GRADE
5
ISBN 978-0-8454-6761-9
Copyright © 2011 The Continental Press, Inc.
No part of this publication may be reproduced in any form or by any
means, electronic, mechanical, photocopying, recording, or otherwise,
without the prior written permission of the publisher. All rights reserved.
Printed in the United States of America.
Table of Contents
Introduction....................................................................................................... 5
Unit 1 Number Sense..........................................................................................7
5.NBT.2
Lesson 1
5.NBT.1
Lesson 2
5.NBT.1, 5.NBT.3.a
Lesson 3
5.NBT.3.b
Lesson 4
5.NBT.4
Lesson 5
Review
Powers of Ten............................................................ 8
Whole-Number Place Value................................... 12
Decimal Place Value................................................ 16
Comparing Decimals...............................................20
Rounding Decimals................................................. 24
Number Sense......................................................... 28
Unit 2 Operations..............................................................................................31
Lesson 1 Multiplying Whole Numbers.................................. 32
5.NBT.6
Lesson 2 Dividing Whole Numbers.......................................36
5.NBT.7
Lesson 3 Adding and Subtracting Decimals.........................40
5.NBT.7
Lesson 4 Multiplying Decimals..............................................44
5.NBT.7
Lesson 5 Dividing Decimals...................................................48
ReviewOperations............................................................... 52
5.NBT.5
Unit 3 Adding and Subtracting Fractions........................................................55
5.NF.1
Equivalent Fractions............................................... 56
Adding and Subtracting Fractions.........................60
Lesson 1
5.NF.1, 5.NF.2
Lesson 2
5.NF.1, 5.NF.2
Lesson 3
Review
Adding and Subtracting Mixed Numbers.............64
Adding and Subtracting Fractions.........................68
Unit 4 Multiplying and Dividing Fractions......................................................71
5.NF.3
Relating Fractions to Division................................ 72
Multiplying Fractions and Whole Numbers.......... 76
Lesson 1
5.NF.4.a, b
Lesson 2
5.NF.4.a, b
Lesson 3
5.NF.5.a, b
Lesson 4
5.NF.7.a, b
Lesson 5
5.NF.6, 5.NF.7.c
Lesson 6
Review
Multiplying Fractions..............................................80
Multiplication and Scale.........................................84
Dividing Fraction and Whole Numbers.................88
Word Problems with Fractions...............................92
Multiplying and Dividing Fractions....................... 96
Unit 5 Algebraic Thinking.................................................................................99
Lesson 1
5.OA.1, 5.OA.2
Lesson 2
5.OA.3
Lesson 3
Review
5.OA.1, 5.OA.2
Writing Expressions............................................. 100
Evaluating Expressions........................................ 104
Patterns and Relationships.................................. 108
Algebraic Thinking............................................... 112
Unit 6 Measurement........................................................................................115
Lesson 1 Customary Units of Measurement....................... 116
5.MD.1
Lesson 2 Metric Units of Measurement..............................120
5.MD.1
Lesson 3 Measurement Conversions................................... 124
5.MD.1
Lesson 4 Measurement Word Problems.............................128
5.MD.2
Lesson 5 Measurement Data...............................................132
ReviewMeasurement........................................................136
5.MD.1
Unit 7 Volume..................................................................................................139
Lesson 1Volume................................................................. 140
5.MD.5.a, b
Lesson 2 Volume of Rectangular Prisms............................ 144
5.MD.5.c
Lesson 3 Volume of Irregular Figures.................................148
ReviewVolume..................................................................152
5.MD.3.a, b; 5.MD.4
Unit 8 Geometry..............................................................................................155
Lesson 1 Coordinate Planes.................................................156
5.G.3, 5.G.4
Lesson 2Triangles................................................................160
5.G.3, 5.G.4
Lesson 3Quadrilaterals...................................................... 164
ReviewGeometry...............................................................168
5.G.1, 5.G.2
Practice Test............................................................................................................... 171
Glossary...................................................................................................................... 183
Finish Line Mathem
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Welc ommon Core State S
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About This Book
Finish Line Mathematics for the Common Core State Standards will
help you prepare for math tests. Each year in math class, you learn new skills and
ideas. This book focuses on the math skills and ideas that are the most important
for each grade. It is important to master the concepts you learn each year because
mathematical ideas and skills build on each other. The things you learn this year will
help you understand and master the skills you will learn next year.
This book has units of related lessons. Each lesson concentrates on one main
math idea. The lesson reviews things you have learned in math class. It provides
explanations and examples. Along the side of each lesson page are reminders to
help you recall what you learned in earlier grades.
After the lesson come three pages of practice problems. The problems are the
same kinds you find on most math tests. The first page has multiple-choice, or
selected-response, problems. Each item has four answers to choose from, and you
must select the best answer. At the top of the page is a sample problem with a
box beneath it that explains how to find the answer. Then there are a number of
problems for you to do on your own.
Constructed-response, or short-answer, items are on the next page. You
must answer these items using your own words. Usually, you will need to show
your work or write an explanation of your answer in these items. This type of
problem helps you demonstrate that you know how to do operations and carry
out procedures. They also show that you understand the skill. Again, the first item
is a sample and its answer is explained. You will complete the rest of the items by
yourself.
The last page has one or two extended-response problems. These items are like
the short writing items, but they have more parts and are often a little harder. The
first part may ask you to solve a problem and show your work. The second may ask
you to explain how you found your answer or why it is correct. This item has a hint
to point you in the right direction.
At the end of each unit is a review section. The problems in it cover all the
different skills and ideas in the lessons of that unit. The review contains multiplechoice, constructed-response, and extended-response items.
A practice test and a glossary appear at the end of the book. The practice test
gives you a chance to try out what you’ve learned. You will need to use all the skills
you have reviewed and practiced in the book on the practice test. The glossary
lists important words and terms along with their definitions to help you remember
them.
Introduction
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5
The Goals of Learning Math
Math is everywhere in the world around you. You use math more than you
probably realize to help you understand and make sense of that world. But what
does it mean to be good at math?
To be good at math, you need to practice certain habits. And you need the
right attitude.
• You make sense of problems and do not give up in solving them. You make
sure you understand the problem before you try to solve it. You form a plan
and then carry out that plan to find an answer. Along the way, you ask
yourself if what you are doing makes sense. And if you do not figure out the
answer on the first try, you try another way.
• You think about numbers using symbols. You can think about a real-life
situation as numbers and operations.
• You draw conclusions about situations and support them with proof. You
use what you know about numbers and operations to provide reasons
for your conclusions and predictions. When you read or hear someone
else’s explanation, you think about it and decide if it makes sense. You ask
questions that help you better understand the ideas.
• You model with mathematics. You represent real-life problems with a
drawing or diagram, a graph, or an equation. You decide if your model
makes sense.
• You use the right tools at the right time. You know how to use rulers,
protractors, calculators, and other tools. More importantly, you know when
to use them.
• You are careful and accurate in your work. You calculate correctly and label
answers. You use the correct symbols and definitions. You choose exactly the
right words for your explanations and descriptions.
• You look for structure in math. You see how different parts of math are
related or connected. You can use an idea you already know to help you
understand a new idea. You make connections between things you have
already learned and new ideas.
• You look for the patterns in math. When you see the patterns, you can find
shortcuts to use that still lead you to the correct answer. You are able to
decide if your shortcut worked or not.
These habits help you master new mathematical ideas so that you can
remember and use them. All of these habits will make math easier to understand
and to do. And that will make it a great tool to use in your everyday life!
6
Introduction
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U n it
1
Number Sense
Lesson 1 Powers of Ten reviews what powers of 10
are and how to use them to multiply and divide numbers
easily.
Lesson 2 Whole-Number Place Value reviews how
to use place value to read, write, and understand whole
numbers.
Lesson 3 Decimal Place Value reviews how to use
place value to read, write, and understand decimals in
standard and expanded forms.
Lesson 4 Comparing Decimals reviews how to use
place value to compare and order decimals.
Lesson 5 Rounding Decimals reviews how to round
decimals to any place.
UNIT 1
Number Sense
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7
Lesson
1
A number written in
exponential form has a
base and an exponent.
It shows repeated
multiplication.
2
3 ← Exponential form
3 3 3 ← Repeated
multiplication
9 ← Standard form
Powers of Ten
5.NBT.2
A power is the result of multiplying a number by itself. The number
that is multiplied is the base. The exponent tells how many times to
use the base as a factor.
102 5 10 3 10 5 100
102, or 10 to the second power, is a power of 10. The number
that is the factor, 10, is the base, and 2 is the exponent.
Powers of 10 are related to the places in our place-value system.
Powers of 10
Any number can be the
base. Any number can
be the exponent.
23 5 2 3 2 3 2
54 5 5 3 5 3 5 3 5
Any number to the zero
power is 1.
30 5 1 8 0 5 1
10 0 5 1
To divide a number by a
power of 10, move the
decimal one place left
for each factor of 10.
123.4 4 10 5 12.34
0.02 4 103 5 0.00002
Factors
101 5 10 3 1
5 10
102 5 10 3 10
5 100
103 5 10 3 100
5 1,000
104 5 10 3 1,000 5 10,000
Word Name
10
Ten
10 3 10
Hundred
10 3 10 3 10
Thousand
10 3 10 3 10 3 10
Ten thousand
Notice that each power of 10 has the same number of zeros as the
exponent. For example, the second power of 10, or 102 , is equal to
100, and 100 has 2 zeros.
To multiply by a power of 10, simply move the decimal point the same
number of places to the right as the number of the exponent.
What is 384.651 3 103?
The exponent is 3. This is the same as multiplying by 10 three
times: 384.651 3 10 3 10 3 10. Each time you multiply by 10,
the decimal point moves one place to the right.
384.651 3 10 5 3,846.51
3,846.51 3 10 5 38,465.1
38,465.1 3 10 5 384,651.0
As a shortcut, you can simply move the decimal point in
384.651 three places to the right.
384.651
3 103 5 384,651.0

8
UNIT 1
Number Sense
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Read each problem. Circle the letter of the best answer.
SAMPLE Which of the following is the fifth power of 10?
A 510
B 50
C 10,000
D 100,000
The correct answer is D. This question asks you to find the fifth
power of 10, that is 10 multiplied by itself five times, or 10 3 10 3
10 3 10 3 10. This equals a 1 followed by five zeros, or 100,000.
Choice A shows 5 with an exponent of 10. This represents 5
multiplied by itself ten times, which is incorrect. Choice B shows 50,
the product of 5 3 10, which is incorrect. Choice C has four zeros
and names the fourth power of 10.
1 Which shows the twelfth power of 10?
5 What is the product of 0.065 3 102?
A1012
A0.00065
B1210
B0.0065
C120
C6.5
D100,000,000,000
D65
2What number is the base in the equation
6 What is the product of 0.1 3 1,000?
7
10 5 100,000?
A7
C 70
B10
D 10,000,000
3What is the exponent in the expression
2.528 3 108?
A0.528
C 8
B2.528
D 10
4 What is the product of 0.22 3 1,000?
A0.00022
C 22
B0.0022
D 220
A0.001
C 3
B0.3
D 100
7 What is the quotient of 34.35 4 103?
A0.03435
C 34.35
B0.3435
D 34,350
8 What is the quotient of 7.985 4 100?
A0.7985
C 7.985
B1
D 798.5
UNIT 1
Number Sense
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9
Read each problem. Write your answer.
SAMPLE Write the seventh power of 10 three ways. Show it as an expression of
factors, in exponential form, and in standard form.
Answer ____________________________________________________
First, show the seventh power of 10 as 10 multiplied seven times
by itself: 10 3 10 3 10 3 10 3 10 3 10 3 10. In exponential form, it
is 107. In standard form, it is written as 10,000,000.
9Find the ninth power of 10. Write it as repeated multiplication, with an
exponent, and as a whole number.
Answer __________________________________________________________________________
_________________________________________________________________________________
10What happens to the decimal point when finding the product of
0.0038 3 104? What is the product?
Answer __________________________________________________________________________
11 What is the product of 1.23456789 3 106?
Answer ________________________
12What happens to the decimal point when finding the quotient for
99,999.99 4 106? What is the quotient?
Answer __________________________________________________________________________
13Venus is the planet that comes nearest to Earth. Its closest position is
about 38,000,000 kilometers from Earth. Write this number expressed
as a product of 38 and a power of 10.
10
Answer ________________________
UNIT 1
Number Sense
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Read the problem. Write your answer to each part.
14Tom is a famous writer whose books earn a lot of money. He did not
make much money on his first books, however.
Part AThe first book Tom wrote earned him only 103 dollars. Write
this number in standard form.
Answer ________________________
Part BTom’s most recent book earned 104 times as much
money as his first book. How much did Tom’s most
recent book earn? Write the amount in standard form.
Explain how you know your answer is correct.
How can you write
“10 times as much” as
an expression using a
standard form number?
4
________________________________________________________
_________________________________________________________________________________
_________________________________________________________________________________
_________________________________________________________________________________
UNIT 1
Number Sense
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11
millions
hundred thousands
ten thousands
thousands
hundreds
tens
ones
The number 125 billion can be written as 125,000,000,000 in
standard form. Standard form represents the sum of the
values of each place.
ten millions
If the value of a place
is 0, it is not included in
the word form.
There are about 125 billion galaxies in the universe.
hundred millions
Three thousand, fortytwo is the word form
of 3,042.
The powers of 10 can help you understand place value in a whole
number. The value of a digit depends on its place in a number.
billions
Numbers can also be
written in word form.
5.NBT.1
ten billions
2
-Number Place Va
e
l
o
h
W
lue
hundred billions
Lesson
1
2
5,
0
0
0,
0
0
0,
0
0
0
What is 125 billion written in expanded form?
When writing a number
in expanded form, it is
not necessary to write
expressions if the value
of a place is 0. If there
is no expression for a
place, its value is 0.
A number in
exponential form
represents repeated
multiplication. The
exponent tells how
many times the base is
used as a factor.
105 ← Exponent
↑
Base
12
Write the value of each place as a multiplication expression
showing the digit times the value of the place.
125 billion 5 (1 3 100,000,000,000) 1 (2 3 10,000,000,000) 1
(5 3 1,000,000,000)
Another way to write expanded form is using exponential form.
125 billion 5 (1 3 1011) 1 (2 3 1010 ) 1 (5 3 109 )
Each place value is 10 times greater than the place value to its right
1
and  
​  10  ​as much as the place value to its left.
Compare the value of the digit 6 in 2,645 and 264.5.
In 2,645 the 6 is in the hundreds place, so its value is 6 3 100 5
600. In 264.5, the 6 is in the tens place, so its value is 6 3 10 5
60. The number 600 is 10 times greater than 60. The number 60
1
is  
​  10  ​of 600.
UNIT 1
Number Sense
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Read each problem. Circle the letter of the best answer.
SAMPLE What is the expanded form of 32,000,000?
A 32 3 100,000,000
B 32 3 10,000,000
C (3 3 10,000,000) 1 (2 3 1,000,000)
D (3 3 100,000) 1 (2 3 10,000)
The correct answer is C. The number shown in standard form
is thirty-two million. There are 3 ten millions, written as 3 3
10,000,000, and 2 millions, written as 2 3 1,000,000. The rest of the
places show zeros, so it is not necessary to write expressions for
them. So the answer is (3 3 10,000,000) 1 (2 3 1,000,000).
1What is the expanded form of six billion,
eight hundred thousand?
4What is the expanded form of 8,004,003
using exponents?
A(6 3 1,000,000,000) 1 (8 3 100,000)
A(8 3 109 ) 1 (4 3 104) 1 (3 3 100 )
B(6 3 100,000,000) 1 (8 3 100,000)
B(8 3 109 ) 1 (4 3 106) 1 (3 3 101)
C(6 3 1,000,000) 1 (8 3 10,000)
C(8 3 107) 1 (4 3 104) 1 (3 3 101)
D(60 3 100,000) 1 (8 3 1,000)
D(8 3 106) 1 (4 3 103) 1 (3 3 100 )
2What is the value of the 5 in 205,316,000?
5What is (1 3 105) 1 (3 3 103) 1 (7 3 102)
in standard form?
A 5 ten thousands
B 5 millions
C 5 ten millions
D 5 billions
A1,030,700
C 103,700
B1,037,000
D 1,370
6What is the number name of (4 3 107) 1
(5 3 106) 1 (4 3 102)?
3In which number does the digit 9 have the
greatest value?
A1,590,020
C 5,200,910
B2,109,500
D 1,905,200
A forty-five million, four thousand
B forty-five million, four hundred
C forty-five million, forty
Dfour million, fifty-four hundred
thousand
UNIT 1
Number Sense
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13
Read each problem. Write your answer.
SAMPLE What is the expanded form of eight hundred seventy-six million?
Answer ______________________________________________________________
Write the number in standard form to identify the place of each
digit: 876,000,000. The digit 8 has a place value of hundred millions,
the 7 a place value of ten millions, and the 6 a place value of
millions. Then write each digit multiplied by its place value and
add the expressions for the digits. Eight hundred seventy-six million
in expanded form is (8 3 100,000,000) 1 (7 3 10,000,000) 1
(6 3 1,000,000).
7Write a ten-digit number with a 7 in the tens million place, a 3 in the
thousands place, and a 0 in the hundreds place.
Answer ____________________________________
8The population of the United States in 2010 was estimated to be
308,400,408. Write this number in expanded form.
_________________________________________________________________________________
_________________________________________________________________________________
9In 2005, Eastside Elementary School was built at a cost of
$18,000,000. The school it replaced was built in 1935 at a cost of
$180,000. Explain how the values of the 8 in each number are related.
_________________________________________________________________________________
_________________________________________________________________________________
10 Jackie wrote 5,406,029 in expanded form this way:
(5 3 1,000,000) 1 (4 3 100,000) 1 (6 3 10,000) 1 (2 3 10)
What two errors did Jackie make?
_________________________________________________________________________________
_________________________________________________________________________________
14
UNIT 1
Number Sense
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Read the problem. Write your answer to each part.
11 Light travels at a speed of 299,792,458 meters per second.
Part AWrite this number in expanded form two ways, 1) with and
2) without exponents.
Answer 1 ________________________________________________________________
_________________________________________________________________________
Answer 2 ________________________________________________________________
_________________________________________________________________________
Part B `There are two 2’s in 299,792,458. Name the place
value of each 2. Explain how their values are related.
________________________________________________________
________________________________________________________
________________________________________________________
Any number to
the 0 power is 1. To
show digits in the
ones place, you can
use 10 0 for the
power of 10.
________________________________________________________
_________________________________________________________________________________
_________________________________________________________________________________
UNIT 1
Number Sense
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15
Lesson
3
A decimal is any
number used in the
decimal system.
235.0 and 0.2 are
both decimals.
Read the decimal point
as “and.”
1.2 is read
“one and two tenths.”
Decimal Place Value
5.NBT.1, 5.NBT.3.a
Our number system is based on the number 10 and called the
decimal system. A decimal point separates whole numbers from a
fractional part of 1. You can use your knowledge of place value to
help you understand the value of decimals.
Kingda Ka, in New Jersey, is the tallest roller coaster in the
world. It is 138.988 meters tall. What is the value of each digit
in this number?
Whole-number values are on the left of the decimal point.
Fractional values are on the right of the decimal point.
hundreds1 3 100
tens3 3 10
ones8 3 1
When you read a
decimal, use the last
place with a digit to
name the fractional
part.
1
tenths9 3 ​  
  ​
10
1
hundredths8 3 ​  
   ​
100
0.5 is “five tenths.”
0.55 is “fifty-five
hundredths.”
0.555 is “five hundred
fifty-five thousandths.”
You can write a zero
after a decimal to fill
places, but the number
does not change.
0.4 is the same as 0.40.
4 tenths is the same as
40 hundredths.
16
1
thousandths8 3 ​  
   ​
1,000
1 3 8.9 8 8
There are 1 hundred, 3 tens, 8 ones, 9 tenths, 8 hundredths,
and 8 thousandths. Read this number as “one hundred thirtyeight and nine hundred eighty-eight thousandths.”
Decimal numbers can be shown in expanded form, just as whole
numbers are.
What is the expanded form of 138.988?
1
1
1
(1 3 100) 1 (3 3 10) 1 (8 3 1) 1 (9 3  
​  10  ​) 1 (8 3  
​  100   ​) 1 (8 3  
​  1,000
   ​)
UNIT 1
Number Sense
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Read each problem. Circle the letter of the best answer.
SAMPLE What is the expanded form of the decimal 53.427?
1
)
1
1
)
1
1
)
1
)
)
1
A (5 3 10) 1 (3 3 1) 1 (4 3  
​  10  ​ 1 (2 3 ​  
   ​ 1 (7 3 ​  
   ​
100
1,000
)
1
)
B (5 3 10) 1 (3 3 1) 1 (4 3 ​  
  ​ 1 (2 3 ​  
  ​ 1 (7 3 ​  
  ​
10
10
10
)
1
)
1
)
C (5 3 10) 1 (3 3 1) 1 (4 3 ​  
  ​ 1 (2 3 ​  
   ​ 1 (7 3 ​  
   ​
10
100
100
1
)
1
)
D (5 3 1) 1 (3 3 1) 1 (4 3 ​  
  ​ 1 (2 3 ​  
   ​ 1 (7 3 ​  
   ​
10
100
1,000
The correct answer is A. The number has 5 tens, 3 ones, 4 tenths,
2 hundredths, and 7 thousandths. In expanded form, these values
1
1
1
  ​ 1 (2 3  
​  100   ​ 1 (7 3  
​  1,000
   ​.
are shown as (5 3 10) 1 (3 3 1) 1 (4 3 ​  
10
)
1Which digit is in the hundredths place of
107.263?
)
)
4Which number shows the standard form of
eighty-one and twelve thousandths?
A1
C 6
A81.12
C 8.112
B0
D 3
B81.012
D 8.12
2What is the expanded form of the decimal
0.64?
5In which number does the 3 have a value
of 3 tenths?
1
)
1
)
A(6 3  
​  10  ​ 1 (4 3  
​  100   ​
)
)
1
1
B(6 3  
​  100   ​ 1 (4 3  
​  10  ​
)
)
1
1
C(6 3  
​  100   ​ 1 (4 3  
​  100   ​
A13.92
C 1.239
B2.391
D 9.123
6What is the standard form of (7 3 100) 1
1
)
D(6 3 1) 1 (4 3  
​  10  ​
A762.1
3What is the place value of the 8 in the
C700.621
1
)
number 5.418?
1
)
1
)
(6 3  
​  10  ​ 1 (2 3 ​  
   ​ 1 (1 3  
​  1,000
   ​?
100
B706.21
D70.621
Aones
C hundredths
Btenths
D thousandths
UNIT 1
Number Sense
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17
Read each problem. Write your answer.
SAMPLE Look at the decimal 0.363. How are the values of the 3’s related in this
number?
Answer ______________________________________________________________
1
The first 3 is in the tenths place and its value is 3 3  
​  10  ​or 0.3. The
1
second 3 is in the thousandths place and its value is 3 3  
​  1,000
   ​ or
0.003. Each place has a value 10 times greater than the place to its
1
right and  
​ 10  ​the value of the place to its left. So, the value of 0.3 is
1
10 3 10 or 100 times greater than 0.003. The value of 0.003 is  
​ 100   ​
the value of 0.3.
7 Write the expanded form of the decimal 7.452.
Answer __________________________________________________________________________
8What is the standard form of the number (2 3 1,000) 1 (5 3 100) 1
1
)
1
)
(3 3  
​  100   ​ 1 (7 3 ​  
   ​?
1,000
Answer ________________________
9 What is the value of the 7 in 2.874?
Answer ________________________
10Write a five-digit number with 9 thousandths, 4 hundredths, and
5 tens.
Answer ________________________
11The fastest roller coaster in the world reaches a speed of up to
205.996 kilometers per hour. How does the value of the digit in the
hundredths place compare to the value of the digit in the tenths place?
18
Answer __________________________________________________________________________
UNIT 1
Number Sense
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Read the problem. Write your answer to each part.
12The longest roller coaster in the world has a length of 415.103 meters.
Part ALabel the place value of each digit in the number 415.103.
Then write 415.103 in expanded form.
4 1 5.1 0 3
Answer _________________________________________________________________
_________________________________________________________________________
Part BThe digit 1 appears twice in the number 415.103.
How do the values of the digits 1 compare?
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
________________________________________________________
A digit in one
place represents 10
times as much as it
represents in the
place to its right
1
  ​of what it
and ​  
10
represents in the
place to its left.
UNIT 1
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19
Lesson
4
Comparing Decimals
5.NBT.3.b
You can use place value and number sense to compare decimals.
The symbols used to
compare numbers are
, is less than
. is greater than
5 is equal to
Annex, or write in,
zeros, if necessary, to
make sure all the place
values are aligned
correctly.
Compare 0.33 and 0.334.
1. Write the numbers in a column, lining up the 0.330
decimal points. Annex a zero to 0.33 so it has 0.334
the same number of places as 0.334.
2. Compare the digits in the same places, 0.330
starting at the left. The digits in the ones place 0.334
are the same.
3. The digits in the tenths place are the same. 0.330
0.334
4. The digits in the hundredths place are the same. 0.330
0.334
5. The digits in the thousandths place differ: 0.330
0 thousandths , 4 thousandths 0.334
So, 0.33 , 0.334
0.33
0.334
You can use a number line to
verify the answer.
0.330
A number with more
decimal places is not
necessarily greater than
a number with fewer
decimal places.
0.340
If you can use place value and number sense to compare two
decimals, you can also use it to put a longer list of decimals in order.
Order 9.2, 8.92, 9.23, and 9.08 from least to greatest.
Compare the digits in the same places: ones, tenths, and
hundredths.
8.92 , 9.08 , 9.2 , 9.23.
From least to greatest, the numbers are: 8.92, 9.08, 9.2, and 9.23.
20
UNIT 1
Number Sense
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Read each problem. Circle the letter of the best answer.
SAMPLE Which decimal is greater than 13.3?
A 1.33
B 13.33
C 13.003 D 13.03
The correct answer is B. Line up the decimals, annex zeros if
necessary, and compare each place value. Choice A has no tens digit,
so it is incorrect. Choices B, C, and D all have the same digits as 13.3
in the tens and ones places, so look at the tenths. In choices B and C,
the tenths digit is a 0 and 0 tenths is less than 3 tenths. Choice B is
greater than 13.3 because it has a 3 in the hundredths place, while
13.3 has an annexed zero in this place.
1Which of the following comparisons is
true?
4Which of the following comparisons is not
true?
A6.092 , 6.029
A3.119 , 3.901
B22.19 . 22.2
B80.06 . 80.60
C10.03 , 10.3
C101.5 5 101.500
D0.773 5 0.77
D0.402 . 0.042
2Which shows the decimals in order from
5Which shows the decimals in order from
greatest to least?
least to greatest?
A 0.905, 0.59, 0.509, 0.095
A 6.2, 6.82, 8.602, 8.6
B 0.59, 0.095, 0.905, 0.509
B 10.95, 10.059, 9.90, 9.09
C 0.509, 0.59, 0.095, 0.905
C 49.08, 49.8, 49.808, 49.88
D 0.095, 0.509, 0.59, 0.905
D 25.373, 25.04, 25.6, 25.201
3Marcy biked 3.4 mi, Natalia biked 3.04 mi,
6 Which decimal is less than 40.2?
Benny biked 3.27 mi and Akira biked
3.72 mi. Who rode the greatest distance?
ABenny
C Marcy
BNatalia
D Akira
A42.002
C 40.202
B40.22
D 40.02
UNIT 1
Number Sense
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21
Read each problem. Write your answer.
SAMPLE The times of four runners in a 100-meter
sprint are shown in the table. What is the
order of runners as they finished the race?
Answer ________________________ Runner
Time (sec)
Ty
12.328
Nam
12.02
Will
12.325
Miguel
12.232
Compare the place values of each decimal. The tens and ones
places are the same. Nam’s time has a zero in the tenths place, so it
is the least time, and Nam is the fastest runner. Miguel has the next
fastest time because his time has a 2 in the tenths place and the
other two times have a 3 in that place. Both Will’s and Ty’s times
have a 2 in the hundredths place. But Will has the next fastest time
because his time has a 5 in the thousandths place, while Ty’s time
has an 8 in the thousandths place.
7Use each of the digits 7, 2, and 5 once to fill in the three boxes and
make the following comparison true.
Answer
.6 , 2. 9
8At the grocery store, Inez bought a loaf of bread for $2.65, a dozen
eggs for $1.99, milk for $2.19, and a small bag of apples for $2.60.
Order the prices from greatest to least.
Answer __________________________________________________________________________
9 Compare 34.509 and 34.59 using ,, ., or 5.
Answer ________________________________________________
10 Explain why 1.06 is less than 1.60.
_________________________________________________________________________________
_________________________________________________________________________________
22
UNIT 1
Number Sense
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Read the problem. Write your answer to each part.
11Marty visited the Insect Museum with his family. At the museum, he
learned that beetles come in many sizes, from tiny to enormous.
In the chart below, Marty recorded some of the sizes of different
beetle species he saw in a museum exhibit.
BEETLE LENGTHS
Beetle Species
Length (inches)
Flower beetle
0.167
Ladybug
0.092
Japanese beetle
0.6
Hercules beetle
6.75
Firefly
0.75
Part AWrite the beetles in order from shortest to longest.
Answer _________________________________________________________________
_________________________________________________________________________
Part BMarty’s sister thinks that the flower beetle is larger than
the firefly. She says that 0.167 is greater than 0.75
because 167 is greater than 75. Explain why Marty’s
sister is incorrect.
________________________________________________________
________________________________________________________
How do the
place values to the
left of a decimal
compare to the
place values to the
right of the decimal?
________________________________________________________
_________________________________________________________________________________
_________________________________________________________________________________
_________________________________________________________________________________
UNIT 1
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23
Lesson
5
Rounding Decimals
5.NBT.4
You round decimals the same way you round whole numbers.
Cassandra has a garden that measures 3.84 meters wide by
4.276 meters long. What is the width of the garden rounded
to the nearest whole number?
When you are asked
to round to the nearest
whole number, you
should round to the
ones place.
Round up if the digit
you are rounding is 5, 6,
7, 8, or 9.
Identify the place you are rounding to, the ones place: 3.84
Look at the digit to the right of the rounding place. In this case,
it is the digit in the tenths place: 3.84
If the digit to the right is 5 or more, change the digit in the
rounding place to the next greatest digit. That is, round up. If
the digit to the right is less than 5, leave the digit in the
rounding place as it is.
To the nearest tenth,
1.45 rounds up to 1.5.
Since 8 . 5, change 3 to 4.
3.84 rounds to 4.
Round down if the digit
is 0, 1, 2, 3, or 4.
To the nearest whole
number, 1.4 rounds
down to 1.
What is the width of the garden rounded to the nearest tenth?
Identify the rounding place: 3.84
Look at the digit to the right in the hundredths place: 3.84
Since 4 , 5, keep 8 as the digit.
3.84 rounds to 3.8.
What is the length of the garden rounded to the nearest
hundredth?
Identify the rounding place: 4.276
Look at the digit to the right in the thousandths place: 4.276
6 . 5, so change 7 to 8.
4.276 rounds to 4.28.
24
UNIT 1
Number Sense
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Read each problem. Circle the letter of the best answer.
SAMPLE Which of the following shows 54.834 rounded to the tenths place?
A 54.8
B 54.83
C 54.9
D 55
The correct answer is A. Find the place you are rounding to, the
tenths: 54.834. The number to the right is 3. Since 3 , 5, keep 8 as
the digit in the tenths place. So 54.834 rounds to 54.8. Choice B is
rounded to the hundredths place, not the tenths place. Choice C is
incorrectly rounded to the tenths place. Choice D is rounded to the
nearest whole number.
1The hummingbird moth is often mistaken
for a very small hummingbird. Clark
measured one of these moths to be
1.545 inches. What is the length of the
moth rounded to the nearest hundredth?
4After a big snowstorm, Toshi measures
2.405 ft of snow outside. What is this
amount rounded to the nearest hundredth?
A 2 ft
B 2.4 ft
A 1.5 inches
C 2.41 ft
B 1.54 inches
D 2.5 ft
C 1.55 inches
D 2 inches
5Which decimal rounds up when rounded to
the hundredths place?
2Which of the following decimals does not
round to 22.3?
A22.28
C 22.34
B22.31
D 22.39
A89.327
B89.372
C89.322
D89.732
3Kaitlin lives 5.38 km from the town library.
What is this distance rounded to the
nearest whole number?
6Lila’s dog weighs 12.95 lb. What is the
dog’s weight rounded to the nearest tenth?
A 5 km
A 12.0 lb
B 5.3 km
B 12.8 lb
C 5.4 km
C 12.9 lb
D 6 km
D 13.0 lb
UNIT 1
Number Sense
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25
Read each problem. Write your answer.
SAMPLE Milo finished a sprinting race in 32.962 seconds. What is Milo’s time
rounded to the nearest hundredth of a second?
Answer ________________________
The digit in the hundredths place is 6. The digit to the right of
the hundredths place is 2. Since 2 , 5, keep the 6 as it is. So 32.962
rounded to the nearest hundredth of a second is 32.96.
7Alberto is 1.371 m tall. His brother Ramon is 1.296 m tall. Rounded to
the nearest tenth of a meter, which brother is closer to 1.3 m in
height? Explain.
_________________________________________________________________________________
_________________________________________________________________________________
8Lian needs 4.82 yards of cloth for a project. The fabric store only sells
the cloth in whole-number amounts. How much fabric does Lian need
to buy?
Answer ________________________
9Explain how 1.556 can be rounded correctly to 2, 1.6, or 1.56.
_________________________________________________________________________________
_________________________________________________________________________________
_________________________________________________________________________________
_________________________________________________________________________________
10When rounding 72.631 to the hundredths place, Jack rounded to
72.64 and Tamar rounded to 72.63. Who is correct? Explain your
answer.
_________________________________________________________________________________
_________________________________________________________________________________
26
UNIT 1
Number Sense
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Read the problem. Write your answer to each part.
11Devon is preparing a pamphlet for a city housing agency. The
pamphlet tells residents about the choices for electric service.
Part ADevon’s pamphlet lists the five companies that supply
electricity in his city. To make the rates easier to
compare, Devon wants to round each rate to the
nearest hundredth.
Complete the table by rounding each decimal to the
hundredths place.
Electricity
Company
Rate Charged
(cents per
kilowatt-hour)
Atlas Energy
12.065
Big City Light
9.138
Connex
9.143
Dynamon Power
9.707
ElectriCo
9.503
Rate Rounded
to the Nearest
Hundredth
What place
do you need to
look at to round
to hundreths?
According to the rounded rates, which company supplies
energy at the lowest rate?
Answer _________________________________________________________________
Part BExplain how two companies that charge different rates can
have the same rate when their rates are rounded to hundredths.
_________________________________________________________________________________
_________________________________________________________________________________
_________________________________________________________________________________
_________________________________________________________________________________
_________________________________________________________________________________
UNIT 1
Number Sense
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27
R e vi e w
Number Sense
Read each problem. Circle the letter of the best answer.
1What is the exponent in the expression
5 What is 30.509 in expanded form?
7
1.425 3 10 ?
A0.425
C 7
B1.425
D 10
2What is the product of 0.011 3 102?
A1.1
B1.11
3What is 4,020,003 in expanded form using
exponents?
A(4 3 109 ) 1 (2 3 107) 1 (3 3 101)
B(4 3 107) 1 (2 3 106) 1 (3 3 101)
C(4 3 106) 1 (2 3 105) 1 (3 3 100 )
D(4 3 106) 1 (2 3 104) 1 (3 3 100 )
4What is the standard form of (9 3 107) 1
(8 3 105) 1 (5 3 104)?
A98,500,000
B90,850,000
C90,085,000
D9,850,000
28
)
1
)
)
1
)
1
B(3 3 10) 1 (5 3  
​  10  ​ 1 (9 3 ​  
   ​
100
1
)
1
)
)
1
)
C(3 3 10) 1 (5 3  
​  100   ​ 1 (9 3 ​  
   ​
1,000
C 0.11
D 0.022
1
A(3 3 10) 1 (5 3  
​  10  ​ 1 (9 3 ​  
   ​
1,000
1
D(3 3 100) 1 (5 3  
​  10  ​ 1 (9 3 ​  
   ​
1,000
6 Which decimal is less than 70.8?
A78.008
C 70.808
B70.88
D 70.08
7Which decimal shows 8.566 rounded to
the hundredths place?
A8.5
C 8.57
B8.56
D 8.6
8The mass of a soccer ball is 0.43 kg. What
is a soccer ball’s mass rounded to the
nearest tenth?
A 0.4 kg
C 0.5 kg
B 0.43 kg
D 1 kg
UNIT 1
Number Sense
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Read each problem. Write your answer.
9What happens to the decimal point in the product of
9.87654321 3 106? What is the product?
Answer __________________________________________________________________________
10Ann wrote 8,102,074 in expanded form as shown here.
(8 3 1,000,000) 1 (1 3 100,000) 1 (2 3 10,000) 1 (7 3 10)
What two errors did she make?
_________________________________________________________________________________
_________________________________________________________________________________
_________________________________________________________________________________
1
1
1
11What is (7 3 1) 1 (5 3  
​  10  ​) 1 (3 3 ​  
   ​ 1 (2 3 ​  
   ​ in standard
100 )
1,000 )
form?
Answer ________________________
12The distances flown by five model
airplanes in a student contest are
shown in the table. What is the
order of the planes from farthest
to shortest distance flown?
Model
Airplane
Distance
in Meters
N–1
16.328
N–2
6.34
N–3
23.59
N–4
12.23
N–5
16.21
Answer __________________________________________________________________________
13When rounding 83.741 to the hundredths place, Karl rounded to 83.75
and Nell rounded to 83.74. Who is correct? Explain your answer.
_________________________________________________________________________________
_________________________________________________________________________________
UNIT 1
Number Sense
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29
Read each problem. Write your answer to each part.
14A solar-powered airplane flew for 24 hours to show that it could
collect enough energy from the sun during the day to stay in the air all
night. The plane weighed 1,587.573 kilograms.
Part A Write 1,587.573 in expanded form.
Answer _________________________________________________________________
_________________________________________________________________________
Part B In 1,587.573, there are two 5’s. How are their values related?
_________________________________________________________________________
_________________________________________________________________________
15Gabriel records the times of the track
runners at the summer meet. He uses
a watch that measures time to the
thousandths place. When Gabriel
posts the runners’ times on the
scoreboard, he rounds each time
to the hundredths place.
Runner
Raw Running
Time (sec)
Eric
58.210
Liz
58.217
Nina
57.896
Pat
59.067
Tamara
58.222
Posted Running
Time Rounded to
Hundredths
Part AComplete the table by rounding the times to the hundredths
place. According to the posted times, which runners took
first, second, and third places in the meet?
Answer ________________________________________________
Part BExplain how two runners can have the same time when their
times are rounded to hundredths.
_________________________________________________________________________
_________________________________________________________________________
_________________________________________________________________________
30
UNIT 1
Number Sense
© The Continental Press, Inc. DUPLICATING THIS MATERIAL IS ILLEGAL.
U n it
2
Operations
Lesson 1 Multiplying Whole Numbers reviews
how to multiply to find the product of two whole
numbers.
Lesson 2 Dividing Whole Numbers reviews how
to divide to find the quotient of two whole numbers.
Lesson 3 Adding and Subtracting Decimals
reviews how to add and subtract decimals.
Lesson 4 Multiplying Decimals reviews how to find
the product of two decimals or of a decimal and a whole
number.
Lesson 5 Dividing Decimals reviews how to find
the quotient of two decimals or of a decimal and a whole
number.
UNIT 2
Operations
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31
Lesson
1
ying Whole Numb
l
p
i
t
l
u
M
ers
5.NBT.5
A rectangular model can help you picture multiplication.
A bakery used 11 dozen eggs one day. How many eggs did the
bakery use?
The commutative
property applies
to multiplication.
Changing the order
of the numbers being
multiplied does not
change the result.
To find the total number of eggs used, multiply the number of
eggs in a dozen, 12, by the number of dozens used: 11 3 12. A
row of 12 dots represents a dozen, and 11 rows show 11 dozen.
Remember that 11 is the same as 10 1 1, so 11 3 12 is the same
as 10 3 12 1 1 3 12.
12
11 3 12 5 132
12 3 11 5 132
11
1 12 12
10 12 120
12 120 132 eggs
In a multiplication
problem, the numbers
you multiply are called
factors. The result is
called the product.
The associative
property applies to
multiplication. The
order that the factors
are grouped in for
multiplication does not
change the result.
(2 3 6) 3 3 5 2 3 (6 5 3)
32
A vertical multiplication problem is a faster way to find a product.
12
311
12
First, multiply 12 by the ones digit of 11.
Multiply 1 3 12. This results in a partial
product of 12.
12
311
12
120
Then multiply 12 by the tens digit of 11.
Multiply 10 3 12. This results in a partial
product of 120.
12
311
12
120
132
Finally, add the partial products to find the
product of 11 3 12.
The product of 11 3 12 is 132.
UNIT 2
Operations
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Read each problem. Circle the letter of the best answer.
SAMPLE The new school gym will be 38 meters long and 27 meters wide. How
many square meters of wood flooring will be needed for the gym?
A 342 m2
B 976 m2
C 926 m2
D 1,026 km2
The correct answer is D. Multiply the length by the width to find
the area of the gym floor: 27 3 38. First, multiply by the ones: 7 3
38 5 266. Multiply by the tens: 20 3 38 5 760. Then add the partial
products: 266 1 760 5 1,026. A total of 1,026 square meters of
flooring are needed.
1 What is 26 3 13?
5An adult human heart beats an average of
A2
C 328
B104
D 338
2A hotel ballroom holds 144 banquet tables.
Each table seats 8 people. How many
people can be seated in the ballroom?
A18
C 1,152
B1,122
D 1,172
3 Find the product of 41 3 19.
A20
C 779
B60
D 800
4On the highway, Bianca’s car can travel
34 miles on a gallon of gas. Its gas tank
holds 17 gallons. How far can the car travel
on a single tank of gas?
A 2 miles
C 578 miles
B 272 miles
D 648 miles
72 times per minute. With each beat it
pumps 70 milliliters of blood. How many
milliliters of blood does the average human
heart pump in 1 minute?
A630
C 4,320
B504
D 5,040
6 Multiply 37 3 73.
A530
C 2,701
B2,121
D 5,329
7A spool holds 45 yards of ribbon. How
many inches of ribbon does it hold?
A405
C 1,590
B540
D 1,620
8The school band is raising money for a trip.
The cost is $137 for each student. There
are 89 students. How much money must
be raised to pay for the trip?
A$979
C $11,693
B$2,169
D $12,193
UNIT 2
Operations
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33
Read each problem. Write your answer.
SAMPLE A school orders 3 cases of blank CDs. Each case holds 16 boxes. Each
box holds 72 CDs. How many CDs did the school order?
Answer ________________________
To find the total number of CDs, first find the number of CDs in
each case. Multiply the number of CDs in each box by the number
of boxes: 16 3 72 5 1,152. Next, multiply the number of CDs per
case by the number of cases the school ordered: 3 3 1,152 5 3,456.
The school ordered 3,456 CDs.
9Scientists are designing a space probe to study Mercury. They would
like the probe to orbit Mercury while the planet makes 3 full
revolutions around the sun. Mercury orbits the sun in 88 days. How
many days will the probe need to orbit Mercury?
Answer ____________
10Derek carefully built a tower out of playing cards. He used 28 full
decks of cards to build it. There are 52 cards in a deck. How many
cards did Derek use to make his tower?
Answer ________________________
11At practice yesterday, Ji Sun ran 18 laps around her school’s track. The
track is 402 meters long. How many meters did Ji Sun run?
Answer ________________________
12On average, a fifth grader blinks 15 times per minute. About how
many times will a fifth grader blink in a day?
34
Answer ________________________
UNIT 2
Operations
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Read the problem. Write your answer to each part.
13A giant panda may eat as much as 14 kilograms of bamboo in a single
day.
Part AHow many kilograms of bamboo will a giant panda eat
in a year (not a leap year)? Explain how you found the
answer.
There are 365
days in a year that
is not a leap year.
_________________________________________________________________________________
_________________________________________________________________________________
_________________________________________________________________________________
Part BTokyo’s Ueno Zoo received a pair of giant pandas from China
in 2011. How many kilograms of bamboo does the zoo need
to have to be prepared to feed the pandas for any single
month? Explain how you found your answer.
_________________________________________________________________________________
_________________________________________________________________________________
_________________________________________________________________________________
_________________________________________________________________________________
_________________________________________________________________________________
_________________________________________________________________________________
UNIT 2
Operations
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35
Lesson
2
Whole Numb
g
n
i
d
i
v
i
D
ers
5.NBT.6
A rectangular model can also help you picture division.
In a division
problem, the number
being divided is the
dividend. The number
you are dividing by is
the divisor. The result is
the quotient.
There are 204 children signed up to play in a weekend softball
league and 17 parent coaches. If each parent coaches 1 team,
how many players will be on each team?
To find the number of players on each team, divide the number
of children by the number of coaches: 204 4 17. Use a star to
represent each child. Arrange 204 stars into 17 columns—one
column for each coach.
17
There are 12 rows, so each
team will have 12 players.
12
A vertical division
problem is also called
long division.
Round the divisor to
estimate where to start
the quotient.
17 is about 20, so the
quotient will start in
the tens place.
36
A vertical division problem is a faster way to find the quotient.
1
17 204
Since 2 cannot be divided by 17, move one
place to the right and divide 20 4 17.
1
17 204
17 3
Multiply: 17 3 1 5 17
Subtract: 20 2 17 5 3
12
17 204
17↓
34
34
Bring down the 4.
Then divide: 34 4 17 5 2
Each team has 12 players.
UNIT 2
Operations
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Read each problem. Circle the letter of the best answer.
SAMPLE At a college graduation, equal rows of chairs are set up for
1,634 seniors. In every row, there are 19 chairs on each side of a
center aisle. How many rows of chairs are there?
A 43
B 58 C 86
D 172
The correct answer is A. There are 19 chairs on
each side of the aisle, so there are 2 3 19 5 38 chairs
in each row. To find the number of rows, divide the
number of seniors by the number of chairs per row:
1,634 4 38. Round 38 to 40 to estimate where to start
the quotient: 40 does not divide into 16, but it does
into 160. So, start in the tens place. Divide, multiply,
subtract, and continue. There will be a total of 43 rows.
1Nicole’s paycheck last week was for $648.
She worked 36 hours that week. How
much is Nicole paid per hour of work?
A$13
C $17
B$14
D $18
2A popular hanging lamp has 96 crystals.
The lamp factory just received a delivery of
2,784 crystals. How many hanging lamps
can be made with the crystals?
A23
C 29
B27
D 33
3Jupiter takes 4,332 Earth days to orbit the
sun. Mercury orbits the sun in 88 Earth
days. How many complete orbits will
Mercury make before Jupiter completes
one orbit?
A48
C 50
B49
D 62
43
38 1,634
1 52 114
114
4Riders in the 2010 Tour de France bicycled
about 3,580 km. The race began on July 4
and ended on July 25. Riders had two days
off during the tour. On average, how many
kilometers did the riders bike on each day
they raced?
A163
C 179
B170
D 188
5An army cook wants to make pudding
for dessert. He has a recipe that calls for
2 cups of milk to make 4 servings. There
are 16 cups in a gallon. How many gallons
of milk does the cook need to make
864 servings of pudding?
A14
C 54
B27
D 108
6 Find 924 4 28.
A27
C 36
B33
D 28,314
UNIT 2
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37
Read each problem. Write your answer.
SAMPLE A river is threatening to flood a small town. Volunteers are filling
sandbags and piling them along a riverbank. Each full sandbag weighs
about 37 pounds. A truck unloads 3,150 pounds of sand. About how
many sandbags can be filled from this load of sand?
Answer ____________
To find the approximate number of sandbags that can be filled,
divide the total pounds of sand by the average weight of a full
sandbag: 3,150 4 37 5 85 with a remainder of 5. The remainder of
5 pounds will not come close to filling a 37-pound bag, so do not
round up from 85 sandbags to 86. About 85 sandbags can be filled
from the truckload of sand.
7The area of a soccer field is 7,140 m2. The width of the field is 68 m.
How many meters long is the field?
Answer ____________
8A scientist using a special camera observes that a hummingbird’s wings
beat 4,680 times a minute. On average, how many times per second
does the bird beat its wings?
Answer ____________
9A group of 266 people is called to jury duty in court. Each jury
includes 12 jurors plus 2 alternates. How many complete juries could
be selected from the pool?
Answer ____________
10Elliot is making flower arrangements for a banquet. The 36 tables will
all have identical centerpieces. Elliott receives a shipment of 48 dozen
white roses. How many roses can he include in each centerpiece?
38
Answer ____________
UNIT 2
Operations
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Read each problem. Write your answer to each part.
11In order to lose 1 pound, a person needs to burn about
3,600 more calories than he or she eats and drinks. A person
who weighs 155 pounds burns about 240 calories per hour by
walking at a moderate pace.
Part AAbout how many minutes would a 155-pound
person have to walk at a moderate pace in order
to lose 1 pound?
How many
minutes are equal
to 1 hour?
Answer ____________
Part B Explain how you found your answer.
_________________________________________________________________________________
_________________________________________________________________________________
_________________________________________________________________________________
_________________________________________________________________________________
12 Lionel weighs 155 pounds. He starts a walking program.
Part AHow many minutes a day would Lionel have to walk at a
moderate pace in order to lose 1 pound in 30 days?
Answer ____________
Part B Explain how you found your answer.
_________________________________________________________________________________
_________________________________________________________________________________
_________________________________________________________________________________
UNIT 2
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39
Lesson
3
Adding and Subtracting
Decimals
5.NBT.7
If you know how to add whole numbers, you can add decimals. The
process is the same. Remember to line up the decimal points in both
numbers vertically so that all the digits are correctly aligned by place
value.
Shana finished the 200-meter race 0.5 second faster than Cindy.
Shana’s time was 25.63 seconds. What was Cindy’s time?
Cindy took more time than Shana. To find Cindy’s time, add
the extra time it took Cindy to finish to Shana’s time. Annex
zeros as needed. Then add from right to left.
When you annex
zeros to the beginning
or the end of a decimal,
it does not change the
value of the decimal.
Add the
hundredths.
25.63
1 0.50
3
Add the tenths. Add the ones.
Regroup the 1.
1
25.63
1 0.50
13
Add the tens.
1
1
25.63
1 0.50
6.13
25.63
1 0.50
26.13
The sum is 26.13. Cindy’s time was 26.13 seconds.
Subtract decimals the same way you subtract whole numbers. Align
the decimal points in order to subtract digits in the same places.
The winner of the 100-meter race finished 0.19 second before
Shana. What was the winner’s time?
If the value of a place
is not great enough to
subtract from, regroup
the next place to the
left.
The winner took less time than Shana. To find the winner’s
time, subtract the extra time it took Shana to finish from
Shana’s time, 25.63 seconds.
Subtract the
hundredths.
Regroup.
5 13
25.63
2 0.19
4
Subtract the
tenths.
5 13
25.63
2 0.19
44
Subtract the
ones.
5 13
25.63
2 0.19
5.44
Subtract the
tens.
5 13
25.63
2 0.19
25.44
The difference is 25.44. The winner’s time was 25.44 seconds.
40
UNIT 2
Operations
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Read each problem. Circle the letter of the best answer.
SAMPLE The table at the right shows average
monthly rainfall data for Buffalo, NY.
How many total inches of rain fall on
Buffalo in the three summer months?
A 9.8
B 9.9
Month
Average
Monthly Rainfall
June
3.8 in.
July
3.1 in.
August
3.9 in.
C 10.8 D 10.9
The correct answer is C. To find the total rainfall for the
summer months, find the sum of the rainfall for all three months:
3.8 1 3.1 1 3.9 5 10.8 in. A total of 10.8 inches of rain fall.
1 Find the sum of 3.28 1 0.41.
A3.321
C 3.79
B3.69
D 7.38
2 Solve 3.76 2 0.32.
A0.44
C 3.44
B0.56
D 5.08
6Tuesday’s low temperature of 50.7°F came
just before dawn. During the day, the
temperature increased 23.9°F. What was
Tuesday’s high temperature?
A26.8°F
C 74.6°F
B73.6°F
D 84.6°F
7A quart of orange juice costs $2.85. Lynn
has a coupon for $0.49 off. With the
coupon, how much will Lynn pay?
342.6 1 8.75 5 h
A1.301
C 50.35
A$2.35
C $2.46
B13.01
D 51.35
B$2.36
D $3.34
8In 2010, the population of California was
417.04 2 4.2 5 h
A12.74
C 13.84
B12.84
D 16.62
39.14 million people. By 2050, it is
expected to reach 59.5 million. By how
many people is the population expected to
grow between 2010 and 2050?
5 Find the difference of 808.3 2 79.82.
A 20.36 million
C 33.19 million
A10.1
C 728.52
B 20.44 million
D 98.64 million
B728.48
D 738.48
UNIT 2
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41
Read each problem. Write your answer.
SAMPLE The table at the right shows
Country
2010 population data for the
three largest countries in
North America.
2010 Population
Canada
33.89 million
Mexico
110.65 million
United States
317.64 million
How many more people live in
the United States than in Mexico and Canada combined?
The problem asks you to find the difference between the
population of the United States and the combined populations of
Canada and Mexico: 317.64 2 (33.89 1 110.65) 5 317.64 2 144.54 5
173.1 million people.
9Justin buys a notebook for $2.99, a highlighter for $0.75, and 2 pens
for $0.20 each. How much money did he spend?
Answer ________________________
10A real estate developer buys three vacant lots next to each other. One
lot is 27.5 meters wide, the next lot is 48 meters wide, and the last lot
is 33.75 meters wide. If the developer combines the lots, how wide
will the combined lot be?
Answer ________________________
11Every month Marina pays $79 for mobile phone service, $59.95 for
cable service, and $45.50 for Internet service. She can switch to a plan
that charges $129.99 a month for all three services. On that plan, how
much money would Marina save monthly?
Answer ________________________
12Explain why you can annex zeros to the beginning and end of a
decimal.
_________________________________________________________________________________
_________________________________________________________________________________
42
UNIT 2
Operations
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Read each problem. Write your answer to each part.
13Amir has $65. On his drive home, he stops at the gas station. Amir
spends $47.72 to fill his gas tank and $3.39 to buy a gallon of milk.
Part AHow much money does Amir have left in his wallet after
he leaves the gas station? Show your work.
Remember to
align digits by place
value and to annex
zeros as needed.
Answer ________________________
Part BIn part A, what operation did you use to find the amount of
money Amir had left? Use the information from the problem,
including your answer to part A, to write a word problem
that is solved by using the inverse operation.
_________________________________________________________________________________
_________________________________________________________________________________
_________________________________________________________________________________
14The Men’s 4 3 100-Meters Freestyle Relay is an Olympic event in
which each member of a 4-person team swims 100 meters. In the
2008 Summer Olympics, a U.S. team won the gold medal with a
combined time of 3 min 8.24 s. The first three swimmers’ times were
47.51 s, 47.02 s, and 47.65 s.
Part A What was the fourth swimmer’s time?
Answer ________________________
Part B Explain how you found your answer.
_________________________________________________________________________________
_________________________________________________________________________________
_________________________________________________________________________________
UNIT 2
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43
Multiplying Decimals
Lesson
4
5.NBT.7
You can use a grid to model multiplication of decimals.
A grid is one kind of
rectangular model.
Find the product of 0.7 3 0.3.
3
10
On a 10-by-10 grid, shade 7 rows
7
one color to represent ​  
  ​, or 0.7.
10
Then shade 3 columns a second
3
color to represent ​  
  ​, or 0.3.
10
0.3
7
10
0.7
There are 21 squares where the shading overlaps. There are
100 squares in the whole grid. So, the overlap area is described
21
by the fraction  
​ 100  ​.
21
0.7 3 0.3 5  
​  100  ​ 5 0.21
When you multiply
decimals, estimate the
answer before you
calculate. This will help
you to make sure you
place the decimal point
in the correct place in
the product.
When you multiply
3.2 3 49.3, you are
multiplying a factor
to the tenths place
by another factor to
tenths.
1
1
1
 
  ​ 5​  
   ​
​  10  ​ 3​  
10
100
So, your product should
be to hundredths.
44
You can multiply decimals the same way you multiply whole numbers.
Baby Alec is 49.3 cm tall. His sister Ava is 3.2 times as tall as he
is. How tall is Ava?
Then multiply
3 3 493.
First multiply
2 3 493.
49.3
33.2
98 6
49.3
33.2
98 6
1479 0
Add the products.
49.3
33.2
98 6
11479 0
1577 6
The product will have the same number of decimal places as
the sum of decimal places in the factors. There are two. So,
place the decimal point two digits from the right: 157.76.
Ava is 157.76 cm tall.
UNIT 2
Operations
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Read each problem. Circle the letter of the best answer.
SAMPLE An old assembly line at an auto plant produces 33.6 cars per hour. A
new, improved assembly line will produce 1.8 times as many cars. How
many cars will the new assembly line produce in an hour?
A 30.24
B 60.38
C 60.48 D 604.8
The correct answer is C. To find the number of cars produced per
hour by the new assembly line, multiply the number produced per
hour by the old assembly line by 1.8: 1.8 3 33.6 5 60.48. Each of
the factors has one digit to the right of the decimal point so the
product has two digits to the right of the decimal point.
1 Multiply 1.4 3 5.2.
6The price of red bell peppers is $2.89 per
A2.60
C 72.8
B7.28
D 728
2 What is 7 3 0.42?
A0.06
C 2.94
B2.84
D 29.4
3 Find the product of 2.4 3 29.6.
pound. Maneesh selects 3 pounds of
peppers. How much will the peppers cost?
A$6.67
C $8.67
B$8.47
D $86.70
7The adult dose of a medication is 260 mg.
The recommended dose for children is
0.75 times the adult dose. What is the
recommended dose for children?
A16.76
C 71.04
A 1.95 mg
C 155 mg
B61.04
D 710.4
B 19.5 mg
D 195 mg
8Dana installed a new toilet that uses
41.3 3 268 5 h
A9.72
C 97.20
B34.84
D 348.4
5 Find the product of 27.9 3 70.8.
A197.532
C 1,975.32
B214.62
D 2,146.2
1.6 gallons of water per flush. Her old toilet
used 3.1 times as much water. How much
water did Dana’s old toilet use per flush?
A 3.96 gal
C 6.40 gal
B 4.96 gal
D 49.6 gal
UNIT 2
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45
Read each problem. Write your answer.
SAMPLE The sneakers Joelle wants to buy normally cost $45. This week the
sneakers are on sale. The sale price is 0.7 times the regular price. What
is the sale price of the sneakers?
Answer ________________________
To find the sale price, multiply the regular price by the rate, 0.7:
0.7 3 45 5 31.5. The factors have a total of one decimal place, so
insert the decimal point one place from the right. But because
money is normally written to hundredths (cents), annex a zero to
the end: $31.50.
9Last August 8.8 inches of rain fell in Miami. This January it received
0.2 times as much rain as in August. How much rain fell in Miami in
January?
Answer ________________________
10The subway fare is $2. The transit authority has proposed a fare
increase. With the increase, the new fare would be 1.15 times the
current fare. What is the proposed new fare?
Answer ________________________
11David takes a photo that is 17 cm wide and enlarges it on the
photocopier. The copy is 1.3 times as wide. Then David takes the
enlarged copy and enlarges it 1.3 times again. How wide is the final
copy of the photo?
Answer ________________________
12Explain where to place the decimal point in the product 28 3 1.32.
_________________________________________________________________________________
_________________________________________________________________________________
_________________________________________________________________________________
46
UNIT 2
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Read each problem. Write your answer to each part.
13Last summer, Arturo planted a vegetable garden that was 18 feet wide
and 24.5 feet long.
Part AWhat was the area of Arturo’s vegetable garden last
summer? Show your work.
Answer ________________________
Before you
multiply decimals,
estimate the
product. Your
estimate will help
you determine if
you have placed the
decimal point in the
correct place in the
product.
Part BThis year, Arturo has less time to garden. His vegetable
garden is only 0.25 times the size of last year’s garden.
What is the area of Arturo’s vegetable garden this
summer? Show your work.
Answer ________________________
14The average person in the United States consumes 0.2 kg of tea per
year. The average person in China consumes 2 times that amount. And
the average person in Turkey consumes 4 times as much tea as the
average person in China.
Part AHow much tea does the average person in Turkey consume
in a year?
Answer ____________
Part BTo find the answer, Eva multiplies 2 3 4 5 8 and then
multiplies 0.2 by 8. Will she get the same answer? Explain
why.
_________________________________________________________________________________
_________________________________________________________________________________
_________________________________________________________________________________
UNIT 2
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47
Lesson
5
Dividing Decimals
5.NBT.7
If you know how to divide whole numbers, you can divide decimals.
At the start of a science experiment, a pea plant was 1.8 cm tall.
Two weeks later, the plant was 4.68 cm tall. How many times
taller was the plant at the end of two weeks?
To find how many times taller the plant was after two weeks,
divide the plant’s end height by its beginning height: 4.68 4 1.8.
In division, the
number you are
dividing is called the
dividend. The number
you are dividing by is
the divisor. And the
result is called the
quotient.
Write the problem vertically, and change the
1.8.  4.68
decimals to whole numbers. To do that, move

the decimal point in the divisor one digit to the right.
Moving the decimal
point one place to
the right is the same
as multiplying by 10.
When both numbers
are multiplied by 10,
the quotient will be
the same as before the
points were moved.
The decimal point in the quotient should go directly above
where you just placed the decimal point in the dividend. Now
you can do the computation to find the quotient.
5.0 4 0.5 5 10
50 4 5 5 10
Multiplication and
division are inverse
operations. Multiplying
by a number is the
inverse of dividing by
that number.
48
Because you moved the decimal point in the
divisor one digit to the right, you must also
move the decimal point in the dividend one
digit to the right.
First, divide:
46 4 18.
2.
18  46.8
Next, multiply:
2 3 18.
Then subtract.
2.
18  46.8
36 10
Bring down
the 8. Divide:
108 4 18.
2.6
18  46.8
36 ↓
10 8
1.8.  4.6.8

Multiply: 6 3 18.
Then subtract.
2.6
18  46.8
36 10 8
10 8
0
After two weeks, the pea plant is 2.6 times as tall as it was
when the experiment began.
You can use multiplication to check your answer.
1.8 3 2.6 5 4.68
UNIT 2
Operations
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Read each problem. Circle the letter of the best answer.
SAMPLE Emilio needs a new desk. He finds one online with a height of
27.3 inches. One centimeter equals 0.39 inch. How many centimeters
tall is the desk?
A 10.65
B 70 C 106.5
D 700
The correct answer is B. To find the height of the desk in
centimeters, divide the height in inches by 0.39: 27.3 4 0.39. First,
change the numbers from decimals to whole numbers. The divisor,
0.39, has two digits to the right of the decimal point, so move the
decimal point in both the divisor and dividend two places to the
right to make 2,730 4 39 5 70. The decimal point in the quotient
goes directly above the decimal point in 2,730.
1 Divide 28 4 1.4.
6The price of plantains is $0.89 per pound.
A2
C 20
B3.92
D 200
2 What is the quotient of 0.36 4 9?
A0.04
C 3.24
B0.4
D 4.0
34.32 4 12 5 h
Kara spends $4.45 on plantains. How many
pounds of plantains did she buy?
A0.5
C 5
B3.96
D 50
7The regular price of a sweater is $32. This
week the sweater is on sale for $20.80.
How many times the regular price is the
sale price?
A0.036
C 3.6
A0.65
C 6.5
B0.36
D 51.84
B1.54
D 15.4
8Workers installed 1,054.9 square yards of
4 Find 666 4 3.7.
A0.18
C 18
B1.8
D 180
5 Solve 885.8 4 20.6.
A4.3
C 430
B43
D 4,300
carpeting in a meeting room. The room is a
rectangle 38.5 yards long. How many yards
wide is the room?
A25.1
C 27.5
B27.4
D 274
UNIT 2
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49
Read each problem. Write your answer.
SAMPLE For adults and school-age children, the recommended amount of
the mineral potassium is 4.7 grams per day. A banana contains
400 milligrams of potassium. How many bananas would a person
have to eat in a day to get the recommended amount of potassium?
Answer ____________
Before you can divide, the measurements need to be in the same
units. There are 1,000 milligrams in 1 gram. Convert 400 milligrams
into grams: 400 4 1,000 5 0.4 g. Now, divide the recommended
daily amount of potassium by the amount in one banana: 4.7 4
0.4 5 11.75 bananas. The divisor had one digit to the right of the
decimal point, so the decimal points in the dividend and the
quotient both were moved one digit to the right.
9China produces about 196 million metric tons of rice per year. The
Philippines produces about 16 million metric tons of rice per year. How
many times the Philippines’ yearly rice production does China produce?
Answer ________________________
10The gas tank in Felix’s car holds 23.5 gallons. Felix can drive 423 miles
on one full tank. How many miles per gallon (mpg) of gas does Felix’s
car get?
Answer ____________
11When you divide a number by a decimal less than 1, is the quotient
less than or greater than the original number?
Answer ________________________
12Traci needs to change $300 U.S. dollars into Euros. The exchange rate is
1 Euro for 1.25 dollars. How many Euros does Traci get for her dollars?
50
Answer ____________
UNIT 2
Operations
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Read each problem. Write your answer to each part.
13In 2010, Cool Game Corporation made a profit of $84.7 million.
The profit made in 2010 was 2.2 times the profit the company
made in 2009.
Part AWhat profit did Cool Game Corporation make in 2009?
Show your work.
Answer ________________________
Part BExplain why you can move the decimal point in the
divisor to the right, so long as you also move the
decimal point in the dividend the same number of
places to the right.
When you move the
decimal point in a number,
you are multiplying by a
power of 10.
_________________________________________________________________________________
_________________________________________________________________________________
_________________________________________________________________________________
_________________________________________________________________________________
14Mori just replaced the showerhead in her bathroom. The old one
used 5.75 gallons per minute (gpm) of water. Her new showerhead
uses 2.5 gpm.
Part AHow many times as much water did Mori’s old
showerhead use?
Answer ________________________
Part BExplain how you found your answer.
_________________________________________________________________________________
_________________________________________________________________________________
_________________________________________________________________________________
UNIT 2
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51
R e vi e w
Operations
Read each problem. Circle the letter of the best answer.
1 Solve 1,026 4 27.
6In August 6.35 inches of rain fell. In
A34.3
C 45
B38
D 27,702
September, 10.2 inches fell. In both
October and November, 5.85 inches fell.
How much rain fell from August through
November?
2Find the difference of 86.19 2 17.2.
A 13.22 inches
C 22.4 inches
A68.99
C 78.99
B 19.07 inches
D 28.25 inches
B69.99
D 84.47
7Last year, the price of a ticket to a baseball
3 What is 19.3 3 40.7?
A70.71
C 585.51
B78.56
D 785.51
4198 4 0.8 5 h
A2.475
C 158.4
B24.75
D 247.5
5Enrico’s paycheck for last week was for
$663. He is paid $17 an hour. How many
hours did Enrico work last week?
A37.25
C 39
B38
D 40
52
game was $48. Many tickets went unsold.
This year ticket price has been reduced to
0.85 times the old price. What is the price
of a ticket to a baseball game this year?
A$34.80
C $40.80
B$40.40
D $56.47
8A cow on a farm produced 165.2 gallons
of milk in 1 week. On average, how many
gallons of milk did the cow produce each
day that week?
A2.36
C 23.6
B22.6
D 226
UNIT 2
Operations
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Read each problem. Write your answer.
9On average, an adult at rest breathes in and out 12 times per minute.
About how many breaths will an adult at rest take in a day?
Answer ________________________
10Together Carla and Yvette drove from New York City to San Diego.
The 2,808-mile trip took them two weeks. Each week, they took one
full day off from driving. On average, how many miles a day did Carla
and Yvette travel on the days that they drove?
Answer ____________
11The table at the right shows 2010 population data
for the three most populous nations in the world.
Do more people live in China or in India and the
United States combined? Show your work.
Country
2010 Population
China
1.34 billion
India
1.18 billion
United States
0.3 billion
Answer ________________________
12Adam reduces a 14-inch long document on the photocopier. The copy
is 0.6 times as long. The type on the reduced document is too small to
read, so Adam enlarges the reduced copy 1.4 times. How long is the
final copy of the document?
Answer ________________________
13Explain where to place the decimal point in the product 1.2 3 98.4.
_________________________________________________________________________________
_________________________________________________________________________________
UNIT 2
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53
Read each problem. Write your answer to each part.
14The regular price of a DVD is $26. This week the DVD is on sale for
$19.50.
Part AHow many times the regular price is the sale price of the
DVD? Show your work.
Answer ________________________
Part BExplain how you knew where to place the decimal point in
your answer to part A.
_________________________________________________________________________________
_________________________________________________________________________________
_________________________________________________________________________________
_________________________________________________________________________________
15At track practice, Trina ran 8.5 miles in 41 minutes 22 seconds.
Part AOn average, how long did it take Trina to run each mile?
Show your work.
Answer ________________________
Part BExplain how you found your answer.
_________________________________________________________________________________
_________________________________________________________________________________
_________________________________________________________________________________
_________________________________________________________________________________
54
UNIT 2
Operations
© The Continental Press, Inc. DUPLICATING THIS MATERIAL IS ILLEGAL.
U n it
3
Adding and Subtracting
Fractions
Lesson 1 Equivalent Fractions reviews how to find
different fractions that name the same value and how to
reduce a fraction to lowest terms.
Lesson 2 Adding and Subtracting Fractions
reviews how to add and subtract fractions with unlike
denominators.
L esson 3 Adding and Subtracting Mixed
Numbers reviews how to add and subtract mixed
numbers with unlike denominators.
UNIT 3
Adding and Subtracting Fractions
© The Continental Press, Inc. DUPLICATING THIS MATERIAL IS ILLEGAL.
55
Equivalent Fractions
Lesson
1
5.NF.1
Two fractions are
equivalent when you
can cross multiply the
fractions and get the
same values.
a
b
c
d
​    ​, then ad 5 bc.
If  
​   ​5  
You can cross multiply
the fractions to
prove that they are
equivalent.
1
Equivalent fractions are fractions that have the same value or
represent the same part of a whole.
To find equivalent fractions in higher terms, multiply the numerator
and denominator by the same nonzero number.
1
Jorge ate ​  
 ​of a sweet potato pie. Find a fraction equivalent
4
1
to  
​  4 ​in higher terms.
Multiply the numerator and denominator by the same nonzero
number.
2
2
1
4
 ​ 
​  4 ​ 5​ 

8
1385432
858
2
8
2
1
4
2
8
To find equivalent fractions in lower terms, divide the numerator and
denominator by the same nonzero number.
6
The greatest common
factor is the largest
factor that two or more
numbers share.
Elise walked ​  
 ​of a mile with her dog. Find a fraction equivalent
9
6
to  
​  9 ​in lower terms.
Divide the numerator and denominator by the same nonzero
number. The number you use must be a factor of both the
numerator and the denominator.
Factors of 10: 1, 2, 5, 10
Factors of 15: 1, 3, 5, 15
Common factors: 1 and 5
GCF: 5
3
6
9
2
3
3
6
9
2
3
To write a fraction in lowest terms, divide the numerator and
denominator by their greatest common factor (GCF). A
fraction is in lowest terms when the numerator and
denominator only have a common factor of 1.
56
UNIT 3
Adding and Subtracting Fractions
© The Continental Press, Inc. DUPLICATING THIS MATERIAL IS ILLEGAL.
Read each problem. Circle the letter of the best answer.
2
SAMPLE Which of the following fractions is equivalent to ​  
 ​?
5
6
4
A​  
  ​
15
B​  
 ​
7
3
4
C​  
 ​
6
D​  
 ​
5
The correct answer is A. To find an equivalent fraction, multiply the
2
numerator and denominator of  
​ 5 ​by the same number to see which
answer choice is a possible product. There is no number that results
3
233
6
in choices B, C, or D. Multiplying by  
​ 3 ​gives choice A: ​  
 
 ​ 5  
​  15  ​.
533
1Which of the following fractions is
10
equivalent to  
​ 14  ​?
10
C  
​ 12  ​
8
5
D  
​ 7 ​
A​  
  ​
28
5
B​  
  ​
14
2Which of the following fractions is not
1
3
4Gita mows  
​  4 ​of a lawn. If her brother
mows the same area of lawn, which
fraction shows how much lawn he
mowed?
3
C  
​ 8 ​
6
7
D  
​ 5 ​
A​  
  ​
12
4
B​  
 ​
8
equivalent to ​  
 ​?
2
3
A​  
 ​
6
4
B​  
 ​
6
4
C  
​ 8 ​
5
D  
​ 10  ​
8
3Jin feeds his cat  
​ 12  ​cup of cat food each
8
day. What is  
​ 12  ​in lowest terms?
2
A​  
 ​
3
4
B​  
 ​
5
4
C  
​ 6 ​
6
D  
​ 10  ​
5Which of the following pairs of fractions
are equivalent?
A​  
  ​ and ​  
 ​
15
8
6
3
C  
​ 9 ​ and ​  
  ​
11
4
12
D  
​ 10  ​ and ​  
 ​
5
B​  
 ​ and  
​  21  ​
7
5
7
6
2
8
6 What is  
​ 32  ​in lowest terms?
2
C  
​ 4 ​
4
D  
​ 3 ​
A​  
 ​
8
B​  
 ​
6
1
2
UNIT 3
Adding and Subtracting Fractions
© The Continental Press, Inc. DUPLICATING THIS MATERIAL IS ILLEGAL.
57
Read each problem. Write your answer.
SAMPLE Yuri and his brother are making brownies in a
square pan. They plan to cut the brownies into
16 equal pieces and give 4 brownies to each of
4 people. In lowest terms, what fraction of the
whole batch will each person receive?
Answer ____________
There are 16 brownies and each person will get 4. Each person
4
receives ​  
  ​of the whole batch. So, you need to find an equivalent
16
4
fraction for ​  
  ​in lowest terms. The greatest common factor of 4
16
4
and 16 is 4. Divide the numerator and denominator of  
​ 16  ​by 4:
444
1
1
​   
 
 ​ 5  
​  4 ​. Each person will receive  
​ 4 ​of the whole batch.
16 4 4
8
10
7Without cross multiplying, how can you tell that ​  
  ​ and  
​  12  ​ are not
12
equivalent fractions?
_________________________________________________________________________________
_________________________________________________________________________________
_________________________________________________________________________________
12
8Write a fraction that is equivalent to  
​ 18  ​and has a denominator that is
less than 10.
Answer ____________
8
9The common housefly grows to about ​  
  ​inch in length. What is this
16
length in lowest form?
58
Answer ________________________
UNIT 3
Adding and Subtracting Fractions
© The Continental Press, Inc. DUPLICATING THIS MATERIAL IS ILLEGAL.
Read the problem. Write your answer to each part.
10Janey and Mika are planting vegetables in a section of their community
garden. They have divided the rectangular plot into equal parts. Mika
drew this diagram of the garden.
3
Part AMika wants to use ​  
 ​of the garden for tomatoes. She
4
decides to put one tomato plant in each section. How can
Mika use equivalent fractions to figure out how many
tomato plants to buy? Explain your answer.
How many
equal sections are
in the garden?
_________________________________________________________________________________
_________________________________________________________________________________
_________________________________________________________________________________
_________________________________________________________________________________
_________________________________________________________________________________
Part BJaney wants to add 2 more equal sections to the garden.
She tells Mika to buy 11 tomato plants if she wants to use
3
​  
 ​of the garden for tomatoes. Is Janey correct? Explain.
4
_________________________________________________________________________________
_________________________________________________________________________________
_________________________________________________________________________________
_________________________________________________________________________________
UNIT 3
Adding and Subtracting Fractions
© The Continental Press, Inc. DUPLICATING THIS MATERIAL IS ILLEGAL.
59
Adding and Subtracting
Fractions
Lesson
2
5.NF.1, 5.NF.2
A multiple is the
product of a whole
number and another
whole number.
Common multiples are
multiples that two or
more numbers share.
The least common
multiple is the smallest
multiple they share.
When talking about
fractions, the least
common multiple is
often called the least
common denominator,
LCD. The LCM and the
LCD are the same.
You can use mental
math to check the
reasonableness of your
answer.
3
1
1
 ​ is ​  
 ​ with ​  
 ​m ore.
​  
4
4
2
1
1
 
​  2 ​ 1​  
 ​ 5 1
2
1
1
and  
​  4 ​m ore is 1​  
 ​.
4
Unlike fractions are fractions with different denominators. To add or
subtract unlike fractions, first find equivalent fractions with the same
denominator. Then you can find the sum or difference.
3
1
Pete needs  
​ 2 ​cup of raisins for a cookie recipe and  
​ 4 ​cup of
raisins for a scone recipe. How many cups of raisins in all does
Pete need?
3
1
 ​ 1  
​  4 ​.
To solve this problem, find ​  
2
Find the least common multiple (LCM) of
both denominators. The LCM of 2 and 4 is 4.
3
1
​  
 ​ 1  
​  4 ​
2
2
Rewrite the fractions as equivalent fractions
using the LCM. You only need to rewrite one
​  1 ​ in this problem.
fraction  
1
2
(2)
2
4
2
2
3
5
2
3
5
Add the numerators.
​  
 ​ 1  
​  4 ​ 5  
​  5 ​
4
Write the sum over the common denominator.
​  
 ​ 1  
​  4 ​5  
​  4 ​
4
If the answer is an improper fraction, rewrite it
as a mixed number in lowest terms.
​  
 ​ 5 1​  
 ​
4
4
5
1
1
Pete needs 1​ 
 ​  cups of raisins.
4
To subtract fractions with different denominators, first find the LCM of
both denominators. Rewrite the fractions using the LCM, subtract the
numerators, and write the difference over the common denominator.
2
1
Find ​  
 ​ 2  
​  2 ​.
3
2
1
2
(2)
1
(3)
4
3
1
​  
 ​ 2  
​  2 ​ 5  
​  3 ​  
​  2 ​ 2  
​  2 ​  
​  3 ​ 5  
​  6 ​ 2  
​  6 ​ 5  
​  6 ​
3
1
The difference is ​  
 ​.
6
60
UNIT 3
Adding and Subtracting Fractions
© The Continental Press, Inc. DUPLICATING THIS MATERIAL IS ILLEGAL.
Read each problem. Circle the letter of the best answer.
5
3
SAMPLE There are  
​ 6 ​gallon of whole milk and  
​ 4 ​gallon of skim milk in the
refrigerator. How many gallons of milk are there in all?
8
8
1
A​  
  ​
10
B​  
  ​
12
7
C​  
  ​
12
D 1​  
  ​
12
The correct answer is D. You need to find how much milk there
is in all, so you need to add. The fractions have unlike denominators,
so find the LCM of 6 and 4: 12. Write equivalent fractions with a
10
9
10
9
​  12  ​. Add the equivalent fractions:  
​ 12  ​ 1  
​  12  ​ 5
denominator of 12:  
​ 12  ​ 1  
19
19
7
 
​ 12  ​as a mixed number: 1​  
  ​.
​  12  ​. Write the improper fraction  
12
5
1
1 What is  
​ 8 ​ 2  
​  2 ​?
1
A​  
 ​
8
B
4Serena is measuring the lengths of beetles
C
1
1​  
 ​
8
4
 
​ 7 ​
1
D  
​ 2 ​
3
5
2 Find the sum of ​  
 ​ and ​  
  ​.
5
12
8
C 1 cm
11
D  
​ 10  ​ cm
B​  
  ​ cm
15
C  
​ 60  ​
4
1
5
2
5 What is  
​ 9 ​ 1  
​  8 ​in lowest terms?
1
B​  
  ​
30
D 1​  
  ​
60
7
3Vik studied  
​  8 ​hour for a test and Lisa
1
studied ​  
 ​hour. How much longer did Vik
2
study than Lisa?
3
1​  
 ​ hours
8
3
3
A​  
 ​ cm
5
1
A​  
  ​
17
A
for a science project. One beetle measures
4
7
 
​  5 ​cm and another measures  
​ 10  ​cm. What
is the difference in the beetles’ lengths?
B​  
 ​ hour
8
1
C  ​ 
 ​ hour
4
2
D  
​ 3 ​ hour
58
C 1​  
  ​
29
11
D  
​ 36  ​
A​  
  ​
72
B​  
  ​
36
7
29
2
6Daniel bought  
​  3 ​pound of cherries. He ate
4
​  
  ​pound of the cherries when he got home.
10
How many pounds of cherries are left?
3
4
C  
​ 5 ​
6
D  
​ 15  ​
A​  
  ​ 15
B​  
  ​ 13
2
UNIT 3
Adding and Subtracting Fractions
© The Continental Press, Inc. DUPLICATING THIS MATERIAL IS ILLEGAL.
61
Read each problem. Write your answer.
15
SAMPLE Beatrice is building a table with legs  
​ 18  ​yard long. Each piece of wood
9
she bought for the table legs measures  
​ 10  ​yard long. How much does
she need to cut from each piece of wood to make the correct size
table legs? Write your answer in lowest terms.
Answer ________________________
Beatrice is cutting down the wood, so subtract to find the answer.
9
15
  ​ 2  
​  18  ​. The LCD for both fractions is 90. Multiply the numerator
Find ​  
10
9
15
  ​by 9 and multiply  
​ 18  ​by 5 to find equivalent
and denominator in ​  
10
81
75
6
6
1
​  90  ​ 5  
​  90  ​. Finally, reduce  
​ 90  ​ to  
​  15  ​.
fractions. Then subtract:  
​ 90  ​ 2  
1
7
7A common denominator for  
​ 6 ​ 1  
​  10  ​is 60. Name the least common
denominator that you can use to add these fractions.
Answer ____________
3
2
1
8Franklin solved this problem: ​  
 ​ 1  
​  7 ​ 5  
​  3 ​. How can you prove that
5
his answer is incorrect using number sense?
_________________________________________________________________________________
_________________________________________________________________________________
_________________________________________________________________________________
4
2
9Xavier needs ​  
 ​cup milk for a cereal and  
​ 3 ​cup milk to drink. How
5
much milk does Xavier need in all? Show your work.
62
Answer ________________________
UNIT 3
Adding and Subtracting Fractions
© The Continental Press, Inc. DUPLICATING THIS MATERIAL IS ILLEGAL.
Read the problem. Write your answer to each part.
10Kaya volunteers at a local animal shelter. She made some notes
about the animal populations in the shelter this week.
3
2
​ 
 ​
 dogs ​ 
 ​
 cats
5
8
rest are other pets,
such as birds,
hamsters, and lizards
Part AWhat fraction of all the animals in the shelter are dogs
and cats? What fraction of all the animals are not dogs
and cats? Explain how you found your answers.
_________________________________________________________________________________
_________________________________________________________________________________
_________________________________________________________________________________
Part BWhy do you need to find common denominators for
fractions before you can find their sums or differences?
________________________________________________________
What does the
numerator in a
fraction represent?
The denominator?
________________________________________________________
_________________________________________________________________________________
_________________________________________________________________________________
_________________________________________________________________________________
UNIT 3
Adding and Subtracting Fractions
© The Continental Press, Inc. DUPLICATING THIS MATERIAL IS ILLEGAL.
63
Adding and Subtracting
Mixed Numbers
Lesson
3
5.NF.1, 5.NF.2
A mixed number
also can be expressed as
an improper fraction.
5
9
5
A mixed number is the sum of a whole number and a proper
5
5
fraction 1​  
 ​ 5 1 1 ​  
 ​. To add mixed numbers, add the fractions
9
9
and then add the whole numbers.
(
)
9
Hector is wrapping two gifts. He uses 1​  
  ​feet of red ribbon
10
3
for one gift and another 2 ​  
 ​feet for the other. How much red
5
ribbon does Hector use?
14
 ​ 5​  
 ​ 1​  
 ​ 5​  
  ​
1​  
9
9
9
9
9
You can also rewrite
both mixed numbers as
improper fractions with
like denominators and
add the numerators.
19
26
45
1
 
  ​ 5​  
  ​ 5 4 ​  
 ​
​  10  ​ 1​  
10
10
2
3
Find 1​  
  ​ 1 2 ​  
 ​.
5
10
Write equivalent fractions
with the lowest common
denominator. The LCD is 10.
Add the equivalent fractions.
9
6
1​  
  ​ 1 2 ​  
  ​
10
10
9
6
15
​  10  ​ 5 1 1 2 1  
​  10  ​
1 1 2 1  
​  10  ​ 1  
Rewrite any improper fractions
5
1 1 2 1 1​  
  ​
10
as a mixed number.
Add the whole numbers.
Rewrite the fraction in
lowest terms.
5
1
4 ​  
  ​ 5 4 ​  
 ​
10
2
1
 ​feet of red ribbon.
Hector will use 4 ​  
2
To subtract mixed numbers, write equivalent fractions with the lowest
common denominator. Subtract the fractions first. Then subtract the
whole numbers.
If you cannot subtract
the second fraction
from the first fraction,
you can also rename
both mixed numbers
as improper fractions
and subtract the
numerators.
3
4
27
22
1
2
Find 4 ​  
 ​ 2 3 ​  
 ​.
2
3
4
You cannot subtract ​  
 ​ from
6
3
3
9
 ​, so rename 4 ​  
 ​ as 3 ​  
 ​.
​  
6
6
6
5
 ​ 2 3​  
 ​ 5  
​  6  ​ 2 ​  
  ​ 5  
​  6 ​
4​  
6
6
6
1
2
9
4
3
4
4 ​  
 ​ 2 3 ​  
 ​ 5 4 ​  
 ​ 2 3 ​  
 ​
2
3
6
6
9
)
4
)
3 ​  
 ​ 2 3 ​  
 ​ 5 (3 1 ​  
 ​ 2 (3 1 ​  
 ​
6
6
6
6
Subtract the fractions.
5
4
(3 2 3) 1  
​  9 ​ 2  
​  6 ​ 5 0 1 ​  
 ​
6
6
Subtract the whole numbers.
(
)
5
The difference is  
​ 6 ​.
64
UNIT 3
Adding and Subtracting Fractions
© The Continental Press, Inc. DUPLICATING THIS MATERIAL IS ILLEGAL.
Read each problem. Circle the letter of the best answer.
5
5
SAMPLE Ivan biked 2 ​  
 ​km on Sunday and 1​  
 ​km on Thursday. How far did
8
6
Ivan bike on these two days?
19
1
A 3 ​  
  ​ km
24
B  
​ 24  ​ km
5
C 3 ​  
 ​ km
7
11
D 4 ​  
  ​ km
24
The correct answer is D. To find out how many kilometers Ivan
biked in all, add. First, rewrite the fractions as equivalent fractions:
533
15
534
20
​   
 
 ​ 5  
​  24  ​ and ​   
 
 ​ 5  
​  24  ​. Add the fractions and rewrite the sum
833
634
15 1 20
35
11
as a mixed number: ​  
   
​ 5  
​  24  ​ 5 1​  
  ​. Add the whole numbers:
24
24
11
11
2 1 1 1 1​  
  ​ 5 4 ​  
  ​.
24
24
7
1Keisha is 4 ​  
  ​feet tall. Her baby brother is
12
3
2 ​  
 ​feet tall. How much taller is Keisha
4
than her brother?
1
A​  
  ​ foot
12
B
5
1​  
 ​ feet
6
1
C 7 ​  
  ​ feet
12
1
D 2 ​  
 ​ feet
2
4
2Three hallways in a house measured 4 ​  
 ​ m,
5
3
1
5 ​  
 ​m, and 7 ​  
 ​m in length. What is the
8
2
difference between the longest and
shortest lengths?
A
3
12 ​  
  ​ m
10
B
4
4 ​  
 ​ m
5
C
7
2 ​  
  ​ m
10
D
3
3 ​  
  ​ m
10
A
B
7
8 ​  
  ​
11
3
calls for 1 cup of pineapple juice, 5 ​  
 ​ cups
4
1
of water, and 1​  
 ​cups of cranberry juice.
3
How many cups of fruit punch will the
recipe make?
5
A
8 ​  
  ​
12
1
C 4 ​  
  ​
12
B
6 ​  
 ​
7
4
D 6 ​  
  ​
12
1
5
3
5 What is 11​  
 ​ 2 7 ​  
 ​?
5
6
13
A
4 ​  
  ​
15
1
C 19 ​  
  ​
30
B
4 ​  
  ​
30
7
D 3 ​  
  ​
12
7
3
1
6 Solve 2 ​  
 ​ 2 1​  
 ​.
8
2
3
4
3 What is 5 ​  
 ​ 1 3 ​  
 ​?
4
7
5
8 ​  
  ​
28
4Jasmine is making fruit punch. The recipe
C
9
8 ​  
  ​
28
D
9
9 ​  
  ​
28
7
7
A​  
 ​
8
C 1​  
 ​
8
1
D 3 ​  
 ​
5
B
1​  
 ​
8
2
UNIT 3
Adding and Subtracting Fractions
© The Continental Press, Inc. DUPLICATING THIS MATERIAL IS ILLEGAL.
65
Read each problem. Write your answer.
SAMPLE Mrs. Chang’s car has enough gas to travel 12 miles. She wants to
2
1
travel 6 ​  
 ​miles to the grocery store and then 4 ​  
 ​miles to the post
3
2
office. Can Mrs. Chang make both trips before filling the car’s gas
tank? Explain.
Answer ____________________________________________________
To answer the question, add the two distances and compare the
3
2
1
4
7
1
 ​ 1 4 ​  
 ​ 5 6 ​  
 ​ 1 4 ​  
 ​ 5 10 ​  
 ​ 5 11​  
 ​.
sum to 12 miles. Add: 6 ​  
3
2
6
6
6
6
1
Compare: 11​  
 ​ , 12. Yes, Mrs. Chang can make both trips before
6
refueling.
7Jeff saw a bicycle from the 1870s at a museum. The bicycle’s front
9
wheel measured 4 ​  
  ​feet in diameter. Jeff’s own bicycle has a front
10
1
wheel that measures 2 ​  
 ​feet in diameter. How much larger was the
2
front wheel of the 1870s bicycle than of Jeff’s bicycle?
Answer ________________________
3
8Write two mixed numbers whose sum is 8 ​  
 ​. Explain how you solved
4
the problem.
_________________________________________________________________________________
_________________________________________________________________________________
_________________________________________________________________________________
2
9A bag contains 5 ​  
 ​kg of sand. The bag has a hole in it. Every
5
2
10 minutes, 1​  
 ​kg of sand escapes from the bag. After 20 minutes,
3
how many kilograms of sand will be in the bag? Explain how you
solved the problem.
_________________________________________________________________________________
_________________________________________________________________________________
_________________________________________________________________________________
66
UNIT 3
Adding and Subtracting Fractions
© The Continental Press, Inc. DUPLICATING THIS MATERIAL IS ILLEGAL.
Read the problem. Write your answer to each part.
10Most large rivers in the United States have monitoring stations that
measure the height of the water in the river. This is done in case of
flooding. After a severe thunderstorm, the water in a particular river
2
 ​feet high. Before the storm, the water in the river measured
is 15 ​  
5
7
  ​ feet.
10 ​  
12
Part AExplain the steps you would take to determine how many
feet the river water rose during the storm.
Do you need to
find a difference or
a sum?
_________________________________________________________________________________
_________________________________________________________________________________
_________________________________________________________________________________
_________________________________________________________________________________
Part BLarissa solved the problem in part A.
35
7
7
2
2
24
15 ​ 
 ​  2 10 
​    ​  
15 ​   ​  5 15 
​    ​ and 10 
​    ​ 5 10 ​    ​

12
12
5
5
60
60
35
24
24
15 
​    ​ 2 10 ​    ​ I have to rename 15 
​    ​.
60
60
60
49
49
35
84
   ​ 2 10 ​    ​ 5 5 
​     ​ The river water rose 5 
​     ​ feet.
15 ​ 
60
60
60
60
Is Larissa’s solution correct? Explain. If her solution is
incorrect, provide the correct answer.
_________________________________________________________________________________
_________________________________________________________________________________
_________________________________________________________________________________
_________________________________________________________________________________
UNIT 3
Adding and Subtracting Fractions
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67
R e vi e w
nd Subtracting
a
g
n
i
d
Ad
Fractions
Read each problem. Circle the letter of the best answer.
1Which of the following fractions is
10
equivalent to  
​ 12  ​?
5
12
A​  
  ​
14
C  
​ 6 ​
1
2
B​  
 ​
6
D  
​ 3 ​
3
1
2Marco spent  
​  3 ​hour exercising,  
​ 4 ​ hour
3
 ​hour doing chores. How
studying, and ​  
5
much time did Marco spend studying and
doing chores?
A
7
2
C  
​ 3 ​ hour
  ​ hours
1​  
20
6
B​  
  ​ hour
20
D
1
1​  
 ​ hours
4
3When Keiko planted a ginkgo tree a few
24
4What is  
​  64  ​in lowest terms?
A
2
6 ​  
 ​
3
B
6 ​  
  ​
24
11
C
1
6 ​  
 ​
6
5
D 7 ​  
  ​
24
C  
​ 7 ​
1
D  
​ 8 ​
3
3
4
5 What is  
​ 5 ​ 2  
​  10  ​?
3
C  
​ 5 ​
7
D  
​ 7 ​
1
A​  
  ​
10
2
B​  
  ​
15
3
6Tanya drank 5 ​  
  ​quarts of milk this week.
10
3
Her brother drank 7 ​  
 ​quarts. How many
4
quarts did they drink in all?
3
C 12 ​  
  ​
10
1
D 13 ​  
  ​
10
A
12 ​  
 ​
7
B
13 ​  
  ​
20
3
1
7Which of the following fractions is not
4
equivalent to ​  
 ​?
7
20
C  
​ 14  ​
24
D  
​ 56  ​
A​  
  ​
28
B​  
  ​
42
68
4
B​  
 ​
2
3
years ago, it was 9 ​  
 ​feet tall. The tree is
8
5
now 15 ​  
 ​feet tall. How many feet has the
6
tree grown?
12
A​  
  ​
32
8
32
UNIT 3
Adding and Subtracting Fractions
© The Continental Press, Inc. DUPLICATING THIS MATERIAL IS ILLEGAL.
Read each problem. Write your answer.
2
1
8Curtis spends  
​  5 ​of his monthly income on rent and ​  
 ​on food. He
3
1
puts  
​  10  ​of his income into a savings account. What fraction of his
monthly income does Curtis spend on rent, food, and savings? Write
your answer in lowest terms.
Answer ____________
5
1
9Kim biked 5 ​  
 ​km to a nearby park. Then she biked another 1​  
 ​ km
8
6
through the park to get to the lake. How far did Kim travel in all?
Answer ________________________
7
10Explain how to find an equivalent fraction for ​  
 ​.
9
_________________________________________________________________________________
_________________________________________________________________________________
1
2
11Aisha is 18 ​  
 ​years old. Her sister is 11​  
 ​years old. How much older is
2
3
Aisha than her sister?
Answer ________________________
9
3
12 What is  
​ 10  ​ 2  
​  8 ​in lowest terms?
Answer ____________
3
13Ari’s cat weighs 20 ​  
 ​pounds. The vet wants the cat to lose
8
1
6 ​  
 ​pounds. What is the cat’s target weight? Show your work.
2
Answer ________________________
UNIT 3
Adding and Subtracting Fractions
© The Continental Press, Inc. DUPLICATING THIS MATERIAL IS ILLEGAL.
69
Read each problem. Write your answer to each part.
14Flora and her father are building a tree house. She needs to nail one
3
piece of wood on top of another. The first piece of wood is 1​  
 ​-inches
8
3
thick and the second is 2 ​  
  -​ inches thick.
16
Part AWhat is the total thickness of the two pieces of wood?
Answer ________________________
Part BCan Flora use 4-inch long nails to nail the two pieces of
wood together? Explain.
_________________________________________________________________________________
_________________________________________________________________________________
_________________________________________________________________________________
15Brandon is planning a small garden. He wants the garden to be
1
1
1
​  
 ​ eggplants,  
​  9 ​peppers, and ​  
 ​cucumbers. The rest will be
6
2
strawberries.
Part AWhat fraction of Brandon’s garden will contain vegetables?
Write your answer in lowest terms.
Answer ____________
Part BWill more of Brandon’s garden contain strawberries or
cucumbers? Explain.
_________________________________________________________________________________
_________________________________________________________________________________
_________________________________________________________________________________
70
UNIT 3
Adding and Subtracting Fractions
© The Continental Press, Inc. DUPLICATING THIS MATERIAL IS ILLEGAL.
U n it
4
ng and Divid
Multiplyi
ing
Fractions
Lesson 1 Relating Fractions to Division reviews
fractions as a form of division of the numerator by the
denominator.
L esson 2 Multiplying Fractions and Whole
Numbers reviews how to multiply fractions and whole
numbers.
Lesson 3 Multiplying Fractions reviews the use of a
rectangular model and a rule to find the product of two
fractions.
Lesson 4 Multiplication and Scale reviews
the relationship of the size of a product to the size of
fractional factors greater than and less than 1.
L esson 5 Dividing Fractions and Whole
Numbers reviews how to divide whole numbers by
fractions.
Lesson 6 Word Problems with Fractions reviews
how to interpret and solve word problems involving
the multiplication and division of fractions and whole
numbers.
UNIT 4
Multiplying and Dividing Fractions
© The Continental Press, Inc. DUPLICATING THIS MATERIAL IS ILLEGAL.
71
Lesson
1
Fractions to Div
g
n
i
t
a
l
ision
Re
The parts of a
fraction must be equal
in size.
Thirds
Not thirds
5.NF.3
A fraction can represent the division of one whole number by another
whole number into equal fractional parts.
Ada, Bart, Cam, and Don have teamed up to paint 3 walls of
the same size at a community center. Each of the 4 students
will paint an equal area. What part of a whole wall does each
person paint?
To find the part each student will paint, divide the number of
walls by the number of students.
Find 3 walls 4 4 students.
To divide 3 by 4, you can use a model. Draw 3 walls. Divide
each wall into 4 equal parts. Label each part with a student’s
name.
Don
Cam
Bart
Ada
Wall 3
Don
Cam
Ada
Bart
Wall 2
Don
Cam
Bart
Ada
Wall 1
Each student is responsible for one part of each of the 3 walls.
1
Each student must contribute 3 parts. Each part is ​  
 ​of a wall.
4
1
1
1
3
Each student paints ​  
 ​ 1  
​  4 ​ 1  
​  4 ​ 5  
​  4 ​of a wall.
4
A numerator is the
number above the
fraction bar in a
fraction.
3 ← The numerator represents the number of walls.
3 4 4 5 ​  
 ​
4 ← The denominator represents the number of
students.
The denominator is
the number below
the fraction bar in a
fraction.
So, the area each student is responsible for painting is equal
3
to  
​  4 ​of a whole wall.
72
UNIT 4
Multiplying and Dividing Fractions
© The Continental Press, Inc. DUPLICATING THIS MATERIAL IS ILLEGAL.
Read each problem. Circle the letter of the best answer.
SAMPLE Levi and four of his friends pool their money to buy sub sandwiches.
They buy 2 large sandwiches. If Levi and each of his friends share the
sandwiches equally, what part of a whole sandwich does each of them
eat?
2
2
A​  
 ​
5
B​  
 ​
4
5
5
C​  
 ​
2
D​  
  ​
10
1
The correct answer is A. Each person will eat  
​ 5 ​of each sandwich,
1
1
2
or ​  
 ​ 1  
​  5 ​ 5  
​  5 ​. This is the same as dividing 2 sandwiches by 5.
5
2
Represent this as 2 4 5 5  
​  5 ​. The 2 is the number divided and the 5
2
is the divisor. So, each friend will eat  
​ 5 ​of a sandwich.
1Gerry is making 21 loaves of bread using
14 cups of milk. Which fraction represents
the amount of milk he will use per loaf?
A
1
1​  
 ​ cups
2
2
B​  
 ​ cup
3
5
D 1​  
 ​
9
3
B​  
 ​
3
D  
​ 7 ​ cup
3
 
​ 17  ​
C
B6
D  
​ 3  ​
17
3Ophelia feeds her two cats 13 ounces of
cat food each day. Assuming each cat eats
an equal amount, how many ounces of
food does each cat eat?
1
C 1​  
 ​
5
A​  
 ​
5
A5
B​  
  ​ 13
3
1
share 17 game tokens. How many tokens
will each of them will get? Ignore the
leftover tokens.
2
party. If the pizzas are shared equally, what
fraction of a pizza would each person get?
C  
​ 3 ​ cup
2Li and her two brothers want to equally
A​  
  ​ 13
4A group of 15 people orders 9 pizzas for a
1
C 6 ​  
 ​
2
2
D 2 ​  
  ​
13
3
4
5Six students want to equally share 4 packs
of pens. Each pack contains 12 pens. How
many pens does each student get?
A12
C 6
B8
D 4
6The distance from New York City to
Philadelphia is about 94 miles. Tony will
bike the same distance each day for 4 days
to travel between the cities. How many
miles will he bike per day?
1
A​  
  ​ 23
B
3
20 ​  
 ​ 4
1
C 4 ​  
  ​
23
1
D 23 ​  
 ​
2
UNIT 4
Multiplying and Dividing Fractions
© The Continental Press, Inc. DUPLICATING THIS MATERIAL IS ILLEGAL.
73
Read each problem. Write your answer.
SAMPLE Abby, Beth, and Cara share the work of mowing 7 lawns of the same
size. Each student mowed exactly the same area. What part of the
total job did Beth and Cara do together?
Answer ____________
Draw a diagram to help you solve the problem.
Draw 7 lawns of the same size and divide each into
thirds. So, the total job consists of 21 equal parts.
Let 21 be the denominator. Beth mowed 7 of those
parts. Cara mowed 7 of those parts. Together they
14
​ 21  ​. In lowest terms, this
mowed 14 of 21 parts, or  
2
is ​  
 ​of the total job.
3
A
A
B
B
C
C
A
A
B
B
C
C
A
A
B
B
C
C
A
B
C
7Each of 4 loaves of bread was cut into 12 equal slices. Two pieces from
each loaf were eaten. What part of a whole loaf does the total
number of slices eaten represent?
Answer ________________________
8Akio and 7 friends will share 3 liters of juice equally. How much juice
does each person get?
Answer ________________________
9Sharon and Fred can make 6 ribbons from 20 inches of fabric. How
many inches of fabric are used to make each ribbon?
74
Answer ____________
UNIT 4
Multiplying and Dividing Fractions
© The Continental Press, Inc. DUPLICATING THIS MATERIAL IS ILLEGAL.
Read the problem. Write your answer to each part.
10Seth enjoys woodworking and sometimes builds things to sell.
Part ASeth made 5 identical birdhouses in 4 hours. What
fraction of an hour did it take Seth to make one
birdhouse? Explain why your answer is correct.
________________________________________________________
________________________________________________________
The number
that is divided is the
numerator. The
number doing the
dividing is the
denominator.
________________________________________________________
_________________________________________________________________________________
_________________________________________________________________________________
_________________________________________________________________________________
Part BSeth made 4 identical shelves in 3 hours. Did it take him
more time or less time than to make one shelf than one
birdhouse? Explain.
_________________________________________________________________________________
_________________________________________________________________________________
_________________________________________________________________________________
UNIT 4
Multiplying and Dividing Fractions
© The Continental Press, Inc. DUPLICATING THIS MATERIAL IS ILLEGAL.
75
ing Fractions a
y
l
p
i
t
l
u
M
nd
Whole Numbers
Lesson
2
5.NF.4.a, b
You can use multiplication to find fractions of whole numbers.
The word of here
means “multiply.”
3
3
 ​of 12 means  
​ 4 ​ 3 12.
​  
4
3
 ​of the sheets
Milena has 20 sheets of fancy paper. She uses ​  
5
for a scrapbook. How many sheets of fancy paper does she
use?
To find the number of sheets Milena uses in her project, find
3
 ​of 20.
​  
5
You can use a model to find the answer.
Draw squares for 20 sheets of paper. Divide them into 5 equal
groups.
The general rule for
multiplying a whole
number by a fraction is
a
b
a3c
b
 ​3 c 5​  
   
​
​  
where c ? 0
Multiplying a fraction
by a whole number
works the same way.
Just reverse the factors.
2
332
6
 ​ 5​  
   
​ 5​  
 ​
3 3​  
7
7
7
You can cancel common
factors as a shortcut.
3
3
4
​  
 ​ 3 20 5​  
 ​ 3 20 5
5
5
Then shade the squares in 3 of the 5 groups.
Count the number of shaded sheets. There are 12. That means
3
 ​of 20 is 12.
that ​  
5
So, Milena uses 12 sheets of fancy paper.
You can also use an equation to find a fraction of a whole number.
Use an equation to find the number of sheets Milena uses.
3
3
1
3 3 4 5 12
76
3
 ​of 20 translates to  
​ 5 ​ 3 20. Multiply the numerator and the
​  
5
whole number. Then divide the product by the denominator.
3 3 20
60
 
​  5 ​ 3 20 5 ​  
   
​ 5 ​  
  ​ 5 12
5
5
Milena uses 12 sheets of fancy paper.
UNIT 4
Multiplying and Dividing Fractions
© The Continental Press, Inc. DUPLICATING THIS MATERIAL IS ILLEGAL.
Read each problem. Circle the letter of the best answer.
3
SAMPLE Rosa has 32 grapes. She eats  
​ 8 ​of them now and saves the rest for
later. Which expression will help Rosa find the number of grapes she
saves for later?
3
A​  
 ​ 1 32
8
3
B32 4  
​  8 ​
3
3
C32 2  
​  8 ​
D​  
 ​ 3 32
8
The correct answer is D. This question asks you to find how many
grapes Rosa saves for later. To do that, you need to subtract a
3
number from 32. That number is equal to  
​ 8 ​of 32. So, you need to
3
find ​  
 ​of 32. This can be found with a multiplication expression,
8
3
 
​  8 ​ 3 32.
4
1A collecting jar holds 45 insects. ​  
 ​of the
5
insects are purple. The rest are blue. How
many insects in the jar are blue?
A41
C 9
B36
D 1
2
2 What is  
​ 9 ​of 63?
A7
C 54
B14
D 129
3There were 400 people at a concert. Half
1
4Jiehae read 75 pages in a book.  
​ 3 ​of the
1
pages discussed birds, and ​  
 ​ discussed
5
reptiles. The remaining pages discussed
mammals. How many pages discussed
reptiles or mammals?
A15
C 40
B25
D 50
3
5 What is  
​ 11  ​of 99?
A9
C 27
B18
D 39
2
of them wore yellow T-shirts, ​  
  ​wore gray
10
T-shirts, and the rest wore orange T-shirts.
How many people wore gray T-shirts?
A80
C 180
B120
D 200
1
6On a menu,  
​ 10  ​of the 60 dishes are
1
1
chicken,  
​  6 ​are beef, ​  
 ​are pork, and the
3
rest are vegetarian. If Sergei doesn’t like
beef or pork, how many dishes can he
choose from on this menu?
A30
C 10
B24
D 6
UNIT 4
Multiplying and Dividing Fractions
© The Continental Press, Inc. DUPLICATING THIS MATERIAL IS ILLEGAL.
77
Read each problem. Write your answer.
2
SAMPLE A science class counted 20 bird nests in the park. ​  
 ​of the nests had
5
3
eggs, and  
​ 10  ​of the nests had chicks. The rest of the nests were
empty. How many nests were empty?
Answer ____________
2
4
Find the number of nests with eggs: 
​  5 ​  3 20 5 8 nests with eggs.
1
2
3
  ​ 3 20 5 6 nests with chicks.
Find the number of nests with chicks: ​ 
10
1
Add to find how many nests with eggs or chicks: 8 1 6 5 14.
Now, subtract to find the number of empty nests: 20 2 14 5 6
empty nests. There were 6 empty nests.
3
7There are 32 bottles on a shelf. Hamilton replaces  
​ 8 ​of the bottles
with cans. How many bottles did Hamilton replace?
Answer ____________
5
8 What is  
​ 6 ​of 72? Show your work.
Answer ____________
2
9There are 100 senators in the United States Senate. At least ​  
 ​of them
3
must vote yes in order to ratify, or approve, a treaty. What is the least
number of yes votes needed to ratify a treaty? Round to the nearest
whole number.
78
Answer ____________
UNIT 4
Multiplying and Dividing Fractions
© The Continental Press, Inc. DUPLICATING THIS MATERIAL IS ILLEGAL.
Read the problem. Write your answer to each part.
10Dora uses the recipe below to make French toast for 12 people.
Recipe for French Toast
(Makes 2 servings)
Ingredients
• 2 eggs
3
• ​ 
 ​  teaspoon sugar
4
1
• ​ 
  ​  teaspoon salt
4
1
• ​ 
  ​  cup milk
2
• 4 slices white bread
Part AAssuming each person will get 1 full serving, how much of
each ingredient does Dora need to make enough French
toast for everyone? Explain your answer.
How many
times greater is 12
servings than 2
servings?
_________________________________________________________________________________
_________________________________________________________________________________
_________________________________________________________________________________
_________________________________________________________________________________
Part BIf Dora uses the same recipe to make French toast for just
herself, how can she find the amount of each ingredient she
needs? Explain.
_________________________________________________________________________________
_________________________________________________________________________________
_________________________________________________________________________________
UNIT 4
Multiplying and Dividing Fractions
© The Continental Press, Inc. DUPLICATING THIS MATERIAL IS ILLEGAL.
79
Multiplying Fractions
Lesson
3
5.NF.4.a, b
You can use a rectangular model to multiply fractions.
3
 ​yard wide
Kendra is weaving a small rug. The rug will be ​  
5
5
by ​  
 ​yard long. What is the area of the rug?
6
3
5
You can also divide
the rectangle into sixths
horizontally and fifths
vertically. You will get
the same product.
Remember that the
commutative property
says you can multiply
two numbers in any
order.
3
5
 
 ​
​  5 ​ 3​  
6
is the same as
5
3
 
 ​
​  6 ​ 3​  
5
5
6
3
5
To find the rug’s area, multiply the length and width: ​  
 ​ 3  
​  6 ​.
5
Draw a rectangular model. Divide a rectangle into fifths
horizontally and sixths vertically.
3
5
 ​of the rectangle one way. Then shade ​  
 ​of the
First, shade ​  
5
6
rectangle another way.
There are 30 units in the rectangle in all. Of the 30 units, 15 are
15
shaded both ways. The fraction that represents this amount is  
​ 30  ​.
15
1
 ​in lowest terms.
You can write  
​ 30  ​ as ​  
2
1
The rug has an area of  
​ 2 ​ yd2.
The general rule for
multiplying fractions is
a
c
ac
​   ​ 3​   ​ 5​    ​

b
d
bd
where b, d ? 0
80
You can also use a rule to find the product of any two fractions.
Multiply the numerators. → 3
5
335
15
 ​ 3  
​  6 ​ 5 ​   
 
 ​ 5  
​  30  ​
​  
5
5
3
6
Multiply the denominators. →
15
1
​  2 ​
Always reduce the product to lowest terms:  
​ 30  ​ 5  
UNIT 4
Multiplying and Dividing Fractions
© The Continental Press, Inc. DUPLICATING THIS MATERIAL IS ILLEGAL.
Read each problem. Circle the letter of the best answer.
7
2
SAMPLE A cornfield is  
​ 3 ​km in size. What is the area of the
​ 9 ​km by  
cornfield?
9
14
A​  
  ​ km2
12
B​  
  ​ km2
27
3
1
C​  
 ​ km2
4
D​  
 ​ km2
2
The correct answer is B. To find the area of the cornfield, multiply
7
2
the field’s dimensions. It is ​  
 ​km long and  
​ 3 ​km wide. Multiply
9
7
2
 
​  9 ​ 3  
​  3 ​. Multiply the numerators and multiply the denominators.
732
14
14
The product is ​  
 
 ​ 5  
​  27  ​. The fraction  
​ 27  ​is in lowest terms.
933
3
2
1What is the product of ​  
 ​ and ​  
 ​in lowest
5
3
terms?
6
A​  
  ​
15
5
B​  
 ​
8
2
C  
​ 5 ​
1
D  
​ 3 ​
3
2
2A track is  
​ 4 ​mile long. Rafael runs ​  
 ​ of
5
the track. What distance does Rafael run?
5
A​  
 ​ mi
9
6
B​  
 ​ mi
9
C
6
​  
  ​ mi
20
3
D  ​ 
  ​ mi
10
4
1
3Multiply  
​  5 ​ 3  
​  8 ​. What is the product in
4The dimensions of a postage stamp are
3
7
 
​  4 ​in. by  
​ 8 ​in. What is the area of the
stamp?
5
C  
​ 16  ​ in2
7
D  
​ 32  ​ in2
A​  
 ​ in2
6
B​  
  ​ in2
11
5
21
2
5Zoe has a piece of fabric that is  
​ 3 ​yd long.
7
She uses ​  
  ​of the piece in a costume.
12
What part of a yard does Zoe use?
3
C  
​ 18  ​
9
D  
​ 36  ​
A​  
 ​ 5
B​  
  ​ 15
7
14
lowest terms?
1
A​  
  ​
10
5
B​  
  ​
13
C
1
 
​ 40  ​
4
D  
​ 40  ​
3
1
6Sam has ​  
 ​of a pizza. He eats ​  
 ​of his
4
3
section. In lowest terms, how much of the
whole pizza does he eat?
1
C  
​ 7 ​
1
D  
​ 12  ​
A​  
 ​
2
 ​
B​  
4
4
3
UNIT 4
Multiplying and Dividing Fractions
© The Continental Press, Inc. DUPLICATING THIS MATERIAL IS ILLEGAL.
81
Read each problem. Write your answer.
3
1
SAMPLE Harry has 1​  
 ​kg of whole-wheat flour. He uses ​  
 ​of the flour to bake
4
2
bread. How much flour did he use?
Answer ________________________
3
1
To find  
​ 4 ​ of 1​  
 ​kg, first change the mixed number to an
2
3
333
9
1
 ​ 5  
​  2 ​. Then multiply: ​  
 
 ​ 5  
​  8 ​. Rewrite the
improper fraction: 1​  
2
432
9
1
1
 ​. Harry used 1​  
 ​kg of flour.
product as a mixed number:  
​ 8 ​ 5 1​  
8
8
3
2
7A window is  
​ 4 ​m high and ​  
 ​of it is covered with frosted glass. What
3
part of a meter is frosted glass? Draw a rectangular model to show
how to find the answer.
Answer ____________
8
10
8A computer screen measures  
​ 12  ​ft by  
​ 12  ​ft. What is the screen’s area
in square feet?
Answer ________________________
9
1
9A landscaper delivered  
​ 10  ​ton of mulch to a job site. Workers spread ​  
 ​
3
of this amount around trees. What amount did the workers spread
around trees? Show your work.
82
Answer ________________________
UNIT 4
Multiplying and Dividing Fractions
© The Continental Press, Inc. DUPLICATING THIS MATERIAL IS ILLEGAL.
Read the problem. Write your answer to each part.
10Saroya needs to cover the floor of a small closet with carpet tiles. Each
carpet tile measures 9 in. by 10 in.
10 in.
9 in.
_______ ft
_______ ft
Part AWhat is each dimension as a fraction of a foot? Label
the drawing above. Then find the area of each tile in
square feet. Explain how you found your answers.
How many
inches are in 1 foot?
_________________________________________________________________________________
_________________________________________________________________________________
_________________________________________________________________________________
_________________________________________________________________________________
Part BThe area of the closet floor is 18 ft2. If Saroya buys 32 tiles,
will she have enough to cover the floor? Explain your answer.
_________________________________________________________________________________
_________________________________________________________________________________
_________________________________________________________________________________
UNIT 4
Multiplying and Dividing Fractions
© The Continental Press, Inc. DUPLICATING THIS MATERIAL IS ILLEGAL.
83
Multiplication and Scale
Lesson
4
5.NF.5.a, b
You can compare
the effect of scale
visually.
100
75
50
25
0
DISTANCE TRAVELED
When you multiply a number by a fraction, you are scaling, or
resizing, that number.
When you multiply a number by a fraction less than 1, the product is
less than the number.
On Monday, Ray drove at 50 miles per hour for 1 hour. He
1
drove 50 miles. On Tuesday, Ray drives at 50 mph for  
​ 2 ​ hour.
1
hr
2
1
1 hr
Does he drive more or less than 50 miles?
1 2 hr
Time (in hours)
General rules for scaling
numbers are
1
b
If  
​  c  ​, 1,
b
)
)
b
If  
​  c  ​. 1,
b
)
then (a 3​  
c  ​ . a
50
1
50 3  
​  2 ​ 5  
​  2  ​ 5 25 miles
then (a 3​  
c  ​ , a
b
If  
​  c  ​5 1,
b
then (a 3​  
c  ​ 5 a
1
Multiply 50 by the fraction ​  
 ​. Because ​  
 ​ , 1, the product
2
2
must be less than 50. By multiplying 50 by a fraction less than 1,
you are scaling down 50.
1
If Ray drives 50 mph for ​  
 ​hour, he drives less than 50 miles.
2
When you multiply a number by a fraction greater than 1, the product
is greater than the number.
1
On Wednesday, Ray drives at 50 mph for 1​  
 ​hours. Does he
2
drive more or less than 50 miles?
1
3
Multiply 50 by the mixed number 1​  
 ​, which equals  
​ 2 ​.
2
3
Because ​  
 ​ . 1, the product must be greater than 50.
2
3
1
150
50 3 1​  
 ​ 5 50 3  
​  2 ​ 5  
​  2   ​ 5 75 miles
2
1
If Ray drives 50 mph for 1​  
 ​hours, he drives more than 50 miles.
2
84
UNIT 4
Multiplying and Dividing Fractions
© The Continental Press, Inc. DUPLICATING THIS MATERIAL IS ILLEGAL.
Read each problem. Circle the letter of the best answer.
SAMPLE Which of the following is true?
3
C
4 3  
​  8 ​ . 5
5
D
8 3 1​  
 ​ , 8
2
A 3 3  
​  8 ​ . 3 B 6 3  
​  6 ​ , 6 7
1
The correct answer is B. To decide which number sentence is true,
compare the expressions on the left and right of each inequality
sign. Choices A and C show a whole number multiplied by a fraction
less than 1, but the products are greater than the number, which is
incorrect. Choice D shows a whole number multiplied by a fraction
greater than 1 with a product that is less than the number. This is
also incorrect. Choice B correctly shows a whole number multiplied
by a fraction less than 1 and a product that is less than the number.
5
1Melinda owns 18 jigsaw puzzles. If ​  
 ​ of
6
them show nature scenes, what is true
about the number that show nature scenes?
A It is 0.
C It is , 18.
B It is . 18.
D It is 18.
2Which of the following is not true?
4
3
4Alice printed one of her photographs. The
print was too small. Which fraction could
she multiply the dimensions of the original
print by to get a larger print?
6
1
C  
​ 8 ​
4
D  
​ 16  ​
A​  
 ​
2
15
 ​
B​  
3
A8 3 2 ​  
 ​ , 8
5
C 11 3  
​  4 ​ , 11
5A barrel holds 16 gallons of rainwater and
5
B3 3 3 ​  
 ​ . 3
6
1
7 3 2 ​  
 ​ . 7
3
is full. Eli uses some water and after three
7
days, the barrel is only  
​ 12  ​full. How much
water is in the barrel?
D
3A soup recipe calls for 2 liters of water.
3
Hiro is making only  
​ 4 ​the amount the
recipe makes. How many liters of water will
Hiro use?
A0
C 2
B more than 2
D less than 2
7
A ,  
​  12  ​ gal
C , 3 gal
B . 8 gal
D . 16 gal
6 Which of the following is true?
3
A2 . 2 3  
​  10  ​
2
B​  
 ​ 3 7 . 7
3
1
C 8 3 2 ​  
 ​ , 6
6
7
D 12 ,  
​  8 ​ 3 12
UNIT 4
Multiplying and Dividing Fractions
© The Continental Press, Inc. DUPLICATING THIS MATERIAL IS ILLEGAL.
85
Read each problem. Write your answer.
3
3
SAMPLE Is 40 times  
​ 5 ​greater than 40? Check by solving 40 3 ​  
 ​.
5
Answer ____________
3
You can decide if 40 3  
​  5 ​will be greater than 40 by remembering
the rules of scaling. A number times a fraction less than 1 will have
3
a product less than the number. Since  
​ 5 ​is a fraction less than 1,
3
3
the product of 40 3  
​  5 ​must be less than 40. Solve: 40 3 ​  
 ​ 5 24,
5
which is less than 40.
5
7Is  
​  6 ​of 30 less than or greater than 30?
Answer ________________________
10
8Is 634 3  
​  9  ​less than, greater than, or equal to 634?
Answer ________________________
1
9A recipe calls for 5 ​  
 ​cups of chicken broth to make 4 servings of
4
soup. Neela wants to make 6 servings of soup. Does she need to use
the amount of broth the recipe calls for, less broth, or more broth?
Answer ________________________
10Gigex Corporation just announced that the company’s sales in
1
November were 2 ​  
 ​times its sales in October. Were sales greater in
3
November or in October?
Answer ________________________
11Dan is making a 4-foot high scale model of a skyscraper. To get the
dimensions for the model, is Dan multiplying the skyscraper dimensions
by a fraction less than 1 or by a fraction greater than 1?
86
Answer ________________________
UNIT 4
Multiplying and Dividing Fractions
© The Continental Press, Inc. DUPLICATING THIS MATERIAL IS ILLEGAL.
Read the problem. Write your answer to each part.
12Michaela drives to and from work five days a week, from Monday
1
through Friday. She uses 1​  
 ​gallons of gasoline to drive to and from
4
work every weekday.
Part AWill Michaela use more than or less than 5 gallons of gas
each week driving to and from work? Explain your answer
without solving the problem.
What is another
way to express the
1
 ​?
mixed number 1​  
4
_________________________________________________________________________________
_________________________________________________________________________________
_________________________________________________________________________________
Part BProve your answer to part A by finding the number of
gallons Michaela will use traveling to and from work in a
week. Show your work.
Answer ________________________
13A jet travels at 504 mph while flying from New York to Los Angeles.
7
Part AIf the jet were flying at ​  
 ​times that speed, would the
8
amount of time the flight takes be shorter or longer?
Answer ________________________
Part BExplain how you know your answer is correct.
_________________________________________________________________________________
_________________________________________________________________________________
_________________________________________________________________________________
_________________________________________________________________________________
UNIT 4
Multiplying and Dividing Fractions
© The Continental Press, Inc. DUPLICATING THIS MATERIAL IS ILLEGAL.
87
Dividing Fractions and
Whole Numbers
Lesson
5
5.NF.7.a, b
When the product
of two fractions equals
1, then the fractions are
reciprocals.
3
4
 
 ​ 5 1
​  4 ​ 3​  
3
Dividing by a fraction is the same as multiplying by the reciprocal of
the fraction.
You can use a model to divide by a fraction.
1
Jan cuts a 6-inch stick of clay into ​  
 ​-inch wide parts. How many
2
1
​  
 ​-inch wide pieces will Jan have after she divides the stick?
2
3
So, ​  
 ​is the reciprocal
4
4
of  
​  3 ​.
A number line is
another kind of model
you can use to divide
fractions.
0 1 2 3 4 5 6
Dividing a whole
number by a fraction
less than 1 results in a
quotient that is greater
than the whole number.
1
2
1
2
1 inch
1
2
1
2
1
2
1
2
1 inch
1
2
1 inch
1
2
1 inch
1
2
1
2
1 inch
1
2
1
2
1 inch
First, draw a rectangle to represent a 6-inch stick. Mark 6 equal
pieces, 1 inch wide, to represent the whole number 6. Then
1
divide each 1-inch wide piece into two ​  
 ​-inch wide pieces.
2
1
Count the total number of ​  
 ​-inch pieces. There are 12. So, Jan
2
1
will have 12 pieces of clay that are  
​ 2 ​inch wide.
You can also use an equation to find the quotient of a whole number
and a fraction. Change the division expression into a multiplication
expression by using the reciprocal of the divisor.
1
Use the reciprocal to find how many ​  
 ​-inch wide pieces Jan
2
will make by dividing a 6-inch piece of clay.
1
Find 6 4  
​  2 ​.
1
 ​. To do that, flip the numerator
First, find the reciprocal of ​  
2
1
2
and denominator. The reciprocal of ​  
 ​ is ​  
 ​or, in simplest
1
2
terms, 2.
Multiply the whole number, 6, by the reciprocal, 2.
1
2
6 4  
​  2 ​ 5 6 3 ​  
 ​ 5 6 3 2 5 12
1
1
So, Jan will have 12 pieces of clay that are  
​ 2 ​inch wide.
88
UNIT 4
Multiplying and Dividing Fractions
© The Continental Press, Inc. DUPLICATING THIS MATERIAL IS ILLEGAL.
Read each problem. Circle the letter of the best answer.
1
SAMPLE What is  
​ 7 ​ 4 6?
6
7
A​  
 ​
7
B​  
 ​
6
1
C​  
  ​
42
D42
The correct answer is C. This question asks you to find the product
of a fraction divided by a whole number. A whole number can be
expressed as an improper fraction with a denominator of 1. The
6
1
1
1
1
whole number 6 is  
​ 1 ​. So, its reciprocal is  
​ 6 ​. Multiply:  
​  7 ​ 3  
​  6 ​ 5  
​  42  ​.
1
The quotient is  
​ 42  ​.
1
1 What is  
​ 3 ​ 4 2?
1
A​  
 ​
5
1
B​  
 ​
6
1
 ​tank of gas while driving for
4Oscar uses ​  
4
C
2
 
​ 3 ​
D 6
3 hours. Which expression shows how
much gas he uses in 1 hour?
1
A​  
 ​ 3 3
4
1
2Isaac bought 4 pizzas for an after-game
party. Each pizza was cut into sixths. How
many slices of pizza did Isaac buy?
A24
C 6
B10
D 4
1
3 How many  
​ 12  ​’s are there in 5?
5
A​  
  ​
12
C 16
2
D 60
B
2 ​  
 ​
5
B3 3  
​  4 ​
1
C  
​ 4 ​ 4 3
1
D 3 4  
​  4 ​
1
5 How many  
​ 7 ​’s are there in 10?
A70
C 10
B56
D 1​  
 ​
7
3
1
6Ilene needs  
​  3 ​of a ball of yarn to knit a
potholder. She has 14 balls of yarn. Which
expression show how many potholders she
can knit?
1
A​  
 ​ 4 14
3
1
B14 4  
​  3 ​
1
C 14 3  
​  3 ​
1
D  
​ 3 ​ 3 14
UNIT 4
Multiplying and Dividing Fractions
© The Continental Press, Inc. DUPLICATING THIS MATERIAL IS ILLEGAL.
89
Read each problem. Write your answer.
1
SAMPLE Phoebe knows a cup is  
​ 16  ​of a gallon. How many cups are there in
3 gallons? Explain how Phoebe can find the quotient.
Answer ____________________________________________________
1
To find the number of cups, or  
​ 16  ​’s, in a gallon, Phoebe should
1
1
divide 3 by  
​ 16  ​: 3 4  
​  16  ​. First, she will need to find the reciprocal of
16
1
the divisor: the reciprocal of  
​ 16  ​ is  
​  1  ​. Then she should multiply:
3 3 16 5 48. There are 48 cups in a gallon.
7Ethan is running a 10-mile race. Refreshment stands have been set up
1
every  
​  3 ​mile along the course. How many refreshment stands will
Ethan have passed by the end of the marathon?
Answer ____________
1
8What is 300 4  
​  5 ​? Show your work.
Answer ________________________
1
9What is  
​  5 ​ 4 3? Show your work.
Answer ____________
10Write and solve a division expression you could use to find how many
nickels are in $7. (Hint: What fraction of a dollar is a nickel?)
90
Answer __________________________________________________________________________
UNIT 4
Multiplying and Dividing Fractions
© The Continental Press, Inc. DUPLICATING THIS MATERIAL IS ILLEGAL.
Read the problem. Write your answer to each part.
11Hal is planting beans in a row that is 20 meters long. He plants a bean
1
seed every  
​ 4 ​ meter.
Part AHow many bean seeds can Hal plant in this row?
Explain your answer.
________________________________________________________
________________________________________________________
Division by a
fraction involves at
least two steps:
finding the reciprocal
of the fraction, and
then multiplication.
________________________________________________________
_________________________________________________________________________________
_________________________________________________________________________________
Part BWould Hal be able to plant more seeds or fewer seeds if
1
he planted a bean seed every  
​ 5 ​meter? How many
1
seeds can he plant if he plants them ​  
 ​meter apart?
5
Show your work.
Answer _________________________________________________________________
UNIT 4
Multiplying and Dividing Fractions
© The Continental Press, Inc. DUPLICATING THIS MATERIAL IS ILLEGAL.
91
lems with Frac
b
o
r
P
d
r
tions
Wo
Lesson
6
5.NF.6, 5.NF.7.c
To write a mixed
number as an improper
fraction, multiply the
denominator and the
whole number. Then
add the numerator
to the sum and write
the product over the
denominator.
1
23 →3
6
2
1
7
1→ 2
Multiplying and dividing fractions and whole numbers can help you to
solve many kinds of problems. Think carefully about the operation
needed.
Some problems call for multiplication.
1
A soup recipe for 4 people calls for 1​  
 ​cups of cream. If Dalia
4
wants to make this soup for a family party of 20 people, how
much cream does she need?
First, find how many servings the recipe makes: 4. Dahlia wants
20 servings, and 20 is 5 times greater than 4 servings. So, Dalia
will need to multiply the amount of cream called for by 5.
5
1
535
25
1
5 3 1​  
 ​ 5 5 3 ​  
 ​ 5 ​   
   
​ 5  
​  4  ​ 5 6 ​  
 ​
4
4
4
4
1
Dalia will need 6 ​  
 ​cups of cream.
4
Some problems call for division.
To solve a problem
involving dividing by
a fraction, use that
fraction’s reciprocal and
multiply.
Dalia makes enough soup for 20 servings. Then she learns that
10 more relatives will attend the party. Will Dalia have enough
2
 ​serving of soup?
soup if she gives each person only ​  
3
2
 ​servings you can get from 20 full
To find the number of ​  
3
2
servings, divide 20 by  
​ 3 ​.
2
3
20 3 3
60
20 4 ​  
 ​ 5 20 3  
​  2 ​ 5 ​  
   
​ 5 ​  
  ​ 5 30
3
2
2
If 10 more people will attend, Dalia will need 30 servings. So,
yes, Dalia will have enough soup for 30 people if she gives
2
everyone ​  
 ​of a serving.
3
92
UNIT 4
Multiplying and Dividing Fractions
© The Continental Press, Inc. DUPLICATING THIS MATERIAL IS ILLEGAL.
Read each problem. Circle the letter of the best answer.
1
SAMPLE On one road there is a mailbox every ​  
  ​kilometer. How many
16
mailboxes will there be on 14 kilometers on the road?
A224
B30
7
C14
D​  
 ​
8
The correct answer is A. This question asks you to find the
1
1
1
  ​. Division by ​  
  ​
number of  
​ 16  ​’s in 14, so you need to divide 14 by ​  
16
16
16
1
​  1  ​ 5
is the same as multiplying by its reciprocal, 16: 14 4  
​  16  ​ 5 14 3  
14 3 16 5 224. There will be 224 mailboxes.
1Jaime has 45 trading cards. He wants to
5
trade  
​  9 ​of them with his friend. How many
trading cards does Jaime want to keep?
A81
C 20
B40
D 10
3
5
piece of wood that measures  
​ 4 ​ft by  
​ 6 ​ ft
for the base of the house. What is the area
of the birdhouse’s base?
C 1 ft
5
2
9
B​  
 ​ ft2
8
3
diagram every  
​ 4 ​page. Which expression
can be used to find the number of
diagrams in the report?
3
C 35 2  
​  4 ​
3
D 35 1  
​  4 ​
A35 4  
​  4 ​
2Selma is building a birdhouse. She uses a
4
A​  
 ​ ft2
5
4A report is 35 pages long. There is a
D  
​ 10  ​ ft2
3Gabe walks at a speed of 3 mph. He runs
B35 3  
​  4 ​
3
3
1
5The average person sleeps ​  
 ​of each day.
3
Assuming there are 365 days in a year, for
about how many days does the average
person sleep in a year?
A8
C 90
B40
D 122
2
at a speed 2 ​  
 ​times faster than his
3
walking speed. What is Gabe’s running
speed?
A
2
5 ​  
 ​ mph
3
B 7 mph
C 8 mph
D 22 mph
3
5
6A lake is  
​ 5 ​km long. Hillary rows ​  
 ​ the
8
length of the lake. What distance does
Hillary row?
8
C  
​ 8 ​ km
2
D  
​ 85  ​ km
A​  
  ​ km
13
B​  
 ​ km
3
3
53
UNIT 4
Multiplying and Dividing Fractions
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93
Read each problem. Write your answer.
1
SAMPLE Frank buys a water bottle that can hold  
​ 2 ​liter. His brother and sister
like the bottle so much they each buy one. How much liquid can Frank
and his siblings carry in their bottles altogether?
Answer ____________
Frank and his siblings have total of 3 bottles, and each holds
1
 
​  2 ​liter of liquid. To find the total amount of liquid, multiply:
3
1
1
3 3  
​  2 ​ 5  
​  2 ​ 5 1​  
 ​liters. So, Frank and his siblings can carry
2
1
1​  
 ​liters altogether.
2
7Mrs. Kwan will teach 8 workshops next week. Each workshop is
1
2 hours long. Mrs. Kwan tries to tell a joke every  
​ 3 ​hour. How many
times will she tell a joke by the end of the last workshop? Show your
work.
Answer ________________________
7
8Andre uses ​  
  ​kilogram of bronze to cast a small sculpture. He receives
12
an order for 15 of these sculptures. How much bronze does he need
to make the sculptures? Show your work.
Answer ________________________
9Georgia adds chlorine tablets to the water in her swimming pool. The
1
label on the container states that she should add 1​  
 ​tablets for every
2
10,000 gallons of water. If Georgia’s pool holds 25,000 gallons of
water, how many tablets should she add? Write your answer as a
mixed number.
94
Answer ________________________
UNIT 4
Multiplying and Dividing Fractions
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Read the problem. Write your answer to each part.
10Mr. Green buys 8 loaves of sliced bread on sale.
3
Part AMr. Green puts  
​ 4 ​of the loaves in the freezer. How many
loaves did he not freeze? Explain how you found your
answer.
What operation
does the word of
signal?
_________________________________________________________________________________
_________________________________________________________________________________
_________________________________________________________________________________
1
Part BMr. Green needs  
​ 6 ​of a loaf to make a sandwich. How
many sandwiches can he make using the bread he did not
freeze? Explain how you found your answer.
_________________________________________________________________________________
_________________________________________________________________________________
_________________________________________________________________________________
UNIT 4
Multiplying and Dividing Fractions
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95
R e vi e w
a
n
g
d Dividin
n
i
y
l
p
i
t
l
u
g
M
Fractions
Read each problem. Circle the letter of the best answer.
8
1
1 What is  
​ 15  ​ 3  
​  2 ​?
4
A​  
  ​
15
3
5Margie eats  
​  8 ​of half of an apple. How
much of a whole apple does she eat?
C 1
9
A16
2
C  
​ 11  ​
1
B​  
  ​
17
D 1​  
  ​
15
1
2 What is  
​ 2 ​ 4 8?
1
3
B​  
 ​
4
3
D  
​ 16  ​
1
6How many ​  
 ​-foot pieces can be cut from
3
1
A​  
 ​
2
C  
​ 16  ​
1
B​  
 ​
8
8
 
​ 16  ​
D
5
3Willow owns 225 books. Of these, ​  
 ​ are
9
mysteries. How many of Willow’s books are
mysteries?
A225
C , 225
B . 225
D 0
4 feet of ribbon?
A12
C 32
B16
D 36
2
7Joe has a collection of 60 insects. If ​  
 ​ of
5
1
the insects are moths, and  
​ 3 ​are beetles,
how many insects are not moths or
beetles?
A16
C 44
B24
D 60
4Anton unpacks 4 dozen glasses from a box.
5
He finds  
​ 12  ​of the glasses are broken. How
many glasses are broken?
A48
C 15
B20
D 4
96
8 Which of the following is true?
A
5
 ​ , 8
(8 3 2 ​  
6)
C
1
 ​ , 5
(5 3 1​  
3)
B
2
 ​ . 2
(2 3 1​  
5)
D
3
​  7 ​) . 7
(7 3  
UNIT 4
Multiplying and Dividing Fractions
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Read each problem. Write your answer.
9Twelve people will share 3 pounds of mixed nuts. If they share equally,
what fraction of a pound will each person get? Write your answer in
lowest terms.
Answer ____________
9
10Sixty votes are required to change a village parking law. If ​  
  ​of the
20
100 people in a village vote to change the law, are there enough votes
to change the law? Explain. If not, tell how many more votes are
needed.
_________________________________________________________________________________
_________________________________________________________________________________
11Mr. Falkner has written a company report every 3 months for the last
2
6 years. If ​  
 ​of the reports show his company earns more money than
3
it spends, how many reports show his company spending more money
than it earns?
Answer ____________
3
12If ​  
 ​cup of flour is used to make 4 individual potpies, how much flour
4
should be used to make 12 potpies?
Answer ________________________
13Logan has 9 pounds of trail mix. He will repackage it in small bags
3
of  
​  5 ​pound each. How many bags can he make? Show your work.
Answer ____________
UNIT 4
Multiplying and Dividing Fractions
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97
Read each problem. Write your answer to each part.
14Angela wants to retile the backsplash in her kitchen
using 4 in. by 4 in. tiles, as shown here.
4 in.
Part AWhat is the area of each tile in square feet?
Explain how you got your answer.
4 in.
_________________________________________________________________________
_________________________________________________________________________
_________________________________________________________________________
Part BIf the area of the backsplash is 24 ft2, how many tiles does
Angela need to cover the entire backsplash? Show your work.
Answer ________________________
15Glucose is a common type of sugar. One glucose molecule is made up
1
1
of 24 atoms. ​  
 ​of the atoms are carbon, ​  
 ​are hydrogen atoms,
4
2
1
and ​  
 ​are oxygen atoms.
4
Part AHow many oxygen atoms are there in 50 glucose molecules?
Answer ________________________
Part BExplain how you found your answer to part A.
_________________________________________________________________________
_________________________________________________________________________
_________________________________________________________________________
_________________________________________________________________________
98
UNIT 4
Multiplying and Dividing Fractions
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U n it
5
Algebraic Thinking
Lesson 1 Writing Expressions reviews how to
translate from words to numerical expressions.
Lesson 2 Evaluating Expressions reviews how to
find a value for a numerical expression that may include
parentheses.
Lesson 3 Patterns and Relationships reviews
input-output tables and other relationships between two
variables.
UNIT 5
Algebraic Thinking
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99
Writing Expressions
Lesson
1
5.OA.1, 5.OA.2
A numerical expression is a grouping of numbers and operation
signs. It shows the value of something.
3 1 9 and 12 4 4 are both numerical expressions.
You can translate key words or phrases in a problem into a numerical
expression. Here are some key words for addition and subtraction.
Pay attention
to the order of the
words when writing a
subtraction or division
expression.
5 less than 14 means
14 2 5. The order of the
numbers is reversed.
The order of the words
does not affect a
multiplication or an
addition expression.
The product of 7 and 4
can be written as 7 3 4
or 4 3 7.
Special words like
double and triple
translate to multiplying
a specific number of
times.
So double 7 translates
to 2 3 7.
Remember that a
fraction represents
division.
One-third of 9 can be
1
written as  
​ 3 ​of 9
9
or ​  
 ​.
3
100
Addition
Subtraction
plus
minus
added to
the difference of
more/greater (than)
less/fewer (than)
increased by
decreased by
the sum of
diminished by
5 added to 10 can be written as 10 1 5.
The difference of 7 and 3 can be written as 7 2 3.
Here are key words for multiplication and division.
Multiplication
times
the product of
double/triple etc.
Division
divided by
the quotient of
[a fraction] of
The product of 6 and 8 translates to 6 3 8.
30 divided by 5 translates to 30 4 5.
Read a word problem carefully to tell which operation comes first.
Ayame has 63 comic books. She gives 3 to a friend. She divides
the rest into 5 equal groups.
Ayame gives 3 away, so you need to subtract: 2 3. Ayame
divides the rest into 5 groups, so divide: 4 5. To find out how
many the rest is, you must first find out how many are left
after Ayame gives some comic books away. You need to
subtract first, so put 2 3 inside parentheses and 4 5 outside:
(63 2 3) 4 5
UNIT 5
Algebraic Thinking
© The Continental Press, Inc. DUPLICATING THIS MATERIAL IS ILLEGAL.
Read each problem. Circle the letter of the best answer.
SAMPLE Which expression shows 14 added to the quotient of 6 divided by 2?
A (2 4 6) 1 14
C
(6 4 2) 1 14
B (6 1 2) 4 14
D
6 4 (2 1 14)
The correct answer is C. The phrase 6 divided by 2 is written 6 4 2.
It represents a quotient, so write it inside parentheses: (6 4 2). The
phrase 14 added to is written 1 14. Since it is added to the
quotient, write (6 4 2) 1 14. Choice A reverses the order of the
numbers being divided. Choice B reverses the operations. Choice D
shows the parentheses around the wrong pair of numbers.
1What operation does “the sum of 4 and
18” indicate?
4Which choice does not translate to 5 2 2?
A the difference of 5 and 2
Aaddition
C multiplication
Bsubtraction
D division
2Leon had 12 seashells in his collection.
After a day at the beach, the number of
shells in his collection doubled. Then he
gave 4 to his sister. Which expression tells
how many seashells Leon had left?
B 5 minus 2
C 2 minus 5
D 2 less than 5
5Lori has 6 bananas. Shane has fewer
bananas than Lori. What operation can you
use to find how many bananas Shane has?
A12 2 2
C (12 3 2) 1 4
Adivision
C multiplication
B(12 1 2) 2 4
D (12 3 2) 2 4
Baddition
D subtraction
3 Which phrase indicates subtraction?
A the product of
B the sum of
6Ginger is 7 years old. Her cousin Amy is
4 years younger than 3 times Ginger’s age.
Which expression can you use to find
Amy’s age?
C the difference of
A(7 3 4) 2 3
C 7 3 (3 2 4)
D the quotient of
B(7 3 3) 2 4
D (7 2 4) 3 3
UNIT 5
Algebraic Thinking
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101
Read each problem. Write your answer.
SAMPLE The drawing shows the number of babies in the
hospital nursery on Monday morning. One-half of
the babies are girls. That day 3 more baby girls are
born and no boys. If no babies leave the hospital,
what expression can you use to find the number of
baby girls in the nursery on Monday night?
HOSPITAL NURSERY
Answer ________________________
First, count the number of babies in the nursery: 18. Now, translate
the key words. One-half means divide by the denominator of the
fraction, 2, so 18 4 2. That day indicates that what came before
should be in parentheses, so (18 4 2). The word more signals
addition, so the expression is (18 4 2) 1 3.
7Write the following numerical expression (8 2 4) 3 3 in words.
Answer __________________________________________________________________________
8How does the expression 7 3 (6 1 5) compare to the expression
6 1 5? Explain how you know.
_________________________________________________________________________________
_________________________________________________________________________________
9A park has 39 visitors on Friday. The number of visitors on Saturday is
5 times that, plus 12. Write an expression you could use to find the
number of visitors to the park on Saturday.
Answer ________________________
10Cleo and Vera are playing a game where they answer questions and
earn points. Cleo answers 16 easy questions that are worth 1 point
each. Vera answers 7 fewer questions than Cleo. Vera’s questions are
worth double Cleo’s, because they are harder. Write an expression you
can use to find how many points Vera earned.
102
Answer ________________________
UNIT 5
Algebraic Thinking
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Read the problem. Write your answer to each part.
11Chen and Jim are comparing their coin collections. Jim has
45 pennies. Chen has 20 fewer coins than Jim but his coins
are nickels. To find the value of Chen’s coins, he must find the
difference of 45 and 20, and then find the product of that
difference and 5.
Part AWhat are the key words in the given information?
What parts of an expression do they indicate?
________________________________________________________
________________________________________________________
________________________________________________________
The names of the
coins are clues to one of
the operations. Pennies
are worth $0.01 and
nickels are worth $0.05.
$0.01 3 5 5 $0.05, so
you know you will need
to multiply by 5.
________________________________________________________
_________________________________________________________________________________
_________________________________________________________________________________
Part BWrite a numerical expression Chen can use to find the value
of his collection. Explain how you found your expression.
_________________________________________________________________________________
_________________________________________________________________________________
_________________________________________________________________________________
UNIT 5
Algebraic Thinking
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103
Evaluating Expressions
Lesson
2
5.OA.1, 5.OA.2
To evaluate a numerical expression means to find the value of it.
When you solve a word problem, you write a numerical expression
and find its value.
Nathan has 14 balloons, but 3 of them pop. How many
balloons does Nathan have left?
Word problems do
not always use familiar
key words to translate
to expressions. When
balloons pop, there
are fewer balloons so
this problem involves
subtraction.
Balloons Nathan has
Balloons that pop
Balloons Nathan has left
First, translate the problem into an expression: 14 2 3. When a
problem asks How many?, How much?, What is the total?, or
similar questions, use an equal sign to make the expression an
equation. Then perform the operation.
14 2 3 5 11 → Nathan has 11 balloons left.
If a problem involves more than one operation, decide which
operation needs to be done first. Place parentheses around that
operation to show that it should be done first.
The words one-half of
signals multiplication.
1
Nathan has 14 balloons. Half of them pop. Then he blows up
6 more balloons. How many balloons does Nathan have now?
1
 
 ​ 3 14
​  2 ​of 14 5​  
2
Because multiplication
and division are inverse
operations, you could
also divide by 2 and get
the same result.
1
14
 
  ​ 5 14 4 2
​  2 ​ 3 14 5​  
2
Always work inside
parentheses first.
104
Balloons Nathan has
Half pop
(14
1
)
3  
​  2 ​ Balloons he blows up
16
1
​  2 ​) 1 6 5 ?
(14 3  
7 1 6 5 13
Nathan has 13 balloons.
UNIT 5
Algebraic Thinking
© The Continental Press, Inc. DUPLICATING THIS MATERIAL IS ILLEGAL.
Read each problem. Circle the letter of the best answer.
SAMPLE Marisol has 64 ounces of lemonade to sell. She divides it into
4-ounce cups. Then she sells 13 cups. How many cups of lemonade
does she have left?
A 3
B 4
C 13
D 16
The correct answer is A. This question asks you to find the
number of cups Marisol has left after dividing the original amount
and selling part of it. The expression for this has two operations:
(64 4 4) 2 13. Do the operation in parentheses first, then subtract:
(64 4 4) 2 13 5 16 2 13 5 3.
1 Evaluate the expression.
25 2 (8 2 7)
A1
C 24
B10
D 40
2A pet store has 24 puppies. One-third of
the puppies are male. How many legs do
all the male puppies in the pet store have?
A4
C 24
B8
D 32
3To evaluate the expression below, which
step should be done first?
4 1 (6 3 7)
A add 4 and (6 3 7)
B multiply 6 3 7
C add 4 1 6
D add 4 1 42
4Jake has $25. He spends $14 on a birthday
present for his mother. Then he earns $9
by mowing his neighbor’s lawn. How much
money does Jake have now?
A$11
C $30
B$20
D $34
5 Evaluate 4 3 (20 2 9).
A11
C 44
B16
D 71
6Troy is 6 years old. His brother Damon is
1 year younger than double Troy’s age.
Their brother Mason is 4 years older than
Damon’s age minus Troy’s age. Which
brother is oldest?
ATroy
BDamon
CMason
D They are the same age.
UNIT 5
Algebraic Thinking
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105
Read each problem. Write your answer.
SAMPLE Greendale Elementary School has 300 students that are divided evenly
among 12 classrooms. Mrs. Mendoza teaches in room 6. On Tuesday,
4 of Mrs. Mendoza’s students are absent. How many students are in
room 6 on Tuesday?
Answer ____________
To find the number of students present, write and evaluate an
expression. There are 300 students and 12 classrooms. The
expression 300 4 12 will tell how many students are in each class.
Subtracting 4 students from that number will tell how many were
in room 6 on Tuesday: (300 4 12) 2 4 5 25 2 4 5 21 students.
7A loaf of bread has 16 slices. Then 2 pieces were made into toast. The
rest were made into sandwiches that use 2 slices of bread each. How
many sandwiches were made from the loaf?
Answer ____________
8Cliff is raising money for charity. He collects 32 donations of $5 each.
Then his dad’s company offers to match the amount Cliff collected,
doubling his total. How much money did Cliff raise in all?
Answer ____________
9Alyssa and Mina are baking for their school’s bake sale. Mina bakes
3 loaves of banana bread that she divides into 12 slices each to sell
individually. Alyssa bakes 48 muffins. How many individual items are
Alyssa and Mina contributing to the bake sale?
Answer ____________
10A train has 5 cars that can each hold 128 people. When the train
leaves the station, all the seats are full. At the first stop, 245 people
get off. How many people are left on the train after the first stop?
106
Answer ____________
UNIT 5
Algebraic Thinking
© The Continental Press, Inc. DUPLICATING THIS MATERIAL IS ILLEGAL.
Read the problem. Write your answer to each part.
11On Monday, one share of stock in a computer company cost $58.
On Tuesday, the value of a share dropped $32. On Wednesday,
the value of a share quadrupled. On Thursday, the value of a
share was $19 less than on Wednesday. On Friday, the value of a
share was one-fifth of what it was on Thursday.
Part AWrite and evaluate an expression to find the value of
Remember to
translate key words.
Quadruple means
“4 times as large,”
so multiply by 4.
One-fifth means
1
 ​ or
multiply by ​  
5
divide by 5.
the stock on Wednesday. Then use your answer to
write and evaluate an expression to find the value
of the stock on Friday.
Wednesday _______________________________________
Friday ____________________________________________
Part BMr. Kwon owns some shares of this stock. He wants to sell it
on the day it has the greatest worth so he will make the
greatest profit. On what day should Mr. Kwon sell his stock?
Explain your answer.
_________________________________________________________________________________
_________________________________________________________________________________
_________________________________________________________________________________
_________________________________________________________________________________
UNIT 5
Algebraic Thinking
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107
You can think of
an input-output table
like a machine. You put
the input number into
one end, the machine
changes the number
by applying the rule,
and then outputs the
changed number.
If you know input
values and the rule, you
can find output values.
If the input is 2 and the
rule is “add 10,” the
output is 2 1 10 5 12.
If you know output
values and the rule,
you need to work
backward.
If the output is 25 and
the rule is “add 10,”
subtract 10:
25 2 10 5 15.
The input value is 15.
There are two sets
of values in an inputoutput table. Each pair
of values in the table
results in one point.
Input values go on the
horizontal axis. Output
values go on the
vertical axis.
5.OA.3
An input-output table shows two sets of numbers that are related
by a rule. The input numbers are one set. The mathematical rule is
applied to the input numbers and the output numbers are the result.
What is the rule
for this table?
IN
0
1
2
3
OUT
2
3
4
5
Compare the input and output values in each column to figure
out the rule. In the example, 0 1 2 5 2, 1 1 2 5 3, 2 1 2 5 4,
and 3 1 2 5 5. Each input value has 2 added to it to get the
output value. So the rule for this table is “add 2.”
y
10
9
8
7
6
5
4
3
2
1
0
0 1 2 3 4 5 6 7 8 9 10 x
You can graph the data
from an input-output table.
The line that connects the
points shows the pattern
made by the points.
INPUT
You can graph the values from
more than one table on the same
graph and compare the rules.
y
The graph to the right shows the
values from the first input-output
table above and the table below.
IN
0
1
2
3
OUT
4
5
6
7
10
9
8
7
6
5
4
3
2
1
0
0 1 2 3 4 5 6 7 8 9 10 x
OUTPUT
3
s and Relationsh
n
r
e
t
t
a
P
ips
OUTPUT
Lesson
INPUT
The rule for the second table is “add 4.”
108
UNIT 5
Algebraic Thinking
© The Continental Press, Inc. DUPLICATING THIS MATERIAL IS ILLEGAL.
Read each problem. Circle the letter of the best answer.
SAMPLE What is the rule for this input-output table?
IN
1
2
3
4
OUT
3
6
9
12
A add 2
B multiply by 3
C
divide by 3
D
subtract 2
The correct answer is B. To find the rule, test each answer choice
against the values. The rule must work for all pairs. Choice A, “add
2,” only works for the first pair, 1 1 2 5 3, but 2 1 2 ? 6. Choices C
and D would not work unless the output numbers were smaller
than the input. The rule in choice B, “multiply by 3,” works for all
pairs: 3 3 1 5 3, 3 3 2 5 6, 3 3 3 5 9, and 3 3 4 5 12.
1Raj is making an input-output table for the
rule “add 5.” What is the output value for
an input value of 2?
A2
C 7
B5
D 10
2What is the rule for the input-output table
4Following the rule, which pair of values
belongs in the table?
IN
20
16
12
8
OUT
5
4
3
2
A 1, 4
C 4, 2
B 4, 1
D 2, 1
below?
IN
10
9
8
7
OUT
3
2
1
0
A subtract 7
C divide by 7
B add 7
D subtract 6
3Katie wants to make an input-output table
with number of weeks as input values and
number of days as output values. What is
the rule for Katie’s input-output table?
A divide by 4
C divide by 7
B add 7
D multiply by 7
5Rockville’s Little League uses the inputoutput table below. The input values are
the number of players that sign up. The
output values are the number of teams
they can make. How many teams can be
made if 108 players sign up?
IN
36
45
54
63
OUT
4
5
6
7
A9
C 12
B10
D 18
UNIT 5
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109
Read each problem. Write your answer.
SAMPLE Roxanne is growing a plant for a science project.
WEEK
She measures it and finds that it grows 3 inches
every week. Write a rule for an input-output table
to track the plant’s height. Then complete the
input-output table.
HEIGHT
(IN.)
1
2
4
6
Answer ____________________________________
10
The input values are the number of weeks. If the plant grows
3 inches a week, then the height will be 3 3 the number of weeks
any given week. The rule is “multiply by 3.” To complete the table,
apply the rule to find the output values. 3 3 1 5 3, 3 3 2 5 6,
3 3 4 5 12, 3 3 6 5 18, 3 3 10 5 30.
6Ron is making an input-output table for the rule “multiply by 2.” What
will be true about the output values, compared to the input values?
Answer __________________________________________________________________________
7Faith is training to run a marathon. She knows she can run one mile in
9 minutes. How would using an input-output table be helpful in
predicting what her final time might be?
_________________________________________________________________________________
_________________________________________________________________________________
_________________________________________________________________________________
8Priya squeezes 4 oranges to get one cup of orange juice. She wants to
make an input-output table to figure out how many cups of juice she
can get out of different numbers of oranges. If the number of oranges
are the input values and the number of cups of juice are the output
values, what rule can she use to complete her table?
110
Answer ____________________________________
UNIT 5
Algebraic Thinking
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Read the problem. Write your answer to each part.
9A baker is making muffins. Each pan holds 12 muffins.
Part AWrite the rule for the input-output table. Then create an
input-output table, using 1, 2, 3, 4, and 6 pans as your input
values.
Answer ____________________________________
IN
OUT
Part BThe baker also makes mini-muffins in pans that can hold
24. Make an input-output table, using the same input
values as part A. Graph the results of both tables on the
same graph.
IN
How can you
change 12 to get
24? How might this
change affect the
number of minimuffins compared
to the number of
muffins?
y
OUT
144
132
120
108
96
84
72
60
48
36
24
12
0
0
1
2
3
4
5
6 x
What do you notice about the numbers of muffins and
mini-muffins?
Answer _________________________________________________________________
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111
R e vi e w
Algebraic Thinking
Read each problem. Circle the letter of the best answer.
1Kerrin is making an input-output table for
the rule “add 5.” What is the output value
for an input value of 5?
A0
C 10
B5
D 25
2Which expression shows the difference of
5Carmen collects $53. Diego collects $61.
They combine the money and donate an
equal amount to each of 3 different
charities. Which expression shows how
much they will donate to each charity?
A53 1 61 4 3
C (53 1 61) 4 3
B3 4 (53 1 61)
D (61 2 53) 4 3
50 and 15 multiplied by 3?
A50 2 15 3 3
C 3 3 50 2 15
B(50 2 15) 3 3
D (15 2 50) 3 3
3What is the value of (38 2 16) 4 2?
A11
C 30
B22
D 44
6Which operation does triple indicate?
Aaddition
C multiplication
Bsubtraction
D division
7Mr. Day owns a bicycle shop. He has
23 tricycles in stock. He has 14 more
bicycles than tricycles. How many wheels
are on all the bicycles at Mr. Day’s store?
4A chef uses the input-output table below
to find how many cups of grated cheese he
can make from a certain number of ounces
of cheese. How many cups of grated
cheese can he get from 36 ounces?
IN
6
12
24
OUT
1
2
4
A18
C 74
B37
D 173
8Soo Ha wants to make an input-output
table using quarts as input values and
gallons as output values. What would be
the rule for Soo Ha’s table?
A3
C 6
A divide by 4
C add 4
B4
D 12
B multiply by 4
D divide by 2
112
UNIT 5
Algebraic Thinking
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Read each problem. Write your answer.
9The school cafeteria is serving pizza for lunch, cut
as shown. The staff serves 15 pizzas and has 4 slices
left over. Write an expression the staff can use to
figure out how many pieces were served.
Answer ____________________________________
10A youth group is going camping. Their tents hold 6 people each.
Write the rule the group can use for an input-output table if the
input values are the numbers of tents and the output values are
the numbers of campers. Then complete the table at the right.
Answer ____________________________________
IN
OUT
1
2
3
5
10
11Barbara earns $9 for every hour that she baby-sits plus $5 per job
for transportation costs. How much does Barbara earn for a 4-hour
baby-sitting job?
Answer ____________
12Write the expression (15 3 3) 2 12 using words.
Answer __________________________________________________________________________
13Juan plays baseball. He gets a hit once in every five at-bats. Explain
how he could use an input-output table to predict how many hits he
will get in 100 at-bats.
_________________________________________________________________________________
_________________________________________________________________________________
_________________________________________________________________________________
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113
Read each problem. Write your answer to each part.
14Ella is playing a video game. She has to solve puzzles to earn points.
There are eight puzzles at every level and six levels in the game. There
are also 15 bonus puzzles she can solve for extra points.
Part AWrite an expression Ella can use to find the total number of
puzzles in the game.
Answer ____________________________________
Part BEach puzzle is worth 1 point. A player needs 50 points to
win the game. Ella solves all the puzzles on 5 of the levels
and 12 of the bonus puzzles. Does she earn enough points
to win? Explain your answer.
_________________________________________________________________________________
_________________________________________________________________________________
_________________________________________________________________________________
_________________________________________________________________________________
15Bill is 13 years old. His sister Bonnie is 2 years older than he is. Bill
wants to make an input-output table to figure out how old his sister
will be when he reaches various ages.
Part AWhat rule can Bill use for his input-output table? What will
be his input and output values?
_________________________________________________________________________________
_________________________________________________________________________________
Part BIn the state where Bill and Bonnie live, a person needs to be
18 years old to get his or her driver’s license. Complete the
input-output table at the right for Bill and Bonnie’s ages.
How old will Bill be when Bonnie gets her driver’s license?
IN
OUT
13
14
16
19
114
Answer ____________________________________
23
UNIT 5
Algebraic Thinking
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U n it
6
Measurement
Lesson 1 Customary Units of Measure reviews
customary units or length, weight, capacity, and time.
Lesson 2 Metric Units of Measure reviews metric
units of length, mass, and capacity.
Lesson 3 Measurement Conversions reviews how
to convert among standard measurement units within a
given measurement system.
Lesson 4 Measurement Word Problems
reviews how to solve multi-step problems involving
measurements.
Lesson 5 Measurement Data reviews how to solve
problems involving measurement information in line plots
by using operations on fractions.
UNIT 6
Measurement
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115
Lesson
1
You can think of
familiar objects to help
you remember the sizes
of units.
An inch is about the
width of a quarter.
A yard is about the
height of a doorknob
from the floor.
Customar y Units of
Measurement
5.MD.1
In the United States, the customary system is used to measure
length, capacity, weight, and time. Below are customary units you may
know and their abbreviations.
Length
inch (in.)
foot (ft)
yard (yd)
mile (mi)
Capacity
cup (C)
fluid ounce (fl oz)
pint (pt)
quart (qt)
gallon (gal)
Weight
ounce (oz)
pound (lb)
ton (T)
Time
Larger units of time
include
decade 5 10 years
century 5 100 years
millennium 5 1,000
years
The most appropriate
unit is the one in which
a measurement can be
expressed using the
smallest whole number
or largest fractional
part or decimal.
second (sec)
minute (min)
hour (hr)
day
week (wk)
month (mo)
year (yr)
Choose the most appropriate unit to measure.
Which unit would be best for measuring the depth of a
swimming pool?
Depth is a linear measurement, so choose a unit of length. The
depth of a pool may vary, but usually is deeper than a person is
tall. An inch is a small unit, about the distance from the tip of
your thumb to the first knuckle. This is too small. A mile is the
distance a person can walk in about 20 minutes, so this is much
too large. The height of a person is usually measured in feet, so
feet would be an appropriate unit for measuring the depth of a
swimming pool.
Helena wants to weigh a banana. What would be a typical
weight, 5 ounces or 5 pounds?
An ounce is a small weight, about the weight of 5 quarters.
There are 16 ounces in a pound, which is about the weight of
package of 4 sticks of butter. A banana is around the same
weight as a stick of butter, so a weight of 5 ounces is more
reasonable than 5 pounds.
116
UNIT 6
Measurement
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Read each problem. Circle the letter of the best answer.
SAMPLE Which of the following is the most likely measurement of the thickness
of a dictionary?
A 4 ft
B 4 lb C 4 in. D 4 qt
The correct answer is C. The thickness of a book is a linear
measurement and requires a unit for length. The unit needs to be a
small one, such as inches. Choice A uses a unit for length, but the
unit is too large—a book cannot be 4 feet thick. Choice B is
incorrect because pounds are a unit of weight. Choice D is incorrect
because a quart is a unit of capacity.
1Which animal’s weight would be the most
appropriate to measure in ounces?
Araccoon
C dog
Bcow
D bird
2Which capacity would be best measured
using gallons?
A a kitchen sink
4The blue whale is the largest animal on
Earth. Which of the following is the most
likely measurement of a blue whale’s
weight?
A 200 tons
B 200 ounces
C 200 miles
D 200 pounds
B a mixing bowl
C a drinking glass
D a juice bottle
3Janelle sat through several movie trailers
before the movie started. Which is the
most appropriate unit for the running time
of the movie trailers?
Aseconds
Bhours
Cpints
Dminutes
5What distance would be represented by a
length of 100 miles?
A from your home to your school
B between New York and Philadelphia
C between Los Angeles and Miami
D from Earth to the moon
6Trinh is making oatmeal for her breakfast.
Which is the most appropriate amount of
water Trinh needs to add?
A 1 ounce
C 1 quart
B 1 cup
D 1 gallon
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117
Read each problem. Write your answer.
SAMPLE Would the capacity of a bathtub more likely be 40 gallons or 40 pints?
Explain your answer.
Answer ____________
The capacity of a bathtub is more likely to be 40 gallons. A
bathtub can hold a large amount of water, so you would want to
use a large unit, such as a gallon. A pint is too small of a unit to use
to measure the capacity of a bathtub.
7What is the most appropriate unit to use to measure the wingspan of
a butterfly?
Answer ________________________
8Would a postal scale used to weigh letters be a good choice to weigh
a box of books? Explain.
_________________________________________________________________________________
_________________________________________________________________________________
9What unit would you use to measure the amount of time it takes you
to sneeze?
Answer ________________________
10Sarah says that her kitten’s water dish can hold 2 gallons of water. Do
you think she is correct? Explain.
_________________________________________________________________________________
_________________________________________________________________________________
118
UNIT 6
Measurement
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Read the problem. Write your answer to each part.
11Carlo is shopping for a few last-minute things for a party. However, his
shopping list is missing all the units of measure!
Shopping List
16 ___________ bottle of lemon juice
2 ____________ bag of potatoes
1 ____________ of milk
3 ___________ fabric for tablecloth
Part AFill in the blanks in Carlo’s list with appropriate units of
measure for each item.
Part BCarlo is making fruit punch for the party. A friend tells
him to use 2 bowls of orange juice and 1 bowl of
pineapple juice, and 1 bowl of cranberry juice to make
the punch. Why is a bowl not an appropriate unit to
measure the amounts of juice? Explain.
Think about the
size of a bowl. Are
all bowls the same
size?
_________________________________________________________________________________
_________________________________________________________________________________
_________________________________________________________________________________
_________________________________________________________________________________
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119
Lesson
2
ts of Measurem
i
n
U
c
i
r
t
ent
Me
The metric system
is also known as the
International System
(SI). In the United
States, the metric
system is widely used
in science, the military,
and industry.
Mass and weight are
not exactly the same,
but you can think of
them that way for the
purposes of measuring
objects.
5.MD.1
Around the world, the metric system is used to measure length,
capacity, and mass.
In the metric system, one base unit is
defined for each type of measurement.
The meter is the base unit for length.
The liter is the base unit for capacity.
And the gram is the base unit for mass.
Very large and very small measurements
are expressed as multiples of ten of the
base unit. Prefixes are added to the base
unit to create the additional units.
Metric
Prefixes
Unit
Multiples
kilo-
1,000
hecto-
100
deca-
10
base unit
1
deci-
0.1
centi-
0.01
milli-
0.001
Below are the metric units used to measure length, capacity, and mass
and their abbreviations.
Length
millimeter (mm) A dime is about 1 mm thick.
centimeter (cm) A fingernail is about 1 cm wide.
meter (m)
A baseball bat is about 1 m in length.
kilometer (km)80 school buses placed end to end are about 1 km.
milliliter (mL)
liter (L)
Capacity
About 20 drops of water is equal to 1 mL.
A regular water bottle holds about 1 L.
gram (g)
kilogram (kg)
Mass
A paper clip weighs about 1 g.
An 8-week-old kitten weighs about 1 kg.
Choose the most appropriate unit to measure.
Would the length of a typical school bus be 11 meters or
11 kilometers?
A meter is short, about the length of a baseball bat. A kilometer
is long, about the distance you can walk in 15 minutes. You can
picture measuring a school bus with a baseball bat, so 11 meters
is a reasonable length.
120
UNIT 6
Measurement
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Read each problem. Circle the letter of the best answer.
SAMPLE What is the most appropriate unit to measure how heavy a bowling
ball is?
A g B cm
C mL
D kg
The correct answer is D. The weight of a bowling ball requires a
unit of mass and the unit needs to be large. Choice A uses a unit of
mass, but a gram is too small to use to find the mass of a bowling
ball. Choice B is incorrect because the centimeter is a unit of length.
Choice C is incorrect because the milliliter is a unit of capacity, not
mass.
1Which of the following is the most likely
measurement of a sink’s capacity?
4Pam’s family is driving from Chicago to
St. Louis. About how far is the trip?
A 68 m
C 68 L
A 472 mm
C 472 m
B 68 mL
D 68 kg
B 472 kg
D 472 km
2Which object is most appropriately
measured using centimeters?
5Which object‘s mass would be most
appropriate to measure in grams?
A the capacity of a jug
A a marble
B the length of a shoe
B a cow
C the weight of a hat
C a person
D the height of a house
D a watermelon
3The wolf spider is a small spider that does
6Dillora is making lemonade for four friends.
not spin a web. Instead, it hunts by hiding
and then attacking insects. Which of the
following is the most likely measurement of
a wolf spider’s mass?
A 5 g
C 5 L
B 5 cm
D 5 kg
Which is the most appropriate amount of
water she needs?
A 20 milliliters
B 2 liters
C 20 grams
D 200 millimeters
UNIT 6
Measurement
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121
Read each problem. Write your answer.
SAMPLE Which of the following would you be more likely to measure using
meters—the height of a fence or the width of a penny?
Answer ________________________
The height of a fence is much longer than the width of a penny.
It is more appropriate to measure the fence in meters. The width of
a penny is much shorter than 1 meter, so you would want to use a
smaller unit when measuring it.
7Would you be more likely to measure the mass of an apple in grams or
kilograms? Explain.
_________________________________________________________________________________
_________________________________________________________________________________
8Is a perfume bottle more likely to contain 40 mL or 40 L of perfume?
Explain.
_________________________________________________________________________________
_________________________________________________________________________________
9What unit would you use to measure the length of a ladybug? Explain.
_________________________________________________________________________________
_________________________________________________________________________________
10Jonah says that his drinking glass can hold 0.5 milliliter. Do you think
he is correct? Explain.
_________________________________________________________________________________
_________________________________________________________________________________
122
UNIT 6
Measurement
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Read the problem. Write your answer to each part.
11Leila created the chart below to help her sister learn about metric
units.
Part AComplete the chart using the items in the box to
indicate the most appropriate unit to use to measure
each item. Use each item only once.
Unit
Item to Measure
gram
kilometer
milliliter
centimeter
When you are
trying to choose an
appropriate unit to
measure something,
you need to determine
the unit that allows
you to express the
measurement with the
smallest whole
number or the largest
fraction or decimal.
liter
meter
kilogram
millimeter
•
•
•
•
•
•
mass of a box of old clothes • length of a bed
width of a pea
• mass of a necklace
distance between two cities
amount of water used by a washing machine
height that a plant grows in a week
amount of liquid medicine to give a pet
Part BLeila’s sister thinks that you can measure the width of a pea
in grams because a pea is rather small. Is she correct?
Explain.
_________________________________________________________________________________
_________________________________________________________________________________
_________________________________________________________________________________
_________________________________________________________________________________
UNIT 6
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123
Lesson
3
ment Conversi
e
r
u
s
a
e
ons
M
Converting among
units in the metric
system is easier than in
the customary system.
Remember that in the
metric system, units are
expressed as multiples
of 10 of the base unit.
This means that you can
convert metric units by
multiplying or dividing
by a power of 10. You
can do that by moving
the decimal point left
or right.
5.MD.1
Use the relationships among units in each measurement system to
convert measures into equivalent measures. Below are the
conversion factors for some customary and metric units.
Customary Units
Length
Capacity
Weight
Time
1 ft 5 12 in.
1 C 5 8 fl oz
1 lb 5 16 oz
1 min 5 60 sec
1 yd 5 36 in.
1 pt 5 2 C
1 T 5 2,000 lb
1 hr 5 60 min
1 yd 5 3 ft
1 qt 5 2 pt
1 day 5 24 hr
1 mi 5 5,280 ft
1 gal 5 4 qt
1 wk 5 7 days
1 mi 5 1,760 yd
1 gal 5 8 pt
1 yr 5 12 mo
1 gal 5 16 C
Metric Units
Length
Capacity
Weight
1 cm 5 10 mm
1 L 5 1,000 mL
1 kg 5 1,000 g
1 m 5 100 cm
When you convert a
larger unit to a smaller
unit, you end up with
more units.
1 m 5 1,000 mm
1 km 5 1,000 m
To convert a larger unit to a smaller unit, multiply.
How many cups equal 4 gallons?
When you convert a
smaller unit to a larger
unit, you end up with
fewer units.
If you are trying to
compare two amounts
with different units, it
doesn’t matter which
unit you choose to
convert. However, it is
usually easier to convert
to the smaller of the
two units.
1 gal 5 16 C
To find the number of cups in
4 gallons, multiply:
4 gal 3 16 5 64 C
1 km 5 1,000 m
To find the number of meters in
8 kilometers, multiply:
8 km 3 1,000 5 8,000 m
To convert a smaller unit to a larger unit, divide.
How many feet are equal to
48 inches?
1 ft 5 12 in.
To find the number of feet in
48 inches, divide:
48 in. 4 12 5 4 ft
124
How many meters are in
8 kilometers?
How many kilograms are
3,000 grams equal to?
1 kg 5 1,000 g
To find the number of kilograms
in 3,000 grams, divide:
3,000 g 4 1,000 5 3 kg
UNIT 6
Measurement
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Read each problem. Circle the letter of the best answer.
SAMPLE How many yards are in 144 inches?
A 108
B 48
C 4
D 5,184
The correct answer is C. You need to convert a small unit, inches,
to a larger unit, yards, so you need to divide. Since 1 yard equals
36 inches, the conversion factor is 36. Divide: 144 4 36 5 4. There
are 4 yards in 144 inches.
1Which of the following conversions is
correct?
A 5,100 g 5 51 kg
B 2,500 g 5 2.5 kg
C 4,000 g 5 40 kg
D 1,500 g 5 150 kg
2It takes Arjun 480 sec to walk home from
school. How many minutes is this?
A8
B10
C16
D28,800
3How many milliliters are equal to 4.5 liters?
A0.45
4Isobel is making 3 quarts of stew. How
many cups of soup is that equal to?
1
A​  
 ​
2
B6
C9
D12
5An elephant weighs 4,000 pounds. How
many tons is that weight?
A1
B2
C4
D2,000
6Which of the following is greater than
40 meters?
B45
A 4 km
C450
B 40 cm
D4,500
C 400 cm
D 4,000 mm
UNIT 6
Measurement
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125
Read each problem. Write your answer.
SAMPLE A rosebush stands 4 feet tall. A lilac bush stands 60 inches tall. Which
bush is taller? Explain.
To compare the heights, you need to convert one of the units so
that both units are the same. You can either convert feet to inches or
inches to feet. Since 1 ft 5 12 in., multiply: 4 3 12 5 48 in. Now,
compare: 48 , 60, so the lilac bush is taller. Or, divide: 60 4 12 5 5 ft.
Compare: 4 , 5, so the lilac bush is taller.
7Kyle needs 48 ounces of flour. Which bag of flour should he buy and
how many of them does he need? Explain.
FLOUR
FLOUR
FLOUR
1 lb
1.5 lb
2 lb
_________________________________________________________________________________
_________________________________________________________________________________
8Danita is working on a new sculpture. It will be 2 m tall when she is
finished. Will the sculpture fit in a room with a 300-cm ceiling?
Explain.
_________________________________________________________________________________
_________________________________________________________________________________
9Is 25 L greater than 20 L 500 mL? Explain.
_________________________________________________________________________________
_________________________________________________________________________________
126
UNIT 6
Measurement
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Read the problem. Write your answer to each part.
10The chart below lists the regulation sizes and weights of the balls
used in various sports.
Type of Ball
Circumference
Weight
golf ball
1.68 in.
1.62 oz
soccer ball
2.25–2.33 ft
0.875–1 lb
baseball
9–9.25 in.
5–5.25 oz
ping pong ball
1.57 in.
0.006 lb
basketball
29.5–30 in.
1.25–1.375 lb
bowling ball
not more than 2.25 ft
not more than 16 lb
tennis ball
2.5–2.625 in.
2–2.17 oz
volleyball
25.6–26.4 in.
9.2–9.9 oz
Circumference is
the distance around
a circle. For a sphere
like a ball, the
circumference is the
distance around the
middle of the ball.
Part AFor each pair listed below, identify which ball is larger or
heavier.
• larger: soccer ball or basketball? ______________________________
• heavier: volleyball or soccer ball? ______________________________
• heavier: ping pong ball or golf ball? ______________________________
Part BNestor is playing tennis with a ball that is 0.2 ft in
circumference and weighs 0.125 lb. Is this a regulation tennis
ball in terms of size and weight? Explain.
_________________________________________________________________________________
_________________________________________________________________________________
_________________________________________________________________________________
_________________________________________________________________________________
_________________________________________________________________________________
UNIT 6
Measurement
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127
Lesson
4
ent Word Prob
m
e
r
u
s
a
lems
Me
There are three
things to do before you
try to solve a multi-step
problem.
1 Identify the key
information in the
problem.
2 Identify what the
problem is asking
you to find.
3 Come up with a
plan to solve the
problem.
Before you try to find
the solution, think
about what you need
to do to solve the
problem. You may want
to create a flowchart or
other graphic organizer
to list the steps.
5.MD.1
You can use what you know about converting units of measure to
help you solve problems.
Often, solving a word problem takes more than one step. To solve a
multi-step problem, plan carefully and then carry out your plan.
Lily makes 3 gallons of ice cream. She wants to divide it equally
into 6 containers. Each container holds 7 cups. Will Lily be able
to fit all the ice cream in the 6 containers?
Solving this problem requires more than one step.
1.
Identify the key information in the problem.
• 3 gallons of ice cream
• divided equally into 6 containers
• each container holds 7 cups
2. Identify what the problem is asking you to find.
Will 3 gallons of ice cream fit into 6 7-cup containers?
3.First, convert 3 gallons to cups, since you know that each
container can hold 7 cups.
1 gal 5 16 C
3 gal 3 16 5 48 C
4.Divide the number of cups of ice cream by the number of
cups each container holds.
48 C 4 7 C 5 6 R6 containers
5.Compare this number to the number of available containers.
6 R6 containers . 6 containers
This means that 3 gallons will fill more than 6 containers with a
7-cup capacity. So, Lily will not be able to fit all the ice cream
into the 6 containers.
128
UNIT 6
Measurement
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Read each problem. Circle the letter of the best answer.
SAMPLE The Eiffel Tower in Paris has three observation decks. One deck is
5,700 cm above the ground, another deck is 115 m above the
ground, and the other deck is 0.275 km from the ground. What
is the difference, in meters, between the highest and lowest
observation decks?
A 58 m
B 218 m C 332 m
D 455 m
The correct answer is B. Convert all the units to meters:
5,700 cm 5 5,700 4 1,000 5 57 m and 0.275 km 5 0.275 3 1,000 5
275 m. The highest deck is 275 m and the lowest is 57 m. Subtract:
275 m 2 57 m 5 218 m.
1Galen is driving to the beach with his dog.
The car weighs 1 T 960 lb, Galen weighs
155 lb, and his dog weighs 320 oz. What is
the combined weight of Galen, his dog,
and the car, in pounds?
A 1,526 lb
C 3,115 lb
B 2,785 lb
D 3,135 lb
2Tilda is mailing two packages. One package
weighs 72 oz and the other weighs 80 oz.
If the shipping company charges $1.50 per
pound, how much will Tilda pay to ship
both packages?
A$1.43
C $21.00
B$14.25
D $75.00
3Alan has a watering can with a capacity of
1 L. He uses 2,500 mL of water for all of his
plants. How many times does Alan have to
fill the watering can to water all his plants?
1
4Gia walked 2 ​  
 ​miles on Monday,
2
1
2,640 yards on Wednesday, and 1​  
 ​ miles
4
on Friday. How far did she walk in all?
A 3 mi
B
1
4 ​  
 ​ mi
4
1
C 5 ​  
 ​ mi
4
D 6 mi
5Jenna takes her laundry to a cleaner that
charges $5.25 per kilogram for washing
and drying. How much will Jenna pay if the
mass of her laundry is 4,000 grams?
A$0.21
C $21.00
B$2.10
D $210.00
6The Kwan family is driving on the Seven
Mile Bridge, a bridge that connects parts
of the Florida Keys. The bridge is about
11,000 m long. If they drive an average
speed 44 km/hr, how long will it take to
travel the length of the bridge?
Aone
C three
A 25 hr
C 0.5 hr
Btwo
D four
B 10 hr
D 0.25 hr
UNIT 6
Measurement
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129
Read each problem. Write your answer.
SAMPLE Gwen is adding a tile border to a wall. The wall is 6 m long and each
tile is 5 cm long. If the tiles come in packs of 20 each, how many
packs does Gwen need to buy?
Answer ____________
First, convert the measure of the wall from meters to centimeters:
6 m 5 600 cm. Divide the length of the wall by the length of one
tile to find out how many tiles Gwen needs: 600 cm 4 5 cm 5 120,
so Gwen needs 120 tiles. If the tiles come in packs of 20, Gwen
needs to buy 120 4 20, or 6 packs of tiles.
7Brett is shopping for bananas.
$3.19
Which is the better buy? Explain.
1,900 g
$3.00
1.5 kg
_________________________________________________________________________________
_________________________________________________________________________________
_________________________________________________________________________________
8At a county fair, Jesse sees a watermelon that weighs 150 lb 14 oz. His
sister tells him that that the world’s largest watermelon weighed in at
268 lb 13 oz. What is the difference in weight between the two
watermelons? Give your answer in pounds and ounces.
Answer __________________________________________________________________________
9Ned uses 120 pints of water to take a 5-minute shower, Sandy uses
100 quarts for a 10-minute shower, and Landon uses 24.8 gallons for
an 8-minute shower. Who uses the most gallons of water per minute?
_________________________________________________________________________________
_________________________________________________________________________________
130
UNIT 6
Measurement
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Read the problem. Write your answer to each part.
10Mr. and Mrs. Perez are redecorating their living room. The dimensions
of one wall in the living room are shown below.
10 ft 6 in.
20 ft 10 in.
Part AMrs. Perez wants to hang a painting in the middle of
the wall. How many inches away from the corner is the
center of the wall? Explain how you found your answer.
Finding the
center along the
length of the wall is
the same as finding
the halfway point
along the length of
the wall. How can
you find the
halfway point?
Answer _________________________________________________________________
_________________________________________________________________________
_________________________________________________________________________
_________________________________________________________________________
_________________________________________________________________________
Part BMr. Perez is hanging a light in the living room. The light will
hang down 1 yd 2 in. from the ceiling. If Mr. Perez is 6 ft
tall, will he be able to walk under the light without hitting
his head? Explain.
Answer _________________________________________________________________
_________________________________________________________________________
_________________________________________________________________________
_________________________________________________________________________
_________________________________________________________________________
UNIT 6
Measurement
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131
Lesson
5
A line plot is
sometimes called a dot
plot and dots are used
instead of X’s.
Intervals on a line plot
should be equal. So use
equivalent fractions
to help you order the
values in the data set.
Measurement Data
5.MD.2
A line plot is used to display a data set. It shows the values of the
data set on a number line. X’s above each value show the frequency,
or the number of times, that value appears in the set.
Mrs. Jackson recorded the heights of her students in the table
below. Make a line plot to display the data set.
Height of Students (feet)
5
4 6
2
4 3
1
4 6
1
5 12
3
4 4
1
3
1
4 2
4 6
2
4 4
1
4 3
7
4 2
4 12
1
5
4 6
3
4 4
1
First, find the least and greatest data values: 4 ​  
 ​ and 5 ​  
  ​.
6
12
Draw a number line from the least to greatest values. Divide it
into equal intervals. All the fractions here can be renamed as
equivalent fractions with a denominator of 12, so divide the
space into twelfths. You can write the equivalent fraction
underneath to make the number line easier to use.
4
1
4 12
2
4 12
3
4 12
4
4 12
5
4 12
1
6
4 12
7
4 12
1
4 6
8
4 12
2
4 2
4 3
9
4 12
3
4 4
11
10
4 12
4 12
5
1
5 12
5
4 6
Plot an X above the number line for each value in the data set.
X
X
4
1
4 12
2
4 12
1
4 6
X
X
3
4 12
4
4 12
5
4 12
6
4 12
1
4 2
X
7
4 12
X
X
8
4 12
2
4 3
X
X
X
9
4 12
3
4 4
X
X
X
11
10
4 12
4 12
5
1
5 12
5
4 6
You can use a line plot to solve problems involving measurement.
1
 ​feet tall or
How many students in Mrs. Jackson’s class are 4 ​  
2
taller?
1
6
The line plot shows that 4 ​  
 ​feet is equal to 4 ​  
  ​, so count the
2
12
X’s above that tick mark and to the right. There are 11 X’s.
So, there are 11 students this height or taller.
132
UNIT 6
Measurement
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Read each problem. Circle the letter of the best answer.
SAMPLE The data set below shows the capacities (in gallons) of buckets that a
store has in stock.
1
4
5 2
1
1
3 4
1
4 2
3 4
1
5 2
1
4 4
1
2
5 2
1
3 2
If you were constructing a line plot for this data set, how many X’s
1
would you write for the value of 5 ​  
 ​ gal?
2
A one
B two
C three
D four
1
The correct answer is C. The value for 5 ​  
 ​is listed three times in
2
the data set. So you would write three X’s for this value.
Use the line plot below to answer
questions 3 and 4.
Use the line plot below to answer
questions 1 and 2.
X
X
0
1
2
1
4
DAILY RAINFALL
X
X
X
X
3
4
1
1 4
1
1
1 2
DISTANCE FROM SCHOOL
X
X
X X
X X X
X
2
10
9
10
X
3
1 4
2
Inches
0
1
10
3
10
4
10
5
10
6
10
7
10
8
10
1
Miles
1What is the total amount of rain that fell
during the week?
A
1
8 ​  
 ​ in.
4
B
7 ​  
 ​ in.
4
1
C 7 in.
1
D 6 ​  
 ​ in.
4
2What is the difference between the
greatest and least amount of rainfall?
1
A​  
 ​ in.
2
3
B​  
 ​ in.
4
3Students living less than 1 mile from school
are allowed to walk. How many students
walk?
Aone
C five
Btwo
D eight
4What fraction of the students live less
3
than  
​  4 ​mile from school?
1
C 1​  
 ​ in.
2
1
C  
​ 8 ​
3
D  
​ 10  ​
A​  
 ​
2
7
3
D 1​  
 ​ in.
4
B​  
 ​
4
7
UNIT 6
Measurement
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133
Read each problem. Write your answer.
SAMPLE The line plot shows the weights, in ounces, of 9 lab mice.
WEIGHTS OF LAB MICE
X
X
X
X
X
X
X
X
5
10
6
10
7
10
8
10
9
10
1
X
1
1 10
2
1 10
Ounces
What is the median weight of the 9 mice?
Answer ____________
The median weight is the weight in the middle. There will be an
equal number of weights to the right and the left of it. There are
9 weights, so the fifth weight is the median. The fifth weight
8
shown is  
​ 10  ​ ounce.
1
1
5The weights of ten newborn kittens are 3 ​  
 ​, 3 ​  
 ​,
4
2
1
3
3
1
1
3 ​  
 ​, 4, 3 ​  
 ​, 2 ​  
 ​, 3, 3, 3 ​  
 ​, 3 ​  
 ​ ounces. Hugo
4
4
2
2
2
made the line plot at the right for the data set.
Is Hugo’s line plot correct? Explain.
X
WEIGHTS OF NEWBORN KITTENS
X
X
X
X
X
X
X
X
X
3
____________________________________
2 4
1
3
1
3 4
3 2
3
3 4
X
4
Ounces
____________________________________
_________________________________________________________________________________
6Seven students hiked the same trail. Each student
wore a pedometer, a device that records the
number of steps taken and miles. The line plot
at the right shows each pedometer's recorded
distance.
PEDOMETER READINGS OF DISTANCES HIKED
X
X
X
X
X
X
X
7
2 8
What is the average distance according to the
pedometer readings? Explain how you determined the answer.
3
1
3 8
1
3 4
3
3 8
Miles
_________________________________________________________________________________
_________________________________________________________________________________
_________________________________________________________________________________
134
UNIT 6
Measurement
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Read the problem. Write your answer to each part.
7Dwayne recorded the amount of water he drank each day for a week.
Dwayne’s Log
1
4
Sunday: 2 ​ 
  ​  L
10
Thursday: 1​  
  ​L
5
1
Monday: 1​  
  ​L 2
8
1
Friday: 2 ​  
   ​L
10
3
Tuesday: 1​  
  ​ L
10
Saturday: 1​  
  ​L
5
1
Wednesday: 2 ​  
  ​L
10
Part AMake a line plot to display the data.
9
Part BIt is recommended that a person drink 1​  
  ​liters of
10
water per day to stay healthy. On average, did Dwayne
drink enough water per day? Explain how you found
your answer.
________________________________________________________
________________________________________________________
How can you
compare the
average amount of
water that Dwayne
drank each day to
the recommended
amount of water he
should drink?
________________________________________________________
________________________________________________________
________________________________________________________
UNIT 6
Measurement
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135
R e vi e w
Measurement
Read each problem. Circle the letter of the best answer.
1Which of the following would be the most
appropriate to measure in feet?
A a coin’s width
C an ocean’s depth
B a car’s length
D a bridge’s height
Use this line plot to answer questions 6
and 7.
X
3
2Kay’s puppy weighs 3 lb 2 oz. How many
HAMSTER WEIGHT
X
X
X
X
2 4
1
3
3 4
C 48
B 34 D 50
3Jerome is going to see a movie with a
running time of 132 minutes. Which of the
following is equal to 132 minutes?
A 2 hr
C 2 hr 12 min
B 2 hr 2 min
D 2 hr 22 min
4Sanjay needs 350 cm of wire. How much
will he pay if the wire is $1.50 per meter?
A$5.25
C $525.00
B$52.50
D $0.53
5Which of the following is the most likely
measurement of a birdbath’s capacity?
A 25 mL
C 150 mL
B 200 L
D 8 L
136
3
3 4
1
4
4 4
1
4 2
Ounces
ounces is this?
A 16 1
3 2
X
6How many hamsters weighed less than
4 ounces?
Aone
C four
Btwo
D five
7What is the average weight of the hamsters
shown in the line plot?
7
C 3 ​  
 ​ oz
7
1
D 1​  
 ​ oz
4
A
3 ​  
  ​ oz
12
B
3 ​  
 ​ oz
2
1
3
8Which of the following is equal to 3 feet
4 inches?
A 12 inches
C 40 inches
B 36 inches
D 44 inches
UNIT 6
Measurement
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Read each problem. Write your answer.
9Is the capacity of a punch bowl more likely to be 3 gal or 3 C?
_________________________________________________________________________________
_________________________________________________________________________________
10Which would you be more likely to measure in grams—the mass of a
coin or the mass of a bag of potting soil? Explain.
_________________________________________________________________________________
_________________________________________________________________________________
11How many meters are equal to 12,500 mm?
Answer ________________________
12Sherman is building a bookcase. The dimensions are shown in the
drawing. If Sherman plans on dividing each shelf into three equal
sections, how many inches wide will each section be? Give your
answer as a decimal rounded to the nearest hundredth.
6 ft 3 in.
Answer ________________________________________________
________________________________________________________
2 ft 2 in.
13Rupert recorded these distances biked daily.
1
5
3
3
2
3
5
5
2
11​  
 ​ 10 ​  
 ​ 9 ​  
 ​ 10 ​  
 ​ 10 ​  
 ​ 10 ​  
 ​ 10 ​  
 ​ 11 9 ​  
 ​ 10 ​  
 ​
4
4
6
6
6
3
6
6
3
Make a line plot for the data set below.
UNIT 6
Measurement
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137
Read each problem. Write your answer to each part.
14Marta is sewing new curtains for her bedroom. She is buying fabric
from a store that uses the metric system.
Part AHow is the metric system organized? How are units in the
metric system related to each other?
_________________________________________________________________________________
_________________________________________________________________________________
_________________________________________________________________________________
Part BIs Marta better off buying fabric that is $1 per meter or
$0.10 per centimeter? Explain.
_________________________________________________________________________________
_________________________________________________________________________________
15Tanner is trying out a new recipe for soup.
Part AIf he uses 6 cups of cream, 3 pints of broth, and 1 quart of
water, will this make 1 gallon of soup? Explain.
_________________________________________________________________________________
_________________________________________________________________________________
Part BWhat unit should Tanner use to determine how many
servings 1 gallon of soup will make? Explain.
_________________________________________________________________________________
_________________________________________________________________________________
_________________________________________________________________________________
138
UNIT 6
Measurement
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U n it
7
Volume
Lesson 1 Volume reviews the concept of volume and
cubic units, focusing on rectangular prisms.
Lesson 2 Volume of Rectangular Prisms reviews
how to use the volume formulas, V 5 l 3 w 3 h and
V 5 b 3 h with rectangular prisms.
Lesson 3 Volume of Irregular Figures reviews
how to find the volume of shapes made up of two
rectangular prisms.
UNIT 7
Volume
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139
Volume
Lesson
1
5.MD.3.a, b; 5.MD.4
A cube is a
rectangular prism with
edges of equal length.
a
Volume is the amount of space inside an object. Volume can be
measured by counting the number of cubic units that can fill the
object. A cubic unit, or 1 unit3, is the amount of space inside a cube
that measures 1 unit on each edge.
1 unit
b
c
In a cube, a 5 b 5 c.
A rectangular prism is
a solid with rectangles
for sides.
1 unit 1 unit
A cube with side lengths measuring 1 unit
each, has a volume of 1 cubic unit, or 1 unit3.
Paulette makes a rectangular prism by placing 3 cubes in a row.
Then she places another 3 cubes in a row next to the first row.
How many cubes did she use to assemble her shape? What is
the volume of the rectangular prism she made?
To find the number of cubes Paulette used to assemble her shape, you
can count the number of cubes in each row, and then add.
3 cubes
3 cubes
6 cubes
So, Paulette used 6 cubes to make her shape.
To find the volume of the rectangular prism, remember that each cube
has a volume of 1 cubic unit. So, if the shape is made up of 6 cubes,
it must have a volume of 6 cubic units.
1 cube 1 unit3
140
6 cubes 6 unit3
UNIT 7
Volume
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Read each problem. Circle the letter of the best answer.
SAMPLE The rectangular prism shown here is made up of cubes.
Each cube has a volume of 1 unit3. How many cubes are
used to make the shape?
A 3 B 10
C 12 D 24
The correct answer is C. This question asks you to find the number
of cubes in the shape. You cannot see all of them, but you can
assume that each layer is the same as the top layer that is visible.
There are 2 rows of 3 cubes, or 6 cubes, in the top layer. There are
2 layers, so 2 3 6 5 12 cubes.
1The rectangular prism below is made up of
4The rectangular prism below is made up of
cubes measuring 1 m on each edge. What
is the volume of the prism?
cubes measuring 1 yd on each edge. What
is the volume of the prism?
A 16 m3
C 4 m3
A 35 yd3
C 15 yd3
B 8 m3
D 2 m3
B 30 yd3
D 10 yd3
2A box can hold exactly 20 blocks. If each
5A truck can hold a volume of 10 m3.
block is exactly 1 in.3, what is the volume
of the box?
Assuming each box is 1 m3, which set of
boxes can fit in the truck?
A 20 in.3
C 10 in.3
A
B 15 in.3
D 5 in.3
3What is the maximum number of 1-cm
3
cubes you can fit into a box with a volume
of 32 cm3?
A1
C 32
B16
D 35
C
B
D
UNIT 7
Volume
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141
Read each problem. Write your answer.
SAMPLE Luca is packing a truck’s cargo area with
5 yards
3
1-yard boxes, as shown here. If the
truck’s cargo area is 2 yards wide by
2 yards high by 5 yards deep, how
many boxes can Luca fit on it?
2 yards
2 yards
Answer ____________
To find the number of boxes Luca can fit into the truck, you can
count the number of 1-yard3 boxes that can fit into the cargo area.
Look at the drawing and count the number of cubes in one layer: 10.
Then multiply by the number of layers: 2 3 10 5 20. Luca can fit
20 boxes on the truck.
6The box shown here was filled with 1-cm square sugar cubes. What is
the volume of the box?
1 cm
3 cm
3 cm
Answer ________________________
7A rectangular prism that is 1 cm high has a volume of 56 cm3. If the
prism is made up of 7 rows of 1-cm3 cubes, how many cubes are in
each row?
Answer ____________
8What is the volume of an aquarium
10 in. long, 6 in. wide, and 10 in. high?
Draw a diagram at the right.
142
Answer ________________________
UNIT 7
Volume
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Read the problem. Write your answer to each part.
9Giles is filling a swimming pool with water. One side of the pool is
20 feet long, another side is 10 feet long, and the depth of the
pool is 4 feet.
4 ft
10 ft
20 ft
Part AExplain how Giles can find the volume of the pool using
a model and cubic units.
The dimensions
are given in feet, so
the cubic units will
be ft3.
_________________________________________________________________________________
_________________________________________________________________________________
_________________________________________________________________________________
Part BGiles’s sister says it is much easier to find the pool’s volume
by multiplying its length by its width, and then multiplying by
its height. Is she correct? Explain your answer.
_________________________________________________________________________________
_________________________________________________________________________________
_________________________________________________________________________________
UNIT 7
Volume
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143
Lesson
2
of Rectangular Pr
e
m
u
l
isms
Vo
A unit block is a
cube that measures
1 unit on each edge.
The base of a
rectangular prism is a
rectangle. So length 3
width is equal to the
area of the rectangle
that is the base. When
you multiply square
units by a third
dimension, the result is
cubic units.
5.MD.5.a, b
The volume of a rectangular prism can be found by multiplying the
length times the width times the height. You can use these formulas:
V5l3w3h
or
V5b3h
Height
1 unit
Width
1 unit
Length
1 unit
In the formulas, l is the prism’s length
w is the prism’s width
h is the prism’s height
bis the prism’s base, which is equal to the prism’s
length times its width, or b 5 l 3 w.
Apollo stacked unit boxes as shown. If he created a stack with
a length of 5 units, a width of 4 units, and a height of 2 units,
what is the volume of the stack?
2 units
5 units
4 units
To find the volume of the stack, first find the base, b, by
multiplying the length, 5, by the width, 4.
b 5 l 3 w 5 5 3 4 5 20 square units
Next, multiply the base by the stack’s height, 2.
V 5 b 3 h 5 20 3 2 5 40 cubic units
So, the volume of the stack is 40 units3.
144
UNIT 7
Volume
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Read each problem. Circle the letter of the best answer.
SAMPLE What is the volume of the rectangular prism
shown here?
5m
3
A 9 m B
10 m3
3
C
12 m
D
20 m3
2m
2m
The correct answer is D. To find the volume of a rectangular
prism with given dimensions, multiply the prism’s length, width,
and height: 2 3 2 3 5 5 4 3 5 5 20. So, the volume is 20 m3.
1
What is the volume of the insect box below?
4 in.
3 in.
6 in.
A 72 in.3
C 18 in.3
B 36 in.3
D 13 in.3
4The floor in Opal’s room is 14 ft by 15 ft.
If her room has an 8-ft ceiling, what is the
volume of the room?
A 1,680 ft3
C 442 ft3
B 1,415 ft3
D 210 ft3
5What is the volume of Hailey’s package
below?
3 cm
2Paige rents a moving truck with a cargo
area measuring 12 ft long, 10 ft wide, and
9 ft tall. She needs exactly 1,000 ft3 of
space to move all her things. Which
statement about the truck’s volume is true?
10 cm
14 cm
A It is 20 ft3 less than Paige needs.
A 1,410 cm3
C 212 cm3
B It is exactly 1,000 ft3.
B 420 cm3
D 143 cm3
C It is 80 ft3 more than Paige needs.
D It is double the volume Paige needs.
3A cereal box is 2 in. wide, 8 in. long, and
12 in. high. What is the maximum volume
of cereal you can fit in the box?
A 12 in.3
C 98 in.3
B 28 in.3
D 192 in.3
6How much water does it take to fill a
rectangular water tank with a base of
100 in.2 and a height of 40 in.?
A 60 in.3
C 1,400 in.3
B 140 in.3
D 4,000 in.3
UNIT 7
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145
Read each problem. Write your answer.
SAMPLE Phil wants to transport 24 stage props. Each prop
has a base of 1 ft2 and is 6 ft tall. If Phil rents a
trailer with a cargo area measuring 6 ft long by
4 ft wide by 6 ft tall, will he be able to transport
all the props in one trip?
6 ft
4 ft
6 ft
Answer ____________
To find the number of props Phil can fit into the van, first find
the volume each prop: V 5 b 3 h 5 1 3 6 5 6 ft3. Next, find the
volume of the trailer: V 5 l 3 w 3 h 5 6 3 4 3 6 5 144 ft3. Then
divide the volume of the van by the volume of each prop:
144 4 6 5 24. Yes, Phil can fit all 24 props into the van.
7Tamika built the owl house shown here using online instructions.
According to the instructions, the house should have a volume of
768 in.3. Do you think Tamika followed the instructions correctly?
Why, or why not?
8 in.
8 in.
12 in.
Answer __________________________________________________________________________
8Jeremy’s footlocker is 33 inches long, 16 inches wide, and 14 inches
high. If he uses half the locker for books, how much space does he
have left?
Answer ________________________
9The associative property for multiplication is stated as a 3 (b 3 c) 5
(a 3 b) 3 c. Find the volume of a 3 in. by 4 in. by 5 in. rectangular
prism using both sides of the equation. Show your work.
146
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Read the problem. Write your answer to each part.
10Mr. Park is opening a new business. He narrows his search to the two
properties shown below.
40 ft
30 ft
50 ft
30 ft
Property A
30 ft
70 ft
Property B
Part AIf Mr. Park needs at least 62,000 ft3 of space for his
business, which property should he rent? Explain your
answer.
The formulas
for volume of
rectangular prisms
are V 5 l 3 w 3 h
and V 5 b 3 h.
_________________________________________________________________________________
_________________________________________________________________________________
_________________________________________________________________________________
Part BAfter two months, Mr. Park needs more space so he decides
to rent the other property, too. How much space will he be
renting in all? Explain how you found your answer.
_________________________________________________________________________________
_________________________________________________________________________________
_________________________________________________________________________________
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Lesson
3
of Irregular Figu
e
m
u
l
o
V
res
When you compose
a figure, you put
together two or more
figures to make one.
When you decompose a
figure, you break apart
a figure into two or
more simpler figures.
The formulas for the
volume of a rectangular
prism are
5.MD.5.c
An irregular figure may be composed of two rectangular prisms. To
find the volume of an irregular figure, find the volume of each prism
and then add their volumes together.
Craig builds a large flower box shaped
like an L to fit on the corner of a deck.
How much soil will he need to fill the
entire box?
4 ft
3 ft
1 ft
1 ft
To find the amount of soil Craig will need, find the volume of
the L-shaped flower box. First, break the figure into two
rectangular prisms.
V5l3w3h
V5b3h
2 ft
4 ft
1 ft
1 ft
1 ft
1 ft
Next, find the volume of each rectangular prism.
4 3 1 3 1 5 4 ft3
2 3 1 3 1 5 2 ft3
Finally, add the volumes of the two prisms together to get the
volume of the whole figure.
41256
So, Craig needs 6 ft3 of soil to fill the flower box.
148
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Read each problem. Circle the letter of the best answer.
SAMPLE What is the volume of the
4 yd
irregular figure shown here?
A 15 yd3
C 10 yd3
B 12 yd3
D 9 yd3
3 yd
1 yd
3 yd
1 yd
1 yd
The correct answer is A. To find the volume of the irregular
figure, find the volume of the individual rectangular prisms that
make up the figure. Multiply to find the volume of the larger prism:
V 5 l 3 w 3 h 5 4 3 1 3 3 5 12 cubic yards. Multiply to find the
volume of the smaller prism: V 5 l 3 w 3 h 5 3 3 1 3 1 5 3 cubic
yards. Then add the volumes together: 12 1 3 5 15. So, the volume
of the figure is 15 yards3.
1What is the volume of the building model
shown below?
1 in.
her cabin. The new room is 10 ft long,
10 ft wide, and 8 ft tall. What is the
volume of the new room?
2 in.
2 in.
2 in.
6 in.
3Mrs. Danvers is building an addition to
2 in.
A 56 in.3
C 20 in.3
B 28 in.3
D 15 in.3
A 880 ft3
C 108 ft3
B 800 ft3
D 98 ft3
4What is the volume of the irregular figure
shown below?
2m 2m
3m
2Wendy is making a custom cage for her
rabbit. She wants to attach a 2 ft by 3 ft by
3 ft cage to a larger cage that measures
4 ft by 4 ft by 3 ft. What is the volume of
the custom cage?
A 18 ft3
C 48 ft3
B 20 ft3
D 66 ft3
4m
4m
4m
A 768 m3
C 76 m3
B 384 m3
D 19 m3
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149
Read each problem. Write your answer.
SAMPLE A company rents the white building shown
100 ft
60 ft
at the right, and wants to double their office
space by renting another building. If they
120 ft
rent the gray building next door, will they
double their space?
40 ft
Answer ____________
40 ft
Doubling the space means multiplying it by 2. So, first find the
volume of the white building: V 5 l 3 w 3 h 5 40 3 60 3 120 5
288,000 ft3. Next, find the volume of the gray building: V 5 l 3 w 3
h 5 (100 2 40) 3 40 3 120 5 60 3 40 3 120 5 288,000 ft3.
Compare the two volumes: 288,000 5 288,000. Yes, the company
will double their space if they rent the gray building next door.
5Aidan and his dad are building the workshop
shown here. It consists of a large studio and a
small tool room. If they need at least 1,000 ft3
for the studio and 400 ft3 for the tool room,
will the workshop be big enough? Why, or
why not?
12 ft
9 ft
12 ft
12 ft
6 ft
6 ft
_________________________________________________________________________________
_________________________________________________________________________________
_________________________________________________________________________________
6Pamela is building an addition to her barn. She needs to add 36 m3 of
space. What is the minimum height necessary to meet her needs if the
floor of the addition measures 12 m2?
150
Answer ____________
UNIT 7
Volume
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Read the problem. Write your answer to each part.
7Mrs. Song is replacing the wooden steps leading to her house with
steps made of poured concrete. The plans for the new steps are
shown below.
10 in.
8 in.
24 in.
30 in.
30 in.
Part AExplain how to find the volume of the new steps.
When finding the
volume of an irregular
figure made up of
rectangular prisms,
you can find the
volume of each prism,
and then add the
volumes together.
_________________________________________________________________________________
_________________________________________________________________________________
_________________________________________________________________________________
Part BWhat volume of concrete will be used to make the steps?
Show your work.
Answer ____________
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R e vi e w
Volume
Read each problem. Circle the letter of the best answer.
1Herman is building a shed. The space is
14 ft long, 10 ft wide, and 8 ft tall. What
is the volume of the shed?
4What is the maximum number of 1-mm3
cubes that can fit into a box with a volume
of 48 mm3?
A 1,418 ft3
C 220 ft3
A1
C 48
B 1,120 ft3
D 122 ft3
B24
D 92
2The rectangular prism below is made of
5What is the volume of Geoff’s reptile tank?
cubes measuring 1 m on each edge. What
is the volume of the prism?
16 in.
12 in.
A 24 m3
3
B 20 m 24 in.
C 14 m3
3
D 10 m
3What is the volume of the building shown
below?
35 ft
C 2,416 in.3
B 1,612 in.3
D 4,608 in.3
6The area of the floor in Edie’s bathroom is
120 ft2. If the ceiling is 9 ft high, what is
the volume of the bathroom?
20 ft
15 ft
25 ft
A 1,224 in.3
20 ft
A 1,415 ft3
C 442 ft3
B 1,080 ft3
D 210 ft3
40 ft
A 70,000 ft3
C 49,400 ft3
B 52,500 ft3
D 41,000 ft3
152
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Read each problem. Write your answer.
7Len made this rectangular stack of 1-yd3 boxes in
15 ft
a warehouse. How many boxes are in the stack?
Answer ____________
Up
Up
Up
Up
Up
Up
Up
Up
Up
Up
6 ft
6 ft
8A rectangular prism that is 2 cm high has a volume of 54 cm3. If the
prism has a length of 9 cm, what is its width?
Answer ____________
9Sunee needs a planter with a volume of at least 2,200 in.3. She buys a
planter with a base of 200 in.2 and a height of 9 in. Did Sunee make a
good choice? Why, or why not?
Answer __________________________________________________________________________
10Mr. Randolph is building an addition to his garage. He is adding
1,920 ft3 of space. What is the minimum height the space can be
if the area of the floor cannot be greater than 240 ft2?
Answer ____________
11These two buildings are for
sale at the same price. Their
30 ft
locations are equally good.
Which is the better bargain? 30 ft
Explain why.
85 ft
90 ft
Lincoln Park
40 ft
25 ft
Jefferson Circle
_________________________________________________________________________________
_________________________________________________________________________________
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153
Read each problem. Write your answer to each part.
12Tamsin made the sculpture here from a 360-cm3 piece
3 cm
3 cm
of solid marble.
4 cm
Part AWhat is the volume of marble in the sculpture?
Explain how you got your answer.
6 cm
________________________________________________
________________________________________________
6 cm
6 cm
_________________________________________________________________________________
_________________________________________________________________________________
Part BBased on your answer to part A, what volume of marble did
Tamsin have to cut away from the original block of marble to
make the sculpture? Show your work.
Answer ________________________
13A company is sending 2,000 1-yd3 boxes overseas in shipping
containers. Each shipping container measures 9 ft high by 9 ft wide by
45 ft long.
Part AHow many boxes can fit into one shipping container?
Answer ____________
Part BHow many shipping containers will the company need to
ship all the boxes? How many boxes are packed in the last
container? Show your work or explain how you found your
answer.
_________________________________________________________________________________
_________________________________________________________________________________
_________________________________________________________________________________
154
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U n it
8
Geometr y
Lesson 1 Coordinate Planes reviews how to use
a coordinate plane to find the locations of points and
distances in a two-dimensional space.
Lesson 2 Triangles reviews how to classify triangles
by side lengths and angle measures.
Lesson 3 Quadrilaterals reviews how to classify
quadrilaterals by side lengths, angle measures,
and parallel sides, and how the different kinds of
quadrilaterals are related.
UNIT 8
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155
Lesson
1
Coordinates are also
called ordered pairs.
That’s because the two
values are always given
in order: x first, then y.
Distance over
↓
(2, 5)
↑
Distance up
Placing a map on a
coordinate plane is
an easy way to find
distances between two
points. If you think of
the units on this plane
as city blocks, you can
add to find how many
blocks Mauricio is from
school. Remember to
stay on the grid lines.
Coordinate Planes
5.G.1, 5.G.2
A coordinate plane is made by two number lines that intersect at
zero. The intersection is called the origin and its coordinates are
(0, 0). The horizontal number line is called the x-axis and the vertical
number line is called the y-axis.
You can use a coordinate plane to find the location of points in a
two-dimensional space.
y
Mauricio’s house is at the origin on
the coordinate plane. His school is at
the location shown. What are the
coordinates of his school?
6
5
4
School
3
2
1
0
1 2 3 4 5 6 x
Mauricio’s
house
y
To find the coordinates of any point,
first read down from the point to the
x-axis. This number is the x-coordinate.
Read over from the point to the y-axis.
This number is the y-coordinate.
6
5
4
School
3
2
1
0
1 2 3 4 5 6 x
Mauricio’s
house
So, the x-coordinate of the school is 4. The y-coordinate is 3.
Coordinates are listed with the x-coordinate followed by the
y-coordinate, so the coordinates of the school are (4, 3).
To locate a point on a coordinate plane, first read over from the origin
to the x-value. Then read up from the x-axis to the y-value.
y
What is the location of (3, 4)?
156
6
5
4
3
2
1
0
1 2 3 4 5 6 x
UNIT 8
Geometry
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Read each problem. Circle the letter of the best answer.
y
SAMPLE The town of Smithville is shown on the
coordinate plane. What are the coordinates
of the firehouse?
A (2, 1)
C (2, 6)
B (1, 3)
D (6, 2)
8
Baseball
7
Firehouse
Field
6
5 Library
Farmer’s
4
Market
Coffee
3
Shop
2
Town
1
Hall
Pond
0 1 2 3 4 5 6 7 8 x
The correct answer is C. To find the coordinates of the firehouse,
find the point for the firehouse. Then read down from the point to
the x-axis to find the x-value: 2. Read over from point to the y-axis
and read the y-value: 6. List the coordinates in (x, y) format: (2, 6).
1 What are the coordinates of the library?
A (5, 3)
C (6, 3)
B (3, 5)
D (3, 6)
2 What is located at (6, 3)?
A town hall
C baseball field
B coffee shop
D farmer’s market
3How many units over from the origin is the
pond?
1
5To get home from the baseball field, Owen
walks 4 units down and 2 units to the
right. What are the coordinates of Owen’s
house?
A (9, 7)
C (2, 3)
B (7, 2)
D (9, 3)
6Lien, Etta, and May are meeting at the
firehouse. Lien is coming from the pond.
Etta is coming from the baseball field. May
is coming from her house at (0, 6). Which
girl will walk farthest?
A​  
 ​ unit
2
C 2 units
ALien
B 1 unit
D 3 units
BEtta
4Along the x-axis, how many units are
between the firehouse and the baseball
field?
A 5 units
C 2 units
B 3 units
D 1​  
 ​ units
2
CMay
D They all walk the same distance.
1
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157
Read each problem. Write your answer.
SAMPLE Ramon works at the guide station in a
state park. He tells a visitor to hike 4 miles
north and 3 miles west from the guide
station to find a waterfall. What are
the coordinates of the waterfall?
Answer ____________
miles
10
9
8
7
6
5
4
3
2
1
0
N
E
W
S
Guide Station
1 2 3 4 5 6 7 8 9 10 miles
To find the coordinates, follow Ramon’s directions on the map.
The guide station is at (3, 4). The compass indicates that north is up
and west is left. The units are miles, so 4 miles north is 4 units up
from the guide station. This gives a y-coordinate of 8. A distance of
3 miles west is 3 units left from the guide station. This gives an
x-coordinate of 0. So the coordinates of the waterfall are (0, 8).
7A wilderness campground is 7 miles east of the guide station. What
are the coordinates of the campground?
Answer ____________
8There are two picnic areas in the park. One is 2 miles south of the
guide station, and the other is 3 miles east of the guide station. What
are the coordinates of the picnic areas?
Answer ________________________
9Rebecca starts her hike at the guide station. She hikes 5 miles east and
2 miles north of the station and then stops for lunch. From her lunch
spot, she hikes 3 miles west and 1 mile north to a pond. From the
pond, she hikes 3 miles east and 2 miles south to a good spot for birdwatching. What are the coordinates of the bird-watching site?
158
Answer ____________
UNIT 8
Geometry
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Read the problem. Write your answer to each part.
10Ed wants to plant pepper plants in a section of his garden. He starts to
mark off this section at one corner, point A. He marks the next corner
four units up at point B. He walks 6 units to the right of point B and
marks the third corner at point C. Then he walks 4 units down to
point D to mark the fourth corner.
y
10
9
8
7
6
5
4
3
2
1
0
1 2 3 4 5 6 7 8 9 10 x
Part AWhat are the coordinates of points A, B, C, and D?
A
Add to or subtract
from the y-coordinate
when moving up or
down. Add to or
subtract from the
x-coordinate when
moving left or right.
Answer _________________________________________________________________
Part BEd decides to grow two different kinds of peppers and
wants to split his pepper section in half. Between what two
points could he draw a line to divide this section in half?
Explain your answer.
_________________________________________________________________________________
_________________________________________________________________________________
_________________________________________________________________________________
_________________________________________________________________________________
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159
Triangles
Lesson
2
5.G.3, 5.G.4
Triangles can be classified by the measures of their angles. Angles are
measured in units called degrees. Triangle classifications are based on
angle measures that are greater than, less than, or equal to 90º.
The angle measures
always add up to 180º,
no matter what kind of
triangle.
The square in the
corner of a right
triangle is the symbol
for a right angle.
15°
80°
30°
35°
25°
65°
140°
60°
Acute
Obtuse
Right
Each of three
angles is less
than 90°.
One angle is
greater than 90°.
One angle
measures
exactly 90°.
You can also classify triangles based on the lengths of their sides.
Equilateral triangles
also have three angles
of equal measure, 60º.
The angles opposite
the equal sides of an
isosceles triangle are of
equal measure.
Scalene triangles also
have angles of three
different measures.
Equilateral
Isosceles
All three sides
have equal
lengths.
At least two
sides are the
same length.
Scalene
None of the
sides are the
same length.
You can combine these methods of classifying to name a triangle
based on both side length and angle measure.
60°
Tick marks show sides
of the same lengths.
Equal lengths are
congruent.
60°
60°
Acute
Equilateral
45°
45°
Right
Isosceles
40°
30°
110°
Obtuse
Scalene
A triangle has one right angle and no equal sides. How is this
triangle classified by angles and sides?
If the triangle has one right angle, it is a right triangle. If it has no
equal sides, it is a scalene triangle. So, it is a right scalene triangle.
160
UNIT 8
Geometry
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Read each problem. Circle the letter of the best answer.
SAMPLE Which choice best describes the triangle shown here?
A right
C right isosceles
B isosceles
D right scalene
The correct answer is C. The square corner shows that it is a right
triangle. The tick marks tell that two sides have equal lengths, so
the triangle is also isosceles. Choice A only classifies the triangle by
angle measure. Choice B only classifies it by side length. Choice D
incorrectly classifies the triangle by side length.
1A triangle’s sides are all of equal length.
4Which choice best describes this triangle?
What else must be true about this triangle?
75°
A It includes a right angle.
B Only two of its angles are equal.
C It includes an obtuse angle.
D All its angles have equal measures.
2A triangle has two congruent sides. The
angle opposite one of the congruent sides
measures 50º. What is the measure of the
angle opposite the other congruent side?
A50º
C 80º
B60º
D 130º
3How is this triangle classified by its angles?
85°
35°
60°
8 in.
18 in.
GO TEAM!
75°
30°
18 in.
Aacute
C acute scalene
Bisosceles
D acute isosceles
5A triangle has sides 5 feet, 13 feet, and
12 feet in length. One angle is a right
angle. Which choice best describes the
triangle?
Aright
C acute
Bscalene
D right scalene
6Toshiro measures the angles of a triangle
and finds they are all different. What else
must be true about this triangle?
A It includes a right angle.
Aright
C obtuse
B It includes an obtuse angle.
Bacute
D scalene
C The sides all have different lengths.
D The sides all have the same length.
UNIT 8
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161
Read each problem. Write your answer.
SAMPLE Mr. Delgado uses the ladder shown to hang
30°
pictures. What kind of triangle is formed by
the sides of the ladder and the ground?
8 ft
8 ft
Answer ________________________
75°
75°
To find the kind of triangle, study the angle measures and side
lengths. All three angles measure less than 90º, so the triangle is
acute. Two of the sides have equal lengths, so the triangle is
isosceles.
7At the beginning of a game of pool,
the balls are organized into a triangle
using a tool called a rack. What kind
of a triangle is the rack shown here?
Answer ________________________________________________
8A triangular television stand fits in the corner of a rectangular
living room, as shown in the diagram. What kind of triangle must
the top of the stand be in order to fit in the corner of the room?
Explain your answer.
Living Room
_________________________________________________________________________________
_________________________________________________________________________________
9The Allegheny and Monongahela Rivers meet in the downtown
N
POINT STATE PARK
area of Pittsburgh, PA, to form the Ohio River. The rivers and a
road form the borders of Point State Park, shown in the diagram.
What kind of triangle is Point State Park?
162
Answer ________________________
UNIT 8
Geometry
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Read the problem. Write your answer to each part.
10A typical baseball field is shown below. The right angle symbols mark
the locations of first base, second base, third base, and home plate.
Foul
pole
Foul
pole
Second base
90 ft
90 ft
Third base
First base
90 ft
90 ft
Home plate
Part ADuring practice, the coach has the players throw the ball
from first to second base, from second base to third base,
and from third base back to first base. What kind of triangle
does the path of the ball make? Explain your answer.
All triangles
that share a corner
with a rectangle or
square are right
triangles.
_________________________________________________________________________________
_________________________________________________________________________________
_________________________________________________________________________________
_________________________________________________________________________________
Part BThe first and third base lines extend into the outfield and
end at the foul poles. The left field foul pole is 318 feet from
home plate. The right field foul pole is 314 feet from home
plate. Is the triangle formed by the first and third base lines
and the dashed line between them the same kind as the
triangle formed in part A? Explain your answer.
_________________________________________________________________________________
_________________________________________________________________________________
_________________________________________________________________________________
_________________________________________________________________________________
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163
Quadrilaterals
Lesson
3
5.G.3, 5.G.4
Congruent angles
have the same measure.
Congruent sides have
equal lengths.
A quadrilateral is a plane figure that has four sides. Quadrilaterals are
classified by their sides and angles.
Parallel sides are always
the same distance
apart.
Square
Rectangle
It has four
equal sides
and four right
angles.
A hierarchy shows
the relationships of
things. Each thing in
a hierarchy shares the
characteristics of the
category above it. So
the levels of a hierarchy
go from most general
characteristics to most
specific.
Parallelogram
It has two pairs of
equal sides and four
right angles.
It has two pairs of
parallel sides and
two pairs of
congruent angles
opposite each other.
Trapezoid
Rhombus
It has four equal sides
and two pairs of
congruent angles
opposite each other.
It has one pair of
parallel sides.
Quadrilaterals are related to each other through a
hierarchy, as shown here. Some quadrilaterals can be
classified in more than one way.
Quadrilateral
Is a square a parallelogram?
Parallelogram
Trapezoid
Rectangle
Rhombus
Look at the diagram. Each quadrilateral in the
diagram can also be classified as the quadrilateral
above it because it meets the standards that
define that figure. So, a square can also be
classified as a rectangle, a rhombus, a
parallelogram, and a quadrilateral.
Square
164
UNIT 8
Geometry
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Read each problem. Circle the letter of the best answer.
SAMPLE Which choice best describes the
quadrilateral shown here?
A rhombus
B parallelogram
C
rectangle
D
square
The correct answer is D. Look at the figure. The marked angles
show that it has four right angles. The tick marks tell that all sides
are congruent. So the quadrilateral is a square.
1Which choice best describes the shape of
4Which of the following statements about
the kite shown here?
quadrilaterals is true?
A All rectangles are squares.
8 in.
60°
120°
8 in.
8 in.
120°
60°
8 in.
Arhombus
C rectangle
Bsquare
D trapezoid
2Only one pair of a quadrilateral’s sides is
parallel. What kind of quadrilateral must
the figure be?
Arectangle
C trapezoid
Bsquare
D parallelogram
3Central Park in New York City is a polygon
with four sides. The opposite sides are
parallel and equal in length. All the angles
are right angles. Which choice best
describes the shape of Central Park?
Aquadrilateral
C square
Brectangle
D rhombus
B All squares are rectangles.
C A rhombus is not a parallelogram.
D A trapezoid is a parallelogram.
5Linh drew the parallelogram shown below.
155°
25°
A
What measure must angle A have?
A25º
C 180º
B155º
D 360º
6If a quadrilateral has at least one pair of
parallel sides, what two types of figures
could it be?
A rectangle or rhombus
B trapezoid or rectangle
C parallelogram or rectangle
D trapezoid or parallelogram
UNIT 8
Geometry
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165
Read each problem. Write your answer.
SAMPLE Mark drew this rhombus. What is the
X
measure of the angle marked X?
Answer ____________
68°
Recall the characteristics of a rhombus. It has four equal sides and
opposite angles that are congruent. So the angle diagonally
opposite to the 68° angle must be congruent, that is, have the
same measure. Angle X measures 68°.
7A typical football field is shown here.
What is the narrowest description of
160 ft
the shape of the football field?
360 ft
G
O
A
L
G
O
A
L
160 ft
360 ft
Answer ________________________
8Diane drew the figure shown. What are all the
terms she could use to describe her drawing?
Answer __________________________________________________________________________
Bridge
9The diagram shows a piece of
climbing equipment at the
playground. It is in the shape
of a trapezoid. What is true
about the bridge and the ground?
166
Ladder
Ladder
Ground
Answer __________________________________________________________________________
UNIT 8
Geometry
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Read the problem. Write your answer to each part.
10Rihana is working on a project for art class. She starts with a square
sheet of paper. Then she draws lines to divide the paper into different
sections. She will paint each quadrilateral a different color. Her project
is shown below, after she has drawn the lines.
Part AHow many quadrilaterals did Rihana make with the lines she
drew? What are they?
_________________________________________________________________________
_________________________________________________________________________
Part BAre all the types of quadrilaterals represented in Rihana’s
art project? Explain your answer.
What shape is
the paper Rihana
started with?
________________________________________________
________________________________________________
_________________________________________________________________________
_________________________________________________________________________
_________________________________________________________________________
_________________________________________________________________________
UNIT 8
Geometry
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167
R e vi e w
Geometr y
Read each problem. Circle the letter of the best answer.
1A point on a coordinate plane is 2 units to
the right of the y-axis and 6 units up from
the x-axis. What are the point’s coordinates?
A (2, 6)
C (6, 12)
B (6, 2)
D (2, 3)
2A quadrilateral has two pairs of equal sides
and two pairs of congruent angles. Which
choice does not match this description?
Asquare
C rectangle
Bparallelogram D trapezoid
3A triangle has side lengths of 6 inches,
10 inches, and 14 inches. One angle
measures 150º. What kind of triangle is it?
A obtuse scalene
B obtuse equilateral
C acute scalene
D obtuse isosceles
5Brad’s house is at (4, 5). He gets to his
friend Stacy’s house by walking 3 blocks to
the left and 2 blocks down. What are the
coordinates of Stacy’s house?
A (7, 3)
C (1, 7)
B (1, 3)
D (7, 7)
6Freddie drew a triangle that has an obtuse
angle. Which of these could Freddie’s
triangle also be?
Aacute
C right
Bscalene
D equilateral
7A quadrilateral has two sides that measure
14 cm and two sides that measure 5 cm.
Two angles measure 110º. The other two
angles measure 70º. The congruent angles
are opposite. The congruent sides are
parallel. What kind of quadrilateral is it?
Asquare
C rectangle
Btrapezoid
D parallelogram
4Which of the following can all rectangles
also be called?
Aparallelograms C squares
Brhombuses
168
D trapezoids
UNIT 8
Geometry
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Read each problem. Write your answer.
Use the coordinate plane at the right to answer questions 8–12.
8This coordinate plane shows a map of a department store, with
the pathways between departments. What are the coordinates of
the Bedding department?
Answer ____________
9What is the best description of the figure made by the path from
Kitchen to Dining to Bedding to Living Room and back to
Kitchen? (Assume the grid lines meet at right angles.)
y
10
9
8
7
6
5
4
3
2
1
0
Dining
Kitchen
Bath Bedding
Living Room
1 2 3 4 5 6 7 8 9 10 x
Answer ________________________
10The distance from Kitchen to Bath is the same as the distance from
Living Room to Bath. What kind of triangle is formed by the paths
between these three departments?
Answer ________________________
11Mrs. Jensen starts in the Living Room department, then goes to the
Bedding department, the Bath department, and the Kitchen
department. Then she remembers an item she forgot and returns to
the Living Room department. What quadrilateral is formed by the
paths Mrs. Jensen followed?
Answer ________________________
12Describe the paths that form a right triangle. (Hint: There are two.)
_________________________________________________________________________________
_________________________________________________________________________________
_________________________________________________________________________________
UNIT 8
Geometry
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169
Read each problem. Write your answer to each part.
13Jessica’s house is at (2, 4). She walks up 3 units and right 3 units to
take her dog to the park. From there, she walks right 3 units and
down 3 units to the pet store to buy her dog a bone.
Part AOn the coordinate plane at the right, mark the
locations of Jessica’s house, the park, and the
pet store. Label each place with its name and
coordinates.
Part BAfter visiting the pet store, Jessica returns home. If
she were to ignore the grid lines and walk directly
to each stop, what figure would her path form?
Explain your answer.
y
10
9
8
7
6
5
4
3
2
1
0
1 2 3 4 5 6 7 8 9 10 x
_________________________________________________________________________________
_________________________________________________________________________________
_________________________________________________________________________________
_________________________________________________________________________________
_________________________________________________________________________________
14Harrison draws a plane figure that has two parallel sides that are
17 centimeters long and two parallel sides that are 23 centimeters
long. All the sides meet at right angles.
Part AWhat are all the names Harrison could use to describe his
figure?
Answer _________________________________________________________________
Part BHarrison draws a line between two opposite corners of his
figure and cuts the figure in half. Describe the two
congruent figures he creates. Explain your answer.
_________________________________________________________________________________
_________________________________________________________________________________
_________________________________________________________________________________
170
UNIT 8
Geometry
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Practice Test
Read each problem. Circle the letter of the best answer.
1 Find 1,062 4 59.
6What is the maximum number of
A17
C 19.7
B18
D 62,658
2Travis is making 5 lasagnas using 12 cups of
cheese. Which number represents the
amount of cheese he will use per lasagna?
5
A​  
  ​ cup
12
B
2
2 ​  
 ​ cups
5
C 7 cups
D 60 cups
3 What is the product of 0.022 3 103?
A0.22
C 22
B2.2
D 66
1-cm3 cubes that can fit into a box with a
volume of 1,200 cm3?
A1
C 1,020
B600
D 1,200
1
4
7At a zoo,  
​ 3 ​of the animals are birds,  
​ 9 ​ are
mammals, and the rest are reptiles and
amphibians. What fraction of animals at
the zoo are reptiles and amphibians?
21
C  
​ 9 ​
5
D  
​ 3 ​
A​  
  ​
27
B​  
  ​
12
2
2
8Which is the expanded form of 4,005,006
4Which of the following is the most
appropriate unit for measuring the length
of a sailboat?
Amile
C foot
Byard
D inch
using exponents?
A(4 3 106) 1 (5 3 104) 1 (6 3 101)
B(4 3 106) 1 (5 3 103) 1 (6 3 100 )
C(4 3 106) 1 (5 3 104) 1 (6 3 101)
D(4 3 109 ) 1 (5 3 105) 1 (6 3 100 )
5Which point is located on the y-axis of a
coordinate plane?
A (6, 0)
C (3, 2)
B (6, 6)
D (0, 6)
Practice Test
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171
Read each problem. Circle the letter of the best answer.
9Mr. Rainer is adding a room to his house.
The new room is 25 ft long, 20 ft wide,
and 12 ft tall. What is the volume of the
room?
3
A 6,012 ft C 3,000 ft
3
3
B 6,000 ft D 2,512 ft
13Jade’s pet snake needs at least 4,000 in.3
of space in its tank. Which tank should
Jade buy?
A
3
10 in.
10 in.
10 Solve 9.8 3 20.7.
B
A35.19
C 351.90
B202.86
D 2,028.6
11What is the standard form of the number
)
)
1
1
(2 3 1) 1 (4 3  
​  10  ​ 1 (9 3 ​  
   ​?
1,000
A2.049
C 2.49
B2.409
D 249
18 in.
12 in.
12 in.
C
12 in.
14 in.
12Point Y is 1 unit down from and 2 units to
the left of point X.
24 in.
24 in.
D
14 in.
y
8
7
6
5
4
3
2
1
0
X
14 in.
14Nobu needs 54 inches of wood to build a
1 2 3 4 5 6 7 8 x
What are the coordinates of point Y?
A (6, 3)
C (5, 1)
B (3, 6)
D (1, 5)
172
20 in.
picture frame. If the wood costs $2.50 a
foot, how much will he pay?
A$11.25
C $100.00
B$0.93
D $11.50
Practice Test
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Read each problem. Circle the letter of the best answer.
15There were 29 tables at a wedding
reception. Each table had 12 seats, and
every seat was full. How many people were
at the wedding reception?
A 77 C 338
B238
D 348
16Which shows the decimals in order from
least to greatest?
A 6.5, 6.13, 8.905, 8.9
B 22.95, 22.059, 21.90, 21.09
C 54.07, 54.7, 54.707, 54.77
D 63.595, 63.06, 63.8, 63.402
19Mila plants a bush at the beginning of the
summer that is 12 inches tall. At the end of
the summer its height has tripled, so she
cuts it down 8 inches. Which expression
shows the height of the bush then?
A (3 1 12) 2 8
B
3 3 (12 2 8)
C
(3 3 12) 2 8
D 3 3 (12) 1 8
20Which number shows 25.254 rounded to
the tenths place?
A30
C 25.25
B25
D 25.3
17 How many tenths are there in 32?
A
1
3 ​  
 ​
5
B
1
32 ​  
  ​
10
C 225
3
21 What is the quotient of ​  
 ​divided by 3?
4
D 320
A​  
 ​
4
18A triangle has two sides of equal length.
Which of the following could the triangle
not be classified as?
Ascalene
1
C 2 ​  
 ​
4
1
4
D 4
B​  
 ​
3
22What is the volume of the irregular figure
shown below?
4 cm
Bright
Cisosceles
Dequilateral
4 cm
5 cm
5 cm
2 cm
2 cm
A 140 cm3
C 108 cm3
B 116 cm3
D 216 cm3
Practice Test
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173
Read each problem. Circle the letter of the best answer.
23How many times must Henry fill a 350-mL
27Jenny made this input-output table.
cup to empty a 10.5-L sink that is filled
with water?
IN
OUT
3
10
A3
C 30
5
12
B300
D 3,000
7
14
12
19
24Kamala drew a figure that has a pair of
congruent parallel sides that measure
3 inches and a pair of congruent parallel
sides that measure 1 inch. The figure has
four right angles. Which choice best
describes Kamala’s figure?
Aquadrilateral
What is the output value for an input value
of 2?
7
A​  
 ​
2
C 9
B5
D 14
C rectangle
Bparallelogram D rhombus
7
28 What is  
​ 12  ​of 120?
25There are 360 people at a museum
A17
C 70
B60
D 206
opening. Half of the people are museum
3
supporters, ​  
 ​are artists, and the rest are
8
museum employees. How many people are
artists?
A36
C 135
B45
D 180
26Yesterday’s low temperature was 65.9°F.
The high temperature was 83°F. What was
the change in temperature during the day?
A5.76°F
C 18.1°F
B17.1°F
D 27.1°F
174
29 Simply the expression below.
17 2 (4 3 3)
A5
C 29
B12
D 39
3
9
30 What is 15 ​  
 ​ 2 6 ​  
  ​in lowest terms?
4
20
1
3
A
10 ​  
 ​
3
C 9 ​  
  ​
10
B
9 ​  
  ​
16
6
D 9 ​  
  ​
10
7
Practice Test
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Read each problem. Write your answer.
4
2
31What is ​  
 ​ of ​  
 ​? Show how to find the answer on the model below.
5
3
Answer ____________
32A business office spent $4,628 on postage last year. On average, how
much did the office spend on postage per week? Show your work.
Answer ____________
5
33Is 20 times  
​ 6 ​greater than or less than 20?
Answer ________________________
34Kent and Terry round 1.604 to the hundredths place. Jeff rounded to
1.61, and Terry rounded to 1.60. Who is correct? Explain your answer.
_________________________________________________________________________________
_________________________________________________________________________________
35Melanie made music playlists for her birthday party. She put 10 songs
on each playlist. During her party, she played 3 whole playlists plus
4 extra songs. Write an expression to represent the number of songs
played during the party.
Answer ____________________________________
Practice Test
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175
Read each problem. Write your answer.
36A carpet is 2.5 m long and 0.75 m wide. What is the area of the
carpet in square meters? Show your work.
Answer ________________________
3
2
7
37Sandra bought ​  
 ​lb of carrots, ​  
 ​lb of celery, and  
​ 8 ​lb of onions.
4
3
How many pounds of vegetables did she buy? Show your work.
Answer ________________________
38On the coordinate plane below, Beatriz’s house is at (1, 7). Beatriz must
walk 4 units to the right and 6 units down to get to her school. Plot
and label the points for Beatriz’s house and the school on the plane.
y
8
7
6
5
4
3
2
1
0
1 2 3 4 5 6 7 8 x
39In the number 13,407.036 there are two 3’s. Explain how the values of
the 3’s are related.
_________________________________________________________________________________
_________________________________________________________________________________
_________________________________________________________________________________
_________________________________________________________________________________
_________________________________________________________________________________
176
Practice Test
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Read each problem. Write your answer.
3
3
1
40Lem says that 12 ​  
 ​ 1 3 ​  
 ​is equal to 15 ​  
 ​. Without adding the
4
4
2
mixed numbers, how can you tell that he is incorrect?
_________________________________________________________________________________
_________________________________________________________________________________
41Michelle cuts a piece of paper to make a greeting card. Two opposite
sides are 8 inches long. The other two opposite sides are 4 inches
long. All the sides meet at right angles.
How can this shape be classified? Name all the ways from most
specific to most general.
Answer __________________________________________________________________________
42Shiro wants to make an input-output table with numbers of meters as
input values and numbers of centimeters as output values. What rule
should Shiro use for his table?
Answer ____________________________________
43Hadasah’s locker is 30 inches tall, 10 inches wide, and 14 inches deep.
If her books take up 1,545 in.3 of space, how much space does she
have left to hang her coat and bookbag?
Answer ________________________
44The table shows the times for five
Swimmers
Time
(in minutes)
Omar
1.194
Craig
1.14
Lance
1.239
Paul
1.21
Serge
1.139
swimmers in the 100-meter race.
List the swimmers in order from
fastest to slowest.
Answer __________________________________________________________________________
Practice Test
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177
Read each problem. Write your answer.
45Warehouse workers need to load 29 boxes of paper onto a truck. Each
box weighs 64 pounds. Can the workers do this safely if the truck has
a capacity of 9,600 pounds? Explain.
_________________________________________________________________________________
_________________________________________________________________________________
3
46Fritz is wrapping 8 identical presents. He needs ​  
 ​m of ribbon to
5
1
wrap each present. He needs an additional  
​ 5 ​m of ribbon to make a
bow. How much ribbon does he need to wrap and make bows for all
8 presents?
Answer ________________________
47Cheryl is planting a row of flowers 10 ft long in her garden. If she
1
plants a flower every  
​ 3 ​ft, how many flowers can she plant? Show
your work.
Answer ____________
48Taariq is shopping for cat litter to fill the litter box shown below.
7 in.
12 in.
16 in.
Explain how Taariq can find the volume of the box using a model and cubic units.
_________________________________________________________________________________
_________________________________________________________________________________
_________________________________________________________________________________
178
Practice Test
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Read each problem. Write your answer.
49Sibyl is making a scale drawing of a bridge on a sheet of paper. The
actual bridge is 200 feet long. To get the dimensions for the drawing, is
Sibyl multiplying by a number greater than 1 or a number less than 1?
Explain.
_________________________________________________________________________________
_________________________________________________________________________________
_________________________________________________________________________________
_________________________________________________________________________________
50Germain’s chemistry class conducted an experiment. The class broke
into groups and each group performed the same experiment. After
observing a chemical reaction, each group recorded the mass of the
sample. The data set below shows the masses that the groups recorded.
1
6 ​  
 ​ 2
7
6 ​  
  ​ 10
2
6 ​  
 ​ 5
1
6 ​  
 ​ 2
1
6 ​  
  ​ 10
2
6 ​  
 ​ 5
2
6 ​  
 ​ 5
3
6 ​  
 ​
5
Make a line plot for the data set.
Practice Test
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179
Read each problem. Write your answer to each part.
51Lucy makes pottery to sell at a flea market. One week, she uses
3
2
6 ​  
 ​kg of clay. The next week, she uses only 2 ​  
 ​kg of clay.
4
3
Part AHow much clay does Lucy use during the two weeks? Show
your work.
Answer _________________________________________________________________
Part BThe clay comes in blocks that weigh 12 kg. How much clay
does Lucy have left after using clay for two weeks? Show
your work.
Answer ________________________
52Mrs. Kendrick buys 32 oz of strawberries, 40 oz of oranges, and 52 oz
of apples.
Part AHow many pounds of fruit does Mrs. Kendrick buy
altogether? Show your work.
Answer ________________________
Part BTo convert a small unit to a larger one, do you multiply or
divide? Explain
_________________________________________________________________________________
_________________________________________________________________________________
180
Practice Test
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Read each problem. Write your answer to each part.
53Roger makes a model of the local library shown below.
Part ARoger uses cubes to make his model. How many cubes does
he use?
Answer ____________
Part BEach cube represents 100 ft3. Explain how you can use this
information and the formula V 5 l 3 w 3 h to find the
volume of the actual building.
_________________________________________________________________________________
_________________________________________________________________________________
_________________________________________________________________________________
_________________________________________________________________________________
_________________________________________________________________________________
Practice Test
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181
GLOSSARY
A
acute angle
an angle that measures less than 90°
acute triangle
a triangle with three acute angles
annex
to fill in with 0s as placeholders
associative allows grouping of numbers with parentheses to be
propertyadded or multiplied: a 1 (b 1 c) 5 (a 1 b) 1 c and
a 3 (b 3 c) 5 (a 3 b) 3 c
B
C
base
t he number that is multiplied by itself in an
exponential expression; example: in 102, the number
10 is the base
common multiple
a multiple that two or more whole numbers share
commutative allows numbers to be added or multiplied in any
propertyorder: a 1 b 5 b 1 a and a 3 b 5 b 3 a
congruent
equal in length, measure, or shape
conversion factor
a number used to change units from one kind to
another
convert
to change
coordinates
rdered pairs of numbers that indicate locations on a
o
coordinate plane
cubic unit
t he amount of space inside a cube that measures
1 unit on each edge
customary system a system of measurement used in the United States.
It includes units of
• length—inch, foot, yard, mile
• capacity—fluid ounce, cup, pint, quart, gallon
• weight— ounce, pound, ton
Glossary
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183
D
E
F
184
data
information in the form of numbers
decimals
umbers with values in places to the left of the
n
decimal point
decimal point
a period that separates whole-number values from
fractional values in numbers written in standard form
decimal system
a system of numbers based on 10
degree
a unit of angle measure; a unit of temperature
measure
denominator
t he number of parts in the whole or set, the number
on the bottom of a fraction
difference
the answer in a subtraction problem
dividend
the number being divided in a division problem
divisor
the number doing the dividing in a division problem
dot plot
a line plot
equal sign
the symbol 5; means the expressions on each side
have the same value
equilateral triangle
a triangle with three sides of the same length and
three 60° angles
equivalent
equal
equivalent fractions
two or more fractions that represent the same value
evaluate
to find the value of an expression
expanded form
a way to write a number in which each digit is
expressed as the product of its face value and a
power of ten
exponent
t he number that tells how many times another
number (the base) is used as a factor; example: in
102, the 2 is the exponent
exponential form
a number written as a base with an exponent
factors
whole numbers that multiply to form a product
frequency
how often something happens
Glossary
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G
greatest common
factor
I
improper fraction a fraction in which the numerator is equal to or
greater than the denominator
the largest of the common factors between two or
more numbers
input-output table a table that shows two sets of values that are related
by a rule
inverse operations operations that undo each other, opposite operations.
Addition and subtraction are inverse operations.
Multiplication and division are inverse operations.
K
L
M
isosceles triangle
a triangle with at least two equal sides
key words
words that indicate operations
least common
the least common multiple shared by two
denominatordenominators
least common multiple
the smallest of the common multiples between two
or more numbers
line plot
a plot in which data is represented by X’s placed over
a number line. Also called a dot plot.
lowest terms
a fraction in which the terms cannot be divided by a
number other than 1; simplest form
metric system
a system of measurement used in most of the world.
It includes units of
• length— millimeter, centimeter, meter, kilometer
• capacity—milliliter, liter
• mass— gram, kilogram mixed number
a whole number plus a fraction
multiples
t he products of a number and nonzero whole
numbers
multi-step problem a problem that requires more than one step to solve
Glossary
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185
N
numerator
t he number of parts talked about, the number on
the top of fraction
an expression containing only numbers and
numerical expressionoperations
Q
186
tenths
hundredths
thousandths
t he value given to the place a digit has in a number;
each place has a value 10 times greater than the
place to its right.
ones
place value
tens
rouping symbols ( ) that indicate an operation
g
should be done first
hundreds
parentheses
thousands
a quadrilateral with two pairs of parallel sides
ten thousands
parallelogram
hundred thousands
the center of a coordinate plane, located at the
intersection of the x- and y-axes, having the
coordinates (0, 0)
millions
origin
ten millions
a triangle with one obtuse angle
hundred millions
obtuse triangle
billions
an angle that measures more than 90° but less
than 180°
ten billions
P
obtuse angle
hundred billions
O
1
2
3,
4
5
6,
7
8
9,
0
1
2
.3
4
5
power
the product of multiplying a number by itself
product
the answer in a multiplication problem
quadrilateral
a polygon with four sides
quotient
the answer in a division problem
Glossary
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R
reciprocal
t he number that multiplies another number for a
product of 1
rectangle
a parallelogram with four right angles
rectangular model a rectangle divided into parts to show fractions or
operations
S
T
U
V
regroup
t o exchange 1 in one place for 10 in the place to its
right, or 10 in one place for 1 in the place to its left;
example: 2 tens can be regrouped as 1 ten and 10 ones
rename
t o name a number in an equivalent form using
different terms
rhombus
a parallelogram with four equal sides
right angle
an angle that measures 90°
right triangle
a triangle with one right angle
round
t o replace a number with a number that tells about
how many or how much
scalene triangle
a triangle with no equal sides
scaling
to resize a number by multiplying by a factor greater
than, equal to, or less than 1
square
a rectangle with four equal sides
standard form
a number written as the sum of the values of its
places
sum
the answer in an addition problem
trapezoid
a quadrilateral with exactly one pair of parallel sides
two-step equation a n equation requiring more than one operation to
solve
unlike fractions
fractions with different denominators
volume
the amount of space inside an object
Glossary
© The Continental Press, Inc. DUPLICATING THIS MATERIAL IS ILLEGAL.
187
W
whole numbers
the counting numbers and 0: 0, 1, 2, 3, …
X
x-axis
the horizontal axis of a coordinate plane
Y
y-axis
the vertical axis of a coordinate plane
188
Glossary
© The Continental Press, Inc. DUPLICATING THIS MATERIAL IS ILLEGAL.
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is less than
symbol
is approximately
equal to symbol
is congruent to
symbol
is perpendicular to
symbol
is parallel to
symbol
denominator
numerator
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is greater than
symbol
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right angle
line of symmetry
square
isosceles triangle
equilateral triangle
parallel lines
scalene triangle
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rectangle
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perpendicular lines
ray
vertex
parallelogram
line segment
rhombus
trapezoid
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intersecting lines
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