SSON 5 Adding and Subtracting Rational Numbers · 2019. 12. 22. · The additive inverse of 26 is 6...
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LESSON 30 Chapter 2: The Number System Duplicating any part of this book is prohibited by law. © 2013 Triumph Learning, LLC UNDERSTAND Every rational number has a location on the number line Numbers to the leſt of 0 on the number line are negative Numbers to the right of 0 are positive Any numbers n and 2n are opposites, or additive inverses The sum of n and 2n is 0 You can use a number line to add or subtract rational numbers It may help you to think of the sum of p 1 q as the number located a distance |q| units from p on the number line Find the sum: 23 1 3 Use a number line 23 is negative, so move 3 units to the leſt of 0, as shown by the blue arrow To add positive 3 to that number, move 3 units to the right as shown by the green arrow The green arrow points to 0, which is the sum ▸ 23 1 3 5 0 This makes sense because 23 and 3 are additive inverses –5 –4 –3 –2 –1 0 1 2 3 4 5 - 3 + 3 5 Adding and Subtracting Rational Numbers Matt has a balance of 2$250 in his bank account He adds $250 Find his new balance The blue arrow shows moving from 0 to a point halfway between 23 and 22 That point shows 2250 The green arrow models adding 250 to that amount The sum is 0 –3 –2 –1 0 1 2 3 - 2.5 + 2.5 ▸ Matt now has a $0 balance in his bank account 2250 and 250 are additive inverses Instead of using a number line to add integers, you can look at the signs of the addends: • If they have the same sign, add their absolute values and give the sum that sign • If they have opposite signs, subtract the lesser absolute value from the greater absolute value Give the sum the sign of the addend with the greater absolute value
Transcript of SSON 5 Adding and Subtracting Rational Numbers · 2019. 12. 22. · The additive inverse of 26 is 6...
LESSON
30 Chapter 2: The Number System
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UNDERSTAND Everyrational numberhasalocationonthenumberline .Numberstotheleftof0onthenumberlinearenegative .Numberstotherightof0arepositive .Anynumbersnand2nareopposites,oradditive inverses .Thesumofnand2nis0 .
Youcanuseanumberlinetoaddorsubtractrationalnumbers .Itmayhelpyoutothinkofthesumofp1q asthenumberlocatedadistance|q|unitsfromp onthenumberline .
Findthesum:2313
Useanumberline .23isnegative,somove3unitstotheleftof0,asshownbythebluearrow .Toaddpositive3tothatnumber,move3unitstotherightasshownbythegreenarrow .Thegreenarrowpointsto0,whichisthesum .
▸ 231350 .Thismakessensebecause23and3areadditiveinverses .
–5 –4 –3 –2 –1 0 1 2 3 4 5
� 3
� 3
5 Adding and Subtracting Rational Numbers
Matthasabalanceof2$2 .50inhisbankaccount .Headds$2 .50 .Findhisnewbalance .
Thebluearrowshowsmovingfrom0toapointhalfwaybetween23and22 .Thatpointshows22 .50 .Thegreenarrowmodelsadding2 .50tothatamount .Thesumis0 .
–3 –2 –1 0 1 2 3
� 2.5
� 2.5
▸Mattnowhasa$0balanceinhisbankaccount .22 .50and2 .50areadditiveinverses .
Insteadofusinganumberlinetoaddintegers,youcanlookatthesignsoftheaddends:
• Iftheyhavethesamesign,addtheirabsolutevaluesandgivethesumthatsign .
• Iftheyhaveoppositesigns,subtractthelesserabsolutevaluefromthegreaterabsolutevalue .Givethesumthesignoftheaddendwiththegreaterabsolutevalue .
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Connect
Lesson 5: Adding and Subtracting Rational Numbers 31
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Usetwodifferentmethodstofindthesum:261(24)
Useanumberlineandtheruleforaddingintegerswiththesamesign .
Youcoulduseanumberlinetodeterminethat(26)1(24)5(210) .
Since26and24havethesamesign,youcouldalsoaddthemlikethis:
|26|1|24|5614510
Since26and24arenegative,givethesumanegativesign .
▸Thesumis210 .Thisshowsthat(2)1(2)5(2) .
� 6
� (�4)
–10 –8 –6 –4 –2 0 2 4 6 8 10
Useanumberlineandtheruleforaddingintegerswithoppositesigns .
Youcoulduseanumberlinetodeterminethat(9)1(23)56 .
Since9and23haveoppositesigns,findtheirdifference .
|9|59 |23|53
92356
Theaddend9hasthegreaterabsolutevalue,sogivethesumapositivesign .
▸Thesumis6 .Thisshowsthatsometimes(1)1(2)5(1) .
� 9
� (�3)
–10 –8 –6 –4 –2 0 2 4 6 8 10
Usetwodifferentmethodstofindthesum:91(23)
Chooseonemethod(numberlineorrulesforaddingintegers) .Useittofindthesum:41(29) .Usethisexampletoshowthatsometimes(1)1(2)5(2) .
TRY
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32 Chapter 2: The Number System
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EXAMPLE A Findthedifference:42(26)
EXAMPLE B Findthedifference:2102(26)
Rewritetheproblemasanadditionproblem .
Theadditiveinverseof26is6 .
So,42(26)5416 .
Since4and6havethesamesign,addtheirabsolutevalues .
|4|1|6|5416510
Bothaddendsarepositive,sotheanswerispositive,10 .
▸42(26)510
Rewritetheproblemasanadditionproblem .
Theadditiveinverseof26is6 .
So,2102(26)521016
Since210and6haveoppositesigns,subtractthelesserabsolutevaluefromthegreaterabsolutevalue .
|210|.|6|because10.6,so:
|210|2|6|5102654
Givetheanswerthesignofthenumberwiththegreaterabsolutevalue .
Since210hasthegreaterabsolutevalue,theansweris24 .
▸ 2102(26)524
Drawamodeltofindthedifference:2325 .
MODEL
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Problem Solving
Lesson 5: Adding and Subtracting Rational Numbers 33
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read
Thehightemperatureinacityonedaywas2 .6°C .Thelowtemperaturewas22 .8°C .Whatwasthetemperaturerangeinthecitythatday?
plan
Thetemperaturerangeforthedayisequaltothedistancebetweenthehigh
temperatureandthelowtemperature .
Thedistancebetweentworationalnumbersonthenumberlineisequaltotheabsolutevalueoftheirdifference .Useabsolutevaluetofindtheanswer .
solve
Findtheabsolutevalueofthedifference:2 .62(22 .8) .
|2 .62(22 .8)|5|2 .61 |5| |5
So,thetemperaturerangeinthecitythatdaywas degreesCelsius .
check
Onewaytocheckthatthedistancebetween2 .6and22 .8isactuallytheansweryoufoundaboveistofindthedistancebetweenthosetwonumbersonthenumberline .
Plotpointsfor2 .6and22 .8onthenumberlinebelow .Thencounttheunitsbetweenthemtocheckthatyouransweriscorrect .
–3 –2 –1 0 1 2 3
Thedistancebetween2 .6and22 .8onthenumberlineis units .
Isthisthesameansweryoufoundabove?
▸Therangeofthetemperaturesonthatdaywas °C .
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Practice
34 Chapter 2: The Number System
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Identify the additive inverse, or opposite, of each number.
1. 7 2. 26
Change the sign of each number to find its opposite.
HINT
3. 9 .53
Rewrite each subtraction expression as an addition expression.
4. 1025
1
5. 211 .1212 .6
1
REMEMBER The expression p 2 q is the same as p 1 (2q ).
6. 1__82 211__5 1
Use the number line to find each sum or difference. Show your work on the number line.
7. 241(25)
–10 –8 –6 –4 –2 0 2 4 6 8 10
9. 62(24)
–10 –8 –6 –4 –2 0 2 4 6 8 10
8. 22110
–10 –8 –6 –4 –2 0 2 4 6 8 10
10. 5211
–10 –8 –6 –4 –2 0 2 4 6 8 10
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Lesson 5: Adding and Subtracting Rational Numbers 35
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The following thermometers show the range in temperatures in a given day. Find the temperature ranges. Show or explain your work.
11.
–10
0
10
20
30
Temperature (in °F)
hightemperature
lowtemperature
12. 543210
–1–2–3–4–5
Temperature (in °C)
hightemperature
lowtemperature
13.
–1
–2
0
1
2
3
Temperature (in °C)
hightemperature
lowtemperature
Choose the best answer.
14. Aheliumatomhas2electrons,eachofwhichhasa21charge .Eachprotonhasa11charge .Overall,theheliumatomhasnocharge .Howmanyprotonsmusttheatomhave?
A. 22 B. 0
C. 2 D. 3
15. OnJanuary1,Rose’sbankbalancewas$200 .Duringthemonth,shewrotechecksfor$115 .25and$350 .00andmadeonedepositof$150 .50 .Whichbestrepresentshercheckingaccountbalanceattheendofthemonth?
A. 2$114 .75 B. 2$114 .25
C. $114 .25 D. $115 .50
Solve.
16. Show Irawritesthatthesumofa1bisanumberthatislocatedadistanceof|b|unitsfromaonthenumberline .Choosefourvaluesforaandbandshowthatthisistrue .Usethefournumberlinesontherighttoshowyourworkandsupportyouranswer .
–5 –4 –3 –2 –1 0 1 2 3 4 5
–5 –4 –3 –2 –1 0 1 2 3 4 5
–5 –4 –3 –2 –1 0 1 2 3 4 5
–5 –4 –3 –2 –1 0 1 2 3 4 5
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