LESSON 1 Number Line: Comparing and Ordering Integersblog.wsd.net/jeelzey/files/2014/08/Math-7-Lessons-1-10.pdf · Number Line: Comparing and Ordering Integers (page 6) ... Which

Saxon Math Course 3 L1-1 Adaptations Lesson 1

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A number line shows numbers in order from least to greatest.

• The number line has zero at the center.Numbers to the right of zero are positive.

Numbers to the left of zero are negative.

• A set is a collection of numbers.Sets are shown inside braces { }. Ellipsis points(…) show that a set is infinite.

• Counting numbers are the numbers we use to count: {1, 2, 3, 4, 5,…}

• Whole numbers are the counting numbers and zero: {0, 1, 2, 3, 4, 5,…}

• Integers are the whole numbers and their opposites: {…, –2, –1, 0, 1, 2,…}

Opposites are two numbers that are the same distance from zero but in opposite directions. 3 and –3 are opposites.

The absolute value of a number is the distance of that number from zero.

| 5 | = 5 | –5 | = 5 The absolute value The absolute value of 5 is 5. of –5 is 5.

The absolute value of a number is always positive.

• We use symbols to compare the values of numbers.

–5 < 4 3 + 2 = 5 0 > –2 –5 is less than 4. 3 plus 2 equals 5. Zero is greater than –2.

• To graph a number, draw a point to correspond to that number on the number line.

Teacher Notes:• Students who have difficulty with

subtraction, multiplication, or division will benefit from working Targeted Practices 1A, 1B, and 1C before Lesson 1.

• Introduce Hint #8, “Positive and Negative Numbers,” and Hint #9, “Comparing Numbers.”

• Refer students to “Number Line” on page 9 and “Number Families” and “Definitions” on page 10 in the Student Reference Guide.

• Post reference chart, “Number Families.”

• A number-line manipulative is available in the Adaptations Manipulative Kit.

Number Line:Comparing andOrdering Integers (page 6)

Practice Set (page 9)

a. Arrange these integers in order from least to greatest: –4, 3, 2, –1, 0

, –1, , ,

b. Which number –4, –1, 0, 2, 3 is an even number but not a whole number? Cross out the odd numbers.

c. Compare: –2 –4 The larger the negative digit, the smaller the number is.

3�3 �1 10 2�2�5 �4 4 5


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d. Graph the numbers in this sequence on a number line: … –4, –2, 0, 2, 4, …

Simplify.

e. | –3 | = f. | 3 | =

g. Write two numbers that are ten units from zero. ,

h. Write an example of a whole number that is not a counting number.

8. Name the type of numbers in problem 7. 9. See page 10 in the Student Reference Guide.

. z is a whole number but

not a c number.

3. whole numbers 4. even numbers 5. Which number is an even number?�5, 3, �2, 1

6. Which whole number is not a counting number?See page 10 in the Student Reference Guide.

7. Write the graphed numbers.

Written Practice (page 9)

{. . . , , , , , . . . }

e integers

{. . . , �2, 0, , 4, , . . .}{ , 1, 2, , 4, . . .}

Practice Set (continued) (page 9)

1. 2. least to greatest

Use a number line.

�5, 3, �2, 1

Use work area. , , ,

Use work area.

5500�5�5

3�3 �1 10 2�2�5 �4 4 5

3�3 �1 10 2�2�5 �4 4 50�6 6


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10. See page 10 in the Student Reference Guide.

. All c numbers are

w numbers.

11. . Integers include c

numbers, their o , and zero.

12. What is the absolute value of 21?

| 21| �

13. | �13 | � 14. | 0 | �

15. 5 �7 16. �3 �2

17. | �3 | | �2 |

____ ____

18.

19. If | n | � 5, then n can be which two numbers? 20. Write two numbers that are five units from zero.

n � , ,

Written Practice (continued) (page 10)

Use work area. Use work area.

Use work area.

3�3 �1 10 2�2�5 �4 4 5

3�3 �1 10 2�2�5 �4 4 5


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25. 10 � � 26. What number is the opposite of �2?

21. 22. Write 2 numbers that are 3 units from 0.

27. �2 � � 2 8. Choose all correct answers.�7

A counting numbers B whole numbers

C integers D none of these

29. Choose all correct answers30



30. Choose all correct answers

1

__ 3



23. 24. What number is the opposite of 10?

Use work area.

Use work area.

,

, ,

Written Practice (continued) (page 10 )

0 5-5 10-10 15-15

3�3 �1 10 2�2�5 �4 4 5 3�3 �1 10 2�2�5 �4 4 5

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The fundamental operations of arithmetic are addition, subtraction, multiplication, and division.

• More than one symbol can show multiplication and division:

Symbols for Multiplication and Division

“three times five” 3 × 5, 3 ∙ 5, 3(5), (3)(5)

“six divided by two” 6 ÷ 2, 2� 6 , 62

• A property of addition or multiplication is something that is always true no matter what numbers are used.

• Properties are written with letters instead of numbers. These letters are called variables.

The letters could represent any number.

Some Properties of Addition and Multiplication

Name of Property Representation Example

Commutative Property a + b = b + a

3 + 4 = 4 + 3

of Addition

Commutative Property a ∙ b = b ∙ a 3 ∙ 4 = 4 ∙ 3

of Multiplication

Associative Property (a + b) + c = a + (b + c) (3 + 4) + 5 = 3 + (4 + 5)

of Addition

Associative Property (a ∙ b) ∙ c = a · (b · c) (3 ∙ 4) ∙ 5 = 3 ∙ (4 ∙ 5)

of Multiplication

Identity Property a + 0 = a 3 + 0 = 3

of Addition

Identity Property a ∙ 1 = a 3 ∙ 1 = 3

of Multiplication

Zero Property a ∙ 0 = 0 3 ∙ 0 = 0

of Multiplication

• The Commutative and Associative Properties do not work with subtraction or division.

Teacher Notes:• Introduce Hint #10,

“Fact Families,” and Hint #11, “Properties of Operations.”

• Refer students to “Properties of Operations” on page 20 in the Student Reference Guide.

• Review “Division” on page 5 in the Student Reference Guide.

• Triangle fact cards are available in the Adaptations Manipulative Kit.

Operations of Arithmetic (page 12)


Name each property illustrated in a–d.

a. 4 · 1 = 4 I Property of

b. 4 + 5 = 5 + 4 C Property of

c. (8 + 6) + 4 = 8 + (6 + 4) A Property of


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Saxon Math Course 3 L 2-6 Adaptations Lesson 2


d. 0 � 5 � 0 Property of M

e. What’s is the difference when the sum of 5 and 7 is subtracted from the product of 5 and 7? (5 � 7) � (5 � 7)

� �

f. See Example 4 on pg. 15

36

�87

87

�36

Answer:

g. What properties did Lee use to simplify his calculations? What are the properties that cover multiplication?

5 � (7 � 8) Given

5 � (8 � 7) Property of

(5 � 8) � 7 Property of

40 � 7 5 � 8 � 40

280 40 � 7 � 280

h. Explain how to check this subtraction problem. What is the opposite of subtraction?

i. Explain how to check this division problem. What is the opposite of division?

For j and k, find the unknown.

j. 12 � m � 48

48

12

m �

k. 12n � 48

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9. See page 10 in the Student Reference Guide.

. Zero is a w number but not a

c number.

10. . All numbers are .


1. ( ) � ( ) � product sum

2. ( ) � ( ) � product sum

{1, , , . . .}

{ , 1, 2, . . .}

3. fact family

30 � � 10

� 10 �

4. fact family

) _____

200 ) _____

200 10 20

30


10 20

200

5. a. What makes 100? ( � ) �

b. The product is .

c. C Property of

A Property of

6. set of counting numbers

Use work area.

7. set of whole numbers 8. set of integers

{. . . , , , , . . .}

Use work area.Use work area.


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11. least to greatest

Use a number line.

0, 1, �2, �3, 4

12. a. | �12 | �

b. | 11 | �

13. See page 20 in the Student Reference Guide.100 � 1 � 100

14. a � 0 � a

15. (5) (0) � 0 16. 5 � (10 � 15) � (5 � 10) �15

17. 10 � 5 � 5 � 10 18. a. The four operations of arithmetic are

(�) , (�) ,

(�) , and (�) .

b. The Commutative and Associative Properties

apply to and .

, , , ,

Property of Multiplication

Zero of Property of Addition



Use work area.

Property of Addition

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21. 22. opposite of 20

25. �5



26. 1

__ 2



Use work area.

23. Which integer is neither positivenor negative?

24. Choose all correct answers. 100




19. | n | � 10| �n | � 10

20. a. 0 �1

b. �2 �3

c. | �2 | | �3 |

, c.

b.

a.


0 1 2 3 4 5–5 –4 –3 –2 –1

Written Practice (continued) (page xx)

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27. 2 8. rearrange.

29. 30. long division

5010� 846 846

� 50105010� 846

780� 49

25 ) ______

5075



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The plot in a word problem tells us what equation to write to solve the problem.

• A formula is an equation written with letters, also called variables.

• Each kind of word problem has a formula.

• Stories about combining have an addition pattern:

some � more � totals � m � t

• Stories about separating have a subtraction pattern:

starting amount � some went away � what is lefts � a � l

• Stories about comparing also have subtraction patterns:

greater � lesser � differenceg � l � d

later � earlier � differencel � e � d

• To solve a word problem:

1. Look for keywords that will help you find the plot of the story:combining, separating, or comparing.Use the key words chart on page 35 in the Student Reference Guide.

2. Write an equation for the problem using the formula and numbers from the story.Use a variable for the missing number

3. Find the missing number.Use the missing numbers chart on page 4 in the Student Reference Guide.

4. Check to see that your answer makes sense.

Teacher Notes:• Introduce Hint #12, “Word

Problem Cues,” Hint #13, “Finding Missing Numbers,” and Hint #14, “Abbreviations and Symbols.”

• Refer students to “Equivalence Table for Units” on page 1, “Time” on page 2, “Missing Numbers” on page 4, and “Word Problem Keywords” on page 35 in the Student Reference Guide.

• Post reference chart, “Word Problem Keywords.”

Addition and SubtractionWord Problems (page 19)


a. What are the three kinds of word problems described in this lesson?

c s c

b. In example 2 on † pg. 21, we solved a word problem to find how much money Alberto spent on

milk and bread. Using the same information, write a word problem that asks how much money Alberto gave to the clerk.

At the store, Alberto bought milk and bread that cost $ . The clerk gave Alberto $

in change. How much money did give to ?

† When you see , refer to your Saxon Math Course 3 textbook.

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c. Write a story problem for this equation.$20.00 � a � $8.45

Abby went to the store with $ . She bought a . The clerk gave her $ in

change. money did Abby spend?

For problems d –f, identify the plot, write an equation, and solve the problem.

d. From 1990 to 2000 the population of Garland increased from 180,635 to 215,768. How many more people lived in Garland in 2000 than in 1990?

plot: equation: � � d

answer: people

e. Binh went to the theater with $20.00 and left the theater with $10.50. How much money did Binh spend at the theater? Explain why your answer is reasonable.

plot: equation: � a �

answer: ; The answer is reasonable because half of $20 is $ . The money left,

$10.50, is a little m than half, so the money spent should be a little l than half.

f. In the three 8th-grade classrooms at Washington school, there are 29 students, 28 students, and 31 students. What is the total number of students in the three classrooms?

plot: equation: � � � t

answer: students

g. Circle the equation that shows how to find how much change a customer should receive from $10.00 for a $6.29 purchase.

A $10.00 � $6.29 � c B $10.00 � $6.29 � c



1. plot:

min sec �

min sec � d

2. plot:

$ � $ � $ � t

d � t �


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7. Sam earned $ mowing yards.

He spent $4.05 on .

How much ?

8. | n | � 3

|�n | � 3

9. least to greatest Use a number line. �6, 5, �4, 3, �2

10. a. �5 1

b. �1 �2

3. plot:

1 ft � in. in. � in. � d

4. plot:

$ � m � $


d � m �

5. plot:

� � d

6. plot:

1 dozen �

� � d

d � d �

Use work area. ,

, , , ,

a.

b.


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c. 17. Commutative Property of Addition

5 � � �

18. Associative Property of Addition

( � ) � �

� ( � )

15. a. What makes 100?

( � ) �

b. C Property of

A Property of

c. ( � ) � =

16. 36 � 17 17 � 36


Use work area.


13. See page 13. 14. See page 13.

a.

b.

a.

b.

11. a. �10 10

b. |�10 | | 10 |

12. |� 5 | (the distance from 0 to � 5 on a number line.)

Use a number line.

a.

b.


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23. Which whole number is not a counting number?

24. opposite of �5

25. 5t � 5

Property of

26. 5 � u � 5

Property of


t � u �

19. 20. rearrange

21. fact family

� �

� �

� �

22. fact family

� �

� �

� �

Use work area.

Use work area.

0 1 2 3 4 5–5 –4 –3 –2 –1

10-15

Use work area.

6 3

9

2 4

8


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27. 4x � 0

Property of

28.

29. 30. long division

x �

$100.00� $90.90

89� $.67

18 ) _______

$72.18


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Some word problems have an equal groups plot.

• Stories about equal groups have a multiplication pattern:

number of groups � number in group � total

n � g � t

• The keywords for equal-groups problems are in each.

• If the missing number is a product, multiply.

Example: There were 24 rows of chairs with 15 chairs in each row. How many chairs were there in all?

n � g � t24 groups � 15 in a group � t24 � 15 � 360

There were 360 chairs in all. To solve this problem, we multiplied.

• If the missing number is a factor, divide.

Example: There were 360 chairs arranged in rows with 15 chairs in each row. How many rows were there in all?

n � g � t

n � 15 in a group � 360 total

2415 )

_____ 360

30060

–600

There were 24 rows. To solve this problem, we divided.

• Some equal-groups problems have a remainder.

Example: Cory sorted 375 quarters into groups of 40 so that he could put them in rolls. How many rolls can Cory fill with the quarters?

n � g � t

n � 40 in a group � 375 total

9 r 1540 )

_____ 375

36015

Cory can make 9 full rolls of quarters and have 15 quarters left over. The problem asks how many rolls Cory can fill. The answer is 9 rolls.

Teacher Note:• Review “Missing Numbers” on page

4 and “Word Problem Keywords” on page 35 in the Student Reference Guide.

Multiplication and Division Word Problems (page 27)

4


1. � g �

) ___

98

2. 1 dozen �

� � t

3. n � � 5000Cancel matching zeros.

5000 _____ 800

) ________

5000 divided by 800 is

with left over.

Only markers

be used.

4. In which equation is the total cost missing?

A c � $1.98 ______ 3

B c � 3 � $1.98

C $1.98 � c � 3 D 3 __ c � $1.98



a. Carver gazed at the ceiling tiles. He saw 30 rows of tiles with 32 tiles in each row. How many ceiling

tiles did Carver see? tiles

� � t offset

b. Four student tickets to the amusement park cost $95.00. The cost of each ticket can be found by solving which of these equations? (Circle one.)

A 4 _______ $95.00

5 t B t __ 4 � $95.00

C 4 x $95 � t D 4t � $95.00

c. Amanda has 632 dimes. How many rolls of 50 dimes can she fill? Explain why your answer is reasonable.

n � � long division

rolls; 632 by 50 is with dimes left over.


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) _____

632

320 � 30

g � t �

n �

7. 8.

5. n � � Cancel matching zeros.

200 ____ 60

) ________

000000

200 divided by 60 is

with left

over. buses willbe used.

6. 306297

9. Jake had $5. He spent all but $ .

did Jake spend?

10. Tajuana paid $ for concert tickets.

Each ticket cost $ . How many

did Tajuana buy?


Written Practice (continued) (page 29)©

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11. least to greatestUse a number line.

�1, 2, �3, � 4, 5

12. a. � 7 � 8

b. 5 � 6

13. a. | �7 | | � 8 |

b. | 11 | | �11 |

14.



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15. | n | � 1

| � n | � 1

16. odd numbers

0 2 4 106 8�10 �4�6�8 �2

, , , , b.

a.

b.

a.

, { … , , , , , , … }�1

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17. Graph the set of odd numbers. 18. See page 4 in the Student Reference Guide.

19. See page 4 in the Student Reference Guide. 20. a. ( � ) �

b. Property of

Property of

21. 12 — 5 5 — 12

22. fact family

� �

� �

� �

23. fact family

� �

� �

� �

24. opposite of 10

0 1 2 3 4 5�5 �4 �3 �2 �1

b.

a.

b.

a.



Use work area.

Use work area.

Use work area.

Use work area.

Use work area.

29. See page 10 in the Student Reference Guide. 30. | n | � 1 __ 2

| �n | � 1 __ 2

27.

� �

28. whole numbers



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25.

� �

26.

� 54 � 48

groups of in. to in.

tickets for { , , , , . . . }

,

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We can use fractions to describe part of a group.

• Fractions have two parts: a numerator (top number) and a denominator (bottom number).

numerator number of parts describeddenominator number of equal parts

• To find a fraction of a group:1. Divide by the denominator.2. Multiply by the numerator.

Example: Two fifths of the 30 questions on the test were multiple-choice. How many questions were multiple-choice?

1. Divide by the denominator: 30 � 5 � 62. Multiply by the numerator: 6 � 2 � 12

12 of the questions were multiple choice.

• One way to compare fractions is to compare each fraction to 1 _ 2 .

• Take half of the denominator and compare it to the numerator.If the numerator is greater than half the denominator, the fraction is greater than 1 _ 2 .If the numerator is less than half the denominator, the fraction is less than 1 _ 2 . If the numerator is equal to half the denominator, the fraction is equal to 1 _ 2 .

Example: Arrange these fractions from least to greatest.

3 __ 6 , 3 __

5 , 3 __

8

Compare each fraction to 1 _ 2 .

3 _ 6 : half of 6 is 3. The numerator is equal to 3, so 3 _ 6 is equal to 1 _ 2 .

3 _ 5 : half of 5 is 2 1 _ 2 . The numerator is more than 2 1 _ 2 , so 3 _ 5 is more than 1 _ 2 .

3 _ 8 : half of 8 is 4. The numerator is less than 4, so 3 _ 8 is less than 1 _ 2 .

3 __ 8 < 3 __

6 < 3 __

5

Teacher Notes:• Introduce Hint #15, “Naming

Fractions/Identifying Fractional Parts,” Hint #16, “Fraction of a Group,” and Hint #18, “Comparing Fractions.”

• Students who are not ready for the abstract nature of fractions will benefit from fraction manipulatives. Fraction tower manipulatives are available in the Adaptations Manipulative Kit. Hint #17, “Fraction Manipulatives,” describes how to make your own fraction tower manipulatives.

• Refer students to “Fraction Terms” on page 12 in the Student Reference Guide.

Fractional Parts (page 31)


a. How many minutes is 1 _ 6 of an hour?

1 hr � minDivide by the denominator.

Multiply by the numerator.

1 __ 3


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b. Three fifths of the 30 questions on the test were multiple-choice. How many multiple-choice questions were there? Explain why your answer is reasonable.

Divide by the denominator. Multiply by the numerator.

questions; 3 _ 5 is than 1 _ 2 , and is greater than 1 _ 2 of 30.

c. Greta drove 288 miles and used 8 gallons of fuel. Greta’s car traveled an average of how many miles

per gallon of fuel?

d. Arrange these fractions from least to greatest. , , Compare each to 1 __ 2 .

5 ___ 10

, 5 __ 6 , 5 ___

12

, , least greatest


1. Divide by the denominator.Multiply by the numerator.

1 _ 4 of a mile

1 mi � yd

2. least to greatestCompare to 1 __ 2 .

, ,

3. 2 _ 3 of 600

Divide by the denominator.Multiply by the numerator.

4. short division

) _______

1 9 2


) _____

288


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5. � � t 6. 1 hr � min

2 1 _ 2 hr � min

500

t �

7. long division

) _____

200

8.

9. 3 __

4 of 8000


10. total

A c � 19 ∙ $2.98 B 19 ______ $2.98

= c

C $2.98 ∙ c = 19 D $2.98 ______ 19

= c

11. offset

x

12. $10


$300 54


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15. 23 16. least to greatest Use a number line.

�5, 7, 4, �3, 0

19. � � 6 20. � � 6

,

, , , , ,

17. Compare to 1 __

2 .

1 __

2

7 ___

15

18. |n| � 6|�n| � 6

,

,

13. Cyndie is microwaving frozen .

Each takes 20 seconds. How

long should she microwave ?

14. Erika had pounds of compost.

She used all but pounds in her

garden. How many of compost

did she use?



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21. Use parentheses.

∙ ∙ = ∙ ∙

22.

25. fact family

� �

� �

� �

26. opposite of 5 �

5 � �

Use work area.

23. What two numbers are 50 unitsfrom zero?

24. fact family

� �

� �

� �

Use work area.

,

6 4

10

Use work area.

Use work area.

0 1 2 3 4 65–6 –4–5 –3 –2 –1



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27. �8 �6 2 8. List some integers and counting numbers.

;

The 1, 2, 3, . . .

are also numbers.

29. List some integers and fractions.

; An example of a f

that is not an is

.

30. 6 ∙ (17 ∙ 50)

6 ∙ ( ∙ )

( ∙ ) ∙ 17

300 ∙

Use work area.

Given

Commutative Property of Multiplication


Multiplied and

Multiplied and

Use work area.

Use work area.




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We measure weight, length, amount of liquids (capacity), and temperature.

• There are two systems of measurement:

The metric system is used throughout the world.

The United States uses both the U.S. Customary Systemand the metric system.

• The U.S. Customary System uses fractions.

• The metric system uses decimal numbers.

• To convert from one unit to another, use five steps.

Example: The 5000-meter run is an Olympic event. How many kilometers is 5000 meters?

1. Name the two units in the problem and write them in a column:

kmm

2. Fill in what you know from the Equivalence Table. Write the amounts next to the units:

kmm

3. Write what you are looking for. Put a question mark in the unknown spot:

km

4. Draw a loop around the two diagonal numbers. The loop should never include the question mark.

km m

5. Multiply the numbers in the loop. Divide by the number outside the loop.

5000 � 1 � 50005000 � 1000 � 5There are 5 km in 5000 m.

Teacher Notes:• Introduce Hint #19, “Converting

Measures and Rate” and Hint # 20, “Measuring Liquids and Capacities of Containers.”

• Refer students to “Liquids” on page 1 and “Proportion (Rate) Problems” on page 19 in the Student Reference Guide.

• Review “Equivalence Table for Units” on page 1 in the Student Reference Guide.

• Post reference chart, “Liquids.”

Converting Measures (page 36)

1 _____ 1000

1 _____ 1000

? _____ 5000

Measure U.S. Customary Metric

Length 12 in. = 1 ft 1000 mm = 1 m 3 ft = 1 yd 100 cm = 1 m 5280 ft = 1 mi 1000 m = 1 km

1 in. = 2.54 cm 1 mi ≈ 1.6 km

Capacity 16 oz = 1 pt 2 pt = 1 qt 1000 mL = 1 L 4 qt = 1 gal

1 qt ≈ 0.95 Liters

Weight/ 16 oz = 1 lb 1000 mg = 1 gMass 2000 lb = 1 ton 1000 g = 1 kg 1000 kg = 1 tonne

2.2 lb ≈ 1 kg 1.1 ton ≈ 1 metric tonne

Equivalent Measures

1 _____ 1000

? _____ 5000


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1. 1 hr � min Multiply the loop.

Divide by the outside number.

hr min

2. 1 pt � oz

Find half of that.

Practice Set (p. 38)

a. A room is 15 feet long and 12 feet wide. What are the length and width of the room in yards?

ft � 1 yd

Multiply the loop. Divide by the outside number.

length width

length: width:

ftyd

b. Nathan is 6 ft 2 in. tall. How many inches tall is Nathan?

First, convert 6 feet to inches. Then add the 2 inches.

1 ft � in.

ft in

c. Seven kilometers is how many meters?

1 km � mMultiply the loop. Divide by the outside number.

km m

1 ___ ? ___ 80

3 __ 1 15 ___

?

1 ___ 12

___ ?

1 ___ ___ ?

3. 1 kg � g

Double that.

4. 3 __ 4

of 300


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5. � � 6. 7. Shade two of the parts.

What fraction is that?

8. ) ___

20 9. Drawing a picturemay help.

10. ) _____

150

11. 1 week � days

1 day � hr Multiply the loop.

day hr

12.

15. �5 4 16. |�2| |�3|

17. 5 |�5|

13. 3 __ 4 of 12 oz.



Use a number line.

Compare 5 __ 7 to 1 __

2 .

�2, 5 __ 7

, 1, 0, 1 __ 2

2318

272 79

2015 12

1 ___ ___ ?

, , , ,

Use work area.


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18. Commutative Property of Addition

� �

�

19. even counting numbers Draw the braces.

20. | 0 | �

21. 22. fact family

� �

� �

� �

26. See p. 10 in the

Student Reference

Guide.

27. ; 3 is a

number and it is an

.

3 12

15

, , , . . .

24. opposite of 100 25. See p. 10 in the

Student Reference

Guide.

23. fact family

� �

� �

� �

28. ; Every number

is greater than zero, and every negative

number is than zero.

29. ; 2 __ 2 is equal to 1. 30. | x | � 7

| �x | � 7

x � ,


Use work area.

0 5–5�5 0 5

5 8

4040

5 8


Use work area.


Use work area.


L E S S O N

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A rate is a relationship between two measures.

• Rates use the word per to mean “in one.”65 miles per hour (65 mph) means “65 miles in one hour.”32 feet per second (32 ft/sec) means “32 feet in one second.”

• To solve rate problems, use the loop method from Lesson 6.

• The average (or mean) describes what number is in the “center” of a group of numbers.

1. Add the numbers.2. Divide by the number of items.

• The average must be between the smallest and largest numbers.Example: Find the average of 5, 1, 3, 5, 4, 8, and 2.

1. Add the numbers. 5 � 1 � 3 � 5 � 4 � 8 � 2 � 282. Divide by the number of items. 28 � 7 � 4

The average is 4. Four is between the smallest number (1) and the largest number (8).

• The median is the middle number when the numbers are put in order.Example: Find the median of 5, 1, 3, 5, 4, 8, and 2.

1. Write the numbers in order. 1, 2, 3, 4, 5, 5, 82. Count the numbers. There are 7 numbers.

Counting from the first number or the last number, 4 is the middle number.

The median is 4. In this group of numbers, the average and the median are the same. This is not always true.

• The mode is the number that occurs most often.Example: Find the mode of 5, 1, 3, 5, 4, 8, and 2.The mode is 5. Five is the only number that occurs more than once in these numbers.

• The range is the difference between the largest and smallest numbers in a group.Example: Find the range of 5, 1, 3, 5, 4, 8, and 2.The range is 7.The largest number is 8 and the smallest number is 1.8 � 1 � 7.

• A line plot is a way to show a group of numbers. Each number is shown by an “X” above anumber line.

Example: Display this group of numbers on a line plot. {5, 1, 3, 5, 4, 8, 2}

Teacher Notes:• Introduce Hint #21,

“Average.”

• Refer students to “Average” on page 7 and “Statistics” on page 23 in the Student Reference Guide.

• Review Hint #19, “Converting Measures and Rate.”

Rates and Average Measures of Central Tendency (page 41)

5 1098760 1 2 3 4

x x x x x x

x

10 2 3 4 5 6 7 8 9 10


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b. How far can Freddy drive in 8 hours at an average speed of 50 miles per hour?

mi hr

c. If a commuter train averages 62 miles per hour between stops that are 18 miles apart, about how

many minutes does it take the train to travel the distance between the two stops? 62 miles per hour is about 60 miles per hour and 60 miles per hour is 1 mile per minute.

mi min

d. If the average number of students in three classrooms is 26, and one of the classrooms has 23 students, then which of the following must be true? (Circle one.) The average must be between the smallest

and largest numbers.

A At least one classroom has fewer than 23 students.

B At least one classroom has more than 23 students and less than 26 students.

C At least one classroom has exactly 26 students.

D At least one classroom has more than 26 students.

e. What is the mean of 84, 92, 92, and 96?

f. The heights of five basketball players are 184 cm, 190 cm, 196 cm, 198 cm, and 202 cm. What is the

average height of the five players?

The price per pound of apples sold at different sold at different grocery stores is reorted below. Use this information to answer problems g–i.

$0.99 $1.99 $1.49 $1.99

$1.49 $0.99 $2.49 $1.49

g. Display the data in a line plot.

___ 1 ? __

8

1 __ 1 ? __

8


a. Alba ran 21 miles in three hours. What was her average speed in miles per hour?

per Multiply the loop.Divide by the outside number.

mi hr

21 ___ 3 ? __

1

$1.50$1.00 $2.00 $2.50


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1. long division

) _____

132

2. 4

__ 5

of 35


3. Convert pounds to ounces.Then add 7 ounces.

1 lb. � oz.

lb. oz.

4. Multiply the loop.Divide by the outside number.

km hr

7. average

8121619

� 20

8. plot:

equation: � � t


1 ___

7 __

?

___ ?

__ 1

5. mi hr

6. See p. 1 in the Student Reference Guide.

___ 1

? __

5

) _____

t �


h. Compute the mean, median, mode, and range of the data.

i. Rudy computed the average price and predicted that he would usually have to pay $1.62 per pound of apples. Why is Rudy’s prediction incorrect?The mode is the most common price for apples. The mode is $ .


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11. 2 terms � yr

plot:

equation: � e �

12. Each bag of weighed pounds.

If there were bags, how much did all

the bags weigh?

13. Ginger gets an allowance of $ each

week. At the end of one week, she had

$ left. How much money did Ginger

spend?


Use a number line.Compare each fraction to

1 __

2 .

0, �1, 2

__ 3

, 1, 2

__ 5

9. plot:

equation: � m �

10. plot:

equation: � l �

15. �11 �10 16. a � b � c

addend � �

d ∙ e � f

factor ∙ �

17. � � � 18. Use parentheses.

� � � � �

e �


, , , ,

Use work area.

Use work area.


m � l �

Use work area.


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19. 20. Rearrange

0�20

21. fact family

� �

� �

� �

22. fact family

� �

� �

� �

23. ; is a number that

is not a number.

24. | �90 | �

25. opposite of 6 26 a. x �


b. y �


27. Rearrange 28.

2 5

7

3 4

12

2020�10,101

$0.79�. 48


88 0

Use work area.

Use work area.

Use work area.

Use work area.

Use work area.


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29. long division

12 ) _______

$60.60

Use work area.


30. 4 · (12 · 75) Given

4 · ( · ) Property

( · ) Property

· Multiplied and

Multiplied and


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A rectangle is a four-sided shape with two dimensions, length ( l ) and width (w).

• The perimeter of a rectangle is the distance around the rectangle.

Perimeter is measured in units of length such as ft, in., cm, and m.

• To find a perimeter, add all sides.The perimeter of the rectangle above is 16 ft.

l � w � l � w � perimeter5 � 3 � 5 � 3 � 16 ft.

• The area of a rectangle is the amount of surface. Area is measured in square units such as ft2, in.2, cm2, and m2.

• Area � length � widthThe area of the rectangle above is 15 ft2.

l � w � Area5 ft � 3 ft � 15 ft2

• To find the perimeter and area of some shapes, we divide the shape into more than one area and find each unknown side length.

1. Subtract to find each unknown side length.h � 12 cm � 5 cm � 7 cmv � 10 cm � 6 cm � 4 cm

2. Perimeter: Add all sides.Perimeter � 5 � 10 � 12 � 6 � 7 � 4 � 44 cm

3. Area: Find the area of each small rectangle. Then add the two areas.

Area of A � 4 cm � 5 cm � 20 cm2

Area of B � 6 cm � 12 cm � 72 cm2

Total area � 20 cm2 � 72 cm2 � 92 cm2

Teacher Notes:• Introduce Hint #22, “Perimeter and

Area Vocabulary” and Hint #23, “Perimeter and Area of Complex Shapes.”

• Refer students to “Perimeter, Area, Volume” on page 16 and “Length and Width” on page 17 in the Student Reference Guide.

• Color tiles for demonstrating perimeter and area are available in the Adaptations Manipulative Kit.

Perimeter and Area (page 47)

5 ft.

3 ft.

12 cm

10 cm6 cm

vh

5 cm

A

B


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a. How many feet of baseboard does Jared need?

Perimeter �

How many tiles will he need?

Area � floor tiles

Find the perimeter and area of each rectangle.

b. P � c. P �

A � A �

d. The length and width of the rectangle in c are three times the length and width of the rectangle in b.

The perimeter of c is how many times the perimeter of b?

The area of c is how many times the area of b?

e. Pete made a rectangle using 12 tiles side by side. This is a 12 � 1 rectangle.

Name two other rectangles Pete can make with 12 tiles. � and � Use square tiles for help.

f. Find the perimeter and area of a room with these dimensions.

x � y � P � A �


4 cm

3 cm

12 cm

9 cm

12 ft

8 ft

13 ft

11 ft

14 ft

3 ft

___ft

___ft


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lb

oz

1. Perimeter: Add all sides.

Area � length � width 2. Label the sides in the figure. Find its perimeter and area

3. The perimeter of 2 is how many timesthe perimeter of 1?

The area of 2 is how many times the area of 1?

4. Use the color tilesfor help.

5. � � t 6. Multiply the loop.

1 ___

___

?

20 yd

15 yd

7. 2 __ 3 of $45

Divide by the denominator.Multiply by the numerator.

8. a. Put in order. , , , ,

, , , , ,

mean: median:

mode: range:

b. Which describes the difference from

greatest to least?

1212121010

9988

� 7

10 ) ___

�

�

Use work area.


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9. � � t 10. ( ) � ( ) � sum difference

11. See p. 1 in the Student Reference Guide. 12. Multiply the loop. Divide by the outside number.

___ ?

__ 1

15. least to greatest.Use a number line.

Compare each fraction to 1 __ 2 .

1 __ 2

, �1, 5 __ 7 , �2, 2 __

6 , 1

16. �100 10

mihr


13. Murat drives miles every day to

.

How many will Murat

drive in 4 days?

14. A carpenter has a board that measures

inches. After cutting, the board

measures inches. How many

did the carpenter

cut off?


, , , ,

19. Commutative Property

� � �

20.


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17. �3 �4 18. | �3 | | �4 |

21. | 15 | � 22. fact family

� �

� �

� �

3 4

7

23. opposite of 3 24. | n | � 5

| �n | � 5

Use work area.

–10 100

Use work area.

Use work area.

,



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27. ; The sum of two

positive integers is a p

i .

28. | x | � 15

| �x | � 15

25. ;

is a

number but is not an

.

26. ; The

contain the

numbers.

29. 30. Three out of four did not order the special. What fraction did?


Use work area. ,

68� 37


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A factor is a counting number that divides evenly into another number.

• Think of factors in pairs that multiply to make the number:6 has two factor pairs: 1 � 6 � 6 and 2 � 3 � 6The factor pairs can be shown with rectangles:

The factors of 6 are 1, 2, 3, and 6.

• A prime number is a counting number greater than 1 that has exactly two factors (one factor pair).

7 has one factor pair: 1 � 7 � 7

The factors of 7 are 1 and 7. Therefore, 7 is a prime number.

• The factors of a prime number are always 1 and the number itself.

Some prime numbers: 2, 3, 5, 7, 11, 13, …

• A counting number that has more than two factors is a composite number.

Any composite number can be written as the product of factors that are prime numbers. This is called prime factorization.

• We can use a factor tree to write the prime factorization of a number:

1. List two factors of the given number under “branches” of the tree. (If you have trouble, start with 2, 3, or 5.)

2. Check if either factor is prime. If the factor is prime, circle it. If it is not, continue to draw branches and factor until each number is prime.3. Write the prime factors in order.

You may have to write some numbers more than once.

Teacher Notes:• Introduce Hint #24, “Factors

of Whole Numbers,” Hint #25, “Tests for Divisibility,” Hint #26, “Prime Factorization Using the Factor Tree,” and Hint #27, “Prime Factorization Using Division by Primes.”

• Refer students to “Factors” and “Tests for Divisibility” on page 5 and “Prime Numbers” on page 9 in the Student Reference Guide.

• Post reference chart, “Primes and Composites.”

1

63

2

1

7

Prime Numbers (page 54)

420

4210

6 752

52

420 � 2 � 2 � 3 � 5 � 7

a. Is 9 a prime or composite number? Use 9 color tiles to make a square.

b. Write the first 10 prime numbers. 2, , , , 11, 13, , , 23, See p. 9 in the Student Reference Guide.

c. Complete this factor tree for 36.

d. Find the prime factors of 60 using division by primes.

60 � � � �

Write the prime factorization of each number in e–g.

e. 25 � �

• Another way to do prime factorization is division by primes:

1. Write the number in a division box.2. Divide by the smallest prime number that is a factor. (Try 2, 3, or 5.)3. Divide that answer by the smallest prime number that is a factor.4. Repeat until the quotient is 1.5. The divisors are the prime factors of the number.

• Tests for divisibility will help you find factors of numbers or tell if a number is prime:


Division by Primes

420 = 2 � 2 � 3 � 5 � 7

17 )

__ 7

5 ) ___

35 3 )

_____ 105

2 ) _____

210 2 )

_____ 420


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36

9

) _____

)

_____

) _____

)

_____ 60

) _____

)

_____ 25

Tests for Divisibility

A number is able to be divided by . . .

2 if the last digit is even. 4 if the last two digits can be divided by 4. 8 if the last three digits can be divided by 8.

5 if the last digit is 0 or 5. 10 if the last digit is 0.

3 if the sum of the digits can be divided by 3. 6 if the number can be divided by 2 and by 3. 9 if the sum of the digits can be divided by 9.

1. 2 __ 3 of


2. Multiply the loop.

m cm


80

___ 1

? ___

2 hours

3 hours

4 hours

4. a. plot:

equation: � m � 320 150

b. average 150 �



2 ) ___

1 ___ 2 __ ?

Use work area.


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f. 100 � � � �

g. 16 � � � �

h. A multi-digit number might be prime if its last digit is .

A 4 B 5 C 6 D 7 See the tests for divisibility.

i. Show that 12 is composite by drawing three different rectangles using 12 squares. Use color tiles for help.

) _____

)

_____

) _____

)

_____ 16

100

10

a.

b.

12 m 15 m

19 m

11. short division

3 ) _____________

$1 2 3 . 4 5


Area � length � width


32 in.

44 in.



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1 ___

5280 _____

?

8.

100 cm– 68 cm

9. $2970.98 � 1429.59

10. Cancel matching zeros.

5280 _____ 30

) ___


ft yd

7. first eight prime numbers

, ,

, ,

, ,

,

5. a. median: Put the numbers in order.

, , , , , ,

b. mean: 86 182 205 214 208 190

�126

c. The is closer to

most of the numbers.


7 ) ___

16. 5 � 4 � 4 � a


17. 17 � 18 � b � 17


18. 20 � c � 20


a b c

19. 21d � 0


20. a. | 9 | �

b. |�12 | |�11 | 21.

a.

b.

13� 5


area � length � width

15.

a. Perimeter: Add all sides.

b. Area = length x width

9 in.

3 in.

y

x

4 in.

6 in.

a.

b.

12 ft

15 ft


Written Practice (continued) (page 58)©

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d �

22. long division

15 ) _____

225

23. Choose all correct answers. 27

A whole number

B counting number

C integer

24. Choose all correct answers.0

A whole number

B counting number

C integer


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29.

48 � � � � �

30. 40 � (23 � 50) Given

40 � ( � ) Property

( � ) � Property

� Multiplied and

Multiplied and

25. Choose all correct answers. �2

A whole number

B counting number

C integer

26. Choose all correct answers. Use tests for divisibility.

5280

A 2 B 3 C 4 D 5

27. See p. 9 in the Student Reference Guide. 28.


) _____

) _____

) _____

) _____

) _____

48

490

10

490 � � � � , , , ,

Use work area.

Teacher Notes:• Introduce Hint #28, “Finding the

Greatest Common Factor,” and Hint #29, “Improper Fractions.”

• Refer students to “Mixed Numbers and Improper Fractions” and “Fraction Families Equivalent Fractions” on page 12 in the Student Reference Guide.

• Review Hint #17, “Fraction Manipulatives,” and Hint #19, “Comparing Fractions.”

• Review “Number Families” on page 10, “Factors” on page 5, and “Fraction Terms” on page 12 in the Student Reference Guide.

10 Rational Numbers Equivalent Fractions (page 60)

100 � 1 � 100100 � 2 � 50


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4 _ 8 3 _

6 2 _

4 1 _

2

10Rational numbers are numbers that can be expressed as a

ratio (division) of two integers.

All integers, whole numbers, and counting numbers are rational numbers.

All fractions are rational numbers.

• Equivalent fractions are different names for the same number.

4 __ 8 , 3 __

6 , 2 __

4 , and 1 __

2 are equivalent fractions.

4 __ 8 , 3 __

6 and 2 __

4 all reduce to 1 __

2 .

• To reduce a fraction:

1. Write the prime factorization of the numerator anddenominator.

2. Cancel all the pairs of factors.

• To form equivalent fractions, multiply by a fraction equal to 1 (same numerator and denominator).

• As a shortcut, use the “loop” method.

Example: Write a fraction equivalent to 1 _ 2 that has a denominator of 100.Multiply the loop. Divide by the outside number.

1 __ 2 = ____

100

50 ____ 100

is the equivalent to 1 __ 2

• Fractions that have the same denominator have common denominators.

5 __ 8 and 7 __

8

Common Denominators Not Common Denominators

∙ ∙ 2 2 ∙ 1

2 = 2

4 3 3

1 2

= 3 6

4 4

1 2

= 4 8

5 __ 8

and 7 ___ 10

48

�21

� 21

21

� 21

� 2�

12

36

�31

2 � 31

�12

24

�21

21

� 2�

12

SXN_M8_SE_L010.indd 63 2/21/06 4:16:08 PM

SM_M8_Ad_L010.indd 51SM_M8_Ad_L010.indd 51 5/7/07 12:37:20 PM5/7/07 12:37:20 PM

• To compare fractions that do not have common denominators, cross-multiply:

Example: Compare: 2 __ 3 3 __

5

2 __ 3

3 __ 5

• An improper (top heavy) fraction is a fraction that is greater than 1 or equal to 1.

• A mixed number is a whole number and a fraction.

• To write an improper fraction as a mixed number or integer: 1. Divide the denominator into the numerator.2. Write the quotient as the whole number.3. Write the remainder as the numerator of the fraction.4. Keep the same denominator.

Example: Express 3 ___ 10

as a mixed number.

The quotient is 3, so that is the whole number. The remainder is 1, so that is the numerator. The denominator stays the same: 3.


Each number in a–c is a member of one or more of the following sets of numbers. Write all the lettersthat apply. A Whole numbers B Integers C Rational numbers

a. 5 b. 2 c. � 2 __ 5

Use prime factorization to reduce each fraction in d–f.

d. 20 ___ 36

� 2 � 2 � 5 __________ 2 � 2 � 3 � 3

�

e. 36 ____ 108

� 2 � 2 � 2 � 2 _____________ 2 � 2 � 2 � 2 � 2

�

f. 75 ____ 100

� 2 � 2 � 2 __________ 2 � 2 � 2 � 2

�

10 ___ 3

� 3 1 __ 3

15 )

__ 5

2 ) ___

10 2 )

___ 20

13 )

__ 3

3 ) __

9 2 )

___ 18

2 ) ___

36

) ___

) ___

) ___

) ___

36

) _____

) _____

) _____

) _____

) _____

108

) ___

) ___

) ___

75

) _____

) _____

) _____

) _____

100

3R13 )

___ 10

9 ___ 1


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10 9 2 � 5 � 103 � 3 � 910 � 9, so 2 __

3 � 3 __

5


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–2 –1 10


Complete each equivalent fraction in g–i.Multiply the loop.Divide by the outside number.

g. 3 __ 5 � ___

20

h. 3 __ 4 � ___

20

i. 1 __ 4 � ____

100

j. Compare: Cross-multiply. 3 __

5 3 __

4

k. Draw and label points on the number line for these numbers: �1, 3 __ 4

, 0, 3 __ 2

, �1 __ 2

l. Convert the improper (top-heavy) fraction to a mixed number. Shade the circles to show that the numbers are equal.

9 __ 4 �

m. Equivalent fractions can be formed by multiplying by a fraction form of 1 (same numerator and denominator). What property of multiplication states that any number multiplied by 1 is equal to the original number?


n. Write a subtraction equation using whole numbers and a difference that is an integer.

� �

R4 )

___ 9

9. 10. Perimeter: Add all sides.

Area� length � width

– 2 – 1 2 1013 ft.

10 ft.

) ___

) ___

22


1. 2.

3. Use Tests for Divisibility. 4.

5. 22 ____ 165

� 2 � 2 ________ 2 � 2 � 2

� 6. 35 ____ 210

� 2 � 2 ________ 2 � 2 � 2

15

1 90

9 10

90 � � � �

) _____

) _____

3 )

_____ 165

165 � � �

) _____

) _____

) _____

165

210

2110

35

5


Divide by the outside number.

2 __ 3 � ___

45

8. 2 __ 5

� ____ 100

Use work area.

Use work area.


© 2

007

Har

cour

t A

chie

ve In

c.


© 2

007

Har

cour

t A

chie

ve In

c.

13. 1 hr � min

2 hr � min

1 __ 2 hr � min

14. 3 __ 4

of 80



17. Dwayne had pounds of apples. After

he made a pie, he had pounds left.

How many of apples

did Dwayne use?

18. Each cost $2. If

the total price was $ , how many

were bought?

178 69

11. Put the numbers in order:

, , , , , ,

median: mode: range:

mean:

848588898278

�82

12. Which measure is the halfway point in order?

7 ) ___

Use work area.


15. � c � 16.


© 2

007

Har

cour

t A

chie

ve In

c.

0, 3 __ 4 , , 1, �1

Convert to a

mixed number.

20. �7 �6


, , , ,

21. |7| | �6| 22. � 3 __ 4

� 1 __ 4

23. � 3 __ 4 � 24. |n| � 7

|�n| � 7

25. sometimes, always, or never?

; Every number is an integer.

26. ; A mixed number has a f part, so

it cannot be a number.

27. ; The set of numbers contains

the .

28. long division

,

) _______

$30.00

29. 30. ( ) � ( ) � product sum

$8.57� 63

$17� 6

$17� 6

4 __ 3 �

� 4 __ 3



Use work area.

Documents

LESSON 1 Number Line: Comparing and Ordering Integersblog.wsd.net/jeelzey/files/2014/08/Math-7-Lessons-1-10.pdf · Number Line: Comparing and Ordering Integers (page 6) ... Which