1. Lesson 11: Absolute Value—Magnitude andDistance
Date: 2/3/15 98
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NYS COMMON CORE MATHEMATICS CURRICULUM 6•3Lesson 11
Lesson 11: Absolute Value—Magnitude and Distance
Student Outcomes
 Students understand the absolutevalueof a number as its distancefromzero on the number line.
 Students use absolutevalueto find the magnitude of a positiveor negative quantity in a real-world situation.
Classwork
OpeningExercise (4 minutes)
For this warm-up exercise, students work individually to record two different rational numbers thatare the same
distancefrom zero. Students find as many examples as possible,and reach a conclusion about what must be true for
every pair of numbers that liethat same distancefrom zero.
Opening Exercise
After two minutes:
 What aresome examples you found (pairs of numbers that arethe samedistancefrom zero)?
 Answers may vary from the following examples: −
1
2
and
1
2
, 8.01 and −8.01, −7 and 7, etc.
 What is the relationship between each pair of numbers?
 They are opposites.
 How do each pair of numbers relate to zero?
 Both numbers in each pair are the same distance from zero.
Discussion(3 minutes)
We justsawthat every number and it oppositeare the same distancefrom zero on the
number line. The absolute value of a number is the distancebetween the number and
zero on a number line.
In other words, a number and its opposite have the same absolute value.
 What is the absolutevalueof 5? 5.
 What is the absolutevalueof −5? 5.
 Both 5 and −5 are five units from zero.
Scaffolding:
Providestudents with a
number lineso they can
physically countthe number of
units between a number and
zero.
MP.8

2. Lesson 11: Absolute Value—Magnitude andDistance
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NYS COMMON CORE MATHEMATICS CURRICULUM 6•3Lesson 11
 What is the absolutevalueof −1?
 1
 What is the absolutevalueof 0?
 0
Example 1 (3 minutes): The Absolute Value of a Number
Example 1: The Absolute ValueofaNumber
The absolute value often iswritten as: |𝟏𝟎|. On the number line,countthenumber ofunitsfrom 𝟏𝟎to 𝟎. How many
unitsis 𝟏𝟎from 0? | 𝟏𝟎|= 𝟏𝟎
What other number hasan absolutevalue of 𝟏𝟎? Why?
|− 𝟏𝟎| = 𝟏𝟎because −𝟏𝟎is 𝟏𝟎units from zero.
Exercises1–3 (4 minutes)
Exercises1–3
Complete thefollowing chart.
Number
Absolute
Value
Number Line Diagram
Different
Number with
the same
Absolute Value
1. −𝟔 |− 𝟔| = 𝟔 𝟔
2. 𝟖 |𝟖| = 𝟖 −𝟖
3. −𝟏 |− 𝟏| = 𝟏 𝟏
1 2 3 4 5 6 7 8 9 10
The absolute value ofanumber isthe distancebetween thenumber and zeroon thenumber line.

3. Lesson 11: Absolute Value—Magnitude andDistance
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NYS COMMON CORE MATHEMATICS CURRICULUM 6•3Lesson 11
Example 2 (3 minutes): UsingAbsolute Value to FindMagnitude
The magnitude of a quantity is found by takingthe absolutevalueof its numerical part.
Example 2: Using Absolute Value to Find Magnitude
Mrs. Owensreceivedacall from her bank because shehad acheckbook balanceof −𝟒𝟓dollars. What wasthe magnitude
of the amount overdrawn?
|− 𝟒𝟓| = 𝟒𝟓 Mrs. Owens overdrew her checking account by $𝟒𝟓.
Exercises4–8 (6 minutes)
Exercises4–8
For each scenario below, use absolutevalue to determine the magnitudeofeach quantity.
4. Mariawas sick with the flu and her weight changeasaresult ofit isrepresented by −𝟒pounds. How much weight
did Marialose?
|− 𝟒|= 𝟒 Maria lost 𝟒pounds.
5. Jeffrey oweshisfriend $𝟓. How much isJeffrey’sdebt?
|− 𝟓|= 𝟓 Jeffrey has a $𝟓debt.
6. The elevation ofNiagaraFalls, which islocated betweenLake Erie and LakeOntario, is 𝟑𝟐𝟔feet. How far isthis
above sealevel?
|𝟑𝟐𝟔|= 𝟑𝟐𝟔 It is 𝟑𝟐𝟔feet abovesea level.
7. How far below zero is −𝟏𝟔degreesCelsius?
|− 𝟏𝟔| = 𝟏𝟔 −𝟏𝟔°𝑪is 𝟏𝟔degrees below zero.
8. Frank received amonthly statement for hiscollege savingsaccount. It listed adeposit of $𝟏𝟎𝟎as+𝟏𝟎𝟎.𝟎𝟎. It
listed awithdrawal of $𝟐𝟓as−𝟐𝟓.𝟎𝟎. The statement showed an overall ending balance of $𝟖𝟑𝟓.𝟓𝟎. How much
money did Frank add to hisaccount that month? How much didhe take out? What isthe total amount Frank has
saved for college?
|𝟏𝟎𝟎|= 𝟏𝟎𝟎 Frank added $𝟏𝟎𝟎to his account.
|− 𝟐𝟓| = 𝟐𝟓 Frank took $𝟐𝟓out ofhis account.
|𝟖𝟑𝟓.𝟓𝟎|= 𝟖𝟑𝟓.𝟓𝟎 The total amount ofFrank’s savings for collegeis $𝟖𝟑𝟓.𝟓𝟎.
The magnitude ofaquantity isfound by taking the absolutevalueofitsnumerical part
MP.6

4. Lesson 11: Absolute Value—Magnitude andDistance
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NYS COMMON CORE MATHEMATICS CURRICULUM 6•3Lesson 11
Exercises9–19 (13 minutes)
Students work independently for 8–10 minutes. Allow3–5 minutes to go over the answers as a whole group.
Exercises9–19
9. Meg isplaying acard game with her friendIona. The cardshave positive and negativenumbers printed on them.
Meg exclaims: “The absolutevalue ofthe number onmy card equals 𝟖!” What isthe number on Meg’scard?
|− 𝟖|= 𝟖 or |𝟖| = 𝟖 Meg either has an 𝟖or a −𝟖on her card.
10. List apositive and negative numberwhose absolute valueisgreater than 𝟑. Explain how to justify your answer
using the number line.
Answers may vary. | − 𝟒| = 𝟒 and |𝟕| = 𝟕; 𝟒 > 𝟑and 𝟕 > 𝟑 . On a number linethedistancefrom zero to
−𝟒 is 𝟒units. So theabsolutevalueof −𝟒is 𝟒. Thenumber 𝟒is to theright of 𝟑on thenumber line, so 𝟒is greater
than 𝟑. Thedistancefrom zero to 𝟕on a number lineis 𝟕units, so theabsolutevalueof 𝟕is 𝟕. Since 𝟕is to theright
of 𝟑 on thenumber line, 𝟕is greater than 𝟑.
11. Which ofthe following situationscan berepresented by theabsolutevalue of 𝟏𝟎? Check all that apply.
The temperatureis 𝟏𝟎degreesbelow zero. Expressthisasan integer.
X Determinethe sizeofHarold’sdebtifhe owes $𝟏𝟎.
X Determinehow far −𝟏𝟎isfrom zero on anumber line.
X 𝟏𝟎degreesishow many degreesabove zero?
12. Juliaused absolute value tofind thedistancebetween 𝟎and 𝟔on anumber line. She then wroteasimilar
statement to represent thedistancebetween 𝟎and −𝟔. Below isherwork. Isit correct? Explain.
|𝟔| = 𝟔 and | − 𝟔| = −𝟔
No. The distanceis 𝟔units whether you go from 𝟎to 𝟔or 𝟎to −𝟔. So theabsolutevalueof−𝟔shouldalso be 𝟔,
but Julia said it was −𝟔.
13. Use absolute value to representtheamount,in dollars, ofa $𝟐𝟑𝟖.𝟐𝟓profit.
| 𝟐𝟑𝟖.𝟐𝟓|= 𝟐𝟑𝟖.𝟐𝟓
14. Judy lost 𝟏𝟓pounds. Use absolute value to represent the number ofpoundsJudy lost.
|− 𝟏𝟓| = 𝟏𝟓
15. In math class, Carl and Angelaare debating aboutintegersand absolute value. Carl said two integerscan have the
same absolute value and Angelasaid oneinteger can have two absolute values. Who isright? Defend your answer.
Carl is right. An integer and its oppositearethe samedistancefrom zero. So, they havethe sameabsolutevalues
becauseabsolutevalueis thedistancebetween thenumber and zero.

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NYS COMMON CORE MATHEMATICS CURRICULUM 6•3Lesson 11
16. Jamie told hismath teacher: “Give meany absolute value, and I can tell you two numbersthat have that absolute
value.” Is Jamie correct? For any given absolutevalue,will therealwaysbe two numbersthat have that absolute
value?
No, Jamieis not correctbecausezero is its own opposite. Only onenumber has an absolutevalue of 𝟎, and that
would be 𝟎.
17. Use a number lineto show why anumber and itsoppositehave thesame absolutevalue.
A number and its oppositearethesamedistancefrom zero, but on oppositesides. An exampleis 𝟓and −𝟓. These
numbers areboth 𝟓units from zero. Their distanceis thesame, so they havethesameabsolutevalue, 𝟓.
18. A bank teller assisted two customerswith transactions. One customer made a $𝟐𝟓.𝟎𝟎withdrawal from asavings
account. The other customer madea $𝟏𝟓deposit. Use absolutevalue to show the size ofeach transaction. Which
transaction involved more money?
|− 𝟐𝟓| = 𝟐𝟓 and |𝟏𝟓| = 𝟏𝟓 The$𝟐𝟓withdrawal involved moremoney.
19. Which isfarther from zero: −𝟕
𝟑
𝟒
or 𝟕
𝟏
𝟐
? Use absolute value to defendyour answer.
The number that isfarther from 0 is −𝟕
𝟑
𝟒
. Thisisbecause | − 𝟕
𝟑
𝟒
| = 𝟕
𝟑
𝟒
and |𝟕
𝟏
𝟐
| = 𝟕
𝟏
𝟐
. Absolutevalueis a
number’s distancefrom zero. I compared theabsolutevalueofeach number to determinewhich was farther from
zero. Theabsolutevalueof −𝟕
𝟑
𝟒
is 𝟕
𝟑
𝟒
. Theabsolutevalueof 𝟕
𝟏
𝟐
is 𝟕
𝟏
𝟐
. Weknow that 𝟕
𝟑
𝟒
is greater than 𝟕
𝟏
𝟐
.
Closing(3 minutes)
 I am thinkingof two numbers. Both numbers have the same absolutevalue. Whatmust be true about the two
numbers?
 The numbers are opposites.
 Can the absolutevalueof a number ever be a negative number? Why or why not?
 No. Absolute value is the distance a number is from zero. If you count the number of units from zero to
the number, the number of units is its absolute value. You could be on the right or left side of zero, but
the number of units you count represents the distance or absolute value, and that will always be a
positive number.
 How can we use absolutevalueto determine magnitude? For instance,how far below zero is −8 degrees?
 Absolute value represents magnitude. This means that −8 degrees is 8 units below zero.
Exit Ticket (6 minutes)
-8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8

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NYS COMMON CORE MATHEMATICS CURRICULUM 6•3Lesson 11
Name ___________________________________________________ Date____________________
Lesson 11: Absolute Value—Magnitude and Distance
Exit Ticket
1. Jessieand his family droveup to a picnic area on a mountain. In the morning, they followed a trail thatled to the
mountain summit, which was 2,000 feet above the picnic area. They then returned to the picnic area for lunch.
After lunch,they hiked on a trail thatled to the mountain overlook, which was 3,500 feet below the picnic area.
a. Locate and label the elevation of the mountain summit and mountain overlook on a vertical number line. The
picnic area represents zero. Writea rational number to represent each location.
picnic area: 0
mountain summit:
mountain overlook:
b. Use absolutevalueto represent the distance on the number lineof each location fromthe
picnic area.
Distancefrom the picnic area to the mountain summit:
Distancefrom the picnic area to the mountain overlook:
c. What is the distancebetween the elevations of the summit and overlook? Use absolutevalue
and your number linefrom part (a) to explain your answer.

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NYS COMMON CORE MATHEMATICS CURRICULUM 6•3Lesson 11
Exit Ticket Sample Solutions
1. Jessie and hisfamily drove up to apicnicareaon amountain. In the morning,they followed atrail that ledto the
mountain summit, which was 𝟐,𝟎𝟎𝟎feet above thepicnicarea. They thenreturned to thepicnicareafor lunch.
After lunch, they hikedon atrail that ledto the mountain overlook, which was 𝟑,𝟓𝟎𝟎feet below the picnic area.
a. Locate and label theelevationofthe mountain summit and mountainoverlook on avertical numberline. The
picnic arearepresentszero. Writearational number to represent each location.
Picnic Area: 0
Mountain Summit: 𝟐,𝟎𝟎𝟎
Mountain Overlook: −𝟑,𝟓𝟎𝟎
b. Use absolute value to representthedistanceon thenumber lineofeach location
from the picnic area.
Distance from the picnicareato the mountain summit: |𝟐, 𝟎𝟎𝟎|= 𝟐,𝟎𝟎𝟎
Distance from the picnicareato the mountain overlook: | − 𝟑,𝟓𝟎𝟎|= 𝟑,𝟓𝟎𝟎
c. What isthe distance between theelevationsofthe summitand overlook? Use
absolute value and your number linefrom Part ato explain your answer
Summit to picnic area andpicnicarea to overlook:𝟐,𝟎𝟎𝟎+ 𝟑, 𝟓𝟎𝟎 = 𝟓,𝟓𝟎𝟎 𝟓,𝟓𝟎𝟎feet
There are 𝟐,𝟎𝟎𝟎units from zero to 𝟐,𝟎𝟎𝟎on thenumber line.
There are 𝟑,𝟓𝟎𝟎units from zero to −𝟑,𝟓𝟎𝟎on thenumber line.
Altogether that equals 𝟓,𝟓𝟎𝟎units, which represents thedistanceon thenumber linebetween thetwo
elevations. So thedifferencein elevations is 𝟓, 𝟓𝟎𝟎feet.
Problem Set Sample Solutions
For each ofthe following two quantities, which hasthe greater magnitude? (Use absolutevalue to defendyour answers.)
1. 𝟑𝟑dollars and −𝟓𝟐dollars
|− 𝟓𝟐| = 𝟓𝟐 |𝟑𝟑|= 𝟑𝟑 𝟓𝟐 > 𝟑𝟑, so −𝟓𝟐 dollars has thegreater magnitude.
2. −𝟏𝟒feet and 𝟐𝟑feet
|− 𝟏𝟒| = 𝟏𝟒 |𝟐𝟑|= 𝟐𝟑 𝟏𝟒 < 𝟐𝟑, so 𝟐𝟑 feet has thegreater magnitude.
3. −𝟐𝟒.𝟔 poundsand −𝟐𝟒.𝟓𝟖pounds
|− 𝟐𝟒.𝟔| = 𝟐𝟒.𝟔 | − 𝟐𝟒.𝟓𝟖|= 𝟐𝟒.𝟓𝟖 𝟐𝟒.𝟔 > 𝟐𝟒.𝟓𝟖, so −𝟐𝟒.𝟔 pounds has thegreater magnitude.

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NYS COMMON CORE MATHEMATICS CURRICULUM 6•3Lesson 11
4. −𝟏𝟏
𝟏
𝟒
degrees and 𝟏𝟏degrees
|− 𝟏𝟏
𝟏
𝟒
| = 𝟏𝟏
𝟏
𝟒
| − 𝟏𝟏|= 𝟏𝟏 𝟏𝟏
𝟏
𝟒
> 𝟏𝟏, so −𝟏𝟏
𝟏
𝟒
degrees has thegreater magnitude.
For questions 5–7, answer trueor false. Iffalse, explain why.
5. The absolute value ofanegative number will alwaysbe apositive number.
True.
6. The absolute value ofany number willalwaysbe a positive number.
False. Zero is theexception sincetheabsolutevalueofzero is zero, and zero is not positive.
7. Positive numberswill alwayshave ahigher absolute valuethan negative numbers.
False. A number and it oppositehavethesameabsolute value.
8. Write aword problem whose solution is: |𝟐𝟎| = 𝟐𝟎.
Answers will vary.
9. Write aword problem whose solution is: | − 𝟕𝟎|= 𝟕𝟎.
Answers will vary.
10. Look at the bank account transactionslisted below and determine which hasthe greatest impacton the account
balance. Explain.
i. A withdrawal of $𝟔𝟎.
ii. A deposit of $𝟓𝟓.
iii. A withdrawal of $𝟓𝟖.𝟓𝟎.
|− 𝟔𝟎| = 𝟔𝟎 |𝟓𝟓|= 𝟓𝟓 | − 𝟓𝟖.𝟓𝟎|= 𝟓𝟖.𝟓𝟎
𝟔𝟎> 𝟓𝟖.𝟓𝟎 > 𝟓𝟓, so a withdrawal of $𝟔𝟎has thegreatest impact ontheaccount balance.