1. Lesson 5: The Oppositeofa Number’s Opposite
Date: 1/22/15 45
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NYS COMMON CORE MATHEMATICS CURRICULUM 6•3Lesson 5
Lesson 5: The Opposite of a Number’s Opposite
Student Outcomes
 Students understand that, for instance,the oppositeof −5 is denoted −(−5) and is equal to 5. In general,
they know that the opposite of the opposite is the original number;e.g., −(−𝑎) = 𝑎.
 Students locateand position oppositenumbers on a number line.
Classwork
OpeningExercises(7 minutes)
Students work independently for 5 minutes to complete the followingreview problems.
Opening Exercises
1. Locate the number −𝟐and itsopposite onthe number line below.
2. Write an integer that representseachofthe following:
a. 𝟗𝟎feet below sealevel
−𝟗𝟎
b. $𝟏𝟎𝟎of debt
−𝟏𝟎𝟎
c. 𝟐℃above zero
𝟐
3. Joe isat the ice cream shop and hishouse is 𝟏𝟎blocksnorth ofthe shop. The park is 𝟏𝟎blockssouth ofthe ice
cream shop. When he isat the icecream shop,isJoe closerto the park or hishouse? How could thenumber zero
be used in thissituation? Explain.
Heis thesamedistancefrom his houseandthepark becausebothare located 𝟏𝟎blocks away fromtheicecream
shop, but in oppositedirections. In this situation, zero represents thelocationoftheicecreamshop.
-8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8

2. Lesson 5: The Oppositeofa Number’s Opposite
Date: 1/22/15 46
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NYS COMMON CORE MATHEMATICS CURRICULUM 6•3Lesson 5
Example 1 (8 minutes): The Opposite of an Opposite of a Number
What is the opposite of the oppositeof 8? How can we illustratethis number on a number line?
 Before startingthe example, allowstudents to discuss their predictionsin groups for one minute. Choose a
few students to sharetheir responses with the rest of the class.
Example 1: The Opposite ofan Opposite ofaNumber
What isthe opposite ofthe oppositeof 𝟖? How can we illustratethisnumber on anumber line?
a. What number is 𝟖unitsto the rightof0? 𝟖
b. How can you illustrate locating theoppositeof 𝟖on thisnumberline? What isthe oppositeof 𝟖? −𝟖
c. Use the same processto locatetheopposite of −𝟖. What isthe opposite of −𝟖? 𝟖
d. The opposite ofan opposite ofanumber is thenumber itself .
Example 2 (8 minutes): Writingthe Opposite of an Opposite of a Number
In this example, students will usethe “−” symbol to indicate“the opposite of a number.” Teacher will display thetask,
“Explain why −(−5) = 5” on the board. A “−” symbol means “the oppositeof a number.”
−(−5) = 5
Sincethe oppositeof 5 is negative 5, and the oppositeof negative 5 is positive5,then −(−5) = 5.
Pose the followingquestions to the class to check for understanding.
 What is the opposite of negative six?
 6
 What is the opposite of the oppositeof 10?
 10
 How would you write the opposite of the oppositeof −12?
 The opposite of −12: −(−12) = 12. The opposite of 12: −(12) = −12. So, −(12) = −12
 What does a “−” symbol mean?
 It can mean the opposite of a number.
 What is the opposite of the oppositeof a debit of $12?
 A debit of $12 is −12. The opposite of −12 is 12, or a credit of $12. The opposite of a credit of $12 is
a debit of $12.
 In general, the oppositeof the opposite of a number is the original number; e.g., −(−𝑎) = 𝑎.
-8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8
Scaffolding:
For visual learners,usethe
curved arrows initially to
develop conceptual
understandingof how to find
the opposite of an opposite of
a number.

3. Lesson 5: The Oppositeofa Number’s Opposite
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NYS COMMON CORE MATHEMATICS CURRICULUM 6•3Lesson 5
Exercise (12 minutes)
Students work in groups of 3–4 to find the oppositeof an oppositeof a set of given numbers. Allow4–5 minutes for
review as a whole class. Distributeeach group member two cards with an integer on it. (Use sticky notes or index
cards.) Students will complete the table usingall thecards in their group.
Exercise
Complete thetableusing thecardsin your group.
Person Card (𝒂) Opposite ofCard (−𝒂) Opposite ofOpposite ofCard −(𝒂)
Jackson 𝟒 −( 𝟒)= −𝟒 −(−𝟒)= 𝟒
DeVonte 𝟏𝟓𝟎 −(𝟏𝟓𝟎)= −𝟏𝟓𝟎 −(−𝟏𝟓𝟎)= 𝟏𝟓𝟎
Cheryl −𝟔 −(−𝟔)= 𝟔 −(𝟔)= −𝟔
Toby −𝟗 −(−𝟗)= 𝟗 −(𝟗)= −𝟗
1. Write the opposite oftheopposite of −𝟏𝟎asan equation.
The oppositeof −𝟏𝟎: −(−𝟏𝟎)= 𝟏𝟎; theoppositeof 𝟏𝟎: −(𝟏𝟎)= −𝟏𝟎. Therefore, (−(−(−𝟏𝟎))) = −𝟏𝟎.
2. In general, the oppositeofthe oppositeofanumber isthe original number.
3. Provide areal-world example ofthisrule. Show your work.
The oppositeoftheoppositeof 𝟏𝟎𝟎feet below sea level is 𝟏𝟎𝟎feet below sea level.
−𝟏𝟎𝟎is 𝟏𝟎𝟎feet below sea level
−(−𝟏𝟎𝟎)= 𝟏𝟎𝟎, theoppositeof −𝟏𝟎𝟎
−(𝟏𝟎𝟎) = −𝟏𝟎𝟎, theoppositeof 𝟏𝟎𝟎
Closing(3 minutes)
 What is the opposite of an opposite of a number? Support your answer with an example.
 The opposite of an opposite of a number is the original number. The opposite of the opposite of
negative 6 is negative 6 because the opposite of −6 is 6. The opposite of 6 is −6.
 What is the relationship between the location of a nonzero number on the number lineand the location of its
oppositeon the number line?
 A number and its opposite are located the same distance from 0 𝑜𝑛 𝑎 𝑛𝑢𝑚𝑏𝑒𝑟 𝑙𝑖𝑛𝑒, but on opposite
sides of 0.
Exit Ticket (7 minutes)

4. Lesson 5: The Oppositeofa Number’s Opposite
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NYS COMMON CORE MATHEMATICS CURRICULUM 6•3Lesson 5
Name ___________________________________________________ Date____________________
Lesson 5: The Opposite of a Number’s Opposite
Exit Ticket
1. Jane completes several example problems that ask her to the find the opposite of the oppositeof a number, and for
each example, the resultis a positivenumber. Jane concludes that when she takes the oppositeof the oppositeof
any number, the resultwill alwaysbepositive. Is Jane correct? Why or why not?
2. To supportyour answer from the previous question,create an example, written as an equation. Illustrateyour
example on the number linebelow.

5. Lesson 5: The Oppositeofa Number’s Opposite
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NYS COMMON CORE MATHEMATICS CURRICULUM 6•3Lesson 5
Exit Ticket Sample Solutions
1. Jane completesseveralexample problemsthat ask her to thefind theopposite oftheopposite ofanumber, and for
each example, theresult isapositive number. Jane concludesthat when she takestheoppositeofthe oppositeof
any number, the resultwill alwaysbe positive. IsJane correct? Why or why not?
Sheis not correct. Theoppositeoftheoppositeofa number is thenumber itself. So, ifJanestarts with a negative
number, shewill end with a negativenumber.
2. To support your answer from the previousquestion, createan example,written asan equation. Illustrateyour
example on the numberlinebelow.
If Janestarts with −𝟕, theoppositeoftheoppositeof−𝟕is written as −(−(−𝟕)) = −𝟕or theoppositeof −𝟕:
−(−𝟕) = 𝟕; theoppositeof 𝟕: −( 𝟕) = −𝟕.
Problem Set Sample Solutions
1. Read each description carefully and writean equation that representsthedescription.
a. The opposite ofnegative seven.
−(−𝟕)= 𝟕
b. The opposite ofthe opposite oftwenty-five.
−(−𝟐𝟓) = 𝟐𝟓
c. The opposite offifteen.
−(𝟏𝟓) = −𝟏𝟓
d. The opposite ofnegative thirty-six.
−(−𝟑𝟔) = 𝟑𝟔
−𝟕 𝟎 𝟕

6. Lesson 5: The Oppositeofa Number’s Opposite
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NYS COMMON CORE MATHEMATICS CURRICULUM 6•3Lesson 5
2. Jose graphed the oppositeofthe opposite of 𝟑on the number line. First,he graphed point 𝑷on thenumber line 𝟑
unitsto the right ofzero. Next, hegraphed the oppositeof 𝑷, 𝟑unitsto the left ofzero and labeledit 𝑲. Finally, he
graphed the oppositeof 𝑲and labeledit 𝑸.
a. Is hisdiagram correct? Ifnot, explain hiserror and correctly locate and label point 𝑸.
Yes, his diagram is correct. It shows that 𝑷is 𝟑becauseit is 𝟑units to theright ofzero. Theoppositeof 𝟑is
−𝟑, which is point 𝑲(𝟑units to theleft ofzero). Theopposite of−𝟑is 𝟑, so point 𝑸 is 𝟑units to theright of
zero.
b. Write the relationship between the points:
𝑷and 𝑲 They areopposites.
𝑲and 𝑸 They areopposites.
𝑷and 𝑸 They arethesame.
3. Read each real-world description. Writetheintegerthat representstheopposite ofthe opposite. Show your work
to support your answer.
a. A temperaturerise of 𝟏𝟓degreesFahrenheit.
−𝟏𝟓is theoppositeof 𝟏𝟓(fall in temperature)
𝟏𝟓is theoppositeof −𝟏𝟓(risein temperature)
−(−𝟏𝟓) = 𝟏𝟓
b. A gain of 𝟓𝟓 yards.
−𝟓𝟓is theoppositeof 𝟓𝟓(loss ofyards)
𝟓𝟓is theoppositeof −𝟓𝟓(gain ofyards)
−(−𝟓𝟓) = 𝟓𝟓
c. A lossof 𝟏𝟎pounds.
𝟏𝟎is theoppositeof −𝟏𝟎(gain of pounds)
−𝟏𝟎is theoppositeof 𝟏𝟎(loss ofpounds)
−( 𝟏𝟎) = −𝟏𝟎
d. A withdrawal of $𝟐,𝟎𝟎𝟎.
𝟐, 𝟎𝟎𝟎is theoppositeof−𝟐,𝟎𝟎𝟎(deposit)
−𝟐, 𝟎𝟎𝟎is theoppositeof 𝟐, 𝟎𝟎𝟎(withdrawal)
−(𝟐,𝟎𝟎𝟎) = −𝟐,𝟎𝟎𝟎
4. Write the integerthat representsthestatement. Locateand labeleach pointon the number line below.
a. The opposite ofagain of 𝟔. __−𝟔_
b. The opposite ofadeposit of $𝟏𝟎. _−𝟏𝟎
c. The opposite ofthe opposite of 𝟎. __ 𝟎
d. The opposite ofthe opposite of 𝟒. ___𝟒
e. The opposite ofthe opposite ofalossof 𝟓. __−𝟓
0
K
Q
P
−𝟏𝟎 −𝟔 −𝟓 𝟎 𝟒