This lesson teaches students about the relationship between addition and subtraction through the use of tape diagrams and algebraic expressions. Students explore identities such as w - x + x = w by building tape diagrams that represent adding and then subtracting the same amount. They recognize that regardless of the numbers used, adding a number and then subtracting the same number will result in the original amount. Students practice writing number sentences to represent identities and evaluating expressions that involve addition and subtraction of the same terms. The lesson aims to help students understand that identities will always be true no matter what numbers are substituted in.

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NYS COMMON CORE MATHEMATICS CURRICULUM 6•4Lesson 1
Lesson 1: The Relationship of Addition and Subtraction
Student Outcomes
 Students build and clarify the relationship of addition and subtraction by evaluatingidentities such as 𝑤 − 𝑥 +
𝑥 = 𝑤 and 𝑤 + 𝑥 − 𝑥 = 𝑤.
Lesson Notes
Teachers need to create squarepieces 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 squaretiles,these can be used in placeof paper squares.
The template for the squares and other shapes used in the lesson areprovided at the end of the lesson.Teachers will
need to cut out the shapes.
FluencyExercise (5 minutes)
White Board Exchange: Multiplication of Decimals
Classwork
OpeningExercises(5 minutes)
Opening Exercises
a. Draw a tape diagram to represent thefollowing expression: 𝟓 + 𝟒.
Answer:
b. Write an expression for eachtape diagram.
i.
ii.
Discuss theanswers with the class. If students struggled with either Opening Exercise, providemore examples before
moving into the discussion.
𝟓 𝟒+
𝟐 𝟐+
𝟑 𝟒+

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NYS COMMON CORE MATHEMATICS CURRICULUM 6•4Lesson 1
Discussion(15 minutes)
Provideeach 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 usingthe squares provided.

 Add two more squares to your tape diagram.

 Write an expression to represent how we created a tape diagramwith 5 squares.

 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.
 3 + 2 − 2
 Evaluate the expression.
 3
 Let’s starta new diagram. This time, create a tape diagramwith 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.
 6 + 4 − 4
 How many tiles areleft on your desk?
 6
 Evaluate the expression.
 6
 How many tiles did we startwith?
 6
 What effect did adding four tiles then subtractingthe four tiles haveon the number of tiles?
 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 westarted with and then subtract 215 tiles? Do you need
to actually add and remove these tiles to know what the resultwill 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.
3 2+

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NYS COMMON CORE MATHEMATICS CURRICULUM 6•4Lesson 1
 What do you notice about the expressions we created with the tape diagrams?
 Possible Answer: When we add one number and then subtract the same number we get our original
number.
 Write a number sentence, usingvariables,to represent what we justdemonstrated with tape diagrams.
Remember that a variableis a letter that represents a number. Use the shapes provided to create tape
diagrams to demonstrate this number sentence.
Providestudents time to work with their partners to write a number sentence.
 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.
Exercises1–5 (12 minutes)
Students will usetheir knowledge gained in the discussion to create another number sentence usingidentities. Allow
students to continueto work with their partners and the squares.
Exercises
1. Predict what will happen when atape diagram hasalarge number ofsquares, some squares
are removed, but then thesame amountofsquaresare added back on.
PossibleAnswer: When somesquares areremoved from a tapediagram, butthen thesame
amount ofsquares areadded back on, thetapediagramwill endup withthe sameamount
ofsquares that it started with.
2. Build atape diagram with 𝟏𝟎squares.
a. Remove 𝟔ofthem. Write an expressionto represent thetape diagram.
𝟏𝟎− 𝟔
b. Add 𝟔squaresonto the tape diagram. Alter the original expression to representthe currenttape diagram.
𝟏𝟎− 𝟔 + 𝟔
Scaffolding:
The exercisecould be
completed as a class if students
are strugglingwith the
concept.
𝑤
𝑤 + 𝑥 − 𝑥
𝑤 + 𝑥 − 𝑥 = 𝑤
𝑤 + 𝑥
MP.2
MP.7

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NYS COMMON CORE MATHEMATICS CURRICULUM 6•4Lesson 1
c. Evaluate the expression.
𝟏𝟎
3. Write anumber sentence,using variables, to represent theidentitieswe demonstratedwith tapediagrams.
PossibleAnswer: 𝒘 − 𝒙 + 𝒙 = 𝒘
4. Using your knowledge ofidentities, fill ineach of theblanks.
a. 𝟒 + 𝟓 − ____ = 𝟒
𝟓
b. 𝟐𝟓− ____ + 𝟏𝟎 = 𝟐𝟓
𝟏𝟎
c. ____ +𝟏𝟔− 𝟏𝟔 = 𝟒𝟓
𝟒𝟓
d. 𝟓𝟔− 𝟐𝟎+ 𝟐𝟎 =_____
𝟓𝟔
5. Using your knowledge ofidentities, fill ineach oftheblanks.
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 valueof 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 ableto evaluate the final expression even when we did not know the amount we were adding
and subtracting?
Exit Ticket (5 minutes)

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NYS COMMON CORE MATHEMATICS CURRICULUM 6•4Lesson 1
Name Date
Lesson 1: The Relationship of Addition and Subtraction
Exit Ticket
1. Draw tape diagrams to represent each of the followingnumber sentences.
a. 3 + 5 − 5 = 3
b. 8 − 2 + 2 = 8
2. Fill in each blank.
a. 65 + _____ −15 = 65
b. _____+𝑔 − 𝑔 = 𝑘
c. 𝑎 + 𝑏 − _____ = 𝑎
d. 367 − 93 + 93 = _____

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NYS COMMON CORE MATHEMATICS CURRICULUM 6•4Lesson 1
Exit Ticket Sample Solutions
1. Draw a seriesoftape diagramsto represent thefollowing numbersentences.
a. 𝟑 + 𝟓 − 𝟓 = 𝟑
b. 𝟖 − 𝟐 + 𝟐 = 𝟖
2. 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
b. 𝟒𝟓𝟎− 𝟐𝟑𝟎+ 𝟐𝟑𝟎 =_____
𝟒𝟓𝟎
c. 𝟏𝟐𝟖𝟗− _____ + 𝟖𝟓𝟔= 𝟏𝟐𝟖𝟗
𝟖𝟓𝟔
2. Why are the number sentences 𝒘− 𝒙 + 𝒙 = 𝒘and 𝒘+ 𝒙 − 𝒙 = 𝒘called identities?
Possible answer: Thesenumber sentences arecalled identities becausethevariables canbereplaced withany
numbers and thesentences will always betrue.

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NYS COMMON CORE MATHEMATICS CURRICULUM 6•4Lesson 1
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 exerciseshould take5–12 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 erasingtheir work. An economical
recommendation is to placecard stock insidesheet protectors to use as the personal whiteboards and to cut sheets of
felt into small squaresto use as erasers.
It is bestto prepare the problems in a way that allows you to reveal them to the class oneat a time. For example, use a
flip chartor PowerPoint presentation; write the problems on the board and cover with paper beforehand, allowingyou
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 beappropriateto verbally call outthe problem to the students.
The teacher reveals or says the firstproblem in the listand announces,“Go.” Students work the problem on their
personal white boards,holdingtheir answers up for the teacher to see as soon as they have them ready. The teacher
gives immediate feedback to each student, pointingand/or makingeye contactwith the student and respondingwith an
affirmation for correctwork such as,“Good job!”, “Yes!”, or “Correct!” For incorrectwork, respond with guidancesuch
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 isable to go
through each problem in quick succession withoutpausingto go through the solution of each problem individually. If
only one or two students have not been ableto get a given problem correct when the rest of the students are finished,it
is appropriateto move the classforward to the next problem without further delay; in this case,find a time to provide
remediation to that student before the next fluency exerciseon this skill is given.

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NYS COMMON CORE MATHEMATICS CURRICULUM 6•4Lesson 1
𝑤 + 𝑥
𝑤 𝑥
𝑤 𝑥
𝑤 + 𝑥
𝑤 𝑥
𝑤 𝑥

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NYS COMMON CORE MATHEMATICS CURRICULUM 6•4Lesson 1