Thank you for participating
in Teach It First!
This Teach It First Kit contains a Common Core Coach, Mathematics teacher lesson
followed by the corresponding student lesson. We are confident that using this
lesson will help you achieve your assessment preparation goals for your entire
class.
The Common Core Coach, Mathematics program is based on the philosophy
that mathematical skills are built on concepts. Math, maybe more than any other
school subject, builds from concept to concept, one on top of the other, over
several years. When students understand concepts and how they connect to
skills, they are better equipped to solve problems that they encounter in the real
world.
This program is 100% aligned to the Common Core State Standards and provides
a set of lessons for each of the five CCSS domains, with each lesson aligning
to one or more standard—together, lessons cover all the domain’s standards.
Concept Lessons begin with an underlying concept that connects directly to the
skill or skills taught in that lesson. Students will use a four-step problem-solving
process—Read, Plan, Solve, Check—to approach any mathematical problem.
Interactive questions follow examples and ask students to discuss a topic, model
a situation, try to solve a problem on their own, or check their work. With this
instructional anchor, you can implement the Common Core State Standards with
confidence.
We are happy to provide you this complimentary sample and would love to know
what you think. Once you have read through this lesson, do what you do best—
present it to your students. Then, don’t forget to complete a quick survey by
going to www.triumphlearning.com/CA/teach-it-first.
Regards,
Triumph Learning
Join the conversation about Common Core today by visiting commoncore.com, the
place where teachers, parents, and experts come together to share best practices and
practical information for successfully implementing Common Core standards in the
classroom.
136 Madison Avenue • New York, NY 10016 • p: 212.652.0215 • f : 212.857.8499 • www.triumphlearning.com
5
Adding and Subtracting Rational
Numbers
Learning Objectives
• Students will use number line models and rules to add signed
integers and other rational numbers.
• Students will subtract rational numbers by rewriting them as
addition problems and then adding the numbers.
Vocabulary
additive inverses addends whose sum is 0
integers the set of positive whole numbers, their opposites,
and zero
rational number a number that can be expressed as the
a
​ b  ​where a and b are integers and b  0
ratio __
Materials
• Fluency Practice, page A5
• Fluency Practice, page A6
Before the Lesson
Draw a number line from 210 to 10 on the board. Explain that the
set of integers includes the positive whole numbers, their opposites,
and 0. Then point out that this number line shows all the integers
from 210 to 10. Ask: Which integers are positive? Which integers
are negative? Remind students that when a number line is read from
left to right, the numbers are ordered from least to greatest. Pose
several questions that ask students to compare the integers shown,
for example: Which is less, 24 or 1? Which is greater, 23 or 26? After
reviewing number lines and how to read them, explain that in this
lesson, students will use number lines as models for the addition and
subtraction of integers and other rational numbers.
You might want to use Fluency Practice, pages A5 and A6, to help
students review addition and subtraction with decimals.
Understand
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.
a. Describe situations in which
opposite quantities combine
to make 0. For example, a
hydrogen atom has 0 charge
because its two constituents
are oppositely charged.
b.Understand p 1 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.
c. Understand subtraction of
rational numbers as adding
the additive inverse, p 2 q 5
p 1 (2q). 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.
Connect
Number lines can be used to help students
understand what it means to add integers and other
rational numbers. Review how to use number lines
to model the sums on the page. After reviewing this
page, ask: Why is each sum on this page equal to 0?
28
Common Core State Standards
Students should indicate that each sum requires
the addition of a number and its opposite, which is
always zero.
To connect the concept to procedural knowledge,
introduce the rules for adding signed numbers.
Duplicating any part of this book is prohibited by law.
LE
SS
O
N
CommonCor
eCoa
c
h,
Ma
t
he
ma
t
i
c
s
,
T
e
a
c
he
rE
di
t
i
on,
Gr
a
de7
TRY MP4 Call on one student to show how to use
a number line model and another student to show
how to use the rules for adding signed integers to
show that 4 1 (29) 5 25. Explain that this shows
that sometimes when a positive integer is added
to a negative integer, the result is negative.
Answers may vary. Possible answer:
4
–10 –8 –6 –4 –2
0
2
4
6
8 10
Examples
CHECK
EXAMPLE A This example requires students to
subtract a negative integer from a positive integer,
which results in the addition of two positive integers.
EXAMPLE B This example requires students to
subtract a negative integer from a negative integer,
which results in the addition of two integers with
different signs.
MODEL MP4 Students should rewrite the
subtraction problem as an addition problem and
then draw a number line model to find the answer.
Answers may vary. Possible answer:
23 2 5 5 23 1 (25), so:
3
–10 –8 –6 –4 –2
0
2
4
6
8 10
Problem Solving
Duplicating any part of this book is prohibited by law.
2.6
2.8
–3
–2
–1
0
1
2
3
Distance 5.4
The distance between 2.6 and 22.8
on the number line is 5.4 units.
Is this the same answer you found
above? yes
▸ The range of the temperatures on that day
was 5.4°C.
Practice
(5)
This is the first time that students are subtracting
signed decimals. Use the Solve step to show how
the rules for adding signed numbers can be applied
to decimals. Reinforce that when students know
two methods to solve a certain type of problem,
they can use one method to solve and the other
method to check.
SOLVE
Domain 2
(9)
|2.6 2 (22.8)| 5 |2.6 1 2.8| 5
|5.4| 5 5.4
So, the temperature range in the city
that day was 5.4 degrees Celsius.
As students are working, pay special attention to
problems 11–13, which require students to read
and use thermometers to solve problems involving
the subtracting of signed numbers. If students
struggle with these problems, help them see that
thermometers are just vertical number lines.
For answers, see pages 92 and 93.
Common Errors
Some students may err when rewriting subtraction
problems as addition problems. If so, review their
answers for questions 4–6 to see what types of
errors they are making.
Students may also make errors when finding sums if
they have misconceptions about addition. Years of
adding only whole numbers may have led students
to believe that addition results in a greater number.
This is why number line models are powerful. They
can help students see why adding two negative
numbers actually results in a sum that is less than
either addend.
29
LE
SS
O
N
CommonCor
eCoa
c
h,
Ma
t
he
ma
t
i
c
s
,
S
t
ude
ntE
di
t
i
on,
Gr
a
de7
5
Adding and Subtracting
Rational Numbers
UNDERSTAND Every rational number has a location on the number line. Numbers to
the left of 0 on the number line are negative. Numbers to the right of 0 are
positive. Any numbers n and 2n are opposites, or additive inverses. The
sum of n and 2n is 0.
You can use a number line to add or subtract rational numbers. It may help you to think of the
sum of p 1 q as the number located a distance |q| units from p on the number line.
Find the sum: 23 1 3
Use a number line.
23 is negative, so move 3 units to the left of 0,
as shown by the blue arrow.
To add positive 3 to that number, move 3 units to the right as shown by the green arrow.
The green arrow points to 0, which is the sum.
3
–5 –4 –3 –2 –1 0
1
2
3
4
5
3
▸ 23 1 3 5 0. This makes sense because 23 and 3 are additive inverses.
Matt has a balance of 2$2.50 in his bank account. He adds $2.50. Find his new balance.
The blue arrow shows moving from 0 to a point halfway between 23 and 22. That point
shows 22.50.
The green arrow models adding 2.50 to that amount. The sum is 0.
2.5
–3
–2
–1
0
1
2
3
▸ Matt now has a $0 balance in his bank account. 22.50 and 2.50 are additive inverses. Instead of using a number line to add integers, you can look at the signs of the addends:
• If they have the same sign, add their absolute values and give the sum that sign.
• I f they have opposite signs, subtract the lesser absolute value from the greater absolute
value. Give the sum the sign of the addend with the greater absolute value.
30 Domain 2: The Number System
Duplicating any part of this book is prohibited by law.
2.5
Connect
Use two different methods to find the sum: 26 1 (24)
Use a number line and the rule for adding integers with the same sign.
6
You could use a number line to determine that (26) 1 (24) 5 (210).
Since 26 and 24 have the same sign, you
could also add them like this:
|26| 1 |24| 5 6 1 4 5 10
–10 –8 –6 –4 –2
0
2
4
6
8 10
6
8 10
(4)
Since 26 and 24 are negative,
give the sum a negative sign.
▸ The sum is 210. This shows that (2) 1 (2) 5 (2).
Use two different methods to find the sum: 9 1 (23)
Use a number line and the rule for adding integers with opposite signs.
You could use a number line to determine that (9) 1 (23) 5 6.
Since 9 and 23 have opposite signs,
find their difference.
|9| 5 9
9
–10 –8 –6 –4 –2
0
2
|23| 5 3
4
(3)
92356
The addend 9 has the greater absolute value, so give the sum a positive sign.
▸ The sum is 6. This shows that sometimes (1) 1 (2) 5 (1).
Duplicating any part of this book is prohibited by law.
TRY
Choose one method (number line or rules for adding integers). Use it to find the sum: 4 1 (29). Use this example to show that sometimes (1) 1 (2) 5 (2).
Lesson 5: Adding and Subtracting Rational Numbers 31
EXAMPLE A Find the difference: 4 2 (26)
Rewrite the problem as an addition problem.
The additive inverse of 26 is 6. So, 4 2 (26) 5 4 1 6.
Since 4 and 6 have the same sign, add their absolute values.
|4| 1 |6| 5 4 1 6 5 10
Both addends are positive, so the answer is positive, 10.
▸ 4 2 (26) 5 10
EXAMPLE B Find the difference: 210 2 (26)
Rewrite the problem as an addition problem.
The additive inverse of 26 is 6. So, 210 2 (26) 5 210 1 6
Since 210 and 6 have opposite signs, subtract the lesser absolute value from the greater
absolute value.
|210| . |6| because 10 . 6, so:
|210| 2 |6| 5 10 2 6 5 4
Give the answer the sign of the number with the greater absolute value.
Since 210 has the greater absolute value, the answer is 24.
M O DE L
Draw a model to find the difference: 23 2 5. 32 Domain 2: The Number System
Duplicating any part of this book is prohibited by law.
▸ 210 2 (26) 5 24
Problem Solving
read
The high temperature in a city one day was 2.6°C. The low temperature was 22.8°C. What was the temperature range in the city that day?
plan
The temperature range for the day is equal to the distance between the high
temperature and the low temperature.
The distance between two rational numbers on the number line is equal to the absolute value of their difference. Use absolute value to find the answer.
solve
Find the absolute value of the difference: 2.6 2 (22.8). |2.6 2 (22.8)| 5 |2.6 1
|5|
|5
So, the temperature range in the city that day was
degrees Celsius.
check
One way to check that the distance between 2.6 and 22.8 is actually
the answer you found above is to find the distance between those two numbers on the number line.
Plot points for 2.6 and 22.8 on the number line below. Then count the
units between them to check that your answer is correct.
Duplicating any part of this book is prohibited by law.
–3
–2
–1
0
1
2
The distance between 2.6 and 22.8 on the number line is
3
units. Is this the same answer you found above?
▸ The range of the temperatures on that day was
°C.
Lesson 5: Adding and Subtracting Rational Numbers 33
Practice
Identify the additive inverse, or opposite, of each number.
7
2.
26
H
IN
T
1.
3.
9.53
Change the sign of
each number to find
its opposite.
Rewrite each subtraction expression as an addition expression.
10 2 5
4.
5.
211.1 2 12.6
1
6.


1
1
__
​ 8 ​  2 ​ 21​ __
5 ​   ​
1
1
REMEMBER The expression p 2 q is the
same as p 1 (2q ).
Use the number line to find each sum or difference. Show your work on the number line.
–10 –8 –6 –4 –2
8.
0
2
4
6
8 10
6 2 (24)
9.
–10 –8 –6 –4 –2
34 22 1 10
–10 –8 –6 –4 –2
0
2
4
6
8 10
0
2
4
6
8 10
10. 5 2 11
0
2
Domain 2: The Number System
4
6
8 10
–10 –8 –6 –4 –2
Duplicating any part of this book is prohibited by law.
24 1 (25)
7.
The following thermometers show the range in temperatures in a given day. Find the
temperature ranges. Show or explain your work.
11.
high
temperature
12.
30
20
high
temperature
10
low
temperature
0
–10
low
temperature
Temperature (in °F)
13.
5
4
3
2
1
0
–1
–2
–3
–4
–5
3
high
temperature
2
1
0
–1
low
temperature
Temperature (in °C)
–2
Temperature (in °C)
Choose the best answer.
14. A helium atom has 2 electrons, each of
which has a 21 charge. Each proton has
a 11 charge. Overall, the helium atom
has no charge. How many protons must
the atom have?
A. 22
B. 0
C.   2
D. 3
15. On January 1, Rose’s bank balance was
$200. During the month, she wrote
checks for $115.25 and $350.00 and
made one deposit of $150.50. Which
best represents her checking account
balance at the end of the month?
A. 2$114.75
B. 2$114.25
C.   $114.25
D.   $115.50
Solve.
Duplicating any part of this book is prohibited by law.
16.
Show Ira writes that the sum of a 1 b is
a number that is located a distance of |b| units from a on the number line. Choose four values for a and b and
show that this is true. Use the four number lines on the right to show your work and support your answer.
–5 –4 –3 –2 –1 0
1
2
3
4
5
–5 –4 –3 –2 –1 0
1
2
3
4
5
–5 –4 –3 –2 –1 0
1
2
3
4
5
–5 –4 –3 –2 –1 0
1
2
3
4
5
Lesson 5: Adding and Subtracting Rational Numbers 35