Understanding Integers ~ Lesson 5
Materials
Subtracting Integers
Students will fluently model addition and subtraction of integers
using strips and on a number line.
Students will learn that subtracting integers is exactly the same
as “adding the opposite.”
Students will be able to model and write matching addition and
subtraction expressions and explain why the two expressions are
equivalent.
Student Whiteboards
Smartboard or
chalkboard
Pre-made number line
Pre-made integer
strips
Practice worksheets
Students need to use
strips repeatedly, slowly
Teaching Actions:
moving from concrete to
Tell students they are going to learn to subtract integers. They will abstract examples. This is
expand on the addition model they learned in Lesson 4. Students especially important when
beginning to perform
will complete the following steps for each problem type:
operations with negative
integers.
1. Represent the problem first as if it were an addition problem,
using integer arrow strips (1 x __ or -1 x __ rectangles).
Through the guided
instruction and practice,
2. Represent the problem as a subtraction problem. Since
students will learn the
subtraction is the inverse of addition, when students encounter
this operation, they will “flip” the second arrow in the model, Note: connection between
addition and subtraction
It is the operation that changes the direction/location of the strip
(rectangle).
3. Cancel out zero pairs to find the answer.
Problem Type: Positive  Positive
ASK: What would the addition problem (5 + 2) look like
when modeled with the strips? Model the strip representation
from the addition of integers lesson.
Deliberately use language
that refers to both distance
and area for the strips.
For example, “A distance
of five from zero” and/or
“five squares from zero”
and/or “a magnitude or
value of 5.” Students
naturally relate area to
value more readily than
length to value.
0
07-16-11
Lesson 5
p. 1
Understanding Integers ~ Lesson 5
Reminders:
SAY: We are now going to subtract the integers. Since
subtraction is the inverse of addition, we must flip (change
the direction of) the top arrow.
52
•Each added arrow
begins at the point end of
the previous arrow.
•Negative arrows point
F lip
left.
•Positive arrows point
right.
•Subtraction changes the
direction/location of the
strip/rectangle.
0
DO:
 Starting at zero, place a +5 arrow on the number line.


Place a +2 arrow at the end of +5 arrow. Flip the +2 arrow to
indicate subtraction (the model now shows this as a -2). Be
sure to line up the second arrow so it starts at the end of the
first arrow.
Highlight the fact that the operation changes the direction of
the strip/rectangle.
Ask: What expression does the model now show? 5 + (-2).
0

Cancel out the zero pairs to find the answer.
Explicitly connect the
arrow in the diagram that
flips the direction of the
integer strip to the
subtraction operation. If
students relate the arrow
to subtraction, they will be
much more likely to
differentiate between the
minus sign as an indicator
of direction and as an
operation. This becomes
important when
discussing equivalence of
expressions and when
multiplying or dividing
integers.
0

The result is 3.
Ask: What do you notice about 5 – 2 and 5 + (-2)? They are
the same model. What do you notice about the answer for 52 and 5 + (-2)? They are the same.
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Lesson 5
p. 2
Understanding Integers ~ Lesson 5
DO: Give another problem of this type: 6 – 3
DO: Have students work with a partner where one shows and
shares how to model 6 + (-3) and the other partner shows and
shares how to model 6 - 3.
ASK groups to share what they notice. The pictures are the
same; the expressions are the same; the operations are different;
the second integer changed from being positive to being
negative, etc. Eventually, you want them to understand that
subtracting is the same as adding the opposite.
Problem Type: Negative  Negative
ASK: What would the addition problem [-5 + (-2)] look like
when modeled with the strips? Model the strip representation
from the addition of integers lesson.
0
SAY: We are now going to subtract those integers.
Remember, since subtraction is the inverse of addition, we
must flip (change the direction of) the top arrow.
-5  (-2)
F lip
0
 DO:
 Starting at zero, place a -5 arrow on the number line.


Place a -2 arrow at the end of -5 arrow. Flip the -2 arrow to
indicate subtraction (the model now shows this as a+2). Be
sure to line up the second arrow so it starts at the end of the
first arrow.
Highlight the fact that the operation changes the direction of
the strip/rectangle.
07-16-11
Lesson 5
p. 3
Understanding Integers ~ Lesson 5
Ask: What expression does this model now show? -5 + (+2)
0

Cancel out the zero pairs to find the answer.
0

The result is -3
Ask: What do you notice about -5 – (-2) and -5 + 2? They are
the same model. What do you notice about the answer for
-5-(-2) and -5 + 2? They are the same.
DO: Give another problem of this type: -6 – (-3)
DO: Have students work with a partner where one shows and
shares how to model -6 + 3 and the other partner shows and
shares how to model -6 – (-3).
ASK
groups to share what they notice. The pictures are the same; the
expressions are the same; the operations are different; the
second integer changed from being negative to being positive,
etc. Eventually, you want them to understand that subtracting is
the same as adding the opposite.
Problem Type: Positive  Negative
ASK: What would the addition problem [5 + (-2)] look like
when modeled with the strips? Model the strip representation
from the addition of integers lesson.
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Lesson 5
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Understanding Integers ~ Lesson 5
0
SAY: We are now going to subtract those integers.
Remember, since subtraction is the inverse of addition, we
must flip (change the direction of) the top arrow.
5  (-2)
F lip
0
DO:



Starting at zero, place a +5 arrow on the number line.
Place a -2 arrow at the end of +5 arrow. Flip the -2 arrow to
indicate subtraction (the model now shows this as a+2). Be
sure to line up the second arrow so it starts at the end of the
first arrow.
Highlight the fact that the operation changes the direction of
the strip/rectangle.
Ask: What expression does this model now show? 5 + (+2)
0

•The result is 7.
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Lesson 5
p. 5
Understanding Integers ~ Lesson 5
Ask: What do you notice about 5 – (-2) and 5 + 2? They are
the same model. What do you notice about the answer for
5-(-2) and 5 + 2. They are the same.
DO: Show another example: 6 – (-3)
DO: Have students work with a partner where one shows and
shares how to model 6 + 3 and the other partner shows and
shares how to model 6 – (-3).
ASK groups to share what they notice. The pictures are the
same; the expressions are the same; the operations are different;
the second integer changed from being negative to being
positive, etc. Eventually, you want them to understand that
subtracting is the same as adding the opposite.
Problem Type: Negative  Positive
ASK: What would the addition problem (-5 + 2) look like
when modeled with the strips? Model the strip representation
from the addition of integers lesson.
0
SAY: We are now going to subtract those integers.
Remember, since subtraction is the inverse of addition, we
must flip (change the direction of) the top arrow.
-5 - 2
F lip
0
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Lesson 5
p. 6
Understanding Integers ~ Lesson 5
DO:


Starting at zero, place a 5 arrow on the number line.
Place a +2 arrow at the end of -5 arrow. Flip the +2 arrow to
indicate subtraction (the model now shows this as a-2). Be
sure to line up the second arrow so it starts at the end of the
first arrow.
Highlight the fact that the operation changes the direction of
the strip/rectangle.

Ask: What expression does this model now show? -5 + (-2)
0

The result is -7.
Ask: What do you notice about -5 – 2 and -5 + (-2)? They are
the same model. What do you notice about the answer for
-5 – 2 and -5 + (-2)? They are the same.
DO: Give another problem of this type: -6 – 3
DO: Have students work with a partner where one shows and
shares how to model -6 + (-3) and the other partner shows and
shares how to model -6 – 3.
ASK groups to share what they notice. The pictures are the
same; the expressions are the same; the operations are different;
the second integer changed from being positive to being
negative, etc. They should be able to articulate why subtracting
and adding the opposite are the same thing.
Students should continue to practice these four problem
types until they are fluent with the model. Students should
also practice writing both possible expressions given the
diagram/strips.
The next two types of problems include subtraction across zero.
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Lesson 5
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Understanding Integers ~ Lesson 5
Problem Type: PositivePositive (across zero)
ASK: What would the addition problem (4 +7) look like
when modeled with the strips? Model the strip representation
from the addition of integers lesson.
0
4-7
SAY: We are now going to subtract those integers.
Remember, since subtraction is the inverse of addition, we
must flip (change the direction of) the top arrow.
F lip
0
DO:



Starting at zero, place a +4 arrow on the number line.
Place a +7 arrow at the end of +4 arrow. Flip the +7 arrow
to indicate subtraction (the model now shows this as a -7).
Be sure to line up the second arrow so it starts at the end of
the first arrow.
Highlight the fact that the operation changes the direction of
the strip/rectangle.
Ask: What expression does this model now show? 4 + (-7)
0
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Lesson 5
p. 8
Understanding Integers ~ Lesson 5
 Cancel out the zero pairs to find the answer.
0

•The result is -3.
DO: Have students work with a partner where one models 3 - 6
and the other models 3 + (-6).
ASK students what they know about these 2 expressions and
why. They have the same value, same model, because
subtraction is the same as adding the opposite.
Problem Type: Negative  Negative across zero.
Provide A LOT of practice
with a mix of subtraction
and addition problems to
make sure students are
not mixing up the rules. (A
math walk could be good
for this). Students who
are struggling should talk
through each part of the
problem as they model it
on with the strips with you
or a classmate.
ASK: What would the addition problem (-4 + -7) look like
when modeled with the strips? Model the strip representation
from the addition of integers lesson.
0
- 4 – (-7)
SAY: We are now going to subtract those integers.
Remember, since subtraction is the inverse of addition, we
must flip (change the direction of) the top arrow.
F lip
0
07-16-11
Lesson 5
p. 9
Understanding Integers ~ Lesson 5
DO:



Starting at zero, place a 4 arrow on the number line.
Place a -7 arrow at the end of -4 arrow. Flip the -7 arrow to
indicate subtraction (the model now shows this as a +7). Be
sure to line up the second arrow so it starts at the end of the
first arrow.
Highlight the fact that the operation changes the direction of
the strip/rectangle.
Ask: What expression does this model now show? -4 + (+7)
0
 Cancel out the zero pairs to find the answer.
0

The result is 3.
DO: Have students work with a partner where one models
-
3 – (-6) and the other models -3 + (+6).
ASK students what they know about these 2 expressions and
why. They have the same value, same model, because
subtraction is the same as adding the opposite.
07-16-11
Lesson 5
p. 10
Understanding Integers ~ Lesson 5
Practice & Assessment:
1. Review the following expression types and have
students model them with integer strips.
Positive minus positive
85
Negative minus negative
8 – (5)
Positive minus negative
8 – (5)
Negative minus positive
8  5
Negative minus negative across zero 5 – (8)
Positive minus positive across zero
58
2. Provide models/diagrams for each type of problem
and have students write both possible expressions for
each.
3. Have the students write all possible equivalent
addition and subtraction expressions using the same
two integers and their opposites that result in the same
difference or sum. For example, 7 – 5 = 2, -5 + 7 = 2,
-5 – (-7) = 2, 7 + (-5) = 2.
4. Have the student use diagrams or words to explain
how they know that 5 – (-7) = 2, 7 + (-5) = 2 are
equivalent even though written differently.
5. Have students to create stories to build understanding.
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Lesson 5
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