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enHanced

Hanced Workbook

Taken from:

enVisionMATH Teacher Resource Masters, Topics 1-20, Grade 5

by Scott Foresman-Addison Wesley

Scott Foresman-Addison Wesley Mathematics, Grade 5:

Review From Last Year Masters


Cover Art: by Luciana Navarro Powell

Taken from:

enVisionMATH Teacher Resource Masters, Topics 1–20, Grade 5


Copyright © 2009 by Pearson Education, Inc.

Published by Scott Foresman-Addison Wesley

Glenview, Illinois 60025

Scott Foresman-Addison Wesley Mathematics Grade 5: Review From Last Year Masters


Copyright © 2004 by Pearson Education, Inc.

Published by Scott Foresman-Addison Wesley

All rights reserved. No part of this book may be reproduced, in any form or by any means, without permission in

writing from the publisher.

This special edition published in cooperation with Pearson Learning Solutions.

All trademarks, service marks, registered trademarks, and registered service marks are the property of their respective

owners and are used herein for identification purposes only.

Pearson Learning Solutions, 501 Boylston Street, Suite 900, Boston, MA 02116

A Pearson Education Company

www.pearsoned.com

000200010270667033

JH

ISBN 10: 0-558-98645-5

ISBN 13: 978-0-558-98645-2

Table of Contents

Review from Last Year ...................................................................................................................... I

Reteaching and Practice

Topic 1 Numeration ............................................................................................................................21

Topic 2 Adding and Subtracting Whole Numbers and Decimals ..................................................... 31

Topic 3 Multiplying Whole Numbers ................................................................................................ 47

Topic 4 Dividing by 1-Digit Divisors ................................................................................................ 63

Topic 5 Dividing by 2-Digit Divisors ................................................................................................ 81

Topic 6 Variables and Expressions .................................................................................................... 97

Topic 7 Multiplying and Dividing Decimals ................................................................................... 109

Topic 8 Shapes ................................................................................................................................ 127

Topic 9 Fractions and Decimals ...................................................................................................... 139

Topic 10 Adding and Subtracting Fractions and Mixed Numbers ...................................................161

Topic 11 Multiplying Fractions and Mixed Numbers ......................................................................175

Topic 12 Perimeter and Area ...........................................................................................................185

Topic 13 Solids ................................................................................................................................ 201

Topic 14 Measurement Units, Time, and Temperature ....................................................................215

Topic 15 Solving and Writing Equations and Inequalities .............................................................. 233

Topic 16 Ratio and Percent .............................................................................................................. 243

Topic 17 Equations and Graphs....................................................................................................... 253

Topic 18 Graphs and Data ............................................................................................................... 263

Topic 19 Transformations, Congruence, and Symmetry ................................................................. 281

Topic 20 Probability ........................................................................................................................ 293

Credits ............................................................................................................................................ 301

Review From Last Year

Corresponds with Student enHanced Workbook pages 1–20

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Reteaching and Practice

Corresponds with Student enHanced Workbook pages 21–300

Name

Place Value

Place-value chart:

Reteaching

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Billions period

Millions period

Thousands period

Ones period

ed

ed

ed

undr

h t en b i l l i ons

b i l l i ons

ands ands

undr undr

t en h t en m i l l i ons

h

m i l l ions

ands eds

t hous t hous undr

m i l l i ons

b i l l i ons

t hous h t ens

6 ,

3 9, 2 5 80 , 1 0 1

o nes

Expanded form: 6,000,000,000 300,000,000 90,000,000 2,000,000

500,000 80,000 100 1

Standard form: 6,392,580,101

Word form: six billion, three hundred ninety-two million,

five hundred eighty thousand, one hundred one

Write the word name for each number and tell the value of the underlined digit.

1. 3,552,308,725

Three billion, five hundred fifty-two

million, three hundred eight thousand,

seven hundred twenty-five; fifty million

2. 843,208,732,833

Eight hundred forty-three billion, two

hundred eight million, seven hundred

thirty-two thousand, eight hundred

thirty-three; 800 billion

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3. Write 2,000,000,000 70,000,000

100,000 70,000 3,000 800

10 in standard form.

4. Number Sense What number

is 100,000,000 more than

5,438,724,022?

2,070,173,810 5,538,724,022

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Place Value

Write the word form for each number and tell the value of the

underlined digit.

1. 34,235,345

2. 19,673,890,004

3. Write 2,430,090 in expanded form.

Practice

1-1

Thirty-four million, two hundred thirty-

five thousand, three hundred forty-five;

five thousand

Nineteen billion, six hundred seventy-

three million, eight hundred ninety

thousand, four; nine billion

2,000,000 400,000 30,000 90

Write each number in standard form.

4. 80,000,000 4,000,000 100 8

5. twenty-nine billion, thirty-two million

6. Number Sense What number is

10,000 less than 337,676?

84,000,108

29,032,000,000

327,676

7. Which number is 164,502,423 decreased by 100,000?

A. 164,402,423 B. 164,501,423 C. 164,512,423 D. 264,502,423

8. Explain It Explain how you would write 423,090,709,000 in word form.

Start at the left and write four hundred

twenty-three. Write billion for the place,

ninety, million for the place, seven

hundred nine, and thousand for the place

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22 Topic 1

Name

Comparing and Ordering

Whole Numbers

Reteaching

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Order these numbers from least to greatest:

4,752,213 5,829,302 4,234,295 4,333,209.

Step 1: Write the numbers,

lining up places. Begin at the

left to find the greatest or

least number.

4,752,213

5,829,302

4,234,295

4,333,209

5,829,302 is the greatest.

Step 2: Write the remaining

numbers, lining up places. Find

the greatest and least of these.

4,752,213 greatest

4,234,295 least

4,333,209

4,752,213 is the greatest of these.

4,234,295 is the least.

Step 3: Write the numbers

from least to greatest.

4,234,295

4,333,209

4,752,213

5,829,302

Complete. Write , , or in each .

1. 7,642 7,843 2.

2,858,534 2,882,201

Order these numbers from least to greatest.

3. 768,265 769,205 739,802

739,802; 768,265; 769,205

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4. Write the areas of each country in

order from greatest to least.

28,748; 28,450;

27,830; 27,750

Country

Area in

Square

Kilometers

Albania 28,748

Burundi 27,830

Solomon Islands 28,450

Haiti 27,750

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Whole Numbers

Complete. Compare the numbers. Use or for each .

1. 23,412 23,098 2.

9,000,000 9,421,090

Practice

1- 2


3. 7,545,999 7,445,999 7,554,000

7,445,999 7,545,999 7,554,000

4. Number Sense What digit could be in the

ten millions place of a number that is less

than 55,000,000 but greater than 25,000,000?

2, 3, 4, or 5

5. Put the trenches in order from the

least depth to the greatest depth.

Kermadec Trench,

Philippine

Trench,

Tonga Trench,

Mariana Trench

Trench

Depths of Major

Ocean Trenches

Depth (in feet)

Philippine Trench 32,995

Mariana Trench 35,840

Kermadec Trench 32,963

Tonga Trench 35,433

6. These numbers are ordered from greatest to least. Which number could be

placed in the second position?

2,643,022 1,764,322 927,322

A 2,743,022 B 1,927,304 C 1,443,322 D 964,322

7. Explain It Explain why 42,678 is greater than 42,067.

Although the digits in the thousands

period are the same, the hundreds

digits differ, and since 6 is greater than

0, 42,678 is the greater number.

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24 Topic 1

Name

Decimal Place Value

Here are different ways to represent 2.753149.

Place-value chart:

Ones Tenths Hundredths Thousandths

Reteaching

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Mill

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2 . 7 5 3 1 4 9

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Expanded Form:

2 0.7 0.05 0.003 0.0001 0.00004 0.000009

Standard form: 2.753149

Word Form: Two and seven hundred fifty-three thousand

one hundred forty-nine millionths

Complete the place-value chart for the following number. Write its word form and

tell the value of the underlined digit.

1. 6.324657

Ten-

thousandths

Hundred-

thou

ousandths

Ten-

Hundred-

Ones Tenths Hundredths Thousandths thousandths

thou

ousandths

Mill

llionths

6

.

3 2 4 6 5 7

Six and three hundred twenty-four thousand

six hundred fifty-seven millionths;

Five hundred-thousandths

usandths

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2. 5 0.1 0.03 0.006 0.0007 0.00002 0.000004

5.136724

3. Two and seven hundred twenty-four thousand three hundred

sixty-five millionths

2.724365

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Decimal Place Value

Write the word form of each number and tell the value of the

underlined digit.

1. 3.100

2. 5.267

3. 2.778


4. 8 0.0 0.05 0.009 0.0006

Practice

1-3

Three and one hundred thousandths;

one tenth

Five and two hundred sixty-seven

thousandths; six hundredths

Two and seven hundred seventy-eight

thousandths; eight thousandths

8.0596

5. 1 0.9 0.08 0.001 0.00002

1.98102

Write two decimals that are equivalent to the given decimal.

6. 5.300 7. 3.7 8. 0.9

5.3, 5.30 3.700, 3.7000 0.90, 0.90000

9. The longest stem on Eli’s

geranium plant is 7.24 inches.

Write 7.24 in word form.

Seven and twenty-

four hundredths

26 Topic 1

10. Explain It The number 4.124 has

two 4s. Why does each 4 have a

different value?

The first 4 is in the

ones place; the

second 4 is in the

thousandths place.

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Decimals

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List the numbers in order from least to greatest:

6.943, 5.229, 6.825, 6.852, 6.779

Step 1: Write the numbers,

lining up places. Begin at the

left to find the greatest or

least number.

6.943

5.229

6.825

6.852

6.779

5.229 is the least.

Step 2: Write the remaining

numbers, lining up places. Find

the greatest and least. Order the

other numbers.

6.943 greatest

6.825 6.825

6.852 6.852

6.779 least

6.779 is the least.

6.943 is the greatest.

6.852 is greater than 6.825.

Step 3: Write the numbers


5.229

6.779

6.825

6.852

6.943

Complete. Write , , or for each .

1. 7.539 7.344 2.

9.202 9.209 3.

0.75 0.750


4. 3.898 3.827 3.779

3.779, 3.827, 3.898

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5. 5.234 5.199 5.002 5.243

5.002, 5.199, 5.234, 5.243

Which had the faster speed?

6. Driver A or Driver D

Driver A

7. Driver C or Driver A

Driver C

Car Racing Winners

Driver

Average Speed

(mph)

Driver A 145.155

Driver B 145.827

Driver C 147.956

Driver D 144.809

Topic 1 27

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Decimals

Write , , or for each .

Practice

1-4

1.

5.424 5.343 2.

0.33 0.330 3.

9.489 9.479

4.

21.012 21.01 5.

223.21 223.199 6.

5.43 5.432


7. 8.37, 8.3, 8.219, 8.129

8. 0.012, 0.100, 0.001, 0.101

9. Number Sense Name three numbers between 0.33 and 0.34.

8.129, 8.219, 8.3, 8.37

0.001, 0.012, 0.100, 0.101

Sample answer: 0.334, 0.335, 0.336

10. Which runner came in first place?

Liz

11. Who ran faster, Amanda or Steve?

Amanda

12. Who ran for the longest time?

Steve

Half-Mile Run

Runner Time (minutes)

Amanda 8.016

Calvin 7.049

Liz 7.03

Steve 8.16

13. Which number is less than 28.43?

A 28.435 B 28.34 C 28.430 D 29.43

14. Explain It Explain why it is not reasonable to say that 4.23 is less than 4.13.

The number 4.23 is greater than 4.13,

because there is a 2 in the tenths place,

and 2 is greater than 1.

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28 Topic 1

Name

Problem Solving: Look for a

Pattern

Reteaching

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Mr. Nagpi works in a machine

shop. In the shop, the drill

bits are kept in a cabinet with

drawers. The drawers are marked

with the diameter of the bits as

shown on the right. Some of

the labels are missing. Help

Mr. Nagpi complete the

drawer labels.

0.10

in.

0.20

in.

0.30

in.

0.12

in.

0.22

in.

0.32

in.

Drill Bits

0.14

in.

0.24

in.

0.34

in.

0.16

in.

0.26

in.

0.36

in.

0.18

in.

0.28

in.

0.38

in.

Read and Understand

What do you know?

What are you trying to find?

Plan and Solve

Find a pattern for the decimals.

1. Look for a pattern to the change

in the tenth-values across a row

or down a column.

2. Look for a pattern to the

change in the hundredthvalues

across a row or down

a column.

3. Use the patterns to complete

the table.

Some drawers are labeled with

decimals.

A way to find the values of the

missing labels

1. The tenth-values are not

increasing across a row. They are

increasing by 1 down a column.

2. The hundredth-values are

increasing by 2 across a row.

They are not increasing down a

column.

3. The missing labels in the third

row are 0.36 in. and 0.38 in.

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Find the pattern in the table. Then fill

in the missing values in the table.

0.20

0.50

0.80

0.21

0.51

0.81

0.22

0.52

0.82

0.23

0.53

0.24

0.54

0.83 0.84

Topic 1 29

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Problem Solving: Look for a

Pattern

Practice

1-5

Determine the pattern and then complete the grids.

1.

0.87 0.88 0.89 0.90

2.

0.12

0.22

0.32

3.

0.22 0.23

0.24 0.25

4.

0.56

0.66

0.76

5. Critical Thinking In a list of numbers, the pattern increases by 0.001 as

you move to the right. If the third number in the list is 0.064, what is the first

number in the list? Explain how you know.

0.062; subtract 0.001 twice to find the

first number in the list.

6. If 5 school buses arrive, each carrying exactly 42 passengers, which

expression would you use to show how many people in all arrived on

the school buses?

A 42 5 B 42 5 C 42 5 D 42 5

7. Explain It Mishell arranged her coins in the following pattern: $0.27, $0.29,

$0.31, $0.33. Explain what her pattern is, and then tell what the next amount

of coins would be.

Pattern increases by $0.02; the next

amount would be $0.35.

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30 Topic 1

Name

Mental Math

There are several ways that you can add and subtract numbers mentally

to solve a problem.

Commutative Property of Addition

You can add two numbers in any order.

15 27 27 15

Reteaching

2-1

Associative Property of Addition

You can change the groupings of addends.

17 (13 10) (17 13) 10

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Compatible numbers are numbers that

are easy to compute mentally.

25 93 75

25 and 75 are compatible because they

are easy to add.

25 93 75 (25 75) 93

100 93 193

With compensation, you adjust one

number to make computations easier and

compensate by changing the other number.

320 190

10 10

⇓

330 200

⇓ 130

Add or subtract mentally.

1,010 730

4,300

1,070

1. 265 410 335 2. 885 155

3. 2,500 1,730 70 4. 1,467 397

5. Number Sense How many more strikeouts

did Pitcher A have than Pitcher C?

152 more strikeouts

6. How many strikeouts did Pitcher B and

Pitcher E have altogether?

498 strikeouts

Pitc

tcher

Strikeout Data

Numb

mber

of Stri

rikeouts

A 372

B 293

C 220

D 175

E 205

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7. How many strikeouts were recorded

by all five pitchers?

1,265 strikeouts

Topic 2 31

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Mental Math

Show how you can use mental math to add or subtract.

1. 70 90 30 2. 350 110

Sample answers:

70 90 30

90 (70 30)

90 100 190

350 110

50

⇓

400

50

Practice

2-1

⇓

160 240

National Monuments

Name State Acres

George Washington Carver Missouri 210

Navajo Arizona 360

Fort Sumter South Carolina 200

Russell Cave Alabama 310

3. How many more acres are there at Navajo

monument than at George Washington

Carver monument?

4. How many acres are there at Fort

Sumter and Russell Cave combined?

5. Fresh Market bought 56 lb of apples in August from a local

orchard. In September, the market purchased an additional

52 lb of apples and 32 lb of strawberries. How many pounds

of fruit did the market buy?

150 more acres

510 acres

A 108 lb B 140 lb C 150 lb D 240 lb

6. Explain It Write the definition and give an example of the

Sample answer:

The Commutative Property of Addition

says you can add numbers in any order.

30 15 2 15 30 2

Commutative Property of Addition.

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32 Topic 2

Name

Rounding Whole Numbers

and Decimals

You can use the number line below to help you round 8,237,650

to the nearest million. Is 8,237,650 closer to 8,000,000 or

9,000,000?

8,237,650 halfway

Reteaching

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8,000,000 8,500,000 9,000,000

8,237,650 is less than halfway to 9,000,000. 8,237,650 is closer to

8,000,000.

The number line can also help you round 7.762 to the nearest

tenth. Is 7.762 closer to 7.7 or 7.8?

halfway 7.762

7.7 7.75 7.8

7.762 is more than halfway to 7.8. 7.762 is closer to 7.8.

Round each number to the place of the underlined digit.

1. 4,725,806

4,700,000

2. 7.049

7

3. 165,023,912

200,000,000

4. 18.692

18.7

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5. Round the number of connected

computers in Year 2 to the nearest

ten million.

40,000,000

6. Number Sense Marc earned $9.37 per hour working

at the library. Round his wage to the nearest ten cents.

$9.40

Number of Computers

Connected to the Internet

Year 1 30,979,376

Year 2 42,199,279

Year 3 63,592,854

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Rounding Whole Numbers

and Decimals

Round each number to the place of the underlined digit.

1. 32.60

2. 489,334,209

32.6

489,000,000

Practice

2-2

3. 324,650

4. 32.073

325,000 32.1

5. Reasoning Name two different numbers that round to 30 when rounded to

the nearest ten.

Sample answer: 29 and 31

In 2000, Italy produced 7,464,000 tons of wheat, and Pakistan

produced 21,079,000 tons of wheat. Round each country’s wheat

production in tons to the nearest hundred thousand.

6. Italy

7. Pakistan

The price of wheat in 1997 was $3.38 per bushel. In 1998, the

price was $2.65 per bushel. Round the price per bushel of wheat

for each year to the nearest tenth of a dollar.

8. 1997

9. 1998

7,500,000 tons

21,100,000 tons

$3.40 per bushel

$2.70 per bushel

10. Number Sense Which number rounds to 15,700,000 when

rounded to the nearest hundred thousand?

A 15,000,000 B 15,579,999 C 15,649,999 D 15,659,999

11. Explain It Write a definition of rounding in your own words.

Sample answer: Rounding helps you

adjust a number to make it easier to use.

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34 Topic 2

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Estimating Sums and

Differences

During one week, Mr. Graham drove a truck to five different towns

to make deliveries. Estimate how far he drove in all.

Mr. Graham’s Mileage Log

Cities

Mileage

Mansley to Mt. Hazel 243

Mt. Hazel to Perkins 303

Perkins to Alberton 279

Alberton to Fort Maynard 277

Fort Maynard to Mansley 352

Reteaching

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To estimate the sum, you can round

each number to the nearest hundred

miles.

243 ⇒

303 ⇒

279 ⇒

277 ⇒

352

⇒

Estimate each sum or difference.

1. 19.7 6.9

200

300

300

300

1,500 400 mi

Mr. Graham drove about 1,500 mi.

About 13

3. 582 169 23

About 770

You can estimate differences in a

similar way.

Estimate 7.25 4.98.

You can round each number to the

nearest whole number.

7.25 ⇒ 7

4.98

⇒ 52

The difference is about 2.

2. 59 43 95

About 200

4. 87.99 52.46

About 36

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5. Estimation Brigid worked 16.75 h. Kevin worked 12.50 h. About how many more

hours did Brigid work than Kevin?

About 4 h

Topic 2 35

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Practice

2-3

Estimating Sums and

Differences

Sample answers are

Estimate each sum or difference. given for 1– 4.

3,300

20

1. 5,602 2,344 2. 7.4 3.1 9.8

3,000

52

3. 2,314 671 4. 54.23 2.39

5. Number Sense Wesley estimated 5.82 4.21 to be about 2. Is this an

overestimate or an underestimate? Explain.

It is an overestimate, because 5.82 was

rounded up and 4.21 was rounded down.

6. Estimate the total precipitation

in inches and the total number

of days with precipitation for

Asheville and Wichita.

About 77 in.;

about 210 days

Average Yearly Precipitation of U.S. Cities

City Inches Days

Asheville, North Carolina 47.71 124

Wichita, Kansas 28.61 85

7. Reasonableness Which numbers should you add to estimate the answer to

this problem: 87,087 98,000?

A 88,000 98,000 C 87,000 98,000

B 85,000 95,000 D 80,000 90,000

8. Explain It Estimate the total weight of two boxes that weigh 9.4 lb and

62.6 lb using rounding and compatible numbers. Which estimate is closer

to the actual total weight? Why?

The rounded numbers are closer to the

srcinal numbers, so the rounded estimate

will be closer to the actual total weight.

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Problem Solving: Draw a

Picture and Write an Equation

A community center is raising funds to buy

a computer. Here is a picture of the sign

they put outside the center. How much

more money must the center raise?

How to write an equation number sentence

for a problem:

New Computer

Fund

Please contribute!

Reteaching

2-4

$1,600

$1,400

$1,200

$1,000

$ 800

$ 600

$ 400

$ 200

$ 0

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One Way

The goal is $1,600.

So far, $1,000 has been raised.

The amount yet to be raised is the

unknown.

Think: The amount raised so far and

the amount yet to be raised will

reach the goal.

Write an equation.

1,000 x 1,600

Think: What number added to 1,000

will result in 1,600?

1,000 600 1,600

The amount yet to be raised is $600.

Another Way

The goal is $1,600.

So far, $1,000 has been raised.

The amount yet to be raised is the

unknown.

Think: The difference between the goal

and what has been raised so far

is the amount yet to be raised.

Write an equation.

1,600 1,000 x

Think: What number will result if 1,000

is subtracted from 1,600?

1,600 1,000 600

The amount yet to be raised is $600.

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A mason needs 22 bricks to make a stoop.

So far he has carried 15 to the site. How many

more bricks must he carry?

Draw a picture. Write an equation. Write a number

sentence. Solve.

Sample: 22 15 x ;

22 15 7; 7 bricks

22 bricks in all

15 x

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Write two different equations; then solve each problem.

1. Dayana picked apples for 2 hours. She picked 28 apples

in the first hour, and at the end of two hours, she had 49.

How many apples did she pick during the second hour?

Practice

2-4

28 a 49

49 28 a

21 apples

2. Dixon bought a pack of pencils and then gave 12 away.

He now has 24 left. How many pencils were in the pack

of pencils that Dixon bought?

Copy and complete the picture. Then write an equation and solve.

3. Rumina is baking 25 muffins for the bake sale. She has

already baked 12. How many more does she need to bake?

25 muffins in all

12 n

12 n 25; 13 muffins

12 p 12 24 24

p

36 pencils

4. Estimation Janet saved 22 dollars one month and 39 dollars

the next month. She wants to buy a bicycle that costs $100.

About how much more money does she need?

A about $40 B about $50 C about $60 D about $70

5. Explain It Stefany ran 2 miles each day for 14 days. How

many miles did she run in 14 days? Explain two different

ways to solve this problem, and then solve.

Sample: Draw a picture or write

an equation; 28 miles

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38 Topic 2

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Adding and Subtracting

ting

Find 35,996 49,801.

Step 1: Write the numbers, lining up places. Add

the ones and then the tens.

t en t housand s

t ho u sand s

h und r ed s

es

t ens n

o

35,996

49,801

97

Reteaching

2-5

Step 2: Continue adding hundreds, thousands and

ten thousands. Regroup as needed.

1 1

35,996

49,801

85,797

Reminder: When the

sum of a column is

greater than 10, write

the ones of the sum

and regroup the tens

onto the next column.

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So 35,996 49,801 85,797.

Find 35,996 17,902.

Step 1: Write the numbers, lining up places.

Subtract the ones, tens, and hundreds.

t en t ho u sand s

t ho u sand s

h und r ed s

es

t ens n

o

35,996

17,902

094

Step 2: Continue by subtracting thousands.

Regroup as needed.

2 15

3/ 5/ , 9 9 6

1 7, 9 0 2

1 8, 0 9 4

Reminder: The 3 in

the ten thousands

place is regrouped to

2 ten thousands and

10 thousands.

So 35,996 17,902 18,094.

Add or subtract

1. 7,502

9,909

2. 64,782

33,925

3. 835,029

26,332

17,411 30,857 808,697

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Myronville School District has 23,081 students, and Saddleton

School District has 45,035 students.

4. Number Sense How many more students are there in Saddleton than in

Myronville?

21,954 more students

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Add or subtract.

1. 29,543

13,976

5. 971,234 55,423

2. 93,210

21,061

3. 369,021

325,310

Practice

2-5

4. 893,887

22,013

43,519 72,149 43,711 915,900

1,026,657

6. Number Sense Is 4,000 a reasonable estimate for the difference of

9,215 5,022? Explain.

Yes, because 9,000 5,000 4,000.

For questions 7 and 8, use the table at right.

7. How many people were employed as

public officials and natural scientists?

1,319,000 people

8. How many more people were

employed as university teachers

than as lawyers and judges?

35,000 more people

People Employed in U.S. by

Occupation in 2000

Occupation Workers

Public officials 753,000

Natural scientists 566,000

University teachers 961,000

Lawyers and judges 926,000

9. Which is the difference between 403,951 and 135,211?

A 200,000 B 221,365 C 268,740 D 539,162

10. Explain It Issac is adding 59,029 and 55,678. Should his answer be greater

than or less than 100,000? Explain how you know.

Greater than 100,000, because an

estimate of 60,000 60,000 120,000

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40 Topic 2

Name

Adding Decimals

In February, Chantell ran a 5K race in 0.6 hour. She ran another

5K race in May in 0.49 hour. What was her combined time for the

two races?

Reteaching

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Step 1: Write the numbers, lining

up the decimal points. Include the

zeros to show place value.

0.60

0.49

You can use decimal squares to

represent this addition problem.

Step 2: Add the hundredths.

0.60

0.49

9

Step 3: Add the tenths.

Remember to write the decimal

point in your answer.

1

0.60

0.49

1.09

Chantell’s combined time for the

two races was 1.09 hours.

Add.

3.32

66.188

1. 2.97 0.35 2. 13.88 7.694

3. 39.488 26.7 4. 88.8 4.277 78.95

5. Number Sense Is 16.7 a reasonable sum for 7.5 9.2? Explain.

21.574

172.027

Yes; 7.5 rounds to 8.0; 9.2 rounds to 9.0;

8.0 9.0 17.0.

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6. How much combined

snowfall was there in

Milwaukee and

Oklahoma City?

105.1 in.

City

Snowfall (inches)

in 2000

Milwaukee, WI 87.8

Baltimore, MD 27.2

Oklahoma City, OK 17.3

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Adding Decimals

Add.

Practice

2-6

1. 58.0 2. 40.5 3. 34.587 4. 43.1000

3.6 22.3 21.098 8.4388

61.6 62.8 55.685 51.5388

23.045 92.62

5. 16.036 7.009 6. 92.30 0.32

7. Number Sense Reilly adds 45.3 and 3.21. Should his sum be greater than or

less than 48? Tell how you know.

The sum should be greater than 48,

because the whole numbers have a sum

of 48 without adding the decimals.

In science class, students weighed different amounts of tin.

Carmen weighed 4.361 g, Kim weighed 2.704 g, Simon weighed

5.295 g, and Angelica weighed 8.537 g.

8. How many grams of tin did Carmen and Angelica have combined?

12.898 g of tin

9. How many grams of tin did Kim and Simon have combined?

7.999 g of tin

10. In December the snowfall was 0.03 in. and in January it was

2.1 in. Which was the total snowfall?

A 3.2 in. B 2.40 in. C 2.13 in. D 0.03 in.

11. Explain It Explain why it is important to line up decimal

numbers by their place value when you add or subtract them.

Sample answer: When adding or

subtracting, you must always add or

subtract digits that have the same place

value.

42 Topic 2

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Name

Subtracting Decimals

Mr. Montoya bought 3.5 lb of ground beef. He used 2.38 lb to

make hamburgers. How much ground beef does he have left?

Step 1: Write the numbers, lining up the decimal points. Include

the zeros to show place value.

Reteaching

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3.50

2.38

You can use decimal squares to represent this subtraction

problem.

Step 2: Subtract the hundredths. Regroup if you need to.

3.50

2.38

2

. 4 10

Step 3: Subtract the tenths and the ones. Remember to write the

decimal point in your answer.

4 10

3.50

2.38

1.12

Mr. Montoya has 1.12 lb of ground beef left over.

Subtract.

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1. 82.7 2. 43.3 3. 7.28

5.59 12.82 4.928

77.11 30.48 2.352

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Subtracting Decimals

Subtract.

1. 92.1

32.6

2. 52.7

36.9

3. 85.76

12.986

5. 8.7 0.3 6. 23.3 1.32

Practice

2-7

4. 32.7

2.328

59.5 15.8 72.774 30.372

8.4 21.98

7. Number answer is

Sense reasonable.

Kelly subtracted 2.3 from 20 and got 17.7. Explain why this

A good estimate is 20 2 18, which

is close to 17.7.

At a local swim meet, the second-place swimmer of the 100-m

freestyle had a time of 9.33 sec. The first-place swimmer’s time

was 1.32 sec faster than the second-place swimmer. The thirdplace

time was 13.65 sec.

8. What was the time for the first-place swimmer?

8.01 sec

9. What was the difference in time between

the second- and third-place swimmers?

4.32 sec

10. Miami’s annual precipitation in 2000 was 61.05 in. Albany’s

was 46.92 in. How much greater was Miami’s precipitation

than Albany’s?

A 107.97 in. B 54.31 in. C 14.93 in. D 14.13 in.

11. Explain It Explain how to subtract 7.6 from 20.39.

Sample answer: Write the numbers,

lining up the decimal points. Write

zeros to show place value. Subtract the

hundredths, the tenths, and the ones.

Place the decimal point in the answer.

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44 Topic 2

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Problem Solving:

Multiple-Step Problems

Kim has a $10 bill, a $20 bill, and 2 $5 gift certificates. She uses

the gift certificates toward the purchase of a CD for $14.00. How

much money does Kim have left after buying the CD?

Reteaching

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Read and Understand

What do you know?


Kim has a ten-dollar bill, a twenty-dollar

bill, and two five-dollar gift certificates.

She uses the 2 certificates toward the

purchase of a CD that costs $14.00.

How much money does Kim have left after

she buys the CD?

Plan and Solve

Answer these hidden questions.

How much money does Kim have? $20.00 $10.00 $30.00

How much are the two $5.00 $5.00 $10.00

certificates worth?

How much cash will Kim need to $14.00 $10.00 $4.00

buy the CD?

Solve the problem.

Write the answer in a

complete sentence.

Look Back and Check

Money cash paid for CD Money left

$30.00 $4.00 $26.00

Kim has $26 left after buying the CD.

Is your answer correct? Yes, $4.00 $26.00 $30.00

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1. You can also find how much money Kim has left by

completing the following expression.

$14.00

$10.00 $20.00 $5.00 $5.00

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Problem Solving:


Solve.

1. Theater tickets for children cost $5. Adult tickets cost $3 more.

If 2 adults and 2 children buy theater tickets, what is the total cost?

$26

Practice

2-8

2. Luis has a $10 bill and three $5 bills. He spends $12.75 on the

entrance fee to an amusement park and $8.50 on snacks.

How much money does he have left?

$3.75

3. Number Sense Alexandra earns $125 from her paper route

each month, but she spends about $20 each month on personal

expenses. To pay for a school trip that costs $800, about how

many months does she need to save money? Explain.

About 8 months, since she can save

about $100 per month

4. Critical Thinking Patty is a member of the environmental club. Each

weekday, she volunteers for 2 hours. On Saturday and Sunday, she

volunteers 3 hours more each day. Which expression shows how to

find the number of hours she volunteers in one week?

A 2 5

B 2 2 2 2 2 5 5

C 2 2 2 3 3

D 2 3 3

5. Explain It Marco’s goal is to eat only 2,000 calories each day. One day

for breakfast he consumed 310 calories, for lunch he consumed 200 more

calories than breakfast, and for dinner he consumed 800. Did he make his

goal? Explain.

Yes: 310 510 800 1,620 calories.

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46 Topic 2

Name

Multiplication Properties

Reteaching

3-1

You can use multiplication properties to help you multiply more

easily.

Associative Property of Multiplication

You can change the grouping of the factors. The product stays

the same.

(3 4) 4 48 3 (4 4) 48

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Factors Product Factors Product

12 48

3 16 48

Commutative Property of Multiplication

You can change the order of the factors. The product stays the same.

7 428

4 7 28


Zero Property of Multiplication

When one of the factors is 0, the product is always 0.

3 0 0 0 3 0


Identity Property of Multiplication

When one of the factors is 1, the product is always the other factor.

Identify the multiplication property or properties used in each equation.

zero

identity

1. 100 0 0 2. 7 2 2 7

3. 1 55 55 4. (6 7) 9 6 (7 9)

commutative

associative

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Reasoning Use the multiplication properties to determine what number must be

in the box.

4

8 0

5. 5 4 5 6. 99 99

7. (3 12) 3 (12 8) 8. 1 0

50

9. 2 2 50 10. (16 ) 25 16 (33 25)

1

33

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Multiplication Properties

rties

In 1 through 5, write the multiplication property used in each

equation.

1. 53 6 6 53

2. 0 374,387 0

3. 5 (11 4) (5 11) 4

4. 42 1 42

5. 14 5 5 14

commutative

zero

associative

identity

commutative

Practice

3-1

6. Reasoning Chan bought 2 large frozen yogurts at $1.50 each and 1 small

bottle of water for $1.00. How much did she pay in total?

$4.00

7. Dan has 4 shelves. He has exactly 10 books on each shelf. Judy has

10 shelves. She has exactly 4 books on each shelf. Who has more books?

Explain.

They have the same number of books.

4 10 10 4

8. Algebra If 3 8 12 8 3 n, what is the value of n?

A 3 B 8 C 12 D 18

9. Explain It Write a definition for the Associative Property of Multiplication in

your own words and explain how you would use it to compute 4 25 27

mentally.

Sample answer: Using the Associative

Property rty of Multiplication you can

group (4 25) and multiply the product

by 27 to get 2,700.

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48 Topic 3

Name

Using Mental Math to Multiply

Reteaching

3- 2

You can also use patterns to multiply mentally.

Fact: 6 8 48

60 8 480 6 80 480

600 8 4,800 60 80 4,800

6,000 8 48,000 600 80 48,000

60,000 8 480,000 6,000 80 480,000

Pattern: Notice that the product is always the digits 48 followed

by the total number of zeros that are in the factors.

Find 30 3 50.

Use the Associative Property of Multiplication to regroup.

(30 50) 3

1,500 3 4,500

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Commutative Property of

Multiplication

You can multiply factors in any

order.

Associative Property of

Multiplication

You can change the grouping of

factors.

15 9 91 5 (820) 5 8 (20 5)

Find each product. Use patterns and properties to compute

mentally.

7,200 32,000

1,000 3,000

1. 80 90 2. 40 800

3. 5 10 20 4. 4 30 25

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5. Number Sense You know that 6 7 42. How can you find 60 700?

Sample answer: Multiply 6 6 7, then place

the total number of zeros in both factors

after the product. So, 60 700 42,000.

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Using Mental Math to Multiply

Use mental math to find each product.

1. 150 20

3,000 0

4. 120 50

6,000

2. 0 50 800

5. 60 70 1

4,200

Practice

3- 2

3. 500 40

20,000

6. 9,000 80

720,000

7. 100 10 1

1,000

8. 1,800 20 0

0

9. 30 20

600

10. 1,400 2,000

2,800,000

11. 7,000 50 1

350,000

12. 1,000 200 30

6,000,000

13. Number Sense A googol is a large number that is the digit one followed by

one hundred zeros. If you multiply a googol by 100, how many zeros will that

product have?

102 zeros

14. Gregorios drives 200 miles per day for 10 days. How many miles did he drive

in all?

2,000 miles

15. Algebra If a b c 0, and a and b are integers greater than 10,

what must c equal?

A 0 B 1 C 2 D 10

16. Explain It SungHee empties her piggy bank and finds that she has 200 quarters,

150 dimes, and 300 pennies. How much money does she have? Explain.

(200 25) (150 10) (300 1)

5,000 1,500 300 6,800 cents, so

she has 68 dollars.

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50 Topic 3

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Estimating Products

Reteaching

3-3

A bus service drives passengers between Milwaukee and Chicago

every day. They travel from city to city a total of 8 times each day. The

distance between the two cities is 89 mi. In the month of February, there

are 28 days. The company’s budget allows for 28,000 total miles for

February. Is 28,000 mi a reasonable budget mileage amount?

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One Way to Estimate

Estimate 28 8 89.

Use rounding.

You can round 89 to 100 and 8 to 10. Then multiply.

28 10 100 280 100 28,000

Because this is an overestimate, there are enough

miles.

Another Way to Estimate

Estimate 28 8 89.

Use compatible numbers.

Replace 28 with 30, 89 with 90, and 8 with 10. 30,

90, and 10 are compatible numbers because they

are close to the actual numbers in the problem

and they are easier to multiply. Now the problem

becomes 30 90 10.

28,000 total miles is a reasonable budget amount.

30 90 = 2,700 Multiply 3 × 9, then place two

zeros after the product.

2,700 10 = 27,000 Multiply 27 1 using

the Identity Property of

Multiplication, then place

three zeros after the product.

In the estimate, we used numbers greater than the

srcinal numbers, so the answer is an overestimate.

Estimate each product. Use rounding or compatible numbers.

18,000 12,000

1. 42 5 90 2. 27 98 4

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Mrs. Carter ordered new supplies for Memorial Hospital.

3. About how much will it cost to purchase

48 electronic thermometers?

About $1,000

4. About how much will it cost to

purchase 96 pillows?

About $2,000

Supplies

Electronic thermometers

Pulse monitors

$19 each

$189 each

Pillows $17 each

Telephones

$19 each

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Estimating Products

Estimate each product.

1. 68 21

4. 99 99

2. 5 101

1,400 500

10,000

5. 87 403

36,000

Practice

3-3

3. 151 21

3,000

6. 19 718

14,000

7. 39 51

2,000

8. 47 29 11

15,000

9. 70 27

2,100

10. 69 21 23

28,000

11. 7 616

4,200

12. 8,880 30

270,000

13. Number Sense Give three numbers whose product is about 9,000.

Possible answer: 99 x 10 x 9

14. About how much would it cost to buy

4 CD/MP3 players and 3 MP3 players?

About $1,100

15. Which is the closest estimate for the

product of 2 19 5?

Electronics Prices

CD player $ 74.00

MP3 player $ 9 9.00

CD/MP3 player $199.00

AM/FM radio $ 29.00

A 1,150 B 200 C 125 D 50

16. Explain It Explain how you know whether an estimate of a product is an

overestimate or an underestimate.

Sample answer: If the numbers are

rounded down, it is an underestimate.

If the numbers are rounded up, it is an

overestimate.

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52 Topic 3

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Multiplying by 1-Digit

Numbers

Mr. McGuire drives his truck 275 miles each day. How far does he

drive in 3 days?

Find 275 3.

Step 1:

Multiply the

ones. Regroup

if necessary.

Step 2:

Multiply the

tens. Regroup

if necessary.

What You Think

3 5 ones 15 ones

Regroup 15 ones as 1 ten

and 5 ones.

What You Think

3 7 tens 21 tens

21 tens 1 ten 22 tens

Regroup as 2 hundreds

and 2 tens.

Reteaching

3-4

What You Write

1

275

3

5

What You Write

2 1

275

3

25

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Step 3:

Multiply the

hundreds.

Regroup if

necessary.

What You Think

3 2 hundreds 6 hundreds

6 hundreds 2 hundreds

8 hundreds

No need to regroup.

Mr. McGuire drives 825 miles in 3 days.

What You Write

2 1

275

3 825

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Find each product. Estimate to check that your

answer is reasonable.

217 116 528

225 816 1,477

1,656 4,239 292

2,128 780 3,084

1. 31 7 2. 29 4 3. 88 6

4. 25 9 5. 102 8 6. 211 7

7. 552 3 8. 471 9 9. 73 4

10. 266 8 11. 390 2 12. 514 6

13. Estimation Estimate the product of 48 and 7. Do you have

an underestimate or overestimate?

Sample answer: 50 7 350;

overestimate

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Multiplying by 1-Digit

Numbers



174

415

1. 58 3 2. 49 8

3. 83 5 4. 95 6

1,092

4,734

5. 273 4 6. 35 8

7. 789 6 8. 643 7

392

570

280

4,501

Practice

3-4

9. 68

2

10. 582

5

13. Xavier painted five portraits and wants to

sell them for 36 dollars each. How much

money will he make if he sells all five?

11. 84

4

$180

12. 926

7

136 2,910 336 6,482

14. A farmer wants to build a square pigpen. The

length of one side of the pen is 13 ft. How

many feet of fencing should the farmer buy?

15. Jasmine wants to buy 4 green bags

for 18 dollars each and 3 purple bags for

15 dollars each. She has 100 dollars.

How much more money does she need?

52 ft

$17

16. Geometry A regular octagon is a figure that has eight sides with equal

lengths. If one side of a regular octagon is 14 inches long, what is the

perimeter of the entire octagon?

A 148 in. B 140 in. C 112 in. D 84 in.

Sample answer: 542 6 is about 3,000;

3,000 is an underestimate, so 2,482 is

not a reasonable answer.

17. Explain It Why is 2,482 not a reasonable answer for 542 6?

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Multiplying 2-Digit by

2-Digit Numbers

Find 43 26.

Step 1:

Multiply by the ones.

Regroup if necessary.

What You Think

6 3 ones 18 ones

Regroup 18 ones as 1 ten

and 8 ones.

What You Write

1

43

Reteaching

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Step 2:

Multiply by the tens.

Regroup if necessary.

6 4 tens 24 tens

24 tens 1 ten 25 tens

Regroup 25 tens as 2 hundreds

and 5 tens.

What You Think

20 3 ones 60 ones

Regroup 60 ones as 6 tens.

258

26

1

43

26

258

860

20 4 tens 80 tens

Regroup 80 tens as 8 hundreds.

Step 3:

Add the partial products.

What You Think

6 43 258

20 43 860

1

43

26

258

860

1,118

partial

products

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Find the product.

1. 38

12

5. 26

21

2. 64

33

6. 73

19

3. 49

27

7. 57

28

4. 85

15

456 2,112 1,323 1,275

8. 91

86

546 1,387 1,596 7,826

9. Number Sense In the problem 62 45, what are the partial products?

310 and 2,480

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Multiplying 2-Digit by

2-Digit Numbers

Find each product. Estimate to check that your answer is

reasonable.

1. 56

34

5. 64

51

2. 45

76

6. 47

30

9. To pay for a sofa, Maddie made a payment

of 64 dollars each month for one year. How

much did the sofa cost ?

3. 35

15

7. 56

19

Practice

3-5

4. 47

94

1,904 3,420 525 4,418

$768

8. 92

49

3,264 1,410 1,064 4,508

10. Geometry To find the volume of a box, you

multiply the length times the width times the

height. What is the volume, in cubic feet, of

a box that is 3 ft long, 8 ft wide, and 16 ft high?

11. Estimation Katie is in charge of buying juice for the teachers’ breakfast party.

If one teacher will drink between 18 and 22 ounces of juice, and there are

32 teachers, which is the best estimate for the amount of juice Katie should buy?

A about 200 ounces

B about 400 ounces

C about 600 ounces

D about 800 ounces

384 cubic feet

12. Explain It Is 7,849 a reasonable answer for 49 49? Why or why not?

Sample answer: No; 49 49 is about

2,500; 2,500 is an overestimate, and

7,849 is not close to 2,500.

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56 Topic 3

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Multiplying Greater Numbers

Reteaching

3-6

Find 128 23.

Estimate: 100 20 2,000

Step 1

Multiply by the

ones. Regroup

as needed.

Step 2

Multiply by the

tens. Regroup

as needed.

Step 3

Add the

products.

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128

23

1

384

2,560

2,944

2

128

3

384

1

128

20

2,560

Because the answer is close to the estimate, the answer is reasonable.

Find the product. Estimate to check if your answer is reasonable.

Problem Multiply by the Ones Multiply by the Tens Add the Products

1. 282

19

7 1

282

9

282

10

1

2,538

2,820

2,538

2,820

5,358

2. 538

46

24,748

2,538

2,820

5,358

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No; if you round 917 to 900 and 33 to

30, the product is 900 30 or 27,000,

so 2,750 is not reasonable.

3. Reasonableness Is 2,750 a reasonable answer for 917 33? Explain.

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Multiplying Greater Numbers



1. 556

34

5. 628

2. 234

75

6. 154

3. 395

76

7. 643

Practice

3-6

4. 483

57

18,904 17,550 30,020 27,531

8. 536

33

35

49

94

20,724 5,390 31,507 50,384

9. Number Sense In a class of 24 students, 13 students sold over 150 raffle

tickets each, and the rest of the class sold about 60 raffle tickets each. The

class goal was to sell 2,000 tickets. Did they reach their goal? Explain.

Sample answer: Yes;

13 150 1,950

and the rest of the class sold more than

50, so they went over their goal of 2,000.

10. Player A’s longest home run distance is 484 ft.

If Player A hits 45 home runs at his longest

distance, what would the total distance be?

11. Player B’s longest home run distance is 500 ft.

There are 5,280 ft in 1 mi. How many home

runs would Player B need to hit at his longest

distance for the total to be greater than 1 mi?

21,780 feet

11 home runs

12. Algebra Which equation shows how you can find the

number of minutes in one year?

A 60 24 365

B 60 60 24

C 60 365

D 60 60 365

13. Explain It Write a real-world problem where you would have to multiply 120

and 75.

Check students’ work.

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58 Topic 3

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Exponents

Reteaching

3-7

You can use exponential notation to write a number that is being

multiplied by itself.

There are two parts in exponential notation. The base tells you

what factor is being multiplied. The exponent tells you how many

of that factor should be multiplied together. The exponent is not a

factor.

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exponent

8 2 = 8 8 The base is 8, so 8 is the factor to be multiplied.

The exponent is 2, so 2 factors of 8 should be

multiplied together.

base

You can write 8 2 in two other forms.

In expanded form, you write out your factors. Since 8 2 means you

multiply two factors of 8, 8 2 in expanded form is 8 8.

In standard form, you write down the product of the factors.

Since 8 8 = 64, 64 is the standard form of 8 2 .

Write in exponential notation.

1. 2 2 2 2. 6 6 6 6 6

Write in expanded form.

3. 1 4 4. 5 3

Write in standard form.

2 3 6 5

1 1 1 1 5 5 5

16 512

5. 2 2 2 2 6. 8 3

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7. A used car lot has 9 lanes for cars and 9 rows for cars in

each lane. What is the exponential notation for the number of

spaces on the lot? Can the owner fit 79 cars on the lot?

9 2 ; Yes. 9 9 81. There are 81 spaces

on the lot.

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Exponents

For questions 1–4, write in exponential notation.

1. 13 13 13 2. 8 8 8 8 8 8

3. 64 64 4. 4 4 4 4

4 4 4 4

Practice

3-7

13 3 8 6

64 2

4 8

For questions 5–8, write in expanded form.

2 2 2 2 2

11 11 11 11

5. 2 5 6. 20 squared

7. 11 4 8. 9 cubed

For questions 9–12, write in standard form.

64

7,776

9. 4 4 4 10. 14 squared

11. 6 5 12. 9 9 9 9

13. Number Sense Which of these numbers, written in

expanded form, is equal to 625?

A 5 5 5 5

B 5 5

C 5 5 5

D 5 5 5 5 5

14. Number Sense Find the number equal to 6 raised to the

second power.

A 18

B 36

C 6

D 12

20 20

9 9 9

196

6,561

15. Explain It Explain what 4 raised to the fourth power means.

4 raised to the fourth power means

four factors of 4 multiplied together:

4 4 4 4.

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60 Topic 3

Name



A hardware store ordered 9 packs of screws from a supplier. Each

pack contains 150 screws. How many screws did the store order?

Read and Understand

What do you know?

Reteaching

3-8

The store ordered nine packs of screws.

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Plan and Solve

Draw a picture of what you know.

Each pack contained 150 screws.

The total number of screws ordered

Write an equation.

Let x the total number of screws.

9 150 x

Multiply. 4

150

9

1,350

The store ordered 1,350 screws.

Look Back and Check

Is your answer reasonable? Yes, 150 10 1,500.

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A state aquarium has display tanks that

each contains 75 fish. Three of these tanks

are at the entrance. How many fish are on

display at the entrance?

Draw a picture. Write an equation. Solve.

Sample: 75 3 x ;

x sh in all

225 fish 75 75 75

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Draw a picture and write an equation. Then solve.

1. When Mary was born, she weighed 8 pounds. When she was

10 years old, she weighed 10 times as much. How much did

she weigh when she was 10 years old?

8 10 n; n 80 lb

Practice

3-8

2. Sandi is 13 years old. Karla is 3 times Sandi’s age. How old

is Karla?

13 3 n; n 39

3. Reasoning Hwong can fit 12 packets of coffee in a small

box and 50 packets of coffee in a large box. Hwong has

10 small boxes and would like to reorganize them into large

boxes. Which boxes should he use? Explain.

Since 10 small boxes is 120 packets

of coffee, he can use 2 large boxes and

2 small boxes.

4. Number Sense Daniel has 12 tennis balls. Manuel has twice

as many tennis balls as Daniel. Kendra has twice as many

balls as Manuel. How many tennis balls do they have in all?

A 24 B 36 C 84 D 96

5. Explain It William travels only on Saturdays and Sundays

and has flown 400 miles this month. Jason travels every

weekday and has flown 500 miles this month. Who travels

more miles per day ? Explain.

William; he travels only two days per

week but has flown almost the same

number of miles this month as Jason.

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62 Topic 3

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Dividing Multiples of 10 and 100

Reteaching

4-1

You can use math facts and patterns to help you divide mentally.

What is 480 6?

You already know that 48 6 8.

480 has one more zero than 48, so

place one more zero in the quotient.

480 6 80.

What is 60,000 6?

60 6 10

60,000 has three more zeros than

6, so place three zeros in the quotient.

60,000 6 10,000.

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Find each quotient. Use mental math.

1. 32 8 2. 320 8 3. 3,200 8 4. 32,000 ÷ 8 =

4

5. 56 7 6. 560 7 7. 5,600 7 8. 56,000 ÷ 7 =

8

9. 15 3 10. 150 3 11. 1,500 3 12. 15,000 ÷ 3 =

5

40

80

50

400

800

500

4,000

8,000

5,000

13. Number Sense Explain how dividing 720 by 9 is like dividing 72 by 9.

You divide 72 by 9. For 720, you add a

zero for the ones place.

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Arlo has a newspaper delivery job. He wants to

wrap each of his newspapers in a plastic bag

to protect them from the rain. The newspapers

are in bundles.

Use mental math to answer the following questions.

14. How many bags will he use for 5 bundles?

15. How many bags will he use for 7 bundles?

16. How many bags will he use for all 12 bundles?

Arlo’s Newspaper Delivery

Number of bundles 12

Number of newspapers

per bundle 9

45 bags

63 bags

108 bags

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Dividing Multiples

of 10 and 100

Use mental math to find each quotient.

1. 27 9

4. 24 4

2. 270 9

3 30

6

5. 240 4

60

3. 2,700 9

300

Practice

4-1

6. 2,400 4

600

7. 720 9

80

8. 140 7

9. 2,100 3

20 700

10. If a bike race covers 120 mi over 6 days and the

cyclists ride the same distance each day, how

many miles does each cyclist ride each day?

20 mi


11. If the vehicles are divided evenly

between the sections, how many

vehicles are in each section?

300 vehicles

Dealership Vehicle Storage

Sections of vehicles ............. 4

Vehicles for sale ............ 1,200

Rows per section ................10

12. If the vehicles are divided evenly between

the rows in each section, how many

vehicles are in each row?

13. Algebra If 160,000 n 4, find n.

30 Vehicles

40,000

14. Find 32,000 8 mentally.

A 4,000 B 400 C 40 D 4

15. Explain It Solve the equation n 50 5,000. Explain your solution.

n 100; Sample answer: Divide each

side by 50.

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64 Topic 4

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Estimating Quotients

Reteaching

4-2

There are several ways to adjust whole numbers to estimate

quotients.

Rounding

Example:

There are 216 students. The school has 8 classrooms.

How many students will be in each classroom?

Estimate 216 8.

Compatible Numbers

Multiplication

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You can use rounding to estimate

a quotient.

Round 216 to the nearest

hundred.

In this case, 216 rounds to 200.

200 8 25

25 students per room is an

underestimate, because 216 was

rounded down to 200.

You can use compatible numbers

to estimate a quotient.

Change 216 to a compatible

number for 8.

Compatible numbers for 8 are

numbers divisible by 8, such

as 160, 240, and 320. Choose

240, because it is the closest

compatible number to 216.

240 8 30

30 students per class is an

overestimate, because 216 was

rounded up to 240.

You can use multiplication to

estimate a quotient.

Think: 8 times what number is

about 216?

8 25 200

8 30 240

216 is between 200 and 240. So

a good estimate is a little more

than 25 and a little less than

30 students per classroom.

Estimate each quotient. You may use any method.

1. 411 2 2. 162 4

400 2 200 160 4 40

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3. Number Sense If you estimate 342 7 by using 350 7 50,

is 50 greater than or less than the exact answer? How did you

decide? Is 50 an overestimate or an underestimate?

It is greater, because you rounded 342

up to 350. So, 50 is an overestimate.

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Estimating Quotients

Estimate each quotient. Tell which method you used.

1. 195 4

2. 283 5

3. 766 8

4. 179 2

5. $395.20 5

6. $31.75 8

7. $247.80 5

50

60

90

90

$80

$4

$50

Practice

4-2

Sample answers

for 1–8.

Multiplication

Rounding

Rounding

Compatible numbers

Rounding

Compatible numbers

Multiplication

8. Reasoning If you use $63.00 9 to estimate $62.59 9, is $7.00 greater

than or less than the exact answer? Explain.

Greater than; 9 7 63, which is

greater than $62.59.

9. A band played 3 concerts and

earned a total of $321.00. The band

earned about the same amount for

each concert. Estimate how much

the band earned each night.

About $100 each

night

11. Which is the closest estimate for 213 4?

10. At a department store, a woman’s

total was $284.00 for 7 items.

Estimate the average cost per item.

About $40.00

A 50 B 40 C 30 D 20

12. Explain It Explain how to estimate 524 9.

Sample answer: Round 524 to 500, 9 to

10, and divide: 500 10 50.

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Problem Solving:

Reasonableness

Reteaching

4-3

After you solve a problem, check to see if your answer is reasonable.

Also check that you answered the right question.

Example:

74 students are going to a special class dinner where they will be

seated 8 to a table. Will 9 tables be enough?

Reasonableness

76 8 9 R2

Answering the right question

The answer is close to 9 tables.

All students must have seats, so there must be

one more table to hold the remaining 2 students,

making 10 tables in all.

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Tell whether each answer is reasonable.

1. Kendra wants to paste 500 photographs into an album, 6 photos

to a page. She figures that she will need about 100 pages.

Six times 100 is 600, much more

space than Kendra needs. Since 500 is

between 480 (6 80) and 540 (6 90),

she needs between 80 and 90 pages.

2. Hwong has 39 muffins. If each of his guests will eat 2 muffins,

Hwong figures that he can serve muffins to 19 guests.

This is reasonable, since 2 19 38.

Tell whether each answer answers the right question.

3. Ivan has a piece of lumber 104 inches long. He is sawing it

into 12-inch lengths to make fence posts. He says he can get

about 9 fence posts out of the board.

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Alth each though

post 104 must 12 be is a closer full 12 to inches, 9 than and

to 8,

therefore e any remainder must be thrown

away. Ivan can only make 8 posts.

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Problem Solving:

Reasonableness

Solve.

1. One tray holds eight sandwiches.

If there are 30 sandwiches in all,

how many trays are needed?

4 trays

Mrs. Favicchio has 72 students in her

science class. The table shows how

many students can use each item of

lab supplies she is ordering.

3. How many packets of pH paper

does she need to order?

8

Practice

4-3

2. There are 53 students on a field

trip. One chaperone is needed

for every 6 students. How many

chaperones are needed?

9

Item

Lab Supplies

Packet of pH paper 10

Case of test tubes 5

Number of

Students

4. How many cases of test tubes

does she need to order?

15

Case of petri dishes 4

5. Algebra A loaf of banana bread serves 6 guests. There will be

47 guests attending the faculty breakfast. Which expression shows

how many loaves are needed to serve them all?

A 47 divided by 6 is 7 R 5, so 7 loaves are needed.

B 47 divided by 6 is 7 R 5, so 8 loaves are needed.

C 47 plus 6 is 53, so 53 loaves are needed.

D 47 minus 6 is 41, so 41 loaves are needed.

6. Explain It You are in line at an amusement park. You count

34 people in front of you. Each rollercoaster fits 11 people.

How many rollercoasters must run before you can get on? Explain.

3 rides must go by, and you will get

on the fourth one.

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Connecting Models and

Symbols

Reteaching

4-4

Divide 138 equally into

3 groups.

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Step 1:

You can model 138 as

13 groups of 10 plus

8 ones.

130 3 40, so each

group will get 4 groups

of 10.

40 3 120

130 120 10, so there

is 1 group of 10 left.

What You Think

What You Write

4

3 1 3 8

1 2

1

Step 2:

There is 1 group of 10 plus

1 group of 8 ones left. You

can model 18 as 18 ones.

18 3 6, so each group

will also get 6 ones. There is

nothing left.

What You Think

What You Write

4 6

3 1 3 8

1 2

1 8

1 8

0

138 3 46

Use models to help you divide.

19 47 26

1. 4 76 2. 2 94 3. 5 130

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34 71 88

4. 7 238 5. 6 426 6. 3 264

7. Algebra If n 3 57, what is the value of n?

171

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Connecting Models

and Symbols

After mowing lawns for one week, John put the money he earned on

the table. There were four $100 bills, three $10 bills, and five $1 bills.

1. If John’s brother borrowed one of the $100 bills and replaced

it with ten $10 bills,

a. how many $100 bills would there be?

b. how many $10 bills would there be?

2. If John needed to divide the money evenly with two

other workers, how much would each person receive?

Practice

4-4

Three $100 bills

Thirteen $10 bills

$145

3. If John needed to divide the money evenly with four

other workers, how much would each person receive?

Complete each division problem. You may use play money or

draw diagrams to help.

4. 5.

4

1

1

3

4

3 6

2

1

1

6

6

0

3

1

6 2

6. If $644.00 is divided equally among 7 people, how much will

each person receive?

A $82.00 B $92.00 C $93.00 D $103.00

1

5

5

1

1

$87

4

2

2

0

7. Explain It Write a story problem using two $100 bills, nine $10 bills, and

seven $1 bills.

Sample answer: Karl borrowed $297

from his sister. He pays her $33 each

week. How many weeks will it take Karl

to pay $297?

70 Topic 4

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Dividing by 1-Digit

Divisors

Reteaching

4-5

Find 362 5.

Step 1: To decide where

to place the first digit in

the quotient, compare

the first digit of the

dividend with the divisor.

Step 2: Divide the

tens. Use multiplication

facts and compatible

numbers.

Think 5 ? 35.

Step 3: Divide the

ones. Use multiplication

facts and compatible

numbers.

Think 5 ? 10.

Step 4: Check by

multiplying.

5 72 360 2 362

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3 5, so the first digit in

the quotient will not go

in the hundreds place.

Now, compare the

first two digits of the

dividend with the divisor.

36 5, so the first digit

in the quotient will go in

the tens place.

Write 7 in the tens place

of the quotient.

Multiply. 5 7 35

7

5 3 6

3 5

1

Subtract. 36 35 1

Compare. 1 5

Bring down the ones.

Write 2 in the ones place

of the quotient.

Multiply. 5 2 10

7 2R2

5 3 6 2

3 5

1 2

1 0

2

Subtract. 12 10 2

Compare. 2 5

There are no more digits

to bring down, so 2 is

the remainder.

Divide. Check by multiplying.

107 R7 35 R4 73

1. 8 863 2. 7 249 3. 5 365

56 249 R1 66

4. 8 448 5. 2 499 6. 6 396

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7. Number Sense How can you tell before you divide 425 by 9 that the first

digit of the quotient is in the tens place?

4 < 9

Topic 4 71

Name

Dividing by 1-Digit

Divisors

Find each quotient.

Practice

4-5

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293 188 R1 143 R3 149 R3

1. 2 586

2. 3 565

3. 5 718

4. 4 599

128 R2 59 23 R3 115 R7

5. 5 642

6. 6 354

7. 9 210

8. 8 927

The Paez family lives in Louisville, Kentucky, and has decided to

take a road trip for their summer vacation.

9. How many miles will the Paez family drive

each day if they decide to take 5 days to

drive 865 mi to Dallas?

10. The Paez family decides they want to drive

996 mi to Boston in 6 days. How many

miles will they drive each day?

11. Reasonableness If a staff of 9 people had to clean a hotel

with 198 rooms, how many rooms would each person have

to clean if they divided the rooms equally?

173 mi each day

166 mi

A 29 B 25 C 23 D 22

12. Explain It Explain how to check the quotient from a division problem.

Sample answer: You need to multiply the

quotient by the divisor and then add the

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remainder. This should equal the dividend.

ation

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Zeros in the Quotient

Reteaching

4-6

Find 816 4.

Step 1: Compare the

first digit of the dividend

with the divisor. 8 4,

so the first digit in the

quotient will go in the

hundreds place.

Divide the hundreds.

Think 4 ? 8.

Write 2 in the hundreds

place of the quotient.

Multiply. 4 2 8

2

4 8 1

8

0 1

Subtract. 8 8 0

Compare. 0 4

Bring down the tens.

Step 2: Compare.

1 4

You cannot divide the

tens, so place 0 in

the tens place of the

quotient.

Bring down the ones.

2 0

4 8 1 6

8

0 1 6

Step 3: Compare. Step 4: Check by

16 4

multiplying.

4 204 816

Divide the ones. Think

4 ? 16.

Write 4 in the ones place

of the quotient.

Multiply. 4 4 16

Subtract. 16 16 0

Compare. 0 4

There are no more digits

to bring down, so the

problem is done.

2 0 4

4 8 1 6

8

0 1 6

1 0

6

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Find each quotient. Check your answers by multiplying.

80 107 R1 227

1. 8 640 2. 3 322 3. 4 908

15 104 160 R5

4. 15 225 5. 6 624 6. 6 965

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7. Explain It Is 593 6 a little less than 10, a little more than 10, a little less

than 100, or a little more than 100? Explain.

A little less than 100;

593 6 98 R5 100.

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Zeros in the Quotient

Find each quotient. Check your answers by multiplying.

70

305 R1

1. 490 7 2. 326 3

3. 916 3 4. 720 2

Practice

4-6

108 R2

360

470 R1 109 R1 105 R5 140 R3

5. 2 941 6. 9 982 7. 7 740 8. 5 703

9. If there are 505 seats in an auditorium

divided equally into 5 sections, how

many seats are in each section?

10. A book company publishes 749 copies of

a novel and distributes them to 7 bookstores.

If each bookstore were to receive the same

number of copies, how many copies would

be sent to each store?

101 seats

107 copies

11. In one year, Dolores and Tom’s four children saved $420 by

recycling cans. When they divided the money equally, how

much money did each child receive?

A $50 B $100 C $105 D $1,500

12. Explain It Explain why estimating before you divide 624 6 helps you place

the first digit in the quotient.

Sample answer: By estimating you

will know if the answer will be in the

hundreds, tens, or ones.

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74 Topic 4

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Understanding Factors

Reteaching

4-7

You can use the divisibility rules to find the factors of a number.

What are the factors of 92?

Possible

Factor

1

2

3

4

5

6

9

Test of Divisibility Result of Test Factors Found

Is 92 a whole

number?

Is 92 an even

number?

Is the sum of the

digits divisible by 3?

Are the last 2 digits

divisible by 4?

Is the last digit 5

or 0?

Is it divisible by both

2 and 3?

Is the sum of the

digits divisible by 9?

Yes. Therefore it is divisible

by 1.


by 2.

No. Therefore it is not

divisible by 3.


by 4.


divisible by 5.


divisible by 6.


divisible by 9.

92 1 92

1 and 92 are factors.

92 2 46


3 is not a factor.

92 4 23


5 is not a factor.

6 is not a factor.

9 is not a factor.

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10 Is the last digit 0? No. Therefore it is not

divisible by 10.

10 is not a factor.

The factors of 92 are 1, 2, 4, 23, 46, and 92.

Find all the factors of each number.

1, 2, 3, 4, 6, 12 1, 5, 7, 35

1, 3, 5, 9, 15, 45 1, 7, 49

1. 12 2. 35

3. 45 4. 49

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5. Number Sense Is 2,340 divisible by 90? How do you know?

Yes. It is divisible by 9 and 10, because

the sum of the digits is divisible by 9,

and the last digit is 0.

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Understanding Factors

List all the factors of each number.

1. 36

2. 90

3. 84

Number Sense A number is divisible by 4 if the last two digits

are divisible by 4. Write yes on the line if the number is divisible

by 4 and no if it is not.

4. 324 5. 634 6. 172

Practice

4-7

1, 2, 3, 4, 6, 9, 12, 18, 36

1, 2, 3, 5, 6, 9, 10, 15, 18, 30, 45, 90

1, 2, 3, 4, 6, 7, 12, 14, 21, 28, 42, 84

Yes

No Yes

7. A class of 80 students is graduating from elementary school.

The teachers need help figuring out how to line up the

students for the ceremony. One row of 80 students would be

too long. What other ways could the students be arranged

for the ceremony?

2 rows of 40, 4 rows of 20, 5 rows of 16,

8 rows of 10

8. A number is divisible by another number when the

after division by that number is 0.

9. Number Sense What factor pair is missing for 45 if you

already know 1 and 45, 5 and 9?

remainder

A 7 and 6 B 8 and 6 C 3 and 15 D. 4 and 12

10. Explain It Explain how to find all the factor pairs of 40.

Sample answer: Follow the rules for

divisibility of 2, 3, 4, 5, 6, 9, and 10, and

check if the number is divisible by

7 and 8.

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76 Topic 4

Name

Prime and Composite

Numbers

Reteaching

4-8

Numbers such as 2, 3, 5, 7, and 11 are prime numbers. A prime

number has only two factors, itself and 1. A whole number that

has more than two factors is called a composite number.

3 is an example of a prime number.

Its only factors are 1 and 3.

8 is a composite number.

Its factors are 1, 2, 4, and 8.

1 3 3 1 8 8

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2 4 8

To find the prime factorization of a number, you can use a

factor tree.

Find the prime factorization of 12.

12 Think of two numbers whose product

is 12. You can use 2 6 or 3 4.

2 is prime. 2 6 6 is not prime, so keep dividing.

2 2 3 All the factors are prime, so you can

stop dividing.

The prime factorization of 12 is 2 2 3.

You can also use divisibility rules to tell whether a number is prime

or composite. In the example above, 12 is an even number, so it is

divisible by 2. Therefore, 12 is a composite number.

Write whether each number is prime or composite.

Prime

1. 17 2. 47

Prime

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Find the prime factorization.

3. 28

227

4. Number Sense The prime factorization of

a number is 2 3 3. What is the number?

18

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Prime and Composite

Numbers

Write whether each number is prime or composite.

1. 21 2. 36 3. 31

4. 87 5. 62 6. 23

Use factor trees to find the prime factorization of each number.

7. 44 8. 63

9. 13 10. 54

Practice

4-8

Composite Composite Prime

Composite Composite Prime

2211

337

11313 2333

11. Number Sense Audrey says that the prime factorization of

42 is 21 2. Is she correct? If not, tell why.

Sample answer: Audrey is not correct. Since

21 can be divided by 7 or 3, 21 is not prime.

12. Is 4,564,282 prime or composite? Explain how you

determined your answer.

Composite; It ends with an even

number, so it is divisible by 2.

13. Which of the following is a prime number?

A 105 B 27 C 19 D 9

14. Explain It Does it matter what two factors you select to

complete a factor tree? Explain.

No; the prime factors will always end

up at the bottom of the tree.

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78 Topic 4

Name



Reteaching

4-9

Mr. Rodriguez needs to store 10 test tubes in racks that hold

4 test tubes apiece. How many racks does he need?

Draw a picture.

4

10 test tubes

? racks

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Write an equation.

test tubes per rack

Let r be the number of test-tube

racks needed.

10 4 r

Solve the problem. r 2 R 2

Think: R 2 means that 2 test tubes are still left. Since they must be in a rack, one

more rack is needed.

Write the answer in a

complete sentence.

Mr. Rodriguez will need

three test-tube racks.

Explain It Mr. Rodriguez could also use racks that hold three test

tubes or racks that hold five test tubes. Which racks should he

use if he wants each rack completely filled? Explain.

He should use the racks that hold 5 test

tubes, because 10 is evenly divisible by 5.

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Topic 4 79

Name



Draw a picture and write an equation. Then solve.

Practice

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1. Tommy paid $39 to fill up the gas tank in his car. If one gallon of

gas costs $3, how many gallons of gas did Tommy put in?

Check students’ diagrams. 39 3

g; g 13; 13 gallons of gas

2. To prepare for the brunch, Ivana needs to place 8 muffins in each

basket. If she has 115 muffins, how many baskets will she need?

Check students’ diagrams. 115 8

b; b 14 R 3; 15 baskets

3. Write a Problem Write a real-world problem that you can solve

by writing an equation. The answer to the problem must be 6.

Answer will vary. Check students’

work.

4. Geometry The perimeter is the distance around an object.

The perimeter of a square is 84 centimeters. What is the

length of one side of the square?

A 75 cm B 42 cm C 21 cm D 14 cm

5. Explain It A perfect score on a quiz is 100. Mrs. Frisoli gives

students 1 point for putting their name on the paper. If there

are only 9 questions on the quiz, how much is each question

worth? Explain how you found your answer.

100 1 99. 99 9 11. Each

question is worth 11 points.

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80 Topic 4

Name

Using Patterns to Divide

Reteaching

5-1

You can use basic facts and patterns to divide mentally.

Using basic facts

What is 350 70?

Think: 350 70 is the same

as 35 tens 7 tens.

35 7 5

So, 350 70 5.

Using patterns


4

6

300

50

What is 5,400 60?

5,400 60 is the same

as 540 6.

54 6 9, so 540 6 90.

So, 5,400 60 90.

1. 280 70 2. 320 40

3. 360 60 4. 7,200 80

5. 9,000 30 6. 4,800 80

7. 2,000 40 8. 5,600 70

8

90

60

80

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9. Number Sense How is dividing 250 by 50 the same as dividing

2,500 by 500?

Sample answer: 250 50 5 and

2,500 500 5. Both problems use

the same basic fact that 25 5 5.

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10. Explain It Explain how you can mentally determine that

35,000 70 500.

You know that 35 7 5. Using math

facts and patterns you also know that

3,500

70

50 and 35,000

70

500.

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Using Patterns to Divide

In 1 through 4, find each quotient. Use mental math.

1. 360 40 36 tens 4 tens

2. 5,400 90 540 tens 9 tens

3. 240 30 24 tens 3 tens

4. 4,800 10 480 tens 1 ten


5. If the vehicles are divided evenly

among the sections, how many

vehicles are in each section?

300 vehicles

9

60

8

480

Practice

5-1

Dealership Vehicle Storage

Sections of vehicles 4

Vehicles for sale 1,200

Rows per section 10

6. If the vehicles are divided evenly among the rows

in each section, how many vehicles are in each row?

30 vehicles

7. Estimation Suppose there are 297 students going on a field trip.

If each schoolbus can carry 58 students, estimate the number

of buses that will be needed to transport all the students.

Since 300

60

60 5, about 5 school buses will

be needed.

8. Algebra If 160,000 n 4, what is the value of n?

A 40 B 400 C 4,000 D 40,000

9. Explain It Solve the equation n 50 5,000. Explain your solution.

n 100; Sample answer: Divide each

side by 50.

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82 Topic 5

Name

Estimating Quotients with

2-Digit Divisors

Reteaching

5-2

You can use compatible numbers to estimate a quotient.

Find 175 32.

Step 1: Find compatible numbers

for 175 and 32.

32 rounds to 30.

Think: 18 can be divided evenly

by 3.

180 is close to 175 and 30 is

close to 32.

Step 2: Divide. Use patterns to

help you, if possible.

Think: 180 30 is the same as

18 tens 3 tens.

18 3 6

So, 180 30 6.

Step 3: Check for

reasonableness.

6 30 180

So, a good estimate of 175 32

is 6.

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180 and 30 are compatible

numbers.

Estimate each quotient using compatible numbers.

1. 298 25

2. 5,391 77

3. 24,303 12

4. 276 42

5. 1,347 54

6. 5,564 91

10

70

2,000

7

27

60

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At Elmer Elementary School, fifth-grade students are

saving money for a summer trip to Washington, D.C.

7. The as great money as the Percy money has saved James is has how saved? many times

about three times as

great

Stud

udent Amou

ount

Save

ved

Percy $125

Emily

George

$ 80

$202

James $ 41

Bertha $159

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Estimating Quotients with

2-Digit Divisors

In 1 through 4, estimate the quotients using compatible numbers.

7

20

1. 566 81 2. 453 93

3. 1,423 69 4. 8,631 10

5

860

Practice

5-2

5. Reasoning If you use $99.00 11 to estimate $98.69 11, is $9.00 greater

than or less than the exact answer? Explain.

Greater than; Sample answer: 11 9

99, which is greater than $98.69.

6. Suppose there are 19 students in a class.

A teacher has 122 pencils and passes

them out to the class. Estimate the

number of pencils each student will receive.

6

7. At a department store, a package of 12

handkerchiefs costs $58.99. Estimate

how much each handkerchief costs.

8. Number Sense Which is the closest estimate for 2,130 33?

A 7 B 17 C 70 D 700

9. Explain It Explain how to estimate 498 12.

$5

Sample answer: Use compatible

numbers: 500 10 50.

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84 Topic 5

Name

Problem Solving:


Reteaching

5-3

Faye is putting together packets of colored beads

to give as gifts. The chart shows the beads she

had on hand yesterday. This morning she bought

4 boxes of yellow beads containing 45 beads each.

How many packets of 60 beads can she put together?

Find the hidden question or questions.

1. How many yellow beads are there? 1. 45 4 180

Trinket

Beads

Color

Number

Red 195

Blue 170

Green 175

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2. How many beads are there in all? 2. 195 170 175 180 720

Solve. 720 60 12

Write the answer in a sentence.

Look Back and Check

Is the answer reasonable?

Faye will make 12 packets.

Yes. Since 60 10 600, the


1. Faye decides to double the number of yellow beads in

the mix. How many packets will she make if she fills

each packet with 60 beads?

15

2. Explain It Faye plans to fill packets with 60 beads after

deciding not to add any yellow beads to the mix.

What hidden question or questions would you have

to ask? Explain.

The hidden question is “How many

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beads are in the mix?”

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Problem Solving:


Write and answer the hidden question. Then solve.

1. Gloria talked on her cell phone for 320 minutes the first

month, 243 minutes the second month, and 489 minutes

the third month. Her payment package does not allow her

to pay per minute; she can only buy packages. If she has

to pay $25 for every 200 minutes, how much did she pay

for the first three months?

2. Each can of paint will cover 450 tiles. Augustin is painting

300 tiles in his bathroom, 675 in his kitchen, and 100 in his

hallway. How many cans of paint does he need to buy?

Practice

5-3

How many total minutes did she use for

the first 3 months? (1,052) How many

“200 minute” packages did she use? (6);

She had to pay $150 for the first three

months.

How many total tiles is Augustin painting?

(1,075) He will need to buy 3 cans of paint.

3. Number Sense The sum of three different numbers is 18. If

every number is a prime number, what are the three numbers?

What prime numbers are less than 18?

(2, 3, 5, 7, 11, 13, 17); 2, 5, and 11 or

2, 3, and 13.

4. Explain It You earn $3 an hour as a waitress. After working

3 hours, you earn $12, $5, and $7 in tips. How much money

did you earn in total? Explain how you found your answer.

3 3 9, and 12 5 7 24,

so you earned $33 total.

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86 Topic 5

Name

Dividing by Multiples of 10

Reteaching

5-4

Find 623 40.

Step 1: Estimate the quotient

using compatible numbers,

600 40 15. Then, divide the

tens.

1

40 623 Divide 62 40 1

40 Multiply 1 40 40

22 Subtract 62 40 22

Compare 22 40

Step 2: Bring down the ones.

Then, divide the ones.

15

40 623

40

223 Divide 223 40 5

200 Multiply 5 40 200

23 Subtract 223 200 23

Step 3: Since 23 40, write 23

as the remainder in the quotient.

15 R23

40 623

40

223

200

23 Compare 23 40

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Complete.

1.

4 R 48 2 2 R15

10

R66

60 288

2.

20 455

3.

80 866

240

40

80

48

55

66

40

15

4. 30 233

7 R23 5. 50 498

9 R48

210

450

23

48

6. Reasoning Celia plans to pack her books in boxes when her family moves.

Each box will hold 20 books. Celia has 97 books. How many boxes will she

need to pack all her books?

5 boxes

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Dividing by Multiples of 10

In 1 through 6, divide.

23 R7 R1 8

10 R13 7 R4

7 R18 58 R7

1. 20 467 2. 40 321

3. 80 813 4. 40 284

5. 90 648 6. 10 587

7. To drive from New York City, NY, to

Los Angeles, CA, you must drive about

2,779 miles. If you drive 60 miles per

hour without stopping, about

how many hours do you have to drive?

Practice

5-4

About 46 hours

8. Reasoning Suppose one bottle of paint can cover 20 tiles.

You have 348 tiles. How many bottles of paint do you need to

buy to cover all 348 tiles? Explain.

18 bottles; Sample answer: You must buy

whole bottles and paint all tiles.

9. A group of 483 students is taking a field trip. One bus is

needed for every 50 students. How many buses are needed?

10 buses

10. Geometry A decagon is a ten-sided figure. If a decagon has

a perimeter of 114 centimeters, how long is each side of the

figure?

A 11.4 cm B 14 cm C 114 cm D 124 cm

11. Explain It To figure out how many hours it will take to drive from his home to

his cousin’s house, a student divides 289 by 60 and estimates that it will take

about 4.5 hours. Explain whether you think this is a reasonable estimate.

Sample answer: yes, because 60 4 240,

and 60 5 300. Since 289 is between 240

and 300, 4.5 is a good estimate.

88 Topic 5

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1-Digit Quotients

Reteaching

5-5

Find 436 53.

To find the answer, first estimate the quotient.

Think: 400 50 8 or 450 50 9

Try 9:

Try 8:

9 Write 9 in the ones place.

53 436 Multiply, 9 53 477.

477 Subtract, 436 477 41.

This estimate is too high.

8 Write 8 in the ones place.

53 436 Multiply, 8 53 424.

424 Subtract, 436 424 12.

12 Compare, 12 53. Write the remainder in the quotient.

436 53 8 R12

Check:

8 53 424

424 12 436

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Complete.

7 R 21 5 R12 30 9 R

1. 32 245 2. 64 332 3. 51 489


R20 4

R66 8

R19 7

4. 49 216 5. 79 698 6. 25 194

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7. Explain It Explain how you know the answer to the problem below

has an error.

2 R86

77 240

154

86

The remainder is greater than

the divisor.

Topic 5 89

Name

1-Digit Quotients

In 1 through 6, find each quotient.

R93

8 R30

R28 5

1. 37 120 2. 39 342 3. 62 338

Practice

5-5

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8 R34

4. 42 284 5. 82 599 6. 55 474

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7. Solomon has $118. He wants to purchase concert tickets for

himself and 5 friends. Each ticket costs $19. Does he have

enough money? Explain.

Yes: 118 19 6 R4, so he has enough

money to buy 6 tickets.

8. Number Sense Which problem will have the greater quotient, 376.0 93 OR

376 93.01? Explain how you know.

376.0 93; Sample answer: The same

number, 376, is divided by a smaller

divisor.

9. Which is 458 73?

A 5 R19 B 5 R20 C 6 R19 D 6 R20

10. Explain It A student solves the problem 354 24. The student finds an

answer of 13 R40. Explain how you can tell that the answer is incorrect

just by looking at the remainder.

Sample answer: The remainder of 40 is

greater than the divisor of 24, so

an error must have been made.

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90 Topic 5

Name

2-Digit Quotients

Reteaching

5-6

Find 866 34.

Step 1: Round the divisor to

the nearest ten. Look at the first

digit in the divisor and the first

digit in the dividend. What basic

division fact is the best estimate

of the quotient of these two

numbers?

34 866 30 866

8 3 2 R2

Step 2: Use this fact to begin the

quotient. Write it over the tens place.

2

34 866

68

186

Multiply, 2 34 68.

Subtract and bring down the

next digit in the dividend.

Step 3: What basic division

fact is the best estimate of the

next division? Use this fact and

write it over the ones place.

25 R16

34 866

68

186

170

16

Multiply, 5 34 170.

Subtract. Compare the

remainder with the divisor.

If the remainder is less than the

divisor, write it in the quotient.

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Check.

25 34 850

850 16 866

Complete.

11 R R3 R

1. 39 437 2. 24 627 3. 26 917


8 26 R7 35

13 R6 11 R24

4. 13 175 5. 44 508

6. Estimation April has 95 baseball cards. She wants to organize them on

pages that hold 18 cards each. She has 5 pages. Does April have enough

pages to organize all her cards?

No, she has room for only 90 cards.

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Topic 5 91

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2-Digit Quotients

In 1 through 6, find each quotient.

29 R7

1. 14 413 2. 29 634

21 R33

3. 35 768 4. 19 401

Practice

5-6

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21 R2

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20 R42

5. 45 942 6. 26 503

7. Reasoning The school student council sponsored a Switch Day where

students were able to switch classes every 20 minutes. The students are in

school for 7 hours. If a student switched as often as possible, how many

times did that student get to visit another classroom? (Hint: There are

60 minutes in 1 hour.)

21 times

19 R9

8. 456 students participated in Switch Day. The students raised money for

charity so that the principal would approve of the day. If the total amount of

money raised was $912, and each student brought in the same amount of

money, how much did each student raise?

$2.00

9. Estimation The total dinner bill at a buffet came out to $589 for

31 people. About how much was the buffet cost per person?

A $15.00 B $20.00 C $22.00 D $25.00

10. Explain It If you have a two-digit divisor and a three-digit dividend, does the

quotient always have the same number of digits?

Sample answer: No, the quotient can

have one or two digits.

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92 Topic 5

Name

Estimating and Dividing

with Greater Numbers

Reteaching

5 -7

Find 8,037 77.

You can use a calculator to divide large numbers.

Step 1: Estimate. Round the

divisor and the dividend.

8,037 77

8,000 80 100

The quotient should be close

to 100.

Step 2: Now, use a calculator to

find the quotient.

8,037 77

104.3766234

Step 3: Round the quotient to

the required place. Remember,

if the digit is 5 or more, add 1 to

the rounding digit. If the digit is

less than 5, leave the rounding

digit alone.

Round the quotient to the

nearest hundredth.

104.3766234 rounded to the

nearest hundredth is 104.38.

This is close to the srcinal

estimate, so the answer is

reasonable.

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Estimate first. Then use a calculator to find the quotient. Round to the nearest

hundredth if necessary.

48.67 50.75 103

1. 78 3,796 2. 51 2,588 3. 38 3,914

202.49 146.20 159.85

4. 37 7,492 5. 46 6,725 6. 62 9,911

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7. Number Sense Is 5,309 26 less than 20, greater than 20 but less

than 200, or greater than 200?

Greater than 200

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Estimating and Dividing

with Greater Numbers

Estimate first. Then use a calculator to find the quotient. Round to

the nearest hundredth if necessary.

Practice

5 - 7

119.32 122.08 364.06 104.70

1. 53 6,324 2. 52 6,348 3. 86 31,309 4. 33 3,455

192.26

88.92

5. 17,496 91 6. 25,214 47

7. 2,312 26 8. 4,895 83

536.47

58.98

The Humphrey family decided to fly from San Francisco to New

York City, and from there to Rome, New Delhi, and finally Tokyo.

9. It took the Humphrey family 6 hours to

travel from San Francisco to New York.

How many kilometers did they travel

per hour?

690 km per hour

Distances by Plane

San Francisco to New York

New York to Rome

Rome to New Delhi

New Delhi to Tokyo

10. During the flight from New Delhi to Tokyo, flight attendants

came through with snacks every 600 km. How many times

did they come through?

9 times

11. Reasoning Use the data from Exercises 9 and 10. When

the family arrived in New Delhi from Rome, the youngest son

asked the pilot how fast he was flying the plane. The pilot

told him about 847 km per hour. How many hours did it take

the family to fly from Rome to New Delhi?

4,140 km

6,907 km

5,929 km

5,857 km

A 5 h B 6 h C 7 h D 8 h

12. Explain It Write a word problem that would require you to

use 5,621 23.

Check students’ problems.

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94 Topic 5

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Problem Solving: Missing or

Extra Information

Reteaching

5-8

Aiko bought 6 red balloons and 11 clear balloons for a party. During

the party, 3 clear balloons burst but none of the red balloons did.

How many clear balloons did Aiko have after the party?

Read and Understand

What do you know?


Plan and Solve

Draw a picture of what you know.

Aiko bought 6 red balloons.

Aiko bought 11 clear balloons.

Three clear balloons burst during

the party.

No red balloons burst during the party.

The number of clear balloons remaining

after the party

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Solve the problem. 11 3 8

Write the answer in a complete

sentence.

Look Back and Check

Is your answer correct? Yes, 8 3 11

Look back at the items listed in “What you know.”

Aiko had 8 clear balloons after the party.

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the number of clear balloons Aiko bought;

the number of clear balloons that burst

1. What information helped you solve the problem?

2. What information did NOT help you solve the problem?

the number of red balloons Aiko bought;

no red balloons burst

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Problem Solving: Missing or

Extra Information

Decide if each problem has extra or missing information.

Solve if possible.

1. It takes 4 hours to drive from Boston to New York. Jordan

has a meeting in New York at 2:00 P.M. Can she arrive at her

meeting on time?

2. Franco hikes 4 miles each day for 5 days. He carries

100 ounces of water with him. It takes him 1 hour to hike

4 miles. How many hours did he hike in 5 days?

3. Write a Problem Write a real-world problem that gives

extra information. Under the problem write what the extra

information is.

Practice

5-8

Missing and when information she is leaving about for where the meeting.

Jordan

He hikes 5 hours. Extra information:

how much water he carries with him

Check students’ work. Answers will vary.

4. Critical Thinking Jorge buys T-shirts for $4 each and paints

designs on them. He sells the designed T-shirts for $7 each.

What information is needed to find how much profit he

makes in one week?

A The price of T-shirts at a store

B The color of the T-shirts that he buys

C The types of designs he draws on the T-shirts

D The number of T-shirts he sells in one week

5. Explain It Krista can type 60 words per minute. She wrote

an essay by hand in 5 hours, and it is now 4 pages long and

has 500 words in it. She wants to type up her essay. About

how long will it take to type her essay? Write what the extra

or missing information is. Then solve if possible.

Extra info: Wrote it in 5 hours and is

4 pages long. It will take her between

8 and 9 minutes to type her essay.

96 Topic 5

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Variables and Expressions

Reteaching

6-1

A variable is a letter or symbol. It represents an unknown amount

that can change.

You can do mathematical operations with variables and numbers.

You can state these operations in word expressions. You can also

state them in algebraic expressions.

Operation

the sum the difference the product

Word expression

of r and 3 between r and 3 of r and 3

Algebraic

expression

Other ways of

saying the word

expression

the quotient

of r and 3

r 3 r 3 3r r 3

3 added to r

3 more than r

3 subtracted from r

3 less than r

r multiplied by 3

3 times r

r divided by 3

3 equal parts of r

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Complete the algebraic expression for each of the following word

expressions.

1. the sum of b and 8

b ____ 8

4. 7 less than z

2. the difference

between m and 6

m ____ 6

5. 2 more than d

z ____ 7

d ____ 2

j ____ 4

Circle the letter of the correct word expression for each algebraic

expression.

3. the quotient of k and 16

7. t 13

a. 13 subtracted from t b. t subtracted from 13

8. 4n a. 4 more than n b. 4 times n

9. 11 s a. 11 more than s b. 11 less than s

k ____ 16

6. j divided by 4

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10. 45a a. the product of a and 45 b. 45 more than a

11. 6 y a. less y 6 b. 6 equal parts of y

12. v 5

a. 5 less than v b. v subtracted from 5

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Variables and Expressions

For questions 1 through 4, use a variable to write an algebraic

expression that represents the word phrase.

1. a number of apples divided into 12 baskets

2. 5 more than s

3. three times the cost for one hat

4. nine fewer than the total number of people

For 5 through 7, translate each algebraic expression into words.

5. 3 w

6. 8x

7. 40 p

Practice

6-1

n 12 or __

n

12

s 5

3 h or 3h3

t 9

Three plus a number w

Eight times a number x

p less than 40

8. Write two different word phrases for the expression t

Samples: A number t divided by 30,

a number t times __

1 , 30 equal parts of t

30 9. Number Sense Do 5 x and x 5 represent the

same expression? Explain.

__

30 .

Yes, because of the Commutative

Property of Addition

10. Algebra Dan is 12 in. taller than Jay. Use x for Jay’s height.

Which expression shows Dan’s height?

A x 12 B x 12 C 12 x D ___ x

12

11. Explain It Explain what the expression 6x means.

Sample: A number x multiplied by 6

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98 Topic 6

Name

Patterns and Expressions

Reteaching

6-2

Chan makes beaded necklaces for gifts. To make a necklace, he

strings beads onto thin ribbon. To tie the necklace, he cuts the

ribbon 10 inches longer than the length he wants to bead. The

chart shows the length of the ribbon for different beaded lengths.

Word Expression

10 inches longer than 11-inch beaded section


10 inches longer than z -inch beaded section

Number

Expression

11 10

14 10

Algebraic

Expression

z 10

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Zoe is going to use Chan’s algebraic expression to make two

necklaces. She wants to make one with a 10-inch beaded

section. The other necklace will have a 12-inch beaded section.

Here is how she evaluated the expression.

Algebraic

Expression

Number

Expression

Word Expression

Evaluated

Expression

z 1 0 10 10

z 1 0 12 10



20

22

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Chan also makes beaded bracelets. The chart

shows the length of the beaded part of the

bracelet. The second column shows the length

of the ribbon he uses to make it.

Beaded

Length (in.)

Ribbon

Length (in.)

5 5 7

6 6 7

8 8 7

1. Write a word expression that describes the pattern of the relationship.

The than ribbon the beaded length section.

is 7 inches longer

Topic 6 99

Name

Patterns and Expressions

In 1 through 4, evaluate each expression for n 3 and n 8.

1. n 10 2. ___ 24

n

13; 18 8; 3

Practice

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3. n 5 4. 36 n

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15; 40 33; 28

Complete each table.

5. 6.

n 0.9 + n

0.5

0.2

0.15

0.1

1.4

1.1

1.05

1.0

n 96 n

1

2

3

4

96

48

32

24

7. Write a Problem Write a situation that can be represented

by the algebraic expression $3.50t .


8. Algebra If a 10, which of the following is the correct

solution for a 0.1?

A 0.01 B 0.1 C 1 D 10

9. Explain It Write one numerical expression and one algebraic

expression. Then explain what the difference between a

numerical and algebraic expression is.

Check students’ work. An algebraic

expression has a variable.

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100 Topic 6

Name

More Patterns and

Expressions

Reteaching

6-3

You can use variables to write and evaluate expressions.

Sunny Rent-A-Car rents cars for $35 per day plus a $12 fee for

cleaning the car when you return it. Write an algebraic expression

for the total cost of a car rental. Evaluate the expression for a

2-day car rental; for a 3-day car rental; for a 5-day car rental.

Step 1. The total cost is the

cleaning fee plus the cost per day

times the number of days. Since

the number of days can change,

it is your variable. Use d for the

number of days.

35d 12

Step 2. To evaluate the

expression, think about counting

out the money to pay for the

car rental. Each rental will have

a stack of $12 for the cleaning

fee. Then, each rental will have

a stack of $35 for each day. A

2-day rental will have 2 stacks

of $35. A 3-day rental will have 3

stacks of $35. A 5-day rental will

have 5 stacks of $35.

Step 3. To solve for the total cost

of the car rentals, multiply the

number of stacks by $35. Then,

add the $12 stack.

2-day rental

(2 35) 12 82

A 2-day rental will cost $82.

3-day rental

(3 35) 12 117


5-day rental

(5 35) 12 187


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Cleaning fee 2-day rental 3-day rental 5-day rental

$12

$35

$35

$35 $35

$35

$35 $35

$35 $35 $35

Write an algebraic expression for each phrase. Let n stand for the

number. Evaluate each expression for n 4.

1. 3 plus 4 times a

number

2. 7 times a number,

minus 2

3. 6 less than a number

times 15

3 4n; 19 7 n 2 ; 26 15n 6; 54

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Topic 6 101

Name

More Patterns and

Expressions

1. Write an algebraic expression to represent the cost of a

concert ticket, h, with a service charge of $6.75.

h $6.75

Practice

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2. Write an algebraic expression to represent the cost of

m gallons of gasoline if each gallon costs $1.45.

m $1.45 or $1.45m

Evaluate each expression for n 3 and n 6.

3. 0.2 n 0.6 1.2 4. n 2.1 0.9 3.9

4 2 38 41

5. ___ 12

n 6. 35 n

Complete each table.

7. 8.

n 0.7 n

0.5

0.2

0.15

0.1

1.2

.9

.85

.8

n 60 n

1

2

3

4

60 30

20

15

9. Explain It What is another way to write the expression 44n?

What is another way to write the expression 44 n?

44 n;

___ 44

n

10. Which is the correct product of n 7 if n $0.25?

A $3.25 B $2.75 C $2.25 D $1.75

11. Write a Problem Write a situation that can be represented

by the algebraic expression $3.25d .

Sample answer: Keri buys d candles at

$3.25 each.

102 Topic 6

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Distributive Property

Reteaching

6-4

Hector’s rock collection is in 7 cases. Each case holds 28 rocks.

How many rocks are in Hector’s collection? You can use the

Distributive Property to find the product of 7 28.

Step 1. Split 28 into 20 8.

7 28 7 (20 8)

Step 2. Multiply 7 times each part of

the sum.

(7 20) (7 8)

140 56

Step 3. Use addition to find the sum.

140 56 196

OR

Step 1. Split 28 into 30 2.

7 28 7 (30 2)

Step 2. Multiply 7 times each part of

the difference.

(7 30) (7 2)

210 14

So, 7 28 196. Hector has 196 rocks in his collection.

Step 3. Use subtraction to find the

difference.

210 14 196

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Rewrite using the Distributive Property. Then find the product.

126 195 882 1,330

1. 3 42 2. 39 5 3. 6 147 4. 19 70

3,618 7,470 9,464 30,510

5. 54 67 6. 90 83 7. 364 26 8. 45 678

Algebra For questions 9 through 12, find the value of n.

9. 4 62 (4 n) (4 2)

10. 79 20 (80 20) ( n 20)

60

1

53

11. 53 118 (53 100) ( n 18)

12. 352 75 ( n 75) (50 75) (2 75)

300

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13. Joey’s drive. There class are is collecting 28 children food in Joey’s for the class. school Each canned childfood

brought in 15 cans of food. Use the Distributive Property to

find out how many cans of food Joey’s class collected.

(15 30 ) (15 2 ) 450 30 420 cans

Topic 6 103

Name

Distributive Property

rty

Use the Distributive Property to multiply mentally.

3,035 1,632

2,940 1,325

2,640

4,180

1. 5 607 2. 16 102

3. 7 420 4. 265 5

5. 44 60 6. 220 19

Practice

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12,600 10,912

7. 45 280 8. 341 32

9. Number Sense Fill in the blanks to show how the

Distributive Property can be used to find 10 147.

10

30 1,470

10 (150 3) (10 150) ( 3)

1,500

10. In 1990, there were 1,133 tornadoes in the U.S. If there were

the same number of tornadoes for 10 years in a row, what

would be the 10-year total?

11,330

11. There were 1,071 tornadoes in the U.S. in 2000. What is the

number of tornadoes multiplied by 20?

21,420

12. If 4 312 4 300 n, which is the value of n?

A 4 B 12 C 48 D 300

13. Explain It Margaret said that she used the Distributive

Property to solve 4 444. Is her answer shown below

correct? Explain.

4 444 4 (400 40 4)

(4 400) (4 40) (4 4)

1,600 160 16 1,776

Margaret correctly used the Distributive

Property to solve 4 444.

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104 Topic 6

Name

Order of Operations

Reteaching

6-5

If you do not use the proper order of operations, you will not get

the correct answer.

Evaluate 8 2 3 6 (1 5).

Step 1. Do the operations inside the

parentheses.

(1 5) 5

8 2 3 6 5

Step 3. Add and subtract in order

from left to right.

4 18 22

22 5 17

So, 8 2 3 6 (1 5) 17

Step 2. Multiply and divide in order

from left to right.

8 2 4 and 3 6 18

4 18 5

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Write which operation should be done first.

multiplication

division

subtraction

addition

and multiplication

109 15

19 92

1. 6 3 2 2. 13 1 4 2

3. 5 (7 2) 1 4. (19 23) (4 5)

For questions 5 through 8, evaluate the expression for x 6 and y 17.

5. 4x 5y 6. 2x (20 y )

7. x 3 y 8. 4y 2 (8x 10)

9. There are 22 students in Natalie’s class. New reading books cost $7 each.

Natalie’s class raised $50 washing cars. If there are 2 teachers in Natalie’s

class, write and evaluate an expression to show how much more each

teacher will have to pay to buy the new reading books.

(22 7 50) 2 $52

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10. Number Sense Carlos solved 20 (2 6) 8 4 29. Is this the correct

answer?

No. Carlos solved the equation from left

to right instead of following the order of

operations.

Topic 6 105

Name

Order of Operations

Use the order of operations to evaluate each expression.

19

26

8

1. 4 4 3 2. 3 6 2 3

3. 24 (8 2) 6 4. (15 11) (25 5)

5. 26 4 5 2 6. 15 (7 7) (5 2)

Practice

6-5

7

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0

7. (8 4) (7 0) 8. 5 (6 3) 10 (8 3)

9. Explain It Which is a true statement, 5 4 1 25 or 3 7 2 17?

Explain your answer.

3 7 2 3 (7 2) 17;

5 4 1 (5 4) 1 21

17

Insert parentheses to make each statement true.

10. 25 5 4 25

11. 7 4 4 2 26

12. 3 5 2 10 6

13. Strategy Practice Insert parentheses in the expression

6 10 2 so that:

a. the expression equals 32.

b. the expression equals (12 1) 2.

14. Solve (25 7) 2 4 2.

25 (5 – 4) 25

7 4 – (4 2) 26

(3 5) 2 10 6

A 18 B 11 C 6 D 5

(6 10) 2

6 (10 2)

15. Write two order-of-operation problems. Then trade with a

classmate and solve the problems.

Sample answers: 2 6 3 (8 8); 14;

(9 20) (24 2) 6; 11

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106 Topic 6

Name

Problem Solving: Act It Out

and Use Reasoning

Reteaching

6-6

In Mackenzie’s class there are 23 students. 14 students have

brown hair. 2 times as many students have blonde hair as have

red hair. How many students have each color hair?

Draw a diagram to show what you know.

There are 9 students left. You know that two times as

many have blonde hair as have red hair. Make a table to

try different numbers and see which pair fits the problem.

red blonde Do numbers add up to 9?

1 1 2 2 does not equal 9

2 2 2 4 does not equal 9

3 3 2 6 equals 9

Since 6 are two times 3 and 6 3 9, this is the correct answer.

So, in Mackenzie’s class, there are 14 students with brown hair, 6

with blonde hair, and 3 with red hair.

23 students total

14 students with

brown hair

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In 1 and 2, solve. Draw a picture and/or table to help find the answer.

1. Jacobson Animal Shelter has half as many cats as dogs.

The shelter has 30 dogs. How many total animals does the

shelter have?

45 animals

2. Summer’s mother gave Summer $20 to share with her 3

brothers equally. How much did each person get?

$5

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Topic 6 107

Name

Problem Solving: Act It Out

and Use Reasoning

1. Christina collects stamps. She has 47 stamps in all. She has

20 stamps from Europe. The number of African stamps is

2 times the number of Asian stamps. How many stamps from

each of these three continents does she have?

20 European, 18 African, 9 Asian

Practice

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2. Write a Problem Write a problem

that can be solved by acting it out

and using reasoning.

Answers will vary.

3. A public pool opened for the

summer. A total of 246 people

came swimming over the first

3 days it was open. On the first

day, 79 came to swim. On the

second day, 104 people swam.

How many people swam on the

third day?

63

4. Marissa earned $480 in the summer. If she earned $40 a

week, how many weeks did she work?

A 48 B 12 C 10 D 9

5. Explain

It How could you use cubes to act out a problem?

Sample answer: You can count out the

total number of cubes in the word

problem and then divide them up

according to the problem to help you

figure out the numbers.

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108 Topic 6

Name

Multiplying Decimals by

10, 100, or 1,000

Reteaching

7-1

You can use patterns to multiply decimals mentally by 10, 100,

and 1,000.

Andrew starts selling his baseball cards for $0.10 each. After

selling 10 cards, he has made $1.00. After selling 100 cards, he

has made $10.00.

$.0.10 $.0.10 $1.00 10

$0.10 100 $10.00

When you multiply by Move the decimal point

10 1 place to the right

100 2 places to the right

1,000 3 places to the right

If Andrew sold 1,000 cards, how much money would he make?

$100.00

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61 3,798

92,300 41.8

For questions 1 through 4, find the product using mental math.

1. 6.1 10 2. 100 37.98

3. 92.3 1,000 4. 0.418 100

5. Myla has an antique flower vase that she bought for $15.75

many years ago. The vase’s value is now 1,000 times as

great. What is the value of the vase?

$15,750

6. Raul can hit a golf ball 26.4 yards. A.J. can hit a golf ball

10 times as far. How far can A.J. hit the ball?

264 yards

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7. Reasonableness Is 0.018 a reasonable answer for 1.8 100?

No. The answer cannot be a smaller number

than the factors unless at least one factor is

between 0 and 1. The answer should be 180.

Topic 7 109

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Multiplying Decimals by

10, 100, or 1,000

Use mental math to find each product.

1. 53.7 10 2. 74.3 100

3. 66.37 1,000 4. 1.03 10

5. 92.5 10 6. 0.8352 100

7. 0.567 100 8. 572.6 1,000

9. 5.8 100 10. 0.21 1,000

11. 6.2 1,000 12. 1.02 10

13. 0.003 1,000 14. 0.002 10

15. 7.03 10 16. 4.06 100

Practice

7-1

537

66,370 7,430

10.3

925 83.52

56.7 572,600

580

6,200

210

10.2

3 .02

70.3 406

17. Algebra Kendra bought 10 gallons of gasoline at $3.26 per

gallon. How much did she pay for the gasoline?

A $326.00 B $32.60 C $1.26 D $0.26

18. Strategy Practice Freddy is helping buy ingredients for

salads for the school spaghetti dinner. He bought 10 pounds

of onions at $0.69 per pound, 100 pounds of tomatoes at

$0.99 pound, 1,000 pounds of bread crumbs at $0.09 per

pound, and 100 pounds of lettuce at $0.69 per pound. Which

of the items he bought cost the most?

A tomatoes B lettuce C bread crumbs D onions

19. Explain It Marco and Suzi each multiplied 0.721 100.

Marco got 7.21 for his product. Suzi got 72.1 for her product.

Which student multiplied correctly? How do you know?

Suzi’s answer was correct. She moved

the decimal the correct number of

spaces to the right.

110 Topic 7

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Multiplying a Decimal

by a Whole Number

Reteaching

7-2

Travis can read a book chapter in 2.6 hours. The book has

18 chapters. How long will it take Travis to read the book?

Step 1. Multiply as with whole

numbers.

2.6

18

208

260

468

Step 2. Count the total decimal

places in both factors.

2.6 1 decimal place

18 0 decimal

places

It will take Travis 46.8 hours to read the book.

Step 3. Since there is a total

of 1 decimal place in

the factors, there is 1

decimal place in the

product.

46.8

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For questions 1 through 3, find the product.

1. 2.3 2. 71.7 3. 0.894

6 12 21

13.8 860.4 18.774

4. Sara is multiplying two factors, one with one decimal place

and one with two decimal places. She says that the product

should have two decimal places. Is this correct? Explain.

This is incorrect. The factors have a

total of three decimal places, so the

product should have three decimal

places.

5. Critical Thinking Light bulbs usually cost $2. They are on

sale for 0.50 of the regular price. What is the sale price?

Is this a better price than if the sale price were 0.35 of the

regular price?

$1; No, 0.35 is a smaller fraction than

0.50; the second price would be better.

Topic 7 111

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Multiplying a Decimal

by a Whole Number

Find each product.

1. 5.4 2. 3.8 3. 0.55 4. 8.19

3 4 8 5

16.2 15.2 4.4 40.95

Insert a decimal point in each answer to make the equation true.

5. 5 6.3 315

6. 3.001 9 27009

Use the table at the right for Exercises 7–9.

7. Which desert accumulates the least

amount of rain in August?

Mojave

8. If each month in Reno had the same

average rainfall as in August, what

would the total number of millimeters

be after 12 months?

2.28 mm

31.5

27.009

Desert

Reno

Sahara

Mojave

Tempe

Practice

7-2

Average Desert

Rainfall in August

Average Rainfall

0.19 mm

0.17 mm

0.1 mm

0.24 mm

9. Explain It In December, the average total rainfall in all of the

deserts together is 0.89 mm. Explain how to use the figures

from the table to write a comparison of the total desert

rainfall in August and December.

In December, the average rainfall is

greater than in August, because 0.89

mm > 0.7 mm.

10. Algebra If 4n 3.60, which is the value of n?

A 0.09 B 0.9 C 9 D 90

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112 Topic 7

Name

Estimating the Product of a

Decimal and a Whole Number

Reteaching

7-3

You can estimate when you are multiplying a decimal by a whole

number to check the reasonableness of your product.

Zane needs to buy 27 lb of roast beef for the company party. The

roast beef costs $2.98 per pound. About how much will the roast

beef cost?

There are two ways to estimate.

Round both numbers

$2.98 27 $2.98 27

Adjust your factors to compatible

numbers you can multiply mentally.

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$3 30 $90 $3 25 $75

The roast beef will cost about $90. The roast beef will cost about $75.

Estimate each product.

Sample answers are given.

20 100

1,800 500

1. 0.8 22 2. 19.3 6

3. 345 5.79 4. 966 0.46

Number Sense Use the chart to answer questions 5 through 7.

5. About how much would it cost for Angelina

and her 4 sisters to get a shampoo and a haircut?

(5 $8) (5 $13)

about $105

Treatment

Cost

Shampoo $7.95

Haircut $12.95

Coloring $18.25

Perm $22.45

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6. Could 3 of the sisters get their hair colored for less than

$100?

Yes; $20 3 $60.

7. Angelina gets 9 haircuts per year. About how much does she

spend on haircuts for the year?

10 $13 about $130

Topic 7 113

Name

Estimating the Product of a

Decimal and a Whole Number

Estimate each product using rounding or compatible numbers.

1. 0.97 312

300

2. 8.02 70

3. 31.04 300

Practice

7-3

4. 0.56 48

560 9,000 25

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5. 0.33 104

30

9. 0.13 7

6. 0.83 12

10. 45.1 5

7. 0.89 51

11. 99.3 92

8. 4.05 11

10 50 44

12. 47.2 93

0.7 225 9,000 4,500

13. Critical Thinking Mr. Webster works 4 days a week at his office and 1 day a

week at home. The distance to Mr. Webster’s office is 23.7 miles. He takes a

different route home, which is 21.8 miles. When Mr. Webster works at home,

he drives to the post office once a day, which is 2.3 miles from his house.

Which piece of information is not important in figuring out how many miles

Mr. Webster drives per week to his office?

A the number of days at the ofce

B the distance to his ofce

C the distance to the post ofce

D the distance from his ofce

14. Strategy Practice Mrs. Smith bought her three children new snowsuits for

winter. Each snowsuit cost $25.99. How much did Mrs. Smith pay in all?

A $259.90 B $77.97 C $51.98 D $25.99

15. Explain It How can estimating be helpful before finding an

actual product?

If your estimate is very different from

your actual product, you know that you

need to check your work.

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114 Topic 7

Name

Multiplying

Two Decimals

Reteaching

7-4

Caroline earns $2.40 per hour for babysitting her brother. She

babysat last night for 3.25 hours. How much did she earn?

First, estimate your product so you can check for reasonableness.

$2.40 3.25

$2 3 6 Caroline earned about $6.00.

Step 1: Multiply as with whole numbers.

3.25

2.40

000

13,000

65,000

78,000

Step 2: Count the total number of decimal places

in both factors.

3.25 2 decimal places

2.40 2 decimal places

So, 4 decimal places total.

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Step 3: Place the decimal point in the product with the correct total number of decimal places following it.

7.8000 $7.80

Caroline earned $7.80 last night. Because $7.80 is close to your

estimate of $6, your answer is reasonable.

Find each product. Check by estimating.

0.92 28.258 1.1956

3.4048 54.498 1.628

1. 0.2 4.6 2. 3.98 7.1 3. 8.54 0.14

4. 0.532 6.4 5. 9.3 5.86 6. 0.37 4.4

7. Critical Thinking Jackie wants to buy a new CD player.

It costs $32.95. She has saved $26 and has a coupon for

30% off the price. Does Jackie have enough money to buy

the CD player?

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Yes.

the CD The player coupon to $23.07.

reduces the price of

Topic 7 115

Name

Multiplying

Two Decimals

Find each product.

Practice

7-4

1. 3.7

0.3

2. 4.4

0.2

3. 0.61

6.8

4. 1.9

0.005

1.11 0.88 4.148 0.0095

5. 0.79 4.3 3.397

6. 0.79 0.005 0.00395

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7. Number Sense The product of 4.7 and 6.5 equals 30.55.

What is the product of 4.7 and 0.65? 4.7 and 65?

3.055, 305.5

8. What would be the gravity in relation to Earth of a

planet with 3.4 times the gravity of Mercury?

1.326

9. The gravity of Venus is 0.35 times that of Jupiter. What

is the gravity of Venus in relation to Earth’s gravity?

0.91

10. How many decimal places are in the product of a number

with decimal places to the thousandths multiplied by a

number with decimal places to the hundredths?

Relative (to Earth)

Surface Gravity

Planet

Gravity

Mercury 0.39

Neptune 1.22

Jupiter 2.6

A 2 B 3 C 4 D 5

11. Explain It Explain how you know the number of decimal

places that should be in the product when you multiply two

decimal numbers together.

The total number of decimal places

in the numbers being multiplied is

the number of decimal places in the

product.

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116 Topic 7

Name

Dividing Decimals

by 10, 100, or 1,000

Reteaching

7-5

You can use place-value patterns when you divide a decimal by

10, 100, or 1,000.

Sanjai has 27.5 lb of clay. If he uses the clay to make 10 bowls,

how much clay will he use for each bowl? What if he makes 100

bowls from the clay? What if he makes 1,000 bowls?

Dividing a number by 10 moves the decimal point one place to the left.

27.5 10 2.75

Dividing a number by 100 moves the decimal point two places to the left.

27.5 100 0.275

Dividing a number by 1,000 moves the decimal point three places

to the left.

27.5 1,000 0.0275

Sanjai will use 2.75 lb for each of 10 bowls, 0.275 lb for each of

100 bowls, and 0.0275 lb for each of 1,000 bowls.

Remember: When you divide a number by 10, 100, or 1,000, your

quotient will be smaller than that number.

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For questions 1 through 6, find the quotient. Use mental math.

1.64

54.09

1. 16.4 10 2. 38.92 100 3. 297.1 100

4. 540.9 ÷ 10 5. 41.628 1,000 6. 0.33 10

7. The city has a section of land 3,694.7 ft long. The city wants

to make 100 equal-sized gardens with this land. How long will

each garden be?

36.947 ft

0.3892

0.041628

8. Reasonableness Connor divided 143.89 by 100. He said his

2.971

0.033

answer was 14.389. Is this a reasonable answer?

No. Connor moved the decimal point

only one place to the left. He should

have moved it two places to the left.

Topic 7 117

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Name

Dividing Decimals

by 10, 100, or 1,000


8.66

0.0199

0.22855

1. 86.6 10 2. 192.5 100

3. 1.99 100 4. 0.87 10

5. 228.55 1,000 6. 0.834 100

0.94335

7. 943.35 1,000 8. 1.25 10

Algebra Write 10, 100, or 1,000 for each n.

Practice

7-5

9. 78.34 n 0.7834 10. 0.32 n 0.032 11. (75.34 25.34) n 5

12. There are 145 children taking swimming

lessons at the pool. If 10 children will be

assigned to each instructor, how many

instructors need to be hired?

13. Ronald ran 534.3 mi in 100 days. If he ran an equal

distance each day, how many miles did he run per day?

A 5 B 5.13 C 5.343 D 6.201

1.925

0.087

0.00834

0.125

n 100 n 10 n 10

15 instructors

14. Explain It Carlos says that 17.43 100 is the

same as 174.3 0.01. Is he correct? Explain.

No, he is not correct. 17.43 100 is

0.1743 and 174.3 0.01 is 1.743.

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118 Topic 7

Name

Dividing a Decimal

by a Whole Number

Reteaching

7-6

Jia needs to cut a board into 3 equal pieces to

make bookshelves. The board is 1.5 yd long.

How long will each bookshelf be?

0 0.5 1.0 1.5

Laying a measuring tape next to the board, Jia

sees that her board can be cut into 3 pieces that are each 0.5 yd

long. Jia also writes out her problem to make sure her answer is

correct.

Step 1. Write a decimal point in the

quotient directly above the decimal

point in the dividend.

Step 2. Divide as you would with whole

numbers.

0.5

3 1.5

15

0

Step 3. Check your quotient by multiplying.

0.5 3 1.5

Sometimes you will need

to add a zero to the right of

the dividend so that you can

continue dividing. Example:

8.1 18

0.45

18 8.10

72

90

90

0

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Since the quotient matches the measurement

that Jia saw on her measuring tape, she knows

that her answer is correct.

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For 1 through 6, find each quotient. Check by multiplying.

0.45

0.03

3.62 0.019

1. 14 6.3 2. 77 2.31 3. 89 2.492

4. 123.08 34 5. 0.57 30 6. 562.86 59

7. A family of five people attends a theme park. They purchase

2 adult tickets for $27.50 each and 3 student tickets for

$12.50 each. If the 5 tickets are purchased with a $100 bill,

how much change do they receive?

$7.50

0.028

9.54

Topic 7 119

Name

Dividing a Decimal

by a Whole Number

Find each quotient.

Practice

7-6

5.3 11.78 0.16 0.89

1. 13 68.9 2. 35 412.3 3. 90 14.4 4. 60 53.4

3.62

5. 123.08 34 6. 0.57 30

9.54

7. 562.86 59 8. 24.4 80

0.019

0.305

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9. John paid $7.99 for 3 boxes of cereal. The tax was $1.69.

Excluding tax, how much did John pay for each box of cereal

if they all were the same price?

$2.10

10. If a package of granola bars with 12 bars costs $3.48,

how much does each granola bar cost?

A 29¢ B 31¢ C 44¢ D $1.00

11. Estimation 64.82 11 is

A a little more than 6. C a little less than 6.

B a little more than 60. D a little less than 60.

12. Explain It Explain how to divide 0.12 by 8.

Sample answer: Write zeros in the ones

and tenths places in the quotient.

Annex a zero in the dividend and

continue dividing. The quotient is 0.015.

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120 Topic 7

Name

Estimation: Decimals Divided

by Whole Numbers

Reteaching

7-7

The merry-go-round in the Smithson Town Park took in $795.60 in the

last six months. The ticket taker says that 442 people rode the merrygo-round

during that time. About how much does it cost to ride?

You can estimate to find the quotient of $795.60 442 either by

rounding or using compatible numbers and mental math.

To round your numbers, cover the digit in the hundreds place with

your hand. Look at the number next to your hand. Since 9 5, you

round $795.60 up to $800. Since 4 5, you round 442 down to 400.

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800 400 2 It costs about $2 to ride.

Or you can use compatible numbers and mental math.

800 400 2 It costs about $2 to ride.

Estimate each quotient.

Sample answers are given.

4 20

4

1. 81.2 19 2. 376.44 22 3. 62.91 16

10 60 150

4. 763.85 82 5. 550.8 9 6. 486.5 3

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7. Number Sense Explain how you would know an error had

been made if you find the quotient for 231.68 16 to be 144.8.

Possible answer: by estimating the

division as 200 20 10, which is

not close to 144.8

8. Reasonableness Is 61.5 15 a little less than 4, a little

more than 4, a little less than 40, or a little more than 40?

a little more than 4

Topic 7 121

Name

Estimation: Decimals

Divided by Whole Numbers

Estimate each quotient. Sample answers are given.

7 3 60

1. 73.5 10 2. 246.78 83 3. 185.7 3

15 6 30

Practice

4. 535.6 35 5. 553.9 90 6. 366.6 12

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5 20 90

9 30 7

7. 35.6 7 8. 86.4 4 9. 270.53 3

10. 839.7 90 11. 93.26 3 12. 77.3 11

13. Joseph is saving $23 a week to

buy a graphing calculator that

costs $275.53. About how many

weeks will it take before he can

buy the calculator?

about 12 weeks

14. Juan works at a health food store

two hours a day, three days a

week. His weekly pay is $73.50.

About how much does Juan make

per hour?

about $12 per hour

15. Reasonableness Which of the following is a reasonable

estimate for the operation 566.3 63?

A about 16 B about 9 C about 4 D about 6

16. Explain It When would you estimate a quotient instead of

finding the exact quotient?

Sample answer: Estimating a quotient

could help you quickly figure out how

many items you can buy with your

money or how long a trip will take.

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122 Topic 7

Name

Dividing a Decimal

by a Decimal

Reteaching

7-8

Sebastian is shipping a box to his cousin. The box weighs 23.6 lb.

The cost to ship the box is $88.50. How much does it cost per

pound to ship the box?

To divide a decimal by a decimal, you need to change the problem

into a simpler type of problem that you already know how to solve.

Step 1: To change this problem, you multiply the divisor

by a power of 10 until it becomes a whole number. 23.6 10 236

Step 2: You must also multiply the dividend by the

same power of 10. Sometimes, you may have to

add zeros as placeholders. 88.50 10 885.0

Step 3: Place a decimal point in the quotient

and divide as you would with whole numbers.

3.75

236 885.00

708

1770

1652

1180

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The box costs $3.75 per pound to ship.

11800

For questions 1 through 6, find each quotient.

1.3 6.1 42

0.007 1.19 1.09

1. 0.104 ÷ 0.08 2. 5.49 ÷ 0.9 3. 50.4 ÷ 1.2

4. 0.427 ÷ 61 5. 0.8449 ÷ 0.71 6. 9.483 ÷ 8.7

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7. Miriam needs to buy new notebooks for school. The notebooks cost

$0.98 each including tax. Miriam’s mother gave her $6.45 and told

her to buy 8 notebooks. Did her mother give her enough money?

No. She can only buy 6 notebooks

with $6.45.

8. Miriam finds a coupon in the store that reduces the price of notebooks to

$0.75 each including tax. How many notebooks can Miriam buy now?

She can buy 8 notebooks.

Topic 7 123

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Dividing a Decimal

by a Decimal

Find each quotient.

1. 0.8 1.84 2. 0.9 2.7 3. 2.5 4.75 4. 1.1 1.21

5. 7.1 6.39 6. 0.8 0.648 7. 1.3 10.725 8. 0.2 0.51

9. 0.07 0.77 10. 4.8 4.32 11. 0.7 8.4 12. 2.3 6.9

13. Chan paid $4.75 for trail mix

that costs $2.50 a pound.

How many pounds of trail mix

did he buy?

1.9 pounds

Practice

7-8

2.3 3 1.9 1.1

0.9 0.81 8.25 2.55

11 0.9 12 3

14. Max’s family car has a gas tank

that holds 12.5 gallons of gas.

It cost $40.62 to completely fill

the tank yesterday. What was the

price of gas per gallon?

$3.25

15. Strategy Practice Strawberries cost $5.99 per pound, and

bananas cost $0.59 per pound. How many pounds of bananas

could you buy for the cost of one pound of strawberries?

A 101.5 pounds B 10.15 pounds C 5.99 pounds D .59 pounds

16. Explain It When dividing a decimal by a decimal, why is

it sometimes necessary to add a zero to the right of the

decimal point in the quotient?

Sample answer: Often, after moving

decimal points to make the numbers

ready for division, you must add a zero

so that a smaller number can be divided

by a larger number.

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124 Topic 7

Name

Problem Solving:


Reteaching

7-9

A multiple-step problem is a problem where you may need more

than one step to find your answer.

Marcie was in a 3-day charity walk. Her friend Gayle said she

would give the charity $1.50 for each mile that Marcie walked.

The first day, Marcie walked 26.42 miles. The second day, Marcie

walked 32.37 miles. The third day, Marcie walked 28.93 miles.

How much money did Gayle give?

Step 1.

Step 2.

Step 3.

Step 4.

Read through the problem again and write a list of what

you already know.

Marcie walked 26.42, 32.37, and 28.93 miles.

Gayle gave $1.50 for each mile.

Write a list of what you need to know.

Total amount Gayle gave

Write a list of the steps to solve the problem.

Find the total number of miles Marcie walked.

Find the amount Gayle gave.

Solve the problem one step at a time.

26.42 32.37 28.93 87.72 total number of miles Marcie walked

87.72 $1.50 $131.58 total amount Gayle gave

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Use the information above to answer Exercise 1.

1. Marcie’s brother Tom was also in the charity walk. He only walked 0.8 as

far as Marcie on the first day, 0.7 as far on the second day, and 0.9 as far

on the third day. How many miles did Tom walk, rounded to the nearest

hundredth of a mile?

69.83 miles

2. Diego is buying fruit at the store. Which costs less:

1 pound of each fruit or 4 pounds of peaches?

4 pounds of peaches

Fruit

Cost pound

per

Apples $0.89

Oranges $1.29

Peaches $0.99

Grapes $1.09

Topic 7 125

Name

Problem Solving:


Write and answer the hidden question or questions

in each problem and then solve the problem.

Write your answer in a complete sentence.

1. Sue bought 2 pairs of jeans and a belt that

cost $6.95. The tax on the items was $5.85.

Jeans

Practice

7-9

Storewide Sale

$29.95 for 1 pair

OR 2 pairs for $55.00

T-shirts $9.95 for 1

OR 3 T-shirts for $25.00

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Sue paid the cashier $70.00. How much money

did Sue receive in change?

What was the total cost of Sue’s items?

$67.80; Sue received $2.20 in change.

2. A recreation department purchased 12 T-shirts for day camp.

The department does not have to pay sales tax. It paid with a

$100.00 bill. How much change did it receive?

What was the cost for 12 T-shirts?

$100.00; The department received no

change, because it paid the exact amount.

3. When Mrs. Johnson saw the sale, she decided to get clothes

for each child in her family. She bought each of her 6 children

a pair of jeans and a T-shirt. She paid $14.35 in sales tax.

How much was Mrs. Johnson’s total bill?

A $94.35 B $119.70 C $229.35 D $253.35

4. Write a Problem Write a two-step problem that contains a hidden

question about buying something at the mall. Tell what the hidden

question is and solve your problem. Use $8.95 somewhere in your

equation. Write your answer in a complete sentence.

Check students’ problems.

5. Explain It What are hidden questions and why are they

important when solving multiple-step problems?

Sample answer: Hidden questions are not

directly stated in the problem. The answers

to the hidden questions give information

needed to solve the problem.

126 Topic 7

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Name

Basic Geometric Ideas

Reteaching

8-1

Points and lines are basic geometric ideas. Lines are sometimes

described by relationships to other lines.

Draw Write Say Description

J

J

point J

J is a point. It shows an

exact location in space.

N

P

O

Q

‹___›

NO ___›

PQ

line NO

ray PQ

‹___›

NO is a line. It is a straight

path of points that goes on

forever in two directions.

___›

PQ is a ray. It goes on

forever in only one direction.

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E

F

G

H

‹___› ‹___›

EF GH

Line EF

is parallel to

line GH.

‹___›

‹___›

EF and GH are parallel

lines. They are the same

distance apart and will not

cross each other.

S

V

U

T

‹___›

SU

intersects

‹___›

VT

Line SU

intersects

line VT .

‹___ SU ›

and

___

‹ ›

VT are intersecting

lines. They pass through the

same point.

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Use the diagram on the right. Name the following.

1. two parallel lines

‹____› ___›

MN , ‹ OP

___›

2. two intersecting rays

___›

GH , JK

J

E

G

M

H

O

N

F

K

P

Topic 8 127

Name

Basic Geometric Ideas

Use the diagram at the right. Name the following.

1. three points

points P , R, , and A

2. a ray

___›

Practice

8-1

Sample answers

given for 1–7.

P J

R

Q

E I F

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P

EJ

3. two intersecting lines but not perpendicular

‹___

___›

__›

RF intersects ‹ NI

4. two parallel lines

5. a line segment

6. two perpendicular lines

‹__

__› ___›

NI ‹ AF

___

JE

‹___

___› ___›

PR ‹ RF

N

A

7. Explain It Can a line segment have two midpoints? Explain.

No, a midpoint is a point halfway between

the endpoints of a line segment. There

cannot be two midpoints.

8. Which type of lines are shown by the figure?

A Congruent

C Perpendicular

A

B

B Parallel

D Curved

9. Draw a picture Draw and label ___ two ____

perpendicular line segments KL and MN .

M

Sample drawing:

L

N

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128 Topic 8

Name

Measuring and

Classifying Angles

Reteaching

8-2

The chart below can help you describe and identify an angle.

180°

MeasureEFG.

90°

F

0°

G

Draw an angle of 65°.

180°

90°

S

E

J

0°

W

Remember to place the 0

mark on one side of the angle.

EFG measures 45°.

Place the center of the protractor on S .

Line up SW with the 0 mark. Place a

point at 65. Label it J . Draw SJ .

Classifying Angles

Acute

between 0 and 90

Right

exactly 90

Obtuse

between 90 and 180

Straight

exactly 180

Classify each angle as acute, right, obtuse, or straight. Then

measure each angle.

1. 2. 3.

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Acute; 60˚

Obtu

tuse; 120˚ Straight; 180˚

Draw an angle with each measure.

4. 100° 5. 170°

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6. Reasoning ABC measures less than 180° but more than 90°.

Is ABC a right, an acute, an obtuse, or a straight angle?

Obtuse; it has a measure between

90˚ and 180˚.

Topic 8 129

Name

Measuring and

Classifying Angles

Classify each angle as acute, right, obtuse, or straight. Then

measure each angle. (Hint: Draw longer sides if necessary.)

Practice

8-2

2.

2

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P

Right angle; 90˚

Draw an angle with each measure.

3. 120° 4. 180°

Acute angle; 30˚

5. Draw an acute angle. Label it with the letters A, B, and C. What is the

measure of the angle?

Check students’

drawings and measures.

6. Which kind of angle is shown in the figure below?

A Acute

B Right

C Obtuse

D Straight

7. Explain It Explain how to use a protractor to measure an angle.

Place center of protractor on angle’s

vertex so the 0˚ mark is on one side of

angle. Read measure where other side

of angle crosses protractor.

130 Topic 8

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Name

Polygons

Reteaching

8-3

A polygon is a closed plane figure made up of line

segments. Common polygons have names that tell

the number of sides the polygon has.

A regular polygon has

sides of equal length and

angles of equal measure.

Triangle

3 sides

Pentagon

5 sides

Octagon

8 sides

60

3 in. 3 in.

60 60

3 in.

Hexagon

6 sides

Open

Figure

Quadrilateral

4 sides

Each side is 3 in. long.

Each angle is 60.

Name each polygon. Then tell if it appears to be a regular

polygon.

Quadrilateral;

not regular

1. 2.

Hexagon;

regular

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3

Pentagon;

not regular

3. 4.

Triangle;

regular

5. Reasoning Shakira sorted

shapes into two different groups.

Use geometric terms to describe

how she sorted the shapes.

The shapes in

Group A

Group B

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Group A are all open

and are not polygons. The shapes in

Group B are closed and are polygons.

Topic 8 131

Name

Polygons

Name each polygon. Then tell if it appears to be a regular

polygon.

1. 2.

Practice

8-3

Octagon;

regular

Quadrilateral;

not regular

3

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3. Name the polygon. Name the vertices.

Quadrilateral;

B, E , H , T

T

H

E

B

4. Which polygon has eight sides?

A quadrilateral B pentagon C hexagon D octagon

5. Explain It Draw two regular polygons and two that are irregular. Use geometric

terms to describe one characteristic of each type.

Sample answer: The regular polygons

have congruent sides. The irregular

polygons have different ent angle measures.

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132 Topic 8

Name

Triangles

Reteaching

8-4

You can classify triangles by the lengths of their sides and the

sizes of their angles.

acute

all angles less

than 90

60

8 cm 8 cm

60 60

8 cm

equilateral

all sides the same

length

This triangle is both

equilateral and acute.

Not all acute triangles

are equilateral.

right

one right

angle

obtuse

one obtuse

angle

16 in. 16 in.

7.5 ft

120

5 ft

4 ft

isosceles

two sides the same

length

scalene

no sides the same

length


isosceles and right.

Not all right triangles

are isosceles.


scalene and obtuse.

Not all obtuse triangles

are scalene.

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4

Remember that the sum of the measures of the angles of a

triangle is 180°.

Classify each triangle by its sides and then by its angles.

1. cm 8 128 cm 8 2. 3.

5 in.

26 26

3 in.

14 cm

4 in.

Isosceles;

Scalene;

obtuse

right

60

3 ft 3 ft

60 60

3 ft

Equilateral;

acute

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The measures of two angles of a triangle are given. Find the

measure of the third angle.

40° 68° 104°

4. 40°, 100°, 5. 14°, 98°, 6. 38°, 38°,

Topic 8 133

Name

Triangles

Classify each triangle by its sides and then by its angles.

12.42 ft

9.5 ft

2.

4 m

60

4 m

Practice

8-4

8 ft

60

4 m

60

right

Scalene triangle

triangle;

Equilateral triangle;

acute triangle

4

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8

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P

The measures of two angles of a triangle are given. Find the

measure of the third angle.

71°

30°

3. 47°, 62°, 4. 29°, 90°,

5. 75°, 75°, 6. 54º, 36°,

61°

90°

7. Judy bought a new tent for a

camping trip. Look at the side

of the tent with the opening to

classify the triangle by

its sides and its angles.

Isosceles triangle; acute triangle

8. Reasonableness Which describes a scalene triangle?

A 4 equal sides B 3 equal sides C 2 equal sides D 0 equal sides

4 ft

3 ft

4 ft

9. Explain It The lengths of two sides of a triangle are 15 in. each.

The third side measures 10 in. What type of triangle is this?

Explain your answer using geometric terms.

It is an isosceles triangle, because two

sides are the same length.

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134 Topic 8

Name

Quadrilaterals

Reteaching

8-5

Quadrilateral Definition Example

Parallelogram

Rectangle

A quadrilateral with

both pairs of opposite

sides parallel and

equal in length

A parallelogram with

four right angles

5 in.

in. 2

in. 2

5 in.

5 ft

ft2

ft 2

Rhombus

Square

A parallelogram with

all sides the same

length

A rectangle with all

sides the same length

5 ft

4 in.

4 in. 4 in.

4 in.

1 ft

1 ft 1 ft

1 ft

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5

Trapezoid

A quadrilateral with

only one pair of

parallel sides

2 in.

in. 2

in. 3

6 in.

Remember that the sum of the measures of the angles

of a quadrilateral is 360 .

Classify each quadrilateral. Be as specific as possible.

1. 6 ft

2. 4 in.

3.

ft 3

ft 3

6 ft

Rectangle

in. 4

in. 4

4 in.

Rhombus

3 ft

7 ft

6 ft

9 ft

Trapezoid

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The measures of three angles of a quadrilateral are given.

Find the measure of the fourth angle.

4. 65°, 150°, 89°, 5. 100°, 80°, 100°,

6. 82°, 78°, 90°,

56° 80°

110°

Topic 8 135

Name

Quadrilaterals

Classify each quadrilateral. Be as specific as possible.

Practice

8-5

6 cm

2.

15 cm

18 cm

18 cm

cm8

cm8

5

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20 cm

Trapezoid

3. 8 in.

4.

4 in.

4 in.

8 in.

Parallelogram

17 cm

17 cm

15 cm

Rectangle

17 cm 17 cm

Rhombus

For 5 and 6, the measures of three angles of a quadrilateral are

given. Find the measure of the fourth angle.

47°

5. 90°, 145°, 78°, 6. 110°, 54°, 100°,

7. Name the vertices of the square to the right.

F , G, A, L

8. Three of the angles of a quadrilateral measure

80°, 100°, and 55°. Which is the measure of the

fourth angle?

A 115° B 120° C 125° D 130°

F

L

G

A

96°

9. Explain It Can a trapezoid have four obtuse angles? Explain.

Sample answer: The figure would not

be able to close the fourth side if all of

the angles were obtuse.

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136 Topic 8

Name

Problem Solving: Make and

Test

Generalizations

Reteaching

8-6

Here is a generalization to be tested: any square can be cut in half

through a diagonal. The result is always two isosceles triangles,

each with a 90 angle.

Test one example of this generalization:

Draw a square, ABCD. Draw a diagonal, AC. Inspect the triangles, ABC

and CDA.

A

B

A

B

A

A

B

D

C

Triangle ABC:

1. AB BC

D

All sides of a square

are equal length.

C

Triangle CDA:

1. CD DA

D

C

All sides of a square

are equal length.

C

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6

2. Angle B 90

All angles of a

square are 90.

2. Angle D 90

All angles of a

square are 90.

Conclusion: Each triangle has two equal sides and contains a right angle. The

generalization is true for the square ABCD.

Repeat for more squares. If for each square the conclusion is the

same, the generalization appears to be correct.

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Explain It Show that the triangles ABC and CDA are congruent.

Sample answer: AB CD and BC DA

(All sides of a square are equal length.)

AC AC. . Two triangles are congruent if

all three sides have matching lengths.

Topic 8 137

Name

Problem Solving: Make and

Test

Generalizations

In 1 through 6, test the generalization and state whether it

appears to be correct or incorrect. If incorrect, give an example

to support why.

1. All triangles have right angles.

2. All rectangles have right angles.

Practice

8-6

Incorrect; ct; some triangles have one right

angle and some triangles don’t have any.

Correct.

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3. Any two triangles can be joined to make a rhombus.

Incorrect; only two congruent isosceles

triangles can be joined to make a rhombus.

4. All rectangles can be cut in half vertically or horizontally to

make two congruent rectangles.

Correct.

5. Intersecting lines are also parallel.

Incorrect; parallel lines will never

intersect.

6. How many whole numbers have exactly three digits? Hint:

999 is the greatest whole number with three digits.

A 890 B 900 C 990 D 999

7. Explain It How can you show that a generalization is

correct?

Sample answer: test it twice to show

that it is correct both times.

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138 Topic 8

Name

Meanings of Fractions

Reteaching

9-1

What fraction of the set of shapes are squares?

Step 1: Find the denominator.

Step 2: Find the numerator.

Step 3: Write the fraction.

How many shapes are there in

the set?

How many squares are there in

the set?

Write the numerator over the

denominator.

There are 5 shapes in the set.

The denominator is the total

number of shapes. So, the

denominator is 5.

There are 3 squares in the set.

The numerator is the number

of squares in the set. So, the

numerator is 3.

Write the fraction that names the shaded part.

1. 2.

3__

5

Numerator

Denominator

3_ of the set are squares.

5

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1

3__

4

___ 4

16

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5__

8

___ 4

14

5. Number Sense If 1_ of a region is not shaded,

5

what part is shaded?

6. Alex has 7 dimes and 3 nickels. What fraction of the

coins are dimes?

__ 4

5

___ 7

10

Topic 9 139

Name

Meanings of Fractions

Write the fraction that names the shaded part.

1.

1__

6

Practice

9-1

2.

3__

In 3 and 4, draw a model to show each fraction.

3. 4_ as part of a set 4. __ 5

8 10

8

Sample answers

given for 3 and 4.

as part of a region

1

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5. Number Sense If __ 17 5 of a region is shaded, what part

is not shaded?

6. Camp Big Trees has 3 red canoes and 4 blue canoes.

What fraction of the canoes are red?

___ 12

17

3__

7

7. In a class of 24 students,13 students are girls. What fraction

of the students are boys?

A

___ 11

13

B

___ 11

24

C 13 ___

24

D

___ 24

11

8. Explain It Trisha says that if 5_ of her pencils are yellow,

7

then 2_ are not yellow. Is she correct? Explain.

7

Sample answer: Yes,

Trisha

is correct

because if 5 out of 7 are yellow, 2 out of

7 must not be yellow.

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140 Topic 9

Name

Fractions and Division

Fractions can represent division. You can write a division

expression as a fraction. For example:

Write a fraction for 5 7.

The first number in the division expression

is the numerator of the fraction. The

second number in the division expression

is the denominator of the fraction.

So, 5 7 5_

7 .

5 7

5__

7

Numerator

Denominator

Reteaching

9-2

Give each answer as a fraction.

___ 3

10

8__

9

___ 6

10

___ 7

12

2__

5

___ 9

13

1. 3 10 2. 7 12 3. 2 3

4. 8 9 5. 2 5 6. 1 6

7. 6 10 8. 9 13 9. 14 16

2__

3

1__

6

___ 14

16

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Reasoning Three congruent circles are each divided into

three equal parts. Use these three circles for 10 through 12.

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10. What part of a whole circle is shown by the white, or

unshaded, area of one circle?

11. What part of a whole circle is shown by the white,

or unshaded, area of two circles?

12. What part of a whole circle is shown by the white,

or unshaded, area of three circles?

1__

3

2__

3

__

3 or 1

Topic 9 141

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Name

Fractions and Division

Give each answer as a fraction. Then graph the answer on the

number line.

1. 3 7

2. 4 9

3. 1 5

3__

7 4 __

91

__

5

4. Use the number line to name the fraction at point A.

0 A 1

0 3

1

7

0 4

1

9

0 1

1

At a golf course, there are 18 holes. Of the 18 holes, 3 are par

threes, 8 are par fours, and 7 are par fives. What fraction of

the holes are

___ 7

18

5. par fives? 6. par threes? 7. par fours?

8. Number Sense Explain how you know that 7 9 is less than 1.

5

___ 3

18

___ 6

16

Practice

9-2

___ 8

18

Sample answer: Because 7 is less than

9, 7 cannot be divided by 9 one time.

9. After school, Chase spends 20 min reading, 30 min

practicing the piano, 15 min cleaning his room, and 40 min

doing his homework. Chase is busy for 105 min. What

fraction of the time does he spend cleaning his room?

____

15

105

10. Venietta read 4 books in 7 weeks. How many books did she

read each week?

A 6__

7

B 4__

7

C 3__

7

11. Explain It In 5 min, Peter completed 2 math problems.

Yvonne says he did 3 _ 5 of a problem each minute. Is she

correct? Explain.

D 2__

7

Sample answer: No, Peter completed

of a problem each minute.

2_

5

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142 Topic 9

Name

Mixed Numbers and

Improper Fractions

Reteaching

9-3

__

4

Example 1: Write 13 as a mixed number.

To write an improper fraction as a mixed number, draw a model.

Draw squares divided into 4 equal parts until you have 13 parts.

1 2

1 2

1 2

1

3 4

3 4

3 4

4

4 1 whole 4

4 1 whole 4

4 1 whole 1

4

The 13 parts make up 3 wholes and 1 4

of a whole.

13

4 31 4

Example 2: Write 5 1_ as an improper fraction.

3

To write a mixed number as an improper fraction, draw a model.

Draw circles divided into 3 equal parts to represent 5 1 3.

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3

1 2

1 2

1 2

1 2

1 2

1

3

3

3

3

3

1 whole 3 3 1 whole 3 3 1 whole 3 3 1 whole 3 3 1 whole 3 3

1

3

3

3 3 3 3 3 3 3 3 3 1 3 16 3

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Write each improper fraction as a mixed number.

1. 8__

3

2. ___ 10

7

Write each mixed number as an improper fraction.

4. 1 2__

5

2 2 __

3

7__

5

5. 4 6__

7

1 3 __

7

___ 34

7

3. 5__

2

6. 2 5__

8

2 1 __

2

___ 21

8

Topic 9 143

Name

Mixed Numbers and

Improper

Fractions

Write an improper fraction and a mixed number for each model.

1.

2.

7__

3 , 212

__

3

___ 10

4 , 222

__

4

Practice

9-3

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Write each improper fraction as a mixed number.

3. ___ 12

7

6. 9__

4

144 Topic 9

4. 7 __

3

7. ___ 29

13

Write each mixed number as an improper fraction.

9. 2 4__

5

12. 7 1__

8

1__

5 7

2__

1 4

___ 14

57

5

___

8

10. 8 7__

9

13. 4 3__

7

2__

1 3

2___

3

13

___ 79

31

9

___

7

5. 5__

2

8. ___ 34

8

11. 3 6__

7

14. 5 1__

4

15. Number Sense Jasmine has 41 lb of dog food to pour into

5 dishes. How many pounds of dog food should she pour in

each dish?

A 4 1__ lb B 81__ lb C 10 lb D 111__

5 5 8 lb

16. Explain It Hank needs 3 quarters to play one video game

each time. If he has 14 quarters, how many times can he

play? Explain.

2 1 __

2

4 2 __

8

___ 27

21

7

___

4

Sample answer: Hank can play the

video game 4 times. He cannot play the

game 4 _ 2 times because the game will

3

not work with only 2 quarters.

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Equivalent Fractions

Reteaching

9-4

The fractions shown below are equivalent.

They all describe the same part of a whole.

3_

5

__ 6

10

__ 12

20

To find equivalent fractions, multiply or divide the numerator and

denominator by the same number.

3__

5 2__

2 ___ 6

10

___ 6

10 2__

2 ___ 12

20

___ 6

10 2__

2 3__

5

___ 12

20 2__

2 ___ 6

10

___ 12

20 4__

4 3__

5

Name two equivalent fractions for each fraction.

1.

3.

1__

3

___ 4

20

2__

6 , __ 3 9

___ 2

10 , __ 1

5

2.

4.

___ 2

12

___ 2

16

Sample answers given

for 1–4.

1__

6 , ___ 4

24

__ 1

8 , ___ 4

32

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Find the missing number to make the fractions equivalent.

5.

4__

7 ___ 8

6. ___

18 4__

6

7.

3__

4 ___

12

8. ___ 15 3__

4

14 12

9 20

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9. Number Sense Are 3_

4

and __ 12 equivalent fractions? Explain.

16

Yes: If you multiply the numerator and

3_

__

12

denominator of

4 by 4, you get

16 .

Topic 9 145

Name

Equivalent Fractions

Name two equivalent fractions for each fraction.

1.

4.

5 ___

15

___ 4

28

1__

3 ; ___ 10

30

___ 2

14 ; __ 1 7

2.

5.

6 ___

36

3 ___

21

1__

6 ; ___ 3

18

___ 6

42 ; __ 1 7

3.

6.

___ 2

12

___ 2

11

Practice

9-4

Sample answers

given for 1–6.

1__

6 ; ___ 4

24

___ 4

22 ; ___ 6

33

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Find the missing number to make the fractions equivalent.

7.

9.

___ 4

13 8__

___ q

54 2__

9

x 26

n 36

q 12 h 40

x 8. ___ 12

30 ___ n

90

11. Renie gave each of six people 1

10

left. Explain how this is true.

__

10. ___ 14

h ___ 7

20

2_

of a veggie pizza. Renie has of the pizza

5

Sample answer: Renie gave away __

6

10 ,

or _ 3 5 , of the pizza, so _ 2 5 , or __

4 , of it is left.

10 12. Which fraction is equivalent to _ 3 7?

A 3__

B

___ 6

6

14

C

3 ___

17

D 7__

7

13. Explain It Jacqueline had four $5 bills. She bought a shirt for

$10. Explain what fraction of her money Jacqueline has left.

Use equivalent fractions.

Sample answer: Four $5 bills equal

$20. After she spent $10, she had __

10

20

or _ 1 , of her money.

2

20 ,

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146 Topic 9

Name


Fractions and Mixed Numbers

Reteaching

9-5

You can compare fractions by finding a

common denominator.

Samantha and her brother Jacob went out

for pizza. Samantha ate 1_ of her pizza. Jacob

2

ate __ 4 of his pizza. Who ate more pizza?

12

Because these fractions have different

denominators, you need to find a common

denominator. Then you can compare them.

Step1.

Write multiples of the two denominators until you get a

common multiple.

2: 2, 4, 6, 8, 10, 12

12: 12, 24, 36, 48, 60 Use 12 as the common denominator.

Step 2. Since you multiply 2 6 to get 12, you must multiply 1 6.

1__

2 ___ 6

12

Step 3.

Compare the fractions with common denominators.

___ 6

12 ___ 4

12

Samantha’s Pizza

So, Samantha ate more pizza.

Remember: If you don’t know the multiples of the denominators, you

can multiply the denominators together to get a common denominator.

Jacob’s pizza

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Compare. Write , , or for each .

1. 2__

3

1__

6

2. 3__ 1__

> 4 > 2

3. 5__

6 <

Order the numbers from least to greatest.

4.

4__

5 , 3__

5 , 3__

4

5. 1 5__

6 , 13__ 6 , 1 ___ 2

12

, ,

, ,

3__

5 , __ 3 4 , __ 4 5

1__

2

12 , 1

, 13 _

6 , 151

_

6

6. Geometry Sofia baked three kinds of pie. Sofia’s Mom told her to

__ 8

4_

3_

bring 16 of the apple pie, 8 of the pecan pie, and 6 of the pumpkin pie to

school to share with her friends. Draw the pies and show which pie will

have the greatest amount brought to school.

All three pies will have equal amounts

brought to school, because __

8

___ 21

24

16 _ 4 8 _ 3 6 .

Topic 9 147

Name


Fractions and Mixed Numbers

Compare the numbers. Write , , or for each .

1. 6__

7

4. 2 4__

5

>

<

6__

8

2 5__

6

2. 4__

9

5. 3 6__

9

<

=

2__

3

3 2__

3

3. 1___

1

10

6. 2__

5

>

>

2__

8

Practice

9-5

1___

1

12

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Order the numbers from least to greatest.

7. 4__

6 , 4__

8 , 3__

4 , 5__

8

8. 4 1__

4 , 41__ 8 , 510 ___

11 , 4 ___ 2

12

9. 1 3__

7 , 13__ 4 , 12__ 4 , 1 ___ 8

14

10. Number Sense How do you know that 5 1_

4

5_

1 4 5 __

5

20

__ 48 , __ 58 , __ 46 , __ 34

4_

1 8 , 4 __

2

12 , 414

_

4 , 5

7 , 121

_

4 , 131

_

4 , 1 __

8

14

1 3 _

20 ; 5 4

__

10 5 __

8

20

is less than 5___

4

10 ?

20 ; 5 __

5

20 < 5 8

__

20

, 510 __

10

11

11. A mechanic uses four wrenches to fix Mrs. Aaron’s car.

The wrenches are different sizes: __ 5 in.,

1_

in.,

1_

in., and __ 7

in.

16 2 4 16

Order the sizes of the wrenches from greatest to least.

1_

2 in.,

in., __

7

16 in., __

5

16 in., _ 1 4 in.

12. Which is greater than 6 1_

3 ?

A 6 1_

B 6 1_

C 6 1_

D 6 1_

6 5 4 2

__

13. Explain It Compare 3 3

22 and 3 2 . Which is greater?

33

How do you know?

3__

3

22 3 __

9

66 3__

2

33 3 __

4

66

__

, which is greater than

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148 Topic 9

Name

Common Factors and

Greatest Common Factor

Reteaching

9-6

The greatest common factor (GCF) of two numbers is the greatest

number that is a factor of both.

Find the greatest common factor of 12 and 18.

You can use arrays to find the factors of 12.

1 12 12 2 12 6

3 4 12

Factors of 12 are 1, 2, 3, 4, 6, and 12.

You can use arrays to find the factors of 18.

1 18 18

2 18 9

3 6 18

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Factors of 18 are 1, 2, 3, 6, 9, and 18.

You can see that the common factors of 12 and 18 are 2, 3, and 6.

The greatest common factor of 12 and 18 is 6.

Find the GCF of each pair of numbers.

9 5

1 8

1. 9, 27 2. 25, 40

3. 7, 36 4. 40, 48

5. Number Sense Can the GCF of 18 and 36 be greater than 18? Explain.

No. No number greater than 18 can

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be a factor of 18.

Topic 9 149

Name

Common Factors and

Greatest Common Factor

Find the GCF of each pair of numbers.

1. 15, 50

4. 18, 32

5. 7, 28

6. 54, 108

Practice

9-6

5 3 5

2. 6, 27

3. 10, 25

2 7 54

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7. 25, 55

8. 14, 48

9. 5 2 81, 135 27

10. Number Sense Can the GCF of 16 and 42 be less than 16? Explain.

Yes;

Sample

answer

wer: The

great

eatestest

common factor will be less than 16,

because 42 is not evenly divisible by 16.

11. A restaurant received a shipment of 42 gal of

orange juice and 18 gal of cranberry juice. The

juice needs to be poured into equal-sized containers.

What is the largest amount of juice that each

container can hold of each kind of juice?

6 gal

12. At a day camp, there are 56 girls and 42 boys.

The campers need to be split into equal groups.

Each has either all girls or all boys. What is the

greatest number of campers each group can have?

13. Which is the GCF of 24 and 64?

A 4 B 8 C 14 D 12

14 campers

14. Explain It Do all even numbers have 2 as a factor? Explain.

Sample answer: Yes, because all even

numbers are divisible by 2

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150 Topic 9

Name

Fractions in Simplest Form

Reteaching

9-7

There are two different ways to write a fraction in simplest form.

Write

___ 20 in simplest form.

24

Divide by Common Factors

• Divide by common factors until

the only common factor is 1.

• You can start by dividing by 2,

since both numbers are even.

______ 20 2

24 2 ___ 10

12

But both 10 and 12 are also even,

so they can be divided by 2.

______ 10 2

12 2 __ 5 6

Divide by the GCF

• First find the GCF of 20 and 24.

20: 1, 2, 4, 5, 10, 20

24: 1, 2, 3, 4, 6, 8, 12, 24

• The common factors of 20 and 24 are 1,

2, and 4. The GCF of 20 and 24 is 4.

• Divide both the numerator and the

denominator by 4.

______ 20 4

24 4 __ 5 6

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• Since 5 and 6 do not have

any common factors, 5_ is the

6

simplest form.

20 ___

24 written in simplest form is 5 __

6 .

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Write each fraction in simplest form.

1. ___ 16

20

4.

7.

8 ___

32

18 ___

21

4_

51_

46_

7

2.

8 ___

16

5. ___ 18

42

8. ___ 24

40

10. Number Sense Explain how you can tell that

___ 31

33

3.

6.

5 ___

10

15 ____

100

9. ___ 55

70

is in simplest form.

Sample answer: 31 is a prime number; 31

and 33 do not have any common factors

except 1, so __

31

is in simplest form.

33 1_

23_

73_

5

1_

__

23

20

__

11

14

Topic 9 151

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Name

14. Explain It Explain how you can find the simplest form of ____ 100

1,000

Practice

9-7

Fractions in Simplest Form

Write each fraction in simplest form.

_____ 3.

___ 9

_____

27

4.

___ 3

1__

5__

_____ 5. ___ 10 _____ 6.

___ 9

3__

_____

15 5

12 6

15 5

1__

___ 5

1__

1.

___ 5

1__

_____ 2.

___ 6

10 2

24

1__

4

1__

3

7. ___ 18 2 _____

9

8. ___ 25 60 _____

12

9. ___ 12 72 _____

6

10. Number Sense Explain how you can tell 4_ is in simplest form.

5

denominator have no common factors

other than 1.

Write in simplest form.

11. What fraction of the problems on

the math test will be word problems?

__ 17 Math Test

➡ 20 Multiple-choice problems

➡ 10 Fill in the blanks

➡

5 Word problems

12. What fraction of the problems on the math

test will be multiple-choice problems?

4__

7

13. Which is the simplest form of 10

82 ?

A 1__

8

B

___ 1

22

C

___ 10

82

D

___ 5

41

Sample answer: The numerator and the

Sample answer: The GCF of 100

and 1,000 is 100. The simplest form

is ________

100100

__

152 Topic 9

1,000100

100 1

10 .

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Name

Tenths and Hundredths

Reteaching

9-8

Fractions can also be named using decimals.

8 out of 10 sections

are shaded.

The fraction is

8

.

10

The word name is

eight tenths.

The decimal is 0.8.

Remember: the first

place to the right of

the decimal is

tenths.

Write 2 as a decimal.

5

Sometimes a fraction

can be rewritten as an

equivalent fraction that

has a denominator of

10 or 100.

2 2 2 4

5 5 2 10

4

10

0.4

2

So, 0.4.

5

3

Write 3 as a

5

decimal.

First write the whole

number.

3

Write the fraction as an

equivalent fraction with

a denominator of 10.

Change the fraction to

a decimal.

3 3 2 6

5 5 2 10 0.6

Write the decimal next

to the whole number

3.6

3

So, 3 3.6.

5

Write 0.07 as a

fraction.

The word name for

0.07 is seven

hundredths.

“Seven” is the

numerator, and

“hundredths” is the

denominator.

7

So, 0.07 .

100

Remember: the

second place to the

right of the decimal is

hundredths.

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Write each fraction or mixed number as a decimal.

1. 1__

2.

___ 6

3. 2 3__

5

25

4

0.2 0.24 2.75 3.9

Write each decimal as a fraction or mixed number.

1____

25

100

____

65

100

5. 1.25 6. 3.29

7. 0.65 8. 5.6

3____

29

100

5___

6

10

4. 3___

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9. Number Sense Dan says 3_ is the same as 3.5. Is he correct? Explain.

5

No. 3.5 is the same as 3 __

10 5 or 3 _ 1 2; _ 3 5 is the

same as 0.6.

Topic 9 153

Name

Tenths and Hundredths

Write a decimal and fraction for the shaded portion of each

model.

1. 2.

Practice

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Write each decimal as either a fraction or a mixed number.

3. 0.6 4. 0.73

5. 6.9 6. 8.57

Write each fraction or mixed number as a decimal.

7.

7 ___

10

9. 7___

2

10. 3____

9

10

100

Use division to change each fraction to a decimal.

11. 4__

5

13.

0.16;

1 ___

50

154 Topic 9

__

4

25 0.7;

__

6

___

73

10

100

6__

9

8___

57

10 100

0.7

8.

____ 33 0.33

100

7.2 3.09

0.8

12. ___ 0.48

12

25

0.02 0.55

14. ___ 11

20

15. Strategy Practice When you convert 0.63 to a fraction,

which of the following could be the first step of the process?

A Since there are 63 hundredths, multiply 0.63 and 100.

B Since there are 63 tenths, divide 0.63 by 10.

C Since there are 63 tenths, place 63 over 10.

__

7

10

D Since there are 63 hundredths, place 63 over 100.

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Name

Thousandths

Reteaching

9-9

Example 1: Write 0.025 as a fraction.

Ones . Tenths Hundredths Thousandths

0 . 0 2 5

You can use a place-value chart to write a decimal as a fraction.

Look at the place-value chart above. The place farthest to the

right that contains a digit tells you the denominator of the fraction.

In this case, it is thousandths. The number written in the placevalue

chart tells you the numerator of the fraction. Here, it is 25.

0.025

25

1,000

Example 2: Write 11 as a decimal.

1,000

Ones . Tenths Hundredths Thousandths

.

You can also use a place-value chart to write a fraction as a

decimal. The denominator tells you the last decimal place in

your number. Here, it is thousandths. The numerator tells you the

decimal itself. Write a 1 in the hundredths place and a 1 in the

thousandths place. Fill in the other places with a 0.

11

1,000 0.011

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Write each decimal as a fraction.

____

2

____

37

1,000 1,000

1. 0.002 2. 0.037 3. 0.099

____

99

1,000

Write each fraction as a decimal.

5

4.

0.005

76

1,000

5.

0.076

40

1,000

6.

0.040

1,000

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7. Explain It Matt reasoned that he can write 9

your answer.

1,000

as 0.9. Is he correct? Explain

No: ____

1,000 9 is read as “nine thousandths,”

which is the decimal 0.009. 0.9 is read

as “nine tenths,” which is the fraction 9

__

10 .

Topic 9 155

Name

Thousandths

Write each decimal as either a fraction or a mixed number.

7

1. 0.007 1,000

2. 0.052

38

3. 0.038 1,000

4. 0.259

20

5. 3.020 1,000

6. 4.926

3 4

52

1,000

259

1,000

926

1,000

Practice

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Write each fraction as a decimal.

7.

9.

11.

73

1,000

854

1,000

5

1,000

12.

996

1,000

Write the numbers in order from least to greatest.

13.

5

1,000 , 0.003, 9

1,000

14. 0.021, 0.845,

0.073 593

8.

0.593

1,000

0.854 11

10.

0.011

1,000

0.005 0.996

99

____

5

____

9

0.003,

1,000 , 1,000

99

0.021,

____

1,000 , 0.845

1,000

15. Look at the model at the right. Write a fraction and a

decimal that the model represents.

____

11

1,000 and 0.011

16. Reasoning In Tasha’s school, 0.600 of the students

participate in a school sport. If there are one thousand

students in Tasha’s school, how many participate

in a school sport?

A 6,000 B 600 C 60 D 6

17. Explain It Explain how knowing that 5 8 0.625 helps you write the

decimal for 4_

5

Sample answer: Knowing that 5 8

0.625 gives the digits to write to the right

of the decimal point.

156 Topic 9

8 .

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Name

Fractions and Decimals

on the Number Line

Reteaching

9-10

Show 0.8, __ 6 , and 0.519 on the same number line.

20

0 0.3 0.519 0.8 1

Step 1: Starting at 0, count 8

tenths to the right. This point is

0.8 or __ 10. 8

Step 2: Change __ 6 to a decimal. Step 3: Estimate the location

20

__ 6

can be thought of as 6 20. of 0.519.

20

6 20 0.3

You know that 0.5 is the same as

Starting at 0, count 3 tenths to 0.500. You also know that 0.6 is

the right. This point is __ 6

20 or 0.3. the same as 0.600.

So, 0.519 is between 0.5 and 0.6.

Show each set of numbers on the same number line.

__

1. 0.1, 6

10 , 0.5

0 0.1 0.5 0.6

1

You know that 0.519 0.550.

So, 0.519 is between 0.500 and

0.550 and closer to 0.500 than

to 0.550.

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0

2.

__ 9

10 , 0.7, __ 4

20

__

50

3. 0.25, 0.40, and 5

0 0.2 0.7 0.9 1

0 0.1 0.25 0.4

1

Name the fraction and decimal for each point.

4. Point A

5. Point B

6. Point C

__

10

0.3 or 3

0.55 or 55

___

100

0.89 or 89

___

100

0 A B C 1

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Topic 9 157

Name

Fractions and Decimals on the

Number Line

Draw a number line to show the set of numbers. Then order the

numbers from least to greatest.

1. 0.75,

___ 8 2__

, 0.2,

10 5

__

10

0.2

2

0.75

5

0.2, _ 2 5 , 0.75, 8

Write a fraction or mixed number in simplest form and a decimal

that name each point.

Practice

9-10

0 1

8

10

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0

Q R S

__

3

10 ;

0.3

1

4_

5 ;

2. Point Q 3. Point R 4. Point S

5. Uma recorded the distances that volunteers walked in the

charity event. Grace walked 1 3_ mi, Wendell walked 1.3 mi,

5

and Simon walked 1__

10 1 mi. Show these distances on a

number line. Who walked the farthest?

Sample drawing:

2

0.8 1

_ 2 5 ; 1.4

1

10

1.3

1

5

Grace

1

2

6. Number Sense Which is a decimal that could go between the

mixed numbers 4 3_

5 and 4__

9 on a number line?

10

A 4.45 B 4.5 C 4.75 D 4.92

7. Explain It Explain how you know that 5.5 is to the right of

5_

1 4 on the number line.

Sample answer: 5 _ 1 4 5.25; 5.25 5.5;

So, 5.5 must be to the right of 5 1_

4 .

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158 Topic 9

Name

Problem Solving:

Writing to Explain

Reteaching

9-11

An environmental scientist is

studying an old apple orchard.

The orchard is shown on the

right. Some of the trees are

infected with mold. Other trees

are infested with beetles. Some

trees are normal.

LEGEND

normal apple tree

apple tree infected with mold

apple tree infested with beetles

The scientist knows that pictures

and symbols can be used to write

a good math explanation. So she

decides to organize her findings

in the chart on the right.

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Use this chart to estimate the fractional part of the orchard that is

infected with mold, using a benchmark fraction that is close to the

actual amount.

A little more than half the grid is covered by trees that are infected

with mold.

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Use this chart to estimate the fractional part of the orchard that is

infested with beetles. Explain how you decided.

One of the eight columns is filled

with trees infested with beetles.

This column represents _ 1 8 of the orchard.

Topic 9 159

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Problem Solving:

Writing to Explain

Estimate the fractional part of the shaded portions below.

Explain how you decided.

1.

2.

3. Draw a square and shade about 1_ of it. How did

8

you decide how much to shade?

4. Draw two rectangles that are different sizes. Shade about 1_ of

each. Are the shaded parts the same amount? Explain. 2

Practice

9-11

The shaded area is about half

1_

1_

of

2, so it is about

4.

About _ 2 , because it is 3 more

than _ 1 , but less than _ 3

2

Check students’ work. Sample:

I shaded half of _ 1

4 .

4

Check students’ drawings. No; the shaded

parts are not the same amount, because

the rectangles are not the same size.

5. Explain It Look at a picture of the American flag.

Approximately what part of the flag is blue? Explain.

__

3

About

16; ; since less than

4

is blue, and white stars are of also the in flag

the

blue area.

1_

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160 Topic 9

Name


ting

Fractions with Like Denominators

Reteaching

10-1

Aisha cut her birthday cake into 10 slices. She and her friends ate

6 slices. Her parents ate 2 slices. What fraction of cake was eaten?

___ 6

10

amount of cake Aisha and her friends ate.

___ 2

amount of cake Aisha’s parents ate.

10

When two fractions have the same denominator, their sum

or difference will also have the same denominator.

Step 1. Add the numerators. Step 2. Write the sum over the denominator.

6 2 8

___ 8

10 4__

5

___ 8

10

Step 3. Simplify the fraction if possible.

So, 4__

5

You can do the same when subtracting fractions with like denominators.

of the cake was eaten.

How much more cake did Aisha and her friends eat than Aisha’s parents?

___ 6

10 ___ 2

10 =

Step 1. Subtract the numerators. Step 2. Place the difference over the

denominator and simplify if possible.

6 2 4

___ 4

10 2__

5

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0

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Find the sum or difference. Simplify your answer.

1. __ 3

8

3__

8

3_

4

2. ___ 11

12

___ 5

12

3. ___ 9

10

___ 4

10

1_

1__

3

2 10

4. 7__

9

2__

9

5_

9

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5. Joachim has 15 medals he won at swim meets. Of the medals, 6 are first

place, 4 are second place, 2 are third place, and 3 are fourth place. What

fraction of Joachim’s medals are first or second place?

6. Algebra Mr. Lucero’s cornbread recipe calls for 3 cups

of cornmeal. He put in 1 2_ cups from one open bag and

2_

3 3 cup

from another open bag. How many cups of cornmeal does

Mr. Lucero need to add when he opens a third bag?

_ 2

3

2_

3 cups

Topic 10 161

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Fractions with Like Denominators

Add or subtract. Simplify if possible.

1. ___ 10

12

___ 8

12

5.

7.

9.

6__

8 5__

8 3__

1__

4 2__

4 3__

2__

5 2__

5 3__

2. 8__

9

5__

9

1 1

_ 2 __ 13

1_

3 4

1_

1 2

1_

2 5

3. ___ 7

10

___ 2

10

8 6. ___ 8

10 ___ 3

10

4 8. ___ 9

11 ___ 1

11

5 10.

10. 7__

8 3__

8

11. What fraction could you add to 4_ to get a sum

7

greater than 1?

any fraction greater than _ 3

4. 2__

3

1__

3

Practice

10-1

9

___ 10 1__ __ 13

2

___ 8

11 __ 1

2

7

12. Reasoning Write three fractions, using 10 as the

denominator, whose sum is 1.

Possible answer: 1

__

10 __

3

10 __

6

10 1

13. Which of the following represents the difference between

two equal fractions?

A 1

B 1__

2

C 1__

4

D 0

14. Explain It In one night, George reads 3 chapters of a book

with 27 chapters. After the second night, he has read a total

of __ 8 of the book. Explain how you would determine the number

of chapters 27 George read the second night. Solve the problem.

__

8

27 __

3

27 __

5

27 __

27

; on the second night,

George 5

27 27 5 chapters

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162 Topic 10

Name

Common Multiples and Least

Common Multiple

Reteaching

10-2

To find the least common denominator (LCD) of two fractions

with different denominators, you need to find the least common

multiple (LCM).

Find the least common multiple and LCD of 3_ 5_

and

4 6 .

Step 1. Find the least common multiple of 4 and 6. Step 2. Make the least common multiple

4

XXXX

4

XXXX

XXXX

4

XXXX

XXXX

6

XXXXXX

6

XXXXXX

6

XXXXXX

XXXXXX

Are there the same number of X’s

in each column?

Since the 1st column has fewer,

add another set of 4 X’s.


in each column?

Since the 2nd column has fewer,



in each column?

Since the 1st column has fewer,


the least common denominator of both

fractions.

3

4

3

3

9

12

5

6

2

2

10

12

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-

2

4

XXXX

XXXX

XXXX

6

XXXXXX

XXXXXX


in each column? Yes.

Since the columns are equal, the number

of X’s is the LCM. The LCM 12.

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Find the least common multiple of each number pair.

6

18

1. 2 and 3 2. 6 and 9 3. 5 and 6 4. 8 and 3

5. Reasonableness Can the LCD of 4_

9

30

and __ 13 be less than 17? Explain.

17

24

No. The LCD for two fractions must be

the LCM of the denominators. The LCM

of two numbers cannot be less than

either of the numbers.

Topic 10 163

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Name

Common Multiples

and Least Common Multiple

Find the LCM of each pair of numbers.

6

24

12

1. 3 and 6 2. 7 and 10

3. 8 and 12 4. 2 and 5

5. 4 and 6 6. 3 and 4

40 42

20

7. 5 and 8 8. 2 and 9

9. 6 and 7 10. 4 and 7

11. 5 and 20 12. 6 and 12

13. Rosario is buying pens for school. Blue pens are sold in

packages of 6. Black pens are sold in packages of 3, and

green pens are sold in packages of 2. What is the least

number of pens she can buy to have equal numbers of

pens in each color?

6 of each

14. Jason’s birthday party punch calls for equal amounts of

pineapple juice and orange juice. Pineapple juice comes in

6-oz cans and orange juice comes in 10-oz cans. What is the

least amount he can mix of each kind of juice without having

any left over?

30 oz of each

70

10

12

18 28

12

Practice

10-2

15. Reasonableness Dawn ordered 4 pizzas each costing

between 8 and 12 dollars. What is a reasonable total cost

of all 4 pizzas?

A less than $24 C between $32 and $48

B between $12 and $24 D about $70

16. Explain It Why is 35 the LCM of 7 and 5?

Sample answer: It is the smallest

number divisible by both 7 and 5.

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164 Topic 10

Name

Adding Fractions with

Unlike Denominators

Reteaching

10-3

Danisha ate 2_ cup of yogurt at breakfast. She ate

1_

cup of yogurt

3 4

at lunch. How much yogurt did she eat today?

You can add fractions with unlike denominators.

Step 1: Find the least common

denominator of the two fractions.

multiples of 3: 3, 6, 9, 12, 15

multiples of 4: 4, 8, 12, 16, 20

2_

3 __ 8 1_

and

12 4 __ 3

12

Step 2: Once you have equivalent

fractions with the same

denominator, add the numerators.

8 3 11

Step 3: Place the sum over

the common denominator and

simplify your fraction if possible.

Danisha ate __ 11 cup of yogurt

today.

12

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For questions 1 through 5, find the sum. Simplify if possible.

1.

4.

4__

5

1__

2

1___

10

3

1__

2 1__

6 __ 3

1___

5

12

2.

5.

4

8__

9

3.

4__

9

__ 5

__ 3

6 4

2__

3 1__

9 __ 5 6

6. Kevin and some friends ordered several pizzas and cut them

into different numbers of slices. Kevin ate 1_ of one pizza,

6

1_ of another, __ 5

4 12 of another, and 1_ of another. Did Kevin eat

3

the equivalent of a whole pizza?

__ 3

1___

13

18

1___

11

18

1___

7

36

7. Cathy spent

10 of an hour on her math homework, 2_ of an

5

hour on her science homework, and 3_ of an hour on her

4

yes: 1 __ 1 6 pizzas

reading homework. How long did Cathy work on homework?

1___

20 9 hours

Topic 10 165

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Name

Adding Fractions with

Unlike Denominators

Find each sum. Simplify if necessary.

1. 2__

9 1__

3

4. 1__

4 2__

3

7. 1__

6 ___ 5

12

10. 3__

4 4__

5

5_

9

__

11

12

7

__

12

1__

11

20

2. 1__

7 ___ 3

21

5.

___ 1

12 4__

6

8. 4__

6 1__

3

11. ___ 11

12 1__

3

2_

7

3_

4

1

1 1 _

4

3. 2__

3 1__

5

6. 1__

2 3__

5

9. 2__

5 1__

8

12. 4__

8 1__

2

Practice

10-3

__

13

15

1__

1

10

21

__

40

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Jeremy collected nickels for one week. He is making stacks of

his nickels to determine how many he has. The thickness of one

nickel is __ 1

16 in.

13. How tall is a stack of 16 nickels?

1 in.

__ 3

8 in.

14. What 1 nickel? is the combined height of 3 nickels, 2 nickels, and

15. Number Sense Which fraction is greatest?

A 5__

6

in. B 7__

9

21

to __?

25 15

C 2__

3

16. Explain It Which equivalent fraction would you have to use

in order to add 3_

5

___

25 ___ 21

25

D

9 ___

12

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166 Topic 10

Name

Subtracting Fractions with

Unlike Denominators

Reteaching

10-4

You can subtract fractions with unlike denominators by using

the least common multiple (LCM) and the least common

denominator (LCD).

Beth wants to exercise for 4_ hour. So far, she has exercised

5

for 2_ hour. What fraction of an hour does she have left to go?

3

Step 1: Find the LCM of 5 and 3.

12

15

10

15

2

15

12

11 1

10

2

9

8

3

4

7 5

6

multiples of 5: 5, 10, 15, 20

multiples of 3: 3, 6, 9, 12, 15

Since 15 is the LCM, it is also

your LCD.

Step 2: Using your LCD, write the

equivalent fractions.

4

5

3

3

12

15

2

3

5

5

10

15

Step 3: Subtract the numerators.

Place the difference over the

LCD. Simplify if possible.

12

15

10

15

2

15

12 10 2

12

11 1

10

2

9

8

3

4

7

6

5

___ 2

hour left

15

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In 1 through 5, find each difference. Simplify if possible.

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1.

3.

3__

4

2__

5

___ 7

20

7 ___

12 1__

1__

3

2.

___ 7

10

1__

5

4 4. __ 5 6 __ 3 8 5. ___ 23

6. Natasha had 7_ gallon of paint. Her brother Ivan took

1_

8 4 gallon

to paint his model boat. Natasha needs at least 1_ gallon to

2

paint her bookshelf. Did Ivan leave her enough paint?

1__

2

___ 11

___ 1

24

24 7__

8 12

Yes. She has __ 5 gallon left.

8

Topic 10 167

Name

Subtracting Fractions with

Unlike Denominators

Find the difference. Simplify if necessary.

1. ___ 10

4.

12 1__

4

___ 7

12 1__

4

7. 4__

8 1__

4

10.

___ 9

15 1__

3

__

7

12

1_

3

1

_ 4

__

4

15

2.

___ 9

10 3__

5

5. 4__

5 1__

3

8.

11.

___ 4

10 1__

5

___ 4

12 1__

6

__

3

10

__

7

15

1

_

1_

6

3. 7__

8 2__

6

6. 2__

3 1__

6

5 9. 7__

9 2__

3

1

12. ___ 14

20 3__

5

__

13

24

1_

2

1

_ 9

__

10

Practice

10-4

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13. The pet shop owner told Jean to fill her new fish tank 3_

4 full

with water. Jean filled it __ 9 full. What fraction of the tank does

12

Jean still need to fill?

0

14. Paul’s dad made a turkey pot pie for dinner on Wednesday.

The family ate 4_ of the pie. On Thursday after school, Paul

8

ate __ 16 2 of the pie for a snack. What fraction of the pie remained?

3__

8

15. Algebra Gracie read 150 pages of a book she got for her

birthday. The book is 227 pages long. Which equation shows

how to find the amount she still needs to read to finish the story?

A 150 n 227 C n 150 227

B 227 150 n D n 150 227

16. Explain It Why do fractions need to have a common

denominator before you add or subtract them?

The denominator tells you what size parts

you are adding or subtracting. The parts

must be the same size when you add or

subtract.

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Adding Mixed Numbers

Reteaching

10-5

Randy talks on the telephone for 2 5_ hours, and then surfs the

6

Internet for 3 3_ hours. How many hours did he spend on the

4

two activities?

Step 1. Write equivalent

fractions with the least common

denominator. You can use fraction

strips to show the equivalent

fractions.

1 1

1 1

3 3

4 9

3

12

2 5

6 10

2

12

1

Step 2. Add the fractions. Then

add the whole numbers.

___ 9

12 ___ 10

12 ___ 19

So, 3 3__

4 25__ 6 519 ___

12

12 3 2 5

Step 3. Simplify the sum if

possible.

___ 19

512 ___ 7

612 hours

In 1 through 6, find each sum. Simplify if possible.

1. 2__

5 6

3 1__

4

2. 1__

3 8

6 3 __

4

3. 5 2__

5

4 1__

2

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6___

12 1

9___

10 9 5___

11 __ 1 81 __ 8 9___

11

14

3

9

12

4. 10 1__

3 7__

9 5.

5. 31__

4 62__

7. Geometry Tirzah wants to put a fence

around her garden. She has

22 yards of fence material.

Does she have enough to go

all the way around the garden?

No. She needs

22 __ 5 yards to

go 6around the

garden.

3 6. 15 __

Tirzah’s garden

3

6 yards

4

7 31__ 2

2

4 yards

3

Topic 10 169

Name

Adding Mixed Numbers

Estimate the sum first. Then add. Simplify if necessary.

1. 7 2__

3 85__ 6

3. 11___

9

10 3 ___ 1

20

5. 5 8__

9 31__ 2

16 _ 1

2

14 __

19

9__

7

20

18

2. 4 3__

4 22__ 5

4. 7 6__

7 52__ 7

6. 21___

11

12 172__ 3

7__

3

20

13 _ 1 7

39 7

__

12

Practice

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7. Number Sense Write two mixed numbers with a sum of 3.

Sample answer: 1 __ 1 4 1 __ 3 4 3

8. What is the total

measure of an average

man’s brain and

heart in kilograms?

1__

7

10 kg

9. What is the total weight of an average

woman’s brain and heart in pounds?

10. What is the sum of the measures of an average

man’s brain and an average woman’s brain in

kilograms?

11. Which is a good comparison of the estimated

sum and the actual sum of 7 7_ __

A Estimated actual

8 211 ? 12

Vital Organ Measures

es

Average woman’s brain 1___

3 kg 24__

10 5 lb

Average man’s brain

Average human heart

C Actual estimated

1 2__

5 kg 3 lb

3 ___

3_

1 2 lb

10 kg ___ 7

10 lb

2__

7

10 kg

B Actual estimated

D Estimated actual

12. Explain It Can the sum of two mixed numbers be equal to 2? Explain why

or why not.

No; Sample answer: It is impossible for

two mixed numbers to equal 2 because

every mixed number is greater than 1.

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170 Topic 10

Name

Subtracting Mixed Numbers

Reteaching

10-6

The Plainville Zoo has had elephants for 12 2_ years. The zoo has

3

had zebras for 5 1_ years. How many years longer has the zoo had

2

elephants?

Step 1. Write equivalent fractions with the least

common denominator. You can use fraction strips.

Step 2. Subtract the fractions. Then subtract the

whole numbers. Simplify the difference if possible.

1

1

1

1

1

1

1

1

12 2 4

3 12

6

1

1

5 1 3

2 5

6

1

1

1

1

1

1

1

6

1

6

1

2

1

6

1

1

6

1

6

1

6

1

6

1 1

3 3

4__

3__ 1__

6 6 6

So, 12 2__

3 51__ 2 71__ 6 years.

12 5 7

Tip: Sometimes you may have to rename a fraction

so you can subtract.

6 rename 5

2 3__

8

8__

8

2 3__

8

3 5__

8

For questions 1 through 4, find the difference. Simplify if possible.

Remember: You may have to rename a fraction in order to subtract.

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1. 43 __

5

2 1__

3

2___

4

15

2. 56 __

7

1 1__

2

4___

5

14

3. 3 4. 65 __

6

1 3__

4

5. Number Sense Rename the number 7 so that you would be

able to find the difference of 7 3 __ 5

12 .

1__

1 4

6___

12

12

5 1__

2

1 1 __

3

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6. Robyn ran 5 3_ miles last week. She ran 4 __ 1

miles this week.

4 10

How many more miles did she run last week?

113 ___ 13 20

Topic 10 171

Name

Subtracting Mixed Numbers

Estimate the difference first. Then subtract. Simplify if necessary.

Practice

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1. 10 3__

4

7 1__

4

3_

1 2

172 Topic 10

2. 7 3__

7

2 ___ 8

21

5 5

3

5. 9__

9 6__

6 6. 4__

4 2 2 __ 3

7. 6 1__

4 31__ 3

9. 8 2__

7 71__ 3

2__

13

2__

18

11

12

__

20

21

5__

1

21

3. 3 4. 17 7__

8

2 2__

3

12 3

12

8. 5 1__

5 37__ 8

10. 2___

9

10 21__ 3

2__

1

1__

13 12

40

__

17

30

5 5 _

8

Strategy Practice The table shows the length and width of several

kinds of bird eggs.

Egg Sizes

14. Explain It Explain why it is necessary to rename 4 1_ if you

4

subtract 3_

4 from it.

11. How much longer is the

Canada goose egg than the

Bird Length Width

raven egg?

1_

Canada goose 3 2__

5 in. 2 ___ 3

10 in.

Robin __ 3

4 in. __

12 in. longer

3 5 in.

12. How much wider is the turtledove

Turtledove 1

egg than the robin egg?

5 in. ___ 9

10 in.

__

3

Raven 1___

9

10 in. 1 ___

in. wider

3

10 10 in.

13. Which is the difference of 21 15

16 183_ 4 ?

A 2___

7

16

B 2___

9

16

C 3___

3

16

D 3___

9

16

Sample answer: You cannot subtract _ 3 4

from _ 1 4 , so you must borrow 1 whole

from the 4 and rename 4 _ 1 4 as 353

_

4 .

1_

3

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Problem Solving:

Try, Check, and Revise

Can you use similar square tiles to cover a floor without cutting

the tiles?

Try

Look at the square tile

and the floor.

2 ft

2 ft

4 ft

Reteaching

8 ft

10-7

Ask yourself:

1. Can I divide the floor into

a whole number of rectangles

that are 2 feet wide?

4 ft

8 ft

1

2

AND

2. Can I divide the floor into

a whole number of rectangles

that are 2 feet long?

8 ft

1 2 3 4

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If the answers are both yes, you

can use the tiles to cover the floor

without cutting them.

Check.

Answer the following questions remembering that you cannot cut

the tiles.

1. Can you completely cover a 4 ft by 8 ft area

with 3 ft by 3 ft tiles?

No

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Yes

No

Topic 10 173

Name

Problem Solving:

Try, Check, and Revise

For questions 1 and 2, suppose you have 2 2 ft, 3 3 ft,

4 4 ft, and 5 5 ft tiles.

1. Which tiles can be used to

cover a 12 12 ft floor?

Practice

10-7

2 2, 3 3, and 4 4

2. Which

cover a

tiles

9

can be used to

9 ft floor?

3 3

3. What size rectangular floor can be completely covered by

using only 3 3 ft tiles OR 5 5 ft tiles? Remember, you

can’t cut tiles or combine the two tile sizes.

Sample: 15 ft 45 ft, 30 ft 60 ft

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4. Adult tickets cost $6 and children’s tickets cost $4.

Mrs. LeCompte says that she paid $30 for tickets, for both

adults and children. How many of each ticket did she buy?

3 adults and 3 children OR 1 adult

and 6 children

5. Reasoning The sum of two odd numbers is 42. They are

both prime numbers, and the difference of the two numbers

is 16. What are the two numbers?

A 20 and 22

B 17 and 25

C 9 and 33

D 13 and 29

6. Explain measures

It 10

Marcy

ft

wants to put tiles on a bathroom floor that

12 ft. What kind of square tiles should she

buy to tile her floor? Explain.

Sample: She can buy 2 2 ft tiles

to tile her floor.

174 Topic 10

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Multiplying Fractions

and Whole Numbers

Reteaching

11-1

You can find the product of a fraction and a whole number.

Tran needs 2__ yard of fabric to sew a pair of shorts. How many

3

yards of fabric will Tran need to sew 6 pairs of shorts?

Step 1: Multiply the numerator by the

whole number.

2 6 12

Remember: In word problems, “of” means “multiply.”

Example: 3__ 3__

of 15

5 5 15

Step 2: Place the product over the

denominator. Simplify if possible.

___ 12

4 yards of fabric

3

In 1 through 4, find each product. Simplify if possible.

1. 1__

3 60

2. 3__ of 32

4

3. __ 78 40

4. 2__ of 35

7

20

24

35

10

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For Exercises 5 through 7, use the table to the right.

Speed

5. What is 1_

Animal

the speed of a cheetah? 10 mi/h

(in mi/h)

7

Cat 30

6. What is 1_ the speed of a cat? 6 mi/h

5 Cheetah 70

7. What is 1_

5 the speed of a jackal? Jackal 35

7 mi/h

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Multiplying Fractions

and Whole Numbers

Find each product.

1. 1__

4

of 96 2.

4__

2.

7

of 28 3.

3.

3__

Practice

11-1

4 72

4. 45 3__

9 5. 7__

56

8 6. 3__

42

7. 1__ of 118 8. 3__ of 56 9.

___ 1

400

2

8

10

10. 84 14

1__

6 11. 64

5

___ 20

16 12. 40

11

___

20

13. 5__

8

of 48 14.

1__

14.

7

of 77 15.

7

15.

4__

5 90

16. 42

___ 3

14 17. 5__

72

8 18. 2__

18

3 =

19. 5__

6

24

15

59

30

9

70

84 20. ___ 11

144 21.

12

21.

6__

22. Strategy Practice Complete the table by writing the product

of each expression in the box below it. Use a pattern to find

each product. Explain the pattern.

1_

2 32 1_

4 32 1_

8 32

23. Reasoning If 1_

2

24. Which is 2_ of 225?

3

32 1__

16

16 32

16 8 4 2

Pattern: divide by 2

16

49

21

11

45

132

of 1 is

1_

, what is

1_

of 2, 3, and 4?

2 2

7 42

1, 1 __ 1 2 , and 2

54

18

40

22

72

12

36

A 75 B 113 C 150 D 450

25. Explain It Explain why 1_

__ 1

2 of 2 1

of 2 equals one whole.

2

__

2 2 __ 2

2 1

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176 Topic 11

Name

Multiplying Two Fractions

Reteaching

11-2

Musa and Karen are riding a bike path that is 4_ mile long. Karen’s

5

bike got a flat tire __ 3 of the way down the path and she had to

10

stop. How many miles did Karen ride?

You can find the product of two fractions by drawing a diagram.

Step 1. Draw a diagram using

shading to represent 4_

5 .

Step 2. Draw lines vertically using dots to

represent __ 3

10 .

4

5

3

10

Step 3. Count the parts of the diagram that are

shaded and dotted. This is the product numerator.

12

Step 5. Simplify if possible.

___ 12

50 ___ 6

25

Another way to find the product:

Step 1. Multiply the numerators: 4 3 12.

Step 4. Count the total number of parts of the

diagram. This is the product denominator.

50

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Step 2. Multiply the denominators: 5 10 50.

Step 3. Simplify if possible:

___ 12

50 ___ 6

25 .

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In 1 through 6, find the product. Simplify if possible.

1. 1__

3 2__

5 2. __ 5

4. 1__

___ 2

15

8 1__

___ 5

32

4 3. __ 5

6 ___ 3

10

6 5. 14 3__ 6.

3 6

2

7

7. Draw a picture Using a diagram, show 3__

7 1__

4 .

3__

1__ 6__

5

2

7

1__

4

___ 9

35

Topic 11 177

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Practice

11-2

Multiplying Two Fractions

Write the multiplication problem that each model represents then

solve. Put your answer in simplest form.

1. 2.

2__

5__

__ 1

___

3 __ 1 6 ___ 2

18 __ 1 9

5

Find each product. Simplify if possible. 7 4 28

3. 7__

8 4__ ___ 7

5 4. 3__

7 2__ 2__

3

10

5. 1__

6 2__ ___

5

1

6. 2__

7

7 1__ ___ 1

4

15

14

7. 2__

9 1__ 1__

2 8. 3__

4 1__ 1__

3

9

4

9. 3__

8 4__ 1__

9 10. 1__

5 5__ 1__

6

6

11. 2__

3 5__

1__

14 12. 6 7 7 6

__

2 1__

3 1__

4

___ 1

9

1__ 24

13. Algebra If 4_

5 2_ , what is ?

5

14. Ms. Shoemaker has 35 students in 2her classroom.

Of those students, 6_ are right-handed, and

4_

7 5 of

the right-handed students have brown eyes. How

many right-handed brown-eyed students are in

Ms. Shoemaker’s class?

24

15. Which does the model represent?

A 3_

8 3_ C 3_

5 5 5_

8

B 7_

8 2_ D 4_

5 8 3_

5

16. Explain It Describe a model that represents 3_

3 4_

4 .

178 Topic 11

Each fraction represents one whole, so

the model would be completely shaded

in for both fract

ractions.

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Multiplying Mixed Numbers

Reteaching

11-3

You can find the product of two mixed numbers.

Millwood City is constructing a new highway through town. The

construction crew can complete 5 3_ miles of road each month.

5

How many miles will they complete in 6 1_

2 months?

Step 1. Round the mixed numbers to whole

numbers so you can make an estimate.

Step 3. Multiply the numerators and the

denominators. Simplify the product if possible.

5 3__

5 6 1__

2

6 7 42

So, they can complete about 42 miles.

Remember to look for common factors.

14

___ 28

5 ___ 13

2 ____ 182

5 362__ 5

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Step 2. Write the mixed numbers

as improper fractions.

5 3__

5 6 1__

2 ___ 28

5 ___ 13

2

Step 4. Compare your product to your estimate to

check for reasonableness.

36 2_ is close to 42, so this answer

5

is reasonable.

The construction crew will complete

36 2_ miles of highway in 6

1_

5 2 months.

For 1 through 6, estimate a product. Then solve for each actual

product. Simplify if possible.

4_

3

3__

3

20 25

8 2 4_

1

4 2

1. 1 3_

4 21_ 2 2. 11_ 5 12_ 3 3. 2 21_

4. 1 2_

5 21_ 4 5. 21_ 2

10 6. 12_

3 1_

7. Using the example above, the new highway will be a total of

54 miles long. Will the highway be finished in 8 months?

No. They will have completed only 44 _ 4 5

miles.

8. Reasonableness Sayed gave an answer of 6 6_ for the

problem 4_

2

7

7 13 _

5

. Using estimates, is this a reasonable

answer?

5

Yes: 4 2 8, and 8 is close to 6 _ 6

7 .

1__

3

Topic 11 179

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Multiplying Mixed Numbers

Estimate the product. Then complete the multiplication.

1. 5 4_

5 7 ______

5

29 7__ 40

Estimate. Then find each product. Simplify.

3. 4 3_

5 2_

3

5. 7 4_

5 21_ 3

7. 2 1_

5 7_

8

9. 1 4_

5 11_ 3 13_ 4

40 __ 3 5

Practice

11-3

1 2. 32_

3 51_ 7 ______ ______

3 7

3___

1

181

__ 5 15

1___

37

40

4__

1 5

4. 6 2 2_

7

6. 3 3_

4 24_ 5

8. 6 1_

3 15_ 6

10. 3_

4 22_ 3 51_ 5

4 p is

11. Algebra Write a mixed number for p so that 3 1_

more than 3 1_

4 .

12. A model house is built on a base that measures 9 1_

4 in.

wide and 8 4_ in. long. What is the total area of the model

5

house’s base?

11 36

36

18

18 __ 6 7

13 __ 5 7

101

__ 2

11 ___ 11

10 __ 2 18

5

Answers will vary. Any mixed number

greater than 1 will work.

812

__ 5 in 2

13. Which is 1 3_ of 1501_

4 2 ?

A 263 B 263 1_

C 263 3_ D 264 3_

8 8 8

14. Explain It Megan’s dog Sparky eats 4 1_ cups of food each

4

day. Explain how Megan can determine how much food to

give Sparky if she needs to feed him only 2_ as much. Solve

3

the problem.

Megan would multiply 4 _ 1 4 c by _ 2 3 to

find 2 _ 5 6 c.

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Relating Division to

Multiplication of Fractions

Reteaching

11-4

How can you divide by a fraction?

Dividing a whole number by a fraction

2 1__

3

1. Two is the sum of one plus one.

2. Each one is the sum of three one-thirds.

3. Count the number of one-thirds.

Check. To divide a whole number by a

fraction, multiply the whole number by the

reciprocal of the fraction.

3 3__

4

1. Three is the sum of one plus one plus

one.

2. Each one is the sum of one three-fourths

and one one-fourth.

3. Count the number of three-fourths.

Check. Multiply the whole number by the

reciprocal of the fraction.

Think: How can I divide two into

one-thirds?

2

6

1

1__

3 1__

3 1__

3

1

1__

3 1__

3 1__

3

2 1__

3 2 3 __

1 2__

1 3 __

1 6 __

1 6

Think: How can I divide three into

three-fourths?

3 1 1 1

4

3__

4 1__ __

4 3

4 1__ __

4 3

4 1__

4

3__

4 3__

4 3__

4 1__

4 1__

4 1__

4

3__

4 3__

4 3__

4 __ 3

4

3

__ 3 4 3 4__

3 __ 3 1 4__

3 ___ 12

3 4

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Draw a picture that shows each division and write the answer.

1. 2 1__

2

4 3

2. 2 2__

3

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Topic 11 181

Name

Relating Division to

Multiplication of Fractions

In 1 and 2, use the picture to find each quotient.

Practice

11-4

1. How many thirds are in 1? 2. How many thirds are in 7?

3 21

In 3 and 4, draw a picture to find each quotient.

3. 3 1__

2

4. 4 1__

8

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6 32

In 5 and 6, use multiplication to find each quotient.

5. 6 1__

3

6. 5

___ 1

10

6 3 18 5 10 50

7. Julie bought 3 yards of cloth to

make holiday napkin rings. If she

needs 3_ of a yard to make each

4

ring, how many rings can she

make?

4

182 Topic 11

8. Reasoning When you divide a

whole number by a fraction with a

numerator of 1, explain how you

can find the quotient.

Sample answer:

You can multiply

the

number whole

by the

denominator of

the fraction.

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Reteaching

11-5

Travis earned 3 stickers for each song he played in his piano lesson.

He received a total of 24 stickers. How many songs did he play?

You can solve a problem like this by drawing a picture and writing

an equation.

Step 1. Write out what you already know.

Travis earned 3 stickers for each

song he played. Travis had

24 stickers at the end of the lesson.

Step 2. Draw a picture to show what you

know.

Step 3. Write out what you are trying to find.

How many songs did Travis play?

Step 4. Write an equation from your drawing.

Since you are dividing Travis’s total

stickers into groups of 3 (stickers

earned per song), this is a

division problem.

24 3 s s number of songs

Travis played

Step 5. Solve the equation.

24 3 8 s 8

So, Travis played 8 songs during his lesson.

Step 6. Check your answer by working backward.

8 3 24: your answer is correct.

Travis’s total stickers

Travis groups earned of 3 stickers

per song

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Draw a picture, write an equation, and solve.

3_

1. Sasha, Rudy, and Mario each have 14 flour. Can they make a recipe for bread cups that of

needs 5 cups of flour?

3 1_

3 4 c; c 5_

1 4 cups

of flour; yes

Sasha’s flour

Rudy’s flour

Mario’s flour

Topic 11 183

Name



Solve each problem. Draw a picture to show the main idea for

each problem. Then write an equation and solve it. Write the

answer in a complete sentence.

1. Bobby has 3 times as many model spaceships as his friend

Sylvester does. Bobby has 21 spaceships. How many model

spaceships does Sylvester have?

Practice

11-5

Check student’s drawings.

21 3 x , x 7; Sylvester has

7 spaceships.

2. Dan saved $463 over the 12 weeks of summer break. He

saved $297 of it during the last 4 weeks. How much did he

save during the first 8 weeks?

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$463 $297 x , x $166; Dan saved

$166 during the first 8 weeks.

3. Strategy Practice Use a separate sheet of paper to show

the main idea for the following problem. Choose the answer

that solves the problem correctly.

A box of peanut-butter crackers was divided evenly among

6 children. Each child got 9 crackers. How many crackers

were in the box?

A 54 B 48 C 39 D 36

4. Explain It Why is it helpful to draw a picture when

attempting to solve an equation?

184 Topic 11

Sample answer: Drawing can give you

a picture of the problem that will help

you create the right equation to solve

the problem.

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Using Customary

Units of Length

Reteaching

12-1

One customary unit of length is the inch. On most rulers it is

divided into fractions to help you measure more precisely.

How to measure length

Measure the length of the pen to the nearest inch, nearest

1_ inch, 1_ inch, and 1_ inch.

2

4

8

012345

INCHES

Step 1: Measure to the

nearest inch.

The length is more than

4 in., but less than 5 in.

It is closer to 5 in.

The pen’s length is

closest to 5 in.

Step 2: Measure to the

nearest 1_

2 inch.


4 1_ in., but less than

2

4 2_ (5) in.

2

It is closer to 4 1_

2 in.


closest to 4 1_

2 in. Step 3: Measure to the

nearest 1_

4 inch.


4 2_

4 (41_ ) in., but less than

2

4 3_

4 in.


2 in.


closest to 4 1_

2 (42_ 4 ) in. Step 4: Measure to the

nearest 1_

8 inch.


4 4_

8 (41_ ) in., but less than

2

4 5_

8 in.


8 in.


closest to 4 5_

8 in.

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Measure to the nearest inch, 1_ inch,

1_

inch, and

1_

inch.

2 4 8

1.

1 in., 1 in., 1 _ 1 4

in., and 11 1_

4 (12 _

8 ) in.

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Use your ruler to draw a line segment of each length.

2. 7__

8

inch 3.

3. 21__

4 inch

Topic 12 185

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Using Customary

Units of Length

Measure each segment to the nearest inch, 1_

2

1_

8 inch.

1.

2.

2 in.; 2 __ 1

2 in.; 1 __ 1

inch,

1_

inch, and

4

3. Reasoning Sarah gave the same answer when asked to

round 4 7_ in. to the nearest

1_

inch and the nearest inch.

8 2

Explain why Sarah is correct.

4. Estimate the length of your thumb. Then use a ruler to find

the actual measure.

2 in.; 212

__

4 in.; 212

__

4 in.

2 in.; 111

__

2 in.; 111

__

2 in.;

Practice

12-1

Sample answer: 5 in. is the nearest

inch and __ 1 4

2 in. to 47 __

8 in.

Answers will vary.

5. Estimation A real motorcycle is 18 times as large as a model

motorcycle. If the model motorcycle is 5 __ 1 in. long, about

16

how long is the real motorcycle?

A 23 in. B 48 in. C 90 in. D 112 in.

6. Explain It If a line is measured as 1 4_ in. long, explain how

8

you could simplify the measurement.

Sample answer: I could simplify 1 __ 4 8

to 111

__

2 by dividing the numerator and

denominator by 4.

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186 Topic 12

Name

Using Metric Units

of Length

Reteaching

12-2

Measurements in the metric system are based on the meter.

Example A

Which metric unit of length would

be most appropriate to measure the

length of a bumblebee?

A bumblebee is very small, so the

millimeter is the most appropriate unit

of length.

Example C

Example B

Write mm, cm, m, or km to complete

the following sentence.

A chair is about 1 tall and a

child’s hand is about 8 wide.

A chair is about 1 m tall, and a child’s

hand is about 8 cm wide.

What is the length of the twig to the nearest centimeter and to

the nearest millimeter?

12345678

CENTIMETERS

Step 1: Measure to the closest

centimeter.

The length of the twig is more than

7 cm but less than 8 cm.

The twig’s length is closest to 8 cm.

Step 2: Measure to the closest

millimeter.


70 mm but less than 80 mm.


75 mm but less than 76 mm.

The twig’s length is closest to 76 mm.

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Which unit would be most appropriate for each measurement?

Write mm, cm, m, or km.

1. distance between two cities 2. width of a room

3. Measure the line segment below to the nearest centimeter and

to the nearest millimeter.

3 cm 31 mm

;

km

m

Topic 12 187

Name

Using Metric Units

of Length

Measure each segment to the nearest centimeter

then to the nearest millimeter.

1.

2.

Number Sense Some of the events at

an upcoming track and field meet are shown

at the right.

3. In which event or events do athletes

travel more than a kilometer?

1,500-m dash

Practice

12-2

5 cm, 54 mm

4 cm, 39mm

Track and Field Events

50-m dash

1,500-m dash

400-m dash

100-m dash

4. In which event or events do athletes travel less than a kilometer?

50-m dash, 100-m dash, 400-m dash

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5. Reasonableness Which unit would be most appropriate for

measuring the distance from Chicago to Miami?

A mm B cm C m D km

6. Explain It List one item in your classroom you would

measure using centimeters and one item in the classroom

you would measure using meters.


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188 Topic 12

Name

Perimeter

Reteaching

12-3

The perimeter is the distance around the outside of a polygon.

You can find the perimeter in two different ways.

Add the lengths of the sides:

Find the perimeter of the figure.

7 cm

Use a formula:

Find the perimeter of the rectangle.

11 cm

cm 3

cm 5

3 cm

4 cm

3 cm

To find the perimeter, add up the sides.

3 3 7 5 3 4 25

So, the perimeter of the figure is 25 cm.

3 cm

Perimeter = (2 length) + (2 width)

P = (2 l ) + (2 w )

P = (2 11) + (2 3)

P = 22 + 6 = 28 cm

So, the perimeter of the rectangle is

28 cm.

Find the perimeter of each figure.

8.9 cm

2.

11 m

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4.2 cm

4 m

26.2 cm 33 m

8 m

7 m

3 m

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3. 3 mm 4 mm

4.

4.7 cm

5 mm

5 mm

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6.5 mm

23.5 mm 18.8 cm

5. Number Sense The perimeter of a square is 24 in.

What is the length of each side?

6 in.

Topic 12 189

Name

Perimeter

Find the perimeter of each figure.

1. 3 cm 3 cm 2. 7 km 3. 1 m 4.

2 m

5 cm

7 km 7 km

3 m

7 km

2 m

1 m

5. Number Sense What is the perimeter of a square if

one of the sides is 3 mi?

2 m

Practice

12-3

5 mm

11 cm 28 km 11 m 25 mm

12 mi

7.5 mm

7.5 mm

5 mm

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Use the dimensions of the football field

shown at the right for 6 and 7.

6. What is the perimeter of the entire

football field including the end

zones?

1,040 ft

7. What is the perimeter of each end zone?

380 ft

ft30

ft 30

300 ft

e

n

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o

160 ft

z

Playing field 160 ft

z

d

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300 ft

ft30

ft 30

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E

8. What is the perimeter of this figure?

A 18 m

B 15 ft

190 Topic 12

C 12 m

D 10 ft

9. Explain It A rectangle has a perimeter of

12 m. If each side is a whole number of meters,

what are the possible dimensions for the length

and width? List them explain your answer.

1 and 5, 2 and 4, 3 and 3, 4 and 2, 5

and 1. If the perimeter is 12 m, the

length and width must add up to 6 m.

6 m

3 m

6 m

3 m

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Area of Squares and

Rectangles

Reteaching

12-4

You can use formulas to find the areas of rectangles and squares.

Find the area of the rectangle.

5 m

Find the area of the square.

42 cm

4 m

42 cm

42 cm

Use this formula for rectangles:

Area length width

A l w

A 5 m 4 m

A 20 square meters 20 m 2

42 cm

Use this formula for squares:

Area side side

A 42 cm 42 cm

A 1,764 square centimeters

1,764 cm 2

Find the area of each figure.

58 ft

2.

5.5 in.

3.2 in.

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3,364 ft 2 17.6 in 2

153.76 m 2

3. a square with a side of 12.4 m

4. a rectangle with a length of 9.7 cm

and a width of 7.3 cm

5. Number Sense If the area of a square

is 81 in 2 , what is the length of one side?

6. What is the area of the tennis court?

195.16 m 2

70.81 cm 2

9 in.

23.8 m

8.2 m

Topic 12 191

Name

Areas of Squares

and Rectangles

Find the area of each figure.

1. 4 cm

2.

s 9.5 mi

Practice

12-4

w 3 cm

s 9.5 mi

s 9.5 mi

3. a rectangle with sides 6.5 km and 3.4 km

4. a square with a side of 10.2 ft

5. a rectangle with sides 9 m and 9.2 m

s 9.5 mi

12 cm 2 90.25 mi 2

22.1 km 2

104.04 ft 2

82.8 m 2

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6. Number Sense Which units would you use to measure the

area of a rectangle with l 1 m and w 34 cm? Explain.

Sample answer: Centimeters; I would

convert the 1 m to centimeters.

7. Which of the following shapes has an area of 34 ft 2 ?

A a square with s 8.5 m

B a rectangle with l 15 ft, w 2 ft

C a square with s 16 ft

D a rectangle with l 17 ft, w 2 ft

8. Explain It The area of a square is 49 m 2 . What is the length

of one of its sides? Explain how you solved this problem.

7 m; Sample answer: The length of each

side of a square is equal. The factors of

49 are 1, 7, 7, and 49, so each side must

be 7 m.

192 Topic 12

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Area of Parallelograms

Reteaching

12-5

The formula used to find the area of a

parallelogram is similar to the one you used

to find the area of a rectangle. Instead of using

length width, use base height.

How to find the area of a

parallelogram:

Find the area of the parallelogram.

10 cm

8 cm

Area base height

A b h

A 10 cm 8 cm

A 80 cm 2

Find the area of each parallelogram.

1.

5 m

Base

(like width)

Height

(like length)

How to find the missing

measurement of a parallelogram:

Area 55 cm 2 base 11 cm

height ? cm

Remember, area base height.

55 cm 2 11 cm ? cm

55 cm 2 11 cm 5 cm

So, the height of the parallelogram

is 5 cm.

70 m 2

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2.3 ft

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11.96 ft 2

5.2 ft

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Find the missing measurement for each parallelogram.

8 in.

17 ft

3. A 72 in 2 , b 9 in., h

4. A 238 ft 2 , b , h 14 ft

Topic 12 193

Name

Area of Parallelograms

Find the area of each parallelogram.

Practice

12-5

1. 2.

3 cm

9 mi

2 mi

1 mm

2 mm

5 cm

15 cm 2 18 mi 2

3. 4.

2 mm 2 9 m 2

6 m

1.5 m

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Algebra Find the missing measurement for the parallelogram.

5. A 34 in 2 , b 17 in., h

6. List three sets of base and height measurements for

parallelograms with areas of 40 square units.

Sample answer: b 2, h 20; b 4,

h 10; b 5, h 8

7. Which is the height of the parallelogram?

A 55 m

B 55.5 m

C 5 m

D 5.5 m

8. Explain It What are a possible base and height for a

parallelogram with an area of 45 ft 2 if the base and height are

a whole number of feet? Explain how you solved this problem.

194 Topic 12

2 in.

A 44 m 2

h ?

b 8 m

Sample answer: 9 ft and 5 ft; 9 and 5 are

factors of 45, so a base of 9 ft and height

of 5 ft would result in an area of 45 ft 2 .

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Name

Area of Triangles

Area of a triangle 1__ base height

2

How to find the area of a triangle:

4 ft

3 ft

A 1__

2 b h

A 1__ 3 ft 4 ft

2

A 1__ 12 ft2

2

A 6 ft 2

How to find the missing

measurement of a triangle:

Reteaching

12-6

Area 100 cm 2 base 40 cm

height ? cm

Remember, Area 1__ base height.

2

100 cm 2 1__ 40 cm ? cm

2

100 cm 2 20 cm ? cm

100 cm 2 20 cm 5 cm

So, the height of the triangle is 5 cm.

Find the area of each triangle.

1. 2. 3.

4 in.

7 in.

2.2 m

5.4 m

14 in 2 5.94 m 2 22.5 cm 2

9 cm

5 cm

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Find the missing measurement for each triangle.

4 in.

10 m

24.32 ft 2

4. A 16 in 2 , b 8 in., h

5. A 20 m 2 , b , h 4 m

6. A , b 6.4 ft, h 7.6 ft

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7. A 14 cm 2 , b 2 cm, h

14 cm

Topic 12 195

Name

Area of Triangles

Find the area of each triangle.

Practice

12-6

1. 2. 3.

10 ft

3.6 yd

7 mm

8 ft

6 yd

13 mm

40 ft 2 10.8 yd 2 45.5 mm 2

4. Number Sense What is the base measurement of

a triangle with an area of 30 m 2 and a height of 10 m?

6 m

Algebra Find the missing measurement for each triangle.

6 mi

45 mm 2

5. A 36 mi 2 , b , h 12 mi

6. A , b 12 mm, h 7.5 mm

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7. List three sets of base and height measurements for triangles

with areas of 30 square units.

Sample answer: b 4, h 15; b 6,

h 10; b 8, h 7.5

8. Which is the height of the triangle?

A 4.5 ft

B 6 ft

C 8 ft

D 9 ft

A 27 ft 2

12 ft

9. Explain It Can you find the base and height measurements

for a triangle if you know that the area is 22 square units?

Explain why or why not.

No; Sample answer: You need to know

at least 2 of the measurements ments of the

base, the height, and the area.

196 Topic 12

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Name

Circles and Circumference

Reteaching

12-7

All the points on a

circle are the same

distance from the

center. The center of

circle C is point C.

A chord has both

endpoints ___ on a circle.

HD is a chord of

circle O.

A central angle is an

angle whose vertex is

the center. ACB is a

central angle.

H

D

O

C

C

B

D

A

A radius goes from

the center to any

point ___ on the circle.

RS is a radius of

circle R.

A diameter is a chord

that passes through

the ___ center of a circle.

DT is a diameter of

circle M.

The circumference

is the distance

around a circle. The

circumference equals

the circle’s diameter

multiplied by

(3.1416). C D

D

R

M

S

T

Use the terms at the top of the page to identify each figure in

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circle F .

1. point F

2. GFS

3. ___

RS

4. ___

UV

5. ___

GF

Center

Central angle

Diameter

Chord

Radius

U

R

F

V

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6. Estimation If a circle has a circumference of three feet,

estimate its diameter to the nearest foot.

1 foot

Topic 12 197

Name

Circles and Circumference

In 1 through 3, use circle X to identify the following.

___

QP

1. a diameter 2. two chords 3. a central angle

In 4 through 9, find the circumference. Use 3.14 for .

Practice

12-7

___

R

RS ___ and QXP ,

QP

QXO, PXO

O

P

X

S Q

4. d 20 in. 5. d 5 yd 6. d 9 cm

62.8 in. 15.7 yd 28.26 cm

7. r 3.20 in. 8. r 13 ft 9. r 20 yd

20.1 in. 81.64 ft 125.6 yd

10. A round swimming pool has a radius of 8 meters.

What is its circumference?

50.24 meters

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11. Estimation The length of the diameter of a circle is

11 centimeters. Is the circumference more or less than

33 centimeters? Explain.

3 11 33, and is greater than 3,

so 11 33

12. Which equation can be used to find the circumference of a

circle with a radius that measures 10 feet?

A C 2 20 C C 1.0

B C 2 10 D C 10

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198 Topic 12

Name

Problem Solving: Draw

a Picture and Make an

Organized List

Reteaching

12 -8

In a garden, a landscaper has to make a small patio using twelve

1-foot-square tiles. The patio must have the smallest possible

perimeter. What should be the dimensions of the patio?

Draw a picture.

1. Construct pictures of rectangles using the

twelve tiles.

2. List the perimeter of each figure in a table.

3. Count tile edges or calculate to find the

perimeter.

A

B

C

D

E

Solve.

Both the 3 4-tile and the

4 3-tile patios have the same

perimeter, 14 feet. This value

is the smallest perimeter in the

table. The landscaper can use

either design.

Rectangle

A

B

C

D

E

F

Length

in feet

12

6

4

3

2

1

Width

in feet

1

2

3

4

6

12

F

Perimeter

in feet

26

16

14

14

16

26

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The landscaper must build another patio with the smallest perimeter

possible using nine 1-foot-square tiles. What should be the dimensions?

3 feet 3 feet

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Topic 12 199

Name

Problem Solving: Draw

a Picture and Make an

Organized List

Draw a picture and make a list to solve.

Practice

12 -8

1. Erica painted a picture of her dog. The picture has an area of 3,600 cm 2

and is square. She has placed the picture in a frame that is 5 cm wide.

What is the perimeter of the picture frame?

280 cm

2. The new playground at Middledale School will be enclosed by a

fence. The playground will be rectangular and will have an area of

225 yd 2 . The number of yards on each side will be a whole number.

What is the least amount of fencing that could be required to enclose

the playground?

60 yd

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3. Reasoning Evan is thinking of a 3-digit odd number that

uses the digit 7 twice. The digit in the tens place is less than

one. What is the number?

A 707

B 717

C 770

D 777

4. Explain It Explain how you solved Exercise 3.


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200 Topic 12

Name

Solids

Reteaching

13-1

The solid’s vertices are: A, B, C, D, E , F , G, and H.

The solid’s edges are: ___

AC , ___

AB , ___

CD , ___

DB , ___

AH , ___

___

BG ,

HG , ___

HE , ___

GF , ___

EF , ___

CE , and ___

DF .

The solid’s faces are: ACEH, BDFG, ABCD,

EFGH, CDEF , and ABHG.

Here are some common solid figures:

C

D

B

A

Edge

Cube Rectangular Prism Cylinder

Cone

Face

E

F

H

G

Vertex

Use the solid at the right for 1 through 3.

___

___

___ ____

____

____

____

OJ , JK , KQ , OQ , ON , NM ,

___

___ ___

___ ___

1. Name the edges.

MQ ,

KL ,

2. Name the faces.

LM ,

JP ,

NP ,

LP

NMQO, KLMQ, JKLP , JONP , JKQO, LMNP

O

N

P

J

Q

M

K

L

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3

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3. Name the vertices.

J , K , L, M , N , O, P , Q

What solid figure does each object resemble?

4. A r t

5. 6.

S u p p li e s

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Soup

Rectangular

prism Cylinder Cone

Topic 13 201

Name

Solids

Practice

13-1

For 1 through 3, use the solid at the right.

A

1. Name the vertices.

A, B, C, D

D

2. Name the faces.

ABC, ABD, ACD, BCD

B

C

___ ___ ___

AB , AC , AD ,

3. Name the edges.

___

BC ,

___

BD ,

___

CD

For 4 through 6, tell which solid figure each object resembles.

4. 5. 6.

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Cube

Cylinder

Rectangular prism

7. Which term best describes the figure at the right?

A Cone

B Triangular prism

C Pyramid

D Rectangular prism

8. Explain It How many vertices does a cone have? Explain.

One; the circular base does not have

any vertices but every point on the circle

meets at one point, the vertex, outside

the base.

202 Topic 13

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Name

Relating Shapes and Solids

Reteaching

13-2

If each net on the left were folded, it would form the figure on the

right.

Top

Side Side Side Side

Bottom

Side

Side

Bottom

Side

Side

What does each net represent?

1. 2.

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Cube

Cylinder

Topic 13 203

Name

Relating Shapes and Solids

For 1 and 2, predict what shape each net will make.

Practice

13-2

1. 2.

Rectangular prism

Square pyramid

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Reasoning For 3 through 5, tell which solid figures could be

made from the descriptions given.

3. A net that has 6 squares

4. A net that has 4 triangles

5. A net that has 2 circles and a rectangle

Cube

Triangular

ar pyramid

Cylinder

6. Which solid can be made by a net that has exactly one circle in it?

A Cone B Cylinder C Sphere D Pyramid

7. Explain It Draw a net for a triangular pyramid.

Explain how you know your diagram is correct.

Sample answer: The

4 equilateral triangles

fold

pyramid.

into a triangular

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204 Topic 13

Name

Surface Area

Reteaching

13-3

The surface area of a rectangular prism is the sum of

the areas of all of its faces.

Danica has a wood block. She wants to paint it. In order

to know how much paint to buy, Danica needs to find

the surface area of the block.

If you could “unfold” this block, it would look

like this:

You can find the area of each face by multiplying

the length times the width.

area of A 15 5 75

area of B 15 8 120

area of C 15 5 75

area of D 15 8 120

area of E 8 5 40

area of F 8 5 40

Adding all the areas together will give the total surface area.

75 120 75 120 40 40 470 in 2

E

15 in.

D

C

B

A

15 in.

8 in.

5 in.

5 in.

F 8 in.

5 in.

8 in.

5 in.

Another way to find the surface area of a rectangular

prism is to use the following formula.

surface area 2( l w) 2( l h) 2( w h)

surface area 2(15 5) + 2(15 8) + 2(5 8)

surface area 150 240 80

surface area 470 in 2

Remember:

l = length

w = width

h = height

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In 1 through 3, find the surface area of each figure.

1. 2. 1.5 mm 3.

3 cm

4 mm

3 cm

3 cm

10.5 mm 12 ft

54 cm 2 127.5 mm 2 358 ft 2

7 ft

5 ft

Topic 13 205

Name

Surface Area

Find the surface area of each rectangular prism.

1. 2.

Practice

13-3

3 m

12 ft

8 m

5 m

12 ft

158 m 2 864 ft 2

12 ft

Strategy Practice Music and computer CDs are often stored in

plastic cases called jewel cases.

3. One size of jewel case is 140 mm 120 mm 4 mm. What

is the surface area of this jewel case?

35,680 mm 2

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4. A jewel case that holds 2 CDs is 140 mm 120 mm 9

mm. What is the surface area of this jewel case?

38,280 mm

2

5. What is the surface area of a rectangular prism with the

dimensions 3 in. by 4 in. by 8 in.?

A 96 in 2 B 112 in 2 C 136 in 2 D 152 in 2

6. Explain It Explain why the formula for finding the surface

area of a rectangular prism is helpful.

Possible answer: the formula gives the

total area of a solid rectangular shape

that needs to be covered or painted.

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206 Topic 13

Name

Views of Solids

Reteaching

13-4

Here is how each figure on the left would look like from the front,

side, and top. In each figure, the number of cubes hidden by the

view is given beneath it.

There are no hidden cubes.

Front Side Top

Front Side Top

There are no hidden cubes.

Front Side Top

There is one hidden cube.

Look at the figure. Label its front,

side, and top views.

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1. 2. 3.

Side

Front

Top

How many cubes are hidden in each figure?

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4. 5. 6.

None One None

Topic 13 207

Name

Views of Solids

For 1 and 2, draw front, side, and top views of each stack of

unit blocks.

Top

1.

Front

Side

Practice

13-4

Front Side Top

2.

3. Reasoning In the figure for Exercise 2, how many blocks

are not visible?

1

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4. In the figure at the right, how many unit

blocks are being used?

A 8

B 9

C 10

D 11

5. Explain It A figure is made from 8 unit blocks. It is 3 units tall.

What is the maximum length the figure could be? Explain.

The maximum length is 6,

because 2 are used to give the

figure height.

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208 Topic 13

Name

Volume

Reteaching

13-5

Volume is a measure of the space inside a solid figure. It is

measured in cubic units. A cubic unit is the volume of a cube

which has edges that are 1 unit.

How to find the volume of a rectangular prism

Counting unit cubes

Using a formula

3 cm

1.2 m

2 cm

4 cm

Count the cubes in each layer: 8 cubes.

Multiply by the number of layers.

8 cubes 3 24 cubes

The volume of each cube is 1 cm 3 .

The volume of the prism is 24 cm 3 .

4 m

2.1m

You know the length, l , the width, w,

and the height, h. Calculate the volume,

V , using the formula V l w h.

w

2 cm

h

l

4 cm

3 cm

V 2 cm 4 cm 3 cm

V 24 cm 3

1. Find the volume of the rectangular prism using a formula.

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10.08 m 3

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Topic 13 209

Name

Volume

Find the volume of each rectangular prism.

1. base area 56 in 2 , height 6 in.

2. base area 32 cm 2 , height 12 cm

3. base area 42 m 2 , height 8 m

Practice

13-5

336 in 3

384 cm 3

336 m 3

4. 5 yd

5.

10 cm

8 cm

2 cm

5 yd

5 yd

125 yd 3 160 cm 3

6. Algebra What is the height of a solid with

a volume of 120 m 3 and base area of 30 m 2 ?

Michael bought some cereal at the grocery store.

4 m

1

3 2 in.

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7. What is the base area of the box?

28 in 2

8. What is the volume of the box?

364 in 3

13 in.

8 in.

9. What is the base area of this figure?

A 3.2 m 2 C 320 m 2

V 320 m 3

B 32 m 2 D 3,200 m 2

10 m

10. Explain It Explain how you would find

the base area of a rectangular prism if

you know the volume and the height.

Sample answer: Divide the volume by

the height to find the base area.

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210 Topic 13

Name

Irregular r Shapes and Solids

Reteaching

13-6

To find the area of an irregular shape, separate it into familiar

shapes. Find the area of each shape and add the areas.

1. To find the area of the figure

at the right, separate the

shape into two rectangles.

(See the dotted line in the

figure.)

5 cm

10 cm

Rectangle A

4 cm 3 cm Rectangle B

8 cm

6 cm

2. Use the formula A w

to find the area of each

rectangle; then add the areas.

Area of Rectangle A Area of Rectangle B

A 10 5 50 A 6 3 18

The area of the shape is 50 18 68 cm 2 .

To find the volume of an irregular solid, separate it into familiar

shapes. Find the volume of each shape and add the volumes.

1. To find the volume of the

figure at the right, separate

the shape into two rectanglar

prisms. (See the dotted line in

the figure.)

2. Use the formula

V w h to find the

volume of each prism; then

add the volumes.

1 ft Prism A

5 ft Prism B

7 ft

2 ft 2 ft

4 ft

3 ft

Volume of Prism A Volume of Prism B

A 1 4 7 28 A 2 4 2 16

The volume of the solid is 28 16 44 ft 3 .

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1. Show two ways of dividing the given solid into two regular solids.

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Topic 13 211

Name

Irregular Shapes and Solids

For 1 and 2, find the area of the irregular shape.

1. 4 cm

2.

m

2 c

5 c m

2 cm

4

c

m

70 yd

70 yd

70 yd

d

0 y 4

Practice

13-6

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5 cm

37 cm 2

For 3 and 4, find the volume of the irregular solid.

2,800 yd 2

3.

4.

2 m

1 m

B 12,000 mm 2 D 14,400 mm 2

6 ft

20 ft

3 m

4 m

5 m

25 ft

54 m 3

16 ft

9,200 ft 3

5. What is the area of this irregular

shape?

6. Explain It When do you find an

area or volume by separating the

figure into smaller figures?

120 mm

60 mm 90 mm

When the figure

200 mm

is irregular

A 1,200 mm 2 C 13,200 mm 2

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212 Topic 13

Name

Problem Solving: Use Objects

and Solve a Simpler Problem

Reteaching

13-7

At a math fair, Willie saw a puzzle about a giant cube made of

identical white smaller cubes. The giant cube was 4 4 4.

It contained 64 smaller cubes. Each of the six faces of the giant

cube was painted red. The puzzle asked, “If the giant cube were

taken apart, how many smaller cubes would have only one face

painted red?” Here is how Willie tried to solve the puzzle.

1. Construct cubes using 8 and

27 smaller cubes. Imagine painting

each of the larger cubes.

2. Classify the smaller cubes. Think:

Where are the cubes located in the

larger cube? How are they painted

differently from each other?

3. Make a table.

corner cube

edge cube

corner cube

face cube

Cube A

Cube B

Location Number Painted Faces Number Painted Faces

Corner 8 3 8

Edge none 12

Face none 6

Center none 1

3210

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Willie organized the data about the 64 smaller cubes in the giant

cube. Use the table above to complete the table below. One set

of data has already been completed.

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Small cubes

Painted Surfaces Location in giant cube Number

3 corner (Think: Same as a 3 3 3.) 8

2 edge (Think: One more than a 3 3 3 on each edge.)

1 face (Think: Three more than a 3 3 3 on each face.)

0 center (Think: The center is now 2 2 2.)

24

24

8

Topic 13 213

Name


and Solve a Simpler Problem

Use objects to help you solve a simpler problem. Use the solution

to help you solve the srcinal problem.

1. Number Sense Six people can be seated at a table. If two

tables are put together, 10 people can be seated. How many

tables are needed to make a long table that will seat

22 people?

5 tables

Practice

13-7

2. Donna is building a large cube that will have 5 layers, each

with 5 rows of 5 small cubes. How many small cubes will the

larger cube contain?

5 5 5 125

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3. Strategy Practice Jerome’s job duties include feeding the

fish. There are 5 kinds of fish that he feeds: guppies, zebra

danios, betas, platys, and neon tetras.

Use the following clues to find the order in which Jerome

feeds them.

• Jerome feeds the guppies third.

• Jerome does not feed the betas right before

or right after the guppies.

• Jerome feeds the zebra danios last.

• Jerome feeds the platys after the betas.

A guppies, zebra danios, betas, platys, and neon tetras

B betas, platys, guppies, neon tetras, zebra danios

C neon tetras, zebra danios, guppies, platys, betas

D betas, guppies, platys, neon tetras, zebra danios

4. Explain It Suppose Ann is placing bowling pins in the

following manner: 1 pin in the first row, 2 pins in the second

row, 3 pins in the third row, and so on. How many pins will

she use if she has 5 rows in her placement? Explain.

1 2 3 4 5 15

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214 Topic 13

Name

Customary Units of Capacity

Reteaching

14-1

Capacity is the measure of an amount of liquid. Common units of

capacity are the gallon (gal), quart (qt), pint (pt), and cup (c).

The chart below shows how capacity units are related.

gallon

q u

a

r t

q

u

a r

t

q u

a

r t

q

u

a r

t

q u

a r

t

Pint

Pint

pint

cup

cup

1 gallon 4 quarts 1 quart 2 pints 1 pint 2 cups

How to estimate volume in different units:

What is the volume estimated in gallons? in quarts?

gallon

q u

a r

t

q

u

a r

t

Step 1: Think: The gallon container

looks about half full.

Step 2: Estimate.

Step 3: Remember how gallons and

quarts are related.

Step 4: Relate 1_ gallon to quarts.

2

Step 5: Simplify.

Step 6: State your estimate in quarts.

The volume of the liquid is about half

the capacity of the container.

The volume of the liquid is about

1_

2 gallon.

1 gal 4 qt

1__ 1__

gal

2 2

1__

2

(4 qt)

(4 qt) 2 qt


2 quarts.

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1_

1. Number Sense How many cups are in 12 quarts of lemonade?

6

Topic 14 215

Name

Customary Units of Capacity

Which unit(s) of capacity would be most reasonable to

measure each?

1. barrel of oil

2. can of applesauce

3. individual carton of orange juice

gal

cup or fl oz

cup or pt

Practice

14-1

Give the amount of liquid in each container. Use a fraction if necessary.

4. 5.

pt

q u

a r

t

2 4

qt

gallon

6. Reasoning Do 1 gallon and 8 pints represent the same amount?

A No, 8 pints is 8 times the amount of 1 gallon.

B No, 1 gallon is equal to 4 quarts.

C Yes, 1 gallon is equal to 4 quarts, which is equal to 8 pints.

D Yes, 8 pints is equal to 8 cups, which is equal to 1 gallon.

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7. Explain It If you only needed 1 c of milk, what is your

best choice at the grocery store—a quart container, a pint

container, or a 1_ gal container? Explain.

2

A pint container (16 fl oz) is closer to a

cup (8 fl oz) than the other measures.

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216 Topic 14

Name

Metric Units of Capacity

Reteaching

14-2

Two common metric units of capacity are the

liter (L) and the milliliter (mL).

1 cm

1 cm

1 cm

1milliter (mL)

How to estimate volume in liters and milliliters:

What is the volume estimated in liters? in milliliters?

1liter (L) 1,000 mL

1liter (L)

Step Think. 1:

Step Estimate. 2:

Step 3: 3 Relate liters to milliliters.

The liter container looks about half full.


1 L 1,000 mL

Step 4: Relate 1_ liter to milliliters. 1_ L 2 2 1_ (1,000 mL) 500 mL

2

1_

2 liter.

Complete.

2.4

900

0.334

230

1. 2,400 mL L

2. 0.9 L mL

3. 334 mL L

4. 0.23 L mL

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Topic 14 217

Name

Metric Units of Capacity

Estimation Which unit of capacity would be most reasonable to

measure each?

1. bottle of water: 1 mL or 1 L

2. bottle cap: 20 mL or 20 L

3. swimming pool: 80,000 mL or 80,000 L

Practice

14-2

1 L

20 mL

80,000 L

Give the amount of liquid in each container.

5.

150 L

1200 mL

800 mL

400 mL

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100 L 200 mL

6. Reasonableness In a science fair project, you test

four different-sized containers of water: 1 L, 2 L, 4 L,

and 5 L. Which of the following expresses these capacities

in milliliters?

A 1 mL, 2 mL, 4 mL, 5 mL

B 10 mL, 20 mL, 40 mL, 50 mL

C 100 mL, 200 mL, 400 mL, 500 mL

D 1,000 mL, 2,000 mL, 4,000 mL, 5,000 mL

7. Explain It Tell whether you would use multiplication or

division to convert milliliters to liters. Explain your answer.

Sample answer: I would divide by

1,000 to convert milliliters to liters,

because 1,000 mL 1 L.

218 Topic 14

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Name

Units of Weight and Mass

Reteaching

14-3

Weight is a measure of how heavy or light something is.

rose: about 1 oz man’s shoe: about 1 lb 500 bricks: about 1 ton

Mass is a measure of the quantity of matter, or stuff, in an object.

1 cm

1 cm

1 cm

M ILK

mL of water: about 1 mg dollar bill: about 1 g liter of milk: about 1 kg

1 L

How Weight

Units are Relate

lated

16 oz 1 lb

2,000 lb 1 T

How Mass Units are Relate

lated

1,000 mg 1 g

1,000 g 1 kg

Which customary weight unit would be best for measuring

each object?

1. elephant 2. hot dog bun 3. loaf of bread

ton

ounce

pound

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Which metric mass unit would be best for measuring the amount

of stuff in each object?

4. dust particle 5. plastic fork 6. dictionary

milligram gram kilogram

7. Estimation Estimate the weight in pounds of 162 oz of hot dogs.

10 lb

Topic 14 219

Name

Units of Weight and Mass

Which customary unit of weight would be best to measure

each weight?

1. newborn baby

2. earrings

pound

ounce

Practice

14-3

Which metric unit of mass would be better to measure

each mass?

3. necktie

4. 5 math textbooks

Which mass or weight is most reasonable for each?

5. laptop computer: 5 lb or 5 mg

6. dozen donuts: 1 oz or 1 kg

gram

kilogram

5 lb

1 kg

7. Reasonableness Which unit would you use to weigh a

garbage truck?

A ton B kilogram C pound D milligram

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8. Explain It Did you know that there is litter in outer space?

Humans exploring space have left behind bags of trash,

bolts, gloves, and pieces of satellites. Suppose there are

about 4,000,000 lb of litter in orbit around Earth. About how

many tons of space litter is this? Explain how you found

your answer.

2,000 T

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220 Topic 14

Name

Converting Customary Units

Reteaching

14-4

1 gal 128 fl oz

1 qt 32 fl oz 1 qt 32 fl oz 1 qt 32 fl oz 1 qt 32 fl oz

1 pt

(16 fl oz)

1 pt

(16 fl oz)

1 pt

(16 fl oz)

1 pt

(16 fl oz)

1 pt

(16 fl oz)

1 pt

(16 fl oz)

1 pt

(16 fl oz)

1 c 1 c 1 c 1 c 1 c 1 c 1 c 1 c 1 c 1 c 1 c 1 c 1 c 1 c 1 c 1 c

1 pt

(16 fl oz)

Converting a capacity measurement from a smaller unit to a larger unit:

4 pints to ____ quarts

You know 2 pt 1 qt.

Think: Quarts are larger than pints. If I convert pints

to quarts, there will be a smaller number of quarts

than there were pints. Operation: divide.

Find 4 2.

4 pt 2 qt

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1 1 1

in. in. in.

1 yd 36 in.

1 ft 12 in. ft 1 12 in. 1 ft

1

in.

1 1

in. in.

1

in.

1 1 1

in. in. in.

1

in.

1 1

in. in.

1

in.

1 1

in. in.

1

in.

1 1

in. in.

1

in.

1 1 1

in. in. in.

1

in.

1 1

in. in.

1

in.

1 1

in. in.

Converting a length measurement from a larger unit to a smaller unit:

2 feet ____ inches

Think: Inches are smaller than feet. If I convert

feet to inches, there will be a larger number of

inches. Operation: multiply.

Complete.

12 in.

1

in.

1 1 1

in. in. in.

1

in.

You know 1 ft 12 in.

Find 2 12.

2 ft 24 in.

16 2,000

32 5

2 15,840

3 3

1. oz 1 lb 2. lb 1 T

3. 8 gal qt 4. 10 c pt

5. 6 ft yd 6. 3 mi ft

7. 48 oz lb 8. 6,000 lb T

9. Estimation A vat has a capacity of 624 fl oz.

Estimate its capacity in cups.

80 c

1 1

in. in.

Topic 14 221

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Name

Converting Customary Units

In 1 through 9, convert each measurement. You may need to

convert more than once.

Practice

14-4

2 125 96,000

64 256 240

13 10 16

1. 3,520 yd mi 2. 10 ft 5 in. in. 3. 3 T oz

4. 4 gal c 5. 2 gal fl oz 6. 6 yd 2 ft in.

7. 6 qt 1 pt pt 8. 40 qt gal 9. 48 ft yd

10. Jenny bought a gallon of orange juice. The recipe for punch

calls for 2 quarts of juice. How many quarts will she have left

over to drink at breakfast?

2 qts left over

11. Algebra Laura has a roll of ribbon. There are 20 yards of

ribbon on the roll. What equation can Laura use to find the

number of inches of ribbon on the roll?

n 20 36

12. Ramona drives a truck weighing 1 ton. She picks up 3 cows

weighing 3,000 pounds total. How much do the truck and the

cows weigh together?

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A 50 lb

B 500 lb

C 5,000 lb

D 50,000 lb

13. Explain It If you want to convert pounds to ounces, do you

multiply or divide?

Sample answer: every pound contains

16 of pounds ounces. by You 16 multiply to find the number of

ounces.

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222 Topic 14

Name

Converting Metric Units

Reteaching

14-5

Converting a capacity measurement from a smaller metric unit to

a larger metric unit:

2,000 milliliters liters

Think: Liters are larger than milliliters. If I convert

milliliters to liters, there will be a smaller number

of liters. Operation: divide.

You know 1,000 mL 1 L.

Find 2,000 1,000.

2,000 mL 2 L

Converting a length measurement from a larger metric unit to a

smaller metric unit:

2 kilometers meters

Think: Meters are shorter than kilometers. If I

convert kilometers to meters, there will be a larger

number of meters. Operation: multiply.

You know 1 km 1,000 m.

Find 2 1,000.

2 km 2,000 m

Complete.

1,000

3,000

500

72,000

1,000

1. 1 kg g 2. 1 g mg

8

14

8

3. 3 L mL 4. 8,000 mL L

5. 5 m cm 6. 14,000 mm m

7. 72 g mg 8. 8,000 g kg

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Topic 14 223

Name

Converting Metric Units

In 1 through 6, convert each measurement.

3 8,000

600 9,800

20 11

1. 3,000 mm m 2. 8 km m

3. 6 m cm 4. 9.8 kg g

Practice

14-5

5. 20,000 mg g 6. 11,000 mL L

In

Use

7 through

, , or

12, compare the measurements.

for each .

7. 4,000 g 5 kg 8. 0.72 L 572 mL 9. 5.12 m 512 cm

10. 8 kg 5,000 g 11. 44 L 44,000 mL 12. 200 m 20,000 mm

13. A watering can holds 4.2 liters. If a small cactus requires 310

mL of water once every 3 weeks, how many of the same

cactus plant can be watered at once without refilling the

watering can?

13 cactus plants

Multiply by 1,000

14. Critical Thinking How do you convert kilometers to meters?

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15. Jeremy’s apartment building is 15 meters tall. How many

centimeters is this?

A 1.5 cm B 50 cm C 1,500 cm D 15,000 cm

16. Explain It Explain how to convert 24 kilograms to grams.

Sample answer: Since 1kg 1,000g,

multiply 24 kg by 1,000. 24 1,000

24,000 grams

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224 Topic 14

Name

Elapsed Time

How to find elapsed time:

Iris left her house at 1:15 P.M. and

arrived at her grandparents’ house at

3:30 P.M. How long did the trip take?

Use a number line to count up.

Start

1:15 P.M.

1:00

P.M.

h1

h1

15

2:00

P.M.

3:00

P.M.

The trip took 2 h 15 min.

min End

3:30

P.M.

Reteaching

14-6

How to use elapsed time to find when

an event began or ended:

Omar and his brothers played floor

hockey for 1 hour and 9 minutes. They

finished playing at 6:30 P.M. At what

time did they begin playing?

You can subtract to find the start time.

End Time Elapsed Time Start Time

6 h 30 min

1 h 9 min

5 h 21 min

So, they began playing at 5:21 P.M.

Find each elapsed time.

1. 8:13 P.M. to 10:00 P.M. 2. 1:24 P.M. to 4:47 P.M.

1 h 47 min 3 h 23 min

3. 3:35 P.M. to 6:09 P.M. 4. 9:55 P.M. to 11:42 P.M.

2 h 34 min 1 h 47 min

Find each start time or end time using the given elapsed time.

5. Start: 5:49 A .M. 6. End: 8:27 P.M.

Elapsed: 5 h 20 min

Elapsed: 4 h 13 min

End: 11:09 A .M. Start: 4:14 P.M.

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Add or subtract.

7. 6 h 31 min

7 h 16 min

8. 7 h 12 min

3 h 30 min

9. 3 h 5 min

8 h 55 min

10. 9 h 5 min

8 h 22 min

13 h 47 min 3 h 42 min 12 h 43 min

Topic 14 225

Name

Elapsed Time

Find each elapsed time.

1. 9:59 P.M. to 10:45 P.M.

2. 1:45 P.M. to 5:38 P.M.

46 min

3 h 53 min

Practice

14-6

3. 8:45 11:36

4.

A.M.

A.M.

12

11 1

10

2

9

8

3

4

7

6

5

12

11 1

10

2

9

8

3

4

7

6

5

P.M.

2 h 51 min 9 h 38 min

P.M.

Find the end time using the given elapsed time.

5. Start: 3:46 P.M. Elapsed: 2 h 20 min

6:06 P.M.

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6. Add. 2 h 45 min

3 h 58 min

The White House Visitor Center is open from 7:30 A .M.

until 4:00 P.M.

7. Add. 6 h 47 min

5 h 28 min

6 h 43 min 12 h 15 min

8. Tara and Miguel got to the Visitor Center when it opened,

and spent 1 hour and 20 minutes there. At what time did

they leave?

9. Jennifer left the Visitor Center at 3:30 P.M. after spending

40 minutes there. At what time did she arrive?

8:50 A .M.

2:50 P.M.

10. A football game lasted 2 hours and 37 minutes. It finished

at 4:22 P.M. When did it start?

A 1:45 P . M . B 1:55 P . M . C 2:45 P . M . D 2:50 P . M .

11. Explain It What is 1 hour and 35 minutes before 4:05 P.M.?

Explain how you solved this problem.

226 Topic 14

Sample answer: 2:30 P.M.; ; I solved this

problem using subtraction.

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Name

Elapsed Time in Other Units

Reteaching

14-7

A ship left Houston, Texas, at 7:10 A .M. on Tuesday. It arrived at

Progreso, Mexico, at 10:15 P.M. on Wednesday. How long did the

voyage take?

Method 1

1. Represent each time on a 24-hour

clock. See the chart below.

Method 2

1. Represent each time on a 48-hour

number line. See the line below.

12 Hour (P.M.)

.)

24 Hour

1:00 2:00 3:00 4:00 5:00 6:00 7:00 8:00 9:00 10:00 11:00 12:00

13:00 14:00 15:00 16:00 17:00 18:00 19:00 20:00 21:00 22:00 23:00 00:00 0246

7:10 46:15

8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48

2. Find the time from departing

Houston until midnight of Tuesday.

24 h 00 min 23 h 60 min

7:10 min 7 h 10 min

16 h 50 min

3. Find the time the boat sailed

on Wednesday until it reached

Progreso.

0 h 00 min

2. Subtract the departing time from

the arrival time.

46 h 15 min

7 h 10 min

39 h 5 min

22 h 15 min

22 h 15 min

4. Add the two times.

16 h 50 min 38 h 65 min

22 h 15 min 1 h 60 min

38 h 65 min 39 h 5 min

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Use a 12-hour clock to represent time. Complete the following

addition to find the sailing time from Houston to Progreso,

Mexico.

12 h 00 min 10 h 15 min

4h 50 min 12 h 00 min

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Topic 14 227

Name

Elapsed Time in Other Units

Find the elapsed time.

1. 9:50 P.M. to 4:00 A .M.

2. 5:15 A .M. to 2:15 P.M.

3. 10:00 P.M. to 4:45 A .M.

4. 6:30 P.M. to 5:10 A .M.

Find the end or start time.

5. Start time: 8:15 A .M.

Elapsed time: 30 h 20 min.

End time: 2:35 P.M.

6 h 10 min

9 h

6 h 45 min

10 h 40 min

7. Number Sense Krishan wakes up at 7:30 A .M. every

morning. He spends 30 minutes getting ready, 6 hours

at school, and 2 hours volunteering. Then he takes the bus

for 15 minutes and arrives at his house. What time does he

arrive at his house?

Practice

14-7

6. End time: 10:30 P.M.

Elapsed time: 13 h 30 min.

Start time: 9:00 A .M.

A 2:00 P.M. B 3:45 P.M. C 4:15 P.M. D 5:45 P.M.

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8. Explain It Emre left for a trip Sunday night at 10:00 P.M.

He returned home Tuesday morning at 6:00 A .M. How many

hours was he away on his trip? Explain how you found

your answer.

24 hours from Sunday to Monday 10 P.M., .,

and another 8 hours to Tues morning.

Total: 24 8 32 hours

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228 Topic 14

Name

Temperature Change

Reteaching

14-8

How to read temperature:

Miami’s average high temperature

in June is 87°F. What is the average

high temperature in degrees Celsius?

°F 87 31 °C

Step 1: Find 87° on the Fahrenheit

scale.

Step 2: Read across to find the

temperature on the Celsius scale.

31°C is the same as 87°F.

How to find changes in temperatures:

Find the change in temperature from

57°F to 18°F.

Step 1: Find the difference between

57°F and 18°F by subtracting.

57°F 18°F 39°F

Step 2: Tell if the difference is an

increase or decrease. Since the second

temperature is less than the first

temperature, there was a decrease.

So, the change is a decrease of 39°F.

Write each temperature in Celsius and Fahrenheit.

°F

55 13 °C 2.

°F 92 33 °C

55°F1

3°C 92°F3

3°C

, ,

Find each change in temperature.

3. 34°F to 89°F 4. 11°C to 26°C

55°F increase

15°C increase

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5. 86°F to 54°F 6. 30°C to 8°C

32°F decrease

22°C decrease

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7. 3°F to 81°F 8. 12°C to 5°C

78°F increase 7°C decrease

9. Number Sense Yesterday’s high temperature was 76°F. The

difference between the high and low temperature was 33°F.

What was yesterday’s low temperature?

43°F

Topic 14 229

Name

Temperature Change

Write each temperature in Celsius and Fahrenheit.

1. °F °C 2. °C 3.

°F

80

70

60

50

20

10

°F

100

40

90

80

30

, , ,

Find each change in temperature.

4. 34°F to 67°F 5. 12°C to 7°C

6. Number Sense Which is a smaller increase in temperature:

a 5°F increase or a 5°C increase?

60

50

40

30

Practice

14-8

70°F 21°C 94°F

34°C

50°F 10°C

33°F increase 5°C decrease

5°F is a smaller increase.

10

0

°C

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Information about the record high temperatures

in four states is shown.

7. What is the difference between the record

high temperature in Florida and the record

high temperature in Alaska in °C?

5°C

8. What is the difference between the record high temperature

in Michigan and the record high temperature in Florida in °F?

Record High

Temperature

State °F °C

Alaska 100 38

Florida 109 43

Michigan 112 44

Hawaii 100 38

3°F

9. What is the difference between 6°C and 12°C?

A 6°C B 12°C C 18°C D 19°C

10. Explain It Which is warmer, 1°F or 1°C? Explain how you

found this answer.

1°C is warmer. Sample answer: You can

tell by looking at the thermometers.

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230 Topic 14

Name

Problem Solving: Make a Table

Reteaching

14-9

A bus company schedules three buses from Elm to Coretown. Starting at

8:05 A .M., the first bus departs Elm with a bus following every 45 minutes.

The trip takes 70 minutes. What time is the third bus scheduled to arrive in

Coretown?

Construct a table of what you know.

Bus Departure Time ( A .M.) Travel Time Arrival Time

1 8:05 min 70

8:05 70 min

2 8:05 45 min 70 min 8:05 45 min 70 min

3 8:50 45 min 45 min 70 min 8:05 45 min 45 min

70 min

Simplify the arrival time of the third bus. 8 h 05 min

8:05 45 min 45 min 70 min 160 min

8:05 160 min 8 h 165 min

Arrival time of third bus in Coretown:

Look back and check:

is your answer reasonable?

State the following times in standard form.

1. Arrival time of first bus at Coretown

2. Departure time of second bus from Elm

10 h 45 min

2 h 120 min

10:45 A .M.

Yes, 3__

4 h 3__

4 h 11__ h is 23__

4 4 h;

8:00 A .M. 2 3__ h is 10:45 A .M.

4

9:15 A .M.

8:50 A .M.

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Topic 14 231

Name

Problem Solving: Make a Table

Make a table to solve the problems.

1. The temperature in the room was 55F at 8:00 A .M. Ed turned

the heat on, and the temperature rose 4F every 30 minutes.

What was the temperature of the room at 11:30 A .M.?

83F

Practice

14-9

2. Charles can type 72 words per minute. He needs to type a

paper with 432 words. How many minutes will it take Charles

to type the paper?

6 minutes

3. Number Sense Yugita wakes up at 5:00 A .M. on Monday.

Each day after that, she wakes up 17 minutes later. What

time will she wake up on Friday?

A 5:17 A .M.

B 5:51 A .M.

C 6:08 A . M .

D 6:25 A .M.

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4. Explain It Train A leaves from New York at 10:15 A .M. and

arrives in New Haven at 12:23 P.M. Train B takes 2 hours to

go from New York to New Haven, but takes a 20-minute rest

break in the middle of the trip. Which train is faster? Explain.

Train A. It takes 2 hours 8 minutes and

Train B takes 2 hours 20 minutes.

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232 Topic 14

Name

Solving Addition and

Subtraction Equations

Reteaching

15-1

Paige gave 6 peaches to Simon. Paige now has 8 peaches. How many peaches

did Paige have to start?

The number of peaches Paige now has equals the number of peaches she had at

the start minus the number she gave to Simon.

So, p 6 8 p number of peaches at the start

To solve this equation, you need to get p alone. You can do this by using an

inverse operation. An inverse operation is an opposite—addition and subtraction,

for example, are inverse operations.

To get p alone, add 6 to both sides of the equation.

Tip: If you add or subtract on one side of the equation, you must do the same on

the other side.

p 6 6 8 6

p 14

So, Paige started out with 14 peaches.

For questions 1 through 4, solve each equation.

a 2 y 23

p 7

c 34

1. a 3 5 2. y 14 9

3. 8 p 15 4. 52 c 18

5. Mary’s sister is 4 years older than Mary. Her sister is 12. Use the equation

m 4 12 to find Mary’s age.

m 8: Mary is eight years old.

6. Explain It Lyndon solved x 18 4. He said the answer was x 4. Explain

why this is correct or incorrect.

This is incorrect because he didn’t add

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18 to both sides of the equation. The

answer should be x 22.

Topic 15 233

Name

Solving Addition and

Subtraction Equations

Solve and check each equation.

1. x 4 16 2. t 8 15

3. m 9 81 4. 7 y 19

5. k 10 25 6. 15 b 50

7. f 18 20 8. w 99 100

9. 75 n 100 10. p 40 0

11. Jennifer has $14. She sold a notebook and pen, and now she

has $18. Solve the equation 14 m 18 to find how much

money Jennifer received by selling the notebook and pen.

$4

Practice

15-1

x 12 t 23

m 90

y 12

k 35

b 35

f 2

w 199

n 25

p 40

12. Kit Carson was born in 1809. He died in 1868. Use the equation 1,809 x

1,868 to find how many years Kit Carson lived.

59 years

13. Strategy Practice Which is the solution for y when

y 6 19?

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A 13 B 15 C 23 D 25

14. Explain It Nellie solved y 3 16. Is her answer correct?

Explain and find the correct answer if she is incorrect. y 3 16

No; Sample answer: Nellie

y 13

used the wrong operation.

Subtract 3

She needed to use addition

to get the variable alone.

The correct answer is y 19.

234 Topic 15

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Solving Multiplication and

Division Equations

Reteaching

15-2

Dan’s computer prints at the rate of 3 pages per minute. How many pages can he

print in 15 minutes?

The number of minutes is equal to the number of pages divided by the rate the

pages are printed.

15 d 3 d number of pages

To solve this equation, you can do an inverse operation. An inverse operation is an

opposite—multiplication and division

are inverse operations.

To get d alone, multiply 3 times both sides of the equation.

Remember: If you multiply or divide one side of an equation, you must do the

same to the other side.

15 3 d 3 3

45 d

Dan can print 45 pages in 15 minutes.

In 1 through 6, solve each equation.

a 4 t 63

x 10 c 25

p 99 y 64

1. 12a 48 2. t 9 7

3. 250 25x 4. 75 3c

5. p 33 3 6. 4 y 16

7. Number Sense A carton of eggs is on sale for $0.49. Use the equation

$0.49x $2.45 to find out how many cartons you can purchase for $2.45.

x 5 cartons

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8. Geometry Shanika has a piece of cardboard with

an area of 63 square inches; one side is 9 inches

long. Write an equation and solve for the value

of the other side.

9y 63, y 7 inches

Area 63 square inches

9 inches

y

Topic 15 235

Name

Solving Multiplication and

Division Equations

Solve each equation.

y 5 c__ c

9 6 54

p 6

w 64

1. 11y 55 2.

3. 150 25p 4. 16 w__

4

k 288 d 3

x 3

m__ m

7 13 91

t 9

b 6

a 5

e 160

5. ___ 36 k 8 6. 13d 39

7. 30 10x 8.

9. 81 9t 10. 5b 30

11. 20 4a 12.

e ___

80 2

Practice

15-2

13. Reasoning Antoinette divides 14. Adam is making a trout dinner for

54 by 9 to solve an equation for y . six people. He buys 48 oz of trout.

One side of the equation is 54. How many ounces of trout will each

Write the equation.

person get?

9y 54 6x 48; x 8 oz each

15. Which operation would you use to solve the equation

19x 646?

A add 19 B subtract 17 C divide by 19 D multiply by 19

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16. Explain It How would you use mental math to find m in the

equation 63 ( 63) m__ 2?

Sample answer: Multiplying and

dividing m by 63 undo each other. So

the equation is m 2.

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236 Topic 15

Name

Inequalities and the

Number Line

Reteaching

15-3

How can you graph an inequality?

An inequality is a mathematical sentence that contains one of the

symbols , , , or . A solution of an inequality is any number

that makes the inequality true.

Graph the inequality x 5.

1. Draw a number line 0–10 with an open

circle around 5. The open circle shows that

5 itself is not a solution.

2. Find and graph three solutions of x 5 on

the number line. Draw an arrow from the

open circle toward the graphed solutions.

The arrow shows that all numbers greater

than 5 are solutions to the inequality.

0 1 2 3 4 5 6 7 8 9 10

0 1 2 3 4 5 6 7 8 9 10

Determine the inequality represented by the following graph.

0 1 2 3 4 5 6 7 8 9 10

1. The closed circle represents the number 4. One value of x can be 4. x 4

2. The arrow represents all numbers less than 4. So x can have any

value less than 4.

x 4

3. The graph shows that x can have a value equal to or less than 4. x 4

Graph each inequality.

1. x 6 2. x 10

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Determine the inequality represented by each graph.

3. 4.

0 1 2 3 4 5 6 7 8 9 10

x 7 x 3

0 1 2 3 4 5 6 7 8 9 10

Topic 15 237

Name

Inequalities and the

Number Line

Name three solutions of each inequality. Then graph each

inequality on a number line.

1. b 5

2. s 2 4

Practice

15-3

0 2 4 6 8 10 0 1 2 3 4 5 6

5, 6, 8

2, 1, 0

3. x 3

4. d 1 6

012345 0 2 4 6 8 10

0, 1, 2, 3 8, 9, 10

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5. Lizette wants to read more

books. Her goal is to read at

least 2 books each month. Let m

represent the number of books

she will read in a year. Use m 24

to graph the number of books she

plans to read on a number line.

20 21 22 23 24 25 26 27

6. Estimation Marcus is making

cookies for his class. There are

26 students, and he wants to

bring 3 cookies per student. On

Sunday, he runs out of time and

decides he must buy at least half

the cookies and will make no

more than half from scratch. Draw

a graph to represent how many

cookies he will make himself.

20 32 34 36 38 40

7. Which sentence is graphed on the line below?

70 71 72 73 74 75 76

A m 74 B m 74 C m 74 D m 74

8. Explain It How do you know whether to use an open circle

or a closed circle when graphing an inequality?

Use an open circle when the inequality

symbol is or . . Use a closed circle

when the inequality symbol is or .

238 Topic 15

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Patterns and Equations

Reteaching

15-4

You can find a rule for a pattern and use that rule to fill in a table.

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b1 st

nd

rd

th

ox b 2 ox b 3 ox 4box

How many apples will be in the 8 th box? Which box will have

39 apples?

Step 1: Draw a table to show the information.

Box number ( b) Number of apples ( a)

1 3

2 6

3 9

4 12

Step 3: Write an equation to express the rule.

number of apples 3 times the box number

a 3b

Step 2: Find a relationship between the two

columns that is the same for each pair. This

relationship is the rule for the pattern.

For these two columns the relationship is to multiply

the box number by 3.

Step 4: Use this equation to find the missing parts

of the table.

a 8 3 39 3b

a 24, 24 apples 39 3 b

in the 8 th box 13 b, 13 th box will have

For questions 1 through 4, write an equation for each table and

solve for the missing values for x and y.

1.

x

0

y

5

3 8

2.

x

16

y

64

9 36

3.

x

y

39 apples

4.

2 4

2

30

40

3

4

48 2

50 5

16 10 80

8

7 12

6 24

10 15

4 16

y x 5 y 4x y x 6 y ___ x

10

x

y

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Patterns and Equations

For 1 through 3, find a rule for each table. Write an equation for

each rule.

1. x y 2. x y

3.

5 15

2 6

11 33

18 9

50 25

12 6

x y

4 –4

8 0

12 4

Practice

15-4

6 18

y 3x

34 17

y __ x

2

16 8

4. Reasoning Write an equation that will give the answer y 5

when x 12.

Possible answer: y x 7

y x 8

5. A farmer sells 200 apples at the market. The next week, he sells 345 apples.

How many more apples did he sell the second week?

200 x 345; 145 more apples

6. In the equation y x 5, which numbers would you use for

x if you wanted y 0?

A numbers 10 B 1, 2, 3, 4 C 6, 7, 8 D 10, 11, 12

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7. Explain It If you know the rule for a table, how can you add

pairs of numbers to the table?

Sample answer: Use values for x not

already in the table and solve for y .

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240 Topic 15

Name



Reteaching

15-5

Salim collects football cards and baseball cards. He has

38 baseball cards. If he has 25 football cards, how many

sports cards does Salim have?

Step 1: Write down what you already know. Step 2: What are you trying to find?

Salim has 25 football cards.

He has 38 baseball cards.

How many sports cards Salim has.

Step 3: Draw a diagram to show the information. Step 4: Write an equation and solve.

Choose a variable to represent what you are trying

38 25 b

to find. b 63

b

38 25

Salim has 63 sports cards.

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For questions 1 through 3, draw a picture or diagram, write an

equation, and solve the equation to answer the question.

1. Hallie spent $28 buying two figurines at the flea

market. If one figurine costs $11.28, how much

did Hallie pay for the other one?

$28.00 x $11.28, x $16.72

2. Cody is selling popcorn for his Cub Scout pack.

His goal is to sell $350. So far, Cody has sold

$284. How much more does Cody need to sell

to meet his goal?

p $284 $350, p $66

3. In Briana’s class there are 5 fewer students than

there were last year. There are 21 students in

Briana’s class. How many students were there

last year?

$28.00

x $11.28

$350.00

$284 p

x 5 21, x 26 students last year

x

5 21

Topic 15 241

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Name



Draw a picture, write, and solve an equation to answer

the question.

Practice

15-5

1. Suki and Amy made a total of 15

homemade holiday cards. Amy

gave away 7 of them. How many

cards did Suki give away?

15 7 p;

p 8 cards for

Suki to give

away

2. Ramon ate 3 more pieces of fruit

today than he did yesterday. Today

he ate 4 pieces. Write an equation to

find out how many pieces of fruit he

ate yesterday.

3 p 4;

p 1 piece

of fruit eaten

yesterday

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3. Critical Thinking A total of 64

children are going on a field trip.

If 14 of the children are girls, how

many are boys?

14 w 64;

w 50 boys

4. Naomi’s class went to the museum.

There are 16 students in her class.

If the total cost of admission for

the class was $96, what does one

admission to the museum cost?

$96 16x ; $6 per

admission

5. Paul is in a 17-kilometer canoe race. He has just reached

the 5-kilometer marker. Which of the following equations can

you use to find out how many more kilometers he needs to

paddle?

A k 5 17 B 5 k 17 C 5 k 17 D 17 5 k

6. Explain It How can you use estimation to decide whether

24.50 multiplied by 4 is close to 100?

Sample answer: 24.50 is about

25, and 25 4 is 100.

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242 Topic 15

Name

Understanding Ratios

Reteaching

16-1

Coach Sanders has 9 basketballs, 6 footballs, and 5 soccer balls

in her equipment bin. What is the ratio of basketballs to footballs?

What is the ratio of soccer balls to total balls?

Using ratios, you can compare parts of a

whole to the whole, some parts to other

parts, or the whole to its parts.

number of number of

basketballs footballs

9 6

Remember There are three different

ways to write the same ratio.

9 to 6 9:6

9__

6

In 1 through 4, write each ratio.

Give each answer in the three

different ratio forms.

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1. small stars to large stars 2. filled stars to total stars

7 to 5, 7:5, __ 7 5

6 to 12, 6:12, ___ 6

12

3. large unfilled stars to small 4. small filled stars to total

unfilled stars

stars

2 to 4, 2:4, __ 2

4

3 to 12, 3:12, ___ 3

12

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Understanding Ratios

Use the chart below in 1 through 4 to write each ratio three ways.

Mr. White’s 3rd-Grade Class (24 Students)

Practice

16-1

Gender: Male 8 Female 16

Eye Color: Blue 6 Brown 4 Hazel 12 Green 2

Hair Color: Blond 5 Red 1 Brown 15 Black 3

1. male students to female students

2. female students to male students

3. red-haired students to all students

4. all students to green-eyed students

5. Reasonableness Is it reasonable to state that the ratio of

male students to female students is the same as the ratio of

male students to all students? Explain.

No; male the students number are of students being compared that the

to

is different.

6. George has 2 sons and 1 daughter. What is the ratio of

daughters to sons?

8 to 16, 8:16, ___ 8

16 (or __ 1

2 )

16 to 8, 16:8, ___ 16

8 (or __ 2 1 )

1 to 24, 1:24, ___ 1

24

24 to 2, 24:2, ___ 24

(or

2 ___ 12

1 )

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A 2 to 1 B 1 to 2 C 3:1 D

7. Explain It The ratio of blue beads to white beads in a

necklace is 3:8. Nancy says that for every 11 beads,

3 are blue. Do you agree? Explain.

Yes; Sample answer: If there e are 3 blue

beads to 8 white beads, that is 11 total

beads, so 3 are blue.

244 Topic 16

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Understanding Percent

Reteaching

16-2

A percent is a special kind of ratio that shows the relationship

between a number and 100. Percent actually means per hundred .

The Hortonville Hoopsters have won 70 games of the last

100 games they have played. What percentage of the games has

the team won?

Using a 100-square grid can help.

There are 70 squares shaded so the ratio is 70 to 100 or ___ 70

___ 70

100 70%.

Remember The % sign is the same thing as saying “out of 100.”

100 .

In 1 through 4, write the ratio and percent of the shaded section on each grid.

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___

25

100

,

100 25% ___

43

, 43%

___

63

100 100 , 63% ___

0

100 , 0%

1. 2. 3. 4.

Number Sense

5. What is “eighty-two percent” written as a fraction in simplest

form?

__

41

50

6. Mystery Bookstore has 100 books on its shelves. Forty-one

are hardbacks and 59 are paperbacks. What percentage of

hardbacks and what percentage of paperbacks are there?

41% hardbacks, 59% paperbacks

Topic 16 245

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Understanding Percent

Write the fraction in lowest terms and the percent that represents

the shaded part of each figure.

1. 2.

Practice

16-2

___ 9

,

25 36% __

13

25 , 52%

3. Strategy Practice In the square, if part A is

1_ of the square and part C is __ 1

of the square,

4 10

what percent of the square is part B?

65%

B

C

4. In Russia, 1_ of the land is covered by forests.

4

What percent of Russia is covered by forest?

What percent of Russia is not covered by forest?

25%; 75%

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__ 3

5. In the United States, of the land is forests and woodland.

10

What percent of the United States is forest and woodland?

30%

6. If 2_ of a figure is shaded, what percent is not shaded?

5

A 20% B 30% C 50% D 60%

A percent is a decimal with the decimal

point moved two places to the right.

7. Explain It Explain how a decimal is related to a percent.

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246 Topic 16

Name

Percents, Fractions,

and Decimals

Reteaching

16-3

At Slider Park, 40% of the children playing were boys. What is this

percent expressed as a fraction and a decimal?

You can change percents into fractions and decimals just as you

can change fractions and decimals into percents.

To change a percent into a fraction, drop the percent sign and

place the number over 100. Simplify the fraction if possible.

40%

____ 40

100 2__

5

To change a fraction to a percent, divide the numerator by the

denominator. Make sure your quotient goes to the hundredths place.

If your quotient has too few digits, add zeros as placeholders. If your

quotient has too many places, round to the hundredths. Then move

the decimal point two places to the right and add the percent sign.

2__

0.40 40%

5

To change a percent to a decimal, drop the percent sign and

move the decimal point two places to the left.

40% 0.40

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In 1 through 3, write each percent as a decimal and a fraction in

simplest form.

0.36,

___ 9

25

0.13,

____

13

100

1. 36% 2. 13% 3. 59%

4. Reasonableness Pilar wrote that 3% 0.30. Is this a

reasonable answer?

No, she forgot to put a zero as a placeholder

in the tenths place. It should be 0.03.

5. On the Pennyville Pigskins football team, two quarterbacks

are competing for the starting job. Jake completes 87% of

0.59,

____

59

100

his

Who

passes.

is the better

Brett only

quarterback?

has 13% of his passes fall incomplete.

They are equal. If 87%

of the passes are complete, then 13% of

the passes are incomplete.

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Topic 16 247

Name

Percents, Fractions,

and Decimals

For questions 1 through 3, write the percent, decimal, and fraction

in simplest form represented by the shaded part of each 100-grid.

1. 2. 3.

Practice

16-3

40%, 0.4, __ 2 50%, 0.5, __ 1 5%, 0.05, ___ 1

5

2

20

For questions 4 through 9, write each percent as a decimal and a

fraction in simplest form.

0.3,

___ 3

10

0.11,

____

11

100

0.6, __ 3 5

1.5, 1 __ 1 2

4. 30% 5. 60% 6. 32%

7. 11% 8. 150% 9. 100%

10. Reasoning If 40% of Jeanne’s friends play kickball on

weekends, what fraction of her friends don’t play kickball?

3__

0.32,

___ 8

25

1.0, 1

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12 eggs

11. If there are 6 eggs in 50% of an egg crate, how many eggs

are in the whole crate?

12. What would you do first to order the following numbers from

least to greatest?

30%, 2_ 8_

, 0.67,

3 9 , 0.7

A Order the B Convert the C Order the D Convert all

decimals. decimals to fractions. numbers to

percents.

decimals.

13. Explain It When writing a percent as a decimal, why do you

move the decimal point 2 places?

Percent means “per hundred.” Each

decimal place is a factor of 10; 100 is

two factors of 10.

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248 Topic 16

Name

Finding Percent of

a Whole Number

Reteaching

16-4

Stefani wants to buy a skirt that costs $28. She received a

coupon in the mail for a 20% discount from the store. How much

will she save?

To find the percent of a whole number, multiply the whole number

by the decimal equivalent of the percent.

28

0.20

$5.60 discount

In 1 through 6, find the percent of each number.

13.94

3.45 252

400 204.4

8.58

1. 17% of 82 2. 3% of 115 3. 42% of 600

4. 50% of 800 5. 73% of 280 6. 39% of 22

7. In the problem above, Stefani is able to save $6 each week. With

the discount, how many weeks will it take for Stefani to save

enough to buy the skirt?

4 weeks

8. Geometry Kirk ordered a pizza with 10 slices. How many

slices would amount to 30% of the pizza?

3 slices

9. Jackson High School’s band sells programs at the State Fair.

They receive 15% of the total amount sold. The band sold

$3,500 worth of programs. How much money did they receive?

$525

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10. Mental Math Using mental math, find 50% of 500. How did you get

your answer?

250: 50% is the same as __ 1 2 , and __ 1

of 500 is 250.

2

Topic 16 249

Name

Finding Percent of

a Whole Number

Find each using mental math.

12

150

22 21

1. 20% of 60 2. 30% of 500

3. 25% of 88 4. 70% of 30

Practice

16-4

5. Reasoning Order these numbers from least to greatest.

0.85, __ 14 , 72%, __ 5 8 , 20%, 0.3

6. What is 40% of 240?

20%, __ 1

4 , 0.3, __ 5 , 72%, 0.85

8

A 48 B 96 C 128 D 960

The table below shows the percent of the population that live in

rural and urban areas of each country. Use the table to answer

7 through 9.

Rural

Urban

Bermuda 0% 100%

Cuba 25% 75%

Guatemala 60% 40%

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7. Out of every 300 people in Cuba, how

many of them live in a rural area?

8. Out of every 1,000 people in Guatemala,

how many live in urban areas?

75

400

9. Explain It If there are 1,241,356 people who live in Bermuda,

how many residents of Bermuda live in urban areas? How

many live in rural areas? Explain your answer.

1,241,356 in urban; 0 in rural; Sample

answer: If 100% are in urban areas,

then 0% are in rural areas.

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250 Topic 16

Name

Problem Solving: Make a

Table and Look for a Pattern

Reteaching

16-5

Sometimes when you are trying to solve a problem, it can help to

make a table and look for a pattern.

Nasser has 5 hours every day between when school ends and

when he goes to bed. He spent 3 hours last night studying. If he

continues like that every night, what percent of his time will be

spent studying?

Step 1. Make a table with the information you already know.

Hours spent studying 3

Total hours 5

Step 2. Fill in the table using equal ratios until you get a

comparison with 100.

Remember You find equal ratios by multiplying or dividing

the top and bottom of the ratio by the same number.

Hours spent studying

3 12 30 48 60

Total hours 5 20 50 80 100

So, Nasser spends 60% of his time studying.

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In 1 and 2, find the percent by completing the table.

1. won 7 out of 10 games

Games won 7

Total games 10 20 50 100

2. ran 13 miles out of 20

Miles run 13

70%

14 35 70

65%

6.5 32.5 65

Total miles 20 10 50 100

3. Denise gets 450 minutes each month to use on her cell

phone. Last month, she used 360 minutes. What percent

of her minutes did Denise use?

80%

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Problem Solving: Make a

Table and Look for a Pattern

For exercises 1 through 4, find each percent by completing each table.

1. 12 out of 40 days were rainy.

Rainy days 12

Total days 40 20 10

2. 2 out of 8 marbles are blue.

Blue 2

Total marbles 8 16 4

3. 32 out of 40 days were windy.

Windy days 32

Total days 40 5 20

4. 16 out of 40 pets on Jack’s street are dogs.

Dogs 16

Total pets 40 80 10

30%

6 3 30

100

25%

4 1 25

100

80%

4 16 80

100

32 4 40

100

40%

5. Write a Problem Write a real-world problem that you can

solve using a table to find a percent.

6. Emmy plans to hike 32 miles this weekend. On Saturday, she

hiked 24 miles. What percent of her goal has Emmy hiked?

Practice

16-5

Sample answer: Coins as a percent of

a dollar or chance of rain

75%

7. Explain It Dave estimated 45% of 87 by finding 50% of

90. Will his estimate be greater than or less than the exact

answer?

Greater than: since 50% 45%

and 90 87, his estimate will be

greater than the exact answer.

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Understanding Integers

Reteaching

17-1

You can use integers to represent word descriptions.

Remember: An integer is a whole number or its opposite

( 7 and 7 are integers). 0 is its own opposite.

Use a positive integer to represent word descriptions that show

an increase.

up 3 floors

3

45 degrees above zero 45

5 steps forward 5

Use a negative integer to represent word descriptions that show

a decrease.

300 feet below sea level 300

cut off 10 inches

10

2 steps backward 2

In 1 through 8, write an integer for each word description.

5 100

3 20

14

2,200

8

15

1. win 5 games 2. earned $3

3. lose 100 points 4. grew 20 inches

5. spent $14 6. 2,200 feet above sea level

7. go down 8 floors 8. 15 minutes before test time

Draw a Picture Draw a number line to show the following:

9. 11

10. 2

0 11

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2

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Topic 17 253

Name

Understanding Integers

Write an integer for each word description.

1. a withdrawal of $50 2. a temperature rise of 14° 3. 10° below zero

50

14

10

Use the number line for 4 through 7. Write the integer for each point.

12 10

8

A C B D

6

4

2

0

2

4

6

8 10 12

8 1 2 7

4. A 5. B 6. C 7. D

Compare. Use , , or for each .

> > >

8. 5 9 9. 8 12 10. 21 26

Write in order from least to greatest.

11

4 4 11

14

8

6

4

0

7

6

11

11. 4, 11, 11, 4 , , ,

12. 6, 6, 0, 14 , , ,

13. 11, 8, 7, 4 , , ,

14. Strategy Practice Which point is farthest to the right on a

number line?

A 6 B 2 C 0 D 2

Practice

17-1

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15. Explain It In Fenland, U.K., the elevation from sea level is

4m. In San Diego, U.S., it is 40 ft. The elevations are given

in different units. Explain how to tell which location has a

greater elevation.

San Diego: any positive elevation

is greater than a negative one.

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254 Topic 17

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Ordered Pairs

Reteaching

17-2

You can locate points on a grid using ordered pairs.

Draw a point on the grid for ( 1, 4).

On a grid, the axis that goes across is called the x -axis. The axis

that goes up and down is called the y -axis. The point where the

x -axis and the y -axis cross is called the srcin.

To locate a point on a grid using

an ordered pair, start at the srcin.

The first number in the ordered pair is

called the x -coordinate. It tells how far to

move to the right (if positive) or to the left

(if negative). Since 1 is positive, move

1 unit to the right.

The second number in the ordered pair is

the y -coordinate. It tells how far to move

up (if positive) or down (if negative). Since

4 is negative, move 4 units down to the

point ( 1, 4).

y -axis

1

4

(1,4)

x -axis

Write the ordered pair for each point.

( 3,

4)

1. B 2. D

( 4,

1)

y

Name the point for each ordered pair.

3. ( 3, 4) 4. ( 2, 2)

5. What is the ordered pair

for the srcin?

(0, 0)

F

G

G

E

5

4

3

2

1

5 4 3 2 1

1

D

2

B

3

4

5

F

1 2 3 4 5

H

C

A

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Topic 17 255

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Ordered Pairs

Write the ordered pair for each point.

1. A

2. B

3. C

4. D

5. E

6. F

(3, 4)

( 2, 3)

(2, 3)

5,

4)

( 7, 1)

(8, 6)

-10

y

+10

Practice

17-2

+8

L

+6

F

G

+4 A

X B

E

+2

-8 -6 -4 -2J

0 +2

H

+4 +6 +8 +10 x

D

K

-2

-4

-6

-8

-10

C

Y

I

Name the point for each ordered pair.

H J L

7. ( 5, 0) 8. ( 1, 1) 9. (0, 7)

I K G

10. ( 6, 5) 11. ( 4, 8) 12. ( 5, 5)

13. Strategy Practice If a taxicab were to start at the point

(0, 0) and drive 6 units left, 3 units down, 1 unit right, and

9 units up, what ordered pair would name the point the

cab would finish at?

( 5, 6)

14. Use the coordinate graph above. Which is the y -coordinate

for point X ?

A 6 B 3 C 3 D 6

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15. Explain It Explain how to graph the ordered pair ( 2, 3).

Begin at the srcin and move

2 places to the left for the x -value

of 2. Then move 3 spaces up for

the y -value of 3.

256 Topic 17

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Distances on Number Lines

and the Coordinate Plane

Reteaching

17-3

How do you find a distance on a number line?

Finding the distance from 2 to 5 on a number line:

1. Plot the integers on a number line.

2. Draw a second number line beneath

the first. Beneath the 2 on the top

number line, start labeling the lower

number line 0, 1, 2. Stop labeling

when you reach 5 on the top line.

Count the units on the lower line.

How do you find a distance on a coordinate grid?

Finding the distance between (1, 3) and (7, 3) on a grid:

1. Plot the points named (1, 3) and

(7, 3) on a coordinate grid.

2. Because the y -coordinate of each

named point is the same, draw a

matching number line parallel to the

x -axis. Next to the point named (1, 3),

start labeling the number line 0, 1,

2, until you reach the point (7, 3).

Count the units on the number line.

The distance between (1, 3) and (7, 3)

is 6.

3 2 1 0 1 2 3 4

5 6 7

0 1 2 3 4 5 6 7

The distance between

2 and 5 is 7.

y

4

2

0 1

0

2 4

(1,3) (7,3)

2 3 4 5 6

x

2 6 8

2

Find the distance between each pair of integers.

7 5

1. 3, 4 2. 2, 3

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5 4 3 2 1 0 1 2 3 4

5

5 4 3 2 1 0 1 2 3 4 5

Find the distance between each pair of points.

4

3. ( 1, 6) and (3, 6) 4. (3, 2) and (3, 5)

7

Topic 17 257

Name

Distances on Number Lines

and the Coordinate Plane

Find the distance between each pair of integers on a number line.

1. 7, 3 2. 7, 1 3. 4, 0

4 units 8 units 4 units

Practice

17-3

4. 0, 5 5. 2, 8 6. 3, 3

5 units 6 units

Find the distance between the points named by each set of

ordered pairs on the coordinate plane.

7. ( 2, 5), ( 2, 1) 8. ( 4, 1), (0, 1)

6 units

6 units 4 units

9 units 1 unit

9. ( 1, 7), ( 1, 2) 10. ( 2, 6), ( 1, 6)

11. At 2:00 P.M., the temperature was

3°C. By 4:00 P.M., the temperature

had dropped to 1°C. What was

the amount of the decrease in

temperature?

4°C

12. Strategy Practice What integer

on a number line is the same

distance from 0 as 4?

4

13. On the coordinate plane, what is the distance between the

points named by ( 2, 6) and ( 3, 4), if you move only

along the lines of the grid?

A 11 B 9 C 7 D 7

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14. Explain It How can you tell if two points lie along the same

grid line just by looking at the ordered pairs?

If two points lie along the same grid

line, then the ordered ed pairs that name

the points either have the same x -value

or the same y -value.

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Graphing Equations

An equation can predict data. The data can

be plotted on a coordinate grid. If the graph

is a straight line, the equation is called

a linear equation.

How to graph an equation:

Step 1

Use the equation to calculate ordered pairs.

Put at least 3 ordered pairs into a table.

Step 2

Choose what to plot on the horizontal

axis and what to plot on the vertical axis.

Choose the starting point and ending

point for each axis.

Choose an interval for each axis. Label and

number the axes.

Step 3

STEP 1

a rs)

n dol gs(i r ni n

E

a

25

20

15

10

5

STEP 2

Hours

Reteaching

17-4

Earnings

0 0

1 5

2 10

3 15

4 20

Ron’s Earnings

STEP 3

0 1 2 3 4

Hours Worked

Graph the data by using the coordinates for each set of data as a

point. Connect all the points in a straight line. Title your graph.

Graph the equation below to show the cost, c, of buying

harmonicas, h. Harmonicas are available online for $3 each,

plus a single shipping charge of $2.

c 3h 2

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h

c

1 5

2 8

3 11

o st

C

12

10

8

6

4

2

0

1 2 3

Harmonicas

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Name

Graphing Equations

In 1 through 4, find the value of y when x = 4, 9, and 15.

1. y 2x 2. y x 4

3. y 4x 4. y x 6

In 5 through 8, make a table of values for each equation and then

graph the equations. Let x 1, 3, 5, and 7.

5. y x 2 6. y 3x

7. y x 1 8. y x 5

9. Reasoning Without drawing the graph, describe what the

graph of x 100 would look like. Explain.

10. Reasonableness Which of the following ordered pairs is

NOT on the graph of the equation y x 9?

A (7, 16) B (15, 24) C (20, 29) D (28, 36)

Practice

17-4

8, 18, 30

0, 5, 11

16, 36, 60 10, 15, 21

3, 5, 7, 9

0, 2, 4, 6

3, 9, 15, 21

6, 8, 10, 12

It would be a vertical line at 100;

every point on that line would have an

x -coordinate of 100.

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11. Explain It A graph contains the ordered pair (2, 4). Write two

different equations that would be possible for this graph.

Explain how you found your answer.

y 2x or y x 2 are both possible

answers because their solutions both

include the ordered pair listed.

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Problem Solving:

Work

Backward

Reteaching

17-5

Sometimes when you are trying to solve a problem, it is easier to

start at the end and work backward.

The Rosewood City train makes its first stop 10 minutes from

the main station. The second stop is 20 minutes farther. The third

stop is 15 minutes after that, and the fourth stop is 15 minutes

after the third stop. The end station is 5 minutes from the fourth

stop. The train reaches the end station at 9:00 A .M. What time did

the train leave the main station?

To solve this kind of problem, you need to work backward.

Think about if the train were actually going backward.

Since you know it is 5 minutes from the end station back to the fourth stop, subtract 5 from

9:00 A .M. Continue subtracting to keep backing the train up until it is back at the main station.

main station

?

1stop 2 stop nd 3 stop rd 4 th stop

end station

9:00 A.M.

10 min 20 min 15 min 15 min 5 min

9:00 0:05 8:55 8:55 – 0:15 8:40 8:40 0:15 8:25

8:25 0:208:05 8:05 0:10 7:55

So, the train left the main station at 7:55 A .M.

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1. Zev has a rosebush in his yard. The first year it grew 3 inches. The second

year it grew 12 inches. Zev then cut it back 5 inches. The third year it grew

twice as much as the first year. The rose bush is now 42 inches tall. How tall

was it when Zev planted it?

26 inches

2. Jesse went on a bike ride. He rode from his house to the library. Then he rode

2 miles from the library to his friend’s house. Then he rode 1 1_ miles to buy a

2

snack at the store. Finally he rode the 3 miles back to his house. He rode a

total of 9 miles. How many miles is it from Jesse’s house to the library?

2_

1 2 miles

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Problem Solving:

Work Backward

Solve each problem by working backward. Write the answers in

complete sentences.

Barbara is refilling her bird feeders and squirrel feeders in her yard.

1. After filling her bird feeders, Barbara has 3 1_ c of mixed

2

birdseed left. The two feeders in the front yard took 4 1_

each. The two feeders in the backyard each took 2 3 _ 4 c. The

two feeders next to the living room window each took 3 1_

4 c.

How much mixed birdseed did Barbara have before filling

the feeders?

24 __ 1 2 c

2. After Barbara fills each of her 4 squirrel feeders with 2 2_

3 c of

peanuts, she has 1 3_ c of peanuts left. How many cups of

4

peanuts did Barbara start with?

12 ___ 5

12 c

2 c

Practice

17-5

3. Strategy Practice Clint spends 1_ hour practicing trumpet,

4

1_

1_

2 hour doing tasks around the house, 12 homework, and 3_ hour cleaning his room.

hour

He is

doing

finished

4

at 7:30 P.M. When did Clint start?

A 4:00 P.M. B 4:15 P.M. C 4:30 P.M. D 5:30 P.M.

4. Write a Problem Write a real-world problem that you can

solve by working backward.

Answers will vary. See students’ work.

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Data from Surveys

Reteaching

18-1

In a survey, each student was asked this question. How many

trees do you have in your yard? Here are the responses: 1, 2, 3, 1,

4, 4, 1, 2, 1, 2, 4.

How to display data collected from surveys:

The data can be displayed in a frequency table or a line plot.

Frequency Table

1. Count the number of times each

different response was made.

1: ////; 2: ///; 3: /; 4: ///

2. Construct a frequency table.

The table lists each response and

its frequency. (The frequency of a

response is how many times it

was made.)

3. Give the frequency table a title

that clearly explains what

information is in the table.

Line Plot

1. Count the number of times each

different response was made.

1: ////; 2: ///; 3: /; 4: ///

2. Construct a line plot. The plot lists

each response along a horizontal

line. The frequencies are stacked

as x’s above each response.

3. Give the line plot a title that

clearly explains what information

is in the plot.

How many Survey trees Question:

are in your yard?

Number of Trees

1

2

3

4

Number

4

3

1

3

y

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F

x

x

Survey Question:

How many trees are

in your yard?

1234

x

x

Number of Trees

x xx

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1. How many students responded to the tree survey?

2. Which number of trees had the greatest responses?

11 1

3. Explain It Describe how you might pick a sample of 50 minivan

owners that represent the minivan owners of your state.

Sample response: Minivan owners living

in four different regions of the state

might be asked to respond.

Topic 18 263

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Data from Surveys

Ms. Chen’s class took a survey on how many minutes it took each

student to get to school. The results are below:

12 14 5 22 18 12 12 6 14 18 12 5 11

1. What are the highest and lowest times?

Practice

18-1

22 and 5

2. Make a line plot

to display the data.

x

Students in Ms. Chen’s Class

x x x x

x

x

5 6 7 8 9 101 11 21 31 41 51 6 17 18 19 20 21 22 23

Minutes it Takes to Get to School

y

c

n

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q

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F

56789

Number of Bags of Groceries

Bought in One Week

Number of Bags

18 students

The number of

bags of groceries bought in one week.

3. How many people responded to the survey

about bags of groceries?

x

x

4. What information was collected about groceries?

x

x

x

x

x

x

x

x

x

5. Use the line plot above. Which number of bags of groceries

were bought most often?

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A 8 B 7 C 6 D 5

6. Explain It Write a survey question that might gather the following

information. “In one school there are 6 sets of twins, 2 sets of triplets, and

one set of quadruplets.”

Sample answer: Are you one of a set of

twins, triplets, or quadruplets?

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Bar Graphs and

Picture Graphs

Reteaching

18-2

A bar graph uses rectangles to show data. A double bar graph

uses two different-colored or shaded bars to show two similar

sets of data. A picture graph uses symbols to show data. You can

use each to compare data.

Look at the information about

the baseball teams in the

frequency table at right. Display

this information as a double bar

graph.

1. Decide on a scale and its

intervals.

Conference Baseball Championships

High School Junior Varsity

Dominican

Phillips

Smith

4

6

5

3

7

4

2. Draw the graph.

Label the axes.

3. Graph the data by

drawing bars of the

correct length and

title the graph.

Conference Girls’ Baseball Championships

Dominican

Step 3

Phillips

Step 2

Junior

Smith Varsity

012345678

Step 1

Number of Championships

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Look at the information about

the junior teams in the frequency

table above. Use this information to

complete the picture graph with the

values for Dominican High School.

Conference Girls’ Junior Baseball Championships

s

p

s hi n

o

p i m

a

o fCh

r

m be u

N

Smith

Phillips

Dominican

= 2 championships

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Bar Graphs and

Picture Graphs

In 1 and 2, answer the questions about the double-bar graph below.

1. How many boys play

indoor soccer? How

many girls play?

12 boys and

14 girls

2. What is the least

popular sport among

girls? Among boys?

Indoor Soccer

Track

Basketball

Swimming

Sports Students Play

Practice

18-2

Number of Students

0 2 4 6 8 10 12 14 16 18

Track

is least popular among girls and

swimming among the boys.

Boys

Girls

3. Frank bought 4 dozen doughnuts for his class. He had 4 left over. Which

shows how to find how many doughnuts Frank gave away?

A (48 2) 4 C (12 2) 4

B (24 2) 12 D (12 4) 4

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4. Explain It Could the data in the bar graph in Exercise 1 be

presented in a picture graph? Explain.

Yes, each game could be represented

by a symbol like a basketball, water,

etc. and a human figure could represent

each participating student. The figures

would be differently colored for boys or

girls.

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Line Graphs

Reteaching

18-3

How to make a line graph:

Step 1

Put the data into a table so that each set of data

has two values.

Step 2

Choose what to plot on the horizontal axis and

what to plot on the vertical axis.

Choose the starting point and ending

point for each axis. (The starting and

ending points must include the

smallest and largest data values.)

Choose an interval for each axis.

Label and number the axes.

Step 3

Graph the data by using the

coordinates for each set of data

as a point. Draw lines to connect

the points.

Step 4

Title your graph.

W e ig htl if t e d( in pounds)

160

140

120

100

80

60

40

20

STEP 1

STEP 4

Months

passed

1 70

2 100

3 100

4 120

5 140

Weight

lifted

pounds)

Dave’s Weightlifting

STEP 3

0 1 2 3 4 5 6

STEP 2

Months Passed

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Make a line graph of the data.

Since the data for the x -axis are

from 1 to 5, the scale can go from 0 to .

Since the data for the y -axis are from

20 to 45, the scale can go from 0 to .

50

n

o rs 40

f Do 30

20

u mbero 10

N

5

50

Blood Drive

0 1 2 3 4 5

Week

Blood Drive

Number

Week

of donors

1 20

2 20

3 30

4 35

5 45

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Topic 18 267

Name

Line Graphs

Display the data in the table below on the coordinate grid.

Practice

18-3

Hour

1 4°

2 8°

3 16°

4 21°

Temperature

(Celsius)

24

) 20

C

e ( 16

tu r e 12

p

ra 8

T

e

m

4

1234

Hour

1. Which hour had the highest temperature?

Hour 4

2. How much higher was the temperature

in Hour 4 than Hour 1?

17 degrees

3. Reasoning If Hour 1 was really 10:00 P.M., do you think the

trend on the line graph would keep increasing? Explain.

No, the temperature because in the should evening,

drop.

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4. Critical Thinking Look at the line graph

at the right. What do you know about

the trend of the housing prices?

A Housing prices increased.

B Housing prices decreased.

C Housing prices increased, then decreased.

D Housing prices decreased, then increased.

5. Explain It In the example above, if housing prices

had stayed the same for all four years, what would

the line graph look like? Explain.

268 Topic 18

s 500

ic

P r e 400

300

u

o

si n g H200

100

1234

Years

A straight horizontal line; it indicates

icates

no change occurred.

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Stem-and-Leaf Plots

Reteaching

18-4

A stem-and-leaf plot is a convenient way to organize data.

A school records the number of

students absent each day. The

records for a two-week period are

shown in the stem-and-leaf plot on

the right.

Number of Students Absent Each Day

Stem

Leaf

1 2 5 6 9

2 1 4 7 8

3 0 3

KEY: 1|2 12

How do you read a stem-and-leaf plot?

1. Identify the stem for each

data value.

2. Identify the leaf for each

data value.

The stems in this plot represent the tens digits

10, 20, and 30.

The leaves in this plot represent the ones digits:

0, 1, 2, 3, 4, 5, 6, 7, 8, and 9.

3. Identify the key. The key shows that each stem-leaf combination

represents a two-digit number.

Read the following and complete the following questions.

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A small card shop records the

number of birthday cards sold

each day. The record is displayed

as the stem-and-leaf plot shown

on the right.

1. Using the key, what is the value of 3|0?

Number of Birthday Cards Sold Each Day

Stem

2. How many days are represented by the plot?

Leaf

2 1 5 6 7 8 9 9

3 0 3

4 2 2 3 4 9

KEY: 2|1 21

3. What is the least number of cards sold in one day?

4. What is the most number of cards sold in one day?

30

14

21

49

Topic 18 269

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Stem-and-Leaf Plots

For 1 through 3, use the stem-and-leaf plot below. It shows the

ages of the 17 people who used the outdoor pool from 6:00 A .M.

to 7:00 A .M. on a Tuesday morning in the summer.

1. How many swimmers were younger

than 30?

3

2. Which age group was swimming the

most at this hour?

people in their

sixties or retirees

ees

3. Why are there two 5’s as leaves next

to the stem 6?

the age 65 appears

in the table twice

4. Make a stem-and-leaf plot of

the data below.

Prices of couch pillows (dollars)

10 75 20 20

37 24 21 9

270 Topic 18

Stem Leaf

0

Ages (years)

1 1

2 3 9

3

4 7

5 5 7 7 9

6 5 5 6 6 7 8 9

7 0 1

Key: 7 | 1 means 71

5. Refer to the stem-and-leaf plot in Exercise 4. Which stem

(or stems) have the most leaves?

A 70 B 9 C 20 D 30

Practice

18-4

Stem Leaf

0 9

1 0

2 0 0 1 4

3 7

7 5

Key: 7 | 5 means 75

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Histograms

Reteaching

18-5

A histogram is a type of bar graph that shows frequency of

data with equal intervals. The equal intervals are shown on the

horizontal axis.

Mr. Sato asked each student in his class which month they

were born in. He put the data into a table. Make a histogram to

determine the two most common birthday months.

Month Jan. Feb. Mar. Apr. May Jun. Jul. Aug. Sept. Oct. Nov. Dec.

Frequency 32 12 4 1 32 5 01 2

Step 1: You want to divide the

data by month, so lay out the

months along the horizontal axis.

Step 2: The number of students

born in each month is the

frequency data, which goes on the

vertical axis.

Step 3: Use the data from the

table to fill in the histogram.

September and May are the most

common birthday months.

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J F M A M J J A S O N D

STEP 1

STEP 3

Months

1. How many people ages 0–19 and

40–59 combined attended the

tournament?

95 people

2. What is the total number of

50 r

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b 30

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N10

0

Spectators at School

Tournament

0–19 20–39 40–59 60–79

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people who attended the

tournament?

135 people

Age (in years)

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Topic 18 271

Name

Histograms

This table shows the results of a class survey to find

out how many pieces of fruit each student ate that week.

Practice

18-5

Amount of

Fruit

0–7 12

8–15 8

16–25 5

Frequency

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1. Complete the histogram below. What percentage of students ate

16–25 pieces of fruit that week?

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272 Topic 18

0–7 8–15 16–25

Pieces of fruit consumed last week

5 out of 25 total students 20%

2. Reasoning Mary says a histogram shows that about

3 times as many people in the 60–79 age group answered a

survey as in the 80–99 age group. How does she know this

from looking at the histogram?

The bar that represents esents ages 60–79

is three times as tall as the bar that

represents ages 80–99.

3. Explain It A political campaign recorded the ages of

100 callers. In a histogram, which data would go on the

horizontal axis and which on the vertical?

The horizontal axis would show equal age

intervals. The vertical axis would show the

numbers of callers in each age bracket.

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Circle Graphs

A circle graph shows 100% of a data set. Each wedge

in the circle represents part of the whole amount.

One hundred people were asked if they owned a pet

and if they did, what kind of pet. The table at the right

shows the data.

From this table, you can create a circle graph.

Reteaching

18-6

Pet

Dog 50

Cat 25

Bird 10

Other 10

No pet 5

Each category is part of the whole. To finish the graph, do the same with

You can find out how big a wedge to the rest of the table.

use for each category by changing the

cat

ratios into fractions.

___ 25

100 1_ , bird ___ 10

4 100 __ 1

10 ,

other

50 people out of 100 had a dog.

___ 10

100 __ 1 , no pet ___ 5

10 100 __ 1

20

___ 50 1_

is .

No pet

Other

100 2

Bird

So, the wedge

Dog

for dog will be

Dog

1_

2 of the circle. Cat

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1. The table below shows what kind of books people said they

had checked out at the library. Using the data, copy the circle

graph and label each section with the correct type of book

and a fraction in simplest form.

Type of book

Fiction 60

Magazine 20

Nonfiction 20

No book 20

2. How many total people were asked about their book choice?

120

Nonfiction

Magazine

1

6

Fiction

1

6 1

1

2

6

No book

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Circle Graphs

1. A bagel shop offers a variety of bagels. One morning, the

following choices were made by the first 20 customers of the

day: plain, 10; poppy seed, 5; sesame seed, 3; multigrain, 2.

Complete the table below.

Practice

18-6

Plain

Poppy Seed

Fraction

Sesame Seed

Multigrain

1_

2

Percent

_ 1

4

__

3

20 __

1

10

50%

25%

15%

10%

2. Copy and complete the circle graph below with the data in

the table above. Label each section with the percent and

fraction of each bagel.

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Poppy

1

, 25%

Plain

Multigrain 4

1

1

2

, 10%

,

10

50%

Sesame

3

20, 15%

3. Number Sense A circle graph is divided into four sections.

One section equals 40%. The other three sections are equal

in size. What percent does each of the other three sections

represent?

20%

4. Reasoning If 10 out of 30 students in a survey chose

ice skating as their favorite sport, what fraction of the

circle should be shaded to represent the students who

chose ice skating?

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Mean

Reteaching

18-7

The mean is the sum of all the values in a set divided by the

number of items in the set. The mean is also called the average.

How to find the mean of a set of data:

Eduardo surveyed 7 of his

friends to find out how many

books they read during the

month. The frequency table

shows the data. What is the

average number of books

read by Eduardo’s friends?

Book Reading

Friend

Number of books read

Jean 2

Raul

Sally

3

8

Jonathan 5

Haley 6

Kristen 3

Owen 1

1. Add the number of books

read by each friend.

2. Divide the sum by the

number of friends.

3. Use the average to answer

the question.

2 3 8 5 6 + 3 1 28

28 ___

7 4

Eduardo’s friends read an average of

4 books during the month.

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1. Find the mean of this set of data: 241, 563, 829, 755.

2. This frequency table shows the number of silver

medals won by American athletes in Summer

Olympic Games between 1972 and 2000. What

is the mean of this set of data?

31

3. Estimation What is the approximate

average of these three

numbers: 9, 18, and 31?

about 20

4. Explain It Explain how you would find the

mean of this set of data: 4, 3, 5.

597

US Silver Medals

Summer Olympics Games

Year

Medals

2000 24

1996 32

1992 34

1988 31

1984 61

1980 0

1976 35

1972 31

Sample

numbers answer: in the set. Add Then the

divide by three.

Topic 18 275

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Mean

Find the mean of each set of data.

1. 2, 5, 9, 4

2. 44, 73, 63

3. 11, 38, 65, 4, 67

4. 3, 6, 3, 7, 8

5. 120, 450, 630

6. 4.2, 5.3, 7.1, 4.0, 11.9

Gene’s bowling scores were as follows: 8, 4, 10, 10, 9, 6, 9.

7. What was his average bowling score?

8. If Gene gets two more strikes (scores of 10),

what is his new average?

5

60

37

5.4

400

6.5

9. Reasoning Krishan wants his quiz average to be at least

90 so that he can get an A in the class. His current quiz

scores are: 80, 100, 85. What does he have to get on his

next quiz to have an average of 90?

A 85 B 90 C 92 D 95

10. Explain It Suppose Krishan’s teacher says that he can

drop one of his test scores. Using his test scores of 80,

100, and 85, which one should he drop, and why? What is

his new average?

Practice

18-7

Drop the lowest score: 80. His new

average would be 92.5.

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276 Topic 18

Name

Median, Mode, and Range

Reteaching

18-8

The median, mode, and range are each numbers that describe a set of data.

Here is Eduardo’s survey of how many

books his friends read last month.

What are the median, mode, and range of

Eduardo’s survey?

Median: The median is the middle number

in a set of data. To find it:

1. Arrange the data in order


2. Locate the middle number.

The median number of books read is 3.

Book Reading

Friend

Number of

books read

Jean 2

Raul 3

Sally 8

Jonathan 5

Haley 6

Kristen 3

Owen 1

1, 2, 3, 3, 5, 6, 8

middle number 3

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Mode: The mode is the data value that occurs most often. To find it:

1. List the data. 1, 2, 3, 3, 5, 6, 8

2. Find the number that occurs most. 3

The mode of the books read by Eduardo’s friends is 3 books.

Range: The range is the difference between the greatest and least values.

To find it:

1. Identify the greatest and least values. 8 and 1

2. Subtract the least from the greatest value. 8 1 7

The range of the books read by Eduardo’s friends is 7 books.

1. Find the median of this data set: 12, 18, 25, 32, 67.

2. Find the mode of this data set: 123, 345, 654, 123, 452, 185.

3. Find the range of this data set: 24, 32, 38, 31, 61, 35, 31.

25

123

37

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Median, Mode, and Range

1. Find the range of this data set: 225 342 288 552 263.

2. Find the median of this data set: 476 234 355 765 470.

3. Find the mode of this data set:

16 7 8 5 16 7 8 4 7 8 16 7.

4. Find the range of this data set:

64 76 46 88 88 43 99 50 55.

5. Reasoning Would the mode change if a 76 were added

to the data in Exercise 4?

Practice

18-8

327

470

7

56

Yes. The mode is now 88. With another

76, the modes would be 76 and 88.

The table below gives the math test scores for Mrs. Jung’s

fifth-grade class.

76

75

76

54

93

88

92

92

80

88

68

70

76

88

88

Test Scores

88

76

72

6. Find the mean of the data.

7. Find the mode of the data.

8. Find the median of the data.

9. What is the range of the data set?

80

88

78

39

10. Find the range of this data set: 247, 366, 785, 998.

A 998 B 781 C 751 D 538

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11. Explain It Will a set of data always have a mode?


No: if every item is different, there is

no mode.

278 Topic 18

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Problem Solving:

Make a Graph

Reteaching

18-9

Choosing which graph to use depends on what kind of data set

you are looking at.

Bar graphs and circle graphs are good graphs to use when you

are comparing data that can be counted. Line graphs are good

at showing changes in the data over time. Histograms divide

the data into intervals and record how many data items fall within

each interval.

There are three ferries that run from Port Shoals to Tiger Island.

187 people were on the 12:30 P.M. ferry, 122 people were on the

3:00 P.M. ferry, and 58 people were on the 5:30 P.M. ferry. Make a

graph for this data set and analyze the data.

This data set is comparing data that can be counted, so a bar

graph or a circle graph would be a good graph choice. Make a

bar graph of the data.

What can you say about the data from this

graph?

The 12:30 P.M. ferry was the most popular.

The 5:30 P.M. ferry was the least popular.

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12:30 P.M. 3:00 P.M. 5:30 P.M.

Ferry Time

1. Each month, Martina measures the height of a tree she

planted. Make a line graph to show the tree’s growth.

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Month

Height (in.)

April 36

May 42

June 60

July 75

August 84

100

.)

80

(in

t 60

h

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0

Tree Growth

Apr. May Jun. Jul. Aug.

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Name

Problem Solving:

Make a Graph

1. In a survey, 100 students from around the country were

asked what news source they preferred. Which news source

is most popular? Make a circle graph to solve the problem.

Practice

18-9

News Source

Number

of Votes

Television 50

Internet 25

Newspaper 15

Radio 10

Students’ Preferred News Source

News-

Paper

Internet

Radio

Television

2. In a survey, 32 students from around the country were asked how they

traveled to and from school. Make a circle graph to show the data.

How Students Get to School

Subway Bus Bike On

Foot

Taxi

On Foot

Bike

Subway

8 16 4 4 0

Bus

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3. If a graph shows that there were 10 people who watched

between 7 and 12 movies, what kind of graph could you be

looking at?

A Circle B Bar C Histogram D Line

4. Explain It Would a line graph be an appropriate graph in

Exercise 3? Why or why not?

Sample answer: No; a line graph is

used to show data that changes

over time. These data do not change

over time.

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Translations

A translation is the motion of a

figure up, down, left, or right.

How to translate a figure:

Translate the triangle to the right

10 units and up 7 units.

y

12

11

10

9

8

7

6

5

4

3

2

1

0

Reteaching

19-1

(13, 11)

(11, 9) (17, 9)

Slide

u

p

C

(3, 4) Slide right (13, 4)

A

B

(1, 2) (7, 2) (11, 2) (17, 2)

1 2 3 4 5 6 7 8 9 101112131415161718

x

1Part

2Part

Translate the triangle to the right 10 units.

Step 1:

Increase the x -coordinates of each vertex of the

triangle by 10. (Do not change the y -coordinate of

each vertex.)

Step 2:

Draw the triangle at its new location.

Translate the triangle up 7 units.

Step 4:

Increase the y -coordinates of each vertex of the

triangle at its new location. (Do not change the

X -coordinate of each vertex.)

Step 5:

Draw the triangle at its final location.

Step 3:

Label each vertex with its ordered pair.

Step 6:

Label each vertex with its ordered pair.

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1. Translate the triangle shown

here up 7 units and to the right

10 units.

12

11

10

9

8

7

6

5

4

3

21

0

y

(3, 11)

(1, 9) (7, 9)

C (3, 4)

A

B

(1, 2) (7, 2)

2. Number sense Vertex A of a figure is located at (18, 6).

The rhombus is translated down 5 units and to the

left 3 units. What is the new location of vertex A?

(13, 11)

(11, 9) (17, 9)

1 2 3 4 5 6 7 8 9 101112131415161718

(15, 1)

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Topic 19 281

Name

Translations

Give the vertical and horizontal translation in units.

Practice

19-1

1. y

2.

6

5

4

3

2

1

0 123456 x

up 3 units

3. y

4.

8

7

6

5

4

3

2

1

0

123456

x

4

3

2

1

0

5

4

3

2

1

0

y

y

123456789

left 5 units

123456

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right down 3 units,

5. Critical Thinking If a vertex of a square moved from

(11, 9) to (6, 4), which of the following describes the

translation?

A 5 left, 5 up C 5 right, 5 up

B 5 left, 5 down D 5 right, 5 down

6. Explain It Explain how the x and y coordinates are related to

the direction an object moves.

282 Topic 19

right up 2 units

2 units,

The x-coordinate

increases when the

object when it moves right left. The and y decreases

-coordinate

increases when an object moves up

and decreases when it moves down.

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Reflections

Reteaching

19-2

A reflection is the mirror image of a figure. The reflection appears

as if the image has been flipped.

How to recognize a reflection:

Reflec

lection acr

cross

a horiz

orizontal

line

The image looks like the figure flipped

top to bottom.

Reflec

lection

across

a vertic

rtical

line

The image looks like the figure

flipped left to right.

Place a checkmark beneath the correct image for each reflection.

1. Reflection across a vertical line 2.

Reflection across a horizontal line

✓

✓

3.

Reflection across a vertical line

✓

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4.

Reflection across a horizontal line

✓

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Topic 19 283

Name

Reflections

Draw the reflection of the object across the reflection line.

1. y

2.

10

9

8

7

6

5

4

32

1

0 1 2468 3 5 7 9 10 x

y

10

9

8

7

6

5

4

32

1

0 1 2468 3 5 7 9 10 x

Tell whether the figures in each pair are related by a translation or

a reflection.

3. 4.

Practice

19-2

Reflection

Reflection or

5. Think About the Process Which of the following gives

enough information to determine whether the figures A and B

have been translated or reflected?

A Figure A is larger than Figure B.

B Figure A and Figure B have the same shape and size.

C Figure A has an identical shape to Figure B.

D Figure A and Figure B are next to each other.

translation

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284 Topic 19

Name

Rotations

Reteaching

19-3

A rotation is a turn that moves a figure about a point.

How to recognize a rotation:

C

A

B

C

D

A

B

Rotation of the triangle around point A

Each vertex rotates the same amount and

in the same direction around A.

Rotation of the triangle around point D

Each vertex rotates the same amount and in the

same direction around D.

1. Place a checkmark beneath the image that is NOT a rotation of the figure

around K .

K

K

K

K

K

2. Place a checkmark beneath the image that is NOT a rotation of the figure

around R.

✓

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R

R

R

R

R

✓

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Topic 19 285

Rotations

Describe whether the figures in each pair are related by a rotation,

a translation, or a reflection.

1. 2.

Practice

19-3

rotation

translation or

reflection

3. If the figure at the right is rotated 90° around the

point shown, which of the figures below could be

the result?

A B C D

4. Explain It Mark says that the figure at the right

is a reflection. Faith says that it is a rotation. Who

is correct? Explain.

They are both correct. It could be a

reflection over a vertical line or it could

be a 180° rotation.

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286 Topic 19

Name

Congruence

Reteaching

19-4

The change of position of a figure is called a transformation.

Figures that are the same size and shape are called congruent figures.

How to recognize a transformation:

2

2

1

1

A translation and a rotation of a

figure produce a transformation.

The size and shape of the figure

do not change.

How to recognize congruent figures:

A rotation and a translation of a

figure produce a transformation.

The size and shape of the figure

do not change.

Congruent

with Figure 1.

Congruent

with Figure 1.

Congruent

with Figure 1.

Congruent figures have the same size and shape.

Not Congruent

with Figure 1.

In each group, cross out the figure that is not congruent with the other figures.

1.

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2.

Topic 19 287

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Name

Congruence

Tell whether the following figures are congruent or not.

1. 2.

Practice

19-4

No

Yes

Identify each transformation. If not a transformation, explain why.

3. 4.

Not; figures are not

the same size or

shape

Rotation

5. Reasonableness Which of the following situations uses an

example of congruent figures?

A A baker uses two sizes of loaf pans.

B The fifth-grade math book is wider than the fourth-grade book.

C You give five friends each a quarter.

D Two pencils are sharpened, the other two are not.

6. Explain It If two triangles are congruent, do they both have the same side

lengths? Explain.

Yes, if the triangles are congruent, they

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are exactly the same size, and all side

lengths are equal.

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Symmetry

Reteaching

19-5

What is line symmetry?

A figure has line symmetry if it can be folded in half so that one side

reflects onto the other side. The fold line is called a line of symmetry.

This figure has line symmetry.

There is only one way to fold it so

that two sides match. It has one

line of symmetry.

What is rotational symmetry?

This figure has line symmetry. It can

be folded into matching halves two

different ways. It has two lines of

symmetry.

A figure has rotational symmetry if it turns onto itself in less than a

full rotation.

This figure has no line

symmetry. It cannot be folded

into matching halves.

This figure has rotational symmetry. The figure

and its rotated shape match after half a rotation.

This figure has no rotational symmetry. The figure

and its rotated shape match only after a full rotation.

Circle the number of each figure that has line symmetry.

1. 2. 3.

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Circle the number of each figure with rotational symmetry.

4. 5. 6.

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Topic 19 289

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Symmetry

For 1 through 4, tell if the figure has line symmetry, rotational

symmetry, or both. If it has line symmetry, how many lines of

symmetry are there? If it has rotational symmetry, what is the

smallest rotation that will rotate the figure onto itself?

1. 2.

Practice

19-5

1 line of symmetry

3. 4.

4

symmetry;

lines of

90º or _ 1 4 turn

180° or __ 1 2 turn

For 5 and 6, copy the figure. Then complete the figure so the

dashed line is a line of symmetry.

5. 6.

2 lines of symmetry;

180° or __ 1 2 turn

7. Does any rectangle rotate onto itself in less than a half-turn?

A rectangle that is a square rotates

onto itself in a 90° or quarter turn.

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8. Draw a Picture Draw a quadrilateral that has neither line

symmetry nor rotational symmetry.

Answers will vary.

290 Topic 19

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Reteaching

19-6

A tetramino is an arrangement of four identical squares

in a plane. The squares are attached to each other edge

to edge. A unique tetramino cannot be matched to another

by rotating or reflecting. Use square tiles to show if the two

figures are unique.

1. Construct the two figures

using tiles.

2. Construct a rotation of the

first figure. Try other rotations.

3. Check for a match. Think: If figures

match, they are not unique. If

figures do not match, figures may

be unique.

4. Construct a reflection of the

first figure.

5. Repeat Step 3.

rotation

reflection

no match

match

6. Write your answer in a complete

sentence.

The two figures are not unique, because

they match by reflection.

1. Show if these two figures are unique tetraminoes.


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The two figures do not match

either by reflection or rotation.

They are unique tetraminoes.

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Topic 19 291

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1. Is the following a pentomino? Explain.

Tell whether the pentominoes in each pair are related by

a reflection or a rotation.

2.

Practice

19-6

Yes; there are 5

identical squares.

Reflection

3.

Rotation

4. How many possible different pentominoes can be formed?

A 3 B 7 C 10 D 12

5. Explain It Use objects to build pentominoes with 1 square in

each row. How many of these kinds of pentominoes can be

built? Explain.

Only 1; using 5 squares, you can only

put one square in each row, and every

pentomino after that will be the same!

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Outcomes

Reteaching

20-1

Almond, Keisha, and Mona are running for student

council president. Barry, Andy, and Maurice are running

for vice-president. Each student has an equal chance

of being elected.

You can use a tree diagram to find all the

possible outcomes. The set of all possible

outcomes is called the sample space.

There are 9 possible outcomes in the

sample space.

President

Almond

Keisha

Mona

Vice-President

Barry

Andy

Maurice

Barry

Andy

Maurice

Barry

Andy

Maurice

Outcome

Almond, Barry

Almond, Andy

Almond, Maurice

Keisha, Barry

Keisha, Andy

Keisha, Maurice

Mona, Barry

Mona, Andy

Mona, Maurice

1. Complete the tree diagram to show the possible outcomes

when Spinner A and Spinner B are spun.

Color

Color

Outcome

Black

White

Spinner A

Black

Blue

Red

Green

Black, Red

Black , Blue

Black

, Green

Green

Red

Blue

Spinner B

White

Red

Blue

Green

White , Red

White

, Blue

White , Green

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2. How many times does the outcome black/green occur

in the tree diagram?

1 time

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Topic 20 293

Name

Outcomes

The coach is trying to decide in what order Jane, Pete, and Lou

will run a relay race.

1. Complete the tree diagram below to show the sample space.

1st

Jane

Pete

Lou

2nd

Jane

Pete

3rd

Pete Lou

Lou Pete

Lou

Jane

Lou

Jane

Pete

Jane

2. How many possible outcomes are there in

the sample space?

6

Practice

20-1

3. After the first runner is chosen, how many

choices are there for the second runner?

2

4. Reasonableness Tom, Bill, John, and Ed are running for

school president. The person in second place automatically

becomes vice-president. How many possible outcomes are

there in the sample space?

A 6 B 9 C 10 D 12

5. Explain It The weather tomorrow could be sunny, cloudy,

rainy, or snowy. Is there a 1 out of 4 chance of the weather

being sunny?

You do not know what the probability

is, because not all the outcomes are

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equally likely.

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Writing Probability

as a Fraction

Reteaching

20-2

The probability of an event is a number that describes the chance

that an event will occur. Probability can be expressed as a fraction.

Probability of

an event

number of favorable outcomes

number of possible outcomes

If Felice spins the spinner, what is the

probability of landing on Orange?

There are 6 possible outcomes (spaces)

and 2 favorable outcomes (Orange spaces).

Probability

(landing on Orange)

number of Orange spaces

number of spaces

The probability of landing on Orange is 2_ or

1_

6 3 .

The probability of landing on Orange can be written

as P(Orange) 1_

3 .

__ 2

6

Orange

Black Green

Orange Blue

Red

Pencil

VCR

Lamp

Television

Radio

Cup

1. Find P(object that does not use electricity)

2. Find P(object that uses electricity)

3. Find P(object used for writing)

2_

6

4_

6

1_

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4. Explain It What does P(radio) mean?

The probability of the pointer stopping

on the segment marked radio in one

spin.

Topic 20 295

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Writing Probability

as a Fraction

Tom put 4 yellow marbles, 2 blue marbles, 6 red marbles,

and 5 black marbles in a bag.

1. Find P(yellow).

2. Find P(blue).

3. Find P(black).

4. Find P(red).

__

4

17

__

2

17

__

17

5

__

6

17

Practice

20-2

A bag contains 12 slips of paper of the same size. Each slip has

one number on it, 1–12.

5. Find P(even number).

6. Find P(a number less than 6).

7. Find P(an odd number).

8. Find P(a number greater than 8).

9. Describe an impossible event.

You will pull out a number greater than 12.

10. A cube has 6 sides and is numbered 1 through 6. If the cube

is tossed, what is the probability that a 3 will be tossed?

__

6

12 or 2

__

5

12

__

6

12 or _ 1 2

__

4

12 or _ 1

3

or 1 _

A 1__

6

B 2__

6

C 3__

6

D 6__

6

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11. Explain It Explain the probability of tossing a prime number

when you toss the cube with numbers 1 through 6.

Prime numbers would be 2, 3, and 5,

so the probability would be _ 3 6 or _ 1 2 .

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Experiments and Predictions

Reteaching

20-3

You can use an experiment to predict outcomes.

5 2

4 3

Spinner A

Compare the chances of

spinning an even number

and spinning an odd

number.

Event: Spinning an

even number

Favorable

outcomes: 2, 4

2 out of 4 possible

outcomes are

favorable, so in 2

out of 4 spins, you

can expect an even

number.

Event: Spinning an

odd number

Favorable

outcomes: 3, 5

2 out of 4 possible

outcomes are

favorable, so in 2

out of 4 spins, you

can expect an odd

number.

Spinning an even

number and

spinning an odd

number are

equally likely

events.

7 5

5 6

Spinner B

Compare the chances of

spinning an even number

and spinning an odd

number.

Event: Spinning an

even number

Favorable

outcomes: 6

1 out of 4 possible

outcomes are

favorable, so in 1

out of 4 spins, you

can expect an even

number.

Event: Spinning an

odd number

Favorable

outcomes: 5, 5, 7

3 out of 4 possible

outcomes are

favorable, so in 3

out of 4 spins, you

can expect an odd

number.

Spinning an even

number is less

likely than

spinning an odd

number. Spinning

an odd number is

more likely than

spinning an even

number.

Think about tossing a standard number cube.

2

1. What are the possible outcome numbers? Are the numbers

equally likely? Explain.

The possible outcome numbers are

1, 2, 3, 4, 5, and 6. They are equally

likely because there is one of each on

the six faces of the number cube.

1

3

2. Number Sense In 6 tosses, how many times would you

expect to toss the number 6?

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Topic 20 297

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Experiments and Predictions

Write each of the following as a fraction in lowest terms.

1. 20 out of 60 2. 16 out of 64 3. 24 out of 60

1__

3

1__

4

2__

5

Practice

20-3

The table below shows data from the woodshop classes at Jones

Elementary School. Students had the choice of making a shelf,

a chair, or a birdhouse.

Project

Number of students

Shelf 15

Chair 25

Birdhouse 20

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4. What is the total number of students

who were taking a woodshop class?

5. What is the probability that someone

will make a chair? a birdhouse?

6. Predict how many students will make a shelf if

there are 180 students in the woodshop class.

7. Reasonableness Using the letters in the word

ELEMENTARY, find the probability of choosing a letter

that is not E.

A

B

C

D

3 ___

10

5 ___

10

___ 7

10

9 ___

10

8. Explain It If 2_ of the 45 customers in the past hour bought a

3

cup of coffee, predict the number of cups that will be sold in

the next 3 hours if sales continue at the same level.

298 Topic 20

60

___ 5 __

12, 1 3

45

90; if 30 are sold in 1 hour, 30 3 90

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Problem Solving: Solve a

Simpler Problem

Reteaching

20-4

A farmer has 4 garden plots in a row. He wants to build paths

connecting each pair of plots. He also doesn’t want the paths to

cross. How many paths must he build?

Read and Understand

What do you know?


There are four plots in a row.

Each pair of plots must be connected

by a path.

The paths cannot cross.

The total number of paths

Plan and Solve

Solve a simpler problem.

1. Construct a picture of two garden plots as shown below.

2. Then draw another picture of three plots. Look for a pattern

for the design and number of paths.

3. Construct a diagram of the four plots.

A farmer has four garden plots arranged in a square. Look at

the following diagrams. Use them to help answer this question.

Can the four garden plots be connected by six paths that do

not cross?

yes

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Problem Solving: Solve a

Simpler Problem

Solve the simpler problems. Use the solutions to

help you solve the srcinal problem.

1. Reggie is designing a triangular magazine rack with

5 shelves. The top shelf will hold 1 magazine. The

second shelf will hold 3 magazines, and the third

shelf will hold 5 magazines. This pattern continues

to the bottom shelf. How many magazines will the

magazine rack hold altogether?

Practice

20-4

Simpler Problem What is the pattern?

Each shelf down holds 2 more

magazines than the shelf above.

7 magazines

How many magazines will the fourth shelf hold?

9 magazines

How many magazines will the bottom shelf hold?

Solution:

The magazine rack will hold 25

magazines.

2. At the deli, you receive 1 free sub after you buy 8 subs.

How many free subs will you receive from the deli if you

buy 24 subs?

3 subs

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3. The chef has 5 different kinds of pasta and 3 different flavors

15 different meals

of sauce. How many different meals is she able to make?

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Credits

Pages I-III Taken from Scott Foresman-Addison Wesley Mathematics, Grade 5:

Review From Last Year Masters by Scott Foresman-Addison Wesley

Pages 21-300

Pages 21-300 Taken from enVisionMATH Teacher Resource Masters,Topics

1-20, Grade 5 by Scott Foresman-Addison Wesley

Credits 301

envisionmath-grade-5-workbook-answer_compress

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