?

Rational Numbers
ESSENTIAL QUESTION
How can you use rational
numbers to solve
You can represent any real-
world quantity that can be
a
written as __b , where a and b
are integers and b ≠ 0, as a
rational number.
MODULE

3
LESSON 3.1

real-world problems? Rational Numbers

and Decimals
7.NS.2b, 7.NS.2d

LESSON 3.2
Adding Rational
Numbers
7.NS.1a, 7.NS.1b,
7.NS.1d, 7.NS.3

LESSON 3.3
Subtracting Rational
Numbers
7.NS.1, 7.NS.1c

LESSON 3.4
Multiplying Rational
Numbers
7.NS.2, 7.NS.2a,
7.NS.2c
© Houghton Mifflin Harcourt Publishing Company • Image Credits: Diego Barbieri/Shutterstock.com

LESSON 3.5
Dividing Rational
Numbers
7.NS.2, 7.NS.2b,
7.NS.2c

LESSON 3.6
Real-World Video Applying Rational
In many competitive sports, scores are given as Number Operations
decimals. For some events, the judges’ scores are 7.NS.3, 7.EE.3
averaged to give the athlete’s final score.
my.hrw.com

my.hrw.com my.hrw.com Math On the Spot Animated Math Personal Math Trainer
Go digital with your Scan with your smart Interactively explore Get immediate
write-in student phone to jump directly key concepts to see feedback and help as
edition, accessible on to the online edition, how math works. you work through
any device. video tutor, and more. practice sets.
57 Module 3
57
Are You Ready? Are YOU Ready?
Complete these exercises to review skills you will need Personal
Assess Readiness for this module. Math Trainer

Online Practice
Use the assessment on this page to determine if students need Multiply Fractions my.hrw.com and Help

intensive or strategic intervention for the module’s prerequisite skills. EXAMPLE 3 _

1 1
_
8
× 49 3 _
_
8
× 49 = ___
8
3 __
× 94 Divide by the common factors.
2 3
3 = _16 Simplify.
2
Response to
1 Intervention Multiply. Write the product in simplest form.
4
9
3
_ 12
__ 5
__ _ or 1_13
1. __
14
× _76 4 2. 3 _
_
5
× 47 35 3. __
8
× 10
11 __
33
12 4. 4
_
9
×3 3

Intervention Enrichment
Operations with Fractions
Access Are You Ready? assessment online, and receive EXAMPLE 2 __
_ 7
÷ 10 10
= _25 × __ Multiply by the reciprocal of the divisor.
5 7
instant scoring, feedback, and customized intervention 2
10
= _25 × __
7
Divide by the common factors.
or enrichment. 1

= _47 Simplify.
Personal Online and Print Resources
Math Trainer Divide.
Online Assessment Skills Intervention worksheets Differentiated Instruction 6
__ 3
_ 3
_
and Intervention 5. _12 ÷ _14 2 13
6. _38 ÷ __
16
13 14
7. _25 ÷ __
15
7 16
8. _49 ÷ __
27
4
• Skill 44 Multiply Fractions • Challenge worksheets 18
__ 6
__ 2
__
my.hrw.com 9. _35 ÷ _56 25 23
10. _14 ÷ __ 23 11. 6 ÷ _35 10 12. _45 ÷ 10 25
• Skill 42 Operations with PRE-AP 24

Fractions Extend the Math PRE-AP Order of Operations

• Skill 51 Order of Operations Lesson Activities in TE EXAMPLE 50 - 3(3 + 1)2 To evaluate, first operate within parentheses.

© Houghton Mifflin Harcourt Publishing Company

50 - 3(4)2 Next simplify exponents.
50 - 3(16) Then multiply and divide from left to right.
Real-World Video Viewing Guide 50 - 48 Finally add and subtract from left to right.
2
After students have watched the video, discuss the following:
Evaluate each expression.
• How do decimals apply to scoring in a diving competition?
• How are decimals used to determine who wins a close race in 13. 21 - 6 ÷ 3 19 14. 18 + (7 - 4) × 3 27 15. 5 + (8 - 3)2 30
swimming and running? 16. 9 + 18 ÷ 3 + 10 25 17. 60 - (3 - 1)4 × 3 12 18. 10 - 16 ÷ 4 × 2 + 6 8

58 Unit 1

PROFESSIONAL DEVELOPMENT VIDEO

my.hrw.com
Author Juli Dixon models successful
Online Teacher Edition Interactive Whiteboards
teaching practices as she explores
Access a full suite of teaching Engage students with interactive
rational numbers in an actual seventh-
resources online—plan, whiteboard-ready lessons and
grade classroom.
present, and manage classes activities.
and assignments.
Personal Math Trainer:
Professional ePlanner Online Assessment and
Development Easily plan your classes and Intervention
access all your resources online. Assign automatically graded
my.hrw.com
homework, quizzes, tests,
Interactive Answers and
and intervention activities.
Solutions
Prepare your students with
Customize answer keys to print
updated practice tests aligned
or display in the classroom.
with Common Core.
Choose to include answers only
or full solutions to all lesson
exercises.

Rational Numbers 58
 EL
Reading Start-Up
Have students complete the activities on this page by working alone
Reading Start-Up Vocabulary
Review Words
Visualize Vocabulary integers (enteros)
or with others. ✔ negative numbers
Use the ✔ words to complete the graphic. You can put more (números negativos)
than one word in each section of the triangle. pattern (patrón)
Strategies for English Learners ✔ positive numbers
(números positivos)
Each lesson in the TE contains specific strategies to help English ✔ whole numbers (números
Integers
Learners of all levels succeed. enteros)
45
Emerging: Students at this level typically progress very quickly, whole numbers, Preview Words
additive inverse (inverso
learning to use English for immediate needs as well as beginning to positive numbers aditivo)
understand and use academic vocabulary and other features of 2, 24, 108 opposite (opuesto)
rational number (número
academic language. positive numbers, racional)
whole numbers
Expanding: Students at this level are challenged to increase their -2, -24, -108
repeating decimal
(decimal periódico)
English skills in more contexts, and learn a greater variety of vocabulary terminating decimal
negative numbers (decimal finito)
and linguistic structures, applying their growing language skills in more
sophisticated ways appropriate to their age and grade level. Understand Vocabulary
Bridging: Students at this level continue to learn and apply a range of Complete the sentences using the preview words.
high-level English language skills in a wide variety of contexts, includ- 1. A decimal number for which the decimals come to an end is a
ing comprehension and production of highly technical texts. terminating decimal.

Active Reading 2. The additive inverse , or opposite , of a number is the

same distance from 0 on a number line as the original number, but on
Integrating Language Arts the other side of 0.

© Houghton Mifflin Harcourt Publishing Company

Students can use these reading and note-taking strategies to help
them organize and understand new concepts and vocabulary.
Active Reading
Additional Resources Layered Book Before beginning the module,
Differentiated Instruction create a layered book to help you learn the concepts
in this module. At the top of the first flap, write the
• Reading Strategies EL title of the module, “Rational Numbers.” Label the
other flaps “Adding,” “Subtracting,” “Multiplying,”
and “Dividing.” As you study each lesson, write
important ideas, such as vocabulary and processes,
on the appropriate flap.

Module 3 59

Focus | Coherence | Rigor

Tracking Your Learning Progression

Before In this module After

Students understand rational Students represent and use rational numbers in a variety Students will:
numbers: of forms: • approximate the value of an
• classify whole numbers, integers, • write rational numbers as decimals irrational number
and rational numbers using a • add, subtract, multiply, and divide rational numbers • describe the relationship between
visual representation such as a fluently sets of real numbers
Venn diagram to describe
relationships between sets of
numbers
• add, subtract, multiply, and divide

59 Module 3
 GETTING READY FOR
GETTING READY FOR Rational Numbers
Rational Numbers Understanding the Standards and the vocabulary terms in the Standards
will help you know exactly what you are expected to learn in this module.

Use the examples on the page to help students know exactly what 7.NS.3
they are expected to learn in this module. Solve real-world and What It Means to You
mathematical problems You will add, subtract, multiply, and divide rational numbers.
involving the four operations
CA Common Core with rational numbers. EXAMPLE 7.NS.3

Standards Key Vocabulary -15 · _23 - 12 ÷ 1 _13

rational number (número · 2 - __
15 _
- __
1 3 1
4
12 ÷ _
3
Write as fractions.
racional)
Content Areas Any number that can be · 2 - __
15 _
- __ 3
12 · _ To divide, multiply by
1 3 1 4 the reciprocal.
expressed as a ratio of two 5 3
·2 ·3
The Number System—7.NS integers. - 15
_____
1·3
- 12
_____
1· 4 Simplify.
1 1
Cluster Apply and extend previous understandings of operations with fractions to add, 10
- __
1
- _91 = -10 - 9 = -19 Multiply.
subtract, multiply, and divide rational numbers.

7.NS.3

Go online to Solve real-world and What It Means to You

mathematical problems You will solve real-world and mathematical problems
see a complete involving the four operations

© Houghton Mifflin Harcourt Publishing Company • Image Credits: ©Ilene MacDonald/Alamy

involving the four operations with rational numbers.
unpacking of the with rational numbers.
EXAMPLE 7.NS.3
CA Common Core
In 1954, the Sunshine Skyway Bridge toll for a car was $1.75. In
Standards. 2012, the toll was _57 of the toll in 1954. What was the toll in 2012?

my.hrw.com 1.75 · _57 = 1_34 · _57 Write the decimal as a fraction.

Write the mixed number as an
= _74 · _57
improper fraction.
1
7· 5
= _____
4·7
Simplify.
1
= _54 = 1.25 Multiply, then write as a decimal.

The Sunshine Skyway Bridge toll for a car was $1.25 in 2012.

Visit my.hrw.com
to see all CA
Common Core
Standards
explained.
my.hrw.com

60 Unit 1

Lesson Lesson Lesson Lesson Lesson Lesson

California Common Core Standards 3.1 3.2 3.3 3.4 3.5 3.6

7.NS.1c Understand subtraction of rational numbers as adding the additive inverse,

p - q = p + (-q). Show that the distance between two rational numbers on the number line is
the absolute value of their difference, and apply this principle in real-world contexts.
7.NS.2 Apply and extend previous understandings of multiplication and division and
of fractions to multiply and divide rational numbers.
7.NS.2d Convert a rational number to a decimal using long division; know that the
decimal form of a rational number terminates in 0s or eventually repeats.
7.NS.3 Solve real-world and mathematical problems involving the four operations
with rational numbers.
7.EE.3 Solve multi-step real-life and mathematical problems posed with positive and
negative rational numbers in any form (whole numbers, fractions, and decimals), using tools
strategically. Apply properties of operations to calculate with numbers in any form; convert
between forms as appropriate; and assess the reasonableness of answers using mental
computation and estimation strategies.

Rational Numbers 60
 LESSON 3.1 Rational Numbers and Decimals

Lesson Support
Content Objective Students will learn to convert rational numbers to decimals.

Language Objective Students will illustrate how to convert a rational number to a decimal.

California Common Core Standards

7.NS.2d Convert a rational number to a decimal using long division; know that the decimal form of a rational number
terminates in 0s or eventually repeats.
7.NS.2b Understand that integers can be divided, provided that the divisor is not zero, and every quotient of integers
p ( -p ) p
(with non-zero divisor) is a rational number. If p and q are integers, then -( _q ) =____ ____
q = ( -q ) . Interpret quotients of rational
numbers by describing real world contexts.
MP.3 Construct viable arguments and critique the reasoning of others.

Focus | Coherence | Rigor

Building Background
Visualize Math Have students work with partner to model
improper fractions. Have each group choose a number of slices;
for example, 8 slices per pizza and 13 slices. Then have students
write the improper fraction and the mixed number and draw a
diagram of their pizza slices in pizza pans.

8 slices per pizza

13 slices = 1 pizza and 5 pizza
8
13 = 15
8 8

Learning Progressions Cluster Connections

In this lesson, students continue to build their understanding of This lesson provides an excellent opportunity to connect ideas
rational numbers, by now including negative rational numbers. in this cluster: Apply and extend previous understandings of
Some key understandings for students are the following: operations with fractions to add, subtract, multiply, and
divide rational numbers.
• A rational number is a number that can be written in the
form __ab when a and b are integers and b ≠ 0. Give students the following prompt: “Shawna wrote the mixed
• Every rational number can be written as a terminal number 5__23 as the decimal 5.23. Shawna made a mistake. What
decimal or a repeating decimal. did she do wrong? What should she do to correct it?”
• To convert a fraction to a decimal, the numerator is the Have students write a response.
dividend and the denominator is the divisor.
Sample answer: Shawna wrote the decimal part using the
In the next lessons, students learn to use the other operations of
numbers from the fraction. She should have
_ divided the
numerator by the denominator. 5__23 = 5.66
subtraction, multiplication, and division with signed rational
numbers.

61A
 PROFESSIONAL DEVELOPMENT
EL
Language Support
California ELD Standards
Emerging 2.I.6c. Reading/viewing closely – Use knowledge of morphology, context, reference materials, and visual cues
to determine the meaning of unknown and multiple-meaning words on familiar topics.
Expanding 2.I.6c. Reading/viewing closely – Use knowledge of morphology, context, reference materials, and visual cues
to determine the meaning of unknown and multiple-meaning words on familiar and new topics.
Bridging 2.I.6c. Reading/viewing closely – Use knowledge of morphology, context, reference materials, and visual
cues to determine the meaning, including figurative and connotative meanings, of unknown and multiple- meaning words on a variety of
new topics.

EL
Linguistic Support

Academic/Content Vocabulary Rules and Patterns

terminating/repeating – This lesson on rational comparisons – Word problems in this lesson that
numbers and decimals describes two types of compare use the -er form of an adjective. This is called
decimals. Help English learners remember the the comparative form. Remind English learners how to
meanings of these words by explaining them with use the comparative form: Jorge is taller than Tomás;
words they already know. Another word for terminate Gina runs faster than María. Display a chart like the
is to stop or end. Terminating decimals stop or end; one below and invite students to add words to it.
they do not go on forever. Repeat means to do
something again and again. Repeating decimals have Adjective Comparative Superlative
digits or patterns of digits that appear again and
again. For Spanish, both terminate and repeat are tall taller tallest
cognates (terminar and repetir). fast faster fastest

good better best

EL
Leveled Strategies for English Learners

Emerging Have students at this level of English proficiency work with a partner to read aloud the
long division steps to demonstrate understanding of how to convert rational numbers to decimals.

Expanding Working in pairs, have English learners at this level of proficiency discuss and take
turns doing the steps in long division to convert rational numbers to decimals.

Bridging Pair students at this level of English proficiency, and have them describe to each other
the steps in long division needed to convert rational numbers to decimals.

The question posed in Math Talk asks students to think about whether decimals
Math Talk with repeating pattern… Model for English learners how to begin their response
with yes/no, I think that… or yes/no, I can think of an example that…

Rational Numbers and Decimals 61B

 LESSON

3.1 Rational Numbers and Decimals

CA Common Core Engage
Standards
ESSENTIAL QUESTION
The student is expected to: How do you convert a rational number to a decimal? Write the rational number in __ab form,
The Number System—7.NS.2d then use long division to divide the numerator by the denominator.
Convert a rational number to a decimal using long Motivate the Lesson
division; know that the decimal form of a rational number Ask: How much money do you have if you have __74 of a dollar? Begin the Explore Activity to
terminates in 0s or eventually repeats. find out.
The Number System—7.NS.2b
Understand that integers can be divided, provided that
the divisor is not zero, and every quotient of integers
(with non-zero divisor) is a rational number. If p and q are Explore
integers, then -(p/q) = (-p)/q = p/(-q). Interpret
quotients of rational numbers by describing real-world EXPLORE ACTIVITY
contexts.
Avoid Common Errors
Mathematical Practices Some students forget that the value of an improper fraction is greater than 1. Remind
MP.3 Logic students that an improper fraction is greater than 1 and is equivalent to a decimal greater
than 1.

Explain
ADDITIONAL EXAMPLE 1 EXAMPLE 1
Write each rational number as a
decimal. Focus on Modeling Mathematical Practices
When they use long division to convert a fraction, some students may divide the larger
A -__
17
40
-0.425 number by the smaller number, regardless of whether the numerator or the denominator is
_
B __
11
24
0.458333... or 0.4583 the larger number. Remind students that the numerator will always be the dividend, and
the denominator will always be the divisor.
Interactive Whiteboard
Interactive example available online Questioning Strategies Mathematical Practices
my.hrw.com
• Can you use long division to show that two fractions are equivalent? Explain. Yes, convert
each fraction to a decimal using long division. If both fractions are equal to the same
decimal, then the fractions are equivalent.
• Which operation does a fraction bar indicate? Does a fraction need to be in simplest terms
to be converted into a decimal using long division? Explain. Division; no, regardless of
whether a fraction is in simplest terms, it can be converted to a decimal using long
division.

Engage with the Whiteboard

In part B, circle the resulting differences in the division problem that show that the
quotient has repeating digits: 31, 13, 31, 13. Explain that once you see that these
differences are recurring, you know that the decimal will continue to repeat forever.

61 Lesson 3.1
 LESSON
Rational Numbers 7.NS.2d EXPLORE ACTIVITY (cont’d)

3.1 and Decimals

Convert a rational number to
a decimal using long division;
know that the decimal
form of a rational number
terminates in 0s or eventually
repeats. Also 7.NS.2b
3. Do you think a mixed number is a rational number? Explain.
A mixed number is a rational number because it can
be rewritten as an improper fraction, which is a ratio
? ESSENTIAL QUESTION
How can you convert a rational number to a decimal?
of two integers.

EXPLORE ACTIVITY 7.NS.2b, 7.NS.2d

Writing Rational Numbers as Decimals
Describing Decimal Forms You can convert a rational number to a decimal using long division. Some decimals
are terminating decimals because the decimals come to an end. Other decimals
of Rational Numbers are repeating decimals because one or more digits repeat infinitely.
Math On the Spot
A rational number is a number that can be written as a ratio of two
integers a and b, where b is not zero. For example, _74 is a rational my.hrw.com
37 EXAMPLE 1 7.NS.2d
number, as is 0.37 because it can be written as the fraction ___100 .

A Use a calculator to find the equivalent decimal form of each fraction.__ Write each rational number as a decimal.
Remember that numbers that repeat can be written as 0.333… or 0.3. 5
A - __
16
0. 3 1 2 5
⎯
Fraction 1
_ 5
_ 2
_ 2
_ 12
__ 1
_ 7
_ 1 6 ⟌ 5. 0 0 0 0
4 8 3 9 5 Divide 5 by 16.
5 8 −4 8
Add a zero after the decimal point.
Decimal 0.2 0.875 No, the 2 0
Equivalent 0.25 0.625 0.666… 0.222… 2.4 Subtract 48 from 50.
number of Use the grid to help you complete the -1 6
B Now find the corresponding fraction of the decimal equivalents given digits in the long division. 4 0
in the last two columns in the table. Write the fractions in simplest form. repeating - 3 2
Add zeros in the dividend and continue 8 0
C Conjecture What do you notice about the digits after the decimal patterns can dividing until the remainder is 0.
1 - 8 0
point in the decimal forms of the fractions? Compare notes with your be different. __11
, 5
or 0.09…, has The decimal equivalent of - __
16 is - 0.3125.
0
neighbor and refine your conjecture if necessary.
2 repeating 13
B __
The digits after the decimal point either repeat or terminate. 33
digits, and _13,
Divide 13 by 33.
or 0.3…, has 1
Reflect Add a zero after the decimal point. 0. 3 9 3 9
© Houghton Mifflin Harcourt Publishing Company

© Houghton Mifflin Harcourt Publishing Company

repeating digit. Subtract 99 from 130. ⎯
1. Consider the decimal 0.101001000100001000001…. Do you think this 3 3 ⟌ 1 3. 0 0 0 0
Use the grid to help you complete the
decimal represents a rational number? Why or why not? −9 9
long division.
Sample answer: No; since the digits after the decimal point do 3 1 0
Math Talk You can stop dividing once you discover a -2 9 7
Mathematical Practices
not terminate or repeat, it does not represent a rational number. repeating pattern in the quotient. 1 3 0
Do you think that decimals
that have repeating patterns -9 9
2. Do you think a negative sign affects whether or not a number is always have the same
Write the quotient with its repeating pattern
3 1 0
a rational number? Use -_85 as an example. number of digits in their and indicate that the repeating numbers
-2 9 7
No; -_85 = -1.6, which is a rational number since the pattern? Explain. continue.
1 3
13
The decimal
___ equivalent of __
33 is 0.3939…,
decimal terminates. Rational numbers can be negative. or 0.39.

Lesson 3.1 61 62 Unit 1

PROFESSIONAL DEVELOPMENT
Integrate Mathematical Math Background
Practices MP.3 A rational number is any number that can be
This lesson provides an opportunity to address written as a fraction. The group of rational
this Mathematical Practice standard. It calls for numbers includes fractions, mixed numbers
students to display, explain, and justify (which can be written as improper fractions), and
mathematical ideas using precise mathematical whole numbers (which can be written as a
language in written or oral communication. As fraction with a denominator of 1). Not all real
students discuss the process of long division, numbers are rational. Some real numbers are
they have the opportunity to use precise irrational numbers, such as pi and the square root
mathematical language such as divisor, of a nonperfect square.
dividend, differences, quotients, and so on.

Rational Numbers and Decimals 62

 YOUR TURN
Focus on Patterns Mathematical Practices
Make sure that students understand that repeating decimals like 0.333… don’t just repeat
once or a few times—they repeat forever.

ADDITIONAL EXAMPLE 2 EXAMPLE 2

Killian bought 3__45 pounds of cashews.
Write 3__45 as a decimal. 3.8 Focus on Reasoning Mathematical Practices
Remind students that if the decimal and the mixed number it is derived from do not have
Interactive Whiteboard the same whole number, they have made a computation error.
Interactive example available online
Questioning Strategies Mathematical Practices
my.hrw.com • If you use long division to convert an improper fraction to a decimal, and the quotient is a
whole number, does this mean that the improper fraction is equivalent to a whole
number? Explain. Yes, because converting by division is simply writing an equivalent form
of the same value.

YOUR TURN
Focus on Technolgoy Mathematical Practices
Point out to students that when the resulting quotient from dividing a numerator by a
denominator is a repeating decimal, a calculator can only show an approximately
equivalent value. The only way to represent the exact equivalent value is to express
_ the
repeating value with a bar above the repeating digits or with ellipses (e.g., 3.3 or 3.333…).

Elaborate
Talk About It
Summarize the Lesson
Ask: Rational numbers can be written as fractions or mixed numbers. How do you
convert a fraction or a mixed number to a decimal? Divide the numerator by the
denominator.

GUIDED PRACTICE
Focus on Patterns Mathematical Practices
When students use a line to indicate a repeating decimal, make sure their line is only over
the part of the decimal that repeats. In Exercise 8, the first few digits in the quotient (1, 4, 2,
and 0) are not part of the repeating pattern, so the bar should not extend over those digits.

Engage with the Whiteboard

For Exercises 4, 5, and 8 with repeating digits, you may want to have students
perform the long division on the whiteboard and circle the recurring differences that
cause the quotient to have repeating digits.

Avoid Common Errors

For Exercises 10–17, remind students that for improper fractions and mixed numbers, the
equivalent decimal number will be greater than 1.

63 Lesson 3.1
 YOUR TURN Guided Practice
Write each rational number as a decimal. Write each rational number as a decimal. Then tell whether each decimal
Personal
4. - _47 - 0.571428... 5. 1
_ 0.333... 6. 9
- __ - 0.45 Math Trainer is a terminating or a repeating decimal. (Explore Activity and Example 1)
3 20 Online Practice
and Help 1. _35 = 0.6 89
2. - ___ = -0.89 4
3. __ = 0.333…
100 12
my.hrw.com
terminating terminating repeating

Writing Mixed Numbers as Decimals 25

4. __ = 0.2525… 5. - _79 = -0.7777… 9
6. - __ = -0.36
99 25
You can convert a mixed number to a decimal by rewriting the fractional part repeating repeating terminating
of the number as a decimal.
1 0.04 25 -0.14204545… 12 0.012
Math On the Spot
7. __
25
= 8. - ___
176
= 9. ____
1,000
=
EXAMPL 2
EXAMPLE 7.NS.2d my.hrw.com terminating repeating terminating
Shawn rode his bike 6 _34 miles to the science museum. Write 6 _34 as a decimal.
My Notes Write each mixed number as a decimal. (Example 2)
STEP 1 Rewrite the fractional part of the 1= -11.166… 9 = 2.9 23 = -8.23
10. - 11 __ 11. 2 ___ 12. - 8 ____
number as a decimal. Science Museum 6 10 100
0.75
⎯
Divide the numerator 3 =
13. 7 ___ 7.2 3 =
14. 54 ___ 54.2727… 1 =
15. - 3 ___ -3.0555…
4⟌ 3.00 by the denominator. 15 11 18
-28
6_34 mi
20 16. Maggie bought 3 _23 lb of apples to make 17. Harry’s dog lost 2 _78 pounds. What is the
-20 some apple pies. What is the weight of the change in the dog’s weight written as a
0 apples written as a decimal? (Example 2) decimal? (Example 2)

STEP 2 Rewrite the mixed number as 3 _23 = 3.666… -2 _78 = -2.875

the sum of the whole part

?
and the decimal part. ESSENTIAL QUESTION CHECK-IN
6 _34 = 6 + _34
3
18. Tom is trying to write __
47 as a decimal. He used long division and divided
= 6 + 0.75 = 6.75 until he got the quotient 0.0638297872, at which point he stopped. Since
3
the decimal doesn’t seem to terminate or repeat, he concluded that __ 47 is
not rational. Do you agree or disagree? Why?
YOUR TURN Disagree; the definition of a rational number is a number
© Houghton Mifflin Harcourt Publishing Company

© Houghton Mifflin Harcourt Publishing Company

7. The change ($) in a stock value was -2 _34 per share. Write -2 _34 as a
that can be written as a ratio of two integers with a
decimal. -2 _34 = -2.75 denominator not equal to zero, which describes __ 3 . Tom
Is the decimal equivalent a terminating or repeating decimal? 47
terminating decimal would need to keep dividing to find a repeating pattern

8. Yvonne bought a watermelon that weighed 7 _13 pounds. Write 7 _13 as

or have the decimal terminate.

a decimal. 7 _13 = 7.333…

Personal
Is the decimal equivalent a terminating or repeating decimal? Math Trainer
Online Practice
repeating decimal and Help
my.hrw.com

Lesson 3.1 63 64 Unit 1

DIFFERENTIATE INSTRUCTION
Cooperative Learning Critical Thinking Additional Resources
Have students discuss different ways to judge if Point out to students that they can use common Differentiated Instruction includes:
their answers are reasonable. benchmark fraction and decimal equivalents • Reading Strategies
along with multiplication to convert fractions to • Success for English Learners EL
Student 1: I compare the numerator to
decimals. For example, tell them that __15 is
the denominator. If the numerator is less • Reteach
equivalent to 0.2. Then ask them to explain how
than half the denominator, the decimal • Challenge PRE-AP
to use multiplication to express __45 as a decimal.
should be less than 0.5. If the numerator is
Possible answer: __15 = 0.2, and __15 × 4 = __45, so to
more than half the denominator, the
convert __45 to a decimal, multiply 0.2 × 4 = 0.8
decimal should be more than 0.5.
Student 2: I compare the numerator to
the denominator, too! If the numerator is
greater than the denominator, the decimal
should be greater than 1. If the numerator
is less than the denominator, the decimal
should be less than 1.

Rational Numbers and Decimals 64

 Personal
Math Trainer
Online Assessment
Evaluate Focus | Coherence | Rigor
and Intervention

Online homework GUIDED AND INDEPENDENT PRACTICE

assignment available 7.NS.2d, 7.NS.2b
my.hrw.com
Concepts & Skills Practice
3.1 LESSON QUIZ Explore Activity Exercises 1–9, 24
Describing Decimal Forms of Rational Numbers
7.NS.2d, 7.NS.2b
Example 1 Exercises 1–9, 19–22, 25, 28
Bob, Tina, and Hadley went Writing Rational Numbers as Decimals
running. Bob ran for __3
hour,
__
4 8
Tina ran for 3 9 hours, and Example 2 Exercises 10–17, 25
Hadley ran for 4___
19
20
hours. Writing Mixed Numbers as Decimals

1. Which running times can be written

as improper fractions?
2. Write the number of hours each Exercise Depth of Knowledge (D.O.K.) Mathematical Practices
person spent running as a decimal.
19–23 2 Skills/Concepts MP.4 Modeling
3. Which decimals are repeating
decimals? 24 3 Strategic Thinking MP.8 Patterns
4. Which decimals are terminating 25 3 Strategic Thinking MP.6 Precision
decimals?
26 2 Skills/Concepts MP.3 Logic
Lesson Quiz available online 27 2 Skills/Concepts MP.6 Precision
my.hrw.com 28 3 Strategic Thinking MP.4 Modeling
29 3 Strategic Thinking MP.7 Using Structure
Answers
1. 3__49 and 4__
19 30 3 Strategic Thinking MP.2 Reasoning
20 _
2. Bob: 0.375; Tina: 3.444… or 3.4; 31 3 Strategic Thinking MP.8 Patterns
Hadley: 4.95
_
3. 3.444… or 3.4
Additional Resources
4. 0.375 and 4.95
Differentiated Instruction includes:
• Leveled Practice Worksheets

65 Lesson 3.1
 Name Class Date
26. Vocabulary A rational number can be written as the ratio of one
integer
3.1 Independent Practice Personal
Math Trainer
terminating
to another and can be represented by a repeating

Online Practice
or decimal.
7.NS.2b, 7.NS.2d and Help
my.hrw.com 7 5
27. Problem Solving Marcus is 5 __ 24
feet tall. Ben is 5 __
16
feet tall. Which of the
Use the table for 19–23. Write each ratio in the two boys is taller? Justify your answer.
Team Sports
form __ba and then as a decimal. Tell whether each Ben is taller because 5.3125 > 5.2916… .
Number of
decimal is a terminating or a repeating decimal. Sport Players
28. Represent Real-World Problems If one store is selling _34 of a bushel of
19. basketball players to football players Baseball 9
5
__ apples for $9, and another store is selling _23 of a bushel of apples for $9,
11
0.4545… repeating Basketball 5
which store has the better deal? Explain your answer.
Football 11
The first store has the better deal because _34 = 0.75, and
20. hockey players to lacrosse players Hockey 6
6
__ 2
_
3 = 0.6 . Since 0.75 is greater than 0.6, the first store is
10
0.6 terminating Lacrosse 10
Polo 4
21. polo players to football players Rugby 15
offering a greater portion of a bushel of apples than the
4
__
11
0.3636… repeating Soccer 11 second store.
22. lacrosse players to rugby players
10
__ FOCUS ON HIGHER ORDER THINKING Work Area
15
0.666… repeating
29. Analyze Relationships You are given a fraction in simplest form. The
23. football players to soccer players
11
__ numerator is not zero. When you write the fraction as a decimal, it is a
11
1 terminating repeating decimal. Which numbers from 1 to 10 could be the denominator?

24. Look for a Pattern Beth said that the ratio of the number of players When the denominator is 3, 6, 7, or 9, the result will be
in any sport to the number of players on a lacrosse team must always be a repeating decimal.
a terminating decimal. Do you agree or disagree? Why?
© Houghton Mifflin Harcourt Publishing Company • Image Credits: Comstock/Getty Images/

Agree; sample answer: to find the ratio, divide the number of players 30. Communicate Mathematical Ideas Julie got 21 of the 23 questions
on her math test correct. She got 29 of the 32 questions on her science
on a given team by 10, the number on a lacrosse team. You can do this test correct. On which test did she get a higher score? Can you compare
21 29
the fractions __ __
23 and 32 by comparing 29 and 21? Explain. How can Julie
by moving the decimal point one place to the left. This leaves you with compare her scores?
a number that terminated one place to the right of the decimal point. Math; Sample answer: No, you should convert the

© Houghton Mifflin Harcourt Publishing Company

25. The change in the water level at the lake was - 4 _78 inches for the month. fractions to decimals because comparing numerators
39
a. What is - 4 _78 written as an improper fraction?
-__
8 only works if both denominators are the same.
Houghton Mifflin Harcourt

b. What is - 4 _78 written as a decimal? -4.875

c. Communicate Mathematical Ideas If the water level at the lake continued 31. Look for a Pattern Look at the decimal 0.121122111222.… If the pattern
to change at the same rate for 3 months in a row, explain how you could continues, is this a repeating decimal? Explain.
estimate the total change in the water level at the end of the 3 month period.
The fraction - _78 is really close to 1, so you could estimate that the water level No; although the digits follow a pattern, the same
combination of digits do not repeat.
changes by approximately -5 inches per month. The total change in the water
level at the end of the 3 month period would be approximately -15 inches.
Lesson 3.1 65 66 Unit 1

EXTEND THE MATH PRE-AP Activity available online my.hrw.com

Identify a specific number being described by the following clues. Explain how you know your
answer is correct.
A. It is a positive integer that is less than 12.
B. All proper fractions with this number as the denominator convert to repeating decimals.
C. Each repeating decimal from Clue B uses the same 6 numbers, in different orders.
From Clue A, the number must be between 1 and 11. From Clue B, we can eliminate all the
even numbers because when you use them as denominators of proper fractions, some
fractions will convert to terminating decimals. For example, __12, __24, __36, __48, and __
5
10
all convert to the
terminating decimal 0.5. The only remaining number that matches Clue C is 7, because
__ __ __ __ __ __
__1 = 0.142857, __2 = 0.285714, __3 = 0.428571, __4 = 0.571428, __5 = 0.714285, and __6 = 0.857412.
7 7 7 7 7 7

Rational Numbers and Decimals 66

 LESSON 3.2 Adding Rational Numbers

Lesson Support
Content Objective Students will learn to add rational numbers.

Language Objective Students will describe how to add rational numbers.

California Common Core Standards

7.NS.1d Apply properties of operations as strategies to add and subtract rational numbers.

7.NS.1a Describe situations in which opposite quantities combine to make 0.

MP.2 Reason abstractly and quantitatively.

Focus | Coherence | Rigor

Building Background -1.5

Visualize Math Have students start at –1.5 on the number

line and identify the point that is 1 unit to the left. Review the
meaning of moving 1 unit to the right, which is add 1, and the -4 -3 -2 -1 0 1 2
meaning of moving 1 one unit to the left, which is subtract 1. 1 unit to the left 1 unit to the right
-2.5 -0.5

Learning Progressions Cluster Connections

In this lesson, students continue to build their understanding This lesson provides an excellent opportunity to connect ideas
of addition as they use the operation of addition with integers. in this cluster: Apply and extend previous understandings of
Some key understandings for students are the following: operations with fractions to add, subtract, multiply, and
divide rational numbers.
• When adding rational numbers with the same sign, the
sum has the same sign as the rational numbers. Give students the following prompt: “Noah says that adding two
• When adding rational numbers with different signs, the numbers always gives a sum greater than either of the addends.
sum has the sign of the rational number with the greater Is that true? What mistake do you think Noah was making?”
absolute value. Encourage students to give examples showing when it is true
• The sum of a number and its additive inverse is 0. and when it is not true. Sample answer: Noah may be thinking
only of whole numbers or only of positive numbers. When you
Students will expand their work with rational numbers to add two whole numbers the sum is always greater than either of
include the other operations of subtraction, multiplication, and the addends. However, when you add two negative numbers or
division. a negative number and a positive number, the sum will not be
greater than both of the addends.

67A
 PROFESSIONAL DEVELOPMENT
EL
Language Support
California ELD Standards
Emerging 2.I.6c. Reading/viewing closely – Use knowledge of morphology, context, reference materials, and visual cues
to determine the meaning of unknown and multiple-meaning words on familiar topics.
Expanding 2.I.6c. Reading/viewing closely – Use knowledge of morphology, context, reference materials, and visual cues
to determine the meaning of unknown and multiple-meaning words on familiar and new topics.
Bridging 2.I.6c. Reading/viewing closely – Use knowledge of morphology, context, reference materials, and visual
cues to determine the meaning, including figurative and connotative meanings, of unknown and multiple- meaning words on a variety of
new topics.

EL
Linguistic Support

Academic/Content Vocabulary Background Knowledge

additive inverse – To help students remember the signal words – Many different English words can be
meaning of additive inverse, point out that add, which used to signal addition and subtraction in word
they already know, is at the root of the word additive. problems. Review with English learners the following
Inverse means opposite or reverse. If you think of signal words that indicate addition or subtraction:
adding numbers on a number line, the additive
• write a check/make a deposit
inverse reverses a number back to zero. When a
number and its additive inverse are added, the • add to/take out
sum is zero. • spend/sell (get income)
• spend/earn (get income)
• hike/hike back (return)

EL
Leveled Strategies for English Learners

Emerging Have student pairs at this level of English proficiency illustrate and label on a number
line how to add rational numbers of the same sign. Repeat for rational numbers of different signs.

Expanding Have English learners at this level of proficiency work in pairs and take turns
illustrating on a number line how to add rational numbers of the same sign. Repeat for rational
numbers of different signs.

Bridging Pair students at this level of English proficiency, and have them use a number line to
illustrate and describe how to add rational numbers of the same sign. Repeat for rational numbers
of different signs.

You can help English learners participate by giving them sentence frames:
Math Talk
The opposite of -3.5 on the number line is _______. We call this the
additive inverse.

Adding Rational Numbers 67B

 LESSON

3.2 Adding Rational Numbers

CA Common Core Engage
Standards
ESSENTIAL QUESTION
The student is expected to: How can you add rational numbers? Sample answer: If the two numbers have the same
The Number System—7.NS.1d sign, you add them together and give the sum the same sign as the two numbers. If the
two numbers have different signs, find the difference of their absolute values and use the
Apply properties of operations as strategies to add and sign of the number with the greater absolute value.
subtract rational numbers.
The Number System—7.NS.1a Motivate the Lesson
Ask: Have you noticed that a measuring cup has tick marks for rational numbers much like
Describe situations in which opposite quantities combine a number line with rational numbers? Begin the Explore Activity to investigate how to use
to make 0. Also 7.NS.1b, 7.NS.3
number lines with rational numbers.
Mathematical Practices
MP.2 Reasoning
Explore
Have students draw a number line to help them find out what number is 7.5 units to the
right of -3 on a number line. Discuss the number lines that students create and the
different methods students have for finding that 4.5 is 7.5 units to the right of -3 on a
number line.

Explain
ADDITIONAL EXAMPLE 1 EXAMPLE 1
Use the number line to find each sum.
Avoid Common Errors
A The temperature in the morning Since both rational numbers in Example 1 have the same sign, students may incorrectly
was -3.5 °F. By noon, the assume that the first addend determines the direction to move on the number line. Remind
temperature had dropped by 1.5 °F. students that the first addend determines where to start on the number line.
What was the final temperature
at noon? -5 °F Questioning Strategies CC Mathematical Practices
• In a real-life context, what could adding a negative number represent? It could represent a
-7 -6 -5 -4 -3 -2 -1 0 loss or a withdrawal.
B Eya bought 1__34 pounds of apples • Why do you take the absolute value of the second addend in part B? The absolute value
and __34 pound of oranges. How many gives you the distance to move to the left.
pounds of fruit did she buy
altogether? 2__12 lb YOUR TURN
Engage with the Whiteboard
1 2 3 Have students directly model the addition problems on the number lines provided.
Be sure they draw the point first, then the arrow, and finally the answer line.
Interactive Whiteboard
Interactive example available online
my.hrw.com

67 Lesson 3.2
DO NOT EDIT--Changes must be made through “File info” DO NOT EDIT--Changes must be made through “File info”
CorrectionKey=D CorrectionKey=A

LESSON
Adding Rational
3.2
7.NS.1d
Reflect
Apply properties of

Numbers operations as strategies to 1. Explain how to determine whether to move right or left on the number
add and subtract rational line when adding rational numbers.
numbers. Also 7.NS.1a,
7.NS.1b, 7.NS.3 Move to the right if the number you are adding is

? EssEntial QuEstion positive and to the left if the number is negative.

How can you add rational numbers?

Adding Rational Numbers with YOUR TURN

the Same Sign Personal

Use a number line to find each sum.
4_12
To add rational numbers with the same sign, apply the rules for adding Math Trainer 2. 3 + 1_12 =
integers. The sum has the same sign as the sign of the rational numbers. Math On the Spot Online Practice
and Help
my.hrw.com my.hrw.com
0 1 2 3 4 5
EXAMPL 1
EXAMPLE 7.NS.1b
3. -2.5 + (-4.5) = -7
A Malachi hikes for 2.5 miles and stops for lunch. Then he hikes for
1.5 more miles. How many miles did he hike altogether? -7 -6 -5 -4 -3 -2 -1 0
STEP 1 Use positive numbers to represent the distance Malachi hiked.
STEP 2 Find 2.5 + 1.5.
Adding Rational Numbers with
STEP 3 Start at 2.5. -5 -4 -3 -2 -1 0 1 2 3 4 5 Different Signs
To add rational numbers with different signs, find the difference of their
STEP 4 Move 1.5 units to the right because the Math On the Spot absolute values. Then use the sign of the rational number with the greater
© Houghton Mifflin Harcourt Publishing Company • Image Credits: ©Science Photo Library/Corbis

second addend is positive. my.hrw.com absolute value.

The result is 4.
EXAMPLE 2 7.NS.1b
Malachi hiked 4 miles.
B Kyle pours out _34 liter of liquid from a beaker. Then he pours out A During the day, the temperature increases by 4.5 degrees. At night,
another _12 liter of liquid. What is the overall change in the amount the temperature decreases by 7.5 degrees. What is the overall change

© Houghton Mifflin Harcourt Publishing Company

of liquid in the beaker? in temperature?
STEP 1 Use negative numbers to represent the amount of change STEP 1 Use a positive number to represent the increase in temperature
each time Kyle pours liquid from the beaker. and a negative number to represent a decrease in temperature.
STEP 2
4
​ _​1 ).
Find - _3 + ​(-
2 STEP 2 Find 4.5 + (-7.5).

STEP 3 Start at -_34. STEP 3 Start at 4.5.

-2 -1 0 -5 -4 -3 -2 -1 0 1 2 3 4 5

STEP 4 Move |​​-_​12 | =​ _12 unit to the left because the second addend STEP 4 Move | -7.5 | = 7.5 units to the left because the second addend
is negative. is negative.
The result is -1_14. The result is -3.
The amount of liquid in the beaker has decreased by 1_14 liters. The temperature decreased by 3 degrees overall.
Lesson 3.2 67 68 Unit 1

7_MCADESE202610_U1M03L2.indd 67 08/09/16 5:58 PM 7_MCAAESE202610_U1M03L2.indd 68 12/04/13 10:53 AM

PROFESSIONAL DEVELOPMENT
Integrate Mathematical Math Background
Practices MP.2 The absolute value of a number is the distance
This lesson provides an opportunity to address that number is from zero, or its length, on a
this Mathematical Practice standard. It calls for number line and is always expressed as a
students to create and use representations to nonnegative number. Absolute value is
organize and communicate mathematical ideas. ­sometimes known as magnitude. The sum a + b
Students use a number line to add two rational can be understood as the combined length, or
numbers that have the same sign and two magnitude, of a segment that is a units long
rational numbers that have different signs. adjoined to a segment that is b units long.

Adding Rational Numbers 68

 ADDITIONAL EXAMPLE 2 EXAMPLE 2
Use a number line to find each sum.
Questioning Strategies CC Mathematical Practices
A Ulysses is climbing a mountain. One • When using a number line, how is finding a sum where the two addends have the same
morning, he hiked 3.5 miles up the sign similar to finding a sum where the two addends have different signs, and how is it
mountain. In the evening, he hiked
different? In both methods, the sign of the second addend determines which direction to
4.5 miles down the mountain. What
was his overall change in altitude move to find the sum on a number line. When the two addends have same sign, the sum
that day? -1 mile also has that sign. When the two addends have different signs, the sum has the sign of the
addend with the greatest absolute value.

-4 -2 0 2 4 Connect Vocabulary EL
In part B, make sure students understand that writing a check reduces the amount of
B Jillian is playing a video game. In money in an account and that making a deposit increases the amount of money in an
one scene, she loses 6.5 health account. Discuss other ways that a bank balance would increase or decrease.
points. In the next scene, she gains
8 health points. What is the overall
increase or decrease to Jillian’s YOUR TURN
health points? increase of 1.5 health Focus on Modeling CC Mathematical Practices
points For Exercise 7, make sure that students can identify the scale being used on the number
line. Point out that there are 4 sections between 0 and 1, so the scale being used for each
mark is __14.
-8 -6 -4 -2 0 2
EXAMPLE 3
Interactive Whiteboard
Interactive example available online Connect Vocabulary EL
Students may incorrectly assume that additive inverse refers to the opposite of addition,
my.hrw.com which is subtraction. Make sure students understand that the additive inverse refers to a
number, rather than an operation.

Integrating Language Arts EL

ADDITIONAL EXAMPLE 3 Encourage English Learners to take notes on new terms or concepts and to write them in
Use a number line to find each sum. familiar language.
A At the start of summer, Joan got a Questioning Strategies CC Mathematical Practices
haircut that took off 1.75 inches. By
• Is the opposite of the additive inverse of 7 equal to 7? Explain. Yes; the opposite of the
the end of the summer, her hair had
grown 1.75 inches. What is the additive inverse of a number is the original number.
overall increase or decrease to • Is there any number on a number line that does not have an opposite? Explain. No; the
Joan’s hair in inches? 0 inches number line goes on forever, so every number that is on a number line will have an
opposite that is on the other side of 0. The opposite of 0 is 0.
-2 -1 0
Animated Math
B Brianna add 2__12 pounds of carrots to Rational Number
her shopping cart. Then she takes Addition
2__12 pounds of carrots out of the
cart. What is the overall increase or Students build fluency with rational number addition in this engaging, fast-paced game.
decrease in pounds of carrots in the
shopping cart? 0 pounds YOUR TURN
Talk About It
Check for Understanding
-4 -2 0 2 4
Ask: What is always true about the sum of a number and its opposite? The sum of a
number and its opposite is always 0.
Interactive Whiteboard
Interactive example available online
my.hrw.com

69 Lesson 3.2
DO NOT EDIT--Changes must be made through “File info” DO NOT EDIT--Changes must be made through “File info”
CorrectionKey=A CorrectionKey=D

B Ernesto writes a check for $2.50. Then he deposits $6 in his

checking account. What is the overall increase or decrease in
Finding the Additive Inverse
the account balance? The opposite, or additive inverse, of a number is the same distance from 0 on
a number line as the original number, but on the other side of 0. Zero is its own
STEP 1 Use a positive number to represent a deposit and a negative Animated additive inverse.
number to represent a withdrawal or a check. Math Math On the Spot
my.hrw.com my.hrw.com
STEP 2 Find -2.5 + 6. EXAMPLE 3 7.NS.1a, 7.NS.1b, 7.NS.1d

STEP 3 Start at -2.5. A A football team loses 3.5 yards on their first play. On the next play,
-5 -4 -3 -2 -1 0 1 2 3 4 5 they gain 3.5 yards. What is the overall increase or decrease in yards?
My Notes
STEP 4 Move | 6 | = 6 units to the right because the second addend Math Talk STEP 1 Use a positive number to represent the gain in yards and a
Mathematical Practices
is positive . negative number to represent the loss in yards.
Explain how to use
a number line to find
The result is 3.5. the additive inverse, or STEP 2 Find -3.5 + 3.5.
opposite, of -3.5.
The account balance will increase by $3.50.
Find a number STEP 3 Start at -3.5.
-5 -4 -3 -2 -1 0 1 2 3 4 5
Reflect that is | -3.5 |
4. Do -3 + 2 and 2 + (-3) have the same sum? Does it matter if the units from 0, STEP 4 Move | 3.5 | = 3.5 units to the right, because the second addend
negative number is the first addend or the second addend? but on the is positive.
The order of the addends does not matter when opposite side The result is 0. This means the overall change is 0 yards.
of 0 from -3.5.
adding a positive and negative rational number.
-3 + 2 and 2 + (-3) both equal -1. My Notes Addition Property of Opposites
The sum of a number and its opposite, or additive inverse, is 0. This
can be written as p + (-p) = 0.
5. Make a Conjecture Do you think the sum of a negative number and a
positive number will always be negative? Explain your reasoning.
No; The sum could be positive if the positive addend
has a greater absolute value than the negative
addend.
YOUR TURN
© Houghton Mifflin Harcourt Publishing Company

© Houghton Mifflin Harcourt Publishing Company

Use a number line to find each sum.
YOUR TURN
Use a number line to find each sum.
9. ( )
2 _12 + -2 _12 = 0 10. -4.5 + 4.5 = 0

6. -8 + 5 = -3
-8 -7 -6 -5 -4 -3 -2 -1 0 1 2 -5 -4 -3 -2 -1 0 1 2 3 4 5 -5 -4 -3 -2 -1 0 1 2 3 4 5
-_14
7. 1
_
2 ( )=
+ -_34 11. Kendrick adds _3 cup of chicken stock to a pot. Then he takes _3 cup
4 4
-1 0 1 of stock out of the pot. What is the overall increase or decrease in
Personal Personal
Math Trainer the amount of chicken stock in the pot?
8. -1 + 7 = 6 Math Trainer
Online Practice Online Practice The overall change is 0 cups.
-3 -2 -1 0 1 2 3 4 5 6 7 8 and Help and Help
my.hrw.com my.hrw.com

Lesson 3.2 69 70 Unit 1

7_MCAAESE202610_U1M03L2.indd 69 16/04/13 1:12 AM 7_MCADESE202610_U1M03L2.indd 70 08/09/16 6:00 PM

DIFFERENTIATE INSTRUCTION
Manipulatives Visual Clues Additional Resources
Demonstrate how to add positive and negative Have students create a word wall listing words Differentiated Instruction includes:
integers using two-color counters. Red counters or phrases that tell when a number should be ••Reading Strategies
represent the positive numbers in an addition positive (like “made a deposit”, or “increase”), ••Success for English Learners EL
problem, and yellow counters represent the and words or phrases that tell when a number
••Reteach
negative numbers in the addition problem. For should be negative (like “spent” or “lost”).
example, to model 3 + (-7), use 3 red counters Students can refer to the list when reading word ••Challenge PRE-AP
and 7 yellow counters. Since a number and its problems to help them write the correct
opposite have a sum of 0, remove the same expression to be simplified.
number of red and yellow counters. Removing
3 red and 3 yellow counters leaves 4 yellow
counters, so 3 + (-7) = -4.

Adding Rational Numbers 70

 ADDITIONAL EXAMPLE 4 EXAMPLE 4
One week, Gerard wrote a check for
Focus on Communication
$8.75, deposited $4.50, and wrote
Discuss with students why numbers with the same sign were grouped together before
another check for $2.50. What was the
adding. Students should realize that grouping numbers with like signs is done for
change to Gerard’s bank account that
convenience but that adding the numbers in any order will give the same answer.
week? -$6.75
Questioning Strategies CC Mathematical Practices
Interactive Whiteboard
Interactive example available online • When adding three or more numbers, numbers with the same sign can be grouped for
convenience. What other ways to group rational numbers might be convenient when
my.hrw.com adding? Sample answer: grouping opposite numbers together or grouping together
numbers whose sum is a whole number
• If you add two positive numbers and one negative number, will the sum always be
positive? Explain. No; the sum can be either positive or negative. For example:
1 + 2 + (-8) = –5.

YOUR TURN
Focus on Modeling CC Mathematical Practices
Have students sketch number lines in the spaces beside Exercises 11 and 12 and show that
their answers are reasonable by modeling each sum on the respective number line.

Elaborate
Talk About It
Summarize the Lesson
Ask: How can you compare and contrast the processes for finding the sum of two
numbers with the same sign, the sum of two numbers with different signs, and the
sum of two numbers that are opposites? The sum of two numbers with the same sign has
the same sign as the two numbers. The sum of two numbers with different signs is the
difference in their absolute values and has the same sign as the number with the greater
absolute value. The sum of two opposites is always 0.

GUIDED PRACTICE
Avoid Common Errors
Exercises 1–6 Remind students that the second addend determines which direction to
move on the number line.
Exercise 3 Caution students to determine the scale of the given number line by counting
the number of sections between whole numbers.

Engage with the Whiteboard

For Exercises 1–6, have students directly model the addition problems on the
number lines provided. Be sure they draw the point first, then the arrow, and finally
the answer line.

71 Lesson 3.2
 Guided Practice
Adding Three or More
Rational Numbers Use a number line to find each sum. (Example 1 and Example 2)
-4.5 5
Recall that the Associative Property of Addition states that if you are adding 1. -3 + (-1.5) = 2. 1.5 + 3.5 =
more than two numbers, you can group any of the numbers together. This Math On the Spot
property can help you add numbers with different signs. my.hrw.com
-5 -4 -3 -2 -1 0 1 2 3 4 5 -5 -4 -3 -2 -1 0 1 2 3 4 5
3
_
EXAMPL 4
EXAMPLE 7.NS.1d, 7.NS.3
3. _14 + _12 = 4 ( )
4. -1_12 + -1 _12 = -3
Tina spent $5.25 on craft supplies to make friendship bracelets. She made
$6.75 and spent an additional $3.25 for supplies on Monday. On Tuesday,
-1 - 0.5 0 0.5 1 -5 -4 -3 -2 -1 0 1 2 3 4 5
she sold an additional $4.50 worth of bracelets. What was Tina’s overall
profit or loss?
5. 3 + (-5) = -2 6. -1.5 + 4 = 2.5
Profit means the
STEP 1 Use negative numbers to represent the
difference between
amount Tina spent and positive numbers income and costs -5 -4 -3 -2 -1 0 1 2 3 4 5 -5 -4 -3 -2 -1 0 1 2 3 4 5
to represent the money Tina earned. is positive.
7. Victor borrowed $21.50 from his mother to go to the theater. A week later,
STEP 2 Find -5.25 + 6.75 + (-3.25) + 4.50. he paid her $21.50 back. How much does he still owe her? (Example 3)
$0
STEP 3 Group numbers with the same sign.
8. Sandra used her debit card to buy lunch for $8.74 on Monday. On
-5.25 + (-3.25) + 6.75 + 4.50 Commutative Property Tuesday, she deposited $8.74 back into her account. What is the overall
(-5.25 + (-3.25)) + (6.75 + 4.50) Associative Property increase or decrease in her bank account? (Example 3)
$0
STEP 4 -8.50 + 11.25 Add the numbers inside the parentheses.
Find each sum without using a number line. (Example 4)
Find the difference of the absolute
values: 11.25 - 8.50 9. 2.75 + (-2) + (-5.25) = -4.5 ( ) ( )
10. -3 + 1 _12 + 2 _12 = 1
Use the sign of the number with the
2.75 greater absolute value. The sum is 11. -12.4 + 9.2 + 1 = -2.2 12. -12 + 8 + 13 = 9
positive.
-_12
13. 4.5 + (-12) + (-4.5) = -12 ( )
14. _14 + - _34 =
© Houghton Mifflin Harcourt Publishing Company

© Houghton Mifflin Harcourt Publishing Company

Tina earned a profit of $2.75.
-2_12 -9_18
15. -4 _12 + 2 = 16. -8 + (-1 _18 ) =

YOUR TURN
? ESSENTIAL QUESTION CHECK-IN
Find each sum.
17. How can you use a number line to find the sum of -4 and 6?
12. -1.5 + 3.5 + 2 = 4
Start at -4. Move 6 units to the right because 6 is
13. ( )
3_14 + (-2) + -2 _14 = -1
Personal
positive. The sum is 2.
14. -2.75 + (-3.25) + 5 = -1 Math Trainer
Online Practice
15. 15 + 8 + (-3) = 20 and Help
my.hrw.com

Lesson 3.2 71 72 Unit 1

DIFFERENTIATE INSTRUCTION
Kinesthetic Experience Integer Bars Integer bars are made from the Use the integer bars to model addition of
Students may find it meaningful to build models graph paper to match the __12 -inch scale of the integers on the number line. Working in pairs
to reinforce their understanding of addition of number line. The integer bars are strips of paper, allows students to have duplicates of each
integers. The number line and integer bars cut to lengths from 1 to 10, matching the integer bar, thereby allowing for the exploration
needed for this activity can be produced by the number line. Label each bar from 1 to 10 and of addition of opposites.
students or provided as a handout. draw an arrow from A to B. Each bar is then
flipped, from left to right, and labeled from (-1) -10 -8 -6 -4 -2 0 2 4 6 8 10
Number Line __14 -inch graph paper (8__12 × 11) to (-10) on each bar, respectively, and an arrow
A B B A
provides a convenient scale for creating a drawn from A to B. The integer bars are drawn 5 -5
number line that ranges from -10 to 10, in correctly if the arrowheads on each strip point to 3 -3
__1 -inch increments. Draw the number line along the same end of the strip, (B). 2 -2
2
the 11-inch edge of the page.

Adding Rational Numbers 72

 Personal
Math Trainer
Online Assessment
Evaluate Focus | Coherence | Rigor
and Intervention

Online homework GUIDED AND INDEPENDENT PRACTICE

assignment available 7.NS.1d, 7.NS.1a, 7.NS.1b, 7.NS.3
my.hrw.com
Concepts & Skills Practice
3.2 LESSON QUIZ Example 1 Exercises 1–4, 20
Adding Rational Numbers with the Same Sign
7.NS.1d, 7.NS.1a,
Example 2 Exercises 5, 6, 19, 21, 24, 25
7.NS.1b, 7.NS.3 Adding Rational Numbers with Different Signs
Find each sum.
Example 3 Exercises 7, 8, 18
1. 8 + (-2.25) = ______ Finding the Additive Inverse
2. -3__12 + (-2__34 ) = ______ Example 4 Exercises 9–16, 22, 23, 26
3. -4.5 + 4.5 = ______ Adding Three or More Rational Numbers
4. 1__14 + 2__12 = ______
5. -9.5 + 6 = ______
6. One month, Julia collected 8.4 Exercise Depth of Knowledge (D.O.K.) Mathematical Practices
gallons of rainwater. That month she
used 5.2 gallons of rainwater to 18–21 2 Skills/Concepts MP.4 Modeling
water her garden and 6.5 gallons 22 MP.4 Modeling
2 Skills/Concepts
of rainwater to water her flowers.
How much was the supply of 23–25 2 Skills/Concepts MP.4 Modeling
rainwater increased or decreased by
the end of the month? 26 2 Skills/Concepts MP.2 Reasoning
27 2 Skills/Concepts MP.7 Using Structure
Lesson Quiz available online
28 2 Skills/Concepts MP.4 Modeling
my.hrw.com
29 3 Strategic Thinking MP.7 Using Structure
30–31 3 Strategic Thinking MP.3 Logic
Answers
1. 5.75
2. -6__14 Additional Resources
Differentiated Instruction includes:
3. 0
• Leveled Practice Worksheets
4. 3__34
5. -3.5 Exercise 31 combines concepts from the California Common Core
6. -3.3 gallons cluster “Apply and extend previous understandings of operations with
fractions to add, subtract, multiply, and divide rational numbers.”

73 Lesson 3.2
 Name Class Date opposite or additive inverse
27. Vocabulary -2 is the of 2.

28. The basketball coach made up a game to play where each player takes
3.2 Independent Practice Personal
Math Trainer 10 shots at the basket. For every basket made, the player gains 10 points.
Online Practice For every basket missed, the player loses 15 points.
7.NS.1a, 7.NS.1b, 7.NS.1d, 7.NS.3 and Help
my.hrw.com
a. The player with the highest score sank 7 baskets and missed 3. What
18. Samuel walks forward 19 steps. He represents this movement with a positive was the highest score?
19. How would he represent the opposite of this number? -19 The highest score was 25 points.
19. Julia spends $2.25 on gas for her lawn mower. She earns $15.00 mowing her
neighbor’s yard. What is Julia’s profit? $12.75 b. The player with the lowest score sank 2 baskets and missed 8. What
was the lowest score?
20. A submarine submerged at a depth of -35.25 meters dives an additional The lowest score was -100 points.
8.5 meters. What is the new depth of the submarine? -43.75 meters
21. Renee hiked for 4 _34 miles. After resting, Renee hiked back along the same c. Write an expression using addition to find out what the score would
route for 3_14 miles. How many more miles does Renee need to hike to be if a player sank 5 baskets and missed 5 baskets.
1_12 miles
return to the place where she started? 10 + 10 + 10 + 10 + 10 + (-15) + (-15) +
22. Geography The average elevation of the city of New Orleans, Louisiana, (-15) + (-15) + (-15)
is 0.5 m below sea level. The highest point in Louisiana is Driskill Mountain
at about 163.5 m higher than New Orleans. How high is Driskill Mountain?
about 163 m
FOCUS ON HIGHER ORDER THINKING Work Area
23. Problem Solving A contestant on a game show has 30 points. She
answers a question correctly to win 15 points. Then she answers a question
29. Communicate Mathematical Ideas Explain the different ways it is
incorrectly and loses 25 points. What is the contestant’s final score?
possible to add two rational numbers and get a negative number.
30 + 15 + (-25) = 20; the final score is 20 points The sum of two negative rational numbers is always
Financial Literacy Use the table for 24–26. Kameh owns Month Income ($) Expenses ($) negative. The sum of a negative and a positive rational
a bakery. He recorded the bakery income and expenses
in a table. January 1,205 1,290.60 number is negative if the absolute value of the negative
February 1,183 1,345.44
24. In which months were the expenses greater than the number is greater than that of the positive number.
March 1,664 1,664.00
income? Name the month and find how much money
June 2,413 2,106.23 30. Explain the Error A student evaluated -4 + x for x = -9 _12 and got an
was lost. Jan: -$85.60, Feb: -$162.44 answer of 5 _12. What might the student have done wrong?
July 2,260 1,958.50
25. In which months was the income greater than the August 2,183 1,845.12 Sample answer: The student might have subtracted the
© Houghton Mifflin Harcourt Publishing Company

© Houghton Mifflin Harcourt Publishing Company

expenses? Name the months and find how much
money was gained. absolute values of the numbers.
June: $306.77, July: $301.50, Aug: $337.88 31. Draw Conclusions Can you find the sum [5.5 + (-2.3)] + (-5.5 + 2.3)
without performing any additions? Explain.
26. Communicate Mathematical Ideas If the bakery started with an extra
$250 from the profits in December, describe how to use the information in Yes; 5.5 and -5.5 are opposites and -2.3 and 2.3 are
the table to figure out the profit or loss of money at the bakery by the end
opposites. [5.5 + (-2.3)] + (-5.5 + 2.3) = 0 because the
of August. Then calculate the profit or loss.
Find the sum of all the money in the “Income” column and $250. Consider the sum of opposites is 0.
Income to be positive. Find the sum of all the money in the “Expenses” column.
Consider the Expenses to be negative. The sum of the income and expenses
will be the profit or loss of the bakery; $948.11
Lesson 3.2 73 74 Unit 1

EXTEND THE MATH PRE-AP Activity available online my.hrw.com

Provide students with the following challenge exercise.

Several business partners have access to a safe, where they keep their savings. One night, Will
took $7.50 from the safe. Carl deposited three times as much as Barry did. Barry deposited
twice the amount that Will took. Belinda took half the amount that Carl deposited. Edgar took
$7.50 less than Belinda. How much did the amount in the safe increase or decrease overall in
one night? Explain.
$15; Will changed the amount in the safe by -$7.50. Barry changed the amount in the safe by
$15.00. Carl changed the amount in the safe by $45.00. Belinda changed the amount in the
safe by $-22.50. Edgar changed the amount in the safe by $–15.00. To find the overall change,
find the sum of all the numbers. -7.5 + 15 + 45 + (-22.5) + (-15) = $15.

Adding Rational Numbers 74

 LESSON 3.3 Subtracting Rational Numbers

Lesson Support
Content Objective Students will learn to subtract rational numbers.

Language Objective Students will explain how to subtract rational numbers.

California Common Core Standards

7.NS.1c Understand subtraction of rational numbers as adding the additive inverse, p - q = p + (-q). Show that the distance
between two rational numbers on the number line is the absolute value of their difference, and apply this principle in real-world contexts.
MP.2 Reason abstractly and quantitatively.

Focus | Coherence | Rigor

Building Background
Eliciting Prior Knowledge Have students locate each of 4 1 12 21
-3 5 -3.1 -2.5 -1 3 -0.5 3 2 3.25
the following rational numbers on a number line and tell
which number has the greatest absolute value and which
-4 -3 -2 -1 0 1 2 3 4
has the least absolute value. Remind students that the
absolute value of a number is its distance from 0.
-3__45 has the greatest absolute value
1__2 , -2.5, -3.1, 2__1 , -3__4 , 3.25, -0.5, -1__1
3 2 5 3 -0.5 has the least absolute value

Learning Progressions Cluster Connections

In this lesson, students apply the skills they have developed This lesson provides an excellent opportunity to connect ideas
subtracting integers to subtracting rational numbers. Some key in this cluster: Apply and extend previous understandings of
understandings for students are the following: operations with fractions to add, subtract, multiply, and
divide rational numbers.
• The Commutative Property does not apply to subtraction.
• To subtract a negative rational number on a number line, Give students the following prompt: “Jared wants to subtract
move in the opposite direction. -4__13 from -5__29 . Jared thinks the difference is going to be less
• Subtraction can be used to find the distance between than -5__29 . Is he right?” Ask students to use a number line help
numbers. them justify their answer.
• The subtraction of a negative number can be rewritten as Sample answer: No; Jared is starting at -5__29 , moving | -4 __13 | to
the adding the opposite number. the right, because he is subtracting a negative.
Students will continue to develop mastery of the operations move -4 __13 to the right
with signed numbers.

-6 -5 -4 -3 -2 -1 0 1 2 3 4
start at -5__29

75A
 PROFESSIONAL DEVELOPMENT
EL
Language Support
California ELD Standards
Emerging 2.I.6c. Reading/viewing closely – Use knowledge of morphology, context, reference materials, and visual cues
to determine the meaning of unknown and multiple-meaning words on familiar topics.
Expanding 2.I.6c. Reading/viewing closely – Use knowledge of morphology, context, reference materials, and visual cues
to determine the meaning of unknown and multiple-meaning words on familiar and new topics.
Bridging 2.I.6c. Reading/viewing closely – Use knowledge of morphology, context, reference materials, and visual
cues to determine the meaning, including figurative and connotative meanings, of unknown and multiple- meaning words on a variety of
new topics.

EL
Linguistic Support

Academic/Content Vocabulary Background Knowledge

Fahrenheit scale – Although the United States superlative form – Word problems in this lesson
primarily uses the Fahrenheit scale, most countries that show the most or least require using the -est
use the Celsius scale for temperature measurement. form of the adjective. This is called the superlative
In this lesson on subtracting rational numbers, you form. Remind English learners how to use the
can help students for whom the Fahrenheit scale is superlative form: hottest week, deepest canyon,
new by pointing out regardless of the measurement tallest tree, and so on.)
scale, the rules for subtracting rational numbers will
be the same. Adjective Comparative Superlative

high higher highest

deep deeper deepest

low lower lowest

EL
Leveled Strategies for English Learners

Emerging Have students at this level of English proficiency work in pairs to draw a thermometer,
and mark changes in temperature showing how to subtract rational numbers to find out the
amount of change in temperature.

Expanding Have students at this level of English proficiency work in pairs to illustrate and discuss
changes in temperature and to demonstrate how to subtract rational numbers.

Bridging Pair students and have them illustrate and describe using a thermometer how a change
in weather is solved by subtracting rational numbers.

Subtracting Rational Numbers 75B

 LESSON

3.3 Subtracting Rational Numbers

CA Common Core Engage
Standards
ESSENTIAL QUESTION
The student is expected to: How do you subtract rational numbers? Sample answer: Move on the number line in the
The Number System—7.NS.1c direction opposite of addition.
Understand subtraction of rational numbers as adding the Motivate the Lesson
additive inverse, p - q = p + (-q). Show that the Ask: How is a thermometer like a number line? Begin the Explore Activity to discuss this.
distance between two rational numbers on the number
line is the absolute value of their difference, and apply
this principle in real-world contexts. Also 7.NS.1
Mathematical Practices Explore
MP.2 Reasoning Compare an analog thermometer with a number line. Discuss with students any similarities
(both show numbers in order) and any differences (numbers on a number line increase as
you move right, but the numbers on a thermometer increase as you move up).

Explain
ADDITIONAL EXAMPLE 1 EXAMPLE 1
A group of campers set up their tent at
Avoid Common Errors
6.25 meters above sea level. They hike
Students may want to rewrite the expression 5.5-7.25 incorrectly as 7.25-5.5. Remind
down a valley and stop 9.5 meters
students that the Commutative Property does not apply to subtraction.
below where they set up camp. What is
the campers’ elevation when they Questioning Strategies Mathematical Practices
stop? -3.25, or 3.25 meters below sea
• When will a difference be positive, and when will it be negative? A difference will be
level
positive when the minuend (first number) is greater than the subtrahend (second
number). The answer will be negative when the minuend is less than the subtrahend.
-4 -2 0 2 4 6 8

Interactive Whiteboard YOUR TURN

Interactive example available online Focus on Modeling Mathematical Practices
Show students that when the subtrahend is a positive rational number, the difference will
my.hrw.com always be less than the minuend. Subtracting a positive number means you move to the
left, and numbers on a number line decrease as you move to the left.
ADDITIONAL EXAMPLE 2
Sheri ended the game with -8.5 EXAMPLE 2
points in a trivia game. On her last turn, Focus on Modeling Mathematical Practices
she lost 6.25 points. How many points Discuss how to select the units to be used on a number line in solving this example.
did she have before her last turn?
-2.25 points Questioning Strategies Mathematical Practices
• When you subtract a negative number, what is always true about the relationship
between the minuend and the difference? The difference is greater than the minuend.
-10 -8 -6 -4 -2 0

Interactive Whiteboard YOUR TURN

Interactive example available online Connect to Daily Life Mathematical Practices
my.hrw.com To help students understand the operation in Exercise 6, present the following situation:
“Tim thought he would be paying $0.25 for a sandwich. However, the cashier would not let
him use his coupon for $1.50 off. How much will Tim’s sandwich cost?”
75 Lesson 3.3
 LESSON
Subtracting Rational 7.NS.1c
Subtracting Negative Rational Numbers
3.3 Numbers
Understand subtraction…
as adding the additive
inverse…. Show that the
distance between two
rational numbers…is the
absolute value of their
To subtract negative rational numbers, move in the opposite direction on the
number line.

Math On the Spot

difference…. Also 7.NS.1
EXAMPLE 2
? my.hrw.com 7.NS.1
ESSENTIAL QUESTION
How do you subtract rational numbers? During the hottest week of the summer, the water level of the Muskrat
River was _56 foot below normal. The following week, the level was _13 foot
below normal. What is the overall change in the water level?
Subtracting Positive Rational Numbers
To subtract rational numbers, you can apply the same rules you use to Subtract to find the difference in water levels.
subtract integers.
STEP 1 Find - _13 - (- _56 ).
Math On the Spot
EXAMPL 1
EXAMPLE 7.NS.1 my.hrw.com STEP 2 Start at - _13.
-1 0 1
The temperature on an outdoor thermometer on Monday was 5.5 °C. STEP 3 Move | -_56 | = _56 to the right because
The temperature on Thursday was 7.25 degrees less than the you are subtracting a negative number.
temperature on Monday. What was the temperature on Thursday?
The result is _12.
Subtract to find the temperature on Thursday.
So, the water level changed 2_1 foot.
STEP 1 Find 5.5 - 7.25.
Reflect
STEP 2 Start at 5.5. - 6 -5 -4 - 3 - 2 - 1 0 1 2 3 4 5 6 4. Work with other students to compare addition of negative numbers
on a number line to subtraction of negative numbers on a number line.
STEP 3 Move | 7.25 | = 7.25
units to the left because you are
Sample answer: When adding a negative number, move to the
subtracting a positive number.

The result is -1.75.

left. When subtracting a negative number, move to the right.

The temperature on Thursday was -1.75 °C. 5. Compare the methods used to solve Example 1 and Example 2.
Both methods used a number line. In Example 1, you
move left from the starting point because you are
YOUR TURN subtracting a positive number. In Example 2, you move
© Houghton Mifflin Harcourt Publishing Company

© Houghton Mifflin Harcourt Publishing Company

Use a number line to find each difference. right because you are subtracting a negative number.
1. -6.5 - 2 = -8.5
- 9 - 8.5 - 8 - 7.5 - 7 - 6.5 - 6 - 5.5 - 5 - 4.5 - 4
YOUR TURN
-_12 Use a number line to find each difference.
2. 1 _12 - 2 =
-1 0 1 2 3 4 6. 0.25 - ( -1.50 ) = 1.75

Personal Personal
-1 0 1 2
3. -2.25 - 5.5 = -7.75 - 10 - 9 - 8 - 7 - 6 - 5 - 4 - 3 - 2 - 1
Math Trainer Math Trainer
0 Online Practice Online Practice 1
_
( )
and Help and Help
7. -_12 - -_34 = 4
my.hrw.com my.hrw.com
-1 0 1

Lesson 3.3 75 76 Unit 1

PROFESSIONAL DEVELOPMENT
Integrate Mathematical Math Background
Practices MP.2 Subtraction is formally defined as addition of the
This lesson provides an opportunity to address opposite, or additive inverse. The Commutative
this Mathematical Practice standard. It calls Property does not apply to subtraction problems,
for students to create and use multiple but when subtraction is written as addition of the
representations to organize, record, and opposite, the Commutative Property does apply.
communicate mathematical ideas. Students use The rational numbers are closed under the
number lines to model subtraction problems operations of addition and subtraction, which
and find their solutions. means that adding or subtracting any two
rational numbers will produce another rational
number.

Subtracting Rational Numbers 76

 EXPLORE ACTIVITY 1
Focus on Modeling Mathematical Practices
Demonstrate to students that p - q = p + (-q) is true by drawing one arrow to represent
-2__12 - 7__12 and another arrow to represent -2__12 + ( -7__12 ). The arrows will both point to -10.

Integrating Language Arts EL

Be sure English learners understand the instructions in Reflect Question 9. You may want to
point out that “conjecture” means a probably true statement based on evidence.

Questioning Strategies Mathematical Practices

• How can you use the number line to check that your answer is reasonable? Identify the
word in the problem that suggests subtraction, like descend. Since you are subtracting a
positive value, the difference will be less than the minuend. Be sure that the numbers get
less and less in the direction that the arrow is pointing.
• Explain how you could change 13 - (-8) to make it easier to evaluate. Change
13 - (-8) to 13 + (8) because subtraction means to add the opposite.

EXPLORE ACTIVITY 2
Integrating Language Arts EL

You may want to pair English learners with a partner for Explore Activity 2 to help them
develop their language skills.

Questioning Strategies Mathematical Practices

• What are some equivalent expressions for the absolute value of the difference between
-11 and -5? | -11 - (-5) |, | -11 + (5) |, | 5 - 11 |, and | 5 + (-11) |

Engage with the Whiteboard

Have students use the space to the right of the number line to model a related
problem: “A cave explorer descended from an elevation of -5 meters to an elevation
of -11 meters. What vertical distance did the explorer descend?” Ask them to explain how
the two models show that | a - b | = | b - a |.

77 Lesson 3.3
 EXPLORE ACTIVITY 2
7.NS.1c
EXPLORE ACTIVITY 1
7.NS.1c

Adding the Opposite Finding the Distance between

Joe is diving 2 _12 feet below sea level. He decides to descend 7 _12 more feet.
How many feet below sea level is he?
Two Numbers
A cave explorer climbed from an elevation of -11 meters to an elevation
STEP 1 Use negative numbers to represent the number of feet below of -5 meters. What vertical distance did the explorer climb?
sea level. There are two ways to find the vertical distance.

STEP 2 Find -2 _12 - 7 _12. A Start at -11 . 0

-1
Count the number of units on the vertical number line up
STEP 3 Start at -2 _12. - 10 - 9 -8 -7 -6 -5 -4 -3 -2 -1 0 to -5. -2
-3
STEP 4 Move | 7 _12 |= 7 _12 units to the left The explorer climbed 6 meters. -4
This means that the vertical distance between -5
because you are subtracting a positive number.
-6
-11 meters and -5 meters is 6 meters.
The result is -10 . -7
You move left on a number line
10 feet below B Find the difference between the two elevations and use -8
Joe is sea level. to add a negative number. You
move the same direction to absolute value to find the distance. -9

Reflect
subtract a positive number.
-11 - (-5) = -6 - 10
- 11
8. Compare the difference -3.5 - 5.8 to the sum -3.5 + (-5.8). Take the absolute value of the difference because
They both equal -9.3. distance traveled is always a nonnegative number.
| -11 - (-5) | = 6

© Houghton Mifflin Harcourt Publishing Company • Image Credits: Robbie Shone/Aurora Photos/Alamy
The vertical distance is 6 meters.
9. Analyze Relationships Work with other students to explain how to
change a subtraction problem into an addition problem.
Reflect
Sample answer: Change the minus sign to a plus,
10. Does it matter which way you subtract the values when finding
and change the second number to its opposite. distance? Explain.
No, it does not matter since you take the absolute
value of the difference.
© Houghton Mifflin Harcourt Publishing Company

11. Would the same methods work if both the numbers were positive?
Adding the Opposite
What if one of the numbers were positive and the other negative?
To subtract a number, add its opposite. This can also be written Yes, because you take the absolute value of the difference.
as p - q = p + (-q).

Distance Between Two Numbers

The distance between two values a and b on a number line is

represented by the absolute value of the difference of a and b.
Distance between a and b = | a - b | or | b - a |.

Lesson 3.3 77 78 Unit 1

DIFFERENTIATE INSTRUCTION
Cognitive Strategies Curriculum Integration Additional Resources
Have students discuss which subtraction Finding the distance between two numbers can Differentiated Instruction includes:
methods they prefer for different kinds of be useful when comparing historical dates. • Reading Strategies
problems. Compare negative and positive numbers to • Success for English Learners EL
historical dates that are before or after 0 B.C.E.
Student 1: When I subtract a negative • Reteach
The number of years between two events can
number from a positive number, like • Challenge PRE-AP
be found using the subtraction methods in this
6 - (-7), I change that to adding the
lesson.
opposite number, which is 6 + 7.
Student 2: When I subtract a negative
number from a negative number, like
-11 - 6, I change that to -11 + (-6) and
use the rules for adding two negative
numbers.

Subtracting Rational Numbers 78

 Elaborate
Talk About It
Summarize the Lesson
Ask: How is subtracting rational numbers similar to adding rational numbers?
Subtraction of rational numbers can be expressed as addition of the opposite.
Ask: How can you tell when the difference will be positive or negative? When subtracting a
positive value, the difference is less than the minuend. When subtracting a negative value,
the difference is greater than the minuend.

GUIDED PRACTICE
Avoid Common Errors
Exercises 5–10 Remind students that the Commutative Property does not apply to the
operation of subtraction.
Exercises 6–10 Students may attempt to use a number line to find the differences. Point
out that the numbers used in these problems make it difficult to use a number line.
Encourage students to express subtraction as addition, and find the sum.

Engage with the Whiteboard

Have students write the equivalent addition expression for each subtraction
expression in Exercises 1–4. Have them use the number line to show that the two
expressions are equivalent.

79 Lesson 3.3
 Guided Practice Name Class Date

Use a number line to find each difference. (Example 1, Example 2 and Explore Activity 1) 3.3 Independent Practice Personal
Math Trainer
1. 5 - (-8) = 13 Online Practice
7.NS.1, 7.NS.1c and Help
my.hrw.com
5 6 7 8 9 10 11 12 13 14 15
16. Science At the beginning of a laboratory experiment, the temperature
2. -3 _12 - 4 _12 = -8 of a substance is -12.6 °C. During the experiment, the temperature of
-9 -8 -7 -6 -5 -4 -3
the substance decreases 7.5 °C. What is the final temperature of the
substance?
3. -7 - 4 = -11 -20.1 °C
-15 -14 -13 -12 -11 -10 -9 -8 -7 -6 -5
17. A diver went 25.65 feet below the surface of the ocean, and then 16.5 feet
further down, he then rose 12.45 feet. Write and solve an expression
4. -0.5 - 3.5 = -4 to find the diver’s new depth.
-6 -5 -4 -3 -2 -1 0 1
-25.65 - 16.5 + 12.45; -29.7 ft; the diver is 29.7 ft
Find each difference. (Explore Activity 1) below the surface.
5. -14 - 22 = -36 6. -12.5 - (-4.8) = -7.7 7. _13 - (-_23 ) = 1
2 _79 78 _12
8. 65 - (-14) = 79 9. -_29 - (-3) = 10. 24 _38 - (-54 _18 ) = Astronomy Use the table for problems 18–19.
Elevations on Planets
Lowest (ft) Highest (ft)
11. A girl is snorkeling 1 meter below sea level and then dives down another 18. How much deeper is the deepest canyon on Mars Earth -36,198 29,035
0.5 meter. How far below sea level is the girl? (Explore Activity 1 1.5 meters than the deepest canyon on Venus? Mars -26,000 70,000
16,500 ft Venus -9,500 35,000
12. The first play of a football game resulted in a loss of 12_12 yards. Then a
penalty resulted in another loss of 5 yards. What is the total loss or gain? 17 __1 yards
2
loss 19. Persevere in Problem Solving What is the difference between Earth’s
(Explore Activity 1) highest mountain and its deepest ocean canyon? What is the difference
13. A climber starts descending from 533 feet above sea level and keeps between Mars’ highest mountain and its deepest canyon? Which
going until she reaches 10 feet below sea level. How many feet did she difference is greater? How much greater is it?
descend? (Explore Activity 2) 543 feet 65,233 ft; 96,000 ft; 96,000 ft (Mars); 30,767 ft
14. Eleni withdrew $45.00 from her savings account. She then used her debit
card to buy groceries for $30.15. What was the total amount Eleni took
took out $75.15
© Houghton Mifflin Harcourt Publishing Company

© Houghton Mifflin Harcourt Publishing Company

out of her account? (Explore Activity 1) 20. A city known for its temperature extremes started the day at -5 degrees
Fahrenheit. The temperature increased by 78 degrees Fahrenheit by

? ESSENTIAL QUESTION CHECK-IN midday, and then dropped 32 degrees by nightfall.

a. What expression can you write to find the temperature at nightfall? -5 + 78 - 32
15. Mandy is trying to subtract 4 - 12, and she has asked you for help. How
would you explain the process of solving the problem to Mandy, using b. What expression can you write to describe the overall change in
a number line? temperature? Hint: Do not include the temperature at the beginning
of the day since you only want to know about how much the
Start at positive 4 on the number line. Then, move 12 places to the left, temperature changed. 78 - 32
because she is subtracting a positive number. She will end on the number c. What is the final temperature at nightfall? What is the overall change
41 °F; 46 °F
-8, which is the answer. in temperature?

Lesson 3.3 79 80 Unit 1

DIFFERENTIATE INSTRUCTION
Kinesthetic Experience Model 4 + 3 = 7 Changing the direction of the 3-bar models the
Integer bars are a physical representation of subtraction operation. The 3 being upside down
3
integers and are introduced as a Differentiate 4 is a visual indication that 3 is now being
Instruction tool in the lesson on addition of subtracted.
integers. The integer bars provide a model that 0 1 2 3 4 5 6 7 8 A flip of the upside down 3 reveals a right-side
students understand and can rely on to interpret
up -3, or what was 4 - 3 is now 4 + (-3).
subtraction of integers. First, students should be
Rotate the 3-bar so that it points to the left and
fluent in addition of integers and be able to use
the 3 is upside down
the bars to model addition. -3
3 4
The introduction of subtraction, using the
integer bars, is accomplished by first modeling 4
0 1 2 3 4 5 6 7 8
addition.
0 1 2 3 4 5 6 7 8

Subtracting Rational Numbers 80

 Personal
Math Trainer
Online Assessment
Evaluate Focus | Coherence | Rigor
and Intervention

Online homework GUIDED AND INDEPENDENT PRACTICE

assignment available 7.NS.1c, 7.NS.1
my.hrw.com
Concepts & Skills Practice
3.3 LESSON QUIZ Example 1 Exercises 2–5, 7, 11, 12, 16, 22, 23
Subtracting Positive Rational Numbers
7.NS.1c, 7.NS.1
Example 2 Exercises 1, 8, 9, 13
Find each difference. Subtracting Negative Rational Numbers
1. -13 - (-5) Explore Activity 1 Exercises 5–12, 14, 16, 24
Adding the Opposite

-15 -13 -11 -9 -7 -5 Explore Activity 2 Exercises 13, 20, 21

Finding the Distance Between Two Numbers
2. -5 - 2.5

Exercise Depth of Knowledge (D.O.K.) Mathematical Practices

-9 -8 -7 -6 -5 -4 -3
16–17 2 Skills/Concepts MP.4 Modeling
3. 6.25 - ( -3.5) =
18 2 Skills/Concepts MP.4 Modeling
4. -23 - 50 =
5. A scuba diver starts at 85.6 meters 19 3 Strategic Thinking MP.4 Modeling
below the surface and descends 20 2 Skills/Concepts MP.4 Modeling
until he reaches 103.2 meters below
sea level. How many meters did he 21 3 Strategic Thinking MP.4 Modeling
descend? 22 3 Strategic Thinking MP.4 Modeling
Lesson Quiz available online 23 3 Strategic Thinking MP.2 Reasoning

my.hrw.com 24 2 Skills/Concepts MP.4 Modeling

25 3 Strategic Thinking MP.7 Using Structure
Answers 26 3 Strategic Thinking MP.4 Modeling
1. -8
27 3 Strategic Thinking MP.3 Logic
2. -7.5
28 3 Strategic Thinking MP.3 Logic
3. 9.75
4. -73
5. 17.6 meters Additional Resources
Differentiated Instruction includes:
• Leveled Practice Worksheets

Exercise 27 combines concepts from the California Common Core

cluster “Apply and extend previous understandings of operations with
fractions to add, subtract, multiply, and divide rational numbers.”

81 Lesson 3.3
 21. Financial Literacy On Monday, your bank account balance was -$12.58.
Because you didn’t realize this, you wrote a check for $30.72 for groceries. FOCUS ON HIGHER ORDER THINKING Work Area
a. What is the new balance in your checking account? -$43.30
25. Look for a Pattern Show how you could use the Commutative Property
b. The bank charges a $25 fee for paying a check on a negative balance. __
7
to simplify the evaluation of the expression -16 - 4_1 - 16
__
5
.
What is the balance in your checking account after this fee? -$68.30 7 - __
Sample answer: -16
___ - 5 = - 16
1 ___
4 16
7
___ + (- 4__1 ) + (- 16
5
___
)=
c. How much money do you need to deposit to bring your account
$68.30 + (- 16
balance back up to $0 after the fee? 7
- 16
___ 5
___
) + (- 4__1 ) = -16
___
( 41 ) = -4__3 + (- 4__1 ) = -1
12 + - __

22. Pamela wants to make some friendship bracelets for her friends. Each
friendship bracelet needs 5.2 inches of string.
a. If Pamela has 20 inches of string, does she have enough to make 26. Problem Solving The temperatures for five days in Kaktovik, Alaska, are
bracelets for 4 of her friends? given below.
No, Pamela would need 20.8 inches of string to make -19.6 °F, -22.5 °F, -20.9 °F, -19.5 °F, -22.4 °F
bracelets for 4 friends. Temperatures for the following week are expected to be approximately
twelve degrees lower each day than the given temperatures. What are the
b. If so, how much string would she had left over? If not, how much highest and lowest temperatures expected for the corresponding 5 days
more string would she need? next week?
She would need 0.8 inch more string.
Lowest: -34.5 °F; Highest: -31.5 °F
23. Jeremy is practicing some tricks on his skateboard. One trick takes him
forward 5 feet, then he flips around and moves backwards 7.2 feet, then
he moves forward again for 2.2 feet. 27. Make a Conjecture Must the difference between two rational numbers
a. What expression could be used to find how far Jeremy is from his be a rational number? Explain.
starting position when he finishes the trick? Sample answer: Yes; since both numbers are rational
5 - 7.2 + 2.2 numbers, each can be written as the ratio of two integers,
b. How far from his starting point is he when he finishes the trick? Explain. for example __ba and __dc . I can rewrite both fractions using
He is exactly where he started because
a common denominator, and then subtract. The result
5 - 7.2 + 2.2 = 0.
will be a fraction, which is a rational number.
24. Esteban has $20 from his allowance. There is a comic book he wishes
© Houghton Mifflin Harcourt Publishing Company

© Houghton Mifflin Harcourt Publishing Company

to buy that costs $4.25, a cereal bar that costs $0.89, and a small remote
control car that costs $10.99. 28. Look for a Pattern Evan said that the difference between two negative
a. Does Esteban have enough to buy everything? numbers must be negative. Was he right? Use examples to illustrate your
answer.
Yes, he does.
No; sometimes the answer is negative [-4 - (-1) = -3].
b. If so, how much will he have left over? If not, how much does he still
need?
But the difference can be positive as well [-5 - (-8) = 3].
Since everything together costs $16.13, he will have
$3.87 left over.

Lesson 3.3 81 82 Unit 1

EXTEND THE MATH PRE-AP Activity available online my.hrw.com

Activity Theresa and Callie are playing a trivia game where you earn points by answering
questions correctly and lose points by answering questions incorrectly. On Theresa’s last three
turns, she answered a question correctly worth 200 points, answered a question incorrectly
worth 325 points, then answered a question incorrectly worth 65 points. Now Theresa has
-675 points. How many points did Theresa have before her last 3 turns? Explain.
-485; on the last three turns, the following amounts were added to Theresa’s score: 200, -325,
and -65. To find her score three turns ago, subtract those values from her current score.
-675 - 200 - (-325) - (-65). Subtracting a negative number is the same as adding its
opposite, so the expression can be rewritten as -675 - 200 + (325) + (65) = -485

Subtracting Rational Numbers 82

 LESSON 3.4 Multiplying Rational Numbers

Lesson Support
Content Objective Students will learn to multiply rational numbers.

Language Objective Students will demonstrate how to multiply rational numbers.

California Common Core Standards

7.NS.2 Apply and extend previous understandings of multiplication and division and of fractions to multiply and divide
rational numbers.
MP.2 Reason abstractly and quantitatively.

Focus | Coherence | Rigor

Building Background
Eliciting Prior Knowledge Have students suggest pairs Positive Products Negative Products
of integers that would have a positive product and pairs that
2 and 3 –1 and 3
would have a negative product. Then ask a volunteer to explain –3 and –5 –3 and 1
how they know whether a pair of integers would have a positive 4 and 1 4 and –5
product or a negative product. –1 and –2 5 and –4
If both integers have the same sign, the product is positive. If the
integers have different signs, the product is negative.

Learning Progressions Cluster Connections

In this lesson, students apply the skills they have developed This lesson provides an excellent opportunity to connect ideas
multiplying integers to multiplying rational numbers. Some key in this cluster: Apply and extend previous understandings of
understandings for students are the following: operations with fractions to add, subtract, multiply, and divide
rational numbers.
• To find the opposite of a number, change the sign of the
number. The opposite of a positive number is negative. Give students the following prompt: “Kim borrows $4.25 from
The opposite of a negative number is positive. each of 3 friends. How much does he owe in all?”
• When multiplying n negative numbers, the product will Have students model this problem on a number line. Then solve
be positive if n is an even number. The product will be the problem using both multiplication and repeated addition.
negative if n is an odd number.
• To show multiplication of a negative number times a 3 × (–4.25) = –12.75
negative number on a number line, first find the product (–4.25) + (–4.25) + (–4.25) = –12.75
of one negative and one positive. Then take the opposite
of the product to account for the second negative
number. -14 -12 -10 -8 -6 -4 -2 0 2
Start at 0. Move 4.25 units to the left 3 times.
Students will next work with division of rational numbers.

83A
 PROFESSIONAL DEVELOPMENT
EL
Language Support
California ELD Standards
Emerging 2.I.1. Exchanging information/ideas – Engage in conversational exchanges and express ideas on familiar topics
by asking and answering yes-no and wh-questions and responding using simple phrases.
Expanding 2.I.1. Exchanging information/ideas – Contribute to class, group, and partner discussions by following
turn-taking rules, asking relevant questions, affirming others, adding relevant information, and paraphrasing key ideas.
Bridging 2.I.1. Exchanging information/ideas – Contribute to class, group, and partner discussions by following turn-taking
rules, asking relevant questions, affirming others, adding relevant information and evidence, paraphrasing key ideas, building on responses, and
providing useful feedback.

EL
Linguistic Support

Academic/Content Vocabulary Multiple-Meaning Words

money – In this module, many word problems require setting Everyday English verbs take on many
understanding how money works in the real world: different meanings depending on context. Students
earning it and/or managing it. Point out to English may need help understanding the phrase “setting a
learners that, in addition to learning to solve word watch.” If possible, demonstrate setting a watch by
problems involving money, these problems provide using a real clock or watch. Remind students that
an opportunity to learn how to make informed and clocks need to be reset for daylight savings time and
effective financial decisions in their lives. sometimes after a power failure.

EL
Leveled Strategies for English Learners

Emerging For students at this English proficiency level, allow students to use their primary
language in peer-to-peer discussion to explain how to multiply rational numbers. Remind them
that they may use the glossary as a tool for referencing words, meanings, and examples.

Expanding Have English learners at this level of proficiency give a real-world example in which
they would need to multiply rational numbers. Have them describe both the problem and the
solution.

Bridging Pair together students at this level of English proficiency, and have them give
a real-world example in which multiplying rational numbers is necessary. Have them describe
the problem and solution, and explain why multiplication of rational numbers is necessary.

The question posed in Math Talk asks students to consider what it means about
Math Talk the product of several rational numbers if one of the factors is zero. Model for
English learners how to begin their response with the following:

If one of the factors is zero, then the product…

Multiplying Rational Numbers 83B

 LESSON

3.4 Multiplying Rational Numbers

CA Common Core Engage
Standards
ESSENTIAL QUESTION
The student is expected to: How do you multiply rational numbers? Sample answer: Multiply the absolute values of the
The Number System—7.NS.2 numbers. If the number of negative factors is even, the product is positive. If the number of
negative factors is odd, the product is negative.
Apply and extend previous understandings of
­multiplication and division and of fractions to multiply Motivate the Lesson
and divide rational numbers. Also 7.NS.2a, 7.NS.2c Ask: How do you multiply -​ __12​by 4? Use a number line or a sketch to try and model an
Mathematical Practices answer before you begin Example 1.
MP.2 Reasoning

Explore
Have students share their number lines and sketches for modeling -​ __12​(4).

Explain
Additional EXAMPLE 1
A section of beach erodes __
​ 34​meter per EXAMPLE 1
year. The beach eroded for 8 years.
Avoid Common Errors     Mathematical Practices
By how many meters did the length
Caution students to place the negative sign carefully. Remind them that the negative sign
of the beach change? -6 m
indicates downward travel, so each __
​ 12​-mile section is negative.
Interactive Whiteboard
Interactive example available online Questioning Strategies     Mathematical Practices
••Is the product of two values with different signs always negative? Explain. Yes, since
my.hrw.com multiplication is repeated addition, the product represents the sum of negative numbers.

YOUR TURN
Talk About It
Check for Understanding
Ask: When using a number line to multiply 2(-3.5), do you start at 0 or -3.5?
Start at 0 on the number line, as they would for the sum (-3.5) + (-3.5).
Additional EXAMPLE 2
Multiply -4(-2.5). 10 EXAMPLE 2
Interactive Whiteboard Questioning Strategies     Mathematical Practices
Interactive example available online ••How do you know whether the sign of the product of two rational numbers is positive or
negative? The sign is positive if the rational numbers have the same sign, and the sign is
my.hrw.com
negative if they have different signs.
Engage with the Whiteboard
Have students work in groups to draw 3.5 groups of -2 on a number line. Use the
space under the number line in Step 2 to show that the result is also -7.

YOUR TURN
Focus on Modeling     Mathematical Practices
Discuss with students the connections between the products in the multiplication
­expression and the direction and movement on the number line.
83 Lesson 3.4
 LESSON
Multiplying Rational 7.NS.2
Multiplying Rational Numbers
3.4 Numbers
Apply and extend
previous understandings
of multiplication...and
of fractions to multiply
...rational numbers. Also
7.NS.2a, 7.NS.2c
with the Same Sign
The rules for the signs of products with the same signs are summarized below.
Math On the Spot

? ESSENTIAL QUESTION
How do you multiply rational numbers?
my.hrw.com Products of Rational Numbers
Sign of Factor p Sign of Factor q Sign of Product pq
+ + +
Multiplying Rational Numbers - - +
with Different Signs
The rules for the signs of products of rational numbers with different signs are You can also use a number line to find the product of rational numbers with
summarized below. Let p and q be rational numbers. Math On the Spot the same signs.
my.hrw.com
Products of Rational Numbers EXAMPLE 2
My Notes 7.NS.2, 7.NS.2a
Sign of Factor p Sign of Factor q Sign of Product pq
Multiply -2(-3.5).
+ - -
STEP 1 First, find the product 2(-3.5).
- + -
+ ( - 3.5) + ( - 3.5)

You can also use the fact that multiplication is repeated addition.
-8 -7 -6 -5 -4 -3 -2 -1 0
STEP 2 Start at 0. Move 3.5 units to the left two times.
EXAMPL 1
EXAMPLE 7.NS.2, 7.NS.2a
STEP 3 The result is -7.
Gina hiked down a canyon and stopped each time she descended
1
_
2
mile to rest. She hiked a total of 4 sections. What is her overall STEP 4 This shows that 2 groups of -3.5 equals -7.
© Houghton Mifflin Harcourt Publishing Company • Image Credits: ©Sebastien Fremont/Fotolia

change in elevation?
So, -2 groups of -3.5 must equal the opposite of -7.
STEP 1 Use a negative number to represent the change in elevation.
STEP 2 ( )
Find 4 -_21 .
-8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8
STEP 3 Start at 0. Move _12 unit to the left 4 times.
STEP 5 -2(-3.5) = 7
The result is -2.

© Houghton Mifflin Harcourt Publishing Company

The overall change is -2 miles. - 3 -2 -1 0 Check: Use the rules for multiplying rational numbers.
Check: Use the rules for multiplying rational numbers. -2(-3.5) = 7 A negative times a negative equals a positive.

( ) ( )
4 -_12 = -_42 A negative times a positive equals a negative.

= -2 ✓ Simplify. YOUR TURN

2. Find -3(-1.25). 3.75
YOUR TURN
Personal Personal
1. Use a number line to find 2(-3.5). -7 Math Trainer Math Trainer
Online Practice Online Practice
and Help and Help -4 -3 -2 -1 0 1 2 3 4
-8 -7 -6 -5 -4 -3 -2 -1 0 my.hrw.com my.hrw.com

Lesson 3.4 83 84 Unit 1

PROFESSIONAL DEVELOPMENT
Integrate Mathematical Math Background
Practices MP.2 For nonnegative values a, b, c, and d, the product
__
a __
· c can be defined as the area of a rectangle
This lesson provides an opportunity to address b d
this Mathematical Practice standard. It calls for with side lengths __ab and __dc . The product of __ab and __dc
students to communicate mathematical ideas can also be defined as a parts of __dc when __dc is
using multiple representations as appropriate. divided into b equal parts.
Students represent a hike down a canyon using a
negative number. Then students use a
multiplication expression to represent the
product of the negative number and a positive
number. Finally, students use a number line to
visualize the product of the two signed numbers.
In this way, students are able to use multiple
representations to model real-world situations.

Multiplying Rational Numbers 84

 ADDITIONAL EXAMPLE 3 EXAMPLE 3
( )( )( )
-__14 -__25 . __
Multiply __12 1
20 Questioning Strategies Mathematical Practices
Interactive Whiteboard • How can you determine what the sign of the product of n negative numbers will be?
Interactive example available online If n is even, the product will be positive, and if n is odd, the product will be negative.
• Would this rule also work for decimals? Yes; since decimals and fractions both represent
my.hrw.com
rational numbers, the rule would be the same.
Talk About It
Check for Understanding
Ask: How do you find the product of three or more rational numbers? First, find
the product of the first two rational numbers. Then multiply the result by the third
rational number.

YOUR TURN
Avoid Common Errors
Students may look for or invent an incorrect rule for product signs, such as “if there are 3 or
more negative signs, the product is negative.” Confirm that students understand the
product sign “flips” with each additional negative factor.

Elaborate
Talk About It
Summarize the Lesson
Have students complete the graphic organizer below.
Let p and q be rational numbers.
Sign of factor p Sign of factor q Sign of pq
+ + +
- - +
+ - -
- + -

GUIDED PRACTICE
Avoid Common Errors
Exercise 7 Remind students that the final answer is the opposite of a negative product, or
positive.
Exercises 11, 12 Remind students to divide both the numerator and denominator by the
same common factors when simplifying.
Engage with the Whiteboard
For Exercises 1–4, have students use the whiteboard to show their multiplication on
the number line.

85 Lesson 3.4
DO NOT EDIT--Changes must be made through “File info” DO NOT EDIT--Changes must be made through “File info”
CorrectionKey=A CorrectionKey=D

Guided Practice
Multiplying More Than
Two Rational Numbers Use a number line to find each product. (Example 1 and Example 2)

-3 _13 -​​_34
(​ ) (​ )
If you multiply three or more rational numbers, you can use a pattern to find
the sign of the product. Math On the Spot
1. 5​ - _​2 = 2. 3​ - _​1 =
3 4
my.hrw.com

EXAMPL 3
EXAMPLE 7.NS.2, 7.NS.2c
-5 -4 -3 -2 -1 0 -2 - 1.5 -1 - 0.5 0
( )(-__12 )(-__35 ).
2
Multiply -__
1 _57
(​ )
3
3. -3​ - _​47 = 4. -​_​34 (-4) = 3
STEP 1 First, find the product of the first two factors. Both factors are
negative, so their product will be positive.

STEP 2 (-_23 ) (-_12 ) = + (_23 · _12) -2 -1 0 1 2 -4 -3 -2 -1 0 1 2 3 4

= _13

5. 4(-3) = -12 6. -1.8(5) = -9 7. -2 (-3.4) = 6.8

STEP 3 Now, multiply the result, which is positive, by the third factor,
which is negative. The product will be negative. 8. 0.54(8) = 4.32 9. -5(-1.2) = 6 10. -2.4(3) = -7.2
STEP 4 _
3 ( ) = _13 (-_35)
1 _
-35
Multiply. (Example 3)
Math Talk
(​ )(​ ) ( )
1
_
-_45
( )( )( ) 1
(​ )
Mathematical Practices
STEP 5 -_23 -_12 -_35 = -_15 11. _1 × _2 × ​_3 = _ × _34 =​ 4 12. -​_​4 -_​3 -_​7 = 12
___ × ​ -_​73 =
Suppose you find the product 2 3 4 3 7 5 3 35
of several rational numbers, 5
-​__
​12
​ _​67) =​
-​_​23(_12​)(-
2
_
Reflect one of which is zero. What
can you say about the 13. - _1 × 5 ×​​​_2​=​ 14. 7
8 3
3. Look for a Pattern You know that the product of two negative product?
numbers is positive, and the product of three negative numbers is 15. The price of one share of Acme Company declined $3.50 per day for
negative. Write a rule for finding the sign of the product of n negative 4 days in a row. What was the overall change in the price of one share?
numbers. The product (Example 1)

If n is even, the sign of the product is positive. If n is will be zero 4(-3.50) = -14; The share price decreased by $14.
since any
odd, the sign of the product is negative. number times 16. In one day, 18 people each withdrew $100 from an ATM machine. What
zero is zero. was the overall change in the amount of money in the ATM machine?
(Example 1)
© Houghton Mifflin Harcourt Publishing Company

© Houghton Mifflin Harcourt Publishing Company

18(-100) = -1,800; The money in the ATM decreased by $1,800.

?
YOUR TURN ESSENTIAL QUESTION CHECK-IN
Find each product.
-_27
4. (-_34)(-_47)(-_23) 17. Explain how you can find the sign of the product of two or more
rational numbers.
2
_
5. (-_23)(-_34)(_45) 5
Personal
Sample answer: Multiply the numbers. Then count the number of

-_12
Math Trainer negative numbers in the product. If the number is even, the product
6. (_23)(-__109 )(_56) Online Practice
and Help
is positive. If the number is odd, the product is negative.
my.hrw.com

Lesson 3.4 85 86 Unit 1

7_MCAAESE202610_U1M03L4.indd 85 4/30/13 2:00 AM 7_MCADESE202610_U1M03L4.indd 86 12/09/16 4:52 PM

DIFFERENTIATE INSTRUCTION
Cooperative Learning Critical Thinking Additional Resources
Have students discuss different ways to apply Point out to students that on a number line they Differentiated Instruction includes:
the rules for multiplying rational numbers. Have show the product of two negative numbers as ••Reading Strategies
them apply the rules as they work together on positive because they are finding the opposite ••Success for English Learners EL
the Guided Practice exercises. of a negative product to the left of 0. Ask
••Reteach
students to examine this claim, discuss why it is
Student 1: I used the rule that if you ••Challenge PRE-AP
true, and justify it with a logical argument.
multiply two rational numbers with the
Sample answer: If the product of a negative
same sign, the product is positive, and if
number and a positive number is shown on a
they have different signs, the product is
number line as a number of groups graphed to
negative.
the left of 0, then the opposite of the product
Student 2: I just count the number of must be an equal number of groups graphed to
negative signs. If there is an odd number the right of 0.
of negatives, the sign of the product is
negative. If there is an even number of
negatives, the sign of the product is
positive.

Multiplying Rational Numbers 86

 Personal
Math Trainer
Online Assessment
Evaluate Focus | Coherence | Rigor
and Intervention

Online homework GUIDED AND INDEPENDENT PRACTICE

assignment available 7.NS.2, 7.NS.2a, 7.NS.2c
my.hrw.com
Concepts & Skills Practice
3.4 LESSON QUIZ Example 1 Exercises 1, 2, 5, 6, 10, 15, 16, 18, 21, 22
Multiplying Rational Numbers with Different
7.NS.2, 7.NS.2a, 7.NS.2c Signs
1. Ronnie cuts __34 -inch sections of Example 2 Exercises 3, 4, 7–9, 19, 20, 23
wood to make a birdhouse roof. He Multiplying Rational Numbers with the Same
cut 36 sections. How many inches Sign
did he cut?
Example 3 Exercises 11–14
Multiply. Write each answer in Multiplying More than Two Rational Numbers
simplest form.
( )
2. 6 -__23
3. -3(-6.2) Exercise Depth of Knowledge (D.O.K.) Mathematical Practices
4. ( )( 2 )( )
-__34 __1 -__23
18 2 Skills/Concepts MP.4 Modeling
5. In one day, Mr. Hanson spent $100
five times with his debit card. How 19 3 Strategic Thinking MP.2 Reasoning
much money was deducted from
20 2 Skills/Concepts MP.4 Modeling
his account?
21 2 Skills/Concepts MP.4 Modeling
Lesson Quiz available online
22 3 Strategic Thinking MP.4 Modeling
my.hrw.com
23 3 Strategic Thinking MP.3 Logic

Answers 24 2 Skills/Concepts MP.4 Modeling

1. 27 in. 25 2 Strategic Thinking MP.4 Modeling
2. -4 26 MP.4 Modeling
2 Skills/Concepts
3. 18.6
27 3 Strategic Thinking MP.2 Reasoning
4. __14
28 3 Strategic Thinking MP.7 Using Structure
5. 5(-100) = -500; $500 was deducted

Additional Resources
Differentiated Instruction includes:
• Leveled Practice Worksheets

87 Lesson 3.4
 Name Class Date
25. The table shows the scoring system for quarterbacks in Quarterback Scoring
Jeremy’s fantasy football league. In one game, Jeremy’s
3.4 Independent Practice Personal
Math Trainer
quarterback had 2 touchdown passes, 16 complete passes,
7 incomplete passes, and 2 interceptions. How many total
Action
Touchdown pass
Points
6
7.NS.2, 7.NS.2a, 7.NS.2c Online Practice points did Jeremy’s quarterback score? Complete pass 0.5
my.hrw.com and Help
13.5 points Incomplete pass −0.5
18. Financial Literacy Sandy has $200 in her 22. Multistep For Home Economics class,
Interception −1.5
bank account. Sandra has 5 cups of flour. She made
3 batches of cookies that each used
a. If she writes 6 checks for exactly FOCUS ON HIGHER ORDER THINKING
1.5 cups of flour. Write and solve an
$19.98, what expression describes the
expression to find the amount of flour
Work Area
change in her bank account?
Sandra has left after making the 3 batches 26. Represent Real-World Problems The ground temperature at Brigham
6(-19.98) of cookies. Airport is 12 °C. The temperature decreases by 6.8 °C for every increase of
1 kilometer above the ground. What is the overall change in temperature
b. What is her account balance after the 5 - 3(1.5) or 3(-1.5) + 5; Sandra has outside a plane flying at an altitude of 5 kilometers above Brigham Airport?
checks are cashed? 0.5 cup, or half a cup of flour left. 5(-6.8) = -34; 12 - 34 = -22; -22 °C
200 - 119.88 = 80.12; $80.12
23. Critique Reasoning In class, Matthew 27. Identify Patterns The product of four numbers, a, b, c, and d, is a
19. Communicating Mathematical stated, “I think that a negative is like negative number. The table shows one combination of positive and
Ideas Explain, in words, how to find the an opposite. That is why multiplying a negative signs of the four numbers that could produce a negative
product of -4(-1.5) using a number line. negative times a negative equals a positive. product. Complete the table to show the seven other possible
Where do you end up? The opposite of negative is positive, so it combinations.
is just like multiplying the opposite of a
Start at 0, then move 1.5 units to negative twice, which is two positives.” Do
the left (because 1.5 is negative) you agree or disagree with this statement? a b c d
What would you say in response to him?
4 times. You are now on -6. Matthew is incorrect; Sample
+ + + -

Then find the opposite of -6, + + - +

answer: Matthew should have
+ - + +
which is 6. said that multiplying by two
- + + +
20. Greg sets his watch for the correct time negatives is like multiplying the
on Wednesday. Exactly one week later, he - - - +
finds that his watch has lost 3 _14 minutes. opposite of a positive twice.
If his watch continues to lose time at the - - + -
The opposite of a positive twice
© Houghton Mifflin Harcourt Publishing Company

© Houghton Mifflin Harcourt Publishing Company

same rate, what will be the overall change - + - -
in time after 8 weeks? brings you back to a positive.
( )
8 -3_14 = -26; 26 minutes behind + - - -

21. A submarine dives below the surface, 24. Kaitlin is on a long car trip. Every time she 28. Reason Abstractly Find two integers whose sum is -7 and whose
heading downward in three moves. If each stops to buy gas, she loses 15 minutes of product is 12. Explain how you found the numbers.
move downward was 325 feet, where is the travel time. If she has to stop 5 times, how -4 and -3; The product is positive so the numbers must
submarine after it is finished diving? late will she be getting to her destination?
have the same sign. Since the sum is negative, I used the
The submarine would be 975 feet 75 minutes or 1 hour and
guess and check strategy to find two negative numbers
below sea level, or -975 feet. 15 minutes
with a sum of -7 and a product of 12.

Lesson 3.4 87 88 Unit 1

EXTEND THE MATH PRE-AP Activity available online my.hrw.com

Activity Write the numbers 2__12 , -__13 , -0.8, __14 , and -1__23 on the board.
1. Ask students to explain how to find which two numbers have the largest product.
Then find the largest product.
2. Ask students to explain how to find which two numbers have the smallest product.
Then find the smallest product.
1. Sample answer: Choose the two numbers with the same sign and the largest absolute
values: -1__23 (-0.8) = 1__13
2. Sample answer: Choose the two numbers with the opposite signs and largest absolute
3 2 ( )
values: -1__2 2__1 = -4__1
6

Multiplying Rational Numbers 88

 LESSON 3.5 Dividing Rational Numbers

Lesson Support
Content Objective Students will learn to divide rational numbers.

Language Objective Students will explain the steps of dividing rational numbers.

California Common Core Standards

7.NS.2 Apply and extend previous understandings of multiplication and division and of fractions to multiply and divide
rational numbers.
7.NS.2b Understand that integers can be divided, provided that the divisor is not zero, and every quotient of integers
p ( -p ) p
(with non-zero divisor) is a rational number. If p and q are integers, then -( _q ) =____ ____
q = ( -q ) . Interpret quotients of rational
numbers by describing real world contexts.
7.NS.2c Apply properties of operations as strategies to multiply and divide rational numbers.
MP.4 Model with mathematics.

Focus | Coherence | Rigor

Building Background
1
Visualize Math Review the meaning of the compound 4
__1 1
fraction ____21 = __12 ÷ __14 . Have students work with a partner to draw 2
4
1
__1 4
a model of __. Remind students that this means “How many __14 s
2
__1
4

There are two 1 s in 1 .

are in __12 ?” 4 2

Learning Progressions Cluster Connections

In this lesson, students apply the skills they have developed This lesson provides an excellent opportunity to connect ideas
dividing integers to dividing rational numbers. Some key in this cluster: Apply and extend previous understandings of
understandings for students are the following: operations with fractions to add, subtract, multiply, and divide
rational numbers.
• When the dividend and the divisor have different signs,
the quotient will be negative. Give students the following prompt: “Malia wants to divide
• When the dividend and the divisor have the same sign, -3__13 by -__56 . What steps should Malia follow to find the
the quotient will be positive. quotient?” Invite a volunteer to explain the steps needed to
• If a quotient is negative, the negative sign can be placed find the quotient.
__1 __5 __
10 __6 __
60
with the divisor, with the dividend, or in front of the Sample answer: -3 3 ÷ (- 6 ) = - 3 × (- 5 ) = 15 = 4
quotient (in fraction form). All of these placements are First, change the mixed number -3__31 to an improper
acceptable and have the same value. fraction. Then multiply by the reciprocal of -__56 . Since both
Students will apply their skills with the operations and rational the dividend and the divisor are negative, the quotient is
numbers to solving real-world problems. positive.

89A
 PROFESSIONAL DEVELOPMENT
EL
Language Support
California ELD Standards
Emerging 2.I.1. Exchanging information/ideas – Engage in conversational exchanges and express ideas on familiar topics
by asking and answering yes-no and wh- questions and responding using simple phrases.
Expanding 2.I.1. Exchanging information/ideas – Contribute to class, group, and partner discussions by following
turn-taking rules, asking relevant questions, affirming others, adding relevant information, and paraphrasing key ideas.
Bridging 2.I.1. Exchanging information/ideas – Contribute to class, group, and partner discussions by following turn-taking
rules, asking relevant questions, affirming others, adding relevant information and evidence, paraphrasing key ideas, building on responses, and
providing useful feedback.

EL
Linguistic Support

Academic/Content Vocabulary Background Knowledge

The spelling rules for the English language are not marathon - Point out to English learners that a
very consistent because there are nearly always marathon is a foot race of 26.2 miles. The marathon
exceptions to a rule. Nearly every sound in the English was one of the original events of the modern
language can be spelled in more than one way. For Olympics in 1896. It is almost the same word
example, although the words rational and quotient in Spanish, maratón.
have the [sh] sound in the middle of them, the letters
real-world - In the Independent Practice, students
SH are not used to spell that sound. Make sure that
are asked to come up with a real-world situation
students understand how to pronounce and spell
involving negative numbers. Brainstorm with
these important vocabulary words.
students to help them think of some examples
involving negative numbers.

EL
Leveled Strategies for English Learners

Emerging Have students with emerging English proficiency signal with thumbs up or down (for
yes or no) to answer the question about whether dividing rational numbers is the same as dividing
integers. Allow students to use examples in the lesson to support their answer.

Expanding Have English learners at this level of English proficiency complete the sentence
frame:

Dividing rational numbers uses the same rules as dividing ______________.

Have students show examples in the lesson to support their answer.

Bridging Have students at this level of English proficiency complete the following sentence
frame to answer the question about how to divide rational numbers. Have them include an
explanation for their answer.

The rules for dividing rational numbers are ______________.

Dividing Rational Numbers 89B

 LESSON

3.5 Dividing Rational Numbers

CA Common Core Engage
Standards
ESSENTIAL QUESTION
The student is expected to: How do you divide rational numbers? Divide the absolute values. If the numbers have
The Number System—7.NS.2 different signs, the quotient is negative. If the numbers have the same sign, the quotient is
positive.
Apply and extend previous understandings of
multiplication and division and of fractions to multiply Motivate the Lesson
and divide rational numbers. Ask: The deeper a diver descends, the more pressure it exerts on his or her body, so it is not
The Number System—7.NS.2b safe to dive too far too fast. How can a diver divide up a 100-ft descent into 5 parts? Begin
Explore Activity 1 to find out.
Understand that integers can be divided, provided that
the divisor is not zero, and every quotient of integers
(with non-zero divisor) is a rationa number. If p and q are
integers, then -(p/q) = (-p)/q = p/(-q). Interpret
quotients of rations numbers by describing real-world Explore
contexts.
The Number System—7.NS.2c EXPLORE ACTIVITY
Apply properties of operations as strategies to multiply Talk About It
and divide rations numbers. Check for Understanding
-36 ___
Ask: Without simplifying, what are two equivalent expressions for ___ 36
? -9 , -__
36
Mathematical Practices 9 9

MP.4 Modeling
Focus on Multiple Representations Mathematical Practices
Point out that the sign of a quotient is not affected by the placement of the negative sign.
For a fractional quotient, the sign can be in the numerator, in the denominator, or outside
the fraction.

Explain
ADDITIONAL EXAMPLE 1 EXAMPLE 1
Find each quotient.
Questioning Strategies Mathematical Practices
Over 6 months, Fiona used her online • How is the sign of the quotient -64.75 related to the statement that Carlos withdrew
banking account to pay a total of $64.75 each month? The negative quotient represents a decrease in the account balance,
$274.50 for her cell phone services. Her or a withdrawal.
cell phone service costs the same
amount each month. How much did
she withdraw from her account each
YOUR TURN
month to pay for the services? -45.75; Talk About It
$45.75 each month Check for Understanding
Ask: How do you compare the rules for finding the sign of a quotient with the
Interactive Whiteboard rules for finding the sign of a product? They are the same. If the signs of the
Interactive example available online numbers you are dividing or multiplying are the same, the quotient or product is positive.
If the signs are different, the quotient or product is negative.
my.hrw.com

89 Lesson 3.5
 LESSON
Dividing Rational 7.NS.2 EXPLORE ACTIVITY (cont’d)

3.5 Numbers
Apply and extend previous
understandings of
multiplication and division
and of fractions to…divide
Reflect
Write two equivalent quotients for each expression.

( )
rational numbers. Also
7.NS.2b, 7.NS.2c -14
____ 14
- __
14 7
? ESSENTIAL QUESTION 1. ___
-7
7 ,
How do you divide rational numbers?
-32
32
__ -( ____
8 )
-32
2. ____
-8
8 ,

EXPLORE ACTIVITY 7.NS.2b

Placement of Negative Signs in Quotients Quotients of Rational Numbers

The rules for dividing rational numbers are the same as dividing integers.
Quotients can have negative signs in different places.
Let p and q be rational numbers.
Math On the Spot EXAMPLE 1 7.NS.2c
my.hrw.com
Quotients of Rational Numbers
Over 5 months, Carlos wrote 5 checks for a total of $323.75 to pay for his
p
Sign of Dividend p Sign of Divisor q Sign of Quotient __
q
cable TV service. His cable bill is the same amount each month. What was
the change in Carlos’s bank account each month to pay for cable?
+ - -
-323.75
Find the quotient: _______
- + - 5
STEP 1 Use a negative number to represent the withdrawal from his
+ + +
account each month.
- - + -323.75
STEP 2 Find _______
5
.
STEP 3 Determine the sign of the quotient.
Are the rational numbers ___, -12, and - ___
12 ____
-4 4
12
4 ( )
equivalent?
The quotient will be negative because the signs are different.
A Find each quotient. Then use the rules in the table to make sure the
sign of the quotient is correct. STEP 4 Divide.
-323.75
( )
_______ = -64.75
12
___ = -3 -12
____ = -3 12
- __ = -3 5
-4 4 4
Carlos withdrew $64.75 each month to pay for cable TV.
B What do you notice about each quotient?
© Houghton Mifflin Harcourt Publishing Company

© Houghton Mifflin Harcourt Publishing Company

The quotients are all the same.
C The rational numbers are / are not equivalent.
YOUR TURN
D Conjecture Explain how the placement of the negative sign in the
rational number affects the sign of the quotients. Find each quotient.
The sign of the quotient is not affected by the 2.8
3. ___ = -0.7 4. -6.64
_____ = 16.6 5. 5.5
-___ = -11
-4 -0.4 0.5
placement of the negative sign. 6. A diver descended 42.56 feet in 11.2 minutes. What was the diver’s average
Personal change in elevation per minute?
Math Trainer
E If p and q are rational numbers and q is not zero, what do you know -3.8 feet per minute
()
Online Practice
p -p p
___
about - __q , ___
q , and -q?
and Help
my.hrw.com
They are equivalent.
Lesson 3.5 89 90 Unit 1

PROFESSIONAL DEVELOPMENT
Integrate Mathematical Math Background
Practices MP.4 Dividing rational numbers is an extension of
This lesson provides an opportunity to address dividing fractions or integers, and multiplying
this Mathematical Practice standard. It calls rational numbers.
for students to apply mathematics to problems Note that division of a decimal by a decimal is
arising in everyday life, society, and the not included in this lesson, that division by 0 is
workplace. At the beginning of the lesson, undefined, and the quotient of 0 and a nonzero
students represent a diver’s depth using a rational number is 0.
negative number. At the end of the lesson,
students relate finding quotients of rational
numbers to consumer economics, such as
paying a cable TV bill. In this way, students are
able to use mathematics to model situations in
everyday life.

Dividing Rational Numbers 90

 ADDITIONAL EXAMPLE 2 EXAMPLE 2
Find each quotient. Questioning Strategies Mathematical Practices
_-3 • When the divisor is between 0 and 1, is the quotient greater than or less than the
A 4 -2
_ dividend? Explain. greater than if the dividend is positive and less than if the dividend is
3
_ negative
8
-4
_ YOUR TURN
B _5 1__5
-7 7
_ Avoid Common Errors
15 Students may not use the correct sign for the quotient. Encourage students to look back at
the signs in the problem to be sure the quotient has the correct sign.
Interactive Whiteboard
Interactive example available online
my.hrw.com
Elaborate
Talk About It
Summarize the Lesson
Have students explain how they divide a fraction by an integer. Write the integer
as a fraction and the division as multiplication by the reciprocal. Use the rule for
multiplying fractions to complete the process.

GUIDED PRACTICE
Avoid Common Errors
Exercise 11 Students may need to review converting mixed numbers to improper
fractions.

Engage with the Whiteboard

Have students write any acceptable equivalent form of answer in the blanks for
Exercises 1–9.

91 Lesson 3.5
 Guided Practice
Complex Fractions a
__
A complex fraction is a fraction that has a fraction b
___ = __ba ÷ __dc Find each quotient. (Explore Activity 1 and 2, Example 1)

( )
c
__
in its numerator, denominator, or both. d
0.72 -0.8
1
_ -_17
1. ____
-0.9
= 2. -___
5
=
Math On the Spot 7
_
EXAMPL 2
EXAMPLE 7.NS.2c, 7.NS.3
my.hrw.com
5

502
7
__
- ___
A Find 10
____
- _15
.
My Notes
56
3. ___
-7
= -8 251
4. ___
4 ( )
÷ -_38 = 3

STEP 1 Determine the sign of the quotient.

The quotient will be negative because the signs are different.
75
5. ___
1=
-375 -91
6. ____ = 7
7 -5
_ -13
__
10 7
STEP 2 Write the complex fraction as division: ____ = __ ÷ -_15
-_15 10 4
-7 3
_ - __
21 12 -400
7. ___ = 8. -____ =
STEP 3 7
Rewrite using multiplication: __
10
× -_51 ( ) Multiply by the
reciprocal.
9
_
4
0.03

STEP 4 7
__
10 ( ) 35
× -_51 = - __
10 Multiply. 9. A water pail in your backyard has a small hole in it. You notice that it
has drained a total of 3.5 liters in 4 days. What is the average change in
= - _72 Simplify. water volume each day? (Example 1)
7
___ -0.875 liter per day
10
____
1
= - _72
-__
5 10. The price of one share of ABC Company decreased a total of $45.75 in
5 days. What was the average change of the price of one share per day?
B Maya wants to divide a _34 -pound box of trail mix into small bags. Each bag (Example 1)
1
will hold __ pound of trail mix. How many bags of trail mix can Maya fill?
12
-$9.15 per day, on average
3
_
STEP 1 Find ___
4
1 .
__ 11. To avoid a storm, a passenger-jet pilot descended 0.44 mile in 0.8
12 minute. What was the plane’s average change of altitude per minute?
STEP 2 Determine the sign of the quotient. (Example 1)
The quotient will be positive because the signs are the same. -0.55 mile per minute
_
3
4
Write the complex fraction as division: ___ _3 __1
1 = 4 ÷ 12 .
STEP 3 __

?
12
ESSENTIAL QUESTION CHECK-IN
STEP 4 Rewrite using multiplication: _43 × __
12
© Houghton Mifflin Harcourt Publishing Company

© Houghton Mifflin Harcourt Publishing Company

1 . Multiply by the reciprocal.
32 ÷ (-2)
12. Explain how you would find the sign of the quotient _________ .
-16 ÷ 4
_
3 __
× 12 = __
36
STEP 5 4 1 4 =9 Multiply. Simplify. First, find the sign of the numerator. It is negative
_3
___
4
because the numbers have different signs. Next, find
1
__
=9
12 the sign of the denominator. It is negative because
Maya can fill 9 bags of trail mix.
the numbers have different signs. So, the quotient is

Personal
positive because the numerator and denominator have
YOUR TURN Math Trainer
Online Practice
the same sign.
5
-__4 and Help
35 -___
__ -_58 -1_35
5
-__
8 12
____ ___5 my.hrw.com
7. ___
6
= 48 8. 2 =
__ 9. 1 =
__
-__ 3 2
7
Lesson 3.5 91 92 Unit 1

DIFFERENTIATE INSTRUCTION
Graphic Organizers Critical Thinking Additional Resources
As students do the lesson, have them create a Point out to students they should analyze each Differentiated Instruction includes:
table containing words and expressions that will real-world problem carefully so that they • Reading Strategies
help them know when a real-world quantity can represent the quantities with signs that make • Success for English Learners EL
be represented by a negative number. Then sense in the problem, either positive or negative.
• Reteach
have them compare their tables with other Ask students to examine why it makes sense to
students. express a diver’s descent, for example, as a • Challenge PRE-AP
negative number compared to his ascent as a
Sample Table positive number. Then ask students to think of
descend decrease and examine other real-world examples of
withdraw lose negative numbers. If the diver’s descent is
expressed as a negative number, then it makes
decline average loss sense that the ascent is a positive number. So,
the sign of the number conveys a sense of
direction to the diver’s movement.

Dividing Rational Numbers 92

 Personal
Math Trainer
Online Assessment
Evaluate Focus | Coherence | Rigor
and Intervention

Online homework GUIDED AND INDEPENDENT PRACTICE

assignment available 7.NS.2, 7.NS.2b, 7.NS.2c
my.hrw.com
Concepts & Skills Practice
3.5 LESSON QUIZ Explore Activity 1 Exercises 1–21, 27
Dividing Rational Numbers
7.NS.2, 7.NS.2b, 7.NS.2c
Explore Activity 2 Exercises 1–21
1. A mountain climber needs to Placement of Negative Signs in Quotients
descend 1,500 feet. He wants to do Example 1 Exercises 1–21, 22–28
it in 4 equal descents. How far Quotients of Rational Numbers
should he travel in each descent?
Divide. Write each answer in
simplest form.
2. 8 ÷ ( -__43 )
Exercise Depth of Knowledge (D.O.K.) Mathematical Practices
___
4 22 3 Strategic Thinking MP.4 Modeling
-5
3. _
-3
___ 23–26 2 Skills/Concepts MP.4 Modeling
10
4. Over 4 months, Ms. Berg wrote 4 27 3 Strategic Thinking MP.7 Using Structure
equal checks totaling $250 to pay 28 MP.4 Modeling
2 Skills/Concepts
for salon services. How much did
she withdraw from her account 29 3 Strategic Thinking MP.4 Modeling
each month to pay for the services?
30 3 Strategic Thinking MP.6 Precision

Lesson Quiz available online 31 3 Strategic Thinking MP.8 Patterns

my.hrw.com
Additional Resources
Differentiated Instruction includes:
Answers • Leveled Practice worksheets
1. -375 ft
2. -6
8
3. _
3
4. -250 ÷ 4 = -62.50; $62.50

93 Lesson 3.5
 Name Class Date
27. Sanderson is having trouble with his assignment. His shown work is as
Personal follows:
3.5 Independent Practice Math Trainer
3
-__
4
Online Practice ___
4 4
× 43 = - __
3 _
= -__ 12
12
= -1
and Help __
7.NS.2, 7.NS.2b, 7.NS.2c my.hrw.com 3
5
13. ___ = -20 23. The running back for the Bulldogs football
However, his answer does not match the answer that his teacher gives
2
-__ him. What is Sanderson’s mistake? Find the correct answer.
8 team carried the ball 9 times for a total loss
32
-__ of 15_34 yards. Find the average change in He forgot to multiply by the reciprocal, and instead just
( )
14. 5_13 ÷ -1_12 = 9 field position on each run.
multiplied by the fraction that was in the denominator.
-1_34 yards 3
-120
15. _____ = 20 -__
-6 The answer should look like this: ___4 = -__3 × __3 = -__
9
24. The 6:00 a.m. temperatures for four 4
__ 4 4 16
4
-__ 6
_ consecutive days in the town of Lincoln 3
16. ___52 = 5
-__ were -12.1 °C, -7.8 °C, -14.3 °C, and
3 28. Science Beginning in 1996, a glacier lost an average of 3.7 meters of
-7.2 °C. What was the average 6:00 a.m.
17. 1.03 ÷ (-10.3) = -0.1 temperature for the four days? thickness each year. Find the total change in its thickness by the end of
2012.
-10.35 °C
-0.4
18. ____ = -0.005 -3.7(17) = -62.9; the glacier has lost 62.9 m of thickness
80
5
_ 25. Multistep A seafood restaurant claims
19. 1 ÷ _95 = 9 an increase of $1,750.00 over its average
profit during a week where it introduced a
FOCUS ON HIGHER ORDER THINKING Work Area
6
4
-1
___ - __
23
special of baked clams.
29. Represent Real-World Problems Describe a real-world situation that
20. ___
23
=
___
24
a. If this is true, how much extra profit can be represented by the quotient -85 ÷ 15. Then find the quotient and
did it receive per day? explain what the quotient means in terms of the real-world situation.
-10.35 4.5
21. ______
-2.3
= $250 Sample answer: The temperature dropped 85° over 15
22. Alex usually runs for 21 hours a week, b. If it had, instead, lost $150 per day, how days. Find the average change in temperature per day.
training for a marathon. If he is unable much money would it have lost for the _
to run for 3 days, describe how to find week? -85 ÷ 15 = -5.6; the average change in temperature
© Houghton Mifflin Harcourt Publishing Company • ©Kelly Harriger/Corbis/HMH

out how many hours of training time he

loses, and write the appropriate integer to $1,050 was about -5.67 degrees per day.
describe how it affects his time.
c. If its total loss was $490 for the week, 30. Construct an Argument Divide 5 by 4. Is your answer a rational number?
If Alex usually runs 21 hours what was its average daily change? Explain.

© Houghton Mifflin Harcourt Publishing Company

per week, then you would -$70 Yes, it is a rational number because 1 _14 can be
need to divide by 7 to find 26. A hot air balloon written as _54, which is a ratio of two integers and the
descended 99.6 meters
out how much time he usually in 12 seconds. What denominator is not zero.
was the balloon’s
spends per day, which is three 31. Critical Thinking Should the quotient of an integer divided by a
average rate of descent
nonzero integer always be a rational number? Why or why not?
hours. Since he is unable to in meters per second?
8.3 meters per Yes, since an integer divided by an integer is a ratio
run for three days, his time is
second of two integers and the denominator is not zero, the
decreased by 9 hours or, -9.
number is rational by definition.

Lesson 3.5 93 94 Unit 1

EXTEND THE MATH PRE-AP Activity available online my.hrw.com

Activity The value of each division expression will show the average weight of a whale in
tons. Complete the table.

Division Weight
Whale
expression (tons)
Blue -186.29 ÷ (-1.3) 143.3
Right 25
-2_18 ÷ ____ 1
44__
-441 10
Gray 432
-___
3
÷ -4 36
Humpback -9
____
3_35 ÷ -73 2
29__
10

Dividing Rational Numbers 94

 LESSON 3.6 Applying Rational Operations

Lesson Support
Content Objective Students will learn to solve problems with rational numbers and choose tools to solve
problems efficiently.

Language Objective Students will describe different forms of rational numbers and explain how to
strategically choose tools to solve problems with rational numbers.

California Common Core Standards

7.EE.3 Solve multi-step real-life and mathematical problems posed with positive and negative rational numbers in any form (whole
numbers, fractions, and decimals), using tools strategically. Apply properties of operations to calculate with numbers in any form; convert
between forms as appropriate; and assess the reasonableness of answers using mental computation and estimation strategies.
MP.1 Make sense of problems and persevere in solving them.

Focus | Coherence | Rigor

Building Background
Eliciting Prior Knowledge Have students give three Estimate to the nearest half Numbers
numbers, at least one of which is a fraction and one a decimal, -5 -4.89, -5.4857, -5__16
that when rounded to the nearest half would round to the given
estimate. Encourage students to explain why each number 12.5 12.43, 12.72, 12__58
rounds to the given estimate. -4__12 -4.6, -4.39, -4__59

Samples answers are given. 3 3.045, 2.803, 3__17

Learning Progressions Cluster Connections

In this lesson, students apply the skills they have developed This lesson provides an excellent opportunity to connect ideas
for performing operations with rational numbers to solving in this cluster: Solve real-life and mathematical problems using
multiple-step problems. Some key understandings for numerical and algebraic expressions and equations.
students are the following:
Give students the following prompt: “How do you know when it
• By checking an answer it is possible to avoid careless is appropriate to use a calculator to solve a problem and when it
errors. is not appropriate?” Have students complete the following table.
• Problems often have numbers presented in different Sample answers are given.
forms, such as integers, decimals, fractions, or percents.
• Before solving a problem with numbers in different forms Use a Calculator Don’t Use a Calculator
it can be a good idea to change all the numbers to the The numbers in a problem are The numbers in a problem are
same form. given as decimals or can be given as fractions that do not
• When using a calculator it is a good idea to pay close converted to terminal or convert easily to decimals.
repeating decimals.
attention to decimal point placement and negative signs.
Students will continue to solve real-world problems throughout
their study of mathematics.

95A
 PROFESSIONAL DEVELOPMENT
EL
Language Support
California ELD Standards
Emerging 2.I.1. Exchanging information/ideas – Engage in conversational exchanges and express ideas on familiar topics
by asking and answering yes-no and wh- questions and responding using simple phrases.
Expanding 2.I.1. Exchanging information/ideas – Contribute to class, group, and partner discussions by following
turn-taking rules, asking relevant questions, affirming others, adding relevant information, and paraphrasing key ideas.
Bridging 2.I.1. Exchanging information/ideas – Contribute to class, group, and partner discussions by following turn-taking
rules, asking relevant questions, affirming others, adding relevant information and evidence, paraphrasing key ideas, building on responses, and
providing useful feedback.

EL
Linguistic Support

Academic/Content Vocabulary Background Knowledge

reasonableness - In this lesson students are asked servings - Many countries, including the United
to assess the reasonableness of answers—to check States, require nutritional labeling on most packaged
to see if their answers are reasonable as a way of foods. Serving size is usually included in the facts on
avoiding careless errors. By adding the -ness to the most labels. However, when following a recipe, most
word reasonable, the word has become a noun. bakers do not think of bags of flour in terms of serving
Although the word reasonableness is not frequently size. They think of the amount needed in a recipe.
used, English learners will likely be familiar with many Point out to English learners that, although serving
other English adjective-to-noun combinations, such size helps to calculate the total amount of flour in
as happy/happiness, kind/kindness, dark/darkness, or a bag, it is not typically thought of as how much flour
aware/awareness. a person would eat in one serving.

EL
Leveled Strategies for English Learners

Emerging When proficiency in English is emerging, ask yes/no questions to see that students
know why checking their answer is a good idea. Review with students several ways that students
can check their answers.

Expanding Have small groups of English learners at this level of proficiency complete the
sentence frame:

It’s a good idea to check my answer because ____________.

I like to check my answer using a ____________.

Bridging Have students discuss with each other why it’s a good idea to check their answers and
the ways to check answers. Remind students to use complete sentences.

To answer the question posed in Math Talk, model for English learners how
Math Talk to begin their response with the following: I would calculate the depth…

Applying Rational Operations 95B

 LESSON

3.6 Applying Rational Number Operations

CA Common Core Engage
Standards
ESSENTIAL QUESTION
The student is expected to: How do you use different forms of rational numbers and strategically choose tools to solve
Expressions and problems? Change all rational numbers in a problem to the same form. Choose a tool that
Equations—7.EE.3 simplifies the solution process and that reflects the mathematical concepts involved.
Solve multi-step real-life and mathematical problems Motivate the Lesson
posed with positive and negative rational numbers in any Ask: How do you center a poster on a wall? Begin the Explore Activity to find out.
form (whole numbers, fractions, and decimals), using
tools strategically. Apply properties of operations to
calculate with numbers in any form; convert between
forms as appropriate; and assess the reasonableness of
answers using mental computation and estimation Explore
strategies. Also 7.NS.3
Focus on Modeling Mathematical Practices
Mathematical Practices
Ask students to measure the width of their desk and their math book. Then have them
MP.1 Problem Solving figure out how far from the edge they should place their math book to center it on
the desk.

Explain
ADDITIONAL EXAMPLE 1 EXAMPLE 1
Hu is installing a new window in his
bedroom wall. He wants to center it
Focus on Modeling Mathematical Practices
horizontally. The window is 40 __12 inches Have students draw a sketch and label the measurements as they work through the
long, and the wall is 105 __14 inches long. example.
How far from each edge of the wall Questioning Strategies Mathematical Practices
should Hu install the window? 32 __38 in. • What if the length of the wall were 121__14 inches? Explain how to find the total length of
the wall not covered by the picture. Rename 121__14 as 120 __54 . Subtract the fractional parts,
Interactive Whiteboard then the whole number parts.
Interactive example available online
• Explain how you can check your answer. Add the length of the wall to the left of the
my.hrw.com picture, the length of the picture, and the length of the wall to the right of the picture,
and confirm that the sum equals the total length of the wall.

YOUR TURN
ADDITIONAL EXAMPLE 2
A 1-gallon container of milk contains
Focus on Modeling Mathematical Practices
32 half-cup servings and costs $3.89. Have students draw a sketch of the situation. Draw a line segment divided into seven parts,
A batch of muffins uses __35 cup of milk. three parts labeled c for commercials and four parts labeled x for entertainment segments.
How many batches can you make if
you use all the milk? What is the cost EXAMPLE 2
of milk for each batch? 53 batches, Engage with the Whiteboard
about $0.07 Have students circle the rational numbers in the problem. Ask them to show how to
convert one of the fractions into a decimal, and a decimal into a fraction.
Interactive Whiteboard
Interactive example available online Questioning Strategies Mathematical Practices
my.hrw.com Why is it easier to divide 19 by 1.25 than by 1__14 ? Sample answer: To divide 19 by 1__14 , I would
have to change 1__14 to an improper fraction and then divide.
95 Lesson 3.6
 LESSON
Applying Rational 7.EE.3
Using Rational Numbers in Any Form
3.6 Number Operations
Solve … problems … with
positive and negative rational
numbers in any form …
using tools strategically. Also
7.NS.3
You have solved problems using integers, positive and negative fractions, and
positive and negative decimals. A single problem may involve rational numbers
in two or more of those forms.
Math On the Spot

? ESSENTIAL QUESTION
How do you use different forms of rational numbers and
strategically choose tools to solve problems?
my.hrw.com
EXAMPLE 2 Problem
Solving
7.EE.3, 7.NS.3

Alana uses 1_14 cups of flour for each batch of

blueberry muffins she makes. She has a 5-pound bag
Assessing Reasonableness of Answers of flour that cost $4.49 and contains seventy-six
1
_
Even when you understand how to solve a problem, you might make a careless My Notes 4 -cup servings. How many batches can Alana make
solving error. You should always check your answer to make sure that it is if she uses all the flour? How much does the flour for
reasonable. one batch cost?
Math On the Spot
my.hrw.com Analyze Information
EXAMPL 1
EXAMPLE 7.EE.3, 7.NS.3
Identify the important information.
Jon is hanging a picture. He wants to center it horizontally on the wall. The • Each batch uses 1_14 cups of flour.
picture is 32 _12 inches long, and the wall is 120 _34 inches long. How far from • Seventy-six _14 -cup servings of flour cost $4.49.
each edge of the wall should he place the picture?
STEP 1 Find the total length of the wall not covered by the 120 34 in. Formulate a Plan
picture. Use logical reasoning to solve the problem. Find the number of cups of
Subtract the whole number 32 12 in. flour that Alana has. Use that information to find the number of batches
120 _34 - 32 _12 = 88 _14 in. parts and then the fractional she can make. Use that information to find the cost of flour for each batch.
parts.
STEP 2 Find the length of the wall on each side of the picture. Justify and Evaluate
Solve
1
_
2 ( )
88 _14 = 44 _18 in. Number of cups of flour in bag:

Jon should place the picture 44 _18 inches from each edge 76 × _14 cup per serving = 19 cups
of the wall. 1 as a
Write 1__
4
Number of batches Alana can make:
STEP 3 Check the answer for reasonableness. decimal.
cups of flour 1.25 cups
The wall is about 120 inches long. The picture is about 30 inches total cups of flour ÷ _________
batch
= 19 cups ÷ _______
1 batch
© Houghton Mifflin Harcourt Publishing Company

© Houghton Mifflin Harcourt Publishing Company

long. The length of wall space left for both sides of the picture is = 19 ÷ 1.25
about 120 - 30 = 90 inches. The length left for each side is about = 15.2
_1 (90) = 45 inches.
2

The answer is reasonable because it is close to the estimate. Alana cannot make 0.2 batch. The recipe calls for one egg, and she cannot
divide one egg into tenths. So, she can make 15 batches.

Cost of flour for each batch: $4.49 ÷ 15 = $0.299, or about $0.30.

YOUR TURN
1. A 30-minute TV program consists of three commercials, each minutes 2_12 Justify and Evaluate
long, and four equal-length entertainment segments. How long is each Personal
Math Trainer A bag contains about 80 quarter cups, or about 20 cups. Each batch uses
entertainment segment?
5 _58 min Online Practice about 1 cup of flour, so there is enough flour for about 20 batches. A bag
and Help
my.hrw.com
costs about $5.00, so the flour for each batch costs about $5.00 ÷ 20 = $0.25.
The answers are close to the estimates, so the answers are reasonable.

Lesson 3.6 95 96 Unit 1

PROFESSIONAL DEVELOPMENT
Integrate Mathematical Math Background
Practices MP.1 Students will solve problems that use rational
This lesson provides an opportunity to address numbers and have more than one operation.
this Mathematical Practice standard. It calls for Some of these problems may involve rational
students to make sense of problems and numbers in different forms. In order to solve a
persevere in solving them. Students solve problem with rational numbers, it is preferable
problems involving rational numbers. Some that all the rational numbers in the problem be
problems involve rational numbers in different changed to the same form. If students are using a
forms and many are multi-step problems. tool such as a calculator, they might find it more
Students may choose tools to help solve straightforward to use the decimal form of a
problems in this lesson. number when the decimal is terminating.

Applying Rational Number Operations 96

 YOUR TURN
Avoid Common Errors
Students may miss that different units of volume are used to describe the amounts of sugar
needed. Make sure that students know and use the conversion 1 cup = 48 teaspoons.

ADDITIONAL EXAMPLE 3 EXAMPLE 3

Mindy has an unpaid balance of $1,250
on her credit card. The monthly finance
Focus on Technology Mathematical Practices
charge is 1__34 %. If she makes no Students may become less careful when they use a calculator in Steps 2 and 3. Remind
additional purchases and does not them that any mistake in inputting a number or operation will lead to an incorrect answer,
make any payments to the credit card so they must still concentrate on decimal placement and the negative signs.
company, what will be the balance
Questioning Strategies Mathematical Practices
after two months? $1,294.13
• Explain why the products in Steps 2 and 3 are rounded to the nearest hundredth.
Interactive Whiteboard The product will be subtracted from a decimal number that is written in hundredths.
Interactive example available online • Why can’t you multiply 1__34 % by 2 and subtract the product from 186.73? The level of the
my.hrw.com lake is not 186.73 meters in both years, the second-year level is 1__34 % less than the first-year
level.

YOUR TURN
Connect Multiple Representations Mathematical Practices
It will be easier to solve this problem if the fraction __35 is written as a decimal. Students
should understand that 12 __35 % = 12.6%. = 0.126.

Elaborate
Talk About It
Summarize the Lesson
Ask: What tool or tools did you use in this lesson to help you solve problems?
Sample answer: I used a calculator and pencil and paper.

GUIDED PRACTICE
Engage with the Whiteboard
Have a student-volunteer fill in the missing values in Steps 1 and 2 of Exercise 1.
Have another volunteer convert each value from decimal to fraction, or fraction
to decimal.

Avoid Common Errors

Exercise 2 Students may not change 2 __38 % to a decimal correctly. Remind them to insert 0
as a placeholder when they move the decimal point two places to the left of 2.

97 Lesson 3.6
 YOUR TURN YOUR TURN
2. A 4-pound bag of sugar contains 454 one-teaspoon servings and costs 3. Three years ago, Jolene bought $750 worth of stock in a software company.
Personal
$3.49. A batch of muffins uses _34 cup of sugar. How many batches can Math Trainer
Personal Since then the value of her purchase has been increasing at an average rate
Math Trainer
you make if you use all the sugar? What is the cost of sugar for each Online Practice
and Help Online Practice
of 12_35 % per year. How much is the stock worth now? $1,070.72
batch? (1 cup = 48 teaspoons) 12 batches; $0.29 per batch
and Help
my.hrw.com
my.hrw.com

Using Tools Strategically Guided Practice

A wide variety of tools are available to help you solve problems. Rulers,
models, calculators, protractors, and software are some of the tools you can 1. Mike hiked to Big Bear Lake in 4.5 hours at an average rate of 3_15 miles per
use in addition to paper and pencil. Choosing tools wisely can help you solve hour. Pedro hiked the same distance at a rate of 3_35 miles per hour. How
problems and increase your understanding of mathematical concepts. Math On the Spot long did it take Pedro to reach the lake? (Example 1 and Example 2)
my.hrw.com

EXAMPL 3
EXAMPLE 7.EE.3, 7.NS.3
STEP 1 Find the distance Mike hiked.
4.5 h × 3_15 or 3.2 miles per hour = 14.4 miles
The depth of Golden Trout Lake has been decreasing in recent years. Two
years ago, the depth of the lake was 186.73 meters. Since then the depth
has been changing at an average rate of -1_34 % per year. What is the depth STEP 2 Find Pedro’s time to hike the same distance.
of the lake today? 3
14.4 miles ÷ 3_ or 3.6 miles per hour = 4 hours
5
STEP 1 Convert the percent to a decimal.
2. Until this year, Greenville had averaged 25.68 inches of rainfall per year for
−1_34% = −1.75% Write the fraction as a decimal. more than a century. This year’s total rainfall showed a change of −2_38 % with
respect to the previous average. How much rain fell this year? (Example 3)
= −0.0175 Move the decimal point two places left.

STEP 2 Find the depth of the lake after one year. Use a calculator to STEP 1 Use a calculator to find this year’s decrease to the nearest
simplify the computations. hundredth.
186.73 × (−0.0175) ≈ −3.27 meters Find the change in depth. 25.68 inches × −0.02375 ≈ −0.61 inches
Math Talk
186.73 − 3.27 = 183.46 meters Find the new depth. Mathematical Practices
How could you write a STEP 2 Find this year’s total rainfall.
STEP 3 Find the depth of the lake after two years. single expression for
© Houghton Mifflin Harcourt Publishing Company

© Houghton Mifflin Harcourt Publishing Company

calculating the depth after 25.68 inches − 0.61 inches ≈ 25.07 inches
183.46 × (−0.0175) ≈ −3.21 meters Find the change in depth. 1 year? after 2 years?

183.46 − 3.21 = 180.25 meters

?
Find the new depth.
ESSENTIAL QUESTION CHECK-IN
STEP 4 Check the answer for reasonableness.
186.73(1 - 0.0175) 3. Why is it important to consider using tools when you are solving a
The original depth was about 190 meters. The depth changed = 186.73(0.9825); problem?
by about −2% per year. Because (−0.02)(190) = −3.8, the depth 186.73(1 - 0.0175)2 Sample answer: Using a calculator to solve a problem that involves
changed by about −4 meters per year or about −8 meters over = 186.73(0.9825)2
two years. So, the new depth was about 182 meters. The answer is complicated arithmetic can help you avoid errors. It can also help
close to the estimate, so it is reasonable.
you to check solutions to any problems you solved by hand.

Lesson 3.6 97 98 Unit 1

DIFFERENTIATE INSTRUCTION
Home Connection Cognitive Strategies Additional Resources
Have students find a recipe from their home, The Distributive Property of Multiplication over Differentiated Instruction includes:
a cook book, a magazine, or an online recipe Addition can be used to multiply a mixed • Reading Strategies
site that contains flour as an ingredient. From number with another rational number. For • Success for English Learners EL
Example 2, there are 19 cups of flour in a example, to multiply 4.5 × 3 __15 , have students
• Reteach
5-pound bag. Have the students compute how rewrite the problem as 4.5 × (3 + __15 ) and then:
many batches of their recipe they can make • Challenge PRE-AP
from a 5-pound bag of flour. 4.5 × (3 + __15 ) = (4.5 × 3) + (4.5 × __15 )
= 13.5 + 0.9
= 14.4

Applying Rational Number Operations 98

 Personal
Math Trainer
Online Assessment
Evaluate Focus | Coherence | Rigor
and Intervention

Online homework GUIDED AND INDEPENDENT PRACTICE

assignment available 7.EE.3
my.hrw.com
Concepts & Skills Practice
3.6 LESSON QUIZ Example 1 Exercises 1, 4–7
Assessing Reasonableness of Answers
7.EE.3
Example 2 Exercises 1, 8–10
1. Matteo is centering a poster on the Using Rational Numbers in Any Form
back of a door that is 32 inches
wide. The poster is 18 __12 inches wide. Example 3 Exercises 2, 3, 10–12
How far from the edge of the door Using Tools Strategically
should Matteo hang the poster?
2. Katelyn bought 3 packages of
ground beef. The packages weighed
2.65 lb, 1.85 lb, and 2.46 lb. She Exercise Depth of Knowledge (D.O.K.) Mathematical Practices
divided the ground beef into
4 2 Skills/Concepts MP.4 Modeling
4 packages, each the same weight.
How many pounds of ground beef 5 2 Skills/Concepts MP.4 Modeling
were in each package?
6 2 Skills/Concepts MP.4 Modeling
3. Two years ago, Paul bought $350
worth of stock in a cell phone 7 2 Skills/Concepts MP.4 Modeling
company. Since then the value of
8 2 Skills/Concepts MP.4 Modeling
his stock has been increasing at an
average rate of 9 __34 % per year. How 9 2 Skills/Concepts MP.4 Modeling
much is the stock worth now?
10 3 Strategic Thinking MP.3 Logic
4. Jasmine finished the bike trail in
2.5 hours at an average rate of 11 3 Strategic Thinking MP.3 Logic
9 __
3
10
miles per hour. Lucy biked 12 3 Strategic Thinking MP.5 Using Tools
the same trail at a rate of 6 __15 miles
per hour. How long did it take Lucy 13 2 Skills/Concepts MP.6 Precision
to bike the trail?
14 2 Skills/Concepts MP.1 Problem Solving
Lesson Quiz available online 15 3 Strategic Thinking MP.7 Using Structure
my.hrw.com 16 3 Strategic Thinking MP.5 Using Tools

Answers Additional Resources

1. 6__34 in. Differentiated Instruction includes:
2. 1.74 lb • Leveled Practice worksheets
3. $421.58
Exercise 14 combines concepts from the California Common Core
4. 3.75 hr
cluster “Apply and extend previous understandings of operations with
fractions to add, subtract, multiply, and divide rational numbers.”

99 Lesson 3.6
 Name Class Date 19
(
For 11–13, use the expression 1.43 × − ___
37
. )
3.6 Independent Practice Personal
Math Trainer
11. Critique Reasoning Jamie says the value of the expression is close to
−0.75. Does Jamie’s estimate seem reasonable? Explain.
7.NS.3, 7.EE.3 Online Practice
and Help
Yes, because the product is negative and about half of 1.5.
my.hrw.com

Solve, using appropriate tools. 12. Find the product. Explain your method.

4. Three rock climbers started a climb with each person carrying Sample answer: approximately -0.7343; use a calculator. Divide
7.8 kilograms of climbing equipment. A fourth climber with no -19 by 37, multiply the quotient by 1.43, then round the product.
equipment joined the group. The group divided the total weight
of climbing equipment equally among the four climbers. How much 13. Does your answer to Exercise 12 justify your answer to Exercise 11?
did each climber carry? 5.85 kg Sample answer: Yes; -0.7343 ≈ -0.75
5. Foster is centering a photo that is 3_12 inches wide on a scrapbook
page that is 12 inches wide. How far from each side of the page FOCUS ON HIGHER ORDER THINKING Work Area
should he put the picture?
4_14 in.
14. Persevere in Problem Solving A scuba diver dove from the surface of the
9
6. Diane serves breakfast to two groups of children at a daycare center. One ocean to an elevation of -79__ 10
feet at a rate of -18.8 feet per minute. After
box of Oaties contains 12 cups of cereal. She needs _13 cup for each younger spending 12.75 minutes at that elevation, the diver ascended to an elevation
9
child and _34 cup for each older child. Today’s group includes 11 younger of -28__
10
feet. The total time for the dive so far was 19_18 minutes. What was
children and 10 older children. Is one box of Oaties enough for everyone? 24 ft/min
the rate of change in the diver’s elevation during the ascent?
Yes; 11 × _13 cup + 10 × _34 cup = 11_16 cups
Explain.
15. Analyze Relationships Describe two ways you could evaluate 37% of
7. The figure shows how the yard lines on a football G 10 20 30 40 50 40 30 20 10 G the sum of 27_35 and 15.9. Tell which method you would use and why.
field are numbered. The goal lines are labeled G. Sample answer: (1) Convert the fraction to a decimal and
A referee was standing on a certain yard line as the
© Houghton Mifflin Harcourt Publishing Company • Image Credits: ©Hemera Technologies/Jupiterimages/

first quarter ended. He walked 41_34 yards to a yard find the sum of 27.6 and 15.9, then multiply the result by
line with the same number as the one he had just
left. How far was the referee from the nearest goal 0.37. (2) Convert the fraction, then use the Distributive
29_18 yd
G 10 20 30 40 50 40 30 20 10 G
line? Property. Multiply both 27.6 and 15.9 by 0.37, then add

In 8–10, a teacher gave a test with 50 questions, each worth the same
the products. The first method; there are fewer steps and
number of points. Donovan got 39 out of 50 questions right. Marci’s score so fewer chances to make errors.
was 10 percentage points higher than Donovan’s.

© Houghton Mifflin Harcourt Publishing Company

16. Represent Real-World Problems Describe a real-world problem you
8. What was Marci’s score? Explain.
39 could solve with the help of a yardstick and a calculator.
88; Donovan’s score was __
50
= 78%, Marci’s score was (78 + 10)% = 88%
Sample answer: You need to know how many gallons of
9. How many more questions did Marci answer correctly? Explain. paint you need to paint a wall. Measure the length and
88 44
5 more; 88% = ___ = __
Getty Images

100 50
.
width in feet of the wall with a yardstick, then find the
10. Explain how you can check your answers for reasonableness. area. Use the calculator to divide the area by the number
Sample answer: Donovan got about 40 out of 50 questions right, or about
of square feet a gallon of the paint covers. Round up rather
80%. So, Marci scored about 90%. 90% × 50 is 45. So Marci answered
than down to the nearest gallon so you have enough paint.
about 45 − 40, or 5 more questions correctly than Donovan.

Lesson 3.6 99 100 Unit 1

EXTEND THE MATH PRE-AP Activity available online my.hrw.com

Challenge In a magic square, all rows, columns, and diagonals have the same sum.
Use fractions, decimals, or percents to complete the magic square.
Accept any other forms of the rational numbers shown.
_3 3 1
3_ 1_
4 8 2
5 1
2_ 1.875 1_
8 8
1 3
2_ _ 3
4 8

Applying Rational Number Operations 100

 MODULE QUIZ

Ready to Go On? Ready Personal

Math Trainer
Assess Mastery 3.1 Rational Numbers and Decimals Online Practice
and Help
Write each mixed number as a decimal. my.hrw.com
Use the assessment on this page to determine if students have _
mastered the concepts and standards covered in this module. 1. 4 _15 4.2 14
2. 12 __
15
12.93 5
3. 5 __
32
5.15625
3.2 Adding Rational Numbers
3
Response to Find each sum.
2
4. 4.5 + 7.1 = 11.6 5. 5 _16 + (-3 _56 ) =
1_13
1 Intervention
3.3 Subtracting Rational Numbers
Find each difference.
Intervention Enrichment 6. - _18 - ( 6 _78 ) = -7 7. 14.2 - (-4.9 ) = 19.1
Access Ready to Go On? assessment online, and receive
3.4 Multiplying Rational Numbers
instant scoring, feedback, and customized intervention
Multiply.
or enrichment.
8. -4 ( __
-2 _45
10 )
7
= 9. -3.2(-5.6 )( 4 ) = 71.68
Personal
Online and Print Resources
Math Trainer 3.5 Dividing Rational Numbers
Online Assessment Differentiated Instruction Differentiated Instruction Find each quotient.
and Intervention 19
10. - __ 38
÷ __ =
- _74 - 32.01
11. ______ = 9.7
• Reteach worksheets • Challenge worksheets 2 7 -3.3
my.hrw.com
• Reading Strategies EL PRE-AP
3.6 Applying Rational Number Operations
• Success for English Extend the Math PRE-AP 12. Luis bought stock at $83.60. The next day, the price increased $15.35. This
new price changed by -4 _34% the following day. What was the final stock
Learners EL Lesson Activities in TE price? Is your answer reasonable? Explain.

© Houghton Mifflin Harcourt Publishing Company

$94.25; yes; price on day 1 ≈ $85, price on day 2 ≈ $100,
price on day 3 ≈ $95.
Additional Resources
ESSENTIAL QUESTION
Assessment Resources
• Leveled Module Quizzes 13. How can you use negative numbers to represent real-world problems?
Sample answer: You can use negative numbers to represent
temperatures below zero or decreases in prices.

Module 3 101

California Common Core Standards

Lesson Exercises Common Core Standards

3.1 1–3 7.NS.2d
3.2 4–5 7.NS.1
3.3 6–7 7.NS.1, 7.NS.1c, 7.NS.1d
3.4 8–9 7.NS.2, 7.NS.2a, 7.NS.2c, 7.EE.3
3.5 10–11 7.NS.2
3.6 12 7.NS.3, 7.EE.3

101 Unit 1 Module 3

 MODULE 3 Personal

Assessment Readiness MIXED REVIEW Math Trainer

Online Practice
Assessment Readiness my.hrw.com and Help

Scoring Guide
Item 3 Award the student 1 point for finding the water level in 1. Consider each expression. Is the value of the expression negative?

April and 1 point for correctly explaining how to use addition and Select Yes or No for expressions A–C.
A. -_12 ÷ (-8) Yes No
subtraction to solve the problem.
B. -_34 × _58 Yes No
Item 4 Award the student 1 point for finding the price of each C. -0.7 - (-0.62) Yes No

package and 1 point for correctly explaining why the answer is 2. Randall had $75 in his bank account. He made 3 withdrawals of $18 each.
reasonable. Choose True or False for each statement.
A. The change in Randall’s balance is -$54. True False
Additional Resources B. The account balance is equal to $75 - 3(-$18). True False
C. Randall now has a negative balance. True False
To assign this assessment online,
Personal login to your Assignment Manager 3. The water level in a lake was 12 inches below normal at the beginning of
Math Trainer March. The water level decreased by 2_14 inches in March and increased by 1_58
at my.hrw.com. inches in April. What was the water level compared to normal at the end of
Online April? Explain how you solved this problem.
Assessment and
my.hrw.com Intervention -12_58 in.; Sample answer: The expression -12 - 2_14 + 1_58
gives the water level compared to normal at the end of
April. Add and subtract from left to right to get -12_58 in.

4. A butcher has 10_34 pounds of ground beef that will be priced at $3.40 per
pound. He divides the meat into 8 equal packages. To the nearest cent, what
will be the price of each package? Explain how you know that your answer is

© Houghton Mifflin Harcourt Publishing Company

reasonable.

$4.57; Sample answer: Each package will weigh more

than 1 pound and less than 2 pounds, so the price of each
package will be between $3.40 and $6.80. So, an answer
of $4.57 is reasonable.

102 Unit 1

California Common Core Standards

Item 3 combines concepts from the California
Items Grade 7 Standards Mathematical Practices Common Core cluster “Apply and extend previous
1 7.NS.1, 7.NS.1c, 7.NS.2, MP.7 understandings of operations with fractions to add,
7.NS.2a subtract, multiply, and divide rational numbers.”

2* 7.NS.2, 7.NS.3 MP.5, MP.2

3 7.NS.1, 7.NS.3 MP.1
4 7.NS.2, 7.NS.3, 7.EE.3 MP.1, MP.6

* Item integrates mixed review concepts from previous modules or a previous course.

Rational Numbers 102