This lesson introduces students to the relationship between addition and subtraction using tape diagrams and algebraic expressions. Students represent addition and subtraction expressions using tape diagrams with squares and explore how adding and then subtracting the same number results in the original amount. They realize this is true regardless of the specific numbers used. Students then write number sentences with variables to represent these identities, like w + x - x = w. The lesson aims to build students' understanding that identities will be true for any numbers substituted into the variables.

1. Lesson 1
NYS COMMON CORE MATHEMATICS CURRICULUM
6•4
Lesson 1: The Relationship of Addition and Subtraction
Student Outcomes

Students build and clarify the relationship of addition and subtraction by evaluating identities such as 𝑤 − 𝑥 +
𝑥 = 𝑤 and 𝑤 + 𝑥 − 𝑥 = 𝑤.
Lesson Notes
Teachers need to create square pieces of paper in order for students to build tape diagrams. Each pair of students will
need ten squares to complete the activities. If the teacher has square tiles, these can be used in place of paper squares.
The template for the squares and other shapes used in the lesson are provided at the end of the lesson. Teachers will
need to cut out the shapes.
Fluency Exercise (5 minutes)
White Board Exchange: Multiplication of Decimals
Classwork
Opening Exercises (5 minutes)
Opening Exercises
a.
Draw a tape diagram to represent the following expression: 𝟓 + 𝟒.
Answer:
𝟓
b.
+
𝟒
Write an expression for each tape diagram.
i.
𝟐 +
𝟐
𝟑
+
ii.
𝟒
Discuss the answers with the class. If students struggled with either Opening Exercise, provide more examples before
moving into the discussion.
Lesson 1:
Date:
© 2013 Common Core, Inc. Some rights reserved. commoncore.org
The Relationship of Addition and Subtraction
3/7/14
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2. Lesson 1
NYS COMMON CORE MATHEMATICS CURRICULUM
6•4
Discussion (15 minutes)
Provide each pair of students with a collection of 10 squares so they can use these squares to create tape diagrams
throughout the lesson.

If each of the squares represents 1 unit, represent the number 3 using the squares provided.


Add two more squares to your tape diagram.

Write an expression to represent how we created a tape diagram with 5 squares.


3 + 2

Remove two squares from the tape diagram.


Alter our original expression 3 + 2 to create an expression that represents what we did with the tape diagram.


Evaluate the expression.


3+2−2
3
Let’s start a new diagram. This time, create a tape diagram with 6 squares.


Use your tiles to demonstrate 6 + 4.


Remove 4 squares from the tape diagram.

Alter our original expression 6 + 4 to create an expression to represent the tape diagram.



How many tiles are left on your desk?


6
What effect did adding four tiles then subtracting the four tiles have on the number of tiles?


6
How many tiles did we start with?


6
Evaluate the expression.


6+4−4
Adding and then subtracting the same number of tiles resulted in the same number that we started
with.
What if I asked you to add 215 tiles to the six tiles we started with and then subtract 215 tiles? Do you need
to actually add and remove these tiles to know what the result will be? Why is that?

We do not actually need to do the addition and subtraction because we now know that it will result in
the same amount of tiles that we started with.
Lesson 1:
Date:
© 2013 Common Core, Inc. Some rights reserved. commoncore.org
The Relationship of Addition and Subtraction
3/7/14
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15

3. Lesson 1
NYS COMMON CORE MATHEMATICS CURRICULUM

What do you notice about the expressions we created with the tape diagrams?


6•4
Possible Answer: When we add one number and then subtract the same number we get our original
number.
Write a number sentence, using variables, to represent what we just demonstrated with tape diagrams.
Remember that a variable is a letter that represents a number. Use the shapes provided to create tape
diagrams to demonstrate this number sentence.
Provide students time to work with their partners to write a number sentence.
MP.2


Possible answer: 𝑤 + 𝑥 − 𝑥 = 𝑤. Emphasize that both 𝑤’s represent the same number and the same
rule applies to the 𝑥’s.
Why is the number sentence 𝑤 + 𝑥 − 𝑥 = 𝑤 called an identity?

Possible answer: The number sentence is called an identity because the variables can be replaced with
any numbers and the sentence will always be true.
𝑤
𝑤+ 𝑥
𝑤+ 𝑥− 𝑥
𝑤+ 𝑥− 𝑥
=
𝑤
Exercises 1–5 (12 minutes)
Students will use their knowledge gained in the discussion to create another number sentence using identities. Allow
students to continue to work with their partners and the squares.
Scaffolding:
Exercises
1.
MP.7
Predict what will happen when a tape diagram has a large number of squares, some squares
are removed, but then the same amount of squares are added back on.
Possible Answer: When some squares are removed from a tape diagram, but then the same
amount of squares are added back on, the tape diagram will end up with the same amount
of squares that it started with.
2.
The exercise could be
completed as a class if students
are struggling with the
concept.
Build a tape diagram with 𝟏𝟎 squares.
a.
Remove 𝟔 of them. Write an expression to represent the tape diagram.
𝟏𝟎 − 𝟔
b.
Add 𝟔 squares onto the tape diagram. Alter the original expression to represent the current tape diagram.
𝟏𝟎 − 𝟔 + 𝟔
Lesson 1:
Date:
© 2013 Common Core, Inc. Some rights reserved. commoncore.org
The Relationship of Addition and Subtraction
3/7/14
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Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
16

4. Lesson 1
NYS COMMON CORE MATHEMATICS CURRICULUM
c.
6•4
Evaluate the expression.
𝟏𝟎
3.
Write a number sentence, using variables, to represent the identities we demonstrated with tape diagrams.
Possible Answer: 𝒘 − 𝒙 + 𝒙 = 𝒘
4.
Using your knowledge of identities, fill in each of the blanks.
a.
𝟒 + 𝟓 − ____ = 𝟒
𝟓
b.
𝟐𝟓 − ____ + 𝟏𝟎 = 𝟐𝟓
𝟏𝟎
c.
____ +𝟏𝟔 − 𝟏𝟔 = 𝟒𝟓
𝟒𝟓
d.
𝟓𝟔 − 𝟐𝟎 + 𝟐𝟎 =_____
𝟓𝟔
5.
Using your knowledge of identities, fill in each of the blanks.
a.
𝒂 + 𝒃 − _____ = 𝒂
𝒃
b.
𝒄 − 𝒅 + 𝒅 = _____
𝒄
c.
𝒆 + _____ − 𝒇 = 𝒆
𝒇
d.
_____ −𝒉 + 𝒉 = 𝒈
𝒈
Closing (3 minutes)

In every problem we did today, why did the final value of the expression equal the initial expression?

Initially, we added an amount and then subtracted the same amount. Later in the lesson, we subtracted an
amount and then added the same amount. Did this alter the outcome?

Why were we able to evaluate the final expression even when we did not know the amount we were adding
and subtracting?
Exit Ticket (5 minutes)
Lesson 1:
Date:
© 2013 Common Core, Inc. Some rights reserved. commoncore.org
The Relationship of Addition and Subtraction
3/7/14
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17

5. Lesson 1
NYS COMMON CORE MATHEMATICS CURRICULUM
Name
6•4
Date
Lesson 1: The Relationship of Addition and Subtraction
Exit Ticket
1.
Draw tape diagrams to represent each of the following number sentences.
a.
b.
2.
3+5−5=3
8−2+2=8
Fill in each blank.
a.
65 + _____ −15 = 65
b.
_____+𝑔 − 𝑔 = 𝑘
c.
𝑎 + 𝑏 − _____ = 𝑎
d.
367 − 93 + 93 = _____
Lesson 1:
Date:
© 2013 Common Core, Inc. Some rights reserved. commoncore.org
The Relationship of Addition and Subtraction
3/7/14
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18

6. Lesson 1
NYS COMMON CORE MATHEMATICS CURRICULUM
6•4
Exit Ticket Sample Solutions
1.
Draw a series of tape diagrams to represent the following number sentences.
a.
b.
2.
𝟑+ 𝟓− 𝟓= 𝟑
𝟖− 𝟐+ 𝟐= 𝟖
Fill in each blank.
a.
𝟔𝟓 + _____ −𝟏𝟓 = 𝟔𝟓
𝟏𝟓
b.
____ +𝒈 − 𝒈 = 𝒌
𝒌
c.
𝒂 + 𝒃 − _____= 𝒂
𝒃
d.
𝟑𝟔𝟕 − 𝟗𝟑 + 𝟗𝟑 = _____
𝟑𝟔𝟕
Problem Set Sample Solutions
1.
Fill in each blank
a.
_____+ 𝟏𝟓 − 𝟏𝟓 = 𝟐𝟏
𝟐𝟏
2.
b.
𝟒𝟓𝟎 − 𝟐𝟑𝟎 + 𝟐𝟑𝟎 =_____
c.
𝟏𝟐𝟖𝟗 − _____ + 𝟖𝟓𝟔 = 𝟏𝟐𝟖𝟗
𝟒𝟓𝟎
𝟖𝟓𝟔
Why are the number sentences 𝒘 − 𝒙 + 𝒙 = 𝒘 and 𝒘 + 𝒙 − 𝒙 = 𝒘 called identities?
Possible answer: These number sentences are called identities because the variables can be replaced with any
numbers and the sentences will always be true.
Lesson 1:
Date:
© 2013 Common Core, Inc. Some rights reserved. commoncore.org
The Relationship of Addition and Subtraction
3/7/14
This work is licensed under a
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
19

7. Lesson 1
NYS COMMON CORE MATHEMATICS CURRICULUM
6•4
White Board Exchange: Multiplication of Decimals
Progression of Exercises:
Answers:
1.
0.5 × 0.5
0.25
2.
0.6 × 0.6
0.36
3.
0.7 × 0.7
0.49
4.
0.5 × 0.6
0.3
5.
1.5 × 1.5
2.25
6.
2.5 × 2.5
6.25
7.
0.25 × 0.25
0.0625
8.
0.1 × 0.1
0.01
9.
0.1 × 123.4
12.34
10. 0.01 × 123.4
1.234
Fluency work such as this exercise should take 5–12 minutes of class.
How to Conduct a White Board Exchange:
All students will need a personal white board, white board marker, and a means of erasing their work. An economical
recommendation is to place card stock inside sheet protectors to use as the personal white boards and to cut sheets of
felt into small squares to use as erasers.
It is best to prepare the problems in a way that allows you to reveal them to the class one at a time. For example, use a
flip chart or PowerPoint presentation; write the problems on the board and cover with paper beforehand, allowing you
to reveal one at a time; or, write only one problem on the board at a time. If the number of digits in the problem is very
low (e.g., 12 divided by 3), it may also be appropriate to verbally call out the problem to the students.
The teacher reveals or says the first problem in the list and announces, “Go.” Students work the problem on their
personal white boards, holding their answers up for the teacher to see as soon as they have them ready. The teacher
gives immediate feedback to each student, pointing and/or making eye contact with the student and responding with an
affirmation for correct work such as, “Good job!”, “Yes!”, or “Correct!” For incorrect work, respond with guidance such
as “Look again!”, “Try again!”, or “Check your work!”
If many students have struggled to get the answer correct, go through the solution of that problem as a class before
moving on to the next problem in the sequence. Fluency in the skill has been established when the class is able to go
through each problem in quick succession without pausing to go through the solution of each problem individually. If
only one or two students have not been able to get a given problem correct when the rest of the students are finished, it
is appropriate to move the class forward to the next problem without further delay; in this case, find a time to provide
remediation to that student before the next fluency exercise on this skill is given.
Lesson 1:
Date:
© 2013 Common Core, Inc. Some rights reserved. commoncore.org
The Relationship of Addition and Subtraction
3/7/14
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Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
20

8. Lesson 1
NYS COMMON CORE MATHEMATICS CURRICULUM
6•4
𝑤+ 𝑥
𝑤
𝑥
𝑤
𝑥
𝑤+ 𝑥
𝑤
𝑥
𝑤
𝑥
Lesson 1:
Date:
© 2013 Common Core, Inc. Some rights reserved. commoncore.org
The Relationship of Addition and Subtraction
3/7/14
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Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
21

9. Lesson 1
NYS COMMON CORE MATHEMATICS CURRICULUM
Lesson 1:
Date:
© 2013 Common Core, Inc. Some rights reserved. commoncore.org
6•4
The Relationship of Addition and Subtraction
3/7/14
This work is licensed under a
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
22