Download 7-TE U1M1 Integers

Adding and
Subtracting Integers
?
ESSENTIAL QUESTION
How can you use addition
and subtraction of integers
to solve real-world
problems?
MODULE
You can represent real-world
quantities with integers, and
then solve the problems
by finding the sums or
differences of the integers.
1
LESSON 1.1
Adding Integers with
the Same Sign
7.NS.1, 7.NS.1b,
7.NS.1d
LESSON 1.2
Adding Integers with
Different Signs
7.NS.1, 7.NS.1b
LESSON 1.3
Subtracting Integers
7.NS.1, 7.NS.1c
LESSON 1.4
Applying Addition
and Subtraction of
Integers
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Images
7.NS.1, 7.NS.1d,
7.NS.3, 7.EE.3
Real-World Video
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3
Module 1
Death Valley contains the lowest point in North America, elevation
–282 feet. The top of Mt. McKinley, elevation 20,320 feet, is the
highest point in North America. To find the difference between
these elevations, you can subtract integers.
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edition, accessible on
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Interactively explore
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you work through
practice sets.
3
Are You Ready?
Are YOU Ready?
Assess Readiness
Complete these exercises to review skills you will need
for this module.
Use the assessment on this page to determine if students need
intensive or strategic intervention for the module’s prerequisite skills.
Understand Integers
2
1
-20
Response to
Intervention
Write an integer to represent each situation.
Intervention
Enrichment
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-27
Access Are You Ready? assessment online, and receive
instant scoring, feedback, and customized intervention
or enrichment.
Online Assessment
and Intervention
Online Practice
and Help
Decide whether the integer is positive
or negative:
descended → negative
Write the integer.
A diver descended 20 meters.
1. an elevator ride
down 27 stories
Personal
Math Trainer
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3. 46 degrees
below zero
4. a gain of 12
yards
-46
700
12
Whole Number Operations
EXAMPLE
Online and Print Resources
3 15
245 - 28
24 5
28
__
217
245 - 28 = 217
Skills Intervention worksheets
Differentiated Instruction
• Skill 33 Understand
Integers
• Challenge worksheets
• Skill 34 Whole Number
Operations
Extend the Math PRE-AP
Lesson Activities in TE
Think:
8>5
Regroup 1 ten as 10 ones.
1 ten + 5 ones = 15 ones
Subtract: 15 - 8 = 7
Find the sum or difference.
5.
PRE-AP
6.
183
+ 78
_
7.
677
-288
_
389
261
8.
1,188
+
902
__
2,090
2,647
-1,885
__
762
Locate Points on a Number Line
• Skill 61 Locate Points on a
Number Line
EXAMPLE
-5
0
5
Graph +2 by starting at 0 and
counting 2 units to the right.
Graph -5 by starting at 0 and
counting 5 units to the left.
Graph each number on the number line.
Real-World Video Viewing Guide
9.
After students have watched the video, discuss the following:
• What are some integers that apply to the topographic map in the
video?
• What integer is represented by sea level? 0
10. -4
7
-10
4
-5
11. -9
0
12. 4
5
© Houghton Mifflin Harcourt Publishing Company
3
EXAMPLE
Personal
Math Trainer
10
Unit 1
PROFESSIONAL DEVELOPMENT VIDEO
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Author Juli Dixon models successful
teaching practices as she explores
adding and subtracting integers in an
actual seventh-grade classroom.
Online Teacher Edition
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resources online—plan,
present, and manage classes
and assignments.
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Interactive Answers and
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or display in the classroom.
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or full solutions to all lesson
exercises.
Interactive Whiteboards
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activities.
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Intervention
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homework, quizzes, tests,
and intervention activities.
Prepare your students with
updated practice tests aligned
with Common Core.
Adding and Subtracting Integers
4
Reading Start-Up
EL
Reading Start-Up
Have students complete the activities on this page by working alone
or with others.
Visualize Vocabulary
Use the ✔ words to fill in the ovals on the graphic. You may
put more than one word in each oval.
Strategies for English Learners
Each lesson in the TE contains specific strategies to help English
Learners of all levels succeed.
Emerging: Students at this level typically progress very quickly,
learning to use English for immediate needs as well as beginning to
understand and use academic vocabulary and other features of
academic language.
Expanding: Students at this level are challenged to increase their
English skills in more contexts, and learn a greater variety of vocabulary
and linguistic structures, applying their growing language skills in more
sophisticated ways appropriate to their age and grade level.
Bridging: Students at this level continue to learn and apply a range of
high-level English language skills in a wide variety of contexts, including comprehension and production of highly technical texts.
Understanding Integers
whole number,
positive number
negative
number
opposites
-50, 50
50
-50
Vocabulary
Review Words
difference (diferencia)
integers (enteros)
✔ negative number (número
negativo)
✔ opposites (opuestos)
✔ positive number (número
positivo)
sum (suma)
✔ whole number (número
entero)
Preview Words
absolute value
(valor absoluto)
additive inverse (inverso
aditivo)
expression (expresión)
model (modelo)
Understand Vocabulary
Complete the sentences using the preview words.
1. The
absolute value
2. The sum of a number and its
of a number gives its distance from zero.
additive inverse
is zero.
Integrating Language Arts
Students can use these reading and note-taking strategies to help
them organize and understand new concepts and vocabulary.
Additional Resources
Differentiated Instruction
• Reading Strategies EL
© Houghton Mifflin Harcourt Publishing Company
Active Reading
Active Reading
Booklet Before beginning the module, create
a booklet to help you learn the concepts in this
module. Write the main idea of each lesson on
each page of the booklet. As you study each
lesson, write important details that support the
main idea, such as vocabulary and processes.
Refer to your finished booklet as you work on
assignments and study for tests.
Module 1
Focus | Coherence | Rigor
Tracking Your Learning Progression
Before
Students understand addition and
subtraction:
• add whole numbers, fractions, and
decimals
• subtract whole numbers, fractions,
and decimals
5
Module 1
In this module
Students represent integer operations with concrete
models and connect the actions with the models to
standardized algorithms:
• add integers fluently
• subtract integers fluently
• solve multi-step problems involving addition and
subtraction of integers
After
Students will connect rational
numbers and integers:
• add rational numbers fluently
• subtract rational numbers fluently
5
GETTING READY FOR
GETTING READY FOR
Adding and Subtracting Integers
Adding and Subtracting
Integers
Understanding the standards and the vocabulary terms in the standards
will help you know exactly what you are expected to learn in this module.
7.NS.1
Apply and extend previous
understandings of addition and
subtraction to add and subtract
rational numbers; represent
addition and subtraction on a
horizontal or vertical number
line diagram.
Use the examples on the page to help students know exactly what
they are expected to learn in this module.
CA Common Core
Standards
You will learn how to use models to add and subtract integers with
the same sign and with different signs.
EXAMPLE 7.NS.1
You will learn how to use models to add and subtract integers with
the same sign and with different signs.
4 + (-7)
Key Vocabulary
additive inverse (inverso
aditivo)
The opposite of a number.
Content Areas
What It Means to You
The Number System—7.NS
Cluster Apply and extend previous understandings of operations with fractions to add,
subtract, multiply, and divide rational numbers.
+(-7)
4
-5 -4 -3 -2 -1
0 1 2 3 4 5
Start at 0.
Move right
4 units.
Then move
left 7 units.
4 + (-7) = -3
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.
Go online to
see a complete
unpacking of the
CA Common Core
Standards.
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Key Vocabulary
integer (entero)
A member of the set of whole
numbers and their opposites.
What It Means to You
You will learn that subtracting an
integer is the same as adding its
additive inverse.
EXAMPLE 7.NS.1c
Find the difference between 3,000 °F
and -250 °F, the temperatures the
space shuttle must endure.
3,000 - (-250)
3,000 + 250 = 3,250
The difference in temperatures the
shuttle must endure is 3,250 °F.
© Houghton Mifflin Harcourt Publishing Company • Image Credits: ©PhotoDisc/Getty Royalty Free
7.NS.1c
Visit my.hrw.com
to see all CA
Common Core
Standards
explained.
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6
California Common Core Standards
Lesson
1.1
Lesson
1.2
Unit 1
Lesson
1.3
Lesson
1.4
7.NS.1 Apply and extend previous understandings of addition
and subtraction to add and subtract rational numbers; represent addition
and subtraction on a horizontal or vertical number line diagram.
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.1d Apply properties of operations as strategies to add and
subtract rational numbers.
7.NS.3 Solve real-world and mathematical problems involving
the four operations with rational numbers.
Adding and Subtracting Integers
6
LESSON
1.1
Adding Integers with the Same Sign
Lesson Support
Content Objective
Language Objective
Students will learn to add integers with the same sign.
Students will explain how to add integers with the same sign.
California Common Core Standards
7.NS.1 Apply and extend previous understandings of addition and subtraction to add and subtract rational numbers; represent
addition and subtraction on a horizontal or vertical number line diagram.
7.NS.1b Understand p + q as the number located a distance | q | from p, in the positive or negative direction depending on whether
q is positive or negative. Show that a number and its opposite have a sum of 0 (are additive inverses). Interpret sums of rational numbers by
describing real-world contexts.
7.NS.1d Apply properties of operations as strategies to add and subtract rational numbers.
MP.5 Use appropriate tools strategically.
Focus | Coherence | Rigor
Building Background
opposites
Eliciting Prior Knowledge Have students work in pairs to
create a word web for integers. Have one or two pairs share their
word web on the board and explain why they chose each of the
terms to be included in the word web.
positive
negative
integers
whole
numbers
number
line
Learning Progressions
Cluster Connections
In this lesson, students continue to develop a unified
understanding of numbers. At Grade 6, students were
introduced to the concept of negative numbers. In this lesson
students begin to apply the operation of addition to negative
integers. Some key understandings for students are the
following:
This lesson provides an excellent opportunity to connect ideas
in this cluster: Apply and extend previous understandings
of operations with fractions to add, subtract, multiply, and
divide rational numbers.
• The absolute value of a number is the distance from zero.
• Addition can be represented on both vertical and
horizontal number lines.
• The sum of an addition expression can be shown as
distance in a positive or negative direction.
This will also lay the foundations for further work with negative
rational numbers in the Grade 8 standards.
7A
Give students the following prompt: “Erik borrows $6 from his
brother and $4 from his mother.” Have students use counters
to model the total amount of money that Erik borrows.
Then have them write an expression using negative numbers.
–4 + -6 = -10
$6 borrowed
$4 borrowed
$10 borrowed
PROFESSIONAL DEVELOPMENT
Language Support
EL
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.
Linguistic Support
EL
Academic/Content Vocabulary
Tools and Resources
Headings and subheadings – It is important for
English learners to become familiar with the nuances
of English, including idioms and multiple meaning
words. Use the textbook to maximize learning by
previewing the design of the pages, including the
headings and sub-headings, so that students can
better understand the intent of each lesson.
Math word problems are often written in the
past tense. While most present tense English
verbs are easily made past tense by adding
-ed to the end of the present tense of the verb
(e.g. walk/walked), some common verbs are irregular
in the past tense. Examples of irregular past tense
verbs in this lesson are made, withdrew, and lost.
Preview lessons to anticipate words that may cause
misunderstandings so that you may address them in
the beginning.
Reflect – English learners may know reflect as related
to a mirror; however, the meaning here is to pause
and think deeply about something.
Leveled Strategies for English Learners
EL
Emerging When English proficiency is limited, allow students to use their primary language in
peer-to-peer discussions, as this encourages higher-level thinking.
Expanding Working in small groups is an excellent way for English learners to deepen concept
knowledge and to simultaneously practice the academic language and vocabulary. Have English
learners work with students of mixed language proficiency.
Bridging Keep in mind that the nuances of language such as humor, multi-meaning words, and
idiomatic expressions can inhibit comprehension of the math concepts when they go unexplained.
Preview the word problem and instructions for any such words or phrases.
Math Talk
This feature is a discussion point in the lesson designed to get students to think
and discuss with others to deepen their understanding and clarify misconceptions. It challenges all students to express their thinking. In order to engage
English learners, elicit responses leveled by English proficiency by using the
recommendations.
Adding Integers with the Same Sign
7B
LESSON
1.1 Adding Integers with the Same Sign
CA Common Core
Standards
The student is expected to:
The Number System—7.NS.1
Apply and extend previous understandings of addition
and subtraction to add and subtract rational numbers;
represent addition and subtraction on a horizontal or
vertical number line diagram.
Engage
ESSENTIAL QUESTION
How do you add integers with the same sign? Sample answer: Add the absolute values
of the integers and use the sign of the integers for the sum.
Motivate the Lesson
Ask: Your team scores -8 and -4 on the first two rounds of a game. How can you use
addition to find your team’s total score? Begin the Explore Activity to find out.
The Number System—7.NS.1b
Understand p + q as the number located a distance |q|
from p, in the positive or negative direction depending on
whether q is positive or negative. Show that a number
and its opposite have a sum of 0 (are additive inverses).
Interpret sums of rational numbers by describing
real-world contexts.
The Number System—7.NS.1d
Explore
EXPLORE ACTIVITY 1
Focus on Modeling
Mathematical Practices
Guide students to understand that each yellow counter represents a positive integer and
has a value of +1. Each red counter represents a negative integer and has a value of -1.
Apply properties of operations as strategies to add and
subtract rational numbers.
Mathematical Practices
MP.5 Using Tools
Explain
EXPLORE ACTIVITY 2
Engage with the Whiteboard
Have students extend the number line shown to -10. Then have them use brackets
and an arrow to model what the thermometer will show when a temperature of
-4 °F drops by 6 degrees.
Mathematical Practices
• What are some advantages and disadvantages of a number-line model over a counter
model? Sample answer: It would be easier to add or subtract larger numbers on a number
line. With counters you don’t need to place a number correctly or decide which way to
move.
Questioning Strategies
• How would you add two positive integers on the vertical number line shown? You would
add them the same way that you add negative integers, except that you would place the
first addend above zero and then move up the number of spaces indicated by the second
addend.
Talk About It
Check for Understanding
Ask: If the temperature is -3 °F, why is a drop of 4 degrees like adding -4 °F
to -3 °F? You are combining the two values, -3 and -4, which is addition.
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.
7
Lesson 1.1
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CorrectionKey=B
1.1
?
Adding Integers with
the Same Sign
7.NS.1
Apply and extend previous
understandings of addition and
subtraction to add and subtract
rational numbers; represent
addition and subtraction on a
horizontal or vertical number line
diagram. Also 7.NS.1b, 7.NS.1d
ESSENTIAL QUESTION
How do you add integers with the same sign?
EXPLORE ACTIVITY 1
EXPLORE ACTIVITY 2
Adding on a Number Line
Just as you can add positive integers on a number line, you can
add negative integers.
The temperature was 2 °F below zero. The temperature
drops by 5 °F. What is the temperature now?
You can use colored counters to add positive integers
and to add negative integers.
6
6
4
4
B Mark the initial temperature on the number line.
2
2
C A drop in temperature of 5° is like adding -5° to
the temperature.
=1
0
0
-2
-2
-4
-4
-6
-6
-8
-8
Count on the number line to find the final
temperature. Mark the temperature now on the
number line.
= -1
Model with two-color counters.
A 3+4
How many counters are there in total?
8
- 2 degrees
Modeling Sums of Integers
with the Same Sign
4 positive counters
8
A What is the initial temperature written as an
integer?
7.NS.1
3 positive counters
7.NS.1, 7.NS.1b
D What is the temperature written as an integer?
- 7 degrees
total number of
counters
The temperature is
7
above / below
7 degrees
Reflect
counters.
2.
© Houghton Mifflin Harcourt Publishing Company
Math Talk
3 negative counters
How many counters are there in total?
Since the counters are negative integers, what is the sum?
the new temperature is - 4 °F.
3.
8
-8
the sign of each
integer
you would start at -2, move 5 units in a negative direction, and get -7 .
Communicate Mathematical Ideas When adding two numbers with
the same sign, what sign do you use for the sum?
4.
the same sign as the addends
Analyze Relationships What are two other negative integers that have
the same sum as - 2 and - 5?
Sample answer: - 3 and - 4
Lesson 1.1
7_MCABESE202610_U1M01L1.indd 7
Communicate Mathematical Ideas How would using a number line
to find the sum 2 + 5 be different from using a number line to find the
sum - 2 + (- 5)?
Instead of starting at 2 and moving 5 units in a positive direction to get 7,
Reflect
1.
What If? Suppose the temperature is -1 °F and drops by 3 °F. Explain
how to use the number line to find the new temperature.
Start at -1. Move 3 units in a negative direction to - 4;
Mathematical Practices
What does the color
of each row of counters
represent?
total number of
counters
-2 + (-5)
Temperature (˚F)
7; sample answer: Find the total number of positive
5 negative counters
+ (-5)
zero.
What is the sum and how do you find it?
B - 5 + (- 3)
-2
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LESSON
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CorrectionKey=B
7
04/11/13 10:54 PM
8
Unit 1
7_MCABESE202610_U1M01L1.indd 8
29/10/13 10:50 PM
professional development
Integrate Mathematical
Practices MP.5
This lesson provides an opportunity to address
this Mathematical Practice standard. It calls for
students to select tools as appropriate, to solve
problems. In Explore Activity 1, students use
counters to add positive and negative integers.
In Explore Activity 2, students use a number line
to add negative integers, and in Example 1,
students work with paper and pencil, using
absolute value to add two same-sign integers.
Math Background
To add means to combine or form the union of
two disjoint sets. For example: (-a) + (-b) =
-(a + b), where a + b is the sum of a and b.
The Closure Property for integer addition states
that if a and b are integers, then a + b is also an
integer.
Addition is commutative: a + b = b + a.
Addition is also associative:
(a + b) + c = a + (b + c).
Adding Integers with the Same Sign 8
ADDITIONAL EXAMPLE 1
Add -4 + (-7). −11
EXAMPLE 1
Connect to Daily Life
Interactive Whiteboard
Interactive example available online
my.hrw.com
Connect the concept of negative integers to a football game, where a loss of 3 yards on a
play is expressed as -3 and a loss of 11 yards on the next play is -11. The total loss on the
two plays is -3 + (-11) = -14.
Mathematical Practices
Point out that -7 + (-6) is the opposite of 7 + 6. The addends in each expression are the
same distance from 0; thus, they are opposites and have the same absolute value.
Focus on Math Connections
Mathematical Practices
• Explain how you know that -5 + (-8) is equal to -8 + (-5). Addition is commutative;
changing the order of the addends does not change the sum.
Questioning Strategies
• If you were to graph -7 + (-6) on a number line, would it be to the left or the right of
-7? Explain. It would be to the left. On a number line, values decrease from right to left;
-11 < -7.
YOUR TURN
Mathematical Practices
Point out that when adding integers with the same sign, there is a pattern for the signs of
the sums. For two negative numbers, the pattern for the signs is (-) + (-) = (-). For two
positive numbers, the pattern for the signs is (+) + (+) = (+).
Focus on Patterns
Elaborate
Talk About It
Summarize the Lesson
Ask: What do you think is the most efficient way to add two integers that have the
same sign? Sample answer: Add the absolute values of each integer, and then use
the sign of the integers for the sum. This method is faster than taking time to draw a
number line or assemble counters.
GUIDED PRACTICE
Engage with the Whiteboard
For Exercises 3–8, have students show each addition on the number lines provided,
and explain how they knew where to begin on the number line. Students can take
turns showing the different steps, such as plotting the first addend, counting spaces to
reach the products, and so on.
Avoid Common Errors
Exercises 9–13, 15–16 Some students may forget to attach the sign of the integers to the
sum after adding. Remind students that the sum of two negative numbers must include
the negative sign.
Exercise 14 Some students may write a negative sign in the sum because all of the
exercises before it were negative. Remind students to look at each exercise carefully to
identify the sign being used.
9
Lesson 1.1
Guided Practice
Adding Integers with a Common Sign
Find each sum. (Explore Activity 1)
To add integers with the same sign, add the absolute values of the integers and
use the sign of the integers for the sum.
7.NS.1, 7.NS.1d
Add -7 + (-6).
my.hrw.com
The signs of both integers are the same.
a. How many counters are there?
Find the absolute values.
The absolute value is always
| -7 | = 7 | -6 | = 6
positive or zero.
STEP 2
Find the sum of the absolute values: 7 + 6 = 13
STEP 3
Use the sign of the integers to write the sum.
-7 + (-6) = -13
Math Talk
Can you use the same
procedure you use to find the
sum of two negative integers
to find the sum of two
positive numbers?
Explain.
The sign of each integer is negative.
Communicate Mathematical Ideas Does the Commutative Property
of Addition apply when you add two negative integers? Explain.
Yes; it doesn’t matter whether you add -7 + (-6) or
-6 + (-7). The sum will still be -13.
6.
Yes; The signs
are the same,
so find the sum
of the absolute
values. The sum
uses the sign of
the integers.
Critical Thinking Choose any two negative integers. Is the sum of the
integers less than or greater than the value of either of the integers?
Will this be true no matter which integers you choose? Explain.
5. -3 + (-7) =
© Houghton Mifflin Harcourt Publishing Company
9.
-48 + (-12) =
-60
13.
-150 + (-1500) =
300
-1650
-3 + (-7) =
-5 -4 -3 -2 -1
- 10 - 9 - 8 - 7 - 6 - 5 - 4 - 3 - 2
-4
8. -6 + (-8) =
- 16
0 1 2 3
- 12
-8
-9
10. -1 + (-10) =
11. -9 + (-1) =
-10
12. -90 + (-20) =
13. -52 + (-48) =
-100
14. 5 + 198 =
?
Find each sum.
109 + 191 =
6. -4 + (-1) =
15. -4 + (-5) + (-6) =
8.
-5 -4 -3 -2 -1
0
-10
9. -5 + (-4) =
YOUR TURN
11.
4. -1 + (-3) =
-4
0 1 2 3
-5
0 1 2 3
-14
-4
0
Find each sum. (Example 1)
negative integers.
-9
-9
c. -2 + (-7) =
-7
-8 -7 -6 -5 -4 -3 -2 -1
-5 -4 -3 -2 -1
integers, so the sum will be less than either of the
-8 + (-1) =
negative
Model each addition problem on the number line to find each sum.
(Explore Activity 2)
Less than; yes; To add negative integers, you move
7.
9
b. Do the counters represent positive or
negative numbers?
-6
c. -5 + (-1) =
7. -2 + (-2) =
in a negative direction on the number line for both
a. How many counters are there?
negative
negative numbers?
3. -5 + (-2) =
Reflect
6
b. Do the counters represent positive or
Mathematical Practices
5.
2. -2 + (-7)
-10
-15
-11
-110
203
16. -50 + (-175) + (-345) =
-570
ESSENTIAL QUESTION CHECK-IN
17. How do you add integers with the same sign?
10.
-32 + (-38) =
-70
12.
-40 + (-105) =
-145
Personal
Math Trainer
14.
-200 + (-800) =
-1000
Online Practice
and Help
© Houghton Mifflin Harcourt Publishing Company
EXAMPL 1
EXAMPLE
STEP 1
1. -5 + (-1)
Math On the Spot
Add their absolute values. Use the sign of the integers
as the sign of the sum.
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Lesson 1.1
9
10
Unit 1
DIFFERENTIATE INSTRUCTION
Kinesthetic Experience
Visual Cues
Additional Resources
Some students may benefit from physically
acting out the addition of integers. Tape or draw
a 6 to -6 number line on the classroom floor,
with numbers about one step apart. Place
students, one at a time, facing the number line.
Guide them to pace out an addition, moving
left to add two negative integers and right to
add two positive ones. For example, to add
-2 + (-3), a student should stand at -2 and
move 3 steps to the left to stand on -5. Invite
students to challenge one another to step out
various additions.
When adding integers, think:
• If the signs are the same, find the sum.
• If the signs are different, find the difference.
Have students predict the sign of the sum of the
following exercises without doing any
calculations.
Differentiated Instruction includes:
• Reading Strategies
• Success for English Learners EL
• Reteach
• Challenge PRE-AP
1. 1 + 4 = ___ 5
2. -9 + (-8) = ___ -17
3. -7 + (-7) = ___ -14
4. 120 + 75 = ___ 195
Adding Integers with the Same Sign
10
Personal
Math Trainer
Online Assessment
and Intervention
Online homework
assignment available
Evaluate
Focus | Coherence | Rigor
GUIDED AND INDEPENDENT PRACTICE
7.NS.1, 7.NS.1b, 7.NS.1d
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1.1 LESSON QUIZ
7.NS.1, 7.NS.1b,
7.NS.1d
Find each sum.
1. -76 + (-124)
Concepts & Skills
Practice
Explore Activity 1
Modeling Sums of Integers with the Same Sign
Exercises 1–2
Explore Activity 2
Adding on a Number Line
Exercises 3–8, 19
Example 1
Adding Integers with a Common Sign
Exercises 9–16, 18, 20–23
2. -12 + (-66) + (-48)
3. A football team receives a 5-yard
penalty on one play and a 10-yard
penalty on the next. Write a sum of
negative integers to represent this
situation.
4. Eli had mini-golf scores of -3, -4,
and -3. What was his total score for
the three rounds?
5. Anna made withdrawals from her
bank account of $60, $85, and $115.
Write and solve an addition
problem that shows her
withdrawals as negative integers.
Lesson Quiz available online
Exercise
Depth of Knowledge (D.O.K.)
Mathematical Practices
2 Skills/Concepts
MP.4 Modeling
20
3 Strategic Thinking
MP.2 Reasoning
21
2 Skills/Concepts
MP.4 Modeling
22
3 Strategic Thinking
MP.7 Using Structure
23
2 Skills/Concepts
MP.4 Modeling
24
3 Strategic Thinking
MP.7 Using Structure
25
3 Strategic Thinking
MP.3 Logic
26
3 Strategic Thinking
MP.7 Using Structure
18–19
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Additional Resources
Answers
1. -200
2. -126
3. -5 + (-10) = -15
4. -10
5. -60 + (-85) + (-115) = -260
11
Lesson 1.1
Differentiated Instruction includes:
• Leveled Practice worksheets
Exercise 18 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.”
Class
Date
1.1 Independent Practice
7.NS.1, 7.NS.1b, 7.NS.1d
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18. Represent Real-World Problems Jane
and Sarah both dive down from the surface
of a pool. Jane first dives down 5 feet, and
then dives down 3 more feet. Sarah first
dives down 3 feet, and then dives down
5 more feet.
a. Multiple Representations Use the
number line to model the equation
-5 + (-3) = -3 + (-5).
Game
Sack yardage
b. Does the order in which you add two
integers with the same sign affect the
sum? Explain.
a. Write a sum of negative integers to show Jan’s withdrawals on
Monday. Find the total amount Jan withdrew.
3
4
-12
-23
-25 + (-45) + (-75) = -145; $145
b. Write a sum of negative integers to show Julie’s withdrawals on
Monday. Find the total amount Julie withdrew.
-35 + (-55) + (-65) = -155; $155
c. Julie and Jan’s brother also withdrew money from his savings account
on Monday. He made three withdrawals and withdrew $10 more than
Julie did. What are three possible amounts he could have withdrawn?
-54
22. Multistep The temperature in Jonestown
and Cooperville was the same at 1:00.
By 2:00, the temperature in Jonestown
dropped 10 degrees, and the temperature
in Cooperville dropped 6 degrees. By 3:00,
the temperature in Jonestown dropped
8 more degrees, and the temperature in
Cooperville dropped 2 more degrees.
Sample answer: $45, $55, and $65
25. Communicate Mathematical Ideas Why might you want to use
the Commutative Property to change the order of the integers in the
following sum before adding?
-80 + (-173) + (-20)
It is easier to add -80 + (-20) first to get -100, and
a. Write an equation that models
the change to the temperature in
Jonestown since 1:00.
-3 + (-5) is also -8.
© Houghton Mifflin Harcourt Publishing Company
2
-5
-3 + (-5)
-8
19. A golfer has the following scores for a
4-day tournament.
Day
1
-14
then add -173 to get -273.
26. Critique Reasoning The absolute value of the sum of two different
integers with the same sign is 8. Pat says there are three pairs of integers
that match this description. Do you agree? Explain.
-10 + (-8) = -18
1
2
3
4
-3
-1
-5
-2
b. Write an equation that models
the change to the temperature in
Cooperville since 1:00.
Disagree; there are three pairs of positive integers: 1 and 7,
2 and 6, and 3 and 5, and three pairs of negative integers:
-6 + (-2) = -8
What was the golfer’s total score for the
tournament?
-11
Work Area
24. Multistep On Monday, Jan made withdrawals of $25, $45, and $75
from her savings account. On the same day, her twin sister Julie made
withdrawals of $35, $55, and $65 from her savings account.
-3 + (-6); -9; the team lost a
-4
Score
FOCUS ON HIGHER ORDER THINKING
total of 9 yards.
0
No; -5 + (-3) is -8 and
-100 + (-75) + (-85) = -260
20. A football team loses 3 yards on one play
and 6 yards on another play. Write a sum
of negative integers to represent this
situation. Find the sum and explain how
it is related to the problem.
-2
-6
Online Practice
and Help
21. When the quarterback is sacked, the team
loses yards. In one game, the quarterback
was sacked four times. What was the total
sack yardage?
2
-5 + (-3)
23. Represent Real-World Problems Julio is playing a trivia game. On his
first turn, he lost 100 points. On his second turn, he lost 75 points. On his
third turn, he lost 85 points. Write a sum of three negative integers that
models the change to Julio’s score after his first three turns.
Personal
Math Trainer
-1 and -7; -2 and -6; -3 and -5. The absolute
c. Where was it colder at 3:00, in
Jonestown or Cooperville?
© Houghton Mifflin Harcourt Publishing Company
Name
value of the sum of any of these six pairs is 8.
Jonestown
Lesson 1.1
EXTEND THE MATH
PRE-AP
11
12
Activity available online
Unit 1
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Activity 1 Use the given integers to make each equation true: 2, 5, 7, -3, -5, -9
1. ___ + ___ = -12 -3, -9
2. ___ + ___ = 12 5, 7
3. ___ + ___ = -14 -5, -9
4. ___ + ___ = 7 2, 5
5. ___ + ___ + ___ = -17 -3, -5, -9
6. ___ + ___ + ___ = 14 2, 5, 7
-2
-10
-3
-6
-5
-4
-7
0
-8
Activity 1 Use the integers 0, -2, -3, -4, -5, -6, -7, -8, and -10 to fill in a 3 × 3
magic square so that every row, column, and diagonal has the same sum. What is the
magic sum? -15
Adding Integers with the Same Sign
12
LESSON
1.2
Adding Integers with the Different Sign
Lesson Support
Content Objective
Language Objective
Students will learn to add integers with different signs.
Students will demonstrate and explain how to add integers with different signs.
California Common Core Standards
7.NS.1 Apply and extend previous understandings of addition and subtraction to add and subtract rational numbers; represent
addition and subtraction on a horizontal or vertical number line diagram.
7.NS.1b Understand p + q as the number located a distance | q | from p in the positive or negative direction depending on whether
q is positive or negative. Show that a number and its opposite have a sum of 0 (are additive inverses). Interpret sums of rational numbers by
describing real-world contexts.
MP.5 Use appropriate tools strategically.
Focus | Coherence | Rigor
Building Background
Visualizing Math Draw a number line from
-10 to 10 on the board. Have students take
turns comparing one positive integer and
one negative integer. Ask them to determine
which integer is farther from zero. Then have
them determine how much farther.
-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1
negative
zero
3
4 5 6
positive
7
8
9 10
Compare -7 and 5;
-7 is 2 units farther from zero than 5.
Learning Progressions
Cluster Connections
In this lesson, students will extend their understanding of
addition with integers to include adding integers with different
signs. Using two-color counters to model integer addition
deepens understanding by making a kinesthetic connection to
the concept of adding integers with different signs. Some key
understandings for students are the following:
This lesson provides an excellent opportunity to connect ideas
in this cluster: Apply and extend previous understandings
of operations with fractions to add, subtract, multiply, and
divide rational numbers.
•
•
•
•
The concept of a zero pair
The absolute value of a number is always positive.
A number and its opposite have a sum of zero.
The Identity Property of Addition, a + 0 = a, is true for all
real numbers, including negative integers.
This will help prepare students for continuing work with integers
and the operation of subtraction.
13A
2
Give students the following prompt: “Omar owes his parents
$10. He earns $3 by washing the dishes. Does he still owe his
parents money? If so, how much?” Have students use both
two-color counters and a number to demonstrate the solution.
Yes, Omar still owes his parents $7; -10 + 3 = -7.
-10 -8 -6 -4 -2 0
Negative
Zero
PROFESSIONAL DEVELOPMENT
Language Support
EL
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.
Linguistic Support
EL
Academic/Content Vocabulary
Multiple Meaning Words
glossary – Point out to students that new vocabulary
is highlighted. They are defined in context where they
are introduced and also in the glossary. Sometimes in
the glossary, there is a visual diagram or other
support, including a Spanish explanation. Students
may also want to find other math vocabulary terms
from past grade levels.
real-world problems – Solving word problems using
students’ knowledge of adding integers with opposite
signs depends on students recognizing words that
signal signed numbers. Look for the words in word
problems intended to cue students to the need for
positive or negative such as gain/loss, earn/spend,
withdraw/deposit, ascend/descend.
Leveled Strategies for English Learners
EL
Emerging At this level of English proficiency, students need lots of time to process their thinking
and response in English. Emerging level students can illustrate, demonstrate, find examples, or help
complete a graphic organizer.
Expanding An excellent way to check for understanding with students at this level is to have
them complete a graphic organizer, list their ideas or answers, write in a math journal, or discuss in
a small group.
Bridging Students at this level of English proficiency benefit from the teacher reiterating by using
a new academic word, and then repeating the idea with a more familiar synonym. In this way
students understand and listen more closely to adopt the new vocabulary.
Math Talk
Help students begin their answer to Math Talk in this lesson with “I chose...”;
then provide them a sentence frame to answer the question.
I chose the integers based on ________.
Adding Integers with the Different Sign
13B
LESSON
1.2 Adding Integers with Different Signs
CA Common Core
Standards
The student is expected to:
The Number System—7.NS.1
Apply and extend previous understandings of addition
and subtraction to add and subtract rational numbers;
represent addition and subtraction on a horizontal or
vertical number line diagram.
The Number System—7.NS.1b
Understand p + q as the number located a distance |q|
from p, in the positive or negative direction depending on
whether q is positive or negative. Show that a number
and its opposite have a sum of 0 (are additive inverses).
Interpret sums of rational numbers by describing
real-world contexts.
Mathematical Practices
MP.5 Using Tools
Engage
ESSENTIAL QUESTION
How do you add integers with different signs? Sample answer: Find the difference between
the absolute values of the two addends and then use the sign of the integer with the
greater absolute value as the sign for the sum.
Motivate the Lesson
Ask: The school had a fundraiser for the band. Your class raised $300, but you spent $28 on
supplies to raise the money. How can you express the actual amount you earned as the
sum of two integers with different signs? Begin the Explore Activity to find out.
Explore
EXPLORE ACTIVITY 1
Focus on Modeling
Mathematical Practices
Direct the students’ attention to the arrows on the models for 3 + 2 = 5 and 3 + (-2) = 1
at the top of the page. Be certain they understand the following:
• The number of units you move on the number line is equal to the absolute value of the
second addend.
• If the second addend is positive, you move to the right on the number line, which is the
positive direction.
• If the second addend is negative, you move to the left on the number line, which is
the negative direction.
Explain
EXPLORE ACTIVITY 2
Focus on Communication
Direct students to use appropriate math terms when referring to the counters. In A, the
yellow counters represent the first addend and the red counters represent the second
addend. A pair of red and yellow counters called a zero pair is removed, leaving the sum.
Engage with the Whiteboard
In B, have students draw a model that shows how to use zero pairs to find the sum of
(-6) + 3. The model should show 6 red counters and 3 yellow counters. There are
three zero pairs circled, leaving 3 red counters to represent the sum -3.
Mathematical Practices
• What does the color of the counters left, after any zero pairs are removed, tell you about
the sign of the sum? If the leftover counters are red, the sum is negative. If they are yellow,
the sum is positive.
Questioning Strategies
13
Lesson 1.2
• When making a model with colored counters, does it make a difference which color
counter is used first? Explain. No. In a model such as this, order is not important. What is
important is having the correct number of each color of counter in order to form zero
pairs and then be able to count the remaining counters.
DO NOT EDIT--Changes must be made through “File info”
CorrectionKey=B
Adding Integers with
Different Signs
1.2
7.NS.1
Apply and extend previous
understandings of addition and
subtraction to add and subtract
rational numbers; represent
addition and subtraction on a
horizontal or vertical number
line diagram. Also 7.NS.1b
ESSENTIAL QUESTION
How do you add integers with different signs?
EXPLORE ACTIVITY 1
EXPLORE ACTIVITY 2
Modeling Sums of Integers with
Different Signs
You can use colored counters to model adding integers with different signs.
When you add a positive integer (yellow counter) and a negative integer
(red counter), the result is 0. One red and one yellow counter form a zero pair.
7.NS.1, 7.NS.1b
Model and find each sum using counters. Part A is modeled for you. For
Part B, follow the steps to model and find the sum using counters.
Adding on a Number Line
3+2=5
-3 -2 -1
Start with 3 positive counters to represent 3.
0 1 2 3 4 5
Form zero pairs.
3 + (−2) = 1
What is left when you remove the zero pairs?
-3 -2 -1
1 positive
0 1 2 3 4 5
Start with
A Model 4 + (-3).
© Houghton Mifflin Harcourt Publishing Company
. Move 5 units to the
positive
6 + (-6) =
right
direction. -7 + 5 =
C Model 6 + (-6).
left
counters to represent adding
-6
3
.
.
What is left when you remove the zero pairs?
-7
6
6 negative counters to represent
3 positive
Form zero pairs.
1
B Model -7 + 5.
the
Add
0 1 2 3 4 5 6 7 8
Start at 4. Move 3 units to the left, or in the negative direction.
Start at
1
B Model -6 + 3.
Model each sum on a number line.
or in the
counter
Find the sum: 3 + (-2) =
The sum of 3 + (-2) is the number that is |-2|
units from 3 in the negative direction.
Start at
The value of a zero pair is
0. Adding or subtracting
0 to any number does
not change its value.
Add 2 negative counters to represent adding -2.
The sum of 3 + 2 is the number that is |2| units
from 3 in the positive direction.
To find the sum of integers with different signs,
such as 3 + (-2), you can start at 3 and
move | -2 | = 2 units in the negative direction.
1 + (-1) = 0
A Model 3 + (-2).
To find the sum of integers with the same sign,
such as 3 + 2, you can start at 3 and
move | 2 | = 2 units in the positive direction.
4 + (-3) =
7.NS.1, 7.NS.1b
. Move
, or in the
6
negative
3 negative
-8 -7 -6 -5 -4 -3 -2 -1
,
0
-2
counters
Find the sum: -6 + 3 =
-3
Reflect
0 1 2 3 4 5 6 7 8
units to
2.
direction.
Make a Prediction Kyle models a sum of two integers. He uses more
negative (red) counters than positive (yellow) counters. What do you
predict about the sign of the sum? Explain.
The sign will be negative. When Kyle forms zero pairs,
0
there will be negative counters left over.
© Houghton Mifflin Harcourt Publishing Company
LESSON
?
DO NOT EDIT--Changes must be made through “File info”
CorrectionKey=A
Reflect
1.
Make a Prediction Predict the sum of -2 + 2. Explain
your prediction and check it using the number line.
-5 -4 -3 -2 -1
0 1 2 3
0; The sum is |2| units from –2 in the positive direction.
Lesson 1.2
7_MCABESE202610_U1M01L2.indd 13
13
31/10/13 10:35 PM
14
Unit 1
7_MCAAESE202610_U1M01L2.indd 14
12/04/13 12:14 AM
PROFESSIONAL DEVELOPMENT
Integrate Mathematical
Practices MP.5
This lesson provides an opportunity to address
this Mathematical Practice standard. It calls for
students to select tools as appropriate, to solve
problems. In Explore Activity 1, students use
number lines to add integers with different signs.
In Explore Activity 2, students use colored
counters to add integers with different signs, and
in Example 1, students work with paper and
pencil, using absolute value to add integers with
different signs.
Math Background
Every integer is a real number. The opposite of
any real number a is -a. The Additive Inverse
Property states that the sum of any real number
and its opposite is 0. So, a + (-a) = 0. Zero is
neither positive nor negative, and zero is its own
opposite.
Zero pairs are formed by combining opposite
integers. When zero is added or subtracted from
any number, that number is unchanged. This
applies to all real numbers and is known as the
Identity Property of Addition, a + 0 = a, or the
Identity Property of Subtraction, a - 0 = a.
Adding Integers with Different Signs
14
ADDITIONAL EXAMPLE 1
Find each sum.
EXAMPLE 1
Mathematical Practices
• In B, why don’t you find the absolute value of each addend to find the sum? The second
addend, 37, is the additive inverse of the first addend, -37. The sum of a number and
its additive inverse (opposite) is 0, so finding the absolute values is not necessary.
Questioning Strategies
A -4 + 18 = ___ 14
B 25 + (-35) = ___ -10
Interactive Whiteboard
Interactive example available online
my.hrw.com
Avoid Common Errors
Some students may forget to record the negative sign in the answer when adding integers
by finding absolute value. Encourage these students to check the sign on every sum.
Focus on Communication
Mathematical Practices
Have students discuss whether number lines or colored counters would be a good way to
model each sum. Ask students to explain their reasoning.
YOUR TURN
Talk About It
Check for Understanding
Ask: Suppose you add two integers and the sum is negative. What does this tell
you about the integers? Either both are negative integers, or they have different
signs and the one with the greater absolute value is a negative integer.
Elaborate
Talk About It
Summarize the Lesson
Ask: How do you add integers with different signs? First, find the absolute value of
each number. Next, find the difference between the absolute values of the two
addends. Finally, use the sign of the integer with the greater absolute value in the sum.
GUIDED PRACTICE
Engage with the Whiteboard
For Exercises 1–4, have students draw arrows above the number lines to show the
addition. Invite students to explain how they knew where to begin the arrows. For
Exercises 5–8, have students circle the zero pairs in each model and then write the sum.
Avoid Common Errors
Exercise 11 Some students may forget to record the negative sign in the answer when
adding integers by finding absolute value. Encourage these students to check the sign on
every sum.
Exercise 12 Remind students that the sum of any real number and its opposite is 0.
Encourage students to use mental math for this type of exercise.
Exercise 13 Remind students that when zero is added to or subtracted from any number,
that number is unchanged. This applies to all real numbers and is known as the Identity
Property of Addition, a + 0 = a and 0 + a = a.
15
Lesson 1.2
EXPLORE ACTIVITY 2 (cont’d)
Guided Practice
Model and find each sum using counters.
3. 5 + (-1)
4
4. 4 + (-6)
-2
5. 1 + (-7)
-6
6. 3 + (-4)
-1
Use a number line to find each sum. (Explore Activity 1)
6
1. 9 + (-3) =
-3 -2 -1
2 3 4 5 6 7 8 9 10
-11
3. -15 + 4 =
Adding Integers
5
2. -2 + 7 =
0 1 2 3 4 5
4. 1 + (-4) =
-3
You have learned how to add integers with the same signs and how to add integers
with different signs. The table below summarizes the rules for adding integers.
Same signs
Add the absolute values of the
integers. Use the common sign for
the sum.
Different signs
Subtract the lesser absolute value
3 + (-5) = -2
from the greater absolute value.
Use the sign of the integer with the -10 + 1 = -9
greater absolute value for the sum.
A number and its
opposite
The sum is 0. The opposite
of any number is called its
additive inverse.
4 + (-4) = 0
-11 + 11 = 0
7.NS.1, 7.NS.1b
Find each sum.
A -11 + 6
© Houghton Mifflin Harcourt Publishing Company
| -11 |
- |6| = 5
-11 + 6 = -5
B
( -37 )
+ 37
( -37 )
+ 37 = 0
my.hrw.com
3+5=8
-2 + (-7) = -9
EXAMPL 1
EXAMPLE
Subtract the lesser absolute value from the greater.
Sample answer:
-7 and 15.
-7 + 15 = 8.
I chose 15
because it was
a positive
integer greater
than the
absolute value
of -7.
Math Talk
Mathematical Practices
Use the sign of the number with the greater
absolute value.
Give an example of two
integers with different signs
whose sum is a positive
number. How did
you choose the
integers?
The sum of a number and its opposite is 0.
- 18
Math On the Spot
Examples
Find each sum.
9. 13 + (-13) =
- 14
- 12
- 10
-5 -4 -3 -2 -1
0 1 2 3
Circle the zero pairs in each model. Find the sum. (Explore Activity 2)
5. -4 + 5 =
1
6. -6 + 6 =
0
7. 2 + (-5) =
-3
8. -3 + 7 =
4
Find each sum. (Example 1)
9. -8 + 14 =
6
10. 7 + (-5) =
11. 5 + (-21) =
-16
12. 14 + (-14) =
13. 0 + (-5) =
-5
14. 32 + (-8) =
?
2
0
24
ESSENTIAL QUESTION CHECK-IN
15. Describe how to find the sums -4 + 2 and -4 + ( -2 ) on a
number line.
To find -4 + 2, start at -4 and move 2 units to the right to -2.
YOUR TURN
7. -51 + 23 =
- 16
To find the sum -4 + (-2), start at -4 and move 2 units to the
-28
0
8. 10 + ( -18 ) =
10. 25 + (-26) =
-8
-1
© Houghton Mifflin Harcourt Publishing Company
Adding Integers
left to -6.
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Math Trainer
Online Practice
and Help
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Lesson 1.2
15
16
Unit 1
DIFFERENTIATE INSTRUCTION
Manipulatives
Critical Thinking
Additional Resources
Some students will benefit from using tiles
marked with a - or + sign rather than
unmarked red and yellow counters. Provide
students with such tiles and demonstrate that
the process is the same as when they are using
red and yellow tiles. For example, represent -3
with three - tiles and 5 with five + tiles. Make
three zero pairs. There are two + tiles left
unpaired, so the sum is 2.
Give each pair of students a set of related
exercises, such as those shown below.
Differentiated Instruction includes:
• Reading Strategies
• Success for English Learners EL
• Reteach
• Challenge PRE-AP
1. -5 + 9 4
2. 5 + (-9) -4
3. -15 + 8 -7
4. 15 + (-8) 7
Have students compare the answers in each set,
and make generalizations about the rules for
adding integers with different signs. Then have
students compare generalizations until they
reach a consensus.
Adding Integers with Different Signs
16
Personal
Math Trainer
Online Assessment
and Intervention
Online homework
assignment available
Evaluate
Focus | Coherence | Rigor
GUIDED AND INDEPENDENT PRACTICE
7.NS.1, 7.NS.1b
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1.2 LESSON QUIZ
7.NS.1, 7.NS.1b
1. Use a number line to show the sum
of 6 + (-7).
2. Sketch 9 red counters and 4 yellow
counters. How many zero pairs can
you make? How many counters do
you have after you remove the zero
pairs? What do they represent?
Concepts & Skills
Practice
Explore Activity 1
Adding on a Number Line
Exercises 1–4
Explore Activity 2
Modeling Sums of Integers with Different Signs
Exercises 5–8
Example 1
Adding Integers
Exercises 9–14, 16–31
Exercise
Depth of Knowledge (D.O.K.)
Mathematical Practices
3. Use absolute value to find the sum
of -12 + 8.
16–25
1 Recall of Information
MP.4 Modeling
4. Find the sum of 15 + (-6).
26–28
2 Skills/Concepts
MP.2 Reasoning
5. A football team lost 5 yards on one
play and gained 12 yards on the
next play. Write a sum of integers to
find the overall change in field
position. Explain your answer.
29
3 Strategic Thinking
MP.4 Modeling
30
3 Strategic Thinking
MP.7 Using Structure
31
3 Strategic Thinking
MP.4 Modeling
6. At 7 A.M. the temperature was -4 °F.
At 9 A.M. it was 8 degrees warmer.
What was the temperature at 9 A.M.?
32
3 Strategic Thinking
MP.7 Using Structure
33
3 Strategic Thinking
MP.3 Logic
34
3 Strategic Thinking
MP.7 Using Structure
Lesson Quiz available online
my.hrw.com
Differentiated Instruction includes:
• Leveled Practice Worksheets
Answers
1. -1
-2
0
2
4
6
2. 4; 5 red counters; the sum -5
3. -4; | -12 | = 12, | 8 | = 8; 12 - 8 = 4;
12 > 8, so the sum is -4.
4. 9; Possible method: | 15 | = 15, | -6 | = 6;
15 - 6 = 9; 15 > 9, so the sum is +9.
5. (-5) + 12 = 7. The team gained 7 yards.
6. 4 °F; (-4) + 8 = 4
17
Additional Resources
Lesson 1.2
Name
Class
Date
1.2 Independent Practice
FOCUS ON HIGHER ORDER THINKING
Personal
Math Trainer
7.NS.1, 7.NS.1b
my.hrw.com
31. Critical Thinking Explain how you could use a number line to show that
-4 + 3 and 3 + (-4) have the same value. Which property of addition
states that these sums are equivalent?
Online Practice
and Help
Find each sum.
Start at -4 and move 3 to the right to reach -1. Start
16. -15 + 71 =
56
17. -53 + 45 =
-8
18. -79 + 79 =
0
19. -25 + 50 =
25
21. 5 + (-100) =
-95
20. 18 + (-32) =
-14
22. -12 + 8 + 7 =
3
24. 15 + (-15) + 200 =
200
at 3 and move 4 to the left to reach -1. The sums are
equivalent by the Commutative Property of Addition.
-7
23. -8 + (-2) + 3 =
32. Represent Real-World Problems Jim is standing beside a pool. He
drops a weight from 4 feet above the surface of the water in the pool.
The weight travels a total distance of 12 feet down before landing on the
bottom of the pool. Explain how you can write a sum of integers to find
the depth of the water.
100
25. -500 + (-600) + 1200 =
Work Area
26. A football team gained 9 yards on one play and then lost 22 yards on the
next. Write a sum of integers to find the overall change in field position.
Explain your answer.
The weight is dropped from 4 feet above the surface.
9 + (-22) = -13. The team lost 13 yards.
Add -12 to represent the distance the weight falls
27. A soccer team is having a car wash. The team spent $55 on supplies.
They earned $275, including tips. The team’s profit is the amount the
team made after paying for supplies. Write a sum of integers that
represents the team’s profit.
before it hits the bottom. 4 + (-12) = -8. The water is
8 feet deep.
-55 + 275 = 220. The team’s profit was $220.
-47 + 47 = 0; $0
Accounts
Regular Checking
33. Communicate Mathematical Ideas Use counters to model two integers
with different signs whose sum is positive. Explain how you know the
sum is positive.
Sign Out
$$
Sample answer: A model with more positive counters
Search transactions
Available Balance
© Houghton Mifflin Harcourt Publishing Company
29. The sum of two integers with different signs is 8. Give two
possible integers that fit this description.
than negative counters represents a sum of two integers
-$47.00
whose sum is positive.
Sample answer: 10 and -2 and 12 and -4
30. Multistep Bart and Sam played a game in which each player earns or
loses points in each turn. A player’s total score after two turns is the sum
of his points earned or lost. The player with the greater score after two
turns wins. Bart earned 123 points and lost 180 points. Sam earned 185
points and lost 255 points. Which person won the game? Explain.
34. Analyze Relationships You know that the sum of -5 and another
integer is a positive integer. What can you conclude about the sign of
the other integer? What can you conclude about the value of the other
integer? Explain.
The sign of the other integer is positive and its value is
Bart won; Bart’s score = 123 + (-180) = -57 points;
© Houghton Mifflin Harcourt Publishing Company
28. As shown in the illustration, Alexa had a negative balance in
her checking account before depositing a $47.00 check. What
is the new balance of Alexa’s checking account?
6 or greater. Sample explanation: If you start at -5 on
Sam’s score = 185 + (-255) = -70 points; -57 > -70,
a number line, you have to move to the right 6 or more
so Bart has the greater score.
units to get a sum that is positive.
Lesson 1.2
EXTEND THE MATH
PRE-AP
17
Activity available online
18
Unit 1
my.hrw.com
Activity Starting with the integer 3 in the upper right-hand corner, use addition to
draw a path horizontally or vertically through the maze to reach the sum 0 shown
below the maze. If the first move is to the left, add 3 + (-6) = -3. Now move down
to get -3 + 7 = 4, or to the left to get -3 + (-1) = -4. Continue in this manner
until you have a number you can add to 4 to get 0.
Start
↓
-3
-1
-6
3
1
-8
7
-5
-4
6
-7
-8
3
4
2
-1
0
Path: 3, left to -6, left to -1, left to -3, down to 1, down to -4, right to 6, down to 4,
down to 0
Adding Integers with Different Signs
18
LESSON
1.3
Subtracting Integers
Lesson Support
Content Objective
Language Objective
Students will learn to subtract integers.
Students will demonstrate how to subtract integers.
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.
7.NS.1 Apply and extend previous understandings of addition and subtraction to add and subtract rational numbers; represent
addition and subtraction on a horizontal or vertical number line diagram.
MP.2 Reason abstractly and quantitatively.
Focus | Coherence | Rigor
Building Background
Eliciting Prior Knowledge Have students work with
partners to develop a graphic organizer to illustrate two
meanings of subtraction: taking away and comparing. Have
students give an example in words of each use of subtraction.
Encourage students to discuss the two uses of subtraction
giving examples of each.
Comparing
Jason has 15 marbles.
He gives 6 away. How
many does he have left?
Ellen has 4 red marbles
and 10 blue marbles.
How many more blue
marbles does she have
than red marbles?
15 - 6 = 9
10 - 4 = 6
Learning Progressions
Cluster Connections
In this lesson, now that students have begun to work with
negative numbers, the operation of subtraction can be thought
of as addition of the opposite. Some key understandings for
students are the following:
This lesson provides an excellent opportunity to connect ideas
in this cluster: Apply and extend previous understandings
of operations with fractions to add, subtract, multiply, and
divide rational numbers.
• To subtract a positive integer on the number line, move
to the left.
• To subtract a negative integer on the number line, move
to the right.
• A subtraction expression can be rewritten as the addition
of the opposite.
• The distance between two integers on the number line is
the absolute value of their difference.
The concepts related to subtraction with integers will be
extended to other negative rational numbers.
19A
Subtraction
Taking Away
Give students the following prompt: “Dena owes her aunt $12.
Her aunt takes away $3 of the debt. How much does Dena still
owe her aunt?”
Have students write the subtraction expression and use two
models, one with zero pairs and one without zero pairs, to show
the solution.
-12 - (-3) = - 9
PROFESSIONAL DEVELOPMENT
Language Support
EL
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.
Linguistic Support
EL
Academic/Content Vocabulary
Multiple Meaning Words
explanations – Students need to understand the
explanations in the textbook in order to use it
effectively. In this lesson, students are asked to rewrite
opposite operations when subtracting integers.
Rewriting operations may be a new use of the word
rewrite because they are being asked to write
it differently, not just write it again.
English learners may already be familiar with the
context of a real-world problem, but may not have
learned the all the English vocabulary. Read over the
exercises assigned ahead of time to find any words
students might not know, such as rose, balance, board,
and chow, and make sure they understand the
intended meanings.
Leveled Strategies for English Learners
EL
Emerging Give students at this level of English proficiency time to process the language. Then
have them work in pairs to demonstrate how to subtract integers using counters.
Expanding Have students at this level model subtraction with integers using counters, and then
list the steps they take to subtract integers.
Bridging Have students at this level model the steps, and then explain how to subtract integers
using counters.
Math Talk
The question posed presents a great opportunity to have students practice
responding in complete sentences. The prompt asks, Why does it make sense
that…? Write out and model for students a sentence frame to use to share their
answer: It makes sense that…
Subtracting Integers
19B
LESSON
1.3 Subtracting Integers
CA Common Core
Standards
The student is expected to:
The Number System—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.
The Number System—7.NS.1
Apply and extend previous understandings of addition
and subtraction to add and subtract rational numbers;
represent addition and subtraction on a horizontal or
vertical number line diagram.
Mathematical Practices
MP.2 Reasoning
Engage
ESSENTIAL QUESTION
How do you subtract integers? Sample answer: Because addition and subtraction
are inverse operations, you can subtract an integer by adding its opposite.
Think: n - 1 = n + (-1).
Motivate the Lesson
Ask: Consider the following situation: You have $10 but want to buy something that costs
$15, so you borrow $5 and have a $5 debt. You could write this as 10 - 15 = -5. How
would you subtract a greater number from a lesser number? Begin the Explore Activity to
find out.
Explore
EXPLORE ACTIVITY 1
Engage with the Whiteboard
Have students model each step of C on the whiteboard. Point out to students that
although they need to remove 5 negative counters, they already have 2 negative
counters, so they only need to add 3 zero pairs.
Explain
EXPLORE ACTIVITY 2
Engage with the Whiteboard
Have students model A on the whiteboard, using counters to show that both
models (counters and number lines) will result in the same answer.
CC Mathematical Practices
• How do you decide which direction the arrow should point? Explain. The arrow points
right if you are adding a positive number because positive numbers increase as you move
to the right of 0. The arrow points left if you are adding a negative number because
negative numbers decrease as you move to the left of 0.
Questioning Strategies
• Can you subtract the numbers in any order? No. Subtraction is not commutative;
a - b = b - a is not true for all real numbers a and b (e.g., 5 - 8 = -3, but 8 - 5 = 3).
Connect Vocabulary
EL
Stress proper mathematical language to avoid confusion regarding the change to addition
of the opposite. Remind students that since addition and subtraction are inverse operations,
the opposite of a number is also referred to as its additive inverse.
19
Lesson 1.3
DO NOT EDIT--Changes must be made through “File info”
CorrectionKey=B
LESSON
1.3
?
DO NOT EDIT--Changes must be made through “File info”
CorrectionKey=A
Subtracting
Integers
EXPLORE ACTIVITY 1 (cont’d)
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.
Also 7.NS.1
Reflect
1.
Yes; to subtract 7 from -4, add 7 zero pairs so you
ESSENTIAL QUESTION
How do you subtract integers?
EXPLORE ACTIVITY 1
Communicate Mathematical Ideas Suppose you want to model the
difference -4 - 7. Do you need to add zero pairs? If so, why? How
many should you add? What is the difference?
have 7 positive counters to take away. You are left with
11 negative counters. So, -4 - 7 = -11.
7.NS.1
Modeling Integer Subtraction
You can use counters to find the difference of two integers. In some cases, you
may need to add zero pairs.
Model and find each difference using counters.
EXPLORE ACTIVITY 2
Subtracting on a Number Line
1 + (-1) = 0
A Model -4 - (-3).
To model the difference 5 - 3 on a number line, you start at 5
and move 3 units to the left. Notice that you model the sum
5 + (-3) in the same way. Subtracting 3 is the same as adding
its opposite, -3.
Start with 4 negative counters to represent -4.
Take away 3 negative counters to represent subtracting -3.
negative counter
Find the difference: -4 - (-3) =
You can use the fact that subtracting a number is the same as
adding its opposite to find a difference of two integers.
-1
B Model 6 - (-3).
Rewrite subtraction as addition of the opposite.
-1 - 5 = - 1 +
© Houghton Mifflin Harcourt Publishing Company
Take away 3 negative counters to represent
subtracting -3.
Start at
9 positive counters
Find the difference: 6 - (-3) =
2 negative
Take away
What is left?
and move
5
units to the left.
-6
-8 -7 -6 -5 -4 -3 -2 -1
0
-8 -7 -6 -5 -4 -3 -2 -1
0
Rewrite subtraction as addition of the opposite.
counters.
counters, so add
3
-7 - (-3) = -7 +
zero pairs.
Start at
counters.
3 positive counters
Find the difference: -2 - (-5) =
-5
B Find -7 - (-3).
5 negative
5 negative
-1
The difference is
9
C Model -2 - (-5).
You need to take away
0 1 2 3 4 5
A Find -1 - 5 on a number line.
You need to take away 3 negative counters,
so add 3 zero pairs.
Start with
-1
Find each difference on a number line.
Start with 6 positive counters to represent 6.
What is left?
5 - 3 = 5 + (-3)
-7
and move
The difference is
3
Lesson 1.3
7_MCABESE202610_U1M01L3.indd 19
19
31/10/13 10:44 PM
20
3
-4
3
units to the
right
© Houghton Mifflin Harcourt Publishing Company
What is left? 1
7.NS.1, 7.NS.1c
.
Unit 1
7_MCAAESE202610_U1M01L3.indd 20
11/04/13 11:53 PM
PROFESSIONAL DEVELOPMENT
Integrate Mathematical
Practices MP.2
This lesson provides an opportunity to address
this Mathematical Practice standard. It calls for
students to create and use representations to
organize, record, and communicate mathematical ideas. In Explore Activity 1, students represent
subtracting integers by using counters. In Explore
Activity 2 and Example 1, students use number
lines to represent subtraction of integers.
Math Background
Subtraction is formally defined as addition of the
opposite, or additive inverse. The integers are
closed under the operations of addition and
subtraction, which means that adding or
subtracting any two integers will produce
another integer. However, unlike addition, there
is no commutative property for subtraction,
because a - b = b - a is not true for all real
numbers a and b (e.g., 5 - 3 = 2, but
3 - 5 = -2).
Also, unlike addition, there is no associative
property for subtraction (e.g., 3 - (6 - 5) = 2,
but (3 - 6) - 5 = -8).
Subtracting Integers
20
ADDITIONAL EXAMPLE 1
The temperature at the start of a
football game was -1 °F. At halftime,
the temperature was -9 °F. Find the
change in temperature.
The temperature decreased by -8 °F.
Interactive Whiteboard
Interactive example available online
my.hrw.com
Animated Math
Subtracting Integers
EXAMPLE 1
Focus on Reasoning
CC Mathematical Practices
Ask: What can you say about the answer when a positive integer is subtracted from a
negative one? Subtracting a positive integer is the same as adding a negative integer, so the
answer will be a negative integer.
CC Mathematical Practices
• If you reverse the order of integers, -5 - 2, will you get the same answer? Explain. No.
Addition is commutative, but subtraction is not commutative. You will get -7.
Questioning Strategies
• What is the relationship between addition and subtraction? What does that tell you about
subtracting a negative number from another number? Addition and subtraction are
inverse operations. Subtracting a negative integer is the same as adding a positive
number.
YOUR TURN
Avoid Common Errors
Students build fluency with integer
subtraction using an interactive
number line.
Some students may have difficulty in rewriting subtraction as the addition of an opposite.
You may want to have them model the subtractions with counters to reinforce that the zero
pair with either the positive or negative counters crossed out represents adding the
opposite.
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Elaborate
Talk About It
Summarize the Lesson
Ask: How would you explain in your own words how to subtract integers? In
particular, how do you determine the sign of the difference? To subtract an integer,
add the opposite of the integer. If the two addends have different signs, apply the rules for
adding integers with unlike signs: subtract the lesser absolute value from the greater, and
use the sign of the addend with the greater absolute value.
GUIDED PRACTICE
Engage with the Whiteboard
For Exercises 1–2, have students draw a model on the whiteboard to represent each
difference. Ask students to explain if they need zero pairs or not. For Exercises 3–4,
have students draw arrows on the number lines to represent each difference.
Avoid Common Errors
Exercises 3–13 Some students may have difficulty in rewriting subtraction as the addition
of an opposite. You may want to have them model the subtractions with counters or with a
number line.
21
Lesson 1.3
EXPLORE ACTIVITY 2 (cont’d)
YOUR TURN
Reflect
2.
Communicate Mathematical Ideas Describe how to find 5 - (-8)
on a number line. If you found the difference using counters, would
you get the same result? Explain.
Find each difference.
Personal
Math Trainer
Rewrite 5 - (-8) as addition of the opposite,
-9
4. -7 - 2 =
Online Practice
and Help
-2
6. 3 - 5 =
my.hrw.com
5. -1 - (-3) =
2
7. -8 - (-4) =
-4
5 + 8 = 13. Start at 5 and move 8 units to the
Guided Practice
right. Yes; start with 5 positive counters and add
8 zero pairs. Take away 8 negative counters, leaving
Explain how to find each difference using counters. (Explore Activity 1)
1. 5 - 8 =
Subtracting Integers by Adding
the Opposite
You can use the fact that subtracting an integer is the same as adding its
opposite to solve problems.
Math On the Spot
-3
Start with 5 positive counters. Add
Start with 5 negative counters.
3 zero pairs and remove 8 positive
Remove 3 negative counters.
counters. 3 negative counters are
2 negative counters are left, so
left, so the difference is -3.
the difference is -2.
Use a number line to find each difference. (Explore Activity 2)
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EXAMPL 1
EXAMPLE
The temperature on Monday was -5 °C. By Tuesday the temperature rose
to -2 °C. Find the change in temperature.
STEP 1
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final temperature - Monday’s temperature = change in temperature
© Houghton Mifflin Harcourt Publishing Company
-2 °C - (-5 °C)
STEP 2
Math Talk
Find the difference.
-2 - (-5) = -2 + 5
-2 + 5 = 3
Mathematical Practices
To subtract -5, add its opposite, 5.
Use the rule for adding integers.
Why does it make sense
that the change in
temperature is a
positive number?
The temperature increased by 3 °C.
Reflect
3.
What If? In Example 1, the temperature rose by 3 °C. Suppose it fell
from -2 °C to -10 °C. Predict whether the change in temperature
would be positive or negative. Then subtract to find the change.
Negative; -10 - (-2) = -10 + 2 = -8; the
The sign of
the answer
tells me the
temperature
increased.
21
0
4. 1 - 4 = 1 +
-4 -3 -2 -1
-4
=
-3
0 1 2 3 4
Solve. (Example 1)
5. 8 - 11 =
-3
7. 15 - 21 =
-6
8. -17 - 1 =
-18
9. 0 - (-5) =
5
10. 1 - (-18) =
19
11. 15 - 1 =
13. 19 - (-19) =
?
6. -3 - (-5) =
14
38
2
12. -3 - (-45) =
42
14. -87 - (-87) =
0
ESSENTIAL QUESTION CHECK-IN
15. How do you subtract an integer from another integer without using a
number line or counters? Give an example.
To subtract an integer, add its opposite. Sample
example: 6 - 8 = 6 + (-8) = -2.
temperature would decrease by 8 °C.
Lesson 1.3
-9
=
-9 -8 -7 -6 -5 -4 -3 -2 -1
Animated
Math
Write a subtraction expression.
(-5)
3. -4 - 5 = -4 +
7.NS.1c, 7.NS.1
-2
2. -5 - (-3) =
© Houghton Mifflin Harcourt Publishing Company
13 positive counters.
22
Unit 1
DIFFERENTIATE INSTRUCTION
Technology
Visual Cues
Have students explore addition and subtraction
of integers on a calculator. Help them to
distinguish between the calculator’s subtraction
key - and the opposite or negative sign (-).
Point out that most calculators color code the
operation keys so that the addition + and
subtraction keys look the same but very different
from the opposite key (-). Encourage students
to try subtracting with the opposite key to
reinforce how the keys are distinct. Then have
students rewrite each difference and check both
expressions on a calculator.
The rule a - b = a + (-b) has important
implications for interpreting expressions on the
number line. Replacing a + sign by a - sign
reverses the direction of the motion. Thus,
3 - (-2) -5 or 3 + 2 + (-5) could be interpreted as: Move 3 places to the right, then move
2 more places to the right, then move 5 places
to the left. The value of the expression is 0.
10 - 6 = 4 becomes 10 + (-6) = 4
2. 7 - (-10) - 12
Have students explain each of the following
expressions as movements on a number line.
1. 5 - 4 - (-6)
Additional Resources
Differentiated Instruction includes:
• Reading Strategies
• Success for English Learners EL
• Reteach
• Challenge PRE-AP
Move 5 places to the right, then move 4 places
to the left, then move 6 places to the right. The
value of the expression is 7.
Move 7 places to the right, then move 10 more
places to the right, then move 12 places to the
left. The value of the expression is 5.
Subtracting Integers
22
Personal
Math Trainer
Online Assessment
and Intervention
Online homework
assignment available
Evaluate
Focus | Coherence | Rigor
GUIDED AND INDEPENDENT PRACTICE
7.NS.1c, 7.NS.1
my.hrw.com
1.3 LESSON QUIZ
7.NS.1c, 7.NS.1
Use a number line to find each
difference.
1. -2 - (-4)
Concepts & Skills
Practice
Explore Activity 1
Modeling Integer Subtraction
Exercises 1–2
Explore Activity 2
Subtracting on a Number Line
Exercises 3–4
Example 1
Subtracting Integers by Adding the Opposite
Exercises 5–14, 16–20
2. -5 - (-4)
3. Explain how to find 5 - 7 by using
counters.
4. Find -14 - 11.
5. At 8 A.M. the temperature was
-14 °F. By noon, the temperature
was 12 °F. Find the difference in
temperature. Did it rise or drop?
6. When Iris subtracted -12 - (-12),
she got a difference of -24. Is her
answer correct? If not, what mistake
did she make?
Lesson Quiz available online
my.hrw.com
Exercise
Depth of Knowledge (D.O.K.)
Mathematical Practices
2 Skills/Concepts
MP.4 Modeling
20
3 Strategic Thinking
MP.8 Patterns
21
2 Skills/Concepts
MP.8 Patterns
22
3 Strategic Thinking
MP.7 Using Structure
23–24
3 Strategic Thinking
MP.3 Logic
25
3 Strategic Thinking
MP.8 Patterns
16–19
Additional Resources
Differentiated Instruction includes:
• Leveled Practice Worksheets
Answers
1. 2
-2
0
2
2. -1
-5 -3 -1 0
3. Start with 5 positive counters. Add
7 negative counters. Then take away
5 zero pairs for a difference of -2.
4. -25
5. The temperature rose 26 degrees.
6. No. The correct answer is 0. Iris may have
added -12 + -12 instead of adding the
opposite of -12.
23
Lesson 1.3
Exercise 24 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.”
Class
Date
1.3 Independent Practice
Personal
Math Trainer
7.NS.1, 7.NS.1c
my.hrw.com
16. Theo had a balance of -$4 in his savings
account. After making a deposit, he has
$25 in his account. What is the overall
change to his account?
$29
21. Analyze Relationships For two months, Nell feeds her cat Diet Chow
brand cat food. Then for the next two months, she feeds her cat Kitty Diet
brand cat food. The table shows the cat’s change in weight over 4 months.
Cat’s Weight
Change (oz)
Online Practice
and Help
20. A scientist conducts three experiments
in which she records the temperature of
some gases that are being heated. The
table shows the initial temperature and the
final temperature for each gas.
17. As shown, Suzi starts her hike at an
elevation below sea level. When she
reaches the end of the hike, she is still
below sea level at -127 feet. What was the
change in elevation from the beginning of
Suzi’s hike to the end of the hike?
Gas
-21 °C
-8 °C
B
-12 °C
12 °C
C
-19 °C
-15 °C
© Houghton Mifflin Harcourt Publishing Company
borrowed $10 more. How much does Susanne owe her
sister in all?
23. Explain the Error When Tom found the difference -11 - (-4), he got
-15. What might Tom have done wrong?
- (-19) = 4;
4 °C increase
Tom found -11 - 4 instead of -11 - (-4). To subtract -4,
b. What If? Suppose the scientist
performs an experiment in which she
cools the three gases. Will the changes
in temperature be positive or negative
for this experiment? Why?
92 °F
19. Cheyenne is playing a board game. Her
score was -275 at the start of her turn, and
at the end of her turn her score was -425.
What was the change in Cheyenne’s score
from the start of her turn to the end of
her turn?
-150 points
Work Area
Sample answer: Susanne owed her sister $4. Then she
- (-12) = 24;
Gas C: -15
-19
22. Represent Real-World Problems Write and solve a word problem that
can be modeled by the difference -4 - 10.
24 °C increase
-127 - (-225) = 98 feet
3
Kitty Diet, Month 4
FOCUS ON HIGHER ORDER THINKING
13 °C increase
18. The record high January temperature
in Austin, Texas, is 90 °F. The record low
January temperature is -2 °F. Find the
difference between the high and low
temperatures.
Kitty Diet, Month 3
Diet Chow and by -16 ounces with Kitty Diet.
Gas A: -8 - (-21) = 13;
Gas B: 12
-18
Diet Chow; The cat’s weight changed by -26 ounces with
a. Write a difference of integers to find
the overall temperature change for
each gas.
Current Elevation:
–225 feet
-8
Diet Chow, Month 2
Which brand of cat food resulted in the greatest weight loss for Nell’s
cat? Explain.
Initial
Final
Temperature Temperature
A
Diet Chow, Month 1
Negative; the final
he should add the opposite of -4: -11 + 4 = -7.
24. Draw Conclusions When you subtract one negative integer from
another, will your answer be greater than or less than the integer you
started with? Explain your reasoning and give an example.
temperature will be less
Your answer will be greater than the integer you started
than the initial temperature
with because when you subtract a negative integer, you
because the gas is cooler. So
add its opposite, a positive integer. For example,
the difference in temperatures
-5 - (-3) = -5 + 3 = -2; -2 > -5.
will be negative.
© Houghton Mifflin Harcourt Publishing Company
Name
25. Look for a Pattern Find the next three terms in the pattern 9, 4, −1, −6,
−11, … . Then describe the pattern.
-16, -21, -26; subtract 5 to get the next term
Lesson 1.3
EXTEND THE MATH
PRE-AP
23
Activity available online
24
Unit 1
my.hrw.com
Activity Have students work in pairs. Each pair needs two -12 to 12 number lines
on grid paper and a 1–6 spinner (or number cube). One partner spins for the first
number, and calls it out as positive or negative. Both partners record the number and
plot it on their own number line. Then the other partner repeats the process to find
the second number. Each partner records the second number and finds the
difference on his or her own number line. Repeat. After three distinct subtractions,
have partners compare and discuss their final answers.
Subtracting Integers
24
LESSON
1.4
Applying Addition and Subtraction of Integers
Lesson Support
Content Objective
Language Objective
Students will learn to solve multistep problems involving addition and subtraction
of integers.
Students will draft a plan for solving multistep problems involving addition and
subtraction of integers.
California Common Core Standards
7.NS.3 Solve real-world and mathematical problems involving the four operations with rational numbers.
MP.1 Make sense of problems and persevere in solving them.
Focus | Coherence | Rigor
Building Background
Multiply to find the cost of the erasers.
Connecting to Every Day Life Work with the class to create
a chain to show the steps for solving a simple problem requiring
more than one step, such as “Jenny buys 12 erasers for 5¢ each
and 12 pencils for 18¢ each. How much more does Jenny spend
on the pencils than on the erasers?” Then have students follow
the steps to solve. 12 × 5 = 60; 12 × 18 = 216; 216 - 60 = 156;
pencils cost $1.56 more than the erasers.
Multiply to find the cost of the pencils.
Subtract the cost of the erasers from the cost of the
pencils to find how much more the pencils cost.
Learning Progressions
Cluster Connections
In this lesson, students apply the skills they have developed
with the operations of addition and subtraction for integers in
problem solving. Some key understandings for students are the
following:
This lesson provides an excellent opportunity to connect ideas
in this cluster: Apply and extend previous understandings
of operations with fractions to add, subtract, multiply, and
divide rational numbers.
• Recognizing cues given in word problems to determine
the operations needed.
• Identifying key words and phrases that indicate direction
and sign, such as ascend and descend.
• Using the Properties of Addition when performing
calculations in problem solving.
Give students the following prompt: “The top of a cliff is 105 feet
above sea level. Bernardo dives down 25 feet from sea level.
Then he dives down another 16 feet. What is the vertical
difference between Bernardo and the top of the cliff?” Have
students complete the table by writing phrases and the
numbers represented and then solve.
Students continue to develop problem-solving skills and utilize
them with rational numbers and irrational numbers.
Phrase
105 feet above sea level
+105
down 25 feet from sea level
-25
down another 16 feet
-16
–25 + –16 = –41; 105 – (–41) = 146
25A
Number
PROFESSIONAL DEVELOPMENT
Language Support
EL
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.
Linguistic Support
EL
Academic/Content Vocabulary
Idioms and Expressions
multi – Help English learners understand that the
prefix multi- means many. This means that it will take
many steps to solve the problem. Other words related
to math that begin with this prefix include: multisided, multi-faceted, multi-lateral, and multi-purpose.
Students may ask about other words that begin with
multi-, such as multiple and multiplication, which are
not hyphenated but also come from the meaning
many.
par – Be sure students understand the concept of
par in golf before they encounter it in a word
problem. Par is considered to be the pre-determined
number of turns needed to get the golf ball into the
hole. Common idiomatic expressions that use the
term par include the following: feeling up to par,
performing up to par, on par, par for the course, and
so on.
Leveled Strategies for English Learners
EL
Emerging Students at this level of English proficiency benefit from peer-to-peer primary
language support when working on problems that require several steps to solve. If possible,
pair students of the same primary language together.
Expanding Working in small groups helps English learners at this level think critically and talk
through the multiple steps needed to solve problems. Structure the small group work to make sure
each student participates by having a task.
Bridging Students at the this level of English proficiency have the language ability to explain
how they solved a multiple-step math problem. Provide a sentence frame, and encourage them to
answer in complete sentences.
In order to solve this problem, I began with _______. Then, I _______. Next, I _______.
Math Talk
Model for students how to begin their complete sentence to answer:
When _______, it means that the football team _______.
Applying Addition and Subtraction of Integers
25B
LESSON
1.4
Applying Addition and Subtraction
of Integers
CA Common Core
Standards
The student is expected to:
The Number System—7.NS.3
Solve real-world and mathematical problems involving
the four operations with rational numbers. Also 7.NS.1,
7.NS.1d, 7.EE.3
Mathematical Practices
MP.1 Problem Solving
Engage
ESSENTIAL QUESTION
How do you solve multistep problems involving addition and subtraction of integers?
Sample answer: Use a problem-solving plan to identify important information, form a plan
to answer the question, find the answer, and check your answer for reasonableness.
Motivate the Lesson
Ask: In a football game, a team had a gain of 10 yards, a loss of 5 yards, a loss of 2 yards, and
a gain of 4 yards in the first 4 plays. How can you use integers to find the team’s change in
position? Begin Example 1 to find out how to solve this problem.
Explore
Engage with the Whiteboard
Write a few examples of saving and spending on the whiteboard. Ask students to
explain how they would know whether a particular combination of saving and
spending results in a total saving or in a total spending. Encourage students to explain their
methods for finding the total.
Explain
ADDITIONAL EXAMPLE 1
Over a period of four hours, the
temperature rose 3 °F, rose 2 °F,
dropped 4 °F, and dropped 1 °F. If the
starting temperature was -2 °F, what
was the temperature after four hours?
-2 °F
Interactive Whiteboard
Interactive example available online
EXAMPLE 1
Mathematical Practices
Point out to students that problems sometimes provide clues and facts that must be used
to answer a question. Encourage students to underline key words or phrases, such as dives
down or swims up, that indicate direction.
Focus on Reasoning
Mathematical Practices
• How could you use a number line to represent this situation? Begin at -5, then move
12 places to the left, then move 8 places to the right. The answer is -9.
Questioning Strategy
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YOUR TURN
ADDITIONAL EXAMPLE 2
What is the total change in Mei’s
checking account if she writes a $35
check for shoes, deposits $50, and then
writes a $55 check for a sweater? The
amount in the account decreased by
$40.
Interactive Whiteboard
Interactive example available online
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25
Lesson 1.4
Connect Vocabulary
EL
Discuss the meanings of ascend (go up; move upward) and descend (go down; move from
higher to lower). Connect these opposite terms with positive and negative integers.
EXAMPLE 2
Connect to Vocabulary
EL
Remind students that a deposit is a credit or a positive number in a checking account, and a
check is a debit or a negative number in a checking account.
Mathematical Practices
• Can you tell what Irene’s checking account balance is from the given information?
No. We don’t know what her starting balance was. We only know how her balance
changed since Monday.
Questioning Strategy
Applying Addition
and Subtraction
of Integers
1.4
?
ESSENTIAL QUESTION
7.NS.3
Applying Properties to Solve Problems
Solve real-world and
mathematical problems
involving the four operations
with rational numbers. Also
7.NS.1, 7.NS.1d, 7.EE.3
You can use properties of addition to solve problems involving integers.
Math On the Spot
EXAMPLE 2
Analyze Information
My Notes
Solving a Multistep Problem
When Irene deposits money, she adds that amount to the account. When
she writes a check, that money is deducted from the account.
You can use what you know about adding and subtracting integers to solve a
multistep problem.
© Houghton Mifflin Harcourt Publishing Company
Starts
-
-5
-
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-160 + 125 + (-40)
Justify and Evaluate
Solve
Sea Level
0
Add the amounts to find the total change in the account. Use properties
of addition to simplify calculations.
–5
– 12
+
-160 + 125 + (-40) = -160 + (-40) + 125
–10
= -200 + 125
–15
= -75
+8
Swims
Dives
+
up
down
12
Use a positive integer for the amount Irene added to the account. Use
negative integers for the checks she wrote. Find the sum.
5
Write an expression.
• The seal starts at 5 feet below the surface,
so its initial position is -5 ft.
STEP 2
Formulate a Plan
Math On the Spot
7.NS.3, 7.NS.1
A seal is swimming in the ocean 5 feet below sea level.
It dives down 12 feet to catch some fish. Then, the seal
swims 8 feet up towards the surface with its catch.
What is the seal’s final elevation relative to sea level?
STEP 1
-5 - 12 + 8 = -17 + 8
= -9
Commutative Property
Associative Property
The amount in the account decreased by $75.
–20
8
Add or subtract from left to right to find the
value of the expression.
7.NS.1d, 7.NS.3, 7.EE.3
Irene has a checking account. On Monday she writes a $160 check for
groceries. Then she deposits $125. Finally she writes another check for $40.
What was the total change in the amount in Irene’s account?
How do you solve multistep problems involving addition and
subtraction of integers?
EXAMPL 1
EXAMPLE
Problem
Solving
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Justify and Evaluate
Irene’s account has $75 less than it did before Monday. This is reasonable
because she wrote checks for $200 but only deposited $125.
This is reasonable
because the seal
swam farther down
than up.
Reflect
2.
Communicative Mathematical Ideas Describe a different way to find
the change in Irene’s account.
Represent the amounts she writes in checks as positive
The seal’s final elevation is 9 feet below sea level.
numbers. 125 - 160 - 40 = -75; a decrease of $75.
YOUR TURN
1.
Anna is in a cave 40 feet below the cave entrance. She descends
13 feet, then ascends 18 feet. Find her new position relative to the
cave entrance.
-40 - 13 + 18; -35; 35 feet below the cave entrance
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LESSON
YOUR TURN
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Lesson 1.4
25
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26
3.
Alex wrote checks on Tuesday for $35 and $45. He also made a deposit in
his checking account of $180. Find the overall change in the amount in his
checking account.
-35 + (-45) + 180 = 100; $100 increase
Unit 1
PROFESSIONAL DEVELOPMENT
Integrate Mathematical
Practices MP.5
This lesson provides an opportunity to address
this Mathematical Practice standard. It which calls
for students to use a problem-solving model…to
determine a solution. Students solve real-world
problems by identifying important information,
formulating a plan, solving the problem, and
evaluating their solution to make sure it is
reasonable.
Math Background
The same properties students learned for whole
number addition are used for adding integers.
The Commutative and Associative properties
allow addition of three or more addends to be
rewritten in the most convenient order. Generally,
this involves first grouping the positives and then
grouping the negatives. Be certain students keep
the sign with its number when they change the
order.
Subtraction is formally defined as addition of the
opposite, or additive inverse. So, once you rewrite
subtraction as the addition of the opposite, you
can regroup the numbers as you would in an
addition problem.
Applying Addition and Subtraction of Integers
26
YOUR TURN
Focus on Communication
How could you use a two-column list to answer this question? Sample answer: Use Debits
and Credits for the headers; under Debits list -35, -45; under Credits list 180. Add to find
the total debits and the total credits. Then add the totals: -80 + 180 = 100
ADDITIONAL EXAMPLE 3
In Round 1 of a game, a student loses
5 points, loses 3 points, gains 17 points,
and loses 7 points. What is the student’s
score at the end of Round 1?
2 points; -5 - 3 + 17 - 7 = -15 + 17
=2
Interactive Whiteboard
Interactive example available online
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EXAMPLE 3
Engage with the Whiteboard
Cover up the solution and have students take turns circling all the important
information in the problem. Ask students to tell whether each number of yards
should be represented by a positive or a negative integer.
Mathematical Practices
• How does using the Associative Property help you simplify an expression involving the
addition of more than two numbers? It permits you to regroup the numbers in a way that
makes it easier to add the numbers.
Questioning Strategy
• If you do not apply the properties to reorder the problem, will the answer be the same?
Explain. Yes, because the gains and losses will remain the same. Applying the properties
simply makes simplifying the expression easier.
YOUR TURN
Avoid Common Errors
Some students may not translate “started out at the surface” as 0. Encourage students to
make a diagram or number line to help visualize the situation.
Elaborate
Talk About It
Summarize the Lesson
Ask: How would you explain in your own words how to solve multistep problems?
First, identify the important information; then make a plan about how to use it to
answer the question. Then check that your answer makes sense. If you write and simplify an
equation, use properties of addition to make the expression easier to simplify.
GUIDED PRACTICE
Engage with the Whiteboard
For Exercises 1–3, have students circle the important information in the problem on
the whiteboard. Then have students make a number line on the whiteboard to
represent each situation.
Avoid Common Errors
Exercises 4–9 Remind students to rewrite subtraction as addition of the opposite before
trying to add or subtract from left to right.
Exercises 10–11 If students have trouble in determining which inequality sign to use, you
may want to remind them that the inequality sign always points to the lesser of two
numbers.
27
Lesson 1.4
Guided Practice
Comparing Values of Expressions
Write an expression. Then find the value of the expression.
(Examples 1, 2, 3)
Sometimes you may want to compare values obtained by adding and
subtracting integers.
Math On the Spot
Problem
Solving
7.NS.3, 7.EE.3
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The Tigers, a football team, must gain 10 yards in the next four plays to
keep possession of the ball. The Tigers lose 12 yards, gain 5 yards, lose
8 yards, and gain 14 yards. Do the Tigers maintain possession of the ball?
-15 + 9 - 12 = -18; 18 feet below sea level
2. The temperature on a winter night was -23 °F. The temperature rose by
5 °F when the sun came up. When the sun set again, the temperature
dropped by 7 °F. Write and evaluate an expression to find the temperature
after the sun set.
Analyze Information
When the team gains yards, add that distance.
When the team loses yards, subtract that distance.
-23 + 5 - 7 = -25; -25 °F
If the total change in yards is greater than or equal to 10,
the team keeps possession of the ball.
3. Jose earned 50 points in a video game. He lost 40 points, earned
87 points, then lost 30 more points. Write and evaluate an expression to
find his final score in the video game.
Formulate a Plan
- 12 + 5 - 8 + 14
The team lost
1 more yard
than it gained.
Justify and Evaluate
Solve
-12 + 5 - 8 + 14
-12 + 5 + (- 8) + 14
To subtract, add the opposite.
-12 + (- 8) + 5 + 14
Commutative Property
(-12 + (- 8)) + (5 + 14)
Associative Property
Math Talk
Mathematical Practices
What does it mean that
the football team had a
total of -1 yard over
four plays?
-20 + 19 = -1
-1 < 10
Compare to 10 yards.
© Houghton Mifflin Harcourt Publishing Company
24
5. 9 - 4 - 17 =
6. 50 - 42 + 10 =
18
7. 6 + 13 + 7 - 5 =
8. 65 + 43 - 11 =
97
9. -35 - 14 + 45 + 31 =
-12
21
27
Determine which expression has a greater value. (Example 3)
(21 - 3 + 8) > (-14 + 31 - 6)
YOUR TURN
Carla: 0 - 20 + 5 - 18 = -33 (33 feet below the surface).
4. -6 + 15 + 15 =
11. 21 - 3 + 8 or -14 + 31 - 6
The football team gained 19 yards and lost 20 yards for a total of -1 yard.
Jim; Jim: -10 - 18 + 5 - 12 = -35 (35 feet below the surface);
Find the value of each expression. (Example 2)
(-12 + 6 - 4) < (-34 - 3 + 39)
Justify and Evaluate
Jim and Carla are scuba diving. Jim started out 10 feet below the
surface. He descended 18 feet, rose 5 feet, and descended 12 more feet.
Then he rested. Carla started out at the surface. She descended 20 feet,
rose 5 feet, and descended another 18 feet. Then she rested. Which
person rested at a greater depth? Explain.
50 - 40 + 87 - 30 = 67 points
10. -12 + 6 - 4 or -34 - 3 + 39
The Tigers gained less than 10 yards, so they do not maintain possession.
4.
1. Tomas works as an underwater photographer. He starts at a position that
is 15 feet below sea level. He rises 9 feet, then descends 12 feet to take a
photo of a coral reef. Write and evaluate an expression to find his position
relative to sea level when he took the photo.
?
ESSENTIAL QUESTION CHECK-IN
12. Explain how you can find the value of the expression -5 + 12 + 10 - 7.
Add and subtract from left to right:
© Houghton Mifflin Harcourt Publishing Company
EXAMPL 3
EXAMPLE
-5 + 12 + 10 - 7 = 7 + 10 - 7 = 17 - 7 = 10.
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Lesson 1.4
27
28
Unit 1
DIFFERENTIATE INSTRUCTION
Cognitive Strategies
Number Sense
Additional Resources
When adding three or more integers, it can be
helpful to follow these steps:
Some advanced students may be able to find
the value of expressions containing several
integers by using mental math. Encourage them
to do so. However, some students may make
careless errors and should be encouraged to
write down partial results. Point out how
students can rearrange terms to make the
mental computations easier. Finally, encourage
students to estimate answers by using
compatible numbers that are easy to add or
subtract.
Differentiated Instruction includes:
• Reading Strategies
• Success for English Learners EL
• Reteach
• Challenge PRE-AP
Step 1: Cross out all opposites.
Step 2: Add all positive integers.
Step 3: Add all negative integers.
Step 4: Add positive and negative sums.
Try it: -2 + 6 + -5 + 2 + 3 + -5
Cross out the -2s; add 3 + 6 to get 9;
add -5 + -5 to get -10; add 9 + (-10)
to get -1.
Applying Addition and Subtraction of Integers
28
Personal
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Online Assessment
and Intervention
Online homework
assignment available
Evaluate
Focus | Coherence | Rigor
GUIDED AND INDEPENDENT PRACTICE
7.NS.3, 7.NS.1, 7.NS.1d, 7.EE.3
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1.4 LESSON QUIZ
7.NS.3, 7.NS.1, 7.NS.1d,
7.EE.3
1. Find the value of the expression
-7 + 11 - 6 + 8.
2. Determine which expression has
the greater value:
40 - 16 + 10 or -20 + 18 + 30?
3. The Possums were on their 20-yard
line. On the next four plays they
gained 1 yard, lost 8 yards, lost 12
yards and gained 24 yards. From
what yard line will they start their
next play?
4. Tanya made a deposit of $250 into
her checking account. Then she
withdrew $55. The next day, she
wrote a check for $145. She had
$350 in her account before any of
these transactions. How much
money is in her account now?
Lesson Quiz available online
Answers
1. 6
2. 40 - 16 + 10: 40 - 16 + 10 = 34 >
-20 + 18 + 30 = 28
3. from the 25-yard line;
20 + 1 - 8 - 12 + 24 = 45 - 20 = 25
4. $400; 350 + 250 - 55 - 145 =
600 - 200 = 400
Lesson 1.4
Practice
Example 1
Using a Problem-Solving Plan
Exercises 1–3, 13–15
Example 2
Applying Properties to Solve Problems
Exercises 4–9, 13–15
Example 3
Comparing Values of Expressions
Exercises 10–11, 16–20
Exercise
Depth of Knowledge (D.O.K.)
Mathematical Practices
13–15
2 Skills/Concepts
MP.1 Problem Solving
16
2 Skills/Concepts
MP.7 Using Structure
17
3 Strategic Thinking
MP.7 Using Structure
2 Skills/Concepts
MP.4 Modeling
21
3 Strategic Thinking
MP.7 Using Structure
22
3 Strategic Thinking
MP.4 Modeling
23
3 Strategic Thinking
MP.7 Using Structure
18–20
Additional Resources
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Concepts & Skills
Differentiated Instruction includes:
• Leveled Practice Worksheets
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Class
Date
1.4 Independent Practice
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13. Sports Cameron is playing 9 holes of golf.
He needs to score a total of at most 15 over
par on the last four holes to beat his best
golf score. On the last four holes, he scores
5 over par, 1 under par, 6 over par, and 1
under par.
a. Before the game ends, Lee answers
a 275-point question correctly, a
70-point question correctly, and a
50-point question incorrectly. Write
and find the value of an expression to
find Lee’s final score.
5-1+6-1=9
b. Is Cameron’s score on the last four
holes over or under par?
-350 + 275 +70 - 50 = -55
over par
b. Barry’s final score is 45. Which player
had the greater final score?
c. Did Cameron beat his best golf score?
3 feet underground
15. Explain the Error Jerome tries to find the
value of the expression 3 - 6 + 5 by first
applying the Commutative Property. He
rewrites the expression as 3 - 5 + 6.
Explain what is wrong with Jerome’s
approach.
The Commutative Property does
not apply to subtraction:
Entering
Leaving
1:00 to 2:00
30
-12
2:00 to 3:00
14
3:00 to 4:00
18
3 - 5 + 6 = 4.
22
August
-53
Leta
-17
-22
18
$51
19. Leta had $45 in her account in May. How much money does she have in
her account in August?
$24
20. Analyze Relationships Whose account had the greatest decrease in
value from May to August?
FOCUS ON HIGHER ORDER THINKING
Carla’s
Work Area
21. Represent Real-World Problems Write
and solve a word problem that matches
the diagram shown.
-9 -8 -7 -6 -5 -4 -3 -2 -1 0
from his brother. He pays his brother back 3 dollars. How
much does Tim still owe his brother? -1 - 6 + 3 = -4; $4
22. Critical Thinking Mary has $10 in savings. She owes her parents $50. She
does some chores and her parents pay her $12. She also gets $25 for her
birthday from her grandmother. Does Mary have enough money to pay
her parents what she owes them? If not, how much more money does she
need? Explain.
No; the total amount she could pay her parents is
10 + 12 + 25 = 47. 50 - 47 = 3. So she still needs $3.
-8
-30
23. Draw Conclusions An expression involves subtracting two numbers
from a positive number. Under what circumstances will the value of the
expression be negative? Give an example.
a. During which hour did more
customers leave than arrive?
The sum of the absolute values of the other two numbers
3:00 to 4:00
must be greater than the value of the first number.
b. There were 75 customers in the store
at 1:00. The store must be emptied of
customers when it closes at 5:00. How
many customers must leave the store
between 4:00 and 5:00?
3 - 6 + 5 = 2 and
July
-18
Tim owes his brother 1 dollar. He borrows 6 more dollars
17. Multistep Rob collects data about how
many customers enter and leave a store
every hour. He records a positive number
for customers entering the store each hour
and a negative number for customers
leaving the store each hour.
14. Herman is standing on a ladder that is
partly in a hole. He starts out on a rung
that is 6 feet under ground, climbs up
14 feet, then climbs down 11 feet.
What is Herman’s final position, relative
to ground level?
June
Carla
18. Carla had $100 in her account in May. How much money does she have in
her account in August?
Barry
yes
© Houghton Mifflin Harcourt Publishing Company
Online Practice
and Help
16. Lee and Barry play a trivia game in which
questions are worth different numbers of
points. If a question is answered correctly,
a player earns points. If a question is
answered incorrectly, the player loses
points. Lee currently has -350 points.
a. Write and find the value of an
expression that gives Cameron’s score
for 4 holes of golf.
The table shows the changes in the values of two friends’ savings accounts
since the previous month.
© Houghton Mifflin Harcourt Publishing Company
Name
Example: 13 - 5 - 10 = -2. | 5 | + | 10 | = 15, which is
greater than 13.
87
Lesson 1.4
7_MCABESE202610_U1M01L4.indd 29
EXTEND THE MATH
29
29/10/13 11:43 PM
PRE-AP
30
Unit 1
7_MCABESE202610_U1M01L4.indd 30
Activity available online
my.hrw.com
Activity Here is how one student subtracted
38 from 54:
The algorithm will always work with integers.
Sample:
54
-3
8
__
–4 ← 4 - 8 = -4
2 0 ← 50 - 30 = 20
__
1 6 ← -4 + 20 = 16
83
-4
5
__
-2 ← 3 - 5 = -2
4 0 ← 80 - 40 = 40
__
3 8 ← -2 + 40 = 38
So, 54 - 38 = 16.
10/31/13 2:38 AM
Do you think this method always works?
Explain. Then try it with another pair of
integers.
Applying Addition and Subtraction of Integers
30
MODULE QUIZ
Ready
Ready to Go On?
Personal
Math Trainer
Assess Mastery
1.1 Adding Integers with the Same Sign
Use the assessment on this page to determine if students have
mastered the concepts and standards covered in this module.
Add.
Online Practice
and Help
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1. -8 + (-6)
-14
2. -4 + (-7)
-11
3. -9 + (-12)
-21
6. 15 + (-8)
7
9. 11 - (-12)
23
1.2 Adding Integers with Different Signs
3
Response to
Intervention
2
1
Add.
4. 5 + (-2)
3
5. -8 + 4
-4
1.3 Subtracting Integers
Intervention
Subtract.
Enrichment
7. 2 - 9
Access Ready to Go On? assessment online, and receive
instant scoring, feedback, and customized intervention
or enrichment.
Personal
Math Trainer
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Differentiated Instruction
• Reteach worksheets
• Challenge worksheets
• Reading Strategies
• Success for English
Learners EL
EL
There are 10 fewer people on the bus.
11. Cate and Elena were playing a card game. The stack of cards in the
middle had 24 cards in it to begin with. Cate added 8 cards to the
stack. Elena then took 12 cards from the stack. Finally, Cate took
9 cards from the stack. How many cards were left in the stack?
PRE-AP
Extend the Math PRE-AP
Lesson Activities in TE
Additional Resources
Assessment Resources includes:
• Leveled Module Quizzes
1
10. A bus makes a stop at 2:30, letting off 15 people and letting on 9. The
bus makes another stop ten minutes later to let off 4 more people.
How many more or fewer people are on the bus after the second stop
compared to the number of people on the bus before the 2:30 stop?
Online and Print Resources
Differentiated Instruction
8. -3 - (-4)
1.4 Applying Addition and Subtraction of Integers
11 cards
ESSENTIAL QUESTION
© Houghton Mifflin Harcourt Publishing Company
Online Assessment
and Intervention
-7
12. Write and solve a word problem that can be modeled by addition of
two negative integers.
Sample answer: Tonya owes her sister $10 and her
friend $5. By how much will her savings change after
she pays them?; -10 + (-5) = -15; $15 decrease
Module 1
California Common Core Standards
31
Common Core Standards
Lesson
Exercises
1.1
1–3
7.NS.1
1.2
4–6
7.NS.1
1.3
7–9
7.NS.1, 7.NS.1c, 7.NS.1d
1.4
10–11
7.NS.3, 7.EE.3
Unit 1 Module 1
31
MODULE 1
MIXED REVIEW
Assessment Readiness
Personal
Math Trainer
Assessment Readiness
my.hrw.com
Online Practice
and Help
Scoring Guide
Item 3 Award the student 1 point for finding the temperature at
9:00 a.m. and 1 point for correctly explaining how to use addition
and subtraction to solve the problem.
Item 4 Award the student 1 point for identifying the winner and
1 point for correctly explaining how to determine that Darren
scored more points than Sherri.
1. Look at each expression. Does it have the same value as -6 - 4?
Select Yes or No for expressions A–C.
A. -6 + (-4)
B. -4 + (-6)
C. 6 + (-4)
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No
No
No
2. Choose True or False for A–C.
A. x = 4 is the solution for x + 4 = 0.
B. x = 24 is the solution for _3x = 8.
C. x = 6 is the solution for 6x = 1
Additional Resources
Personal
Math Trainer
Yes
Yes
Yes
True
True
True
False
False
False
3. At 3:00 a.m., the temperature is –5 °F. Between 3:00 a.m. and 6:00 a.m.,
the temperature drops by 12 °F. Between 6:00 a.m. and 9:00 a.m., the
temperature rises by 4 °F. What is the temperature at 9:00 a.m.? Explain how
you solved this problem.
Online
Assessment and
Intervention
-13 °F; Sample answer: The expression -5 - 12 + 4
represents the temperature at 9:00 a.m.. Add or subtract
from left to right: -5 - 12 + 4 = -17 + 4 = -13.
Round
Sherri’s Points
1
35
-10
2
-20
15
3
-5
15
© Houghton Mifflin Harcourt Publishing Company
4. Sherri and Darren are playing a board game. The table shows the number
of points each player scores in 3 rounds. If the player with the greater total
score wins, who is the winner? Explain how you know.
Darren’s Points
Darren; Sample answer: Sherri scored 35 + (-20) + (-5) =
10 points, and Darren scored -10 + 15 + 15 = 20 points.
32
Unit 1
California Common Core Standards
Items
Grade 7 Standards
Mathematical Practices
1
7.NS.1, 7.NS.1c, 7.NS.1d
MP.7
2*
6.EE.5
MP.1
3
7.NS.1, 7.NS.3
MP.1
4
7.NS.1, 7.NS.3
MP.1
Item 4 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.”
* Item integrates mixed review concepts from previous modules or a previous course.
Adding and Subtracting Integers
32