practice book - delta education · 5.2 writing vertical records .....p36 5.3 writing shorter...

Practice Book

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ContentsChapter 1 Algebra: Machines and Puzzles1.1 Introducing This Year’s

Mathematics ........................................P1

1.2 Investigating Cross Number Puzzles ...P2

1.3 Investigating Input-Output Tables .......P3

1.4 Connecting Input-Output Machines and Puzzles .......................... P4

1.5 Introducing Negative Outputs .............P5

1.6 Determining Rules Using Two Operations ...........................................P6

1.7 Multiplying Cross Number Puzzles ......P7

Chapter 2 Multiplication and Large Numbers2.1 Finding Patterns in the Multiplication

Table .....................................................P8

2.2 Splitting Area Models ..........................P9

2.3 Doubling and Adding ......................... P10

2.4 Multiplying by Multiples of 10 .......... P11

2.5 Working with Large Numbers ............ P12

2.6 Estimating Products ...........................P13

2.7 Estimating in Various Ways ...............P14

2.8 Discovering a Useful Multiplication Pattern ............................................... P15

2.9 Extending the MultiplicationPattern ............................................... P16

2.10 Investigating Why the PatternWorks ................................................. P17

2.11 Finding Products of Large Factors .....P18

Chapter 3 Factoring and Prime Numbers3.1 Investigating Mystery Number

Puzzles ................................................ P19

3.2 Factoring ............................................P20

3.3 Finding Common Factors ...................P21

3.4 Investigating Prime and Composite Numbers .............................................P22

3.5 Writing a Number as the Product of Prime Factors .................................P23

3.6 Investigating Divisibility by 2, 5, and 10 ................................................P24

3.7 Investigating Divisibility by 3, 6, and 9 ..................................................P25

Chapter 4 Equivalence and Comparison of Fractions4.1 Investigating the Result of Two

Operations .........................................P26

4.2 Investigating the Order of Two Operations ......................................... P27

4.3 Finding Equivalent Fractions .............P28

4.4 Equivalent Fractions Using Dot Sketches .............................................P29

4.5 Strategies for Comparing Fractions ...P30

4.6 Comparing Fractions Using Common Denominators .....................P31

4.7 Area Models and Number Lines .........P32

4.8 Numbers Greater Than 1 ...................P33

4.9 Equivalent Fractions GreaterThan 1 ................................................P34

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ContentsChapter 5 Recording Multi-Digit Multiplication5.1 Multiplying Multi-Digit Numbers ......P35

5.2 Writing Vertical Records ....................P36

5.3 Writing Shorter Records .................... P37

5.4 Using Square Number Differences ....P38

5.5 Multiplying Large Numbers ...............P39

Chapter 6 Grids and Graphs6.1 Making Figures on a Coordinate

Grid ....................................................P40

6.2 Translating Figures on a Grid ............. P41

6.3 Reflecting Figures on a Grid ..............P42

6.4 Rotating Figures on a Grid .................P43

6.5 More About Transformations ............P44

6.6 Graphing with Negative Numbers .....P45

6.7 Moving on a Coordinate Grid ............P46

6.8 Graphing Data .................................... P47

6.9 What is Typical? .................................P48

6.10 Another Way of Describing What’s Typical ................................................P49

6.11 Reading Graphs and Tables ...............P50

Chapter 7 Decimals7.1 Investigating Decimals ...................... P51

7.2 Comparing and Ordering Decimals ...P52

7.3 Large and Small Numbers ..................P53

7.4 Connecting Decimals to Fractions .....P54

7.5 Connecting Decimals to Other Fractions ............................................P55

7.6 Estimating Decimals Using Familiar Fractions ............................................P56

7.7 Estimating Decimals UsingRounding ............................................P57

7.8 Adding with Decimals ........................P58

7.9 Subtracting with Decimals ................P59

7.10 Adding and Subtracting Decimals .....P60

7.11 Multiplying with Decimals .................P61

Chapter 8 Developing a Division Algorithm8.1 Exploring Missing Factors ..................P62

8.2 Connecting Multiplication and Division ..............................................P63

8.3 Dividing Using Multiplication and the Area Model ...........................P64

8.4 Recording the Steps in Division .........P65

8.5 Dividing and Recording Division Efficiently ...........................................P66

8.6 Using Multiplication to Check Division ..............................................P67

8.7 Investigating Remainders ..................P68

8.8 Interpreting Remainders in Word Problems ............................................P69

8.9 Another Option for Interpreting Remainders ........................................P70

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Chapter 9 Attributes of Two-Dimensional Figures9.1 Investigating Angles ..........................P71

9.2 Classifying Angles and Triangles .......P72

9.3 Constructing Triangles .......................P73

9.4 Constructing Similar Triangles ........... P74

9.5 Angles Formed by IntersectingLines ...................................................P75

9.6 Angles Formed by a Line Intersecting Parallel Lines ...................................... P76

9.7 Comparing and Classifying Quadrilaterals ....................................P77

9.8 Investigating Quadrilaterals ..............P78

Chapter 10 Area and Perimeter10.1 Length and Perimeter ........................P79

10.2 Perimeter Formulas ............................P80

10.3 Area of Parallelograms ......................P81

10.4 Measuring to Find Areas of Parallelograms ...................................P82

10.5 Area of Triangles and Trapezoids ......P83

10.6 Area and Perimeter of Other Polygons .............................................P84

Chapter 11 Fraction Computation11.1 Adding and Subtracting Fractions

with Like Denominators .....................P85

11.2 More Adding and Subtracting Fractions with Like Denominators .....P86

11.3 Stories About Adding and Subtracting Fractions .........................P87

11.4 Adding and Subtracting Unlike Things .................................................P88

11.5 Adding and Subtracting Fractions with Unlike Denominators .................P89

11.6 Stories with Fractions ........................P90

11.7 Using Area to Multiply Fractions .......P91

11.8 Using Other Models to Multiply Fractions ............................................P92

11.9 Fractions of Quantities ......................P93

11.10 Stories About Multiplying Fractions ............................................P94

Chapter 12 Three-Dimensional Geometry12.1 Transforming Two-Dimensional Nets

into Three-Dimensional Figures ........P95

12.2 Describing Three-Dimensional Figures ................................................P96

12.3 Sorting Three-Dimensional Figures ...P97

12.4 Volume of Rectangular Prisms ...........P98

12.5 Volume of Prisms ...............................P99

12.6 Area of Nets .....................................P100

12.7 Surface Area of Polyhedra ...............P101

12.8 Comparing Volume and Surface Area ..................................... P102

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ContentsChapter 13 Fun with Algebra13.1 Introducing Mobiles .........................P103

13.2 Balancing Mobiles ............................P104

13.3 Equations for Mobiles ......................P105

13.4 Balance Puzzles ................................P106

13.5 Number Tricks ..................................P107

13.6 Making Diagrams .............................P108

13.7 Equations for Stories .......................P109

Chapter 14 Data and Probability14.1 Conducting a Probability

Experiment ....................................... P110

14.2 Finding Probabilities ........................ P111

14.3 Sampling Experiments ..................... P112

14.4 Another Sampling Experiment ........ P113

14.5 Introducing Percents ........................ P114

14.6 Circle Graphs .................................... P115

Chapter 15 Graphing15.1 Graphing .......................................... P116

15.2 Graphing Capacity Conversions ....... P117

15.3 Changing the Scale of Graphs ......... P118

15.4 Graphing Change Over Time ............ P119

15.5 Graphing the Story of a Trip ............P120

15.6 Graphing Temperature Conversions ...................................... P121

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Practice Book

These pages provide additional practice for each lesson in the chapter. The exercises are used to reinforce the skills being taught in each lesson.

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PracticeLesson 1

Introducing This Year’s Mathematics Use the table to make an organized list of all possible combinations of dimes, nickels, and pennies that make 28¢.

DIMES NICKELS PENNIES

2 1 3

2 0 8

1 3 3

1 2 8

1 1 13

1 0 18

0 5 3

0 4 8

0 3 13

0 2 18

0 1 23

0 0 28

Which combination of coins is NOT worth 49¢?

A. 4 dimes, 1 nickel, 4 penniesB. 3 dimes, 3 nickels, 4 penniesC. 2 dimes, 5 nickels, 4 penniesD. ! 1 dime, 4 nickels, 4 pennies

Sandy has 2 quarters, 1 dime, and 4 pennies. How much money does she have?

A. 39¢ C. 60¢B. 59¢ D. ! 64¢

Order may vary.

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Chapter 1 Practice Book P1

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PracticeLesson 2

John leaves his house at 7:30 A.M. He arrives at school at 8:25 A.M. How long does it take John to get to school? Explain how you found your answer.

55 minutes; Possible explanation: I thought about a clock.

It is 30 minutes from 7:30 A.M. to 8:00 A.M. and 25 minutes

from 8:00 A.M. to 8:25 A.M. 30 minutes ! 25 minutes "

55 minutes

Investigating Cross Number PuzzlesComplete each Cross Number Puzzle by filling in numbers that make amounts on both sides of the thick line the same.

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P2 Practice Book Chapter 1


PracticeLesson 3

Which expression does NOT equal 16?

A. 4 ! 4 B. ! 36 " 2 C. 9 # 7 D. 21 $ 5

Investigating Input-Output TablesComplete the tables.

INPUT 6 10 4 8 12 20

Add 4 10 14 8 12 16 24

Multiply by 2 20 28 16 24 32 48

Subtract 8 12 20 8 16 24 40

MACHINE OUTPUT 12 20 8 16 24 40

INPUT 6 10 4 8 12 20

Divide by 2 3 5 2 4 6 10

Multiply by 4 12 20 8 16 24 40

MACHINE OUTPUT 12 20 8 16 24 40

INPUT 6 10 4 8 12

Add 5 11 15 9 13 17

Double 22 30 18 26 34

Subtract 10 12 20 8 16 24

MACHINE OUTPUT 12 20 8 16 24

MAKE YOUR OWN.Many correct answers are possible for last column.

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PracticeLesson 4

Connecting Input-Output Machines and PuzzlesThe numbers in the puzzle on the right are all double the numbers on the left. Complete the pairs of Cross Number Puzzles.

Write the numbers in order from greatest to least. Explain how you decided how to order the numbers.

648,831 684,301 684,299

684,301; 684,299; 648,831. Possible explanation: I compared

the digits of the same place value beginning with the

hundred thousands. All three numbers have 6 in the hundred

thousands place, so I looked at the ten thousands place.

648,831 is the least number. The other two numbers have the

same digit in the ten thousands and thousands places. In the

hundreds place, 2 is less than 3, so 684,299 is less than 684,301.

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PracticeLesson 5

The temperature was 97!F at 2:00 P.M. Then a thunderstorm rolled in and the temperature dropped 16!F. After the storm, the temperature rose 8!F. What was the temperature then? Explain your answer.

89!F; Possible explanation: I drew a thermometer.

I started at 97!F, went down 16!F, and then up 8!F to 89!F.

Introducing Negative OutputsEach of the tables was made using one of the rules below. For each table, write the letter of the rule that was used to create it. Then complete the table.

Rule A: Multiply the input by 3. Rule B: Subtract 6 from the input.

Rule C: Multiply the input by itself.

Rule B

INPUT 7 15 50 6 5 4 0

OUTPUT 1 9 44 0 "1 "2 "6

Rule C

INPUT 3 5 8 10 1 6 7

OUTPUT 9 25 64 100 1 36 49

Rule A

INPUT 3 5 8 10 7 4 11

OUTPUT 9 15 24 30 21 12 33

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PracticeLesson 6

Determining Rules Using Two OperationsComplete the tables. Some of the rules use two operations.

INPUT 3 9 2 10 5 7 0 6 8 11

OUTPUT 15 45 10 50 25 35 0 30 40 55

INPUT 18 9 14 11 29 129 100 114 99 155

OUTPUT 9 0 5 2 20 120 91 105 90 146

INPUT 6 10 4 8 5 12 3 11 7 9

OUTPUT 10 18 6 14 8 22 4 20 12 16

INPUT 4 7 3 9 0 11 8 10 12 6

OUTPUT 16 25 13 31 4 37 28 34 40 22

Marti used a rule to make this list of numbers.

1, 3, 7, 13, 21, !

If she uses the same rule to continue the list, which number would come next?

A. 23 C. 33B. ! 31 D. 37

What is the value of c in the equation 63 ! c " 29?

A. ! 34

B. 44

C. 46

D. 92

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PracticeLesson 7

Multiplying Cross Number PuzzlesComplete all the puzzles.

Which number sentence is correct?

A. 2 ! 9 " 7B. 2 ! 9 " 0C. 2 ! 9 " 11D. ! 2 ! 9 " !7

Jacob left for school with 4 boxes of pencils. Each box had 12 pencils. At school, he gave 6 pencils to each of his 4 friends. Which number sentence below can be used to find the remaining number of pencils?

A. (4 # 12) $ (6 # 4) " !

B. (4 $ 12) ! (6 $ 4) " !

C. (12 # 6) ! (6 # 4) " !

D. ! (4 # 12) ! (6 # 4) " !

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Finding Patterns in the Multiplication Table

Complete the table.

Notice the pattern of the squares that are shaded in the designs below.

How many squares would be shaded in Design 4? Explain how you know.

20 squares; there would be a 4-by-4 array of 16 shaded squares

in the center and a shaded square in each corner. 16 ! 4 " 20

PracticeLesson 1

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Splitting Area ModelsComplete the area models and puzzles.

! 5

8 407 3515 75

! 10 6 16

4 40 24 64

! 7

6 429 6315 105

! 10 8 18

11 110 88 198

! 2 4 6

3 6 12 185 10 20 308 16 32 48

! 4 6 10

7 28 42 7010 40 60 10017 68 102 170

Which is NOT a way to describe a 13 ! 14 area model?

A. (10 " 3) ! (10 " 4) D. (12 " 1) ! (13 " 1)B. (5 " 8) ! (6 " 8) E. (13 ! 9) " (13 ! 5)C. ! (10 ! 3) " (10 ! 4)

PracticeLesson 2

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Doubling and AddingFill in the columns by adding or doubling.

! 39

1 392 783 1174 1565 1956 2347 2738 3129 35110 390

! 39

11 42912 46813 50714 54615 58516 62417 66318 70219 74120 780

Complete.

1 ! 39 "

39 , so 10 ! 39 "

390 2 ! 39 "

78 , so 20 ! 39 "

780

PracticeLesson 3

An adult takes 180 breaths in 15 minutes and a baby takes 300. How many more breaths than an adult will a baby take in one hour? Explain how you know.

480 more breaths; Possible explanation: One hour is 4 times

longer than 15 minutes; in one hour an adult takes 4 ! 180, or 720

breaths and a baby takes 4 ! 300, or 1,200, breaths. 1,200 " 720 # 480

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Multiplying by Multiples of 10Complete the puzzles.

! 50 40 90

6 300 240 54010 500 400 90016 800 640 1,440

! 60 5

20 1,200 100 1,30030 1,800 150 1,95050 3,000 250 3,250

Complete the number sentences.

9 ! 5 " 45 5 ! 2 " 10

90 ! 5 " 450

20 ! 5 " 100

50 ! 9 " 450

50 ! 2 " 100

90 ! 50 " 4,500

20 ! 50 " 1,000

PracticeLesson 4

Marco had 1,400 baseball cards and 50 football cards in his collection. After selling some cards to his brother, he had 1,274 cards left. How many cards did he sell to his brother? Explain how you solved the problem.

176 cards; Possible explanation: I added the number of

baseball cards and football cards, and then subtracted the

number of cards he had left from the sum.

1,400 " 50 # 1,450; 1,450 $ 1,274 # 176

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Working with Large NumbersWrite numbers to match the words.

Twelve million, forty-nine thousand, nine hundred two

12,049,902

Two hundred fourteen billion, five hundred million, seven hundred seventeen thousand, twelve

214,500,717,012

Six hundred eight million, eight

608,000,008

Compare the numbers and order them from least to greatest by writing 1, 2, or 3 in the boxes.

3 29,642,831,076

2 29,642,813,076

1 29,624,831,760

PracticeLesson 5

Which expression does NOThave a value equal to 2,000?

A. 100 ! 10 ! 2B. ! 50 ! 400C. 20 ! 100D. 40 ! 50

Which shows another way to write 10,000?

A. 10 ! 10 ! 10B. 100 ! 10C. ! 100 ! 100D. 1 ! 100 ! 10

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PracticeLesson 6

Estimating ProductsComplete the number sentences.

200 ! 9 " 1,800 4 ! 800 " 3,200

200 ! 90 " 18,000 40 ! 800 " 32,000

200 ! 900 " 180,000 400 ! 800 " 320,000

60 ! 40 " 2,400 7 ! 60 " 420

600 ! 400 " 240,000 70 ! 600 " 42,000

60 ! 400 " 24,000 70 ! 60 " 4,200

Solve the problem.

Fifty students each spend about 300 hours a year studying and doing homework. About how many hours per year do they spend altogether?

15,000 hours

A large bag of potato chips costs $0.75 more than a small bag. If ! represents the price of the large bag, which expression shows the price of a small bag?

A. ! ! # $0.75 C. ! $ $0.75B. $0.75 # ! D. $0.75 $ !

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Estimating in Various WaysEstimate the products. Estimates may vary.

16 ! 46 34 ! 29

Estimate: Estimate:

20 ! 50 " 1,000 33 ! 30 " 990

31 ! 24 55 ! 78

Estimate: Estimate:

30 ! 25 " 750 60 ! 80 " 4,800

Natalee wants to buy 24 cupcakes at tomorrow’s bake sale. She knows that each cupcake will cost 32¢. How much money should she bring to be sure she has enough money? Show your work and explain your answer.

Estimate:

24 ! 40 " 960

She should bring 960 cents, or $9.60. For the estimate

I rounded 32 up to 40 to make sure she had enough money.

PracticeLesson 7

Which list shows the common factors of 12 and 30?

A. 1, 2, 3, 4, 5, 6, 10, 12, 15, 30B. 2, 4, 6, 10, 12C. 1, 3, 5, 15D. ! 1, 2, 3, 6

Which pair of signs would make this sentence true?

12 ! 1 " 12 ! 1

A. #, $B. $, !C. %, &D. ! !, '

Estimates and explanations will vary.

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PracticeLesson 8

Discovering a Useful Multiplication PatternComplete the diagrams and number sentences.

8 ! 8 " 64 10 ! 10 " 100

7 ! 9 " 63 9 ! 11 " 99

12 ! 12 " 144 16 ! 16 " 256

11 ! 13 " 143 15 ! 17 " 255

Jake makes $21 each day he works. He plans to work 112 days this year. At the end of the year, will he have earned the $2,000 he wants to save for a trip to visit his grandmother? Estimate to solve and explain your answer.

20 ! 112 " 2,240; yes; Possible explanation: I adjusted the

factors, so that the estimated product is less than the

actual product. The actual product will be more than 2,240.

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PracticeLesson 9

Extending the Multiplication PatternComplete the diagram and tables.

Steps Away 11 ! 11 "

1 10 ! 12 " 120

2 9 ! 13 " 117

3 8 ! 14 " 112

4 7 ! 15 " 105

Steps Away 7 ! 7 "

3 4 ! 10 " 40

Steps Away 19 ! 19 " 361

2 17 ! 21 " 357

A certain number is multiplied by 3. The product is 8 less than 35.

What is the number?

A. 8 C. 7B. 5 D. ! 9

A certain odd number is less than 10. If it is multiplied by 6, and 6 is added to the product, the result is 60.

What is the number?

A. ! 9 C. 7B. 5 D. 3

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An eagle beats its wings 150 times in a minute. A hummingbird beats its wings 30 times as fast as an eagle. How many times does a hummingbird beat its wings in one minute? Explain how you know?

4,500 times; I need to multiply 150 x 30. First I multiply 150 ! 10

(1,500) and then I multiply that product by 3: 3 ! 1,500 " 4,500.

PracticeLesson 10

Investigating Why the Pattern WorksFill in the missing numbers.

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PracticeLesson 11

Finding Products of Large FactorsFill in the missing numbers.

(20 ! 30) " 25 # 625 (30 ! 40) " 25 # 1,225

( 50 ! 60 ) " 25 # 3,025 ( 40 ! 50 ) " 25 # 2,025

Which group contains a number that is NOT a square number?

A. ! 121, 11, 25, 4B. 16, 49, 1, 144C. 36, 16, 0, 4D. 100, 49, 9, 25

Which group of words correctly describes the number 25?

A. multiple of 5, multiple of 20B. odd, multiple of 10C. prime, square, oddD. ! odd, square, composite

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Investigating Mystery Number PuzzlesSolve the puzzles. The boxes below the clues show you the number of digits in the solution.

Clues Workspace

Puzzle A

! Multiple of 9 less than 81

! Even

! Difference between the digits ! 5

7 2

Puzzle B

! Multiple of 20 greater than 80, but less than 300

! Sum of the digits is even

! Sum of the digits is a 2-digitnumber

2 8 0

Ms. Nichols wanted to put the same number of computers into 3 classrooms. She had a total of 84 computers. Which statement is true?

A. She cannot put the same number of computers into each classroom.

B. She can put 29 computers into each classroom.C. She can put 43 computers into each classroom.D. " She can put the same number of computers into each

classroom.

PracticeLesson 1

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PracticeLesson 2

FactoringWrite all the factors of each product in the diagram. Connect pairs of factors.

Gayle is shading squares with multiples on the grid.

If she shades all the squares with multiples of 2,

how many squares will she shade? 50





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PracticeLesson 3

Finding Common Factors• To solve these puzzles, you may need to make more

than one list of numbers.

• Read all the clues for each puzzle before you begin.

• The boxes below the clues show you the number of digits in the solution.

• Some puzzles have more than one solution.

Clues Workspace

Puzzle A

! Odd

! Common factor of 12 and 18

3

Puzzle B

! Less than 200

! Sum of the digits ! 6

! Product of the digits ! 0

! Each factor of 75 is its factor too

1 5 0

1 or

Which number is NOT a common multiple of 8 and 5?

A. 80B. 0C. " 140D. 200

Lois arrived at the library at 9:30 A.M. She spent 35 minutes in the magazine section, 48 minutes in the fiction section, and 1 hour and 15 minutes in the biography section. What time did Lois leave the library?

12:08 P.M.

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PracticeLesson 4

Investigating Prime and Composite NumbersList the factors. Write P for Prime, C for Composite, or N for Neither.

Number Factors P, C, or N

40

23

49

1

100

Which group contains all of the factors of 18?

A. 1, 18B. 1, 2, 6, 9, 18C. ! 1, 2, 3, 6, 9, 18D. 1, 3, 6, 9, 18

Kenji and John drive 270 miles using 9 gallons of gas. How many miles do they drive on one gallon of gas?

30 miles

C

C

C

P

N

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44 ! 2 ! 2 ! 11

72 ! 2 ! 2 ! 2 ! 3 ! 3

28 ! 2 ! 2 ! 7

144 ! 2 ! 2 ! 2 ! 2 ! 3 ! 3

The factor trees and the order of the factors in the sentences may vary.

PracticeLesson 5

Writing a Number as the Product of Prime FactorsDraw factor trees and circle the prime factors. Write number sentences with the prime factors.

Which number is divisible by 2, 3, 5, 6, and 10?

A. 48,405B. ! 45,840C. 36,315D. 63,550

A bead factory divides 54,000 beads evenly into 6 containers. How many beads are in each container? Are there any beads left over?

9,000 beads; no

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PracticeLesson 6

Investigating Divisibility by 2, 5, and 10Write yes or no.

Is it divisible by 2?

128 yes 1,046 yes 2,468 yes

465 no 1,298 yes 788 yes

How do you know? Possible answer: Numbers divisible by 2 have

ones digits of 0, 2, 4, 6, or 8.


110 yes 65 yes 105 yes

42 no 1,040 yes 6,630 yes


ones digits of 0 or 5.


425 no 1,250 yes 16,802 no

760 yes 405 no 21,970 yes


a ones digit of 0.

Mr. Ruiz used a copy machine to print 395 pages. The machine stapled them into packets of 5 pages each. How many pages were left over?

A. ! 0 B. 2 C. 3 D. 4

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PracticeLesson 7

Investigating Divisibility by 3, 6, and 9Write yes or no.

Is the number divisible by 3?

102 yes 473 no 780 yes

312 yes 561 yes 803 no

How can you tell if a number is divisible by 3? Numbers are divisible

by 3 if the sum of the digits is divisible by 3.


333 yes 612 yes 3,210 no

945 yes 514 no 4,959 yes


by 9 if the sum of the digits is divisible by 9.


501 no 840 yes 4,545 no

102 yes 134 no 5,454 yes


by 6 if the number is even and divisible by 3.

The number 8,955 is NOT divisible by

A. 3 C. 9B. 5 D. ! 10

On Friday, Saturday, and Sunday, a total of 630 newspapers were delivered. If the same number of newspapers were delivered each day, how many newspapers were delivered on Sunday?

210 newspapers

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PracticeLesson 1

Investigating the Result of Two OperationsWrite the outputs.

Example:

What are all the common factors of 24 and 36?

A. 1, 2, 4, 24, 36 B. 1, 2, 3, 6, 12 C. 1, 2, 3, 4, 12 D. ! 1, 2, 3, 4, 6, 12

Which group shows common multiples of 6 and 4?

A. 1, 6, 4, 12 B. ! 36, 12, 24 C. 1, 12, 18 D. 12, 18, 24, 36

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PracticeLesson 2

Investigating the Order of Two OperationsRecord the outputs.

Record the missing numbers.

If you multiply 12 by 3 and divide the result by 4, which statement is NOT true?

A. You can either multiply 12 by 3 first or divide 12 by 4 first and still get the correct answer.

B. The correct answer is 9.

C. You can divide 12 by 4 and then multiply the result by 3 to get the correct answer.

D. ! The correct answer is 4.

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Finding Equivalent Fractions• Check (!) the fraction machines that produce the

result shown.• Cross out (") the fraction machines that do not.• Fill in the boxes on the left with the smallest

numbers that produce the result shown.

PracticeLesson 3

Mackenzie used 12 feet of ribbon to wrap a gift. Tyler used twice as much ribbon to wrap 4 small gifts. He used the same amount of ribbon for each gift. How much ribbon did Tyler use for each gift?

A. 24 feet C. ! 6 feetB. 8 feet D. 4 feet

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PracticeLesson 4

Equivalent Fractions Using Dot SketchesUse dot sketches to find equivalent fractions.

5 _ 6 ! 15 _ 18 3 _ 5 !

15 _ 25

Find any equivalent fraction with a dot sketch.

2 _ 7 ! 6 _

21

7 _ 8 !

14 _

16

2 _ 5 ! 8 _

20

4 _ 7 !

12 _

21

Which fraction is equivalent to 2 _ 9 ?

A. 1 _ 18 C. ! 6 _ 27

B. 1 _ 3 D. 9 _ 2

What fraction is the simplest form of 15 _ 25 ?

A. 6 _ 10 C. 12 _ 20

B. ! 3 _ 5 D. 9 _ 15

Answers will vary.

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Strategies for Comparing FractionsCompare the fractions. Write !, ", or #.

How did you figure it out? Choose one or more.

! Compared each fraction to 1 __ 2 . Choices will vary.

11 _ 12

" 3 _ 8 ! Figured out which fraction is closer to 1.

! Recognized equivalent fractions.

! Something else:



5 _ 6

" 4 _ 10 ! Figured out which fraction is closer to 1.


! Something else:



3 _ 4

# 6 _ 8 ! Figured out which fraction is closer to 1.


! Something else:

PracticeLesson 5

Damon wrote this riddle. Find the answerto the riddle. Explain the strategy you used. 7 __ 14 ; Possible explanation:

The denominator has to be

numerator, doubled the number for the denominator and

and checking until I found numbers with a sum of 21.

I am a fraction equivalent to 2 _ 4 .The sum of my numerator and my denominator is 21.What fraction am I?

twice the numerator, so I guessed a number for the

then checked to see if the sum was 21. I kept guessing

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Comparing Fractions Using Common DenominatorsFor each pair of fractions: • Write an equivalent pair of fractions, but with a

common denominator.• Use dot sketches to make equivalent fractions,

if you wish.• Write !, ", or # between the fractions.

Example:

5 _ 8 3 _ 4 1 _ 4 2 _ 6

5 _ 8 ! 6 _ 8

3 _ 12 !

4 _ 12

2 _ 3 3 _ 5 5 _ 6 6 _ 8

10 _

15

"

9 _

15

20 _

24

"

18 _

24

7 _ 8 2 _ 3 3 _ 4 4 _ 5

21 _

24

"

16 _

24

15 _

20

!

16 _

20

PracticeLesson 6

If used, dot sketches may be drawn and shaded in various ways.

Common denominators and fractions may vary.

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PracticeLesson 7

Area Models and Number LinesShade the sketches for the fractions.

1 _ 4 4 _ 6 2 _ 6 2 _ 12

1 _ 3 3 _ 12 1 _ 6 2 _ 3

Write the fractions from Problems 1–8 as pairsof equivalent fractions.

1 _

4 !

3 _

12

4 _

6 !

2 _

3

2 _

6 !

1 _

3

2 _

12 !

1 _

6

Jake hiked 3 _ 4 mile around the pond. Marcia hiked 3 _ 5 mile to the cabin. Who hiked farther? Explain how you know.

Jake; Possible explanation; 3 _ 4 is greater than 3 _ 5 . Both

fractions have the same numerator, so the fraction with

the smaller denominator is larger.

Order may vary.

The sections shaded may vary.

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PracticeLesson 8

Numbers Greater Than 1 Write the numbers at their locations on the number line. If two numbers label the same point, write one above the line and the other below.

7 _ 3 13 _ 4 2 1 _ 3 8 _ 3 5 _ 4 1 _ 2 1 1 _ 4

0 1 2 3

1__2

7__3

8__3

5__4

1__4

1 1__3

2

13__4

Solve the problem.

Small paper cups at the water machine hold 1 _ 4 cup of water. Erika was very thirsty and filled her cup eleven times. How much water did she drink? Explain how you know.

11 __ 4 ! 2 3 _ 4 cups of water; Possible explanation: I know that

4 _ 4 ! 1 and 8 _ 4 ! 2, so 11 __ 4 ! 2 3 _ 4

Katie has $8 in her wallet. She has 1 _ 2 of that amount in her pocket and 1 _ 4 of that amount in her hand. How much money does she have in all? Explain how you know.$14; Possible explanation: 1 _ 2 of $8 is $4 and 1 _ 4 of

$8 is $2. $8 " $4 " $2 ! $14

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PracticeLesson 9

Equivalent Fractions Greater Than 1 Draw lines to match the equivalent numbers.

9 _ 8

3 2 _ 5

2 2 _ 8

6 8 _ 20

6 4 _ 10

1 2 _ 16

18 _ 8

3 6 _ 15

Write equivalent fractions or mixed numbers.

8 1 _ 3 ! 25

_

3 ! 8

2

_ 6 6 3 _ 4 !

27

_ 4 !

6 6

_ 8

38 _ 6 !

6 2

_ 6 !

6 1

_ 3 43 _ 8 !

5 3

_ 8 !

5 9

_ 24

Look at the hexagon covered with 3 different shapes. Which statement is NOT true?

A. ! The triangle is 1 _ 3 of the hexagon.

B. The trapezoid is 1 _ 2 of the hexagon.

C. The rhombus is 1 _ 3 of the hexagon.

D. The triangle is 1 _ 6 of the hexagon.

Answers will vary.

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PracticeLesson 1

Multiplying Multi-Digit NumbersComplete the multiplication sentences after splitting and completing an area model or completing a puzzle.

39 ! 52 " 2,028 61 ! 48 " 2,928

54 ! 76 " 4,104 82 ! 44 " 3,608

Which fraction represents the shaded part?

A. 1 _ 3 C. 2 _ 6 B. 1 _ 4 D. ! 1 _ 6

Which fraction represents the shaded part?

A. 2 _ 3 C. ! 2 _ 8 B. 2 _ 5 D. 1 _ 3

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PracticeLesson 2

Writing Vertical Records Fill in the puzzle. Then complete the multiplication records.

Write the partial products on the area models. Then complete the multiplication records.

Write a word problem that can be represented by the number sentence 19 ! 36 " ? . Then solve your problem.Problems will vary. Possible problem: Each of the 19 students in thefi fth-grade class baked three dozen cookies. How many cookies werebaked? 684 cookies

Order of partial products may vary.

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Writing Shorter Records Complete the puzzle and record.

Complete the area model. Then complete the puzzle and record.

PracticeLesson 3

Which of the following is NOT equal to 86 ! 24?

A. (80 ! 24) " (6 ! 24)B. ! (80 ! 6) " (20 ! 4)C. (20 ! 86) " (4 ! 86)D. (24 ! 6) " (24 ! 80)

Which will NOT produce the same result as 81 ! 69?

A. 69 ! 81B. (80 " 1) ! (9 " 60)C. ! (60 " 9) " (80 " 1)D. (81 ! 60) " (81 ! 9)

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PracticeLesson 4

Using Square Number DifferencesComplete the tables.

a 7 9 12 40 50a2 49 81 144 1,600 2,500

b 8 10 12 15 30b2 ! 1 63 99 143 224 899

c 4 3 5 20 11

(c " 2) # (c ! 2) 12 5 21 396 117c2 ! 4 12 5 21 396 117

A certain pair of numbers have a sum of 25 and a difference of 9. The numbers must be:

A. 5, 5 C. 25, 9B. ! 17, 8 D. 9, 16

The square of one number is added to the square of another number. The sum is 41. The numbers could be:

A. 6, 2 C. ! 5, 4B. 40, 1 D. 3, 5

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Multiplying Large NumbersComplete the area model, puzzle, and record.

PracticeLesson 5

The Gomez family is planning a party at a restaurant. They are inviting 51 adults and 26 children. If the cost is $49 per adult and $24 per child, how much should they expect to spend on the party? Explain how you found your answer.

$3,123; Possible explanation: I need to fi nd the sum of what it costs

for the adults (51 ! $49) and what it costs for children (26 ! $24).

I can use the pattern of squares to fi nd the two products:

51 ! $49 " $2,499; 26 ! $24 " $624; $2,499 # $624 " $3,123.

Partial products and their order may vary.

267 ! 48 " 12,816

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Making Figures on a Coordinate Grid Plot each point, label it, and then connect A ! B ! C ! D ! E ! A.

Name A B C D E

Coordinates (1,2) (3,4) (4,3) (5,1) (3,1)

Complete the table for the rule given.

Name A B C D E

Coordinates (x,y) (1,2) (3,4) (4,3) (5,1) (3,1)

New Ordered Pair:Add 7 to the First Coordinate (x ! 7, y)

(8,2) (10,4) (11,3) (12,1) (10,1)

Plot the points whose coordinates are given in the new ordered pairs. Connect the new points: A ! B ! C ! D ! E ! A.

PracticeLesson 1

Jessica added 3 _ 4 cup of pineapple, 2 _ 3 cup of chopped almonds, and 3 _ 5 cup of dried cranberries to a salad. Did she add more pineapple or dried cranberries? Explain how you know.more pineapple; 3 _ 4 " 3 _ 5 ; Possible explanation: 3 _ 4 and 3 _ 5 have

the same numerators, so the fraction with the smaller

denominator is the greater fraction.

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MNENL07AWK5X_PB_C06_040-050_V4.indd 40MNENL07AWK5X_PB_C06_040-050_V4.indd 40 12/8/06 4:25:18 PM12/8/06 4:25:18 PM

Translating Figures on a Grid In the first row of the table below, record the coordinates of each vertex of Figure F.

Slide Figure F five spaces down. Draw it, and record the new coordinates and the rule in the table. Label the new image Figure G.

Slide Figure G three spaces to the right. Draw it and record the new coordinates and the rule in the table. Label the new image Figure H.

Order of columns may vary. Rule

F (1,7) (2,7) (2,5) (1,5) (0,6) (x,y)

G (1,2) (2,2) (2,0) (1,0) (0,1) (x, y ! 5)H (4,2) (5,2) (5,0) (4,0) (3,1) (x " 3, y ! 5)

Which 2 figures look congruent? Explain how you could check to make sure they are congruent.

Figures Q and R. Possible answer: I could trace Figure R and

then put the tracing on top of Figure Q to see if they match.

If they are the same size and shape, they are congruent.

PracticeLesson 2

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Reflecting Figures on a Grid The vertices of a figure are given in the table below. Plot and label each vertex.

Use a straightedge to connect A � B � C � D � E � F � G � A.

Reflect the figure over the dotted, horizontal line. Plot each new vertex, draw the figure, and write its coordinates in the table.

VerticesOriginal Figure

New Figure

A (1,3) (1,5)B (2,4) (2,4)C (4,4) (4,4)D (5,3) (5,5)E (5,1) (5,7)F (3,2) (3,6)G (1,1) (1,7)

Aaron made this map showing some locations in his neighborhood.

Which ordered pair represents the location of the school?

A. (2,3) C. � (4,5)

B. (5,2) D. (5,4)

What is located at (2,3)?

A. � Library C. School

B. Park D. Home

PracticeLesson 3

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Rotating Figures on a GridFigures B, C, and D are rotations of Figure A around (4,4).

Complete the table of coordinates for Figures C and D.

A B C D

(4,6) (2,4) (4,2) (6,4)(3,6) (2,3) (5,2) (6,5)(3,5) (3,3) (5,3) (5,5)(1,5) (3,1) (7,3) (5,7)

(1,4) (4,1) (7,4) (4,7)

(4,4) (4,4) (4,4) (4,4)

Draw and label Figures B and D on the grid.

Does the diagram show a translation, a reflection, or a rotation? If it is a rotation,show the point around which the figure is rotated. If it is a reflection, show the line over which it is reflected. If it is a translation, givedirections to tell how much to add to or subtractfrom each coordinate.

a refl ection

PracticeLesson 4

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More About Transformations List the coordinates of Figure A’s vertices in the table.

Draw any reflection of Figure A and label it B. List the coordinates of its vertices.

Draw a translation of Figure A and label it C. Record its vertices.

Rotate Figure A and label the result D. Record D’s vertices.

A B C D

(5,5)(5,8)(8,9)(7,7)(7,5)

Which group shows all the numbers that are common factors of 24 and 30?

A. ! 1, 2, 3, 6B. 1, 2, 3, 5, 6C. 1, 2, 3, 4, 6, 8, 12, 24D. 1, 2, 3, 5, 6, 10, 15, 30

Which is the greatest common factor of 24 and 30?

A. 3B. ! 6C. 24D. 30

Figures B, C, and D will vary.

PracticeLesson 5

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Graphing with Negative NumbersFor each coordinate pair, write the letter that labels the point.

Rebecca had only a 1 _ 4 -measuring cup to measure flour for a muffin recipe. She filled the measuring cup six times. How much flour did she measure?

A. 1 1 _ 4 cups C. 1 3 _ 4 cups

B. ! 1 1 _ 2 cups D. 2 1 _ 2 cups

Jake ordered a pizza for lunch and ate 3 _ 8 of the pizza. He took the rest of the pizza home. What part of the pizza did he take home?

A. 3 _ 8 C. ! 5 _

8

B. 1 _ 2 D. 3 _

4

(4,3) C

(5,0) D

(3,4) B

(3,!4) F

(0,!5) G

(4,!3) E

(!4,!3) I

(!5,0) J

(!3,!4) H

(!3,4) L

(0,5) A

(!4,3) K

PracticeLesson 6

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Draw the following segments.

Which fraction is closest to 1 __ 2 ? Explain how you decided.

5 _ 9 ; possible answer: I can think about

decimals for the fractions; 5 _ 9 ! 0.5555 which is closer to 0.5

than the decimals for the other three fractions.

Moving on a Coordinate Grid

("4,6) to ("3,5)

("4,2) to ("2,1)

("4,"2) to ("2,"2)

("4,"5) to ("2,"5)

(4,6) to (4,3)

(4,1) to (4,0)

(3,"2) to (3,"4)

(2,6) to (2,3)

(2,2) to (2,"1)

("1,6) to (1,6)

("1,2) to (1,2)

("1,"2) to (1,"2)

("2,0) to ("4,"1)

(0,"1) to (0,2)

("1,6) to ("1,3)

("4,2) to ("4,"1)

("3,5) to ("2,6)

("1,"1) to (1,"1)

(2,2) to (4,1)

("1,3) to (1,3)

(0,"2) to (0,"5)

("3,3) to ("3,5)

(4,0) to (2,"1)

("3,"2) to ("3,"5)

(2,3) to (4,3)

("2,0) to ("2,1)

(1,6) to (1,3)

Draw a big dot at (3,"5)

PracticeLesson 7

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Graphing DataA group of students wondered ABOUT how many raisins are in a small box. They counted the number of raisins in each of 17 boxes. Here are the numbers they found:

37, 33, 35, 36, 38, 34, 35, 38, 35, 37, 35, 33, 35, 35, 36, 37, 40.

Make a line plot for the data.

31 37 41 423330 32 3934 38 403635

What is the greatest (maximum) number of raisins found? 40

What is the least (minimum) number of raisins found? 33

What is the difference (range) between the greatestnumber of raisins and the least number of raisins in a box? 7

What is the number of raisins that was found the most often (the mode)? 35

Summer camp runs for ten weeks. The campers are served 3 meals a day. How many meals are served in the ten weeks?

A. 30 meals C. ! 210 mealsB. 150 meals D. 250 meals

Each week at summer camp costs $79 per person. If 27 girls and 23 boys attend the camp, what is the total cost?

A. $1,817 C. $3,590B. $2,133 D. ! $3,950

PracticeLesson 8

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What Is Typical?

1 30 2 4 65

Use the line plot to decide if the statements are true or false.

The title might be “Ages of Fifth Grade Boys’ Mothers.” false

The range is 6. true

Both the mode and median are 2. true

The title could be “Number of Servings of Fruit and Vegetables in a Day.” true

Derek drew a triangle on a grid. The vertices of the triangle are (1,2), (3,2), and (2,!1). If he translates the triangle 2 spaces to the left and 3 spaces down, what will the coordinates of the new triangle be? Explain how you know.

(!1,!1), (1,!1), (0,!4); Possible explanation: If the triangle

moves 2 spaces to the left and 3 spaces down, each vertex

will move the same.

PracticeLesson 9

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Another Way of Describing What’s TypicalAnswer as many questions as you can. If the graph or table does not provide a way to figure out the answer, write “Cannot tell.”

Morgan made a graph to show the ages of children in her neighborhood that were in kindergarten through fifth grade.

CHILDREN’S AGES5 76 118 109

• How many children does the graph represent? 10

• What is the median age? 7

• How many people live in the U.S.?

cannot tell

• What is the median population of the 9 most populous U.S. cities?

1,479,339

Which statement is NOT true for this data set?

10, 12, 14, 8, 14

A. The mode is greater than the minimum.B. ! The median is greater than the mode.C. The mode is the same as the maximum.D. The range is 6.

PracticeLesson 10

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Reading Graphs and TablesBob took a survey to find out which pets some first graders preferred.

0 5 10 15 20 25

Dog

Cat

Bird

Fish

Other

Which choice is the mode? dog

How many 1st graders were surveyed? 50

How many more people chose cats than birds? 10

PracticeLesson 11

The line plot shows some students’ spelling scores.

How many students had a score of 80 or better?

A. 3 students B. 4 students C. 12 students D. ! 13 students

Which score did 4 students receive?

A. 85 B. ! 90 C. 95 D. 100

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Investigating DecimalsWrite any number that comes between the two given numbers.

10 10.5 11 8.7 8.75 8.8

0.5 0.55 0.6 9.18 9.185 9.19

Write the number that comes halfway between the two given numbers.

Which number could not be a common denominator for fractions with denominators of 6 and 8?

A. 24 C. 96B. ! 12 D. 48

The table shows the prices for tickets to a museum. How much will it cost for a class of 23 students?

Tickets 3 7 11 15

Prices $8.25 $19.25 $30.25 $41.25

A. $57.25 C. ! $63.25B. $62.25 D. $68.50

Answers will vary. Possible answers are given.

PracticeLesson 1

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MNENL07AWK5X_PB_C07_P51-P61_V5.indd P51MNENL07AWK5X_PB_C07_P51-P61_V5.indd P51 12/12/06 3:37:31 PM12/12/06 3:37:31 PM

Comparing and Ordering DecimalsWrite !, ", or # to complete the number sentences.

5.2 ! 5.18 17.04 #

17.040 29.604 "

29.8

63.406 " 63.60 89.8 !

89.088 1.976 "

19.760

360.48 " 360.481 46.55 #

46.550 101.6 !

101.59

Write the numbers in order from least to greatest.

12.34 2.413 42.31 32.41 2.341 4.123 32.24

2.341

2.413

4.123

12.34

32.24

32.41

42.31

Which is not a fraction for the shaded part?

A. 1 __ 3

B. 2 __ 6

C. 3 __ 9

D. ! 3 __ 6

On Sunday, Ben started an exercise program by lifting weights. On Monday, he went for a run. He will run every third day and lift weights every fifth day. On which day of the week will he do both activities together for the first time?

A. Tuesday C. ThursdayB. ! Wednesday D. Friday

PracticeLesson 2

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Large and Small NumbersWrite the numbers in order from greatest to least.

23,540,610 23,450,601 23,450,061 23,456,100

23,540,610

23,456,100

23,450,601

23,450,061

5.5 5.1 5.8 5.21 5.12

5.8

5.5

5.21

5.12

5.1

10.05 10.005 10.500 10.055

10.500

10.055

10.05

10.005

Which fraction matches the shaded part of the sketch?

A. ! 2 ___ 12

C. 2 __ 8

B. 2 ___ 10

D. 2 __ 6

A room has 31 rows of 31 chairs. One row is added, and one chair is removed from each row. Which is the only expression that does not show how many chairs there will be?

A. (31 ! 1) " (31 # 1)B. 31 " 31 # 1C. ! (31 # 1) " 31D. 32 " 30

PracticeLesson 3

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Connecting Decimals to FractionsFill in the fraction notation (above the picture) and decimal notation (below the picture) to match the blocks.

Example

Lucie swam 50 meters in more than 31.5 seconds, but less than 31.6 seconds. Name three different answers for how long she could have taken. Explain your answer.

31.52 seconds, 31.55 seconds, 31.575 seconds; 31.52, 31.55,

and 31.575 are all greater than 31.5 and less than 31.6.

Answers will vary. Possible answers:

PracticeLesson 4

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Connecting Decimals to Other FractionsWrite equivalent fractions and decimals.

1 __ 5 !

2

10 ! 0.2 4 __

5 !

8

10 ! 0.8

1 __ 4 !

25

10 ! 0.25 3 __

4 !

75

100 ! 0.75

1 ___ 20

!

5

100 ! 0.05 3 ___

20 !

15

100 ! 0.15

Write the mixed numbers above the number line and the matching decimals below.

Erika has $15.09. Which could NOT be true?

A. She has 12 whole dollars, 26 tenths of a dollar, and 49 hundredths of a dollar.

B. She has 12 dollars, 26 dimes, and 49 pennies.

C. She has 12 dollars, 10 quarters, 5 dimes, and 9 pennies.

D. ! She has 15 dollars, and 9 tenths of a dollar.

Jackie has 24 markers and 40 pencils to put into bags. Each bag must have the same number of markers and the same number of pencils. What is the greatest number of bags she can fill if she uses all the markers and pencils?

A. 2 C. ! 8B. 3 D. 12

PracticeLesson 5

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Estimating Decimals Using Familiar FractionsMake designs by shading in some of the hundredths. Record the fractions and decimals.

Which point is incorrectly labeled on the number line?

A. ! 1 __ 8 C. 1 1 __

4

B. 1 __ 2 D. 1 3 __

4

A bug is sitting at the point above 1 3 _ 4 on the number line. It starts crawling toward 0 at the rate of 1 _ 4 unit every 10 seconds. How long will it take it to reach 0?

A. 50 secondsB. 1 minute C. ! 1 minute 10 secondsD. 1 minute 20 seconds

100

! 0.

100

! 0.

Answers will vary.

Answers will vary.

PracticeLesson 6

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Estimating Decimals Using RoundingRound each number to the nearest whole number.

42.83 43

160.09 160

109.6 110

Round each number to the nearest tenth.

2.03 2.0

8.75 8.8

16.98 17.0

Round each number to the nearest hundredth.

4.616 4.62

9.002 9.00

12.123 12.12

Ryan bought four items that cost $12.29, $16.45, $1.99, and $9.49.

Which is the best estimate of the amount he paid?

A. between $33 and $35B. between $35 and $37C. between $37 and $39D. ! between $39 and $41

A factory produced 5,712 snacks. Which is the only way the snacks cannot be packaged if they are all used?

A. ! packages of 9 B. packages of 6 C. packages of 3D. packages of 2

PracticeLesson 7

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Adding with DecimalsComplete the number sentences.

6 ! 4.6 " 10.6 5.3 ! 2 " 7.3

6.7 ! 4 " 10.7 5 ! 2.8 " 7.8

6.7 ! 4.6 " 11.3 5.3 ! 2.8

_

8.1

53 ! 28

_

81

0.53 ! 0.28

_

0.81

67 ! 46 " 113

0.67 ! 0.46 " 1.13

4.6 ! 3 " 7.6 46 ! 38 " 84

4 ! 3.8 " 7.8 0.46 ! 0.38

_

0.84

0.046 ! 0.038

_

0.084

0.46 ! 3.8

_

4.26

4.6 ! 3.8 " 8.4

4.6 ! 0.38 " 4.98

Renee’s kitchen floor is a rectangle greater than 60 square feet, but less than 70 square feet in area. Each dimension of the floor is greater than 6 feet and the floor is perfectly tiled with 1-foot square tiles, none of which have been cut. What could the area be? Explain.

Possible answers: 63 sq ft or 64 sq ft; since each dimension

is a whole number greater than 6 feet, and 9 ! 9 is already

more than 70, at least one of the dimensions must be 7 or 8.

7 by 9 and 8 by 8 are the ones that work.

PracticeLesson 8

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Subtracting with DecimalsComplete the number sentences.

6.7 ! 4 " 2.7 5.3 ! 2 " 3.3

6 ! 4.6 " 1.4 5.3 ! 2.8 " 2.5

6.7 ! 4.6 " 2.1 53 ! 28

_

25

0.53 ! 0.28

_

0.25

5 ! 2.8

_

2.2

67 ! 46 " 21

0.67 ! 0.46 " 0.21

4.4 ! 3 " 1.4 8.2 ! 6 " 2.2

4 ! 3.3 " 0.7 8 ! 6.8 " 1.2

4.4 ! 3.3 " 1.1 8.2 ! 6.8

_

1.4

82 ! 68

_

14

0.82 ! 0.68

_

0.14

44 ! 33 " 11

0.44 ! 0.33 " 0.11

Seven of the 56 musicians in the Somer School band are drummers. The same fraction of the Euclid School band are drummers. Euclid has 10 drummers. How many musicians are in the Euclid School band? Explain how you found your answer.80 musicians; Possible answer: 7

__ 56 , or 1

_ 8 , of the musicians in the

Somer School band are drummers. If Euclid School has 10 drummers,

then they must have 80 musicians because 10

__ 80 is equivalent to 1

_ 8 .

PracticeLesson 9

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Adding and Subtracting DecimalsAdd or subtract.

2.5 ! 3.7 " 6.2 25.9 # 17.82

_

8.08

8.16 ! 1.3 " 9.46

5.2 ! 0.85 " 6.05 0.973 ! 3.6458

_

4.6188

12.00 # 2.5 " 9.5

9.9 # 6.09 " 3.81

Estimate. Estimates will vary.

5.007 ! 6.8395 11.8 or 12

17.631 # 5.9 11.7

2.83 # 0.009 2.8

8.025 ! 1.75 10

1.56 ! 1.47 3

35.72 ! 64.082 100

20.85 # 9.999 11

56.987 # 42.9 14

Explain how to determine if 0.087 is equivalent to 0.0807.

PracticeLesson 10

by comparing the digits in the same place-value positions beginning

at the left. The digits in the ones, tenths, and hundreds places are

the same. In the thousandths place, 7 > 0, so 0.087 > 0.0807.

Possible answer:

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Multiplying with DecimalsFirst, circle the best estimate.Then, calculate an exact answer.

6.2 ! 8

_

49.6

closer to

48

4.8

4.6 ! 0.7

_

3.22

closer to

28

2.8

0.53 ! 6

_

3.18

closer to

3

30

2.41 ! 3.3

_

7.953

closer to

60

6

0.36 ! 9

_

3.24

closer to

0.4

4

17.3 ! 0.3

_

5.19

closer to

50

5

29.6 ! 2.1

_

62.16

closer to

6

60

0.67 ! 16.3

_

10.921

closer to

1.2

12

Which of these sentences is true?

A. 1 __ 2 " 0.6

B. ! 0.23 # 1 __ 4

C. 1 __ 3 # 0.3

D. 3 __ 4 " 0.812

Oranges are packed 144 to a crate and apples 96 to a crate. A truck can carry 90 crates of oranges and 60 crates of apples. What is the maximum number of pieces of fruit the truck can carry?

A. 16,820 C. 17,780B. 17,620 D. ! 18,720

PracticeLesson 11

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PracticeLesson 1

Exploring Missing Factors

Complete the puzzles and number sentences. Use one stamp from Group A and one from Group B.

A B

! 20

3 60 24 84

A B

! 2

5 200 10 210

3 ! 28 " 84 5 ! 42 " 210

A B

! 7

6 420 42 462

A B

! 7

10 100 70 170

4 40 28 6814 140 98 238

6 ! 77 " 462

The Haskell family has been driving at 60 miles per hour for three hours. They still have 45 miles to go before arriving at the beach. How many miles is the whole trip to the beach? Explain how you found the answer.

225 miles; to fi nd the number of miles they drove in the

fi rst 3 hours, I multiplied 60 miles by 3: 60 ! 3 " 180.

Then I added the miles to go: 180 # 45 " 225.

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Connecting Multiplication and Division Each of these rectangles should be labeled with its area (inside) and the lengths of its sides. Fill in the missing values.

Fill in the missing numbers.

! 7

9 63

! 7

6 42

9

9 8 1

1 0

11 1 1 0

1 2

12 1 4 4

1 4

10 1 4 0

2 0

13 2 6 0

Use each problem to help you with related problems.

1 212 1 4 4

2 4

6 1 4 4

4 8

6 2 8 8

1 0

11 1 1 0

2 0

11 2 2 0

3 0

11 3 3 0

1 5

22 3 3 0

Fifteen minutes after the time shown on the

clock, Marcie began dinner. She finished dinner at 6:10 P.M. How long did she spend eating dinner?

A. 25 minutes C. ! 35 minutesB. 30 minutes D. 40 minutes

212 2 4

1 012 1 2 0

PracticeLesson 2

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Dividing Using Multiplication and the Area ModelThis time there are nineteen rows. How many squares are there per row? To make your work easier, list some useful multiples of 19 or use multiples of 20 to estimate.

A. 240 C. 350 B. 300 D. ! 420

Shira has less than 500 pennies. She can divide themevenly into 2 piles, 3 piles, 4 piles, 5 piles, 6 piles, or7 piles. How many pennies does she have?

PracticeLesson 3

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Recording the Steps in Division Complete the table of multiples of 27.

! 1 2 4 5 8 10 20 40 50 80

27 27 54 108 135 216 270 540 1,080 1,350 2,160

Complete the area model and division record.

Solve these problems on a separate sheet of paper.

2 327 6 2 1

3 4

27 9 1 8

8 1

27 2, 1 8 7

6 7

27 1, 8 0 9

Hamburgers come in packages of 6, and hamburger buns come in packages of 8. If Shane buys 5 packages of hamburgers and enough buns, what is the least number of buns he will have left? Explain how you know.2 buns; Shane buys 5 ! 6, or 30, hamburgers. Buns come in packages of 8; so the number of buns he buys is a multiple of 8.

32 is the smallest multiple of 8 that is greater than 30.

PracticeLesson 4

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Which of these problems has the greatest quotient? Try to figure this out without actually calculating the quotients.

A. ! 27 9 7 2 C. 27 9 4 5

B. 36 9 7 2 D. 36 9 0 0

Which of these problems has the greatest divisor?

A. 4 0

! 8 0 0

C. 5 0

! 8 0 0

B. 4 1

! 8 2 0

D. ! 3 2

! 8 0 0

PracticeLesson 5

Dividing and Recording Division Efficiently

Complete the table of multiples of 31.

! 1 2 3 4 5 6 7 8 9

31 31 62 93 124 155 186 217 248 279

Complete the area model and division record.

Solve these problems on a separate sheet of paper.

2 131 6 5 1

2 9

31 8 9 9

3 9

31 1, 2 0 9

5 731 1, 7 6 7

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A large cardboard box is sitting on a table. The area of one side of the box is 3 square feet. The height of the box is 3 feet. Mark all statements that could be true.

A. The volume is 9 square feet. C. ! The volume is 4 1 _ 2 cubic feet.B. The box is a cube. D. ! The dimensions of the base

are 3 feet by 1 foot.

PracticeLesson 6

Using Multiplication to Check Division Complete the table of multiples of 47¢, or $0.47. Save work by doubling and adding.

! 1 2 3 4 5 6 7 8 9

47¢ $0.47 $0.94 $1.41 $1.88 $2.35 $2.82 $3.29 $3.76 $4.23

Use the multiples to compute the cost of different numbers of items that cost 47¢ each.

40 at 47¢ each $ 18.80 30 at 47¢ each $ 14.10

5 at 47¢ each $ 2.35 6 at 47¢ each $ 2.82

45 at 47¢ each $21.15 36 at 47¢ each $16.92

20 at 47¢ each $ 9.40 90 at 47¢ each $42.30

6 at 47¢ each $ 2.82 9 at 47¢ each $ 4.23

26 at 47¢ each $ 12.22 99 at 47¢ each $46.53

How many 47¢ items can be bought for the three amounts shown below? Divide to find out. If you need more room, do the work on a separate sheet of paper, and write the summaries here.

1 7

$0.47 $ 7. 9 9

3 6

$0.47 $ 1 6. 9 2

4 5

$0.47 $ 2 1. 1 5

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Investigating RemaindersLook at the example to see how it is labeled. Fill in the missing numbers and number sentences for the other area models.

EXAMPLE

25 ! 4 " 6 1 __ 4

4 # 6 1 __ 4 " 25

16 ! 3 " 5 1 _ 3

3 #

5 1 _ 3 " 16

14 ! 3 " 4 2 _ 3

3 # 4 2 _ 3 " 14

16 ! 5 " 3 1 _ 5

5 # 3 1 _ 5 " 16

33 ! 5 " 6 3 _ 5

5 # 6 3 _ 5 " 33

11 ! 4 " 2 3 _ 4

4 # 2 3 _ 4 " 11

Ben bought 4 packets of stamps. Each packet had 100 stamps in it. He mounted the same number of stamps on each of 5 pages. How many stamps did he mount on each page? Explain how you know.

80 stamps; Ben bought 4 # 100, or 400, stamps in all. To fi nd

the number on each page, I divided 400 by 5. 400 ! 5 " 80

PracticeLesson 7

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Interpreting Remainders in Word ProblemsRead the stories and solve the problems by drawing a diagram or writing a record. What do you do about the remainder—ignore it or include it as a fraction or decimal?

You would not believe how hungry Lydia, Arthur, Ray, andKaty are today! If they share their 5 small pizzas equally,how much pizza will each get?

1 1

__ 4 pizzas

What should you do with the remainder? show it as a fraction

Graham is unloading a box of twelve dozen Many methods are possible. One such method is shown.paperback books onto a bookshelf. Each shelf can

hold 25 books. How many shelves will these books completely fill? 5 shelves

25 25 25

25 25 19

What should you do with the remainder? ignore it

If there are 7 yards of ribbon in a full roll, how many feet of ribbon are on 5 full rolls?

A. 35 C. ! 105B. 12 D. 21

Which would give the best estimate for 77 ! 93?

A. 80 ! 100 C. 70 ! 100B. 70 ! 90 D. ! 80 ! 90

Many methods are possible. One such method is shown.

PracticeLesson 8

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Another Option for Interpreting RemaindersSolve. Decide what to do when there is a remainder—ignore it (round down), include it as a fraction or a decimal, or round up. Show your work.

Some mini-vans can carry 7 people. How many 7-person mini-vans will be needed to take 18 people to a museum?

27 1 8 ! 1 4

4Solution: 3

mini-vans

What should you do about the remainder? round up

There are 350 seats in the auditorium where the fifth-grade graduation will be held. If each of the 58 fifth graders gets an equal number of tickets, how many will each fifth grader get?

658 3 5 0

! 3 4 8 2Solution: 6

tickets

What should you do about the remainder? ignore it

A class of fifth graders sold homemade cheese pizzasas a fundraiser. They sold 20 pizzas and made $165.If the price of each pizza was the same, how muchdid each pizza cost?

8 5 __ 20 " 8 1 _ 4 20 1 6 5

! 1 6 0 5Solution: $8.25

What should you do about the remainder? include it as a decimal

PracticeLesson 9

Alvin had fewer than 100 pennies. He found he could divide them evenly into 2 piles, 3 piles, 4 piles, 5 piles, or 6 piles. How many pennies did he have? Explain how you know.

60 pennies; Possible explanation: 60 is the only number

less than 100 that is divisible by 2, 3, 4, 5, and 6.

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PracticeLesson 1

Investigating AnglesTell whether each marked angle looks acute, right, or obtuse.

acute

acute

obtuse

right

right

obtuse

14 ! 288 " 4,032. Which of the following is false?

A. 1.4 ! 2.88 " 4.032B. 14 ! 28.8 " 403.2C. 1.4 ! 28.8 " 40.32D. ! 0.14 ! 0.288 " 0.4032

26 ! 317 " 8,242. Which of the following is false?

A. 2.6 ! 3.17 " 8.242B. ! 0.26 ! 0.317 " 0.8242C. 2.6 ! 31.7 " 82.42D. 26 ! 31.7 " 824.2

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In the theater, there are 20 rows of seats with 15 seats in each row. If 155 adults and 76 children attend a performance at the theater, how many seats will be empty? Explain how you know.

69 seats; I multiplied to fi nd the total number of seats:

20 ! 15 " 300. Then I subtracted the number of people

who attended the performance: 300 # (155 $ 76) " 69

Classifying Angles and Triangles Classify each triangle by the lengths of its sides or measures of its angles. Be as specific as possible. The measurements tell how the triangles should be drawn, but the pictures are not correct. Do not judge anything by the way it looks.

A B 5 5

8

C 6.3 6.3

6.3

D 18 8

18

E 3 5

4

F 1.9 1.9

1.2

G 60°

30° 90°

H 30°

30° 120°

I

110°

50° 20°

J 65°

45° 70°

K 40°

60° 80°

L

160°

10° 10°

PracticeLesson 2

Scalene Isosceles Equilateral

A XB XC XD XE XF X

Acute Obtuse Right

G XH XI XJ XK XL X

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PracticeLesson 3

Constructing TrianglesUse a straightedge to draw a line to make the angles.

1 2 4 5

10 170

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90 10080

11070 120

60 13050 14040 15030

16020

17010

180

180

AO

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90 10080

11070 120

60 13050 14040 15030

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180

180BO

measure of ! A: 110! measure of ! B: 55!

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60 13050 14040 15030

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90 10080

11070 120

60 13050 14040 15030

16020

17010

180

180DO

measure of ! C: 20! measure of ! D: 90!

Which is a reasonable way to approximate the value of 42.319 " 19.8?

A. ! 42 " 20B. (42 " 19) # (319 " 8)C. 42,319 " 198D. 423 " 198

Which is a reasonable estimatefor 13.079 $ 4.82?

A. 130B. ! 65C. 480D. 18

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Constructing Similar TrianglesUse a straightedge to draw a line to make the angles.

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90 10080

11070 120

60 13050 14040 15030

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180

180AO

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90 10080

11070 120

60 13050 14040 15030

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180

180BO

measure of ! A: 100! measure of ! B: 45!

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11070 120

60 13050 14040 15030

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17010

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90 10080

11070 120

60 13050 14040 15030

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17010

180

180DO

measure of ! C: 90! measure of ! D: 30!

Record the angle measures.

PracticeLesson 4

Maria drew a right triangle. She said one angle was right, one angle was acute, and one angle was obtuse.Is this possible? Explain how you know.

No; Possible explanation: The sum of the

measures of a triangle is 180!. The right angle

The right angle is 90!, so that leaves only 90!for the other two angles to share. The other two angles must both be acute.

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PracticeLesson 5

Angles Formed by Intersecting LinesUse your knowledge of straight angles and opposite angles to find the missing angle measures. (No protractors, please!)

Read, but do NOT solve this problem.

There are 36 students to seat in the library. If each table holds 8 students, what is the fewest number of tables needed for all to sit?

If there is a remainder, what should you do about it? Explain.

You should round the quotient up to the next whole

number. It does make sense to include the remainder as

a fraction or decimal and if you ignore the remainder,

you will not have enough room for all students.

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PracticeLesson 6

Angles Formed by a Line Intersecting Parallel LinesWithout using a protractor, use your knowledge about Zs, straight angles, and opposite angles to figure out the missing angle measures. Do you see other Zs?

r ! s

i ! j

l ! m

o ! p

Which of the following is NOT true?

A. x ! y ! z " 180#

B. If x " 90#, then y ! z " 90#

C. If x " 90#, then y " 90# $ zD. ! x ! y ! z " 90#

y x

z

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In !RST, the measure of "S is 40! and the measure of "T is 50!. What is the measure of "R? Explain how you know.

90°; The sum of the measures of the angles

of a triangle is 180°. So, the measure of ! R is

180° ! (40° " 50°) # 180° ! 90° # 90°.

PracticeLesson 7

Comparing and Classifying QuadrilateralsCongruent sides and angles have the same markings.Circle ALL the names that match each quadrilateral.

rectangle parallelogram square rhombus

trapezoid rhombus rectangle parallelogram

rhombus

rectangle parallelogram rhombus

square parallelogram rectangle

square

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A triangle with sides measuring 3 cm, 5 cm, and 3 cm must be:

A. ! isoscelesB. acuteC. scaleneD. equilateral

A certain quadrilateral has only 2 lines of symmetry, 2 pairs of parallel sides, and 4 right angles. It must be a:

A. squareB. rhombusC. ! rectangleD. trapezoid

PracticeLesson 8

Investigating QuadrilateralsWithout using a protractor, use your knowledge about Zs, straight angles, opposite angles, and the sum of the measures of the angles in triangles and quadrilaterals to find the missing measurements.

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PracticeLesson 1

Length and PerimeterMeasure the sides of each figure to the nearest centimeter. Record the perimeter in cm.

A

BC

D E

G

F

AB 4 cm BC 4 cm EF 3 cm FG 5 cm

CD 4 cm DA 4 cm GE 6 cm

Perimeter 16 cm Perimeter 14 cm

P

O

MN

L

HI 7 cm I J 3 cm LM 3 cm MN 5 cm

JK 3 cm KH 2 cm NO 4 cm OP 3 cm

Perimeter 15 cm PL 3 cm Perimeter 18 cm

JK

I

H

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Perimeter FormulasFind the perimeter of each parallelogram.

29 29

99

33

Perimeter 116 units

Perimeter 264 units

339

112

99

99

Perimeter 902 units

Perimeter 396 units

PracticeLesson 2

Taylor used 64 feet of fencing to enclose a square pen for his dog. How long is each side of the pen? Explain how you know.

16 ft; Possible explanation: The four sides of a square all

have the same length. So, each side is 64 ! 4, or 16 feet.

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PracticeLesson 3

Area of ParallelogramsFind the area of each parallelogram.Record the area in square centimeters (sq cm).

Base 3 cm Base 3 cm

Height 3 cm Height 2 cm

Area 9 sq cm Area 6 sq cm

Base 2 cm Base 3 cm


Area 2 sq cm

Area 9 sq cm

Solve the problem.

A rectangular field measures 12 feet by 8 feet. A farmer needs to know the area in order to buy seed. What is the area? 96 sq ft

The area of the large shaded rectangle is 1. What fractioncan you write for the area of the shaded triangle? Explain.

Area is 1

1 __ 12 ; Possible explanation: If you divide the rectangle into

small triangles like the shaded one, there will be 12 smalltriangles, so each triangle is 1 __ 12 of the rectangle.

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PracticeLesson 4

Measuring to Find Areas of ParallelogramsCut out the ruler on the right, if needed. Measure the sides and height of each parallelogram to the nearest cm. Record the area and perimeter for each figure.

Base BC 5 cm

Height 4 cm

Area 20 sq cm

Perimeter 20 cm

Base EF 6 cm

Height 4 cm

Area 24 sq cm

Perimeter 22 sq cm

E

H

G

F

A

C

B

D

Mario noticed that if he put 4 stickers on each page of his sticker album, he would have 2 left over. If he put 3 on each page, there would also be 2 left over. Mario had more than 10 stickers, but fewer than 30 stickers. How many stickers did he have?

A. 12 or 26 stickers C. 14 or 21 stickersB. ! 14 or 26 stickers D. 13 or 25 stickers

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Mr. Howe’s rectangular garden has an area of 24 square feet. One of these could NOT be the length of the fence around the garden. Circle it. Explain how you know.

16 feet 20 feet 22 feet 28 feet

Possible explanation: For 20 feet of fencing it would be 6 ftby 4 ft, for 22 ft: 8 ft by 3 ft, and for 28 ft: 12 ft by 2 ft. So, 16 ft is the one that is not possible.

PracticeLesson 5

Area of Triangles and TrapezoidsFind the area and perimeter of each figure usingthe (approximate) measures given.

CB

A F

E

D

Base BC 2 cm Base DF 2 cm


Side AB 2 cm Side DE 3 cm

Side AC 3 cm Side EF 4 cm



Base GJ 2 cm Base KL 2 cm

Base HI 4 cm Base MN 4 cm


Side GH 2 cm Side KN 4 cm

Side J I 3 cm Side LM 3 cm



M

N

L

K

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Area and Perimeter of Other PolygonsUse measurements to the nearest centimeter to find the perimeter. Use a ruler to draw lines showing how you would split each polygon into triangles to find its area.

Perimeter 25 cm

Perimeter

17 cm

PracticeLesson 6

Answers may vary.

Which measurement is NOT needed to find the area ofthe trapezoid?

y

z

w

x

A. w C. ! zB. y D. x

Which polygon does NOT have at least 2 lines of symmetry?

A. square C. ! right triangleB. equilateral triangle D. rectangle

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Adding and Subtracting Fractions with Like DenominatorsShade the bars to show the sums. Complete the number sentences. Change improper fractions to mixed numbers.

Use the pictures to complete the number sentences.

PracticeLesson 1

The base of a parallelogram is two times its height. If the base is 12 centimeters, what is the area? Explain.

72 sq cm; the formula for the area of a parallelogram is

A ! b " h. The height is 6 cm, so A ! 12 " 6 ! 72 sq cm.

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More Adding and Subtracting Fractions with Like DenominatorsWrite fractions to complete the number sentences.

Round the numbers 49.03 and 29.95 to the nearest tenth. What is the difference between the rounded numbers?

A. 19.1 C. ! 19B. 20.9 D. 20

Which number is NOT equivalentto 4 8 _ 9 ?

A. 44 _ 9 C. 4 16 _ 18

B. ! 3 16 _ 18 D. 88 _ 18

PracticeLesson 2

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Stories About Adding and Subtracting FractionsYou may use the picture to help you solve both Problems 1 and 2.

Solve both problems and write number sentences to match the solutions.

Felicia spent 6 _ 12 of the daylight hours in school and 4 _ 12 of them on homework, soccer, and household chores. What fraction of the daylight hours might she use in any way she pleases?

2 _ 12

Number sentence(s):

6 _ 12 ! 4 _ 12 " 10 _ 12 10 _ 12 ! 2 _ 12 " 12 _ 12

Erik set out 7 _ 12 of a dozen donuts as snacks for his friends. What fraction of the dozen did he leave for later?

5 _ 12

Number sentence(s):

12 _ 12 # 7 _ 12 " 5 _ 12

Which is NOT the same as 2 _ 6 ! 2 _ 6 ?

A. ! 4 _ 12 C. 4 _ 6

B. 1 _ 3 ! 1 _ 3 D. 2 _ 3

Which number would NOT be a common denominator for 6 and 9?

A. ! 12 C. 18B. 54 D. 36

PracticeLesson 3

Number sentences will vary.

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Adding and Subtracting Unlike ThingsConversion Key

1 lb ! 16 oz 1 hr ! 60 min 1 L ! 1,000 mL 1 m ! 100 cm

1 yd ! 3 ft 1 min ! 60 sec 1 km ! 1,000 m 1 cm ! 10 mm

Complete the number sentences using the conversion key above.

8 yd " 15 ft ! 13 yd 3 lb # 14 oz ! 34 oz

4 m " 12 cm ! 412 cm 2 hr # 80 min ! 40 min

144 in. # 2 yd ! 72 in. 12 m " 4,000 cm ! 5,200 cm

3 hr " 120 min ! 300 min 3,000 mL # 1 L ! 2 L

Jewell has 40 feet of fencing to put around a garden. What are the dimensions of the garden with the largest possible area? Explain.

10 ft ! 10 ft; Possible explanation: the largest area will be

a square. If the total length of fencing is 40 ft, then each

side of the square will be 40 ft " 4 # 10 ft.

PracticeLesson 4

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Adding and Subtracting Fractions with Unlike DenominatorsAdd or subtract fractions of an hour and find the number of minutes.

1 _ 2 of an hour ! 30 min, or 30 _ 60 of an hour




PracticeLesson 5

Josie has a rectangular piece of paper that is 8 inches by 10 inches. She cuts the rectangle into two congruent triangles. What is the area of each triangle? Explain.

40 sq in.; Possible explanation: the area of the rectangle is

8 in. ! 10 in., or 80 sq in. So, the area of each triangle

is 80 sq in. " 2, or 40 sq in.

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PracticeLesson 6

The sum of 3 _ 5 ! 2 _ 3 is . . .

A. less than 1 _ 2 C. ! more than 1

B. 5 _ 8 D. 6 _ 15

Which of the following is NOT equal to 1 _ 2 ?

A. 1 _ 3 ! 1 _ 6 C. 2 _ 7 ! 3 _ 14

B. 7 _ 10 " 1 _ 5 D. ! 3 _ 4 " 1 _ 8

Stories with Fractions

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Using Area to Multiply FractionsFill in the blanks and find the shaded area to multiply the fractions.

Add.

PracticeLesson 7

Trey says that all parallelograms are rectangles. Do you agree? Explain.

No; Possible explanation: not all parallelograms have four

right angles. Some parallelograms are rectangles.

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Using Other Models to Multiply FractionsUse the dot sketches to complete the sentences.

PracticeLesson 8

1 square meter has been divided into 12 equal parts as shown. Which of the following is NOT equal to the area of the shaded rectangle?

A. 1 _ 2 sq m C. ! 17 _ 12 sq m

B. 3 _ 4 m ! 2 _ 3 sq m D. 6 _ 12 sq m

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Fractions of Quantities The input is 10 cents. Write the outputs (the number of cents) in the white boxes.

Complete each sentence.

or any equivalent fraction or any equivalent fraction

Sage divided 48,288 by 48 and got a quotient of 106. She was worried that she might have made a mistake. All of these are reasonable ways to check her answer EXCEPT:

A. Use a calculator to multiply 48 ! 106 .B. Round 48 to 50 and 106 to 100, multiply 50 ! 100, and

compare the product to 48,288.C. Multiply 48 ! 100 and compare it to the dividend 48,288.D. ! Use a calculator to multiply 48 ! 48,288.

PracticeLesson 9

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Stories About Multiplying Fractions Judy wanted to plant a flower garden in the corner of her front yard. She told her parents that she would need a section that was one-sixth by one-fourth of the front yard.

What fraction of the front yard did she need for her flower garden?

1 _ 24

Complete the number sentences.

Use another sheet of paper to make sketches if you wish.

Mackenzie wants to lay floor tile in a kitchen that measures 12 ft by14 ft and a hallway that measures 4 ft by 12 ft. What is the total areato be covered? Explain.

216 sq ft; Possible explanation: I found the area of the

kitchen: 12 ft ! 14 ft " 168 sq ft, and the area of the

hallway: 4 ft ! 12 ft " 48 sq ft; and then found the

sum of the two areas : 168 sq ft # 48 sq ft " 216 sq ft.

PracticeLesson 10

Show how you got your answer.

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PracticeLesson 1

Transforming Two-Dimensional Nets into Three-Dimensional Figures

Cut out, fold, and tape each net to make three congruent pyramids.

Try to fit the three pyramids together to make a cube. It can be done. Try to figure out how!

If the volume of the cube is about3 cu in., what is the volume of oneof the pyramids?

1 cu in.

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PracticeLesson 2

Describing Three-Dimensional Figures Cut out the net and assemble the three-dimensional figure.

Complete the chart and the sentences. F ! V " 11

F ! V # E "

2

Faces 5

Vertices 6

Edges 9

Write two different prime numbers. Explain how you know the numbers are prime.

Responses will vary. Possible numbers: 3 and 5; a prime

number has only two factors: 1 and itself.

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PracticeLesson 3

Sorting Three-Dimensional Figures Cut out each net and fold it along the dotted lines to make a three-dimensional figure.

Fit these two figures together (by matching two congruent faces together, one from each figure) to make a new figure.

See what new figures you can make.

How many ways can you combine these to make a prism? How many ways can you combine them to make a pyramid?

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PracticeLesson 4

Volume of Rectangular PrismsFind the area of the base and the volume of each of these rectangular prisms built out of centimeter cubes.

Area Area Areaof base: 6 sq cm of base: 8 sq cm of base: 9 sq cm

Volume: 24 cu cm Volume: 16 cu cm Volume: 27 cu cm

Area Area Areaof base: 10 sq cm of base: 12 sq cm of base: 5 sq cm

Volume: 10 cu cm Volume: 24 cu cm Volume: 15 cu cm

! 1 cu cm

Which of the following is NOT true about this rectangular prism?

A. It has more vertices than faces.B. It has 3 pairs of parallel faces. C. ! Its volume is 8 cubic units.D. It has 2 congruent, square

faces.

What is the area of the square base?

A. 2 sq units B. ! 4 sq units C. 8 sq unitsD. 16 sq units

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PracticeLesson 5

Volume of PrismsEach diagram shows the base of a triangular prism. Use the dimensions to compute the volume.

Height of Prism: 5 Height of Prism: 4

Volume: 30 cu units Volume: 48 cu units

Height of Prism: 5

Height of Prism: 8

Volume: 48 cu units Volume: 22 1 _ 2 cu units

What is the area of the base?

A. ! 12 sq cm B. 12 cu cm C. 24 sq cmD. 24 cu cm

What is the volume of the prism?

A. 240 sq cmB. 240 cu cm C. 120 sq cmD. ! 120 cu cm

The height of a triangular prism is 10 cm. Its triangular base has a height of 4 cm and a length of 6 cm.

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PracticeLesson 6

Area of NetsThe answers are given. Write questions to match.

Answers Questions

Find the height of this two-dimensional figure and the length of its base, and then multiply those two numbers.

How do you fi nd the area of a parallelogram?

Find the areas of all the faces and then add them up.

How do you fi nd the total area of a net?

orHow do you fi nd the

surface area of a solid?

It is a three-dimensional figure with a base that could be any polygon. All the other faces are triangles that meet at a common vertex.

What is a pyramid?

Find the height of this three-dimensional figure, and find the length and width of the rectangular base. Multiply those three numbers.

How do you fi nd the volume of a rectangular prism?

Measure the base and height of this two-dimensional figure, multiply those two numbers, and then take half the result.

How do you fi nd the area of a triangle?

How many edges does a rectangular prism have? Explain what an edge of a prism is.

12 edges; An edge of a prism is where two faces of

the prism meet.

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PracticeLesson 7

Surface Area of PolyhedraPuzzle it out.

I am a rectangular prism. My volume is 30 cu cm. My shorter dimensions are 2 cm and 3 cm.

What is my longest dimension? 5 cm

I am a triangle. A parallelogram whose base and height are the same as mine has an area of 9 sq in.

What is my area? 4 1 _ 2 sq in.

I am a triangular prism. My volume is 24 cu cm. My surface area is 60 sq cm. My height is 4 cm. I am cut into two congruent, triangular prisms, each 2 cm high.

What is the volume of each? 12 cu cm

I am a trapezoid. My area is 5 sq cm. The lengths of my bases are 1 cm and 3 cm.

What is my height? 2 1 _ 2 cm

Workspace

Mabel drew this trapezoid.

Which two line segments appear to be parallel?

A. a and b C. a and dB. a and c D. ! b and d

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PracticeLesson 8

Comparing Volume and Surface AreaUse Activity Master 110: Net J to help you complete this page.

What is the area of Net J? 24 sq in.

Explain why the surface area of the three-dimensional figure you make from this net should be the same as the area of the net.

The surface is made from the net. The whole net is used and no more

paper is used.

How many faces does the net have? 10

Explain why the number of faces on the three-dimensional figure will be the same as the number of faces on the net.All the faces on the net become faces on the three-dimensional fi gure.

How many edges does the net have? 23

Explain why the number of edges on the three-dimensional figure will not be the same as the number of edges on the net.Some edges on the net get taped together to make a single edge

of the three-dimensional fi gure, so there are fewer edges on the

three-dimensional fi gure.

How many vertices are on the net? 14

Explain why the number of vertices on the three-dimensional figure will not be the same as the number of vertices on the net.When the net is taped, some of its vertices end up in the same place on

the three-dimensional fi gure, so they become one vertex instead of two

or more. The three-dimensional fi gure has fewer vertices.

NOTE: You can cut out the net and build the three-dimensional figure to help you answer the questions above.

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PracticeLesson 1

Introducing MobilesWrite yes or no at each arm of the mobile to show if it is balanced. Write the total weights.

5 5 6 4

yes

yes no

Total Weight: 20

6 6 4 4

4

yes

yes

no

Total Weight: 24

Which group shows equivalent numbers?

A. 5 __ 2 , 2 5 ___

10 , 2 1 __

2 , 2.05 C. 1 2 __

3 , 1 6 __

9 , 1.23, 15 ___

9

B. ! 3 1 __ 5 , 3 5 ___

25 , 3.2, 16 ___

5 D. 4.025, 4 25 ____

100 , 17 ___

4 , 4 1 __

4

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PracticeLesson 2

Balancing MobilesMartina makes mobiles that balance perfectly! Find the weights of each shape in these mobiles so they all balance.

Total Weight: 12

3 6

4 2

Total Weight: 16

15 3

Total Weight: 40

Suki earns the same amount of money (x) each weekday for doing chores. If she does her chores every weekday without forgetting, she gets an extra amount (y) when she gets paid. Which expression shows how much money she can make in a week?

A. 7x ! 7yB. 7x ! yC. 5x " yD. ! 5x ! y

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Equations for MobilesCircle equations that agree with each mobile. Then write the weight of each shape.

Total Weight: 24

Total Weight: 40

2T ! S 2C ! T 2S ! H " C H " C ! S

3C ! S 2T " S ! 3C S ! 2T " H 2C " 2H ! 2S

! 3

! 2

! 5

! 3

! 6

! 2

! 1

PracticeLesson 3

Which point is incorrectly labeled on the number line? What is a correct label for the point? Explain how you know.

0.25 2.500

310 2

5__3

6__9

6 _ 9 is incorrectly labeled. 4 _ 3 is a correct label. The segment between

1 and 2 is divided into 3 equal parts; the fi rst dot after 1 is at 1 1 _ 3 or 4

_ 3 .

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PracticeLesson 4

Balance PuzzlesSolve these balance puzzles.

Write an equation for this puzzle. Use t for triangle, c for circle, and s for square.

2s ! c " s ! 3t

Which shows the prime factorization product of the prime factors of 80?

A. 2 ! 2 ! 4 ! 5 C. 2 ! 2 ! 2 ! 5B. 2 ! 2 ! 3 ! 5 D. ! 2 ! 2 ! 2 ! 2 ! 5

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Number TricksMaxie invented this number trick.

Complete the chart, picking a starting number for yourself.

Words Diagram Shorthand Number

Pick a number. N

Double it. 2N

Add 7. 2N ! 7

Multiply by 3. 6N ! 21

Subtract 11. 6N ! 10

Divide by 2. 3N ! 5

Barry said his final result was 26. Find his starting number, and explain how you found it.

His starting number was 7. Explanations may vary.

Possible explanation: I worked backward: 26 " 2 # 52;

52 ! 11 # 63; 63 $ 3 # 21; 21 % 7 # 14; 14 $ 2 # 7

Carlos practices piano for 35 minutes every day. How much time will he spend practicing in the 31 days of May? Explain how you found your answer.

1,085 minutes; I multiplied 35 " 31. 35 " 30 # 1,050

1,050 ! 35 # 1,085

PracticeLesson 5

Answers in this column

will vary.

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Making DiagramsMatch the situation to the diagram that illustrates it.

Steven has stilts that make him a ! 100

sseem 4 feet taller than he really is!

Boxes of Lovitt’s Lemon Cookies b !

4

shave four rows of cookies in them.There are 100 cookies in a box.

A football field is 100 yards long c ! 100

sand many yards wide.

Pavarti walked 100 yards along d ! a straight road, stopped to peta cat, then walked a little further.

Draw a diagram that illustrates this situation.

Karl likes to bowl. He remembers l

42that a lane is 42 inches wide, but he forgets how long the lane is.

PracticeLesson 6

Edmund is 5 years older than Felicia and 6 years older than Gil. If E represents Edmund’s age, F represents Felicia’s age, and G represents Gil’s age, which equation is NOT true?

A. E ! G " 6 C. G ! E # 6B. F ! G " 1 D. ! F ! E " 5

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Equations for Stories

Jackie went to the pet store to look at the iguanas and birds.

She noticed the animals had 24 feet all together. Complete the table to show the possible combinations of birds and iguanas.

Birds 0 2 4 6 8 10 12

Iguanas 6 5 4 3 2 1 0

If there are also 18 eyes, how many of each animal was there? 6 birds and 3 iguanas

On another day, there were B birds and I iguanas.

Write an equation that gives the number of eyes, E, for the birds and iguanas. E ! 2B " 2I or E ! 2(B " I)

Write an equation that gives the number of feet, F. F ! 2B " 4I or F ! 2(B " 2I)

This rectangular prism is made of 1 cm cubes. What is its volume? Explain how you found the volume.

120 cu cm; I multiplied the length

(6 cm) # the width (5 cm) # height

(4 cm). 6 cm # 5 cm # 4 cm !

120 cu cm

PracticeLesson 7

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MNENL07AWK5X_PB_C13_103-109_V5.indd P109 1/4/07 3:34:25 PM

Conducting a Probability ExperimentThe table shows the results of spinning an unequal color spinner.

What are the possible outcomes? green, blue, red, yellow

How many spins were made? 25

Based on this experiment, write fractions to describe the probability of spinning:

Green 6 _ 25

Blue 8 _ 25

Red 4 _ 25

Yellow 7 _ 25

Write a fraction (based on these results) to show the probability of spinning blue OR red.

12 _ 25

The formula, or rule, Volume ! 1 _ 2 " (base length 1 # base length 2) " height describes the process for finding:

A. the volume of a prism C. ! the area of a trapezoid

B. the area of a triangle D. the volume of a pyramid

What is the area of this triangle

A. ! 6 sq m C. 12 sq m B. 7.5 sq m D. 15 sq m

PracticeLesson 1

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Finding ProbabilitiesLena and four of her friends each drew a card at random from this deck 20 times. The table shows their results.

Results of Card DrawTRIAL 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

Card Number 10 6 8 12 20 16 14 20 2 8 6 16 20 14 10 4 4 12 10 2

Card Number 18 4 8 2 10 14 8 14 16 12 2 18 16 20 4 15 14 12 2 10

Card Number 4 10 6 12 16 20 2 6 20 14 20 8 18 12 8 18 16 10 4 12

Card Number 14 12 2 20 10 2 18 6 18 10 18 4 12 14 4 8 16 6 18 8

Card Number 6 2 12 8 8 16 14 12 14 4 20 6 20 18 8 10 18 4 16 6

List 4 possible events. Then use the table above to write a fraction that describes the probability of eachof the events.

Event Experimental Probability

Even number _ 100 100

PracticeLesson 2

If you draw a card from the deck above, what is the probability of drawing a card where the sum of the digits is odd? Explain how you know.

1 _ 2 ; Possible explanation: The sum of the digits on fi ve of

the ten cards is odd, so the probability is 5 _ 10 , or 1 _ 2 .

Answers will vary.

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PracticeLesson 3

Sampling ExperimentsFive people performed a

Andrea Bobby Carrie David Elizabeth

3 5 3 4 0

9 10 11 7 12

8 5 6 9 8

sampling experiment. Each one drew a card from a bag, recorded the figure that was drawn on the card, put the card back in the bag, and drew again.

This table shows the data collected by all 5 people.

What fraction of the figures would you estimate are s? 15 _ 100

or 1 _ 6


or 1 _ 2


or 1 _ 3

If there are 100 figures in the bag, about how many are . . .

. . . s? 15 . . . s? 49 . . . s? 36

Look at the spinner. Which statement is true?

A. If you spin 100 times on this spinner, you are likely to land on green about 75 times.

B. ! The likelihood of landing on either red OR blue is the same as landing on green.

C. If you land on green 50 times and spin 50 more times, you are not likely to land on green again.

Answers will vary. Accept other estimates that may not be equivalent fractions but are close to them.

For #4, answers depend on estimate used. If 1 _ 6 , 1 _ 2 , and 1 _ 3 are used, then the expected numbers are 17 for s, 50 for s, and 36 for s.

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Another Sampling ExperimentThe list below shows a random sample drawn from a population of 100, all people in Littletown, who watched TV show A, B, C, or N (nothing) at 8:00 on a Wednesday night.

N, B, C, A, A, C, B, C, C, N, A, B, N, C, A, N, C, B, A, N

Record the fraction of the sample that watched each show.

Show A: 5 _ 20 ,

or 1 _ 4 Show B:

4 _ 20 , or 1 _ 5

Show C: 6 _ 20 ,

or 3 _ 10 N(none):

5 _ 20 , or 1 _ 4

Use the experimental results in Problem 1 to predict the fractions of the entire population that watched each show.

Show A: 25 _ 100 ,

or 5 _ 20 , or 1 _ 4 Show B:

20 _ 100 , or 4 _ 20 , or 1 _ 5

Show C: 30 _ 100 , or 6 _ 20 , or 3 _ 10

N(none): 25 _ 100 ,

or 3 _ 10 , or 1 _ 4

PracticeLesson 4

For which of the following situations would you compute 3 _ 4 ! 1 _ 2 ?

A. A sandwich with 3 _ 4 lb of ham C. ! There is 3 _ 4 of a pizza on the table and 1 _ 2 lb of swiss cheese and 2 people are sharing it.

B. You have traveled 1 _ 2 a mile and D. Ryan has 3 _ 4 as many marbles asthe whole trip is 3 _ 4 of a mile. Jake. I have 1 _ 2 as many marbles as Jake.

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Introducing PercentsMake designs by shading in some of the hundredths. Record the fraction and percent for the shaded part of the large square.

_ 100 ! %

_ 100 ! %

One point is incorrectly labeled on the number line. Which point is it? Explain how you know what the label should be.

1 _ 8 ; Possible explanation: The point should be labeled

1 _ 4 because it is 1 _ 4 of the distance between 0 and 1.

PracticeLesson 5

Answers will vary.

Answers will vary.

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Circle GraphsSixty fifth graders were surveyed to find out their favorite types of books to read. The results are summarized in this graph.

Which did more fifth graders prefer to read about: sports or science?

science

True or false? About a third of the fifth graders chose fiction as their favorite reading material.

true

About how many fifth graders preferred to read about sports?

15

About what fraction of the students preferred reading about sports or history?

35 _ 100 , or 7 _ 20

Which of the following is NOT equivalent to 2 _ 8 of 360!?

A. ! 1 _ 4 " 90! B. 90! C. 4 _ 8 " 180! D. 1 _ 4 of 360!

Chris made a set of ten cards for multiples of 2 from 2 to 20. He drew one card at random from the deck. What is the probability that he drew a card that is a multiple of 3?A. 2 _ 10 B. ! 3 _ 10 C. 4 _ 10 D. 5 _ 10

PracticeLesson 6

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Graphing Jake measures the temperature every day at 2 P.M. The temperature on Monday was 65!F. On Thursday the temperature was 69.5!F and on Sunday it was 74!F. Jake said the temperature increased by a constant amount each day. Assuming that Jake was correct, fill in the chart, and then graph the temperatures for the week.

PracticeLesson 1

Kaylee picked a marble from a bag. After noting the color she put it back in the bag and drew again. After ten draws she made this table.

Color Red White Green Black

Number of draws 3 1 2 4

Based on these results, what is the experimental probability of drawing a black marble from the bag?

A. 1 _ 2 B. 4 _ 5 C. ! 2 _ 5 D. 3 _ 10

Monday 65!F

Tuesday 66.5!F

Wednesday 68!F

Thursday 69.5!F

Friday 71!F

Saturday 72.5!F

Sunday 74!F

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The manager of the hardware store wants to string lights around the window. How many feet of lights will he need to outline the 4 sides of the window? Explain how you found the answer.

28 feet; I added the lengths of the

four sides of the window. 6 ft ! 8 ft ! 6 ft ! 8 ft " 28 ft

PracticeLesson 2

Graphing Capacity ConversionsFill in each conversion table and graph the points.

Quarts Gallons

4 1

8 2

16 4

12 3

Pints Quarts

2 1

8 4

6 3

3 1 1 _ 2

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Changing the Scale of GraphsComplete each table and make a graph to show the conversion. Choose an appropriate scale and number the axes accordingly. Other answers are possible.

Kilograms Grams

1 1,000

2 2,000

3 3,000

6 6,000

7 7,000

8 8,000

Pounds Ounces

1 16

2 32

1 _ 2 8

1 1 _ 2 24

3 48

PracticeLesson 3

Find two equivalent fractions for 3 __ 6 and explain how you did it.

Possible explanation: You can get 1 _ 2 by dividing the

numerator and the denominator by 3; you can get 6 __ 12 by

multiplying the numerator and denominator by 2.

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PracticeLesson 4

Graphing Change Over TimeThis graph shows how far Tom went on his bike ride and how long it took him.

Complete the table.

Time (in minutes)

Distance (in miles)

10 2 1 _ 2

20 5

40 10

60 15

80 20

100 25

140 35

How fast did Tom ride? 15 miles per hour

It took Francesca half an hour to ride 5 miles. Did she ride faster or slower than Tom? slower

How long will it take Francesca to go 15 miles? 1 1 _ 2 hours, or 90 min

A photocopy machine takes 20 minutes to print 180 pages. This represents 2 _ 5 of a large job. Explain how you would find the length of time needed to print the entire job.

Possible answer: If it takes 20 minutes to print 2 _ 5 of the job,

then it takes 10 minutes to print 1 _ 5 of the job. So, it must

take 5 ! 10, or 50 minutes, to print the whole job. We don’t

need to think about the number of pages at all.

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Graphing the Story of a TripThe Callahan family went on a trip in their car. They changed speed at 4 points along the way, but kept a constant speed between one point and the next.

Complete the table and graph of the Callahans’ trip.

Point Time on clock

Distance from Start

A 1:00 0

B 1:45 40

C 2:15 60

D 3:15 140

E 4:00 160

How long did it take them to drive from Point C to Point E?

1 hour and 45 minutes, or 105 minutes

Were they driving faster between Point B and Point C, or between Point C and Point D? Explain how you know.

between C and D; The line segment from C to D is steeper.

PracticeLesson 5

A restaurant has tables that seat 4 people. When the restaurant is full, it holds 152 people. How many people are in the restaurant if half of the tables are full and half have two people at them? Explain how you found your answer.

114 people; Possible explanation: There are 38 tables in the

restaurant because 4 ! 38 " 154. 19 tables have 4 people at

them (which is 76 people) and 19 tables have 2 people at

them (which is 38 people). 76 # 38 " 114

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Graphing Temperature Conversions

Use this table to make a graph of how the temperature changed over the day.

Time 12:00 1:00 3:00 5:00 8:00

Temperature 4!C 3!C 1!C !1!C !4!C

If the temperature keeps following this pattern, what will the temperature be at 9:00 P.M.?

!5"C

PracticeLesson 6

Matt turned on the oven. Ten minutes later, the temperature in the oven had risen by 113!F and was now 181!F. What was the temperature in the oven before Matt turned it on?

A. 72!F C. 294!FB. 78!F D. ! 68!F

Which could be a rule for the Nth number in this pattern.

!3, !1, 1, 3, 5

A. N " 4 C. ! 2N " 5B. N " 3 D. 3N " 6

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practice book - delta education · 5.2 writing vertical records .....p36 5.3 writing shorter...

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