Sorting It All OutSorting It All Out Mathematical TopicsMathematical Topics Subsets of real numbers and the relationships Subsets of real numbers and the relationships

between these subsets between these subsets Similarities and differences between rational and Similarities and differences between rational and

irrational numbers irrational numbers Convert equivalent forms of rational numbers Convert equivalent forms of rational numbers Venn diagrams Venn diagrams Terminating decimals and their fractional Terminating decimals and their fractional

equivalents equivalents Repeating decimals and their fractional equivalents Repeating decimals and their fractional equivalents Ordering real numbers Ordering real numbers

Required MaterialsRequired Materials Calculator Calculator Overhead of the given Venn diagram Overhead of the given Venn diagram

STANDARD/ELEMENTSSTANDARD/ELEMENTSM8N1 Students will understand different representations of numbers including M8N1 Students will understand different representations of numbers including

square roots, exponents, and scientific notation. square roots, exponents, and scientific notation. Elements: Elements:

a. Find square roots of perfect squares.a. Find square roots of perfect squares.b. Recognize the (positive) square root of a number as a b. Recognize the (positive) square root of a number as a length of a side of a square with a given area.length of a side of a square with a given area.c. Recognize square roots as points and as lengths on a c. Recognize square roots as points and as lengths on a number line.number line.d. Understand that the square root of 0 is 0 and that every d. Understand that the square root of 0 is 0 and that every positive number has two square roots that are opposite in positive number has two square roots that are opposite in sign.sign.e. Recognize and use the radical symbol to denote the e. Recognize and use the radical symbol to denote the positive square root of a positive number.positive square root of a positive number.f. Estimate square roots of positive numbers.f. Estimate square roots of positive numbers.g. Simplify, add, subtract, multiply, and divide expressions g. Simplify, add, subtract, multiply, and divide expressions containing square roots.containing square roots.h. Distinguish between rational and irrational numbers.h. Distinguish between rational and irrational numbers.i. Simplify expressions containing integer exponents.i. Simplify expressions containing integer exponents.j. Express and use numbers in scientific notation.j. Express and use numbers in scientific notation.

Work PeriodWork Period

Complete odd Complete odd numbers on 2-1 numbers on 2-1 changing fractions to changing fractions to a decimal and a decimal and decimal to a fractiondecimal to a fraction

SpongeSponge

Rational or Rational or Irrational numbers Irrational numbers in fractional formin fractional form

   

Rational or Rational or Irrational numbers Irrational numbers in decimal formin decimal form

   

1/31/3

3/83/8

5/115/11

What pattern do you What pattern do you see?see?

Home WorkHome Work1Rational Numbers have a decimal 1Rational Numbers have a decimal

expansion that a.) terminates or b.) expansion that a.) terminates or b.) doesn’t terminatedoesn’t terminate

a.) a.) 3/43/4 b.) 2/3b.) 2/3 True or FalseTrue or False Every integer is a rational numberEvery integer is a rational number Every rational number is a whole numberEvery rational number is a whole number Every natural number is a whole numberEvery natural number is a whole number d.) 3 is an element of the rational numbersd.) 3 is an element of the rational numbersExpress the following rational numbers as Express the following rational numbers as

decimals:decimals:a.) a.) 5/115/11 b.) b.) 10/11 10/11 c).c). --19/1000019/10000

ClosingClosing

Determine which numbers in the set Determine which numbers in the set are natural numbers, whole numbers, are natural numbers, whole numbers, integers, rational numbers, irrational integers, rational numbers, irrational numbers, and real numbers. numbers, and real numbers.

4,89.3,0,16,5,7.0,

2,3

2

WORK PERIODWORK PERIOD8.8. Use long division, without a calculator, to write each of the Use long division, without a calculator, to write each of the

following rational numbers as a decimal.following rational numbers as a decimal.

a. a.

b. b.

c.c.

9.9. Describe any patterns you observe in Question 8.Describe any patterns you observe in Question 8.

Sorting it All OutSorting it All Out Prerequisite KnowledgePrerequisite Knowledge Equivalent representations of simple fractions Equivalent representations of simple fractions

and decimals and decimals Knowledge of Knowledge of Meaning of the square root symbol Meaning of the square root symbol Basic idea of Venn diagrams Basic idea of Venn diagrams Basic equation-solving with one variable Basic equation-solving with one variable Concept of sets and subsets Concept of sets and subsets Definition of an even number Definition of an even number Number lines Number lines

VENN DIAGRAMVENN DIAGRAM

IntroductionIntroductionYour teacher is going to use a Venn diagram like the Your teacher is going to use a Venn diagram like the

one shown below to help you understand the one shown below to help you understand the relationships between different types of numbers. relationships between different types of numbers. Your goal is to figure out the description for each of Your goal is to figure out the description for each of the circles and regions of the diagram.the circles and regions of the diagram. You will gain You will gain information by suggesting numbers and seeing information by suggesting numbers and seeing their correct placement in the Venn diagram. their correct placement in the Venn diagram.

Begin by calling out a number for your teacher to place into Begin by calling out a number for your teacher to place into the Venn diagram. Listen carefully to the number suggestions the Venn diagram. Listen carefully to the number suggestions given by your classmates and try to determine how your given by your classmates and try to determine how your teacher decides where to place each number in the diagram. teacher decides where to place each number in the diagram.

As this activity progresses, see if you can guess where a As this activity progresses, see if you can guess where a number will be placed in the Venn diagram before your number will be placed in the Venn diagram before your teacher shows the class. teacher shows the class.

As an extra challenge, try to suggest a number that will be As an extra challenge, try to suggest a number that will be placed in a region of the Venn diagram that does not yet have placed in a region of the Venn diagram that does not yet have many (or any) numbers. many (or any) numbers.

Think about what each region of the Venn diagram might Think about what each region of the Venn diagram might represent. Be ready to give a description for each region in represent. Be ready to give a description for each region in the diagram. the diagram.

Throughout this Unit, you will discover which of your predictions and descriptions are correct, and you will learn to identify and describe different sets of numbers.

HomeworkHomework

Rational Numbers have a decimal Rational Numbers have a decimal expansion that a.) terminates or b.) expansion that a.) terminates or b.) doesn’t terminatedoesn’t terminate

3/43/4 2/32/3 EXAMPLE: True or FalseEXAMPLE: True or False Every integer is a rational numberEvery integer is a rational number Every rational number is a whole numberEvery rational number is a whole number Every natural number is a whole numberEvery natural number is a whole number d.) 3 is an element of the rational d.) 3 is an element of the rational

numbersnumbers

WORKPERIODWORKPERIOD

ClosingClosing

Which number is rationalWhich number is rational

A. A.

Records remain that show that many ancient Records remain that show that many ancient cultures used different systems of writing cultures used different systems of writing numbers. Most of these cultures used numbers. Most of these cultures used representations for the numbers representations for the numbers

1, 2, 3, 4, 5, . . . 1, 2, 3, 4, 5, . . . This set of numbers is named the set of This set of numbers is named the set of

natural numbers or set of counting numbers natural numbers or set of counting numbers and is often represented by the symboland is often represented by the symbol NN. .

1.1. Why do you think the set of numbers Why do you think the set of numbers {1, {1, 2, 3, 4, 5, …} is named the 2, 3, 4, 5, …} is named the set of set of naturalnatural or or counting numberscounting numbers??

Ancient cultures did not have a concept Ancient cultures did not have a concept of the number 0. One of the first of the number 0. One of the first cultures to have the full use of the cultures to have the full use of the concept of 0 was the Mayan culture. If concept of 0 was the Mayan culture. If you add the number 0 to the set of you add the number 0 to the set of natural numbers, you form the natural numbers, you form the set of set of whole numberswhole numbers. The set of whole . The set of whole numbers is often represented by the numbers is often represented by the symbol symbol WW..

2.2. Think about the set of whole Think about the set of whole numbers numbers and and the set of natural the set of natural numbers.numbers.

a.a. What are some similarities between What are some similarities between the the set of whole numbers and the set of set of whole numbers and the set of

natural numbers?natural numbers?

b.b. What are some differences between What are some differences between the the set of whole numbers and the set of set of whole numbers and the set of

natural numbers?natural numbers?

The numbers –3 and 3 are examples of The numbers –3 and 3 are examples of opposites. On the number line, these opposites. On the number line, these two numbers are the same distance two numbers are the same distance from 0, in opposite directions.from 0, in opposite directions.

One of the greatest accomplishments of ancient One of the greatest accomplishments of ancient Chinese mathematicians was recorded about Chinese mathematicians was recorded about 2000 years ago. The ancient Chinese 2000 years ago. The ancient Chinese mathematicians are noted for their use of mathematicians are noted for their use of negative numbers. The Chinese performed negative numbers. The Chinese performed computations by manipulating counting rods – computations by manipulating counting rods – short rods approximately 10 centimeters long short rods approximately 10 centimeters long – on a table or counting board. Red rods – on a table or counting board. Red rods represented positive numbers and black rods represented positive numbers and black rods represented opposite or negative numbers. represented opposite or negative numbers.

3.3. When the set of whole numbers is When the set of whole numbers is combined combined with the numbers’ opposites, the with the numbers’ opposites, the new set of new set of numbers formed is made up numbers formed is made up

of integers. The of integers. The symbol symbol ZZ is often used is often used to represent the set of to represent the set of integers.integers.

a.a. What is the opposite of 52?What is the opposite of 52?

b.b. What is the opposite of −312?What is the opposite of −312?

c.c. What is the opposite of 0?What is the opposite of 0?

In the Venn diagram in the Introduction, In the Venn diagram in the Introduction, the rectangle shown by region D the rectangle shown by region D contains what are referred to as the contains what are referred to as the set set of rational numbersof rational numbers. These numbers . These numbers can all be written as a fraction or ratio can all be written as a fraction or ratio of two integers. The set of rational of two integers. The set of rational numbers is often represented by the numbers is often represented by the symbol symbol QQ..

5.5. Explain why natural numbers, whole Explain why natural numbers, whole numbers, and integers are all subsets numbers, and integers are all subsets of the set of rational numbers.of the set of rational numbers.

6.6. Look back at the numbers your Look back at the numbers your teacher placed in the original Venn teacher placed in the original Venn diagram. diagram.

a.a. List the numbers that were List the numbers that were placed in placed in region D, but not in region D, but not in regions A, B, or C.regions A, B, or C.

b.b. Describe those numbers you Describe those numbers you listed in listed in Part (a).Part (a).

Some of the numbers written in region Some of the numbers written in region D may have been expressed as the D may have been expressed as the ratio of two integersratio of two integers and are and are rational numbers. Other numbers in rational numbers. Other numbers in region D may have been written as region D may have been written as decimals.decimals.

7.7.Some of the decimals written in region D Some of the decimals written in region D may have been terminating (or ending) may have been terminating (or ending) decimals. In order for these decimals to be decimals. In order for these decimals to be rational numbers, it must be possible to rational numbers, it must be possible to express these numbers as the ratio of two express these numbers as the ratio of two integers. integers.

a.a. Write 0.35 as a ratio of two integers. Write 0.35 as a ratio of two integers. Express Express your answer in lowest terms.your answer in lowest terms.

b.b. Write 2.004 as a ratio of two integers. Write 2.004 as a ratio of two integers. Express Express your answer in lowest terms.your answer in lowest terms.

c.c. Can all terminating (ending) decimals Can all terminating (ending) decimals be be written as fractions? Explain and give written as fractions? Explain and give

examples.examples.

REVIEWREVIEW

WORK PERIODWORK PERIOD

10. Based on the patterns you listed in 10. Based on the patterns you listed in Question 9, complete the following Question 9, complete the following table and then use a calculator to table and then use a calculator to verify your results. verify your results.

Because the repeating decimals shown in the Because the repeating decimals shown in the table above can be expressed as fractions or table above can be expressed as fractions or the ratio of two integers, these repeating the ratio of two integers, these repeating decimals are rational numbers. The following decimals are rational numbers. The following questions will demonstrate how a repeating questions will demonstrate how a repeating decimal can be changed to a fraction.decimal can be changed to a fraction.

11. Let 11. Let

a.a. Complete the equations below.Complete the equations below.

  10  10xx = =

    xx = =

b.b. Subtract Subtract xx from 10 from 10xx and show the results and show the results from both sides of the equations from from both sides of the equations from Part Part (a).(a).

c.c. Solve the resulting equation in Part (b) for Solve the resulting equation in Part (b) for xx..

12.12. Let Let

a.a. Complete the equations below.Complete the equations below.

 100 100xx = = xx = =

b.b. Subtract Subtract xx from 100 from 100xx and show the and show the results results from both sides of the equations from Part (a).from both sides of the equations from Part (a).

c.c. Solve the resulting equation in Part (b) for Solve the resulting equation in Part (b) for xx..

13.13. Do you think that all repeating decimals can Do you think that all repeating decimals can be written as fractions? Explain your be written as fractions? Explain your

reasoning.reasoning.

Part IIIPart III

We know that all natural numbers, whole We know that all natural numbers, whole numbers, integers, fractions, terminating numbers, integers, fractions, terminating decimals, and repeating decimals can be decimals, and repeating decimals can be expressed as the ratio of two integers and expressed as the ratio of two integers and thus are part of the set of rational thus are part of the set of rational numbers. The ancient Greeks once numbers. The ancient Greeks once believed that all numbers were rational believed that all numbers were rational numbers. They also believed that rational numbers. They also believed that rational numbers would fill in all of the points on numbers would fill in all of the points on the number line.the number line.

14.14. Use the number line to plot Use the number line to plot each each rational number given rational number given below.below.

aa. 1.5. 1.5

b. -6/4b. -6/4

c. 0.2c. 0.2

d. .5d. .5

        

.         

.

SPONGE

15.15. Find the coordinate of point Find the coordinate of point AA by by finding the length of the finding the length of the

hypotenuse of the right hypotenuse of the right triangle.triangle.

In Question 15, you found that the In Question 15, you found that the coordinate of point A is . The coordinate of point A is . The number cannot be written as the number cannot be written as the ratio of two integers. This means that ratio of two integers. This means that there are more numbers than the there are more numbers than the rational numbers represented by points rational numbers represented by points on the number line. The Pythagoreans, on the number line. The Pythagoreans, the students and followers of the students and followers of Pythagoras, were among the first to Pythagoras, were among the first to prove that is not rational. prove that is not rational.

The proof caused quite a crisis in the The proof caused quite a crisis in the world of mathematicians in ancient world of mathematicians in ancient Greece, as it had been thought that Greece, as it had been thought that all numbers were rational. The all numbers were rational. The Pythagoreans used a method called Pythagoreans used a method called reductio ad absurdumreductio ad absurdum, proof by , proof by contradiction, to argue that could contradiction, to argue that could not be written as the ratio of two not be written as the ratio of two integers.integers.

16.16. First assume that is a rational First assume that is a rational number. Thus can be written as the number. Thus can be written as the ratio of two integers, , where ratio of two integers, , where aa and and bb are integers that share no common are integers that share no common factors. In other words, is a reduced factors. In other words, is a reduced fraction.fraction.

a.a. Square both sides of the equation Square both sides of the equation and write the result below.and write the result below.

b.b. Multiply both sides of the Multiply both sides of the resulting resulting equation from Part (a) by equation from Part (a) by the the denominator of the fraction. What is denominator of the fraction. What is

the result?the result?

c.c. Because , Because , aa22 must be must be an an even number. Explain why even number. Explain why this is this is true.true.

d. Because d. Because aa is an even number, is an even number,

for some integer for some integer pp. Explain . Explain why why this is true.this is true.

e.e. In Part (a), you determined that In Part (a), you determined that . Substitute 2 . Substitute 2pp for a in this for a in this equation.equation.

f. Use the resulting equation from Part f. Use the resulting equation from Part (e) to show that (e) to show that bb must be an even must be an even number. Show your work.number. Show your work.

g. Why does the fact that both g. Why does the fact that both aa and and bb are even numbers contradict the are even numbers contradict the original statement in Question 16? original statement in Question 16? Explain your reasoning.Explain your reasoning.

Because a contradiction to the fact that Because a contradiction to the fact that aa and and bb share no common factors was reached, the share no common factors was reached, the original assumption that is a rational original assumption that is a rational number must be false. The number is not number must be false. The number is not a rational number. It belongs to a set of a rational number. It belongs to a set of numbers known as the numbers known as the set of irrational set of irrational numbersnumbers. The set of irrational numbers is . The set of irrational numbers is often represented by the symbol .often represented by the symbol .

17. Can there be a terminating or 17. Can there be a terminating or repeating repeating decimal representation of decimal representation of ? Explain why ? Explain why or why not.or why not.

18.18. Give some additional examples of Give some additional examples of irrational numbers.irrational numbers.

The set of rational numbers, along The set of rational numbers, along with the set of irrational with the set of irrational numbers, complete the number numbers, complete the number line. The set containing all of the line. The set containing all of the rational and irrational numbers is rational and irrational numbers is called the set of real numbers. called the set of real numbers. The set of real numbers is often The set of real numbers is often represented by the symbol R.represented by the symbol R.

19. The Venn diagram below represents 19. The Venn diagram below represents all real numbers. Correctly label each all real numbers. Correctly label each region of the diagram with the symbol region of the diagram with the symbol or name of a subset of the set of real or name of a subset of the set of real numbers.numbers.

20. Use the Venn diagram above to help 20. Use the Venn diagram above to help complete the table below by placing a complete the table below by placing a check-mark under the name of each set of check-mark under the name of each set of numbers to which the given number numbers to which the given number belongs.belongs.

  

Sponge Aug 20Sponge Aug 20

21. Name one number for each of the following 21. Name one number for each of the following criteria.criteria.

a. A whole number but not a natural numbera. A whole number but not a natural number

b. An integer but not a whole numberb. An integer but not a whole number

c. An integer and a natural numberc. An integer and a natural number

d. An irrational numberd. An irrational number

e. A rational number but not an integere. A rational number but not an integer

Scoring CriteriaScoring CriteriaScoringScoring

CriteriaCriteriaExceedExceed

ExpectationExpectationMeetMeet

ExpectationExpectationDid not Did not meetmeet

ExpectationExpectation

ProbleProblemm

solvingsolving

Gives Gives multiple multiple correct correct examples examples for each set for each set of criteria.of criteria.

Gives correct Gives correct examples for examples for at least four at least four of the five of the five sets of sets of criteria.criteria.

Gives Gives correct correct examples examples for fewer for fewer than four of than four of the five sets the five sets of criteria.of criteria.