SECONDARY MATH I // MODULE 2

LINEAR & EXPONENTIAL FUNCTIONS – 2.5

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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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LINEAR & EXPONENTIAL FUNCTIONS – 2.5

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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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LINEAR & EXPONENTIAL FUNCTIONS – 2.5

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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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LINEAR & EXPONENTIAL FUNCTIONS – 2.5

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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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LINEAR & EXPONENTIAL FUNCTIONS – 2.5

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9. Sione:Iwonderifwecouldusethispatterntographlines,thinkingofthestartingpointandusingtheslope.Tryitwiththeequation:! ! = −2 ! + 1 − 3.Startingpoint:Slope:

Graph:

10. Zacwonders,“Whatisitaboutlinesthatmakesthiswork?”HowwouldyouanswerZac?

11. Couldyouusethispatterntowritetheequationofanylinearfunction?Whyorwhynot?

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