Questioning Strategies Mathematical Practices bull How can you verify that radic
_ 28 is between 52 and 53 5 2 2 = 2704 and 5 3 2 = 2809
bull Explain how to determine which number is greater 5 _
5 or 55 When the repeating decimal is rounded to the nearest tenth or hundredth you can see that it is greater
Connect to Daily LifeDiscuss how measuring across a canyon might involve different methods than measuring along a road Explain that measurements like these are often done using calculations that approximate the distance
YOUR TURNFocus on Critical Thinking Mathematical PracticesDiscuss with students which number is greater 3
_ 45 or 3450 3
_ 45 or 3455 and why Explain
that 3 _
45 can be written out as 34545hellipMake sure they understand that 3 _
45 is greater than 345 but less than 3455
ElaborateTalk About ItSummarize the Lesson
Ask How can you order two numbers in different forms whose decimal approxi-mations appear to be equal Approximate one or both numbers to an additional
number of decimal places
GUIDED PRACTICEEngage with the Whiteboard
Have students place and label additional points on the number line in Exercise 9 Allow the points to be in any format other than decimal
Avoid Common ErrorsExercises 3ndash4 Caution students to read the problem carefully so that they do not misread the problem as the same numbers combined by addition on each side of the circleExercise 10 Remind students that the calculations have units
myhrwcom
23 Lesson 13
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
0 05 1 15 2 25 3 35 4 45 5 55 6 65 7
2πradic3
Compare Write lt gt or = (Example 1)
1 radic_
3 + 2 radic_
3 + 3 2 radic_
8 + 17 radic_
11 + 15
3 radic_
6 + 5 6 + radic_
5 4 radic_
9 + 3 9 + radic_
3
5 radic_
17 - 3 -2 + radic_
5 6 12 - radic_
2 14 - radic_
8
7 radic_
7 + 2 radic_
10 - 1 8 radic_
17 + 3 3 + radic_
11
9 Order radic_
3 2π and 15 from least to greatest Then graph them on the number line (Example 2)
radic_
3 is between and so radic_
3 asymp
π asymp 314 so 2π asymp
From least to greatest the numbers are
10 Four people have found the perimeter of a forest using different methods Their results are given in the table Order their calculations from greatest to least (Example 3)
11 Explain how to order a set of real numbers
CHECK-INESSENTIAL QUESTION
Forest Perimeter (km)
Leon Mika Jason Ashley
radic_
17 - 2 1 +thinsp π __ 2 12 ___ 5 25
Guided Practice
17
15
1 + π _ 2 km 25 km 12 __ 5 km radic_
17 - 2 km
2π radic
_ 3
18 175
628
Sample answer Convert each number to a decimal
equivalent using estimation to find equivalents for
irrational numbers Graph each number on a number line
Read the numbers from left to right for least to greatest
Read the numbers from right to left for greatest to least
lt gt
lt lt
ltgt
gt gt
24 Unit 1
copy H
ough
ton
Miff
lin H
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ublis
hing
Com
pany
bull Im
age C
redi
ts copy
Elena
Eliss
eeva
Alam
y Im
ages
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
8_MCAAESE206984_U1M01L3indd 24 41613 448 AM
My Notes
5 52 54 56 58 6
radic28 5 12
23455
Math TrainerOnline Practice
and Help
Personal
myhrwcom
Math On the Spot
myhrwcom
Ordering Real Numbers in a Real-World Context Calculations and estimations in the real world may differ It can be important to know not only which are the most accurate but which give the greatest or least values depending upon the context
Four people have found the distance in kilometers across a canyon using different methods Their results are given in the table Order the distances from greatest to least
Distance Across Quarry Canyon (km)
Juana Lee Ann Ryne Jackson
radic_
28 23 __ 4 5 _
5 5 1 _ 2
Write each value as a decimal
radic_
28 is between 52 and 53 Since 53 2 = 2809 an approximate value for radic
_ 28 is 53
23 __ 4 = 575
5 _
5 is 5555hellip so 5 _
5 to the nearest hundredth is 556
5 1 _ 2 = 55
Plot radic_
28 23 __ 4 5 _
5 and 5 1 _ 2 on a number line
From greatest to least the distances are
23 __ 4 km 5 _
5 km 5 1 _ 2 km radic_
28 km
EXAMPLEXAMPLE 3
STEP 1
STEP 2
7 Four people have found the distance in miles across a crater using different methods Their results are given below
Jonathan 10 __ 3 Elaine 3 _
45 Joseacute 3 1 _ 2 Lashonda radic_
10
Order the distances from greatest to least
YOUR TURN
8NS2
3 1 _ 2 mi 3 _
45 mi 10 __ 3 mi radic_
10 mi
23Lesson 13
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ough
ton
Miff
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pany
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8_MCAAESE206984_U1M01L3indd 23 41613 447 AM
ModelingPlace papers around the room with the numbers from 1 to 5 one per sheet Give each student a card showing a number between 1 and 5 in different forms Have students place his or her card between the correct integers and decide where the number goes in relation to any numbers already placed
Multiple RepresentationsGive students a vertical number line which some students might find easier to use than a horizontal one Have them decide whether to place points for rational and irrational numbers above or below existing points
Additional ResourcesDifferentiated Instruction includes bull Reading Strategies bull Success for English Learners EL
bull Reteach bull Challenge PRE-AP
DIFFERENTIATE INSTRUCTION
Ordering Real Numbers 24
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
Math TrainerOnline Assessment
and Intervention
Personal
Online homework assignment available
myhrwcom
myhrwcom
Lesson Quiz available online
13 LESSON QUIZ
1 Compare Write lt gt or =
radic_
95 - 5 radic_
62 - 2
2 Order 105 radic_
105 and 3π + 1 from greatest to least
3 A length in centimeters is calculated differently by four different people Order their calculations from least to greatest
KD 11 __ 2 cm Silvio 5 __ 3 π cm
Paula 5 _
4 cm Luis radic_
33 cm
EvaluateGUIDED AND INDEPENDENT PRACTICE
Concepts amp Skills Practice
Example 1Comparing Irrational Numbers
Exercises 1ndash8
Example 2Ordering Real Numbers
Exercises 9 12ndash15 18ndash21
Example 3Ordering Real Numbers in a Real-World Context
Exercises 10 16ndash17
Additional ResourcesDifferentiated Instruction includes bull Leveled Practice worksheets
Focus | Coherence | Rigor
Exercise Depth of Knowledge (DOK) Mathematical Practices
12ndash15 1 Recall of Information MP5 Using Tools
16 2 SkillsConcepts MP2 Reasoning
17 2 SkillsConcepts MP6 Precision
18ndash21 2 SkillsConcepts MP2 Reasoning
22 3 Strategic Thinking MP4 Modeling
23ndash24 3 Strategic Thinking MP3 Logic
8NS2
8NS2
Answers1 radic
_ 95 - 5 lt radic
_ 62 - 2
2 radic_
105 3π + 1 105
3 Silvio 5 __ 3 π cm Paula 5 _
4 cm
KD 11
__ 2 cm Luis radic_
33 cm
25 Lesson 13
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
Work Area
3140 3141 3142 3143
314 π227
20 A teacher asks his students to write the numbers shown in order from least to greatest Paul thinks the numbers are already in order Sandra thinks the order should be reversed Who is right
21 Math History There is a famous irrational number called Eulerrsquos number symbolized with an e Like π its decimal form never ends or repeats The first few digits of e are 27182818284
a Between which two square roots of integers could you find this number
b Between which two square roots of integers can you find π
22 Analyze Relationships There are several approximations used for π including 314 and 22 __ 7 π is approximately 314159265358979
a Label π and the two approximations on the number line
b Which of the two approximations is a better estimate for π Explain
c Find a whole number x so that the ratio x ___ 113 is a better estimate for π
than the two given approximations
23 Communicate Mathematical Ideas If a set of six numbers that include both rational and irrational numbers is graphed on a number line what is the fewest number of distinct points that need to be graphed Explain
24 Critique Reasoning Jill says that 12 _
6 is less than 1263 Explain her error
FOCUS ON HIGHER ORDER THINKING
radic_
115 115 ___ 11 and 105624
between radic_
7 asymp 265 and radic_
8 asymp 283
between radic_
9 = 3 and radic_
10 asymp 316
22 __ 7 it is closer to π on the number line
She did not consider the repeating digit 1266
2 rational numbers can have the same location and
irrational numbers can have the same location but they
cannot share a location
355
Neither student is correct The answer
should be 115 ___ 11 105624 radic_
115
Unit 126
copy H
ough
ton M
ifflin
Har
cour
t Pub
lishin
g Com
pany
Imag
e Cre
dits
copy3D
Stoc
kiSt
ockP
hoto
com
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
8_MCAAESE206984_U1M01L3indd 26 210513 801 AM
Personal Math Trainer
Online Practice and Helpmyhrwcom
Name Class Date
Independent Practice
16 Your sister is considering two different shapes for her garden One is a square with side lengths of 35 meters and the other is a circle with a diameter of 4 meters
a Find the area of the square
b Find the area of the circle
c Compare your answers from parts a and b Which garden would give your sister the most space to plant
17 Winnie measured the length of her fatherrsquos ranch four times and got four different distances Her measurements are shown in the table
a To estimate the actual length Winnie first approximated each distance to the nearest hundredth Then she averaged the four numbers Using a calculator find Winniersquos estimate
b Winniersquos father estimated the distance across his ranch to be radic_
56 km How does this distance compare to Winniersquos estimate
Give an example of each type of number
18 a real number between radic_
13 and radic_
14
19 an irrational number between 5 and 7
Order the numbers from least to greatest
12 radic_
7 2 radic_
8 ___ 2 13 radic_
10 π 35
14 radic_
220 -10 radic_
100 115 15 radic_
8 -375 3 9 _ 4
Distance Across Fatherrsquos Ranch (km)
1 2 3 4
radic_
60 58 __ 8 7 _
3 7 3 _ 5
138NS2
radic_
8 ___ 2 2 radic_
7
-10 radic_
100 115 radic_
220
radic_
60 asymp 775 58 __ 8 = 725 7 _
3 asymp 733 7 3 _ 5 = 760 so the average
π radic_
10 35
-375 9 _ 4 radic_
8 3
is 74825 km
1225 m2
4π m2 or approximately 126 m2
They are nearly identical radic_
56 is approximately 74833hellip
The circle would give her more space to plant because it has a
larger area
Sample answer 37
Sample answer radic_
31
25Lesson 13
copy H
ough
ton
Miff
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hing
Com
pany
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
8_MCAAESE206984_U1M01L3indd 25 41613 448 AM
Activity available online myhrwcomEXTEND THE MATH PRE-AP
Activity Have students investigate whether there are infinitely many numbers between two numbers by giving examples for each of the following
bull Between any two rational numbers there is at least one other rational number Sample answer 45 is between 41 and 48
bull Between any two irrational numbers there is at least one rational number Sample answer 45 is between radic
_ 11 and radic
_ 29
bull Between any two rational numbers there is at least one irrational number Sample answer radic
_ 11 is between 31 and 36
bull Between any two irrational numbers there is at least one irrational number Sample answer radic
_ 17 is between radic
_ 11 and radic
_ 29
Ordering Real Numbers 26
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
ReadyMath Trainer
Online Practiceand Help
Personal
myhrwcom
Module Quiz
11ensp RationalenspandenspIrrationalenspNumbersWrite each fraction as a decimal or each decimal as a fraction
1 7__20 2 1___
27 3 17_8
Solve each equation for x
4 x2=81 5 x3=343 6 x2= 1___100
7 Asquarepatiohasanareaof200squarefeetHowlongiseachside
ofthepatiotothenearesttenth
12ensp SetsenspofenspRealenspNumbersWrite all names that apply to each number
8 121____radic
____121
9 π__2
10 TellwhetherthestatementldquoAllintegersarerationalnumbersrdquoistrueorfalseExplainyourchoice
13ensp OrderingenspRealenspNumbersCompare Write lt gt or =
11 radic__
8+3 8+radic__
3 12 radic__
5+11emsp emsp emsp 5+radic___
11
Order the numbers from least to greatest
13 radic___
99π29__
8 14 radic___
1__251_40__
2
15 Howarerealnumbersusedtodescribereal-worldsituations
ESSENTIAL QUESTION
035
9-9
141ft
7 1__10- 1__10
14__11 1875
wholeintegerrationalreal
Trueintegerscanbewrittenasthequotientoftwointegers
SampleanswerRealnumberssuchastherational
π29__
8radic___
99
irrationalreal
lt gt
number1_4candescribeamountsusedincooking
radic___
1__250__
21_4
27Module1
copy H
ough
ton
Miff
lin H
arco
urt P
ublis
hing
Com
pany
DONOTEDIT--ChangesmustbemadethroughldquoFileinfordquoCorrectionKey=A
8_MCAAESE206984_U1M01RTindd 27 41513 1113 PM
Math TrainerOnline Assessment
and Intervention
Personal
myhrwcom
1
2
3 Response toIntervention
Intervention Enrichment
Access Ready to Go On assessment online and receive instant scoring feedback and customized intervention or enrichment
Online and Print Resources
Differentiated Instruction
bull Reteach worksheets
bull Reading Strategies EL
bull Success for English Learners EL
Differentiated Instruction
bull Challenge worksheets PRE-AP
Extend the Math PRE-AP
Lesson Activities in TE
Additional ResourcesAssessment Resources includes bull Leveled Module Quizzes
Ready to Go OnAssess MasteryUse the assessment on this page to determine if students have mastered the concepts and standards covered in this module
California Common Core Standards
Lesson Exercises Common Core Standards
11 1ndash7 8NS1 8NS2 8EE2
12 8ndash10 8NS1
13 11ndash14 8NS2
27 Unit 1 Module 1
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
Personal Math Trainer
Online Practice and HelpmyhrwcomAssessment Readiness
Module 1 MIXed ReVIeW
1 Look at each number Is the number between 2π and radic___
52
Select Yes or No for expressions AndashC
A 6 2 _ 3 Yes No
B 5π __ 2 Yes No
C 3 radic__
5 Yes No
2 Consider the number - 11 __ 15
Choose True or False for each statement
A The number is rational True False
B The number can be written as True Falsea repeating decimal
C The number is less than ndash08 True False
3 The volume of a cube is given by V = x3 where x is the length of an edge of the cube A cube-shaped end table has a volume of 3 3 _ 8 cubic feet What is the length of an edge of the end table Explain how you solved this problem
4 A student says that radic___
83 is greater than 29 __ 3 Is the student correct Justify your
reasoning
1 1 _ 2 ft Sample answer The equation x3 = 3 3 _ 8 can be used
to find the edge length in feet To solve the equation
write the mixed number as a fraction greater than 1
x3 = 27 __ 8 Then take the cube root of both sides x = 3 _ 2 = 1 1 _ 2
No Sample answer radic___
83 asymp 91 and 29 __ 3 = 9
__ 6
Because 91 lt 9 __
6 radic___
83 lt 29 __ 3
28 Unit 1
copy H
ough
ton
Miff
lin H
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ublis
hing
Com
pany
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=A
8_MCAAESE206984_U1M01RTindd 28 240413 946 AM
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Online Assessment and
Interventionmyhrwcom
Scoring GuideItem 3 Award the student 1 point for finding the edge length of the cube and 1 point for correctly explaining how to use a cube root to solve the problem
Item 4 Award the student 1 point for determining that the student is incorrect and 1 point for correctly justifying the reasoning for this conclusion
Additional ResourcesTo assign this assessment online login to your Assignment Manager at myhrwcom
Assessment Readiness
California Common Core Standards
Items Grade 8 Standards Mathematical Practices
1 8NS2 MP7
2 7NS2b 7NS2d 8NS1 MP7
3 8EE2 MP1 MP4
4 8NS1 8NS2 MP3
Item integrates mixed review concepts from previous modules or a previous course
Item 4 combines concepts from the California Common Core cluster ldquoKnow that there are numbers that are not rational and approximate them by rational numbersrdquo
Real Numbers 28
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
Focus on Technology Mathematical PracticesPoint out the importance of entering a repeating decimal correctly when using a graphing calculator to convert the decimal to a fraction The decimal 0
_ 59 must be entered as
0595959595959 not 059
YOUR TURNFocus on Math ConnectionsMake sure students understand that the place value of the last digit in Exercises 4 and 6 determines the denominator of the corresponding fraction or mixed number So for Exercise 4 the place value hundredths gives a denominator of 100 and for Exercise 6 the place value tenths gives a denominator of 10
EXAMPLE 3Questioning Strategies Mathematical Practices bull How can a solution of an equation of the form x 2 = p be negative if p is a positive number Since the square of a negative number is positive a negative number is also a solution of x 2 equals a positive number
bull When is a solution of an equation of the form x 3 = p larger than p The solution is larger than p if p is a number between 0 and 1
Focus on Math Connections Make sure students understand the difference in finding radic
_ 121 and solving x 2 = 121 The
symbol radic_
indicates the positive or principal square root only while the equation x 2 = 121 has two roots the principal square root and its opposite
YOUR TURNAvoid Common ErrorsTo avoid sign errors in Exercise 9 make sure that students understand that the cube of a negative number is not a positive number Therefore -8 is not a solution of x 3 = 512
Talk About ItCheck for Understanding
Ask Kris predicts that there are two real solutions for Exercises 7 and 8 and that there are three real solutions for Exercises 9 and 10 Is his prediction correct
Explain His prediction is correct for Exercises 7 and 8 because there are two numbers whose squares are the same positive number given in the exercises His prediction is not correct for Exercises 9 and 10 however because there is only one real number whose cube is the same positive number given in the exercises
EXPLORE ACTIVITYQuestioning Strategies Mathematical Practices bull Compare the values for 13 2 and 13 2 The digits are the same but 13 2 has two decimal places (169) while 13 2 has none (169)
bull How do you know whether radic_
2 will be closer to 1 or closer to 2 It will be closer to 1 because 2 is between the perfect squares of 1 and 4 but closer to 1 than it is to 4
Connect Vocabulary EL
Explain to students that the word irrational when used as an ordinary word in English means without logic or reason In mathematics when we say that a number is irrational it means only that the number cannot be written as the quotient of two integers
Engage with the WhiteboardHave students extend the number line in both directions and label the locations of the whole numbers 1 and 2 These are the roots of the consecutive perfect squares
1 and 4 used to estimate radic_
7
Interactive Whiteboard Interactive example available online
ADDITIONAL EXAMPLE 3Solve each equation for x
A x 2 = 324 18 -18
B x 2 = 25 ___ 144 5 __ 12 - 5 __ 12
C 343 = x 3 7
D x 3 = 125 ___ 512 5 __ 8
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9 Lesson 11
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Math TrainerOnline Practice
and Help
Personal
myhrwcom
EXPLORE ACTIVITY
lt 2 lt
radic_
lt radic
_ 2 lt
radic_
lt radic
_ 2 lt
The solution is 9
The solution is 2 _ 5
C
D
729 = x 3
3 radic_ 729 = 3 radic
_ x 3
3 radic_ 729 = x
9 = x
x 3 = 8 ___ 125
3 radic_
x 3 =thinsp 3 radic_ 8 ___ 125
x =thinsp 3 radic_ 8 ___ 125
x = 2 _ 5
Solve each equation for x
YOUR TURN
7 x 2 = 196 8 x 2 = 9 ___ 256
9 x 3 = 512 10 x 3 = 64 ___ 343
Estimating Irrational NumbersIrrational numbers are numbers that are not rational In other words they cannot be written in the form a _ b where a and b are integers and b is not 0 Square roots of perfect squares are rational numbers Square roots of numbers that are not perfect squares are irrational Some equations like those in Example 3 involve square roots of numbers that are not perfect squares
x 2 = 2 x = plusmn radic_
2
Estimate the value of radic_
2
Find two consecutive perfect squares that 2 is between Complete the inequality by writing these perfect squares in the boxes
Now take the square root of each number
Simplify the square roots of perfect squares
radic_
2 is between and
A
B
C
8NS2 8EE2
Solve for x by taking the cube root of both sides
Solve for x by taking the cube root of both sides
Apply the definition of cube root
Think What number cubed equals 729
Apply the definition of cube root
Think What number cubed equals 8 ____ 125
radic_
2 is irrational
x = plusmn14 x = plusmn 3 __ 16
x = 8 x = 4 _ 7
1 2
1 4
1 4
1 2
Unit 110
copy H
ough
ton
Miff
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ublis
hing
Com
pany
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8_MCAAESE206984_U1M01L1indd 10 41613 1211 AM
Math TrainerOnline Practice
and Help
Personal
myhrwcom
Math On the Spot
myhrwcom
Write each decimal as a fraction in simplest form
YOUR TURN
Finding Square Roots and Cube RootsThe square root of a positive number p is x if x 2 = p There are two square roots for every positive number For example the square roots of 36 are 6 and minus6 because 6 2 = 36 and (minus6) 2 = 36 The square roots of 1 __ 25 are 1 _ 5 and minus 1 _ 5 You can write the square roots of 1 __ 25 as plusmn 1 _ 5 The symbol radic
_ 5 indicates the positive
or principal square root
A number that is a perfect square has square roots that are integers The number 81 is a perfect square because its square roots are 9 and minus9
The cube root of a positive number p is x if x 3 = p There is one cube root for every positive number For example the cube root of 8 is 2 because 2 3 = 8 The cube root of 1 __ 27 is 1 _ 3 because ( 1 _ 3 )
3
= 1 __ 27 The symbol 3 radic_ 1 indicates the
cube root
A number that is a perfect cube has a cube root that is an integer The number 125 is a perfect cube because its cube root is 5
Solve each equation for x
The solutions are 11 and minus11
The solutions are 4 __ 13 and minus 4 __ 13
EXAMPLEXAMPLE 3
A x 2 = 121
x 2 = 121
x = plusmn radic_
121
x = plusmn11
B x 2 = 16 ___ 169
x 2 = 16 ___ 169
x = plusmn radic_
16 ___ 169
x = plusmn 4 __ 13
4 012 5 0 _
57 6 14
Can you square an integer and get a negative number
What does this indicate about whether negative
numbers have square roots
Math TalkMathematical Practices
8EE2
Solve for x by taking the square root of both sides
Apply the definition of square root
Think What numbers squared equal 121
Solve for x by taking the square root of both sides
Apply the definition of square root
Think What numbers squared equal 16 ____ 169
3 __ 25 19 __ 33 1 2 _ 5
No the square of a positive integer is positive the square of a negative integer is positive and the square of 0 is 0 So negative numbers do not have (real) square roots
9Lesson 11
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ough
ton
Miff
lin H
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Com
pany
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8_MCAAESE206984_U1M01L1indd 9 41913 240 PM
Critical ThinkingIn the Explore Activity students estimated the location of radic
_ 2 on a number line Ask students
whether they think that it is possible to locate more precisely the point that represents radic
_ 2 In
other words can you graph irrational numbers exactly on a number line along with rational numbers Students should understand that radic
_ 2
is a real number and all real numbers can be located on a real number line A more precise estimate will allow more precise placement on a number line
The Modeling note tells one way to do this
ModelingHave students use a ruler to represent a number line with a unit that is one inch long Have them draw a square with a side of one inch and draw the diagonal to make two isosceles triangles Lead students to understand that the length of the diagonal (or hypotenuse) is radic
_ 2
Have them copy the length of their diagonal onto their ruler or number line starting at zero The end point of the diagonal represents the exact point for the irrational number radic
_ 2 on a
number line
Additional ResourcesDifferentiated Instruction includes bull Reading Strategies bull Success for English Learners EL
bull Reteach bull Challenge PRE-AP
DIFFERENTIATE INSTRUCTION
Rational and Irrational Numbers 10
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
ElaborateTalk About ItSummarize the Lesson
Ask If someone claims that a certain number is irrational but you know it is actually rational how could you prove to that person that the number is rational
You could find a fraction equal to the number such that the number is the ratio of two integers with the denominator not equal to zero
GUIDED PRACTICEEngage with the Whiteboard
Have students plot each number in Exercises 16ndash18 on a number line Students should label each point with the irrational number written as a radical and as a
decimal
Avoid Common ErrorsExercises 1ndash6 To avoid reversing the order of the dividend and divisor tell students to start at the top of the fraction and read the bar as ldquodivided byrdquo
Focus on TechnologyHave students use a calculator to investigate the decimal equivalents of such fractions as 1 __ 9 2 __ 9 8 __ 9 and 1 __ 11 2 __ 11 10
__ 11 Ask them to describe the patterns they find as a result of these investigations
11 Lesson 11
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
Guided Practice
7 0675 8 56 9 044
10 0 _
4
10x =
x =
11 0 _
26
100x =
x =
12 0 _
325
1000x =
x =
Solve each equation for x (Example 3 and Explore Activity)
- x
-
_______________
x =
- x
-
___________________
x =
- x
-
_______________________
x =
Write each fraction or mixed number as a decimal (Example 1)
1 2 _ 5 2 8 _ 9 3 3 3 _ 4
4 7 __ 10 5 2 3 _ 8 6 5 _ 6
Write each decimal as a fraction or mixed number in simplest form (Example 2)
13 x 2 = 17 14 x 2 = 25 ___ 289 15 x 3 = 216
Approximate each irrational number to one decimal place without a calculator
x = plusmn radic__
asymp plusmn x = 3
radic__
=
(Explore Activity)
16 radic_
5 asymp
17 radic_
3 asymp
18 radic_
10 asymp
19 What is the difference between rational and irrational numbers
CHECK-INESSENTIAL QUESTION
x = plusmn radic__
__________ = plusmn _____
4 _
4
0 _
4
4 99
6216
269
41 25 5
17289
17
22 17 32
04
07
27__40
26 __ 99 325 ___ 999 4 _ 9
11__255 3_5
0 _
8
2375
375
08 _
3
26 _
26
0 _
26
325 _
325
0 _
325
999 325
Rational numbers can be written in the form a __ b where
a and b are integers and b ne 0 Irrational numbers cannot
be written in this form
Unit 112
copy H
ough
ton
Miff
lin H
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Com
pany
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8_MCAAESE206984_U1M01L1indd 12 41613 1211 AM
11 12 13 14 15
radic2 asymp 14
141 142 143 144 145
radic2 asymp 141
0 1 2 3 4
radic2 asymp 15
Estimate that radic_
2 asymp 15
To find a better estimate first choose some numbers between 1 and 2 and square them For example choose 13 14 and 15
1 3 2 = 1 4 2 = 1 5 2 =
Is radic_
2 between 13 and 14 How do you know
Is radic_
2 between 14 and 15 How do you know
2 is closer to than to so radic_
2 asymp
Locate and label this value on the number line
Reflect 11 How could you find an even better estimate of radic
_ 2
12 Find a better estimate of radic_
2
1 41 2 = 1 42 2 = 1 43 2 =
2 is closer to than to so radic_
2 asymp
Draw a number line and locate and label your estimate
13 Solve x 2 = 7 Write your answer as a radical expression Then estimate to one decimal place
D
E
F
No 2 is not between 169 and 196
Yes 2 is between 196 and 225
196
19881
19881
225
20164
20164
14
141
20449
169 196 225
Test the squares of numbers between 14 and 15
x = plusmn radic_
7 x asymp plusmn26
11Lesson 11
copy H
ough
ton
Miff
lin H
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ublis
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Com
pany
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8_MCAAESE206984_U1M01L1indd 11 41613 1211 AM
Rational and Irrational Numbers 12
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Math TrainerOnline Assessment
and Intervention
Personal
Online homework assignment available
myhrwcom
EvaluateGUIDED AND INDEPENDENT PRACTICE
Concepts amp Skills Practice
Example 1Expressing Rational Numbers as Decimals
Exercises 1ndash6 20ndash21 24ndash25
Example 2Expressing Decimals as Rational Numbers
Exercises 7ndash12 22ndash23 26ndash27
Example 3Finding Square Roots and Cube Roots
Exercises 13ndash15 28 30ndash31 35
Explore ActivityEstimating Irrational Numbers
Exercises 13 16ndash18 29 32ndash34
Additional ResourcesDifferentiated Instruction includes bull Leveled Practice worksheets
Lesson Quiz available online
11 LESSON QUIZ
1 Write as a decimal 2 5 __ 8 1 7 __ 12
2 Write as a fraction 034 1 _
24
3 Solve x 2 = 9 __ 49 for x
4 Solve x 3 = 216 for x
5 Estimate the value of radic_
13 to one decimal place without using a calculator
myhrwcom
Focus | Coherence | Rigor
Exercise Depth of Knowledge (DOK) Mathematical Practices
20ndash27 2 SkillsConcepts MP4 Modeling
28 3 Strategic Thinking MP4 Modeling
29ndash32 2 SkillsConcepts MP6 Precision
33 3 Strategic Thinking MP7 Using Structure
34 2 SkillsConcepts MP3 Logic
35 2 SkillsConcepts MP4 Modeling
36 3 Strategic Thinking MP3 Logic
37 3 Strategic Thinking MP7 Using Structure
38 3 Strategic Thinking MP2 Reasoning
8NS1 8NS2 8EE2
8NS1 8NS2 8EE2
Answers1 2625 158
_ 3
2 17 __ 50 1 8 __ 33
3 x = plusmn 3 __ 7
4 x = 6
5 36
13 Lesson 11
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
Work Area
33 Analyze Relationships To find radic_
15 Beau found 3 2 = 9 and 4 2 = 16 He said that since 15 is between 9 and 16 radic
_ 15 must be between 3 and 4 He
thinks a good estimate for radic_
15 is 3 + 4 ____ 2 = 35 Is Beaursquos estimate high low
or correct Explain
34 Justify Reasoning What is a good estimate for the solution to the equation x 3 = 95 How did you come up with your estimate
35 The volume of a sphere is 36π f t 3 What is the radius of the sphere Use the formula V = 4 _ 3 π r 3 to find your answer
36 Draw Conclusions Can you find the cube root of a negative number If so is it positive or negative Explain your reasoning
37 Make a Conjecture Evaluate and compare the following expressions
radic_
4 __ 25 and radic
_ 4 ____
radic_
25 radic
_
16 __ 81 and radic_
16 ____
radic_
81 radic
_
36 __ 49 and radic_
36 ____
radic_
49
Use your results to make a conjecture about a division rule for square roots Since division is multiplication by the reciprocal make a conjecture about a multiplication rule for square roots
38 Persevere in Problem Solving The difference between the solutions to the equation x 2 = a is 30 What is a Show that your answer is correct
FOCUS ON HIGHER ORDER THINKING
His estimate is low because 15 is very close to 16
so radic_
15 is very close to radic_
16 or 4 A better estimate
would be 38 or 39
Sample answer about 45 4 3 = 64 and 5 3 = 125
Because 95 is about halfway between 64 and 125 try 45
45 3 = 91125 which is a good estimate
3 feet
Yes the cube root of a negative number is negative
because a negative number cubed is always negative
and a nonnegative number cubed is always nonnegative
radic_
4 __ 25 = 2 _ 5 = radic
_ 4 ____
radic_
25 radic
_
16 __ 81 = 4 _ 9 = radic_
16 ____
radic_
81 radic
_
36 __ 49 = 6 _ 7 = radic_
36 ____
radic_
49
225 the solutions to x 2 = a are x = plusmn15 and
radic_
a ___
radic_
b = radic
_ a __
b radic
_ a radic
_ b = radic
_ a b
15 - (-15) = 30
Unit 114
copy H
ough
ton
Miff
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ublis
hing
Com
pany
bull copy
Ilen
e Mac
Dona
ldA
lamy I
mag
es
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=B
8_MCABESE206984_U1M01L1indd 14 102913 1142 PM
Personal Math Trainer
Online Practice and Helpmyhrwcom
Name Class Date
Independent Practice11
20 A 7 __ 16 -inch-long bolt is used in a machine What is this length written as a decimal
21 The weight of an object on the moon is 1 _ 6 its weight on Earth Write 1 _ 6 as a decimal
22 The distance to the nearest gas station is 2 4 _ 5 kilometers What is this distance written as a decimal
23 A baseball pitcher has pitched 98 2 _ 3 innings What is the number of innings written as a decimal
24 A heartbeat takes 08 second How many seconds is this written as a fraction
25 There are 262 miles in a marathon Write the number of miles using a fraction
26 The average score on a biology test was 72
_ 1 Write the average score using a
fraction
27 The metal in a penny is worth about 0505 cent How many cents is this written as a fraction
28 Multistep An artist wants to frame a square painting with an area of 400 square inches She wants to know the length of the wood trim that is needed to go around the painting
a If x is the length of one side of the painting what equation can you set up to find the length of a side How many solutions does the equation have
b Do all of the solutions that you found make sense in the context of the problem Explain
c What is the length of the wood trim needed to go around the painting
Solve each equation for x Write your answers as radical expressions Then estimate to one decimal place if necessary
29 x 2 = 14 30 x 3 = 1331
31 x 2 = 144 32 x 2 = 29
8NS1 8NS2 8EE2
04375 in 01 _6
28 km 98 _6 innings
x 2 = 400 x = plusmnthinsp20 the equation has 2 solutions
x = 20 makes sense but x = -20 doesnrsquot because a
painting cannot have a side length of -20 inches
4 times 20 = 80 inches
x = plusmn radic_
14 asymp plusmn37
x = plusmn radic_
144 = plusmn12 x = plusmn radic_
29 asymp plusmn54
x = 3 radic_ 1331 = 11
4_5 second 26 1_5 mi
72 1 _ 9 101 ___ 200 cent
13Lesson 11
copy H
ough
ton
Miff
lin H
arco
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ublis
hing
Com
pany
bull copy
Phot
odisc
Get
ty Im
ages
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
8_MCAAESE206984_U1M01L1indd 13 41613 1211 AM
myhrwcomActivity available onlineEXTEND THE MATH PRE-AP
Activity Write radic_
09 on the board and invite students to conjecture what the value might be Have them check their conjectures by squaring Invite them to suggest ways to estimate radic
_ 09 As a hint point out that 09 is close to 10 and so they might
use that to help guide their estimates Lead them to see that since 092 is 081 and 102 is 1 the value of radic
_ 09 is greater than 09 and less than 10 Try squaring 095 to get
09025 A good estimate for radic_
09 is 095
Rational and Irrational Numbers 14
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=B
Integers
Rational Numbers IrrationalNumbers
Real Numbers
WholeNumbers
-3-4-5 -2-1 1 2 3 50 4
23
34-4 -π -1 25
radic2
Lesson Support Content Objective Students will learn to describe relationships between sets of numbers
Language Objective Students will explain how to describe relationships between sets of real numbers
LESSON 12 Sets of Real Numbers
Building BackgroundEliciting Prior Knowledge Have students draw a number line from -5 to 5 Ask them to plot points on the number line to approximate the location of rational and irrational numbers such as -1 3 __ 4 25 -4 2 __ 3 radic
_ 2 and -π
Learning ProgressionsIn this lesson students clarify their understanding of the real number system They characterize sets and subsets of the real numbers They also identify sets for real-world situations Important understandings for students include the following
bull Identify all of the possible subsets of the real numbers for a given number
bull Decide whether a statement about a subset of the real numbers is true or false
bull Identify the set of numbers that best describes a real-world situation
Understanding the relationships among the sets of numbers that make up the real numbers is essential as students are introduced to different forms of numbers throughout the school year This lesson provides a foundation for the comparing and ordering of real numbers in the next lesson
Cluster ConnectionsThis lesson provides an excellent opportunity to connect ideas in this cluster Know that there are numbers that are not rational and approximate them by rational numbers Have students copy this diagram which relates the sets of real numbers
Ask students to complete the diagram by writing three examples for each set of numbers Have students share examples and explain how they knew each number they selected belonged in the appropriate set Answers may vary Check studentsrsquo work
Focus | Coherence | Rigor
California Common Core Standards
8NS1 Know that numbers that are not rational are called irrational Understand informally that every number has a decimal expansion for rational numbers show that the decimal expansion repeats eventually and convert a decimal expansion which repeats eventually into a rational number
MP7 Look for and make use of structure
15A
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
Math Talk
Language Support EL
PROFESSIONAL DEVELOPMENT
Linguistic Support EL
AcademicContent Vocabulary
Venn diagrams ndash Students need descriptive language to describe the categories that the different areas and colors of a Venn diagram represent the concept of a set and how sets are distinct or can overlap Use sentence frames such as
The big oval represents __________The darklight blue color in the middle of the
big ovals represents __________These sets overlap because __________
In this way students have the language and structure to identify the criteria that distinguish a set and to explain the abstract representation Also point out the use of the prefix sub- meaning ldquounderrdquo in the term subset
Rules and Patterns
Abbreviations ndash In this lesson the abbreviation mph is used Be sure to point out that mph stands for miles per hour and is used to give units in a rate of speed Students may also have seen mpg (miles per gallon) which gives the units in a rate of fuel efficiency
Borrowed Words ndash Terminology used in baseball such as inning and pitcher may require some explanation Spanish as well as some other languages have borrowed these terms from English so some students may be familiar with these words already Despite this whenever a word is critical to students understanding the word problem it is best to explain the meaning
Leveled Strategies for English Learners
Emerging Allow students to indicate true or false orally in Guided Practice Exercises 9 and 10
Expanding Have students use sentence frames to describe the meaning of regions and colors used in a Venn diagram Then give them similar sentence frames orally and have them draw and shade a Venn diagram based on the oral prompts
Bridging Have students work in groups to draw a Venn diagram to represent sets based on real-world examples in the lesson
To help students answer the question posed in Math Talk provide a sentence frame for their answer
The numbers between 31 and 39 on a number line are __________ because __________
EL
California ELD Standards
Emerging 2II5 Modifying to add details ndash Expand sentences with simple adverbials to provide details about a familiar activity or process
Expanding 2II5 Modifying to add details ndash Expand sentences with adverbials to provide details about a familiar or new activity or process
Bridging 2II5 Modifying to add details ndash Expand sentences with increasingly complex adverbials to provide details about a variety of familiar and new activities and processes
Sets of Real Numbers 15B
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
12L E S S O N
Sets of Real Numbers
EngageESSENTIAL QUESTION
How can you describe relationships between sets of real numbers Sample answer Describe them as two different sets or one set as being a subset of another
Motivate the LessonAsk How many different types of tigers can you name How does the set of Bengal tigers relate to the set of tigers
ExplorePoint to different locations in the Animals diagram and ask for examples for that classification Do the same for the Real Numbers diagram Students should understand that everything within a region is part of the set for example both -3 and 2 are integers
ExplainEXAMPLE 1
Questioning Strategies Mathematical Practices bull In A why is 5 not a perfect square It does not have rational numbers as its square roots
bull Can the number in B be written as a fraction Why or why not Yes it is a terminating decimal so it is a rational number
Engage with the WhiteboardHave students place the numbers in Example 1 and Additional Example 1 in the Venn diagram for numbers
YOUR TURNAvoid Common ErrorsBe sure that students read Exercise 2 carefully before answering The number given in the problem 10 is the area not the side length
EXAMPLE 2Questioning Strategies Mathematical Practices bull What two major sets are the real numbers composed of rational and irrational numbers
bull What is the location of the set of whole numbers in the Venn diagram in relation to the set of rational numbers Explain Inside it whole numbers are rational numbers
Focus on Reasoning Mathematical PracticesRemind students that it takes only one counterexample to show that a statement is false
Interactive Whiteboard Interactive example available online
ADDITIONAL EXAMPLE 1Write all names that apply to each number
A -10integer rational real
B 12 _ 3
whole integer rational real
myhrwcom
Interactive Whiteboard Interactive example available online
ADDITIONAL EXAMPLE 2Tell whether the given statement is true or false Explain your choice
No integers are whole numbers
False every whole number is also an integer
myhrwcom
Animated MathClassifying Numbers
Students build fluency in classifying numbers in this engaging fast-paced game
myhrwcom
CA Common CoreStandards
The student is expected to
The Number Systemmdash8NS1
Know that numbers that are not rational are called irrational Understand informally that every number has a decimal expansion for rational numbers show that the decimal expansion repeats eventually and convert a decimal expansion which repeats eventually into a rational numberMathematical Practices
MP7 Using Structure
The student is expected to
15 Lesson 12
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=B
Math TrainerOnline Practice
and Help
Personal
myhrwcom
Math TrainerOnline Practice
and Help
Personal
myhrwcom
Math On the Spotmyhrwcom
Understanding Sets and Subsets of Real NumbersBy understanding which sets are subsets of types of numbers you can verify whether statements about the relationships between sets are true or false
Tell whether the given statement is true or false Explain your choice
All irrational numbers are real numbers
True Every irrational number is included in the set of real numbers The irrational numbers are a subset of the real numbers
No rational numbers are whole numbers
False A whole number can be written as a fraction with a denominator of 1 so every whole number is included in the set of rational numbers The whole numbers are a subset of the rational numbers
EXAMPLE 2
A
B
Write all names that apply to each number
1 A baseball pitcher has pitched 12 2 _ 3 innings
2 The length of the side of a square that has an
area of 10 square yards
YOUR TURN
Tell whether the given statement is true or false Explain your choice
3 All rational numbers are integers
4 Some irrational numbers are integers
YOUR TURN
Give an example of a rational number that is a
whole number Show that the number is both whole
and rational
Math TalkMathematical Practices
Give an example of a
8NS1
False Every integer is a rational number but not every
False Real numbers are either rational or irrational numbers
Integers are rational numbers so no integers are irrational numbers
rational real
irrational real
Sample answer 8 8 = 8_
1
and -thinsp 5 _ 2 are not integers
rational number is an integer Rational numbers such as 3 _ 5
Unit 116
copy H
ough
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ublis
hing
Com
pany
bull Im
age C
redi
ts D
igita
l Im
age c
opyr
ight
copy20
04 Ey
ewire
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8_MCAAESE206984_U1M01L2indd 16 41613 136 AM
Math On the Spot
myhrwcom
Vertebrates
Birds
Passerines
Animals
Integers
Rational Numbers IrrationalNumbers
Real Numbers
WholeNumbers
1
45
3
0
274
67
radic4
-
-3
-2
-1
03
radic2
radic17
radic11-
π
Animated Math
myhrwcom
Classifying Real NumbersBiologists classify animals based on shared characteristics A cardinal is an animal a vertebrate a bird and a passerine
You already know that the set of rational numbers consists of whole numbers integers and fractions The set of real numbers consists of the set of rational numbers and the set of irrational numbers
Write all names that apply to each number
radic_
5 irrational real
ndash1784rational real
whole integer rational real
EXAMPLEXAMPLE 1
A
B
C radic_ 81 ____ 9
L E S S O N
12Sets of Real Numbers
ESSENTIAL QUESTIONHow can you describe relationships between sets of real numbers
Passerines such as the cardinal are also called ldquoperching birdsrdquo
What types of numbers are between 31 and 39 on a
number line
Math TalkMathematical Practices
What types of numbers are
8NS1
8NS1
Know that numbers that are not rational are called irrational Understand informally that every number has a decimal expansion for rational numbers show that the decimal expansion repeats eventually and convert a decimal expansion which repeats eventually into a relation number
ndash1784 is a terminating decimal
5 is a whole number that is not a perfect square
radic_
81 _____ 9 = 9 __ 9 = 1 rational irrational real
15Lesson 12
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ough
ton
Miff
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ublis
hing
Com
pany
bull Im
age C
redi
ts copy
Wiki
med
ia Co
mm
ons
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=B
8_MCABESE206984_U1M01L2indd 15 061113 1144 AM
PROFESSIONAL DEVELOPMENT
Math BackgroundThe relationships between sets of numbers extend to include complex numbers A complex number can be written as a sum of a real number a and an imaginary number bi
a + bi
An imaginary number is a special number that when squared gives a negative value When you square a real number you get a nonnegative number When you square an imaginary number you get a negative value The imaginary unit is i
i = radic_
-1
Integrate Mathematical Practices MP7
This lesson provides an opportunity to address this Mathematical Practices standard It calls for students to discern structure to connect and communicate mathematical ideas
Students use a Venn diagram to structure relationships between sets of numbers They connect and communicate mathematical ideas when they make logical statements about the sets and describe which set best describes numbers applied to real-life situations
Sets of Real Numbers 16
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=B
YOUR TURNAvoid Common ErrorsStudents may see the word ldquoAllldquo or rdquoNordquo in Exercises 3 and 4 and immediately assume that any absolute statements like these are false Remind them that there are true statements that begin with these words and encourage them to provide examples
EXAMPLE 3Questioning Strategies Mathematical Practices bull In A how does the phrase ldquonumber of rdquo give you a clue about the number classification It indicates a counting number
bull What is the relationship between the circumference of a circle and the diameter The circumference is diameter times π
Focus on Critical Thinking Mathematical PracticesIn B suppose the diameters in inches were 25
__ π 28 __ π
31 __ π and so on What set of numbers would
best describe the circumferences Explain Whole numbers the circumferences would be the whole numbers 25 28 31 and so on
YOUR TURNFocus on Critical Thinking Mathematical PracticesHave students compare and contrast the classification of numbers in the answers in Exercises 5 and 6
ElaborateTalk About ItSummarize the Lesson
Ask What are some ways that number sets can be related Sets may be subsets of other sets or they may be separate from other sets
GUIDED PRACTICEEngage with the Whiteboard
Have students place the numbers in Exercises 1ndashthinsp8 in the Venn diagram for numbers at the beginning of the lesson
Integrating Language Arts EL
Encourage English learners to ask for clarification on any terms or phrases that they do not understand
Avoid Common ErrorsExercise 7 Remind students that a repeating decimal is a rational numberExercises 9ndash10 Remind students that it only takes one counterexample to show that a statement is false
Interactive Whiteboard Interactive example available online
ADDITIONAL EXAMPLE 3Identify the set of numbers that best describes the situation Explain your choice
A the amount of time that has passed since midnight
The set of real numbers time is continuous so the amount of time can be rational or irrational
B the number of tickets sold to a basketball game
The set of whole numbers the number of tickets sold may be 0 or a counting number
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17 Lesson 12
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
1IN
116 inch
Guided Practice
Write all names that apply to each number (Example 1)
1 7 _ 8 2 radic_
36
3 radic_
24 4 075
5 0 6 - radic_ 100
7 5 _
45 8 - 18 __ 6
Tell whether the given statement is true or false Explain your choice (Example 2)
9 All whole numbers are rational numbers
10 No irrational numbers are whole numbers
Identify the set of numbers that best describes each situation Explain your choice (Example 3)
11 the change in the value of an account when given to the nearest dollar
12 the markings on a standard ruler
13 What are some ways to describe the relationships between sets of numbers
CHECK-INESSENTIAL QUESTION
rational real
rational real
True Whole numbers are rational numbers
Rational numbers the ruler is marked every 1 __ 16 th inch
Sample answer Describe one set as being a subset of
another or show their relationships in a Venn diagram
Integers the change can be a whole dollar amount
and can be positive negative or zero
True Whole numbers are a subset of the set of rational numbers
and can be written as a ratio of the whole number to 1
irrational real
whole integer rational real
whole integer rational real
rational real
integer rational real
integer rational real
Unit 118
copy H
ough
ton
Miff
lin H
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ublis
hing
Com
pany
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
8_MCAAESE206984_U1M01L2indd 18 41613 136 AM
My Notes
Math TrainerOnline Practice
and Help
Personal
myhrwcom
Math On the Spot
myhrwcom
Identifying Sets for Real-World SituationsReal numbers can be used to represent real-world quantities Highways have posted speed limit signs that are represented by natural numbers such as 55 mph Integers appear on thermometers Rational numbers are used in many daily activities including cooking For example ingredients in a recipe are often given in fractional amounts such as 2 _ 3 cup flour
Identify the set of numbers that best describes each situation Explain your choice
the number of people wearing glasses in a room
The set of whole numbers best describes the situation The number of people wearing glasses may be 0 or a counting number
the circumference of a flying disk has a diameter of 8 9 10 11 or 14 inches
The set of irrational numbers best describes the situation Each circumference would be a product of π and the diameter and any multiple of π is irrational
EXAMPLEXAMPLE 3
A
B
Identify the set of numbers that best describes the situation Explain your choice
5 the amount of water in a glass as it evaporates
6 the weight of a person in pounds
YOUR TURN
8NS1
Rational numbers a personrsquos weight can be a decimal
such as 835 pounds
Real numbers the amount can be any number greater
than 0
17Lesson 12
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ough
ton
Miff
lin H
arco
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ublis
hing
Com
pany
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
8_MCAAESE206984_U1M01L2indd 17 41613 520 AM
Graphic OrganizersGive students a list of numbers (including terminating and repeating decimals fractions integers and rational and irrational square roots) and a graphic organizer as shown below
Real Numbers
Rational numbers Irrational numbers
Integer numbers
Whole numbers
Ask students to write each number in the list in the correct section of the organizer
Number SensePoint out to students that knowing the types of numbers to expect in different situations can alert them to incorrect math as well as to impossible situations For example 135 shots made in basketballs is not possible but an average number of shots can equal 135
Additional ResourcesDifferentiated Instruction includes bull Reading Strategies bull Success for English Learners EL
bull Reteach bull Challenge PRE-AP
DIFFERENTIATE INSTRUCTION
Sets of Real Numbers 18
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
Math TrainerOnline Assessment
and Intervention
Personal
Online homework assignment available
myhrwcom
Lesson Quiz available online
12 LESSON QUIZ
1 Write all the names that apply to the number
2 Tell whether the given statement is true or false Explain your choice All numbers between 1 and 2 are rational numbers
3 Identify the set of numbers that best describes the situation Explain your choiceThe choices on a survey question change the total points for the survey by -2 -1 0 1 or 2 points
-1 _
5
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Additional ResourcesDifferentiated Instruction includes bull Leveled Practice worksheets
EvaluateGUIDED AND INDEPENDENT PRACTICE
Concepts amp Skills Practice
Example 1Classifying Real Numbers
Exercises 1ndash8 14ndash19 22ndash24
Example 2Understanding Sets and Subsets of Real Numbers
Exercises 9ndash10
Example 3Identifying Sets for Real-World Situations
Exercises 11ndash12 20ndash21 25
Focus | Coherence | Rigor
Exercise Depth of Knowledge (DOK) Mathematical Practices
14ndash19 2 SkillsConcepts MP7 Using Structure
20ndash21 2 SkillsConcepts MP6 Precision
22ndash23 2 SkillsConcepts MP3 Logic
24 1 Recall of Information MP7 Using Structure
25 2 SkillsConcepts MP2 Reasoning
26ndash27 3 Strategic Thinking MP3 Logic
28 3 Strategic Thinking MP8 Patterns
29 3 Strategic Thinking MP3 Logic
8NS1
8NS1
Exercise 29 combines concepts from the California Common Core cluster ldquoKnow that there are numbers that are not rational and approximate them by rational numbersrdquo
Answers1 rational real
2 False radic_
2 is an example of an irrational number between 1 and 2
3 Integers each number is an integer but only three are whole numbers
19 Lesson 12
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
Work Area
π mi23 Critique Reasoning The circumference of a circular region is shown
What type of number best describes the diameter of the circle Explain
your answer
24 Critical Thinking A number is not an integer What type of number can it be
25 A grocery store has a shelf with half-gallon containers of milk What type of number best represents the total number of gallons
26 Explain the Error Katie said ldquoNegative numbers are integersrdquo What was her error
27 Justify Reasoning Can you ever use a calculator to determine if a number is rational or irrational Explain
28 Draw Conclusions The decimal 0 _
3 represents 1 _ 3 What type of number best describes 0
_ 9 which is 3 middot 0
_ 3 Explain
29 Communicate Mathematical Ideas Irrational numbers can never be precisely represented in decimal form Why is this
FOCUS ON HIGHER ORDER THINKING
It can be a rational number that is not an integer or an irrational number
rational number
The set of negative numbers also includes non-integer
rational numbers and irrational numbers
Sample answer If the calculator shows a decimal that
terminates in fewer digits than what the calculator screen
allows then you can tell that the number is rational If not
you cannot tell from the calculator display whether the
number terminates because you see a limited number
of digits It may be a repeating decimal (rational) or
non-terminating non-repeating decimal (irrational)
Whole 3 middot 0 _
3 represents 3 middot 1 _ 3 = 1 so 0 _
9 is exactly 1
Sample answer In decimal form irrational numbers never
terminate and never repeat Therefore no matter how
many decimal places you include the number will never
be precisely represented There are always more digits
Whole the diameter is π _ π = 1 mile
Unit 120
copy H
ough
ton
Miff
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ublis
hing
Com
pany
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
8_MCAAESE206984_U1M01L2indd 20 120413 909 PM
Integers
Rational Numbers Irrational Numbers
Real Numbers
Whole Numbers
257
radic16
166
radic9
128 radic50
Personal Math Trainer
Online Practice and Helpmyhrwcom
Name Class Date
Independent Practice
Identify the set of numbers that best describes each situation Explain your choice
20 the height of an airplane as it descends to an airport runway
21 the score with respect to par of several golfers 2 ndash 3 5 0 ndash 1
22 Critique Reasoning Ronald states that the number 1 __ 11 is not rational because when converted into a decimal it does not terminate Nathaniel says it is rational because it is a fraction Which boy is correct Explain
12
14 - radic_
9 15 257
16 radic_
50 17 8 1 _ 2
18 166 19 radic_
16
Write all names that apply to each number Then place the numbers in the correct location on the Venn diagram
8NS1
Real numbers the height can be any number greater than zero
integer rational real whole integer rational real
whole integer rational real
irrational real
rational real
rational real
Integers the scores are counting numbers their
opposites and zero
Nathaniel is correct A rational number is a number that can be written as a fraction and 1 __ 11 is a fraction
19Lesson 12
copy H
ough
ton
Miff
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pany
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8_MCAAESE206984_U1M01L2indd 19 41613 136 AM
myhrwcomActivity available onlineEXTEND THE MATH PRE-AP
Activity Have students consider the concept of restricted domain for the sets of numbers that describe situations For example the number of sisters a person has can best be described by whole numbers but no one has ever had 1500 sisters An area code is an integer or whole number between 200 and 999
Have students use a source such as the Guinness Book of World Records and give examples of sets of numbers that describe situations where the domain is restricted Ask whether the restriction may be changed in the future
Sets of Real Numbers 20
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
-3-4-5 -2-1 1 2 3 50 4
12-4 -radic5
Lesson Support Content Objective Students will learn to order a set of real numbers
Language Objective Students will show and describe how to order a set of real numbers
LESSON 13 Ordering Real Numbers
Building BackgroundEliciting Prior Knowledge Have students draw a number line to compare a rational number and an irrational number such as - radic
_ 5 and -4 1 __ 2 Ask them to explain how
they approximated the irrational number on the number line Then have them identify the greater and the lesser real number Repeat with several other pairs of real numbers in different forms
Learning ProgressionsIn this lesson students order a set of real numbers They use rational approximations to compare the sizes of irrational numbers They also order numbers for real-world situations Important understandings for students include the following
bull Compare irrational numbers bull Estimate the value of expressions with irrational numbers bull Order a set of real numbers bull Order real numbers in a real-world context
Work with real numbers continues throughout Grade 8 and into high school This lesson provides students with a foundation for understanding the relative sizes of numbers in different forms in the real number system
Cluster ConnectionsThis lesson provides an excellent opportunity to connect ideas in this cluster Know that there are numbers that are not rational and approximate them by rational numbers Tell students that there is a special number called the golden ratio with applications in mathematics geometry art and architecture The golden ratio is called phi and is represented by the Greek letter ϕ It includes an irrational number in its definition
Have students explain why the golden ratio is irrational Ask them to find the two whole numbers the golden ratio lies between Then challenge them to approximate the golden ratio to the nearest tenth It is irrational because it includes an irrational number in its definition It lies between 1 and 2 To the nearest tenth ϕ = 16
ϕ = 1 + radic_
5 _ 2
Focus | Coherence | Rigor
California Common Core Standards
8NS2 Use rational approximations of irrational numbers to compare the size of irrational numbers locate them approximately on a number line diagram and estimate the value of expressions (eg π 2 )
MP4 Model with mathematics
21A
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
Math Talk
Language Support EL
PROFESSIONAL DEVELOPMENT
Linguistic Support EL
AcademicContent Vocabulary
Post a chart like this to remind students of the regular comparative forms of adjectives that use the -er and -est suffixes Add to the chart for terms that appear in examples and exercises in each lesson Include any irregular verb forms
Background Knowledge
Go On ndash the title of the module review or quiz is Ready to Go On This title uses an idiomatic expression In this context to go on means ldquoto move aheadrdquo or ldquoto proceedrdquo It is different from the use of go on that means having enough facts to use meaningfully as in having enough to go on Also the intonation used in pronouncing an expression can give it different meanings For example when the speaker emphasizes the word on he or she might be expressing disbelief as in ldquoGo ON Yoursquore kidding rightrdquo Discuss with students other ways that the phrase go on may be used
Leveled Strategies for English Learners
Emerging Label points on a number line with the terms used in ordering greater greatest less lesser least Use sentence frames to insert the correct terms
Expanding Have students give two or three complete sentences to compare the placement of numbers on a number line using the correct forms of the comparative and superlative adjectives
Bridging Have students work in pairs with one student giving directions to the other in complete sentences to order numbers on a number line
To help students answer the question posed in Math Talk make sure that students have a command of the forms for making comparisons and the superlative and the concept of opposite order so that the focus is on the math concept instead of the language skills needed to describe and explain order
EL
Adjective Comparative Superlative
Far Farther Farthest
Large Larger Largest
Great Greater Greatest
Some Less Least
Some More Most
California ELD Standards
Emerging 2I8 Analyzing language choices ndash Explain how phrasing or different common words with similar meanings produce different effects on the audience
Expanding 2I8 Analyzing language choices ndash Explain how phrasing or different words with similar meanings or figurative language produce shades of meaning and different effects on the audience
Bridging 2I8 Analyzing language choices ndash Explain how phrasing or different words with similar meanings or figurative language produce shades of meaning nuances and different effects on the audience
Ordering Real Numbers 21B
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
13L E S S O N
Ordering Real Numbers
Interactive Whiteboard Interactive example available online
ADDITIONAL EXAMPLE 1Compare Write lt gt or =
A radic_
8 - 2 4 - radic_
8 lt
B radic_
20 + 1 3 + radic_
2 gt
EngageESSENTIAL QUESTION
How do you order a set of real numbers Sample answer Find their approximate decimal values and order them
Motivate the LessonAsk What kind of numbers are you comparing when you compare the price of gasoline at two different gas stations
ExploreGive students two rational numbers and ask them to name a number between them Repeat a few times and then give them two irrational numbers and ask them to name a number between them
ExplainEXAMPLE 1
Questioning Strategies Mathematical Practices bull Which is greater the difference between 5 and 3 or the difference between radic
_ 5 and radic
_ 3
The difference between 5 and 3 is 2 the difference between radic_
5 and radic_
3 is approximately 1 So the difference between 5 and 3 is greater
Avoid Common ErrorsCaution students to read the problem carefully and think about what the radical sign means so that they do not misread the problem and answer that the two sides are equal
YOUR TURNFocus on TechnologyCalculators should not be used at this point because developing number sense is the goal
EXAMPLE 2Questioning Strategies Mathematical Practices bull How do you determine whether radic
_ 22 is less than or greater than 45 The square of 45 is
2025 which is less than 22 so the square root of 22 must be greater than 45
Engage with the WhiteboardHave students graph and label various real numbers between 42 and 44 and between 47 and 5
YOUR TURNFocus on Modeling Mathematical PracticesHave students label the integers on the number line with their equivalent square root For example 1 2 and 3 on the number line would be labeled radic
_ 1 radic
_ 4 and radic
_ 9
Interactive Whiteboard Interactive example available online
ADDITIONAL EXAMPLE 2Order 3π radic
_ 10 and 325 from greatest
to least
3π 325 radic_
10
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CA Common CoreStandards
The student is expected to
The Number Systemmdash8NS2
Use rational approximations of irrational numbers to compare the size of irrational numbers locate them approximately on a number line diagram and estimate the value of expressions (eg π 2 )
Mathematical Practices
MP4 Modeling
The student is expected to
21 Lesson 13
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
Math TrainerOnline Practice
and Help
Personal
myhrwcom
Math On the Spotmyhrwcom
0 05 1 15 2 25 3 35 4
radic5radic3
π2
8 85 9 95 10 105 11 115 12
radic75
4 42 44 46 48 5
radic224 12π + 1
Ordering Real Numbers You can compare and order real numbers and list them from least to greatest
Order radic_
22 π + 1 and 4 1 _ 2 from least to greatest
First approximate radic_
22
radic_
22 is between 4 and 5 Since you donrsquot know where it falls between 4 and 5 you need to find a better estimate for radic
_ 22 so
you can compare it to 4 1 _ 2
Since 22 is closer to 25 than 16 use squares of numbers between 45 and 5 to find a better estimate of radic
_ 22
45 2 = 2025 46 2 = 2116 47 2 = 2209 48 2 = 2304
Since 47 2 = 2209 an approximate value for radic_
22 is 47
An approximate value of π is 314 So an approximate value of π +1 is 414
Plot radic_
22 π + 1 and 4 1 _ 2 on a number line
Read the numbers from left to right to place them in order from least to greatest
From least to greatest the numbers are π + 1 4 1 _ 2 and radic_
22
EXAMPLE 2
STEP 1
STEP 2
Order the numbers from least to greatest Then graph them on the number line
YOUR TURN
5 radic_
5 25 radic_
3
6 π 2 10 radic_
75
If real numbers a b and c are in order from least to greatest what is the order
of their opposites from least to greatest
Explain
Math TalkMathematical Practices
8NS2
radic_
3 radic_
5 25
radic_
75 π2 10
Math Talk answer -c -b -a -c is farthest to the left on a number line -b is in the middle and -a is farthest to the right
Unit 122
copy H
ough
ton
Miff
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ublis
hing
Com
pany
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
8_MCAAESE206984_U1M01L3indd 22 41613 447 AM
My Notes
Math TrainerOnline Practice
and Help
Personal
myhrwcom
Math On the Spot
myhrwcom
Comparing Irrational NumbersBetween any two real numbers is another real number To compare and order real numbers you can approximate irrational numbers as decimals
Compare radic_
3 + 5 3 + radic_
5 Write lt gt or =
First approximate radic_
3
radic_
3 is between 1 and 2
Next approximate radic_
5
radic_
5 is between 2 and 3
Then use your approximations to simplify the expressions
radic_
3 + 5 is between 6 and 7
3 + radic_
5 is between 5 and 6
So radic_
3 + 5 gt 3 + radic_
5
Reflect1 If 7 + radic
_ 5 is equal to radic
_ 5 plus a number what do you know about the
number Why
2 What are the closest two integers that radic_
300 is between
EXAMPLEXAMPLE 1
STEP 1
STEP 2
Compare Write lt gt or =
YOUR TURN
3 radic_
2 + 4 2 + radic_
4 4 radic_
12 + 6 12 + radic_
6
L E S S O N
13 Ordering Real Numbers
ESSENTIAL QUESTIONHow do you order a set of real numbers
8NS2
Use rational approximations of irrational numbers to compare the size of irrational numbers locate them approximately on a number line diagram and estimate the value of expressions (eg π 2 )
8NS2
Use perfect squares to estimate square roots
1 2 = 1 2 2 = 4 3 2 = 9
The number is 7 both expressions must equal 7 + radic_
5
17 and 18
gt lt
21Lesson 13
copy H
ough
ton
Miff
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ublis
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Com
pany
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
8_MCAAESE206984_U1M01L3indd 21 41913 246 PM
PROFESSIONAL DEVELOPMENT
Math BackgroundIn this lesson students estimate irrational numbers in the form of square roots of nonper-fect squares by finding two perfect squares between which the number falls A more precise method involves repeated division For example to find radic
_ 28 find a whole number whose perfect
square is close to 28 such as 5 Divide 28 by that number 28 divide 5 = 56 Find the average of the quotient and divisor 5 + 56
_____ 2 = 53 Continue dividing 28 by each result and averaging until you get the desired accuracy
Integrate Mathematical Practices MP4
This lesson provides an opportunity to address this Mathematical Practices standard It calls for students to model relationships using multiple representations including diagrams graphs and language as appropriate Students use multiple representations when they use number lines to estimate the locations of and order rational and irrational numbers given as symbols
Ordering Real Numbers 22
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
Interactive Whiteboard Interactive example available online
ADDITIONAL EXAMPLE 3The diameter of a meteorite in millimeters is calculated by four different methods Order the results from least to greatest
Joe radic_
18 mm Lisa 13 __ 3 mm
Pablo 46 mm Julien 4π __ 3 mm
Julien 4π __ 3 mm Lisa 13 __ 3 mm
Joe radic_
18 mm Pablo 46 mm
EXAMPLE 3Questioning Strategies Mathematical Practices bull How can you verify that radic
_ 28 is between 52 and 53 5 2 2 = 2704 and 5 3 2 = 2809
bull Explain how to determine which number is greater 5 _
5 or 55 When the repeating decimal is rounded to the nearest tenth or hundredth you can see that it is greater
Connect to Daily LifeDiscuss how measuring across a canyon might involve different methods than measuring along a road Explain that measurements like these are often done using calculations that approximate the distance
YOUR TURNFocus on Critical Thinking Mathematical PracticesDiscuss with students which number is greater 3
_ 45 or 3450 3
_ 45 or 3455 and why Explain
that 3 _
45 can be written out as 34545hellipMake sure they understand that 3 _
45 is greater than 345 but less than 3455
ElaborateTalk About ItSummarize the Lesson
Ask How can you order two numbers in different forms whose decimal approxi-mations appear to be equal Approximate one or both numbers to an additional
number of decimal places
GUIDED PRACTICEEngage with the Whiteboard
Have students place and label additional points on the number line in Exercise 9 Allow the points to be in any format other than decimal
Avoid Common ErrorsExercises 3ndash4 Caution students to read the problem carefully so that they do not misread the problem as the same numbers combined by addition on each side of the circleExercise 10 Remind students that the calculations have units
myhrwcom
23 Lesson 13
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
0 05 1 15 2 25 3 35 4 45 5 55 6 65 7
2πradic3
Compare Write lt gt or = (Example 1)
1 radic_
3 + 2 radic_
3 + 3 2 radic_
8 + 17 radic_
11 + 15
3 radic_
6 + 5 6 + radic_
5 4 radic_
9 + 3 9 + radic_
3
5 radic_
17 - 3 -2 + radic_
5 6 12 - radic_
2 14 - radic_
8
7 radic_
7 + 2 radic_
10 - 1 8 radic_
17 + 3 3 + radic_
11
9 Order radic_
3 2π and 15 from least to greatest Then graph them on the number line (Example 2)
radic_
3 is between and so radic_
3 asymp
π asymp 314 so 2π asymp
From least to greatest the numbers are
10 Four people have found the perimeter of a forest using different methods Their results are given in the table Order their calculations from greatest to least (Example 3)
11 Explain how to order a set of real numbers
CHECK-INESSENTIAL QUESTION
Forest Perimeter (km)
Leon Mika Jason Ashley
radic_
17 - 2 1 +thinsp π __ 2 12 ___ 5 25
Guided Practice
17
15
1 + π _ 2 km 25 km 12 __ 5 km radic_
17 - 2 km
2π radic
_ 3
18 175
628
Sample answer Convert each number to a decimal
equivalent using estimation to find equivalents for
irrational numbers Graph each number on a number line
Read the numbers from left to right for least to greatest
Read the numbers from right to left for greatest to least
lt gt
lt lt
ltgt
gt gt
24 Unit 1
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ough
ton
Miff
lin H
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ublis
hing
Com
pany
bull Im
age C
redi
ts copy
Elena
Eliss
eeva
Alam
y Im
ages
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
8_MCAAESE206984_U1M01L3indd 24 41613 448 AM
My Notes
5 52 54 56 58 6
radic28 5 12
23455
Math TrainerOnline Practice
and Help
Personal
myhrwcom
Math On the Spot
myhrwcom
Ordering Real Numbers in a Real-World Context Calculations and estimations in the real world may differ It can be important to know not only which are the most accurate but which give the greatest or least values depending upon the context
Four people have found the distance in kilometers across a canyon using different methods Their results are given in the table Order the distances from greatest to least
Distance Across Quarry Canyon (km)
Juana Lee Ann Ryne Jackson
radic_
28 23 __ 4 5 _
5 5 1 _ 2
Write each value as a decimal
radic_
28 is between 52 and 53 Since 53 2 = 2809 an approximate value for radic
_ 28 is 53
23 __ 4 = 575
5 _
5 is 5555hellip so 5 _
5 to the nearest hundredth is 556
5 1 _ 2 = 55
Plot radic_
28 23 __ 4 5 _
5 and 5 1 _ 2 on a number line
From greatest to least the distances are
23 __ 4 km 5 _
5 km 5 1 _ 2 km radic_
28 km
EXAMPLEXAMPLE 3
STEP 1
STEP 2
7 Four people have found the distance in miles across a crater using different methods Their results are given below
Jonathan 10 __ 3 Elaine 3 _
45 Joseacute 3 1 _ 2 Lashonda radic_
10
Order the distances from greatest to least
YOUR TURN
8NS2
3 1 _ 2 mi 3 _
45 mi 10 __ 3 mi radic_
10 mi
23Lesson 13
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ough
ton
Miff
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pany
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8_MCAAESE206984_U1M01L3indd 23 41613 447 AM
ModelingPlace papers around the room with the numbers from 1 to 5 one per sheet Give each student a card showing a number between 1 and 5 in different forms Have students place his or her card between the correct integers and decide where the number goes in relation to any numbers already placed
Multiple RepresentationsGive students a vertical number line which some students might find easier to use than a horizontal one Have them decide whether to place points for rational and irrational numbers above or below existing points
Additional ResourcesDifferentiated Instruction includes bull Reading Strategies bull Success for English Learners EL
bull Reteach bull Challenge PRE-AP
DIFFERENTIATE INSTRUCTION
Ordering Real Numbers 24
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
Math TrainerOnline Assessment
and Intervention
Personal
Online homework assignment available
myhrwcom
myhrwcom
Lesson Quiz available online
13 LESSON QUIZ
1 Compare Write lt gt or =
radic_
95 - 5 radic_
62 - 2
2 Order 105 radic_
105 and 3π + 1 from greatest to least
3 A length in centimeters is calculated differently by four different people Order their calculations from least to greatest
KD 11 __ 2 cm Silvio 5 __ 3 π cm
Paula 5 _
4 cm Luis radic_
33 cm
EvaluateGUIDED AND INDEPENDENT PRACTICE
Concepts amp Skills Practice
Example 1Comparing Irrational Numbers
Exercises 1ndash8
Example 2Ordering Real Numbers
Exercises 9 12ndash15 18ndash21
Example 3Ordering Real Numbers in a Real-World Context
Exercises 10 16ndash17
Additional ResourcesDifferentiated Instruction includes bull Leveled Practice worksheets
Focus | Coherence | Rigor
Exercise Depth of Knowledge (DOK) Mathematical Practices
12ndash15 1 Recall of Information MP5 Using Tools
16 2 SkillsConcepts MP2 Reasoning
17 2 SkillsConcepts MP6 Precision
18ndash21 2 SkillsConcepts MP2 Reasoning
22 3 Strategic Thinking MP4 Modeling
23ndash24 3 Strategic Thinking MP3 Logic
8NS2
8NS2
Answers1 radic
_ 95 - 5 lt radic
_ 62 - 2
2 radic_
105 3π + 1 105
3 Silvio 5 __ 3 π cm Paula 5 _
4 cm
KD 11
__ 2 cm Luis radic_
33 cm
25 Lesson 13
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
Work Area
3140 3141 3142 3143
314 π227
20 A teacher asks his students to write the numbers shown in order from least to greatest Paul thinks the numbers are already in order Sandra thinks the order should be reversed Who is right
21 Math History There is a famous irrational number called Eulerrsquos number symbolized with an e Like π its decimal form never ends or repeats The first few digits of e are 27182818284
a Between which two square roots of integers could you find this number
b Between which two square roots of integers can you find π
22 Analyze Relationships There are several approximations used for π including 314 and 22 __ 7 π is approximately 314159265358979
a Label π and the two approximations on the number line
b Which of the two approximations is a better estimate for π Explain
c Find a whole number x so that the ratio x ___ 113 is a better estimate for π
than the two given approximations
23 Communicate Mathematical Ideas If a set of six numbers that include both rational and irrational numbers is graphed on a number line what is the fewest number of distinct points that need to be graphed Explain
24 Critique Reasoning Jill says that 12 _
6 is less than 1263 Explain her error
FOCUS ON HIGHER ORDER THINKING
radic_
115 115 ___ 11 and 105624
between radic_
7 asymp 265 and radic_
8 asymp 283
between radic_
9 = 3 and radic_
10 asymp 316
22 __ 7 it is closer to π on the number line
She did not consider the repeating digit 1266
2 rational numbers can have the same location and
irrational numbers can have the same location but they
cannot share a location
355
Neither student is correct The answer
should be 115 ___ 11 105624 radic_
115
Unit 126
copy H
ough
ton M
ifflin
Har
cour
t Pub
lishin
g Com
pany
Imag
e Cre
dits
copy3D
Stoc
kiSt
ockP
hoto
com
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
8_MCAAESE206984_U1M01L3indd 26 210513 801 AM
Personal Math Trainer
Online Practice and Helpmyhrwcom
Name Class Date
Independent Practice
16 Your sister is considering two different shapes for her garden One is a square with side lengths of 35 meters and the other is a circle with a diameter of 4 meters
a Find the area of the square
b Find the area of the circle
c Compare your answers from parts a and b Which garden would give your sister the most space to plant
17 Winnie measured the length of her fatherrsquos ranch four times and got four different distances Her measurements are shown in the table
a To estimate the actual length Winnie first approximated each distance to the nearest hundredth Then she averaged the four numbers Using a calculator find Winniersquos estimate
b Winniersquos father estimated the distance across his ranch to be radic_
56 km How does this distance compare to Winniersquos estimate
Give an example of each type of number
18 a real number between radic_
13 and radic_
14
19 an irrational number between 5 and 7
Order the numbers from least to greatest
12 radic_
7 2 radic_
8 ___ 2 13 radic_
10 π 35
14 radic_
220 -10 radic_
100 115 15 radic_
8 -375 3 9 _ 4
Distance Across Fatherrsquos Ranch (km)
1 2 3 4
radic_
60 58 __ 8 7 _
3 7 3 _ 5
138NS2
radic_
8 ___ 2 2 radic_
7
-10 radic_
100 115 radic_
220
radic_
60 asymp 775 58 __ 8 = 725 7 _
3 asymp 733 7 3 _ 5 = 760 so the average
π radic_
10 35
-375 9 _ 4 radic_
8 3
is 74825 km
1225 m2
4π m2 or approximately 126 m2
They are nearly identical radic_
56 is approximately 74833hellip
The circle would give her more space to plant because it has a
larger area
Sample answer 37
Sample answer radic_
31
25Lesson 13
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ough
ton
Miff
lin H
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pany
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8_MCAAESE206984_U1M01L3indd 25 41613 448 AM
Activity available online myhrwcomEXTEND THE MATH PRE-AP
Activity Have students investigate whether there are infinitely many numbers between two numbers by giving examples for each of the following
bull Between any two rational numbers there is at least one other rational number Sample answer 45 is between 41 and 48
bull Between any two irrational numbers there is at least one rational number Sample answer 45 is between radic
_ 11 and radic
_ 29
bull Between any two rational numbers there is at least one irrational number Sample answer radic
_ 11 is between 31 and 36
bull Between any two irrational numbers there is at least one irrational number Sample answer radic
_ 17 is between radic
_ 11 and radic
_ 29
Ordering Real Numbers 26
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
ReadyMath Trainer
Online Practiceand Help
Personal
myhrwcom
Module Quiz
11ensp RationalenspandenspIrrationalenspNumbersWrite each fraction as a decimal or each decimal as a fraction
1 7__20 2 1___
27 3 17_8
Solve each equation for x
4 x2=81 5 x3=343 6 x2= 1___100
7 Asquarepatiohasanareaof200squarefeetHowlongiseachside
ofthepatiotothenearesttenth
12ensp SetsenspofenspRealenspNumbersWrite all names that apply to each number
8 121____radic
____121
9 π__2
10 TellwhetherthestatementldquoAllintegersarerationalnumbersrdquoistrueorfalseExplainyourchoice
13ensp OrderingenspRealenspNumbersCompare Write lt gt or =
11 radic__
8+3 8+radic__
3 12 radic__
5+11emsp emsp emsp 5+radic___
11
Order the numbers from least to greatest
13 radic___
99π29__
8 14 radic___
1__251_40__
2
15 Howarerealnumbersusedtodescribereal-worldsituations
ESSENTIAL QUESTION
035
9-9
141ft
7 1__10- 1__10
14__11 1875
wholeintegerrationalreal
Trueintegerscanbewrittenasthequotientoftwointegers
SampleanswerRealnumberssuchastherational
π29__
8radic___
99
irrationalreal
lt gt
number1_4candescribeamountsusedincooking
radic___
1__250__
21_4
27Module1
copy H
ough
ton
Miff
lin H
arco
urt P
ublis
hing
Com
pany
DONOTEDIT--ChangesmustbemadethroughldquoFileinfordquoCorrectionKey=A
8_MCAAESE206984_U1M01RTindd 27 41513 1113 PM
Math TrainerOnline Assessment
and Intervention
Personal
myhrwcom
1
2
3 Response toIntervention
Intervention Enrichment
Access Ready to Go On assessment online and receive instant scoring feedback and customized intervention or enrichment
Online and Print Resources
Differentiated Instruction
bull Reteach worksheets
bull Reading Strategies EL
bull Success for English Learners EL
Differentiated Instruction
bull Challenge worksheets PRE-AP
Extend the Math PRE-AP
Lesson Activities in TE
Additional ResourcesAssessment Resources includes bull Leveled Module Quizzes
Ready to Go OnAssess MasteryUse the assessment on this page to determine if students have mastered the concepts and standards covered in this module
California Common Core Standards
Lesson Exercises Common Core Standards
11 1ndash7 8NS1 8NS2 8EE2
12 8ndash10 8NS1
13 11ndash14 8NS2
27 Unit 1 Module 1
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
Personal Math Trainer
Online Practice and HelpmyhrwcomAssessment Readiness
Module 1 MIXed ReVIeW
1 Look at each number Is the number between 2π and radic___
52
Select Yes or No for expressions AndashC
A 6 2 _ 3 Yes No
B 5π __ 2 Yes No
C 3 radic__
5 Yes No
2 Consider the number - 11 __ 15
Choose True or False for each statement
A The number is rational True False
B The number can be written as True Falsea repeating decimal
C The number is less than ndash08 True False
3 The volume of a cube is given by V = x3 where x is the length of an edge of the cube A cube-shaped end table has a volume of 3 3 _ 8 cubic feet What is the length of an edge of the end table Explain how you solved this problem
4 A student says that radic___
83 is greater than 29 __ 3 Is the student correct Justify your
reasoning
1 1 _ 2 ft Sample answer The equation x3 = 3 3 _ 8 can be used
to find the edge length in feet To solve the equation
write the mixed number as a fraction greater than 1
x3 = 27 __ 8 Then take the cube root of both sides x = 3 _ 2 = 1 1 _ 2
No Sample answer radic___
83 asymp 91 and 29 __ 3 = 9
__ 6
Because 91 lt 9 __
6 radic___
83 lt 29 __ 3
28 Unit 1
copy H
ough
ton
Miff
lin H
arco
urt P
ublis
hing
Com
pany
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=A
8_MCAAESE206984_U1M01RTindd 28 240413 946 AM
Personal Math Trainer
Online Assessment and
Interventionmyhrwcom
Scoring GuideItem 3 Award the student 1 point for finding the edge length of the cube and 1 point for correctly explaining how to use a cube root to solve the problem
Item 4 Award the student 1 point for determining that the student is incorrect and 1 point for correctly justifying the reasoning for this conclusion
Additional ResourcesTo assign this assessment online login to your Assignment Manager at myhrwcom
Assessment Readiness
California Common Core Standards
Items Grade 8 Standards Mathematical Practices
1 8NS2 MP7
2 7NS2b 7NS2d 8NS1 MP7
3 8EE2 MP1 MP4
4 8NS1 8NS2 MP3
Item integrates mixed review concepts from previous modules or a previous course
Item 4 combines concepts from the California Common Core cluster ldquoKnow that there are numbers that are not rational and approximate them by rational numbersrdquo
Real Numbers 28
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
Math TrainerOnline Practice
and Help
Personal
myhrwcom
EXPLORE ACTIVITY
lt 2 lt
radic_
lt radic
_ 2 lt
radic_
lt radic
_ 2 lt
The solution is 9
The solution is 2 _ 5
C
D
729 = x 3
3 radic_ 729 = 3 radic
_ x 3
3 radic_ 729 = x
9 = x
x 3 = 8 ___ 125
3 radic_
x 3 =thinsp 3 radic_ 8 ___ 125
x =thinsp 3 radic_ 8 ___ 125
x = 2 _ 5
Solve each equation for x
YOUR TURN
7 x 2 = 196 8 x 2 = 9 ___ 256
9 x 3 = 512 10 x 3 = 64 ___ 343
Estimating Irrational NumbersIrrational numbers are numbers that are not rational In other words they cannot be written in the form a _ b where a and b are integers and b is not 0 Square roots of perfect squares are rational numbers Square roots of numbers that are not perfect squares are irrational Some equations like those in Example 3 involve square roots of numbers that are not perfect squares
x 2 = 2 x = plusmn radic_
2
Estimate the value of radic_
2
Find two consecutive perfect squares that 2 is between Complete the inequality by writing these perfect squares in the boxes
Now take the square root of each number
Simplify the square roots of perfect squares
radic_
2 is between and
A
B
C
8NS2 8EE2
Solve for x by taking the cube root of both sides
Solve for x by taking the cube root of both sides
Apply the definition of cube root
Think What number cubed equals 729
Apply the definition of cube root
Think What number cubed equals 8 ____ 125
radic_
2 is irrational
x = plusmn14 x = plusmn 3 __ 16
x = 8 x = 4 _ 7
1 2
1 4
1 4
1 2
Unit 110
copy H
ough
ton
Miff
lin H
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ublis
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pany
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
8_MCAAESE206984_U1M01L1indd 10 41613 1211 AM
Math TrainerOnline Practice
and Help
Personal
myhrwcom
Math On the Spot
myhrwcom
Write each decimal as a fraction in simplest form
YOUR TURN
Finding Square Roots and Cube RootsThe square root of a positive number p is x if x 2 = p There are two square roots for every positive number For example the square roots of 36 are 6 and minus6 because 6 2 = 36 and (minus6) 2 = 36 The square roots of 1 __ 25 are 1 _ 5 and minus 1 _ 5 You can write the square roots of 1 __ 25 as plusmn 1 _ 5 The symbol radic
_ 5 indicates the positive
or principal square root
A number that is a perfect square has square roots that are integers The number 81 is a perfect square because its square roots are 9 and minus9
The cube root of a positive number p is x if x 3 = p There is one cube root for every positive number For example the cube root of 8 is 2 because 2 3 = 8 The cube root of 1 __ 27 is 1 _ 3 because ( 1 _ 3 )
3
= 1 __ 27 The symbol 3 radic_ 1 indicates the
cube root
A number that is a perfect cube has a cube root that is an integer The number 125 is a perfect cube because its cube root is 5
Solve each equation for x
The solutions are 11 and minus11
The solutions are 4 __ 13 and minus 4 __ 13
EXAMPLEXAMPLE 3
A x 2 = 121
x 2 = 121
x = plusmn radic_
121
x = plusmn11
B x 2 = 16 ___ 169
x 2 = 16 ___ 169
x = plusmn radic_
16 ___ 169
x = plusmn 4 __ 13
4 012 5 0 _
57 6 14
Can you square an integer and get a negative number
What does this indicate about whether negative
numbers have square roots
Math TalkMathematical Practices
8EE2
Solve for x by taking the square root of both sides
Apply the definition of square root
Think What numbers squared equal 121
Solve for x by taking the square root of both sides
Apply the definition of square root
Think What numbers squared equal 16 ____ 169
3 __ 25 19 __ 33 1 2 _ 5
No the square of a positive integer is positive the square of a negative integer is positive and the square of 0 is 0 So negative numbers do not have (real) square roots
9Lesson 11
copy H
ough
ton
Miff
lin H
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pany
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8_MCAAESE206984_U1M01L1indd 9 41913 240 PM
Critical ThinkingIn the Explore Activity students estimated the location of radic
_ 2 on a number line Ask students
whether they think that it is possible to locate more precisely the point that represents radic
_ 2 In
other words can you graph irrational numbers exactly on a number line along with rational numbers Students should understand that radic
_ 2
is a real number and all real numbers can be located on a real number line A more precise estimate will allow more precise placement on a number line
The Modeling note tells one way to do this
ModelingHave students use a ruler to represent a number line with a unit that is one inch long Have them draw a square with a side of one inch and draw the diagonal to make two isosceles triangles Lead students to understand that the length of the diagonal (or hypotenuse) is radic
_ 2
Have them copy the length of their diagonal onto their ruler or number line starting at zero The end point of the diagonal represents the exact point for the irrational number radic
_ 2 on a
number line
Additional ResourcesDifferentiated Instruction includes bull Reading Strategies bull Success for English Learners EL
bull Reteach bull Challenge PRE-AP
DIFFERENTIATE INSTRUCTION
Rational and Irrational Numbers 10
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
ElaborateTalk About ItSummarize the Lesson
Ask If someone claims that a certain number is irrational but you know it is actually rational how could you prove to that person that the number is rational
You could find a fraction equal to the number such that the number is the ratio of two integers with the denominator not equal to zero
GUIDED PRACTICEEngage with the Whiteboard
Have students plot each number in Exercises 16ndash18 on a number line Students should label each point with the irrational number written as a radical and as a
decimal
Avoid Common ErrorsExercises 1ndash6 To avoid reversing the order of the dividend and divisor tell students to start at the top of the fraction and read the bar as ldquodivided byrdquo
Focus on TechnologyHave students use a calculator to investigate the decimal equivalents of such fractions as 1 __ 9 2 __ 9 8 __ 9 and 1 __ 11 2 __ 11 10
__ 11 Ask them to describe the patterns they find as a result of these investigations
11 Lesson 11
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
Guided Practice
7 0675 8 56 9 044
10 0 _
4
10x =
x =
11 0 _
26
100x =
x =
12 0 _
325
1000x =
x =
Solve each equation for x (Example 3 and Explore Activity)
- x
-
_______________
x =
- x
-
___________________
x =
- x
-
_______________________
x =
Write each fraction or mixed number as a decimal (Example 1)
1 2 _ 5 2 8 _ 9 3 3 3 _ 4
4 7 __ 10 5 2 3 _ 8 6 5 _ 6
Write each decimal as a fraction or mixed number in simplest form (Example 2)
13 x 2 = 17 14 x 2 = 25 ___ 289 15 x 3 = 216
Approximate each irrational number to one decimal place without a calculator
x = plusmn radic__
asymp plusmn x = 3
radic__
=
(Explore Activity)
16 radic_
5 asymp
17 radic_
3 asymp
18 radic_
10 asymp
19 What is the difference between rational and irrational numbers
CHECK-INESSENTIAL QUESTION
x = plusmn radic__
__________ = plusmn _____
4 _
4
0 _
4
4 99
6216
269
41 25 5
17289
17
22 17 32
04
07
27__40
26 __ 99 325 ___ 999 4 _ 9
11__255 3_5
0 _
8
2375
375
08 _
3
26 _
26
0 _
26
325 _
325
0 _
325
999 325
Rational numbers can be written in the form a __ b where
a and b are integers and b ne 0 Irrational numbers cannot
be written in this form
Unit 112
copy H
ough
ton
Miff
lin H
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pany
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
8_MCAAESE206984_U1M01L1indd 12 41613 1211 AM
11 12 13 14 15
radic2 asymp 14
141 142 143 144 145
radic2 asymp 141
0 1 2 3 4
radic2 asymp 15
Estimate that radic_
2 asymp 15
To find a better estimate first choose some numbers between 1 and 2 and square them For example choose 13 14 and 15
1 3 2 = 1 4 2 = 1 5 2 =
Is radic_
2 between 13 and 14 How do you know
Is radic_
2 between 14 and 15 How do you know
2 is closer to than to so radic_
2 asymp
Locate and label this value on the number line
Reflect 11 How could you find an even better estimate of radic
_ 2
12 Find a better estimate of radic_
2
1 41 2 = 1 42 2 = 1 43 2 =
2 is closer to than to so radic_
2 asymp
Draw a number line and locate and label your estimate
13 Solve x 2 = 7 Write your answer as a radical expression Then estimate to one decimal place
D
E
F
No 2 is not between 169 and 196
Yes 2 is between 196 and 225
196
19881
19881
225
20164
20164
14
141
20449
169 196 225
Test the squares of numbers between 14 and 15
x = plusmn radic_
7 x asymp plusmn26
11Lesson 11
copy H
ough
ton
Miff
lin H
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ublis
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Com
pany
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
8_MCAAESE206984_U1M01L1indd 11 41613 1211 AM
Rational and Irrational Numbers 12
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
Math TrainerOnline Assessment
and Intervention
Personal
Online homework assignment available
myhrwcom
EvaluateGUIDED AND INDEPENDENT PRACTICE
Concepts amp Skills Practice
Example 1Expressing Rational Numbers as Decimals
Exercises 1ndash6 20ndash21 24ndash25
Example 2Expressing Decimals as Rational Numbers
Exercises 7ndash12 22ndash23 26ndash27
Example 3Finding Square Roots and Cube Roots
Exercises 13ndash15 28 30ndash31 35
Explore ActivityEstimating Irrational Numbers
Exercises 13 16ndash18 29 32ndash34
Additional ResourcesDifferentiated Instruction includes bull Leveled Practice worksheets
Lesson Quiz available online
11 LESSON QUIZ
1 Write as a decimal 2 5 __ 8 1 7 __ 12
2 Write as a fraction 034 1 _
24
3 Solve x 2 = 9 __ 49 for x
4 Solve x 3 = 216 for x
5 Estimate the value of radic_
13 to one decimal place without using a calculator
myhrwcom
Focus | Coherence | Rigor
Exercise Depth of Knowledge (DOK) Mathematical Practices
20ndash27 2 SkillsConcepts MP4 Modeling
28 3 Strategic Thinking MP4 Modeling
29ndash32 2 SkillsConcepts MP6 Precision
33 3 Strategic Thinking MP7 Using Structure
34 2 SkillsConcepts MP3 Logic
35 2 SkillsConcepts MP4 Modeling
36 3 Strategic Thinking MP3 Logic
37 3 Strategic Thinking MP7 Using Structure
38 3 Strategic Thinking MP2 Reasoning
8NS1 8NS2 8EE2
8NS1 8NS2 8EE2
Answers1 2625 158
_ 3
2 17 __ 50 1 8 __ 33
3 x = plusmn 3 __ 7
4 x = 6
5 36
13 Lesson 11
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
Work Area
33 Analyze Relationships To find radic_
15 Beau found 3 2 = 9 and 4 2 = 16 He said that since 15 is between 9 and 16 radic
_ 15 must be between 3 and 4 He
thinks a good estimate for radic_
15 is 3 + 4 ____ 2 = 35 Is Beaursquos estimate high low
or correct Explain
34 Justify Reasoning What is a good estimate for the solution to the equation x 3 = 95 How did you come up with your estimate
35 The volume of a sphere is 36π f t 3 What is the radius of the sphere Use the formula V = 4 _ 3 π r 3 to find your answer
36 Draw Conclusions Can you find the cube root of a negative number If so is it positive or negative Explain your reasoning
37 Make a Conjecture Evaluate and compare the following expressions
radic_
4 __ 25 and radic
_ 4 ____
radic_
25 radic
_
16 __ 81 and radic_
16 ____
radic_
81 radic
_
36 __ 49 and radic_
36 ____
radic_
49
Use your results to make a conjecture about a division rule for square roots Since division is multiplication by the reciprocal make a conjecture about a multiplication rule for square roots
38 Persevere in Problem Solving The difference between the solutions to the equation x 2 = a is 30 What is a Show that your answer is correct
FOCUS ON HIGHER ORDER THINKING
His estimate is low because 15 is very close to 16
so radic_
15 is very close to radic_
16 or 4 A better estimate
would be 38 or 39
Sample answer about 45 4 3 = 64 and 5 3 = 125
Because 95 is about halfway between 64 and 125 try 45
45 3 = 91125 which is a good estimate
3 feet
Yes the cube root of a negative number is negative
because a negative number cubed is always negative
and a nonnegative number cubed is always nonnegative
radic_
4 __ 25 = 2 _ 5 = radic
_ 4 ____
radic_
25 radic
_
16 __ 81 = 4 _ 9 = radic_
16 ____
radic_
81 radic
_
36 __ 49 = 6 _ 7 = radic_
36 ____
radic_
49
225 the solutions to x 2 = a are x = plusmn15 and
radic_
a ___
radic_
b = radic
_ a __
b radic
_ a radic
_ b = radic
_ a b
15 - (-15) = 30
Unit 114
copy H
ough
ton
Miff
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ublis
hing
Com
pany
bull copy
Ilen
e Mac
Dona
ldA
lamy I
mag
es
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=B
8_MCABESE206984_U1M01L1indd 14 102913 1142 PM
Personal Math Trainer
Online Practice and Helpmyhrwcom
Name Class Date
Independent Practice11
20 A 7 __ 16 -inch-long bolt is used in a machine What is this length written as a decimal
21 The weight of an object on the moon is 1 _ 6 its weight on Earth Write 1 _ 6 as a decimal
22 The distance to the nearest gas station is 2 4 _ 5 kilometers What is this distance written as a decimal
23 A baseball pitcher has pitched 98 2 _ 3 innings What is the number of innings written as a decimal
24 A heartbeat takes 08 second How many seconds is this written as a fraction
25 There are 262 miles in a marathon Write the number of miles using a fraction
26 The average score on a biology test was 72
_ 1 Write the average score using a
fraction
27 The metal in a penny is worth about 0505 cent How many cents is this written as a fraction
28 Multistep An artist wants to frame a square painting with an area of 400 square inches She wants to know the length of the wood trim that is needed to go around the painting
a If x is the length of one side of the painting what equation can you set up to find the length of a side How many solutions does the equation have
b Do all of the solutions that you found make sense in the context of the problem Explain
c What is the length of the wood trim needed to go around the painting
Solve each equation for x Write your answers as radical expressions Then estimate to one decimal place if necessary
29 x 2 = 14 30 x 3 = 1331
31 x 2 = 144 32 x 2 = 29
8NS1 8NS2 8EE2
04375 in 01 _6
28 km 98 _6 innings
x 2 = 400 x = plusmnthinsp20 the equation has 2 solutions
x = 20 makes sense but x = -20 doesnrsquot because a
painting cannot have a side length of -20 inches
4 times 20 = 80 inches
x = plusmn radic_
14 asymp plusmn37
x = plusmn radic_
144 = plusmn12 x = plusmn radic_
29 asymp plusmn54
x = 3 radic_ 1331 = 11
4_5 second 26 1_5 mi
72 1 _ 9 101 ___ 200 cent
13Lesson 11
copy H
ough
ton
Miff
lin H
arco
urt P
ublis
hing
Com
pany
bull copy
Phot
odisc
Get
ty Im
ages
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
8_MCAAESE206984_U1M01L1indd 13 41613 1211 AM
myhrwcomActivity available onlineEXTEND THE MATH PRE-AP
Activity Write radic_
09 on the board and invite students to conjecture what the value might be Have them check their conjectures by squaring Invite them to suggest ways to estimate radic
_ 09 As a hint point out that 09 is close to 10 and so they might
use that to help guide their estimates Lead them to see that since 092 is 081 and 102 is 1 the value of radic
_ 09 is greater than 09 and less than 10 Try squaring 095 to get
09025 A good estimate for radic_
09 is 095
Rational and Irrational Numbers 14
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=B
Integers
Rational Numbers IrrationalNumbers
Real Numbers
WholeNumbers
-3-4-5 -2-1 1 2 3 50 4
23
34-4 -π -1 25
radic2
Lesson Support Content Objective Students will learn to describe relationships between sets of numbers
Language Objective Students will explain how to describe relationships between sets of real numbers
LESSON 12 Sets of Real Numbers
Building BackgroundEliciting Prior Knowledge Have students draw a number line from -5 to 5 Ask them to plot points on the number line to approximate the location of rational and irrational numbers such as -1 3 __ 4 25 -4 2 __ 3 radic
_ 2 and -π
Learning ProgressionsIn this lesson students clarify their understanding of the real number system They characterize sets and subsets of the real numbers They also identify sets for real-world situations Important understandings for students include the following
bull Identify all of the possible subsets of the real numbers for a given number
bull Decide whether a statement about a subset of the real numbers is true or false
bull Identify the set of numbers that best describes a real-world situation
Understanding the relationships among the sets of numbers that make up the real numbers is essential as students are introduced to different forms of numbers throughout the school year This lesson provides a foundation for the comparing and ordering of real numbers in the next lesson
Cluster ConnectionsThis lesson provides an excellent opportunity to connect ideas in this cluster Know that there are numbers that are not rational and approximate them by rational numbers Have students copy this diagram which relates the sets of real numbers
Ask students to complete the diagram by writing three examples for each set of numbers Have students share examples and explain how they knew each number they selected belonged in the appropriate set Answers may vary Check studentsrsquo work
Focus | Coherence | Rigor
California Common Core Standards
8NS1 Know that numbers that are not rational are called irrational Understand informally that every number has a decimal expansion for rational numbers show that the decimal expansion repeats eventually and convert a decimal expansion which repeats eventually into a rational number
MP7 Look for and make use of structure
15A
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
Math Talk
Language Support EL
PROFESSIONAL DEVELOPMENT
Linguistic Support EL
AcademicContent Vocabulary
Venn diagrams ndash Students need descriptive language to describe the categories that the different areas and colors of a Venn diagram represent the concept of a set and how sets are distinct or can overlap Use sentence frames such as
The big oval represents __________The darklight blue color in the middle of the
big ovals represents __________These sets overlap because __________
In this way students have the language and structure to identify the criteria that distinguish a set and to explain the abstract representation Also point out the use of the prefix sub- meaning ldquounderrdquo in the term subset
Rules and Patterns
Abbreviations ndash In this lesson the abbreviation mph is used Be sure to point out that mph stands for miles per hour and is used to give units in a rate of speed Students may also have seen mpg (miles per gallon) which gives the units in a rate of fuel efficiency
Borrowed Words ndash Terminology used in baseball such as inning and pitcher may require some explanation Spanish as well as some other languages have borrowed these terms from English so some students may be familiar with these words already Despite this whenever a word is critical to students understanding the word problem it is best to explain the meaning
Leveled Strategies for English Learners
Emerging Allow students to indicate true or false orally in Guided Practice Exercises 9 and 10
Expanding Have students use sentence frames to describe the meaning of regions and colors used in a Venn diagram Then give them similar sentence frames orally and have them draw and shade a Venn diagram based on the oral prompts
Bridging Have students work in groups to draw a Venn diagram to represent sets based on real-world examples in the lesson
To help students answer the question posed in Math Talk provide a sentence frame for their answer
The numbers between 31 and 39 on a number line are __________ because __________
EL
California ELD Standards
Emerging 2II5 Modifying to add details ndash Expand sentences with simple adverbials to provide details about a familiar activity or process
Expanding 2II5 Modifying to add details ndash Expand sentences with adverbials to provide details about a familiar or new activity or process
Bridging 2II5 Modifying to add details ndash Expand sentences with increasingly complex adverbials to provide details about a variety of familiar and new activities and processes
Sets of Real Numbers 15B
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
12L E S S O N
Sets of Real Numbers
EngageESSENTIAL QUESTION
How can you describe relationships between sets of real numbers Sample answer Describe them as two different sets or one set as being a subset of another
Motivate the LessonAsk How many different types of tigers can you name How does the set of Bengal tigers relate to the set of tigers
ExplorePoint to different locations in the Animals diagram and ask for examples for that classification Do the same for the Real Numbers diagram Students should understand that everything within a region is part of the set for example both -3 and 2 are integers
ExplainEXAMPLE 1
Questioning Strategies Mathematical Practices bull In A why is 5 not a perfect square It does not have rational numbers as its square roots
bull Can the number in B be written as a fraction Why or why not Yes it is a terminating decimal so it is a rational number
Engage with the WhiteboardHave students place the numbers in Example 1 and Additional Example 1 in the Venn diagram for numbers
YOUR TURNAvoid Common ErrorsBe sure that students read Exercise 2 carefully before answering The number given in the problem 10 is the area not the side length
EXAMPLE 2Questioning Strategies Mathematical Practices bull What two major sets are the real numbers composed of rational and irrational numbers
bull What is the location of the set of whole numbers in the Venn diagram in relation to the set of rational numbers Explain Inside it whole numbers are rational numbers
Focus on Reasoning Mathematical PracticesRemind students that it takes only one counterexample to show that a statement is false
Interactive Whiteboard Interactive example available online
ADDITIONAL EXAMPLE 1Write all names that apply to each number
A -10integer rational real
B 12 _ 3
whole integer rational real
myhrwcom
Interactive Whiteboard Interactive example available online
ADDITIONAL EXAMPLE 2Tell whether the given statement is true or false Explain your choice
No integers are whole numbers
False every whole number is also an integer
myhrwcom
Animated MathClassifying Numbers
Students build fluency in classifying numbers in this engaging fast-paced game
myhrwcom
CA Common CoreStandards
The student is expected to
The Number Systemmdash8NS1
Know that numbers that are not rational are called irrational Understand informally that every number has a decimal expansion for rational numbers show that the decimal expansion repeats eventually and convert a decimal expansion which repeats eventually into a rational numberMathematical Practices
MP7 Using Structure
The student is expected to
15 Lesson 12
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=B
Math TrainerOnline Practice
and Help
Personal
myhrwcom
Math TrainerOnline Practice
and Help
Personal
myhrwcom
Math On the Spotmyhrwcom
Understanding Sets and Subsets of Real NumbersBy understanding which sets are subsets of types of numbers you can verify whether statements about the relationships between sets are true or false
Tell whether the given statement is true or false Explain your choice
All irrational numbers are real numbers
True Every irrational number is included in the set of real numbers The irrational numbers are a subset of the real numbers
No rational numbers are whole numbers
False A whole number can be written as a fraction with a denominator of 1 so every whole number is included in the set of rational numbers The whole numbers are a subset of the rational numbers
EXAMPLE 2
A
B
Write all names that apply to each number
1 A baseball pitcher has pitched 12 2 _ 3 innings
2 The length of the side of a square that has an
area of 10 square yards
YOUR TURN
Tell whether the given statement is true or false Explain your choice
3 All rational numbers are integers
4 Some irrational numbers are integers
YOUR TURN
Give an example of a rational number that is a
whole number Show that the number is both whole
and rational
Math TalkMathematical Practices
Give an example of a
8NS1
False Every integer is a rational number but not every
False Real numbers are either rational or irrational numbers
Integers are rational numbers so no integers are irrational numbers
rational real
irrational real
Sample answer 8 8 = 8_
1
and -thinsp 5 _ 2 are not integers
rational number is an integer Rational numbers such as 3 _ 5
Unit 116
copy H
ough
ton
Miff
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hing
Com
pany
bull Im
age C
redi
ts D
igita
l Im
age c
opyr
ight
copy20
04 Ey
ewire
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8_MCAAESE206984_U1M01L2indd 16 41613 136 AM
Math On the Spot
myhrwcom
Vertebrates
Birds
Passerines
Animals
Integers
Rational Numbers IrrationalNumbers
Real Numbers
WholeNumbers
1
45
3
0
274
67
radic4
-
-3
-2
-1
03
radic2
radic17
radic11-
π
Animated Math
myhrwcom
Classifying Real NumbersBiologists classify animals based on shared characteristics A cardinal is an animal a vertebrate a bird and a passerine
You already know that the set of rational numbers consists of whole numbers integers and fractions The set of real numbers consists of the set of rational numbers and the set of irrational numbers
Write all names that apply to each number
radic_
5 irrational real
ndash1784rational real
whole integer rational real
EXAMPLEXAMPLE 1
A
B
C radic_ 81 ____ 9
L E S S O N
12Sets of Real Numbers
ESSENTIAL QUESTIONHow can you describe relationships between sets of real numbers
Passerines such as the cardinal are also called ldquoperching birdsrdquo
What types of numbers are between 31 and 39 on a
number line
Math TalkMathematical Practices
What types of numbers are
8NS1
8NS1
Know that numbers that are not rational are called irrational Understand informally that every number has a decimal expansion for rational numbers show that the decimal expansion repeats eventually and convert a decimal expansion which repeats eventually into a relation number
ndash1784 is a terminating decimal
5 is a whole number that is not a perfect square
radic_
81 _____ 9 = 9 __ 9 = 1 rational irrational real
15Lesson 12
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ough
ton
Miff
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Com
pany
bull Im
age C
redi
ts copy
Wiki
med
ia Co
mm
ons
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=B
8_MCABESE206984_U1M01L2indd 15 061113 1144 AM
PROFESSIONAL DEVELOPMENT
Math BackgroundThe relationships between sets of numbers extend to include complex numbers A complex number can be written as a sum of a real number a and an imaginary number bi
a + bi
An imaginary number is a special number that when squared gives a negative value When you square a real number you get a nonnegative number When you square an imaginary number you get a negative value The imaginary unit is i
i = radic_
-1
Integrate Mathematical Practices MP7
This lesson provides an opportunity to address this Mathematical Practices standard It calls for students to discern structure to connect and communicate mathematical ideas
Students use a Venn diagram to structure relationships between sets of numbers They connect and communicate mathematical ideas when they make logical statements about the sets and describe which set best describes numbers applied to real-life situations
Sets of Real Numbers 16
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=B
YOUR TURNAvoid Common ErrorsStudents may see the word ldquoAllldquo or rdquoNordquo in Exercises 3 and 4 and immediately assume that any absolute statements like these are false Remind them that there are true statements that begin with these words and encourage them to provide examples
EXAMPLE 3Questioning Strategies Mathematical Practices bull In A how does the phrase ldquonumber of rdquo give you a clue about the number classification It indicates a counting number
bull What is the relationship between the circumference of a circle and the diameter The circumference is diameter times π
Focus on Critical Thinking Mathematical PracticesIn B suppose the diameters in inches were 25
__ π 28 __ π
31 __ π and so on What set of numbers would
best describe the circumferences Explain Whole numbers the circumferences would be the whole numbers 25 28 31 and so on
YOUR TURNFocus on Critical Thinking Mathematical PracticesHave students compare and contrast the classification of numbers in the answers in Exercises 5 and 6
ElaborateTalk About ItSummarize the Lesson
Ask What are some ways that number sets can be related Sets may be subsets of other sets or they may be separate from other sets
GUIDED PRACTICEEngage with the Whiteboard
Have students place the numbers in Exercises 1ndashthinsp8 in the Venn diagram for numbers at the beginning of the lesson
Integrating Language Arts EL
Encourage English learners to ask for clarification on any terms or phrases that they do not understand
Avoid Common ErrorsExercise 7 Remind students that a repeating decimal is a rational numberExercises 9ndash10 Remind students that it only takes one counterexample to show that a statement is false
Interactive Whiteboard Interactive example available online
ADDITIONAL EXAMPLE 3Identify the set of numbers that best describes the situation Explain your choice
A the amount of time that has passed since midnight
The set of real numbers time is continuous so the amount of time can be rational or irrational
B the number of tickets sold to a basketball game
The set of whole numbers the number of tickets sold may be 0 or a counting number
myhrwcom
17 Lesson 12
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
1IN
116 inch
Guided Practice
Write all names that apply to each number (Example 1)
1 7 _ 8 2 radic_
36
3 radic_
24 4 075
5 0 6 - radic_ 100
7 5 _
45 8 - 18 __ 6
Tell whether the given statement is true or false Explain your choice (Example 2)
9 All whole numbers are rational numbers
10 No irrational numbers are whole numbers
Identify the set of numbers that best describes each situation Explain your choice (Example 3)
11 the change in the value of an account when given to the nearest dollar
12 the markings on a standard ruler
13 What are some ways to describe the relationships between sets of numbers
CHECK-INESSENTIAL QUESTION
rational real
rational real
True Whole numbers are rational numbers
Rational numbers the ruler is marked every 1 __ 16 th inch
Sample answer Describe one set as being a subset of
another or show their relationships in a Venn diagram
Integers the change can be a whole dollar amount
and can be positive negative or zero
True Whole numbers are a subset of the set of rational numbers
and can be written as a ratio of the whole number to 1
irrational real
whole integer rational real
whole integer rational real
rational real
integer rational real
integer rational real
Unit 118
copy H
ough
ton
Miff
lin H
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Com
pany
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
8_MCAAESE206984_U1M01L2indd 18 41613 136 AM
My Notes
Math TrainerOnline Practice
and Help
Personal
myhrwcom
Math On the Spot
myhrwcom
Identifying Sets for Real-World SituationsReal numbers can be used to represent real-world quantities Highways have posted speed limit signs that are represented by natural numbers such as 55 mph Integers appear on thermometers Rational numbers are used in many daily activities including cooking For example ingredients in a recipe are often given in fractional amounts such as 2 _ 3 cup flour
Identify the set of numbers that best describes each situation Explain your choice
the number of people wearing glasses in a room
The set of whole numbers best describes the situation The number of people wearing glasses may be 0 or a counting number
the circumference of a flying disk has a diameter of 8 9 10 11 or 14 inches
The set of irrational numbers best describes the situation Each circumference would be a product of π and the diameter and any multiple of π is irrational
EXAMPLEXAMPLE 3
A
B
Identify the set of numbers that best describes the situation Explain your choice
5 the amount of water in a glass as it evaporates
6 the weight of a person in pounds
YOUR TURN
8NS1
Rational numbers a personrsquos weight can be a decimal
such as 835 pounds
Real numbers the amount can be any number greater
than 0
17Lesson 12
copy H
ough
ton
Miff
lin H
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ublis
hing
Com
pany
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
8_MCAAESE206984_U1M01L2indd 17 41613 520 AM
Graphic OrganizersGive students a list of numbers (including terminating and repeating decimals fractions integers and rational and irrational square roots) and a graphic organizer as shown below
Real Numbers
Rational numbers Irrational numbers
Integer numbers
Whole numbers
Ask students to write each number in the list in the correct section of the organizer
Number SensePoint out to students that knowing the types of numbers to expect in different situations can alert them to incorrect math as well as to impossible situations For example 135 shots made in basketballs is not possible but an average number of shots can equal 135
Additional ResourcesDifferentiated Instruction includes bull Reading Strategies bull Success for English Learners EL
bull Reteach bull Challenge PRE-AP
DIFFERENTIATE INSTRUCTION
Sets of Real Numbers 18
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
Math TrainerOnline Assessment
and Intervention
Personal
Online homework assignment available
myhrwcom
Lesson Quiz available online
12 LESSON QUIZ
1 Write all the names that apply to the number
2 Tell whether the given statement is true or false Explain your choice All numbers between 1 and 2 are rational numbers
3 Identify the set of numbers that best describes the situation Explain your choiceThe choices on a survey question change the total points for the survey by -2 -1 0 1 or 2 points
-1 _
5
myhrwcom
Additional ResourcesDifferentiated Instruction includes bull Leveled Practice worksheets
EvaluateGUIDED AND INDEPENDENT PRACTICE
Concepts amp Skills Practice
Example 1Classifying Real Numbers
Exercises 1ndash8 14ndash19 22ndash24
Example 2Understanding Sets and Subsets of Real Numbers
Exercises 9ndash10
Example 3Identifying Sets for Real-World Situations
Exercises 11ndash12 20ndash21 25
Focus | Coherence | Rigor
Exercise Depth of Knowledge (DOK) Mathematical Practices
14ndash19 2 SkillsConcepts MP7 Using Structure
20ndash21 2 SkillsConcepts MP6 Precision
22ndash23 2 SkillsConcepts MP3 Logic
24 1 Recall of Information MP7 Using Structure
25 2 SkillsConcepts MP2 Reasoning
26ndash27 3 Strategic Thinking MP3 Logic
28 3 Strategic Thinking MP8 Patterns
29 3 Strategic Thinking MP3 Logic
8NS1
8NS1
Exercise 29 combines concepts from the California Common Core cluster ldquoKnow that there are numbers that are not rational and approximate them by rational numbersrdquo
Answers1 rational real
2 False radic_
2 is an example of an irrational number between 1 and 2
3 Integers each number is an integer but only three are whole numbers
19 Lesson 12
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
Work Area
π mi23 Critique Reasoning The circumference of a circular region is shown
What type of number best describes the diameter of the circle Explain
your answer
24 Critical Thinking A number is not an integer What type of number can it be
25 A grocery store has a shelf with half-gallon containers of milk What type of number best represents the total number of gallons
26 Explain the Error Katie said ldquoNegative numbers are integersrdquo What was her error
27 Justify Reasoning Can you ever use a calculator to determine if a number is rational or irrational Explain
28 Draw Conclusions The decimal 0 _
3 represents 1 _ 3 What type of number best describes 0
_ 9 which is 3 middot 0
_ 3 Explain
29 Communicate Mathematical Ideas Irrational numbers can never be precisely represented in decimal form Why is this
FOCUS ON HIGHER ORDER THINKING
It can be a rational number that is not an integer or an irrational number
rational number
The set of negative numbers also includes non-integer
rational numbers and irrational numbers
Sample answer If the calculator shows a decimal that
terminates in fewer digits than what the calculator screen
allows then you can tell that the number is rational If not
you cannot tell from the calculator display whether the
number terminates because you see a limited number
of digits It may be a repeating decimal (rational) or
non-terminating non-repeating decimal (irrational)
Whole 3 middot 0 _
3 represents 3 middot 1 _ 3 = 1 so 0 _
9 is exactly 1
Sample answer In decimal form irrational numbers never
terminate and never repeat Therefore no matter how
many decimal places you include the number will never
be precisely represented There are always more digits
Whole the diameter is π _ π = 1 mile
Unit 120
copy H
ough
ton
Miff
lin H
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ublis
hing
Com
pany
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
8_MCAAESE206984_U1M01L2indd 20 120413 909 PM
Integers
Rational Numbers Irrational Numbers
Real Numbers
Whole Numbers
257
radic16
166
radic9
128 radic50
Personal Math Trainer
Online Practice and Helpmyhrwcom
Name Class Date
Independent Practice
Identify the set of numbers that best describes each situation Explain your choice
20 the height of an airplane as it descends to an airport runway
21 the score with respect to par of several golfers 2 ndash 3 5 0 ndash 1
22 Critique Reasoning Ronald states that the number 1 __ 11 is not rational because when converted into a decimal it does not terminate Nathaniel says it is rational because it is a fraction Which boy is correct Explain
12
14 - radic_
9 15 257
16 radic_
50 17 8 1 _ 2
18 166 19 radic_
16
Write all names that apply to each number Then place the numbers in the correct location on the Venn diagram
8NS1
Real numbers the height can be any number greater than zero
integer rational real whole integer rational real
whole integer rational real
irrational real
rational real
rational real
Integers the scores are counting numbers their
opposites and zero
Nathaniel is correct A rational number is a number that can be written as a fraction and 1 __ 11 is a fraction
19Lesson 12
copy H
ough
ton
Miff
lin H
arco
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ublis
hing
Com
pany
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
8_MCAAESE206984_U1M01L2indd 19 41613 136 AM
myhrwcomActivity available onlineEXTEND THE MATH PRE-AP
Activity Have students consider the concept of restricted domain for the sets of numbers that describe situations For example the number of sisters a person has can best be described by whole numbers but no one has ever had 1500 sisters An area code is an integer or whole number between 200 and 999
Have students use a source such as the Guinness Book of World Records and give examples of sets of numbers that describe situations where the domain is restricted Ask whether the restriction may be changed in the future
Sets of Real Numbers 20
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
-3-4-5 -2-1 1 2 3 50 4
12-4 -radic5
Lesson Support Content Objective Students will learn to order a set of real numbers
Language Objective Students will show and describe how to order a set of real numbers
LESSON 13 Ordering Real Numbers
Building BackgroundEliciting Prior Knowledge Have students draw a number line to compare a rational number and an irrational number such as - radic
_ 5 and -4 1 __ 2 Ask them to explain how
they approximated the irrational number on the number line Then have them identify the greater and the lesser real number Repeat with several other pairs of real numbers in different forms
Learning ProgressionsIn this lesson students order a set of real numbers They use rational approximations to compare the sizes of irrational numbers They also order numbers for real-world situations Important understandings for students include the following
bull Compare irrational numbers bull Estimate the value of expressions with irrational numbers bull Order a set of real numbers bull Order real numbers in a real-world context
Work with real numbers continues throughout Grade 8 and into high school This lesson provides students with a foundation for understanding the relative sizes of numbers in different forms in the real number system
Cluster ConnectionsThis lesson provides an excellent opportunity to connect ideas in this cluster Know that there are numbers that are not rational and approximate them by rational numbers Tell students that there is a special number called the golden ratio with applications in mathematics geometry art and architecture The golden ratio is called phi and is represented by the Greek letter ϕ It includes an irrational number in its definition
Have students explain why the golden ratio is irrational Ask them to find the two whole numbers the golden ratio lies between Then challenge them to approximate the golden ratio to the nearest tenth It is irrational because it includes an irrational number in its definition It lies between 1 and 2 To the nearest tenth ϕ = 16
ϕ = 1 + radic_
5 _ 2
Focus | Coherence | Rigor
California Common Core Standards
8NS2 Use rational approximations of irrational numbers to compare the size of irrational numbers locate them approximately on a number line diagram and estimate the value of expressions (eg π 2 )
MP4 Model with mathematics
21A
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
Math Talk
Language Support EL
PROFESSIONAL DEVELOPMENT
Linguistic Support EL
AcademicContent Vocabulary
Post a chart like this to remind students of the regular comparative forms of adjectives that use the -er and -est suffixes Add to the chart for terms that appear in examples and exercises in each lesson Include any irregular verb forms
Background Knowledge
Go On ndash the title of the module review or quiz is Ready to Go On This title uses an idiomatic expression In this context to go on means ldquoto move aheadrdquo or ldquoto proceedrdquo It is different from the use of go on that means having enough facts to use meaningfully as in having enough to go on Also the intonation used in pronouncing an expression can give it different meanings For example when the speaker emphasizes the word on he or she might be expressing disbelief as in ldquoGo ON Yoursquore kidding rightrdquo Discuss with students other ways that the phrase go on may be used
Leveled Strategies for English Learners
Emerging Label points on a number line with the terms used in ordering greater greatest less lesser least Use sentence frames to insert the correct terms
Expanding Have students give two or three complete sentences to compare the placement of numbers on a number line using the correct forms of the comparative and superlative adjectives
Bridging Have students work in pairs with one student giving directions to the other in complete sentences to order numbers on a number line
To help students answer the question posed in Math Talk make sure that students have a command of the forms for making comparisons and the superlative and the concept of opposite order so that the focus is on the math concept instead of the language skills needed to describe and explain order
EL
Adjective Comparative Superlative
Far Farther Farthest
Large Larger Largest
Great Greater Greatest
Some Less Least
Some More Most
California ELD Standards
Emerging 2I8 Analyzing language choices ndash Explain how phrasing or different common words with similar meanings produce different effects on the audience
Expanding 2I8 Analyzing language choices ndash Explain how phrasing or different words with similar meanings or figurative language produce shades of meaning and different effects on the audience
Bridging 2I8 Analyzing language choices ndash Explain how phrasing or different words with similar meanings or figurative language produce shades of meaning nuances and different effects on the audience
Ordering Real Numbers 21B
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
13L E S S O N
Ordering Real Numbers
Interactive Whiteboard Interactive example available online
ADDITIONAL EXAMPLE 1Compare Write lt gt or =
A radic_
8 - 2 4 - radic_
8 lt
B radic_
20 + 1 3 + radic_
2 gt
EngageESSENTIAL QUESTION
How do you order a set of real numbers Sample answer Find their approximate decimal values and order them
Motivate the LessonAsk What kind of numbers are you comparing when you compare the price of gasoline at two different gas stations
ExploreGive students two rational numbers and ask them to name a number between them Repeat a few times and then give them two irrational numbers and ask them to name a number between them
ExplainEXAMPLE 1
Questioning Strategies Mathematical Practices bull Which is greater the difference between 5 and 3 or the difference between radic
_ 5 and radic
_ 3
The difference between 5 and 3 is 2 the difference between radic_
5 and radic_
3 is approximately 1 So the difference between 5 and 3 is greater
Avoid Common ErrorsCaution students to read the problem carefully and think about what the radical sign means so that they do not misread the problem and answer that the two sides are equal
YOUR TURNFocus on TechnologyCalculators should not be used at this point because developing number sense is the goal
EXAMPLE 2Questioning Strategies Mathematical Practices bull How do you determine whether radic
_ 22 is less than or greater than 45 The square of 45 is
2025 which is less than 22 so the square root of 22 must be greater than 45
Engage with the WhiteboardHave students graph and label various real numbers between 42 and 44 and between 47 and 5
YOUR TURNFocus on Modeling Mathematical PracticesHave students label the integers on the number line with their equivalent square root For example 1 2 and 3 on the number line would be labeled radic
_ 1 radic
_ 4 and radic
_ 9
Interactive Whiteboard Interactive example available online
ADDITIONAL EXAMPLE 2Order 3π radic
_ 10 and 325 from greatest
to least
3π 325 radic_
10
myhrwcom
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CA Common CoreStandards
The student is expected to
The Number Systemmdash8NS2
Use rational approximations of irrational numbers to compare the size of irrational numbers locate them approximately on a number line diagram and estimate the value of expressions (eg π 2 )
Mathematical Practices
MP4 Modeling
The student is expected to
21 Lesson 13
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
Math TrainerOnline Practice
and Help
Personal
myhrwcom
Math On the Spotmyhrwcom
0 05 1 15 2 25 3 35 4
radic5radic3
π2
8 85 9 95 10 105 11 115 12
radic75
4 42 44 46 48 5
radic224 12π + 1
Ordering Real Numbers You can compare and order real numbers and list them from least to greatest
Order radic_
22 π + 1 and 4 1 _ 2 from least to greatest
First approximate radic_
22
radic_
22 is between 4 and 5 Since you donrsquot know where it falls between 4 and 5 you need to find a better estimate for radic
_ 22 so
you can compare it to 4 1 _ 2
Since 22 is closer to 25 than 16 use squares of numbers between 45 and 5 to find a better estimate of radic
_ 22
45 2 = 2025 46 2 = 2116 47 2 = 2209 48 2 = 2304
Since 47 2 = 2209 an approximate value for radic_
22 is 47
An approximate value of π is 314 So an approximate value of π +1 is 414
Plot radic_
22 π + 1 and 4 1 _ 2 on a number line
Read the numbers from left to right to place them in order from least to greatest
From least to greatest the numbers are π + 1 4 1 _ 2 and radic_
22
EXAMPLE 2
STEP 1
STEP 2
Order the numbers from least to greatest Then graph them on the number line
YOUR TURN
5 radic_
5 25 radic_
3
6 π 2 10 radic_
75
If real numbers a b and c are in order from least to greatest what is the order
of their opposites from least to greatest
Explain
Math TalkMathematical Practices
8NS2
radic_
3 radic_
5 25
radic_
75 π2 10
Math Talk answer -c -b -a -c is farthest to the left on a number line -b is in the middle and -a is farthest to the right
Unit 122
copy H
ough
ton
Miff
lin H
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ublis
hing
Com
pany
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
8_MCAAESE206984_U1M01L3indd 22 41613 447 AM
My Notes
Math TrainerOnline Practice
and Help
Personal
myhrwcom
Math On the Spot
myhrwcom
Comparing Irrational NumbersBetween any two real numbers is another real number To compare and order real numbers you can approximate irrational numbers as decimals
Compare radic_
3 + 5 3 + radic_
5 Write lt gt or =
First approximate radic_
3
radic_
3 is between 1 and 2
Next approximate radic_
5
radic_
5 is between 2 and 3
Then use your approximations to simplify the expressions
radic_
3 + 5 is between 6 and 7
3 + radic_
5 is between 5 and 6
So radic_
3 + 5 gt 3 + radic_
5
Reflect1 If 7 + radic
_ 5 is equal to radic
_ 5 plus a number what do you know about the
number Why
2 What are the closest two integers that radic_
300 is between
EXAMPLEXAMPLE 1
STEP 1
STEP 2
Compare Write lt gt or =
YOUR TURN
3 radic_
2 + 4 2 + radic_
4 4 radic_
12 + 6 12 + radic_
6
L E S S O N
13 Ordering Real Numbers
ESSENTIAL QUESTIONHow do you order a set of real numbers
8NS2
Use rational approximations of irrational numbers to compare the size of irrational numbers locate them approximately on a number line diagram and estimate the value of expressions (eg π 2 )
8NS2
Use perfect squares to estimate square roots
1 2 = 1 2 2 = 4 3 2 = 9
The number is 7 both expressions must equal 7 + radic_
5
17 and 18
gt lt
21Lesson 13
copy H
ough
ton
Miff
lin H
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ublis
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Com
pany
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
8_MCAAESE206984_U1M01L3indd 21 41913 246 PM
PROFESSIONAL DEVELOPMENT
Math BackgroundIn this lesson students estimate irrational numbers in the form of square roots of nonper-fect squares by finding two perfect squares between which the number falls A more precise method involves repeated division For example to find radic
_ 28 find a whole number whose perfect
square is close to 28 such as 5 Divide 28 by that number 28 divide 5 = 56 Find the average of the quotient and divisor 5 + 56
_____ 2 = 53 Continue dividing 28 by each result and averaging until you get the desired accuracy
Integrate Mathematical Practices MP4
This lesson provides an opportunity to address this Mathematical Practices standard It calls for students to model relationships using multiple representations including diagrams graphs and language as appropriate Students use multiple representations when they use number lines to estimate the locations of and order rational and irrational numbers given as symbols
Ordering Real Numbers 22
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
Interactive Whiteboard Interactive example available online
ADDITIONAL EXAMPLE 3The diameter of a meteorite in millimeters is calculated by four different methods Order the results from least to greatest
Joe radic_
18 mm Lisa 13 __ 3 mm
Pablo 46 mm Julien 4π __ 3 mm
Julien 4π __ 3 mm Lisa 13 __ 3 mm
Joe radic_
18 mm Pablo 46 mm
EXAMPLE 3Questioning Strategies Mathematical Practices bull How can you verify that radic
_ 28 is between 52 and 53 5 2 2 = 2704 and 5 3 2 = 2809
bull Explain how to determine which number is greater 5 _
5 or 55 When the repeating decimal is rounded to the nearest tenth or hundredth you can see that it is greater
Connect to Daily LifeDiscuss how measuring across a canyon might involve different methods than measuring along a road Explain that measurements like these are often done using calculations that approximate the distance
YOUR TURNFocus on Critical Thinking Mathematical PracticesDiscuss with students which number is greater 3
_ 45 or 3450 3
_ 45 or 3455 and why Explain
that 3 _
45 can be written out as 34545hellipMake sure they understand that 3 _
45 is greater than 345 but less than 3455
ElaborateTalk About ItSummarize the Lesson
Ask How can you order two numbers in different forms whose decimal approxi-mations appear to be equal Approximate one or both numbers to an additional
number of decimal places
GUIDED PRACTICEEngage with the Whiteboard
Have students place and label additional points on the number line in Exercise 9 Allow the points to be in any format other than decimal
Avoid Common ErrorsExercises 3ndash4 Caution students to read the problem carefully so that they do not misread the problem as the same numbers combined by addition on each side of the circleExercise 10 Remind students that the calculations have units
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23 Lesson 13
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
0 05 1 15 2 25 3 35 4 45 5 55 6 65 7
2πradic3
Compare Write lt gt or = (Example 1)
1 radic_
3 + 2 radic_
3 + 3 2 radic_
8 + 17 radic_
11 + 15
3 radic_
6 + 5 6 + radic_
5 4 radic_
9 + 3 9 + radic_
3
5 radic_
17 - 3 -2 + radic_
5 6 12 - radic_
2 14 - radic_
8
7 radic_
7 + 2 radic_
10 - 1 8 radic_
17 + 3 3 + radic_
11
9 Order radic_
3 2π and 15 from least to greatest Then graph them on the number line (Example 2)
radic_
3 is between and so radic_
3 asymp
π asymp 314 so 2π asymp
From least to greatest the numbers are
10 Four people have found the perimeter of a forest using different methods Their results are given in the table Order their calculations from greatest to least (Example 3)
11 Explain how to order a set of real numbers
CHECK-INESSENTIAL QUESTION
Forest Perimeter (km)
Leon Mika Jason Ashley
radic_
17 - 2 1 +thinsp π __ 2 12 ___ 5 25
Guided Practice
17
15
1 + π _ 2 km 25 km 12 __ 5 km radic_
17 - 2 km
2π radic
_ 3
18 175
628
Sample answer Convert each number to a decimal
equivalent using estimation to find equivalents for
irrational numbers Graph each number on a number line
Read the numbers from left to right for least to greatest
Read the numbers from right to left for greatest to least
lt gt
lt lt
ltgt
gt gt
24 Unit 1
copy H
ough
ton
Miff
lin H
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urt P
ublis
hing
Com
pany
bull Im
age C
redi
ts copy
Elena
Eliss
eeva
Alam
y Im
ages
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
8_MCAAESE206984_U1M01L3indd 24 41613 448 AM
My Notes
5 52 54 56 58 6
radic28 5 12
23455
Math TrainerOnline Practice
and Help
Personal
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Math On the Spot
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Ordering Real Numbers in a Real-World Context Calculations and estimations in the real world may differ It can be important to know not only which are the most accurate but which give the greatest or least values depending upon the context
Four people have found the distance in kilometers across a canyon using different methods Their results are given in the table Order the distances from greatest to least
Distance Across Quarry Canyon (km)
Juana Lee Ann Ryne Jackson
radic_
28 23 __ 4 5 _
5 5 1 _ 2
Write each value as a decimal
radic_
28 is between 52 and 53 Since 53 2 = 2809 an approximate value for radic
_ 28 is 53
23 __ 4 = 575
5 _
5 is 5555hellip so 5 _
5 to the nearest hundredth is 556
5 1 _ 2 = 55
Plot radic_
28 23 __ 4 5 _
5 and 5 1 _ 2 on a number line
From greatest to least the distances are
23 __ 4 km 5 _
5 km 5 1 _ 2 km radic_
28 km
EXAMPLEXAMPLE 3
STEP 1
STEP 2
7 Four people have found the distance in miles across a crater using different methods Their results are given below
Jonathan 10 __ 3 Elaine 3 _
45 Joseacute 3 1 _ 2 Lashonda radic_
10
Order the distances from greatest to least
YOUR TURN
8NS2
3 1 _ 2 mi 3 _
45 mi 10 __ 3 mi radic_
10 mi
23Lesson 13
copy H
ough
ton
Miff
lin H
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Com
pany
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8_MCAAESE206984_U1M01L3indd 23 41613 447 AM
ModelingPlace papers around the room with the numbers from 1 to 5 one per sheet Give each student a card showing a number between 1 and 5 in different forms Have students place his or her card between the correct integers and decide where the number goes in relation to any numbers already placed
Multiple RepresentationsGive students a vertical number line which some students might find easier to use than a horizontal one Have them decide whether to place points for rational and irrational numbers above or below existing points
Additional ResourcesDifferentiated Instruction includes bull Reading Strategies bull Success for English Learners EL
bull Reteach bull Challenge PRE-AP
DIFFERENTIATE INSTRUCTION
Ordering Real Numbers 24
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
Math TrainerOnline Assessment
and Intervention
Personal
Online homework assignment available
myhrwcom
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Lesson Quiz available online
13 LESSON QUIZ
1 Compare Write lt gt or =
radic_
95 - 5 radic_
62 - 2
2 Order 105 radic_
105 and 3π + 1 from greatest to least
3 A length in centimeters is calculated differently by four different people Order their calculations from least to greatest
KD 11 __ 2 cm Silvio 5 __ 3 π cm
Paula 5 _
4 cm Luis radic_
33 cm
EvaluateGUIDED AND INDEPENDENT PRACTICE
Concepts amp Skills Practice
Example 1Comparing Irrational Numbers
Exercises 1ndash8
Example 2Ordering Real Numbers
Exercises 9 12ndash15 18ndash21
Example 3Ordering Real Numbers in a Real-World Context
Exercises 10 16ndash17
Additional ResourcesDifferentiated Instruction includes bull Leveled Practice worksheets
Focus | Coherence | Rigor
Exercise Depth of Knowledge (DOK) Mathematical Practices
12ndash15 1 Recall of Information MP5 Using Tools
16 2 SkillsConcepts MP2 Reasoning
17 2 SkillsConcepts MP6 Precision
18ndash21 2 SkillsConcepts MP2 Reasoning
22 3 Strategic Thinking MP4 Modeling
23ndash24 3 Strategic Thinking MP3 Logic
8NS2
8NS2
Answers1 radic
_ 95 - 5 lt radic
_ 62 - 2
2 radic_
105 3π + 1 105
3 Silvio 5 __ 3 π cm Paula 5 _
4 cm
KD 11
__ 2 cm Luis radic_
33 cm
25 Lesson 13
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
Work Area
3140 3141 3142 3143
314 π227
20 A teacher asks his students to write the numbers shown in order from least to greatest Paul thinks the numbers are already in order Sandra thinks the order should be reversed Who is right
21 Math History There is a famous irrational number called Eulerrsquos number symbolized with an e Like π its decimal form never ends or repeats The first few digits of e are 27182818284
a Between which two square roots of integers could you find this number
b Between which two square roots of integers can you find π
22 Analyze Relationships There are several approximations used for π including 314 and 22 __ 7 π is approximately 314159265358979
a Label π and the two approximations on the number line
b Which of the two approximations is a better estimate for π Explain
c Find a whole number x so that the ratio x ___ 113 is a better estimate for π
than the two given approximations
23 Communicate Mathematical Ideas If a set of six numbers that include both rational and irrational numbers is graphed on a number line what is the fewest number of distinct points that need to be graphed Explain
24 Critique Reasoning Jill says that 12 _
6 is less than 1263 Explain her error
FOCUS ON HIGHER ORDER THINKING
radic_
115 115 ___ 11 and 105624
between radic_
7 asymp 265 and radic_
8 asymp 283
between radic_
9 = 3 and radic_
10 asymp 316
22 __ 7 it is closer to π on the number line
She did not consider the repeating digit 1266
2 rational numbers can have the same location and
irrational numbers can have the same location but they
cannot share a location
355
Neither student is correct The answer
should be 115 ___ 11 105624 radic_
115
Unit 126
copy H
ough
ton M
ifflin
Har
cour
t Pub
lishin
g Com
pany
Imag
e Cre
dits
copy3D
Stoc
kiSt
ockP
hoto
com
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
8_MCAAESE206984_U1M01L3indd 26 210513 801 AM
Personal Math Trainer
Online Practice and Helpmyhrwcom
Name Class Date
Independent Practice
16 Your sister is considering two different shapes for her garden One is a square with side lengths of 35 meters and the other is a circle with a diameter of 4 meters
a Find the area of the square
b Find the area of the circle
c Compare your answers from parts a and b Which garden would give your sister the most space to plant
17 Winnie measured the length of her fatherrsquos ranch four times and got four different distances Her measurements are shown in the table
a To estimate the actual length Winnie first approximated each distance to the nearest hundredth Then she averaged the four numbers Using a calculator find Winniersquos estimate
b Winniersquos father estimated the distance across his ranch to be radic_
56 km How does this distance compare to Winniersquos estimate
Give an example of each type of number
18 a real number between radic_
13 and radic_
14
19 an irrational number between 5 and 7
Order the numbers from least to greatest
12 radic_
7 2 radic_
8 ___ 2 13 radic_
10 π 35
14 radic_
220 -10 radic_
100 115 15 radic_
8 -375 3 9 _ 4
Distance Across Fatherrsquos Ranch (km)
1 2 3 4
radic_
60 58 __ 8 7 _
3 7 3 _ 5
138NS2
radic_
8 ___ 2 2 radic_
7
-10 radic_
100 115 radic_
220
radic_
60 asymp 775 58 __ 8 = 725 7 _
3 asymp 733 7 3 _ 5 = 760 so the average
π radic_
10 35
-375 9 _ 4 radic_
8 3
is 74825 km
1225 m2
4π m2 or approximately 126 m2
They are nearly identical radic_
56 is approximately 74833hellip
The circle would give her more space to plant because it has a
larger area
Sample answer 37
Sample answer radic_
31
25Lesson 13
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Miff
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pany
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8_MCAAESE206984_U1M01L3indd 25 41613 448 AM
Activity available online myhrwcomEXTEND THE MATH PRE-AP
Activity Have students investigate whether there are infinitely many numbers between two numbers by giving examples for each of the following
bull Between any two rational numbers there is at least one other rational number Sample answer 45 is between 41 and 48
bull Between any two irrational numbers there is at least one rational number Sample answer 45 is between radic
_ 11 and radic
_ 29
bull Between any two rational numbers there is at least one irrational number Sample answer radic
_ 11 is between 31 and 36
bull Between any two irrational numbers there is at least one irrational number Sample answer radic
_ 17 is between radic
_ 11 and radic
_ 29
Ordering Real Numbers 26
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
ReadyMath Trainer
Online Practiceand Help
Personal
myhrwcom
Module Quiz
11ensp RationalenspandenspIrrationalenspNumbersWrite each fraction as a decimal or each decimal as a fraction
1 7__20 2 1___
27 3 17_8
Solve each equation for x
4 x2=81 5 x3=343 6 x2= 1___100
7 Asquarepatiohasanareaof200squarefeetHowlongiseachside
ofthepatiotothenearesttenth
12ensp SetsenspofenspRealenspNumbersWrite all names that apply to each number
8 121____radic
____121
9 π__2
10 TellwhetherthestatementldquoAllintegersarerationalnumbersrdquoistrueorfalseExplainyourchoice
13ensp OrderingenspRealenspNumbersCompare Write lt gt or =
11 radic__
8+3 8+radic__
3 12 radic__
5+11emsp emsp emsp 5+radic___
11
Order the numbers from least to greatest
13 radic___
99π29__
8 14 radic___
1__251_40__
2
15 Howarerealnumbersusedtodescribereal-worldsituations
ESSENTIAL QUESTION
035
9-9
141ft
7 1__10- 1__10
14__11 1875
wholeintegerrationalreal
Trueintegerscanbewrittenasthequotientoftwointegers
SampleanswerRealnumberssuchastherational
π29__
8radic___
99
irrationalreal
lt gt
number1_4candescribeamountsusedincooking
radic___
1__250__
21_4
27Module1
copy H
ough
ton
Miff
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Com
pany
DONOTEDIT--ChangesmustbemadethroughldquoFileinfordquoCorrectionKey=A
8_MCAAESE206984_U1M01RTindd 27 41513 1113 PM
Math TrainerOnline Assessment
and Intervention
Personal
myhrwcom
1
2
3 Response toIntervention
Intervention Enrichment
Access Ready to Go On assessment online and receive instant scoring feedback and customized intervention or enrichment
Online and Print Resources
Differentiated Instruction
bull Reteach worksheets
bull Reading Strategies EL
bull Success for English Learners EL
Differentiated Instruction
bull Challenge worksheets PRE-AP
Extend the Math PRE-AP
Lesson Activities in TE
Additional ResourcesAssessment Resources includes bull Leveled Module Quizzes
Ready to Go OnAssess MasteryUse the assessment on this page to determine if students have mastered the concepts and standards covered in this module
California Common Core Standards
Lesson Exercises Common Core Standards
11 1ndash7 8NS1 8NS2 8EE2
12 8ndash10 8NS1
13 11ndash14 8NS2
27 Unit 1 Module 1
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
Personal Math Trainer
Online Practice and HelpmyhrwcomAssessment Readiness
Module 1 MIXed ReVIeW
1 Look at each number Is the number between 2π and radic___
52
Select Yes or No for expressions AndashC
A 6 2 _ 3 Yes No
B 5π __ 2 Yes No
C 3 radic__
5 Yes No
2 Consider the number - 11 __ 15
Choose True or False for each statement
A The number is rational True False
B The number can be written as True Falsea repeating decimal
C The number is less than ndash08 True False
3 The volume of a cube is given by V = x3 where x is the length of an edge of the cube A cube-shaped end table has a volume of 3 3 _ 8 cubic feet What is the length of an edge of the end table Explain how you solved this problem
4 A student says that radic___
83 is greater than 29 __ 3 Is the student correct Justify your
reasoning
1 1 _ 2 ft Sample answer The equation x3 = 3 3 _ 8 can be used
to find the edge length in feet To solve the equation
write the mixed number as a fraction greater than 1
x3 = 27 __ 8 Then take the cube root of both sides x = 3 _ 2 = 1 1 _ 2
No Sample answer radic___
83 asymp 91 and 29 __ 3 = 9
__ 6
Because 91 lt 9 __
6 radic___
83 lt 29 __ 3
28 Unit 1
copy H
ough
ton
Miff
lin H
arco
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ublis
hing
Com
pany
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=A
8_MCAAESE206984_U1M01RTindd 28 240413 946 AM
Personal Math Trainer
Online Assessment and
Interventionmyhrwcom
Scoring GuideItem 3 Award the student 1 point for finding the edge length of the cube and 1 point for correctly explaining how to use a cube root to solve the problem
Item 4 Award the student 1 point for determining that the student is incorrect and 1 point for correctly justifying the reasoning for this conclusion
Additional ResourcesTo assign this assessment online login to your Assignment Manager at myhrwcom
Assessment Readiness
California Common Core Standards
Items Grade 8 Standards Mathematical Practices
1 8NS2 MP7
2 7NS2b 7NS2d 8NS1 MP7
3 8EE2 MP1 MP4
4 8NS1 8NS2 MP3
Item integrates mixed review concepts from previous modules or a previous course
Item 4 combines concepts from the California Common Core cluster ldquoKnow that there are numbers that are not rational and approximate them by rational numbersrdquo
Real Numbers 28
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
ElaborateTalk About ItSummarize the Lesson
Ask If someone claims that a certain number is irrational but you know it is actually rational how could you prove to that person that the number is rational
You could find a fraction equal to the number such that the number is the ratio of two integers with the denominator not equal to zero
GUIDED PRACTICEEngage with the Whiteboard
Have students plot each number in Exercises 16ndash18 on a number line Students should label each point with the irrational number written as a radical and as a
decimal
Avoid Common ErrorsExercises 1ndash6 To avoid reversing the order of the dividend and divisor tell students to start at the top of the fraction and read the bar as ldquodivided byrdquo
Focus on TechnologyHave students use a calculator to investigate the decimal equivalents of such fractions as 1 __ 9 2 __ 9 8 __ 9 and 1 __ 11 2 __ 11 10
__ 11 Ask them to describe the patterns they find as a result of these investigations
11 Lesson 11
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
Guided Practice
7 0675 8 56 9 044
10 0 _
4
10x =
x =
11 0 _
26
100x =
x =
12 0 _
325
1000x =
x =
Solve each equation for x (Example 3 and Explore Activity)
- x
-
_______________
x =
- x
-
___________________
x =
- x
-
_______________________
x =
Write each fraction or mixed number as a decimal (Example 1)
1 2 _ 5 2 8 _ 9 3 3 3 _ 4
4 7 __ 10 5 2 3 _ 8 6 5 _ 6
Write each decimal as a fraction or mixed number in simplest form (Example 2)
13 x 2 = 17 14 x 2 = 25 ___ 289 15 x 3 = 216
Approximate each irrational number to one decimal place without a calculator
x = plusmn radic__
asymp plusmn x = 3
radic__
=
(Explore Activity)
16 radic_
5 asymp
17 radic_
3 asymp
18 radic_
10 asymp
19 What is the difference between rational and irrational numbers
CHECK-INESSENTIAL QUESTION
x = plusmn radic__
__________ = plusmn _____
4 _
4
0 _
4
4 99
6216
269
41 25 5
17289
17
22 17 32
04
07
27__40
26 __ 99 325 ___ 999 4 _ 9
11__255 3_5
0 _
8
2375
375
08 _
3
26 _
26
0 _
26
325 _
325
0 _
325
999 325
Rational numbers can be written in the form a __ b where
a and b are integers and b ne 0 Irrational numbers cannot
be written in this form
Unit 112
copy H
ough
ton
Miff
lin H
arco
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ublis
hing
Com
pany
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
8_MCAAESE206984_U1M01L1indd 12 41613 1211 AM
11 12 13 14 15
radic2 asymp 14
141 142 143 144 145
radic2 asymp 141
0 1 2 3 4
radic2 asymp 15
Estimate that radic_
2 asymp 15
To find a better estimate first choose some numbers between 1 and 2 and square them For example choose 13 14 and 15
1 3 2 = 1 4 2 = 1 5 2 =
Is radic_
2 between 13 and 14 How do you know
Is radic_
2 between 14 and 15 How do you know
2 is closer to than to so radic_
2 asymp
Locate and label this value on the number line
Reflect 11 How could you find an even better estimate of radic
_ 2
12 Find a better estimate of radic_
2
1 41 2 = 1 42 2 = 1 43 2 =
2 is closer to than to so radic_
2 asymp
Draw a number line and locate and label your estimate
13 Solve x 2 = 7 Write your answer as a radical expression Then estimate to one decimal place
D
E
F
No 2 is not between 169 and 196
Yes 2 is between 196 and 225
196
19881
19881
225
20164
20164
14
141
20449
169 196 225
Test the squares of numbers between 14 and 15
x = plusmn radic_
7 x asymp plusmn26
11Lesson 11
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ough
ton
Miff
lin H
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urt P
ublis
hing
Com
pany
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
8_MCAAESE206984_U1M01L1indd 11 41613 1211 AM
Rational and Irrational Numbers 12
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
Math TrainerOnline Assessment
and Intervention
Personal
Online homework assignment available
myhrwcom
EvaluateGUIDED AND INDEPENDENT PRACTICE
Concepts amp Skills Practice
Example 1Expressing Rational Numbers as Decimals
Exercises 1ndash6 20ndash21 24ndash25
Example 2Expressing Decimals as Rational Numbers
Exercises 7ndash12 22ndash23 26ndash27
Example 3Finding Square Roots and Cube Roots
Exercises 13ndash15 28 30ndash31 35
Explore ActivityEstimating Irrational Numbers
Exercises 13 16ndash18 29 32ndash34
Additional ResourcesDifferentiated Instruction includes bull Leveled Practice worksheets
Lesson Quiz available online
11 LESSON QUIZ
1 Write as a decimal 2 5 __ 8 1 7 __ 12
2 Write as a fraction 034 1 _
24
3 Solve x 2 = 9 __ 49 for x
4 Solve x 3 = 216 for x
5 Estimate the value of radic_
13 to one decimal place without using a calculator
myhrwcom
Focus | Coherence | Rigor
Exercise Depth of Knowledge (DOK) Mathematical Practices
20ndash27 2 SkillsConcepts MP4 Modeling
28 3 Strategic Thinking MP4 Modeling
29ndash32 2 SkillsConcepts MP6 Precision
33 3 Strategic Thinking MP7 Using Structure
34 2 SkillsConcepts MP3 Logic
35 2 SkillsConcepts MP4 Modeling
36 3 Strategic Thinking MP3 Logic
37 3 Strategic Thinking MP7 Using Structure
38 3 Strategic Thinking MP2 Reasoning
8NS1 8NS2 8EE2
8NS1 8NS2 8EE2
Answers1 2625 158
_ 3
2 17 __ 50 1 8 __ 33
3 x = plusmn 3 __ 7
4 x = 6
5 36
13 Lesson 11
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
Work Area
33 Analyze Relationships To find radic_
15 Beau found 3 2 = 9 and 4 2 = 16 He said that since 15 is between 9 and 16 radic
_ 15 must be between 3 and 4 He
thinks a good estimate for radic_
15 is 3 + 4 ____ 2 = 35 Is Beaursquos estimate high low
or correct Explain
34 Justify Reasoning What is a good estimate for the solution to the equation x 3 = 95 How did you come up with your estimate
35 The volume of a sphere is 36π f t 3 What is the radius of the sphere Use the formula V = 4 _ 3 π r 3 to find your answer
36 Draw Conclusions Can you find the cube root of a negative number If so is it positive or negative Explain your reasoning
37 Make a Conjecture Evaluate and compare the following expressions
radic_
4 __ 25 and radic
_ 4 ____
radic_
25 radic
_
16 __ 81 and radic_
16 ____
radic_
81 radic
_
36 __ 49 and radic_
36 ____
radic_
49
Use your results to make a conjecture about a division rule for square roots Since division is multiplication by the reciprocal make a conjecture about a multiplication rule for square roots
38 Persevere in Problem Solving The difference between the solutions to the equation x 2 = a is 30 What is a Show that your answer is correct
FOCUS ON HIGHER ORDER THINKING
His estimate is low because 15 is very close to 16
so radic_
15 is very close to radic_
16 or 4 A better estimate
would be 38 or 39
Sample answer about 45 4 3 = 64 and 5 3 = 125
Because 95 is about halfway between 64 and 125 try 45
45 3 = 91125 which is a good estimate
3 feet
Yes the cube root of a negative number is negative
because a negative number cubed is always negative
and a nonnegative number cubed is always nonnegative
radic_
4 __ 25 = 2 _ 5 = radic
_ 4 ____
radic_
25 radic
_
16 __ 81 = 4 _ 9 = radic_
16 ____
radic_
81 radic
_
36 __ 49 = 6 _ 7 = radic_
36 ____
radic_
49
225 the solutions to x 2 = a are x = plusmn15 and
radic_
a ___
radic_
b = radic
_ a __
b radic
_ a radic
_ b = radic
_ a b
15 - (-15) = 30
Unit 114
copy H
ough
ton
Miff
lin H
arco
urt P
ublis
hing
Com
pany
bull copy
Ilen
e Mac
Dona
ldA
lamy I
mag
es
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=B
8_MCABESE206984_U1M01L1indd 14 102913 1142 PM
Personal Math Trainer
Online Practice and Helpmyhrwcom
Name Class Date
Independent Practice11
20 A 7 __ 16 -inch-long bolt is used in a machine What is this length written as a decimal
21 The weight of an object on the moon is 1 _ 6 its weight on Earth Write 1 _ 6 as a decimal
22 The distance to the nearest gas station is 2 4 _ 5 kilometers What is this distance written as a decimal
23 A baseball pitcher has pitched 98 2 _ 3 innings What is the number of innings written as a decimal
24 A heartbeat takes 08 second How many seconds is this written as a fraction
25 There are 262 miles in a marathon Write the number of miles using a fraction
26 The average score on a biology test was 72
_ 1 Write the average score using a
fraction
27 The metal in a penny is worth about 0505 cent How many cents is this written as a fraction
28 Multistep An artist wants to frame a square painting with an area of 400 square inches She wants to know the length of the wood trim that is needed to go around the painting
a If x is the length of one side of the painting what equation can you set up to find the length of a side How many solutions does the equation have
b Do all of the solutions that you found make sense in the context of the problem Explain
c What is the length of the wood trim needed to go around the painting
Solve each equation for x Write your answers as radical expressions Then estimate to one decimal place if necessary
29 x 2 = 14 30 x 3 = 1331
31 x 2 = 144 32 x 2 = 29
8NS1 8NS2 8EE2
04375 in 01 _6
28 km 98 _6 innings
x 2 = 400 x = plusmnthinsp20 the equation has 2 solutions
x = 20 makes sense but x = -20 doesnrsquot because a
painting cannot have a side length of -20 inches
4 times 20 = 80 inches
x = plusmn radic_
14 asymp plusmn37
x = plusmn radic_
144 = plusmn12 x = plusmn radic_
29 asymp plusmn54
x = 3 radic_ 1331 = 11
4_5 second 26 1_5 mi
72 1 _ 9 101 ___ 200 cent
13Lesson 11
copy H
ough
ton
Miff
lin H
arco
urt P
ublis
hing
Com
pany
bull copy
Phot
odisc
Get
ty Im
ages
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
8_MCAAESE206984_U1M01L1indd 13 41613 1211 AM
myhrwcomActivity available onlineEXTEND THE MATH PRE-AP
Activity Write radic_
09 on the board and invite students to conjecture what the value might be Have them check their conjectures by squaring Invite them to suggest ways to estimate radic
_ 09 As a hint point out that 09 is close to 10 and so they might
use that to help guide their estimates Lead them to see that since 092 is 081 and 102 is 1 the value of radic
_ 09 is greater than 09 and less than 10 Try squaring 095 to get
09025 A good estimate for radic_
09 is 095
Rational and Irrational Numbers 14
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=B
Integers
Rational Numbers IrrationalNumbers
Real Numbers
WholeNumbers
-3-4-5 -2-1 1 2 3 50 4
23
34-4 -π -1 25
radic2
Lesson Support Content Objective Students will learn to describe relationships between sets of numbers
Language Objective Students will explain how to describe relationships between sets of real numbers
LESSON 12 Sets of Real Numbers
Building BackgroundEliciting Prior Knowledge Have students draw a number line from -5 to 5 Ask them to plot points on the number line to approximate the location of rational and irrational numbers such as -1 3 __ 4 25 -4 2 __ 3 radic
_ 2 and -π
Learning ProgressionsIn this lesson students clarify their understanding of the real number system They characterize sets and subsets of the real numbers They also identify sets for real-world situations Important understandings for students include the following
bull Identify all of the possible subsets of the real numbers for a given number
bull Decide whether a statement about a subset of the real numbers is true or false
bull Identify the set of numbers that best describes a real-world situation
Understanding the relationships among the sets of numbers that make up the real numbers is essential as students are introduced to different forms of numbers throughout the school year This lesson provides a foundation for the comparing and ordering of real numbers in the next lesson
Cluster ConnectionsThis lesson provides an excellent opportunity to connect ideas in this cluster Know that there are numbers that are not rational and approximate them by rational numbers Have students copy this diagram which relates the sets of real numbers
Ask students to complete the diagram by writing three examples for each set of numbers Have students share examples and explain how they knew each number they selected belonged in the appropriate set Answers may vary Check studentsrsquo work
Focus | Coherence | Rigor
California Common Core Standards
8NS1 Know that numbers that are not rational are called irrational Understand informally that every number has a decimal expansion for rational numbers show that the decimal expansion repeats eventually and convert a decimal expansion which repeats eventually into a rational number
MP7 Look for and make use of structure
15A
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
Math Talk
Language Support EL
PROFESSIONAL DEVELOPMENT
Linguistic Support EL
AcademicContent Vocabulary
Venn diagrams ndash Students need descriptive language to describe the categories that the different areas and colors of a Venn diagram represent the concept of a set and how sets are distinct or can overlap Use sentence frames such as
The big oval represents __________The darklight blue color in the middle of the
big ovals represents __________These sets overlap because __________
In this way students have the language and structure to identify the criteria that distinguish a set and to explain the abstract representation Also point out the use of the prefix sub- meaning ldquounderrdquo in the term subset
Rules and Patterns
Abbreviations ndash In this lesson the abbreviation mph is used Be sure to point out that mph stands for miles per hour and is used to give units in a rate of speed Students may also have seen mpg (miles per gallon) which gives the units in a rate of fuel efficiency
Borrowed Words ndash Terminology used in baseball such as inning and pitcher may require some explanation Spanish as well as some other languages have borrowed these terms from English so some students may be familiar with these words already Despite this whenever a word is critical to students understanding the word problem it is best to explain the meaning
Leveled Strategies for English Learners
Emerging Allow students to indicate true or false orally in Guided Practice Exercises 9 and 10
Expanding Have students use sentence frames to describe the meaning of regions and colors used in a Venn diagram Then give them similar sentence frames orally and have them draw and shade a Venn diagram based on the oral prompts
Bridging Have students work in groups to draw a Venn diagram to represent sets based on real-world examples in the lesson
To help students answer the question posed in Math Talk provide a sentence frame for their answer
The numbers between 31 and 39 on a number line are __________ because __________
EL
California ELD Standards
Emerging 2II5 Modifying to add details ndash Expand sentences with simple adverbials to provide details about a familiar activity or process
Expanding 2II5 Modifying to add details ndash Expand sentences with adverbials to provide details about a familiar or new activity or process
Bridging 2II5 Modifying to add details ndash Expand sentences with increasingly complex adverbials to provide details about a variety of familiar and new activities and processes
Sets of Real Numbers 15B
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
12L E S S O N
Sets of Real Numbers
EngageESSENTIAL QUESTION
How can you describe relationships between sets of real numbers Sample answer Describe them as two different sets or one set as being a subset of another
Motivate the LessonAsk How many different types of tigers can you name How does the set of Bengal tigers relate to the set of tigers
ExplorePoint to different locations in the Animals diagram and ask for examples for that classification Do the same for the Real Numbers diagram Students should understand that everything within a region is part of the set for example both -3 and 2 are integers
ExplainEXAMPLE 1
Questioning Strategies Mathematical Practices bull In A why is 5 not a perfect square It does not have rational numbers as its square roots
bull Can the number in B be written as a fraction Why or why not Yes it is a terminating decimal so it is a rational number
Engage with the WhiteboardHave students place the numbers in Example 1 and Additional Example 1 in the Venn diagram for numbers
YOUR TURNAvoid Common ErrorsBe sure that students read Exercise 2 carefully before answering The number given in the problem 10 is the area not the side length
EXAMPLE 2Questioning Strategies Mathematical Practices bull What two major sets are the real numbers composed of rational and irrational numbers
bull What is the location of the set of whole numbers in the Venn diagram in relation to the set of rational numbers Explain Inside it whole numbers are rational numbers
Focus on Reasoning Mathematical PracticesRemind students that it takes only one counterexample to show that a statement is false
Interactive Whiteboard Interactive example available online
ADDITIONAL EXAMPLE 1Write all names that apply to each number
A -10integer rational real
B 12 _ 3
whole integer rational real
myhrwcom
Interactive Whiteboard Interactive example available online
ADDITIONAL EXAMPLE 2Tell whether the given statement is true or false Explain your choice
No integers are whole numbers
False every whole number is also an integer
myhrwcom
Animated MathClassifying Numbers
Students build fluency in classifying numbers in this engaging fast-paced game
myhrwcom
CA Common CoreStandards
The student is expected to
The Number Systemmdash8NS1
Know that numbers that are not rational are called irrational Understand informally that every number has a decimal expansion for rational numbers show that the decimal expansion repeats eventually and convert a decimal expansion which repeats eventually into a rational numberMathematical Practices
MP7 Using Structure
The student is expected to
15 Lesson 12
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=B
Math TrainerOnline Practice
and Help
Personal
myhrwcom
Math TrainerOnline Practice
and Help
Personal
myhrwcom
Math On the Spotmyhrwcom
Understanding Sets and Subsets of Real NumbersBy understanding which sets are subsets of types of numbers you can verify whether statements about the relationships between sets are true or false
Tell whether the given statement is true or false Explain your choice
All irrational numbers are real numbers
True Every irrational number is included in the set of real numbers The irrational numbers are a subset of the real numbers
No rational numbers are whole numbers
False A whole number can be written as a fraction with a denominator of 1 so every whole number is included in the set of rational numbers The whole numbers are a subset of the rational numbers
EXAMPLE 2
A
B
Write all names that apply to each number
1 A baseball pitcher has pitched 12 2 _ 3 innings
2 The length of the side of a square that has an
area of 10 square yards
YOUR TURN
Tell whether the given statement is true or false Explain your choice
3 All rational numbers are integers
4 Some irrational numbers are integers
YOUR TURN
Give an example of a rational number that is a
whole number Show that the number is both whole
and rational
Math TalkMathematical Practices
Give an example of a
8NS1
False Every integer is a rational number but not every
False Real numbers are either rational or irrational numbers
Integers are rational numbers so no integers are irrational numbers
rational real
irrational real
Sample answer 8 8 = 8_
1
and -thinsp 5 _ 2 are not integers
rational number is an integer Rational numbers such as 3 _ 5
Unit 116
copy H
ough
ton
Miff
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hing
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pany
bull Im
age C
redi
ts D
igita
l Im
age c
opyr
ight
copy20
04 Ey
ewire
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8_MCAAESE206984_U1M01L2indd 16 41613 136 AM
Math On the Spot
myhrwcom
Vertebrates
Birds
Passerines
Animals
Integers
Rational Numbers IrrationalNumbers
Real Numbers
WholeNumbers
1
45
3
0
274
67
radic4
-
-3
-2
-1
03
radic2
radic17
radic11-
π
Animated Math
myhrwcom
Classifying Real NumbersBiologists classify animals based on shared characteristics A cardinal is an animal a vertebrate a bird and a passerine
You already know that the set of rational numbers consists of whole numbers integers and fractions The set of real numbers consists of the set of rational numbers and the set of irrational numbers
Write all names that apply to each number
radic_
5 irrational real
ndash1784rational real
whole integer rational real
EXAMPLEXAMPLE 1
A
B
C radic_ 81 ____ 9
L E S S O N
12Sets of Real Numbers
ESSENTIAL QUESTIONHow can you describe relationships between sets of real numbers
Passerines such as the cardinal are also called ldquoperching birdsrdquo
What types of numbers are between 31 and 39 on a
number line
Math TalkMathematical Practices
What types of numbers are
8NS1
8NS1
Know that numbers that are not rational are called irrational Understand informally that every number has a decimal expansion for rational numbers show that the decimal expansion repeats eventually and convert a decimal expansion which repeats eventually into a relation number
ndash1784 is a terminating decimal
5 is a whole number that is not a perfect square
radic_
81 _____ 9 = 9 __ 9 = 1 rational irrational real
15Lesson 12
copy H
ough
ton
Miff
lin H
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ublis
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Com
pany
bull Im
age C
redi
ts copy
Wiki
med
ia Co
mm
ons
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=B
8_MCABESE206984_U1M01L2indd 15 061113 1144 AM
PROFESSIONAL DEVELOPMENT
Math BackgroundThe relationships between sets of numbers extend to include complex numbers A complex number can be written as a sum of a real number a and an imaginary number bi
a + bi
An imaginary number is a special number that when squared gives a negative value When you square a real number you get a nonnegative number When you square an imaginary number you get a negative value The imaginary unit is i
i = radic_
-1
Integrate Mathematical Practices MP7
This lesson provides an opportunity to address this Mathematical Practices standard It calls for students to discern structure to connect and communicate mathematical ideas
Students use a Venn diagram to structure relationships between sets of numbers They connect and communicate mathematical ideas when they make logical statements about the sets and describe which set best describes numbers applied to real-life situations
Sets of Real Numbers 16
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=B
YOUR TURNAvoid Common ErrorsStudents may see the word ldquoAllldquo or rdquoNordquo in Exercises 3 and 4 and immediately assume that any absolute statements like these are false Remind them that there are true statements that begin with these words and encourage them to provide examples
EXAMPLE 3Questioning Strategies Mathematical Practices bull In A how does the phrase ldquonumber of rdquo give you a clue about the number classification It indicates a counting number
bull What is the relationship between the circumference of a circle and the diameter The circumference is diameter times π
Focus on Critical Thinking Mathematical PracticesIn B suppose the diameters in inches were 25
__ π 28 __ π
31 __ π and so on What set of numbers would
best describe the circumferences Explain Whole numbers the circumferences would be the whole numbers 25 28 31 and so on
YOUR TURNFocus on Critical Thinking Mathematical PracticesHave students compare and contrast the classification of numbers in the answers in Exercises 5 and 6
ElaborateTalk About ItSummarize the Lesson
Ask What are some ways that number sets can be related Sets may be subsets of other sets or they may be separate from other sets
GUIDED PRACTICEEngage with the Whiteboard
Have students place the numbers in Exercises 1ndashthinsp8 in the Venn diagram for numbers at the beginning of the lesson
Integrating Language Arts EL
Encourage English learners to ask for clarification on any terms or phrases that they do not understand
Avoid Common ErrorsExercise 7 Remind students that a repeating decimal is a rational numberExercises 9ndash10 Remind students that it only takes one counterexample to show that a statement is false
Interactive Whiteboard Interactive example available online
ADDITIONAL EXAMPLE 3Identify the set of numbers that best describes the situation Explain your choice
A the amount of time that has passed since midnight
The set of real numbers time is continuous so the amount of time can be rational or irrational
B the number of tickets sold to a basketball game
The set of whole numbers the number of tickets sold may be 0 or a counting number
myhrwcom
17 Lesson 12
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
1IN
116 inch
Guided Practice
Write all names that apply to each number (Example 1)
1 7 _ 8 2 radic_
36
3 radic_
24 4 075
5 0 6 - radic_ 100
7 5 _
45 8 - 18 __ 6
Tell whether the given statement is true or false Explain your choice (Example 2)
9 All whole numbers are rational numbers
10 No irrational numbers are whole numbers
Identify the set of numbers that best describes each situation Explain your choice (Example 3)
11 the change in the value of an account when given to the nearest dollar
12 the markings on a standard ruler
13 What are some ways to describe the relationships between sets of numbers
CHECK-INESSENTIAL QUESTION
rational real
rational real
True Whole numbers are rational numbers
Rational numbers the ruler is marked every 1 __ 16 th inch
Sample answer Describe one set as being a subset of
another or show their relationships in a Venn diagram
Integers the change can be a whole dollar amount
and can be positive negative or zero
True Whole numbers are a subset of the set of rational numbers
and can be written as a ratio of the whole number to 1
irrational real
whole integer rational real
whole integer rational real
rational real
integer rational real
integer rational real
Unit 118
copy H
ough
ton
Miff
lin H
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ublis
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pany
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
8_MCAAESE206984_U1M01L2indd 18 41613 136 AM
My Notes
Math TrainerOnline Practice
and Help
Personal
myhrwcom
Math On the Spot
myhrwcom
Identifying Sets for Real-World SituationsReal numbers can be used to represent real-world quantities Highways have posted speed limit signs that are represented by natural numbers such as 55 mph Integers appear on thermometers Rational numbers are used in many daily activities including cooking For example ingredients in a recipe are often given in fractional amounts such as 2 _ 3 cup flour
Identify the set of numbers that best describes each situation Explain your choice
the number of people wearing glasses in a room
The set of whole numbers best describes the situation The number of people wearing glasses may be 0 or a counting number
the circumference of a flying disk has a diameter of 8 9 10 11 or 14 inches
The set of irrational numbers best describes the situation Each circumference would be a product of π and the diameter and any multiple of π is irrational
EXAMPLEXAMPLE 3
A
B
Identify the set of numbers that best describes the situation Explain your choice
5 the amount of water in a glass as it evaporates
6 the weight of a person in pounds
YOUR TURN
8NS1
Rational numbers a personrsquos weight can be a decimal
such as 835 pounds
Real numbers the amount can be any number greater
than 0
17Lesson 12
copy H
ough
ton
Miff
lin H
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ublis
hing
Com
pany
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
8_MCAAESE206984_U1M01L2indd 17 41613 520 AM
Graphic OrganizersGive students a list of numbers (including terminating and repeating decimals fractions integers and rational and irrational square roots) and a graphic organizer as shown below
Real Numbers
Rational numbers Irrational numbers
Integer numbers
Whole numbers
Ask students to write each number in the list in the correct section of the organizer
Number SensePoint out to students that knowing the types of numbers to expect in different situations can alert them to incorrect math as well as to impossible situations For example 135 shots made in basketballs is not possible but an average number of shots can equal 135
Additional ResourcesDifferentiated Instruction includes bull Reading Strategies bull Success for English Learners EL
bull Reteach bull Challenge PRE-AP
DIFFERENTIATE INSTRUCTION
Sets of Real Numbers 18
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
Math TrainerOnline Assessment
and Intervention
Personal
Online homework assignment available
myhrwcom
Lesson Quiz available online
12 LESSON QUIZ
1 Write all the names that apply to the number
2 Tell whether the given statement is true or false Explain your choice All numbers between 1 and 2 are rational numbers
3 Identify the set of numbers that best describes the situation Explain your choiceThe choices on a survey question change the total points for the survey by -2 -1 0 1 or 2 points
-1 _
5
myhrwcom
Additional ResourcesDifferentiated Instruction includes bull Leveled Practice worksheets
EvaluateGUIDED AND INDEPENDENT PRACTICE
Concepts amp Skills Practice
Example 1Classifying Real Numbers
Exercises 1ndash8 14ndash19 22ndash24
Example 2Understanding Sets and Subsets of Real Numbers
Exercises 9ndash10
Example 3Identifying Sets for Real-World Situations
Exercises 11ndash12 20ndash21 25
Focus | Coherence | Rigor
Exercise Depth of Knowledge (DOK) Mathematical Practices
14ndash19 2 SkillsConcepts MP7 Using Structure
20ndash21 2 SkillsConcepts MP6 Precision
22ndash23 2 SkillsConcepts MP3 Logic
24 1 Recall of Information MP7 Using Structure
25 2 SkillsConcepts MP2 Reasoning
26ndash27 3 Strategic Thinking MP3 Logic
28 3 Strategic Thinking MP8 Patterns
29 3 Strategic Thinking MP3 Logic
8NS1
8NS1
Exercise 29 combines concepts from the California Common Core cluster ldquoKnow that there are numbers that are not rational and approximate them by rational numbersrdquo
Answers1 rational real
2 False radic_
2 is an example of an irrational number between 1 and 2
3 Integers each number is an integer but only three are whole numbers
19 Lesson 12
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
Work Area
π mi23 Critique Reasoning The circumference of a circular region is shown
What type of number best describes the diameter of the circle Explain
your answer
24 Critical Thinking A number is not an integer What type of number can it be
25 A grocery store has a shelf with half-gallon containers of milk What type of number best represents the total number of gallons
26 Explain the Error Katie said ldquoNegative numbers are integersrdquo What was her error
27 Justify Reasoning Can you ever use a calculator to determine if a number is rational or irrational Explain
28 Draw Conclusions The decimal 0 _
3 represents 1 _ 3 What type of number best describes 0
_ 9 which is 3 middot 0
_ 3 Explain
29 Communicate Mathematical Ideas Irrational numbers can never be precisely represented in decimal form Why is this
FOCUS ON HIGHER ORDER THINKING
It can be a rational number that is not an integer or an irrational number
rational number
The set of negative numbers also includes non-integer
rational numbers and irrational numbers
Sample answer If the calculator shows a decimal that
terminates in fewer digits than what the calculator screen
allows then you can tell that the number is rational If not
you cannot tell from the calculator display whether the
number terminates because you see a limited number
of digits It may be a repeating decimal (rational) or
non-terminating non-repeating decimal (irrational)
Whole 3 middot 0 _
3 represents 3 middot 1 _ 3 = 1 so 0 _
9 is exactly 1
Sample answer In decimal form irrational numbers never
terminate and never repeat Therefore no matter how
many decimal places you include the number will never
be precisely represented There are always more digits
Whole the diameter is π _ π = 1 mile
Unit 120
copy H
ough
ton
Miff
lin H
arco
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ublis
hing
Com
pany
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
8_MCAAESE206984_U1M01L2indd 20 120413 909 PM
Integers
Rational Numbers Irrational Numbers
Real Numbers
Whole Numbers
257
radic16
166
radic9
128 radic50
Personal Math Trainer
Online Practice and Helpmyhrwcom
Name Class Date
Independent Practice
Identify the set of numbers that best describes each situation Explain your choice
20 the height of an airplane as it descends to an airport runway
21 the score with respect to par of several golfers 2 ndash 3 5 0 ndash 1
22 Critique Reasoning Ronald states that the number 1 __ 11 is not rational because when converted into a decimal it does not terminate Nathaniel says it is rational because it is a fraction Which boy is correct Explain
12
14 - radic_
9 15 257
16 radic_
50 17 8 1 _ 2
18 166 19 radic_
16
Write all names that apply to each number Then place the numbers in the correct location on the Venn diagram
8NS1
Real numbers the height can be any number greater than zero
integer rational real whole integer rational real
whole integer rational real
irrational real
rational real
rational real
Integers the scores are counting numbers their
opposites and zero
Nathaniel is correct A rational number is a number that can be written as a fraction and 1 __ 11 is a fraction
19Lesson 12
copy H
ough
ton
Miff
lin H
arco
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ublis
hing
Com
pany
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
8_MCAAESE206984_U1M01L2indd 19 41613 136 AM
myhrwcomActivity available onlineEXTEND THE MATH PRE-AP
Activity Have students consider the concept of restricted domain for the sets of numbers that describe situations For example the number of sisters a person has can best be described by whole numbers but no one has ever had 1500 sisters An area code is an integer or whole number between 200 and 999
Have students use a source such as the Guinness Book of World Records and give examples of sets of numbers that describe situations where the domain is restricted Ask whether the restriction may be changed in the future
Sets of Real Numbers 20
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
-3-4-5 -2-1 1 2 3 50 4
12-4 -radic5
Lesson Support Content Objective Students will learn to order a set of real numbers
Language Objective Students will show and describe how to order a set of real numbers
LESSON 13 Ordering Real Numbers
Building BackgroundEliciting Prior Knowledge Have students draw a number line to compare a rational number and an irrational number such as - radic
_ 5 and -4 1 __ 2 Ask them to explain how
they approximated the irrational number on the number line Then have them identify the greater and the lesser real number Repeat with several other pairs of real numbers in different forms
Learning ProgressionsIn this lesson students order a set of real numbers They use rational approximations to compare the sizes of irrational numbers They also order numbers for real-world situations Important understandings for students include the following
bull Compare irrational numbers bull Estimate the value of expressions with irrational numbers bull Order a set of real numbers bull Order real numbers in a real-world context
Work with real numbers continues throughout Grade 8 and into high school This lesson provides students with a foundation for understanding the relative sizes of numbers in different forms in the real number system
Cluster ConnectionsThis lesson provides an excellent opportunity to connect ideas in this cluster Know that there are numbers that are not rational and approximate them by rational numbers Tell students that there is a special number called the golden ratio with applications in mathematics geometry art and architecture The golden ratio is called phi and is represented by the Greek letter ϕ It includes an irrational number in its definition
Have students explain why the golden ratio is irrational Ask them to find the two whole numbers the golden ratio lies between Then challenge them to approximate the golden ratio to the nearest tenth It is irrational because it includes an irrational number in its definition It lies between 1 and 2 To the nearest tenth ϕ = 16
ϕ = 1 + radic_
5 _ 2
Focus | Coherence | Rigor
California Common Core Standards
8NS2 Use rational approximations of irrational numbers to compare the size of irrational numbers locate them approximately on a number line diagram and estimate the value of expressions (eg π 2 )
MP4 Model with mathematics
21A
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
Math Talk
Language Support EL
PROFESSIONAL DEVELOPMENT
Linguistic Support EL
AcademicContent Vocabulary
Post a chart like this to remind students of the regular comparative forms of adjectives that use the -er and -est suffixes Add to the chart for terms that appear in examples and exercises in each lesson Include any irregular verb forms
Background Knowledge
Go On ndash the title of the module review or quiz is Ready to Go On This title uses an idiomatic expression In this context to go on means ldquoto move aheadrdquo or ldquoto proceedrdquo It is different from the use of go on that means having enough facts to use meaningfully as in having enough to go on Also the intonation used in pronouncing an expression can give it different meanings For example when the speaker emphasizes the word on he or she might be expressing disbelief as in ldquoGo ON Yoursquore kidding rightrdquo Discuss with students other ways that the phrase go on may be used
Leveled Strategies for English Learners
Emerging Label points on a number line with the terms used in ordering greater greatest less lesser least Use sentence frames to insert the correct terms
Expanding Have students give two or three complete sentences to compare the placement of numbers on a number line using the correct forms of the comparative and superlative adjectives
Bridging Have students work in pairs with one student giving directions to the other in complete sentences to order numbers on a number line
To help students answer the question posed in Math Talk make sure that students have a command of the forms for making comparisons and the superlative and the concept of opposite order so that the focus is on the math concept instead of the language skills needed to describe and explain order
EL
Adjective Comparative Superlative
Far Farther Farthest
Large Larger Largest
Great Greater Greatest
Some Less Least
Some More Most
California ELD Standards
Emerging 2I8 Analyzing language choices ndash Explain how phrasing or different common words with similar meanings produce different effects on the audience
Expanding 2I8 Analyzing language choices ndash Explain how phrasing or different words with similar meanings or figurative language produce shades of meaning and different effects on the audience
Bridging 2I8 Analyzing language choices ndash Explain how phrasing or different words with similar meanings or figurative language produce shades of meaning nuances and different effects on the audience
Ordering Real Numbers 21B
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
13L E S S O N
Ordering Real Numbers
Interactive Whiteboard Interactive example available online
ADDITIONAL EXAMPLE 1Compare Write lt gt or =
A radic_
8 - 2 4 - radic_
8 lt
B radic_
20 + 1 3 + radic_
2 gt
EngageESSENTIAL QUESTION
How do you order a set of real numbers Sample answer Find their approximate decimal values and order them
Motivate the LessonAsk What kind of numbers are you comparing when you compare the price of gasoline at two different gas stations
ExploreGive students two rational numbers and ask them to name a number between them Repeat a few times and then give them two irrational numbers and ask them to name a number between them
ExplainEXAMPLE 1
Questioning Strategies Mathematical Practices bull Which is greater the difference between 5 and 3 or the difference between radic
_ 5 and radic
_ 3
The difference between 5 and 3 is 2 the difference between radic_
5 and radic_
3 is approximately 1 So the difference between 5 and 3 is greater
Avoid Common ErrorsCaution students to read the problem carefully and think about what the radical sign means so that they do not misread the problem and answer that the two sides are equal
YOUR TURNFocus on TechnologyCalculators should not be used at this point because developing number sense is the goal
EXAMPLE 2Questioning Strategies Mathematical Practices bull How do you determine whether radic
_ 22 is less than or greater than 45 The square of 45 is
2025 which is less than 22 so the square root of 22 must be greater than 45
Engage with the WhiteboardHave students graph and label various real numbers between 42 and 44 and between 47 and 5
YOUR TURNFocus on Modeling Mathematical PracticesHave students label the integers on the number line with their equivalent square root For example 1 2 and 3 on the number line would be labeled radic
_ 1 radic
_ 4 and radic
_ 9
Interactive Whiteboard Interactive example available online
ADDITIONAL EXAMPLE 2Order 3π radic
_ 10 and 325 from greatest
to least
3π 325 radic_
10
myhrwcom
myhrwcom
CA Common CoreStandards
The student is expected to
The Number Systemmdash8NS2
Use rational approximations of irrational numbers to compare the size of irrational numbers locate them approximately on a number line diagram and estimate the value of expressions (eg π 2 )
Mathematical Practices
MP4 Modeling
The student is expected to
21 Lesson 13
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
Math TrainerOnline Practice
and Help
Personal
myhrwcom
Math On the Spotmyhrwcom
0 05 1 15 2 25 3 35 4
radic5radic3
π2
8 85 9 95 10 105 11 115 12
radic75
4 42 44 46 48 5
radic224 12π + 1
Ordering Real Numbers You can compare and order real numbers and list them from least to greatest
Order radic_
22 π + 1 and 4 1 _ 2 from least to greatest
First approximate radic_
22
radic_
22 is between 4 and 5 Since you donrsquot know where it falls between 4 and 5 you need to find a better estimate for radic
_ 22 so
you can compare it to 4 1 _ 2
Since 22 is closer to 25 than 16 use squares of numbers between 45 and 5 to find a better estimate of radic
_ 22
45 2 = 2025 46 2 = 2116 47 2 = 2209 48 2 = 2304
Since 47 2 = 2209 an approximate value for radic_
22 is 47
An approximate value of π is 314 So an approximate value of π +1 is 414
Plot radic_
22 π + 1 and 4 1 _ 2 on a number line
Read the numbers from left to right to place them in order from least to greatest
From least to greatest the numbers are π + 1 4 1 _ 2 and radic_
22
EXAMPLE 2
STEP 1
STEP 2
Order the numbers from least to greatest Then graph them on the number line
YOUR TURN
5 radic_
5 25 radic_
3
6 π 2 10 radic_
75
If real numbers a b and c are in order from least to greatest what is the order
of their opposites from least to greatest
Explain
Math TalkMathematical Practices
8NS2
radic_
3 radic_
5 25
radic_
75 π2 10
Math Talk answer -c -b -a -c is farthest to the left on a number line -b is in the middle and -a is farthest to the right
Unit 122
copy H
ough
ton
Miff
lin H
arco
urt P
ublis
hing
Com
pany
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
8_MCAAESE206984_U1M01L3indd 22 41613 447 AM
My Notes
Math TrainerOnline Practice
and Help
Personal
myhrwcom
Math On the Spot
myhrwcom
Comparing Irrational NumbersBetween any two real numbers is another real number To compare and order real numbers you can approximate irrational numbers as decimals
Compare radic_
3 + 5 3 + radic_
5 Write lt gt or =
First approximate radic_
3
radic_
3 is between 1 and 2
Next approximate radic_
5
radic_
5 is between 2 and 3
Then use your approximations to simplify the expressions
radic_
3 + 5 is between 6 and 7
3 + radic_
5 is between 5 and 6
So radic_
3 + 5 gt 3 + radic_
5
Reflect1 If 7 + radic
_ 5 is equal to radic
_ 5 plus a number what do you know about the
number Why
2 What are the closest two integers that radic_
300 is between
EXAMPLEXAMPLE 1
STEP 1
STEP 2
Compare Write lt gt or =
YOUR TURN
3 radic_
2 + 4 2 + radic_
4 4 radic_
12 + 6 12 + radic_
6
L E S S O N
13 Ordering Real Numbers
ESSENTIAL QUESTIONHow do you order a set of real numbers
8NS2
Use rational approximations of irrational numbers to compare the size of irrational numbers locate them approximately on a number line diagram and estimate the value of expressions (eg π 2 )
8NS2
Use perfect squares to estimate square roots
1 2 = 1 2 2 = 4 3 2 = 9
The number is 7 both expressions must equal 7 + radic_
5
17 and 18
gt lt
21Lesson 13
copy H
ough
ton
Miff
lin H
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urt P
ublis
hing
Com
pany
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
8_MCAAESE206984_U1M01L3indd 21 41913 246 PM
PROFESSIONAL DEVELOPMENT
Math BackgroundIn this lesson students estimate irrational numbers in the form of square roots of nonper-fect squares by finding two perfect squares between which the number falls A more precise method involves repeated division For example to find radic
_ 28 find a whole number whose perfect
square is close to 28 such as 5 Divide 28 by that number 28 divide 5 = 56 Find the average of the quotient and divisor 5 + 56
_____ 2 = 53 Continue dividing 28 by each result and averaging until you get the desired accuracy
Integrate Mathematical Practices MP4
This lesson provides an opportunity to address this Mathematical Practices standard It calls for students to model relationships using multiple representations including diagrams graphs and language as appropriate Students use multiple representations when they use number lines to estimate the locations of and order rational and irrational numbers given as symbols
Ordering Real Numbers 22
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
Interactive Whiteboard Interactive example available online
ADDITIONAL EXAMPLE 3The diameter of a meteorite in millimeters is calculated by four different methods Order the results from least to greatest
Joe radic_
18 mm Lisa 13 __ 3 mm
Pablo 46 mm Julien 4π __ 3 mm
Julien 4π __ 3 mm Lisa 13 __ 3 mm
Joe radic_
18 mm Pablo 46 mm
EXAMPLE 3Questioning Strategies Mathematical Practices bull How can you verify that radic
_ 28 is between 52 and 53 5 2 2 = 2704 and 5 3 2 = 2809
bull Explain how to determine which number is greater 5 _
5 or 55 When the repeating decimal is rounded to the nearest tenth or hundredth you can see that it is greater
Connect to Daily LifeDiscuss how measuring across a canyon might involve different methods than measuring along a road Explain that measurements like these are often done using calculations that approximate the distance
YOUR TURNFocus on Critical Thinking Mathematical PracticesDiscuss with students which number is greater 3
_ 45 or 3450 3
_ 45 or 3455 and why Explain
that 3 _
45 can be written out as 34545hellipMake sure they understand that 3 _
45 is greater than 345 but less than 3455
ElaborateTalk About ItSummarize the Lesson
Ask How can you order two numbers in different forms whose decimal approxi-mations appear to be equal Approximate one or both numbers to an additional
number of decimal places
GUIDED PRACTICEEngage with the Whiteboard
Have students place and label additional points on the number line in Exercise 9 Allow the points to be in any format other than decimal
Avoid Common ErrorsExercises 3ndash4 Caution students to read the problem carefully so that they do not misread the problem as the same numbers combined by addition on each side of the circleExercise 10 Remind students that the calculations have units
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23 Lesson 13
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
0 05 1 15 2 25 3 35 4 45 5 55 6 65 7
2πradic3
Compare Write lt gt or = (Example 1)
1 radic_
3 + 2 radic_
3 + 3 2 radic_
8 + 17 radic_
11 + 15
3 radic_
6 + 5 6 + radic_
5 4 radic_
9 + 3 9 + radic_
3
5 radic_
17 - 3 -2 + radic_
5 6 12 - radic_
2 14 - radic_
8
7 radic_
7 + 2 radic_
10 - 1 8 radic_
17 + 3 3 + radic_
11
9 Order radic_
3 2π and 15 from least to greatest Then graph them on the number line (Example 2)
radic_
3 is between and so radic_
3 asymp
π asymp 314 so 2π asymp
From least to greatest the numbers are
10 Four people have found the perimeter of a forest using different methods Their results are given in the table Order their calculations from greatest to least (Example 3)
11 Explain how to order a set of real numbers
CHECK-INESSENTIAL QUESTION
Forest Perimeter (km)
Leon Mika Jason Ashley
radic_
17 - 2 1 +thinsp π __ 2 12 ___ 5 25
Guided Practice
17
15
1 + π _ 2 km 25 km 12 __ 5 km radic_
17 - 2 km
2π radic
_ 3
18 175
628
Sample answer Convert each number to a decimal
equivalent using estimation to find equivalents for
irrational numbers Graph each number on a number line
Read the numbers from left to right for least to greatest
Read the numbers from right to left for greatest to least
lt gt
lt lt
ltgt
gt gt
24 Unit 1
copy H
ough
ton
Miff
lin H
arco
urt P
ublis
hing
Com
pany
bull Im
age C
redi
ts copy
Elena
Eliss
eeva
Alam
y Im
ages
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
8_MCAAESE206984_U1M01L3indd 24 41613 448 AM
My Notes
5 52 54 56 58 6
radic28 5 12
23455
Math TrainerOnline Practice
and Help
Personal
myhrwcom
Math On the Spot
myhrwcom
Ordering Real Numbers in a Real-World Context Calculations and estimations in the real world may differ It can be important to know not only which are the most accurate but which give the greatest or least values depending upon the context
Four people have found the distance in kilometers across a canyon using different methods Their results are given in the table Order the distances from greatest to least
Distance Across Quarry Canyon (km)
Juana Lee Ann Ryne Jackson
radic_
28 23 __ 4 5 _
5 5 1 _ 2
Write each value as a decimal
radic_
28 is between 52 and 53 Since 53 2 = 2809 an approximate value for radic
_ 28 is 53
23 __ 4 = 575
5 _
5 is 5555hellip so 5 _
5 to the nearest hundredth is 556
5 1 _ 2 = 55
Plot radic_
28 23 __ 4 5 _
5 and 5 1 _ 2 on a number line
From greatest to least the distances are
23 __ 4 km 5 _
5 km 5 1 _ 2 km radic_
28 km
EXAMPLEXAMPLE 3
STEP 1
STEP 2
7 Four people have found the distance in miles across a crater using different methods Their results are given below
Jonathan 10 __ 3 Elaine 3 _
45 Joseacute 3 1 _ 2 Lashonda radic_
10
Order the distances from greatest to least
YOUR TURN
8NS2
3 1 _ 2 mi 3 _
45 mi 10 __ 3 mi radic_
10 mi
23Lesson 13
copy H
ough
ton
Miff
lin H
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ublis
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Com
pany
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8_MCAAESE206984_U1M01L3indd 23 41613 447 AM
ModelingPlace papers around the room with the numbers from 1 to 5 one per sheet Give each student a card showing a number between 1 and 5 in different forms Have students place his or her card between the correct integers and decide where the number goes in relation to any numbers already placed
Multiple RepresentationsGive students a vertical number line which some students might find easier to use than a horizontal one Have them decide whether to place points for rational and irrational numbers above or below existing points
Additional ResourcesDifferentiated Instruction includes bull Reading Strategies bull Success for English Learners EL
bull Reteach bull Challenge PRE-AP
DIFFERENTIATE INSTRUCTION
Ordering Real Numbers 24
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
Math TrainerOnline Assessment
and Intervention
Personal
Online homework assignment available
myhrwcom
myhrwcom
Lesson Quiz available online
13 LESSON QUIZ
1 Compare Write lt gt or =
radic_
95 - 5 radic_
62 - 2
2 Order 105 radic_
105 and 3π + 1 from greatest to least
3 A length in centimeters is calculated differently by four different people Order their calculations from least to greatest
KD 11 __ 2 cm Silvio 5 __ 3 π cm
Paula 5 _
4 cm Luis radic_
33 cm
EvaluateGUIDED AND INDEPENDENT PRACTICE
Concepts amp Skills Practice
Example 1Comparing Irrational Numbers
Exercises 1ndash8
Example 2Ordering Real Numbers
Exercises 9 12ndash15 18ndash21
Example 3Ordering Real Numbers in a Real-World Context
Exercises 10 16ndash17
Additional ResourcesDifferentiated Instruction includes bull Leveled Practice worksheets
Focus | Coherence | Rigor
Exercise Depth of Knowledge (DOK) Mathematical Practices
12ndash15 1 Recall of Information MP5 Using Tools
16 2 SkillsConcepts MP2 Reasoning
17 2 SkillsConcepts MP6 Precision
18ndash21 2 SkillsConcepts MP2 Reasoning
22 3 Strategic Thinking MP4 Modeling
23ndash24 3 Strategic Thinking MP3 Logic
8NS2
8NS2
Answers1 radic
_ 95 - 5 lt radic
_ 62 - 2
2 radic_
105 3π + 1 105
3 Silvio 5 __ 3 π cm Paula 5 _
4 cm
KD 11
__ 2 cm Luis radic_
33 cm
25 Lesson 13
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
Work Area
3140 3141 3142 3143
314 π227
20 A teacher asks his students to write the numbers shown in order from least to greatest Paul thinks the numbers are already in order Sandra thinks the order should be reversed Who is right
21 Math History There is a famous irrational number called Eulerrsquos number symbolized with an e Like π its decimal form never ends or repeats The first few digits of e are 27182818284
a Between which two square roots of integers could you find this number
b Between which two square roots of integers can you find π
22 Analyze Relationships There are several approximations used for π including 314 and 22 __ 7 π is approximately 314159265358979
a Label π and the two approximations on the number line
b Which of the two approximations is a better estimate for π Explain
c Find a whole number x so that the ratio x ___ 113 is a better estimate for π
than the two given approximations
23 Communicate Mathematical Ideas If a set of six numbers that include both rational and irrational numbers is graphed on a number line what is the fewest number of distinct points that need to be graphed Explain
24 Critique Reasoning Jill says that 12 _
6 is less than 1263 Explain her error
FOCUS ON HIGHER ORDER THINKING
radic_
115 115 ___ 11 and 105624
between radic_
7 asymp 265 and radic_
8 asymp 283
between radic_
9 = 3 and radic_
10 asymp 316
22 __ 7 it is closer to π on the number line
She did not consider the repeating digit 1266
2 rational numbers can have the same location and
irrational numbers can have the same location but they
cannot share a location
355
Neither student is correct The answer
should be 115 ___ 11 105624 radic_
115
Unit 126
copy H
ough
ton M
ifflin
Har
cour
t Pub
lishin
g Com
pany
Imag
e Cre
dits
copy3D
Stoc
kiSt
ockP
hoto
com
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
8_MCAAESE206984_U1M01L3indd 26 210513 801 AM
Personal Math Trainer
Online Practice and Helpmyhrwcom
Name Class Date
Independent Practice
16 Your sister is considering two different shapes for her garden One is a square with side lengths of 35 meters and the other is a circle with a diameter of 4 meters
a Find the area of the square
b Find the area of the circle
c Compare your answers from parts a and b Which garden would give your sister the most space to plant
17 Winnie measured the length of her fatherrsquos ranch four times and got four different distances Her measurements are shown in the table
a To estimate the actual length Winnie first approximated each distance to the nearest hundredth Then she averaged the four numbers Using a calculator find Winniersquos estimate
b Winniersquos father estimated the distance across his ranch to be radic_
56 km How does this distance compare to Winniersquos estimate
Give an example of each type of number
18 a real number between radic_
13 and radic_
14
19 an irrational number between 5 and 7
Order the numbers from least to greatest
12 radic_
7 2 radic_
8 ___ 2 13 radic_
10 π 35
14 radic_
220 -10 radic_
100 115 15 radic_
8 -375 3 9 _ 4
Distance Across Fatherrsquos Ranch (km)
1 2 3 4
radic_
60 58 __ 8 7 _
3 7 3 _ 5
138NS2
radic_
8 ___ 2 2 radic_
7
-10 radic_
100 115 radic_
220
radic_
60 asymp 775 58 __ 8 = 725 7 _
3 asymp 733 7 3 _ 5 = 760 so the average
π radic_
10 35
-375 9 _ 4 radic_
8 3
is 74825 km
1225 m2
4π m2 or approximately 126 m2
They are nearly identical radic_
56 is approximately 74833hellip
The circle would give her more space to plant because it has a
larger area
Sample answer 37
Sample answer radic_
31
25Lesson 13
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ough
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Miff
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pany
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
8_MCAAESE206984_U1M01L3indd 25 41613 448 AM
Activity available online myhrwcomEXTEND THE MATH PRE-AP
Activity Have students investigate whether there are infinitely many numbers between two numbers by giving examples for each of the following
bull Between any two rational numbers there is at least one other rational number Sample answer 45 is between 41 and 48
bull Between any two irrational numbers there is at least one rational number Sample answer 45 is between radic
_ 11 and radic
_ 29
bull Between any two rational numbers there is at least one irrational number Sample answer radic
_ 11 is between 31 and 36
bull Between any two irrational numbers there is at least one irrational number Sample answer radic
_ 17 is between radic
_ 11 and radic
_ 29
Ordering Real Numbers 26
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
ReadyMath Trainer
Online Practiceand Help
Personal
myhrwcom
Module Quiz
11ensp RationalenspandenspIrrationalenspNumbersWrite each fraction as a decimal or each decimal as a fraction
1 7__20 2 1___
27 3 17_8
Solve each equation for x
4 x2=81 5 x3=343 6 x2= 1___100
7 Asquarepatiohasanareaof200squarefeetHowlongiseachside
ofthepatiotothenearesttenth
12ensp SetsenspofenspRealenspNumbersWrite all names that apply to each number
8 121____radic
____121
9 π__2
10 TellwhetherthestatementldquoAllintegersarerationalnumbersrdquoistrueorfalseExplainyourchoice
13ensp OrderingenspRealenspNumbersCompare Write lt gt or =
11 radic__
8+3 8+radic__
3 12 radic__
5+11emsp emsp emsp 5+radic___
11
Order the numbers from least to greatest
13 radic___
99π29__
8 14 radic___
1__251_40__
2
15 Howarerealnumbersusedtodescribereal-worldsituations
ESSENTIAL QUESTION
035
9-9
141ft
7 1__10- 1__10
14__11 1875
wholeintegerrationalreal
Trueintegerscanbewrittenasthequotientoftwointegers
SampleanswerRealnumberssuchastherational
π29__
8radic___
99
irrationalreal
lt gt
number1_4candescribeamountsusedincooking
radic___
1__250__
21_4
27Module1
copy H
ough
ton
Miff
lin H
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ublis
hing
Com
pany
DONOTEDIT--ChangesmustbemadethroughldquoFileinfordquoCorrectionKey=A
8_MCAAESE206984_U1M01RTindd 27 41513 1113 PM
Math TrainerOnline Assessment
and Intervention
Personal
myhrwcom
1
2
3 Response toIntervention
Intervention Enrichment
Access Ready to Go On assessment online and receive instant scoring feedback and customized intervention or enrichment
Online and Print Resources
Differentiated Instruction
bull Reteach worksheets
bull Reading Strategies EL
bull Success for English Learners EL
Differentiated Instruction
bull Challenge worksheets PRE-AP
Extend the Math PRE-AP
Lesson Activities in TE
Additional ResourcesAssessment Resources includes bull Leveled Module Quizzes
Ready to Go OnAssess MasteryUse the assessment on this page to determine if students have mastered the concepts and standards covered in this module
California Common Core Standards
Lesson Exercises Common Core Standards
11 1ndash7 8NS1 8NS2 8EE2
12 8ndash10 8NS1
13 11ndash14 8NS2
27 Unit 1 Module 1
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
Personal Math Trainer
Online Practice and HelpmyhrwcomAssessment Readiness
Module 1 MIXed ReVIeW
1 Look at each number Is the number between 2π and radic___
52
Select Yes or No for expressions AndashC
A 6 2 _ 3 Yes No
B 5π __ 2 Yes No
C 3 radic__
5 Yes No
2 Consider the number - 11 __ 15
Choose True or False for each statement
A The number is rational True False
B The number can be written as True Falsea repeating decimal
C The number is less than ndash08 True False
3 The volume of a cube is given by V = x3 where x is the length of an edge of the cube A cube-shaped end table has a volume of 3 3 _ 8 cubic feet What is the length of an edge of the end table Explain how you solved this problem
4 A student says that radic___
83 is greater than 29 __ 3 Is the student correct Justify your
reasoning
1 1 _ 2 ft Sample answer The equation x3 = 3 3 _ 8 can be used
to find the edge length in feet To solve the equation
write the mixed number as a fraction greater than 1
x3 = 27 __ 8 Then take the cube root of both sides x = 3 _ 2 = 1 1 _ 2
No Sample answer radic___
83 asymp 91 and 29 __ 3 = 9
__ 6
Because 91 lt 9 __
6 radic___
83 lt 29 __ 3
28 Unit 1
copy H
ough
ton
Miff
lin H
arco
urt P
ublis
hing
Com
pany
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=A
8_MCAAESE206984_U1M01RTindd 28 240413 946 AM
Personal Math Trainer
Online Assessment and
Interventionmyhrwcom
Scoring GuideItem 3 Award the student 1 point for finding the edge length of the cube and 1 point for correctly explaining how to use a cube root to solve the problem
Item 4 Award the student 1 point for determining that the student is incorrect and 1 point for correctly justifying the reasoning for this conclusion
Additional ResourcesTo assign this assessment online login to your Assignment Manager at myhrwcom
Assessment Readiness
California Common Core Standards
Items Grade 8 Standards Mathematical Practices
1 8NS2 MP7
2 7NS2b 7NS2d 8NS1 MP7
3 8EE2 MP1 MP4
4 8NS1 8NS2 MP3
Item integrates mixed review concepts from previous modules or a previous course
Item 4 combines concepts from the California Common Core cluster ldquoKnow that there are numbers that are not rational and approximate them by rational numbersrdquo
Real Numbers 28
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
Guided Practice
7 0675 8 56 9 044
10 0 _
4
10x =
x =
11 0 _
26
100x =
x =
12 0 _
325
1000x =
x =
Solve each equation for x (Example 3 and Explore Activity)
- x
-
_______________
x =
- x
-
___________________
x =
- x
-
_______________________
x =
Write each fraction or mixed number as a decimal (Example 1)
1 2 _ 5 2 8 _ 9 3 3 3 _ 4
4 7 __ 10 5 2 3 _ 8 6 5 _ 6
Write each decimal as a fraction or mixed number in simplest form (Example 2)
13 x 2 = 17 14 x 2 = 25 ___ 289 15 x 3 = 216
Approximate each irrational number to one decimal place without a calculator
x = plusmn radic__
asymp plusmn x = 3
radic__
=
(Explore Activity)
16 radic_
5 asymp
17 radic_
3 asymp
18 radic_
10 asymp
19 What is the difference between rational and irrational numbers
CHECK-INESSENTIAL QUESTION
x = plusmn radic__
__________ = plusmn _____
4 _
4
0 _
4
4 99
6216
269
41 25 5
17289
17
22 17 32
04
07
27__40
26 __ 99 325 ___ 999 4 _ 9
11__255 3_5
0 _
8
2375
375
08 _
3
26 _
26
0 _
26
325 _
325
0 _
325
999 325
Rational numbers can be written in the form a __ b where
a and b are integers and b ne 0 Irrational numbers cannot
be written in this form
Unit 112
copy H
ough
ton
Miff
lin H
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urt P
ublis
hing
Com
pany
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
8_MCAAESE206984_U1M01L1indd 12 41613 1211 AM
11 12 13 14 15
radic2 asymp 14
141 142 143 144 145
radic2 asymp 141
0 1 2 3 4
radic2 asymp 15
Estimate that radic_
2 asymp 15
To find a better estimate first choose some numbers between 1 and 2 and square them For example choose 13 14 and 15
1 3 2 = 1 4 2 = 1 5 2 =
Is radic_
2 between 13 and 14 How do you know
Is radic_
2 between 14 and 15 How do you know
2 is closer to than to so radic_
2 asymp
Locate and label this value on the number line
Reflect 11 How could you find an even better estimate of radic
_ 2
12 Find a better estimate of radic_
2
1 41 2 = 1 42 2 = 1 43 2 =
2 is closer to than to so radic_
2 asymp
Draw a number line and locate and label your estimate
13 Solve x 2 = 7 Write your answer as a radical expression Then estimate to one decimal place
D
E
F
No 2 is not between 169 and 196
Yes 2 is between 196 and 225
196
19881
19881
225
20164
20164
14
141
20449
169 196 225
Test the squares of numbers between 14 and 15
x = plusmn radic_
7 x asymp plusmn26
11Lesson 11
copy H
ough
ton
Miff
lin H
arco
urt P
ublis
hing
Com
pany
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
8_MCAAESE206984_U1M01L1indd 11 41613 1211 AM
Rational and Irrational Numbers 12
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
Math TrainerOnline Assessment
and Intervention
Personal
Online homework assignment available
myhrwcom
EvaluateGUIDED AND INDEPENDENT PRACTICE
Concepts amp Skills Practice
Example 1Expressing Rational Numbers as Decimals
Exercises 1ndash6 20ndash21 24ndash25
Example 2Expressing Decimals as Rational Numbers
Exercises 7ndash12 22ndash23 26ndash27
Example 3Finding Square Roots and Cube Roots
Exercises 13ndash15 28 30ndash31 35
Explore ActivityEstimating Irrational Numbers
Exercises 13 16ndash18 29 32ndash34
Additional ResourcesDifferentiated Instruction includes bull Leveled Practice worksheets
Lesson Quiz available online
11 LESSON QUIZ
1 Write as a decimal 2 5 __ 8 1 7 __ 12
2 Write as a fraction 034 1 _
24
3 Solve x 2 = 9 __ 49 for x
4 Solve x 3 = 216 for x
5 Estimate the value of radic_
13 to one decimal place without using a calculator
myhrwcom
Focus | Coherence | Rigor
Exercise Depth of Knowledge (DOK) Mathematical Practices
20ndash27 2 SkillsConcepts MP4 Modeling
28 3 Strategic Thinking MP4 Modeling
29ndash32 2 SkillsConcepts MP6 Precision
33 3 Strategic Thinking MP7 Using Structure
34 2 SkillsConcepts MP3 Logic
35 2 SkillsConcepts MP4 Modeling
36 3 Strategic Thinking MP3 Logic
37 3 Strategic Thinking MP7 Using Structure
38 3 Strategic Thinking MP2 Reasoning
8NS1 8NS2 8EE2
8NS1 8NS2 8EE2
Answers1 2625 158
_ 3
2 17 __ 50 1 8 __ 33
3 x = plusmn 3 __ 7
4 x = 6
5 36
13 Lesson 11
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
Work Area
33 Analyze Relationships To find radic_
15 Beau found 3 2 = 9 and 4 2 = 16 He said that since 15 is between 9 and 16 radic
_ 15 must be between 3 and 4 He
thinks a good estimate for radic_
15 is 3 + 4 ____ 2 = 35 Is Beaursquos estimate high low
or correct Explain
34 Justify Reasoning What is a good estimate for the solution to the equation x 3 = 95 How did you come up with your estimate
35 The volume of a sphere is 36π f t 3 What is the radius of the sphere Use the formula V = 4 _ 3 π r 3 to find your answer
36 Draw Conclusions Can you find the cube root of a negative number If so is it positive or negative Explain your reasoning
37 Make a Conjecture Evaluate and compare the following expressions
radic_
4 __ 25 and radic
_ 4 ____
radic_
25 radic
_
16 __ 81 and radic_
16 ____
radic_
81 radic
_
36 __ 49 and radic_
36 ____
radic_
49
Use your results to make a conjecture about a division rule for square roots Since division is multiplication by the reciprocal make a conjecture about a multiplication rule for square roots
38 Persevere in Problem Solving The difference between the solutions to the equation x 2 = a is 30 What is a Show that your answer is correct
FOCUS ON HIGHER ORDER THINKING
His estimate is low because 15 is very close to 16
so radic_
15 is very close to radic_
16 or 4 A better estimate
would be 38 or 39
Sample answer about 45 4 3 = 64 and 5 3 = 125
Because 95 is about halfway between 64 and 125 try 45
45 3 = 91125 which is a good estimate
3 feet
Yes the cube root of a negative number is negative
because a negative number cubed is always negative
and a nonnegative number cubed is always nonnegative
radic_
4 __ 25 = 2 _ 5 = radic
_ 4 ____
radic_
25 radic
_
16 __ 81 = 4 _ 9 = radic_
16 ____
radic_
81 radic
_
36 __ 49 = 6 _ 7 = radic_
36 ____
radic_
49
225 the solutions to x 2 = a are x = plusmn15 and
radic_
a ___
radic_
b = radic
_ a __
b radic
_ a radic
_ b = radic
_ a b
15 - (-15) = 30
Unit 114
copy H
ough
ton
Miff
lin H
arco
urt P
ublis
hing
Com
pany
bull copy
Ilen
e Mac
Dona
ldA
lamy I
mag
es
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=B
8_MCABESE206984_U1M01L1indd 14 102913 1142 PM
Personal Math Trainer
Online Practice and Helpmyhrwcom
Name Class Date
Independent Practice11
20 A 7 __ 16 -inch-long bolt is used in a machine What is this length written as a decimal
21 The weight of an object on the moon is 1 _ 6 its weight on Earth Write 1 _ 6 as a decimal
22 The distance to the nearest gas station is 2 4 _ 5 kilometers What is this distance written as a decimal
23 A baseball pitcher has pitched 98 2 _ 3 innings What is the number of innings written as a decimal
24 A heartbeat takes 08 second How many seconds is this written as a fraction
25 There are 262 miles in a marathon Write the number of miles using a fraction
26 The average score on a biology test was 72
_ 1 Write the average score using a
fraction
27 The metal in a penny is worth about 0505 cent How many cents is this written as a fraction
28 Multistep An artist wants to frame a square painting with an area of 400 square inches She wants to know the length of the wood trim that is needed to go around the painting
a If x is the length of one side of the painting what equation can you set up to find the length of a side How many solutions does the equation have
b Do all of the solutions that you found make sense in the context of the problem Explain
c What is the length of the wood trim needed to go around the painting
Solve each equation for x Write your answers as radical expressions Then estimate to one decimal place if necessary
29 x 2 = 14 30 x 3 = 1331
31 x 2 = 144 32 x 2 = 29
8NS1 8NS2 8EE2
04375 in 01 _6
28 km 98 _6 innings
x 2 = 400 x = plusmnthinsp20 the equation has 2 solutions
x = 20 makes sense but x = -20 doesnrsquot because a
painting cannot have a side length of -20 inches
4 times 20 = 80 inches
x = plusmn radic_
14 asymp plusmn37
x = plusmn radic_
144 = plusmn12 x = plusmn radic_
29 asymp plusmn54
x = 3 radic_ 1331 = 11
4_5 second 26 1_5 mi
72 1 _ 9 101 ___ 200 cent
13Lesson 11
copy H
ough
ton
Miff
lin H
arco
urt P
ublis
hing
Com
pany
bull copy
Phot
odisc
Get
ty Im
ages
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
8_MCAAESE206984_U1M01L1indd 13 41613 1211 AM
myhrwcomActivity available onlineEXTEND THE MATH PRE-AP
Activity Write radic_
09 on the board and invite students to conjecture what the value might be Have them check their conjectures by squaring Invite them to suggest ways to estimate radic
_ 09 As a hint point out that 09 is close to 10 and so they might
use that to help guide their estimates Lead them to see that since 092 is 081 and 102 is 1 the value of radic
_ 09 is greater than 09 and less than 10 Try squaring 095 to get
09025 A good estimate for radic_
09 is 095
Rational and Irrational Numbers 14
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=B
Integers
Rational Numbers IrrationalNumbers
Real Numbers
WholeNumbers
-3-4-5 -2-1 1 2 3 50 4
23
34-4 -π -1 25
radic2
Lesson Support Content Objective Students will learn to describe relationships between sets of numbers
Language Objective Students will explain how to describe relationships between sets of real numbers
LESSON 12 Sets of Real Numbers
Building BackgroundEliciting Prior Knowledge Have students draw a number line from -5 to 5 Ask them to plot points on the number line to approximate the location of rational and irrational numbers such as -1 3 __ 4 25 -4 2 __ 3 radic
_ 2 and -π
Learning ProgressionsIn this lesson students clarify their understanding of the real number system They characterize sets and subsets of the real numbers They also identify sets for real-world situations Important understandings for students include the following
bull Identify all of the possible subsets of the real numbers for a given number
bull Decide whether a statement about a subset of the real numbers is true or false
bull Identify the set of numbers that best describes a real-world situation
Understanding the relationships among the sets of numbers that make up the real numbers is essential as students are introduced to different forms of numbers throughout the school year This lesson provides a foundation for the comparing and ordering of real numbers in the next lesson
Cluster ConnectionsThis lesson provides an excellent opportunity to connect ideas in this cluster Know that there are numbers that are not rational and approximate them by rational numbers Have students copy this diagram which relates the sets of real numbers
Ask students to complete the diagram by writing three examples for each set of numbers Have students share examples and explain how they knew each number they selected belonged in the appropriate set Answers may vary Check studentsrsquo work
Focus | Coherence | Rigor
California Common Core Standards
8NS1 Know that numbers that are not rational are called irrational Understand informally that every number has a decimal expansion for rational numbers show that the decimal expansion repeats eventually and convert a decimal expansion which repeats eventually into a rational number
MP7 Look for and make use of structure
15A
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
Math Talk
Language Support EL
PROFESSIONAL DEVELOPMENT
Linguistic Support EL
AcademicContent Vocabulary
Venn diagrams ndash Students need descriptive language to describe the categories that the different areas and colors of a Venn diagram represent the concept of a set and how sets are distinct or can overlap Use sentence frames such as
The big oval represents __________The darklight blue color in the middle of the
big ovals represents __________These sets overlap because __________
In this way students have the language and structure to identify the criteria that distinguish a set and to explain the abstract representation Also point out the use of the prefix sub- meaning ldquounderrdquo in the term subset
Rules and Patterns
Abbreviations ndash In this lesson the abbreviation mph is used Be sure to point out that mph stands for miles per hour and is used to give units in a rate of speed Students may also have seen mpg (miles per gallon) which gives the units in a rate of fuel efficiency
Borrowed Words ndash Terminology used in baseball such as inning and pitcher may require some explanation Spanish as well as some other languages have borrowed these terms from English so some students may be familiar with these words already Despite this whenever a word is critical to students understanding the word problem it is best to explain the meaning
Leveled Strategies for English Learners
Emerging Allow students to indicate true or false orally in Guided Practice Exercises 9 and 10
Expanding Have students use sentence frames to describe the meaning of regions and colors used in a Venn diagram Then give them similar sentence frames orally and have them draw and shade a Venn diagram based on the oral prompts
Bridging Have students work in groups to draw a Venn diagram to represent sets based on real-world examples in the lesson
To help students answer the question posed in Math Talk provide a sentence frame for their answer
The numbers between 31 and 39 on a number line are __________ because __________
EL
California ELD Standards
Emerging 2II5 Modifying to add details ndash Expand sentences with simple adverbials to provide details about a familiar activity or process
Expanding 2II5 Modifying to add details ndash Expand sentences with adverbials to provide details about a familiar or new activity or process
Bridging 2II5 Modifying to add details ndash Expand sentences with increasingly complex adverbials to provide details about a variety of familiar and new activities and processes
Sets of Real Numbers 15B
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
12L E S S O N
Sets of Real Numbers
EngageESSENTIAL QUESTION
How can you describe relationships between sets of real numbers Sample answer Describe them as two different sets or one set as being a subset of another
Motivate the LessonAsk How many different types of tigers can you name How does the set of Bengal tigers relate to the set of tigers
ExplorePoint to different locations in the Animals diagram and ask for examples for that classification Do the same for the Real Numbers diagram Students should understand that everything within a region is part of the set for example both -3 and 2 are integers
ExplainEXAMPLE 1
Questioning Strategies Mathematical Practices bull In A why is 5 not a perfect square It does not have rational numbers as its square roots
bull Can the number in B be written as a fraction Why or why not Yes it is a terminating decimal so it is a rational number
Engage with the WhiteboardHave students place the numbers in Example 1 and Additional Example 1 in the Venn diagram for numbers
YOUR TURNAvoid Common ErrorsBe sure that students read Exercise 2 carefully before answering The number given in the problem 10 is the area not the side length
EXAMPLE 2Questioning Strategies Mathematical Practices bull What two major sets are the real numbers composed of rational and irrational numbers
bull What is the location of the set of whole numbers in the Venn diagram in relation to the set of rational numbers Explain Inside it whole numbers are rational numbers
Focus on Reasoning Mathematical PracticesRemind students that it takes only one counterexample to show that a statement is false
Interactive Whiteboard Interactive example available online
ADDITIONAL EXAMPLE 1Write all names that apply to each number
A -10integer rational real
B 12 _ 3
whole integer rational real
myhrwcom
Interactive Whiteboard Interactive example available online
ADDITIONAL EXAMPLE 2Tell whether the given statement is true or false Explain your choice
No integers are whole numbers
False every whole number is also an integer
myhrwcom
Animated MathClassifying Numbers
Students build fluency in classifying numbers in this engaging fast-paced game
myhrwcom
CA Common CoreStandards
The student is expected to
The Number Systemmdash8NS1
Know that numbers that are not rational are called irrational Understand informally that every number has a decimal expansion for rational numbers show that the decimal expansion repeats eventually and convert a decimal expansion which repeats eventually into a rational numberMathematical Practices
MP7 Using Structure
The student is expected to
15 Lesson 12
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=B
Math TrainerOnline Practice
and Help
Personal
myhrwcom
Math TrainerOnline Practice
and Help
Personal
myhrwcom
Math On the Spotmyhrwcom
Understanding Sets and Subsets of Real NumbersBy understanding which sets are subsets of types of numbers you can verify whether statements about the relationships between sets are true or false
Tell whether the given statement is true or false Explain your choice
All irrational numbers are real numbers
True Every irrational number is included in the set of real numbers The irrational numbers are a subset of the real numbers
No rational numbers are whole numbers
False A whole number can be written as a fraction with a denominator of 1 so every whole number is included in the set of rational numbers The whole numbers are a subset of the rational numbers
EXAMPLE 2
A
B
Write all names that apply to each number
1 A baseball pitcher has pitched 12 2 _ 3 innings
2 The length of the side of a square that has an
area of 10 square yards
YOUR TURN
Tell whether the given statement is true or false Explain your choice
3 All rational numbers are integers
4 Some irrational numbers are integers
YOUR TURN
Give an example of a rational number that is a
whole number Show that the number is both whole
and rational
Math TalkMathematical Practices
Give an example of a
8NS1
False Every integer is a rational number but not every
False Real numbers are either rational or irrational numbers
Integers are rational numbers so no integers are irrational numbers
rational real
irrational real
Sample answer 8 8 = 8_
1
and -thinsp 5 _ 2 are not integers
rational number is an integer Rational numbers such as 3 _ 5
Unit 116
copy H
ough
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Miff
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Com
pany
bull Im
age C
redi
ts D
igita
l Im
age c
opyr
ight
copy20
04 Ey
ewire
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
8_MCAAESE206984_U1M01L2indd 16 41613 136 AM
Math On the Spot
myhrwcom
Vertebrates
Birds
Passerines
Animals
Integers
Rational Numbers IrrationalNumbers
Real Numbers
WholeNumbers
1
45
3
0
274
67
radic4
-
-3
-2
-1
03
radic2
radic17
radic11-
π
Animated Math
myhrwcom
Classifying Real NumbersBiologists classify animals based on shared characteristics A cardinal is an animal a vertebrate a bird and a passerine
You already know that the set of rational numbers consists of whole numbers integers and fractions The set of real numbers consists of the set of rational numbers and the set of irrational numbers
Write all names that apply to each number
radic_
5 irrational real
ndash1784rational real
whole integer rational real
EXAMPLEXAMPLE 1
A
B
C radic_ 81 ____ 9
L E S S O N
12Sets of Real Numbers
ESSENTIAL QUESTIONHow can you describe relationships between sets of real numbers
Passerines such as the cardinal are also called ldquoperching birdsrdquo
What types of numbers are between 31 and 39 on a
number line
Math TalkMathematical Practices
What types of numbers are
8NS1
8NS1
Know that numbers that are not rational are called irrational Understand informally that every number has a decimal expansion for rational numbers show that the decimal expansion repeats eventually and convert a decimal expansion which repeats eventually into a relation number
ndash1784 is a terminating decimal
5 is a whole number that is not a perfect square
radic_
81 _____ 9 = 9 __ 9 = 1 rational irrational real
15Lesson 12
copy H
ough
ton
Miff
lin H
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ublis
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Com
pany
bull Im
age C
redi
ts copy
Wiki
med
ia Co
mm
ons
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=B
8_MCABESE206984_U1M01L2indd 15 061113 1144 AM
PROFESSIONAL DEVELOPMENT
Math BackgroundThe relationships between sets of numbers extend to include complex numbers A complex number can be written as a sum of a real number a and an imaginary number bi
a + bi
An imaginary number is a special number that when squared gives a negative value When you square a real number you get a nonnegative number When you square an imaginary number you get a negative value The imaginary unit is i
i = radic_
-1
Integrate Mathematical Practices MP7
This lesson provides an opportunity to address this Mathematical Practices standard It calls for students to discern structure to connect and communicate mathematical ideas
Students use a Venn diagram to structure relationships between sets of numbers They connect and communicate mathematical ideas when they make logical statements about the sets and describe which set best describes numbers applied to real-life situations
Sets of Real Numbers 16
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=B
YOUR TURNAvoid Common ErrorsStudents may see the word ldquoAllldquo or rdquoNordquo in Exercises 3 and 4 and immediately assume that any absolute statements like these are false Remind them that there are true statements that begin with these words and encourage them to provide examples
EXAMPLE 3Questioning Strategies Mathematical Practices bull In A how does the phrase ldquonumber of rdquo give you a clue about the number classification It indicates a counting number
bull What is the relationship between the circumference of a circle and the diameter The circumference is diameter times π
Focus on Critical Thinking Mathematical PracticesIn B suppose the diameters in inches were 25
__ π 28 __ π
31 __ π and so on What set of numbers would
best describe the circumferences Explain Whole numbers the circumferences would be the whole numbers 25 28 31 and so on
YOUR TURNFocus on Critical Thinking Mathematical PracticesHave students compare and contrast the classification of numbers in the answers in Exercises 5 and 6
ElaborateTalk About ItSummarize the Lesson
Ask What are some ways that number sets can be related Sets may be subsets of other sets or they may be separate from other sets
GUIDED PRACTICEEngage with the Whiteboard
Have students place the numbers in Exercises 1ndashthinsp8 in the Venn diagram for numbers at the beginning of the lesson
Integrating Language Arts EL
Encourage English learners to ask for clarification on any terms or phrases that they do not understand
Avoid Common ErrorsExercise 7 Remind students that a repeating decimal is a rational numberExercises 9ndash10 Remind students that it only takes one counterexample to show that a statement is false
Interactive Whiteboard Interactive example available online
ADDITIONAL EXAMPLE 3Identify the set of numbers that best describes the situation Explain your choice
A the amount of time that has passed since midnight
The set of real numbers time is continuous so the amount of time can be rational or irrational
B the number of tickets sold to a basketball game
The set of whole numbers the number of tickets sold may be 0 or a counting number
myhrwcom
17 Lesson 12
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
1IN
116 inch
Guided Practice
Write all names that apply to each number (Example 1)
1 7 _ 8 2 radic_
36
3 radic_
24 4 075
5 0 6 - radic_ 100
7 5 _
45 8 - 18 __ 6
Tell whether the given statement is true or false Explain your choice (Example 2)
9 All whole numbers are rational numbers
10 No irrational numbers are whole numbers
Identify the set of numbers that best describes each situation Explain your choice (Example 3)
11 the change in the value of an account when given to the nearest dollar
12 the markings on a standard ruler
13 What are some ways to describe the relationships between sets of numbers
CHECK-INESSENTIAL QUESTION
rational real
rational real
True Whole numbers are rational numbers
Rational numbers the ruler is marked every 1 __ 16 th inch
Sample answer Describe one set as being a subset of
another or show their relationships in a Venn diagram
Integers the change can be a whole dollar amount
and can be positive negative or zero
True Whole numbers are a subset of the set of rational numbers
and can be written as a ratio of the whole number to 1
irrational real
whole integer rational real
whole integer rational real
rational real
integer rational real
integer rational real
Unit 118
copy H
ough
ton
Miff
lin H
arco
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ublis
hing
Com
pany
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
8_MCAAESE206984_U1M01L2indd 18 41613 136 AM
My Notes
Math TrainerOnline Practice
and Help
Personal
myhrwcom
Math On the Spot
myhrwcom
Identifying Sets for Real-World SituationsReal numbers can be used to represent real-world quantities Highways have posted speed limit signs that are represented by natural numbers such as 55 mph Integers appear on thermometers Rational numbers are used in many daily activities including cooking For example ingredients in a recipe are often given in fractional amounts such as 2 _ 3 cup flour
Identify the set of numbers that best describes each situation Explain your choice
the number of people wearing glasses in a room
The set of whole numbers best describes the situation The number of people wearing glasses may be 0 or a counting number
the circumference of a flying disk has a diameter of 8 9 10 11 or 14 inches
The set of irrational numbers best describes the situation Each circumference would be a product of π and the diameter and any multiple of π is irrational
EXAMPLEXAMPLE 3
A
B
Identify the set of numbers that best describes the situation Explain your choice
5 the amount of water in a glass as it evaporates
6 the weight of a person in pounds
YOUR TURN
8NS1
Rational numbers a personrsquos weight can be a decimal
such as 835 pounds
Real numbers the amount can be any number greater
than 0
17Lesson 12
copy H
ough
ton
Miff
lin H
arco
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ublis
hing
Com
pany
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
8_MCAAESE206984_U1M01L2indd 17 41613 520 AM
Graphic OrganizersGive students a list of numbers (including terminating and repeating decimals fractions integers and rational and irrational square roots) and a graphic organizer as shown below
Real Numbers
Rational numbers Irrational numbers
Integer numbers
Whole numbers
Ask students to write each number in the list in the correct section of the organizer
Number SensePoint out to students that knowing the types of numbers to expect in different situations can alert them to incorrect math as well as to impossible situations For example 135 shots made in basketballs is not possible but an average number of shots can equal 135
Additional ResourcesDifferentiated Instruction includes bull Reading Strategies bull Success for English Learners EL
bull Reteach bull Challenge PRE-AP
DIFFERENTIATE INSTRUCTION
Sets of Real Numbers 18
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
Math TrainerOnline Assessment
and Intervention
Personal
Online homework assignment available
myhrwcom
Lesson Quiz available online
12 LESSON QUIZ
1 Write all the names that apply to the number
2 Tell whether the given statement is true or false Explain your choice All numbers between 1 and 2 are rational numbers
3 Identify the set of numbers that best describes the situation Explain your choiceThe choices on a survey question change the total points for the survey by -2 -1 0 1 or 2 points
-1 _
5
myhrwcom
Additional ResourcesDifferentiated Instruction includes bull Leveled Practice worksheets
EvaluateGUIDED AND INDEPENDENT PRACTICE
Concepts amp Skills Practice
Example 1Classifying Real Numbers
Exercises 1ndash8 14ndash19 22ndash24
Example 2Understanding Sets and Subsets of Real Numbers
Exercises 9ndash10
Example 3Identifying Sets for Real-World Situations
Exercises 11ndash12 20ndash21 25
Focus | Coherence | Rigor
Exercise Depth of Knowledge (DOK) Mathematical Practices
14ndash19 2 SkillsConcepts MP7 Using Structure
20ndash21 2 SkillsConcepts MP6 Precision
22ndash23 2 SkillsConcepts MP3 Logic
24 1 Recall of Information MP7 Using Structure
25 2 SkillsConcepts MP2 Reasoning
26ndash27 3 Strategic Thinking MP3 Logic
28 3 Strategic Thinking MP8 Patterns
29 3 Strategic Thinking MP3 Logic
8NS1
8NS1
Exercise 29 combines concepts from the California Common Core cluster ldquoKnow that there are numbers that are not rational and approximate them by rational numbersrdquo
Answers1 rational real
2 False radic_
2 is an example of an irrational number between 1 and 2
3 Integers each number is an integer but only three are whole numbers
19 Lesson 12
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
Work Area
π mi23 Critique Reasoning The circumference of a circular region is shown
What type of number best describes the diameter of the circle Explain
your answer
24 Critical Thinking A number is not an integer What type of number can it be
25 A grocery store has a shelf with half-gallon containers of milk What type of number best represents the total number of gallons
26 Explain the Error Katie said ldquoNegative numbers are integersrdquo What was her error
27 Justify Reasoning Can you ever use a calculator to determine if a number is rational or irrational Explain
28 Draw Conclusions The decimal 0 _
3 represents 1 _ 3 What type of number best describes 0
_ 9 which is 3 middot 0
_ 3 Explain
29 Communicate Mathematical Ideas Irrational numbers can never be precisely represented in decimal form Why is this
FOCUS ON HIGHER ORDER THINKING
It can be a rational number that is not an integer or an irrational number
rational number
The set of negative numbers also includes non-integer
rational numbers and irrational numbers
Sample answer If the calculator shows a decimal that
terminates in fewer digits than what the calculator screen
allows then you can tell that the number is rational If not
you cannot tell from the calculator display whether the
number terminates because you see a limited number
of digits It may be a repeating decimal (rational) or
non-terminating non-repeating decimal (irrational)
Whole 3 middot 0 _
3 represents 3 middot 1 _ 3 = 1 so 0 _
9 is exactly 1
Sample answer In decimal form irrational numbers never
terminate and never repeat Therefore no matter how
many decimal places you include the number will never
be precisely represented There are always more digits
Whole the diameter is π _ π = 1 mile
Unit 120
copy H
ough
ton
Miff
lin H
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ublis
hing
Com
pany
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
8_MCAAESE206984_U1M01L2indd 20 120413 909 PM
Integers
Rational Numbers Irrational Numbers
Real Numbers
Whole Numbers
257
radic16
166
radic9
128 radic50
Personal Math Trainer
Online Practice and Helpmyhrwcom
Name Class Date
Independent Practice
Identify the set of numbers that best describes each situation Explain your choice
20 the height of an airplane as it descends to an airport runway
21 the score with respect to par of several golfers 2 ndash 3 5 0 ndash 1
22 Critique Reasoning Ronald states that the number 1 __ 11 is not rational because when converted into a decimal it does not terminate Nathaniel says it is rational because it is a fraction Which boy is correct Explain
12
14 - radic_
9 15 257
16 radic_
50 17 8 1 _ 2
18 166 19 radic_
16
Write all names that apply to each number Then place the numbers in the correct location on the Venn diagram
8NS1
Real numbers the height can be any number greater than zero
integer rational real whole integer rational real
whole integer rational real
irrational real
rational real
rational real
Integers the scores are counting numbers their
opposites and zero
Nathaniel is correct A rational number is a number that can be written as a fraction and 1 __ 11 is a fraction
19Lesson 12
copy H
ough
ton
Miff
lin H
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urt P
ublis
hing
Com
pany
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
8_MCAAESE206984_U1M01L2indd 19 41613 136 AM
myhrwcomActivity available onlineEXTEND THE MATH PRE-AP
Activity Have students consider the concept of restricted domain for the sets of numbers that describe situations For example the number of sisters a person has can best be described by whole numbers but no one has ever had 1500 sisters An area code is an integer or whole number between 200 and 999
Have students use a source such as the Guinness Book of World Records and give examples of sets of numbers that describe situations where the domain is restricted Ask whether the restriction may be changed in the future
Sets of Real Numbers 20
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
-3-4-5 -2-1 1 2 3 50 4
12-4 -radic5
Lesson Support Content Objective Students will learn to order a set of real numbers
Language Objective Students will show and describe how to order a set of real numbers
LESSON 13 Ordering Real Numbers
Building BackgroundEliciting Prior Knowledge Have students draw a number line to compare a rational number and an irrational number such as - radic
_ 5 and -4 1 __ 2 Ask them to explain how
they approximated the irrational number on the number line Then have them identify the greater and the lesser real number Repeat with several other pairs of real numbers in different forms
Learning ProgressionsIn this lesson students order a set of real numbers They use rational approximations to compare the sizes of irrational numbers They also order numbers for real-world situations Important understandings for students include the following
bull Compare irrational numbers bull Estimate the value of expressions with irrational numbers bull Order a set of real numbers bull Order real numbers in a real-world context
Work with real numbers continues throughout Grade 8 and into high school This lesson provides students with a foundation for understanding the relative sizes of numbers in different forms in the real number system
Cluster ConnectionsThis lesson provides an excellent opportunity to connect ideas in this cluster Know that there are numbers that are not rational and approximate them by rational numbers Tell students that there is a special number called the golden ratio with applications in mathematics geometry art and architecture The golden ratio is called phi and is represented by the Greek letter ϕ It includes an irrational number in its definition
Have students explain why the golden ratio is irrational Ask them to find the two whole numbers the golden ratio lies between Then challenge them to approximate the golden ratio to the nearest tenth It is irrational because it includes an irrational number in its definition It lies between 1 and 2 To the nearest tenth ϕ = 16
ϕ = 1 + radic_
5 _ 2
Focus | Coherence | Rigor
California Common Core Standards
8NS2 Use rational approximations of irrational numbers to compare the size of irrational numbers locate them approximately on a number line diagram and estimate the value of expressions (eg π 2 )
MP4 Model with mathematics
21A
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
Math Talk
Language Support EL
PROFESSIONAL DEVELOPMENT
Linguistic Support EL
AcademicContent Vocabulary
Post a chart like this to remind students of the regular comparative forms of adjectives that use the -er and -est suffixes Add to the chart for terms that appear in examples and exercises in each lesson Include any irregular verb forms
Background Knowledge
Go On ndash the title of the module review or quiz is Ready to Go On This title uses an idiomatic expression In this context to go on means ldquoto move aheadrdquo or ldquoto proceedrdquo It is different from the use of go on that means having enough facts to use meaningfully as in having enough to go on Also the intonation used in pronouncing an expression can give it different meanings For example when the speaker emphasizes the word on he or she might be expressing disbelief as in ldquoGo ON Yoursquore kidding rightrdquo Discuss with students other ways that the phrase go on may be used
Leveled Strategies for English Learners
Emerging Label points on a number line with the terms used in ordering greater greatest less lesser least Use sentence frames to insert the correct terms
Expanding Have students give two or three complete sentences to compare the placement of numbers on a number line using the correct forms of the comparative and superlative adjectives
Bridging Have students work in pairs with one student giving directions to the other in complete sentences to order numbers on a number line
To help students answer the question posed in Math Talk make sure that students have a command of the forms for making comparisons and the superlative and the concept of opposite order so that the focus is on the math concept instead of the language skills needed to describe and explain order
EL
Adjective Comparative Superlative
Far Farther Farthest
Large Larger Largest
Great Greater Greatest
Some Less Least
Some More Most
California ELD Standards
Emerging 2I8 Analyzing language choices ndash Explain how phrasing or different common words with similar meanings produce different effects on the audience
Expanding 2I8 Analyzing language choices ndash Explain how phrasing or different words with similar meanings or figurative language produce shades of meaning and different effects on the audience
Bridging 2I8 Analyzing language choices ndash Explain how phrasing or different words with similar meanings or figurative language produce shades of meaning nuances and different effects on the audience
Ordering Real Numbers 21B
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
13L E S S O N
Ordering Real Numbers
Interactive Whiteboard Interactive example available online
ADDITIONAL EXAMPLE 1Compare Write lt gt or =
A radic_
8 - 2 4 - radic_
8 lt
B radic_
20 + 1 3 + radic_
2 gt
EngageESSENTIAL QUESTION
How do you order a set of real numbers Sample answer Find their approximate decimal values and order them
Motivate the LessonAsk What kind of numbers are you comparing when you compare the price of gasoline at two different gas stations
ExploreGive students two rational numbers and ask them to name a number between them Repeat a few times and then give them two irrational numbers and ask them to name a number between them
ExplainEXAMPLE 1
Questioning Strategies Mathematical Practices bull Which is greater the difference between 5 and 3 or the difference between radic
_ 5 and radic
_ 3
The difference between 5 and 3 is 2 the difference between radic_
5 and radic_
3 is approximately 1 So the difference between 5 and 3 is greater
Avoid Common ErrorsCaution students to read the problem carefully and think about what the radical sign means so that they do not misread the problem and answer that the two sides are equal
YOUR TURNFocus on TechnologyCalculators should not be used at this point because developing number sense is the goal
EXAMPLE 2Questioning Strategies Mathematical Practices bull How do you determine whether radic
_ 22 is less than or greater than 45 The square of 45 is
2025 which is less than 22 so the square root of 22 must be greater than 45
Engage with the WhiteboardHave students graph and label various real numbers between 42 and 44 and between 47 and 5
YOUR TURNFocus on Modeling Mathematical PracticesHave students label the integers on the number line with their equivalent square root For example 1 2 and 3 on the number line would be labeled radic
_ 1 radic
_ 4 and radic
_ 9
Interactive Whiteboard Interactive example available online
ADDITIONAL EXAMPLE 2Order 3π radic
_ 10 and 325 from greatest
to least
3π 325 radic_
10
myhrwcom
myhrwcom
CA Common CoreStandards
The student is expected to
The Number Systemmdash8NS2
Use rational approximations of irrational numbers to compare the size of irrational numbers locate them approximately on a number line diagram and estimate the value of expressions (eg π 2 )
Mathematical Practices
MP4 Modeling
The student is expected to
21 Lesson 13
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
Math TrainerOnline Practice
and Help
Personal
myhrwcom
Math On the Spotmyhrwcom
0 05 1 15 2 25 3 35 4
radic5radic3
π2
8 85 9 95 10 105 11 115 12
radic75
4 42 44 46 48 5
radic224 12π + 1
Ordering Real Numbers You can compare and order real numbers and list them from least to greatest
Order radic_
22 π + 1 and 4 1 _ 2 from least to greatest
First approximate radic_
22
radic_
22 is between 4 and 5 Since you donrsquot know where it falls between 4 and 5 you need to find a better estimate for radic
_ 22 so
you can compare it to 4 1 _ 2
Since 22 is closer to 25 than 16 use squares of numbers between 45 and 5 to find a better estimate of radic
_ 22
45 2 = 2025 46 2 = 2116 47 2 = 2209 48 2 = 2304
Since 47 2 = 2209 an approximate value for radic_
22 is 47
An approximate value of π is 314 So an approximate value of π +1 is 414
Plot radic_
22 π + 1 and 4 1 _ 2 on a number line
Read the numbers from left to right to place them in order from least to greatest
From least to greatest the numbers are π + 1 4 1 _ 2 and radic_
22
EXAMPLE 2
STEP 1
STEP 2
Order the numbers from least to greatest Then graph them on the number line
YOUR TURN
5 radic_
5 25 radic_
3
6 π 2 10 radic_
75
If real numbers a b and c are in order from least to greatest what is the order
of their opposites from least to greatest
Explain
Math TalkMathematical Practices
8NS2
radic_
3 radic_
5 25
radic_
75 π2 10
Math Talk answer -c -b -a -c is farthest to the left on a number line -b is in the middle and -a is farthest to the right
Unit 122
copy H
ough
ton
Miff
lin H
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hing
Com
pany
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
8_MCAAESE206984_U1M01L3indd 22 41613 447 AM
My Notes
Math TrainerOnline Practice
and Help
Personal
myhrwcom
Math On the Spot
myhrwcom
Comparing Irrational NumbersBetween any two real numbers is another real number To compare and order real numbers you can approximate irrational numbers as decimals
Compare radic_
3 + 5 3 + radic_
5 Write lt gt or =
First approximate radic_
3
radic_
3 is between 1 and 2
Next approximate radic_
5
radic_
5 is between 2 and 3
Then use your approximations to simplify the expressions
radic_
3 + 5 is between 6 and 7
3 + radic_
5 is between 5 and 6
So radic_
3 + 5 gt 3 + radic_
5
Reflect1 If 7 + radic
_ 5 is equal to radic
_ 5 plus a number what do you know about the
number Why
2 What are the closest two integers that radic_
300 is between
EXAMPLEXAMPLE 1
STEP 1
STEP 2
Compare Write lt gt or =
YOUR TURN
3 radic_
2 + 4 2 + radic_
4 4 radic_
12 + 6 12 + radic_
6
L E S S O N
13 Ordering Real Numbers
ESSENTIAL QUESTIONHow do you order a set of real numbers
8NS2
Use rational approximations of irrational numbers to compare the size of irrational numbers locate them approximately on a number line diagram and estimate the value of expressions (eg π 2 )
8NS2
Use perfect squares to estimate square roots
1 2 = 1 2 2 = 4 3 2 = 9
The number is 7 both expressions must equal 7 + radic_
5
17 and 18
gt lt
21Lesson 13
copy H
ough
ton
Miff
lin H
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ublis
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Com
pany
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
8_MCAAESE206984_U1M01L3indd 21 41913 246 PM
PROFESSIONAL DEVELOPMENT
Math BackgroundIn this lesson students estimate irrational numbers in the form of square roots of nonper-fect squares by finding two perfect squares between which the number falls A more precise method involves repeated division For example to find radic
_ 28 find a whole number whose perfect
square is close to 28 such as 5 Divide 28 by that number 28 divide 5 = 56 Find the average of the quotient and divisor 5 + 56
_____ 2 = 53 Continue dividing 28 by each result and averaging until you get the desired accuracy
Integrate Mathematical Practices MP4
This lesson provides an opportunity to address this Mathematical Practices standard It calls for students to model relationships using multiple representations including diagrams graphs and language as appropriate Students use multiple representations when they use number lines to estimate the locations of and order rational and irrational numbers given as symbols
Ordering Real Numbers 22
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
Interactive Whiteboard Interactive example available online
ADDITIONAL EXAMPLE 3The diameter of a meteorite in millimeters is calculated by four different methods Order the results from least to greatest
Joe radic_
18 mm Lisa 13 __ 3 mm
Pablo 46 mm Julien 4π __ 3 mm
Julien 4π __ 3 mm Lisa 13 __ 3 mm
Joe radic_
18 mm Pablo 46 mm
EXAMPLE 3Questioning Strategies Mathematical Practices bull How can you verify that radic
_ 28 is between 52 and 53 5 2 2 = 2704 and 5 3 2 = 2809
bull Explain how to determine which number is greater 5 _
5 or 55 When the repeating decimal is rounded to the nearest tenth or hundredth you can see that it is greater
Connect to Daily LifeDiscuss how measuring across a canyon might involve different methods than measuring along a road Explain that measurements like these are often done using calculations that approximate the distance
YOUR TURNFocus on Critical Thinking Mathematical PracticesDiscuss with students which number is greater 3
_ 45 or 3450 3
_ 45 or 3455 and why Explain
that 3 _
45 can be written out as 34545hellipMake sure they understand that 3 _
45 is greater than 345 but less than 3455
ElaborateTalk About ItSummarize the Lesson
Ask How can you order two numbers in different forms whose decimal approxi-mations appear to be equal Approximate one or both numbers to an additional
number of decimal places
GUIDED PRACTICEEngage with the Whiteboard
Have students place and label additional points on the number line in Exercise 9 Allow the points to be in any format other than decimal
Avoid Common ErrorsExercises 3ndash4 Caution students to read the problem carefully so that they do not misread the problem as the same numbers combined by addition on each side of the circleExercise 10 Remind students that the calculations have units
myhrwcom
23 Lesson 13
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
0 05 1 15 2 25 3 35 4 45 5 55 6 65 7
2πradic3
Compare Write lt gt or = (Example 1)
1 radic_
3 + 2 radic_
3 + 3 2 radic_
8 + 17 radic_
11 + 15
3 radic_
6 + 5 6 + radic_
5 4 radic_
9 + 3 9 + radic_
3
5 radic_
17 - 3 -2 + radic_
5 6 12 - radic_
2 14 - radic_
8
7 radic_
7 + 2 radic_
10 - 1 8 radic_
17 + 3 3 + radic_
11
9 Order radic_
3 2π and 15 from least to greatest Then graph them on the number line (Example 2)
radic_
3 is between and so radic_
3 asymp
π asymp 314 so 2π asymp
From least to greatest the numbers are
10 Four people have found the perimeter of a forest using different methods Their results are given in the table Order their calculations from greatest to least (Example 3)
11 Explain how to order a set of real numbers
CHECK-INESSENTIAL QUESTION
Forest Perimeter (km)
Leon Mika Jason Ashley
radic_
17 - 2 1 +thinsp π __ 2 12 ___ 5 25
Guided Practice
17
15
1 + π _ 2 km 25 km 12 __ 5 km radic_
17 - 2 km
2π radic
_ 3
18 175
628
Sample answer Convert each number to a decimal
equivalent using estimation to find equivalents for
irrational numbers Graph each number on a number line
Read the numbers from left to right for least to greatest
Read the numbers from right to left for greatest to least
lt gt
lt lt
ltgt
gt gt
24 Unit 1
copy H
ough
ton
Miff
lin H
arco
urt P
ublis
hing
Com
pany
bull Im
age C
redi
ts copy
Elena
Eliss
eeva
Alam
y Im
ages
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
8_MCAAESE206984_U1M01L3indd 24 41613 448 AM
My Notes
5 52 54 56 58 6
radic28 5 12
23455
Math TrainerOnline Practice
and Help
Personal
myhrwcom
Math On the Spot
myhrwcom
Ordering Real Numbers in a Real-World Context Calculations and estimations in the real world may differ It can be important to know not only which are the most accurate but which give the greatest or least values depending upon the context
Four people have found the distance in kilometers across a canyon using different methods Their results are given in the table Order the distances from greatest to least
Distance Across Quarry Canyon (km)
Juana Lee Ann Ryne Jackson
radic_
28 23 __ 4 5 _
5 5 1 _ 2
Write each value as a decimal
radic_
28 is between 52 and 53 Since 53 2 = 2809 an approximate value for radic
_ 28 is 53
23 __ 4 = 575
5 _
5 is 5555hellip so 5 _
5 to the nearest hundredth is 556
5 1 _ 2 = 55
Plot radic_
28 23 __ 4 5 _
5 and 5 1 _ 2 on a number line
From greatest to least the distances are
23 __ 4 km 5 _
5 km 5 1 _ 2 km radic_
28 km
EXAMPLEXAMPLE 3
STEP 1
STEP 2
7 Four people have found the distance in miles across a crater using different methods Their results are given below
Jonathan 10 __ 3 Elaine 3 _
45 Joseacute 3 1 _ 2 Lashonda radic_
10
Order the distances from greatest to least
YOUR TURN
8NS2
3 1 _ 2 mi 3 _
45 mi 10 __ 3 mi radic_
10 mi
23Lesson 13
copy H
ough
ton
Miff
lin H
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Com
pany
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8_MCAAESE206984_U1M01L3indd 23 41613 447 AM
ModelingPlace papers around the room with the numbers from 1 to 5 one per sheet Give each student a card showing a number between 1 and 5 in different forms Have students place his or her card between the correct integers and decide where the number goes in relation to any numbers already placed
Multiple RepresentationsGive students a vertical number line which some students might find easier to use than a horizontal one Have them decide whether to place points for rational and irrational numbers above or below existing points
Additional ResourcesDifferentiated Instruction includes bull Reading Strategies bull Success for English Learners EL
bull Reteach bull Challenge PRE-AP
DIFFERENTIATE INSTRUCTION
Ordering Real Numbers 24
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
Math TrainerOnline Assessment
and Intervention
Personal
Online homework assignment available
myhrwcom
myhrwcom
Lesson Quiz available online
13 LESSON QUIZ
1 Compare Write lt gt or =
radic_
95 - 5 radic_
62 - 2
2 Order 105 radic_
105 and 3π + 1 from greatest to least
3 A length in centimeters is calculated differently by four different people Order their calculations from least to greatest
KD 11 __ 2 cm Silvio 5 __ 3 π cm
Paula 5 _
4 cm Luis radic_
33 cm
EvaluateGUIDED AND INDEPENDENT PRACTICE
Concepts amp Skills Practice
Example 1Comparing Irrational Numbers
Exercises 1ndash8
Example 2Ordering Real Numbers
Exercises 9 12ndash15 18ndash21
Example 3Ordering Real Numbers in a Real-World Context
Exercises 10 16ndash17
Additional ResourcesDifferentiated Instruction includes bull Leveled Practice worksheets
Focus | Coherence | Rigor
Exercise Depth of Knowledge (DOK) Mathematical Practices
12ndash15 1 Recall of Information MP5 Using Tools
16 2 SkillsConcepts MP2 Reasoning
17 2 SkillsConcepts MP6 Precision
18ndash21 2 SkillsConcepts MP2 Reasoning
22 3 Strategic Thinking MP4 Modeling
23ndash24 3 Strategic Thinking MP3 Logic
8NS2
8NS2
Answers1 radic
_ 95 - 5 lt radic
_ 62 - 2
2 radic_
105 3π + 1 105
3 Silvio 5 __ 3 π cm Paula 5 _
4 cm
KD 11
__ 2 cm Luis radic_
33 cm
25 Lesson 13
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
Work Area
3140 3141 3142 3143
314 π227
20 A teacher asks his students to write the numbers shown in order from least to greatest Paul thinks the numbers are already in order Sandra thinks the order should be reversed Who is right
21 Math History There is a famous irrational number called Eulerrsquos number symbolized with an e Like π its decimal form never ends or repeats The first few digits of e are 27182818284
a Between which two square roots of integers could you find this number
b Between which two square roots of integers can you find π
22 Analyze Relationships There are several approximations used for π including 314 and 22 __ 7 π is approximately 314159265358979
a Label π and the two approximations on the number line
b Which of the two approximations is a better estimate for π Explain
c Find a whole number x so that the ratio x ___ 113 is a better estimate for π
than the two given approximations
23 Communicate Mathematical Ideas If a set of six numbers that include both rational and irrational numbers is graphed on a number line what is the fewest number of distinct points that need to be graphed Explain
24 Critique Reasoning Jill says that 12 _
6 is less than 1263 Explain her error
FOCUS ON HIGHER ORDER THINKING
radic_
115 115 ___ 11 and 105624
between radic_
7 asymp 265 and radic_
8 asymp 283
between radic_
9 = 3 and radic_
10 asymp 316
22 __ 7 it is closer to π on the number line
She did not consider the repeating digit 1266
2 rational numbers can have the same location and
irrational numbers can have the same location but they
cannot share a location
355
Neither student is correct The answer
should be 115 ___ 11 105624 radic_
115
Unit 126
copy H
ough
ton M
ifflin
Har
cour
t Pub
lishin
g Com
pany
Imag
e Cre
dits
copy3D
Stoc
kiSt
ockP
hoto
com
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
8_MCAAESE206984_U1M01L3indd 26 210513 801 AM
Personal Math Trainer
Online Practice and Helpmyhrwcom
Name Class Date
Independent Practice
16 Your sister is considering two different shapes for her garden One is a square with side lengths of 35 meters and the other is a circle with a diameter of 4 meters
a Find the area of the square
b Find the area of the circle
c Compare your answers from parts a and b Which garden would give your sister the most space to plant
17 Winnie measured the length of her fatherrsquos ranch four times and got four different distances Her measurements are shown in the table
a To estimate the actual length Winnie first approximated each distance to the nearest hundredth Then she averaged the four numbers Using a calculator find Winniersquos estimate
b Winniersquos father estimated the distance across his ranch to be radic_
56 km How does this distance compare to Winniersquos estimate
Give an example of each type of number
18 a real number between radic_
13 and radic_
14
19 an irrational number between 5 and 7
Order the numbers from least to greatest
12 radic_
7 2 radic_
8 ___ 2 13 radic_
10 π 35
14 radic_
220 -10 radic_
100 115 15 radic_
8 -375 3 9 _ 4
Distance Across Fatherrsquos Ranch (km)
1 2 3 4
radic_
60 58 __ 8 7 _
3 7 3 _ 5
138NS2
radic_
8 ___ 2 2 radic_
7
-10 radic_
100 115 radic_
220
radic_
60 asymp 775 58 __ 8 = 725 7 _
3 asymp 733 7 3 _ 5 = 760 so the average
π radic_
10 35
-375 9 _ 4 radic_
8 3
is 74825 km
1225 m2
4π m2 or approximately 126 m2
They are nearly identical radic_
56 is approximately 74833hellip
The circle would give her more space to plant because it has a
larger area
Sample answer 37
Sample answer radic_
31
25Lesson 13
copy H
ough
ton
Miff
lin H
arco
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ublis
hing
Com
pany
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
8_MCAAESE206984_U1M01L3indd 25 41613 448 AM
Activity available online myhrwcomEXTEND THE MATH PRE-AP
Activity Have students investigate whether there are infinitely many numbers between two numbers by giving examples for each of the following
bull Between any two rational numbers there is at least one other rational number Sample answer 45 is between 41 and 48
bull Between any two irrational numbers there is at least one rational number Sample answer 45 is between radic
_ 11 and radic
_ 29
bull Between any two rational numbers there is at least one irrational number Sample answer radic
_ 11 is between 31 and 36
bull Between any two irrational numbers there is at least one irrational number Sample answer radic
_ 17 is between radic
_ 11 and radic
_ 29
Ordering Real Numbers 26
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
ReadyMath Trainer
Online Practiceand Help
Personal
myhrwcom
Module Quiz
11ensp RationalenspandenspIrrationalenspNumbersWrite each fraction as a decimal or each decimal as a fraction
1 7__20 2 1___
27 3 17_8
Solve each equation for x
4 x2=81 5 x3=343 6 x2= 1___100
7 Asquarepatiohasanareaof200squarefeetHowlongiseachside
ofthepatiotothenearesttenth
12ensp SetsenspofenspRealenspNumbersWrite all names that apply to each number
8 121____radic
____121
9 π__2
10 TellwhetherthestatementldquoAllintegersarerationalnumbersrdquoistrueorfalseExplainyourchoice
13ensp OrderingenspRealenspNumbersCompare Write lt gt or =
11 radic__
8+3 8+radic__
3 12 radic__
5+11emsp emsp emsp 5+radic___
11
Order the numbers from least to greatest
13 radic___
99π29__
8 14 radic___
1__251_40__
2
15 Howarerealnumbersusedtodescribereal-worldsituations
ESSENTIAL QUESTION
035
9-9
141ft
7 1__10- 1__10
14__11 1875
wholeintegerrationalreal
Trueintegerscanbewrittenasthequotientoftwointegers
SampleanswerRealnumberssuchastherational
π29__
8radic___
99
irrationalreal
lt gt
number1_4candescribeamountsusedincooking
radic___
1__250__
21_4
27Module1
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ough
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Miff
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hing
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pany
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8_MCAAESE206984_U1M01RTindd 27 41513 1113 PM
Math TrainerOnline Assessment
and Intervention
Personal
myhrwcom
1
2
3 Response toIntervention
Intervention Enrichment
Access Ready to Go On assessment online and receive instant scoring feedback and customized intervention or enrichment
Online and Print Resources
Differentiated Instruction
bull Reteach worksheets
bull Reading Strategies EL
bull Success for English Learners EL
Differentiated Instruction
bull Challenge worksheets PRE-AP
Extend the Math PRE-AP
Lesson Activities in TE
Additional ResourcesAssessment Resources includes bull Leveled Module Quizzes
Ready to Go OnAssess MasteryUse the assessment on this page to determine if students have mastered the concepts and standards covered in this module
California Common Core Standards
Lesson Exercises Common Core Standards
11 1ndash7 8NS1 8NS2 8EE2
12 8ndash10 8NS1
13 11ndash14 8NS2
27 Unit 1 Module 1
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
Personal Math Trainer
Online Practice and HelpmyhrwcomAssessment Readiness
Module 1 MIXed ReVIeW
1 Look at each number Is the number between 2π and radic___
52
Select Yes or No for expressions AndashC
A 6 2 _ 3 Yes No
B 5π __ 2 Yes No
C 3 radic__
5 Yes No
2 Consider the number - 11 __ 15
Choose True or False for each statement
A The number is rational True False
B The number can be written as True Falsea repeating decimal
C The number is less than ndash08 True False
3 The volume of a cube is given by V = x3 where x is the length of an edge of the cube A cube-shaped end table has a volume of 3 3 _ 8 cubic feet What is the length of an edge of the end table Explain how you solved this problem
4 A student says that radic___
83 is greater than 29 __ 3 Is the student correct Justify your
reasoning
1 1 _ 2 ft Sample answer The equation x3 = 3 3 _ 8 can be used
to find the edge length in feet To solve the equation
write the mixed number as a fraction greater than 1
x3 = 27 __ 8 Then take the cube root of both sides x = 3 _ 2 = 1 1 _ 2
No Sample answer radic___
83 asymp 91 and 29 __ 3 = 9
__ 6
Because 91 lt 9 __
6 radic___
83 lt 29 __ 3
28 Unit 1
copy H
ough
ton
Miff
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pany
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=A
8_MCAAESE206984_U1M01RTindd 28 240413 946 AM
Personal Math Trainer
Online Assessment and
Interventionmyhrwcom
Scoring GuideItem 3 Award the student 1 point for finding the edge length of the cube and 1 point for correctly explaining how to use a cube root to solve the problem
Item 4 Award the student 1 point for determining that the student is incorrect and 1 point for correctly justifying the reasoning for this conclusion
Additional ResourcesTo assign this assessment online login to your Assignment Manager at myhrwcom
Assessment Readiness
California Common Core Standards
Items Grade 8 Standards Mathematical Practices
1 8NS2 MP7
2 7NS2b 7NS2d 8NS1 MP7
3 8EE2 MP1 MP4
4 8NS1 8NS2 MP3
Item integrates mixed review concepts from previous modules or a previous course
Item 4 combines concepts from the California Common Core cluster ldquoKnow that there are numbers that are not rational and approximate them by rational numbersrdquo
Real Numbers 28
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
Math TrainerOnline Assessment
and Intervention
Personal
Online homework assignment available
myhrwcom
EvaluateGUIDED AND INDEPENDENT PRACTICE
Concepts amp Skills Practice
Example 1Expressing Rational Numbers as Decimals
Exercises 1ndash6 20ndash21 24ndash25
Example 2Expressing Decimals as Rational Numbers
Exercises 7ndash12 22ndash23 26ndash27
Example 3Finding Square Roots and Cube Roots
Exercises 13ndash15 28 30ndash31 35
Explore ActivityEstimating Irrational Numbers
Exercises 13 16ndash18 29 32ndash34
Additional ResourcesDifferentiated Instruction includes bull Leveled Practice worksheets
Lesson Quiz available online
11 LESSON QUIZ
1 Write as a decimal 2 5 __ 8 1 7 __ 12
2 Write as a fraction 034 1 _
24
3 Solve x 2 = 9 __ 49 for x
4 Solve x 3 = 216 for x
5 Estimate the value of radic_
13 to one decimal place without using a calculator
myhrwcom
Focus | Coherence | Rigor
Exercise Depth of Knowledge (DOK) Mathematical Practices
20ndash27 2 SkillsConcepts MP4 Modeling
28 3 Strategic Thinking MP4 Modeling
29ndash32 2 SkillsConcepts MP6 Precision
33 3 Strategic Thinking MP7 Using Structure
34 2 SkillsConcepts MP3 Logic
35 2 SkillsConcepts MP4 Modeling
36 3 Strategic Thinking MP3 Logic
37 3 Strategic Thinking MP7 Using Structure
38 3 Strategic Thinking MP2 Reasoning
8NS1 8NS2 8EE2
8NS1 8NS2 8EE2
Answers1 2625 158
_ 3
2 17 __ 50 1 8 __ 33
3 x = plusmn 3 __ 7
4 x = 6
5 36
13 Lesson 11
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
Work Area
33 Analyze Relationships To find radic_
15 Beau found 3 2 = 9 and 4 2 = 16 He said that since 15 is between 9 and 16 radic
_ 15 must be between 3 and 4 He
thinks a good estimate for radic_
15 is 3 + 4 ____ 2 = 35 Is Beaursquos estimate high low
or correct Explain
34 Justify Reasoning What is a good estimate for the solution to the equation x 3 = 95 How did you come up with your estimate
35 The volume of a sphere is 36π f t 3 What is the radius of the sphere Use the formula V = 4 _ 3 π r 3 to find your answer
36 Draw Conclusions Can you find the cube root of a negative number If so is it positive or negative Explain your reasoning
37 Make a Conjecture Evaluate and compare the following expressions
radic_
4 __ 25 and radic
_ 4 ____
radic_
25 radic
_
16 __ 81 and radic_
16 ____
radic_
81 radic
_
36 __ 49 and radic_
36 ____
radic_
49
Use your results to make a conjecture about a division rule for square roots Since division is multiplication by the reciprocal make a conjecture about a multiplication rule for square roots
38 Persevere in Problem Solving The difference between the solutions to the equation x 2 = a is 30 What is a Show that your answer is correct
FOCUS ON HIGHER ORDER THINKING
His estimate is low because 15 is very close to 16
so radic_
15 is very close to radic_
16 or 4 A better estimate
would be 38 or 39
Sample answer about 45 4 3 = 64 and 5 3 = 125
Because 95 is about halfway between 64 and 125 try 45
45 3 = 91125 which is a good estimate
3 feet
Yes the cube root of a negative number is negative
because a negative number cubed is always negative
and a nonnegative number cubed is always nonnegative
radic_
4 __ 25 = 2 _ 5 = radic
_ 4 ____
radic_
25 radic
_
16 __ 81 = 4 _ 9 = radic_
16 ____
radic_
81 radic
_
36 __ 49 = 6 _ 7 = radic_
36 ____
radic_
49
225 the solutions to x 2 = a are x = plusmn15 and
radic_
a ___
radic_
b = radic
_ a __
b radic
_ a radic
_ b = radic
_ a b
15 - (-15) = 30
Unit 114
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ton
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hing
Com
pany
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Ilen
e Mac
Dona
ldA
lamy I
mag
es
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=B
8_MCABESE206984_U1M01L1indd 14 102913 1142 PM
Personal Math Trainer
Online Practice and Helpmyhrwcom
Name Class Date
Independent Practice11
20 A 7 __ 16 -inch-long bolt is used in a machine What is this length written as a decimal
21 The weight of an object on the moon is 1 _ 6 its weight on Earth Write 1 _ 6 as a decimal
22 The distance to the nearest gas station is 2 4 _ 5 kilometers What is this distance written as a decimal
23 A baseball pitcher has pitched 98 2 _ 3 innings What is the number of innings written as a decimal
24 A heartbeat takes 08 second How many seconds is this written as a fraction
25 There are 262 miles in a marathon Write the number of miles using a fraction
26 The average score on a biology test was 72
_ 1 Write the average score using a
fraction
27 The metal in a penny is worth about 0505 cent How many cents is this written as a fraction
28 Multistep An artist wants to frame a square painting with an area of 400 square inches She wants to know the length of the wood trim that is needed to go around the painting
a If x is the length of one side of the painting what equation can you set up to find the length of a side How many solutions does the equation have
b Do all of the solutions that you found make sense in the context of the problem Explain
c What is the length of the wood trim needed to go around the painting
Solve each equation for x Write your answers as radical expressions Then estimate to one decimal place if necessary
29 x 2 = 14 30 x 3 = 1331
31 x 2 = 144 32 x 2 = 29
8NS1 8NS2 8EE2
04375 in 01 _6
28 km 98 _6 innings
x 2 = 400 x = plusmnthinsp20 the equation has 2 solutions
x = 20 makes sense but x = -20 doesnrsquot because a
painting cannot have a side length of -20 inches
4 times 20 = 80 inches
x = plusmn radic_
14 asymp plusmn37
x = plusmn radic_
144 = plusmn12 x = plusmn radic_
29 asymp plusmn54
x = 3 radic_ 1331 = 11
4_5 second 26 1_5 mi
72 1 _ 9 101 ___ 200 cent
13Lesson 11
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ough
ton
Miff
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hing
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pany
bull copy
Phot
odisc
Get
ty Im
ages
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
8_MCAAESE206984_U1M01L1indd 13 41613 1211 AM
myhrwcomActivity available onlineEXTEND THE MATH PRE-AP
Activity Write radic_
09 on the board and invite students to conjecture what the value might be Have them check their conjectures by squaring Invite them to suggest ways to estimate radic
_ 09 As a hint point out that 09 is close to 10 and so they might
use that to help guide their estimates Lead them to see that since 092 is 081 and 102 is 1 the value of radic
_ 09 is greater than 09 and less than 10 Try squaring 095 to get
09025 A good estimate for radic_
09 is 095
Rational and Irrational Numbers 14
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=B
Integers
Rational Numbers IrrationalNumbers
Real Numbers
WholeNumbers
-3-4-5 -2-1 1 2 3 50 4
23
34-4 -π -1 25
radic2
Lesson Support Content Objective Students will learn to describe relationships between sets of numbers
Language Objective Students will explain how to describe relationships between sets of real numbers
LESSON 12 Sets of Real Numbers
Building BackgroundEliciting Prior Knowledge Have students draw a number line from -5 to 5 Ask them to plot points on the number line to approximate the location of rational and irrational numbers such as -1 3 __ 4 25 -4 2 __ 3 radic
_ 2 and -π
Learning ProgressionsIn this lesson students clarify their understanding of the real number system They characterize sets and subsets of the real numbers They also identify sets for real-world situations Important understandings for students include the following
bull Identify all of the possible subsets of the real numbers for a given number
bull Decide whether a statement about a subset of the real numbers is true or false
bull Identify the set of numbers that best describes a real-world situation
Understanding the relationships among the sets of numbers that make up the real numbers is essential as students are introduced to different forms of numbers throughout the school year This lesson provides a foundation for the comparing and ordering of real numbers in the next lesson
Cluster ConnectionsThis lesson provides an excellent opportunity to connect ideas in this cluster Know that there are numbers that are not rational and approximate them by rational numbers Have students copy this diagram which relates the sets of real numbers
Ask students to complete the diagram by writing three examples for each set of numbers Have students share examples and explain how they knew each number they selected belonged in the appropriate set Answers may vary Check studentsrsquo work
Focus | Coherence | Rigor
California Common Core Standards
8NS1 Know that numbers that are not rational are called irrational Understand informally that every number has a decimal expansion for rational numbers show that the decimal expansion repeats eventually and convert a decimal expansion which repeats eventually into a rational number
MP7 Look for and make use of structure
15A
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
Math Talk
Language Support EL
PROFESSIONAL DEVELOPMENT
Linguistic Support EL
AcademicContent Vocabulary
Venn diagrams ndash Students need descriptive language to describe the categories that the different areas and colors of a Venn diagram represent the concept of a set and how sets are distinct or can overlap Use sentence frames such as
The big oval represents __________The darklight blue color in the middle of the
big ovals represents __________These sets overlap because __________
In this way students have the language and structure to identify the criteria that distinguish a set and to explain the abstract representation Also point out the use of the prefix sub- meaning ldquounderrdquo in the term subset
Rules and Patterns
Abbreviations ndash In this lesson the abbreviation mph is used Be sure to point out that mph stands for miles per hour and is used to give units in a rate of speed Students may also have seen mpg (miles per gallon) which gives the units in a rate of fuel efficiency
Borrowed Words ndash Terminology used in baseball such as inning and pitcher may require some explanation Spanish as well as some other languages have borrowed these terms from English so some students may be familiar with these words already Despite this whenever a word is critical to students understanding the word problem it is best to explain the meaning
Leveled Strategies for English Learners
Emerging Allow students to indicate true or false orally in Guided Practice Exercises 9 and 10
Expanding Have students use sentence frames to describe the meaning of regions and colors used in a Venn diagram Then give them similar sentence frames orally and have them draw and shade a Venn diagram based on the oral prompts
Bridging Have students work in groups to draw a Venn diagram to represent sets based on real-world examples in the lesson
To help students answer the question posed in Math Talk provide a sentence frame for their answer
The numbers between 31 and 39 on a number line are __________ because __________
EL
California ELD Standards
Emerging 2II5 Modifying to add details ndash Expand sentences with simple adverbials to provide details about a familiar activity or process
Expanding 2II5 Modifying to add details ndash Expand sentences with adverbials to provide details about a familiar or new activity or process
Bridging 2II5 Modifying to add details ndash Expand sentences with increasingly complex adverbials to provide details about a variety of familiar and new activities and processes
Sets of Real Numbers 15B
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
12L E S S O N
Sets of Real Numbers
EngageESSENTIAL QUESTION
How can you describe relationships between sets of real numbers Sample answer Describe them as two different sets or one set as being a subset of another
Motivate the LessonAsk How many different types of tigers can you name How does the set of Bengal tigers relate to the set of tigers
ExplorePoint to different locations in the Animals diagram and ask for examples for that classification Do the same for the Real Numbers diagram Students should understand that everything within a region is part of the set for example both -3 and 2 are integers
ExplainEXAMPLE 1
Questioning Strategies Mathematical Practices bull In A why is 5 not a perfect square It does not have rational numbers as its square roots
bull Can the number in B be written as a fraction Why or why not Yes it is a terminating decimal so it is a rational number
Engage with the WhiteboardHave students place the numbers in Example 1 and Additional Example 1 in the Venn diagram for numbers
YOUR TURNAvoid Common ErrorsBe sure that students read Exercise 2 carefully before answering The number given in the problem 10 is the area not the side length
EXAMPLE 2Questioning Strategies Mathematical Practices bull What two major sets are the real numbers composed of rational and irrational numbers
bull What is the location of the set of whole numbers in the Venn diagram in relation to the set of rational numbers Explain Inside it whole numbers are rational numbers
Focus on Reasoning Mathematical PracticesRemind students that it takes only one counterexample to show that a statement is false
Interactive Whiteboard Interactive example available online
ADDITIONAL EXAMPLE 1Write all names that apply to each number
A -10integer rational real
B 12 _ 3
whole integer rational real
myhrwcom
Interactive Whiteboard Interactive example available online
ADDITIONAL EXAMPLE 2Tell whether the given statement is true or false Explain your choice
No integers are whole numbers
False every whole number is also an integer
myhrwcom
Animated MathClassifying Numbers
Students build fluency in classifying numbers in this engaging fast-paced game
myhrwcom
CA Common CoreStandards
The student is expected to
The Number Systemmdash8NS1
Know that numbers that are not rational are called irrational Understand informally that every number has a decimal expansion for rational numbers show that the decimal expansion repeats eventually and convert a decimal expansion which repeats eventually into a rational numberMathematical Practices
MP7 Using Structure
The student is expected to
15 Lesson 12
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=B
Math TrainerOnline Practice
and Help
Personal
myhrwcom
Math TrainerOnline Practice
and Help
Personal
myhrwcom
Math On the Spotmyhrwcom
Understanding Sets and Subsets of Real NumbersBy understanding which sets are subsets of types of numbers you can verify whether statements about the relationships between sets are true or false
Tell whether the given statement is true or false Explain your choice
All irrational numbers are real numbers
True Every irrational number is included in the set of real numbers The irrational numbers are a subset of the real numbers
No rational numbers are whole numbers
False A whole number can be written as a fraction with a denominator of 1 so every whole number is included in the set of rational numbers The whole numbers are a subset of the rational numbers
EXAMPLE 2
A
B
Write all names that apply to each number
1 A baseball pitcher has pitched 12 2 _ 3 innings
2 The length of the side of a square that has an
area of 10 square yards
YOUR TURN
Tell whether the given statement is true or false Explain your choice
3 All rational numbers are integers
4 Some irrational numbers are integers
YOUR TURN
Give an example of a rational number that is a
whole number Show that the number is both whole
and rational
Math TalkMathematical Practices
Give an example of a
8NS1
False Every integer is a rational number but not every
False Real numbers are either rational or irrational numbers
Integers are rational numbers so no integers are irrational numbers
rational real
irrational real
Sample answer 8 8 = 8_
1
and -thinsp 5 _ 2 are not integers
rational number is an integer Rational numbers such as 3 _ 5
Unit 116
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pany
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age C
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igita
l Im
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opyr
ight
copy20
04 Ey
ewire
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8_MCAAESE206984_U1M01L2indd 16 41613 136 AM
Math On the Spot
myhrwcom
Vertebrates
Birds
Passerines
Animals
Integers
Rational Numbers IrrationalNumbers
Real Numbers
WholeNumbers
1
45
3
0
274
67
radic4
-
-3
-2
-1
03
radic2
radic17
radic11-
π
Animated Math
myhrwcom
Classifying Real NumbersBiologists classify animals based on shared characteristics A cardinal is an animal a vertebrate a bird and a passerine
You already know that the set of rational numbers consists of whole numbers integers and fractions The set of real numbers consists of the set of rational numbers and the set of irrational numbers
Write all names that apply to each number
radic_
5 irrational real
ndash1784rational real
whole integer rational real
EXAMPLEXAMPLE 1
A
B
C radic_ 81 ____ 9
L E S S O N
12Sets of Real Numbers
ESSENTIAL QUESTIONHow can you describe relationships between sets of real numbers
Passerines such as the cardinal are also called ldquoperching birdsrdquo
What types of numbers are between 31 and 39 on a
number line
Math TalkMathematical Practices
What types of numbers are
8NS1
8NS1
Know that numbers that are not rational are called irrational Understand informally that every number has a decimal expansion for rational numbers show that the decimal expansion repeats eventually and convert a decimal expansion which repeats eventually into a relation number
ndash1784 is a terminating decimal
5 is a whole number that is not a perfect square
radic_
81 _____ 9 = 9 __ 9 = 1 rational irrational real
15Lesson 12
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pany
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Wiki
med
ia Co
mm
ons
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=B
8_MCABESE206984_U1M01L2indd 15 061113 1144 AM
PROFESSIONAL DEVELOPMENT
Math BackgroundThe relationships between sets of numbers extend to include complex numbers A complex number can be written as a sum of a real number a and an imaginary number bi
a + bi
An imaginary number is a special number that when squared gives a negative value When you square a real number you get a nonnegative number When you square an imaginary number you get a negative value The imaginary unit is i
i = radic_
-1
Integrate Mathematical Practices MP7
This lesson provides an opportunity to address this Mathematical Practices standard It calls for students to discern structure to connect and communicate mathematical ideas
Students use a Venn diagram to structure relationships between sets of numbers They connect and communicate mathematical ideas when they make logical statements about the sets and describe which set best describes numbers applied to real-life situations
Sets of Real Numbers 16
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=B
YOUR TURNAvoid Common ErrorsStudents may see the word ldquoAllldquo or rdquoNordquo in Exercises 3 and 4 and immediately assume that any absolute statements like these are false Remind them that there are true statements that begin with these words and encourage them to provide examples
EXAMPLE 3Questioning Strategies Mathematical Practices bull In A how does the phrase ldquonumber of rdquo give you a clue about the number classification It indicates a counting number
bull What is the relationship between the circumference of a circle and the diameter The circumference is diameter times π
Focus on Critical Thinking Mathematical PracticesIn B suppose the diameters in inches were 25
__ π 28 __ π
31 __ π and so on What set of numbers would
best describe the circumferences Explain Whole numbers the circumferences would be the whole numbers 25 28 31 and so on
YOUR TURNFocus on Critical Thinking Mathematical PracticesHave students compare and contrast the classification of numbers in the answers in Exercises 5 and 6
ElaborateTalk About ItSummarize the Lesson
Ask What are some ways that number sets can be related Sets may be subsets of other sets or they may be separate from other sets
GUIDED PRACTICEEngage with the Whiteboard
Have students place the numbers in Exercises 1ndashthinsp8 in the Venn diagram for numbers at the beginning of the lesson
Integrating Language Arts EL
Encourage English learners to ask for clarification on any terms or phrases that they do not understand
Avoid Common ErrorsExercise 7 Remind students that a repeating decimal is a rational numberExercises 9ndash10 Remind students that it only takes one counterexample to show that a statement is false
Interactive Whiteboard Interactive example available online
ADDITIONAL EXAMPLE 3Identify the set of numbers that best describes the situation Explain your choice
A the amount of time that has passed since midnight
The set of real numbers time is continuous so the amount of time can be rational or irrational
B the number of tickets sold to a basketball game
The set of whole numbers the number of tickets sold may be 0 or a counting number
myhrwcom
17 Lesson 12
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
1IN
116 inch
Guided Practice
Write all names that apply to each number (Example 1)
1 7 _ 8 2 radic_
36
3 radic_
24 4 075
5 0 6 - radic_ 100
7 5 _
45 8 - 18 __ 6
Tell whether the given statement is true or false Explain your choice (Example 2)
9 All whole numbers are rational numbers
10 No irrational numbers are whole numbers
Identify the set of numbers that best describes each situation Explain your choice (Example 3)
11 the change in the value of an account when given to the nearest dollar
12 the markings on a standard ruler
13 What are some ways to describe the relationships between sets of numbers
CHECK-INESSENTIAL QUESTION
rational real
rational real
True Whole numbers are rational numbers
Rational numbers the ruler is marked every 1 __ 16 th inch
Sample answer Describe one set as being a subset of
another or show their relationships in a Venn diagram
Integers the change can be a whole dollar amount
and can be positive negative or zero
True Whole numbers are a subset of the set of rational numbers
and can be written as a ratio of the whole number to 1
irrational real
whole integer rational real
whole integer rational real
rational real
integer rational real
integer rational real
Unit 118
copy H
ough
ton
Miff
lin H
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pany
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
8_MCAAESE206984_U1M01L2indd 18 41613 136 AM
My Notes
Math TrainerOnline Practice
and Help
Personal
myhrwcom
Math On the Spot
myhrwcom
Identifying Sets for Real-World SituationsReal numbers can be used to represent real-world quantities Highways have posted speed limit signs that are represented by natural numbers such as 55 mph Integers appear on thermometers Rational numbers are used in many daily activities including cooking For example ingredients in a recipe are often given in fractional amounts such as 2 _ 3 cup flour
Identify the set of numbers that best describes each situation Explain your choice
the number of people wearing glasses in a room
The set of whole numbers best describes the situation The number of people wearing glasses may be 0 or a counting number
the circumference of a flying disk has a diameter of 8 9 10 11 or 14 inches
The set of irrational numbers best describes the situation Each circumference would be a product of π and the diameter and any multiple of π is irrational
EXAMPLEXAMPLE 3
A
B
Identify the set of numbers that best describes the situation Explain your choice
5 the amount of water in a glass as it evaporates
6 the weight of a person in pounds
YOUR TURN
8NS1
Rational numbers a personrsquos weight can be a decimal
such as 835 pounds
Real numbers the amount can be any number greater
than 0
17Lesson 12
copy H
ough
ton
Miff
lin H
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ublis
hing
Com
pany
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
8_MCAAESE206984_U1M01L2indd 17 41613 520 AM
Graphic OrganizersGive students a list of numbers (including terminating and repeating decimals fractions integers and rational and irrational square roots) and a graphic organizer as shown below
Real Numbers
Rational numbers Irrational numbers
Integer numbers
Whole numbers
Ask students to write each number in the list in the correct section of the organizer
Number SensePoint out to students that knowing the types of numbers to expect in different situations can alert them to incorrect math as well as to impossible situations For example 135 shots made in basketballs is not possible but an average number of shots can equal 135
Additional ResourcesDifferentiated Instruction includes bull Reading Strategies bull Success for English Learners EL
bull Reteach bull Challenge PRE-AP
DIFFERENTIATE INSTRUCTION
Sets of Real Numbers 18
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
Math TrainerOnline Assessment
and Intervention
Personal
Online homework assignment available
myhrwcom
Lesson Quiz available online
12 LESSON QUIZ
1 Write all the names that apply to the number
2 Tell whether the given statement is true or false Explain your choice All numbers between 1 and 2 are rational numbers
3 Identify the set of numbers that best describes the situation Explain your choiceThe choices on a survey question change the total points for the survey by -2 -1 0 1 or 2 points
-1 _
5
myhrwcom
Additional ResourcesDifferentiated Instruction includes bull Leveled Practice worksheets
EvaluateGUIDED AND INDEPENDENT PRACTICE
Concepts amp Skills Practice
Example 1Classifying Real Numbers
Exercises 1ndash8 14ndash19 22ndash24
Example 2Understanding Sets and Subsets of Real Numbers
Exercises 9ndash10
Example 3Identifying Sets for Real-World Situations
Exercises 11ndash12 20ndash21 25
Focus | Coherence | Rigor
Exercise Depth of Knowledge (DOK) Mathematical Practices
14ndash19 2 SkillsConcepts MP7 Using Structure
20ndash21 2 SkillsConcepts MP6 Precision
22ndash23 2 SkillsConcepts MP3 Logic
24 1 Recall of Information MP7 Using Structure
25 2 SkillsConcepts MP2 Reasoning
26ndash27 3 Strategic Thinking MP3 Logic
28 3 Strategic Thinking MP8 Patterns
29 3 Strategic Thinking MP3 Logic
8NS1
8NS1
Exercise 29 combines concepts from the California Common Core cluster ldquoKnow that there are numbers that are not rational and approximate them by rational numbersrdquo
Answers1 rational real
2 False radic_
2 is an example of an irrational number between 1 and 2
3 Integers each number is an integer but only three are whole numbers
19 Lesson 12
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
Work Area
π mi23 Critique Reasoning The circumference of a circular region is shown
What type of number best describes the diameter of the circle Explain
your answer
24 Critical Thinking A number is not an integer What type of number can it be
25 A grocery store has a shelf with half-gallon containers of milk What type of number best represents the total number of gallons
26 Explain the Error Katie said ldquoNegative numbers are integersrdquo What was her error
27 Justify Reasoning Can you ever use a calculator to determine if a number is rational or irrational Explain
28 Draw Conclusions The decimal 0 _
3 represents 1 _ 3 What type of number best describes 0
_ 9 which is 3 middot 0
_ 3 Explain
29 Communicate Mathematical Ideas Irrational numbers can never be precisely represented in decimal form Why is this
FOCUS ON HIGHER ORDER THINKING
It can be a rational number that is not an integer or an irrational number
rational number
The set of negative numbers also includes non-integer
rational numbers and irrational numbers
Sample answer If the calculator shows a decimal that
terminates in fewer digits than what the calculator screen
allows then you can tell that the number is rational If not
you cannot tell from the calculator display whether the
number terminates because you see a limited number
of digits It may be a repeating decimal (rational) or
non-terminating non-repeating decimal (irrational)
Whole 3 middot 0 _
3 represents 3 middot 1 _ 3 = 1 so 0 _
9 is exactly 1
Sample answer In decimal form irrational numbers never
terminate and never repeat Therefore no matter how
many decimal places you include the number will never
be precisely represented There are always more digits
Whole the diameter is π _ π = 1 mile
Unit 120
copy H
ough
ton
Miff
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ublis
hing
Com
pany
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
8_MCAAESE206984_U1M01L2indd 20 120413 909 PM
Integers
Rational Numbers Irrational Numbers
Real Numbers
Whole Numbers
257
radic16
166
radic9
128 radic50
Personal Math Trainer
Online Practice and Helpmyhrwcom
Name Class Date
Independent Practice
Identify the set of numbers that best describes each situation Explain your choice
20 the height of an airplane as it descends to an airport runway
21 the score with respect to par of several golfers 2 ndash 3 5 0 ndash 1
22 Critique Reasoning Ronald states that the number 1 __ 11 is not rational because when converted into a decimal it does not terminate Nathaniel says it is rational because it is a fraction Which boy is correct Explain
12
14 - radic_
9 15 257
16 radic_
50 17 8 1 _ 2
18 166 19 radic_
16
Write all names that apply to each number Then place the numbers in the correct location on the Venn diagram
8NS1
Real numbers the height can be any number greater than zero
integer rational real whole integer rational real
whole integer rational real
irrational real
rational real
rational real
Integers the scores are counting numbers their
opposites and zero
Nathaniel is correct A rational number is a number that can be written as a fraction and 1 __ 11 is a fraction
19Lesson 12
copy H
ough
ton
Miff
lin H
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urt P
ublis
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Com
pany
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
8_MCAAESE206984_U1M01L2indd 19 41613 136 AM
myhrwcomActivity available onlineEXTEND THE MATH PRE-AP
Activity Have students consider the concept of restricted domain for the sets of numbers that describe situations For example the number of sisters a person has can best be described by whole numbers but no one has ever had 1500 sisters An area code is an integer or whole number between 200 and 999
Have students use a source such as the Guinness Book of World Records and give examples of sets of numbers that describe situations where the domain is restricted Ask whether the restriction may be changed in the future
Sets of Real Numbers 20
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
-3-4-5 -2-1 1 2 3 50 4
12-4 -radic5
Lesson Support Content Objective Students will learn to order a set of real numbers
Language Objective Students will show and describe how to order a set of real numbers
LESSON 13 Ordering Real Numbers
Building BackgroundEliciting Prior Knowledge Have students draw a number line to compare a rational number and an irrational number such as - radic
_ 5 and -4 1 __ 2 Ask them to explain how
they approximated the irrational number on the number line Then have them identify the greater and the lesser real number Repeat with several other pairs of real numbers in different forms
Learning ProgressionsIn this lesson students order a set of real numbers They use rational approximations to compare the sizes of irrational numbers They also order numbers for real-world situations Important understandings for students include the following
bull Compare irrational numbers bull Estimate the value of expressions with irrational numbers bull Order a set of real numbers bull Order real numbers in a real-world context
Work with real numbers continues throughout Grade 8 and into high school This lesson provides students with a foundation for understanding the relative sizes of numbers in different forms in the real number system
Cluster ConnectionsThis lesson provides an excellent opportunity to connect ideas in this cluster Know that there are numbers that are not rational and approximate them by rational numbers Tell students that there is a special number called the golden ratio with applications in mathematics geometry art and architecture The golden ratio is called phi and is represented by the Greek letter ϕ It includes an irrational number in its definition
Have students explain why the golden ratio is irrational Ask them to find the two whole numbers the golden ratio lies between Then challenge them to approximate the golden ratio to the nearest tenth It is irrational because it includes an irrational number in its definition It lies between 1 and 2 To the nearest tenth ϕ = 16
ϕ = 1 + radic_
5 _ 2
Focus | Coherence | Rigor
California Common Core Standards
8NS2 Use rational approximations of irrational numbers to compare the size of irrational numbers locate them approximately on a number line diagram and estimate the value of expressions (eg π 2 )
MP4 Model with mathematics
21A
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
Math Talk
Language Support EL
PROFESSIONAL DEVELOPMENT
Linguistic Support EL
AcademicContent Vocabulary
Post a chart like this to remind students of the regular comparative forms of adjectives that use the -er and -est suffixes Add to the chart for terms that appear in examples and exercises in each lesson Include any irregular verb forms
Background Knowledge
Go On ndash the title of the module review or quiz is Ready to Go On This title uses an idiomatic expression In this context to go on means ldquoto move aheadrdquo or ldquoto proceedrdquo It is different from the use of go on that means having enough facts to use meaningfully as in having enough to go on Also the intonation used in pronouncing an expression can give it different meanings For example when the speaker emphasizes the word on he or she might be expressing disbelief as in ldquoGo ON Yoursquore kidding rightrdquo Discuss with students other ways that the phrase go on may be used
Leveled Strategies for English Learners
Emerging Label points on a number line with the terms used in ordering greater greatest less lesser least Use sentence frames to insert the correct terms
Expanding Have students give two or three complete sentences to compare the placement of numbers on a number line using the correct forms of the comparative and superlative adjectives
Bridging Have students work in pairs with one student giving directions to the other in complete sentences to order numbers on a number line
To help students answer the question posed in Math Talk make sure that students have a command of the forms for making comparisons and the superlative and the concept of opposite order so that the focus is on the math concept instead of the language skills needed to describe and explain order
EL
Adjective Comparative Superlative
Far Farther Farthest
Large Larger Largest
Great Greater Greatest
Some Less Least
Some More Most
California ELD Standards
Emerging 2I8 Analyzing language choices ndash Explain how phrasing or different common words with similar meanings produce different effects on the audience
Expanding 2I8 Analyzing language choices ndash Explain how phrasing or different words with similar meanings or figurative language produce shades of meaning and different effects on the audience
Bridging 2I8 Analyzing language choices ndash Explain how phrasing or different words with similar meanings or figurative language produce shades of meaning nuances and different effects on the audience
Ordering Real Numbers 21B
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
13L E S S O N
Ordering Real Numbers
Interactive Whiteboard Interactive example available online
ADDITIONAL EXAMPLE 1Compare Write lt gt or =
A radic_
8 - 2 4 - radic_
8 lt
B radic_
20 + 1 3 + radic_
2 gt
EngageESSENTIAL QUESTION
How do you order a set of real numbers Sample answer Find their approximate decimal values and order them
Motivate the LessonAsk What kind of numbers are you comparing when you compare the price of gasoline at two different gas stations
ExploreGive students two rational numbers and ask them to name a number between them Repeat a few times and then give them two irrational numbers and ask them to name a number between them
ExplainEXAMPLE 1
Questioning Strategies Mathematical Practices bull Which is greater the difference between 5 and 3 or the difference between radic
_ 5 and radic
_ 3
The difference between 5 and 3 is 2 the difference between radic_
5 and radic_
3 is approximately 1 So the difference between 5 and 3 is greater
Avoid Common ErrorsCaution students to read the problem carefully and think about what the radical sign means so that they do not misread the problem and answer that the two sides are equal
YOUR TURNFocus on TechnologyCalculators should not be used at this point because developing number sense is the goal
EXAMPLE 2Questioning Strategies Mathematical Practices bull How do you determine whether radic
_ 22 is less than or greater than 45 The square of 45 is
2025 which is less than 22 so the square root of 22 must be greater than 45
Engage with the WhiteboardHave students graph and label various real numbers between 42 and 44 and between 47 and 5
YOUR TURNFocus on Modeling Mathematical PracticesHave students label the integers on the number line with their equivalent square root For example 1 2 and 3 on the number line would be labeled radic
_ 1 radic
_ 4 and radic
_ 9
Interactive Whiteboard Interactive example available online
ADDITIONAL EXAMPLE 2Order 3π radic
_ 10 and 325 from greatest
to least
3π 325 radic_
10
myhrwcom
myhrwcom
CA Common CoreStandards
The student is expected to
The Number Systemmdash8NS2
Use rational approximations of irrational numbers to compare the size of irrational numbers locate them approximately on a number line diagram and estimate the value of expressions (eg π 2 )
Mathematical Practices
MP4 Modeling
The student is expected to
21 Lesson 13
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
Math TrainerOnline Practice
and Help
Personal
myhrwcom
Math On the Spotmyhrwcom
0 05 1 15 2 25 3 35 4
radic5radic3
π2
8 85 9 95 10 105 11 115 12
radic75
4 42 44 46 48 5
radic224 12π + 1
Ordering Real Numbers You can compare and order real numbers and list them from least to greatest
Order radic_
22 π + 1 and 4 1 _ 2 from least to greatest
First approximate radic_
22
radic_
22 is between 4 and 5 Since you donrsquot know where it falls between 4 and 5 you need to find a better estimate for radic
_ 22 so
you can compare it to 4 1 _ 2
Since 22 is closer to 25 than 16 use squares of numbers between 45 and 5 to find a better estimate of radic
_ 22
45 2 = 2025 46 2 = 2116 47 2 = 2209 48 2 = 2304
Since 47 2 = 2209 an approximate value for radic_
22 is 47
An approximate value of π is 314 So an approximate value of π +1 is 414
Plot radic_
22 π + 1 and 4 1 _ 2 on a number line
Read the numbers from left to right to place them in order from least to greatest
From least to greatest the numbers are π + 1 4 1 _ 2 and radic_
22
EXAMPLE 2
STEP 1
STEP 2
Order the numbers from least to greatest Then graph them on the number line
YOUR TURN
5 radic_
5 25 radic_
3
6 π 2 10 radic_
75
If real numbers a b and c are in order from least to greatest what is the order
of their opposites from least to greatest
Explain
Math TalkMathematical Practices
8NS2
radic_
3 radic_
5 25
radic_
75 π2 10
Math Talk answer -c -b -a -c is farthest to the left on a number line -b is in the middle and -a is farthest to the right
Unit 122
copy H
ough
ton
Miff
lin H
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pany
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
8_MCAAESE206984_U1M01L3indd 22 41613 447 AM
My Notes
Math TrainerOnline Practice
and Help
Personal
myhrwcom
Math On the Spot
myhrwcom
Comparing Irrational NumbersBetween any two real numbers is another real number To compare and order real numbers you can approximate irrational numbers as decimals
Compare radic_
3 + 5 3 + radic_
5 Write lt gt or =
First approximate radic_
3
radic_
3 is between 1 and 2
Next approximate radic_
5
radic_
5 is between 2 and 3
Then use your approximations to simplify the expressions
radic_
3 + 5 is between 6 and 7
3 + radic_
5 is between 5 and 6
So radic_
3 + 5 gt 3 + radic_
5
Reflect1 If 7 + radic
_ 5 is equal to radic
_ 5 plus a number what do you know about the
number Why
2 What are the closest two integers that radic_
300 is between
EXAMPLEXAMPLE 1
STEP 1
STEP 2
Compare Write lt gt or =
YOUR TURN
3 radic_
2 + 4 2 + radic_
4 4 radic_
12 + 6 12 + radic_
6
L E S S O N
13 Ordering Real Numbers
ESSENTIAL QUESTIONHow do you order a set of real numbers
8NS2
Use rational approximations of irrational numbers to compare the size of irrational numbers locate them approximately on a number line diagram and estimate the value of expressions (eg π 2 )
8NS2
Use perfect squares to estimate square roots
1 2 = 1 2 2 = 4 3 2 = 9
The number is 7 both expressions must equal 7 + radic_
5
17 and 18
gt lt
21Lesson 13
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ough
ton
Miff
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ublis
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pany
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
8_MCAAESE206984_U1M01L3indd 21 41913 246 PM
PROFESSIONAL DEVELOPMENT
Math BackgroundIn this lesson students estimate irrational numbers in the form of square roots of nonper-fect squares by finding two perfect squares between which the number falls A more precise method involves repeated division For example to find radic
_ 28 find a whole number whose perfect
square is close to 28 such as 5 Divide 28 by that number 28 divide 5 = 56 Find the average of the quotient and divisor 5 + 56
_____ 2 = 53 Continue dividing 28 by each result and averaging until you get the desired accuracy
Integrate Mathematical Practices MP4
This lesson provides an opportunity to address this Mathematical Practices standard It calls for students to model relationships using multiple representations including diagrams graphs and language as appropriate Students use multiple representations when they use number lines to estimate the locations of and order rational and irrational numbers given as symbols
Ordering Real Numbers 22
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
Interactive Whiteboard Interactive example available online
ADDITIONAL EXAMPLE 3The diameter of a meteorite in millimeters is calculated by four different methods Order the results from least to greatest
Joe radic_
18 mm Lisa 13 __ 3 mm
Pablo 46 mm Julien 4π __ 3 mm
Julien 4π __ 3 mm Lisa 13 __ 3 mm
Joe radic_
18 mm Pablo 46 mm
EXAMPLE 3Questioning Strategies Mathematical Practices bull How can you verify that radic
_ 28 is between 52 and 53 5 2 2 = 2704 and 5 3 2 = 2809
bull Explain how to determine which number is greater 5 _
5 or 55 When the repeating decimal is rounded to the nearest tenth or hundredth you can see that it is greater
Connect to Daily LifeDiscuss how measuring across a canyon might involve different methods than measuring along a road Explain that measurements like these are often done using calculations that approximate the distance
YOUR TURNFocus on Critical Thinking Mathematical PracticesDiscuss with students which number is greater 3
_ 45 or 3450 3
_ 45 or 3455 and why Explain
that 3 _
45 can be written out as 34545hellipMake sure they understand that 3 _
45 is greater than 345 but less than 3455
ElaborateTalk About ItSummarize the Lesson
Ask How can you order two numbers in different forms whose decimal approxi-mations appear to be equal Approximate one or both numbers to an additional
number of decimal places
GUIDED PRACTICEEngage with the Whiteboard
Have students place and label additional points on the number line in Exercise 9 Allow the points to be in any format other than decimal
Avoid Common ErrorsExercises 3ndash4 Caution students to read the problem carefully so that they do not misread the problem as the same numbers combined by addition on each side of the circleExercise 10 Remind students that the calculations have units
myhrwcom
23 Lesson 13
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
0 05 1 15 2 25 3 35 4 45 5 55 6 65 7
2πradic3
Compare Write lt gt or = (Example 1)
1 radic_
3 + 2 radic_
3 + 3 2 radic_
8 + 17 radic_
11 + 15
3 radic_
6 + 5 6 + radic_
5 4 radic_
9 + 3 9 + radic_
3
5 radic_
17 - 3 -2 + radic_
5 6 12 - radic_
2 14 - radic_
8
7 radic_
7 + 2 radic_
10 - 1 8 radic_
17 + 3 3 + radic_
11
9 Order radic_
3 2π and 15 from least to greatest Then graph them on the number line (Example 2)
radic_
3 is between and so radic_
3 asymp
π asymp 314 so 2π asymp
From least to greatest the numbers are
10 Four people have found the perimeter of a forest using different methods Their results are given in the table Order their calculations from greatest to least (Example 3)
11 Explain how to order a set of real numbers
CHECK-INESSENTIAL QUESTION
Forest Perimeter (km)
Leon Mika Jason Ashley
radic_
17 - 2 1 +thinsp π __ 2 12 ___ 5 25
Guided Practice
17
15
1 + π _ 2 km 25 km 12 __ 5 km radic_
17 - 2 km
2π radic
_ 3
18 175
628
Sample answer Convert each number to a decimal
equivalent using estimation to find equivalents for
irrational numbers Graph each number on a number line
Read the numbers from left to right for least to greatest
Read the numbers from right to left for greatest to least
lt gt
lt lt
ltgt
gt gt
24 Unit 1
copy H
ough
ton
Miff
lin H
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ublis
hing
Com
pany
bull Im
age C
redi
ts copy
Elena
Eliss
eeva
Alam
y Im
ages
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
8_MCAAESE206984_U1M01L3indd 24 41613 448 AM
My Notes
5 52 54 56 58 6
radic28 5 12
23455
Math TrainerOnline Practice
and Help
Personal
myhrwcom
Math On the Spot
myhrwcom
Ordering Real Numbers in a Real-World Context Calculations and estimations in the real world may differ It can be important to know not only which are the most accurate but which give the greatest or least values depending upon the context
Four people have found the distance in kilometers across a canyon using different methods Their results are given in the table Order the distances from greatest to least
Distance Across Quarry Canyon (km)
Juana Lee Ann Ryne Jackson
radic_
28 23 __ 4 5 _
5 5 1 _ 2
Write each value as a decimal
radic_
28 is between 52 and 53 Since 53 2 = 2809 an approximate value for radic
_ 28 is 53
23 __ 4 = 575
5 _
5 is 5555hellip so 5 _
5 to the nearest hundredth is 556
5 1 _ 2 = 55
Plot radic_
28 23 __ 4 5 _
5 and 5 1 _ 2 on a number line
From greatest to least the distances are
23 __ 4 km 5 _
5 km 5 1 _ 2 km radic_
28 km
EXAMPLEXAMPLE 3
STEP 1
STEP 2
7 Four people have found the distance in miles across a crater using different methods Their results are given below
Jonathan 10 __ 3 Elaine 3 _
45 Joseacute 3 1 _ 2 Lashonda radic_
10
Order the distances from greatest to least
YOUR TURN
8NS2
3 1 _ 2 mi 3 _
45 mi 10 __ 3 mi radic_
10 mi
23Lesson 13
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ough
ton
Miff
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pany
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8_MCAAESE206984_U1M01L3indd 23 41613 447 AM
ModelingPlace papers around the room with the numbers from 1 to 5 one per sheet Give each student a card showing a number between 1 and 5 in different forms Have students place his or her card between the correct integers and decide where the number goes in relation to any numbers already placed
Multiple RepresentationsGive students a vertical number line which some students might find easier to use than a horizontal one Have them decide whether to place points for rational and irrational numbers above or below existing points
Additional ResourcesDifferentiated Instruction includes bull Reading Strategies bull Success for English Learners EL
bull Reteach bull Challenge PRE-AP
DIFFERENTIATE INSTRUCTION
Ordering Real Numbers 24
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
Math TrainerOnline Assessment
and Intervention
Personal
Online homework assignment available
myhrwcom
myhrwcom
Lesson Quiz available online
13 LESSON QUIZ
1 Compare Write lt gt or =
radic_
95 - 5 radic_
62 - 2
2 Order 105 radic_
105 and 3π + 1 from greatest to least
3 A length in centimeters is calculated differently by four different people Order their calculations from least to greatest
KD 11 __ 2 cm Silvio 5 __ 3 π cm
Paula 5 _
4 cm Luis radic_
33 cm
EvaluateGUIDED AND INDEPENDENT PRACTICE
Concepts amp Skills Practice
Example 1Comparing Irrational Numbers
Exercises 1ndash8
Example 2Ordering Real Numbers
Exercises 9 12ndash15 18ndash21
Example 3Ordering Real Numbers in a Real-World Context
Exercises 10 16ndash17
Additional ResourcesDifferentiated Instruction includes bull Leveled Practice worksheets
Focus | Coherence | Rigor
Exercise Depth of Knowledge (DOK) Mathematical Practices
12ndash15 1 Recall of Information MP5 Using Tools
16 2 SkillsConcepts MP2 Reasoning
17 2 SkillsConcepts MP6 Precision
18ndash21 2 SkillsConcepts MP2 Reasoning
22 3 Strategic Thinking MP4 Modeling
23ndash24 3 Strategic Thinking MP3 Logic
8NS2
8NS2
Answers1 radic
_ 95 - 5 lt radic
_ 62 - 2
2 radic_
105 3π + 1 105
3 Silvio 5 __ 3 π cm Paula 5 _
4 cm
KD 11
__ 2 cm Luis radic_
33 cm
25 Lesson 13
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
Work Area
3140 3141 3142 3143
314 π227
20 A teacher asks his students to write the numbers shown in order from least to greatest Paul thinks the numbers are already in order Sandra thinks the order should be reversed Who is right
21 Math History There is a famous irrational number called Eulerrsquos number symbolized with an e Like π its decimal form never ends or repeats The first few digits of e are 27182818284
a Between which two square roots of integers could you find this number
b Between which two square roots of integers can you find π
22 Analyze Relationships There are several approximations used for π including 314 and 22 __ 7 π is approximately 314159265358979
a Label π and the two approximations on the number line
b Which of the two approximations is a better estimate for π Explain
c Find a whole number x so that the ratio x ___ 113 is a better estimate for π
than the two given approximations
23 Communicate Mathematical Ideas If a set of six numbers that include both rational and irrational numbers is graphed on a number line what is the fewest number of distinct points that need to be graphed Explain
24 Critique Reasoning Jill says that 12 _
6 is less than 1263 Explain her error
FOCUS ON HIGHER ORDER THINKING
radic_
115 115 ___ 11 and 105624
between radic_
7 asymp 265 and radic_
8 asymp 283
between radic_
9 = 3 and radic_
10 asymp 316
22 __ 7 it is closer to π on the number line
She did not consider the repeating digit 1266
2 rational numbers can have the same location and
irrational numbers can have the same location but they
cannot share a location
355
Neither student is correct The answer
should be 115 ___ 11 105624 radic_
115
Unit 126
copy H
ough
ton M
ifflin
Har
cour
t Pub
lishin
g Com
pany
Imag
e Cre
dits
copy3D
Stoc
kiSt
ockP
hoto
com
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
8_MCAAESE206984_U1M01L3indd 26 210513 801 AM
Personal Math Trainer
Online Practice and Helpmyhrwcom
Name Class Date
Independent Practice
16 Your sister is considering two different shapes for her garden One is a square with side lengths of 35 meters and the other is a circle with a diameter of 4 meters
a Find the area of the square
b Find the area of the circle
c Compare your answers from parts a and b Which garden would give your sister the most space to plant
17 Winnie measured the length of her fatherrsquos ranch four times and got four different distances Her measurements are shown in the table
a To estimate the actual length Winnie first approximated each distance to the nearest hundredth Then she averaged the four numbers Using a calculator find Winniersquos estimate
b Winniersquos father estimated the distance across his ranch to be radic_
56 km How does this distance compare to Winniersquos estimate
Give an example of each type of number
18 a real number between radic_
13 and radic_
14
19 an irrational number between 5 and 7
Order the numbers from least to greatest
12 radic_
7 2 radic_
8 ___ 2 13 radic_
10 π 35
14 radic_
220 -10 radic_
100 115 15 radic_
8 -375 3 9 _ 4
Distance Across Fatherrsquos Ranch (km)
1 2 3 4
radic_
60 58 __ 8 7 _
3 7 3 _ 5
138NS2
radic_
8 ___ 2 2 radic_
7
-10 radic_
100 115 radic_
220
radic_
60 asymp 775 58 __ 8 = 725 7 _
3 asymp 733 7 3 _ 5 = 760 so the average
π radic_
10 35
-375 9 _ 4 radic_
8 3
is 74825 km
1225 m2
4π m2 or approximately 126 m2
They are nearly identical radic_
56 is approximately 74833hellip
The circle would give her more space to plant because it has a
larger area
Sample answer 37
Sample answer radic_
31
25Lesson 13
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ough
ton
Miff
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pany
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
8_MCAAESE206984_U1M01L3indd 25 41613 448 AM
Activity available online myhrwcomEXTEND THE MATH PRE-AP
Activity Have students investigate whether there are infinitely many numbers between two numbers by giving examples for each of the following
bull Between any two rational numbers there is at least one other rational number Sample answer 45 is between 41 and 48
bull Between any two irrational numbers there is at least one rational number Sample answer 45 is between radic
_ 11 and radic
_ 29
bull Between any two rational numbers there is at least one irrational number Sample answer radic
_ 11 is between 31 and 36
bull Between any two irrational numbers there is at least one irrational number Sample answer radic
_ 17 is between radic
_ 11 and radic
_ 29
Ordering Real Numbers 26
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
ReadyMath Trainer
Online Practiceand Help
Personal
myhrwcom
Module Quiz
11ensp RationalenspandenspIrrationalenspNumbersWrite each fraction as a decimal or each decimal as a fraction
1 7__20 2 1___
27 3 17_8
Solve each equation for x
4 x2=81 5 x3=343 6 x2= 1___100
7 Asquarepatiohasanareaof200squarefeetHowlongiseachside
ofthepatiotothenearesttenth
12ensp SetsenspofenspRealenspNumbersWrite all names that apply to each number
8 121____radic
____121
9 π__2
10 TellwhetherthestatementldquoAllintegersarerationalnumbersrdquoistrueorfalseExplainyourchoice
13ensp OrderingenspRealenspNumbersCompare Write lt gt or =
11 radic__
8+3 8+radic__
3 12 radic__
5+11emsp emsp emsp 5+radic___
11
Order the numbers from least to greatest
13 radic___
99π29__
8 14 radic___
1__251_40__
2
15 Howarerealnumbersusedtodescribereal-worldsituations
ESSENTIAL QUESTION
035
9-9
141ft
7 1__10- 1__10
14__11 1875
wholeintegerrationalreal
Trueintegerscanbewrittenasthequotientoftwointegers
SampleanswerRealnumberssuchastherational
π29__
8radic___
99
irrationalreal
lt gt
number1_4candescribeamountsusedincooking
radic___
1__250__
21_4
27Module1
copy H
ough
ton
Miff
lin H
arco
urt P
ublis
hing
Com
pany
DONOTEDIT--ChangesmustbemadethroughldquoFileinfordquoCorrectionKey=A
8_MCAAESE206984_U1M01RTindd 27 41513 1113 PM
Math TrainerOnline Assessment
and Intervention
Personal
myhrwcom
1
2
3 Response toIntervention
Intervention Enrichment
Access Ready to Go On assessment online and receive instant scoring feedback and customized intervention or enrichment
Online and Print Resources
Differentiated Instruction
bull Reteach worksheets
bull Reading Strategies EL
bull Success for English Learners EL
Differentiated Instruction
bull Challenge worksheets PRE-AP
Extend the Math PRE-AP
Lesson Activities in TE
Additional ResourcesAssessment Resources includes bull Leveled Module Quizzes
Ready to Go OnAssess MasteryUse the assessment on this page to determine if students have mastered the concepts and standards covered in this module
California Common Core Standards
Lesson Exercises Common Core Standards
11 1ndash7 8NS1 8NS2 8EE2
12 8ndash10 8NS1
13 11ndash14 8NS2
27 Unit 1 Module 1
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
Personal Math Trainer
Online Practice and HelpmyhrwcomAssessment Readiness
Module 1 MIXed ReVIeW
1 Look at each number Is the number between 2π and radic___
52
Select Yes or No for expressions AndashC
A 6 2 _ 3 Yes No
B 5π __ 2 Yes No
C 3 radic__
5 Yes No
2 Consider the number - 11 __ 15
Choose True or False for each statement
A The number is rational True False
B The number can be written as True Falsea repeating decimal
C The number is less than ndash08 True False
3 The volume of a cube is given by V = x3 where x is the length of an edge of the cube A cube-shaped end table has a volume of 3 3 _ 8 cubic feet What is the length of an edge of the end table Explain how you solved this problem
4 A student says that radic___
83 is greater than 29 __ 3 Is the student correct Justify your
reasoning
1 1 _ 2 ft Sample answer The equation x3 = 3 3 _ 8 can be used
to find the edge length in feet To solve the equation
write the mixed number as a fraction greater than 1
x3 = 27 __ 8 Then take the cube root of both sides x = 3 _ 2 = 1 1 _ 2
No Sample answer radic___
83 asymp 91 and 29 __ 3 = 9
__ 6
Because 91 lt 9 __
6 radic___
83 lt 29 __ 3
28 Unit 1
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ough
ton
Miff
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hing
Com
pany
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=A
8_MCAAESE206984_U1M01RTindd 28 240413 946 AM
Personal Math Trainer
Online Assessment and
Interventionmyhrwcom
Scoring GuideItem 3 Award the student 1 point for finding the edge length of the cube and 1 point for correctly explaining how to use a cube root to solve the problem
Item 4 Award the student 1 point for determining that the student is incorrect and 1 point for correctly justifying the reasoning for this conclusion
Additional ResourcesTo assign this assessment online login to your Assignment Manager at myhrwcom
Assessment Readiness
California Common Core Standards
Items Grade 8 Standards Mathematical Practices
1 8NS2 MP7
2 7NS2b 7NS2d 8NS1 MP7
3 8EE2 MP1 MP4
4 8NS1 8NS2 MP3
Item integrates mixed review concepts from previous modules or a previous course
Item 4 combines concepts from the California Common Core cluster ldquoKnow that there are numbers that are not rational and approximate them by rational numbersrdquo
Real Numbers 28
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
Work Area
33 Analyze Relationships To find radic_
15 Beau found 3 2 = 9 and 4 2 = 16 He said that since 15 is between 9 and 16 radic
_ 15 must be between 3 and 4 He
thinks a good estimate for radic_
15 is 3 + 4 ____ 2 = 35 Is Beaursquos estimate high low
or correct Explain
34 Justify Reasoning What is a good estimate for the solution to the equation x 3 = 95 How did you come up with your estimate
35 The volume of a sphere is 36π f t 3 What is the radius of the sphere Use the formula V = 4 _ 3 π r 3 to find your answer
36 Draw Conclusions Can you find the cube root of a negative number If so is it positive or negative Explain your reasoning
37 Make a Conjecture Evaluate and compare the following expressions
radic_
4 __ 25 and radic
_ 4 ____
radic_
25 radic
_
16 __ 81 and radic_
16 ____
radic_
81 radic
_
36 __ 49 and radic_
36 ____
radic_
49
Use your results to make a conjecture about a division rule for square roots Since division is multiplication by the reciprocal make a conjecture about a multiplication rule for square roots
38 Persevere in Problem Solving The difference between the solutions to the equation x 2 = a is 30 What is a Show that your answer is correct
FOCUS ON HIGHER ORDER THINKING
His estimate is low because 15 is very close to 16
so radic_
15 is very close to radic_
16 or 4 A better estimate
would be 38 or 39
Sample answer about 45 4 3 = 64 and 5 3 = 125
Because 95 is about halfway between 64 and 125 try 45
45 3 = 91125 which is a good estimate
3 feet
Yes the cube root of a negative number is negative
because a negative number cubed is always negative
and a nonnegative number cubed is always nonnegative
radic_
4 __ 25 = 2 _ 5 = radic
_ 4 ____
radic_
25 radic
_
16 __ 81 = 4 _ 9 = radic_
16 ____
radic_
81 radic
_
36 __ 49 = 6 _ 7 = radic_
36 ____
radic_
49
225 the solutions to x 2 = a are x = plusmn15 and
radic_
a ___
radic_
b = radic
_ a __
b radic
_ a radic
_ b = radic
_ a b
15 - (-15) = 30
Unit 114
copy H
ough
ton
Miff
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hing
Com
pany
bull copy
Ilen
e Mac
Dona
ldA
lamy I
mag
es
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=B
8_MCABESE206984_U1M01L1indd 14 102913 1142 PM
Personal Math Trainer
Online Practice and Helpmyhrwcom
Name Class Date
Independent Practice11
20 A 7 __ 16 -inch-long bolt is used in a machine What is this length written as a decimal
21 The weight of an object on the moon is 1 _ 6 its weight on Earth Write 1 _ 6 as a decimal
22 The distance to the nearest gas station is 2 4 _ 5 kilometers What is this distance written as a decimal
23 A baseball pitcher has pitched 98 2 _ 3 innings What is the number of innings written as a decimal
24 A heartbeat takes 08 second How many seconds is this written as a fraction
25 There are 262 miles in a marathon Write the number of miles using a fraction
26 The average score on a biology test was 72
_ 1 Write the average score using a
fraction
27 The metal in a penny is worth about 0505 cent How many cents is this written as a fraction
28 Multistep An artist wants to frame a square painting with an area of 400 square inches She wants to know the length of the wood trim that is needed to go around the painting
a If x is the length of one side of the painting what equation can you set up to find the length of a side How many solutions does the equation have
b Do all of the solutions that you found make sense in the context of the problem Explain
c What is the length of the wood trim needed to go around the painting
Solve each equation for x Write your answers as radical expressions Then estimate to one decimal place if necessary
29 x 2 = 14 30 x 3 = 1331
31 x 2 = 144 32 x 2 = 29
8NS1 8NS2 8EE2
04375 in 01 _6
28 km 98 _6 innings
x 2 = 400 x = plusmnthinsp20 the equation has 2 solutions
x = 20 makes sense but x = -20 doesnrsquot because a
painting cannot have a side length of -20 inches
4 times 20 = 80 inches
x = plusmn radic_
14 asymp plusmn37
x = plusmn radic_
144 = plusmn12 x = plusmn radic_
29 asymp plusmn54
x = 3 radic_ 1331 = 11
4_5 second 26 1_5 mi
72 1 _ 9 101 ___ 200 cent
13Lesson 11
copy H
ough
ton
Miff
lin H
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urt P
ublis
hing
Com
pany
bull copy
Phot
odisc
Get
ty Im
ages
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
8_MCAAESE206984_U1M01L1indd 13 41613 1211 AM
myhrwcomActivity available onlineEXTEND THE MATH PRE-AP
Activity Write radic_
09 on the board and invite students to conjecture what the value might be Have them check their conjectures by squaring Invite them to suggest ways to estimate radic
_ 09 As a hint point out that 09 is close to 10 and so they might
use that to help guide their estimates Lead them to see that since 092 is 081 and 102 is 1 the value of radic
_ 09 is greater than 09 and less than 10 Try squaring 095 to get
09025 A good estimate for radic_
09 is 095
Rational and Irrational Numbers 14
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=B
Integers
Rational Numbers IrrationalNumbers
Real Numbers
WholeNumbers
-3-4-5 -2-1 1 2 3 50 4
23
34-4 -π -1 25
radic2
Lesson Support Content Objective Students will learn to describe relationships between sets of numbers
Language Objective Students will explain how to describe relationships between sets of real numbers
LESSON 12 Sets of Real Numbers
Building BackgroundEliciting Prior Knowledge Have students draw a number line from -5 to 5 Ask them to plot points on the number line to approximate the location of rational and irrational numbers such as -1 3 __ 4 25 -4 2 __ 3 radic
_ 2 and -π
Learning ProgressionsIn this lesson students clarify their understanding of the real number system They characterize sets and subsets of the real numbers They also identify sets for real-world situations Important understandings for students include the following
bull Identify all of the possible subsets of the real numbers for a given number
bull Decide whether a statement about a subset of the real numbers is true or false
bull Identify the set of numbers that best describes a real-world situation
Understanding the relationships among the sets of numbers that make up the real numbers is essential as students are introduced to different forms of numbers throughout the school year This lesson provides a foundation for the comparing and ordering of real numbers in the next lesson
Cluster ConnectionsThis lesson provides an excellent opportunity to connect ideas in this cluster Know that there are numbers that are not rational and approximate them by rational numbers Have students copy this diagram which relates the sets of real numbers
Ask students to complete the diagram by writing three examples for each set of numbers Have students share examples and explain how they knew each number they selected belonged in the appropriate set Answers may vary Check studentsrsquo work
Focus | Coherence | Rigor
California Common Core Standards
8NS1 Know that numbers that are not rational are called irrational Understand informally that every number has a decimal expansion for rational numbers show that the decimal expansion repeats eventually and convert a decimal expansion which repeats eventually into a rational number
MP7 Look for and make use of structure
15A
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
Math Talk
Language Support EL
PROFESSIONAL DEVELOPMENT
Linguistic Support EL
AcademicContent Vocabulary
Venn diagrams ndash Students need descriptive language to describe the categories that the different areas and colors of a Venn diagram represent the concept of a set and how sets are distinct or can overlap Use sentence frames such as
The big oval represents __________The darklight blue color in the middle of the
big ovals represents __________These sets overlap because __________
In this way students have the language and structure to identify the criteria that distinguish a set and to explain the abstract representation Also point out the use of the prefix sub- meaning ldquounderrdquo in the term subset
Rules and Patterns
Abbreviations ndash In this lesson the abbreviation mph is used Be sure to point out that mph stands for miles per hour and is used to give units in a rate of speed Students may also have seen mpg (miles per gallon) which gives the units in a rate of fuel efficiency
Borrowed Words ndash Terminology used in baseball such as inning and pitcher may require some explanation Spanish as well as some other languages have borrowed these terms from English so some students may be familiar with these words already Despite this whenever a word is critical to students understanding the word problem it is best to explain the meaning
Leveled Strategies for English Learners
Emerging Allow students to indicate true or false orally in Guided Practice Exercises 9 and 10
Expanding Have students use sentence frames to describe the meaning of regions and colors used in a Venn diagram Then give them similar sentence frames orally and have them draw and shade a Venn diagram based on the oral prompts
Bridging Have students work in groups to draw a Venn diagram to represent sets based on real-world examples in the lesson
To help students answer the question posed in Math Talk provide a sentence frame for their answer
The numbers between 31 and 39 on a number line are __________ because __________
EL
California ELD Standards
Emerging 2II5 Modifying to add details ndash Expand sentences with simple adverbials to provide details about a familiar activity or process
Expanding 2II5 Modifying to add details ndash Expand sentences with adverbials to provide details about a familiar or new activity or process
Bridging 2II5 Modifying to add details ndash Expand sentences with increasingly complex adverbials to provide details about a variety of familiar and new activities and processes
Sets of Real Numbers 15B
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
12L E S S O N
Sets of Real Numbers
EngageESSENTIAL QUESTION
How can you describe relationships between sets of real numbers Sample answer Describe them as two different sets or one set as being a subset of another
Motivate the LessonAsk How many different types of tigers can you name How does the set of Bengal tigers relate to the set of tigers
ExplorePoint to different locations in the Animals diagram and ask for examples for that classification Do the same for the Real Numbers diagram Students should understand that everything within a region is part of the set for example both -3 and 2 are integers
ExplainEXAMPLE 1
Questioning Strategies Mathematical Practices bull In A why is 5 not a perfect square It does not have rational numbers as its square roots
bull Can the number in B be written as a fraction Why or why not Yes it is a terminating decimal so it is a rational number
Engage with the WhiteboardHave students place the numbers in Example 1 and Additional Example 1 in the Venn diagram for numbers
YOUR TURNAvoid Common ErrorsBe sure that students read Exercise 2 carefully before answering The number given in the problem 10 is the area not the side length
EXAMPLE 2Questioning Strategies Mathematical Practices bull What two major sets are the real numbers composed of rational and irrational numbers
bull What is the location of the set of whole numbers in the Venn diagram in relation to the set of rational numbers Explain Inside it whole numbers are rational numbers
Focus on Reasoning Mathematical PracticesRemind students that it takes only one counterexample to show that a statement is false
Interactive Whiteboard Interactive example available online
ADDITIONAL EXAMPLE 1Write all names that apply to each number
A -10integer rational real
B 12 _ 3
whole integer rational real
myhrwcom
Interactive Whiteboard Interactive example available online
ADDITIONAL EXAMPLE 2Tell whether the given statement is true or false Explain your choice
No integers are whole numbers
False every whole number is also an integer
myhrwcom
Animated MathClassifying Numbers
Students build fluency in classifying numbers in this engaging fast-paced game
myhrwcom
CA Common CoreStandards
The student is expected to
The Number Systemmdash8NS1
Know that numbers that are not rational are called irrational Understand informally that every number has a decimal expansion for rational numbers show that the decimal expansion repeats eventually and convert a decimal expansion which repeats eventually into a rational numberMathematical Practices
MP7 Using Structure
The student is expected to
15 Lesson 12
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=B
Math TrainerOnline Practice
and Help
Personal
myhrwcom
Math TrainerOnline Practice
and Help
Personal
myhrwcom
Math On the Spotmyhrwcom
Understanding Sets and Subsets of Real NumbersBy understanding which sets are subsets of types of numbers you can verify whether statements about the relationships between sets are true or false
Tell whether the given statement is true or false Explain your choice
All irrational numbers are real numbers
True Every irrational number is included in the set of real numbers The irrational numbers are a subset of the real numbers
No rational numbers are whole numbers
False A whole number can be written as a fraction with a denominator of 1 so every whole number is included in the set of rational numbers The whole numbers are a subset of the rational numbers
EXAMPLE 2
A
B
Write all names that apply to each number
1 A baseball pitcher has pitched 12 2 _ 3 innings
2 The length of the side of a square that has an
area of 10 square yards
YOUR TURN
Tell whether the given statement is true or false Explain your choice
3 All rational numbers are integers
4 Some irrational numbers are integers
YOUR TURN
Give an example of a rational number that is a
whole number Show that the number is both whole
and rational
Math TalkMathematical Practices
Give an example of a
8NS1
False Every integer is a rational number but not every
False Real numbers are either rational or irrational numbers
Integers are rational numbers so no integers are irrational numbers
rational real
irrational real
Sample answer 8 8 = 8_
1
and -thinsp 5 _ 2 are not integers
rational number is an integer Rational numbers such as 3 _ 5
Unit 116
copy H
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pany
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age C
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igita
l Im
age c
opyr
ight
copy20
04 Ey
ewire
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8_MCAAESE206984_U1M01L2indd 16 41613 136 AM
Math On the Spot
myhrwcom
Vertebrates
Birds
Passerines
Animals
Integers
Rational Numbers IrrationalNumbers
Real Numbers
WholeNumbers
1
45
3
0
274
67
radic4
-
-3
-2
-1
03
radic2
radic17
radic11-
π
Animated Math
myhrwcom
Classifying Real NumbersBiologists classify animals based on shared characteristics A cardinal is an animal a vertebrate a bird and a passerine
You already know that the set of rational numbers consists of whole numbers integers and fractions The set of real numbers consists of the set of rational numbers and the set of irrational numbers
Write all names that apply to each number
radic_
5 irrational real
ndash1784rational real
whole integer rational real
EXAMPLEXAMPLE 1
A
B
C radic_ 81 ____ 9
L E S S O N
12Sets of Real Numbers
ESSENTIAL QUESTIONHow can you describe relationships between sets of real numbers
Passerines such as the cardinal are also called ldquoperching birdsrdquo
What types of numbers are between 31 and 39 on a
number line
Math TalkMathematical Practices
What types of numbers are
8NS1
8NS1
Know that numbers that are not rational are called irrational Understand informally that every number has a decimal expansion for rational numbers show that the decimal expansion repeats eventually and convert a decimal expansion which repeats eventually into a relation number
ndash1784 is a terminating decimal
5 is a whole number that is not a perfect square
radic_
81 _____ 9 = 9 __ 9 = 1 rational irrational real
15Lesson 12
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pany
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age C
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ts copy
Wiki
med
ia Co
mm
ons
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=B
8_MCABESE206984_U1M01L2indd 15 061113 1144 AM
PROFESSIONAL DEVELOPMENT
Math BackgroundThe relationships between sets of numbers extend to include complex numbers A complex number can be written as a sum of a real number a and an imaginary number bi
a + bi
An imaginary number is a special number that when squared gives a negative value When you square a real number you get a nonnegative number When you square an imaginary number you get a negative value The imaginary unit is i
i = radic_
-1
Integrate Mathematical Practices MP7
This lesson provides an opportunity to address this Mathematical Practices standard It calls for students to discern structure to connect and communicate mathematical ideas
Students use a Venn diagram to structure relationships between sets of numbers They connect and communicate mathematical ideas when they make logical statements about the sets and describe which set best describes numbers applied to real-life situations
Sets of Real Numbers 16
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=B
YOUR TURNAvoid Common ErrorsStudents may see the word ldquoAllldquo or rdquoNordquo in Exercises 3 and 4 and immediately assume that any absolute statements like these are false Remind them that there are true statements that begin with these words and encourage them to provide examples
EXAMPLE 3Questioning Strategies Mathematical Practices bull In A how does the phrase ldquonumber of rdquo give you a clue about the number classification It indicates a counting number
bull What is the relationship between the circumference of a circle and the diameter The circumference is diameter times π
Focus on Critical Thinking Mathematical PracticesIn B suppose the diameters in inches were 25
__ π 28 __ π
31 __ π and so on What set of numbers would
best describe the circumferences Explain Whole numbers the circumferences would be the whole numbers 25 28 31 and so on
YOUR TURNFocus on Critical Thinking Mathematical PracticesHave students compare and contrast the classification of numbers in the answers in Exercises 5 and 6
ElaborateTalk About ItSummarize the Lesson
Ask What are some ways that number sets can be related Sets may be subsets of other sets or they may be separate from other sets
GUIDED PRACTICEEngage with the Whiteboard
Have students place the numbers in Exercises 1ndashthinsp8 in the Venn diagram for numbers at the beginning of the lesson
Integrating Language Arts EL
Encourage English learners to ask for clarification on any terms or phrases that they do not understand
Avoid Common ErrorsExercise 7 Remind students that a repeating decimal is a rational numberExercises 9ndash10 Remind students that it only takes one counterexample to show that a statement is false
Interactive Whiteboard Interactive example available online
ADDITIONAL EXAMPLE 3Identify the set of numbers that best describes the situation Explain your choice
A the amount of time that has passed since midnight
The set of real numbers time is continuous so the amount of time can be rational or irrational
B the number of tickets sold to a basketball game
The set of whole numbers the number of tickets sold may be 0 or a counting number
myhrwcom
17 Lesson 12
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
1IN
116 inch
Guided Practice
Write all names that apply to each number (Example 1)
1 7 _ 8 2 radic_
36
3 radic_
24 4 075
5 0 6 - radic_ 100
7 5 _
45 8 - 18 __ 6
Tell whether the given statement is true or false Explain your choice (Example 2)
9 All whole numbers are rational numbers
10 No irrational numbers are whole numbers
Identify the set of numbers that best describes each situation Explain your choice (Example 3)
11 the change in the value of an account when given to the nearest dollar
12 the markings on a standard ruler
13 What are some ways to describe the relationships between sets of numbers
CHECK-INESSENTIAL QUESTION
rational real
rational real
True Whole numbers are rational numbers
Rational numbers the ruler is marked every 1 __ 16 th inch
Sample answer Describe one set as being a subset of
another or show their relationships in a Venn diagram
Integers the change can be a whole dollar amount
and can be positive negative or zero
True Whole numbers are a subset of the set of rational numbers
and can be written as a ratio of the whole number to 1
irrational real
whole integer rational real
whole integer rational real
rational real
integer rational real
integer rational real
Unit 118
copy H
ough
ton
Miff
lin H
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pany
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8_MCAAESE206984_U1M01L2indd 18 41613 136 AM
My Notes
Math TrainerOnline Practice
and Help
Personal
myhrwcom
Math On the Spot
myhrwcom
Identifying Sets for Real-World SituationsReal numbers can be used to represent real-world quantities Highways have posted speed limit signs that are represented by natural numbers such as 55 mph Integers appear on thermometers Rational numbers are used in many daily activities including cooking For example ingredients in a recipe are often given in fractional amounts such as 2 _ 3 cup flour
Identify the set of numbers that best describes each situation Explain your choice
the number of people wearing glasses in a room
The set of whole numbers best describes the situation The number of people wearing glasses may be 0 or a counting number
the circumference of a flying disk has a diameter of 8 9 10 11 or 14 inches
The set of irrational numbers best describes the situation Each circumference would be a product of π and the diameter and any multiple of π is irrational
EXAMPLEXAMPLE 3
A
B
Identify the set of numbers that best describes the situation Explain your choice
5 the amount of water in a glass as it evaporates
6 the weight of a person in pounds
YOUR TURN
8NS1
Rational numbers a personrsquos weight can be a decimal
such as 835 pounds
Real numbers the amount can be any number greater
than 0
17Lesson 12
copy H
ough
ton
Miff
lin H
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hing
Com
pany
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
8_MCAAESE206984_U1M01L2indd 17 41613 520 AM
Graphic OrganizersGive students a list of numbers (including terminating and repeating decimals fractions integers and rational and irrational square roots) and a graphic organizer as shown below
Real Numbers
Rational numbers Irrational numbers
Integer numbers
Whole numbers
Ask students to write each number in the list in the correct section of the organizer
Number SensePoint out to students that knowing the types of numbers to expect in different situations can alert them to incorrect math as well as to impossible situations For example 135 shots made in basketballs is not possible but an average number of shots can equal 135
Additional ResourcesDifferentiated Instruction includes bull Reading Strategies bull Success for English Learners EL
bull Reteach bull Challenge PRE-AP
DIFFERENTIATE INSTRUCTION
Sets of Real Numbers 18
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
Math TrainerOnline Assessment
and Intervention
Personal
Online homework assignment available
myhrwcom
Lesson Quiz available online
12 LESSON QUIZ
1 Write all the names that apply to the number
2 Tell whether the given statement is true or false Explain your choice All numbers between 1 and 2 are rational numbers
3 Identify the set of numbers that best describes the situation Explain your choiceThe choices on a survey question change the total points for the survey by -2 -1 0 1 or 2 points
-1 _
5
myhrwcom
Additional ResourcesDifferentiated Instruction includes bull Leveled Practice worksheets
EvaluateGUIDED AND INDEPENDENT PRACTICE
Concepts amp Skills Practice
Example 1Classifying Real Numbers
Exercises 1ndash8 14ndash19 22ndash24
Example 2Understanding Sets and Subsets of Real Numbers
Exercises 9ndash10
Example 3Identifying Sets for Real-World Situations
Exercises 11ndash12 20ndash21 25
Focus | Coherence | Rigor
Exercise Depth of Knowledge (DOK) Mathematical Practices
14ndash19 2 SkillsConcepts MP7 Using Structure
20ndash21 2 SkillsConcepts MP6 Precision
22ndash23 2 SkillsConcepts MP3 Logic
24 1 Recall of Information MP7 Using Structure
25 2 SkillsConcepts MP2 Reasoning
26ndash27 3 Strategic Thinking MP3 Logic
28 3 Strategic Thinking MP8 Patterns
29 3 Strategic Thinking MP3 Logic
8NS1
8NS1
Exercise 29 combines concepts from the California Common Core cluster ldquoKnow that there are numbers that are not rational and approximate them by rational numbersrdquo
Answers1 rational real
2 False radic_
2 is an example of an irrational number between 1 and 2
3 Integers each number is an integer but only three are whole numbers
19 Lesson 12
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
Work Area
π mi23 Critique Reasoning The circumference of a circular region is shown
What type of number best describes the diameter of the circle Explain
your answer
24 Critical Thinking A number is not an integer What type of number can it be
25 A grocery store has a shelf with half-gallon containers of milk What type of number best represents the total number of gallons
26 Explain the Error Katie said ldquoNegative numbers are integersrdquo What was her error
27 Justify Reasoning Can you ever use a calculator to determine if a number is rational or irrational Explain
28 Draw Conclusions The decimal 0 _
3 represents 1 _ 3 What type of number best describes 0
_ 9 which is 3 middot 0
_ 3 Explain
29 Communicate Mathematical Ideas Irrational numbers can never be precisely represented in decimal form Why is this
FOCUS ON HIGHER ORDER THINKING
It can be a rational number that is not an integer or an irrational number
rational number
The set of negative numbers also includes non-integer
rational numbers and irrational numbers
Sample answer If the calculator shows a decimal that
terminates in fewer digits than what the calculator screen
allows then you can tell that the number is rational If not
you cannot tell from the calculator display whether the
number terminates because you see a limited number
of digits It may be a repeating decimal (rational) or
non-terminating non-repeating decimal (irrational)
Whole 3 middot 0 _
3 represents 3 middot 1 _ 3 = 1 so 0 _
9 is exactly 1
Sample answer In decimal form irrational numbers never
terminate and never repeat Therefore no matter how
many decimal places you include the number will never
be precisely represented There are always more digits
Whole the diameter is π _ π = 1 mile
Unit 120
copy H
ough
ton
Miff
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hing
Com
pany
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
8_MCAAESE206984_U1M01L2indd 20 120413 909 PM
Integers
Rational Numbers Irrational Numbers
Real Numbers
Whole Numbers
257
radic16
166
radic9
128 radic50
Personal Math Trainer
Online Practice and Helpmyhrwcom
Name Class Date
Independent Practice
Identify the set of numbers that best describes each situation Explain your choice
20 the height of an airplane as it descends to an airport runway
21 the score with respect to par of several golfers 2 ndash 3 5 0 ndash 1
22 Critique Reasoning Ronald states that the number 1 __ 11 is not rational because when converted into a decimal it does not terminate Nathaniel says it is rational because it is a fraction Which boy is correct Explain
12
14 - radic_
9 15 257
16 radic_
50 17 8 1 _ 2
18 166 19 radic_
16
Write all names that apply to each number Then place the numbers in the correct location on the Venn diagram
8NS1
Real numbers the height can be any number greater than zero
integer rational real whole integer rational real
whole integer rational real
irrational real
rational real
rational real
Integers the scores are counting numbers their
opposites and zero
Nathaniel is correct A rational number is a number that can be written as a fraction and 1 __ 11 is a fraction
19Lesson 12
copy H
ough
ton
Miff
lin H
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urt P
ublis
hing
Com
pany
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
8_MCAAESE206984_U1M01L2indd 19 41613 136 AM
myhrwcomActivity available onlineEXTEND THE MATH PRE-AP
Activity Have students consider the concept of restricted domain for the sets of numbers that describe situations For example the number of sisters a person has can best be described by whole numbers but no one has ever had 1500 sisters An area code is an integer or whole number between 200 and 999
Have students use a source such as the Guinness Book of World Records and give examples of sets of numbers that describe situations where the domain is restricted Ask whether the restriction may be changed in the future
Sets of Real Numbers 20
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
-3-4-5 -2-1 1 2 3 50 4
12-4 -radic5
Lesson Support Content Objective Students will learn to order a set of real numbers
Language Objective Students will show and describe how to order a set of real numbers
LESSON 13 Ordering Real Numbers
Building BackgroundEliciting Prior Knowledge Have students draw a number line to compare a rational number and an irrational number such as - radic
_ 5 and -4 1 __ 2 Ask them to explain how
they approximated the irrational number on the number line Then have them identify the greater and the lesser real number Repeat with several other pairs of real numbers in different forms
Learning ProgressionsIn this lesson students order a set of real numbers They use rational approximations to compare the sizes of irrational numbers They also order numbers for real-world situations Important understandings for students include the following
bull Compare irrational numbers bull Estimate the value of expressions with irrational numbers bull Order a set of real numbers bull Order real numbers in a real-world context
Work with real numbers continues throughout Grade 8 and into high school This lesson provides students with a foundation for understanding the relative sizes of numbers in different forms in the real number system
Cluster ConnectionsThis lesson provides an excellent opportunity to connect ideas in this cluster Know that there are numbers that are not rational and approximate them by rational numbers Tell students that there is a special number called the golden ratio with applications in mathematics geometry art and architecture The golden ratio is called phi and is represented by the Greek letter ϕ It includes an irrational number in its definition
Have students explain why the golden ratio is irrational Ask them to find the two whole numbers the golden ratio lies between Then challenge them to approximate the golden ratio to the nearest tenth It is irrational because it includes an irrational number in its definition It lies between 1 and 2 To the nearest tenth ϕ = 16
ϕ = 1 + radic_
5 _ 2
Focus | Coherence | Rigor
California Common Core Standards
8NS2 Use rational approximations of irrational numbers to compare the size of irrational numbers locate them approximately on a number line diagram and estimate the value of expressions (eg π 2 )
MP4 Model with mathematics
21A
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
Math Talk
Language Support EL
PROFESSIONAL DEVELOPMENT
Linguistic Support EL
AcademicContent Vocabulary
Post a chart like this to remind students of the regular comparative forms of adjectives that use the -er and -est suffixes Add to the chart for terms that appear in examples and exercises in each lesson Include any irregular verb forms
Background Knowledge
Go On ndash the title of the module review or quiz is Ready to Go On This title uses an idiomatic expression In this context to go on means ldquoto move aheadrdquo or ldquoto proceedrdquo It is different from the use of go on that means having enough facts to use meaningfully as in having enough to go on Also the intonation used in pronouncing an expression can give it different meanings For example when the speaker emphasizes the word on he or she might be expressing disbelief as in ldquoGo ON Yoursquore kidding rightrdquo Discuss with students other ways that the phrase go on may be used
Leveled Strategies for English Learners
Emerging Label points on a number line with the terms used in ordering greater greatest less lesser least Use sentence frames to insert the correct terms
Expanding Have students give two or three complete sentences to compare the placement of numbers on a number line using the correct forms of the comparative and superlative adjectives
Bridging Have students work in pairs with one student giving directions to the other in complete sentences to order numbers on a number line
To help students answer the question posed in Math Talk make sure that students have a command of the forms for making comparisons and the superlative and the concept of opposite order so that the focus is on the math concept instead of the language skills needed to describe and explain order
EL
Adjective Comparative Superlative
Far Farther Farthest
Large Larger Largest
Great Greater Greatest
Some Less Least
Some More Most
California ELD Standards
Emerging 2I8 Analyzing language choices ndash Explain how phrasing or different common words with similar meanings produce different effects on the audience
Expanding 2I8 Analyzing language choices ndash Explain how phrasing or different words with similar meanings or figurative language produce shades of meaning and different effects on the audience
Bridging 2I8 Analyzing language choices ndash Explain how phrasing or different words with similar meanings or figurative language produce shades of meaning nuances and different effects on the audience
Ordering Real Numbers 21B
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
13L E S S O N
Ordering Real Numbers
Interactive Whiteboard Interactive example available online
ADDITIONAL EXAMPLE 1Compare Write lt gt or =
A radic_
8 - 2 4 - radic_
8 lt
B radic_
20 + 1 3 + radic_
2 gt
EngageESSENTIAL QUESTION
How do you order a set of real numbers Sample answer Find their approximate decimal values and order them
Motivate the LessonAsk What kind of numbers are you comparing when you compare the price of gasoline at two different gas stations
ExploreGive students two rational numbers and ask them to name a number between them Repeat a few times and then give them two irrational numbers and ask them to name a number between them
ExplainEXAMPLE 1
Questioning Strategies Mathematical Practices bull Which is greater the difference between 5 and 3 or the difference between radic
_ 5 and radic
_ 3
The difference between 5 and 3 is 2 the difference between radic_
5 and radic_
3 is approximately 1 So the difference between 5 and 3 is greater
Avoid Common ErrorsCaution students to read the problem carefully and think about what the radical sign means so that they do not misread the problem and answer that the two sides are equal
YOUR TURNFocus on TechnologyCalculators should not be used at this point because developing number sense is the goal
EXAMPLE 2Questioning Strategies Mathematical Practices bull How do you determine whether radic
_ 22 is less than or greater than 45 The square of 45 is
2025 which is less than 22 so the square root of 22 must be greater than 45
Engage with the WhiteboardHave students graph and label various real numbers between 42 and 44 and between 47 and 5
YOUR TURNFocus on Modeling Mathematical PracticesHave students label the integers on the number line with their equivalent square root For example 1 2 and 3 on the number line would be labeled radic
_ 1 radic
_ 4 and radic
_ 9
Interactive Whiteboard Interactive example available online
ADDITIONAL EXAMPLE 2Order 3π radic
_ 10 and 325 from greatest
to least
3π 325 radic_
10
myhrwcom
myhrwcom
CA Common CoreStandards
The student is expected to
The Number Systemmdash8NS2
Use rational approximations of irrational numbers to compare the size of irrational numbers locate them approximately on a number line diagram and estimate the value of expressions (eg π 2 )
Mathematical Practices
MP4 Modeling
The student is expected to
21 Lesson 13
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
Math TrainerOnline Practice
and Help
Personal
myhrwcom
Math On the Spotmyhrwcom
0 05 1 15 2 25 3 35 4
radic5radic3
π2
8 85 9 95 10 105 11 115 12
radic75
4 42 44 46 48 5
radic224 12π + 1
Ordering Real Numbers You can compare and order real numbers and list them from least to greatest
Order radic_
22 π + 1 and 4 1 _ 2 from least to greatest
First approximate radic_
22
radic_
22 is between 4 and 5 Since you donrsquot know where it falls between 4 and 5 you need to find a better estimate for radic
_ 22 so
you can compare it to 4 1 _ 2
Since 22 is closer to 25 than 16 use squares of numbers between 45 and 5 to find a better estimate of radic
_ 22
45 2 = 2025 46 2 = 2116 47 2 = 2209 48 2 = 2304
Since 47 2 = 2209 an approximate value for radic_
22 is 47
An approximate value of π is 314 So an approximate value of π +1 is 414
Plot radic_
22 π + 1 and 4 1 _ 2 on a number line
Read the numbers from left to right to place them in order from least to greatest
From least to greatest the numbers are π + 1 4 1 _ 2 and radic_
22
EXAMPLE 2
STEP 1
STEP 2
Order the numbers from least to greatest Then graph them on the number line
YOUR TURN
5 radic_
5 25 radic_
3
6 π 2 10 radic_
75
If real numbers a b and c are in order from least to greatest what is the order
of their opposites from least to greatest
Explain
Math TalkMathematical Practices
8NS2
radic_
3 radic_
5 25
radic_
75 π2 10
Math Talk answer -c -b -a -c is farthest to the left on a number line -b is in the middle and -a is farthest to the right
Unit 122
copy H
ough
ton
Miff
lin H
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urt P
ublis
hing
Com
pany
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
8_MCAAESE206984_U1M01L3indd 22 41613 447 AM
My Notes
Math TrainerOnline Practice
and Help
Personal
myhrwcom
Math On the Spot
myhrwcom
Comparing Irrational NumbersBetween any two real numbers is another real number To compare and order real numbers you can approximate irrational numbers as decimals
Compare radic_
3 + 5 3 + radic_
5 Write lt gt or =
First approximate radic_
3
radic_
3 is between 1 and 2
Next approximate radic_
5
radic_
5 is between 2 and 3
Then use your approximations to simplify the expressions
radic_
3 + 5 is between 6 and 7
3 + radic_
5 is between 5 and 6
So radic_
3 + 5 gt 3 + radic_
5
Reflect1 If 7 + radic
_ 5 is equal to radic
_ 5 plus a number what do you know about the
number Why
2 What are the closest two integers that radic_
300 is between
EXAMPLEXAMPLE 1
STEP 1
STEP 2
Compare Write lt gt or =
YOUR TURN
3 radic_
2 + 4 2 + radic_
4 4 radic_
12 + 6 12 + radic_
6
L E S S O N
13 Ordering Real Numbers
ESSENTIAL QUESTIONHow do you order a set of real numbers
8NS2
Use rational approximations of irrational numbers to compare the size of irrational numbers locate them approximately on a number line diagram and estimate the value of expressions (eg π 2 )
8NS2
Use perfect squares to estimate square roots
1 2 = 1 2 2 = 4 3 2 = 9
The number is 7 both expressions must equal 7 + radic_
5
17 and 18
gt lt
21Lesson 13
copy H
ough
ton
Miff
lin H
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pany
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8_MCAAESE206984_U1M01L3indd 21 41913 246 PM
PROFESSIONAL DEVELOPMENT
Math BackgroundIn this lesson students estimate irrational numbers in the form of square roots of nonper-fect squares by finding two perfect squares between which the number falls A more precise method involves repeated division For example to find radic
_ 28 find a whole number whose perfect
square is close to 28 such as 5 Divide 28 by that number 28 divide 5 = 56 Find the average of the quotient and divisor 5 + 56
_____ 2 = 53 Continue dividing 28 by each result and averaging until you get the desired accuracy
Integrate Mathematical Practices MP4
This lesson provides an opportunity to address this Mathematical Practices standard It calls for students to model relationships using multiple representations including diagrams graphs and language as appropriate Students use multiple representations when they use number lines to estimate the locations of and order rational and irrational numbers given as symbols
Ordering Real Numbers 22
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
Interactive Whiteboard Interactive example available online
ADDITIONAL EXAMPLE 3The diameter of a meteorite in millimeters is calculated by four different methods Order the results from least to greatest
Joe radic_
18 mm Lisa 13 __ 3 mm
Pablo 46 mm Julien 4π __ 3 mm
Julien 4π __ 3 mm Lisa 13 __ 3 mm
Joe radic_
18 mm Pablo 46 mm
EXAMPLE 3Questioning Strategies Mathematical Practices bull How can you verify that radic
_ 28 is between 52 and 53 5 2 2 = 2704 and 5 3 2 = 2809
bull Explain how to determine which number is greater 5 _
5 or 55 When the repeating decimal is rounded to the nearest tenth or hundredth you can see that it is greater
Connect to Daily LifeDiscuss how measuring across a canyon might involve different methods than measuring along a road Explain that measurements like these are often done using calculations that approximate the distance
YOUR TURNFocus on Critical Thinking Mathematical PracticesDiscuss with students which number is greater 3
_ 45 or 3450 3
_ 45 or 3455 and why Explain
that 3 _
45 can be written out as 34545hellipMake sure they understand that 3 _
45 is greater than 345 but less than 3455
ElaborateTalk About ItSummarize the Lesson
Ask How can you order two numbers in different forms whose decimal approxi-mations appear to be equal Approximate one or both numbers to an additional
number of decimal places
GUIDED PRACTICEEngage with the Whiteboard
Have students place and label additional points on the number line in Exercise 9 Allow the points to be in any format other than decimal
Avoid Common ErrorsExercises 3ndash4 Caution students to read the problem carefully so that they do not misread the problem as the same numbers combined by addition on each side of the circleExercise 10 Remind students that the calculations have units
myhrwcom
23 Lesson 13
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
0 05 1 15 2 25 3 35 4 45 5 55 6 65 7
2πradic3
Compare Write lt gt or = (Example 1)
1 radic_
3 + 2 radic_
3 + 3 2 radic_
8 + 17 radic_
11 + 15
3 radic_
6 + 5 6 + radic_
5 4 radic_
9 + 3 9 + radic_
3
5 radic_
17 - 3 -2 + radic_
5 6 12 - radic_
2 14 - radic_
8
7 radic_
7 + 2 radic_
10 - 1 8 radic_
17 + 3 3 + radic_
11
9 Order radic_
3 2π and 15 from least to greatest Then graph them on the number line (Example 2)
radic_
3 is between and so radic_
3 asymp
π asymp 314 so 2π asymp
From least to greatest the numbers are
10 Four people have found the perimeter of a forest using different methods Their results are given in the table Order their calculations from greatest to least (Example 3)
11 Explain how to order a set of real numbers
CHECK-INESSENTIAL QUESTION
Forest Perimeter (km)
Leon Mika Jason Ashley
radic_
17 - 2 1 +thinsp π __ 2 12 ___ 5 25
Guided Practice
17
15
1 + π _ 2 km 25 km 12 __ 5 km radic_
17 - 2 km
2π radic
_ 3
18 175
628
Sample answer Convert each number to a decimal
equivalent using estimation to find equivalents for
irrational numbers Graph each number on a number line
Read the numbers from left to right for least to greatest
Read the numbers from right to left for greatest to least
lt gt
lt lt
ltgt
gt gt
24 Unit 1
copy H
ough
ton
Miff
lin H
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ublis
hing
Com
pany
bull Im
age C
redi
ts copy
Elena
Eliss
eeva
Alam
y Im
ages
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
8_MCAAESE206984_U1M01L3indd 24 41613 448 AM
My Notes
5 52 54 56 58 6
radic28 5 12
23455
Math TrainerOnline Practice
and Help
Personal
myhrwcom
Math On the Spot
myhrwcom
Ordering Real Numbers in a Real-World Context Calculations and estimations in the real world may differ It can be important to know not only which are the most accurate but which give the greatest or least values depending upon the context
Four people have found the distance in kilometers across a canyon using different methods Their results are given in the table Order the distances from greatest to least
Distance Across Quarry Canyon (km)
Juana Lee Ann Ryne Jackson
radic_
28 23 __ 4 5 _
5 5 1 _ 2
Write each value as a decimal
radic_
28 is between 52 and 53 Since 53 2 = 2809 an approximate value for radic
_ 28 is 53
23 __ 4 = 575
5 _
5 is 5555hellip so 5 _
5 to the nearest hundredth is 556
5 1 _ 2 = 55
Plot radic_
28 23 __ 4 5 _
5 and 5 1 _ 2 on a number line
From greatest to least the distances are
23 __ 4 km 5 _
5 km 5 1 _ 2 km radic_
28 km
EXAMPLEXAMPLE 3
STEP 1
STEP 2
7 Four people have found the distance in miles across a crater using different methods Their results are given below
Jonathan 10 __ 3 Elaine 3 _
45 Joseacute 3 1 _ 2 Lashonda radic_
10
Order the distances from greatest to least
YOUR TURN
8NS2
3 1 _ 2 mi 3 _
45 mi 10 __ 3 mi radic_
10 mi
23Lesson 13
copy H
ough
ton
Miff
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pany
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8_MCAAESE206984_U1M01L3indd 23 41613 447 AM
ModelingPlace papers around the room with the numbers from 1 to 5 one per sheet Give each student a card showing a number between 1 and 5 in different forms Have students place his or her card between the correct integers and decide where the number goes in relation to any numbers already placed
Multiple RepresentationsGive students a vertical number line which some students might find easier to use than a horizontal one Have them decide whether to place points for rational and irrational numbers above or below existing points
Additional ResourcesDifferentiated Instruction includes bull Reading Strategies bull Success for English Learners EL
bull Reteach bull Challenge PRE-AP
DIFFERENTIATE INSTRUCTION
Ordering Real Numbers 24
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
Math TrainerOnline Assessment
and Intervention
Personal
Online homework assignment available
myhrwcom
myhrwcom
Lesson Quiz available online
13 LESSON QUIZ
1 Compare Write lt gt or =
radic_
95 - 5 radic_
62 - 2
2 Order 105 radic_
105 and 3π + 1 from greatest to least
3 A length in centimeters is calculated differently by four different people Order their calculations from least to greatest
KD 11 __ 2 cm Silvio 5 __ 3 π cm
Paula 5 _
4 cm Luis radic_
33 cm
EvaluateGUIDED AND INDEPENDENT PRACTICE
Concepts amp Skills Practice
Example 1Comparing Irrational Numbers
Exercises 1ndash8
Example 2Ordering Real Numbers
Exercises 9 12ndash15 18ndash21
Example 3Ordering Real Numbers in a Real-World Context
Exercises 10 16ndash17
Additional ResourcesDifferentiated Instruction includes bull Leveled Practice worksheets
Focus | Coherence | Rigor
Exercise Depth of Knowledge (DOK) Mathematical Practices
12ndash15 1 Recall of Information MP5 Using Tools
16 2 SkillsConcepts MP2 Reasoning
17 2 SkillsConcepts MP6 Precision
18ndash21 2 SkillsConcepts MP2 Reasoning
22 3 Strategic Thinking MP4 Modeling
23ndash24 3 Strategic Thinking MP3 Logic
8NS2
8NS2
Answers1 radic
_ 95 - 5 lt radic
_ 62 - 2
2 radic_
105 3π + 1 105
3 Silvio 5 __ 3 π cm Paula 5 _
4 cm
KD 11
__ 2 cm Luis radic_
33 cm
25 Lesson 13
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
Work Area
3140 3141 3142 3143
314 π227
20 A teacher asks his students to write the numbers shown in order from least to greatest Paul thinks the numbers are already in order Sandra thinks the order should be reversed Who is right
21 Math History There is a famous irrational number called Eulerrsquos number symbolized with an e Like π its decimal form never ends or repeats The first few digits of e are 27182818284
a Between which two square roots of integers could you find this number
b Between which two square roots of integers can you find π
22 Analyze Relationships There are several approximations used for π including 314 and 22 __ 7 π is approximately 314159265358979
a Label π and the two approximations on the number line
b Which of the two approximations is a better estimate for π Explain
c Find a whole number x so that the ratio x ___ 113 is a better estimate for π
than the two given approximations
23 Communicate Mathematical Ideas If a set of six numbers that include both rational and irrational numbers is graphed on a number line what is the fewest number of distinct points that need to be graphed Explain
24 Critique Reasoning Jill says that 12 _
6 is less than 1263 Explain her error
FOCUS ON HIGHER ORDER THINKING
radic_
115 115 ___ 11 and 105624
between radic_
7 asymp 265 and radic_
8 asymp 283
between radic_
9 = 3 and radic_
10 asymp 316
22 __ 7 it is closer to π on the number line
She did not consider the repeating digit 1266
2 rational numbers can have the same location and
irrational numbers can have the same location but they
cannot share a location
355
Neither student is correct The answer
should be 115 ___ 11 105624 radic_
115
Unit 126
copy H
ough
ton M
ifflin
Har
cour
t Pub
lishin
g Com
pany
Imag
e Cre
dits
copy3D
Stoc
kiSt
ockP
hoto
com
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
8_MCAAESE206984_U1M01L3indd 26 210513 801 AM
Personal Math Trainer
Online Practice and Helpmyhrwcom
Name Class Date
Independent Practice
16 Your sister is considering two different shapes for her garden One is a square with side lengths of 35 meters and the other is a circle with a diameter of 4 meters
a Find the area of the square
b Find the area of the circle
c Compare your answers from parts a and b Which garden would give your sister the most space to plant
17 Winnie measured the length of her fatherrsquos ranch four times and got four different distances Her measurements are shown in the table
a To estimate the actual length Winnie first approximated each distance to the nearest hundredth Then she averaged the four numbers Using a calculator find Winniersquos estimate
b Winniersquos father estimated the distance across his ranch to be radic_
56 km How does this distance compare to Winniersquos estimate
Give an example of each type of number
18 a real number between radic_
13 and radic_
14
19 an irrational number between 5 and 7
Order the numbers from least to greatest
12 radic_
7 2 radic_
8 ___ 2 13 radic_
10 π 35
14 radic_
220 -10 radic_
100 115 15 radic_
8 -375 3 9 _ 4
Distance Across Fatherrsquos Ranch (km)
1 2 3 4
radic_
60 58 __ 8 7 _
3 7 3 _ 5
138NS2
radic_
8 ___ 2 2 radic_
7
-10 radic_
100 115 radic_
220
radic_
60 asymp 775 58 __ 8 = 725 7 _
3 asymp 733 7 3 _ 5 = 760 so the average
π radic_
10 35
-375 9 _ 4 radic_
8 3
is 74825 km
1225 m2
4π m2 or approximately 126 m2
They are nearly identical radic_
56 is approximately 74833hellip
The circle would give her more space to plant because it has a
larger area
Sample answer 37
Sample answer radic_
31
25Lesson 13
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ough
ton
Miff
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ublis
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pany
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
8_MCAAESE206984_U1M01L3indd 25 41613 448 AM
Activity available online myhrwcomEXTEND THE MATH PRE-AP
Activity Have students investigate whether there are infinitely many numbers between two numbers by giving examples for each of the following
bull Between any two rational numbers there is at least one other rational number Sample answer 45 is between 41 and 48
bull Between any two irrational numbers there is at least one rational number Sample answer 45 is between radic
_ 11 and radic
_ 29
bull Between any two rational numbers there is at least one irrational number Sample answer radic
_ 11 is between 31 and 36
bull Between any two irrational numbers there is at least one irrational number Sample answer radic
_ 17 is between radic
_ 11 and radic
_ 29
Ordering Real Numbers 26
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
ReadyMath Trainer
Online Practiceand Help
Personal
myhrwcom
Module Quiz
11ensp RationalenspandenspIrrationalenspNumbersWrite each fraction as a decimal or each decimal as a fraction
1 7__20 2 1___
27 3 17_8
Solve each equation for x
4 x2=81 5 x3=343 6 x2= 1___100
7 Asquarepatiohasanareaof200squarefeetHowlongiseachside
ofthepatiotothenearesttenth
12ensp SetsenspofenspRealenspNumbersWrite all names that apply to each number
8 121____radic
____121
9 π__2
10 TellwhetherthestatementldquoAllintegersarerationalnumbersrdquoistrueorfalseExplainyourchoice
13ensp OrderingenspRealenspNumbersCompare Write lt gt or =
11 radic__
8+3 8+radic__
3 12 radic__
5+11emsp emsp emsp 5+radic___
11
Order the numbers from least to greatest
13 radic___
99π29__
8 14 radic___
1__251_40__
2
15 Howarerealnumbersusedtodescribereal-worldsituations
ESSENTIAL QUESTION
035
9-9
141ft
7 1__10- 1__10
14__11 1875
wholeintegerrationalreal
Trueintegerscanbewrittenasthequotientoftwointegers
SampleanswerRealnumberssuchastherational
π29__
8radic___
99
irrationalreal
lt gt
number1_4candescribeamountsusedincooking
radic___
1__250__
21_4
27Module1
copy H
ough
ton
Miff
lin H
arco
urt P
ublis
hing
Com
pany
DONOTEDIT--ChangesmustbemadethroughldquoFileinfordquoCorrectionKey=A
8_MCAAESE206984_U1M01RTindd 27 41513 1113 PM
Math TrainerOnline Assessment
and Intervention
Personal
myhrwcom
1
2
3 Response toIntervention
Intervention Enrichment
Access Ready to Go On assessment online and receive instant scoring feedback and customized intervention or enrichment
Online and Print Resources
Differentiated Instruction
bull Reteach worksheets
bull Reading Strategies EL
bull Success for English Learners EL
Differentiated Instruction
bull Challenge worksheets PRE-AP
Extend the Math PRE-AP
Lesson Activities in TE
Additional ResourcesAssessment Resources includes bull Leveled Module Quizzes
Ready to Go OnAssess MasteryUse the assessment on this page to determine if students have mastered the concepts and standards covered in this module
California Common Core Standards
Lesson Exercises Common Core Standards
11 1ndash7 8NS1 8NS2 8EE2
12 8ndash10 8NS1
13 11ndash14 8NS2
27 Unit 1 Module 1
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
Personal Math Trainer
Online Practice and HelpmyhrwcomAssessment Readiness
Module 1 MIXed ReVIeW
1 Look at each number Is the number between 2π and radic___
52
Select Yes or No for expressions AndashC
A 6 2 _ 3 Yes No
B 5π __ 2 Yes No
C 3 radic__
5 Yes No
2 Consider the number - 11 __ 15
Choose True or False for each statement
A The number is rational True False
B The number can be written as True Falsea repeating decimal
C The number is less than ndash08 True False
3 The volume of a cube is given by V = x3 where x is the length of an edge of the cube A cube-shaped end table has a volume of 3 3 _ 8 cubic feet What is the length of an edge of the end table Explain how you solved this problem
4 A student says that radic___
83 is greater than 29 __ 3 Is the student correct Justify your
reasoning
1 1 _ 2 ft Sample answer The equation x3 = 3 3 _ 8 can be used
to find the edge length in feet To solve the equation
write the mixed number as a fraction greater than 1
x3 = 27 __ 8 Then take the cube root of both sides x = 3 _ 2 = 1 1 _ 2
No Sample answer radic___
83 asymp 91 and 29 __ 3 = 9
__ 6
Because 91 lt 9 __
6 radic___
83 lt 29 __ 3
28 Unit 1
copy H
ough
ton
Miff
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ublis
hing
Com
pany
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=A
8_MCAAESE206984_U1M01RTindd 28 240413 946 AM
Personal Math Trainer
Online Assessment and
Interventionmyhrwcom
Scoring GuideItem 3 Award the student 1 point for finding the edge length of the cube and 1 point for correctly explaining how to use a cube root to solve the problem
Item 4 Award the student 1 point for determining that the student is incorrect and 1 point for correctly justifying the reasoning for this conclusion
Additional ResourcesTo assign this assessment online login to your Assignment Manager at myhrwcom
Assessment Readiness
California Common Core Standards
Items Grade 8 Standards Mathematical Practices
1 8NS2 MP7
2 7NS2b 7NS2d 8NS1 MP7
3 8EE2 MP1 MP4
4 8NS1 8NS2 MP3
Item integrates mixed review concepts from previous modules or a previous course
Item 4 combines concepts from the California Common Core cluster ldquoKnow that there are numbers that are not rational and approximate them by rational numbersrdquo
Real Numbers 28
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
Integers
Rational Numbers IrrationalNumbers
Real Numbers
WholeNumbers
-3-4-5 -2-1 1 2 3 50 4
23
34-4 -π -1 25
radic2
Lesson Support Content Objective Students will learn to describe relationships between sets of numbers
Language Objective Students will explain how to describe relationships between sets of real numbers
LESSON 12 Sets of Real Numbers
Building BackgroundEliciting Prior Knowledge Have students draw a number line from -5 to 5 Ask them to plot points on the number line to approximate the location of rational and irrational numbers such as -1 3 __ 4 25 -4 2 __ 3 radic
_ 2 and -π
Learning ProgressionsIn this lesson students clarify their understanding of the real number system They characterize sets and subsets of the real numbers They also identify sets for real-world situations Important understandings for students include the following
bull Identify all of the possible subsets of the real numbers for a given number
bull Decide whether a statement about a subset of the real numbers is true or false
bull Identify the set of numbers that best describes a real-world situation
Understanding the relationships among the sets of numbers that make up the real numbers is essential as students are introduced to different forms of numbers throughout the school year This lesson provides a foundation for the comparing and ordering of real numbers in the next lesson
Cluster ConnectionsThis lesson provides an excellent opportunity to connect ideas in this cluster Know that there are numbers that are not rational and approximate them by rational numbers Have students copy this diagram which relates the sets of real numbers
Ask students to complete the diagram by writing three examples for each set of numbers Have students share examples and explain how they knew each number they selected belonged in the appropriate set Answers may vary Check studentsrsquo work
Focus | Coherence | Rigor
California Common Core Standards
8NS1 Know that numbers that are not rational are called irrational Understand informally that every number has a decimal expansion for rational numbers show that the decimal expansion repeats eventually and convert a decimal expansion which repeats eventually into a rational number
MP7 Look for and make use of structure
15A
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
Math Talk
Language Support EL
PROFESSIONAL DEVELOPMENT
Linguistic Support EL
AcademicContent Vocabulary
Venn diagrams ndash Students need descriptive language to describe the categories that the different areas and colors of a Venn diagram represent the concept of a set and how sets are distinct or can overlap Use sentence frames such as
The big oval represents __________The darklight blue color in the middle of the
big ovals represents __________These sets overlap because __________
In this way students have the language and structure to identify the criteria that distinguish a set and to explain the abstract representation Also point out the use of the prefix sub- meaning ldquounderrdquo in the term subset
Rules and Patterns
Abbreviations ndash In this lesson the abbreviation mph is used Be sure to point out that mph stands for miles per hour and is used to give units in a rate of speed Students may also have seen mpg (miles per gallon) which gives the units in a rate of fuel efficiency
Borrowed Words ndash Terminology used in baseball such as inning and pitcher may require some explanation Spanish as well as some other languages have borrowed these terms from English so some students may be familiar with these words already Despite this whenever a word is critical to students understanding the word problem it is best to explain the meaning
Leveled Strategies for English Learners
Emerging Allow students to indicate true or false orally in Guided Practice Exercises 9 and 10
Expanding Have students use sentence frames to describe the meaning of regions and colors used in a Venn diagram Then give them similar sentence frames orally and have them draw and shade a Venn diagram based on the oral prompts
Bridging Have students work in groups to draw a Venn diagram to represent sets based on real-world examples in the lesson
To help students answer the question posed in Math Talk provide a sentence frame for their answer
The numbers between 31 and 39 on a number line are __________ because __________
EL
California ELD Standards
Emerging 2II5 Modifying to add details ndash Expand sentences with simple adverbials to provide details about a familiar activity or process
Expanding 2II5 Modifying to add details ndash Expand sentences with adverbials to provide details about a familiar or new activity or process
Bridging 2II5 Modifying to add details ndash Expand sentences with increasingly complex adverbials to provide details about a variety of familiar and new activities and processes
Sets of Real Numbers 15B
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
12L E S S O N
Sets of Real Numbers
EngageESSENTIAL QUESTION
How can you describe relationships between sets of real numbers Sample answer Describe them as two different sets or one set as being a subset of another
Motivate the LessonAsk How many different types of tigers can you name How does the set of Bengal tigers relate to the set of tigers
ExplorePoint to different locations in the Animals diagram and ask for examples for that classification Do the same for the Real Numbers diagram Students should understand that everything within a region is part of the set for example both -3 and 2 are integers
ExplainEXAMPLE 1
Questioning Strategies Mathematical Practices bull In A why is 5 not a perfect square It does not have rational numbers as its square roots
bull Can the number in B be written as a fraction Why or why not Yes it is a terminating decimal so it is a rational number
Engage with the WhiteboardHave students place the numbers in Example 1 and Additional Example 1 in the Venn diagram for numbers
YOUR TURNAvoid Common ErrorsBe sure that students read Exercise 2 carefully before answering The number given in the problem 10 is the area not the side length
EXAMPLE 2Questioning Strategies Mathematical Practices bull What two major sets are the real numbers composed of rational and irrational numbers
bull What is the location of the set of whole numbers in the Venn diagram in relation to the set of rational numbers Explain Inside it whole numbers are rational numbers
Focus on Reasoning Mathematical PracticesRemind students that it takes only one counterexample to show that a statement is false
Interactive Whiteboard Interactive example available online
ADDITIONAL EXAMPLE 1Write all names that apply to each number
A -10integer rational real
B 12 _ 3
whole integer rational real
myhrwcom
Interactive Whiteboard Interactive example available online
ADDITIONAL EXAMPLE 2Tell whether the given statement is true or false Explain your choice
No integers are whole numbers
False every whole number is also an integer
myhrwcom
Animated MathClassifying Numbers
Students build fluency in classifying numbers in this engaging fast-paced game
myhrwcom
CA Common CoreStandards
The student is expected to
The Number Systemmdash8NS1
Know that numbers that are not rational are called irrational Understand informally that every number has a decimal expansion for rational numbers show that the decimal expansion repeats eventually and convert a decimal expansion which repeats eventually into a rational numberMathematical Practices
MP7 Using Structure
The student is expected to
15 Lesson 12
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=B
Math TrainerOnline Practice
and Help
Personal
myhrwcom
Math TrainerOnline Practice
and Help
Personal
myhrwcom
Math On the Spotmyhrwcom
Understanding Sets and Subsets of Real NumbersBy understanding which sets are subsets of types of numbers you can verify whether statements about the relationships between sets are true or false
Tell whether the given statement is true or false Explain your choice
All irrational numbers are real numbers
True Every irrational number is included in the set of real numbers The irrational numbers are a subset of the real numbers
No rational numbers are whole numbers
False A whole number can be written as a fraction with a denominator of 1 so every whole number is included in the set of rational numbers The whole numbers are a subset of the rational numbers
EXAMPLE 2
A
B
Write all names that apply to each number
1 A baseball pitcher has pitched 12 2 _ 3 innings
2 The length of the side of a square that has an
area of 10 square yards
YOUR TURN
Tell whether the given statement is true or false Explain your choice
3 All rational numbers are integers
4 Some irrational numbers are integers
YOUR TURN
Give an example of a rational number that is a
whole number Show that the number is both whole
and rational
Math TalkMathematical Practices
Give an example of a
8NS1
False Every integer is a rational number but not every
False Real numbers are either rational or irrational numbers
Integers are rational numbers so no integers are irrational numbers
rational real
irrational real
Sample answer 8 8 = 8_
1
and -thinsp 5 _ 2 are not integers
rational number is an integer Rational numbers such as 3 _ 5
Unit 116
copy H
ough
ton
Miff
lin H
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ublis
hing
Com
pany
bull Im
age C
redi
ts D
igita
l Im
age c
opyr
ight
copy20
04 Ey
ewire
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
8_MCAAESE206984_U1M01L2indd 16 41613 136 AM
Math On the Spot
myhrwcom
Vertebrates
Birds
Passerines
Animals
Integers
Rational Numbers IrrationalNumbers
Real Numbers
WholeNumbers
1
45
3
0
274
67
radic4
-
-3
-2
-1
03
radic2
radic17
radic11-
π
Animated Math
myhrwcom
Classifying Real NumbersBiologists classify animals based on shared characteristics A cardinal is an animal a vertebrate a bird and a passerine
You already know that the set of rational numbers consists of whole numbers integers and fractions The set of real numbers consists of the set of rational numbers and the set of irrational numbers
Write all names that apply to each number
radic_
5 irrational real
ndash1784rational real
whole integer rational real
EXAMPLEXAMPLE 1
A
B
C radic_ 81 ____ 9
L E S S O N
12Sets of Real Numbers
ESSENTIAL QUESTIONHow can you describe relationships between sets of real numbers
Passerines such as the cardinal are also called ldquoperching birdsrdquo
What types of numbers are between 31 and 39 on a
number line
Math TalkMathematical Practices
What types of numbers are
8NS1
8NS1
Know that numbers that are not rational are called irrational Understand informally that every number has a decimal expansion for rational numbers show that the decimal expansion repeats eventually and convert a decimal expansion which repeats eventually into a relation number
ndash1784 is a terminating decimal
5 is a whole number that is not a perfect square
radic_
81 _____ 9 = 9 __ 9 = 1 rational irrational real
15Lesson 12
copy H
ough
ton
Miff
lin H
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ublis
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Com
pany
bull Im
age C
redi
ts copy
Wiki
med
ia Co
mm
ons
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=B
8_MCABESE206984_U1M01L2indd 15 061113 1144 AM
PROFESSIONAL DEVELOPMENT
Math BackgroundThe relationships between sets of numbers extend to include complex numbers A complex number can be written as a sum of a real number a and an imaginary number bi
a + bi
An imaginary number is a special number that when squared gives a negative value When you square a real number you get a nonnegative number When you square an imaginary number you get a negative value The imaginary unit is i
i = radic_
-1
Integrate Mathematical Practices MP7
This lesson provides an opportunity to address this Mathematical Practices standard It calls for students to discern structure to connect and communicate mathematical ideas
Students use a Venn diagram to structure relationships between sets of numbers They connect and communicate mathematical ideas when they make logical statements about the sets and describe which set best describes numbers applied to real-life situations
Sets of Real Numbers 16
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=B
YOUR TURNAvoid Common ErrorsStudents may see the word ldquoAllldquo or rdquoNordquo in Exercises 3 and 4 and immediately assume that any absolute statements like these are false Remind them that there are true statements that begin with these words and encourage them to provide examples
EXAMPLE 3Questioning Strategies Mathematical Practices bull In A how does the phrase ldquonumber of rdquo give you a clue about the number classification It indicates a counting number
bull What is the relationship between the circumference of a circle and the diameter The circumference is diameter times π
Focus on Critical Thinking Mathematical PracticesIn B suppose the diameters in inches were 25
__ π 28 __ π
31 __ π and so on What set of numbers would
best describe the circumferences Explain Whole numbers the circumferences would be the whole numbers 25 28 31 and so on
YOUR TURNFocus on Critical Thinking Mathematical PracticesHave students compare and contrast the classification of numbers in the answers in Exercises 5 and 6
ElaborateTalk About ItSummarize the Lesson
Ask What are some ways that number sets can be related Sets may be subsets of other sets or they may be separate from other sets
GUIDED PRACTICEEngage with the Whiteboard
Have students place the numbers in Exercises 1ndashthinsp8 in the Venn diagram for numbers at the beginning of the lesson
Integrating Language Arts EL
Encourage English learners to ask for clarification on any terms or phrases that they do not understand
Avoid Common ErrorsExercise 7 Remind students that a repeating decimal is a rational numberExercises 9ndash10 Remind students that it only takes one counterexample to show that a statement is false
Interactive Whiteboard Interactive example available online
ADDITIONAL EXAMPLE 3Identify the set of numbers that best describes the situation Explain your choice
A the amount of time that has passed since midnight
The set of real numbers time is continuous so the amount of time can be rational or irrational
B the number of tickets sold to a basketball game
The set of whole numbers the number of tickets sold may be 0 or a counting number
myhrwcom
17 Lesson 12
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
1IN
116 inch
Guided Practice
Write all names that apply to each number (Example 1)
1 7 _ 8 2 radic_
36
3 radic_
24 4 075
5 0 6 - radic_ 100
7 5 _
45 8 - 18 __ 6
Tell whether the given statement is true or false Explain your choice (Example 2)
9 All whole numbers are rational numbers
10 No irrational numbers are whole numbers
Identify the set of numbers that best describes each situation Explain your choice (Example 3)
11 the change in the value of an account when given to the nearest dollar
12 the markings on a standard ruler
13 What are some ways to describe the relationships between sets of numbers
CHECK-INESSENTIAL QUESTION
rational real
rational real
True Whole numbers are rational numbers
Rational numbers the ruler is marked every 1 __ 16 th inch
Sample answer Describe one set as being a subset of
another or show their relationships in a Venn diagram
Integers the change can be a whole dollar amount
and can be positive negative or zero
True Whole numbers are a subset of the set of rational numbers
and can be written as a ratio of the whole number to 1
irrational real
whole integer rational real
whole integer rational real
rational real
integer rational real
integer rational real
Unit 118
copy H
ough
ton
Miff
lin H
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ublis
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Com
pany
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
8_MCAAESE206984_U1M01L2indd 18 41613 136 AM
My Notes
Math TrainerOnline Practice
and Help
Personal
myhrwcom
Math On the Spot
myhrwcom
Identifying Sets for Real-World SituationsReal numbers can be used to represent real-world quantities Highways have posted speed limit signs that are represented by natural numbers such as 55 mph Integers appear on thermometers Rational numbers are used in many daily activities including cooking For example ingredients in a recipe are often given in fractional amounts such as 2 _ 3 cup flour
Identify the set of numbers that best describes each situation Explain your choice
the number of people wearing glasses in a room
The set of whole numbers best describes the situation The number of people wearing glasses may be 0 or a counting number
the circumference of a flying disk has a diameter of 8 9 10 11 or 14 inches
The set of irrational numbers best describes the situation Each circumference would be a product of π and the diameter and any multiple of π is irrational
EXAMPLEXAMPLE 3
A
B
Identify the set of numbers that best describes the situation Explain your choice
5 the amount of water in a glass as it evaporates
6 the weight of a person in pounds
YOUR TURN
8NS1
Rational numbers a personrsquos weight can be a decimal
such as 835 pounds
Real numbers the amount can be any number greater
than 0
17Lesson 12
copy H
ough
ton
Miff
lin H
arco
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ublis
hing
Com
pany
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
8_MCAAESE206984_U1M01L2indd 17 41613 520 AM
Graphic OrganizersGive students a list of numbers (including terminating and repeating decimals fractions integers and rational and irrational square roots) and a graphic organizer as shown below
Real Numbers
Rational numbers Irrational numbers
Integer numbers
Whole numbers
Ask students to write each number in the list in the correct section of the organizer
Number SensePoint out to students that knowing the types of numbers to expect in different situations can alert them to incorrect math as well as to impossible situations For example 135 shots made in basketballs is not possible but an average number of shots can equal 135
Additional ResourcesDifferentiated Instruction includes bull Reading Strategies bull Success for English Learners EL
bull Reteach bull Challenge PRE-AP
DIFFERENTIATE INSTRUCTION
Sets of Real Numbers 18
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
Math TrainerOnline Assessment
and Intervention
Personal
Online homework assignment available
myhrwcom
Lesson Quiz available online
12 LESSON QUIZ
1 Write all the names that apply to the number
2 Tell whether the given statement is true or false Explain your choice All numbers between 1 and 2 are rational numbers
3 Identify the set of numbers that best describes the situation Explain your choiceThe choices on a survey question change the total points for the survey by -2 -1 0 1 or 2 points
-1 _
5
myhrwcom
Additional ResourcesDifferentiated Instruction includes bull Leveled Practice worksheets
EvaluateGUIDED AND INDEPENDENT PRACTICE
Concepts amp Skills Practice
Example 1Classifying Real Numbers
Exercises 1ndash8 14ndash19 22ndash24
Example 2Understanding Sets and Subsets of Real Numbers
Exercises 9ndash10
Example 3Identifying Sets for Real-World Situations
Exercises 11ndash12 20ndash21 25
Focus | Coherence | Rigor
Exercise Depth of Knowledge (DOK) Mathematical Practices
14ndash19 2 SkillsConcepts MP7 Using Structure
20ndash21 2 SkillsConcepts MP6 Precision
22ndash23 2 SkillsConcepts MP3 Logic
24 1 Recall of Information MP7 Using Structure
25 2 SkillsConcepts MP2 Reasoning
26ndash27 3 Strategic Thinking MP3 Logic
28 3 Strategic Thinking MP8 Patterns
29 3 Strategic Thinking MP3 Logic
8NS1
8NS1
Exercise 29 combines concepts from the California Common Core cluster ldquoKnow that there are numbers that are not rational and approximate them by rational numbersrdquo
Answers1 rational real
2 False radic_
2 is an example of an irrational number between 1 and 2
3 Integers each number is an integer but only three are whole numbers
19 Lesson 12
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
Work Area
π mi23 Critique Reasoning The circumference of a circular region is shown
What type of number best describes the diameter of the circle Explain
your answer
24 Critical Thinking A number is not an integer What type of number can it be
25 A grocery store has a shelf with half-gallon containers of milk What type of number best represents the total number of gallons
26 Explain the Error Katie said ldquoNegative numbers are integersrdquo What was her error
27 Justify Reasoning Can you ever use a calculator to determine if a number is rational or irrational Explain
28 Draw Conclusions The decimal 0 _
3 represents 1 _ 3 What type of number best describes 0
_ 9 which is 3 middot 0
_ 3 Explain
29 Communicate Mathematical Ideas Irrational numbers can never be precisely represented in decimal form Why is this
FOCUS ON HIGHER ORDER THINKING
It can be a rational number that is not an integer or an irrational number
rational number
The set of negative numbers also includes non-integer
rational numbers and irrational numbers
Sample answer If the calculator shows a decimal that
terminates in fewer digits than what the calculator screen
allows then you can tell that the number is rational If not
you cannot tell from the calculator display whether the
number terminates because you see a limited number
of digits It may be a repeating decimal (rational) or
non-terminating non-repeating decimal (irrational)
Whole 3 middot 0 _
3 represents 3 middot 1 _ 3 = 1 so 0 _
9 is exactly 1
Sample answer In decimal form irrational numbers never
terminate and never repeat Therefore no matter how
many decimal places you include the number will never
be precisely represented There are always more digits
Whole the diameter is π _ π = 1 mile
Unit 120
copy H
ough
ton
Miff
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hing
Com
pany
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
8_MCAAESE206984_U1M01L2indd 20 120413 909 PM
Integers
Rational Numbers Irrational Numbers
Real Numbers
Whole Numbers
257
radic16
166
radic9
128 radic50
Personal Math Trainer
Online Practice and Helpmyhrwcom
Name Class Date
Independent Practice
Identify the set of numbers that best describes each situation Explain your choice
20 the height of an airplane as it descends to an airport runway
21 the score with respect to par of several golfers 2 ndash 3 5 0 ndash 1
22 Critique Reasoning Ronald states that the number 1 __ 11 is not rational because when converted into a decimal it does not terminate Nathaniel says it is rational because it is a fraction Which boy is correct Explain
12
14 - radic_
9 15 257
16 radic_
50 17 8 1 _ 2
18 166 19 radic_
16
Write all names that apply to each number Then place the numbers in the correct location on the Venn diagram
8NS1
Real numbers the height can be any number greater than zero
integer rational real whole integer rational real
whole integer rational real
irrational real
rational real
rational real
Integers the scores are counting numbers their
opposites and zero
Nathaniel is correct A rational number is a number that can be written as a fraction and 1 __ 11 is a fraction
19Lesson 12
copy H
ough
ton
Miff
lin H
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pany
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
8_MCAAESE206984_U1M01L2indd 19 41613 136 AM
myhrwcomActivity available onlineEXTEND THE MATH PRE-AP
Activity Have students consider the concept of restricted domain for the sets of numbers that describe situations For example the number of sisters a person has can best be described by whole numbers but no one has ever had 1500 sisters An area code is an integer or whole number between 200 and 999
Have students use a source such as the Guinness Book of World Records and give examples of sets of numbers that describe situations where the domain is restricted Ask whether the restriction may be changed in the future
Sets of Real Numbers 20
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
-3-4-5 -2-1 1 2 3 50 4
12-4 -radic5
Lesson Support Content Objective Students will learn to order a set of real numbers
Language Objective Students will show and describe how to order a set of real numbers
LESSON 13 Ordering Real Numbers
Building BackgroundEliciting Prior Knowledge Have students draw a number line to compare a rational number and an irrational number such as - radic
_ 5 and -4 1 __ 2 Ask them to explain how
they approximated the irrational number on the number line Then have them identify the greater and the lesser real number Repeat with several other pairs of real numbers in different forms
Learning ProgressionsIn this lesson students order a set of real numbers They use rational approximations to compare the sizes of irrational numbers They also order numbers for real-world situations Important understandings for students include the following
bull Compare irrational numbers bull Estimate the value of expressions with irrational numbers bull Order a set of real numbers bull Order real numbers in a real-world context
Work with real numbers continues throughout Grade 8 and into high school This lesson provides students with a foundation for understanding the relative sizes of numbers in different forms in the real number system
Cluster ConnectionsThis lesson provides an excellent opportunity to connect ideas in this cluster Know that there are numbers that are not rational and approximate them by rational numbers Tell students that there is a special number called the golden ratio with applications in mathematics geometry art and architecture The golden ratio is called phi and is represented by the Greek letter ϕ It includes an irrational number in its definition
Have students explain why the golden ratio is irrational Ask them to find the two whole numbers the golden ratio lies between Then challenge them to approximate the golden ratio to the nearest tenth It is irrational because it includes an irrational number in its definition It lies between 1 and 2 To the nearest tenth ϕ = 16
ϕ = 1 + radic_
5 _ 2
Focus | Coherence | Rigor
California Common Core Standards
8NS2 Use rational approximations of irrational numbers to compare the size of irrational numbers locate them approximately on a number line diagram and estimate the value of expressions (eg π 2 )
MP4 Model with mathematics
21A
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
Math Talk
Language Support EL
PROFESSIONAL DEVELOPMENT
Linguistic Support EL
AcademicContent Vocabulary
Post a chart like this to remind students of the regular comparative forms of adjectives that use the -er and -est suffixes Add to the chart for terms that appear in examples and exercises in each lesson Include any irregular verb forms
Background Knowledge
Go On ndash the title of the module review or quiz is Ready to Go On This title uses an idiomatic expression In this context to go on means ldquoto move aheadrdquo or ldquoto proceedrdquo It is different from the use of go on that means having enough facts to use meaningfully as in having enough to go on Also the intonation used in pronouncing an expression can give it different meanings For example when the speaker emphasizes the word on he or she might be expressing disbelief as in ldquoGo ON Yoursquore kidding rightrdquo Discuss with students other ways that the phrase go on may be used
Leveled Strategies for English Learners
Emerging Label points on a number line with the terms used in ordering greater greatest less lesser least Use sentence frames to insert the correct terms
Expanding Have students give two or three complete sentences to compare the placement of numbers on a number line using the correct forms of the comparative and superlative adjectives
Bridging Have students work in pairs with one student giving directions to the other in complete sentences to order numbers on a number line
To help students answer the question posed in Math Talk make sure that students have a command of the forms for making comparisons and the superlative and the concept of opposite order so that the focus is on the math concept instead of the language skills needed to describe and explain order
EL
Adjective Comparative Superlative
Far Farther Farthest
Large Larger Largest
Great Greater Greatest
Some Less Least
Some More Most
California ELD Standards
Emerging 2I8 Analyzing language choices ndash Explain how phrasing or different common words with similar meanings produce different effects on the audience
Expanding 2I8 Analyzing language choices ndash Explain how phrasing or different words with similar meanings or figurative language produce shades of meaning and different effects on the audience
Bridging 2I8 Analyzing language choices ndash Explain how phrasing or different words with similar meanings or figurative language produce shades of meaning nuances and different effects on the audience
Ordering Real Numbers 21B
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
13L E S S O N
Ordering Real Numbers
Interactive Whiteboard Interactive example available online
ADDITIONAL EXAMPLE 1Compare Write lt gt or =
A radic_
8 - 2 4 - radic_
8 lt
B radic_
20 + 1 3 + radic_
2 gt
EngageESSENTIAL QUESTION
How do you order a set of real numbers Sample answer Find their approximate decimal values and order them
Motivate the LessonAsk What kind of numbers are you comparing when you compare the price of gasoline at two different gas stations
ExploreGive students two rational numbers and ask them to name a number between them Repeat a few times and then give them two irrational numbers and ask them to name a number between them
ExplainEXAMPLE 1
Questioning Strategies Mathematical Practices bull Which is greater the difference between 5 and 3 or the difference between radic
_ 5 and radic
_ 3
The difference between 5 and 3 is 2 the difference between radic_
5 and radic_
3 is approximately 1 So the difference between 5 and 3 is greater
Avoid Common ErrorsCaution students to read the problem carefully and think about what the radical sign means so that they do not misread the problem and answer that the two sides are equal
YOUR TURNFocus on TechnologyCalculators should not be used at this point because developing number sense is the goal
EXAMPLE 2Questioning Strategies Mathematical Practices bull How do you determine whether radic
_ 22 is less than or greater than 45 The square of 45 is
2025 which is less than 22 so the square root of 22 must be greater than 45
Engage with the WhiteboardHave students graph and label various real numbers between 42 and 44 and between 47 and 5
YOUR TURNFocus on Modeling Mathematical PracticesHave students label the integers on the number line with their equivalent square root For example 1 2 and 3 on the number line would be labeled radic
_ 1 radic
_ 4 and radic
_ 9
Interactive Whiteboard Interactive example available online
ADDITIONAL EXAMPLE 2Order 3π radic
_ 10 and 325 from greatest
to least
3π 325 radic_
10
myhrwcom
myhrwcom
CA Common CoreStandards
The student is expected to
The Number Systemmdash8NS2
Use rational approximations of irrational numbers to compare the size of irrational numbers locate them approximately on a number line diagram and estimate the value of expressions (eg π 2 )
Mathematical Practices
MP4 Modeling
The student is expected to
21 Lesson 13
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
Math TrainerOnline Practice
and Help
Personal
myhrwcom
Math On the Spotmyhrwcom
0 05 1 15 2 25 3 35 4
radic5radic3
π2
8 85 9 95 10 105 11 115 12
radic75
4 42 44 46 48 5
radic224 12π + 1
Ordering Real Numbers You can compare and order real numbers and list them from least to greatest
Order radic_
22 π + 1 and 4 1 _ 2 from least to greatest
First approximate radic_
22
radic_
22 is between 4 and 5 Since you donrsquot know where it falls between 4 and 5 you need to find a better estimate for radic
_ 22 so
you can compare it to 4 1 _ 2
Since 22 is closer to 25 than 16 use squares of numbers between 45 and 5 to find a better estimate of radic
_ 22
45 2 = 2025 46 2 = 2116 47 2 = 2209 48 2 = 2304
Since 47 2 = 2209 an approximate value for radic_
22 is 47
An approximate value of π is 314 So an approximate value of π +1 is 414
Plot radic_
22 π + 1 and 4 1 _ 2 on a number line
Read the numbers from left to right to place them in order from least to greatest
From least to greatest the numbers are π + 1 4 1 _ 2 and radic_
22
EXAMPLE 2
STEP 1
STEP 2
Order the numbers from least to greatest Then graph them on the number line
YOUR TURN
5 radic_
5 25 radic_
3
6 π 2 10 radic_
75
If real numbers a b and c are in order from least to greatest what is the order
of their opposites from least to greatest
Explain
Math TalkMathematical Practices
8NS2
radic_
3 radic_
5 25
radic_
75 π2 10
Math Talk answer -c -b -a -c is farthest to the left on a number line -b is in the middle and -a is farthest to the right
Unit 122
copy H
ough
ton
Miff
lin H
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hing
Com
pany
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
8_MCAAESE206984_U1M01L3indd 22 41613 447 AM
My Notes
Math TrainerOnline Practice
and Help
Personal
myhrwcom
Math On the Spot
myhrwcom
Comparing Irrational NumbersBetween any two real numbers is another real number To compare and order real numbers you can approximate irrational numbers as decimals
Compare radic_
3 + 5 3 + radic_
5 Write lt gt or =
First approximate radic_
3
radic_
3 is between 1 and 2
Next approximate radic_
5
radic_
5 is between 2 and 3
Then use your approximations to simplify the expressions
radic_
3 + 5 is between 6 and 7
3 + radic_
5 is between 5 and 6
So radic_
3 + 5 gt 3 + radic_
5
Reflect1 If 7 + radic
_ 5 is equal to radic
_ 5 plus a number what do you know about the
number Why
2 What are the closest two integers that radic_
300 is between
EXAMPLEXAMPLE 1
STEP 1
STEP 2
Compare Write lt gt or =
YOUR TURN
3 radic_
2 + 4 2 + radic_
4 4 radic_
12 + 6 12 + radic_
6
L E S S O N
13 Ordering Real Numbers
ESSENTIAL QUESTIONHow do you order a set of real numbers
8NS2
Use rational approximations of irrational numbers to compare the size of irrational numbers locate them approximately on a number line diagram and estimate the value of expressions (eg π 2 )
8NS2
Use perfect squares to estimate square roots
1 2 = 1 2 2 = 4 3 2 = 9
The number is 7 both expressions must equal 7 + radic_
5
17 and 18
gt lt
21Lesson 13
copy H
ough
ton
Miff
lin H
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ublis
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Com
pany
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
8_MCAAESE206984_U1M01L3indd 21 41913 246 PM
PROFESSIONAL DEVELOPMENT
Math BackgroundIn this lesson students estimate irrational numbers in the form of square roots of nonper-fect squares by finding two perfect squares between which the number falls A more precise method involves repeated division For example to find radic
_ 28 find a whole number whose perfect
square is close to 28 such as 5 Divide 28 by that number 28 divide 5 = 56 Find the average of the quotient and divisor 5 + 56
_____ 2 = 53 Continue dividing 28 by each result and averaging until you get the desired accuracy
Integrate Mathematical Practices MP4
This lesson provides an opportunity to address this Mathematical Practices standard It calls for students to model relationships using multiple representations including diagrams graphs and language as appropriate Students use multiple representations when they use number lines to estimate the locations of and order rational and irrational numbers given as symbols
Ordering Real Numbers 22
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
Interactive Whiteboard Interactive example available online
ADDITIONAL EXAMPLE 3The diameter of a meteorite in millimeters is calculated by four different methods Order the results from least to greatest
Joe radic_
18 mm Lisa 13 __ 3 mm
Pablo 46 mm Julien 4π __ 3 mm
Julien 4π __ 3 mm Lisa 13 __ 3 mm
Joe radic_
18 mm Pablo 46 mm
EXAMPLE 3Questioning Strategies Mathematical Practices bull How can you verify that radic
_ 28 is between 52 and 53 5 2 2 = 2704 and 5 3 2 = 2809
bull Explain how to determine which number is greater 5 _
5 or 55 When the repeating decimal is rounded to the nearest tenth or hundredth you can see that it is greater
Connect to Daily LifeDiscuss how measuring across a canyon might involve different methods than measuring along a road Explain that measurements like these are often done using calculations that approximate the distance
YOUR TURNFocus on Critical Thinking Mathematical PracticesDiscuss with students which number is greater 3
_ 45 or 3450 3
_ 45 or 3455 and why Explain
that 3 _
45 can be written out as 34545hellipMake sure they understand that 3 _
45 is greater than 345 but less than 3455
ElaborateTalk About ItSummarize the Lesson
Ask How can you order two numbers in different forms whose decimal approxi-mations appear to be equal Approximate one or both numbers to an additional
number of decimal places
GUIDED PRACTICEEngage with the Whiteboard
Have students place and label additional points on the number line in Exercise 9 Allow the points to be in any format other than decimal
Avoid Common ErrorsExercises 3ndash4 Caution students to read the problem carefully so that they do not misread the problem as the same numbers combined by addition on each side of the circleExercise 10 Remind students that the calculations have units
myhrwcom
23 Lesson 13
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
0 05 1 15 2 25 3 35 4 45 5 55 6 65 7
2πradic3
Compare Write lt gt or = (Example 1)
1 radic_
3 + 2 radic_
3 + 3 2 radic_
8 + 17 radic_
11 + 15
3 radic_
6 + 5 6 + radic_
5 4 radic_
9 + 3 9 + radic_
3
5 radic_
17 - 3 -2 + radic_
5 6 12 - radic_
2 14 - radic_
8
7 radic_
7 + 2 radic_
10 - 1 8 radic_
17 + 3 3 + radic_
11
9 Order radic_
3 2π and 15 from least to greatest Then graph them on the number line (Example 2)
radic_
3 is between and so radic_
3 asymp
π asymp 314 so 2π asymp
From least to greatest the numbers are
10 Four people have found the perimeter of a forest using different methods Their results are given in the table Order their calculations from greatest to least (Example 3)
11 Explain how to order a set of real numbers
CHECK-INESSENTIAL QUESTION
Forest Perimeter (km)
Leon Mika Jason Ashley
radic_
17 - 2 1 +thinsp π __ 2 12 ___ 5 25
Guided Practice
17
15
1 + π _ 2 km 25 km 12 __ 5 km radic_
17 - 2 km
2π radic
_ 3
18 175
628
Sample answer Convert each number to a decimal
equivalent using estimation to find equivalents for
irrational numbers Graph each number on a number line
Read the numbers from left to right for least to greatest
Read the numbers from right to left for greatest to least
lt gt
lt lt
ltgt
gt gt
24 Unit 1
copy H
ough
ton
Miff
lin H
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ublis
hing
Com
pany
bull Im
age C
redi
ts copy
Elena
Eliss
eeva
Alam
y Im
ages
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
8_MCAAESE206984_U1M01L3indd 24 41613 448 AM
My Notes
5 52 54 56 58 6
radic28 5 12
23455
Math TrainerOnline Practice
and Help
Personal
myhrwcom
Math On the Spot
myhrwcom
Ordering Real Numbers in a Real-World Context Calculations and estimations in the real world may differ It can be important to know not only which are the most accurate but which give the greatest or least values depending upon the context
Four people have found the distance in kilometers across a canyon using different methods Their results are given in the table Order the distances from greatest to least
Distance Across Quarry Canyon (km)
Juana Lee Ann Ryne Jackson
radic_
28 23 __ 4 5 _
5 5 1 _ 2
Write each value as a decimal
radic_
28 is between 52 and 53 Since 53 2 = 2809 an approximate value for radic
_ 28 is 53
23 __ 4 = 575
5 _
5 is 5555hellip so 5 _
5 to the nearest hundredth is 556
5 1 _ 2 = 55
Plot radic_
28 23 __ 4 5 _
5 and 5 1 _ 2 on a number line
From greatest to least the distances are
23 __ 4 km 5 _
5 km 5 1 _ 2 km radic_
28 km
EXAMPLEXAMPLE 3
STEP 1
STEP 2
7 Four people have found the distance in miles across a crater using different methods Their results are given below
Jonathan 10 __ 3 Elaine 3 _
45 Joseacute 3 1 _ 2 Lashonda radic_
10
Order the distances from greatest to least
YOUR TURN
8NS2
3 1 _ 2 mi 3 _
45 mi 10 __ 3 mi radic_
10 mi
23Lesson 13
copy H
ough
ton
Miff
lin H
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ublis
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Com
pany
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8_MCAAESE206984_U1M01L3indd 23 41613 447 AM
ModelingPlace papers around the room with the numbers from 1 to 5 one per sheet Give each student a card showing a number between 1 and 5 in different forms Have students place his or her card between the correct integers and decide where the number goes in relation to any numbers already placed
Multiple RepresentationsGive students a vertical number line which some students might find easier to use than a horizontal one Have them decide whether to place points for rational and irrational numbers above or below existing points
Additional ResourcesDifferentiated Instruction includes bull Reading Strategies bull Success for English Learners EL
bull Reteach bull Challenge PRE-AP
DIFFERENTIATE INSTRUCTION
Ordering Real Numbers 24
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
Math TrainerOnline Assessment
and Intervention
Personal
Online homework assignment available
myhrwcom
myhrwcom
Lesson Quiz available online
13 LESSON QUIZ
1 Compare Write lt gt or =
radic_
95 - 5 radic_
62 - 2
2 Order 105 radic_
105 and 3π + 1 from greatest to least
3 A length in centimeters is calculated differently by four different people Order their calculations from least to greatest
KD 11 __ 2 cm Silvio 5 __ 3 π cm
Paula 5 _
4 cm Luis radic_
33 cm
EvaluateGUIDED AND INDEPENDENT PRACTICE
Concepts amp Skills Practice
Example 1Comparing Irrational Numbers
Exercises 1ndash8
Example 2Ordering Real Numbers
Exercises 9 12ndash15 18ndash21
Example 3Ordering Real Numbers in a Real-World Context
Exercises 10 16ndash17
Additional ResourcesDifferentiated Instruction includes bull Leveled Practice worksheets
Focus | Coherence | Rigor
Exercise Depth of Knowledge (DOK) Mathematical Practices
12ndash15 1 Recall of Information MP5 Using Tools
16 2 SkillsConcepts MP2 Reasoning
17 2 SkillsConcepts MP6 Precision
18ndash21 2 SkillsConcepts MP2 Reasoning
22 3 Strategic Thinking MP4 Modeling
23ndash24 3 Strategic Thinking MP3 Logic
8NS2
8NS2
Answers1 radic
_ 95 - 5 lt radic
_ 62 - 2
2 radic_
105 3π + 1 105
3 Silvio 5 __ 3 π cm Paula 5 _
4 cm
KD 11
__ 2 cm Luis radic_
33 cm
25 Lesson 13
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
Work Area
3140 3141 3142 3143
314 π227
20 A teacher asks his students to write the numbers shown in order from least to greatest Paul thinks the numbers are already in order Sandra thinks the order should be reversed Who is right
21 Math History There is a famous irrational number called Eulerrsquos number symbolized with an e Like π its decimal form never ends or repeats The first few digits of e are 27182818284
a Between which two square roots of integers could you find this number
b Between which two square roots of integers can you find π
22 Analyze Relationships There are several approximations used for π including 314 and 22 __ 7 π is approximately 314159265358979
a Label π and the two approximations on the number line
b Which of the two approximations is a better estimate for π Explain
c Find a whole number x so that the ratio x ___ 113 is a better estimate for π
than the two given approximations
23 Communicate Mathematical Ideas If a set of six numbers that include both rational and irrational numbers is graphed on a number line what is the fewest number of distinct points that need to be graphed Explain
24 Critique Reasoning Jill says that 12 _
6 is less than 1263 Explain her error
FOCUS ON HIGHER ORDER THINKING
radic_
115 115 ___ 11 and 105624
between radic_
7 asymp 265 and radic_
8 asymp 283
between radic_
9 = 3 and radic_
10 asymp 316
22 __ 7 it is closer to π on the number line
She did not consider the repeating digit 1266
2 rational numbers can have the same location and
irrational numbers can have the same location but they
cannot share a location
355
Neither student is correct The answer
should be 115 ___ 11 105624 radic_
115
Unit 126
copy H
ough
ton M
ifflin
Har
cour
t Pub
lishin
g Com
pany
Imag
e Cre
dits
copy3D
Stoc
kiSt
ockP
hoto
com
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
8_MCAAESE206984_U1M01L3indd 26 210513 801 AM
Personal Math Trainer
Online Practice and Helpmyhrwcom
Name Class Date
Independent Practice
16 Your sister is considering two different shapes for her garden One is a square with side lengths of 35 meters and the other is a circle with a diameter of 4 meters
a Find the area of the square
b Find the area of the circle
c Compare your answers from parts a and b Which garden would give your sister the most space to plant
17 Winnie measured the length of her fatherrsquos ranch four times and got four different distances Her measurements are shown in the table
a To estimate the actual length Winnie first approximated each distance to the nearest hundredth Then she averaged the four numbers Using a calculator find Winniersquos estimate
b Winniersquos father estimated the distance across his ranch to be radic_
56 km How does this distance compare to Winniersquos estimate
Give an example of each type of number
18 a real number between radic_
13 and radic_
14
19 an irrational number between 5 and 7
Order the numbers from least to greatest
12 radic_
7 2 radic_
8 ___ 2 13 radic_
10 π 35
14 radic_
220 -10 radic_
100 115 15 radic_
8 -375 3 9 _ 4
Distance Across Fatherrsquos Ranch (km)
1 2 3 4
radic_
60 58 __ 8 7 _
3 7 3 _ 5
138NS2
radic_
8 ___ 2 2 radic_
7
-10 radic_
100 115 radic_
220
radic_
60 asymp 775 58 __ 8 = 725 7 _
3 asymp 733 7 3 _ 5 = 760 so the average
π radic_
10 35
-375 9 _ 4 radic_
8 3
is 74825 km
1225 m2
4π m2 or approximately 126 m2
They are nearly identical radic_
56 is approximately 74833hellip
The circle would give her more space to plant because it has a
larger area
Sample answer 37
Sample answer radic_
31
25Lesson 13
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ough
ton
Miff
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pany
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8_MCAAESE206984_U1M01L3indd 25 41613 448 AM
Activity available online myhrwcomEXTEND THE MATH PRE-AP
Activity Have students investigate whether there are infinitely many numbers between two numbers by giving examples for each of the following
bull Between any two rational numbers there is at least one other rational number Sample answer 45 is between 41 and 48
bull Between any two irrational numbers there is at least one rational number Sample answer 45 is between radic
_ 11 and radic
_ 29
bull Between any two rational numbers there is at least one irrational number Sample answer radic
_ 11 is between 31 and 36
bull Between any two irrational numbers there is at least one irrational number Sample answer radic
_ 17 is between radic
_ 11 and radic
_ 29
Ordering Real Numbers 26
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
ReadyMath Trainer
Online Practiceand Help
Personal
myhrwcom
Module Quiz
11ensp RationalenspandenspIrrationalenspNumbersWrite each fraction as a decimal or each decimal as a fraction
1 7__20 2 1___
27 3 17_8
Solve each equation for x
4 x2=81 5 x3=343 6 x2= 1___100
7 Asquarepatiohasanareaof200squarefeetHowlongiseachside
ofthepatiotothenearesttenth
12ensp SetsenspofenspRealenspNumbersWrite all names that apply to each number
8 121____radic
____121
9 π__2
10 TellwhetherthestatementldquoAllintegersarerationalnumbersrdquoistrueorfalseExplainyourchoice
13ensp OrderingenspRealenspNumbersCompare Write lt gt or =
11 radic__
8+3 8+radic__
3 12 radic__
5+11emsp emsp emsp 5+radic___
11
Order the numbers from least to greatest
13 radic___
99π29__
8 14 radic___
1__251_40__
2
15 Howarerealnumbersusedtodescribereal-worldsituations
ESSENTIAL QUESTION
035
9-9
141ft
7 1__10- 1__10
14__11 1875
wholeintegerrationalreal
Trueintegerscanbewrittenasthequotientoftwointegers
SampleanswerRealnumberssuchastherational
π29__
8radic___
99
irrationalreal
lt gt
number1_4candescribeamountsusedincooking
radic___
1__250__
21_4
27Module1
copy H
ough
ton
Miff
lin H
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urt P
ublis
hing
Com
pany
DONOTEDIT--ChangesmustbemadethroughldquoFileinfordquoCorrectionKey=A
8_MCAAESE206984_U1M01RTindd 27 41513 1113 PM
Math TrainerOnline Assessment
and Intervention
Personal
myhrwcom
1
2
3 Response toIntervention
Intervention Enrichment
Access Ready to Go On assessment online and receive instant scoring feedback and customized intervention or enrichment
Online and Print Resources
Differentiated Instruction
bull Reteach worksheets
bull Reading Strategies EL
bull Success for English Learners EL
Differentiated Instruction
bull Challenge worksheets PRE-AP
Extend the Math PRE-AP
Lesson Activities in TE
Additional ResourcesAssessment Resources includes bull Leveled Module Quizzes
Ready to Go OnAssess MasteryUse the assessment on this page to determine if students have mastered the concepts and standards covered in this module
California Common Core Standards
Lesson Exercises Common Core Standards
11 1ndash7 8NS1 8NS2 8EE2
12 8ndash10 8NS1
13 11ndash14 8NS2
27 Unit 1 Module 1
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
Personal Math Trainer
Online Practice and HelpmyhrwcomAssessment Readiness
Module 1 MIXed ReVIeW
1 Look at each number Is the number between 2π and radic___
52
Select Yes or No for expressions AndashC
A 6 2 _ 3 Yes No
B 5π __ 2 Yes No
C 3 radic__
5 Yes No
2 Consider the number - 11 __ 15
Choose True or False for each statement
A The number is rational True False
B The number can be written as True Falsea repeating decimal
C The number is less than ndash08 True False
3 The volume of a cube is given by V = x3 where x is the length of an edge of the cube A cube-shaped end table has a volume of 3 3 _ 8 cubic feet What is the length of an edge of the end table Explain how you solved this problem
4 A student says that radic___
83 is greater than 29 __ 3 Is the student correct Justify your
reasoning
1 1 _ 2 ft Sample answer The equation x3 = 3 3 _ 8 can be used
to find the edge length in feet To solve the equation
write the mixed number as a fraction greater than 1
x3 = 27 __ 8 Then take the cube root of both sides x = 3 _ 2 = 1 1 _ 2
No Sample answer radic___
83 asymp 91 and 29 __ 3 = 9
__ 6
Because 91 lt 9 __
6 radic___
83 lt 29 __ 3
28 Unit 1
copy H
ough
ton
Miff
lin H
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ublis
hing
Com
pany
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=A
8_MCAAESE206984_U1M01RTindd 28 240413 946 AM
Personal Math Trainer
Online Assessment and
Interventionmyhrwcom
Scoring GuideItem 3 Award the student 1 point for finding the edge length of the cube and 1 point for correctly explaining how to use a cube root to solve the problem
Item 4 Award the student 1 point for determining that the student is incorrect and 1 point for correctly justifying the reasoning for this conclusion
Additional ResourcesTo assign this assessment online login to your Assignment Manager at myhrwcom
Assessment Readiness
California Common Core Standards
Items Grade 8 Standards Mathematical Practices
1 8NS2 MP7
2 7NS2b 7NS2d 8NS1 MP7
3 8EE2 MP1 MP4
4 8NS1 8NS2 MP3
Item integrates mixed review concepts from previous modules or a previous course
Item 4 combines concepts from the California Common Core cluster ldquoKnow that there are numbers that are not rational and approximate them by rational numbersrdquo
Real Numbers 28
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
Math Talk
Language Support EL
PROFESSIONAL DEVELOPMENT
Linguistic Support EL
AcademicContent Vocabulary
Venn diagrams ndash Students need descriptive language to describe the categories that the different areas and colors of a Venn diagram represent the concept of a set and how sets are distinct or can overlap Use sentence frames such as
The big oval represents __________The darklight blue color in the middle of the
big ovals represents __________These sets overlap because __________
In this way students have the language and structure to identify the criteria that distinguish a set and to explain the abstract representation Also point out the use of the prefix sub- meaning ldquounderrdquo in the term subset
Rules and Patterns
Abbreviations ndash In this lesson the abbreviation mph is used Be sure to point out that mph stands for miles per hour and is used to give units in a rate of speed Students may also have seen mpg (miles per gallon) which gives the units in a rate of fuel efficiency
Borrowed Words ndash Terminology used in baseball such as inning and pitcher may require some explanation Spanish as well as some other languages have borrowed these terms from English so some students may be familiar with these words already Despite this whenever a word is critical to students understanding the word problem it is best to explain the meaning
Leveled Strategies for English Learners
Emerging Allow students to indicate true or false orally in Guided Practice Exercises 9 and 10
Expanding Have students use sentence frames to describe the meaning of regions and colors used in a Venn diagram Then give them similar sentence frames orally and have them draw and shade a Venn diagram based on the oral prompts
Bridging Have students work in groups to draw a Venn diagram to represent sets based on real-world examples in the lesson
To help students answer the question posed in Math Talk provide a sentence frame for their answer
The numbers between 31 and 39 on a number line are __________ because __________
EL
California ELD Standards
Emerging 2II5 Modifying to add details ndash Expand sentences with simple adverbials to provide details about a familiar activity or process
Expanding 2II5 Modifying to add details ndash Expand sentences with adverbials to provide details about a familiar or new activity or process
Bridging 2II5 Modifying to add details ndash Expand sentences with increasingly complex adverbials to provide details about a variety of familiar and new activities and processes
Sets of Real Numbers 15B
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
12L E S S O N
Sets of Real Numbers
EngageESSENTIAL QUESTION
How can you describe relationships between sets of real numbers Sample answer Describe them as two different sets or one set as being a subset of another
Motivate the LessonAsk How many different types of tigers can you name How does the set of Bengal tigers relate to the set of tigers
ExplorePoint to different locations in the Animals diagram and ask for examples for that classification Do the same for the Real Numbers diagram Students should understand that everything within a region is part of the set for example both -3 and 2 are integers
ExplainEXAMPLE 1
Questioning Strategies Mathematical Practices bull In A why is 5 not a perfect square It does not have rational numbers as its square roots
bull Can the number in B be written as a fraction Why or why not Yes it is a terminating decimal so it is a rational number
Engage with the WhiteboardHave students place the numbers in Example 1 and Additional Example 1 in the Venn diagram for numbers
YOUR TURNAvoid Common ErrorsBe sure that students read Exercise 2 carefully before answering The number given in the problem 10 is the area not the side length
EXAMPLE 2Questioning Strategies Mathematical Practices bull What two major sets are the real numbers composed of rational and irrational numbers
bull What is the location of the set of whole numbers in the Venn diagram in relation to the set of rational numbers Explain Inside it whole numbers are rational numbers
Focus on Reasoning Mathematical PracticesRemind students that it takes only one counterexample to show that a statement is false
Interactive Whiteboard Interactive example available online
ADDITIONAL EXAMPLE 1Write all names that apply to each number
A -10integer rational real
B 12 _ 3
whole integer rational real
myhrwcom
Interactive Whiteboard Interactive example available online
ADDITIONAL EXAMPLE 2Tell whether the given statement is true or false Explain your choice
No integers are whole numbers
False every whole number is also an integer
myhrwcom
Animated MathClassifying Numbers
Students build fluency in classifying numbers in this engaging fast-paced game
myhrwcom
CA Common CoreStandards
The student is expected to
The Number Systemmdash8NS1
Know that numbers that are not rational are called irrational Understand informally that every number has a decimal expansion for rational numbers show that the decimal expansion repeats eventually and convert a decimal expansion which repeats eventually into a rational numberMathematical Practices
MP7 Using Structure
The student is expected to
15 Lesson 12
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=B
Math TrainerOnline Practice
and Help
Personal
myhrwcom
Math TrainerOnline Practice
and Help
Personal
myhrwcom
Math On the Spotmyhrwcom
Understanding Sets and Subsets of Real NumbersBy understanding which sets are subsets of types of numbers you can verify whether statements about the relationships between sets are true or false
Tell whether the given statement is true or false Explain your choice
All irrational numbers are real numbers
True Every irrational number is included in the set of real numbers The irrational numbers are a subset of the real numbers
No rational numbers are whole numbers
False A whole number can be written as a fraction with a denominator of 1 so every whole number is included in the set of rational numbers The whole numbers are a subset of the rational numbers
EXAMPLE 2
A
B
Write all names that apply to each number
1 A baseball pitcher has pitched 12 2 _ 3 innings
2 The length of the side of a square that has an
area of 10 square yards
YOUR TURN
Tell whether the given statement is true or false Explain your choice
3 All rational numbers are integers
4 Some irrational numbers are integers
YOUR TURN
Give an example of a rational number that is a
whole number Show that the number is both whole
and rational
Math TalkMathematical Practices
Give an example of a
8NS1
False Every integer is a rational number but not every
False Real numbers are either rational or irrational numbers
Integers are rational numbers so no integers are irrational numbers
rational real
irrational real
Sample answer 8 8 = 8_
1
and -thinsp 5 _ 2 are not integers
rational number is an integer Rational numbers such as 3 _ 5
Unit 116
copy H
ough
ton
Miff
lin H
arco
urt P
ublis
hing
Com
pany
bull Im
age C
redi
ts D
igita
l Im
age c
opyr
ight
copy20
04 Ey
ewire
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
8_MCAAESE206984_U1M01L2indd 16 41613 136 AM
Math On the Spot
myhrwcom
Vertebrates
Birds
Passerines
Animals
Integers
Rational Numbers IrrationalNumbers
Real Numbers
WholeNumbers
1
45
3
0
274
67
radic4
-
-3
-2
-1
03
radic2
radic17
radic11-
π
Animated Math
myhrwcom
Classifying Real NumbersBiologists classify animals based on shared characteristics A cardinal is an animal a vertebrate a bird and a passerine
You already know that the set of rational numbers consists of whole numbers integers and fractions The set of real numbers consists of the set of rational numbers and the set of irrational numbers
Write all names that apply to each number
radic_
5 irrational real
ndash1784rational real
whole integer rational real
EXAMPLEXAMPLE 1
A
B
C radic_ 81 ____ 9
L E S S O N
12Sets of Real Numbers
ESSENTIAL QUESTIONHow can you describe relationships between sets of real numbers
Passerines such as the cardinal are also called ldquoperching birdsrdquo
What types of numbers are between 31 and 39 on a
number line
Math TalkMathematical Practices
What types of numbers are
8NS1
8NS1
Know that numbers that are not rational are called irrational Understand informally that every number has a decimal expansion for rational numbers show that the decimal expansion repeats eventually and convert a decimal expansion which repeats eventually into a relation number
ndash1784 is a terminating decimal
5 is a whole number that is not a perfect square
radic_
81 _____ 9 = 9 __ 9 = 1 rational irrational real
15Lesson 12
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ough
ton
Miff
lin H
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urt P
ublis
hing
Com
pany
bull Im
age C
redi
ts copy
Wiki
med
ia Co
mm
ons
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=B
8_MCABESE206984_U1M01L2indd 15 061113 1144 AM
PROFESSIONAL DEVELOPMENT
Math BackgroundThe relationships between sets of numbers extend to include complex numbers A complex number can be written as a sum of a real number a and an imaginary number bi
a + bi
An imaginary number is a special number that when squared gives a negative value When you square a real number you get a nonnegative number When you square an imaginary number you get a negative value The imaginary unit is i
i = radic_
-1
Integrate Mathematical Practices MP7
This lesson provides an opportunity to address this Mathematical Practices standard It calls for students to discern structure to connect and communicate mathematical ideas
Students use a Venn diagram to structure relationships between sets of numbers They connect and communicate mathematical ideas when they make logical statements about the sets and describe which set best describes numbers applied to real-life situations
Sets of Real Numbers 16
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=B
YOUR TURNAvoid Common ErrorsStudents may see the word ldquoAllldquo or rdquoNordquo in Exercises 3 and 4 and immediately assume that any absolute statements like these are false Remind them that there are true statements that begin with these words and encourage them to provide examples
EXAMPLE 3Questioning Strategies Mathematical Practices bull In A how does the phrase ldquonumber of rdquo give you a clue about the number classification It indicates a counting number
bull What is the relationship between the circumference of a circle and the diameter The circumference is diameter times π
Focus on Critical Thinking Mathematical PracticesIn B suppose the diameters in inches were 25
__ π 28 __ π
31 __ π and so on What set of numbers would
best describe the circumferences Explain Whole numbers the circumferences would be the whole numbers 25 28 31 and so on
YOUR TURNFocus on Critical Thinking Mathematical PracticesHave students compare and contrast the classification of numbers in the answers in Exercises 5 and 6
ElaborateTalk About ItSummarize the Lesson
Ask What are some ways that number sets can be related Sets may be subsets of other sets or they may be separate from other sets
GUIDED PRACTICEEngage with the Whiteboard
Have students place the numbers in Exercises 1ndashthinsp8 in the Venn diagram for numbers at the beginning of the lesson
Integrating Language Arts EL
Encourage English learners to ask for clarification on any terms or phrases that they do not understand
Avoid Common ErrorsExercise 7 Remind students that a repeating decimal is a rational numberExercises 9ndash10 Remind students that it only takes one counterexample to show that a statement is false
Interactive Whiteboard Interactive example available online
ADDITIONAL EXAMPLE 3Identify the set of numbers that best describes the situation Explain your choice
A the amount of time that has passed since midnight
The set of real numbers time is continuous so the amount of time can be rational or irrational
B the number of tickets sold to a basketball game
The set of whole numbers the number of tickets sold may be 0 or a counting number
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17 Lesson 12
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
1IN
116 inch
Guided Practice
Write all names that apply to each number (Example 1)
1 7 _ 8 2 radic_
36
3 radic_
24 4 075
5 0 6 - radic_ 100
7 5 _
45 8 - 18 __ 6
Tell whether the given statement is true or false Explain your choice (Example 2)
9 All whole numbers are rational numbers
10 No irrational numbers are whole numbers
Identify the set of numbers that best describes each situation Explain your choice (Example 3)
11 the change in the value of an account when given to the nearest dollar
12 the markings on a standard ruler
13 What are some ways to describe the relationships between sets of numbers
CHECK-INESSENTIAL QUESTION
rational real
rational real
True Whole numbers are rational numbers
Rational numbers the ruler is marked every 1 __ 16 th inch
Sample answer Describe one set as being a subset of
another or show their relationships in a Venn diagram
Integers the change can be a whole dollar amount
and can be positive negative or zero
True Whole numbers are a subset of the set of rational numbers
and can be written as a ratio of the whole number to 1
irrational real
whole integer rational real
whole integer rational real
rational real
integer rational real
integer rational real
Unit 118
copy H
ough
ton
Miff
lin H
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ublis
hing
Com
pany
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
8_MCAAESE206984_U1M01L2indd 18 41613 136 AM
My Notes
Math TrainerOnline Practice
and Help
Personal
myhrwcom
Math On the Spot
myhrwcom
Identifying Sets for Real-World SituationsReal numbers can be used to represent real-world quantities Highways have posted speed limit signs that are represented by natural numbers such as 55 mph Integers appear on thermometers Rational numbers are used in many daily activities including cooking For example ingredients in a recipe are often given in fractional amounts such as 2 _ 3 cup flour
Identify the set of numbers that best describes each situation Explain your choice
the number of people wearing glasses in a room
The set of whole numbers best describes the situation The number of people wearing glasses may be 0 or a counting number
the circumference of a flying disk has a diameter of 8 9 10 11 or 14 inches
The set of irrational numbers best describes the situation Each circumference would be a product of π and the diameter and any multiple of π is irrational
EXAMPLEXAMPLE 3
A
B
Identify the set of numbers that best describes the situation Explain your choice
5 the amount of water in a glass as it evaporates
6 the weight of a person in pounds
YOUR TURN
8NS1
Rational numbers a personrsquos weight can be a decimal
such as 835 pounds
Real numbers the amount can be any number greater
than 0
17Lesson 12
copy H
ough
ton
Miff
lin H
arco
urt P
ublis
hing
Com
pany
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
8_MCAAESE206984_U1M01L2indd 17 41613 520 AM
Graphic OrganizersGive students a list of numbers (including terminating and repeating decimals fractions integers and rational and irrational square roots) and a graphic organizer as shown below
Real Numbers
Rational numbers Irrational numbers
Integer numbers
Whole numbers
Ask students to write each number in the list in the correct section of the organizer
Number SensePoint out to students that knowing the types of numbers to expect in different situations can alert them to incorrect math as well as to impossible situations For example 135 shots made in basketballs is not possible but an average number of shots can equal 135
Additional ResourcesDifferentiated Instruction includes bull Reading Strategies bull Success for English Learners EL
bull Reteach bull Challenge PRE-AP
DIFFERENTIATE INSTRUCTION
Sets of Real Numbers 18
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
Math TrainerOnline Assessment
and Intervention
Personal
Online homework assignment available
myhrwcom
Lesson Quiz available online
12 LESSON QUIZ
1 Write all the names that apply to the number
2 Tell whether the given statement is true or false Explain your choice All numbers between 1 and 2 are rational numbers
3 Identify the set of numbers that best describes the situation Explain your choiceThe choices on a survey question change the total points for the survey by -2 -1 0 1 or 2 points
-1 _
5
myhrwcom
Additional ResourcesDifferentiated Instruction includes bull Leveled Practice worksheets
EvaluateGUIDED AND INDEPENDENT PRACTICE
Concepts amp Skills Practice
Example 1Classifying Real Numbers
Exercises 1ndash8 14ndash19 22ndash24
Example 2Understanding Sets and Subsets of Real Numbers
Exercises 9ndash10
Example 3Identifying Sets for Real-World Situations
Exercises 11ndash12 20ndash21 25
Focus | Coherence | Rigor
Exercise Depth of Knowledge (DOK) Mathematical Practices
14ndash19 2 SkillsConcepts MP7 Using Structure
20ndash21 2 SkillsConcepts MP6 Precision
22ndash23 2 SkillsConcepts MP3 Logic
24 1 Recall of Information MP7 Using Structure
25 2 SkillsConcepts MP2 Reasoning
26ndash27 3 Strategic Thinking MP3 Logic
28 3 Strategic Thinking MP8 Patterns
29 3 Strategic Thinking MP3 Logic
8NS1
8NS1
Exercise 29 combines concepts from the California Common Core cluster ldquoKnow that there are numbers that are not rational and approximate them by rational numbersrdquo
Answers1 rational real
2 False radic_
2 is an example of an irrational number between 1 and 2
3 Integers each number is an integer but only three are whole numbers
19 Lesson 12
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
Work Area
π mi23 Critique Reasoning The circumference of a circular region is shown
What type of number best describes the diameter of the circle Explain
your answer
24 Critical Thinking A number is not an integer What type of number can it be
25 A grocery store has a shelf with half-gallon containers of milk What type of number best represents the total number of gallons
26 Explain the Error Katie said ldquoNegative numbers are integersrdquo What was her error
27 Justify Reasoning Can you ever use a calculator to determine if a number is rational or irrational Explain
28 Draw Conclusions The decimal 0 _
3 represents 1 _ 3 What type of number best describes 0
_ 9 which is 3 middot 0
_ 3 Explain
29 Communicate Mathematical Ideas Irrational numbers can never be precisely represented in decimal form Why is this
FOCUS ON HIGHER ORDER THINKING
It can be a rational number that is not an integer or an irrational number
rational number
The set of negative numbers also includes non-integer
rational numbers and irrational numbers
Sample answer If the calculator shows a decimal that
terminates in fewer digits than what the calculator screen
allows then you can tell that the number is rational If not
you cannot tell from the calculator display whether the
number terminates because you see a limited number
of digits It may be a repeating decimal (rational) or
non-terminating non-repeating decimal (irrational)
Whole 3 middot 0 _
3 represents 3 middot 1 _ 3 = 1 so 0 _
9 is exactly 1
Sample answer In decimal form irrational numbers never
terminate and never repeat Therefore no matter how
many decimal places you include the number will never
be precisely represented There are always more digits
Whole the diameter is π _ π = 1 mile
Unit 120
copy H
ough
ton
Miff
lin H
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ublis
hing
Com
pany
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
8_MCAAESE206984_U1M01L2indd 20 120413 909 PM
Integers
Rational Numbers Irrational Numbers
Real Numbers
Whole Numbers
257
radic16
166
radic9
128 radic50
Personal Math Trainer
Online Practice and Helpmyhrwcom
Name Class Date
Independent Practice
Identify the set of numbers that best describes each situation Explain your choice
20 the height of an airplane as it descends to an airport runway
21 the score with respect to par of several golfers 2 ndash 3 5 0 ndash 1
22 Critique Reasoning Ronald states that the number 1 __ 11 is not rational because when converted into a decimal it does not terminate Nathaniel says it is rational because it is a fraction Which boy is correct Explain
12
14 - radic_
9 15 257
16 radic_
50 17 8 1 _ 2
18 166 19 radic_
16
Write all names that apply to each number Then place the numbers in the correct location on the Venn diagram
8NS1
Real numbers the height can be any number greater than zero
integer rational real whole integer rational real
whole integer rational real
irrational real
rational real
rational real
Integers the scores are counting numbers their
opposites and zero
Nathaniel is correct A rational number is a number that can be written as a fraction and 1 __ 11 is a fraction
19Lesson 12
copy H
ough
ton
Miff
lin H
arco
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ublis
hing
Com
pany
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
8_MCAAESE206984_U1M01L2indd 19 41613 136 AM
myhrwcomActivity available onlineEXTEND THE MATH PRE-AP
Activity Have students consider the concept of restricted domain for the sets of numbers that describe situations For example the number of sisters a person has can best be described by whole numbers but no one has ever had 1500 sisters An area code is an integer or whole number between 200 and 999
Have students use a source such as the Guinness Book of World Records and give examples of sets of numbers that describe situations where the domain is restricted Ask whether the restriction may be changed in the future
Sets of Real Numbers 20
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
-3-4-5 -2-1 1 2 3 50 4
12-4 -radic5
Lesson Support Content Objective Students will learn to order a set of real numbers
Language Objective Students will show and describe how to order a set of real numbers
LESSON 13 Ordering Real Numbers
Building BackgroundEliciting Prior Knowledge Have students draw a number line to compare a rational number and an irrational number such as - radic
_ 5 and -4 1 __ 2 Ask them to explain how
they approximated the irrational number on the number line Then have them identify the greater and the lesser real number Repeat with several other pairs of real numbers in different forms
Learning ProgressionsIn this lesson students order a set of real numbers They use rational approximations to compare the sizes of irrational numbers They also order numbers for real-world situations Important understandings for students include the following
bull Compare irrational numbers bull Estimate the value of expressions with irrational numbers bull Order a set of real numbers bull Order real numbers in a real-world context
Work with real numbers continues throughout Grade 8 and into high school This lesson provides students with a foundation for understanding the relative sizes of numbers in different forms in the real number system
Cluster ConnectionsThis lesson provides an excellent opportunity to connect ideas in this cluster Know that there are numbers that are not rational and approximate them by rational numbers Tell students that there is a special number called the golden ratio with applications in mathematics geometry art and architecture The golden ratio is called phi and is represented by the Greek letter ϕ It includes an irrational number in its definition
Have students explain why the golden ratio is irrational Ask them to find the two whole numbers the golden ratio lies between Then challenge them to approximate the golden ratio to the nearest tenth It is irrational because it includes an irrational number in its definition It lies between 1 and 2 To the nearest tenth ϕ = 16
ϕ = 1 + radic_
5 _ 2
Focus | Coherence | Rigor
California Common Core Standards
8NS2 Use rational approximations of irrational numbers to compare the size of irrational numbers locate them approximately on a number line diagram and estimate the value of expressions (eg π 2 )
MP4 Model with mathematics
21A
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
Math Talk
Language Support EL
PROFESSIONAL DEVELOPMENT
Linguistic Support EL
AcademicContent Vocabulary
Post a chart like this to remind students of the regular comparative forms of adjectives that use the -er and -est suffixes Add to the chart for terms that appear in examples and exercises in each lesson Include any irregular verb forms
Background Knowledge
Go On ndash the title of the module review or quiz is Ready to Go On This title uses an idiomatic expression In this context to go on means ldquoto move aheadrdquo or ldquoto proceedrdquo It is different from the use of go on that means having enough facts to use meaningfully as in having enough to go on Also the intonation used in pronouncing an expression can give it different meanings For example when the speaker emphasizes the word on he or she might be expressing disbelief as in ldquoGo ON Yoursquore kidding rightrdquo Discuss with students other ways that the phrase go on may be used
Leveled Strategies for English Learners
Emerging Label points on a number line with the terms used in ordering greater greatest less lesser least Use sentence frames to insert the correct terms
Expanding Have students give two or three complete sentences to compare the placement of numbers on a number line using the correct forms of the comparative and superlative adjectives
Bridging Have students work in pairs with one student giving directions to the other in complete sentences to order numbers on a number line
To help students answer the question posed in Math Talk make sure that students have a command of the forms for making comparisons and the superlative and the concept of opposite order so that the focus is on the math concept instead of the language skills needed to describe and explain order
EL
Adjective Comparative Superlative
Far Farther Farthest
Large Larger Largest
Great Greater Greatest
Some Less Least
Some More Most
California ELD Standards
Emerging 2I8 Analyzing language choices ndash Explain how phrasing or different common words with similar meanings produce different effects on the audience
Expanding 2I8 Analyzing language choices ndash Explain how phrasing or different words with similar meanings or figurative language produce shades of meaning and different effects on the audience
Bridging 2I8 Analyzing language choices ndash Explain how phrasing or different words with similar meanings or figurative language produce shades of meaning nuances and different effects on the audience
Ordering Real Numbers 21B
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
13L E S S O N
Ordering Real Numbers
Interactive Whiteboard Interactive example available online
ADDITIONAL EXAMPLE 1Compare Write lt gt or =
A radic_
8 - 2 4 - radic_
8 lt
B radic_
20 + 1 3 + radic_
2 gt
EngageESSENTIAL QUESTION
How do you order a set of real numbers Sample answer Find their approximate decimal values and order them
Motivate the LessonAsk What kind of numbers are you comparing when you compare the price of gasoline at two different gas stations
ExploreGive students two rational numbers and ask them to name a number between them Repeat a few times and then give them two irrational numbers and ask them to name a number between them
ExplainEXAMPLE 1
Questioning Strategies Mathematical Practices bull Which is greater the difference between 5 and 3 or the difference between radic
_ 5 and radic
_ 3
The difference between 5 and 3 is 2 the difference between radic_
5 and radic_
3 is approximately 1 So the difference between 5 and 3 is greater
Avoid Common ErrorsCaution students to read the problem carefully and think about what the radical sign means so that they do not misread the problem and answer that the two sides are equal
YOUR TURNFocus on TechnologyCalculators should not be used at this point because developing number sense is the goal
EXAMPLE 2Questioning Strategies Mathematical Practices bull How do you determine whether radic
_ 22 is less than or greater than 45 The square of 45 is
2025 which is less than 22 so the square root of 22 must be greater than 45
Engage with the WhiteboardHave students graph and label various real numbers between 42 and 44 and between 47 and 5
YOUR TURNFocus on Modeling Mathematical PracticesHave students label the integers on the number line with their equivalent square root For example 1 2 and 3 on the number line would be labeled radic
_ 1 radic
_ 4 and radic
_ 9
Interactive Whiteboard Interactive example available online
ADDITIONAL EXAMPLE 2Order 3π radic
_ 10 and 325 from greatest
to least
3π 325 radic_
10
myhrwcom
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CA Common CoreStandards
The student is expected to
The Number Systemmdash8NS2
Use rational approximations of irrational numbers to compare the size of irrational numbers locate them approximately on a number line diagram and estimate the value of expressions (eg π 2 )
Mathematical Practices
MP4 Modeling
The student is expected to
21 Lesson 13
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
Math TrainerOnline Practice
and Help
Personal
myhrwcom
Math On the Spotmyhrwcom
0 05 1 15 2 25 3 35 4
radic5radic3
π2
8 85 9 95 10 105 11 115 12
radic75
4 42 44 46 48 5
radic224 12π + 1
Ordering Real Numbers You can compare and order real numbers and list them from least to greatest
Order radic_
22 π + 1 and 4 1 _ 2 from least to greatest
First approximate radic_
22
radic_
22 is between 4 and 5 Since you donrsquot know where it falls between 4 and 5 you need to find a better estimate for radic
_ 22 so
you can compare it to 4 1 _ 2
Since 22 is closer to 25 than 16 use squares of numbers between 45 and 5 to find a better estimate of radic
_ 22
45 2 = 2025 46 2 = 2116 47 2 = 2209 48 2 = 2304
Since 47 2 = 2209 an approximate value for radic_
22 is 47
An approximate value of π is 314 So an approximate value of π +1 is 414
Plot radic_
22 π + 1 and 4 1 _ 2 on a number line
Read the numbers from left to right to place them in order from least to greatest
From least to greatest the numbers are π + 1 4 1 _ 2 and radic_
22
EXAMPLE 2
STEP 1
STEP 2
Order the numbers from least to greatest Then graph them on the number line
YOUR TURN
5 radic_
5 25 radic_
3
6 π 2 10 radic_
75
If real numbers a b and c are in order from least to greatest what is the order
of their opposites from least to greatest
Explain
Math TalkMathematical Practices
8NS2
radic_
3 radic_
5 25
radic_
75 π2 10
Math Talk answer -c -b -a -c is farthest to the left on a number line -b is in the middle and -a is farthest to the right
Unit 122
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ough
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Miff
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pany
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
8_MCAAESE206984_U1M01L3indd 22 41613 447 AM
My Notes
Math TrainerOnline Practice
and Help
Personal
myhrwcom
Math On the Spot
myhrwcom
Comparing Irrational NumbersBetween any two real numbers is another real number To compare and order real numbers you can approximate irrational numbers as decimals
Compare radic_
3 + 5 3 + radic_
5 Write lt gt or =
First approximate radic_
3
radic_
3 is between 1 and 2
Next approximate radic_
5
radic_
5 is between 2 and 3
Then use your approximations to simplify the expressions
radic_
3 + 5 is between 6 and 7
3 + radic_
5 is between 5 and 6
So radic_
3 + 5 gt 3 + radic_
5
Reflect1 If 7 + radic
_ 5 is equal to radic
_ 5 plus a number what do you know about the
number Why
2 What are the closest two integers that radic_
300 is between
EXAMPLEXAMPLE 1
STEP 1
STEP 2
Compare Write lt gt or =
YOUR TURN
3 radic_
2 + 4 2 + radic_
4 4 radic_
12 + 6 12 + radic_
6
L E S S O N
13 Ordering Real Numbers
ESSENTIAL QUESTIONHow do you order a set of real numbers
8NS2
Use rational approximations of irrational numbers to compare the size of irrational numbers locate them approximately on a number line diagram and estimate the value of expressions (eg π 2 )
8NS2
Use perfect squares to estimate square roots
1 2 = 1 2 2 = 4 3 2 = 9
The number is 7 both expressions must equal 7 + radic_
5
17 and 18
gt lt
21Lesson 13
copy H
ough
ton
Miff
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Com
pany
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
8_MCAAESE206984_U1M01L3indd 21 41913 246 PM
PROFESSIONAL DEVELOPMENT
Math BackgroundIn this lesson students estimate irrational numbers in the form of square roots of nonper-fect squares by finding two perfect squares between which the number falls A more precise method involves repeated division For example to find radic
_ 28 find a whole number whose perfect
square is close to 28 such as 5 Divide 28 by that number 28 divide 5 = 56 Find the average of the quotient and divisor 5 + 56
_____ 2 = 53 Continue dividing 28 by each result and averaging until you get the desired accuracy
Integrate Mathematical Practices MP4
This lesson provides an opportunity to address this Mathematical Practices standard It calls for students to model relationships using multiple representations including diagrams graphs and language as appropriate Students use multiple representations when they use number lines to estimate the locations of and order rational and irrational numbers given as symbols
Ordering Real Numbers 22
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
Interactive Whiteboard Interactive example available online
ADDITIONAL EXAMPLE 3The diameter of a meteorite in millimeters is calculated by four different methods Order the results from least to greatest
Joe radic_
18 mm Lisa 13 __ 3 mm
Pablo 46 mm Julien 4π __ 3 mm
Julien 4π __ 3 mm Lisa 13 __ 3 mm
Joe radic_
18 mm Pablo 46 mm
EXAMPLE 3Questioning Strategies Mathematical Practices bull How can you verify that radic
_ 28 is between 52 and 53 5 2 2 = 2704 and 5 3 2 = 2809
bull Explain how to determine which number is greater 5 _
5 or 55 When the repeating decimal is rounded to the nearest tenth or hundredth you can see that it is greater
Connect to Daily LifeDiscuss how measuring across a canyon might involve different methods than measuring along a road Explain that measurements like these are often done using calculations that approximate the distance
YOUR TURNFocus on Critical Thinking Mathematical PracticesDiscuss with students which number is greater 3
_ 45 or 3450 3
_ 45 or 3455 and why Explain
that 3 _
45 can be written out as 34545hellipMake sure they understand that 3 _
45 is greater than 345 but less than 3455
ElaborateTalk About ItSummarize the Lesson
Ask How can you order two numbers in different forms whose decimal approxi-mations appear to be equal Approximate one or both numbers to an additional
number of decimal places
GUIDED PRACTICEEngage with the Whiteboard
Have students place and label additional points on the number line in Exercise 9 Allow the points to be in any format other than decimal
Avoid Common ErrorsExercises 3ndash4 Caution students to read the problem carefully so that they do not misread the problem as the same numbers combined by addition on each side of the circleExercise 10 Remind students that the calculations have units
myhrwcom
23 Lesson 13
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
0 05 1 15 2 25 3 35 4 45 5 55 6 65 7
2πradic3
Compare Write lt gt or = (Example 1)
1 radic_
3 + 2 radic_
3 + 3 2 radic_
8 + 17 radic_
11 + 15
3 radic_
6 + 5 6 + radic_
5 4 radic_
9 + 3 9 + radic_
3
5 radic_
17 - 3 -2 + radic_
5 6 12 - radic_
2 14 - radic_
8
7 radic_
7 + 2 radic_
10 - 1 8 radic_
17 + 3 3 + radic_
11
9 Order radic_
3 2π and 15 from least to greatest Then graph them on the number line (Example 2)
radic_
3 is between and so radic_
3 asymp
π asymp 314 so 2π asymp
From least to greatest the numbers are
10 Four people have found the perimeter of a forest using different methods Their results are given in the table Order their calculations from greatest to least (Example 3)
11 Explain how to order a set of real numbers
CHECK-INESSENTIAL QUESTION
Forest Perimeter (km)
Leon Mika Jason Ashley
radic_
17 - 2 1 +thinsp π __ 2 12 ___ 5 25
Guided Practice
17
15
1 + π _ 2 km 25 km 12 __ 5 km radic_
17 - 2 km
2π radic
_ 3
18 175
628
Sample answer Convert each number to a decimal
equivalent using estimation to find equivalents for
irrational numbers Graph each number on a number line
Read the numbers from left to right for least to greatest
Read the numbers from right to left for greatest to least
lt gt
lt lt
ltgt
gt gt
24 Unit 1
copy H
ough
ton
Miff
lin H
arco
urt P
ublis
hing
Com
pany
bull Im
age C
redi
ts copy
Elena
Eliss
eeva
Alam
y Im
ages
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
8_MCAAESE206984_U1M01L3indd 24 41613 448 AM
My Notes
5 52 54 56 58 6
radic28 5 12
23455
Math TrainerOnline Practice
and Help
Personal
myhrwcom
Math On the Spot
myhrwcom
Ordering Real Numbers in a Real-World Context Calculations and estimations in the real world may differ It can be important to know not only which are the most accurate but which give the greatest or least values depending upon the context
Four people have found the distance in kilometers across a canyon using different methods Their results are given in the table Order the distances from greatest to least
Distance Across Quarry Canyon (km)
Juana Lee Ann Ryne Jackson
radic_
28 23 __ 4 5 _
5 5 1 _ 2
Write each value as a decimal
radic_
28 is between 52 and 53 Since 53 2 = 2809 an approximate value for radic
_ 28 is 53
23 __ 4 = 575
5 _
5 is 5555hellip so 5 _
5 to the nearest hundredth is 556
5 1 _ 2 = 55
Plot radic_
28 23 __ 4 5 _
5 and 5 1 _ 2 on a number line
From greatest to least the distances are
23 __ 4 km 5 _
5 km 5 1 _ 2 km radic_
28 km
EXAMPLEXAMPLE 3
STEP 1
STEP 2
7 Four people have found the distance in miles across a crater using different methods Their results are given below
Jonathan 10 __ 3 Elaine 3 _
45 Joseacute 3 1 _ 2 Lashonda radic_
10
Order the distances from greatest to least
YOUR TURN
8NS2
3 1 _ 2 mi 3 _
45 mi 10 __ 3 mi radic_
10 mi
23Lesson 13
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ough
ton
Miff
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pany
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
8_MCAAESE206984_U1M01L3indd 23 41613 447 AM
ModelingPlace papers around the room with the numbers from 1 to 5 one per sheet Give each student a card showing a number between 1 and 5 in different forms Have students place his or her card between the correct integers and decide where the number goes in relation to any numbers already placed
Multiple RepresentationsGive students a vertical number line which some students might find easier to use than a horizontal one Have them decide whether to place points for rational and irrational numbers above or below existing points
Additional ResourcesDifferentiated Instruction includes bull Reading Strategies bull Success for English Learners EL
bull Reteach bull Challenge PRE-AP
DIFFERENTIATE INSTRUCTION
Ordering Real Numbers 24
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
Math TrainerOnline Assessment
and Intervention
Personal
Online homework assignment available
myhrwcom
myhrwcom
Lesson Quiz available online
13 LESSON QUIZ
1 Compare Write lt gt or =
radic_
95 - 5 radic_
62 - 2
2 Order 105 radic_
105 and 3π + 1 from greatest to least
3 A length in centimeters is calculated differently by four different people Order their calculations from least to greatest
KD 11 __ 2 cm Silvio 5 __ 3 π cm
Paula 5 _
4 cm Luis radic_
33 cm
EvaluateGUIDED AND INDEPENDENT PRACTICE
Concepts amp Skills Practice
Example 1Comparing Irrational Numbers
Exercises 1ndash8
Example 2Ordering Real Numbers
Exercises 9 12ndash15 18ndash21
Example 3Ordering Real Numbers in a Real-World Context
Exercises 10 16ndash17
Additional ResourcesDifferentiated Instruction includes bull Leveled Practice worksheets
Focus | Coherence | Rigor
Exercise Depth of Knowledge (DOK) Mathematical Practices
12ndash15 1 Recall of Information MP5 Using Tools
16 2 SkillsConcepts MP2 Reasoning
17 2 SkillsConcepts MP6 Precision
18ndash21 2 SkillsConcepts MP2 Reasoning
22 3 Strategic Thinking MP4 Modeling
23ndash24 3 Strategic Thinking MP3 Logic
8NS2
8NS2
Answers1 radic
_ 95 - 5 lt radic
_ 62 - 2
2 radic_
105 3π + 1 105
3 Silvio 5 __ 3 π cm Paula 5 _
4 cm
KD 11
__ 2 cm Luis radic_
33 cm
25 Lesson 13
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
Work Area
3140 3141 3142 3143
314 π227
20 A teacher asks his students to write the numbers shown in order from least to greatest Paul thinks the numbers are already in order Sandra thinks the order should be reversed Who is right
21 Math History There is a famous irrational number called Eulerrsquos number symbolized with an e Like π its decimal form never ends or repeats The first few digits of e are 27182818284
a Between which two square roots of integers could you find this number
b Between which two square roots of integers can you find π
22 Analyze Relationships There are several approximations used for π including 314 and 22 __ 7 π is approximately 314159265358979
a Label π and the two approximations on the number line
b Which of the two approximations is a better estimate for π Explain
c Find a whole number x so that the ratio x ___ 113 is a better estimate for π
than the two given approximations
23 Communicate Mathematical Ideas If a set of six numbers that include both rational and irrational numbers is graphed on a number line what is the fewest number of distinct points that need to be graphed Explain
24 Critique Reasoning Jill says that 12 _
6 is less than 1263 Explain her error
FOCUS ON HIGHER ORDER THINKING
radic_
115 115 ___ 11 and 105624
between radic_
7 asymp 265 and radic_
8 asymp 283
between radic_
9 = 3 and radic_
10 asymp 316
22 __ 7 it is closer to π on the number line
She did not consider the repeating digit 1266
2 rational numbers can have the same location and
irrational numbers can have the same location but they
cannot share a location
355
Neither student is correct The answer
should be 115 ___ 11 105624 radic_
115
Unit 126
copy H
ough
ton M
ifflin
Har
cour
t Pub
lishin
g Com
pany
Imag
e Cre
dits
copy3D
Stoc
kiSt
ockP
hoto
com
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
8_MCAAESE206984_U1M01L3indd 26 210513 801 AM
Personal Math Trainer
Online Practice and Helpmyhrwcom
Name Class Date
Independent Practice
16 Your sister is considering two different shapes for her garden One is a square with side lengths of 35 meters and the other is a circle with a diameter of 4 meters
a Find the area of the square
b Find the area of the circle
c Compare your answers from parts a and b Which garden would give your sister the most space to plant
17 Winnie measured the length of her fatherrsquos ranch four times and got four different distances Her measurements are shown in the table
a To estimate the actual length Winnie first approximated each distance to the nearest hundredth Then she averaged the four numbers Using a calculator find Winniersquos estimate
b Winniersquos father estimated the distance across his ranch to be radic_
56 km How does this distance compare to Winniersquos estimate
Give an example of each type of number
18 a real number between radic_
13 and radic_
14
19 an irrational number between 5 and 7
Order the numbers from least to greatest
12 radic_
7 2 radic_
8 ___ 2 13 radic_
10 π 35
14 radic_
220 -10 radic_
100 115 15 radic_
8 -375 3 9 _ 4
Distance Across Fatherrsquos Ranch (km)
1 2 3 4
radic_
60 58 __ 8 7 _
3 7 3 _ 5
138NS2
radic_
8 ___ 2 2 radic_
7
-10 radic_
100 115 radic_
220
radic_
60 asymp 775 58 __ 8 = 725 7 _
3 asymp 733 7 3 _ 5 = 760 so the average
π radic_
10 35
-375 9 _ 4 radic_
8 3
is 74825 km
1225 m2
4π m2 or approximately 126 m2
They are nearly identical radic_
56 is approximately 74833hellip
The circle would give her more space to plant because it has a
larger area
Sample answer 37
Sample answer radic_
31
25Lesson 13
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ough
ton
Miff
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ublis
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pany
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
8_MCAAESE206984_U1M01L3indd 25 41613 448 AM
Activity available online myhrwcomEXTEND THE MATH PRE-AP
Activity Have students investigate whether there are infinitely many numbers between two numbers by giving examples for each of the following
bull Between any two rational numbers there is at least one other rational number Sample answer 45 is between 41 and 48
bull Between any two irrational numbers there is at least one rational number Sample answer 45 is between radic
_ 11 and radic
_ 29
bull Between any two rational numbers there is at least one irrational number Sample answer radic
_ 11 is between 31 and 36
bull Between any two irrational numbers there is at least one irrational number Sample answer radic
_ 17 is between radic
_ 11 and radic
_ 29
Ordering Real Numbers 26
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
ReadyMath Trainer
Online Practiceand Help
Personal
myhrwcom
Module Quiz
11ensp RationalenspandenspIrrationalenspNumbersWrite each fraction as a decimal or each decimal as a fraction
1 7__20 2 1___
27 3 17_8
Solve each equation for x
4 x2=81 5 x3=343 6 x2= 1___100
7 Asquarepatiohasanareaof200squarefeetHowlongiseachside
ofthepatiotothenearesttenth
12ensp SetsenspofenspRealenspNumbersWrite all names that apply to each number
8 121____radic
____121
9 π__2
10 TellwhetherthestatementldquoAllintegersarerationalnumbersrdquoistrueorfalseExplainyourchoice
13ensp OrderingenspRealenspNumbersCompare Write lt gt or =
11 radic__
8+3 8+radic__
3 12 radic__
5+11emsp emsp emsp 5+radic___
11
Order the numbers from least to greatest
13 radic___
99π29__
8 14 radic___
1__251_40__
2
15 Howarerealnumbersusedtodescribereal-worldsituations
ESSENTIAL QUESTION
035
9-9
141ft
7 1__10- 1__10
14__11 1875
wholeintegerrationalreal
Trueintegerscanbewrittenasthequotientoftwointegers
SampleanswerRealnumberssuchastherational
π29__
8radic___
99
irrationalreal
lt gt
number1_4candescribeamountsusedincooking
radic___
1__250__
21_4
27Module1
copy H
ough
ton
Miff
lin H
arco
urt P
ublis
hing
Com
pany
DONOTEDIT--ChangesmustbemadethroughldquoFileinfordquoCorrectionKey=A
8_MCAAESE206984_U1M01RTindd 27 41513 1113 PM
Math TrainerOnline Assessment
and Intervention
Personal
myhrwcom
1
2
3 Response toIntervention
Intervention Enrichment
Access Ready to Go On assessment online and receive instant scoring feedback and customized intervention or enrichment
Online and Print Resources
Differentiated Instruction
bull Reteach worksheets
bull Reading Strategies EL
bull Success for English Learners EL
Differentiated Instruction
bull Challenge worksheets PRE-AP
Extend the Math PRE-AP
Lesson Activities in TE
Additional ResourcesAssessment Resources includes bull Leveled Module Quizzes
Ready to Go OnAssess MasteryUse the assessment on this page to determine if students have mastered the concepts and standards covered in this module
California Common Core Standards
Lesson Exercises Common Core Standards
11 1ndash7 8NS1 8NS2 8EE2
12 8ndash10 8NS1
13 11ndash14 8NS2
27 Unit 1 Module 1
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
Personal Math Trainer
Online Practice and HelpmyhrwcomAssessment Readiness
Module 1 MIXed ReVIeW
1 Look at each number Is the number between 2π and radic___
52
Select Yes or No for expressions AndashC
A 6 2 _ 3 Yes No
B 5π __ 2 Yes No
C 3 radic__
5 Yes No
2 Consider the number - 11 __ 15
Choose True or False for each statement
A The number is rational True False
B The number can be written as True Falsea repeating decimal
C The number is less than ndash08 True False
3 The volume of a cube is given by V = x3 where x is the length of an edge of the cube A cube-shaped end table has a volume of 3 3 _ 8 cubic feet What is the length of an edge of the end table Explain how you solved this problem
4 A student says that radic___
83 is greater than 29 __ 3 Is the student correct Justify your
reasoning
1 1 _ 2 ft Sample answer The equation x3 = 3 3 _ 8 can be used
to find the edge length in feet To solve the equation
write the mixed number as a fraction greater than 1
x3 = 27 __ 8 Then take the cube root of both sides x = 3 _ 2 = 1 1 _ 2
No Sample answer radic___
83 asymp 91 and 29 __ 3 = 9
__ 6
Because 91 lt 9 __
6 radic___
83 lt 29 __ 3
28 Unit 1
copy H
ough
ton
Miff
lin H
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Com
pany
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=A
8_MCAAESE206984_U1M01RTindd 28 240413 946 AM
Personal Math Trainer
Online Assessment and
Interventionmyhrwcom
Scoring GuideItem 3 Award the student 1 point for finding the edge length of the cube and 1 point for correctly explaining how to use a cube root to solve the problem
Item 4 Award the student 1 point for determining that the student is incorrect and 1 point for correctly justifying the reasoning for this conclusion
Additional ResourcesTo assign this assessment online login to your Assignment Manager at myhrwcom
Assessment Readiness
California Common Core Standards
Items Grade 8 Standards Mathematical Practices
1 8NS2 MP7
2 7NS2b 7NS2d 8NS1 MP7
3 8EE2 MP1 MP4
4 8NS1 8NS2 MP3
Item integrates mixed review concepts from previous modules or a previous course
Item 4 combines concepts from the California Common Core cluster ldquoKnow that there are numbers that are not rational and approximate them by rational numbersrdquo
Real Numbers 28
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
12L E S S O N
Sets of Real Numbers
EngageESSENTIAL QUESTION
How can you describe relationships between sets of real numbers Sample answer Describe them as two different sets or one set as being a subset of another
Motivate the LessonAsk How many different types of tigers can you name How does the set of Bengal tigers relate to the set of tigers
ExplorePoint to different locations in the Animals diagram and ask for examples for that classification Do the same for the Real Numbers diagram Students should understand that everything within a region is part of the set for example both -3 and 2 are integers
ExplainEXAMPLE 1
Questioning Strategies Mathematical Practices bull In A why is 5 not a perfect square It does not have rational numbers as its square roots
bull Can the number in B be written as a fraction Why or why not Yes it is a terminating decimal so it is a rational number
Engage with the WhiteboardHave students place the numbers in Example 1 and Additional Example 1 in the Venn diagram for numbers
YOUR TURNAvoid Common ErrorsBe sure that students read Exercise 2 carefully before answering The number given in the problem 10 is the area not the side length
EXAMPLE 2Questioning Strategies Mathematical Practices bull What two major sets are the real numbers composed of rational and irrational numbers
bull What is the location of the set of whole numbers in the Venn diagram in relation to the set of rational numbers Explain Inside it whole numbers are rational numbers
Focus on Reasoning Mathematical PracticesRemind students that it takes only one counterexample to show that a statement is false
Interactive Whiteboard Interactive example available online
ADDITIONAL EXAMPLE 1Write all names that apply to each number
A -10integer rational real
B 12 _ 3
whole integer rational real
myhrwcom
Interactive Whiteboard Interactive example available online
ADDITIONAL EXAMPLE 2Tell whether the given statement is true or false Explain your choice
No integers are whole numbers
False every whole number is also an integer
myhrwcom
Animated MathClassifying Numbers
Students build fluency in classifying numbers in this engaging fast-paced game
myhrwcom
CA Common CoreStandards
The student is expected to
The Number Systemmdash8NS1
Know that numbers that are not rational are called irrational Understand informally that every number has a decimal expansion for rational numbers show that the decimal expansion repeats eventually and convert a decimal expansion which repeats eventually into a rational numberMathematical Practices
MP7 Using Structure
The student is expected to
15 Lesson 12
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=B
Math TrainerOnline Practice
and Help
Personal
myhrwcom
Math TrainerOnline Practice
and Help
Personal
myhrwcom
Math On the Spotmyhrwcom
Understanding Sets and Subsets of Real NumbersBy understanding which sets are subsets of types of numbers you can verify whether statements about the relationships between sets are true or false
Tell whether the given statement is true or false Explain your choice
All irrational numbers are real numbers
True Every irrational number is included in the set of real numbers The irrational numbers are a subset of the real numbers
No rational numbers are whole numbers
False A whole number can be written as a fraction with a denominator of 1 so every whole number is included in the set of rational numbers The whole numbers are a subset of the rational numbers
EXAMPLE 2
A
B
Write all names that apply to each number
1 A baseball pitcher has pitched 12 2 _ 3 innings
2 The length of the side of a square that has an
area of 10 square yards
YOUR TURN
Tell whether the given statement is true or false Explain your choice
3 All rational numbers are integers
4 Some irrational numbers are integers
YOUR TURN
Give an example of a rational number that is a
whole number Show that the number is both whole
and rational
Math TalkMathematical Practices
Give an example of a
8NS1
False Every integer is a rational number but not every
False Real numbers are either rational or irrational numbers
Integers are rational numbers so no integers are irrational numbers
rational real
irrational real
Sample answer 8 8 = 8_
1
and -thinsp 5 _ 2 are not integers
rational number is an integer Rational numbers such as 3 _ 5
Unit 116
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ough
ton
Miff
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hing
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pany
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age C
redi
ts D
igita
l Im
age c
opyr
ight
copy20
04 Ey
ewire
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
8_MCAAESE206984_U1M01L2indd 16 41613 136 AM
Math On the Spot
myhrwcom
Vertebrates
Birds
Passerines
Animals
Integers
Rational Numbers IrrationalNumbers
Real Numbers
WholeNumbers
1
45
3
0
274
67
radic4
-
-3
-2
-1
03
radic2
radic17
radic11-
π
Animated Math
myhrwcom
Classifying Real NumbersBiologists classify animals based on shared characteristics A cardinal is an animal a vertebrate a bird and a passerine
You already know that the set of rational numbers consists of whole numbers integers and fractions The set of real numbers consists of the set of rational numbers and the set of irrational numbers
Write all names that apply to each number
radic_
5 irrational real
ndash1784rational real
whole integer rational real
EXAMPLEXAMPLE 1
A
B
C radic_ 81 ____ 9
L E S S O N
12Sets of Real Numbers
ESSENTIAL QUESTIONHow can you describe relationships between sets of real numbers
Passerines such as the cardinal are also called ldquoperching birdsrdquo
What types of numbers are between 31 and 39 on a
number line
Math TalkMathematical Practices
What types of numbers are
8NS1
8NS1
Know that numbers that are not rational are called irrational Understand informally that every number has a decimal expansion for rational numbers show that the decimal expansion repeats eventually and convert a decimal expansion which repeats eventually into a relation number
ndash1784 is a terminating decimal
5 is a whole number that is not a perfect square
radic_
81 _____ 9 = 9 __ 9 = 1 rational irrational real
15Lesson 12
copy H
ough
ton
Miff
lin H
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hing
Com
pany
bull Im
age C
redi
ts copy
Wiki
med
ia Co
mm
ons
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=B
8_MCABESE206984_U1M01L2indd 15 061113 1144 AM
PROFESSIONAL DEVELOPMENT
Math BackgroundThe relationships between sets of numbers extend to include complex numbers A complex number can be written as a sum of a real number a and an imaginary number bi
a + bi
An imaginary number is a special number that when squared gives a negative value When you square a real number you get a nonnegative number When you square an imaginary number you get a negative value The imaginary unit is i
i = radic_
-1
Integrate Mathematical Practices MP7
This lesson provides an opportunity to address this Mathematical Practices standard It calls for students to discern structure to connect and communicate mathematical ideas
Students use a Venn diagram to structure relationships between sets of numbers They connect and communicate mathematical ideas when they make logical statements about the sets and describe which set best describes numbers applied to real-life situations
Sets of Real Numbers 16
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=B
YOUR TURNAvoid Common ErrorsStudents may see the word ldquoAllldquo or rdquoNordquo in Exercises 3 and 4 and immediately assume that any absolute statements like these are false Remind them that there are true statements that begin with these words and encourage them to provide examples
EXAMPLE 3Questioning Strategies Mathematical Practices bull In A how does the phrase ldquonumber of rdquo give you a clue about the number classification It indicates a counting number
bull What is the relationship between the circumference of a circle and the diameter The circumference is diameter times π
Focus on Critical Thinking Mathematical PracticesIn B suppose the diameters in inches were 25
__ π 28 __ π
31 __ π and so on What set of numbers would
best describe the circumferences Explain Whole numbers the circumferences would be the whole numbers 25 28 31 and so on
YOUR TURNFocus on Critical Thinking Mathematical PracticesHave students compare and contrast the classification of numbers in the answers in Exercises 5 and 6
ElaborateTalk About ItSummarize the Lesson
Ask What are some ways that number sets can be related Sets may be subsets of other sets or they may be separate from other sets
GUIDED PRACTICEEngage with the Whiteboard
Have students place the numbers in Exercises 1ndashthinsp8 in the Venn diagram for numbers at the beginning of the lesson
Integrating Language Arts EL
Encourage English learners to ask for clarification on any terms or phrases that they do not understand
Avoid Common ErrorsExercise 7 Remind students that a repeating decimal is a rational numberExercises 9ndash10 Remind students that it only takes one counterexample to show that a statement is false
Interactive Whiteboard Interactive example available online
ADDITIONAL EXAMPLE 3Identify the set of numbers that best describes the situation Explain your choice
A the amount of time that has passed since midnight
The set of real numbers time is continuous so the amount of time can be rational or irrational
B the number of tickets sold to a basketball game
The set of whole numbers the number of tickets sold may be 0 or a counting number
myhrwcom
17 Lesson 12
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
1IN
116 inch
Guided Practice
Write all names that apply to each number (Example 1)
1 7 _ 8 2 radic_
36
3 radic_
24 4 075
5 0 6 - radic_ 100
7 5 _
45 8 - 18 __ 6
Tell whether the given statement is true or false Explain your choice (Example 2)
9 All whole numbers are rational numbers
10 No irrational numbers are whole numbers
Identify the set of numbers that best describes each situation Explain your choice (Example 3)
11 the change in the value of an account when given to the nearest dollar
12 the markings on a standard ruler
13 What are some ways to describe the relationships between sets of numbers
CHECK-INESSENTIAL QUESTION
rational real
rational real
True Whole numbers are rational numbers
Rational numbers the ruler is marked every 1 __ 16 th inch
Sample answer Describe one set as being a subset of
another or show their relationships in a Venn diagram
Integers the change can be a whole dollar amount
and can be positive negative or zero
True Whole numbers are a subset of the set of rational numbers
and can be written as a ratio of the whole number to 1
irrational real
whole integer rational real
whole integer rational real
rational real
integer rational real
integer rational real
Unit 118
copy H
ough
ton
Miff
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pany
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
8_MCAAESE206984_U1M01L2indd 18 41613 136 AM
My Notes
Math TrainerOnline Practice
and Help
Personal
myhrwcom
Math On the Spot
myhrwcom
Identifying Sets for Real-World SituationsReal numbers can be used to represent real-world quantities Highways have posted speed limit signs that are represented by natural numbers such as 55 mph Integers appear on thermometers Rational numbers are used in many daily activities including cooking For example ingredients in a recipe are often given in fractional amounts such as 2 _ 3 cup flour
Identify the set of numbers that best describes each situation Explain your choice
the number of people wearing glasses in a room
The set of whole numbers best describes the situation The number of people wearing glasses may be 0 or a counting number
the circumference of a flying disk has a diameter of 8 9 10 11 or 14 inches
The set of irrational numbers best describes the situation Each circumference would be a product of π and the diameter and any multiple of π is irrational
EXAMPLEXAMPLE 3
A
B
Identify the set of numbers that best describes the situation Explain your choice
5 the amount of water in a glass as it evaporates
6 the weight of a person in pounds
YOUR TURN
8NS1
Rational numbers a personrsquos weight can be a decimal
such as 835 pounds
Real numbers the amount can be any number greater
than 0
17Lesson 12
copy H
ough
ton
Miff
lin H
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ublis
hing
Com
pany
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
8_MCAAESE206984_U1M01L2indd 17 41613 520 AM
Graphic OrganizersGive students a list of numbers (including terminating and repeating decimals fractions integers and rational and irrational square roots) and a graphic organizer as shown below
Real Numbers
Rational numbers Irrational numbers
Integer numbers
Whole numbers
Ask students to write each number in the list in the correct section of the organizer
Number SensePoint out to students that knowing the types of numbers to expect in different situations can alert them to incorrect math as well as to impossible situations For example 135 shots made in basketballs is not possible but an average number of shots can equal 135
Additional ResourcesDifferentiated Instruction includes bull Reading Strategies bull Success for English Learners EL
bull Reteach bull Challenge PRE-AP
DIFFERENTIATE INSTRUCTION
Sets of Real Numbers 18
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
Math TrainerOnline Assessment
and Intervention
Personal
Online homework assignment available
myhrwcom
Lesson Quiz available online
12 LESSON QUIZ
1 Write all the names that apply to the number
2 Tell whether the given statement is true or false Explain your choice All numbers between 1 and 2 are rational numbers
3 Identify the set of numbers that best describes the situation Explain your choiceThe choices on a survey question change the total points for the survey by -2 -1 0 1 or 2 points
-1 _
5
myhrwcom
Additional ResourcesDifferentiated Instruction includes bull Leveled Practice worksheets
EvaluateGUIDED AND INDEPENDENT PRACTICE
Concepts amp Skills Practice
Example 1Classifying Real Numbers
Exercises 1ndash8 14ndash19 22ndash24
Example 2Understanding Sets and Subsets of Real Numbers
Exercises 9ndash10
Example 3Identifying Sets for Real-World Situations
Exercises 11ndash12 20ndash21 25
Focus | Coherence | Rigor
Exercise Depth of Knowledge (DOK) Mathematical Practices
14ndash19 2 SkillsConcepts MP7 Using Structure
20ndash21 2 SkillsConcepts MP6 Precision
22ndash23 2 SkillsConcepts MP3 Logic
24 1 Recall of Information MP7 Using Structure
25 2 SkillsConcepts MP2 Reasoning
26ndash27 3 Strategic Thinking MP3 Logic
28 3 Strategic Thinking MP8 Patterns
29 3 Strategic Thinking MP3 Logic
8NS1
8NS1
Exercise 29 combines concepts from the California Common Core cluster ldquoKnow that there are numbers that are not rational and approximate them by rational numbersrdquo
Answers1 rational real
2 False radic_
2 is an example of an irrational number between 1 and 2
3 Integers each number is an integer but only three are whole numbers
19 Lesson 12
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
Work Area
π mi23 Critique Reasoning The circumference of a circular region is shown
What type of number best describes the diameter of the circle Explain
your answer
24 Critical Thinking A number is not an integer What type of number can it be
25 A grocery store has a shelf with half-gallon containers of milk What type of number best represents the total number of gallons
26 Explain the Error Katie said ldquoNegative numbers are integersrdquo What was her error
27 Justify Reasoning Can you ever use a calculator to determine if a number is rational or irrational Explain
28 Draw Conclusions The decimal 0 _
3 represents 1 _ 3 What type of number best describes 0
_ 9 which is 3 middot 0
_ 3 Explain
29 Communicate Mathematical Ideas Irrational numbers can never be precisely represented in decimal form Why is this
FOCUS ON HIGHER ORDER THINKING
It can be a rational number that is not an integer or an irrational number
rational number
The set of negative numbers also includes non-integer
rational numbers and irrational numbers
Sample answer If the calculator shows a decimal that
terminates in fewer digits than what the calculator screen
allows then you can tell that the number is rational If not
you cannot tell from the calculator display whether the
number terminates because you see a limited number
of digits It may be a repeating decimal (rational) or
non-terminating non-repeating decimal (irrational)
Whole 3 middot 0 _
3 represents 3 middot 1 _ 3 = 1 so 0 _
9 is exactly 1
Sample answer In decimal form irrational numbers never
terminate and never repeat Therefore no matter how
many decimal places you include the number will never
be precisely represented There are always more digits
Whole the diameter is π _ π = 1 mile
Unit 120
copy H
ough
ton
Miff
lin H
arco
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ublis
hing
Com
pany
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
8_MCAAESE206984_U1M01L2indd 20 120413 909 PM
Integers
Rational Numbers Irrational Numbers
Real Numbers
Whole Numbers
257
radic16
166
radic9
128 radic50
Personal Math Trainer
Online Practice and Helpmyhrwcom
Name Class Date
Independent Practice
Identify the set of numbers that best describes each situation Explain your choice
20 the height of an airplane as it descends to an airport runway
21 the score with respect to par of several golfers 2 ndash 3 5 0 ndash 1
22 Critique Reasoning Ronald states that the number 1 __ 11 is not rational because when converted into a decimal it does not terminate Nathaniel says it is rational because it is a fraction Which boy is correct Explain
12
14 - radic_
9 15 257
16 radic_
50 17 8 1 _ 2
18 166 19 radic_
16
Write all names that apply to each number Then place the numbers in the correct location on the Venn diagram
8NS1
Real numbers the height can be any number greater than zero
integer rational real whole integer rational real
whole integer rational real
irrational real
rational real
rational real
Integers the scores are counting numbers their
opposites and zero
Nathaniel is correct A rational number is a number that can be written as a fraction and 1 __ 11 is a fraction
19Lesson 12
copy H
ough
ton
Miff
lin H
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urt P
ublis
hing
Com
pany
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
8_MCAAESE206984_U1M01L2indd 19 41613 136 AM
myhrwcomActivity available onlineEXTEND THE MATH PRE-AP
Activity Have students consider the concept of restricted domain for the sets of numbers that describe situations For example the number of sisters a person has can best be described by whole numbers but no one has ever had 1500 sisters An area code is an integer or whole number between 200 and 999
Have students use a source such as the Guinness Book of World Records and give examples of sets of numbers that describe situations where the domain is restricted Ask whether the restriction may be changed in the future
Sets of Real Numbers 20
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
-3-4-5 -2-1 1 2 3 50 4
12-4 -radic5
Lesson Support Content Objective Students will learn to order a set of real numbers
Language Objective Students will show and describe how to order a set of real numbers
LESSON 13 Ordering Real Numbers
Building BackgroundEliciting Prior Knowledge Have students draw a number line to compare a rational number and an irrational number such as - radic
_ 5 and -4 1 __ 2 Ask them to explain how
they approximated the irrational number on the number line Then have them identify the greater and the lesser real number Repeat with several other pairs of real numbers in different forms
Learning ProgressionsIn this lesson students order a set of real numbers They use rational approximations to compare the sizes of irrational numbers They also order numbers for real-world situations Important understandings for students include the following
bull Compare irrational numbers bull Estimate the value of expressions with irrational numbers bull Order a set of real numbers bull Order real numbers in a real-world context
Work with real numbers continues throughout Grade 8 and into high school This lesson provides students with a foundation for understanding the relative sizes of numbers in different forms in the real number system
Cluster ConnectionsThis lesson provides an excellent opportunity to connect ideas in this cluster Know that there are numbers that are not rational and approximate them by rational numbers Tell students that there is a special number called the golden ratio with applications in mathematics geometry art and architecture The golden ratio is called phi and is represented by the Greek letter ϕ It includes an irrational number in its definition
Have students explain why the golden ratio is irrational Ask them to find the two whole numbers the golden ratio lies between Then challenge them to approximate the golden ratio to the nearest tenth It is irrational because it includes an irrational number in its definition It lies between 1 and 2 To the nearest tenth ϕ = 16
ϕ = 1 + radic_
5 _ 2
Focus | Coherence | Rigor
California Common Core Standards
8NS2 Use rational approximations of irrational numbers to compare the size of irrational numbers locate them approximately on a number line diagram and estimate the value of expressions (eg π 2 )
MP4 Model with mathematics
21A
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
Math Talk
Language Support EL
PROFESSIONAL DEVELOPMENT
Linguistic Support EL
AcademicContent Vocabulary
Post a chart like this to remind students of the regular comparative forms of adjectives that use the -er and -est suffixes Add to the chart for terms that appear in examples and exercises in each lesson Include any irregular verb forms
Background Knowledge
Go On ndash the title of the module review or quiz is Ready to Go On This title uses an idiomatic expression In this context to go on means ldquoto move aheadrdquo or ldquoto proceedrdquo It is different from the use of go on that means having enough facts to use meaningfully as in having enough to go on Also the intonation used in pronouncing an expression can give it different meanings For example when the speaker emphasizes the word on he or she might be expressing disbelief as in ldquoGo ON Yoursquore kidding rightrdquo Discuss with students other ways that the phrase go on may be used
Leveled Strategies for English Learners
Emerging Label points on a number line with the terms used in ordering greater greatest less lesser least Use sentence frames to insert the correct terms
Expanding Have students give two or three complete sentences to compare the placement of numbers on a number line using the correct forms of the comparative and superlative adjectives
Bridging Have students work in pairs with one student giving directions to the other in complete sentences to order numbers on a number line
To help students answer the question posed in Math Talk make sure that students have a command of the forms for making comparisons and the superlative and the concept of opposite order so that the focus is on the math concept instead of the language skills needed to describe and explain order
EL
Adjective Comparative Superlative
Far Farther Farthest
Large Larger Largest
Great Greater Greatest
Some Less Least
Some More Most
California ELD Standards
Emerging 2I8 Analyzing language choices ndash Explain how phrasing or different common words with similar meanings produce different effects on the audience
Expanding 2I8 Analyzing language choices ndash Explain how phrasing or different words with similar meanings or figurative language produce shades of meaning and different effects on the audience
Bridging 2I8 Analyzing language choices ndash Explain how phrasing or different words with similar meanings or figurative language produce shades of meaning nuances and different effects on the audience
Ordering Real Numbers 21B
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
13L E S S O N
Ordering Real Numbers
Interactive Whiteboard Interactive example available online
ADDITIONAL EXAMPLE 1Compare Write lt gt or =
A radic_
8 - 2 4 - radic_
8 lt
B radic_
20 + 1 3 + radic_
2 gt
EngageESSENTIAL QUESTION
How do you order a set of real numbers Sample answer Find their approximate decimal values and order them
Motivate the LessonAsk What kind of numbers are you comparing when you compare the price of gasoline at two different gas stations
ExploreGive students two rational numbers and ask them to name a number between them Repeat a few times and then give them two irrational numbers and ask them to name a number between them
ExplainEXAMPLE 1
Questioning Strategies Mathematical Practices bull Which is greater the difference between 5 and 3 or the difference between radic
_ 5 and radic
_ 3
The difference between 5 and 3 is 2 the difference between radic_
5 and radic_
3 is approximately 1 So the difference between 5 and 3 is greater
Avoid Common ErrorsCaution students to read the problem carefully and think about what the radical sign means so that they do not misread the problem and answer that the two sides are equal
YOUR TURNFocus on TechnologyCalculators should not be used at this point because developing number sense is the goal
EXAMPLE 2Questioning Strategies Mathematical Practices bull How do you determine whether radic
_ 22 is less than or greater than 45 The square of 45 is
2025 which is less than 22 so the square root of 22 must be greater than 45
Engage with the WhiteboardHave students graph and label various real numbers between 42 and 44 and between 47 and 5
YOUR TURNFocus on Modeling Mathematical PracticesHave students label the integers on the number line with their equivalent square root For example 1 2 and 3 on the number line would be labeled radic
_ 1 radic
_ 4 and radic
_ 9
Interactive Whiteboard Interactive example available online
ADDITIONAL EXAMPLE 2Order 3π radic
_ 10 and 325 from greatest
to least
3π 325 radic_
10
myhrwcom
myhrwcom
CA Common CoreStandards
The student is expected to
The Number Systemmdash8NS2
Use rational approximations of irrational numbers to compare the size of irrational numbers locate them approximately on a number line diagram and estimate the value of expressions (eg π 2 )
Mathematical Practices
MP4 Modeling
The student is expected to
21 Lesson 13
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
Math TrainerOnline Practice
and Help
Personal
myhrwcom
Math On the Spotmyhrwcom
0 05 1 15 2 25 3 35 4
radic5radic3
π2
8 85 9 95 10 105 11 115 12
radic75
4 42 44 46 48 5
radic224 12π + 1
Ordering Real Numbers You can compare and order real numbers and list them from least to greatest
Order radic_
22 π + 1 and 4 1 _ 2 from least to greatest
First approximate radic_
22
radic_
22 is between 4 and 5 Since you donrsquot know where it falls between 4 and 5 you need to find a better estimate for radic
_ 22 so
you can compare it to 4 1 _ 2
Since 22 is closer to 25 than 16 use squares of numbers between 45 and 5 to find a better estimate of radic
_ 22
45 2 = 2025 46 2 = 2116 47 2 = 2209 48 2 = 2304
Since 47 2 = 2209 an approximate value for radic_
22 is 47
An approximate value of π is 314 So an approximate value of π +1 is 414
Plot radic_
22 π + 1 and 4 1 _ 2 on a number line
Read the numbers from left to right to place them in order from least to greatest
From least to greatest the numbers are π + 1 4 1 _ 2 and radic_
22
EXAMPLE 2
STEP 1
STEP 2
Order the numbers from least to greatest Then graph them on the number line
YOUR TURN
5 radic_
5 25 radic_
3
6 π 2 10 radic_
75
If real numbers a b and c are in order from least to greatest what is the order
of their opposites from least to greatest
Explain
Math TalkMathematical Practices
8NS2
radic_
3 radic_
5 25
radic_
75 π2 10
Math Talk answer -c -b -a -c is farthest to the left on a number line -b is in the middle and -a is farthest to the right
Unit 122
copy H
ough
ton
Miff
lin H
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urt P
ublis
hing
Com
pany
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
8_MCAAESE206984_U1M01L3indd 22 41613 447 AM
My Notes
Math TrainerOnline Practice
and Help
Personal
myhrwcom
Math On the Spot
myhrwcom
Comparing Irrational NumbersBetween any two real numbers is another real number To compare and order real numbers you can approximate irrational numbers as decimals
Compare radic_
3 + 5 3 + radic_
5 Write lt gt or =
First approximate radic_
3
radic_
3 is between 1 and 2
Next approximate radic_
5
radic_
5 is between 2 and 3
Then use your approximations to simplify the expressions
radic_
3 + 5 is between 6 and 7
3 + radic_
5 is between 5 and 6
So radic_
3 + 5 gt 3 + radic_
5
Reflect1 If 7 + radic
_ 5 is equal to radic
_ 5 plus a number what do you know about the
number Why
2 What are the closest two integers that radic_
300 is between
EXAMPLEXAMPLE 1
STEP 1
STEP 2
Compare Write lt gt or =
YOUR TURN
3 radic_
2 + 4 2 + radic_
4 4 radic_
12 + 6 12 + radic_
6
L E S S O N
13 Ordering Real Numbers
ESSENTIAL QUESTIONHow do you order a set of real numbers
8NS2
Use rational approximations of irrational numbers to compare the size of irrational numbers locate them approximately on a number line diagram and estimate the value of expressions (eg π 2 )
8NS2
Use perfect squares to estimate square roots
1 2 = 1 2 2 = 4 3 2 = 9
The number is 7 both expressions must equal 7 + radic_
5
17 and 18
gt lt
21Lesson 13
copy H
ough
ton
Miff
lin H
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ublis
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pany
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8_MCAAESE206984_U1M01L3indd 21 41913 246 PM
PROFESSIONAL DEVELOPMENT
Math BackgroundIn this lesson students estimate irrational numbers in the form of square roots of nonper-fect squares by finding two perfect squares between which the number falls A more precise method involves repeated division For example to find radic
_ 28 find a whole number whose perfect
square is close to 28 such as 5 Divide 28 by that number 28 divide 5 = 56 Find the average of the quotient and divisor 5 + 56
_____ 2 = 53 Continue dividing 28 by each result and averaging until you get the desired accuracy
Integrate Mathematical Practices MP4
This lesson provides an opportunity to address this Mathematical Practices standard It calls for students to model relationships using multiple representations including diagrams graphs and language as appropriate Students use multiple representations when they use number lines to estimate the locations of and order rational and irrational numbers given as symbols
Ordering Real Numbers 22
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
Interactive Whiteboard Interactive example available online
ADDITIONAL EXAMPLE 3The diameter of a meteorite in millimeters is calculated by four different methods Order the results from least to greatest
Joe radic_
18 mm Lisa 13 __ 3 mm
Pablo 46 mm Julien 4π __ 3 mm
Julien 4π __ 3 mm Lisa 13 __ 3 mm
Joe radic_
18 mm Pablo 46 mm
EXAMPLE 3Questioning Strategies Mathematical Practices bull How can you verify that radic
_ 28 is between 52 and 53 5 2 2 = 2704 and 5 3 2 = 2809
bull Explain how to determine which number is greater 5 _
5 or 55 When the repeating decimal is rounded to the nearest tenth or hundredth you can see that it is greater
Connect to Daily LifeDiscuss how measuring across a canyon might involve different methods than measuring along a road Explain that measurements like these are often done using calculations that approximate the distance
YOUR TURNFocus on Critical Thinking Mathematical PracticesDiscuss with students which number is greater 3
_ 45 or 3450 3
_ 45 or 3455 and why Explain
that 3 _
45 can be written out as 34545hellipMake sure they understand that 3 _
45 is greater than 345 but less than 3455
ElaborateTalk About ItSummarize the Lesson
Ask How can you order two numbers in different forms whose decimal approxi-mations appear to be equal Approximate one or both numbers to an additional
number of decimal places
GUIDED PRACTICEEngage with the Whiteboard
Have students place and label additional points on the number line in Exercise 9 Allow the points to be in any format other than decimal
Avoid Common ErrorsExercises 3ndash4 Caution students to read the problem carefully so that they do not misread the problem as the same numbers combined by addition on each side of the circleExercise 10 Remind students that the calculations have units
myhrwcom
23 Lesson 13
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
0 05 1 15 2 25 3 35 4 45 5 55 6 65 7
2πradic3
Compare Write lt gt or = (Example 1)
1 radic_
3 + 2 radic_
3 + 3 2 radic_
8 + 17 radic_
11 + 15
3 radic_
6 + 5 6 + radic_
5 4 radic_
9 + 3 9 + radic_
3
5 radic_
17 - 3 -2 + radic_
5 6 12 - radic_
2 14 - radic_
8
7 radic_
7 + 2 radic_
10 - 1 8 radic_
17 + 3 3 + radic_
11
9 Order radic_
3 2π and 15 from least to greatest Then graph them on the number line (Example 2)
radic_
3 is between and so radic_
3 asymp
π asymp 314 so 2π asymp
From least to greatest the numbers are
10 Four people have found the perimeter of a forest using different methods Their results are given in the table Order their calculations from greatest to least (Example 3)
11 Explain how to order a set of real numbers
CHECK-INESSENTIAL QUESTION
Forest Perimeter (km)
Leon Mika Jason Ashley
radic_
17 - 2 1 +thinsp π __ 2 12 ___ 5 25
Guided Practice
17
15
1 + π _ 2 km 25 km 12 __ 5 km radic_
17 - 2 km
2π radic
_ 3
18 175
628
Sample answer Convert each number to a decimal
equivalent using estimation to find equivalents for
irrational numbers Graph each number on a number line
Read the numbers from left to right for least to greatest
Read the numbers from right to left for greatest to least
lt gt
lt lt
ltgt
gt gt
24 Unit 1
copy H
ough
ton
Miff
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ublis
hing
Com
pany
bull Im
age C
redi
ts copy
Elena
Eliss
eeva
Alam
y Im
ages
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
8_MCAAESE206984_U1M01L3indd 24 41613 448 AM
My Notes
5 52 54 56 58 6
radic28 5 12
23455
Math TrainerOnline Practice
and Help
Personal
myhrwcom
Math On the Spot
myhrwcom
Ordering Real Numbers in a Real-World Context Calculations and estimations in the real world may differ It can be important to know not only which are the most accurate but which give the greatest or least values depending upon the context
Four people have found the distance in kilometers across a canyon using different methods Their results are given in the table Order the distances from greatest to least
Distance Across Quarry Canyon (km)
Juana Lee Ann Ryne Jackson
radic_
28 23 __ 4 5 _
5 5 1 _ 2
Write each value as a decimal
radic_
28 is between 52 and 53 Since 53 2 = 2809 an approximate value for radic
_ 28 is 53
23 __ 4 = 575
5 _
5 is 5555hellip so 5 _
5 to the nearest hundredth is 556
5 1 _ 2 = 55
Plot radic_
28 23 __ 4 5 _
5 and 5 1 _ 2 on a number line
From greatest to least the distances are
23 __ 4 km 5 _
5 km 5 1 _ 2 km radic_
28 km
EXAMPLEXAMPLE 3
STEP 1
STEP 2
7 Four people have found the distance in miles across a crater using different methods Their results are given below
Jonathan 10 __ 3 Elaine 3 _
45 Joseacute 3 1 _ 2 Lashonda radic_
10
Order the distances from greatest to least
YOUR TURN
8NS2
3 1 _ 2 mi 3 _
45 mi 10 __ 3 mi radic_
10 mi
23Lesson 13
copy H
ough
ton
Miff
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pany
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8_MCAAESE206984_U1M01L3indd 23 41613 447 AM
ModelingPlace papers around the room with the numbers from 1 to 5 one per sheet Give each student a card showing a number between 1 and 5 in different forms Have students place his or her card between the correct integers and decide where the number goes in relation to any numbers already placed
Multiple RepresentationsGive students a vertical number line which some students might find easier to use than a horizontal one Have them decide whether to place points for rational and irrational numbers above or below existing points
Additional ResourcesDifferentiated Instruction includes bull Reading Strategies bull Success for English Learners EL
bull Reteach bull Challenge PRE-AP
DIFFERENTIATE INSTRUCTION
Ordering Real Numbers 24
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
Math TrainerOnline Assessment
and Intervention
Personal
Online homework assignment available
myhrwcom
myhrwcom
Lesson Quiz available online
13 LESSON QUIZ
1 Compare Write lt gt or =
radic_
95 - 5 radic_
62 - 2
2 Order 105 radic_
105 and 3π + 1 from greatest to least
3 A length in centimeters is calculated differently by four different people Order their calculations from least to greatest
KD 11 __ 2 cm Silvio 5 __ 3 π cm
Paula 5 _
4 cm Luis radic_
33 cm
EvaluateGUIDED AND INDEPENDENT PRACTICE
Concepts amp Skills Practice
Example 1Comparing Irrational Numbers
Exercises 1ndash8
Example 2Ordering Real Numbers
Exercises 9 12ndash15 18ndash21
Example 3Ordering Real Numbers in a Real-World Context
Exercises 10 16ndash17
Additional ResourcesDifferentiated Instruction includes bull Leveled Practice worksheets
Focus | Coherence | Rigor
Exercise Depth of Knowledge (DOK) Mathematical Practices
12ndash15 1 Recall of Information MP5 Using Tools
16 2 SkillsConcepts MP2 Reasoning
17 2 SkillsConcepts MP6 Precision
18ndash21 2 SkillsConcepts MP2 Reasoning
22 3 Strategic Thinking MP4 Modeling
23ndash24 3 Strategic Thinking MP3 Logic
8NS2
8NS2
Answers1 radic
_ 95 - 5 lt radic
_ 62 - 2
2 radic_
105 3π + 1 105
3 Silvio 5 __ 3 π cm Paula 5 _
4 cm
KD 11
__ 2 cm Luis radic_
33 cm
25 Lesson 13
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
Work Area
3140 3141 3142 3143
314 π227
20 A teacher asks his students to write the numbers shown in order from least to greatest Paul thinks the numbers are already in order Sandra thinks the order should be reversed Who is right
21 Math History There is a famous irrational number called Eulerrsquos number symbolized with an e Like π its decimal form never ends or repeats The first few digits of e are 27182818284
a Between which two square roots of integers could you find this number
b Between which two square roots of integers can you find π
22 Analyze Relationships There are several approximations used for π including 314 and 22 __ 7 π is approximately 314159265358979
a Label π and the two approximations on the number line
b Which of the two approximations is a better estimate for π Explain
c Find a whole number x so that the ratio x ___ 113 is a better estimate for π
than the two given approximations
23 Communicate Mathematical Ideas If a set of six numbers that include both rational and irrational numbers is graphed on a number line what is the fewest number of distinct points that need to be graphed Explain
24 Critique Reasoning Jill says that 12 _
6 is less than 1263 Explain her error
FOCUS ON HIGHER ORDER THINKING
radic_
115 115 ___ 11 and 105624
between radic_
7 asymp 265 and radic_
8 asymp 283
between radic_
9 = 3 and radic_
10 asymp 316
22 __ 7 it is closer to π on the number line
She did not consider the repeating digit 1266
2 rational numbers can have the same location and
irrational numbers can have the same location but they
cannot share a location
355
Neither student is correct The answer
should be 115 ___ 11 105624 radic_
115
Unit 126
copy H
ough
ton M
ifflin
Har
cour
t Pub
lishin
g Com
pany
Imag
e Cre
dits
copy3D
Stoc
kiSt
ockP
hoto
com
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
8_MCAAESE206984_U1M01L3indd 26 210513 801 AM
Personal Math Trainer
Online Practice and Helpmyhrwcom
Name Class Date
Independent Practice
16 Your sister is considering two different shapes for her garden One is a square with side lengths of 35 meters and the other is a circle with a diameter of 4 meters
a Find the area of the square
b Find the area of the circle
c Compare your answers from parts a and b Which garden would give your sister the most space to plant
17 Winnie measured the length of her fatherrsquos ranch four times and got four different distances Her measurements are shown in the table
a To estimate the actual length Winnie first approximated each distance to the nearest hundredth Then she averaged the four numbers Using a calculator find Winniersquos estimate
b Winniersquos father estimated the distance across his ranch to be radic_
56 km How does this distance compare to Winniersquos estimate
Give an example of each type of number
18 a real number between radic_
13 and radic_
14
19 an irrational number between 5 and 7
Order the numbers from least to greatest
12 radic_
7 2 radic_
8 ___ 2 13 radic_
10 π 35
14 radic_
220 -10 radic_
100 115 15 radic_
8 -375 3 9 _ 4
Distance Across Fatherrsquos Ranch (km)
1 2 3 4
radic_
60 58 __ 8 7 _
3 7 3 _ 5
138NS2
radic_
8 ___ 2 2 radic_
7
-10 radic_
100 115 radic_
220
radic_
60 asymp 775 58 __ 8 = 725 7 _
3 asymp 733 7 3 _ 5 = 760 so the average
π radic_
10 35
-375 9 _ 4 radic_
8 3
is 74825 km
1225 m2
4π m2 or approximately 126 m2
They are nearly identical radic_
56 is approximately 74833hellip
The circle would give her more space to plant because it has a
larger area
Sample answer 37
Sample answer radic_
31
25Lesson 13
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ough
ton
Miff
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pany
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
8_MCAAESE206984_U1M01L3indd 25 41613 448 AM
Activity available online myhrwcomEXTEND THE MATH PRE-AP
Activity Have students investigate whether there are infinitely many numbers between two numbers by giving examples for each of the following
bull Between any two rational numbers there is at least one other rational number Sample answer 45 is between 41 and 48
bull Between any two irrational numbers there is at least one rational number Sample answer 45 is between radic
_ 11 and radic
_ 29
bull Between any two rational numbers there is at least one irrational number Sample answer radic
_ 11 is between 31 and 36
bull Between any two irrational numbers there is at least one irrational number Sample answer radic
_ 17 is between radic
_ 11 and radic
_ 29
Ordering Real Numbers 26
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
ReadyMath Trainer
Online Practiceand Help
Personal
myhrwcom
Module Quiz
11ensp RationalenspandenspIrrationalenspNumbersWrite each fraction as a decimal or each decimal as a fraction
1 7__20 2 1___
27 3 17_8
Solve each equation for x
4 x2=81 5 x3=343 6 x2= 1___100
7 Asquarepatiohasanareaof200squarefeetHowlongiseachside
ofthepatiotothenearesttenth
12ensp SetsenspofenspRealenspNumbersWrite all names that apply to each number
8 121____radic
____121
9 π__2
10 TellwhetherthestatementldquoAllintegersarerationalnumbersrdquoistrueorfalseExplainyourchoice
13ensp OrderingenspRealenspNumbersCompare Write lt gt or =
11 radic__
8+3 8+radic__
3 12 radic__
5+11emsp emsp emsp 5+radic___
11
Order the numbers from least to greatest
13 radic___
99π29__
8 14 radic___
1__251_40__
2
15 Howarerealnumbersusedtodescribereal-worldsituations
ESSENTIAL QUESTION
035
9-9
141ft
7 1__10- 1__10
14__11 1875
wholeintegerrationalreal
Trueintegerscanbewrittenasthequotientoftwointegers
SampleanswerRealnumberssuchastherational
π29__
8radic___
99
irrationalreal
lt gt
number1_4candescribeamountsusedincooking
radic___
1__250__
21_4
27Module1
copy H
ough
ton
Miff
lin H
arco
urt P
ublis
hing
Com
pany
DONOTEDIT--ChangesmustbemadethroughldquoFileinfordquoCorrectionKey=A
8_MCAAESE206984_U1M01RTindd 27 41513 1113 PM
Math TrainerOnline Assessment
and Intervention
Personal
myhrwcom
1
2
3 Response toIntervention
Intervention Enrichment
Access Ready to Go On assessment online and receive instant scoring feedback and customized intervention or enrichment
Online and Print Resources
Differentiated Instruction
bull Reteach worksheets
bull Reading Strategies EL
bull Success for English Learners EL
Differentiated Instruction
bull Challenge worksheets PRE-AP
Extend the Math PRE-AP
Lesson Activities in TE
Additional ResourcesAssessment Resources includes bull Leveled Module Quizzes
Ready to Go OnAssess MasteryUse the assessment on this page to determine if students have mastered the concepts and standards covered in this module
California Common Core Standards
Lesson Exercises Common Core Standards
11 1ndash7 8NS1 8NS2 8EE2
12 8ndash10 8NS1
13 11ndash14 8NS2
27 Unit 1 Module 1
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
Personal Math Trainer
Online Practice and HelpmyhrwcomAssessment Readiness
Module 1 MIXed ReVIeW
1 Look at each number Is the number between 2π and radic___
52
Select Yes or No for expressions AndashC
A 6 2 _ 3 Yes No
B 5π __ 2 Yes No
C 3 radic__
5 Yes No
2 Consider the number - 11 __ 15
Choose True or False for each statement
A The number is rational True False
B The number can be written as True Falsea repeating decimal
C The number is less than ndash08 True False
3 The volume of a cube is given by V = x3 where x is the length of an edge of the cube A cube-shaped end table has a volume of 3 3 _ 8 cubic feet What is the length of an edge of the end table Explain how you solved this problem
4 A student says that radic___
83 is greater than 29 __ 3 Is the student correct Justify your
reasoning
1 1 _ 2 ft Sample answer The equation x3 = 3 3 _ 8 can be used
to find the edge length in feet To solve the equation
write the mixed number as a fraction greater than 1
x3 = 27 __ 8 Then take the cube root of both sides x = 3 _ 2 = 1 1 _ 2
No Sample answer radic___
83 asymp 91 and 29 __ 3 = 9
__ 6
Because 91 lt 9 __
6 radic___
83 lt 29 __ 3
28 Unit 1
copy H
ough
ton
Miff
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ublis
hing
Com
pany
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=A
8_MCAAESE206984_U1M01RTindd 28 240413 946 AM
Personal Math Trainer
Online Assessment and
Interventionmyhrwcom
Scoring GuideItem 3 Award the student 1 point for finding the edge length of the cube and 1 point for correctly explaining how to use a cube root to solve the problem
Item 4 Award the student 1 point for determining that the student is incorrect and 1 point for correctly justifying the reasoning for this conclusion
Additional ResourcesTo assign this assessment online login to your Assignment Manager at myhrwcom
Assessment Readiness
California Common Core Standards
Items Grade 8 Standards Mathematical Practices
1 8NS2 MP7
2 7NS2b 7NS2d 8NS1 MP7
3 8EE2 MP1 MP4
4 8NS1 8NS2 MP3
Item integrates mixed review concepts from previous modules or a previous course
Item 4 combines concepts from the California Common Core cluster ldquoKnow that there are numbers that are not rational and approximate them by rational numbersrdquo
Real Numbers 28
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
Math TrainerOnline Practice
and Help
Personal
myhrwcom
Math TrainerOnline Practice
and Help
Personal
myhrwcom
Math On the Spotmyhrwcom
Understanding Sets and Subsets of Real NumbersBy understanding which sets are subsets of types of numbers you can verify whether statements about the relationships between sets are true or false
Tell whether the given statement is true or false Explain your choice
All irrational numbers are real numbers
True Every irrational number is included in the set of real numbers The irrational numbers are a subset of the real numbers
No rational numbers are whole numbers
False A whole number can be written as a fraction with a denominator of 1 so every whole number is included in the set of rational numbers The whole numbers are a subset of the rational numbers
EXAMPLE 2
A
B
Write all names that apply to each number
1 A baseball pitcher has pitched 12 2 _ 3 innings
2 The length of the side of a square that has an
area of 10 square yards
YOUR TURN
Tell whether the given statement is true or false Explain your choice
3 All rational numbers are integers
4 Some irrational numbers are integers
YOUR TURN
Give an example of a rational number that is a
whole number Show that the number is both whole
and rational
Math TalkMathematical Practices
Give an example of a
8NS1
False Every integer is a rational number but not every
False Real numbers are either rational or irrational numbers
Integers are rational numbers so no integers are irrational numbers
rational real
irrational real
Sample answer 8 8 = 8_
1
and -thinsp 5 _ 2 are not integers
rational number is an integer Rational numbers such as 3 _ 5
Unit 116
copy H
ough
ton
Miff
lin H
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ublis
hing
Com
pany
bull Im
age C
redi
ts D
igita
l Im
age c
opyr
ight
copy20
04 Ey
ewire
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
8_MCAAESE206984_U1M01L2indd 16 41613 136 AM
Math On the Spot
myhrwcom
Vertebrates
Birds
Passerines
Animals
Integers
Rational Numbers IrrationalNumbers
Real Numbers
WholeNumbers
1
45
3
0
274
67
radic4
-
-3
-2
-1
03
radic2
radic17
radic11-
π
Animated Math
myhrwcom
Classifying Real NumbersBiologists classify animals based on shared characteristics A cardinal is an animal a vertebrate a bird and a passerine
You already know that the set of rational numbers consists of whole numbers integers and fractions The set of real numbers consists of the set of rational numbers and the set of irrational numbers
Write all names that apply to each number
radic_
5 irrational real
ndash1784rational real
whole integer rational real
EXAMPLEXAMPLE 1
A
B
C radic_ 81 ____ 9
L E S S O N
12Sets of Real Numbers
ESSENTIAL QUESTIONHow can you describe relationships between sets of real numbers
Passerines such as the cardinal are also called ldquoperching birdsrdquo
What types of numbers are between 31 and 39 on a
number line
Math TalkMathematical Practices
What types of numbers are
8NS1
8NS1
Know that numbers that are not rational are called irrational Understand informally that every number has a decimal expansion for rational numbers show that the decimal expansion repeats eventually and convert a decimal expansion which repeats eventually into a relation number
ndash1784 is a terminating decimal
5 is a whole number that is not a perfect square
radic_
81 _____ 9 = 9 __ 9 = 1 rational irrational real
15Lesson 12
copy H
ough
ton
Miff
lin H
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ublis
hing
Com
pany
bull Im
age C
redi
ts copy
Wiki
med
ia Co
mm
ons
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=B
8_MCABESE206984_U1M01L2indd 15 061113 1144 AM
PROFESSIONAL DEVELOPMENT
Math BackgroundThe relationships between sets of numbers extend to include complex numbers A complex number can be written as a sum of a real number a and an imaginary number bi
a + bi
An imaginary number is a special number that when squared gives a negative value When you square a real number you get a nonnegative number When you square an imaginary number you get a negative value The imaginary unit is i
i = radic_
-1
Integrate Mathematical Practices MP7
This lesson provides an opportunity to address this Mathematical Practices standard It calls for students to discern structure to connect and communicate mathematical ideas
Students use a Venn diagram to structure relationships between sets of numbers They connect and communicate mathematical ideas when they make logical statements about the sets and describe which set best describes numbers applied to real-life situations
Sets of Real Numbers 16
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=B
YOUR TURNAvoid Common ErrorsStudents may see the word ldquoAllldquo or rdquoNordquo in Exercises 3 and 4 and immediately assume that any absolute statements like these are false Remind them that there are true statements that begin with these words and encourage them to provide examples
EXAMPLE 3Questioning Strategies Mathematical Practices bull In A how does the phrase ldquonumber of rdquo give you a clue about the number classification It indicates a counting number
bull What is the relationship between the circumference of a circle and the diameter The circumference is diameter times π
Focus on Critical Thinking Mathematical PracticesIn B suppose the diameters in inches were 25
__ π 28 __ π
31 __ π and so on What set of numbers would
best describe the circumferences Explain Whole numbers the circumferences would be the whole numbers 25 28 31 and so on
YOUR TURNFocus on Critical Thinking Mathematical PracticesHave students compare and contrast the classification of numbers in the answers in Exercises 5 and 6
ElaborateTalk About ItSummarize the Lesson
Ask What are some ways that number sets can be related Sets may be subsets of other sets or they may be separate from other sets
GUIDED PRACTICEEngage with the Whiteboard
Have students place the numbers in Exercises 1ndashthinsp8 in the Venn diagram for numbers at the beginning of the lesson
Integrating Language Arts EL
Encourage English learners to ask for clarification on any terms or phrases that they do not understand
Avoid Common ErrorsExercise 7 Remind students that a repeating decimal is a rational numberExercises 9ndash10 Remind students that it only takes one counterexample to show that a statement is false
Interactive Whiteboard Interactive example available online
ADDITIONAL EXAMPLE 3Identify the set of numbers that best describes the situation Explain your choice
A the amount of time that has passed since midnight
The set of real numbers time is continuous so the amount of time can be rational or irrational
B the number of tickets sold to a basketball game
The set of whole numbers the number of tickets sold may be 0 or a counting number
myhrwcom
17 Lesson 12
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
1IN
116 inch
Guided Practice
Write all names that apply to each number (Example 1)
1 7 _ 8 2 radic_
36
3 radic_
24 4 075
5 0 6 - radic_ 100
7 5 _
45 8 - 18 __ 6
Tell whether the given statement is true or false Explain your choice (Example 2)
9 All whole numbers are rational numbers
10 No irrational numbers are whole numbers
Identify the set of numbers that best describes each situation Explain your choice (Example 3)
11 the change in the value of an account when given to the nearest dollar
12 the markings on a standard ruler
13 What are some ways to describe the relationships between sets of numbers
CHECK-INESSENTIAL QUESTION
rational real
rational real
True Whole numbers are rational numbers
Rational numbers the ruler is marked every 1 __ 16 th inch
Sample answer Describe one set as being a subset of
another or show their relationships in a Venn diagram
Integers the change can be a whole dollar amount
and can be positive negative or zero
True Whole numbers are a subset of the set of rational numbers
and can be written as a ratio of the whole number to 1
irrational real
whole integer rational real
whole integer rational real
rational real
integer rational real
integer rational real
Unit 118
copy H
ough
ton
Miff
lin H
arco
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ublis
hing
Com
pany
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
8_MCAAESE206984_U1M01L2indd 18 41613 136 AM
My Notes
Math TrainerOnline Practice
and Help
Personal
myhrwcom
Math On the Spot
myhrwcom
Identifying Sets for Real-World SituationsReal numbers can be used to represent real-world quantities Highways have posted speed limit signs that are represented by natural numbers such as 55 mph Integers appear on thermometers Rational numbers are used in many daily activities including cooking For example ingredients in a recipe are often given in fractional amounts such as 2 _ 3 cup flour
Identify the set of numbers that best describes each situation Explain your choice
the number of people wearing glasses in a room
The set of whole numbers best describes the situation The number of people wearing glasses may be 0 or a counting number
the circumference of a flying disk has a diameter of 8 9 10 11 or 14 inches
The set of irrational numbers best describes the situation Each circumference would be a product of π and the diameter and any multiple of π is irrational
EXAMPLEXAMPLE 3
A
B
Identify the set of numbers that best describes the situation Explain your choice
5 the amount of water in a glass as it evaporates
6 the weight of a person in pounds
YOUR TURN
8NS1
Rational numbers a personrsquos weight can be a decimal
such as 835 pounds
Real numbers the amount can be any number greater
than 0
17Lesson 12
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ough
ton
Miff
lin H
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ublis
hing
Com
pany
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
8_MCAAESE206984_U1M01L2indd 17 41613 520 AM
Graphic OrganizersGive students a list of numbers (including terminating and repeating decimals fractions integers and rational and irrational square roots) and a graphic organizer as shown below
Real Numbers
Rational numbers Irrational numbers
Integer numbers
Whole numbers
Ask students to write each number in the list in the correct section of the organizer
Number SensePoint out to students that knowing the types of numbers to expect in different situations can alert them to incorrect math as well as to impossible situations For example 135 shots made in basketballs is not possible but an average number of shots can equal 135
Additional ResourcesDifferentiated Instruction includes bull Reading Strategies bull Success for English Learners EL
bull Reteach bull Challenge PRE-AP
DIFFERENTIATE INSTRUCTION
Sets of Real Numbers 18
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
Math TrainerOnline Assessment
and Intervention
Personal
Online homework assignment available
myhrwcom
Lesson Quiz available online
12 LESSON QUIZ
1 Write all the names that apply to the number
2 Tell whether the given statement is true or false Explain your choice All numbers between 1 and 2 are rational numbers
3 Identify the set of numbers that best describes the situation Explain your choiceThe choices on a survey question change the total points for the survey by -2 -1 0 1 or 2 points
-1 _
5
myhrwcom
Additional ResourcesDifferentiated Instruction includes bull Leveled Practice worksheets
EvaluateGUIDED AND INDEPENDENT PRACTICE
Concepts amp Skills Practice
Example 1Classifying Real Numbers
Exercises 1ndash8 14ndash19 22ndash24
Example 2Understanding Sets and Subsets of Real Numbers
Exercises 9ndash10
Example 3Identifying Sets for Real-World Situations
Exercises 11ndash12 20ndash21 25
Focus | Coherence | Rigor
Exercise Depth of Knowledge (DOK) Mathematical Practices
14ndash19 2 SkillsConcepts MP7 Using Structure
20ndash21 2 SkillsConcepts MP6 Precision
22ndash23 2 SkillsConcepts MP3 Logic
24 1 Recall of Information MP7 Using Structure
25 2 SkillsConcepts MP2 Reasoning
26ndash27 3 Strategic Thinking MP3 Logic
28 3 Strategic Thinking MP8 Patterns
29 3 Strategic Thinking MP3 Logic
8NS1
8NS1
Exercise 29 combines concepts from the California Common Core cluster ldquoKnow that there are numbers that are not rational and approximate them by rational numbersrdquo
Answers1 rational real
2 False radic_
2 is an example of an irrational number between 1 and 2
3 Integers each number is an integer but only three are whole numbers
19 Lesson 12
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
Work Area
π mi23 Critique Reasoning The circumference of a circular region is shown
What type of number best describes the diameter of the circle Explain
your answer
24 Critical Thinking A number is not an integer What type of number can it be
25 A grocery store has a shelf with half-gallon containers of milk What type of number best represents the total number of gallons
26 Explain the Error Katie said ldquoNegative numbers are integersrdquo What was her error
27 Justify Reasoning Can you ever use a calculator to determine if a number is rational or irrational Explain
28 Draw Conclusions The decimal 0 _
3 represents 1 _ 3 What type of number best describes 0
_ 9 which is 3 middot 0
_ 3 Explain
29 Communicate Mathematical Ideas Irrational numbers can never be precisely represented in decimal form Why is this
FOCUS ON HIGHER ORDER THINKING
It can be a rational number that is not an integer or an irrational number
rational number
The set of negative numbers also includes non-integer
rational numbers and irrational numbers
Sample answer If the calculator shows a decimal that
terminates in fewer digits than what the calculator screen
allows then you can tell that the number is rational If not
you cannot tell from the calculator display whether the
number terminates because you see a limited number
of digits It may be a repeating decimal (rational) or
non-terminating non-repeating decimal (irrational)
Whole 3 middot 0 _
3 represents 3 middot 1 _ 3 = 1 so 0 _
9 is exactly 1
Sample answer In decimal form irrational numbers never
terminate and never repeat Therefore no matter how
many decimal places you include the number will never
be precisely represented There are always more digits
Whole the diameter is π _ π = 1 mile
Unit 120
copy H
ough
ton
Miff
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ublis
hing
Com
pany
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
8_MCAAESE206984_U1M01L2indd 20 120413 909 PM
Integers
Rational Numbers Irrational Numbers
Real Numbers
Whole Numbers
257
radic16
166
radic9
128 radic50
Personal Math Trainer
Online Practice and Helpmyhrwcom
Name Class Date
Independent Practice
Identify the set of numbers that best describes each situation Explain your choice
20 the height of an airplane as it descends to an airport runway
21 the score with respect to par of several golfers 2 ndash 3 5 0 ndash 1
22 Critique Reasoning Ronald states that the number 1 __ 11 is not rational because when converted into a decimal it does not terminate Nathaniel says it is rational because it is a fraction Which boy is correct Explain
12
14 - radic_
9 15 257
16 radic_
50 17 8 1 _ 2
18 166 19 radic_
16
Write all names that apply to each number Then place the numbers in the correct location on the Venn diagram
8NS1
Real numbers the height can be any number greater than zero
integer rational real whole integer rational real
whole integer rational real
irrational real
rational real
rational real
Integers the scores are counting numbers their
opposites and zero
Nathaniel is correct A rational number is a number that can be written as a fraction and 1 __ 11 is a fraction
19Lesson 12
copy H
ough
ton
Miff
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Com
pany
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
8_MCAAESE206984_U1M01L2indd 19 41613 136 AM
myhrwcomActivity available onlineEXTEND THE MATH PRE-AP
Activity Have students consider the concept of restricted domain for the sets of numbers that describe situations For example the number of sisters a person has can best be described by whole numbers but no one has ever had 1500 sisters An area code is an integer or whole number between 200 and 999
Have students use a source such as the Guinness Book of World Records and give examples of sets of numbers that describe situations where the domain is restricted Ask whether the restriction may be changed in the future
Sets of Real Numbers 20
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
-3-4-5 -2-1 1 2 3 50 4
12-4 -radic5
Lesson Support Content Objective Students will learn to order a set of real numbers
Language Objective Students will show and describe how to order a set of real numbers
LESSON 13 Ordering Real Numbers
Building BackgroundEliciting Prior Knowledge Have students draw a number line to compare a rational number and an irrational number such as - radic
_ 5 and -4 1 __ 2 Ask them to explain how
they approximated the irrational number on the number line Then have them identify the greater and the lesser real number Repeat with several other pairs of real numbers in different forms
Learning ProgressionsIn this lesson students order a set of real numbers They use rational approximations to compare the sizes of irrational numbers They also order numbers for real-world situations Important understandings for students include the following
bull Compare irrational numbers bull Estimate the value of expressions with irrational numbers bull Order a set of real numbers bull Order real numbers in a real-world context
Work with real numbers continues throughout Grade 8 and into high school This lesson provides students with a foundation for understanding the relative sizes of numbers in different forms in the real number system
Cluster ConnectionsThis lesson provides an excellent opportunity to connect ideas in this cluster Know that there are numbers that are not rational and approximate them by rational numbers Tell students that there is a special number called the golden ratio with applications in mathematics geometry art and architecture The golden ratio is called phi and is represented by the Greek letter ϕ It includes an irrational number in its definition
Have students explain why the golden ratio is irrational Ask them to find the two whole numbers the golden ratio lies between Then challenge them to approximate the golden ratio to the nearest tenth It is irrational because it includes an irrational number in its definition It lies between 1 and 2 To the nearest tenth ϕ = 16
ϕ = 1 + radic_
5 _ 2
Focus | Coherence | Rigor
California Common Core Standards
8NS2 Use rational approximations of irrational numbers to compare the size of irrational numbers locate them approximately on a number line diagram and estimate the value of expressions (eg π 2 )
MP4 Model with mathematics
21A
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
Math Talk
Language Support EL
PROFESSIONAL DEVELOPMENT
Linguistic Support EL
AcademicContent Vocabulary
Post a chart like this to remind students of the regular comparative forms of adjectives that use the -er and -est suffixes Add to the chart for terms that appear in examples and exercises in each lesson Include any irregular verb forms
Background Knowledge
Go On ndash the title of the module review or quiz is Ready to Go On This title uses an idiomatic expression In this context to go on means ldquoto move aheadrdquo or ldquoto proceedrdquo It is different from the use of go on that means having enough facts to use meaningfully as in having enough to go on Also the intonation used in pronouncing an expression can give it different meanings For example when the speaker emphasizes the word on he or she might be expressing disbelief as in ldquoGo ON Yoursquore kidding rightrdquo Discuss with students other ways that the phrase go on may be used
Leveled Strategies for English Learners
Emerging Label points on a number line with the terms used in ordering greater greatest less lesser least Use sentence frames to insert the correct terms
Expanding Have students give two or three complete sentences to compare the placement of numbers on a number line using the correct forms of the comparative and superlative adjectives
Bridging Have students work in pairs with one student giving directions to the other in complete sentences to order numbers on a number line
To help students answer the question posed in Math Talk make sure that students have a command of the forms for making comparisons and the superlative and the concept of opposite order so that the focus is on the math concept instead of the language skills needed to describe and explain order
EL
Adjective Comparative Superlative
Far Farther Farthest
Large Larger Largest
Great Greater Greatest
Some Less Least
Some More Most
California ELD Standards
Emerging 2I8 Analyzing language choices ndash Explain how phrasing or different common words with similar meanings produce different effects on the audience
Expanding 2I8 Analyzing language choices ndash Explain how phrasing or different words with similar meanings or figurative language produce shades of meaning and different effects on the audience
Bridging 2I8 Analyzing language choices ndash Explain how phrasing or different words with similar meanings or figurative language produce shades of meaning nuances and different effects on the audience
Ordering Real Numbers 21B
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
13L E S S O N
Ordering Real Numbers
Interactive Whiteboard Interactive example available online
ADDITIONAL EXAMPLE 1Compare Write lt gt or =
A radic_
8 - 2 4 - radic_
8 lt
B radic_
20 + 1 3 + radic_
2 gt
EngageESSENTIAL QUESTION
How do you order a set of real numbers Sample answer Find their approximate decimal values and order them
Motivate the LessonAsk What kind of numbers are you comparing when you compare the price of gasoline at two different gas stations
ExploreGive students two rational numbers and ask them to name a number between them Repeat a few times and then give them two irrational numbers and ask them to name a number between them
ExplainEXAMPLE 1
Questioning Strategies Mathematical Practices bull Which is greater the difference between 5 and 3 or the difference between radic
_ 5 and radic
_ 3
The difference between 5 and 3 is 2 the difference between radic_
5 and radic_
3 is approximately 1 So the difference between 5 and 3 is greater
Avoid Common ErrorsCaution students to read the problem carefully and think about what the radical sign means so that they do not misread the problem and answer that the two sides are equal
YOUR TURNFocus on TechnologyCalculators should not be used at this point because developing number sense is the goal
EXAMPLE 2Questioning Strategies Mathematical Practices bull How do you determine whether radic
_ 22 is less than or greater than 45 The square of 45 is
2025 which is less than 22 so the square root of 22 must be greater than 45
Engage with the WhiteboardHave students graph and label various real numbers between 42 and 44 and between 47 and 5
YOUR TURNFocus on Modeling Mathematical PracticesHave students label the integers on the number line with their equivalent square root For example 1 2 and 3 on the number line would be labeled radic
_ 1 radic
_ 4 and radic
_ 9
Interactive Whiteboard Interactive example available online
ADDITIONAL EXAMPLE 2Order 3π radic
_ 10 and 325 from greatest
to least
3π 325 radic_
10
myhrwcom
myhrwcom
CA Common CoreStandards
The student is expected to
The Number Systemmdash8NS2
Use rational approximations of irrational numbers to compare the size of irrational numbers locate them approximately on a number line diagram and estimate the value of expressions (eg π 2 )
Mathematical Practices
MP4 Modeling
The student is expected to
21 Lesson 13
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
Math TrainerOnline Practice
and Help
Personal
myhrwcom
Math On the Spotmyhrwcom
0 05 1 15 2 25 3 35 4
radic5radic3
π2
8 85 9 95 10 105 11 115 12
radic75
4 42 44 46 48 5
radic224 12π + 1
Ordering Real Numbers You can compare and order real numbers and list them from least to greatest
Order radic_
22 π + 1 and 4 1 _ 2 from least to greatest
First approximate radic_
22
radic_
22 is between 4 and 5 Since you donrsquot know where it falls between 4 and 5 you need to find a better estimate for radic
_ 22 so
you can compare it to 4 1 _ 2
Since 22 is closer to 25 than 16 use squares of numbers between 45 and 5 to find a better estimate of radic
_ 22
45 2 = 2025 46 2 = 2116 47 2 = 2209 48 2 = 2304
Since 47 2 = 2209 an approximate value for radic_
22 is 47
An approximate value of π is 314 So an approximate value of π +1 is 414
Plot radic_
22 π + 1 and 4 1 _ 2 on a number line
Read the numbers from left to right to place them in order from least to greatest
From least to greatest the numbers are π + 1 4 1 _ 2 and radic_
22
EXAMPLE 2
STEP 1
STEP 2
Order the numbers from least to greatest Then graph them on the number line
YOUR TURN
5 radic_
5 25 radic_
3
6 π 2 10 radic_
75
If real numbers a b and c are in order from least to greatest what is the order
of their opposites from least to greatest
Explain
Math TalkMathematical Practices
8NS2
radic_
3 radic_
5 25
radic_
75 π2 10
Math Talk answer -c -b -a -c is farthest to the left on a number line -b is in the middle and -a is farthest to the right
Unit 122
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ough
ton
Miff
lin H
arco
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hing
Com
pany
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
8_MCAAESE206984_U1M01L3indd 22 41613 447 AM
My Notes
Math TrainerOnline Practice
and Help
Personal
myhrwcom
Math On the Spot
myhrwcom
Comparing Irrational NumbersBetween any two real numbers is another real number To compare and order real numbers you can approximate irrational numbers as decimals
Compare radic_
3 + 5 3 + radic_
5 Write lt gt or =
First approximate radic_
3
radic_
3 is between 1 and 2
Next approximate radic_
5
radic_
5 is between 2 and 3
Then use your approximations to simplify the expressions
radic_
3 + 5 is between 6 and 7
3 + radic_
5 is between 5 and 6
So radic_
3 + 5 gt 3 + radic_
5
Reflect1 If 7 + radic
_ 5 is equal to radic
_ 5 plus a number what do you know about the
number Why
2 What are the closest two integers that radic_
300 is between
EXAMPLEXAMPLE 1
STEP 1
STEP 2
Compare Write lt gt or =
YOUR TURN
3 radic_
2 + 4 2 + radic_
4 4 radic_
12 + 6 12 + radic_
6
L E S S O N
13 Ordering Real Numbers
ESSENTIAL QUESTIONHow do you order a set of real numbers
8NS2
Use rational approximations of irrational numbers to compare the size of irrational numbers locate them approximately on a number line diagram and estimate the value of expressions (eg π 2 )
8NS2
Use perfect squares to estimate square roots
1 2 = 1 2 2 = 4 3 2 = 9
The number is 7 both expressions must equal 7 + radic_
5
17 and 18
gt lt
21Lesson 13
copy H
ough
ton
Miff
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hing
Com
pany
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
8_MCAAESE206984_U1M01L3indd 21 41913 246 PM
PROFESSIONAL DEVELOPMENT
Math BackgroundIn this lesson students estimate irrational numbers in the form of square roots of nonper-fect squares by finding two perfect squares between which the number falls A more precise method involves repeated division For example to find radic
_ 28 find a whole number whose perfect
square is close to 28 such as 5 Divide 28 by that number 28 divide 5 = 56 Find the average of the quotient and divisor 5 + 56
_____ 2 = 53 Continue dividing 28 by each result and averaging until you get the desired accuracy
Integrate Mathematical Practices MP4
This lesson provides an opportunity to address this Mathematical Practices standard It calls for students to model relationships using multiple representations including diagrams graphs and language as appropriate Students use multiple representations when they use number lines to estimate the locations of and order rational and irrational numbers given as symbols
Ordering Real Numbers 22
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
Interactive Whiteboard Interactive example available online
ADDITIONAL EXAMPLE 3The diameter of a meteorite in millimeters is calculated by four different methods Order the results from least to greatest
Joe radic_
18 mm Lisa 13 __ 3 mm
Pablo 46 mm Julien 4π __ 3 mm
Julien 4π __ 3 mm Lisa 13 __ 3 mm
Joe radic_
18 mm Pablo 46 mm
EXAMPLE 3Questioning Strategies Mathematical Practices bull How can you verify that radic
_ 28 is between 52 and 53 5 2 2 = 2704 and 5 3 2 = 2809
bull Explain how to determine which number is greater 5 _
5 or 55 When the repeating decimal is rounded to the nearest tenth or hundredth you can see that it is greater
Connect to Daily LifeDiscuss how measuring across a canyon might involve different methods than measuring along a road Explain that measurements like these are often done using calculations that approximate the distance
YOUR TURNFocus on Critical Thinking Mathematical PracticesDiscuss with students which number is greater 3
_ 45 or 3450 3
_ 45 or 3455 and why Explain
that 3 _
45 can be written out as 34545hellipMake sure they understand that 3 _
45 is greater than 345 but less than 3455
ElaborateTalk About ItSummarize the Lesson
Ask How can you order two numbers in different forms whose decimal approxi-mations appear to be equal Approximate one or both numbers to an additional
number of decimal places
GUIDED PRACTICEEngage with the Whiteboard
Have students place and label additional points on the number line in Exercise 9 Allow the points to be in any format other than decimal
Avoid Common ErrorsExercises 3ndash4 Caution students to read the problem carefully so that they do not misread the problem as the same numbers combined by addition on each side of the circleExercise 10 Remind students that the calculations have units
myhrwcom
23 Lesson 13
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
0 05 1 15 2 25 3 35 4 45 5 55 6 65 7
2πradic3
Compare Write lt gt or = (Example 1)
1 radic_
3 + 2 radic_
3 + 3 2 radic_
8 + 17 radic_
11 + 15
3 radic_
6 + 5 6 + radic_
5 4 radic_
9 + 3 9 + radic_
3
5 radic_
17 - 3 -2 + radic_
5 6 12 - radic_
2 14 - radic_
8
7 radic_
7 + 2 radic_
10 - 1 8 radic_
17 + 3 3 + radic_
11
9 Order radic_
3 2π and 15 from least to greatest Then graph them on the number line (Example 2)
radic_
3 is between and so radic_
3 asymp
π asymp 314 so 2π asymp
From least to greatest the numbers are
10 Four people have found the perimeter of a forest using different methods Their results are given in the table Order their calculations from greatest to least (Example 3)
11 Explain how to order a set of real numbers
CHECK-INESSENTIAL QUESTION
Forest Perimeter (km)
Leon Mika Jason Ashley
radic_
17 - 2 1 +thinsp π __ 2 12 ___ 5 25
Guided Practice
17
15
1 + π _ 2 km 25 km 12 __ 5 km radic_
17 - 2 km
2π radic
_ 3
18 175
628
Sample answer Convert each number to a decimal
equivalent using estimation to find equivalents for
irrational numbers Graph each number on a number line
Read the numbers from left to right for least to greatest
Read the numbers from right to left for greatest to least
lt gt
lt lt
ltgt
gt gt
24 Unit 1
copy H
ough
ton
Miff
lin H
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urt P
ublis
hing
Com
pany
bull Im
age C
redi
ts copy
Elena
Eliss
eeva
Alam
y Im
ages
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
8_MCAAESE206984_U1M01L3indd 24 41613 448 AM
My Notes
5 52 54 56 58 6
radic28 5 12
23455
Math TrainerOnline Practice
and Help
Personal
myhrwcom
Math On the Spot
myhrwcom
Ordering Real Numbers in a Real-World Context Calculations and estimations in the real world may differ It can be important to know not only which are the most accurate but which give the greatest or least values depending upon the context
Four people have found the distance in kilometers across a canyon using different methods Their results are given in the table Order the distances from greatest to least
Distance Across Quarry Canyon (km)
Juana Lee Ann Ryne Jackson
radic_
28 23 __ 4 5 _
5 5 1 _ 2
Write each value as a decimal
radic_
28 is between 52 and 53 Since 53 2 = 2809 an approximate value for radic
_ 28 is 53
23 __ 4 = 575
5 _
5 is 5555hellip so 5 _
5 to the nearest hundredth is 556
5 1 _ 2 = 55
Plot radic_
28 23 __ 4 5 _
5 and 5 1 _ 2 on a number line
From greatest to least the distances are
23 __ 4 km 5 _
5 km 5 1 _ 2 km radic_
28 km
EXAMPLEXAMPLE 3
STEP 1
STEP 2
7 Four people have found the distance in miles across a crater using different methods Their results are given below
Jonathan 10 __ 3 Elaine 3 _
45 Joseacute 3 1 _ 2 Lashonda radic_
10
Order the distances from greatest to least
YOUR TURN
8NS2
3 1 _ 2 mi 3 _
45 mi 10 __ 3 mi radic_
10 mi
23Lesson 13
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ton
Miff
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pany
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8_MCAAESE206984_U1M01L3indd 23 41613 447 AM
ModelingPlace papers around the room with the numbers from 1 to 5 one per sheet Give each student a card showing a number between 1 and 5 in different forms Have students place his or her card between the correct integers and decide where the number goes in relation to any numbers already placed
Multiple RepresentationsGive students a vertical number line which some students might find easier to use than a horizontal one Have them decide whether to place points for rational and irrational numbers above or below existing points
Additional ResourcesDifferentiated Instruction includes bull Reading Strategies bull Success for English Learners EL
bull Reteach bull Challenge PRE-AP
DIFFERENTIATE INSTRUCTION
Ordering Real Numbers 24
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
Math TrainerOnline Assessment
and Intervention
Personal
Online homework assignment available
myhrwcom
myhrwcom
Lesson Quiz available online
13 LESSON QUIZ
1 Compare Write lt gt or =
radic_
95 - 5 radic_
62 - 2
2 Order 105 radic_
105 and 3π + 1 from greatest to least
3 A length in centimeters is calculated differently by four different people Order their calculations from least to greatest
KD 11 __ 2 cm Silvio 5 __ 3 π cm
Paula 5 _
4 cm Luis radic_
33 cm
EvaluateGUIDED AND INDEPENDENT PRACTICE
Concepts amp Skills Practice
Example 1Comparing Irrational Numbers
Exercises 1ndash8
Example 2Ordering Real Numbers
Exercises 9 12ndash15 18ndash21
Example 3Ordering Real Numbers in a Real-World Context
Exercises 10 16ndash17
Additional ResourcesDifferentiated Instruction includes bull Leveled Practice worksheets
Focus | Coherence | Rigor
Exercise Depth of Knowledge (DOK) Mathematical Practices
12ndash15 1 Recall of Information MP5 Using Tools
16 2 SkillsConcepts MP2 Reasoning
17 2 SkillsConcepts MP6 Precision
18ndash21 2 SkillsConcepts MP2 Reasoning
22 3 Strategic Thinking MP4 Modeling
23ndash24 3 Strategic Thinking MP3 Logic
8NS2
8NS2
Answers1 radic
_ 95 - 5 lt radic
_ 62 - 2
2 radic_
105 3π + 1 105
3 Silvio 5 __ 3 π cm Paula 5 _
4 cm
KD 11
__ 2 cm Luis radic_
33 cm
25 Lesson 13
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
Work Area
3140 3141 3142 3143
314 π227
20 A teacher asks his students to write the numbers shown in order from least to greatest Paul thinks the numbers are already in order Sandra thinks the order should be reversed Who is right
21 Math History There is a famous irrational number called Eulerrsquos number symbolized with an e Like π its decimal form never ends or repeats The first few digits of e are 27182818284
a Between which two square roots of integers could you find this number
b Between which two square roots of integers can you find π
22 Analyze Relationships There are several approximations used for π including 314 and 22 __ 7 π is approximately 314159265358979
a Label π and the two approximations on the number line
b Which of the two approximations is a better estimate for π Explain
c Find a whole number x so that the ratio x ___ 113 is a better estimate for π
than the two given approximations
23 Communicate Mathematical Ideas If a set of six numbers that include both rational and irrational numbers is graphed on a number line what is the fewest number of distinct points that need to be graphed Explain
24 Critique Reasoning Jill says that 12 _
6 is less than 1263 Explain her error
FOCUS ON HIGHER ORDER THINKING
radic_
115 115 ___ 11 and 105624
between radic_
7 asymp 265 and radic_
8 asymp 283
between radic_
9 = 3 and radic_
10 asymp 316
22 __ 7 it is closer to π on the number line
She did not consider the repeating digit 1266
2 rational numbers can have the same location and
irrational numbers can have the same location but they
cannot share a location
355
Neither student is correct The answer
should be 115 ___ 11 105624 radic_
115
Unit 126
copy H
ough
ton M
ifflin
Har
cour
t Pub
lishin
g Com
pany
Imag
e Cre
dits
copy3D
Stoc
kiSt
ockP
hoto
com
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
8_MCAAESE206984_U1M01L3indd 26 210513 801 AM
Personal Math Trainer
Online Practice and Helpmyhrwcom
Name Class Date
Independent Practice
16 Your sister is considering two different shapes for her garden One is a square with side lengths of 35 meters and the other is a circle with a diameter of 4 meters
a Find the area of the square
b Find the area of the circle
c Compare your answers from parts a and b Which garden would give your sister the most space to plant
17 Winnie measured the length of her fatherrsquos ranch four times and got four different distances Her measurements are shown in the table
a To estimate the actual length Winnie first approximated each distance to the nearest hundredth Then she averaged the four numbers Using a calculator find Winniersquos estimate
b Winniersquos father estimated the distance across his ranch to be radic_
56 km How does this distance compare to Winniersquos estimate
Give an example of each type of number
18 a real number between radic_
13 and radic_
14
19 an irrational number between 5 and 7
Order the numbers from least to greatest
12 radic_
7 2 radic_
8 ___ 2 13 radic_
10 π 35
14 radic_
220 -10 radic_
100 115 15 radic_
8 -375 3 9 _ 4
Distance Across Fatherrsquos Ranch (km)
1 2 3 4
radic_
60 58 __ 8 7 _
3 7 3 _ 5
138NS2
radic_
8 ___ 2 2 radic_
7
-10 radic_
100 115 radic_
220
radic_
60 asymp 775 58 __ 8 = 725 7 _
3 asymp 733 7 3 _ 5 = 760 so the average
π radic_
10 35
-375 9 _ 4 radic_
8 3
is 74825 km
1225 m2
4π m2 or approximately 126 m2
They are nearly identical radic_
56 is approximately 74833hellip
The circle would give her more space to plant because it has a
larger area
Sample answer 37
Sample answer radic_
31
25Lesson 13
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ough
ton
Miff
lin H
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ublis
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Com
pany
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
8_MCAAESE206984_U1M01L3indd 25 41613 448 AM
Activity available online myhrwcomEXTEND THE MATH PRE-AP
Activity Have students investigate whether there are infinitely many numbers between two numbers by giving examples for each of the following
bull Between any two rational numbers there is at least one other rational number Sample answer 45 is between 41 and 48
bull Between any two irrational numbers there is at least one rational number Sample answer 45 is between radic
_ 11 and radic
_ 29
bull Between any two rational numbers there is at least one irrational number Sample answer radic
_ 11 is between 31 and 36
bull Between any two irrational numbers there is at least one irrational number Sample answer radic
_ 17 is between radic
_ 11 and radic
_ 29
Ordering Real Numbers 26
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
ReadyMath Trainer
Online Practiceand Help
Personal
myhrwcom
Module Quiz
11ensp RationalenspandenspIrrationalenspNumbersWrite each fraction as a decimal or each decimal as a fraction
1 7__20 2 1___
27 3 17_8
Solve each equation for x
4 x2=81 5 x3=343 6 x2= 1___100
7 Asquarepatiohasanareaof200squarefeetHowlongiseachside
ofthepatiotothenearesttenth
12ensp SetsenspofenspRealenspNumbersWrite all names that apply to each number
8 121____radic
____121
9 π__2
10 TellwhetherthestatementldquoAllintegersarerationalnumbersrdquoistrueorfalseExplainyourchoice
13ensp OrderingenspRealenspNumbersCompare Write lt gt or =
11 radic__
8+3 8+radic__
3 12 radic__
5+11emsp emsp emsp 5+radic___
11
Order the numbers from least to greatest
13 radic___
99π29__
8 14 radic___
1__251_40__
2
15 Howarerealnumbersusedtodescribereal-worldsituations
ESSENTIAL QUESTION
035
9-9
141ft
7 1__10- 1__10
14__11 1875
wholeintegerrationalreal
Trueintegerscanbewrittenasthequotientoftwointegers
SampleanswerRealnumberssuchastherational
π29__
8radic___
99
irrationalreal
lt gt
number1_4candescribeamountsusedincooking
radic___
1__250__
21_4
27Module1
copy H
ough
ton
Miff
lin H
arco
urt P
ublis
hing
Com
pany
DONOTEDIT--ChangesmustbemadethroughldquoFileinfordquoCorrectionKey=A
8_MCAAESE206984_U1M01RTindd 27 41513 1113 PM
Math TrainerOnline Assessment
and Intervention
Personal
myhrwcom
1
2
3 Response toIntervention
Intervention Enrichment
Access Ready to Go On assessment online and receive instant scoring feedback and customized intervention or enrichment
Online and Print Resources
Differentiated Instruction
bull Reteach worksheets
bull Reading Strategies EL
bull Success for English Learners EL
Differentiated Instruction
bull Challenge worksheets PRE-AP
Extend the Math PRE-AP
Lesson Activities in TE
Additional ResourcesAssessment Resources includes bull Leveled Module Quizzes
Ready to Go OnAssess MasteryUse the assessment on this page to determine if students have mastered the concepts and standards covered in this module
California Common Core Standards
Lesson Exercises Common Core Standards
11 1ndash7 8NS1 8NS2 8EE2
12 8ndash10 8NS1
13 11ndash14 8NS2
27 Unit 1 Module 1
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
Personal Math Trainer
Online Practice and HelpmyhrwcomAssessment Readiness
Module 1 MIXed ReVIeW
1 Look at each number Is the number between 2π and radic___
52
Select Yes or No for expressions AndashC
A 6 2 _ 3 Yes No
B 5π __ 2 Yes No
C 3 radic__
5 Yes No
2 Consider the number - 11 __ 15
Choose True or False for each statement
A The number is rational True False
B The number can be written as True Falsea repeating decimal
C The number is less than ndash08 True False
3 The volume of a cube is given by V = x3 where x is the length of an edge of the cube A cube-shaped end table has a volume of 3 3 _ 8 cubic feet What is the length of an edge of the end table Explain how you solved this problem
4 A student says that radic___
83 is greater than 29 __ 3 Is the student correct Justify your
reasoning
1 1 _ 2 ft Sample answer The equation x3 = 3 3 _ 8 can be used
to find the edge length in feet To solve the equation
write the mixed number as a fraction greater than 1
x3 = 27 __ 8 Then take the cube root of both sides x = 3 _ 2 = 1 1 _ 2
No Sample answer radic___
83 asymp 91 and 29 __ 3 = 9
__ 6
Because 91 lt 9 __
6 radic___
83 lt 29 __ 3
28 Unit 1
copy H
ough
ton
Miff
lin H
arco
urt P
ublis
hing
Com
pany
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=A
8_MCAAESE206984_U1M01RTindd 28 240413 946 AM
Personal Math Trainer
Online Assessment and
Interventionmyhrwcom
Scoring GuideItem 3 Award the student 1 point for finding the edge length of the cube and 1 point for correctly explaining how to use a cube root to solve the problem
Item 4 Award the student 1 point for determining that the student is incorrect and 1 point for correctly justifying the reasoning for this conclusion
Additional ResourcesTo assign this assessment online login to your Assignment Manager at myhrwcom
Assessment Readiness
California Common Core Standards
Items Grade 8 Standards Mathematical Practices
1 8NS2 MP7
2 7NS2b 7NS2d 8NS1 MP7
3 8EE2 MP1 MP4
4 8NS1 8NS2 MP3
Item integrates mixed review concepts from previous modules or a previous course
Item 4 combines concepts from the California Common Core cluster ldquoKnow that there are numbers that are not rational and approximate them by rational numbersrdquo
Real Numbers 28
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
YOUR TURNAvoid Common ErrorsStudents may see the word ldquoAllldquo or rdquoNordquo in Exercises 3 and 4 and immediately assume that any absolute statements like these are false Remind them that there are true statements that begin with these words and encourage them to provide examples
EXAMPLE 3Questioning Strategies Mathematical Practices bull In A how does the phrase ldquonumber of rdquo give you a clue about the number classification It indicates a counting number
bull What is the relationship between the circumference of a circle and the diameter The circumference is diameter times π
Focus on Critical Thinking Mathematical PracticesIn B suppose the diameters in inches were 25
__ π 28 __ π
31 __ π and so on What set of numbers would
best describe the circumferences Explain Whole numbers the circumferences would be the whole numbers 25 28 31 and so on
YOUR TURNFocus on Critical Thinking Mathematical PracticesHave students compare and contrast the classification of numbers in the answers in Exercises 5 and 6
ElaborateTalk About ItSummarize the Lesson
Ask What are some ways that number sets can be related Sets may be subsets of other sets or they may be separate from other sets
GUIDED PRACTICEEngage with the Whiteboard
Have students place the numbers in Exercises 1ndashthinsp8 in the Venn diagram for numbers at the beginning of the lesson
Integrating Language Arts EL
Encourage English learners to ask for clarification on any terms or phrases that they do not understand
Avoid Common ErrorsExercise 7 Remind students that a repeating decimal is a rational numberExercises 9ndash10 Remind students that it only takes one counterexample to show that a statement is false
Interactive Whiteboard Interactive example available online
ADDITIONAL EXAMPLE 3Identify the set of numbers that best describes the situation Explain your choice
A the amount of time that has passed since midnight
The set of real numbers time is continuous so the amount of time can be rational or irrational
B the number of tickets sold to a basketball game
The set of whole numbers the number of tickets sold may be 0 or a counting number
myhrwcom
17 Lesson 12
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
1IN
116 inch
Guided Practice
Write all names that apply to each number (Example 1)
1 7 _ 8 2 radic_
36
3 radic_
24 4 075
5 0 6 - radic_ 100
7 5 _
45 8 - 18 __ 6
Tell whether the given statement is true or false Explain your choice (Example 2)
9 All whole numbers are rational numbers
10 No irrational numbers are whole numbers
Identify the set of numbers that best describes each situation Explain your choice (Example 3)
11 the change in the value of an account when given to the nearest dollar
12 the markings on a standard ruler
13 What are some ways to describe the relationships between sets of numbers
CHECK-INESSENTIAL QUESTION
rational real
rational real
True Whole numbers are rational numbers
Rational numbers the ruler is marked every 1 __ 16 th inch
Sample answer Describe one set as being a subset of
another or show their relationships in a Venn diagram
Integers the change can be a whole dollar amount
and can be positive negative or zero
True Whole numbers are a subset of the set of rational numbers
and can be written as a ratio of the whole number to 1
irrational real
whole integer rational real
whole integer rational real
rational real
integer rational real
integer rational real
Unit 118
copy H
ough
ton
Miff
lin H
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pany
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
8_MCAAESE206984_U1M01L2indd 18 41613 136 AM
My Notes
Math TrainerOnline Practice
and Help
Personal
myhrwcom
Math On the Spot
myhrwcom
Identifying Sets for Real-World SituationsReal numbers can be used to represent real-world quantities Highways have posted speed limit signs that are represented by natural numbers such as 55 mph Integers appear on thermometers Rational numbers are used in many daily activities including cooking For example ingredients in a recipe are often given in fractional amounts such as 2 _ 3 cup flour
Identify the set of numbers that best describes each situation Explain your choice
the number of people wearing glasses in a room
The set of whole numbers best describes the situation The number of people wearing glasses may be 0 or a counting number
the circumference of a flying disk has a diameter of 8 9 10 11 or 14 inches
The set of irrational numbers best describes the situation Each circumference would be a product of π and the diameter and any multiple of π is irrational
EXAMPLEXAMPLE 3
A
B
Identify the set of numbers that best describes the situation Explain your choice
5 the amount of water in a glass as it evaporates
6 the weight of a person in pounds
YOUR TURN
8NS1
Rational numbers a personrsquos weight can be a decimal
such as 835 pounds
Real numbers the amount can be any number greater
than 0
17Lesson 12
copy H
ough
ton
Miff
lin H
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ublis
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Com
pany
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
8_MCAAESE206984_U1M01L2indd 17 41613 520 AM
Graphic OrganizersGive students a list of numbers (including terminating and repeating decimals fractions integers and rational and irrational square roots) and a graphic organizer as shown below
Real Numbers
Rational numbers Irrational numbers
Integer numbers
Whole numbers
Ask students to write each number in the list in the correct section of the organizer
Number SensePoint out to students that knowing the types of numbers to expect in different situations can alert them to incorrect math as well as to impossible situations For example 135 shots made in basketballs is not possible but an average number of shots can equal 135
Additional ResourcesDifferentiated Instruction includes bull Reading Strategies bull Success for English Learners EL
bull Reteach bull Challenge PRE-AP
DIFFERENTIATE INSTRUCTION
Sets of Real Numbers 18
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
Math TrainerOnline Assessment
and Intervention
Personal
Online homework assignment available
myhrwcom
Lesson Quiz available online
12 LESSON QUIZ
1 Write all the names that apply to the number
2 Tell whether the given statement is true or false Explain your choice All numbers between 1 and 2 are rational numbers
3 Identify the set of numbers that best describes the situation Explain your choiceThe choices on a survey question change the total points for the survey by -2 -1 0 1 or 2 points
-1 _
5
myhrwcom
Additional ResourcesDifferentiated Instruction includes bull Leveled Practice worksheets
EvaluateGUIDED AND INDEPENDENT PRACTICE
Concepts amp Skills Practice
Example 1Classifying Real Numbers
Exercises 1ndash8 14ndash19 22ndash24
Example 2Understanding Sets and Subsets of Real Numbers
Exercises 9ndash10
Example 3Identifying Sets for Real-World Situations
Exercises 11ndash12 20ndash21 25
Focus | Coherence | Rigor
Exercise Depth of Knowledge (DOK) Mathematical Practices
14ndash19 2 SkillsConcepts MP7 Using Structure
20ndash21 2 SkillsConcepts MP6 Precision
22ndash23 2 SkillsConcepts MP3 Logic
24 1 Recall of Information MP7 Using Structure
25 2 SkillsConcepts MP2 Reasoning
26ndash27 3 Strategic Thinking MP3 Logic
28 3 Strategic Thinking MP8 Patterns
29 3 Strategic Thinking MP3 Logic
8NS1
8NS1
Exercise 29 combines concepts from the California Common Core cluster ldquoKnow that there are numbers that are not rational and approximate them by rational numbersrdquo
Answers1 rational real
2 False radic_
2 is an example of an irrational number between 1 and 2
3 Integers each number is an integer but only three are whole numbers
19 Lesson 12
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
Work Area
π mi23 Critique Reasoning The circumference of a circular region is shown
What type of number best describes the diameter of the circle Explain
your answer
24 Critical Thinking A number is not an integer What type of number can it be
25 A grocery store has a shelf with half-gallon containers of milk What type of number best represents the total number of gallons
26 Explain the Error Katie said ldquoNegative numbers are integersrdquo What was her error
27 Justify Reasoning Can you ever use a calculator to determine if a number is rational or irrational Explain
28 Draw Conclusions The decimal 0 _
3 represents 1 _ 3 What type of number best describes 0
_ 9 which is 3 middot 0
_ 3 Explain
29 Communicate Mathematical Ideas Irrational numbers can never be precisely represented in decimal form Why is this
FOCUS ON HIGHER ORDER THINKING
It can be a rational number that is not an integer or an irrational number
rational number
The set of negative numbers also includes non-integer
rational numbers and irrational numbers
Sample answer If the calculator shows a decimal that
terminates in fewer digits than what the calculator screen
allows then you can tell that the number is rational If not
you cannot tell from the calculator display whether the
number terminates because you see a limited number
of digits It may be a repeating decimal (rational) or
non-terminating non-repeating decimal (irrational)
Whole 3 middot 0 _
3 represents 3 middot 1 _ 3 = 1 so 0 _
9 is exactly 1
Sample answer In decimal form irrational numbers never
terminate and never repeat Therefore no matter how
many decimal places you include the number will never
be precisely represented There are always more digits
Whole the diameter is π _ π = 1 mile
Unit 120
copy H
ough
ton
Miff
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pany
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
8_MCAAESE206984_U1M01L2indd 20 120413 909 PM
Integers
Rational Numbers Irrational Numbers
Real Numbers
Whole Numbers
257
radic16
166
radic9
128 radic50
Personal Math Trainer
Online Practice and Helpmyhrwcom
Name Class Date
Independent Practice
Identify the set of numbers that best describes each situation Explain your choice
20 the height of an airplane as it descends to an airport runway
21 the score with respect to par of several golfers 2 ndash 3 5 0 ndash 1
22 Critique Reasoning Ronald states that the number 1 __ 11 is not rational because when converted into a decimal it does not terminate Nathaniel says it is rational because it is a fraction Which boy is correct Explain
12
14 - radic_
9 15 257
16 radic_
50 17 8 1 _ 2
18 166 19 radic_
16
Write all names that apply to each number Then place the numbers in the correct location on the Venn diagram
8NS1
Real numbers the height can be any number greater than zero
integer rational real whole integer rational real
whole integer rational real
irrational real
rational real
rational real
Integers the scores are counting numbers their
opposites and zero
Nathaniel is correct A rational number is a number that can be written as a fraction and 1 __ 11 is a fraction
19Lesson 12
copy H
ough
ton
Miff
lin H
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Com
pany
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
8_MCAAESE206984_U1M01L2indd 19 41613 136 AM
myhrwcomActivity available onlineEXTEND THE MATH PRE-AP
Activity Have students consider the concept of restricted domain for the sets of numbers that describe situations For example the number of sisters a person has can best be described by whole numbers but no one has ever had 1500 sisters An area code is an integer or whole number between 200 and 999
Have students use a source such as the Guinness Book of World Records and give examples of sets of numbers that describe situations where the domain is restricted Ask whether the restriction may be changed in the future
Sets of Real Numbers 20
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
-3-4-5 -2-1 1 2 3 50 4
12-4 -radic5
Lesson Support Content Objective Students will learn to order a set of real numbers
Language Objective Students will show and describe how to order a set of real numbers
LESSON 13 Ordering Real Numbers
Building BackgroundEliciting Prior Knowledge Have students draw a number line to compare a rational number and an irrational number such as - radic
_ 5 and -4 1 __ 2 Ask them to explain how
they approximated the irrational number on the number line Then have them identify the greater and the lesser real number Repeat with several other pairs of real numbers in different forms
Learning ProgressionsIn this lesson students order a set of real numbers They use rational approximations to compare the sizes of irrational numbers They also order numbers for real-world situations Important understandings for students include the following
bull Compare irrational numbers bull Estimate the value of expressions with irrational numbers bull Order a set of real numbers bull Order real numbers in a real-world context
Work with real numbers continues throughout Grade 8 and into high school This lesson provides students with a foundation for understanding the relative sizes of numbers in different forms in the real number system
Cluster ConnectionsThis lesson provides an excellent opportunity to connect ideas in this cluster Know that there are numbers that are not rational and approximate them by rational numbers Tell students that there is a special number called the golden ratio with applications in mathematics geometry art and architecture The golden ratio is called phi and is represented by the Greek letter ϕ It includes an irrational number in its definition
Have students explain why the golden ratio is irrational Ask them to find the two whole numbers the golden ratio lies between Then challenge them to approximate the golden ratio to the nearest tenth It is irrational because it includes an irrational number in its definition It lies between 1 and 2 To the nearest tenth ϕ = 16
ϕ = 1 + radic_
5 _ 2
Focus | Coherence | Rigor
California Common Core Standards
8NS2 Use rational approximations of irrational numbers to compare the size of irrational numbers locate them approximately on a number line diagram and estimate the value of expressions (eg π 2 )
MP4 Model with mathematics
21A
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
Math Talk
Language Support EL
PROFESSIONAL DEVELOPMENT
Linguistic Support EL
AcademicContent Vocabulary
Post a chart like this to remind students of the regular comparative forms of adjectives that use the -er and -est suffixes Add to the chart for terms that appear in examples and exercises in each lesson Include any irregular verb forms
Background Knowledge
Go On ndash the title of the module review or quiz is Ready to Go On This title uses an idiomatic expression In this context to go on means ldquoto move aheadrdquo or ldquoto proceedrdquo It is different from the use of go on that means having enough facts to use meaningfully as in having enough to go on Also the intonation used in pronouncing an expression can give it different meanings For example when the speaker emphasizes the word on he or she might be expressing disbelief as in ldquoGo ON Yoursquore kidding rightrdquo Discuss with students other ways that the phrase go on may be used
Leveled Strategies for English Learners
Emerging Label points on a number line with the terms used in ordering greater greatest less lesser least Use sentence frames to insert the correct terms
Expanding Have students give two or three complete sentences to compare the placement of numbers on a number line using the correct forms of the comparative and superlative adjectives
Bridging Have students work in pairs with one student giving directions to the other in complete sentences to order numbers on a number line
To help students answer the question posed in Math Talk make sure that students have a command of the forms for making comparisons and the superlative and the concept of opposite order so that the focus is on the math concept instead of the language skills needed to describe and explain order
EL
Adjective Comparative Superlative
Far Farther Farthest
Large Larger Largest
Great Greater Greatest
Some Less Least
Some More Most
California ELD Standards
Emerging 2I8 Analyzing language choices ndash Explain how phrasing or different common words with similar meanings produce different effects on the audience
Expanding 2I8 Analyzing language choices ndash Explain how phrasing or different words with similar meanings or figurative language produce shades of meaning and different effects on the audience
Bridging 2I8 Analyzing language choices ndash Explain how phrasing or different words with similar meanings or figurative language produce shades of meaning nuances and different effects on the audience
Ordering Real Numbers 21B
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
13L E S S O N
Ordering Real Numbers
Interactive Whiteboard Interactive example available online
ADDITIONAL EXAMPLE 1Compare Write lt gt or =
A radic_
8 - 2 4 - radic_
8 lt
B radic_
20 + 1 3 + radic_
2 gt
EngageESSENTIAL QUESTION
How do you order a set of real numbers Sample answer Find their approximate decimal values and order them
Motivate the LessonAsk What kind of numbers are you comparing when you compare the price of gasoline at two different gas stations
ExploreGive students two rational numbers and ask them to name a number between them Repeat a few times and then give them two irrational numbers and ask them to name a number between them
ExplainEXAMPLE 1
Questioning Strategies Mathematical Practices bull Which is greater the difference between 5 and 3 or the difference between radic
_ 5 and radic
_ 3
The difference between 5 and 3 is 2 the difference between radic_
5 and radic_
3 is approximately 1 So the difference between 5 and 3 is greater
Avoid Common ErrorsCaution students to read the problem carefully and think about what the radical sign means so that they do not misread the problem and answer that the two sides are equal
YOUR TURNFocus on TechnologyCalculators should not be used at this point because developing number sense is the goal
EXAMPLE 2Questioning Strategies Mathematical Practices bull How do you determine whether radic
_ 22 is less than or greater than 45 The square of 45 is
2025 which is less than 22 so the square root of 22 must be greater than 45
Engage with the WhiteboardHave students graph and label various real numbers between 42 and 44 and between 47 and 5
YOUR TURNFocus on Modeling Mathematical PracticesHave students label the integers on the number line with their equivalent square root For example 1 2 and 3 on the number line would be labeled radic
_ 1 radic
_ 4 and radic
_ 9
Interactive Whiteboard Interactive example available online
ADDITIONAL EXAMPLE 2Order 3π radic
_ 10 and 325 from greatest
to least
3π 325 radic_
10
myhrwcom
myhrwcom
CA Common CoreStandards
The student is expected to
The Number Systemmdash8NS2
Use rational approximations of irrational numbers to compare the size of irrational numbers locate them approximately on a number line diagram and estimate the value of expressions (eg π 2 )
Mathematical Practices
MP4 Modeling
The student is expected to
21 Lesson 13
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
Math TrainerOnline Practice
and Help
Personal
myhrwcom
Math On the Spotmyhrwcom
0 05 1 15 2 25 3 35 4
radic5radic3
π2
8 85 9 95 10 105 11 115 12
radic75
4 42 44 46 48 5
radic224 12π + 1
Ordering Real Numbers You can compare and order real numbers and list them from least to greatest
Order radic_
22 π + 1 and 4 1 _ 2 from least to greatest
First approximate radic_
22
radic_
22 is between 4 and 5 Since you donrsquot know where it falls between 4 and 5 you need to find a better estimate for radic
_ 22 so
you can compare it to 4 1 _ 2
Since 22 is closer to 25 than 16 use squares of numbers between 45 and 5 to find a better estimate of radic
_ 22
45 2 = 2025 46 2 = 2116 47 2 = 2209 48 2 = 2304
Since 47 2 = 2209 an approximate value for radic_
22 is 47
An approximate value of π is 314 So an approximate value of π +1 is 414
Plot radic_
22 π + 1 and 4 1 _ 2 on a number line
Read the numbers from left to right to place them in order from least to greatest
From least to greatest the numbers are π + 1 4 1 _ 2 and radic_
22
EXAMPLE 2
STEP 1
STEP 2
Order the numbers from least to greatest Then graph them on the number line
YOUR TURN
5 radic_
5 25 radic_
3
6 π 2 10 radic_
75
If real numbers a b and c are in order from least to greatest what is the order
of their opposites from least to greatest
Explain
Math TalkMathematical Practices
8NS2
radic_
3 radic_
5 25
radic_
75 π2 10
Math Talk answer -c -b -a -c is farthest to the left on a number line -b is in the middle and -a is farthest to the right
Unit 122
copy H
ough
ton
Miff
lin H
arco
urt P
ublis
hing
Com
pany
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
8_MCAAESE206984_U1M01L3indd 22 41613 447 AM
My Notes
Math TrainerOnline Practice
and Help
Personal
myhrwcom
Math On the Spot
myhrwcom
Comparing Irrational NumbersBetween any two real numbers is another real number To compare and order real numbers you can approximate irrational numbers as decimals
Compare radic_
3 + 5 3 + radic_
5 Write lt gt or =
First approximate radic_
3
radic_
3 is between 1 and 2
Next approximate radic_
5
radic_
5 is between 2 and 3
Then use your approximations to simplify the expressions
radic_
3 + 5 is between 6 and 7
3 + radic_
5 is between 5 and 6
So radic_
3 + 5 gt 3 + radic_
5
Reflect1 If 7 + radic
_ 5 is equal to radic
_ 5 plus a number what do you know about the
number Why
2 What are the closest two integers that radic_
300 is between
EXAMPLEXAMPLE 1
STEP 1
STEP 2
Compare Write lt gt or =
YOUR TURN
3 radic_
2 + 4 2 + radic_
4 4 radic_
12 + 6 12 + radic_
6
L E S S O N
13 Ordering Real Numbers
ESSENTIAL QUESTIONHow do you order a set of real numbers
8NS2
Use rational approximations of irrational numbers to compare the size of irrational numbers locate them approximately on a number line diagram and estimate the value of expressions (eg π 2 )
8NS2
Use perfect squares to estimate square roots
1 2 = 1 2 2 = 4 3 2 = 9
The number is 7 both expressions must equal 7 + radic_
5
17 and 18
gt lt
21Lesson 13
copy H
ough
ton
Miff
lin H
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urt P
ublis
hing
Com
pany
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
8_MCAAESE206984_U1M01L3indd 21 41913 246 PM
PROFESSIONAL DEVELOPMENT
Math BackgroundIn this lesson students estimate irrational numbers in the form of square roots of nonper-fect squares by finding two perfect squares between which the number falls A more precise method involves repeated division For example to find radic
_ 28 find a whole number whose perfect
square is close to 28 such as 5 Divide 28 by that number 28 divide 5 = 56 Find the average of the quotient and divisor 5 + 56
_____ 2 = 53 Continue dividing 28 by each result and averaging until you get the desired accuracy
Integrate Mathematical Practices MP4
This lesson provides an opportunity to address this Mathematical Practices standard It calls for students to model relationships using multiple representations including diagrams graphs and language as appropriate Students use multiple representations when they use number lines to estimate the locations of and order rational and irrational numbers given as symbols
Ordering Real Numbers 22
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
Interactive Whiteboard Interactive example available online
ADDITIONAL EXAMPLE 3The diameter of a meteorite in millimeters is calculated by four different methods Order the results from least to greatest
Joe radic_
18 mm Lisa 13 __ 3 mm
Pablo 46 mm Julien 4π __ 3 mm
Julien 4π __ 3 mm Lisa 13 __ 3 mm
Joe radic_
18 mm Pablo 46 mm
EXAMPLE 3Questioning Strategies Mathematical Practices bull How can you verify that radic
_ 28 is between 52 and 53 5 2 2 = 2704 and 5 3 2 = 2809
bull Explain how to determine which number is greater 5 _
5 or 55 When the repeating decimal is rounded to the nearest tenth or hundredth you can see that it is greater
Connect to Daily LifeDiscuss how measuring across a canyon might involve different methods than measuring along a road Explain that measurements like these are often done using calculations that approximate the distance
YOUR TURNFocus on Critical Thinking Mathematical PracticesDiscuss with students which number is greater 3
_ 45 or 3450 3
_ 45 or 3455 and why Explain
that 3 _
45 can be written out as 34545hellipMake sure they understand that 3 _
45 is greater than 345 but less than 3455
ElaborateTalk About ItSummarize the Lesson
Ask How can you order two numbers in different forms whose decimal approxi-mations appear to be equal Approximate one or both numbers to an additional
number of decimal places
GUIDED PRACTICEEngage with the Whiteboard
Have students place and label additional points on the number line in Exercise 9 Allow the points to be in any format other than decimal
Avoid Common ErrorsExercises 3ndash4 Caution students to read the problem carefully so that they do not misread the problem as the same numbers combined by addition on each side of the circleExercise 10 Remind students that the calculations have units
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23 Lesson 13
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
0 05 1 15 2 25 3 35 4 45 5 55 6 65 7
2πradic3
Compare Write lt gt or = (Example 1)
1 radic_
3 + 2 radic_
3 + 3 2 radic_
8 + 17 radic_
11 + 15
3 radic_
6 + 5 6 + radic_
5 4 radic_
9 + 3 9 + radic_
3
5 radic_
17 - 3 -2 + radic_
5 6 12 - radic_
2 14 - radic_
8
7 radic_
7 + 2 radic_
10 - 1 8 radic_
17 + 3 3 + radic_
11
9 Order radic_
3 2π and 15 from least to greatest Then graph them on the number line (Example 2)
radic_
3 is between and so radic_
3 asymp
π asymp 314 so 2π asymp
From least to greatest the numbers are
10 Four people have found the perimeter of a forest using different methods Their results are given in the table Order their calculations from greatest to least (Example 3)
11 Explain how to order a set of real numbers
CHECK-INESSENTIAL QUESTION
Forest Perimeter (km)
Leon Mika Jason Ashley
radic_
17 - 2 1 +thinsp π __ 2 12 ___ 5 25
Guided Practice
17
15
1 + π _ 2 km 25 km 12 __ 5 km radic_
17 - 2 km
2π radic
_ 3
18 175
628
Sample answer Convert each number to a decimal
equivalent using estimation to find equivalents for
irrational numbers Graph each number on a number line
Read the numbers from left to right for least to greatest
Read the numbers from right to left for greatest to least
lt gt
lt lt
ltgt
gt gt
24 Unit 1
copy H
ough
ton
Miff
lin H
arco
urt P
ublis
hing
Com
pany
bull Im
age C
redi
ts copy
Elena
Eliss
eeva
Alam
y Im
ages
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
8_MCAAESE206984_U1M01L3indd 24 41613 448 AM
My Notes
5 52 54 56 58 6
radic28 5 12
23455
Math TrainerOnline Practice
and Help
Personal
myhrwcom
Math On the Spot
myhrwcom
Ordering Real Numbers in a Real-World Context Calculations and estimations in the real world may differ It can be important to know not only which are the most accurate but which give the greatest or least values depending upon the context
Four people have found the distance in kilometers across a canyon using different methods Their results are given in the table Order the distances from greatest to least
Distance Across Quarry Canyon (km)
Juana Lee Ann Ryne Jackson
radic_
28 23 __ 4 5 _
5 5 1 _ 2
Write each value as a decimal
radic_
28 is between 52 and 53 Since 53 2 = 2809 an approximate value for radic
_ 28 is 53
23 __ 4 = 575
5 _
5 is 5555hellip so 5 _
5 to the nearest hundredth is 556
5 1 _ 2 = 55
Plot radic_
28 23 __ 4 5 _
5 and 5 1 _ 2 on a number line
From greatest to least the distances are
23 __ 4 km 5 _
5 km 5 1 _ 2 km radic_
28 km
EXAMPLEXAMPLE 3
STEP 1
STEP 2
7 Four people have found the distance in miles across a crater using different methods Their results are given below
Jonathan 10 __ 3 Elaine 3 _
45 Joseacute 3 1 _ 2 Lashonda radic_
10
Order the distances from greatest to least
YOUR TURN
8NS2
3 1 _ 2 mi 3 _
45 mi 10 __ 3 mi radic_
10 mi
23Lesson 13
copy H
ough
ton
Miff
lin H
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ublis
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Com
pany
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8_MCAAESE206984_U1M01L3indd 23 41613 447 AM
ModelingPlace papers around the room with the numbers from 1 to 5 one per sheet Give each student a card showing a number between 1 and 5 in different forms Have students place his or her card between the correct integers and decide where the number goes in relation to any numbers already placed
Multiple RepresentationsGive students a vertical number line which some students might find easier to use than a horizontal one Have them decide whether to place points for rational and irrational numbers above or below existing points
Additional ResourcesDifferentiated Instruction includes bull Reading Strategies bull Success for English Learners EL
bull Reteach bull Challenge PRE-AP
DIFFERENTIATE INSTRUCTION
Ordering Real Numbers 24
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
Math TrainerOnline Assessment
and Intervention
Personal
Online homework assignment available
myhrwcom
myhrwcom
Lesson Quiz available online
13 LESSON QUIZ
1 Compare Write lt gt or =
radic_
95 - 5 radic_
62 - 2
2 Order 105 radic_
105 and 3π + 1 from greatest to least
3 A length in centimeters is calculated differently by four different people Order their calculations from least to greatest
KD 11 __ 2 cm Silvio 5 __ 3 π cm
Paula 5 _
4 cm Luis radic_
33 cm
EvaluateGUIDED AND INDEPENDENT PRACTICE
Concepts amp Skills Practice
Example 1Comparing Irrational Numbers
Exercises 1ndash8
Example 2Ordering Real Numbers
Exercises 9 12ndash15 18ndash21
Example 3Ordering Real Numbers in a Real-World Context
Exercises 10 16ndash17
Additional ResourcesDifferentiated Instruction includes bull Leveled Practice worksheets
Focus | Coherence | Rigor
Exercise Depth of Knowledge (DOK) Mathematical Practices
12ndash15 1 Recall of Information MP5 Using Tools
16 2 SkillsConcepts MP2 Reasoning
17 2 SkillsConcepts MP6 Precision
18ndash21 2 SkillsConcepts MP2 Reasoning
22 3 Strategic Thinking MP4 Modeling
23ndash24 3 Strategic Thinking MP3 Logic
8NS2
8NS2
Answers1 radic
_ 95 - 5 lt radic
_ 62 - 2
2 radic_
105 3π + 1 105
3 Silvio 5 __ 3 π cm Paula 5 _
4 cm
KD 11
__ 2 cm Luis radic_
33 cm
25 Lesson 13
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
Work Area
3140 3141 3142 3143
314 π227
20 A teacher asks his students to write the numbers shown in order from least to greatest Paul thinks the numbers are already in order Sandra thinks the order should be reversed Who is right
21 Math History There is a famous irrational number called Eulerrsquos number symbolized with an e Like π its decimal form never ends or repeats The first few digits of e are 27182818284
a Between which two square roots of integers could you find this number
b Between which two square roots of integers can you find π
22 Analyze Relationships There are several approximations used for π including 314 and 22 __ 7 π is approximately 314159265358979
a Label π and the two approximations on the number line
b Which of the two approximations is a better estimate for π Explain
c Find a whole number x so that the ratio x ___ 113 is a better estimate for π
than the two given approximations
23 Communicate Mathematical Ideas If a set of six numbers that include both rational and irrational numbers is graphed on a number line what is the fewest number of distinct points that need to be graphed Explain
24 Critique Reasoning Jill says that 12 _
6 is less than 1263 Explain her error
FOCUS ON HIGHER ORDER THINKING
radic_
115 115 ___ 11 and 105624
between radic_
7 asymp 265 and radic_
8 asymp 283
between radic_
9 = 3 and radic_
10 asymp 316
22 __ 7 it is closer to π on the number line
She did not consider the repeating digit 1266
2 rational numbers can have the same location and
irrational numbers can have the same location but they
cannot share a location
355
Neither student is correct The answer
should be 115 ___ 11 105624 radic_
115
Unit 126
copy H
ough
ton M
ifflin
Har
cour
t Pub
lishin
g Com
pany
Imag
e Cre
dits
copy3D
Stoc
kiSt
ockP
hoto
com
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
8_MCAAESE206984_U1M01L3indd 26 210513 801 AM
Personal Math Trainer
Online Practice and Helpmyhrwcom
Name Class Date
Independent Practice
16 Your sister is considering two different shapes for her garden One is a square with side lengths of 35 meters and the other is a circle with a diameter of 4 meters
a Find the area of the square
b Find the area of the circle
c Compare your answers from parts a and b Which garden would give your sister the most space to plant
17 Winnie measured the length of her fatherrsquos ranch four times and got four different distances Her measurements are shown in the table
a To estimate the actual length Winnie first approximated each distance to the nearest hundredth Then she averaged the four numbers Using a calculator find Winniersquos estimate
b Winniersquos father estimated the distance across his ranch to be radic_
56 km How does this distance compare to Winniersquos estimate
Give an example of each type of number
18 a real number between radic_
13 and radic_
14
19 an irrational number between 5 and 7
Order the numbers from least to greatest
12 radic_
7 2 radic_
8 ___ 2 13 radic_
10 π 35
14 radic_
220 -10 radic_
100 115 15 radic_
8 -375 3 9 _ 4
Distance Across Fatherrsquos Ranch (km)
1 2 3 4
radic_
60 58 __ 8 7 _
3 7 3 _ 5
138NS2
radic_
8 ___ 2 2 radic_
7
-10 radic_
100 115 radic_
220
radic_
60 asymp 775 58 __ 8 = 725 7 _
3 asymp 733 7 3 _ 5 = 760 so the average
π radic_
10 35
-375 9 _ 4 radic_
8 3
is 74825 km
1225 m2
4π m2 or approximately 126 m2
They are nearly identical radic_
56 is approximately 74833hellip
The circle would give her more space to plant because it has a
larger area
Sample answer 37
Sample answer radic_
31
25Lesson 13
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ough
ton
Miff
lin H
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pany
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8_MCAAESE206984_U1M01L3indd 25 41613 448 AM
Activity available online myhrwcomEXTEND THE MATH PRE-AP
Activity Have students investigate whether there are infinitely many numbers between two numbers by giving examples for each of the following
bull Between any two rational numbers there is at least one other rational number Sample answer 45 is between 41 and 48
bull Between any two irrational numbers there is at least one rational number Sample answer 45 is between radic
_ 11 and radic
_ 29
bull Between any two rational numbers there is at least one irrational number Sample answer radic
_ 11 is between 31 and 36
bull Between any two irrational numbers there is at least one irrational number Sample answer radic
_ 17 is between radic
_ 11 and radic
_ 29
Ordering Real Numbers 26
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
ReadyMath Trainer
Online Practiceand Help
Personal
myhrwcom
Module Quiz
11ensp RationalenspandenspIrrationalenspNumbersWrite each fraction as a decimal or each decimal as a fraction
1 7__20 2 1___
27 3 17_8
Solve each equation for x
4 x2=81 5 x3=343 6 x2= 1___100
7 Asquarepatiohasanareaof200squarefeetHowlongiseachside
ofthepatiotothenearesttenth
12ensp SetsenspofenspRealenspNumbersWrite all names that apply to each number
8 121____radic
____121
9 π__2
10 TellwhetherthestatementldquoAllintegersarerationalnumbersrdquoistrueorfalseExplainyourchoice
13ensp OrderingenspRealenspNumbersCompare Write lt gt or =
11 radic__
8+3 8+radic__
3 12 radic__
5+11emsp emsp emsp 5+radic___
11
Order the numbers from least to greatest
13 radic___
99π29__
8 14 radic___
1__251_40__
2
15 Howarerealnumbersusedtodescribereal-worldsituations
ESSENTIAL QUESTION
035
9-9
141ft
7 1__10- 1__10
14__11 1875
wholeintegerrationalreal
Trueintegerscanbewrittenasthequotientoftwointegers
SampleanswerRealnumberssuchastherational
π29__
8radic___
99
irrationalreal
lt gt
number1_4candescribeamountsusedincooking
radic___
1__250__
21_4
27Module1
copy H
ough
ton
Miff
lin H
arco
urt P
ublis
hing
Com
pany
DONOTEDIT--ChangesmustbemadethroughldquoFileinfordquoCorrectionKey=A
8_MCAAESE206984_U1M01RTindd 27 41513 1113 PM
Math TrainerOnline Assessment
and Intervention
Personal
myhrwcom
1
2
3 Response toIntervention
Intervention Enrichment
Access Ready to Go On assessment online and receive instant scoring feedback and customized intervention or enrichment
Online and Print Resources
Differentiated Instruction
bull Reteach worksheets
bull Reading Strategies EL
bull Success for English Learners EL
Differentiated Instruction
bull Challenge worksheets PRE-AP
Extend the Math PRE-AP
Lesson Activities in TE
Additional ResourcesAssessment Resources includes bull Leveled Module Quizzes
Ready to Go OnAssess MasteryUse the assessment on this page to determine if students have mastered the concepts and standards covered in this module
California Common Core Standards
Lesson Exercises Common Core Standards
11 1ndash7 8NS1 8NS2 8EE2
12 8ndash10 8NS1
13 11ndash14 8NS2
27 Unit 1 Module 1
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
Personal Math Trainer
Online Practice and HelpmyhrwcomAssessment Readiness
Module 1 MIXed ReVIeW
1 Look at each number Is the number between 2π and radic___
52
Select Yes or No for expressions AndashC
A 6 2 _ 3 Yes No
B 5π __ 2 Yes No
C 3 radic__
5 Yes No
2 Consider the number - 11 __ 15
Choose True or False for each statement
A The number is rational True False
B The number can be written as True Falsea repeating decimal
C The number is less than ndash08 True False
3 The volume of a cube is given by V = x3 where x is the length of an edge of the cube A cube-shaped end table has a volume of 3 3 _ 8 cubic feet What is the length of an edge of the end table Explain how you solved this problem
4 A student says that radic___
83 is greater than 29 __ 3 Is the student correct Justify your
reasoning
1 1 _ 2 ft Sample answer The equation x3 = 3 3 _ 8 can be used
to find the edge length in feet To solve the equation
write the mixed number as a fraction greater than 1
x3 = 27 __ 8 Then take the cube root of both sides x = 3 _ 2 = 1 1 _ 2
No Sample answer radic___
83 asymp 91 and 29 __ 3 = 9
__ 6
Because 91 lt 9 __
6 radic___
83 lt 29 __ 3
28 Unit 1
copy H
ough
ton
Miff
lin H
arco
urt P
ublis
hing
Com
pany
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=A
8_MCAAESE206984_U1M01RTindd 28 240413 946 AM
Personal Math Trainer
Online Assessment and
Interventionmyhrwcom
Scoring GuideItem 3 Award the student 1 point for finding the edge length of the cube and 1 point for correctly explaining how to use a cube root to solve the problem
Item 4 Award the student 1 point for determining that the student is incorrect and 1 point for correctly justifying the reasoning for this conclusion
Additional ResourcesTo assign this assessment online login to your Assignment Manager at myhrwcom
Assessment Readiness
California Common Core Standards
Items Grade 8 Standards Mathematical Practices
1 8NS2 MP7
2 7NS2b 7NS2d 8NS1 MP7
3 8EE2 MP1 MP4
4 8NS1 8NS2 MP3
Item integrates mixed review concepts from previous modules or a previous course
Item 4 combines concepts from the California Common Core cluster ldquoKnow that there are numbers that are not rational and approximate them by rational numbersrdquo
Real Numbers 28
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
1IN
116 inch
Guided Practice
Write all names that apply to each number (Example 1)
1 7 _ 8 2 radic_
36
3 radic_
24 4 075
5 0 6 - radic_ 100
7 5 _
45 8 - 18 __ 6
Tell whether the given statement is true or false Explain your choice (Example 2)
9 All whole numbers are rational numbers
10 No irrational numbers are whole numbers
Identify the set of numbers that best describes each situation Explain your choice (Example 3)
11 the change in the value of an account when given to the nearest dollar
12 the markings on a standard ruler
13 What are some ways to describe the relationships between sets of numbers
CHECK-INESSENTIAL QUESTION
rational real
rational real
True Whole numbers are rational numbers
Rational numbers the ruler is marked every 1 __ 16 th inch
Sample answer Describe one set as being a subset of
another or show their relationships in a Venn diagram
Integers the change can be a whole dollar amount
and can be positive negative or zero
True Whole numbers are a subset of the set of rational numbers
and can be written as a ratio of the whole number to 1
irrational real
whole integer rational real
whole integer rational real
rational real
integer rational real
integer rational real
Unit 118
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pany
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
8_MCAAESE206984_U1M01L2indd 18 41613 136 AM
My Notes
Math TrainerOnline Practice
and Help
Personal
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Math On the Spot
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Identifying Sets for Real-World SituationsReal numbers can be used to represent real-world quantities Highways have posted speed limit signs that are represented by natural numbers such as 55 mph Integers appear on thermometers Rational numbers are used in many daily activities including cooking For example ingredients in a recipe are often given in fractional amounts such as 2 _ 3 cup flour
Identify the set of numbers that best describes each situation Explain your choice
the number of people wearing glasses in a room
The set of whole numbers best describes the situation The number of people wearing glasses may be 0 or a counting number
the circumference of a flying disk has a diameter of 8 9 10 11 or 14 inches
The set of irrational numbers best describes the situation Each circumference would be a product of π and the diameter and any multiple of π is irrational
EXAMPLEXAMPLE 3
A
B
Identify the set of numbers that best describes the situation Explain your choice
5 the amount of water in a glass as it evaporates
6 the weight of a person in pounds
YOUR TURN
8NS1
Rational numbers a personrsquos weight can be a decimal
such as 835 pounds
Real numbers the amount can be any number greater
than 0
17Lesson 12
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ough
ton
Miff
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Com
pany
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
8_MCAAESE206984_U1M01L2indd 17 41613 520 AM
Graphic OrganizersGive students a list of numbers (including terminating and repeating decimals fractions integers and rational and irrational square roots) and a graphic organizer as shown below
Real Numbers
Rational numbers Irrational numbers
Integer numbers
Whole numbers
Ask students to write each number in the list in the correct section of the organizer
Number SensePoint out to students that knowing the types of numbers to expect in different situations can alert them to incorrect math as well as to impossible situations For example 135 shots made in basketballs is not possible but an average number of shots can equal 135
Additional ResourcesDifferentiated Instruction includes bull Reading Strategies bull Success for English Learners EL
bull Reteach bull Challenge PRE-AP
DIFFERENTIATE INSTRUCTION
Sets of Real Numbers 18
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
Math TrainerOnline Assessment
and Intervention
Personal
Online homework assignment available
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Lesson Quiz available online
12 LESSON QUIZ
1 Write all the names that apply to the number
2 Tell whether the given statement is true or false Explain your choice All numbers between 1 and 2 are rational numbers
3 Identify the set of numbers that best describes the situation Explain your choiceThe choices on a survey question change the total points for the survey by -2 -1 0 1 or 2 points
-1 _
5
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Additional ResourcesDifferentiated Instruction includes bull Leveled Practice worksheets
EvaluateGUIDED AND INDEPENDENT PRACTICE
Concepts amp Skills Practice
Example 1Classifying Real Numbers
Exercises 1ndash8 14ndash19 22ndash24
Example 2Understanding Sets and Subsets of Real Numbers
Exercises 9ndash10
Example 3Identifying Sets for Real-World Situations
Exercises 11ndash12 20ndash21 25
Focus | Coherence | Rigor
Exercise Depth of Knowledge (DOK) Mathematical Practices
14ndash19 2 SkillsConcepts MP7 Using Structure
20ndash21 2 SkillsConcepts MP6 Precision
22ndash23 2 SkillsConcepts MP3 Logic
24 1 Recall of Information MP7 Using Structure
25 2 SkillsConcepts MP2 Reasoning
26ndash27 3 Strategic Thinking MP3 Logic
28 3 Strategic Thinking MP8 Patterns
29 3 Strategic Thinking MP3 Logic
8NS1
8NS1
Exercise 29 combines concepts from the California Common Core cluster ldquoKnow that there are numbers that are not rational and approximate them by rational numbersrdquo
Answers1 rational real
2 False radic_
2 is an example of an irrational number between 1 and 2
3 Integers each number is an integer but only three are whole numbers
19 Lesson 12
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
Work Area
π mi23 Critique Reasoning The circumference of a circular region is shown
What type of number best describes the diameter of the circle Explain
your answer
24 Critical Thinking A number is not an integer What type of number can it be
25 A grocery store has a shelf with half-gallon containers of milk What type of number best represents the total number of gallons
26 Explain the Error Katie said ldquoNegative numbers are integersrdquo What was her error
27 Justify Reasoning Can you ever use a calculator to determine if a number is rational or irrational Explain
28 Draw Conclusions The decimal 0 _
3 represents 1 _ 3 What type of number best describes 0
_ 9 which is 3 middot 0
_ 3 Explain
29 Communicate Mathematical Ideas Irrational numbers can never be precisely represented in decimal form Why is this
FOCUS ON HIGHER ORDER THINKING
It can be a rational number that is not an integer or an irrational number
rational number
The set of negative numbers also includes non-integer
rational numbers and irrational numbers
Sample answer If the calculator shows a decimal that
terminates in fewer digits than what the calculator screen
allows then you can tell that the number is rational If not
you cannot tell from the calculator display whether the
number terminates because you see a limited number
of digits It may be a repeating decimal (rational) or
non-terminating non-repeating decimal (irrational)
Whole 3 middot 0 _
3 represents 3 middot 1 _ 3 = 1 so 0 _
9 is exactly 1
Sample answer In decimal form irrational numbers never
terminate and never repeat Therefore no matter how
many decimal places you include the number will never
be precisely represented There are always more digits
Whole the diameter is π _ π = 1 mile
Unit 120
copy H
ough
ton
Miff
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Com
pany
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
8_MCAAESE206984_U1M01L2indd 20 120413 909 PM
Integers
Rational Numbers Irrational Numbers
Real Numbers
Whole Numbers
257
radic16
166
radic9
128 radic50
Personal Math Trainer
Online Practice and Helpmyhrwcom
Name Class Date
Independent Practice
Identify the set of numbers that best describes each situation Explain your choice
20 the height of an airplane as it descends to an airport runway
21 the score with respect to par of several golfers 2 ndash 3 5 0 ndash 1
22 Critique Reasoning Ronald states that the number 1 __ 11 is not rational because when converted into a decimal it does not terminate Nathaniel says it is rational because it is a fraction Which boy is correct Explain
12
14 - radic_
9 15 257
16 radic_
50 17 8 1 _ 2
18 166 19 radic_
16
Write all names that apply to each number Then place the numbers in the correct location on the Venn diagram
8NS1
Real numbers the height can be any number greater than zero
integer rational real whole integer rational real
whole integer rational real
irrational real
rational real
rational real
Integers the scores are counting numbers their
opposites and zero
Nathaniel is correct A rational number is a number that can be written as a fraction and 1 __ 11 is a fraction
19Lesson 12
copy H
ough
ton
Miff
lin H
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pany
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8_MCAAESE206984_U1M01L2indd 19 41613 136 AM
myhrwcomActivity available onlineEXTEND THE MATH PRE-AP
Activity Have students consider the concept of restricted domain for the sets of numbers that describe situations For example the number of sisters a person has can best be described by whole numbers but no one has ever had 1500 sisters An area code is an integer or whole number between 200 and 999
Have students use a source such as the Guinness Book of World Records and give examples of sets of numbers that describe situations where the domain is restricted Ask whether the restriction may be changed in the future
Sets of Real Numbers 20
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
-3-4-5 -2-1 1 2 3 50 4
12-4 -radic5
Lesson Support Content Objective Students will learn to order a set of real numbers
Language Objective Students will show and describe how to order a set of real numbers
LESSON 13 Ordering Real Numbers
Building BackgroundEliciting Prior Knowledge Have students draw a number line to compare a rational number and an irrational number such as - radic
_ 5 and -4 1 __ 2 Ask them to explain how
they approximated the irrational number on the number line Then have them identify the greater and the lesser real number Repeat with several other pairs of real numbers in different forms
Learning ProgressionsIn this lesson students order a set of real numbers They use rational approximations to compare the sizes of irrational numbers They also order numbers for real-world situations Important understandings for students include the following
bull Compare irrational numbers bull Estimate the value of expressions with irrational numbers bull Order a set of real numbers bull Order real numbers in a real-world context
Work with real numbers continues throughout Grade 8 and into high school This lesson provides students with a foundation for understanding the relative sizes of numbers in different forms in the real number system
Cluster ConnectionsThis lesson provides an excellent opportunity to connect ideas in this cluster Know that there are numbers that are not rational and approximate them by rational numbers Tell students that there is a special number called the golden ratio with applications in mathematics geometry art and architecture The golden ratio is called phi and is represented by the Greek letter ϕ It includes an irrational number in its definition
Have students explain why the golden ratio is irrational Ask them to find the two whole numbers the golden ratio lies between Then challenge them to approximate the golden ratio to the nearest tenth It is irrational because it includes an irrational number in its definition It lies between 1 and 2 To the nearest tenth ϕ = 16
ϕ = 1 + radic_
5 _ 2
Focus | Coherence | Rigor
California Common Core Standards
8NS2 Use rational approximations of irrational numbers to compare the size of irrational numbers locate them approximately on a number line diagram and estimate the value of expressions (eg π 2 )
MP4 Model with mathematics
21A
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
Math Talk
Language Support EL
PROFESSIONAL DEVELOPMENT
Linguistic Support EL
AcademicContent Vocabulary
Post a chart like this to remind students of the regular comparative forms of adjectives that use the -er and -est suffixes Add to the chart for terms that appear in examples and exercises in each lesson Include any irregular verb forms
Background Knowledge
Go On ndash the title of the module review or quiz is Ready to Go On This title uses an idiomatic expression In this context to go on means ldquoto move aheadrdquo or ldquoto proceedrdquo It is different from the use of go on that means having enough facts to use meaningfully as in having enough to go on Also the intonation used in pronouncing an expression can give it different meanings For example when the speaker emphasizes the word on he or she might be expressing disbelief as in ldquoGo ON Yoursquore kidding rightrdquo Discuss with students other ways that the phrase go on may be used
Leveled Strategies for English Learners
Emerging Label points on a number line with the terms used in ordering greater greatest less lesser least Use sentence frames to insert the correct terms
Expanding Have students give two or three complete sentences to compare the placement of numbers on a number line using the correct forms of the comparative and superlative adjectives
Bridging Have students work in pairs with one student giving directions to the other in complete sentences to order numbers on a number line
To help students answer the question posed in Math Talk make sure that students have a command of the forms for making comparisons and the superlative and the concept of opposite order so that the focus is on the math concept instead of the language skills needed to describe and explain order
EL
Adjective Comparative Superlative
Far Farther Farthest
Large Larger Largest
Great Greater Greatest
Some Less Least
Some More Most
California ELD Standards
Emerging 2I8 Analyzing language choices ndash Explain how phrasing or different common words with similar meanings produce different effects on the audience
Expanding 2I8 Analyzing language choices ndash Explain how phrasing or different words with similar meanings or figurative language produce shades of meaning and different effects on the audience
Bridging 2I8 Analyzing language choices ndash Explain how phrasing or different words with similar meanings or figurative language produce shades of meaning nuances and different effects on the audience
Ordering Real Numbers 21B
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
13L E S S O N
Ordering Real Numbers
Interactive Whiteboard Interactive example available online
ADDITIONAL EXAMPLE 1Compare Write lt gt or =
A radic_
8 - 2 4 - radic_
8 lt
B radic_
20 + 1 3 + radic_
2 gt
EngageESSENTIAL QUESTION
How do you order a set of real numbers Sample answer Find their approximate decimal values and order them
Motivate the LessonAsk What kind of numbers are you comparing when you compare the price of gasoline at two different gas stations
ExploreGive students two rational numbers and ask them to name a number between them Repeat a few times and then give them two irrational numbers and ask them to name a number between them
ExplainEXAMPLE 1
Questioning Strategies Mathematical Practices bull Which is greater the difference between 5 and 3 or the difference between radic
_ 5 and radic
_ 3
The difference between 5 and 3 is 2 the difference between radic_
5 and radic_
3 is approximately 1 So the difference between 5 and 3 is greater
Avoid Common ErrorsCaution students to read the problem carefully and think about what the radical sign means so that they do not misread the problem and answer that the two sides are equal
YOUR TURNFocus on TechnologyCalculators should not be used at this point because developing number sense is the goal
EXAMPLE 2Questioning Strategies Mathematical Practices bull How do you determine whether radic
_ 22 is less than or greater than 45 The square of 45 is
2025 which is less than 22 so the square root of 22 must be greater than 45
Engage with the WhiteboardHave students graph and label various real numbers between 42 and 44 and between 47 and 5
YOUR TURNFocus on Modeling Mathematical PracticesHave students label the integers on the number line with their equivalent square root For example 1 2 and 3 on the number line would be labeled radic
_ 1 radic
_ 4 and radic
_ 9
Interactive Whiteboard Interactive example available online
ADDITIONAL EXAMPLE 2Order 3π radic
_ 10 and 325 from greatest
to least
3π 325 radic_
10
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CA Common CoreStandards
The student is expected to
The Number Systemmdash8NS2
Use rational approximations of irrational numbers to compare the size of irrational numbers locate them approximately on a number line diagram and estimate the value of expressions (eg π 2 )
Mathematical Practices
MP4 Modeling
The student is expected to
21 Lesson 13
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
Math TrainerOnline Practice
and Help
Personal
myhrwcom
Math On the Spotmyhrwcom
0 05 1 15 2 25 3 35 4
radic5radic3
π2
8 85 9 95 10 105 11 115 12
radic75
4 42 44 46 48 5
radic224 12π + 1
Ordering Real Numbers You can compare and order real numbers and list them from least to greatest
Order radic_
22 π + 1 and 4 1 _ 2 from least to greatest
First approximate radic_
22
radic_
22 is between 4 and 5 Since you donrsquot know where it falls between 4 and 5 you need to find a better estimate for radic
_ 22 so
you can compare it to 4 1 _ 2
Since 22 is closer to 25 than 16 use squares of numbers between 45 and 5 to find a better estimate of radic
_ 22
45 2 = 2025 46 2 = 2116 47 2 = 2209 48 2 = 2304
Since 47 2 = 2209 an approximate value for radic_
22 is 47
An approximate value of π is 314 So an approximate value of π +1 is 414
Plot radic_
22 π + 1 and 4 1 _ 2 on a number line
Read the numbers from left to right to place them in order from least to greatest
From least to greatest the numbers are π + 1 4 1 _ 2 and radic_
22
EXAMPLE 2
STEP 1
STEP 2
Order the numbers from least to greatest Then graph them on the number line
YOUR TURN
5 radic_
5 25 radic_
3
6 π 2 10 radic_
75
If real numbers a b and c are in order from least to greatest what is the order
of their opposites from least to greatest
Explain
Math TalkMathematical Practices
8NS2
radic_
3 radic_
5 25
radic_
75 π2 10
Math Talk answer -c -b -a -c is farthest to the left on a number line -b is in the middle and -a is farthest to the right
Unit 122
copy H
ough
ton
Miff
lin H
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pany
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
8_MCAAESE206984_U1M01L3indd 22 41613 447 AM
My Notes
Math TrainerOnline Practice
and Help
Personal
myhrwcom
Math On the Spot
myhrwcom
Comparing Irrational NumbersBetween any two real numbers is another real number To compare and order real numbers you can approximate irrational numbers as decimals
Compare radic_
3 + 5 3 + radic_
5 Write lt gt or =
First approximate radic_
3
radic_
3 is between 1 and 2
Next approximate radic_
5
radic_
5 is between 2 and 3
Then use your approximations to simplify the expressions
radic_
3 + 5 is between 6 and 7
3 + radic_
5 is between 5 and 6
So radic_
3 + 5 gt 3 + radic_
5
Reflect1 If 7 + radic
_ 5 is equal to radic
_ 5 plus a number what do you know about the
number Why
2 What are the closest two integers that radic_
300 is between
EXAMPLEXAMPLE 1
STEP 1
STEP 2
Compare Write lt gt or =
YOUR TURN
3 radic_
2 + 4 2 + radic_
4 4 radic_
12 + 6 12 + radic_
6
L E S S O N
13 Ordering Real Numbers
ESSENTIAL QUESTIONHow do you order a set of real numbers
8NS2
Use rational approximations of irrational numbers to compare the size of irrational numbers locate them approximately on a number line diagram and estimate the value of expressions (eg π 2 )
8NS2
Use perfect squares to estimate square roots
1 2 = 1 2 2 = 4 3 2 = 9
The number is 7 both expressions must equal 7 + radic_
5
17 and 18
gt lt
21Lesson 13
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ough
ton
Miff
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pany
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8_MCAAESE206984_U1M01L3indd 21 41913 246 PM
PROFESSIONAL DEVELOPMENT
Math BackgroundIn this lesson students estimate irrational numbers in the form of square roots of nonper-fect squares by finding two perfect squares between which the number falls A more precise method involves repeated division For example to find radic
_ 28 find a whole number whose perfect
square is close to 28 such as 5 Divide 28 by that number 28 divide 5 = 56 Find the average of the quotient and divisor 5 + 56
_____ 2 = 53 Continue dividing 28 by each result and averaging until you get the desired accuracy
Integrate Mathematical Practices MP4
This lesson provides an opportunity to address this Mathematical Practices standard It calls for students to model relationships using multiple representations including diagrams graphs and language as appropriate Students use multiple representations when they use number lines to estimate the locations of and order rational and irrational numbers given as symbols
Ordering Real Numbers 22
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
Interactive Whiteboard Interactive example available online
ADDITIONAL EXAMPLE 3The diameter of a meteorite in millimeters is calculated by four different methods Order the results from least to greatest
Joe radic_
18 mm Lisa 13 __ 3 mm
Pablo 46 mm Julien 4π __ 3 mm
Julien 4π __ 3 mm Lisa 13 __ 3 mm
Joe radic_
18 mm Pablo 46 mm
EXAMPLE 3Questioning Strategies Mathematical Practices bull How can you verify that radic
_ 28 is between 52 and 53 5 2 2 = 2704 and 5 3 2 = 2809
bull Explain how to determine which number is greater 5 _
5 or 55 When the repeating decimal is rounded to the nearest tenth or hundredth you can see that it is greater
Connect to Daily LifeDiscuss how measuring across a canyon might involve different methods than measuring along a road Explain that measurements like these are often done using calculations that approximate the distance
YOUR TURNFocus on Critical Thinking Mathematical PracticesDiscuss with students which number is greater 3
_ 45 or 3450 3
_ 45 or 3455 and why Explain
that 3 _
45 can be written out as 34545hellipMake sure they understand that 3 _
45 is greater than 345 but less than 3455
ElaborateTalk About ItSummarize the Lesson
Ask How can you order two numbers in different forms whose decimal approxi-mations appear to be equal Approximate one or both numbers to an additional
number of decimal places
GUIDED PRACTICEEngage with the Whiteboard
Have students place and label additional points on the number line in Exercise 9 Allow the points to be in any format other than decimal
Avoid Common ErrorsExercises 3ndash4 Caution students to read the problem carefully so that they do not misread the problem as the same numbers combined by addition on each side of the circleExercise 10 Remind students that the calculations have units
myhrwcom
23 Lesson 13
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
0 05 1 15 2 25 3 35 4 45 5 55 6 65 7
2πradic3
Compare Write lt gt or = (Example 1)
1 radic_
3 + 2 radic_
3 + 3 2 radic_
8 + 17 radic_
11 + 15
3 radic_
6 + 5 6 + radic_
5 4 radic_
9 + 3 9 + radic_
3
5 radic_
17 - 3 -2 + radic_
5 6 12 - radic_
2 14 - radic_
8
7 radic_
7 + 2 radic_
10 - 1 8 radic_
17 + 3 3 + radic_
11
9 Order radic_
3 2π and 15 from least to greatest Then graph them on the number line (Example 2)
radic_
3 is between and so radic_
3 asymp
π asymp 314 so 2π asymp
From least to greatest the numbers are
10 Four people have found the perimeter of a forest using different methods Their results are given in the table Order their calculations from greatest to least (Example 3)
11 Explain how to order a set of real numbers
CHECK-INESSENTIAL QUESTION
Forest Perimeter (km)
Leon Mika Jason Ashley
radic_
17 - 2 1 +thinsp π __ 2 12 ___ 5 25
Guided Practice
17
15
1 + π _ 2 km 25 km 12 __ 5 km radic_
17 - 2 km
2π radic
_ 3
18 175
628
Sample answer Convert each number to a decimal
equivalent using estimation to find equivalents for
irrational numbers Graph each number on a number line
Read the numbers from left to right for least to greatest
Read the numbers from right to left for greatest to least
lt gt
lt lt
ltgt
gt gt
24 Unit 1
copy H
ough
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Miff
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ublis
hing
Com
pany
bull Im
age C
redi
ts copy
Elena
Eliss
eeva
Alam
y Im
ages
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
8_MCAAESE206984_U1M01L3indd 24 41613 448 AM
My Notes
5 52 54 56 58 6
radic28 5 12
23455
Math TrainerOnline Practice
and Help
Personal
myhrwcom
Math On the Spot
myhrwcom
Ordering Real Numbers in a Real-World Context Calculations and estimations in the real world may differ It can be important to know not only which are the most accurate but which give the greatest or least values depending upon the context
Four people have found the distance in kilometers across a canyon using different methods Their results are given in the table Order the distances from greatest to least
Distance Across Quarry Canyon (km)
Juana Lee Ann Ryne Jackson
radic_
28 23 __ 4 5 _
5 5 1 _ 2
Write each value as a decimal
radic_
28 is between 52 and 53 Since 53 2 = 2809 an approximate value for radic
_ 28 is 53
23 __ 4 = 575
5 _
5 is 5555hellip so 5 _
5 to the nearest hundredth is 556
5 1 _ 2 = 55
Plot radic_
28 23 __ 4 5 _
5 and 5 1 _ 2 on a number line
From greatest to least the distances are
23 __ 4 km 5 _
5 km 5 1 _ 2 km radic_
28 km
EXAMPLEXAMPLE 3
STEP 1
STEP 2
7 Four people have found the distance in miles across a crater using different methods Their results are given below
Jonathan 10 __ 3 Elaine 3 _
45 Joseacute 3 1 _ 2 Lashonda radic_
10
Order the distances from greatest to least
YOUR TURN
8NS2
3 1 _ 2 mi 3 _
45 mi 10 __ 3 mi radic_
10 mi
23Lesson 13
copy H
ough
ton
Miff
lin H
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ublis
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Com
pany
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
8_MCAAESE206984_U1M01L3indd 23 41613 447 AM
ModelingPlace papers around the room with the numbers from 1 to 5 one per sheet Give each student a card showing a number between 1 and 5 in different forms Have students place his or her card between the correct integers and decide where the number goes in relation to any numbers already placed
Multiple RepresentationsGive students a vertical number line which some students might find easier to use than a horizontal one Have them decide whether to place points for rational and irrational numbers above or below existing points
Additional ResourcesDifferentiated Instruction includes bull Reading Strategies bull Success for English Learners EL
bull Reteach bull Challenge PRE-AP
DIFFERENTIATE INSTRUCTION
Ordering Real Numbers 24
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
Math TrainerOnline Assessment
and Intervention
Personal
Online homework assignment available
myhrwcom
myhrwcom
Lesson Quiz available online
13 LESSON QUIZ
1 Compare Write lt gt or =
radic_
95 - 5 radic_
62 - 2
2 Order 105 radic_
105 and 3π + 1 from greatest to least
3 A length in centimeters is calculated differently by four different people Order their calculations from least to greatest
KD 11 __ 2 cm Silvio 5 __ 3 π cm
Paula 5 _
4 cm Luis radic_
33 cm
EvaluateGUIDED AND INDEPENDENT PRACTICE
Concepts amp Skills Practice
Example 1Comparing Irrational Numbers
Exercises 1ndash8
Example 2Ordering Real Numbers
Exercises 9 12ndash15 18ndash21
Example 3Ordering Real Numbers in a Real-World Context
Exercises 10 16ndash17
Additional ResourcesDifferentiated Instruction includes bull Leveled Practice worksheets
Focus | Coherence | Rigor
Exercise Depth of Knowledge (DOK) Mathematical Practices
12ndash15 1 Recall of Information MP5 Using Tools
16 2 SkillsConcepts MP2 Reasoning
17 2 SkillsConcepts MP6 Precision
18ndash21 2 SkillsConcepts MP2 Reasoning
22 3 Strategic Thinking MP4 Modeling
23ndash24 3 Strategic Thinking MP3 Logic
8NS2
8NS2
Answers1 radic
_ 95 - 5 lt radic
_ 62 - 2
2 radic_
105 3π + 1 105
3 Silvio 5 __ 3 π cm Paula 5 _
4 cm
KD 11
__ 2 cm Luis radic_
33 cm
25 Lesson 13
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
Work Area
3140 3141 3142 3143
314 π227
20 A teacher asks his students to write the numbers shown in order from least to greatest Paul thinks the numbers are already in order Sandra thinks the order should be reversed Who is right
21 Math History There is a famous irrational number called Eulerrsquos number symbolized with an e Like π its decimal form never ends or repeats The first few digits of e are 27182818284
a Between which two square roots of integers could you find this number
b Between which two square roots of integers can you find π
22 Analyze Relationships There are several approximations used for π including 314 and 22 __ 7 π is approximately 314159265358979
a Label π and the two approximations on the number line
b Which of the two approximations is a better estimate for π Explain
c Find a whole number x so that the ratio x ___ 113 is a better estimate for π
than the two given approximations
23 Communicate Mathematical Ideas If a set of six numbers that include both rational and irrational numbers is graphed on a number line what is the fewest number of distinct points that need to be graphed Explain
24 Critique Reasoning Jill says that 12 _
6 is less than 1263 Explain her error
FOCUS ON HIGHER ORDER THINKING
radic_
115 115 ___ 11 and 105624
between radic_
7 asymp 265 and radic_
8 asymp 283
between radic_
9 = 3 and radic_
10 asymp 316
22 __ 7 it is closer to π on the number line
She did not consider the repeating digit 1266
2 rational numbers can have the same location and
irrational numbers can have the same location but they
cannot share a location
355
Neither student is correct The answer
should be 115 ___ 11 105624 radic_
115
Unit 126
copy H
ough
ton M
ifflin
Har
cour
t Pub
lishin
g Com
pany
Imag
e Cre
dits
copy3D
Stoc
kiSt
ockP
hoto
com
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
8_MCAAESE206984_U1M01L3indd 26 210513 801 AM
Personal Math Trainer
Online Practice and Helpmyhrwcom
Name Class Date
Independent Practice
16 Your sister is considering two different shapes for her garden One is a square with side lengths of 35 meters and the other is a circle with a diameter of 4 meters
a Find the area of the square
b Find the area of the circle
c Compare your answers from parts a and b Which garden would give your sister the most space to plant
17 Winnie measured the length of her fatherrsquos ranch four times and got four different distances Her measurements are shown in the table
a To estimate the actual length Winnie first approximated each distance to the nearest hundredth Then she averaged the four numbers Using a calculator find Winniersquos estimate
b Winniersquos father estimated the distance across his ranch to be radic_
56 km How does this distance compare to Winniersquos estimate
Give an example of each type of number
18 a real number between radic_
13 and radic_
14
19 an irrational number between 5 and 7
Order the numbers from least to greatest
12 radic_
7 2 radic_
8 ___ 2 13 radic_
10 π 35
14 radic_
220 -10 radic_
100 115 15 radic_
8 -375 3 9 _ 4
Distance Across Fatherrsquos Ranch (km)
1 2 3 4
radic_
60 58 __ 8 7 _
3 7 3 _ 5
138NS2
radic_
8 ___ 2 2 radic_
7
-10 radic_
100 115 radic_
220
radic_
60 asymp 775 58 __ 8 = 725 7 _
3 asymp 733 7 3 _ 5 = 760 so the average
π radic_
10 35
-375 9 _ 4 radic_
8 3
is 74825 km
1225 m2
4π m2 or approximately 126 m2
They are nearly identical radic_
56 is approximately 74833hellip
The circle would give her more space to plant because it has a
larger area
Sample answer 37
Sample answer radic_
31
25Lesson 13
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ough
ton
Miff
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ublis
hing
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pany
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
8_MCAAESE206984_U1M01L3indd 25 41613 448 AM
Activity available online myhrwcomEXTEND THE MATH PRE-AP
Activity Have students investigate whether there are infinitely many numbers between two numbers by giving examples for each of the following
bull Between any two rational numbers there is at least one other rational number Sample answer 45 is between 41 and 48
bull Between any two irrational numbers there is at least one rational number Sample answer 45 is between radic
_ 11 and radic
_ 29
bull Between any two rational numbers there is at least one irrational number Sample answer radic
_ 11 is between 31 and 36
bull Between any two irrational numbers there is at least one irrational number Sample answer radic
_ 17 is between radic
_ 11 and radic
_ 29
Ordering Real Numbers 26
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
ReadyMath Trainer
Online Practiceand Help
Personal
myhrwcom
Module Quiz
11ensp RationalenspandenspIrrationalenspNumbersWrite each fraction as a decimal or each decimal as a fraction
1 7__20 2 1___
27 3 17_8
Solve each equation for x
4 x2=81 5 x3=343 6 x2= 1___100
7 Asquarepatiohasanareaof200squarefeetHowlongiseachside
ofthepatiotothenearesttenth
12ensp SetsenspofenspRealenspNumbersWrite all names that apply to each number
8 121____radic
____121
9 π__2
10 TellwhetherthestatementldquoAllintegersarerationalnumbersrdquoistrueorfalseExplainyourchoice
13ensp OrderingenspRealenspNumbersCompare Write lt gt or =
11 radic__
8+3 8+radic__
3 12 radic__
5+11emsp emsp emsp 5+radic___
11
Order the numbers from least to greatest
13 radic___
99π29__
8 14 radic___
1__251_40__
2
15 Howarerealnumbersusedtodescribereal-worldsituations
ESSENTIAL QUESTION
035
9-9
141ft
7 1__10- 1__10
14__11 1875
wholeintegerrationalreal
Trueintegerscanbewrittenasthequotientoftwointegers
SampleanswerRealnumberssuchastherational
π29__
8radic___
99
irrationalreal
lt gt
number1_4candescribeamountsusedincooking
radic___
1__250__
21_4
27Module1
copy H
ough
ton
Miff
lin H
arco
urt P
ublis
hing
Com
pany
DONOTEDIT--ChangesmustbemadethroughldquoFileinfordquoCorrectionKey=A
8_MCAAESE206984_U1M01RTindd 27 41513 1113 PM
Math TrainerOnline Assessment
and Intervention
Personal
myhrwcom
1
2
3 Response toIntervention
Intervention Enrichment
Access Ready to Go On assessment online and receive instant scoring feedback and customized intervention or enrichment
Online and Print Resources
Differentiated Instruction
bull Reteach worksheets
bull Reading Strategies EL
bull Success for English Learners EL
Differentiated Instruction
bull Challenge worksheets PRE-AP
Extend the Math PRE-AP
Lesson Activities in TE
Additional ResourcesAssessment Resources includes bull Leveled Module Quizzes
Ready to Go OnAssess MasteryUse the assessment on this page to determine if students have mastered the concepts and standards covered in this module
California Common Core Standards
Lesson Exercises Common Core Standards
11 1ndash7 8NS1 8NS2 8EE2
12 8ndash10 8NS1
13 11ndash14 8NS2
27 Unit 1 Module 1
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
Personal Math Trainer
Online Practice and HelpmyhrwcomAssessment Readiness
Module 1 MIXed ReVIeW
1 Look at each number Is the number between 2π and radic___
52
Select Yes or No for expressions AndashC
A 6 2 _ 3 Yes No
B 5π __ 2 Yes No
C 3 radic__
5 Yes No
2 Consider the number - 11 __ 15
Choose True or False for each statement
A The number is rational True False
B The number can be written as True Falsea repeating decimal
C The number is less than ndash08 True False
3 The volume of a cube is given by V = x3 where x is the length of an edge of the cube A cube-shaped end table has a volume of 3 3 _ 8 cubic feet What is the length of an edge of the end table Explain how you solved this problem
4 A student says that radic___
83 is greater than 29 __ 3 Is the student correct Justify your
reasoning
1 1 _ 2 ft Sample answer The equation x3 = 3 3 _ 8 can be used
to find the edge length in feet To solve the equation
write the mixed number as a fraction greater than 1
x3 = 27 __ 8 Then take the cube root of both sides x = 3 _ 2 = 1 1 _ 2
No Sample answer radic___
83 asymp 91 and 29 __ 3 = 9
__ 6
Because 91 lt 9 __
6 radic___
83 lt 29 __ 3
28 Unit 1
copy H
ough
ton
Miff
lin H
arco
urt P
ublis
hing
Com
pany
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=A
8_MCAAESE206984_U1M01RTindd 28 240413 946 AM
Personal Math Trainer
Online Assessment and
Interventionmyhrwcom
Scoring GuideItem 3 Award the student 1 point for finding the edge length of the cube and 1 point for correctly explaining how to use a cube root to solve the problem
Item 4 Award the student 1 point for determining that the student is incorrect and 1 point for correctly justifying the reasoning for this conclusion
Additional ResourcesTo assign this assessment online login to your Assignment Manager at myhrwcom
Assessment Readiness
California Common Core Standards
Items Grade 8 Standards Mathematical Practices
1 8NS2 MP7
2 7NS2b 7NS2d 8NS1 MP7
3 8EE2 MP1 MP4
4 8NS1 8NS2 MP3
Item integrates mixed review concepts from previous modules or a previous course
Item 4 combines concepts from the California Common Core cluster ldquoKnow that there are numbers that are not rational and approximate them by rational numbersrdquo
Real Numbers 28
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
Math TrainerOnline Assessment
and Intervention
Personal
Online homework assignment available
myhrwcom
Lesson Quiz available online
12 LESSON QUIZ
1 Write all the names that apply to the number
2 Tell whether the given statement is true or false Explain your choice All numbers between 1 and 2 are rational numbers
3 Identify the set of numbers that best describes the situation Explain your choiceThe choices on a survey question change the total points for the survey by -2 -1 0 1 or 2 points
-1 _
5
myhrwcom
Additional ResourcesDifferentiated Instruction includes bull Leveled Practice worksheets
EvaluateGUIDED AND INDEPENDENT PRACTICE
Concepts amp Skills Practice
Example 1Classifying Real Numbers
Exercises 1ndash8 14ndash19 22ndash24
Example 2Understanding Sets and Subsets of Real Numbers
Exercises 9ndash10
Example 3Identifying Sets for Real-World Situations
Exercises 11ndash12 20ndash21 25
Focus | Coherence | Rigor
Exercise Depth of Knowledge (DOK) Mathematical Practices
14ndash19 2 SkillsConcepts MP7 Using Structure
20ndash21 2 SkillsConcepts MP6 Precision
22ndash23 2 SkillsConcepts MP3 Logic
24 1 Recall of Information MP7 Using Structure
25 2 SkillsConcepts MP2 Reasoning
26ndash27 3 Strategic Thinking MP3 Logic
28 3 Strategic Thinking MP8 Patterns
29 3 Strategic Thinking MP3 Logic
8NS1
8NS1
Exercise 29 combines concepts from the California Common Core cluster ldquoKnow that there are numbers that are not rational and approximate them by rational numbersrdquo
Answers1 rational real
2 False radic_
2 is an example of an irrational number between 1 and 2
3 Integers each number is an integer but only three are whole numbers
19 Lesson 12
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
Work Area
π mi23 Critique Reasoning The circumference of a circular region is shown
What type of number best describes the diameter of the circle Explain
your answer
24 Critical Thinking A number is not an integer What type of number can it be
25 A grocery store has a shelf with half-gallon containers of milk What type of number best represents the total number of gallons
26 Explain the Error Katie said ldquoNegative numbers are integersrdquo What was her error
27 Justify Reasoning Can you ever use a calculator to determine if a number is rational or irrational Explain
28 Draw Conclusions The decimal 0 _
3 represents 1 _ 3 What type of number best describes 0
_ 9 which is 3 middot 0
_ 3 Explain
29 Communicate Mathematical Ideas Irrational numbers can never be precisely represented in decimal form Why is this
FOCUS ON HIGHER ORDER THINKING
It can be a rational number that is not an integer or an irrational number
rational number
The set of negative numbers also includes non-integer
rational numbers and irrational numbers
Sample answer If the calculator shows a decimal that
terminates in fewer digits than what the calculator screen
allows then you can tell that the number is rational If not
you cannot tell from the calculator display whether the
number terminates because you see a limited number
of digits It may be a repeating decimal (rational) or
non-terminating non-repeating decimal (irrational)
Whole 3 middot 0 _
3 represents 3 middot 1 _ 3 = 1 so 0 _
9 is exactly 1
Sample answer In decimal form irrational numbers never
terminate and never repeat Therefore no matter how
many decimal places you include the number will never
be precisely represented There are always more digits
Whole the diameter is π _ π = 1 mile
Unit 120
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ough
ton
Miff
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hing
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pany
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
8_MCAAESE206984_U1M01L2indd 20 120413 909 PM
Integers
Rational Numbers Irrational Numbers
Real Numbers
Whole Numbers
257
radic16
166
radic9
128 radic50
Personal Math Trainer
Online Practice and Helpmyhrwcom
Name Class Date
Independent Practice
Identify the set of numbers that best describes each situation Explain your choice
20 the height of an airplane as it descends to an airport runway
21 the score with respect to par of several golfers 2 ndash 3 5 0 ndash 1
22 Critique Reasoning Ronald states that the number 1 __ 11 is not rational because when converted into a decimal it does not terminate Nathaniel says it is rational because it is a fraction Which boy is correct Explain
12
14 - radic_
9 15 257
16 radic_
50 17 8 1 _ 2
18 166 19 radic_
16
Write all names that apply to each number Then place the numbers in the correct location on the Venn diagram
8NS1
Real numbers the height can be any number greater than zero
integer rational real whole integer rational real
whole integer rational real
irrational real
rational real
rational real
Integers the scores are counting numbers their
opposites and zero
Nathaniel is correct A rational number is a number that can be written as a fraction and 1 __ 11 is a fraction
19Lesson 12
copy H
ough
ton
Miff
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ublis
hing
Com
pany
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
8_MCAAESE206984_U1M01L2indd 19 41613 136 AM
myhrwcomActivity available onlineEXTEND THE MATH PRE-AP
Activity Have students consider the concept of restricted domain for the sets of numbers that describe situations For example the number of sisters a person has can best be described by whole numbers but no one has ever had 1500 sisters An area code is an integer or whole number between 200 and 999
Have students use a source such as the Guinness Book of World Records and give examples of sets of numbers that describe situations where the domain is restricted Ask whether the restriction may be changed in the future
Sets of Real Numbers 20
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
-3-4-5 -2-1 1 2 3 50 4
12-4 -radic5
Lesson Support Content Objective Students will learn to order a set of real numbers
Language Objective Students will show and describe how to order a set of real numbers
LESSON 13 Ordering Real Numbers
Building BackgroundEliciting Prior Knowledge Have students draw a number line to compare a rational number and an irrational number such as - radic
_ 5 and -4 1 __ 2 Ask them to explain how
they approximated the irrational number on the number line Then have them identify the greater and the lesser real number Repeat with several other pairs of real numbers in different forms
Learning ProgressionsIn this lesson students order a set of real numbers They use rational approximations to compare the sizes of irrational numbers They also order numbers for real-world situations Important understandings for students include the following
bull Compare irrational numbers bull Estimate the value of expressions with irrational numbers bull Order a set of real numbers bull Order real numbers in a real-world context
Work with real numbers continues throughout Grade 8 and into high school This lesson provides students with a foundation for understanding the relative sizes of numbers in different forms in the real number system
Cluster ConnectionsThis lesson provides an excellent opportunity to connect ideas in this cluster Know that there are numbers that are not rational and approximate them by rational numbers Tell students that there is a special number called the golden ratio with applications in mathematics geometry art and architecture The golden ratio is called phi and is represented by the Greek letter ϕ It includes an irrational number in its definition
Have students explain why the golden ratio is irrational Ask them to find the two whole numbers the golden ratio lies between Then challenge them to approximate the golden ratio to the nearest tenth It is irrational because it includes an irrational number in its definition It lies between 1 and 2 To the nearest tenth ϕ = 16
ϕ = 1 + radic_
5 _ 2
Focus | Coherence | Rigor
California Common Core Standards
8NS2 Use rational approximations of irrational numbers to compare the size of irrational numbers locate them approximately on a number line diagram and estimate the value of expressions (eg π 2 )
MP4 Model with mathematics
21A
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
Math Talk
Language Support EL
PROFESSIONAL DEVELOPMENT
Linguistic Support EL
AcademicContent Vocabulary
Post a chart like this to remind students of the regular comparative forms of adjectives that use the -er and -est suffixes Add to the chart for terms that appear in examples and exercises in each lesson Include any irregular verb forms
Background Knowledge
Go On ndash the title of the module review or quiz is Ready to Go On This title uses an idiomatic expression In this context to go on means ldquoto move aheadrdquo or ldquoto proceedrdquo It is different from the use of go on that means having enough facts to use meaningfully as in having enough to go on Also the intonation used in pronouncing an expression can give it different meanings For example when the speaker emphasizes the word on he or she might be expressing disbelief as in ldquoGo ON Yoursquore kidding rightrdquo Discuss with students other ways that the phrase go on may be used
Leveled Strategies for English Learners
Emerging Label points on a number line with the terms used in ordering greater greatest less lesser least Use sentence frames to insert the correct terms
Expanding Have students give two or three complete sentences to compare the placement of numbers on a number line using the correct forms of the comparative and superlative adjectives
Bridging Have students work in pairs with one student giving directions to the other in complete sentences to order numbers on a number line
To help students answer the question posed in Math Talk make sure that students have a command of the forms for making comparisons and the superlative and the concept of opposite order so that the focus is on the math concept instead of the language skills needed to describe and explain order
EL
Adjective Comparative Superlative
Far Farther Farthest
Large Larger Largest
Great Greater Greatest
Some Less Least
Some More Most
California ELD Standards
Emerging 2I8 Analyzing language choices ndash Explain how phrasing or different common words with similar meanings produce different effects on the audience
Expanding 2I8 Analyzing language choices ndash Explain how phrasing or different words with similar meanings or figurative language produce shades of meaning and different effects on the audience
Bridging 2I8 Analyzing language choices ndash Explain how phrasing or different words with similar meanings or figurative language produce shades of meaning nuances and different effects on the audience
Ordering Real Numbers 21B
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
13L E S S O N
Ordering Real Numbers
Interactive Whiteboard Interactive example available online
ADDITIONAL EXAMPLE 1Compare Write lt gt or =
A radic_
8 - 2 4 - radic_
8 lt
B radic_
20 + 1 3 + radic_
2 gt
EngageESSENTIAL QUESTION
How do you order a set of real numbers Sample answer Find their approximate decimal values and order them
Motivate the LessonAsk What kind of numbers are you comparing when you compare the price of gasoline at two different gas stations
ExploreGive students two rational numbers and ask them to name a number between them Repeat a few times and then give them two irrational numbers and ask them to name a number between them
ExplainEXAMPLE 1
Questioning Strategies Mathematical Practices bull Which is greater the difference between 5 and 3 or the difference between radic
_ 5 and radic
_ 3
The difference between 5 and 3 is 2 the difference between radic_
5 and radic_
3 is approximately 1 So the difference between 5 and 3 is greater
Avoid Common ErrorsCaution students to read the problem carefully and think about what the radical sign means so that they do not misread the problem and answer that the two sides are equal
YOUR TURNFocus on TechnologyCalculators should not be used at this point because developing number sense is the goal
EXAMPLE 2Questioning Strategies Mathematical Practices bull How do you determine whether radic
_ 22 is less than or greater than 45 The square of 45 is
2025 which is less than 22 so the square root of 22 must be greater than 45
Engage with the WhiteboardHave students graph and label various real numbers between 42 and 44 and between 47 and 5
YOUR TURNFocus on Modeling Mathematical PracticesHave students label the integers on the number line with their equivalent square root For example 1 2 and 3 on the number line would be labeled radic
_ 1 radic
_ 4 and radic
_ 9
Interactive Whiteboard Interactive example available online
ADDITIONAL EXAMPLE 2Order 3π radic
_ 10 and 325 from greatest
to least
3π 325 radic_
10
myhrwcom
myhrwcom
CA Common CoreStandards
The student is expected to
The Number Systemmdash8NS2
Use rational approximations of irrational numbers to compare the size of irrational numbers locate them approximately on a number line diagram and estimate the value of expressions (eg π 2 )
Mathematical Practices
MP4 Modeling
The student is expected to
21 Lesson 13
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
Math TrainerOnline Practice
and Help
Personal
myhrwcom
Math On the Spotmyhrwcom
0 05 1 15 2 25 3 35 4
radic5radic3
π2
8 85 9 95 10 105 11 115 12
radic75
4 42 44 46 48 5
radic224 12π + 1
Ordering Real Numbers You can compare and order real numbers and list them from least to greatest
Order radic_
22 π + 1 and 4 1 _ 2 from least to greatest
First approximate radic_
22
radic_
22 is between 4 and 5 Since you donrsquot know where it falls between 4 and 5 you need to find a better estimate for radic
_ 22 so
you can compare it to 4 1 _ 2
Since 22 is closer to 25 than 16 use squares of numbers between 45 and 5 to find a better estimate of radic
_ 22
45 2 = 2025 46 2 = 2116 47 2 = 2209 48 2 = 2304
Since 47 2 = 2209 an approximate value for radic_
22 is 47
An approximate value of π is 314 So an approximate value of π +1 is 414
Plot radic_
22 π + 1 and 4 1 _ 2 on a number line
Read the numbers from left to right to place them in order from least to greatest
From least to greatest the numbers are π + 1 4 1 _ 2 and radic_
22
EXAMPLE 2
STEP 1
STEP 2
Order the numbers from least to greatest Then graph them on the number line
YOUR TURN
5 radic_
5 25 radic_
3
6 π 2 10 radic_
75
If real numbers a b and c are in order from least to greatest what is the order
of their opposites from least to greatest
Explain
Math TalkMathematical Practices
8NS2
radic_
3 radic_
5 25
radic_
75 π2 10
Math Talk answer -c -b -a -c is farthest to the left on a number line -b is in the middle and -a is farthest to the right
Unit 122
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pany
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
8_MCAAESE206984_U1M01L3indd 22 41613 447 AM
My Notes
Math TrainerOnline Practice
and Help
Personal
myhrwcom
Math On the Spot
myhrwcom
Comparing Irrational NumbersBetween any two real numbers is another real number To compare and order real numbers you can approximate irrational numbers as decimals
Compare radic_
3 + 5 3 + radic_
5 Write lt gt or =
First approximate radic_
3
radic_
3 is between 1 and 2
Next approximate radic_
5
radic_
5 is between 2 and 3
Then use your approximations to simplify the expressions
radic_
3 + 5 is between 6 and 7
3 + radic_
5 is between 5 and 6
So radic_
3 + 5 gt 3 + radic_
5
Reflect1 If 7 + radic
_ 5 is equal to radic
_ 5 plus a number what do you know about the
number Why
2 What are the closest two integers that radic_
300 is between
EXAMPLEXAMPLE 1
STEP 1
STEP 2
Compare Write lt gt or =
YOUR TURN
3 radic_
2 + 4 2 + radic_
4 4 radic_
12 + 6 12 + radic_
6
L E S S O N
13 Ordering Real Numbers
ESSENTIAL QUESTIONHow do you order a set of real numbers
8NS2
Use rational approximations of irrational numbers to compare the size of irrational numbers locate them approximately on a number line diagram and estimate the value of expressions (eg π 2 )
8NS2
Use perfect squares to estimate square roots
1 2 = 1 2 2 = 4 3 2 = 9
The number is 7 both expressions must equal 7 + radic_
5
17 and 18
gt lt
21Lesson 13
copy H
ough
ton
Miff
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Com
pany
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
8_MCAAESE206984_U1M01L3indd 21 41913 246 PM
PROFESSIONAL DEVELOPMENT
Math BackgroundIn this lesson students estimate irrational numbers in the form of square roots of nonper-fect squares by finding two perfect squares between which the number falls A more precise method involves repeated division For example to find radic
_ 28 find a whole number whose perfect
square is close to 28 such as 5 Divide 28 by that number 28 divide 5 = 56 Find the average of the quotient and divisor 5 + 56
_____ 2 = 53 Continue dividing 28 by each result and averaging until you get the desired accuracy
Integrate Mathematical Practices MP4
This lesson provides an opportunity to address this Mathematical Practices standard It calls for students to model relationships using multiple representations including diagrams graphs and language as appropriate Students use multiple representations when they use number lines to estimate the locations of and order rational and irrational numbers given as symbols
Ordering Real Numbers 22
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
Interactive Whiteboard Interactive example available online
ADDITIONAL EXAMPLE 3The diameter of a meteorite in millimeters is calculated by four different methods Order the results from least to greatest
Joe radic_
18 mm Lisa 13 __ 3 mm
Pablo 46 mm Julien 4π __ 3 mm
Julien 4π __ 3 mm Lisa 13 __ 3 mm
Joe radic_
18 mm Pablo 46 mm
EXAMPLE 3Questioning Strategies Mathematical Practices bull How can you verify that radic
_ 28 is between 52 and 53 5 2 2 = 2704 and 5 3 2 = 2809
bull Explain how to determine which number is greater 5 _
5 or 55 When the repeating decimal is rounded to the nearest tenth or hundredth you can see that it is greater
Connect to Daily LifeDiscuss how measuring across a canyon might involve different methods than measuring along a road Explain that measurements like these are often done using calculations that approximate the distance
YOUR TURNFocus on Critical Thinking Mathematical PracticesDiscuss with students which number is greater 3
_ 45 or 3450 3
_ 45 or 3455 and why Explain
that 3 _
45 can be written out as 34545hellipMake sure they understand that 3 _
45 is greater than 345 but less than 3455
ElaborateTalk About ItSummarize the Lesson
Ask How can you order two numbers in different forms whose decimal approxi-mations appear to be equal Approximate one or both numbers to an additional
number of decimal places
GUIDED PRACTICEEngage with the Whiteboard
Have students place and label additional points on the number line in Exercise 9 Allow the points to be in any format other than decimal
Avoid Common ErrorsExercises 3ndash4 Caution students to read the problem carefully so that they do not misread the problem as the same numbers combined by addition on each side of the circleExercise 10 Remind students that the calculations have units
myhrwcom
23 Lesson 13
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
0 05 1 15 2 25 3 35 4 45 5 55 6 65 7
2πradic3
Compare Write lt gt or = (Example 1)
1 radic_
3 + 2 radic_
3 + 3 2 radic_
8 + 17 radic_
11 + 15
3 radic_
6 + 5 6 + radic_
5 4 radic_
9 + 3 9 + radic_
3
5 radic_
17 - 3 -2 + radic_
5 6 12 - radic_
2 14 - radic_
8
7 radic_
7 + 2 radic_
10 - 1 8 radic_
17 + 3 3 + radic_
11
9 Order radic_
3 2π and 15 from least to greatest Then graph them on the number line (Example 2)
radic_
3 is between and so radic_
3 asymp
π asymp 314 so 2π asymp
From least to greatest the numbers are
10 Four people have found the perimeter of a forest using different methods Their results are given in the table Order their calculations from greatest to least (Example 3)
11 Explain how to order a set of real numbers
CHECK-INESSENTIAL QUESTION
Forest Perimeter (km)
Leon Mika Jason Ashley
radic_
17 - 2 1 +thinsp π __ 2 12 ___ 5 25
Guided Practice
17
15
1 + π _ 2 km 25 km 12 __ 5 km radic_
17 - 2 km
2π radic
_ 3
18 175
628
Sample answer Convert each number to a decimal
equivalent using estimation to find equivalents for
irrational numbers Graph each number on a number line
Read the numbers from left to right for least to greatest
Read the numbers from right to left for greatest to least
lt gt
lt lt
ltgt
gt gt
24 Unit 1
copy H
ough
ton
Miff
lin H
arco
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ublis
hing
Com
pany
bull Im
age C
redi
ts copy
Elena
Eliss
eeva
Alam
y Im
ages
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
8_MCAAESE206984_U1M01L3indd 24 41613 448 AM
My Notes
5 52 54 56 58 6
radic28 5 12
23455
Math TrainerOnline Practice
and Help
Personal
myhrwcom
Math On the Spot
myhrwcom
Ordering Real Numbers in a Real-World Context Calculations and estimations in the real world may differ It can be important to know not only which are the most accurate but which give the greatest or least values depending upon the context
Four people have found the distance in kilometers across a canyon using different methods Their results are given in the table Order the distances from greatest to least
Distance Across Quarry Canyon (km)
Juana Lee Ann Ryne Jackson
radic_
28 23 __ 4 5 _
5 5 1 _ 2
Write each value as a decimal
radic_
28 is between 52 and 53 Since 53 2 = 2809 an approximate value for radic
_ 28 is 53
23 __ 4 = 575
5 _
5 is 5555hellip so 5 _
5 to the nearest hundredth is 556
5 1 _ 2 = 55
Plot radic_
28 23 __ 4 5 _
5 and 5 1 _ 2 on a number line
From greatest to least the distances are
23 __ 4 km 5 _
5 km 5 1 _ 2 km radic_
28 km
EXAMPLEXAMPLE 3
STEP 1
STEP 2
7 Four people have found the distance in miles across a crater using different methods Their results are given below
Jonathan 10 __ 3 Elaine 3 _
45 Joseacute 3 1 _ 2 Lashonda radic_
10
Order the distances from greatest to least
YOUR TURN
8NS2
3 1 _ 2 mi 3 _
45 mi 10 __ 3 mi radic_
10 mi
23Lesson 13
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ough
ton
Miff
lin H
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pany
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8_MCAAESE206984_U1M01L3indd 23 41613 447 AM
ModelingPlace papers around the room with the numbers from 1 to 5 one per sheet Give each student a card showing a number between 1 and 5 in different forms Have students place his or her card between the correct integers and decide where the number goes in relation to any numbers already placed
Multiple RepresentationsGive students a vertical number line which some students might find easier to use than a horizontal one Have them decide whether to place points for rational and irrational numbers above or below existing points
Additional ResourcesDifferentiated Instruction includes bull Reading Strategies bull Success for English Learners EL
bull Reteach bull Challenge PRE-AP
DIFFERENTIATE INSTRUCTION
Ordering Real Numbers 24
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
Math TrainerOnline Assessment
and Intervention
Personal
Online homework assignment available
myhrwcom
myhrwcom
Lesson Quiz available online
13 LESSON QUIZ
1 Compare Write lt gt or =
radic_
95 - 5 radic_
62 - 2
2 Order 105 radic_
105 and 3π + 1 from greatest to least
3 A length in centimeters is calculated differently by four different people Order their calculations from least to greatest
KD 11 __ 2 cm Silvio 5 __ 3 π cm
Paula 5 _
4 cm Luis radic_
33 cm
EvaluateGUIDED AND INDEPENDENT PRACTICE
Concepts amp Skills Practice
Example 1Comparing Irrational Numbers
Exercises 1ndash8
Example 2Ordering Real Numbers
Exercises 9 12ndash15 18ndash21
Example 3Ordering Real Numbers in a Real-World Context
Exercises 10 16ndash17
Additional ResourcesDifferentiated Instruction includes bull Leveled Practice worksheets
Focus | Coherence | Rigor
Exercise Depth of Knowledge (DOK) Mathematical Practices
12ndash15 1 Recall of Information MP5 Using Tools
16 2 SkillsConcepts MP2 Reasoning
17 2 SkillsConcepts MP6 Precision
18ndash21 2 SkillsConcepts MP2 Reasoning
22 3 Strategic Thinking MP4 Modeling
23ndash24 3 Strategic Thinking MP3 Logic
8NS2
8NS2
Answers1 radic
_ 95 - 5 lt radic
_ 62 - 2
2 radic_
105 3π + 1 105
3 Silvio 5 __ 3 π cm Paula 5 _
4 cm
KD 11
__ 2 cm Luis radic_
33 cm
25 Lesson 13
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
Work Area
3140 3141 3142 3143
314 π227
20 A teacher asks his students to write the numbers shown in order from least to greatest Paul thinks the numbers are already in order Sandra thinks the order should be reversed Who is right
21 Math History There is a famous irrational number called Eulerrsquos number symbolized with an e Like π its decimal form never ends or repeats The first few digits of e are 27182818284
a Between which two square roots of integers could you find this number
b Between which two square roots of integers can you find π
22 Analyze Relationships There are several approximations used for π including 314 and 22 __ 7 π is approximately 314159265358979
a Label π and the two approximations on the number line
b Which of the two approximations is a better estimate for π Explain
c Find a whole number x so that the ratio x ___ 113 is a better estimate for π
than the two given approximations
23 Communicate Mathematical Ideas If a set of six numbers that include both rational and irrational numbers is graphed on a number line what is the fewest number of distinct points that need to be graphed Explain
24 Critique Reasoning Jill says that 12 _
6 is less than 1263 Explain her error
FOCUS ON HIGHER ORDER THINKING
radic_
115 115 ___ 11 and 105624
between radic_
7 asymp 265 and radic_
8 asymp 283
between radic_
9 = 3 and radic_
10 asymp 316
22 __ 7 it is closer to π on the number line
She did not consider the repeating digit 1266
2 rational numbers can have the same location and
irrational numbers can have the same location but they
cannot share a location
355
Neither student is correct The answer
should be 115 ___ 11 105624 radic_
115
Unit 126
copy H
ough
ton M
ifflin
Har
cour
t Pub
lishin
g Com
pany
Imag
e Cre
dits
copy3D
Stoc
kiSt
ockP
hoto
com
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
8_MCAAESE206984_U1M01L3indd 26 210513 801 AM
Personal Math Trainer
Online Practice and Helpmyhrwcom
Name Class Date
Independent Practice
16 Your sister is considering two different shapes for her garden One is a square with side lengths of 35 meters and the other is a circle with a diameter of 4 meters
a Find the area of the square
b Find the area of the circle
c Compare your answers from parts a and b Which garden would give your sister the most space to plant
17 Winnie measured the length of her fatherrsquos ranch four times and got four different distances Her measurements are shown in the table
a To estimate the actual length Winnie first approximated each distance to the nearest hundredth Then she averaged the four numbers Using a calculator find Winniersquos estimate
b Winniersquos father estimated the distance across his ranch to be radic_
56 km How does this distance compare to Winniersquos estimate
Give an example of each type of number
18 a real number between radic_
13 and radic_
14
19 an irrational number between 5 and 7
Order the numbers from least to greatest
12 radic_
7 2 radic_
8 ___ 2 13 radic_
10 π 35
14 radic_
220 -10 radic_
100 115 15 radic_
8 -375 3 9 _ 4
Distance Across Fatherrsquos Ranch (km)
1 2 3 4
radic_
60 58 __ 8 7 _
3 7 3 _ 5
138NS2
radic_
8 ___ 2 2 radic_
7
-10 radic_
100 115 radic_
220
radic_
60 asymp 775 58 __ 8 = 725 7 _
3 asymp 733 7 3 _ 5 = 760 so the average
π radic_
10 35
-375 9 _ 4 radic_
8 3
is 74825 km
1225 m2
4π m2 or approximately 126 m2
They are nearly identical radic_
56 is approximately 74833hellip
The circle would give her more space to plant because it has a
larger area
Sample answer 37
Sample answer radic_
31
25Lesson 13
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ough
ton
Miff
lin H
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ublis
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pany
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
8_MCAAESE206984_U1M01L3indd 25 41613 448 AM
Activity available online myhrwcomEXTEND THE MATH PRE-AP
Activity Have students investigate whether there are infinitely many numbers between two numbers by giving examples for each of the following
bull Between any two rational numbers there is at least one other rational number Sample answer 45 is between 41 and 48
bull Between any two irrational numbers there is at least one rational number Sample answer 45 is between radic
_ 11 and radic
_ 29
bull Between any two rational numbers there is at least one irrational number Sample answer radic
_ 11 is between 31 and 36
bull Between any two irrational numbers there is at least one irrational number Sample answer radic
_ 17 is between radic
_ 11 and radic
_ 29
Ordering Real Numbers 26
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
ReadyMath Trainer
Online Practiceand Help
Personal
myhrwcom
Module Quiz
11ensp RationalenspandenspIrrationalenspNumbersWrite each fraction as a decimal or each decimal as a fraction
1 7__20 2 1___
27 3 17_8
Solve each equation for x
4 x2=81 5 x3=343 6 x2= 1___100
7 Asquarepatiohasanareaof200squarefeetHowlongiseachside
ofthepatiotothenearesttenth
12ensp SetsenspofenspRealenspNumbersWrite all names that apply to each number
8 121____radic
____121
9 π__2
10 TellwhetherthestatementldquoAllintegersarerationalnumbersrdquoistrueorfalseExplainyourchoice
13ensp OrderingenspRealenspNumbersCompare Write lt gt or =
11 radic__
8+3 8+radic__
3 12 radic__
5+11emsp emsp emsp 5+radic___
11
Order the numbers from least to greatest
13 radic___
99π29__
8 14 radic___
1__251_40__
2
15 Howarerealnumbersusedtodescribereal-worldsituations
ESSENTIAL QUESTION
035
9-9
141ft
7 1__10- 1__10
14__11 1875
wholeintegerrationalreal
Trueintegerscanbewrittenasthequotientoftwointegers
SampleanswerRealnumberssuchastherational
π29__
8radic___
99
irrationalreal
lt gt
number1_4candescribeamountsusedincooking
radic___
1__250__
21_4
27Module1
copy H
ough
ton
Miff
lin H
arco
urt P
ublis
hing
Com
pany
DONOTEDIT--ChangesmustbemadethroughldquoFileinfordquoCorrectionKey=A
8_MCAAESE206984_U1M01RTindd 27 41513 1113 PM
Math TrainerOnline Assessment
and Intervention
Personal
myhrwcom
1
2
3 Response toIntervention
Intervention Enrichment
Access Ready to Go On assessment online and receive instant scoring feedback and customized intervention or enrichment
Online and Print Resources
Differentiated Instruction
bull Reteach worksheets
bull Reading Strategies EL
bull Success for English Learners EL
Differentiated Instruction
bull Challenge worksheets PRE-AP
Extend the Math PRE-AP
Lesson Activities in TE
Additional ResourcesAssessment Resources includes bull Leveled Module Quizzes
Ready to Go OnAssess MasteryUse the assessment on this page to determine if students have mastered the concepts and standards covered in this module
California Common Core Standards
Lesson Exercises Common Core Standards
11 1ndash7 8NS1 8NS2 8EE2
12 8ndash10 8NS1
13 11ndash14 8NS2
27 Unit 1 Module 1
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
Personal Math Trainer
Online Practice and HelpmyhrwcomAssessment Readiness
Module 1 MIXed ReVIeW
1 Look at each number Is the number between 2π and radic___
52
Select Yes or No for expressions AndashC
A 6 2 _ 3 Yes No
B 5π __ 2 Yes No
C 3 radic__
5 Yes No
2 Consider the number - 11 __ 15
Choose True or False for each statement
A The number is rational True False
B The number can be written as True Falsea repeating decimal
C The number is less than ndash08 True False
3 The volume of a cube is given by V = x3 where x is the length of an edge of the cube A cube-shaped end table has a volume of 3 3 _ 8 cubic feet What is the length of an edge of the end table Explain how you solved this problem
4 A student says that radic___
83 is greater than 29 __ 3 Is the student correct Justify your
reasoning
1 1 _ 2 ft Sample answer The equation x3 = 3 3 _ 8 can be used
to find the edge length in feet To solve the equation
write the mixed number as a fraction greater than 1
x3 = 27 __ 8 Then take the cube root of both sides x = 3 _ 2 = 1 1 _ 2
No Sample answer radic___
83 asymp 91 and 29 __ 3 = 9
__ 6
Because 91 lt 9 __
6 radic___
83 lt 29 __ 3
28 Unit 1
copy H
ough
ton
Miff
lin H
arco
urt P
ublis
hing
Com
pany
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=A
8_MCAAESE206984_U1M01RTindd 28 240413 946 AM
Personal Math Trainer
Online Assessment and
Interventionmyhrwcom
Scoring GuideItem 3 Award the student 1 point for finding the edge length of the cube and 1 point for correctly explaining how to use a cube root to solve the problem
Item 4 Award the student 1 point for determining that the student is incorrect and 1 point for correctly justifying the reasoning for this conclusion
Additional ResourcesTo assign this assessment online login to your Assignment Manager at myhrwcom
Assessment Readiness
California Common Core Standards
Items Grade 8 Standards Mathematical Practices
1 8NS2 MP7
2 7NS2b 7NS2d 8NS1 MP7
3 8EE2 MP1 MP4
4 8NS1 8NS2 MP3
Item integrates mixed review concepts from previous modules or a previous course
Item 4 combines concepts from the California Common Core cluster ldquoKnow that there are numbers that are not rational and approximate them by rational numbersrdquo
Real Numbers 28
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
Work Area
π mi23 Critique Reasoning The circumference of a circular region is shown
What type of number best describes the diameter of the circle Explain
your answer
24 Critical Thinking A number is not an integer What type of number can it be
25 A grocery store has a shelf with half-gallon containers of milk What type of number best represents the total number of gallons
26 Explain the Error Katie said ldquoNegative numbers are integersrdquo What was her error
27 Justify Reasoning Can you ever use a calculator to determine if a number is rational or irrational Explain
28 Draw Conclusions The decimal 0 _
3 represents 1 _ 3 What type of number best describes 0
_ 9 which is 3 middot 0
_ 3 Explain
29 Communicate Mathematical Ideas Irrational numbers can never be precisely represented in decimal form Why is this
FOCUS ON HIGHER ORDER THINKING
It can be a rational number that is not an integer or an irrational number
rational number
The set of negative numbers also includes non-integer
rational numbers and irrational numbers
Sample answer If the calculator shows a decimal that
terminates in fewer digits than what the calculator screen
allows then you can tell that the number is rational If not
you cannot tell from the calculator display whether the
number terminates because you see a limited number
of digits It may be a repeating decimal (rational) or
non-terminating non-repeating decimal (irrational)
Whole 3 middot 0 _
3 represents 3 middot 1 _ 3 = 1 so 0 _
9 is exactly 1
Sample answer In decimal form irrational numbers never
terminate and never repeat Therefore no matter how
many decimal places you include the number will never
be precisely represented There are always more digits
Whole the diameter is π _ π = 1 mile
Unit 120
copy H
ough
ton
Miff
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ublis
hing
Com
pany
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
8_MCAAESE206984_U1M01L2indd 20 120413 909 PM
Integers
Rational Numbers Irrational Numbers
Real Numbers
Whole Numbers
257
radic16
166
radic9
128 radic50
Personal Math Trainer
Online Practice and Helpmyhrwcom
Name Class Date
Independent Practice
Identify the set of numbers that best describes each situation Explain your choice
20 the height of an airplane as it descends to an airport runway
21 the score with respect to par of several golfers 2 ndash 3 5 0 ndash 1
22 Critique Reasoning Ronald states that the number 1 __ 11 is not rational because when converted into a decimal it does not terminate Nathaniel says it is rational because it is a fraction Which boy is correct Explain
12
14 - radic_
9 15 257
16 radic_
50 17 8 1 _ 2
18 166 19 radic_
16
Write all names that apply to each number Then place the numbers in the correct location on the Venn diagram
8NS1
Real numbers the height can be any number greater than zero
integer rational real whole integer rational real
whole integer rational real
irrational real
rational real
rational real
Integers the scores are counting numbers their
opposites and zero
Nathaniel is correct A rational number is a number that can be written as a fraction and 1 __ 11 is a fraction
19Lesson 12
copy H
ough
ton
Miff
lin H
arco
urt P
ublis
hing
Com
pany
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
8_MCAAESE206984_U1M01L2indd 19 41613 136 AM
myhrwcomActivity available onlineEXTEND THE MATH PRE-AP
Activity Have students consider the concept of restricted domain for the sets of numbers that describe situations For example the number of sisters a person has can best be described by whole numbers but no one has ever had 1500 sisters An area code is an integer or whole number between 200 and 999
Have students use a source such as the Guinness Book of World Records and give examples of sets of numbers that describe situations where the domain is restricted Ask whether the restriction may be changed in the future
Sets of Real Numbers 20
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
-3-4-5 -2-1 1 2 3 50 4
12-4 -radic5
Lesson Support Content Objective Students will learn to order a set of real numbers
Language Objective Students will show and describe how to order a set of real numbers
LESSON 13 Ordering Real Numbers
Building BackgroundEliciting Prior Knowledge Have students draw a number line to compare a rational number and an irrational number such as - radic
_ 5 and -4 1 __ 2 Ask them to explain how
they approximated the irrational number on the number line Then have them identify the greater and the lesser real number Repeat with several other pairs of real numbers in different forms
Learning ProgressionsIn this lesson students order a set of real numbers They use rational approximations to compare the sizes of irrational numbers They also order numbers for real-world situations Important understandings for students include the following
bull Compare irrational numbers bull Estimate the value of expressions with irrational numbers bull Order a set of real numbers bull Order real numbers in a real-world context
Work with real numbers continues throughout Grade 8 and into high school This lesson provides students with a foundation for understanding the relative sizes of numbers in different forms in the real number system
Cluster ConnectionsThis lesson provides an excellent opportunity to connect ideas in this cluster Know that there are numbers that are not rational and approximate them by rational numbers Tell students that there is a special number called the golden ratio with applications in mathematics geometry art and architecture The golden ratio is called phi and is represented by the Greek letter ϕ It includes an irrational number in its definition
Have students explain why the golden ratio is irrational Ask them to find the two whole numbers the golden ratio lies between Then challenge them to approximate the golden ratio to the nearest tenth It is irrational because it includes an irrational number in its definition It lies between 1 and 2 To the nearest tenth ϕ = 16
ϕ = 1 + radic_
5 _ 2
Focus | Coherence | Rigor
California Common Core Standards
8NS2 Use rational approximations of irrational numbers to compare the size of irrational numbers locate them approximately on a number line diagram and estimate the value of expressions (eg π 2 )
MP4 Model with mathematics
21A
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
Math Talk
Language Support EL
PROFESSIONAL DEVELOPMENT
Linguistic Support EL
AcademicContent Vocabulary
Post a chart like this to remind students of the regular comparative forms of adjectives that use the -er and -est suffixes Add to the chart for terms that appear in examples and exercises in each lesson Include any irregular verb forms
Background Knowledge
Go On ndash the title of the module review or quiz is Ready to Go On This title uses an idiomatic expression In this context to go on means ldquoto move aheadrdquo or ldquoto proceedrdquo It is different from the use of go on that means having enough facts to use meaningfully as in having enough to go on Also the intonation used in pronouncing an expression can give it different meanings For example when the speaker emphasizes the word on he or she might be expressing disbelief as in ldquoGo ON Yoursquore kidding rightrdquo Discuss with students other ways that the phrase go on may be used
Leveled Strategies for English Learners
Emerging Label points on a number line with the terms used in ordering greater greatest less lesser least Use sentence frames to insert the correct terms
Expanding Have students give two or three complete sentences to compare the placement of numbers on a number line using the correct forms of the comparative and superlative adjectives
Bridging Have students work in pairs with one student giving directions to the other in complete sentences to order numbers on a number line
To help students answer the question posed in Math Talk make sure that students have a command of the forms for making comparisons and the superlative and the concept of opposite order so that the focus is on the math concept instead of the language skills needed to describe and explain order
EL
Adjective Comparative Superlative
Far Farther Farthest
Large Larger Largest
Great Greater Greatest
Some Less Least
Some More Most
California ELD Standards
Emerging 2I8 Analyzing language choices ndash Explain how phrasing or different common words with similar meanings produce different effects on the audience
Expanding 2I8 Analyzing language choices ndash Explain how phrasing or different words with similar meanings or figurative language produce shades of meaning and different effects on the audience
Bridging 2I8 Analyzing language choices ndash Explain how phrasing or different words with similar meanings or figurative language produce shades of meaning nuances and different effects on the audience
Ordering Real Numbers 21B
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
13L E S S O N
Ordering Real Numbers
Interactive Whiteboard Interactive example available online
ADDITIONAL EXAMPLE 1Compare Write lt gt or =
A radic_
8 - 2 4 - radic_
8 lt
B radic_
20 + 1 3 + radic_
2 gt
EngageESSENTIAL QUESTION
How do you order a set of real numbers Sample answer Find their approximate decimal values and order them
Motivate the LessonAsk What kind of numbers are you comparing when you compare the price of gasoline at two different gas stations
ExploreGive students two rational numbers and ask them to name a number between them Repeat a few times and then give them two irrational numbers and ask them to name a number between them
ExplainEXAMPLE 1
Questioning Strategies Mathematical Practices bull Which is greater the difference between 5 and 3 or the difference between radic
_ 5 and radic
_ 3
The difference between 5 and 3 is 2 the difference between radic_
5 and radic_
3 is approximately 1 So the difference between 5 and 3 is greater
Avoid Common ErrorsCaution students to read the problem carefully and think about what the radical sign means so that they do not misread the problem and answer that the two sides are equal
YOUR TURNFocus on TechnologyCalculators should not be used at this point because developing number sense is the goal
EXAMPLE 2Questioning Strategies Mathematical Practices bull How do you determine whether radic
_ 22 is less than or greater than 45 The square of 45 is
2025 which is less than 22 so the square root of 22 must be greater than 45
Engage with the WhiteboardHave students graph and label various real numbers between 42 and 44 and between 47 and 5
YOUR TURNFocus on Modeling Mathematical PracticesHave students label the integers on the number line with their equivalent square root For example 1 2 and 3 on the number line would be labeled radic
_ 1 radic
_ 4 and radic
_ 9
Interactive Whiteboard Interactive example available online
ADDITIONAL EXAMPLE 2Order 3π radic
_ 10 and 325 from greatest
to least
3π 325 radic_
10
myhrwcom
myhrwcom
CA Common CoreStandards
The student is expected to
The Number Systemmdash8NS2
Use rational approximations of irrational numbers to compare the size of irrational numbers locate them approximately on a number line diagram and estimate the value of expressions (eg π 2 )
Mathematical Practices
MP4 Modeling
The student is expected to
21 Lesson 13
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
Math TrainerOnline Practice
and Help
Personal
myhrwcom
Math On the Spotmyhrwcom
0 05 1 15 2 25 3 35 4
radic5radic3
π2
8 85 9 95 10 105 11 115 12
radic75
4 42 44 46 48 5
radic224 12π + 1
Ordering Real Numbers You can compare and order real numbers and list them from least to greatest
Order radic_
22 π + 1 and 4 1 _ 2 from least to greatest
First approximate radic_
22
radic_
22 is between 4 and 5 Since you donrsquot know where it falls between 4 and 5 you need to find a better estimate for radic
_ 22 so
you can compare it to 4 1 _ 2
Since 22 is closer to 25 than 16 use squares of numbers between 45 and 5 to find a better estimate of radic
_ 22
45 2 = 2025 46 2 = 2116 47 2 = 2209 48 2 = 2304
Since 47 2 = 2209 an approximate value for radic_
22 is 47
An approximate value of π is 314 So an approximate value of π +1 is 414
Plot radic_
22 π + 1 and 4 1 _ 2 on a number line
Read the numbers from left to right to place them in order from least to greatest
From least to greatest the numbers are π + 1 4 1 _ 2 and radic_
22
EXAMPLE 2
STEP 1
STEP 2
Order the numbers from least to greatest Then graph them on the number line
YOUR TURN
5 radic_
5 25 radic_
3
6 π 2 10 radic_
75
If real numbers a b and c are in order from least to greatest what is the order
of their opposites from least to greatest
Explain
Math TalkMathematical Practices
8NS2
radic_
3 radic_
5 25
radic_
75 π2 10
Math Talk answer -c -b -a -c is farthest to the left on a number line -b is in the middle and -a is farthest to the right
Unit 122
copy H
ough
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Miff
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pany
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8_MCAAESE206984_U1M01L3indd 22 41613 447 AM
My Notes
Math TrainerOnline Practice
and Help
Personal
myhrwcom
Math On the Spot
myhrwcom
Comparing Irrational NumbersBetween any two real numbers is another real number To compare and order real numbers you can approximate irrational numbers as decimals
Compare radic_
3 + 5 3 + radic_
5 Write lt gt or =
First approximate radic_
3
radic_
3 is between 1 and 2
Next approximate radic_
5
radic_
5 is between 2 and 3
Then use your approximations to simplify the expressions
radic_
3 + 5 is between 6 and 7
3 + radic_
5 is between 5 and 6
So radic_
3 + 5 gt 3 + radic_
5
Reflect1 If 7 + radic
_ 5 is equal to radic
_ 5 plus a number what do you know about the
number Why
2 What are the closest two integers that radic_
300 is between
EXAMPLEXAMPLE 1
STEP 1
STEP 2
Compare Write lt gt or =
YOUR TURN
3 radic_
2 + 4 2 + radic_
4 4 radic_
12 + 6 12 + radic_
6
L E S S O N
13 Ordering Real Numbers
ESSENTIAL QUESTIONHow do you order a set of real numbers
8NS2
Use rational approximations of irrational numbers to compare the size of irrational numbers locate them approximately on a number line diagram and estimate the value of expressions (eg π 2 )
8NS2
Use perfect squares to estimate square roots
1 2 = 1 2 2 = 4 3 2 = 9
The number is 7 both expressions must equal 7 + radic_
5
17 and 18
gt lt
21Lesson 13
copy H
ough
ton
Miff
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pany
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
8_MCAAESE206984_U1M01L3indd 21 41913 246 PM
PROFESSIONAL DEVELOPMENT
Math BackgroundIn this lesson students estimate irrational numbers in the form of square roots of nonper-fect squares by finding two perfect squares between which the number falls A more precise method involves repeated division For example to find radic
_ 28 find a whole number whose perfect
square is close to 28 such as 5 Divide 28 by that number 28 divide 5 = 56 Find the average of the quotient and divisor 5 + 56
_____ 2 = 53 Continue dividing 28 by each result and averaging until you get the desired accuracy
Integrate Mathematical Practices MP4
This lesson provides an opportunity to address this Mathematical Practices standard It calls for students to model relationships using multiple representations including diagrams graphs and language as appropriate Students use multiple representations when they use number lines to estimate the locations of and order rational and irrational numbers given as symbols
Ordering Real Numbers 22
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
Interactive Whiteboard Interactive example available online
ADDITIONAL EXAMPLE 3The diameter of a meteorite in millimeters is calculated by four different methods Order the results from least to greatest
Joe radic_
18 mm Lisa 13 __ 3 mm
Pablo 46 mm Julien 4π __ 3 mm
Julien 4π __ 3 mm Lisa 13 __ 3 mm
Joe radic_
18 mm Pablo 46 mm
EXAMPLE 3Questioning Strategies Mathematical Practices bull How can you verify that radic
_ 28 is between 52 and 53 5 2 2 = 2704 and 5 3 2 = 2809
bull Explain how to determine which number is greater 5 _
5 or 55 When the repeating decimal is rounded to the nearest tenth or hundredth you can see that it is greater
Connect to Daily LifeDiscuss how measuring across a canyon might involve different methods than measuring along a road Explain that measurements like these are often done using calculations that approximate the distance
YOUR TURNFocus on Critical Thinking Mathematical PracticesDiscuss with students which number is greater 3
_ 45 or 3450 3
_ 45 or 3455 and why Explain
that 3 _
45 can be written out as 34545hellipMake sure they understand that 3 _
45 is greater than 345 but less than 3455
ElaborateTalk About ItSummarize the Lesson
Ask How can you order two numbers in different forms whose decimal approxi-mations appear to be equal Approximate one or both numbers to an additional
number of decimal places
GUIDED PRACTICEEngage with the Whiteboard
Have students place and label additional points on the number line in Exercise 9 Allow the points to be in any format other than decimal
Avoid Common ErrorsExercises 3ndash4 Caution students to read the problem carefully so that they do not misread the problem as the same numbers combined by addition on each side of the circleExercise 10 Remind students that the calculations have units
myhrwcom
23 Lesson 13
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
0 05 1 15 2 25 3 35 4 45 5 55 6 65 7
2πradic3
Compare Write lt gt or = (Example 1)
1 radic_
3 + 2 radic_
3 + 3 2 radic_
8 + 17 radic_
11 + 15
3 radic_
6 + 5 6 + radic_
5 4 radic_
9 + 3 9 + radic_
3
5 radic_
17 - 3 -2 + radic_
5 6 12 - radic_
2 14 - radic_
8
7 radic_
7 + 2 radic_
10 - 1 8 radic_
17 + 3 3 + radic_
11
9 Order radic_
3 2π and 15 from least to greatest Then graph them on the number line (Example 2)
radic_
3 is between and so radic_
3 asymp
π asymp 314 so 2π asymp
From least to greatest the numbers are
10 Four people have found the perimeter of a forest using different methods Their results are given in the table Order their calculations from greatest to least (Example 3)
11 Explain how to order a set of real numbers
CHECK-INESSENTIAL QUESTION
Forest Perimeter (km)
Leon Mika Jason Ashley
radic_
17 - 2 1 +thinsp π __ 2 12 ___ 5 25
Guided Practice
17
15
1 + π _ 2 km 25 km 12 __ 5 km radic_
17 - 2 km
2π radic
_ 3
18 175
628
Sample answer Convert each number to a decimal
equivalent using estimation to find equivalents for
irrational numbers Graph each number on a number line
Read the numbers from left to right for least to greatest
Read the numbers from right to left for greatest to least
lt gt
lt lt
ltgt
gt gt
24 Unit 1
copy H
ough
ton
Miff
lin H
arco
urt P
ublis
hing
Com
pany
bull Im
age C
redi
ts copy
Elena
Eliss
eeva
Alam
y Im
ages
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
8_MCAAESE206984_U1M01L3indd 24 41613 448 AM
My Notes
5 52 54 56 58 6
radic28 5 12
23455
Math TrainerOnline Practice
and Help
Personal
myhrwcom
Math On the Spot
myhrwcom
Ordering Real Numbers in a Real-World Context Calculations and estimations in the real world may differ It can be important to know not only which are the most accurate but which give the greatest or least values depending upon the context
Four people have found the distance in kilometers across a canyon using different methods Their results are given in the table Order the distances from greatest to least
Distance Across Quarry Canyon (km)
Juana Lee Ann Ryne Jackson
radic_
28 23 __ 4 5 _
5 5 1 _ 2
Write each value as a decimal
radic_
28 is between 52 and 53 Since 53 2 = 2809 an approximate value for radic
_ 28 is 53
23 __ 4 = 575
5 _
5 is 5555hellip so 5 _
5 to the nearest hundredth is 556
5 1 _ 2 = 55
Plot radic_
28 23 __ 4 5 _
5 and 5 1 _ 2 on a number line
From greatest to least the distances are
23 __ 4 km 5 _
5 km 5 1 _ 2 km radic_
28 km
EXAMPLEXAMPLE 3
STEP 1
STEP 2
7 Four people have found the distance in miles across a crater using different methods Their results are given below
Jonathan 10 __ 3 Elaine 3 _
45 Joseacute 3 1 _ 2 Lashonda radic_
10
Order the distances from greatest to least
YOUR TURN
8NS2
3 1 _ 2 mi 3 _
45 mi 10 __ 3 mi radic_
10 mi
23Lesson 13
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ough
ton
Miff
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pany
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8_MCAAESE206984_U1M01L3indd 23 41613 447 AM
ModelingPlace papers around the room with the numbers from 1 to 5 one per sheet Give each student a card showing a number between 1 and 5 in different forms Have students place his or her card between the correct integers and decide where the number goes in relation to any numbers already placed
Multiple RepresentationsGive students a vertical number line which some students might find easier to use than a horizontal one Have them decide whether to place points for rational and irrational numbers above or below existing points
Additional ResourcesDifferentiated Instruction includes bull Reading Strategies bull Success for English Learners EL
bull Reteach bull Challenge PRE-AP
DIFFERENTIATE INSTRUCTION
Ordering Real Numbers 24
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
Math TrainerOnline Assessment
and Intervention
Personal
Online homework assignment available
myhrwcom
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Lesson Quiz available online
13 LESSON QUIZ
1 Compare Write lt gt or =
radic_
95 - 5 radic_
62 - 2
2 Order 105 radic_
105 and 3π + 1 from greatest to least
3 A length in centimeters is calculated differently by four different people Order their calculations from least to greatest
KD 11 __ 2 cm Silvio 5 __ 3 π cm
Paula 5 _
4 cm Luis radic_
33 cm
EvaluateGUIDED AND INDEPENDENT PRACTICE
Concepts amp Skills Practice
Example 1Comparing Irrational Numbers
Exercises 1ndash8
Example 2Ordering Real Numbers
Exercises 9 12ndash15 18ndash21
Example 3Ordering Real Numbers in a Real-World Context
Exercises 10 16ndash17
Additional ResourcesDifferentiated Instruction includes bull Leveled Practice worksheets
Focus | Coherence | Rigor
Exercise Depth of Knowledge (DOK) Mathematical Practices
12ndash15 1 Recall of Information MP5 Using Tools
16 2 SkillsConcepts MP2 Reasoning
17 2 SkillsConcepts MP6 Precision
18ndash21 2 SkillsConcepts MP2 Reasoning
22 3 Strategic Thinking MP4 Modeling
23ndash24 3 Strategic Thinking MP3 Logic
8NS2
8NS2
Answers1 radic
_ 95 - 5 lt radic
_ 62 - 2
2 radic_
105 3π + 1 105
3 Silvio 5 __ 3 π cm Paula 5 _
4 cm
KD 11
__ 2 cm Luis radic_
33 cm
25 Lesson 13
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
Work Area
3140 3141 3142 3143
314 π227
20 A teacher asks his students to write the numbers shown in order from least to greatest Paul thinks the numbers are already in order Sandra thinks the order should be reversed Who is right
21 Math History There is a famous irrational number called Eulerrsquos number symbolized with an e Like π its decimal form never ends or repeats The first few digits of e are 27182818284
a Between which two square roots of integers could you find this number
b Between which two square roots of integers can you find π
22 Analyze Relationships There are several approximations used for π including 314 and 22 __ 7 π is approximately 314159265358979
a Label π and the two approximations on the number line
b Which of the two approximations is a better estimate for π Explain
c Find a whole number x so that the ratio x ___ 113 is a better estimate for π
than the two given approximations
23 Communicate Mathematical Ideas If a set of six numbers that include both rational and irrational numbers is graphed on a number line what is the fewest number of distinct points that need to be graphed Explain
24 Critique Reasoning Jill says that 12 _
6 is less than 1263 Explain her error
FOCUS ON HIGHER ORDER THINKING
radic_
115 115 ___ 11 and 105624
between radic_
7 asymp 265 and radic_
8 asymp 283
between radic_
9 = 3 and radic_
10 asymp 316
22 __ 7 it is closer to π on the number line
She did not consider the repeating digit 1266
2 rational numbers can have the same location and
irrational numbers can have the same location but they
cannot share a location
355
Neither student is correct The answer
should be 115 ___ 11 105624 radic_
115
Unit 126
copy H
ough
ton M
ifflin
Har
cour
t Pub
lishin
g Com
pany
Imag
e Cre
dits
copy3D
Stoc
kiSt
ockP
hoto
com
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
8_MCAAESE206984_U1M01L3indd 26 210513 801 AM
Personal Math Trainer
Online Practice and Helpmyhrwcom
Name Class Date
Independent Practice
16 Your sister is considering two different shapes for her garden One is a square with side lengths of 35 meters and the other is a circle with a diameter of 4 meters
a Find the area of the square
b Find the area of the circle
c Compare your answers from parts a and b Which garden would give your sister the most space to plant
17 Winnie measured the length of her fatherrsquos ranch four times and got four different distances Her measurements are shown in the table
a To estimate the actual length Winnie first approximated each distance to the nearest hundredth Then she averaged the four numbers Using a calculator find Winniersquos estimate
b Winniersquos father estimated the distance across his ranch to be radic_
56 km How does this distance compare to Winniersquos estimate
Give an example of each type of number
18 a real number between radic_
13 and radic_
14
19 an irrational number between 5 and 7
Order the numbers from least to greatest
12 radic_
7 2 radic_
8 ___ 2 13 radic_
10 π 35
14 radic_
220 -10 radic_
100 115 15 radic_
8 -375 3 9 _ 4
Distance Across Fatherrsquos Ranch (km)
1 2 3 4
radic_
60 58 __ 8 7 _
3 7 3 _ 5
138NS2
radic_
8 ___ 2 2 radic_
7
-10 radic_
100 115 radic_
220
radic_
60 asymp 775 58 __ 8 = 725 7 _
3 asymp 733 7 3 _ 5 = 760 so the average
π radic_
10 35
-375 9 _ 4 radic_
8 3
is 74825 km
1225 m2
4π m2 or approximately 126 m2
They are nearly identical radic_
56 is approximately 74833hellip
The circle would give her more space to plant because it has a
larger area
Sample answer 37
Sample answer radic_
31
25Lesson 13
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ough
ton
Miff
lin H
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Com
pany
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8_MCAAESE206984_U1M01L3indd 25 41613 448 AM
Activity available online myhrwcomEXTEND THE MATH PRE-AP
Activity Have students investigate whether there are infinitely many numbers between two numbers by giving examples for each of the following
bull Between any two rational numbers there is at least one other rational number Sample answer 45 is between 41 and 48
bull Between any two irrational numbers there is at least one rational number Sample answer 45 is between radic
_ 11 and radic
_ 29
bull Between any two rational numbers there is at least one irrational number Sample answer radic
_ 11 is between 31 and 36
bull Between any two irrational numbers there is at least one irrational number Sample answer radic
_ 17 is between radic
_ 11 and radic
_ 29
Ordering Real Numbers 26
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
ReadyMath Trainer
Online Practiceand Help
Personal
myhrwcom
Module Quiz
11ensp RationalenspandenspIrrationalenspNumbersWrite each fraction as a decimal or each decimal as a fraction
1 7__20 2 1___
27 3 17_8
Solve each equation for x
4 x2=81 5 x3=343 6 x2= 1___100
7 Asquarepatiohasanareaof200squarefeetHowlongiseachside
ofthepatiotothenearesttenth
12ensp SetsenspofenspRealenspNumbersWrite all names that apply to each number
8 121____radic
____121
9 π__2
10 TellwhetherthestatementldquoAllintegersarerationalnumbersrdquoistrueorfalseExplainyourchoice
13ensp OrderingenspRealenspNumbersCompare Write lt gt or =
11 radic__
8+3 8+radic__
3 12 radic__
5+11emsp emsp emsp 5+radic___
11
Order the numbers from least to greatest
13 radic___
99π29__
8 14 radic___
1__251_40__
2
15 Howarerealnumbersusedtodescribereal-worldsituations
ESSENTIAL QUESTION
035
9-9
141ft
7 1__10- 1__10
14__11 1875
wholeintegerrationalreal
Trueintegerscanbewrittenasthequotientoftwointegers
SampleanswerRealnumberssuchastherational
π29__
8radic___
99
irrationalreal
lt gt
number1_4candescribeamountsusedincooking
radic___
1__250__
21_4
27Module1
copy H
ough
ton
Miff
lin H
arco
urt P
ublis
hing
Com
pany
DONOTEDIT--ChangesmustbemadethroughldquoFileinfordquoCorrectionKey=A
8_MCAAESE206984_U1M01RTindd 27 41513 1113 PM
Math TrainerOnline Assessment
and Intervention
Personal
myhrwcom
1
2
3 Response toIntervention
Intervention Enrichment
Access Ready to Go On assessment online and receive instant scoring feedback and customized intervention or enrichment
Online and Print Resources
Differentiated Instruction
bull Reteach worksheets
bull Reading Strategies EL
bull Success for English Learners EL
Differentiated Instruction
bull Challenge worksheets PRE-AP
Extend the Math PRE-AP
Lesson Activities in TE
Additional ResourcesAssessment Resources includes bull Leveled Module Quizzes
Ready to Go OnAssess MasteryUse the assessment on this page to determine if students have mastered the concepts and standards covered in this module
California Common Core Standards
Lesson Exercises Common Core Standards
11 1ndash7 8NS1 8NS2 8EE2
12 8ndash10 8NS1
13 11ndash14 8NS2
27 Unit 1 Module 1
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
Personal Math Trainer
Online Practice and HelpmyhrwcomAssessment Readiness
Module 1 MIXed ReVIeW
1 Look at each number Is the number between 2π and radic___
52
Select Yes or No for expressions AndashC
A 6 2 _ 3 Yes No
B 5π __ 2 Yes No
C 3 radic__
5 Yes No
2 Consider the number - 11 __ 15
Choose True or False for each statement
A The number is rational True False
B The number can be written as True Falsea repeating decimal
C The number is less than ndash08 True False
3 The volume of a cube is given by V = x3 where x is the length of an edge of the cube A cube-shaped end table has a volume of 3 3 _ 8 cubic feet What is the length of an edge of the end table Explain how you solved this problem
4 A student says that radic___
83 is greater than 29 __ 3 Is the student correct Justify your
reasoning
1 1 _ 2 ft Sample answer The equation x3 = 3 3 _ 8 can be used
to find the edge length in feet To solve the equation
write the mixed number as a fraction greater than 1
x3 = 27 __ 8 Then take the cube root of both sides x = 3 _ 2 = 1 1 _ 2
No Sample answer radic___
83 asymp 91 and 29 __ 3 = 9
__ 6
Because 91 lt 9 __
6 radic___
83 lt 29 __ 3
28 Unit 1
copy H
ough
ton
Miff
lin H
arco
urt P
ublis
hing
Com
pany
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=A
8_MCAAESE206984_U1M01RTindd 28 240413 946 AM
Personal Math Trainer
Online Assessment and
Interventionmyhrwcom
Scoring GuideItem 3 Award the student 1 point for finding the edge length of the cube and 1 point for correctly explaining how to use a cube root to solve the problem
Item 4 Award the student 1 point for determining that the student is incorrect and 1 point for correctly justifying the reasoning for this conclusion
Additional ResourcesTo assign this assessment online login to your Assignment Manager at myhrwcom
Assessment Readiness
California Common Core Standards
Items Grade 8 Standards Mathematical Practices
1 8NS2 MP7
2 7NS2b 7NS2d 8NS1 MP7
3 8EE2 MP1 MP4
4 8NS1 8NS2 MP3
Item integrates mixed review concepts from previous modules or a previous course
Item 4 combines concepts from the California Common Core cluster ldquoKnow that there are numbers that are not rational and approximate them by rational numbersrdquo
Real Numbers 28
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
-3-4-5 -2-1 1 2 3 50 4
12-4 -radic5
Lesson Support Content Objective Students will learn to order a set of real numbers
Language Objective Students will show and describe how to order a set of real numbers
LESSON 13 Ordering Real Numbers
Building BackgroundEliciting Prior Knowledge Have students draw a number line to compare a rational number and an irrational number such as - radic
_ 5 and -4 1 __ 2 Ask them to explain how
they approximated the irrational number on the number line Then have them identify the greater and the lesser real number Repeat with several other pairs of real numbers in different forms
Learning ProgressionsIn this lesson students order a set of real numbers They use rational approximations to compare the sizes of irrational numbers They also order numbers for real-world situations Important understandings for students include the following
bull Compare irrational numbers bull Estimate the value of expressions with irrational numbers bull Order a set of real numbers bull Order real numbers in a real-world context
Work with real numbers continues throughout Grade 8 and into high school This lesson provides students with a foundation for understanding the relative sizes of numbers in different forms in the real number system
Cluster ConnectionsThis lesson provides an excellent opportunity to connect ideas in this cluster Know that there are numbers that are not rational and approximate them by rational numbers Tell students that there is a special number called the golden ratio with applications in mathematics geometry art and architecture The golden ratio is called phi and is represented by the Greek letter ϕ It includes an irrational number in its definition
Have students explain why the golden ratio is irrational Ask them to find the two whole numbers the golden ratio lies between Then challenge them to approximate the golden ratio to the nearest tenth It is irrational because it includes an irrational number in its definition It lies between 1 and 2 To the nearest tenth ϕ = 16
ϕ = 1 + radic_
5 _ 2
Focus | Coherence | Rigor
California Common Core Standards
8NS2 Use rational approximations of irrational numbers to compare the size of irrational numbers locate them approximately on a number line diagram and estimate the value of expressions (eg π 2 )
MP4 Model with mathematics
21A
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
Math Talk
Language Support EL
PROFESSIONAL DEVELOPMENT
Linguistic Support EL
AcademicContent Vocabulary
Post a chart like this to remind students of the regular comparative forms of adjectives that use the -er and -est suffixes Add to the chart for terms that appear in examples and exercises in each lesson Include any irregular verb forms
Background Knowledge
Go On ndash the title of the module review or quiz is Ready to Go On This title uses an idiomatic expression In this context to go on means ldquoto move aheadrdquo or ldquoto proceedrdquo It is different from the use of go on that means having enough facts to use meaningfully as in having enough to go on Also the intonation used in pronouncing an expression can give it different meanings For example when the speaker emphasizes the word on he or she might be expressing disbelief as in ldquoGo ON Yoursquore kidding rightrdquo Discuss with students other ways that the phrase go on may be used
Leveled Strategies for English Learners
Emerging Label points on a number line with the terms used in ordering greater greatest less lesser least Use sentence frames to insert the correct terms
Expanding Have students give two or three complete sentences to compare the placement of numbers on a number line using the correct forms of the comparative and superlative adjectives
Bridging Have students work in pairs with one student giving directions to the other in complete sentences to order numbers on a number line
To help students answer the question posed in Math Talk make sure that students have a command of the forms for making comparisons and the superlative and the concept of opposite order so that the focus is on the math concept instead of the language skills needed to describe and explain order
EL
Adjective Comparative Superlative
Far Farther Farthest
Large Larger Largest
Great Greater Greatest
Some Less Least
Some More Most
California ELD Standards
Emerging 2I8 Analyzing language choices ndash Explain how phrasing or different common words with similar meanings produce different effects on the audience
Expanding 2I8 Analyzing language choices ndash Explain how phrasing or different words with similar meanings or figurative language produce shades of meaning and different effects on the audience
Bridging 2I8 Analyzing language choices ndash Explain how phrasing or different words with similar meanings or figurative language produce shades of meaning nuances and different effects on the audience
Ordering Real Numbers 21B
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
13L E S S O N
Ordering Real Numbers
Interactive Whiteboard Interactive example available online
ADDITIONAL EXAMPLE 1Compare Write lt gt or =
A radic_
8 - 2 4 - radic_
8 lt
B radic_
20 + 1 3 + radic_
2 gt
EngageESSENTIAL QUESTION
How do you order a set of real numbers Sample answer Find their approximate decimal values and order them
Motivate the LessonAsk What kind of numbers are you comparing when you compare the price of gasoline at two different gas stations
ExploreGive students two rational numbers and ask them to name a number between them Repeat a few times and then give them two irrational numbers and ask them to name a number between them
ExplainEXAMPLE 1
Questioning Strategies Mathematical Practices bull Which is greater the difference between 5 and 3 or the difference between radic
_ 5 and radic
_ 3
The difference between 5 and 3 is 2 the difference between radic_
5 and radic_
3 is approximately 1 So the difference between 5 and 3 is greater
Avoid Common ErrorsCaution students to read the problem carefully and think about what the radical sign means so that they do not misread the problem and answer that the two sides are equal
YOUR TURNFocus on TechnologyCalculators should not be used at this point because developing number sense is the goal
EXAMPLE 2Questioning Strategies Mathematical Practices bull How do you determine whether radic
_ 22 is less than or greater than 45 The square of 45 is
2025 which is less than 22 so the square root of 22 must be greater than 45
Engage with the WhiteboardHave students graph and label various real numbers between 42 and 44 and between 47 and 5
YOUR TURNFocus on Modeling Mathematical PracticesHave students label the integers on the number line with their equivalent square root For example 1 2 and 3 on the number line would be labeled radic
_ 1 radic
_ 4 and radic
_ 9
Interactive Whiteboard Interactive example available online
ADDITIONAL EXAMPLE 2Order 3π radic
_ 10 and 325 from greatest
to least
3π 325 radic_
10
myhrwcom
myhrwcom
CA Common CoreStandards
The student is expected to
The Number Systemmdash8NS2
Use rational approximations of irrational numbers to compare the size of irrational numbers locate them approximately on a number line diagram and estimate the value of expressions (eg π 2 )
Mathematical Practices
MP4 Modeling
The student is expected to
21 Lesson 13
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
Math TrainerOnline Practice
and Help
Personal
myhrwcom
Math On the Spotmyhrwcom
0 05 1 15 2 25 3 35 4
radic5radic3
π2
8 85 9 95 10 105 11 115 12
radic75
4 42 44 46 48 5
radic224 12π + 1
Ordering Real Numbers You can compare and order real numbers and list them from least to greatest
Order radic_
22 π + 1 and 4 1 _ 2 from least to greatest
First approximate radic_
22
radic_
22 is between 4 and 5 Since you donrsquot know where it falls between 4 and 5 you need to find a better estimate for radic
_ 22 so
you can compare it to 4 1 _ 2
Since 22 is closer to 25 than 16 use squares of numbers between 45 and 5 to find a better estimate of radic
_ 22
45 2 = 2025 46 2 = 2116 47 2 = 2209 48 2 = 2304
Since 47 2 = 2209 an approximate value for radic_
22 is 47
An approximate value of π is 314 So an approximate value of π +1 is 414
Plot radic_
22 π + 1 and 4 1 _ 2 on a number line
Read the numbers from left to right to place them in order from least to greatest
From least to greatest the numbers are π + 1 4 1 _ 2 and radic_
22
EXAMPLE 2
STEP 1
STEP 2
Order the numbers from least to greatest Then graph them on the number line
YOUR TURN
5 radic_
5 25 radic_
3
6 π 2 10 radic_
75
If real numbers a b and c are in order from least to greatest what is the order
of their opposites from least to greatest
Explain
Math TalkMathematical Practices
8NS2
radic_
3 radic_
5 25
radic_
75 π2 10
Math Talk answer -c -b -a -c is farthest to the left on a number line -b is in the middle and -a is farthest to the right
Unit 122
copy H
ough
ton
Miff
lin H
arco
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ublis
hing
Com
pany
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8_MCAAESE206984_U1M01L3indd 22 41613 447 AM
My Notes
Math TrainerOnline Practice
and Help
Personal
myhrwcom
Math On the Spot
myhrwcom
Comparing Irrational NumbersBetween any two real numbers is another real number To compare and order real numbers you can approximate irrational numbers as decimals
Compare radic_
3 + 5 3 + radic_
5 Write lt gt or =
First approximate radic_
3
radic_
3 is between 1 and 2
Next approximate radic_
5
radic_
5 is between 2 and 3
Then use your approximations to simplify the expressions
radic_
3 + 5 is between 6 and 7
3 + radic_
5 is between 5 and 6
So radic_
3 + 5 gt 3 + radic_
5
Reflect1 If 7 + radic
_ 5 is equal to radic
_ 5 plus a number what do you know about the
number Why
2 What are the closest two integers that radic_
300 is between
EXAMPLEXAMPLE 1
STEP 1
STEP 2
Compare Write lt gt or =
YOUR TURN
3 radic_
2 + 4 2 + radic_
4 4 radic_
12 + 6 12 + radic_
6
L E S S O N
13 Ordering Real Numbers
ESSENTIAL QUESTIONHow do you order a set of real numbers
8NS2
Use rational approximations of irrational numbers to compare the size of irrational numbers locate them approximately on a number line diagram and estimate the value of expressions (eg π 2 )
8NS2
Use perfect squares to estimate square roots
1 2 = 1 2 2 = 4 3 2 = 9
The number is 7 both expressions must equal 7 + radic_
5
17 and 18
gt lt
21Lesson 13
copy H
ough
ton
Miff
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Com
pany
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8_MCAAESE206984_U1M01L3indd 21 41913 246 PM
PROFESSIONAL DEVELOPMENT
Math BackgroundIn this lesson students estimate irrational numbers in the form of square roots of nonper-fect squares by finding two perfect squares between which the number falls A more precise method involves repeated division For example to find radic
_ 28 find a whole number whose perfect
square is close to 28 such as 5 Divide 28 by that number 28 divide 5 = 56 Find the average of the quotient and divisor 5 + 56
_____ 2 = 53 Continue dividing 28 by each result and averaging until you get the desired accuracy
Integrate Mathematical Practices MP4
This lesson provides an opportunity to address this Mathematical Practices standard It calls for students to model relationships using multiple representations including diagrams graphs and language as appropriate Students use multiple representations when they use number lines to estimate the locations of and order rational and irrational numbers given as symbols
Ordering Real Numbers 22
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
Interactive Whiteboard Interactive example available online
ADDITIONAL EXAMPLE 3The diameter of a meteorite in millimeters is calculated by four different methods Order the results from least to greatest
Joe radic_
18 mm Lisa 13 __ 3 mm
Pablo 46 mm Julien 4π __ 3 mm
Julien 4π __ 3 mm Lisa 13 __ 3 mm
Joe radic_
18 mm Pablo 46 mm
EXAMPLE 3Questioning Strategies Mathematical Practices bull How can you verify that radic
_ 28 is between 52 and 53 5 2 2 = 2704 and 5 3 2 = 2809
bull Explain how to determine which number is greater 5 _
5 or 55 When the repeating decimal is rounded to the nearest tenth or hundredth you can see that it is greater
Connect to Daily LifeDiscuss how measuring across a canyon might involve different methods than measuring along a road Explain that measurements like these are often done using calculations that approximate the distance
YOUR TURNFocus on Critical Thinking Mathematical PracticesDiscuss with students which number is greater 3
_ 45 or 3450 3
_ 45 or 3455 and why Explain
that 3 _
45 can be written out as 34545hellipMake sure they understand that 3 _
45 is greater than 345 but less than 3455
ElaborateTalk About ItSummarize the Lesson
Ask How can you order two numbers in different forms whose decimal approxi-mations appear to be equal Approximate one or both numbers to an additional
number of decimal places
GUIDED PRACTICEEngage with the Whiteboard
Have students place and label additional points on the number line in Exercise 9 Allow the points to be in any format other than decimal
Avoid Common ErrorsExercises 3ndash4 Caution students to read the problem carefully so that they do not misread the problem as the same numbers combined by addition on each side of the circleExercise 10 Remind students that the calculations have units
myhrwcom
23 Lesson 13
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
0 05 1 15 2 25 3 35 4 45 5 55 6 65 7
2πradic3
Compare Write lt gt or = (Example 1)
1 radic_
3 + 2 radic_
3 + 3 2 radic_
8 + 17 radic_
11 + 15
3 radic_
6 + 5 6 + radic_
5 4 radic_
9 + 3 9 + radic_
3
5 radic_
17 - 3 -2 + radic_
5 6 12 - radic_
2 14 - radic_
8
7 radic_
7 + 2 radic_
10 - 1 8 radic_
17 + 3 3 + radic_
11
9 Order radic_
3 2π and 15 from least to greatest Then graph them on the number line (Example 2)
radic_
3 is between and so radic_
3 asymp
π asymp 314 so 2π asymp
From least to greatest the numbers are
10 Four people have found the perimeter of a forest using different methods Their results are given in the table Order their calculations from greatest to least (Example 3)
11 Explain how to order a set of real numbers
CHECK-INESSENTIAL QUESTION
Forest Perimeter (km)
Leon Mika Jason Ashley
radic_
17 - 2 1 +thinsp π __ 2 12 ___ 5 25
Guided Practice
17
15
1 + π _ 2 km 25 km 12 __ 5 km radic_
17 - 2 km
2π radic
_ 3
18 175
628
Sample answer Convert each number to a decimal
equivalent using estimation to find equivalents for
irrational numbers Graph each number on a number line
Read the numbers from left to right for least to greatest
Read the numbers from right to left for greatest to least
lt gt
lt lt
ltgt
gt gt
24 Unit 1
copy H
ough
ton
Miff
lin H
arco
urt P
ublis
hing
Com
pany
bull Im
age C
redi
ts copy
Elena
Eliss
eeva
Alam
y Im
ages
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
8_MCAAESE206984_U1M01L3indd 24 41613 448 AM
My Notes
5 52 54 56 58 6
radic28 5 12
23455
Math TrainerOnline Practice
and Help
Personal
myhrwcom
Math On the Spot
myhrwcom
Ordering Real Numbers in a Real-World Context Calculations and estimations in the real world may differ It can be important to know not only which are the most accurate but which give the greatest or least values depending upon the context
Four people have found the distance in kilometers across a canyon using different methods Their results are given in the table Order the distances from greatest to least
Distance Across Quarry Canyon (km)
Juana Lee Ann Ryne Jackson
radic_
28 23 __ 4 5 _
5 5 1 _ 2
Write each value as a decimal
radic_
28 is between 52 and 53 Since 53 2 = 2809 an approximate value for radic
_ 28 is 53
23 __ 4 = 575
5 _
5 is 5555hellip so 5 _
5 to the nearest hundredth is 556
5 1 _ 2 = 55
Plot radic_
28 23 __ 4 5 _
5 and 5 1 _ 2 on a number line
From greatest to least the distances are
23 __ 4 km 5 _
5 km 5 1 _ 2 km radic_
28 km
EXAMPLEXAMPLE 3
STEP 1
STEP 2
7 Four people have found the distance in miles across a crater using different methods Their results are given below
Jonathan 10 __ 3 Elaine 3 _
45 Joseacute 3 1 _ 2 Lashonda radic_
10
Order the distances from greatest to least
YOUR TURN
8NS2
3 1 _ 2 mi 3 _
45 mi 10 __ 3 mi radic_
10 mi
23Lesson 13
copy H
ough
ton
Miff
lin H
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ublis
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Com
pany
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8_MCAAESE206984_U1M01L3indd 23 41613 447 AM
ModelingPlace papers around the room with the numbers from 1 to 5 one per sheet Give each student a card showing a number between 1 and 5 in different forms Have students place his or her card between the correct integers and decide where the number goes in relation to any numbers already placed
Multiple RepresentationsGive students a vertical number line which some students might find easier to use than a horizontal one Have them decide whether to place points for rational and irrational numbers above or below existing points
Additional ResourcesDifferentiated Instruction includes bull Reading Strategies bull Success for English Learners EL
bull Reteach bull Challenge PRE-AP
DIFFERENTIATE INSTRUCTION
Ordering Real Numbers 24
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
Math TrainerOnline Assessment
and Intervention
Personal
Online homework assignment available
myhrwcom
myhrwcom
Lesson Quiz available online
13 LESSON QUIZ
1 Compare Write lt gt or =
radic_
95 - 5 radic_
62 - 2
2 Order 105 radic_
105 and 3π + 1 from greatest to least
3 A length in centimeters is calculated differently by four different people Order their calculations from least to greatest
KD 11 __ 2 cm Silvio 5 __ 3 π cm
Paula 5 _
4 cm Luis radic_
33 cm
EvaluateGUIDED AND INDEPENDENT PRACTICE
Concepts amp Skills Practice
Example 1Comparing Irrational Numbers
Exercises 1ndash8
Example 2Ordering Real Numbers
Exercises 9 12ndash15 18ndash21
Example 3Ordering Real Numbers in a Real-World Context
Exercises 10 16ndash17
Additional ResourcesDifferentiated Instruction includes bull Leveled Practice worksheets
Focus | Coherence | Rigor
Exercise Depth of Knowledge (DOK) Mathematical Practices
12ndash15 1 Recall of Information MP5 Using Tools
16 2 SkillsConcepts MP2 Reasoning
17 2 SkillsConcepts MP6 Precision
18ndash21 2 SkillsConcepts MP2 Reasoning
22 3 Strategic Thinking MP4 Modeling
23ndash24 3 Strategic Thinking MP3 Logic
8NS2
8NS2
Answers1 radic
_ 95 - 5 lt radic
_ 62 - 2
2 radic_
105 3π + 1 105
3 Silvio 5 __ 3 π cm Paula 5 _
4 cm
KD 11
__ 2 cm Luis radic_
33 cm
25 Lesson 13
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
Work Area
3140 3141 3142 3143
314 π227
20 A teacher asks his students to write the numbers shown in order from least to greatest Paul thinks the numbers are already in order Sandra thinks the order should be reversed Who is right
21 Math History There is a famous irrational number called Eulerrsquos number symbolized with an e Like π its decimal form never ends or repeats The first few digits of e are 27182818284
a Between which two square roots of integers could you find this number
b Between which two square roots of integers can you find π
22 Analyze Relationships There are several approximations used for π including 314 and 22 __ 7 π is approximately 314159265358979
a Label π and the two approximations on the number line
b Which of the two approximations is a better estimate for π Explain
c Find a whole number x so that the ratio x ___ 113 is a better estimate for π
than the two given approximations
23 Communicate Mathematical Ideas If a set of six numbers that include both rational and irrational numbers is graphed on a number line what is the fewest number of distinct points that need to be graphed Explain
24 Critique Reasoning Jill says that 12 _
6 is less than 1263 Explain her error
FOCUS ON HIGHER ORDER THINKING
radic_
115 115 ___ 11 and 105624
between radic_
7 asymp 265 and radic_
8 asymp 283
between radic_
9 = 3 and radic_
10 asymp 316
22 __ 7 it is closer to π on the number line
She did not consider the repeating digit 1266
2 rational numbers can have the same location and
irrational numbers can have the same location but they
cannot share a location
355
Neither student is correct The answer
should be 115 ___ 11 105624 radic_
115
Unit 126
copy H
ough
ton M
ifflin
Har
cour
t Pub
lishin
g Com
pany
Imag
e Cre
dits
copy3D
Stoc
kiSt
ockP
hoto
com
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
8_MCAAESE206984_U1M01L3indd 26 210513 801 AM
Personal Math Trainer
Online Practice and Helpmyhrwcom
Name Class Date
Independent Practice
16 Your sister is considering two different shapes for her garden One is a square with side lengths of 35 meters and the other is a circle with a diameter of 4 meters
a Find the area of the square
b Find the area of the circle
c Compare your answers from parts a and b Which garden would give your sister the most space to plant
17 Winnie measured the length of her fatherrsquos ranch four times and got four different distances Her measurements are shown in the table
a To estimate the actual length Winnie first approximated each distance to the nearest hundredth Then she averaged the four numbers Using a calculator find Winniersquos estimate
b Winniersquos father estimated the distance across his ranch to be radic_
56 km How does this distance compare to Winniersquos estimate
Give an example of each type of number
18 a real number between radic_
13 and radic_
14
19 an irrational number between 5 and 7
Order the numbers from least to greatest
12 radic_
7 2 radic_
8 ___ 2 13 radic_
10 π 35
14 radic_
220 -10 radic_
100 115 15 radic_
8 -375 3 9 _ 4
Distance Across Fatherrsquos Ranch (km)
1 2 3 4
radic_
60 58 __ 8 7 _
3 7 3 _ 5
138NS2
radic_
8 ___ 2 2 radic_
7
-10 radic_
100 115 radic_
220
radic_
60 asymp 775 58 __ 8 = 725 7 _
3 asymp 733 7 3 _ 5 = 760 so the average
π radic_
10 35
-375 9 _ 4 radic_
8 3
is 74825 km
1225 m2
4π m2 or approximately 126 m2
They are nearly identical radic_
56 is approximately 74833hellip
The circle would give her more space to plant because it has a
larger area
Sample answer 37
Sample answer radic_
31
25Lesson 13
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ough
ton
Miff
lin H
arco
urt P
ublis
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Com
pany
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
8_MCAAESE206984_U1M01L3indd 25 41613 448 AM
Activity available online myhrwcomEXTEND THE MATH PRE-AP
Activity Have students investigate whether there are infinitely many numbers between two numbers by giving examples for each of the following
bull Between any two rational numbers there is at least one other rational number Sample answer 45 is between 41 and 48
bull Between any two irrational numbers there is at least one rational number Sample answer 45 is between radic
_ 11 and radic
_ 29
bull Between any two rational numbers there is at least one irrational number Sample answer radic
_ 11 is between 31 and 36
bull Between any two irrational numbers there is at least one irrational number Sample answer radic
_ 17 is between radic
_ 11 and radic
_ 29
Ordering Real Numbers 26
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
ReadyMath Trainer
Online Practiceand Help
Personal
myhrwcom
Module Quiz
11ensp RationalenspandenspIrrationalenspNumbersWrite each fraction as a decimal or each decimal as a fraction
1 7__20 2 1___
27 3 17_8
Solve each equation for x
4 x2=81 5 x3=343 6 x2= 1___100
7 Asquarepatiohasanareaof200squarefeetHowlongiseachside
ofthepatiotothenearesttenth
12ensp SetsenspofenspRealenspNumbersWrite all names that apply to each number
8 121____radic
____121
9 π__2
10 TellwhetherthestatementldquoAllintegersarerationalnumbersrdquoistrueorfalseExplainyourchoice
13ensp OrderingenspRealenspNumbersCompare Write lt gt or =
11 radic__
8+3 8+radic__
3 12 radic__
5+11emsp emsp emsp 5+radic___
11
Order the numbers from least to greatest
13 radic___
99π29__
8 14 radic___
1__251_40__
2
15 Howarerealnumbersusedtodescribereal-worldsituations
ESSENTIAL QUESTION
035
9-9
141ft
7 1__10- 1__10
14__11 1875
wholeintegerrationalreal
Trueintegerscanbewrittenasthequotientoftwointegers
SampleanswerRealnumberssuchastherational
π29__
8radic___
99
irrationalreal
lt gt
number1_4candescribeamountsusedincooking
radic___
1__250__
21_4
27Module1
copy H
ough
ton
Miff
lin H
arco
urt P
ublis
hing
Com
pany
DONOTEDIT--ChangesmustbemadethroughldquoFileinfordquoCorrectionKey=A
8_MCAAESE206984_U1M01RTindd 27 41513 1113 PM
Math TrainerOnline Assessment
and Intervention
Personal
myhrwcom
1
2
3 Response toIntervention
Intervention Enrichment
Access Ready to Go On assessment online and receive instant scoring feedback and customized intervention or enrichment
Online and Print Resources
Differentiated Instruction
bull Reteach worksheets
bull Reading Strategies EL
bull Success for English Learners EL
Differentiated Instruction
bull Challenge worksheets PRE-AP
Extend the Math PRE-AP
Lesson Activities in TE
Additional ResourcesAssessment Resources includes bull Leveled Module Quizzes
Ready to Go OnAssess MasteryUse the assessment on this page to determine if students have mastered the concepts and standards covered in this module
California Common Core Standards
Lesson Exercises Common Core Standards
11 1ndash7 8NS1 8NS2 8EE2
12 8ndash10 8NS1
13 11ndash14 8NS2
27 Unit 1 Module 1
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
Personal Math Trainer
Online Practice and HelpmyhrwcomAssessment Readiness
Module 1 MIXed ReVIeW
1 Look at each number Is the number between 2π and radic___
52
Select Yes or No for expressions AndashC
A 6 2 _ 3 Yes No
B 5π __ 2 Yes No
C 3 radic__
5 Yes No
2 Consider the number - 11 __ 15
Choose True or False for each statement
A The number is rational True False
B The number can be written as True Falsea repeating decimal
C The number is less than ndash08 True False
3 The volume of a cube is given by V = x3 where x is the length of an edge of the cube A cube-shaped end table has a volume of 3 3 _ 8 cubic feet What is the length of an edge of the end table Explain how you solved this problem
4 A student says that radic___
83 is greater than 29 __ 3 Is the student correct Justify your
reasoning
1 1 _ 2 ft Sample answer The equation x3 = 3 3 _ 8 can be used
to find the edge length in feet To solve the equation
write the mixed number as a fraction greater than 1
x3 = 27 __ 8 Then take the cube root of both sides x = 3 _ 2 = 1 1 _ 2
No Sample answer radic___
83 asymp 91 and 29 __ 3 = 9
__ 6
Because 91 lt 9 __
6 radic___
83 lt 29 __ 3
28 Unit 1
copy H
ough
ton
Miff
lin H
arco
urt P
ublis
hing
Com
pany
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=A
8_MCAAESE206984_U1M01RTindd 28 240413 946 AM
Personal Math Trainer
Online Assessment and
Interventionmyhrwcom
Scoring GuideItem 3 Award the student 1 point for finding the edge length of the cube and 1 point for correctly explaining how to use a cube root to solve the problem
Item 4 Award the student 1 point for determining that the student is incorrect and 1 point for correctly justifying the reasoning for this conclusion
Additional ResourcesTo assign this assessment online login to your Assignment Manager at myhrwcom
Assessment Readiness
California Common Core Standards
Items Grade 8 Standards Mathematical Practices
1 8NS2 MP7
2 7NS2b 7NS2d 8NS1 MP7
3 8EE2 MP1 MP4
4 8NS1 8NS2 MP3
Item integrates mixed review concepts from previous modules or a previous course
Item 4 combines concepts from the California Common Core cluster ldquoKnow that there are numbers that are not rational and approximate them by rational numbersrdquo
Real Numbers 28
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
Math Talk
Language Support EL
PROFESSIONAL DEVELOPMENT
Linguistic Support EL
AcademicContent Vocabulary
Post a chart like this to remind students of the regular comparative forms of adjectives that use the -er and -est suffixes Add to the chart for terms that appear in examples and exercises in each lesson Include any irregular verb forms
Background Knowledge
Go On ndash the title of the module review or quiz is Ready to Go On This title uses an idiomatic expression In this context to go on means ldquoto move aheadrdquo or ldquoto proceedrdquo It is different from the use of go on that means having enough facts to use meaningfully as in having enough to go on Also the intonation used in pronouncing an expression can give it different meanings For example when the speaker emphasizes the word on he or she might be expressing disbelief as in ldquoGo ON Yoursquore kidding rightrdquo Discuss with students other ways that the phrase go on may be used
Leveled Strategies for English Learners
Emerging Label points on a number line with the terms used in ordering greater greatest less lesser least Use sentence frames to insert the correct terms
Expanding Have students give two or three complete sentences to compare the placement of numbers on a number line using the correct forms of the comparative and superlative adjectives
Bridging Have students work in pairs with one student giving directions to the other in complete sentences to order numbers on a number line
To help students answer the question posed in Math Talk make sure that students have a command of the forms for making comparisons and the superlative and the concept of opposite order so that the focus is on the math concept instead of the language skills needed to describe and explain order
EL
Adjective Comparative Superlative
Far Farther Farthest
Large Larger Largest
Great Greater Greatest
Some Less Least
Some More Most
California ELD Standards
Emerging 2I8 Analyzing language choices ndash Explain how phrasing or different common words with similar meanings produce different effects on the audience
Expanding 2I8 Analyzing language choices ndash Explain how phrasing or different words with similar meanings or figurative language produce shades of meaning and different effects on the audience
Bridging 2I8 Analyzing language choices ndash Explain how phrasing or different words with similar meanings or figurative language produce shades of meaning nuances and different effects on the audience
Ordering Real Numbers 21B
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
13L E S S O N
Ordering Real Numbers
Interactive Whiteboard Interactive example available online
ADDITIONAL EXAMPLE 1Compare Write lt gt or =
A radic_
8 - 2 4 - radic_
8 lt
B radic_
20 + 1 3 + radic_
2 gt
EngageESSENTIAL QUESTION
How do you order a set of real numbers Sample answer Find their approximate decimal values and order them
Motivate the LessonAsk What kind of numbers are you comparing when you compare the price of gasoline at two different gas stations
ExploreGive students two rational numbers and ask them to name a number between them Repeat a few times and then give them two irrational numbers and ask them to name a number between them
ExplainEXAMPLE 1
Questioning Strategies Mathematical Practices bull Which is greater the difference between 5 and 3 or the difference between radic
_ 5 and radic
_ 3
The difference between 5 and 3 is 2 the difference between radic_
5 and radic_
3 is approximately 1 So the difference between 5 and 3 is greater
Avoid Common ErrorsCaution students to read the problem carefully and think about what the radical sign means so that they do not misread the problem and answer that the two sides are equal
YOUR TURNFocus on TechnologyCalculators should not be used at this point because developing number sense is the goal
EXAMPLE 2Questioning Strategies Mathematical Practices bull How do you determine whether radic
_ 22 is less than or greater than 45 The square of 45 is
2025 which is less than 22 so the square root of 22 must be greater than 45
Engage with the WhiteboardHave students graph and label various real numbers between 42 and 44 and between 47 and 5
YOUR TURNFocus on Modeling Mathematical PracticesHave students label the integers on the number line with their equivalent square root For example 1 2 and 3 on the number line would be labeled radic
_ 1 radic
_ 4 and radic
_ 9
Interactive Whiteboard Interactive example available online
ADDITIONAL EXAMPLE 2Order 3π radic
_ 10 and 325 from greatest
to least
3π 325 radic_
10
myhrwcom
myhrwcom
CA Common CoreStandards
The student is expected to
The Number Systemmdash8NS2
Use rational approximations of irrational numbers to compare the size of irrational numbers locate them approximately on a number line diagram and estimate the value of expressions (eg π 2 )
Mathematical Practices
MP4 Modeling
The student is expected to
21 Lesson 13
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
Math TrainerOnline Practice
and Help
Personal
myhrwcom
Math On the Spotmyhrwcom
0 05 1 15 2 25 3 35 4
radic5radic3
π2
8 85 9 95 10 105 11 115 12
radic75
4 42 44 46 48 5
radic224 12π + 1
Ordering Real Numbers You can compare and order real numbers and list them from least to greatest
Order radic_
22 π + 1 and 4 1 _ 2 from least to greatest
First approximate radic_
22
radic_
22 is between 4 and 5 Since you donrsquot know where it falls between 4 and 5 you need to find a better estimate for radic
_ 22 so
you can compare it to 4 1 _ 2
Since 22 is closer to 25 than 16 use squares of numbers between 45 and 5 to find a better estimate of radic
_ 22
45 2 = 2025 46 2 = 2116 47 2 = 2209 48 2 = 2304
Since 47 2 = 2209 an approximate value for radic_
22 is 47
An approximate value of π is 314 So an approximate value of π +1 is 414
Plot radic_
22 π + 1 and 4 1 _ 2 on a number line
Read the numbers from left to right to place them in order from least to greatest
From least to greatest the numbers are π + 1 4 1 _ 2 and radic_
22
EXAMPLE 2
STEP 1
STEP 2
Order the numbers from least to greatest Then graph them on the number line
YOUR TURN
5 radic_
5 25 radic_
3
6 π 2 10 radic_
75
If real numbers a b and c are in order from least to greatest what is the order
of their opposites from least to greatest
Explain
Math TalkMathematical Practices
8NS2
radic_
3 radic_
5 25
radic_
75 π2 10
Math Talk answer -c -b -a -c is farthest to the left on a number line -b is in the middle and -a is farthest to the right
Unit 122
copy H
ough
ton
Miff
lin H
arco
urt P
ublis
hing
Com
pany
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
8_MCAAESE206984_U1M01L3indd 22 41613 447 AM
My Notes
Math TrainerOnline Practice
and Help
Personal
myhrwcom
Math On the Spot
myhrwcom
Comparing Irrational NumbersBetween any two real numbers is another real number To compare and order real numbers you can approximate irrational numbers as decimals
Compare radic_
3 + 5 3 + radic_
5 Write lt gt or =
First approximate radic_
3
radic_
3 is between 1 and 2
Next approximate radic_
5
radic_
5 is between 2 and 3
Then use your approximations to simplify the expressions
radic_
3 + 5 is between 6 and 7
3 + radic_
5 is between 5 and 6
So radic_
3 + 5 gt 3 + radic_
5
Reflect1 If 7 + radic
_ 5 is equal to radic
_ 5 plus a number what do you know about the
number Why
2 What are the closest two integers that radic_
300 is between
EXAMPLEXAMPLE 1
STEP 1
STEP 2
Compare Write lt gt or =
YOUR TURN
3 radic_
2 + 4 2 + radic_
4 4 radic_
12 + 6 12 + radic_
6
L E S S O N
13 Ordering Real Numbers
ESSENTIAL QUESTIONHow do you order a set of real numbers
8NS2
Use rational approximations of irrational numbers to compare the size of irrational numbers locate them approximately on a number line diagram and estimate the value of expressions (eg π 2 )
8NS2
Use perfect squares to estimate square roots
1 2 = 1 2 2 = 4 3 2 = 9
The number is 7 both expressions must equal 7 + radic_
5
17 and 18
gt lt
21Lesson 13
copy H
ough
ton
Miff
lin H
arco
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ublis
hing
Com
pany
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
8_MCAAESE206984_U1M01L3indd 21 41913 246 PM
PROFESSIONAL DEVELOPMENT
Math BackgroundIn this lesson students estimate irrational numbers in the form of square roots of nonper-fect squares by finding two perfect squares between which the number falls A more precise method involves repeated division For example to find radic
_ 28 find a whole number whose perfect
square is close to 28 such as 5 Divide 28 by that number 28 divide 5 = 56 Find the average of the quotient and divisor 5 + 56
_____ 2 = 53 Continue dividing 28 by each result and averaging until you get the desired accuracy
Integrate Mathematical Practices MP4
This lesson provides an opportunity to address this Mathematical Practices standard It calls for students to model relationships using multiple representations including diagrams graphs and language as appropriate Students use multiple representations when they use number lines to estimate the locations of and order rational and irrational numbers given as symbols
Ordering Real Numbers 22
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
Interactive Whiteboard Interactive example available online
ADDITIONAL EXAMPLE 3The diameter of a meteorite in millimeters is calculated by four different methods Order the results from least to greatest
Joe radic_
18 mm Lisa 13 __ 3 mm
Pablo 46 mm Julien 4π __ 3 mm
Julien 4π __ 3 mm Lisa 13 __ 3 mm
Joe radic_
18 mm Pablo 46 mm
EXAMPLE 3Questioning Strategies Mathematical Practices bull How can you verify that radic
_ 28 is between 52 and 53 5 2 2 = 2704 and 5 3 2 = 2809
bull Explain how to determine which number is greater 5 _
5 or 55 When the repeating decimal is rounded to the nearest tenth or hundredth you can see that it is greater
Connect to Daily LifeDiscuss how measuring across a canyon might involve different methods than measuring along a road Explain that measurements like these are often done using calculations that approximate the distance
YOUR TURNFocus on Critical Thinking Mathematical PracticesDiscuss with students which number is greater 3
_ 45 or 3450 3
_ 45 or 3455 and why Explain
that 3 _
45 can be written out as 34545hellipMake sure they understand that 3 _
45 is greater than 345 but less than 3455
ElaborateTalk About ItSummarize the Lesson
Ask How can you order two numbers in different forms whose decimal approxi-mations appear to be equal Approximate one or both numbers to an additional
number of decimal places
GUIDED PRACTICEEngage with the Whiteboard
Have students place and label additional points on the number line in Exercise 9 Allow the points to be in any format other than decimal
Avoid Common ErrorsExercises 3ndash4 Caution students to read the problem carefully so that they do not misread the problem as the same numbers combined by addition on each side of the circleExercise 10 Remind students that the calculations have units
myhrwcom
23 Lesson 13
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
0 05 1 15 2 25 3 35 4 45 5 55 6 65 7
2πradic3
Compare Write lt gt or = (Example 1)
1 radic_
3 + 2 radic_
3 + 3 2 radic_
8 + 17 radic_
11 + 15
3 radic_
6 + 5 6 + radic_
5 4 radic_
9 + 3 9 + radic_
3
5 radic_
17 - 3 -2 + radic_
5 6 12 - radic_
2 14 - radic_
8
7 radic_
7 + 2 radic_
10 - 1 8 radic_
17 + 3 3 + radic_
11
9 Order radic_
3 2π and 15 from least to greatest Then graph them on the number line (Example 2)
radic_
3 is between and so radic_
3 asymp
π asymp 314 so 2π asymp
From least to greatest the numbers are
10 Four people have found the perimeter of a forest using different methods Their results are given in the table Order their calculations from greatest to least (Example 3)
11 Explain how to order a set of real numbers
CHECK-INESSENTIAL QUESTION
Forest Perimeter (km)
Leon Mika Jason Ashley
radic_
17 - 2 1 +thinsp π __ 2 12 ___ 5 25
Guided Practice
17
15
1 + π _ 2 km 25 km 12 __ 5 km radic_
17 - 2 km
2π radic
_ 3
18 175
628
Sample answer Convert each number to a decimal
equivalent using estimation to find equivalents for
irrational numbers Graph each number on a number line
Read the numbers from left to right for least to greatest
Read the numbers from right to left for greatest to least
lt gt
lt lt
ltgt
gt gt
24 Unit 1
copy H
ough
ton
Miff
lin H
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urt P
ublis
hing
Com
pany
bull Im
age C
redi
ts copy
Elena
Eliss
eeva
Alam
y Im
ages
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
8_MCAAESE206984_U1M01L3indd 24 41613 448 AM
My Notes
5 52 54 56 58 6
radic28 5 12
23455
Math TrainerOnline Practice
and Help
Personal
myhrwcom
Math On the Spot
myhrwcom
Ordering Real Numbers in a Real-World Context Calculations and estimations in the real world may differ It can be important to know not only which are the most accurate but which give the greatest or least values depending upon the context
Four people have found the distance in kilometers across a canyon using different methods Their results are given in the table Order the distances from greatest to least
Distance Across Quarry Canyon (km)
Juana Lee Ann Ryne Jackson
radic_
28 23 __ 4 5 _
5 5 1 _ 2
Write each value as a decimal
radic_
28 is between 52 and 53 Since 53 2 = 2809 an approximate value for radic
_ 28 is 53
23 __ 4 = 575
5 _
5 is 5555hellip so 5 _
5 to the nearest hundredth is 556
5 1 _ 2 = 55
Plot radic_
28 23 __ 4 5 _
5 and 5 1 _ 2 on a number line
From greatest to least the distances are
23 __ 4 km 5 _
5 km 5 1 _ 2 km radic_
28 km
EXAMPLEXAMPLE 3
STEP 1
STEP 2
7 Four people have found the distance in miles across a crater using different methods Their results are given below
Jonathan 10 __ 3 Elaine 3 _
45 Joseacute 3 1 _ 2 Lashonda radic_
10
Order the distances from greatest to least
YOUR TURN
8NS2
3 1 _ 2 mi 3 _
45 mi 10 __ 3 mi radic_
10 mi
23Lesson 13
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ough
ton
Miff
lin H
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ublis
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Com
pany
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8_MCAAESE206984_U1M01L3indd 23 41613 447 AM
ModelingPlace papers around the room with the numbers from 1 to 5 one per sheet Give each student a card showing a number between 1 and 5 in different forms Have students place his or her card between the correct integers and decide where the number goes in relation to any numbers already placed
Multiple RepresentationsGive students a vertical number line which some students might find easier to use than a horizontal one Have them decide whether to place points for rational and irrational numbers above or below existing points
Additional ResourcesDifferentiated Instruction includes bull Reading Strategies bull Success for English Learners EL
bull Reteach bull Challenge PRE-AP
DIFFERENTIATE INSTRUCTION
Ordering Real Numbers 24
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
Math TrainerOnline Assessment
and Intervention
Personal
Online homework assignment available
myhrwcom
myhrwcom
Lesson Quiz available online
13 LESSON QUIZ
1 Compare Write lt gt or =
radic_
95 - 5 radic_
62 - 2
2 Order 105 radic_
105 and 3π + 1 from greatest to least
3 A length in centimeters is calculated differently by four different people Order their calculations from least to greatest
KD 11 __ 2 cm Silvio 5 __ 3 π cm
Paula 5 _
4 cm Luis radic_
33 cm
EvaluateGUIDED AND INDEPENDENT PRACTICE
Concepts amp Skills Practice
Example 1Comparing Irrational Numbers
Exercises 1ndash8
Example 2Ordering Real Numbers
Exercises 9 12ndash15 18ndash21
Example 3Ordering Real Numbers in a Real-World Context
Exercises 10 16ndash17
Additional ResourcesDifferentiated Instruction includes bull Leveled Practice worksheets
Focus | Coherence | Rigor
Exercise Depth of Knowledge (DOK) Mathematical Practices
12ndash15 1 Recall of Information MP5 Using Tools
16 2 SkillsConcepts MP2 Reasoning
17 2 SkillsConcepts MP6 Precision
18ndash21 2 SkillsConcepts MP2 Reasoning
22 3 Strategic Thinking MP4 Modeling
23ndash24 3 Strategic Thinking MP3 Logic
8NS2
8NS2
Answers1 radic
_ 95 - 5 lt radic
_ 62 - 2
2 radic_
105 3π + 1 105
3 Silvio 5 __ 3 π cm Paula 5 _
4 cm
KD 11
__ 2 cm Luis radic_
33 cm
25 Lesson 13
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
Work Area
3140 3141 3142 3143
314 π227
20 A teacher asks his students to write the numbers shown in order from least to greatest Paul thinks the numbers are already in order Sandra thinks the order should be reversed Who is right
21 Math History There is a famous irrational number called Eulerrsquos number symbolized with an e Like π its decimal form never ends or repeats The first few digits of e are 27182818284
a Between which two square roots of integers could you find this number
b Between which two square roots of integers can you find π
22 Analyze Relationships There are several approximations used for π including 314 and 22 __ 7 π is approximately 314159265358979
a Label π and the two approximations on the number line
b Which of the two approximations is a better estimate for π Explain
c Find a whole number x so that the ratio x ___ 113 is a better estimate for π
than the two given approximations
23 Communicate Mathematical Ideas If a set of six numbers that include both rational and irrational numbers is graphed on a number line what is the fewest number of distinct points that need to be graphed Explain
24 Critique Reasoning Jill says that 12 _
6 is less than 1263 Explain her error
FOCUS ON HIGHER ORDER THINKING
radic_
115 115 ___ 11 and 105624
between radic_
7 asymp 265 and radic_
8 asymp 283
between radic_
9 = 3 and radic_
10 asymp 316
22 __ 7 it is closer to π on the number line
She did not consider the repeating digit 1266
2 rational numbers can have the same location and
irrational numbers can have the same location but they
cannot share a location
355
Neither student is correct The answer
should be 115 ___ 11 105624 radic_
115
Unit 126
copy H
ough
ton M
ifflin
Har
cour
t Pub
lishin
g Com
pany
Imag
e Cre
dits
copy3D
Stoc
kiSt
ockP
hoto
com
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
8_MCAAESE206984_U1M01L3indd 26 210513 801 AM
Personal Math Trainer
Online Practice and Helpmyhrwcom
Name Class Date
Independent Practice
16 Your sister is considering two different shapes for her garden One is a square with side lengths of 35 meters and the other is a circle with a diameter of 4 meters
a Find the area of the square
b Find the area of the circle
c Compare your answers from parts a and b Which garden would give your sister the most space to plant
17 Winnie measured the length of her fatherrsquos ranch four times and got four different distances Her measurements are shown in the table
a To estimate the actual length Winnie first approximated each distance to the nearest hundredth Then she averaged the four numbers Using a calculator find Winniersquos estimate
b Winniersquos father estimated the distance across his ranch to be radic_
56 km How does this distance compare to Winniersquos estimate
Give an example of each type of number
18 a real number between radic_
13 and radic_
14
19 an irrational number between 5 and 7
Order the numbers from least to greatest
12 radic_
7 2 radic_
8 ___ 2 13 radic_
10 π 35
14 radic_
220 -10 radic_
100 115 15 radic_
8 -375 3 9 _ 4
Distance Across Fatherrsquos Ranch (km)
1 2 3 4
radic_
60 58 __ 8 7 _
3 7 3 _ 5
138NS2
radic_
8 ___ 2 2 radic_
7
-10 radic_
100 115 radic_
220
radic_
60 asymp 775 58 __ 8 = 725 7 _
3 asymp 733 7 3 _ 5 = 760 so the average
π radic_
10 35
-375 9 _ 4 radic_
8 3
is 74825 km
1225 m2
4π m2 or approximately 126 m2
They are nearly identical radic_
56 is approximately 74833hellip
The circle would give her more space to plant because it has a
larger area
Sample answer 37
Sample answer radic_
31
25Lesson 13
copy H
ough
ton
Miff
lin H
arco
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ublis
hing
Com
pany
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8_MCAAESE206984_U1M01L3indd 25 41613 448 AM
Activity available online myhrwcomEXTEND THE MATH PRE-AP
Activity Have students investigate whether there are infinitely many numbers between two numbers by giving examples for each of the following
bull Between any two rational numbers there is at least one other rational number Sample answer 45 is between 41 and 48
bull Between any two irrational numbers there is at least one rational number Sample answer 45 is between radic
_ 11 and radic
_ 29
bull Between any two rational numbers there is at least one irrational number Sample answer radic
_ 11 is between 31 and 36
bull Between any two irrational numbers there is at least one irrational number Sample answer radic
_ 17 is between radic
_ 11 and radic
_ 29
Ordering Real Numbers 26
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
ReadyMath Trainer
Online Practiceand Help
Personal
myhrwcom
Module Quiz
11ensp RationalenspandenspIrrationalenspNumbersWrite each fraction as a decimal or each decimal as a fraction
1 7__20 2 1___
27 3 17_8
Solve each equation for x
4 x2=81 5 x3=343 6 x2= 1___100
7 Asquarepatiohasanareaof200squarefeetHowlongiseachside
ofthepatiotothenearesttenth
12ensp SetsenspofenspRealenspNumbersWrite all names that apply to each number
8 121____radic
____121
9 π__2
10 TellwhetherthestatementldquoAllintegersarerationalnumbersrdquoistrueorfalseExplainyourchoice
13ensp OrderingenspRealenspNumbersCompare Write lt gt or =
11 radic__
8+3 8+radic__
3 12 radic__
5+11emsp emsp emsp 5+radic___
11
Order the numbers from least to greatest
13 radic___
99π29__
8 14 radic___
1__251_40__
2
15 Howarerealnumbersusedtodescribereal-worldsituations
ESSENTIAL QUESTION
035
9-9
141ft
7 1__10- 1__10
14__11 1875
wholeintegerrationalreal
Trueintegerscanbewrittenasthequotientoftwointegers
SampleanswerRealnumberssuchastherational
π29__
8radic___
99
irrationalreal
lt gt
number1_4candescribeamountsusedincooking
radic___
1__250__
21_4
27Module1
copy H
ough
ton
Miff
lin H
arco
urt P
ublis
hing
Com
pany
DONOTEDIT--ChangesmustbemadethroughldquoFileinfordquoCorrectionKey=A
8_MCAAESE206984_U1M01RTindd 27 41513 1113 PM
Math TrainerOnline Assessment
and Intervention
Personal
myhrwcom
1
2
3 Response toIntervention
Intervention Enrichment
Access Ready to Go On assessment online and receive instant scoring feedback and customized intervention or enrichment
Online and Print Resources
Differentiated Instruction
bull Reteach worksheets
bull Reading Strategies EL
bull Success for English Learners EL
Differentiated Instruction
bull Challenge worksheets PRE-AP
Extend the Math PRE-AP
Lesson Activities in TE
Additional ResourcesAssessment Resources includes bull Leveled Module Quizzes
Ready to Go OnAssess MasteryUse the assessment on this page to determine if students have mastered the concepts and standards covered in this module
California Common Core Standards
Lesson Exercises Common Core Standards
11 1ndash7 8NS1 8NS2 8EE2
12 8ndash10 8NS1
13 11ndash14 8NS2
27 Unit 1 Module 1
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
Personal Math Trainer
Online Practice and HelpmyhrwcomAssessment Readiness
Module 1 MIXed ReVIeW
1 Look at each number Is the number between 2π and radic___
52
Select Yes or No for expressions AndashC
A 6 2 _ 3 Yes No
B 5π __ 2 Yes No
C 3 radic__
5 Yes No
2 Consider the number - 11 __ 15
Choose True or False for each statement
A The number is rational True False
B The number can be written as True Falsea repeating decimal
C The number is less than ndash08 True False
3 The volume of a cube is given by V = x3 where x is the length of an edge of the cube A cube-shaped end table has a volume of 3 3 _ 8 cubic feet What is the length of an edge of the end table Explain how you solved this problem
4 A student says that radic___
83 is greater than 29 __ 3 Is the student correct Justify your
reasoning
1 1 _ 2 ft Sample answer The equation x3 = 3 3 _ 8 can be used
to find the edge length in feet To solve the equation
write the mixed number as a fraction greater than 1
x3 = 27 __ 8 Then take the cube root of both sides x = 3 _ 2 = 1 1 _ 2
No Sample answer radic___
83 asymp 91 and 29 __ 3 = 9
__ 6
Because 91 lt 9 __
6 radic___
83 lt 29 __ 3
28 Unit 1
copy H
ough
ton
Miff
lin H
arco
urt P
ublis
hing
Com
pany
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=A
8_MCAAESE206984_U1M01RTindd 28 240413 946 AM
Personal Math Trainer
Online Assessment and
Interventionmyhrwcom
Scoring GuideItem 3 Award the student 1 point for finding the edge length of the cube and 1 point for correctly explaining how to use a cube root to solve the problem
Item 4 Award the student 1 point for determining that the student is incorrect and 1 point for correctly justifying the reasoning for this conclusion
Additional ResourcesTo assign this assessment online login to your Assignment Manager at myhrwcom
Assessment Readiness
California Common Core Standards
Items Grade 8 Standards Mathematical Practices
1 8NS2 MP7
2 7NS2b 7NS2d 8NS1 MP7
3 8EE2 MP1 MP4
4 8NS1 8NS2 MP3
Item integrates mixed review concepts from previous modules or a previous course
Item 4 combines concepts from the California Common Core cluster ldquoKnow that there are numbers that are not rational and approximate them by rational numbersrdquo
Real Numbers 28
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
13L E S S O N
Ordering Real Numbers
Interactive Whiteboard Interactive example available online
ADDITIONAL EXAMPLE 1Compare Write lt gt or =
A radic_
8 - 2 4 - radic_
8 lt
B radic_
20 + 1 3 + radic_
2 gt
EngageESSENTIAL QUESTION
How do you order a set of real numbers Sample answer Find their approximate decimal values and order them
Motivate the LessonAsk What kind of numbers are you comparing when you compare the price of gasoline at two different gas stations
ExploreGive students two rational numbers and ask them to name a number between them Repeat a few times and then give them two irrational numbers and ask them to name a number between them
ExplainEXAMPLE 1
Questioning Strategies Mathematical Practices bull Which is greater the difference between 5 and 3 or the difference between radic
_ 5 and radic
_ 3
The difference between 5 and 3 is 2 the difference between radic_
5 and radic_
3 is approximately 1 So the difference between 5 and 3 is greater
Avoid Common ErrorsCaution students to read the problem carefully and think about what the radical sign means so that they do not misread the problem and answer that the two sides are equal
YOUR TURNFocus on TechnologyCalculators should not be used at this point because developing number sense is the goal
EXAMPLE 2Questioning Strategies Mathematical Practices bull How do you determine whether radic
_ 22 is less than or greater than 45 The square of 45 is
2025 which is less than 22 so the square root of 22 must be greater than 45
Engage with the WhiteboardHave students graph and label various real numbers between 42 and 44 and between 47 and 5
YOUR TURNFocus on Modeling Mathematical PracticesHave students label the integers on the number line with their equivalent square root For example 1 2 and 3 on the number line would be labeled radic
_ 1 radic
_ 4 and radic
_ 9
Interactive Whiteboard Interactive example available online
ADDITIONAL EXAMPLE 2Order 3π radic
_ 10 and 325 from greatest
to least
3π 325 radic_
10
myhrwcom
myhrwcom
CA Common CoreStandards
The student is expected to
The Number Systemmdash8NS2
Use rational approximations of irrational numbers to compare the size of irrational numbers locate them approximately on a number line diagram and estimate the value of expressions (eg π 2 )
Mathematical Practices
MP4 Modeling
The student is expected to
21 Lesson 13
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
Math TrainerOnline Practice
and Help
Personal
myhrwcom
Math On the Spotmyhrwcom
0 05 1 15 2 25 3 35 4
radic5radic3
π2
8 85 9 95 10 105 11 115 12
radic75
4 42 44 46 48 5
radic224 12π + 1
Ordering Real Numbers You can compare and order real numbers and list them from least to greatest
Order radic_
22 π + 1 and 4 1 _ 2 from least to greatest
First approximate radic_
22
radic_
22 is between 4 and 5 Since you donrsquot know where it falls between 4 and 5 you need to find a better estimate for radic
_ 22 so
you can compare it to 4 1 _ 2
Since 22 is closer to 25 than 16 use squares of numbers between 45 and 5 to find a better estimate of radic
_ 22
45 2 = 2025 46 2 = 2116 47 2 = 2209 48 2 = 2304
Since 47 2 = 2209 an approximate value for radic_
22 is 47
An approximate value of π is 314 So an approximate value of π +1 is 414
Plot radic_
22 π + 1 and 4 1 _ 2 on a number line
Read the numbers from left to right to place them in order from least to greatest
From least to greatest the numbers are π + 1 4 1 _ 2 and radic_
22
EXAMPLE 2
STEP 1
STEP 2
Order the numbers from least to greatest Then graph them on the number line
YOUR TURN
5 radic_
5 25 radic_
3
6 π 2 10 radic_
75
If real numbers a b and c are in order from least to greatest what is the order
of their opposites from least to greatest
Explain
Math TalkMathematical Practices
8NS2
radic_
3 radic_
5 25
radic_
75 π2 10
Math Talk answer -c -b -a -c is farthest to the left on a number line -b is in the middle and -a is farthest to the right
Unit 122
copy H
ough
ton
Miff
lin H
arco
urt P
ublis
hing
Com
pany
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
8_MCAAESE206984_U1M01L3indd 22 41613 447 AM
My Notes
Math TrainerOnline Practice
and Help
Personal
myhrwcom
Math On the Spot
myhrwcom
Comparing Irrational NumbersBetween any two real numbers is another real number To compare and order real numbers you can approximate irrational numbers as decimals
Compare radic_
3 + 5 3 + radic_
5 Write lt gt or =
First approximate radic_
3
radic_
3 is between 1 and 2
Next approximate radic_
5
radic_
5 is between 2 and 3
Then use your approximations to simplify the expressions
radic_
3 + 5 is between 6 and 7
3 + radic_
5 is between 5 and 6
So radic_
3 + 5 gt 3 + radic_
5
Reflect1 If 7 + radic
_ 5 is equal to radic
_ 5 plus a number what do you know about the
number Why
2 What are the closest two integers that radic_
300 is between
EXAMPLEXAMPLE 1
STEP 1
STEP 2
Compare Write lt gt or =
YOUR TURN
3 radic_
2 + 4 2 + radic_
4 4 radic_
12 + 6 12 + radic_
6
L E S S O N
13 Ordering Real Numbers
ESSENTIAL QUESTIONHow do you order a set of real numbers
8NS2
Use rational approximations of irrational numbers to compare the size of irrational numbers locate them approximately on a number line diagram and estimate the value of expressions (eg π 2 )
8NS2
Use perfect squares to estimate square roots
1 2 = 1 2 2 = 4 3 2 = 9
The number is 7 both expressions must equal 7 + radic_
5
17 and 18
gt lt
21Lesson 13
copy H
ough
ton
Miff
lin H
arco
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ublis
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Com
pany
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
8_MCAAESE206984_U1M01L3indd 21 41913 246 PM
PROFESSIONAL DEVELOPMENT
Math BackgroundIn this lesson students estimate irrational numbers in the form of square roots of nonper-fect squares by finding two perfect squares between which the number falls A more precise method involves repeated division For example to find radic
_ 28 find a whole number whose perfect
square is close to 28 such as 5 Divide 28 by that number 28 divide 5 = 56 Find the average of the quotient and divisor 5 + 56
_____ 2 = 53 Continue dividing 28 by each result and averaging until you get the desired accuracy
Integrate Mathematical Practices MP4
This lesson provides an opportunity to address this Mathematical Practices standard It calls for students to model relationships using multiple representations including diagrams graphs and language as appropriate Students use multiple representations when they use number lines to estimate the locations of and order rational and irrational numbers given as symbols
Ordering Real Numbers 22
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
Interactive Whiteboard Interactive example available online
ADDITIONAL EXAMPLE 3The diameter of a meteorite in millimeters is calculated by four different methods Order the results from least to greatest
Joe radic_
18 mm Lisa 13 __ 3 mm
Pablo 46 mm Julien 4π __ 3 mm
Julien 4π __ 3 mm Lisa 13 __ 3 mm
Joe radic_
18 mm Pablo 46 mm
EXAMPLE 3Questioning Strategies Mathematical Practices bull How can you verify that radic
_ 28 is between 52 and 53 5 2 2 = 2704 and 5 3 2 = 2809
bull Explain how to determine which number is greater 5 _
5 or 55 When the repeating decimal is rounded to the nearest tenth or hundredth you can see that it is greater
Connect to Daily LifeDiscuss how measuring across a canyon might involve different methods than measuring along a road Explain that measurements like these are often done using calculations that approximate the distance
YOUR TURNFocus on Critical Thinking Mathematical PracticesDiscuss with students which number is greater 3
_ 45 or 3450 3
_ 45 or 3455 and why Explain
that 3 _
45 can be written out as 34545hellipMake sure they understand that 3 _
45 is greater than 345 but less than 3455
ElaborateTalk About ItSummarize the Lesson
Ask How can you order two numbers in different forms whose decimal approxi-mations appear to be equal Approximate one or both numbers to an additional
number of decimal places
GUIDED PRACTICEEngage with the Whiteboard
Have students place and label additional points on the number line in Exercise 9 Allow the points to be in any format other than decimal
Avoid Common ErrorsExercises 3ndash4 Caution students to read the problem carefully so that they do not misread the problem as the same numbers combined by addition on each side of the circleExercise 10 Remind students that the calculations have units
myhrwcom
23 Lesson 13
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
0 05 1 15 2 25 3 35 4 45 5 55 6 65 7
2πradic3
Compare Write lt gt or = (Example 1)
1 radic_
3 + 2 radic_
3 + 3 2 radic_
8 + 17 radic_
11 + 15
3 radic_
6 + 5 6 + radic_
5 4 radic_
9 + 3 9 + radic_
3
5 radic_
17 - 3 -2 + radic_
5 6 12 - radic_
2 14 - radic_
8
7 radic_
7 + 2 radic_
10 - 1 8 radic_
17 + 3 3 + radic_
11
9 Order radic_
3 2π and 15 from least to greatest Then graph them on the number line (Example 2)
radic_
3 is between and so radic_
3 asymp
π asymp 314 so 2π asymp
From least to greatest the numbers are
10 Four people have found the perimeter of a forest using different methods Their results are given in the table Order their calculations from greatest to least (Example 3)
11 Explain how to order a set of real numbers
CHECK-INESSENTIAL QUESTION
Forest Perimeter (km)
Leon Mika Jason Ashley
radic_
17 - 2 1 +thinsp π __ 2 12 ___ 5 25
Guided Practice
17
15
1 + π _ 2 km 25 km 12 __ 5 km radic_
17 - 2 km
2π radic
_ 3
18 175
628
Sample answer Convert each number to a decimal
equivalent using estimation to find equivalents for
irrational numbers Graph each number on a number line
Read the numbers from left to right for least to greatest
Read the numbers from right to left for greatest to least
lt gt
lt lt
ltgt
gt gt
24 Unit 1
copy H
ough
ton
Miff
lin H
arco
urt P
ublis
hing
Com
pany
bull Im
age C
redi
ts copy
Elena
Eliss
eeva
Alam
y Im
ages
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
8_MCAAESE206984_U1M01L3indd 24 41613 448 AM
My Notes
5 52 54 56 58 6
radic28 5 12
23455
Math TrainerOnline Practice
and Help
Personal
myhrwcom
Math On the Spot
myhrwcom
Ordering Real Numbers in a Real-World Context Calculations and estimations in the real world may differ It can be important to know not only which are the most accurate but which give the greatest or least values depending upon the context
Four people have found the distance in kilometers across a canyon using different methods Their results are given in the table Order the distances from greatest to least
Distance Across Quarry Canyon (km)
Juana Lee Ann Ryne Jackson
radic_
28 23 __ 4 5 _
5 5 1 _ 2
Write each value as a decimal
radic_
28 is between 52 and 53 Since 53 2 = 2809 an approximate value for radic
_ 28 is 53
23 __ 4 = 575
5 _
5 is 5555hellip so 5 _
5 to the nearest hundredth is 556
5 1 _ 2 = 55
Plot radic_
28 23 __ 4 5 _
5 and 5 1 _ 2 on a number line
From greatest to least the distances are
23 __ 4 km 5 _
5 km 5 1 _ 2 km radic_
28 km
EXAMPLEXAMPLE 3
STEP 1
STEP 2
7 Four people have found the distance in miles across a crater using different methods Their results are given below
Jonathan 10 __ 3 Elaine 3 _
45 Joseacute 3 1 _ 2 Lashonda radic_
10
Order the distances from greatest to least
YOUR TURN
8NS2
3 1 _ 2 mi 3 _
45 mi 10 __ 3 mi radic_
10 mi
23Lesson 13
copy H
ough
ton
Miff
lin H
arco
urt P
ublis
hing
Com
pany
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
8_MCAAESE206984_U1M01L3indd 23 41613 447 AM
ModelingPlace papers around the room with the numbers from 1 to 5 one per sheet Give each student a card showing a number between 1 and 5 in different forms Have students place his or her card between the correct integers and decide where the number goes in relation to any numbers already placed
Multiple RepresentationsGive students a vertical number line which some students might find easier to use than a horizontal one Have them decide whether to place points for rational and irrational numbers above or below existing points
Additional ResourcesDifferentiated Instruction includes bull Reading Strategies bull Success for English Learners EL
bull Reteach bull Challenge PRE-AP
DIFFERENTIATE INSTRUCTION
Ordering Real Numbers 24
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
Math TrainerOnline Assessment
and Intervention
Personal
Online homework assignment available
myhrwcom
myhrwcom
Lesson Quiz available online
13 LESSON QUIZ
1 Compare Write lt gt or =
radic_
95 - 5 radic_
62 - 2
2 Order 105 radic_
105 and 3π + 1 from greatest to least
3 A length in centimeters is calculated differently by four different people Order their calculations from least to greatest
KD 11 __ 2 cm Silvio 5 __ 3 π cm
Paula 5 _
4 cm Luis radic_
33 cm
EvaluateGUIDED AND INDEPENDENT PRACTICE
Concepts amp Skills Practice
Example 1Comparing Irrational Numbers
Exercises 1ndash8
Example 2Ordering Real Numbers
Exercises 9 12ndash15 18ndash21
Example 3Ordering Real Numbers in a Real-World Context
Exercises 10 16ndash17
Additional ResourcesDifferentiated Instruction includes bull Leveled Practice worksheets
Focus | Coherence | Rigor
Exercise Depth of Knowledge (DOK) Mathematical Practices
12ndash15 1 Recall of Information MP5 Using Tools
16 2 SkillsConcepts MP2 Reasoning
17 2 SkillsConcepts MP6 Precision
18ndash21 2 SkillsConcepts MP2 Reasoning
22 3 Strategic Thinking MP4 Modeling
23ndash24 3 Strategic Thinking MP3 Logic
8NS2
8NS2
Answers1 radic
_ 95 - 5 lt radic
_ 62 - 2
2 radic_
105 3π + 1 105
3 Silvio 5 __ 3 π cm Paula 5 _
4 cm
KD 11
__ 2 cm Luis radic_
33 cm
25 Lesson 13
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
Work Area
3140 3141 3142 3143
314 π227
20 A teacher asks his students to write the numbers shown in order from least to greatest Paul thinks the numbers are already in order Sandra thinks the order should be reversed Who is right
21 Math History There is a famous irrational number called Eulerrsquos number symbolized with an e Like π its decimal form never ends or repeats The first few digits of e are 27182818284
a Between which two square roots of integers could you find this number
b Between which two square roots of integers can you find π
22 Analyze Relationships There are several approximations used for π including 314 and 22 __ 7 π is approximately 314159265358979
a Label π and the two approximations on the number line
b Which of the two approximations is a better estimate for π Explain
c Find a whole number x so that the ratio x ___ 113 is a better estimate for π
than the two given approximations
23 Communicate Mathematical Ideas If a set of six numbers that include both rational and irrational numbers is graphed on a number line what is the fewest number of distinct points that need to be graphed Explain
24 Critique Reasoning Jill says that 12 _
6 is less than 1263 Explain her error
FOCUS ON HIGHER ORDER THINKING
radic_
115 115 ___ 11 and 105624
between radic_
7 asymp 265 and radic_
8 asymp 283
between radic_
9 = 3 and radic_
10 asymp 316
22 __ 7 it is closer to π on the number line
She did not consider the repeating digit 1266
2 rational numbers can have the same location and
irrational numbers can have the same location but they
cannot share a location
355
Neither student is correct The answer
should be 115 ___ 11 105624 radic_
115
Unit 126
copy H
ough
ton M
ifflin
Har
cour
t Pub
lishin
g Com
pany
Imag
e Cre
dits
copy3D
Stoc
kiSt
ockP
hoto
com
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
8_MCAAESE206984_U1M01L3indd 26 210513 801 AM
Personal Math Trainer
Online Practice and Helpmyhrwcom
Name Class Date
Independent Practice
16 Your sister is considering two different shapes for her garden One is a square with side lengths of 35 meters and the other is a circle with a diameter of 4 meters
a Find the area of the square
b Find the area of the circle
c Compare your answers from parts a and b Which garden would give your sister the most space to plant
17 Winnie measured the length of her fatherrsquos ranch four times and got four different distances Her measurements are shown in the table
a To estimate the actual length Winnie first approximated each distance to the nearest hundredth Then she averaged the four numbers Using a calculator find Winniersquos estimate
b Winniersquos father estimated the distance across his ranch to be radic_
56 km How does this distance compare to Winniersquos estimate
Give an example of each type of number
18 a real number between radic_
13 and radic_
14
19 an irrational number between 5 and 7
Order the numbers from least to greatest
12 radic_
7 2 radic_
8 ___ 2 13 radic_
10 π 35
14 radic_
220 -10 radic_
100 115 15 radic_
8 -375 3 9 _ 4
Distance Across Fatherrsquos Ranch (km)
1 2 3 4
radic_
60 58 __ 8 7 _
3 7 3 _ 5
138NS2
radic_
8 ___ 2 2 radic_
7
-10 radic_
100 115 radic_
220
radic_
60 asymp 775 58 __ 8 = 725 7 _
3 asymp 733 7 3 _ 5 = 760 so the average
π radic_
10 35
-375 9 _ 4 radic_
8 3
is 74825 km
1225 m2
4π m2 or approximately 126 m2
They are nearly identical radic_
56 is approximately 74833hellip
The circle would give her more space to plant because it has a
larger area
Sample answer 37
Sample answer radic_
31
25Lesson 13
copy H
ough
ton
Miff
lin H
arco
urt P
ublis
hing
Com
pany
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
8_MCAAESE206984_U1M01L3indd 25 41613 448 AM
Activity available online myhrwcomEXTEND THE MATH PRE-AP
Activity Have students investigate whether there are infinitely many numbers between two numbers by giving examples for each of the following
bull Between any two rational numbers there is at least one other rational number Sample answer 45 is between 41 and 48
bull Between any two irrational numbers there is at least one rational number Sample answer 45 is between radic
_ 11 and radic
_ 29
bull Between any two rational numbers there is at least one irrational number Sample answer radic
_ 11 is between 31 and 36
bull Between any two irrational numbers there is at least one irrational number Sample answer radic
_ 17 is between radic
_ 11 and radic
_ 29
Ordering Real Numbers 26
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
ReadyMath Trainer
Online Practiceand Help
Personal
myhrwcom
Module Quiz
11ensp RationalenspandenspIrrationalenspNumbersWrite each fraction as a decimal or each decimal as a fraction
1 7__20 2 1___
27 3 17_8
Solve each equation for x
4 x2=81 5 x3=343 6 x2= 1___100
7 Asquarepatiohasanareaof200squarefeetHowlongiseachside
ofthepatiotothenearesttenth
12ensp SetsenspofenspRealenspNumbersWrite all names that apply to each number
8 121____radic
____121
9 π__2
10 TellwhetherthestatementldquoAllintegersarerationalnumbersrdquoistrueorfalseExplainyourchoice
13ensp OrderingenspRealenspNumbersCompare Write lt gt or =
11 radic__
8+3 8+radic__
3 12 radic__
5+11emsp emsp emsp 5+radic___
11
Order the numbers from least to greatest
13 radic___
99π29__
8 14 radic___
1__251_40__
2
15 Howarerealnumbersusedtodescribereal-worldsituations
ESSENTIAL QUESTION
035
9-9
141ft
7 1__10- 1__10
14__11 1875
wholeintegerrationalreal
Trueintegerscanbewrittenasthequotientoftwointegers
SampleanswerRealnumberssuchastherational
π29__
8radic___
99
irrationalreal
lt gt
number1_4candescribeamountsusedincooking
radic___
1__250__
21_4
27Module1
copy H
ough
ton
Miff
lin H
arco
urt P
ublis
hing
Com
pany
DONOTEDIT--ChangesmustbemadethroughldquoFileinfordquoCorrectionKey=A
8_MCAAESE206984_U1M01RTindd 27 41513 1113 PM
Math TrainerOnline Assessment
and Intervention
Personal
myhrwcom
1
2
3 Response toIntervention
Intervention Enrichment
Access Ready to Go On assessment online and receive instant scoring feedback and customized intervention or enrichment
Online and Print Resources
Differentiated Instruction
bull Reteach worksheets
bull Reading Strategies EL
bull Success for English Learners EL
Differentiated Instruction
bull Challenge worksheets PRE-AP
Extend the Math PRE-AP
Lesson Activities in TE
Additional ResourcesAssessment Resources includes bull Leveled Module Quizzes
Ready to Go OnAssess MasteryUse the assessment on this page to determine if students have mastered the concepts and standards covered in this module
California Common Core Standards
Lesson Exercises Common Core Standards
11 1ndash7 8NS1 8NS2 8EE2
12 8ndash10 8NS1
13 11ndash14 8NS2
27 Unit 1 Module 1
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
Personal Math Trainer
Online Practice and HelpmyhrwcomAssessment Readiness
Module 1 MIXed ReVIeW
1 Look at each number Is the number between 2π and radic___
52
Select Yes or No for expressions AndashC
A 6 2 _ 3 Yes No
B 5π __ 2 Yes No
C 3 radic__
5 Yes No
2 Consider the number - 11 __ 15
Choose True or False for each statement
A The number is rational True False
B The number can be written as True Falsea repeating decimal
C The number is less than ndash08 True False
3 The volume of a cube is given by V = x3 where x is the length of an edge of the cube A cube-shaped end table has a volume of 3 3 _ 8 cubic feet What is the length of an edge of the end table Explain how you solved this problem
4 A student says that radic___
83 is greater than 29 __ 3 Is the student correct Justify your
reasoning
1 1 _ 2 ft Sample answer The equation x3 = 3 3 _ 8 can be used
to find the edge length in feet To solve the equation
write the mixed number as a fraction greater than 1
x3 = 27 __ 8 Then take the cube root of both sides x = 3 _ 2 = 1 1 _ 2
No Sample answer radic___
83 asymp 91 and 29 __ 3 = 9
__ 6
Because 91 lt 9 __
6 radic___
83 lt 29 __ 3
28 Unit 1
copy H
ough
ton
Miff
lin H
arco
urt P
ublis
hing
Com
pany
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=A
8_MCAAESE206984_U1M01RTindd 28 240413 946 AM
Personal Math Trainer
Online Assessment and
Interventionmyhrwcom
Scoring GuideItem 3 Award the student 1 point for finding the edge length of the cube and 1 point for correctly explaining how to use a cube root to solve the problem
Item 4 Award the student 1 point for determining that the student is incorrect and 1 point for correctly justifying the reasoning for this conclusion
Additional ResourcesTo assign this assessment online login to your Assignment Manager at myhrwcom
Assessment Readiness
California Common Core Standards
Items Grade 8 Standards Mathematical Practices
1 8NS2 MP7
2 7NS2b 7NS2d 8NS1 MP7
3 8EE2 MP1 MP4
4 8NS1 8NS2 MP3
Item integrates mixed review concepts from previous modules or a previous course
Item 4 combines concepts from the California Common Core cluster ldquoKnow that there are numbers that are not rational and approximate them by rational numbersrdquo
Real Numbers 28
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
Math TrainerOnline Practice
and Help
Personal
myhrwcom
Math On the Spotmyhrwcom
0 05 1 15 2 25 3 35 4
radic5radic3
π2
8 85 9 95 10 105 11 115 12
radic75
4 42 44 46 48 5
radic224 12π + 1
Ordering Real Numbers You can compare and order real numbers and list them from least to greatest
Order radic_
22 π + 1 and 4 1 _ 2 from least to greatest
First approximate radic_
22
radic_
22 is between 4 and 5 Since you donrsquot know where it falls between 4 and 5 you need to find a better estimate for radic
_ 22 so
you can compare it to 4 1 _ 2
Since 22 is closer to 25 than 16 use squares of numbers between 45 and 5 to find a better estimate of radic
_ 22
45 2 = 2025 46 2 = 2116 47 2 = 2209 48 2 = 2304
Since 47 2 = 2209 an approximate value for radic_
22 is 47
An approximate value of π is 314 So an approximate value of π +1 is 414
Plot radic_
22 π + 1 and 4 1 _ 2 on a number line
Read the numbers from left to right to place them in order from least to greatest
From least to greatest the numbers are π + 1 4 1 _ 2 and radic_
22
EXAMPLE 2
STEP 1
STEP 2
Order the numbers from least to greatest Then graph them on the number line
YOUR TURN
5 radic_
5 25 radic_
3
6 π 2 10 radic_
75
If real numbers a b and c are in order from least to greatest what is the order
of their opposites from least to greatest
Explain
Math TalkMathematical Practices
8NS2
radic_
3 radic_
5 25
radic_
75 π2 10
Math Talk answer -c -b -a -c is farthest to the left on a number line -b is in the middle and -a is farthest to the right
Unit 122
copy H
ough
ton
Miff
lin H
arco
urt P
ublis
hing
Com
pany
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
8_MCAAESE206984_U1M01L3indd 22 41613 447 AM
My Notes
Math TrainerOnline Practice
and Help
Personal
myhrwcom
Math On the Spot
myhrwcom
Comparing Irrational NumbersBetween any two real numbers is another real number To compare and order real numbers you can approximate irrational numbers as decimals
Compare radic_
3 + 5 3 + radic_
5 Write lt gt or =
First approximate radic_
3
radic_
3 is between 1 and 2
Next approximate radic_
5
radic_
5 is between 2 and 3
Then use your approximations to simplify the expressions
radic_
3 + 5 is between 6 and 7
3 + radic_
5 is between 5 and 6
So radic_
3 + 5 gt 3 + radic_
5
Reflect1 If 7 + radic
_ 5 is equal to radic
_ 5 plus a number what do you know about the
number Why
2 What are the closest two integers that radic_
300 is between
EXAMPLEXAMPLE 1
STEP 1
STEP 2
Compare Write lt gt or =
YOUR TURN
3 radic_
2 + 4 2 + radic_
4 4 radic_
12 + 6 12 + radic_
6
L E S S O N
13 Ordering Real Numbers
ESSENTIAL QUESTIONHow do you order a set of real numbers
8NS2
Use rational approximations of irrational numbers to compare the size of irrational numbers locate them approximately on a number line diagram and estimate the value of expressions (eg π 2 )
8NS2
Use perfect squares to estimate square roots
1 2 = 1 2 2 = 4 3 2 = 9
The number is 7 both expressions must equal 7 + radic_
5
17 and 18
gt lt
21Lesson 13
copy H
ough
ton
Miff
lin H
arco
urt P
ublis
hing
Com
pany
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
8_MCAAESE206984_U1M01L3indd 21 41913 246 PM
PROFESSIONAL DEVELOPMENT
Math BackgroundIn this lesson students estimate irrational numbers in the form of square roots of nonper-fect squares by finding two perfect squares between which the number falls A more precise method involves repeated division For example to find radic
_ 28 find a whole number whose perfect
square is close to 28 such as 5 Divide 28 by that number 28 divide 5 = 56 Find the average of the quotient and divisor 5 + 56
_____ 2 = 53 Continue dividing 28 by each result and averaging until you get the desired accuracy
Integrate Mathematical Practices MP4
This lesson provides an opportunity to address this Mathematical Practices standard It calls for students to model relationships using multiple representations including diagrams graphs and language as appropriate Students use multiple representations when they use number lines to estimate the locations of and order rational and irrational numbers given as symbols
Ordering Real Numbers 22
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
Interactive Whiteboard Interactive example available online
ADDITIONAL EXAMPLE 3The diameter of a meteorite in millimeters is calculated by four different methods Order the results from least to greatest
Joe radic_
18 mm Lisa 13 __ 3 mm
Pablo 46 mm Julien 4π __ 3 mm
Julien 4π __ 3 mm Lisa 13 __ 3 mm
Joe radic_
18 mm Pablo 46 mm
EXAMPLE 3Questioning Strategies Mathematical Practices bull How can you verify that radic
_ 28 is between 52 and 53 5 2 2 = 2704 and 5 3 2 = 2809
bull Explain how to determine which number is greater 5 _
5 or 55 When the repeating decimal is rounded to the nearest tenth or hundredth you can see that it is greater
Connect to Daily LifeDiscuss how measuring across a canyon might involve different methods than measuring along a road Explain that measurements like these are often done using calculations that approximate the distance
YOUR TURNFocus on Critical Thinking Mathematical PracticesDiscuss with students which number is greater 3
_ 45 or 3450 3
_ 45 or 3455 and why Explain
that 3 _
45 can be written out as 34545hellipMake sure they understand that 3 _
45 is greater than 345 but less than 3455
ElaborateTalk About ItSummarize the Lesson
Ask How can you order two numbers in different forms whose decimal approxi-mations appear to be equal Approximate one or both numbers to an additional
number of decimal places
GUIDED PRACTICEEngage with the Whiteboard
Have students place and label additional points on the number line in Exercise 9 Allow the points to be in any format other than decimal
Avoid Common ErrorsExercises 3ndash4 Caution students to read the problem carefully so that they do not misread the problem as the same numbers combined by addition on each side of the circleExercise 10 Remind students that the calculations have units
myhrwcom
23 Lesson 13
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
0 05 1 15 2 25 3 35 4 45 5 55 6 65 7
2πradic3
Compare Write lt gt or = (Example 1)
1 radic_
3 + 2 radic_
3 + 3 2 radic_
8 + 17 radic_
11 + 15
3 radic_
6 + 5 6 + radic_
5 4 radic_
9 + 3 9 + radic_
3
5 radic_
17 - 3 -2 + radic_
5 6 12 - radic_
2 14 - radic_
8
7 radic_
7 + 2 radic_
10 - 1 8 radic_
17 + 3 3 + radic_
11
9 Order radic_
3 2π and 15 from least to greatest Then graph them on the number line (Example 2)
radic_
3 is between and so radic_
3 asymp
π asymp 314 so 2π asymp
From least to greatest the numbers are
10 Four people have found the perimeter of a forest using different methods Their results are given in the table Order their calculations from greatest to least (Example 3)
11 Explain how to order a set of real numbers
CHECK-INESSENTIAL QUESTION
Forest Perimeter (km)
Leon Mika Jason Ashley
radic_
17 - 2 1 +thinsp π __ 2 12 ___ 5 25
Guided Practice
17
15
1 + π _ 2 km 25 km 12 __ 5 km radic_
17 - 2 km
2π radic
_ 3
18 175
628
Sample answer Convert each number to a decimal
equivalent using estimation to find equivalents for
irrational numbers Graph each number on a number line
Read the numbers from left to right for least to greatest
Read the numbers from right to left for greatest to least
lt gt
lt lt
ltgt
gt gt
24 Unit 1
copy H
ough
ton
Miff
lin H
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urt P
ublis
hing
Com
pany
bull Im
age C
redi
ts copy
Elena
Eliss
eeva
Alam
y Im
ages
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
8_MCAAESE206984_U1M01L3indd 24 41613 448 AM
My Notes
5 52 54 56 58 6
radic28 5 12
23455
Math TrainerOnline Practice
and Help
Personal
myhrwcom
Math On the Spot
myhrwcom
Ordering Real Numbers in a Real-World Context Calculations and estimations in the real world may differ It can be important to know not only which are the most accurate but which give the greatest or least values depending upon the context
Four people have found the distance in kilometers across a canyon using different methods Their results are given in the table Order the distances from greatest to least
Distance Across Quarry Canyon (km)
Juana Lee Ann Ryne Jackson
radic_
28 23 __ 4 5 _
5 5 1 _ 2
Write each value as a decimal
radic_
28 is between 52 and 53 Since 53 2 = 2809 an approximate value for radic
_ 28 is 53
23 __ 4 = 575
5 _
5 is 5555hellip so 5 _
5 to the nearest hundredth is 556
5 1 _ 2 = 55
Plot radic_
28 23 __ 4 5 _
5 and 5 1 _ 2 on a number line
From greatest to least the distances are
23 __ 4 km 5 _
5 km 5 1 _ 2 km radic_
28 km
EXAMPLEXAMPLE 3
STEP 1
STEP 2
7 Four people have found the distance in miles across a crater using different methods Their results are given below
Jonathan 10 __ 3 Elaine 3 _
45 Joseacute 3 1 _ 2 Lashonda radic_
10
Order the distances from greatest to least
YOUR TURN
8NS2
3 1 _ 2 mi 3 _
45 mi 10 __ 3 mi radic_
10 mi
23Lesson 13
copy H
ough
ton
Miff
lin H
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ublis
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Com
pany
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
8_MCAAESE206984_U1M01L3indd 23 41613 447 AM
ModelingPlace papers around the room with the numbers from 1 to 5 one per sheet Give each student a card showing a number between 1 and 5 in different forms Have students place his or her card between the correct integers and decide where the number goes in relation to any numbers already placed
Multiple RepresentationsGive students a vertical number line which some students might find easier to use than a horizontal one Have them decide whether to place points for rational and irrational numbers above or below existing points
Additional ResourcesDifferentiated Instruction includes bull Reading Strategies bull Success for English Learners EL
bull Reteach bull Challenge PRE-AP
DIFFERENTIATE INSTRUCTION
Ordering Real Numbers 24
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
Math TrainerOnline Assessment
and Intervention
Personal
Online homework assignment available
myhrwcom
myhrwcom
Lesson Quiz available online
13 LESSON QUIZ
1 Compare Write lt gt or =
radic_
95 - 5 radic_
62 - 2
2 Order 105 radic_
105 and 3π + 1 from greatest to least
3 A length in centimeters is calculated differently by four different people Order their calculations from least to greatest
KD 11 __ 2 cm Silvio 5 __ 3 π cm
Paula 5 _
4 cm Luis radic_
33 cm
EvaluateGUIDED AND INDEPENDENT PRACTICE
Concepts amp Skills Practice
Example 1Comparing Irrational Numbers
Exercises 1ndash8
Example 2Ordering Real Numbers
Exercises 9 12ndash15 18ndash21
Example 3Ordering Real Numbers in a Real-World Context
Exercises 10 16ndash17
Additional ResourcesDifferentiated Instruction includes bull Leveled Practice worksheets
Focus | Coherence | Rigor
Exercise Depth of Knowledge (DOK) Mathematical Practices
12ndash15 1 Recall of Information MP5 Using Tools
16 2 SkillsConcepts MP2 Reasoning
17 2 SkillsConcepts MP6 Precision
18ndash21 2 SkillsConcepts MP2 Reasoning
22 3 Strategic Thinking MP4 Modeling
23ndash24 3 Strategic Thinking MP3 Logic
8NS2
8NS2
Answers1 radic
_ 95 - 5 lt radic
_ 62 - 2
2 radic_
105 3π + 1 105
3 Silvio 5 __ 3 π cm Paula 5 _
4 cm
KD 11
__ 2 cm Luis radic_
33 cm
25 Lesson 13
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
Work Area
3140 3141 3142 3143
314 π227
20 A teacher asks his students to write the numbers shown in order from least to greatest Paul thinks the numbers are already in order Sandra thinks the order should be reversed Who is right
21 Math History There is a famous irrational number called Eulerrsquos number symbolized with an e Like π its decimal form never ends or repeats The first few digits of e are 27182818284
a Between which two square roots of integers could you find this number
b Between which two square roots of integers can you find π
22 Analyze Relationships There are several approximations used for π including 314 and 22 __ 7 π is approximately 314159265358979
a Label π and the two approximations on the number line
b Which of the two approximations is a better estimate for π Explain
c Find a whole number x so that the ratio x ___ 113 is a better estimate for π
than the two given approximations
23 Communicate Mathematical Ideas If a set of six numbers that include both rational and irrational numbers is graphed on a number line what is the fewest number of distinct points that need to be graphed Explain
24 Critique Reasoning Jill says that 12 _
6 is less than 1263 Explain her error
FOCUS ON HIGHER ORDER THINKING
radic_
115 115 ___ 11 and 105624
between radic_
7 asymp 265 and radic_
8 asymp 283
between radic_
9 = 3 and radic_
10 asymp 316
22 __ 7 it is closer to π on the number line
She did not consider the repeating digit 1266
2 rational numbers can have the same location and
irrational numbers can have the same location but they
cannot share a location
355
Neither student is correct The answer
should be 115 ___ 11 105624 radic_
115
Unit 126
copy H
ough
ton M
ifflin
Har
cour
t Pub
lishin
g Com
pany
Imag
e Cre
dits
copy3D
Stoc
kiSt
ockP
hoto
com
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
8_MCAAESE206984_U1M01L3indd 26 210513 801 AM
Personal Math Trainer
Online Practice and Helpmyhrwcom
Name Class Date
Independent Practice
16 Your sister is considering two different shapes for her garden One is a square with side lengths of 35 meters and the other is a circle with a diameter of 4 meters
a Find the area of the square
b Find the area of the circle
c Compare your answers from parts a and b Which garden would give your sister the most space to plant
17 Winnie measured the length of her fatherrsquos ranch four times and got four different distances Her measurements are shown in the table
a To estimate the actual length Winnie first approximated each distance to the nearest hundredth Then she averaged the four numbers Using a calculator find Winniersquos estimate
b Winniersquos father estimated the distance across his ranch to be radic_
56 km How does this distance compare to Winniersquos estimate
Give an example of each type of number
18 a real number between radic_
13 and radic_
14
19 an irrational number between 5 and 7
Order the numbers from least to greatest
12 radic_
7 2 radic_
8 ___ 2 13 radic_
10 π 35
14 radic_
220 -10 radic_
100 115 15 radic_
8 -375 3 9 _ 4
Distance Across Fatherrsquos Ranch (km)
1 2 3 4
radic_
60 58 __ 8 7 _
3 7 3 _ 5
138NS2
radic_
8 ___ 2 2 radic_
7
-10 radic_
100 115 radic_
220
radic_
60 asymp 775 58 __ 8 = 725 7 _
3 asymp 733 7 3 _ 5 = 760 so the average
π radic_
10 35
-375 9 _ 4 radic_
8 3
is 74825 km
1225 m2
4π m2 or approximately 126 m2
They are nearly identical radic_
56 is approximately 74833hellip
The circle would give her more space to plant because it has a
larger area
Sample answer 37
Sample answer radic_
31
25Lesson 13
copy H
ough
ton
Miff
lin H
arco
urt P
ublis
hing
Com
pany
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
8_MCAAESE206984_U1M01L3indd 25 41613 448 AM
Activity available online myhrwcomEXTEND THE MATH PRE-AP
Activity Have students investigate whether there are infinitely many numbers between two numbers by giving examples for each of the following
bull Between any two rational numbers there is at least one other rational number Sample answer 45 is between 41 and 48
bull Between any two irrational numbers there is at least one rational number Sample answer 45 is between radic
_ 11 and radic
_ 29
bull Between any two rational numbers there is at least one irrational number Sample answer radic
_ 11 is between 31 and 36
bull Between any two irrational numbers there is at least one irrational number Sample answer radic
_ 17 is between radic
_ 11 and radic
_ 29
Ordering Real Numbers 26
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
ReadyMath Trainer
Online Practiceand Help
Personal
myhrwcom
Module Quiz
11ensp RationalenspandenspIrrationalenspNumbersWrite each fraction as a decimal or each decimal as a fraction
1 7__20 2 1___
27 3 17_8
Solve each equation for x
4 x2=81 5 x3=343 6 x2= 1___100
7 Asquarepatiohasanareaof200squarefeetHowlongiseachside
ofthepatiotothenearesttenth
12ensp SetsenspofenspRealenspNumbersWrite all names that apply to each number
8 121____radic
____121
9 π__2
10 TellwhetherthestatementldquoAllintegersarerationalnumbersrdquoistrueorfalseExplainyourchoice
13ensp OrderingenspRealenspNumbersCompare Write lt gt or =
11 radic__
8+3 8+radic__
3 12 radic__
5+11emsp emsp emsp 5+radic___
11
Order the numbers from least to greatest
13 radic___
99π29__
8 14 radic___
1__251_40__
2
15 Howarerealnumbersusedtodescribereal-worldsituations
ESSENTIAL QUESTION
035
9-9
141ft
7 1__10- 1__10
14__11 1875
wholeintegerrationalreal
Trueintegerscanbewrittenasthequotientoftwointegers
SampleanswerRealnumberssuchastherational
π29__
8radic___
99
irrationalreal
lt gt
number1_4candescribeamountsusedincooking
radic___
1__250__
21_4
27Module1
copy H
ough
ton
Miff
lin H
arco
urt P
ublis
hing
Com
pany
DONOTEDIT--ChangesmustbemadethroughldquoFileinfordquoCorrectionKey=A
8_MCAAESE206984_U1M01RTindd 27 41513 1113 PM
Math TrainerOnline Assessment
and Intervention
Personal
myhrwcom
1
2
3 Response toIntervention
Intervention Enrichment
Access Ready to Go On assessment online and receive instant scoring feedback and customized intervention or enrichment
Online and Print Resources
Differentiated Instruction
bull Reteach worksheets
bull Reading Strategies EL
bull Success for English Learners EL
Differentiated Instruction
bull Challenge worksheets PRE-AP
Extend the Math PRE-AP
Lesson Activities in TE
Additional ResourcesAssessment Resources includes bull Leveled Module Quizzes
Ready to Go OnAssess MasteryUse the assessment on this page to determine if students have mastered the concepts and standards covered in this module
California Common Core Standards
Lesson Exercises Common Core Standards
11 1ndash7 8NS1 8NS2 8EE2
12 8ndash10 8NS1
13 11ndash14 8NS2
27 Unit 1 Module 1
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
Personal Math Trainer
Online Practice and HelpmyhrwcomAssessment Readiness
Module 1 MIXed ReVIeW
1 Look at each number Is the number between 2π and radic___
52
Select Yes or No for expressions AndashC
A 6 2 _ 3 Yes No
B 5π __ 2 Yes No
C 3 radic__
5 Yes No
2 Consider the number - 11 __ 15
Choose True or False for each statement
A The number is rational True False
B The number can be written as True Falsea repeating decimal
C The number is less than ndash08 True False
3 The volume of a cube is given by V = x3 where x is the length of an edge of the cube A cube-shaped end table has a volume of 3 3 _ 8 cubic feet What is the length of an edge of the end table Explain how you solved this problem
4 A student says that radic___
83 is greater than 29 __ 3 Is the student correct Justify your
reasoning
1 1 _ 2 ft Sample answer The equation x3 = 3 3 _ 8 can be used
to find the edge length in feet To solve the equation
write the mixed number as a fraction greater than 1
x3 = 27 __ 8 Then take the cube root of both sides x = 3 _ 2 = 1 1 _ 2
No Sample answer radic___
83 asymp 91 and 29 __ 3 = 9
__ 6
Because 91 lt 9 __
6 radic___
83 lt 29 __ 3
28 Unit 1
copy H
ough
ton
Miff
lin H
arco
urt P
ublis
hing
Com
pany
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=A
8_MCAAESE206984_U1M01RTindd 28 240413 946 AM
Personal Math Trainer
Online Assessment and
Interventionmyhrwcom
Scoring GuideItem 3 Award the student 1 point for finding the edge length of the cube and 1 point for correctly explaining how to use a cube root to solve the problem
Item 4 Award the student 1 point for determining that the student is incorrect and 1 point for correctly justifying the reasoning for this conclusion
Additional ResourcesTo assign this assessment online login to your Assignment Manager at myhrwcom
Assessment Readiness
California Common Core Standards
Items Grade 8 Standards Mathematical Practices
1 8NS2 MP7
2 7NS2b 7NS2d 8NS1 MP7
3 8EE2 MP1 MP4
4 8NS1 8NS2 MP3
Item integrates mixed review concepts from previous modules or a previous course
Item 4 combines concepts from the California Common Core cluster ldquoKnow that there are numbers that are not rational and approximate them by rational numbersrdquo
Real Numbers 28
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
Interactive Whiteboard Interactive example available online
ADDITIONAL EXAMPLE 3The diameter of a meteorite in millimeters is calculated by four different methods Order the results from least to greatest
Joe radic_
18 mm Lisa 13 __ 3 mm
Pablo 46 mm Julien 4π __ 3 mm
Julien 4π __ 3 mm Lisa 13 __ 3 mm
Joe radic_
18 mm Pablo 46 mm
EXAMPLE 3Questioning Strategies Mathematical Practices bull How can you verify that radic
_ 28 is between 52 and 53 5 2 2 = 2704 and 5 3 2 = 2809
bull Explain how to determine which number is greater 5 _
5 or 55 When the repeating decimal is rounded to the nearest tenth or hundredth you can see that it is greater
Connect to Daily LifeDiscuss how measuring across a canyon might involve different methods than measuring along a road Explain that measurements like these are often done using calculations that approximate the distance
YOUR TURNFocus on Critical Thinking Mathematical PracticesDiscuss with students which number is greater 3
_ 45 or 3450 3
_ 45 or 3455 and why Explain
that 3 _
45 can be written out as 34545hellipMake sure they understand that 3 _
45 is greater than 345 but less than 3455
ElaborateTalk About ItSummarize the Lesson
Ask How can you order two numbers in different forms whose decimal approxi-mations appear to be equal Approximate one or both numbers to an additional
number of decimal places
GUIDED PRACTICEEngage with the Whiteboard
Have students place and label additional points on the number line in Exercise 9 Allow the points to be in any format other than decimal
Avoid Common ErrorsExercises 3ndash4 Caution students to read the problem carefully so that they do not misread the problem as the same numbers combined by addition on each side of the circleExercise 10 Remind students that the calculations have units
myhrwcom
23 Lesson 13
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
0 05 1 15 2 25 3 35 4 45 5 55 6 65 7
2πradic3
Compare Write lt gt or = (Example 1)
1 radic_
3 + 2 radic_
3 + 3 2 radic_
8 + 17 radic_
11 + 15
3 radic_
6 + 5 6 + radic_
5 4 radic_
9 + 3 9 + radic_
3
5 radic_
17 - 3 -2 + radic_
5 6 12 - radic_
2 14 - radic_
8
7 radic_
7 + 2 radic_
10 - 1 8 radic_
17 + 3 3 + radic_
11
9 Order radic_
3 2π and 15 from least to greatest Then graph them on the number line (Example 2)
radic_
3 is between and so radic_
3 asymp
π asymp 314 so 2π asymp
From least to greatest the numbers are
10 Four people have found the perimeter of a forest using different methods Their results are given in the table Order their calculations from greatest to least (Example 3)
11 Explain how to order a set of real numbers
CHECK-INESSENTIAL QUESTION
Forest Perimeter (km)
Leon Mika Jason Ashley
radic_
17 - 2 1 +thinsp π __ 2 12 ___ 5 25
Guided Practice
17
15
1 + π _ 2 km 25 km 12 __ 5 km radic_
17 - 2 km
2π radic
_ 3
18 175
628
Sample answer Convert each number to a decimal
equivalent using estimation to find equivalents for
irrational numbers Graph each number on a number line
Read the numbers from left to right for least to greatest
Read the numbers from right to left for greatest to least
lt gt
lt lt
ltgt
gt gt
24 Unit 1
copy H
ough
ton
Miff
lin H
arco
urt P
ublis
hing
Com
pany
bull Im
age C
redi
ts copy
Elena
Eliss
eeva
Alam
y Im
ages
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
8_MCAAESE206984_U1M01L3indd 24 41613 448 AM
My Notes
5 52 54 56 58 6
radic28 5 12
23455
Math TrainerOnline Practice
and Help
Personal
myhrwcom
Math On the Spot
myhrwcom
Ordering Real Numbers in a Real-World Context Calculations and estimations in the real world may differ It can be important to know not only which are the most accurate but which give the greatest or least values depending upon the context
Four people have found the distance in kilometers across a canyon using different methods Their results are given in the table Order the distances from greatest to least
Distance Across Quarry Canyon (km)
Juana Lee Ann Ryne Jackson
radic_
28 23 __ 4 5 _
5 5 1 _ 2
Write each value as a decimal
radic_
28 is between 52 and 53 Since 53 2 = 2809 an approximate value for radic
_ 28 is 53
23 __ 4 = 575
5 _
5 is 5555hellip so 5 _
5 to the nearest hundredth is 556
5 1 _ 2 = 55
Plot radic_
28 23 __ 4 5 _
5 and 5 1 _ 2 on a number line
From greatest to least the distances are
23 __ 4 km 5 _
5 km 5 1 _ 2 km radic_
28 km
EXAMPLEXAMPLE 3
STEP 1
STEP 2
7 Four people have found the distance in miles across a crater using different methods Their results are given below
Jonathan 10 __ 3 Elaine 3 _
45 Joseacute 3 1 _ 2 Lashonda radic_
10
Order the distances from greatest to least
YOUR TURN
8NS2
3 1 _ 2 mi 3 _
45 mi 10 __ 3 mi radic_
10 mi
23Lesson 13
copy H
ough
ton
Miff
lin H
arco
urt P
ublis
hing
Com
pany
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
8_MCAAESE206984_U1M01L3indd 23 41613 447 AM
ModelingPlace papers around the room with the numbers from 1 to 5 one per sheet Give each student a card showing a number between 1 and 5 in different forms Have students place his or her card between the correct integers and decide where the number goes in relation to any numbers already placed
Multiple RepresentationsGive students a vertical number line which some students might find easier to use than a horizontal one Have them decide whether to place points for rational and irrational numbers above or below existing points
Additional ResourcesDifferentiated Instruction includes bull Reading Strategies bull Success for English Learners EL
bull Reteach bull Challenge PRE-AP
DIFFERENTIATE INSTRUCTION
Ordering Real Numbers 24
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
Math TrainerOnline Assessment
and Intervention
Personal
Online homework assignment available
myhrwcom
myhrwcom
Lesson Quiz available online
13 LESSON QUIZ
1 Compare Write lt gt or =
radic_
95 - 5 radic_
62 - 2
2 Order 105 radic_
105 and 3π + 1 from greatest to least
3 A length in centimeters is calculated differently by four different people Order their calculations from least to greatest
KD 11 __ 2 cm Silvio 5 __ 3 π cm
Paula 5 _
4 cm Luis radic_
33 cm
EvaluateGUIDED AND INDEPENDENT PRACTICE
Concepts amp Skills Practice
Example 1Comparing Irrational Numbers
Exercises 1ndash8
Example 2Ordering Real Numbers
Exercises 9 12ndash15 18ndash21
Example 3Ordering Real Numbers in a Real-World Context
Exercises 10 16ndash17
Additional ResourcesDifferentiated Instruction includes bull Leveled Practice worksheets
Focus | Coherence | Rigor
Exercise Depth of Knowledge (DOK) Mathematical Practices
12ndash15 1 Recall of Information MP5 Using Tools
16 2 SkillsConcepts MP2 Reasoning
17 2 SkillsConcepts MP6 Precision
18ndash21 2 SkillsConcepts MP2 Reasoning
22 3 Strategic Thinking MP4 Modeling
23ndash24 3 Strategic Thinking MP3 Logic
8NS2
8NS2
Answers1 radic
_ 95 - 5 lt radic
_ 62 - 2
2 radic_
105 3π + 1 105
3 Silvio 5 __ 3 π cm Paula 5 _
4 cm
KD 11
__ 2 cm Luis radic_
33 cm
25 Lesson 13
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
Work Area
3140 3141 3142 3143
314 π227
20 A teacher asks his students to write the numbers shown in order from least to greatest Paul thinks the numbers are already in order Sandra thinks the order should be reversed Who is right
21 Math History There is a famous irrational number called Eulerrsquos number symbolized with an e Like π its decimal form never ends or repeats The first few digits of e are 27182818284
a Between which two square roots of integers could you find this number
b Between which two square roots of integers can you find π
22 Analyze Relationships There are several approximations used for π including 314 and 22 __ 7 π is approximately 314159265358979
a Label π and the two approximations on the number line
b Which of the two approximations is a better estimate for π Explain
c Find a whole number x so that the ratio x ___ 113 is a better estimate for π
than the two given approximations
23 Communicate Mathematical Ideas If a set of six numbers that include both rational and irrational numbers is graphed on a number line what is the fewest number of distinct points that need to be graphed Explain
24 Critique Reasoning Jill says that 12 _
6 is less than 1263 Explain her error
FOCUS ON HIGHER ORDER THINKING
radic_
115 115 ___ 11 and 105624
between radic_
7 asymp 265 and radic_
8 asymp 283
between radic_
9 = 3 and radic_
10 asymp 316
22 __ 7 it is closer to π on the number line
She did not consider the repeating digit 1266
2 rational numbers can have the same location and
irrational numbers can have the same location but they
cannot share a location
355
Neither student is correct The answer
should be 115 ___ 11 105624 radic_
115
Unit 126
copy H
ough
ton M
ifflin
Har
cour
t Pub
lishin
g Com
pany
Imag
e Cre
dits
copy3D
Stoc
kiSt
ockP
hoto
com
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
8_MCAAESE206984_U1M01L3indd 26 210513 801 AM
Personal Math Trainer
Online Practice and Helpmyhrwcom
Name Class Date
Independent Practice
16 Your sister is considering two different shapes for her garden One is a square with side lengths of 35 meters and the other is a circle with a diameter of 4 meters
a Find the area of the square
b Find the area of the circle
c Compare your answers from parts a and b Which garden would give your sister the most space to plant
17 Winnie measured the length of her fatherrsquos ranch four times and got four different distances Her measurements are shown in the table
a To estimate the actual length Winnie first approximated each distance to the nearest hundredth Then she averaged the four numbers Using a calculator find Winniersquos estimate
b Winniersquos father estimated the distance across his ranch to be radic_
56 km How does this distance compare to Winniersquos estimate
Give an example of each type of number
18 a real number between radic_
13 and radic_
14
19 an irrational number between 5 and 7
Order the numbers from least to greatest
12 radic_
7 2 radic_
8 ___ 2 13 radic_
10 π 35
14 radic_
220 -10 radic_
100 115 15 radic_
8 -375 3 9 _ 4
Distance Across Fatherrsquos Ranch (km)
1 2 3 4
radic_
60 58 __ 8 7 _
3 7 3 _ 5
138NS2
radic_
8 ___ 2 2 radic_
7
-10 radic_
100 115 radic_
220
radic_
60 asymp 775 58 __ 8 = 725 7 _
3 asymp 733 7 3 _ 5 = 760 so the average
π radic_
10 35
-375 9 _ 4 radic_
8 3
is 74825 km
1225 m2
4π m2 or approximately 126 m2
They are nearly identical radic_
56 is approximately 74833hellip
The circle would give her more space to plant because it has a
larger area
Sample answer 37
Sample answer radic_
31
25Lesson 13
copy H
ough
ton
Miff
lin H
arco
urt P
ublis
hing
Com
pany
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
8_MCAAESE206984_U1M01L3indd 25 41613 448 AM
Activity available online myhrwcomEXTEND THE MATH PRE-AP
Activity Have students investigate whether there are infinitely many numbers between two numbers by giving examples for each of the following
bull Between any two rational numbers there is at least one other rational number Sample answer 45 is between 41 and 48
bull Between any two irrational numbers there is at least one rational number Sample answer 45 is between radic
_ 11 and radic
_ 29
bull Between any two rational numbers there is at least one irrational number Sample answer radic
_ 11 is between 31 and 36
bull Between any two irrational numbers there is at least one irrational number Sample answer radic
_ 17 is between radic
_ 11 and radic
_ 29
Ordering Real Numbers 26
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
ReadyMath Trainer
Online Practiceand Help
Personal
myhrwcom
Module Quiz
11ensp RationalenspandenspIrrationalenspNumbersWrite each fraction as a decimal or each decimal as a fraction
1 7__20 2 1___
27 3 17_8
Solve each equation for x
4 x2=81 5 x3=343 6 x2= 1___100
7 Asquarepatiohasanareaof200squarefeetHowlongiseachside
ofthepatiotothenearesttenth
12ensp SetsenspofenspRealenspNumbersWrite all names that apply to each number
8 121____radic
____121
9 π__2
10 TellwhetherthestatementldquoAllintegersarerationalnumbersrdquoistrueorfalseExplainyourchoice
13ensp OrderingenspRealenspNumbersCompare Write lt gt or =
11 radic__
8+3 8+radic__
3 12 radic__
5+11emsp emsp emsp 5+radic___
11
Order the numbers from least to greatest
13 radic___
99π29__
8 14 radic___
1__251_40__
2
15 Howarerealnumbersusedtodescribereal-worldsituations
ESSENTIAL QUESTION
035
9-9
141ft
7 1__10- 1__10
14__11 1875
wholeintegerrationalreal
Trueintegerscanbewrittenasthequotientoftwointegers
SampleanswerRealnumberssuchastherational
π29__
8radic___
99
irrationalreal
lt gt
number1_4candescribeamountsusedincooking
radic___
1__250__
21_4
27Module1
copy H
ough
ton
Miff
lin H
arco
urt P
ublis
hing
Com
pany
DONOTEDIT--ChangesmustbemadethroughldquoFileinfordquoCorrectionKey=A
8_MCAAESE206984_U1M01RTindd 27 41513 1113 PM
Math TrainerOnline Assessment
and Intervention
Personal
myhrwcom
1
2
3 Response toIntervention
Intervention Enrichment
Access Ready to Go On assessment online and receive instant scoring feedback and customized intervention or enrichment
Online and Print Resources
Differentiated Instruction
bull Reteach worksheets
bull Reading Strategies EL
bull Success for English Learners EL
Differentiated Instruction
bull Challenge worksheets PRE-AP
Extend the Math PRE-AP
Lesson Activities in TE
Additional ResourcesAssessment Resources includes bull Leveled Module Quizzes
Ready to Go OnAssess MasteryUse the assessment on this page to determine if students have mastered the concepts and standards covered in this module
California Common Core Standards
Lesson Exercises Common Core Standards
11 1ndash7 8NS1 8NS2 8EE2
12 8ndash10 8NS1
13 11ndash14 8NS2
27 Unit 1 Module 1
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
Personal Math Trainer
Online Practice and HelpmyhrwcomAssessment Readiness
Module 1 MIXed ReVIeW
1 Look at each number Is the number between 2π and radic___
52
Select Yes or No for expressions AndashC
A 6 2 _ 3 Yes No
B 5π __ 2 Yes No
C 3 radic__
5 Yes No
2 Consider the number - 11 __ 15
Choose True or False for each statement
A The number is rational True False
B The number can be written as True Falsea repeating decimal
C The number is less than ndash08 True False
3 The volume of a cube is given by V = x3 where x is the length of an edge of the cube A cube-shaped end table has a volume of 3 3 _ 8 cubic feet What is the length of an edge of the end table Explain how you solved this problem
4 A student says that radic___
83 is greater than 29 __ 3 Is the student correct Justify your
reasoning
1 1 _ 2 ft Sample answer The equation x3 = 3 3 _ 8 can be used
to find the edge length in feet To solve the equation
write the mixed number as a fraction greater than 1
x3 = 27 __ 8 Then take the cube root of both sides x = 3 _ 2 = 1 1 _ 2
No Sample answer radic___
83 asymp 91 and 29 __ 3 = 9
__ 6
Because 91 lt 9 __
6 radic___
83 lt 29 __ 3
28 Unit 1
copy H
ough
ton
Miff
lin H
arco
urt P
ublis
hing
Com
pany
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=A
8_MCAAESE206984_U1M01RTindd 28 240413 946 AM
Personal Math Trainer
Online Assessment and
Interventionmyhrwcom
Scoring GuideItem 3 Award the student 1 point for finding the edge length of the cube and 1 point for correctly explaining how to use a cube root to solve the problem
Item 4 Award the student 1 point for determining that the student is incorrect and 1 point for correctly justifying the reasoning for this conclusion
Additional ResourcesTo assign this assessment online login to your Assignment Manager at myhrwcom
Assessment Readiness
California Common Core Standards
Items Grade 8 Standards Mathematical Practices
1 8NS2 MP7
2 7NS2b 7NS2d 8NS1 MP7
3 8EE2 MP1 MP4
4 8NS1 8NS2 MP3
Item integrates mixed review concepts from previous modules or a previous course
Item 4 combines concepts from the California Common Core cluster ldquoKnow that there are numbers that are not rational and approximate them by rational numbersrdquo
Real Numbers 28
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
0 05 1 15 2 25 3 35 4 45 5 55 6 65 7
2πradic3
Compare Write lt gt or = (Example 1)
1 radic_
3 + 2 radic_
3 + 3 2 radic_
8 + 17 radic_
11 + 15
3 radic_
6 + 5 6 + radic_
5 4 radic_
9 + 3 9 + radic_
3
5 radic_
17 - 3 -2 + radic_
5 6 12 - radic_
2 14 - radic_
8
7 radic_
7 + 2 radic_
10 - 1 8 radic_
17 + 3 3 + radic_
11
9 Order radic_
3 2π and 15 from least to greatest Then graph them on the number line (Example 2)
radic_
3 is between and so radic_
3 asymp
π asymp 314 so 2π asymp
From least to greatest the numbers are
10 Four people have found the perimeter of a forest using different methods Their results are given in the table Order their calculations from greatest to least (Example 3)
11 Explain how to order a set of real numbers
CHECK-INESSENTIAL QUESTION
Forest Perimeter (km)
Leon Mika Jason Ashley
radic_
17 - 2 1 +thinsp π __ 2 12 ___ 5 25
Guided Practice
17
15
1 + π _ 2 km 25 km 12 __ 5 km radic_
17 - 2 km
2π radic
_ 3
18 175
628
Sample answer Convert each number to a decimal
equivalent using estimation to find equivalents for
irrational numbers Graph each number on a number line
Read the numbers from left to right for least to greatest
Read the numbers from right to left for greatest to least
lt gt
lt lt
ltgt
gt gt
24 Unit 1
copy H
ough
ton
Miff
lin H
arco
urt P
ublis
hing
Com
pany
bull Im
age C
redi
ts copy
Elena
Eliss
eeva
Alam
y Im
ages
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
8_MCAAESE206984_U1M01L3indd 24 41613 448 AM
My Notes
5 52 54 56 58 6
radic28 5 12
23455
Math TrainerOnline Practice
and Help
Personal
myhrwcom
Math On the Spot
myhrwcom
Ordering Real Numbers in a Real-World Context Calculations and estimations in the real world may differ It can be important to know not only which are the most accurate but which give the greatest or least values depending upon the context
Four people have found the distance in kilometers across a canyon using different methods Their results are given in the table Order the distances from greatest to least
Distance Across Quarry Canyon (km)
Juana Lee Ann Ryne Jackson
radic_
28 23 __ 4 5 _
5 5 1 _ 2
Write each value as a decimal
radic_
28 is between 52 and 53 Since 53 2 = 2809 an approximate value for radic
_ 28 is 53
23 __ 4 = 575
5 _
5 is 5555hellip so 5 _
5 to the nearest hundredth is 556
5 1 _ 2 = 55
Plot radic_
28 23 __ 4 5 _
5 and 5 1 _ 2 on a number line
From greatest to least the distances are
23 __ 4 km 5 _
5 km 5 1 _ 2 km radic_
28 km
EXAMPLEXAMPLE 3
STEP 1
STEP 2
7 Four people have found the distance in miles across a crater using different methods Their results are given below
Jonathan 10 __ 3 Elaine 3 _
45 Joseacute 3 1 _ 2 Lashonda radic_
10
Order the distances from greatest to least
YOUR TURN
8NS2
3 1 _ 2 mi 3 _
45 mi 10 __ 3 mi radic_
10 mi
23Lesson 13
copy H
ough
ton
Miff
lin H
arco
urt P
ublis
hing
Com
pany
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
8_MCAAESE206984_U1M01L3indd 23 41613 447 AM
ModelingPlace papers around the room with the numbers from 1 to 5 one per sheet Give each student a card showing a number between 1 and 5 in different forms Have students place his or her card between the correct integers and decide where the number goes in relation to any numbers already placed
Multiple RepresentationsGive students a vertical number line which some students might find easier to use than a horizontal one Have them decide whether to place points for rational and irrational numbers above or below existing points
Additional ResourcesDifferentiated Instruction includes bull Reading Strategies bull Success for English Learners EL
bull Reteach bull Challenge PRE-AP
DIFFERENTIATE INSTRUCTION
Ordering Real Numbers 24
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
Math TrainerOnline Assessment
and Intervention
Personal
Online homework assignment available
myhrwcom
myhrwcom
Lesson Quiz available online
13 LESSON QUIZ
1 Compare Write lt gt or =
radic_
95 - 5 radic_
62 - 2
2 Order 105 radic_
105 and 3π + 1 from greatest to least
3 A length in centimeters is calculated differently by four different people Order their calculations from least to greatest
KD 11 __ 2 cm Silvio 5 __ 3 π cm
Paula 5 _
4 cm Luis radic_
33 cm
EvaluateGUIDED AND INDEPENDENT PRACTICE
Concepts amp Skills Practice
Example 1Comparing Irrational Numbers
Exercises 1ndash8
Example 2Ordering Real Numbers
Exercises 9 12ndash15 18ndash21
Example 3Ordering Real Numbers in a Real-World Context
Exercises 10 16ndash17
Additional ResourcesDifferentiated Instruction includes bull Leveled Practice worksheets
Focus | Coherence | Rigor
Exercise Depth of Knowledge (DOK) Mathematical Practices
12ndash15 1 Recall of Information MP5 Using Tools
16 2 SkillsConcepts MP2 Reasoning
17 2 SkillsConcepts MP6 Precision
18ndash21 2 SkillsConcepts MP2 Reasoning
22 3 Strategic Thinking MP4 Modeling
23ndash24 3 Strategic Thinking MP3 Logic
8NS2
8NS2
Answers1 radic
_ 95 - 5 lt radic
_ 62 - 2
2 radic_
105 3π + 1 105
3 Silvio 5 __ 3 π cm Paula 5 _
4 cm
KD 11
__ 2 cm Luis radic_
33 cm
25 Lesson 13
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
Work Area
3140 3141 3142 3143
314 π227
20 A teacher asks his students to write the numbers shown in order from least to greatest Paul thinks the numbers are already in order Sandra thinks the order should be reversed Who is right
21 Math History There is a famous irrational number called Eulerrsquos number symbolized with an e Like π its decimal form never ends or repeats The first few digits of e are 27182818284
a Between which two square roots of integers could you find this number
b Between which two square roots of integers can you find π
22 Analyze Relationships There are several approximations used for π including 314 and 22 __ 7 π is approximately 314159265358979
a Label π and the two approximations on the number line
b Which of the two approximations is a better estimate for π Explain
c Find a whole number x so that the ratio x ___ 113 is a better estimate for π
than the two given approximations
23 Communicate Mathematical Ideas If a set of six numbers that include both rational and irrational numbers is graphed on a number line what is the fewest number of distinct points that need to be graphed Explain
24 Critique Reasoning Jill says that 12 _
6 is less than 1263 Explain her error
FOCUS ON HIGHER ORDER THINKING
radic_
115 115 ___ 11 and 105624
between radic_
7 asymp 265 and radic_
8 asymp 283
between radic_
9 = 3 and radic_
10 asymp 316
22 __ 7 it is closer to π on the number line
She did not consider the repeating digit 1266
2 rational numbers can have the same location and
irrational numbers can have the same location but they
cannot share a location
355
Neither student is correct The answer
should be 115 ___ 11 105624 radic_
115
Unit 126
copy H
ough
ton M
ifflin
Har
cour
t Pub
lishin
g Com
pany
Imag
e Cre
dits
copy3D
Stoc
kiSt
ockP
hoto
com
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
8_MCAAESE206984_U1M01L3indd 26 210513 801 AM
Personal Math Trainer
Online Practice and Helpmyhrwcom
Name Class Date
Independent Practice
16 Your sister is considering two different shapes for her garden One is a square with side lengths of 35 meters and the other is a circle with a diameter of 4 meters
a Find the area of the square
b Find the area of the circle
c Compare your answers from parts a and b Which garden would give your sister the most space to plant
17 Winnie measured the length of her fatherrsquos ranch four times and got four different distances Her measurements are shown in the table
a To estimate the actual length Winnie first approximated each distance to the nearest hundredth Then she averaged the four numbers Using a calculator find Winniersquos estimate
b Winniersquos father estimated the distance across his ranch to be radic_
56 km How does this distance compare to Winniersquos estimate
Give an example of each type of number
18 a real number between radic_
13 and radic_
14
19 an irrational number between 5 and 7
Order the numbers from least to greatest
12 radic_
7 2 radic_
8 ___ 2 13 radic_
10 π 35
14 radic_
220 -10 radic_
100 115 15 radic_
8 -375 3 9 _ 4
Distance Across Fatherrsquos Ranch (km)
1 2 3 4
radic_
60 58 __ 8 7 _
3 7 3 _ 5
138NS2
radic_
8 ___ 2 2 radic_
7
-10 radic_
100 115 radic_
220
radic_
60 asymp 775 58 __ 8 = 725 7 _
3 asymp 733 7 3 _ 5 = 760 so the average
π radic_
10 35
-375 9 _ 4 radic_
8 3
is 74825 km
1225 m2
4π m2 or approximately 126 m2
They are nearly identical radic_
56 is approximately 74833hellip
The circle would give her more space to plant because it has a
larger area
Sample answer 37
Sample answer radic_
31
25Lesson 13
copy H
ough
ton
Miff
lin H
arco
urt P
ublis
hing
Com
pany
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
8_MCAAESE206984_U1M01L3indd 25 41613 448 AM
Activity available online myhrwcomEXTEND THE MATH PRE-AP
Activity Have students investigate whether there are infinitely many numbers between two numbers by giving examples for each of the following
bull Between any two rational numbers there is at least one other rational number Sample answer 45 is between 41 and 48
bull Between any two irrational numbers there is at least one rational number Sample answer 45 is between radic
_ 11 and radic
_ 29
bull Between any two rational numbers there is at least one irrational number Sample answer radic
_ 11 is between 31 and 36
bull Between any two irrational numbers there is at least one irrational number Sample answer radic
_ 17 is between radic
_ 11 and radic
_ 29
Ordering Real Numbers 26
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
ReadyMath Trainer
Online Practiceand Help
Personal
myhrwcom
Module Quiz
11ensp RationalenspandenspIrrationalenspNumbersWrite each fraction as a decimal or each decimal as a fraction
1 7__20 2 1___
27 3 17_8
Solve each equation for x
4 x2=81 5 x3=343 6 x2= 1___100
7 Asquarepatiohasanareaof200squarefeetHowlongiseachside
ofthepatiotothenearesttenth
12ensp SetsenspofenspRealenspNumbersWrite all names that apply to each number
8 121____radic
____121
9 π__2
10 TellwhetherthestatementldquoAllintegersarerationalnumbersrdquoistrueorfalseExplainyourchoice
13ensp OrderingenspRealenspNumbersCompare Write lt gt or =
11 radic__
8+3 8+radic__
3 12 radic__
5+11emsp emsp emsp 5+radic___
11
Order the numbers from least to greatest
13 radic___
99π29__
8 14 radic___
1__251_40__
2
15 Howarerealnumbersusedtodescribereal-worldsituations
ESSENTIAL QUESTION
035
9-9
141ft
7 1__10- 1__10
14__11 1875
wholeintegerrationalreal
Trueintegerscanbewrittenasthequotientoftwointegers
SampleanswerRealnumberssuchastherational
π29__
8radic___
99
irrationalreal
lt gt
number1_4candescribeamountsusedincooking
radic___
1__250__
21_4
27Module1
copy H
ough
ton
Miff
lin H
arco
urt P
ublis
hing
Com
pany
DONOTEDIT--ChangesmustbemadethroughldquoFileinfordquoCorrectionKey=A
8_MCAAESE206984_U1M01RTindd 27 41513 1113 PM
Math TrainerOnline Assessment
and Intervention
Personal
myhrwcom
1
2
3 Response toIntervention
Intervention Enrichment
Access Ready to Go On assessment online and receive instant scoring feedback and customized intervention or enrichment
Online and Print Resources
Differentiated Instruction
bull Reteach worksheets
bull Reading Strategies EL
bull Success for English Learners EL
Differentiated Instruction
bull Challenge worksheets PRE-AP
Extend the Math PRE-AP
Lesson Activities in TE
Additional ResourcesAssessment Resources includes bull Leveled Module Quizzes
Ready to Go OnAssess MasteryUse the assessment on this page to determine if students have mastered the concepts and standards covered in this module
California Common Core Standards
Lesson Exercises Common Core Standards
11 1ndash7 8NS1 8NS2 8EE2
12 8ndash10 8NS1
13 11ndash14 8NS2
27 Unit 1 Module 1
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
Personal Math Trainer
Online Practice and HelpmyhrwcomAssessment Readiness
Module 1 MIXed ReVIeW
1 Look at each number Is the number between 2π and radic___
52
Select Yes or No for expressions AndashC
A 6 2 _ 3 Yes No
B 5π __ 2 Yes No
C 3 radic__
5 Yes No
2 Consider the number - 11 __ 15
Choose True or False for each statement
A The number is rational True False
B The number can be written as True Falsea repeating decimal
C The number is less than ndash08 True False
3 The volume of a cube is given by V = x3 where x is the length of an edge of the cube A cube-shaped end table has a volume of 3 3 _ 8 cubic feet What is the length of an edge of the end table Explain how you solved this problem
4 A student says that radic___
83 is greater than 29 __ 3 Is the student correct Justify your
reasoning
1 1 _ 2 ft Sample answer The equation x3 = 3 3 _ 8 can be used
to find the edge length in feet To solve the equation
write the mixed number as a fraction greater than 1
x3 = 27 __ 8 Then take the cube root of both sides x = 3 _ 2 = 1 1 _ 2
No Sample answer radic___
83 asymp 91 and 29 __ 3 = 9
__ 6
Because 91 lt 9 __
6 radic___
83 lt 29 __ 3
28 Unit 1
copy H
ough
ton
Miff
lin H
arco
urt P
ublis
hing
Com
pany
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=A
8_MCAAESE206984_U1M01RTindd 28 240413 946 AM
Personal Math Trainer
Online Assessment and
Interventionmyhrwcom
Scoring GuideItem 3 Award the student 1 point for finding the edge length of the cube and 1 point for correctly explaining how to use a cube root to solve the problem
Item 4 Award the student 1 point for determining that the student is incorrect and 1 point for correctly justifying the reasoning for this conclusion
Additional ResourcesTo assign this assessment online login to your Assignment Manager at myhrwcom
Assessment Readiness
California Common Core Standards
Items Grade 8 Standards Mathematical Practices
1 8NS2 MP7
2 7NS2b 7NS2d 8NS1 MP7
3 8EE2 MP1 MP4
4 8NS1 8NS2 MP3
Item integrates mixed review concepts from previous modules or a previous course
Item 4 combines concepts from the California Common Core cluster ldquoKnow that there are numbers that are not rational and approximate them by rational numbersrdquo
Real Numbers 28
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
Math TrainerOnline Assessment
and Intervention
Personal
Online homework assignment available
myhrwcom
myhrwcom
Lesson Quiz available online
13 LESSON QUIZ
1 Compare Write lt gt or =
radic_
95 - 5 radic_
62 - 2
2 Order 105 radic_
105 and 3π + 1 from greatest to least
3 A length in centimeters is calculated differently by four different people Order their calculations from least to greatest
KD 11 __ 2 cm Silvio 5 __ 3 π cm
Paula 5 _
4 cm Luis radic_
33 cm
EvaluateGUIDED AND INDEPENDENT PRACTICE
Concepts amp Skills Practice
Example 1Comparing Irrational Numbers
Exercises 1ndash8
Example 2Ordering Real Numbers
Exercises 9 12ndash15 18ndash21
Example 3Ordering Real Numbers in a Real-World Context
Exercises 10 16ndash17
Additional ResourcesDifferentiated Instruction includes bull Leveled Practice worksheets
Focus | Coherence | Rigor
Exercise Depth of Knowledge (DOK) Mathematical Practices
12ndash15 1 Recall of Information MP5 Using Tools
16 2 SkillsConcepts MP2 Reasoning
17 2 SkillsConcepts MP6 Precision
18ndash21 2 SkillsConcepts MP2 Reasoning
22 3 Strategic Thinking MP4 Modeling
23ndash24 3 Strategic Thinking MP3 Logic
8NS2
8NS2
Answers1 radic
_ 95 - 5 lt radic
_ 62 - 2
2 radic_
105 3π + 1 105
3 Silvio 5 __ 3 π cm Paula 5 _
4 cm
KD 11
__ 2 cm Luis radic_
33 cm
25 Lesson 13
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
Work Area
3140 3141 3142 3143
314 π227
20 A teacher asks his students to write the numbers shown in order from least to greatest Paul thinks the numbers are already in order Sandra thinks the order should be reversed Who is right
21 Math History There is a famous irrational number called Eulerrsquos number symbolized with an e Like π its decimal form never ends or repeats The first few digits of e are 27182818284
a Between which two square roots of integers could you find this number
b Between which two square roots of integers can you find π
22 Analyze Relationships There are several approximations used for π including 314 and 22 __ 7 π is approximately 314159265358979
a Label π and the two approximations on the number line
b Which of the two approximations is a better estimate for π Explain
c Find a whole number x so that the ratio x ___ 113 is a better estimate for π
than the two given approximations
23 Communicate Mathematical Ideas If a set of six numbers that include both rational and irrational numbers is graphed on a number line what is the fewest number of distinct points that need to be graphed Explain
24 Critique Reasoning Jill says that 12 _
6 is less than 1263 Explain her error
FOCUS ON HIGHER ORDER THINKING
radic_
115 115 ___ 11 and 105624
between radic_
7 asymp 265 and radic_
8 asymp 283
between radic_
9 = 3 and radic_
10 asymp 316
22 __ 7 it is closer to π on the number line
She did not consider the repeating digit 1266
2 rational numbers can have the same location and
irrational numbers can have the same location but they
cannot share a location
355
Neither student is correct The answer
should be 115 ___ 11 105624 radic_
115
Unit 126
copy H
ough
ton M
ifflin
Har
cour
t Pub
lishin
g Com
pany
Imag
e Cre
dits
copy3D
Stoc
kiSt
ockP
hoto
com
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
8_MCAAESE206984_U1M01L3indd 26 210513 801 AM
Personal Math Trainer
Online Practice and Helpmyhrwcom
Name Class Date
Independent Practice
16 Your sister is considering two different shapes for her garden One is a square with side lengths of 35 meters and the other is a circle with a diameter of 4 meters
a Find the area of the square
b Find the area of the circle
c Compare your answers from parts a and b Which garden would give your sister the most space to plant
17 Winnie measured the length of her fatherrsquos ranch four times and got four different distances Her measurements are shown in the table
a To estimate the actual length Winnie first approximated each distance to the nearest hundredth Then she averaged the four numbers Using a calculator find Winniersquos estimate
b Winniersquos father estimated the distance across his ranch to be radic_
56 km How does this distance compare to Winniersquos estimate
Give an example of each type of number
18 a real number between radic_
13 and radic_
14
19 an irrational number between 5 and 7
Order the numbers from least to greatest
12 radic_
7 2 radic_
8 ___ 2 13 radic_
10 π 35
14 radic_
220 -10 radic_
100 115 15 radic_
8 -375 3 9 _ 4
Distance Across Fatherrsquos Ranch (km)
1 2 3 4
radic_
60 58 __ 8 7 _
3 7 3 _ 5
138NS2
radic_
8 ___ 2 2 radic_
7
-10 radic_
100 115 radic_
220
radic_
60 asymp 775 58 __ 8 = 725 7 _
3 asymp 733 7 3 _ 5 = 760 so the average
π radic_
10 35
-375 9 _ 4 radic_
8 3
is 74825 km
1225 m2
4π m2 or approximately 126 m2
They are nearly identical radic_
56 is approximately 74833hellip
The circle would give her more space to plant because it has a
larger area
Sample answer 37
Sample answer radic_
31
25Lesson 13
copy H
ough
ton
Miff
lin H
arco
urt P
ublis
hing
Com
pany
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
8_MCAAESE206984_U1M01L3indd 25 41613 448 AM
Activity available online myhrwcomEXTEND THE MATH PRE-AP
Activity Have students investigate whether there are infinitely many numbers between two numbers by giving examples for each of the following
bull Between any two rational numbers there is at least one other rational number Sample answer 45 is between 41 and 48
bull Between any two irrational numbers there is at least one rational number Sample answer 45 is between radic
_ 11 and radic
_ 29
bull Between any two rational numbers there is at least one irrational number Sample answer radic
_ 11 is between 31 and 36
bull Between any two irrational numbers there is at least one irrational number Sample answer radic
_ 17 is between radic
_ 11 and radic
_ 29
Ordering Real Numbers 26
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
ReadyMath Trainer
Online Practiceand Help
Personal
myhrwcom
Module Quiz
11ensp RationalenspandenspIrrationalenspNumbersWrite each fraction as a decimal or each decimal as a fraction
1 7__20 2 1___
27 3 17_8
Solve each equation for x
4 x2=81 5 x3=343 6 x2= 1___100
7 Asquarepatiohasanareaof200squarefeetHowlongiseachside
ofthepatiotothenearesttenth
12ensp SetsenspofenspRealenspNumbersWrite all names that apply to each number
8 121____radic
____121
9 π__2
10 TellwhetherthestatementldquoAllintegersarerationalnumbersrdquoistrueorfalseExplainyourchoice
13ensp OrderingenspRealenspNumbersCompare Write lt gt or =
11 radic__
8+3 8+radic__
3 12 radic__
5+11emsp emsp emsp 5+radic___
11
Order the numbers from least to greatest
13 radic___
99π29__
8 14 radic___
1__251_40__
2
15 Howarerealnumbersusedtodescribereal-worldsituations
ESSENTIAL QUESTION
035
9-9
141ft
7 1__10- 1__10
14__11 1875
wholeintegerrationalreal
Trueintegerscanbewrittenasthequotientoftwointegers
SampleanswerRealnumberssuchastherational
π29__
8radic___
99
irrationalreal
lt gt
number1_4candescribeamountsusedincooking
radic___
1__250__
21_4
27Module1
copy H
ough
ton
Miff
lin H
arco
urt P
ublis
hing
Com
pany
DONOTEDIT--ChangesmustbemadethroughldquoFileinfordquoCorrectionKey=A
8_MCAAESE206984_U1M01RTindd 27 41513 1113 PM
Math TrainerOnline Assessment
and Intervention
Personal
myhrwcom
1
2
3 Response toIntervention
Intervention Enrichment
Access Ready to Go On assessment online and receive instant scoring feedback and customized intervention or enrichment
Online and Print Resources
Differentiated Instruction
bull Reteach worksheets
bull Reading Strategies EL
bull Success for English Learners EL
Differentiated Instruction
bull Challenge worksheets PRE-AP
Extend the Math PRE-AP
Lesson Activities in TE
Additional ResourcesAssessment Resources includes bull Leveled Module Quizzes
Ready to Go OnAssess MasteryUse the assessment on this page to determine if students have mastered the concepts and standards covered in this module
California Common Core Standards
Lesson Exercises Common Core Standards
11 1ndash7 8NS1 8NS2 8EE2
12 8ndash10 8NS1
13 11ndash14 8NS2
27 Unit 1 Module 1
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
Personal Math Trainer
Online Practice and HelpmyhrwcomAssessment Readiness
Module 1 MIXed ReVIeW
1 Look at each number Is the number between 2π and radic___
52
Select Yes or No for expressions AndashC
A 6 2 _ 3 Yes No
B 5π __ 2 Yes No
C 3 radic__
5 Yes No
2 Consider the number - 11 __ 15
Choose True or False for each statement
A The number is rational True False
B The number can be written as True Falsea repeating decimal
C The number is less than ndash08 True False
3 The volume of a cube is given by V = x3 where x is the length of an edge of the cube A cube-shaped end table has a volume of 3 3 _ 8 cubic feet What is the length of an edge of the end table Explain how you solved this problem
4 A student says that radic___
83 is greater than 29 __ 3 Is the student correct Justify your
reasoning
1 1 _ 2 ft Sample answer The equation x3 = 3 3 _ 8 can be used
to find the edge length in feet To solve the equation
write the mixed number as a fraction greater than 1
x3 = 27 __ 8 Then take the cube root of both sides x = 3 _ 2 = 1 1 _ 2
No Sample answer radic___
83 asymp 91 and 29 __ 3 = 9
__ 6
Because 91 lt 9 __
6 radic___
83 lt 29 __ 3
28 Unit 1
copy H
ough
ton
Miff
lin H
arco
urt P
ublis
hing
Com
pany
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=A
8_MCAAESE206984_U1M01RTindd 28 240413 946 AM
Personal Math Trainer
Online Assessment and
Interventionmyhrwcom
Scoring GuideItem 3 Award the student 1 point for finding the edge length of the cube and 1 point for correctly explaining how to use a cube root to solve the problem
Item 4 Award the student 1 point for determining that the student is incorrect and 1 point for correctly justifying the reasoning for this conclusion
Additional ResourcesTo assign this assessment online login to your Assignment Manager at myhrwcom
Assessment Readiness
California Common Core Standards
Items Grade 8 Standards Mathematical Practices
1 8NS2 MP7
2 7NS2b 7NS2d 8NS1 MP7
3 8EE2 MP1 MP4
4 8NS1 8NS2 MP3
Item integrates mixed review concepts from previous modules or a previous course
Item 4 combines concepts from the California Common Core cluster ldquoKnow that there are numbers that are not rational and approximate them by rational numbersrdquo
Real Numbers 28
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
Work Area
3140 3141 3142 3143
314 π227
20 A teacher asks his students to write the numbers shown in order from least to greatest Paul thinks the numbers are already in order Sandra thinks the order should be reversed Who is right
21 Math History There is a famous irrational number called Eulerrsquos number symbolized with an e Like π its decimal form never ends or repeats The first few digits of e are 27182818284
a Between which two square roots of integers could you find this number
b Between which two square roots of integers can you find π
22 Analyze Relationships There are several approximations used for π including 314 and 22 __ 7 π is approximately 314159265358979
a Label π and the two approximations on the number line
b Which of the two approximations is a better estimate for π Explain
c Find a whole number x so that the ratio x ___ 113 is a better estimate for π
than the two given approximations
23 Communicate Mathematical Ideas If a set of six numbers that include both rational and irrational numbers is graphed on a number line what is the fewest number of distinct points that need to be graphed Explain
24 Critique Reasoning Jill says that 12 _
6 is less than 1263 Explain her error
FOCUS ON HIGHER ORDER THINKING
radic_
115 115 ___ 11 and 105624
between radic_
7 asymp 265 and radic_
8 asymp 283
between radic_
9 = 3 and radic_
10 asymp 316
22 __ 7 it is closer to π on the number line
She did not consider the repeating digit 1266
2 rational numbers can have the same location and
irrational numbers can have the same location but they
cannot share a location
355
Neither student is correct The answer
should be 115 ___ 11 105624 radic_
115
Unit 126
copy H
ough
ton M
ifflin
Har
cour
t Pub
lishin
g Com
pany
Imag
e Cre
dits
copy3D
Stoc
kiSt
ockP
hoto
com
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
8_MCAAESE206984_U1M01L3indd 26 210513 801 AM
Personal Math Trainer
Online Practice and Helpmyhrwcom
Name Class Date
Independent Practice
16 Your sister is considering two different shapes for her garden One is a square with side lengths of 35 meters and the other is a circle with a diameter of 4 meters
a Find the area of the square
b Find the area of the circle
c Compare your answers from parts a and b Which garden would give your sister the most space to plant
17 Winnie measured the length of her fatherrsquos ranch four times and got four different distances Her measurements are shown in the table
a To estimate the actual length Winnie first approximated each distance to the nearest hundredth Then she averaged the four numbers Using a calculator find Winniersquos estimate
b Winniersquos father estimated the distance across his ranch to be radic_
56 km How does this distance compare to Winniersquos estimate
Give an example of each type of number
18 a real number between radic_
13 and radic_
14
19 an irrational number between 5 and 7
Order the numbers from least to greatest
12 radic_
7 2 radic_
8 ___ 2 13 radic_
10 π 35
14 radic_
220 -10 radic_
100 115 15 radic_
8 -375 3 9 _ 4
Distance Across Fatherrsquos Ranch (km)
1 2 3 4
radic_
60 58 __ 8 7 _
3 7 3 _ 5
138NS2
radic_
8 ___ 2 2 radic_
7
-10 radic_
100 115 radic_
220
radic_
60 asymp 775 58 __ 8 = 725 7 _
3 asymp 733 7 3 _ 5 = 760 so the average
π radic_
10 35
-375 9 _ 4 radic_
8 3
is 74825 km
1225 m2
4π m2 or approximately 126 m2
They are nearly identical radic_
56 is approximately 74833hellip
The circle would give her more space to plant because it has a
larger area
Sample answer 37
Sample answer radic_
31
25Lesson 13
copy H
ough
ton
Miff
lin H
arco
urt P
ublis
hing
Com
pany
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
8_MCAAESE206984_U1M01L3indd 25 41613 448 AM
Activity available online myhrwcomEXTEND THE MATH PRE-AP
Activity Have students investigate whether there are infinitely many numbers between two numbers by giving examples for each of the following
bull Between any two rational numbers there is at least one other rational number Sample answer 45 is between 41 and 48
bull Between any two irrational numbers there is at least one rational number Sample answer 45 is between radic
_ 11 and radic
_ 29
bull Between any two rational numbers there is at least one irrational number Sample answer radic
_ 11 is between 31 and 36
bull Between any two irrational numbers there is at least one irrational number Sample answer radic
_ 17 is between radic
_ 11 and radic
_ 29
Ordering Real Numbers 26
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
ReadyMath Trainer
Online Practiceand Help
Personal
myhrwcom
Module Quiz
11ensp RationalenspandenspIrrationalenspNumbersWrite each fraction as a decimal or each decimal as a fraction
1 7__20 2 1___
27 3 17_8
Solve each equation for x
4 x2=81 5 x3=343 6 x2= 1___100
7 Asquarepatiohasanareaof200squarefeetHowlongiseachside
ofthepatiotothenearesttenth
12ensp SetsenspofenspRealenspNumbersWrite all names that apply to each number
8 121____radic
____121
9 π__2
10 TellwhetherthestatementldquoAllintegersarerationalnumbersrdquoistrueorfalseExplainyourchoice
13ensp OrderingenspRealenspNumbersCompare Write lt gt or =
11 radic__
8+3 8+radic__
3 12 radic__
5+11emsp emsp emsp 5+radic___
11
Order the numbers from least to greatest
13 radic___
99π29__
8 14 radic___
1__251_40__
2
15 Howarerealnumbersusedtodescribereal-worldsituations
ESSENTIAL QUESTION
035
9-9
141ft
7 1__10- 1__10
14__11 1875
wholeintegerrationalreal
Trueintegerscanbewrittenasthequotientoftwointegers
SampleanswerRealnumberssuchastherational
π29__
8radic___
99
irrationalreal
lt gt
number1_4candescribeamountsusedincooking
radic___
1__250__
21_4
27Module1
copy H
ough
ton
Miff
lin H
arco
urt P
ublis
hing
Com
pany
DONOTEDIT--ChangesmustbemadethroughldquoFileinfordquoCorrectionKey=A
8_MCAAESE206984_U1M01RTindd 27 41513 1113 PM
Math TrainerOnline Assessment
and Intervention
Personal
myhrwcom
1
2
3 Response toIntervention
Intervention Enrichment
Access Ready to Go On assessment online and receive instant scoring feedback and customized intervention or enrichment
Online and Print Resources
Differentiated Instruction
bull Reteach worksheets
bull Reading Strategies EL
bull Success for English Learners EL
Differentiated Instruction
bull Challenge worksheets PRE-AP
Extend the Math PRE-AP
Lesson Activities in TE
Additional ResourcesAssessment Resources includes bull Leveled Module Quizzes
Ready to Go OnAssess MasteryUse the assessment on this page to determine if students have mastered the concepts and standards covered in this module
California Common Core Standards
Lesson Exercises Common Core Standards
11 1ndash7 8NS1 8NS2 8EE2
12 8ndash10 8NS1
13 11ndash14 8NS2
27 Unit 1 Module 1
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
Personal Math Trainer
Online Practice and HelpmyhrwcomAssessment Readiness
Module 1 MIXed ReVIeW
1 Look at each number Is the number between 2π and radic___
52
Select Yes or No for expressions AndashC
A 6 2 _ 3 Yes No
B 5π __ 2 Yes No
C 3 radic__
5 Yes No
2 Consider the number - 11 __ 15
Choose True or False for each statement
A The number is rational True False
B The number can be written as True Falsea repeating decimal
C The number is less than ndash08 True False
3 The volume of a cube is given by V = x3 where x is the length of an edge of the cube A cube-shaped end table has a volume of 3 3 _ 8 cubic feet What is the length of an edge of the end table Explain how you solved this problem
4 A student says that radic___
83 is greater than 29 __ 3 Is the student correct Justify your
reasoning
1 1 _ 2 ft Sample answer The equation x3 = 3 3 _ 8 can be used
to find the edge length in feet To solve the equation
write the mixed number as a fraction greater than 1
x3 = 27 __ 8 Then take the cube root of both sides x = 3 _ 2 = 1 1 _ 2
No Sample answer radic___
83 asymp 91 and 29 __ 3 = 9
__ 6
Because 91 lt 9 __
6 radic___
83 lt 29 __ 3
28 Unit 1
copy H
ough
ton
Miff
lin H
arco
urt P
ublis
hing
Com
pany
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=A
8_MCAAESE206984_U1M01RTindd 28 240413 946 AM
Personal Math Trainer
Online Assessment and
Interventionmyhrwcom
Scoring GuideItem 3 Award the student 1 point for finding the edge length of the cube and 1 point for correctly explaining how to use a cube root to solve the problem
Item 4 Award the student 1 point for determining that the student is incorrect and 1 point for correctly justifying the reasoning for this conclusion
Additional ResourcesTo assign this assessment online login to your Assignment Manager at myhrwcom
Assessment Readiness
California Common Core Standards
Items Grade 8 Standards Mathematical Practices
1 8NS2 MP7
2 7NS2b 7NS2d 8NS1 MP7
3 8EE2 MP1 MP4
4 8NS1 8NS2 MP3
Item integrates mixed review concepts from previous modules or a previous course
Item 4 combines concepts from the California Common Core cluster ldquoKnow that there are numbers that are not rational and approximate them by rational numbersrdquo
Real Numbers 28
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
ReadyMath Trainer
Online Practiceand Help
Personal
myhrwcom
Module Quiz
11ensp RationalenspandenspIrrationalenspNumbersWrite each fraction as a decimal or each decimal as a fraction
1 7__20 2 1___
27 3 17_8
Solve each equation for x
4 x2=81 5 x3=343 6 x2= 1___100
7 Asquarepatiohasanareaof200squarefeetHowlongiseachside
ofthepatiotothenearesttenth
12ensp SetsenspofenspRealenspNumbersWrite all names that apply to each number
8 121____radic
____121
9 π__2
10 TellwhetherthestatementldquoAllintegersarerationalnumbersrdquoistrueorfalseExplainyourchoice
13ensp OrderingenspRealenspNumbersCompare Write lt gt or =
11 radic__
8+3 8+radic__
3 12 radic__
5+11emsp emsp emsp 5+radic___
11
Order the numbers from least to greatest
13 radic___
99π29__
8 14 radic___
1__251_40__
2
15 Howarerealnumbersusedtodescribereal-worldsituations
ESSENTIAL QUESTION
035
9-9
141ft
7 1__10- 1__10
14__11 1875
wholeintegerrationalreal
Trueintegerscanbewrittenasthequotientoftwointegers
SampleanswerRealnumberssuchastherational
π29__
8radic___
99
irrationalreal
lt gt
number1_4candescribeamountsusedincooking
radic___
1__250__
21_4
27Module1
copy H
ough
ton
Miff
lin H
arco
urt P
ublis
hing
Com
pany
DONOTEDIT--ChangesmustbemadethroughldquoFileinfordquoCorrectionKey=A
8_MCAAESE206984_U1M01RTindd 27 41513 1113 PM
Math TrainerOnline Assessment
and Intervention
Personal
myhrwcom
1
2
3 Response toIntervention
Intervention Enrichment
Access Ready to Go On assessment online and receive instant scoring feedback and customized intervention or enrichment
Online and Print Resources
Differentiated Instruction
bull Reteach worksheets
bull Reading Strategies EL
bull Success for English Learners EL
Differentiated Instruction
bull Challenge worksheets PRE-AP
Extend the Math PRE-AP
Lesson Activities in TE
Additional ResourcesAssessment Resources includes bull Leveled Module Quizzes
Ready to Go OnAssess MasteryUse the assessment on this page to determine if students have mastered the concepts and standards covered in this module
California Common Core Standards
Lesson Exercises Common Core Standards
11 1ndash7 8NS1 8NS2 8EE2
12 8ndash10 8NS1
13 11ndash14 8NS2
27 Unit 1 Module 1
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
Personal Math Trainer
Online Practice and HelpmyhrwcomAssessment Readiness
Module 1 MIXed ReVIeW
1 Look at each number Is the number between 2π and radic___
52
Select Yes or No for expressions AndashC
A 6 2 _ 3 Yes No
B 5π __ 2 Yes No
C 3 radic__
5 Yes No
2 Consider the number - 11 __ 15
Choose True or False for each statement
A The number is rational True False
B The number can be written as True Falsea repeating decimal
C The number is less than ndash08 True False
3 The volume of a cube is given by V = x3 where x is the length of an edge of the cube A cube-shaped end table has a volume of 3 3 _ 8 cubic feet What is the length of an edge of the end table Explain how you solved this problem
4 A student says that radic___
83 is greater than 29 __ 3 Is the student correct Justify your
reasoning
1 1 _ 2 ft Sample answer The equation x3 = 3 3 _ 8 can be used
to find the edge length in feet To solve the equation
write the mixed number as a fraction greater than 1
x3 = 27 __ 8 Then take the cube root of both sides x = 3 _ 2 = 1 1 _ 2
No Sample answer radic___
83 asymp 91 and 29 __ 3 = 9
__ 6
Because 91 lt 9 __
6 radic___
83 lt 29 __ 3
28 Unit 1
copy H
ough
ton
Miff
lin H
arco
urt P
ublis
hing
Com
pany
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=A
8_MCAAESE206984_U1M01RTindd 28 240413 946 AM
Personal Math Trainer
Online Assessment and
Interventionmyhrwcom
Scoring GuideItem 3 Award the student 1 point for finding the edge length of the cube and 1 point for correctly explaining how to use a cube root to solve the problem
Item 4 Award the student 1 point for determining that the student is incorrect and 1 point for correctly justifying the reasoning for this conclusion
Additional ResourcesTo assign this assessment online login to your Assignment Manager at myhrwcom
Assessment Readiness
California Common Core Standards
Items Grade 8 Standards Mathematical Practices
1 8NS2 MP7
2 7NS2b 7NS2d 8NS1 MP7
3 8EE2 MP1 MP4
4 8NS1 8NS2 MP3
Item integrates mixed review concepts from previous modules or a previous course
Item 4 combines concepts from the California Common Core cluster ldquoKnow that there are numbers that are not rational and approximate them by rational numbersrdquo
Real Numbers 28
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A
Personal Math Trainer
Online Practice and HelpmyhrwcomAssessment Readiness
Module 1 MIXed ReVIeW
1 Look at each number Is the number between 2π and radic___
52
Select Yes or No for expressions AndashC
A 6 2 _ 3 Yes No
B 5π __ 2 Yes No
C 3 radic__
5 Yes No
2 Consider the number - 11 __ 15
Choose True or False for each statement
A The number is rational True False
B The number can be written as True Falsea repeating decimal
C The number is less than ndash08 True False
3 The volume of a cube is given by V = x3 where x is the length of an edge of the cube A cube-shaped end table has a volume of 3 3 _ 8 cubic feet What is the length of an edge of the end table Explain how you solved this problem
4 A student says that radic___
83 is greater than 29 __ 3 Is the student correct Justify your
reasoning
1 1 _ 2 ft Sample answer The equation x3 = 3 3 _ 8 can be used
to find the edge length in feet To solve the equation
write the mixed number as a fraction greater than 1
x3 = 27 __ 8 Then take the cube root of both sides x = 3 _ 2 = 1 1 _ 2
No Sample answer radic___
83 asymp 91 and 29 __ 3 = 9
__ 6
Because 91 lt 9 __
6 radic___
83 lt 29 __ 3
28 Unit 1
copy H
ough
ton
Miff
lin H
arco
urt P
ublis
hing
Com
pany
DO NOT EDIT--Changes must be made through ldquoFile infordquo CorrectionKey=A
8_MCAAESE206984_U1M01RTindd 28 240413 946 AM
Personal Math Trainer
Online Assessment and
Interventionmyhrwcom
Scoring GuideItem 3 Award the student 1 point for finding the edge length of the cube and 1 point for correctly explaining how to use a cube root to solve the problem
Item 4 Award the student 1 point for determining that the student is incorrect and 1 point for correctly justifying the reasoning for this conclusion
Additional ResourcesTo assign this assessment online login to your Assignment Manager at myhrwcom
Assessment Readiness
California Common Core Standards
Items Grade 8 Standards Mathematical Practices
1 8NS2 MP7
2 7NS2b 7NS2d 8NS1 MP7
3 8EE2 MP1 MP4
4 8NS1 8NS2 MP3
Item integrates mixed review concepts from previous modules or a previous course
Item 4 combines concepts from the California Common Core cluster ldquoKnow that there are numbers that are not rational and approximate them by rational numbersrdquo
Real Numbers 28
DO NOT EDIT--Changes must be made through ldquoFile infordquoCorrectionKey=A