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SECONDARY MATH I // MODULE 2
LINEAR & EXPONENTIAL FUNCTIONS – 2.5
Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org
2.5 Making My Point
A Solidify Understanding Task
ZacandSionewereworkingonpredictingthenumberofquiltblocksinthispattern:
Whentheycomparedtheirresults,theyhadaninterestingdiscussion:
Zac:Igot! = 6! + 1 becauseInoticedthat6blockswereaddedeachtimesothepatternmusthavestartedwith1blockatn=0.
Sione:Igot! = 6 ! − 1 + 7becauseInoticedthatatn=1therewere7blocksandatn=2therewere13,soIusedmytabletoseethatIcouldgetthenumberofblocksbytakingonelessthanthen,multiplyingby6(becausethereare6newblocksineachfigure)andthenadding7becausethat’showmanyblocksinthefirstfigure.Here’smytable:
1 2 3 n7 13 19 6 ! − 1 + 7
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SECONDARY MATH I // MODULE 2
LINEAR & EXPONENTIAL FUNCTIONS – 2.5
Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org
1. WhatdoyouthinkaboutthestrategiesthatZacandSioneused?Areeitherofthemcorrect?Whyorwhynot?Useasmanyrepresentationsasyoucantosupportyouranswer.
ThenextproblemZacandSioneworkedonwastowritetheequationofthelineshownonthegraphbelow.
Whentheywerefinished,hereistheconversationtheyhadabouthowtheygottheirequations:
Sione:Itwashardformetotellwherethegraphcrossedtheyaxis,soIfoundtwopointsthatIcouldreadeasily,(-9,2)and(-15,5).Ifiguredoutthattheslopewas-!!andmadeatableandcheckeditagainstthegraph.Here’smytable:
x -15 -13 -11 -9 n
f(x) 5 4 3 2
− 12 ! + 9 + 2
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SECONDARY MATH I // MODULE 2
LINEAR & EXPONENTIAL FUNCTIONS – 2.5
Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org
Iwassurprisedtonoticethatthepatternwastostartwiththen,add9,multiplybytheslopeandthenadd2.
Igottheequation:! ! = − !! ! + 9 + 2.
Zac:Hey—IthinkIdidsomethingsimilar,butIusedthepoints,(7,-6)and(9,-7).
Iendedupwiththeequation:! ! = − !! ! − 9 − 7.Oneofusmustbewrongbecauseyours
saysthatyouadd9tothenandminesaysthatyousubtract9.Howcanwebothberight?
2. Whatdoyousay?Cantheybothberight?Showsomemathematicalworktosupportyourthinking.
Zac:Myequationmademewonderiftherewassomethingspecialaboutthepoint(9,-7)sinceitseemedtoappearinmyequation! ! = − !
! ! − 9 − 7whenIlookedatthenumberpattern.NowI’mnoticingsomethinginteresting—thesamethingseemstohappenwithyourequation,! ! = − !
! ! + 9 + 2andthepoint(-9,2)
3. DescribethepatternthatZacisnoticing.
4. FindanotherpointonthelinegivenaboveandwritetheequationthatwouldcomefromZac’spattern.
5. Whatwouldthepatternlooklikewiththepoint(a,b)ifyouknewthattheslopeofthelinewasm?
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SECONDARY MATH I // MODULE 2
LINEAR & EXPONENTIAL FUNCTIONS – 2.5
Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org
6. Zacchallengesyoutousethepatternhenoticedtowritetheequationoflinethathasaslopeof3andcontainsthepoint(2,-1).What’syouranswer?
Showawaytochecktoseeifyourequationiscorrect.
7. Sionechallengesyoutousethepatterntowritetheequationofthelinegraphedbelow,usingthepoint(5,4).
Showawaytochecktoseeifyourequationiscorrect.
8. Zac:“I’llbetyoucan’tusethepatterntowritetheequationofthelinethroughthepoints(1,-3)and(3,-5).Tryit!”
Showawaytochecktoseeifyourequationiscorrect.
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SECONDARY MATH I // MODULE 2
LINEAR & EXPONENTIAL FUNCTIONS – 2.5
Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org
9. Sione:Iwonderifwecouldusethispatterntographlines,thinkingofthestartingpointandusingtheslope.Tryitwiththeequation:! ! = −2 ! + 1 − 3.Startingpoint:Slope:
Graph:
10. Zacwonders,“Whatisitaboutlinesthatmakesthiswork?”HowwouldyouanswerZac?
11. Couldyouusethispatterntowritetheequationofanylinearfunction?Whyorwhynot?
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