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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
by Scott Foresman-Addison Wesley
Cover Art: by Luciana Navarro Powell
Taken from:
enVisionMATH Teacher Resource Masters, Topics 1–20, Grade 5
by Scott Foresman-Addison Wesley
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
by Scott Foresman-Addison Wesley
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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III
Reteaching and Practice
Corresponds with Student enHanced Workbook pages 21–300
Name
Place Value
Place-value chart:
Reteaching
1-1
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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
Topic 1 21
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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
1- 2
R
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1
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2
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
Topic 1 23
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Comparing and Ordering
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
Order these numbers from least to greatest.
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
1-3
Mill
llionths
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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Write each number in standard form.
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
Topic 1 25
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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
Write each number in standard form.
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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Name
Comparing and Ordering
Decimals
Reteaching
1-4
R
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1
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4
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
from least to greatest.
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
Order these numbers from least to greatest.
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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Comparing and Ordering
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
Order these numbers from least to greatest.
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
1-5
R
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1
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5
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
R
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2
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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
2-2
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2
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2
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
Topic 2 33
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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
Name
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
2-3
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3
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.
36 Topic 2
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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
Topic 2 37
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Problem Solving: Draw a
Picture and Write an Equation
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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Adding and Subtracting
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
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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
Topic 2 41
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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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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
2-8
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Read and Understand
What do you know?
What are you trying to find?
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:
Multiple-Step Problems
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
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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
Factors Product Factors Product
Zero Property of Multiplication
When one of the factors is 0, the product is always 0.
3 0 0 0 3 0
Factors Product Factors Product
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
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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
Find each product. Estimate to check that your
answer is reasonable.
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?
54 Topic 3
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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
3-5
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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
Name
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.
Topic 3 57
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Multiplying Greater Numbers
Find each product. Estimate to check that your
answer is reasonable.
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.
Topic 3 59
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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
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Problem Solving: Draw a
Picture and Write an Equation
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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What are you trying to find?
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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Problem Solving: Draw a
Picture and Write an Equation
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
Name
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
Use mental math to answer the following questions.
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
Name
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.
Topic 4 65
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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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66 Topic 4
Name
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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68 Topic 4
Name
Connecting Models and
Symbols
Reteaching
4-4
Divide 138 equally into
3 groups.
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4
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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Name
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
5
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remainder. This should equal the dividend.
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72 Topic 4
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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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Name
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
Name
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.
Yes. Therefore it is divisible
by 2.
No. Therefore it is not
divisible by 3.
Yes. Therefore it is divisible
by 4.
No. Therefore it is not
divisible by 5.
No. Therefore it is not
divisible by 6.
No. Therefore it is not
divisible by 9.
92 1 92
1 and 92 are factors.
92 2 46
2 and 46 are factors.
3 is not a factor.
92 4 23
4 and 23 are factors.
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.
Topic 4 75
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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
Problem Solving: Draw a
Picture and Write an Equation
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
Problem Solving: Draw a
Picture and Write an Equation
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
Find each quotient. Use mental math.
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.
Topic 5 81
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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
Use mental math to answer the following questions.
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
Topic 5 83
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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:
Multiple-Step Problems
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
answer is reasonable.
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?”
Topic 5 85
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Problem Solving:
Multiple-Step Problems
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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Topic 5 87
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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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Name
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
Divide. Check by multiplying.
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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6 R32 7 R25
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
Divide. Check by multiplying.
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
Name
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
21 R25
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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Name
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
Name
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?
What are you trying to find?
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
Topic 5 95
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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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Name
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
Topic 6 97
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Name
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 14-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
10 inches longer than 10-inch beaded section
10 inches longer than 12-inch beaded section
20
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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 .
Check students’ work.
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
A 3-day rental will cost $117.
5-day rental
(5 35) 12 187
A 5-day rental will cost $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
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7
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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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Name
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
Find each quotient. Use mental math.
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:
Multiple-Step Problems
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:
Multiple-Step Problems
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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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.
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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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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
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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
This triangle is both
isosceles and right.
Not all right triangles
are isosceles.
This triangle is both
scalene and obtuse.
Not all obtuse triangles
are scalene.
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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
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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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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
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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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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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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
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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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Name
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
Comparing and Ordering
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
Comparing and Ordering
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___
9
10
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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
Adding and Subtracting
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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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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Adding and Subtracting
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.
Are there the same number of X’s
in each column?
Since the 2nd column has fewer,
add another set of 6 X’s.
Are there the same number of X’s
in each column?
Since the 1st column has fewer,
add another set of 4 X’s.
the least common denominator of both
fractions.
3
4
3
3
9
12
5
6
2
2
10
12
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4
XXXX
XXXX
XXXX
6
XXXXXX
XXXXXX
Are there the same number of X’s
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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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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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
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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
10-6
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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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2. Can you completely cover a 3 ft by 6 ft area
with 3 ft by 3 ft tiles?
3. Can you completely cover a 3 ft by 6 ft area
with 2 ft by 2 ft tiles?
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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Topic 11 175
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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
1
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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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180 Topic 11
Name
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__
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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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Problem Solving: Draw a
Picture and Write an Equation
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
Problem Solving: Draw a
Picture and Write an Equation
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.
The length is more than
4 1_ in., but less than
2
4 2_ (5) in.
2
It is closer to 4 1_
2 in.
The pen’s length is
closest to 4 1_
2 in. Step 3: Measure to the
nearest 1_
4 inch.
The length is more than
4 2_
4 (41_ ) in., but less than
2
4 3_
4 in.
It is closer to 4 1_
2 in.
The pen’s length is
closest to 4 1_
2 (42_ 4 ) in. Step 4: Measure to the
nearest 1_
8 inch.
The length is more than
4 4_
8 (41_ ) in., but less than
2
4 5_
8 in.
It is closer to 4 5_
8 in.
The pen’s length is
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.
The length of the twig is more than
70 mm but less than 80 mm.
The length of the twig is more than
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.
Check students’ work.
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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. 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
o
o
160 ft
z
Playing field 160 ft
z
d
n
E
300 ft
ft30
ft 30
e
n
d
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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.
2.3 ft
14 m
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.
Check students’ work.
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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. 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
Problem Solving: Use Objects
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
The volume of the liquid is about
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.
The volume of the liquid is about
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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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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Name
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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0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 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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Name
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
Topic 15 239
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Name
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
Problem Solving: Draw a
Picture and Write an Equation
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
Problem Solving: Draw a
Picture and Write an Equation
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
Topic 16 243
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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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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.
252 Topic 16
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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
0
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
Name
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.
258 Topic 17
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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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Topic 17 259
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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260 Topic 17
Name
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
Topic 17 261
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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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262 Topic 17
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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
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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
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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?
264 Topic 18
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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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Topic 18 265
Name
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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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
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Name
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
m
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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)
g
1
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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?
ts
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tu
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6
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0
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
Topic 18 273
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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.
8
8.44
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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
from least to greatest.
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
Topic 18 277
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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?
Explain your answer.
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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p 200
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f
o 100
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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
ig 40
e
H 20
0
Tree Growth
Apr. May Jun. Jul. Aug.
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Topic 18 279
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.
280 Topic 18
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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
x
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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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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.
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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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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.
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Problem Solving: Use Objects
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.
Explain your answer.
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The two figures do not match
either by reflection or rotation.
They are unique tetraminoes.
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Problem Solving: Use Objects
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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292 Topic 19
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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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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.
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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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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
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___ 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?
What are you trying to find?
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