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

©201

2www.flickr.com

/pho

tos/tedd

ylam

bec

CCBYCa

milleKing

https://flic.kr/p/hRFp

27

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

28

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?

29

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.

30

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?

31

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– Teacher Notes

A Solidify Understanding Task

Purpose:Thisisthefirsttaskoftwothatfocusonunderstandingandusingvariousnotationsforlinearfunctions.Thetaskinvolvesstudentsinthinkingaboutacontextwherestudentshaveselectedtheindexintwodifferentways,thusgettingtwodifferent,butequivalentequations.Theideaisextendedsothatstudentscanseetherelationshipexpressedinpoint/slopeformoftheequationoftheline.CoreStandardsFocus:

A-SSE.1 Interpretexpressionsthatrepresentaquantityintermsofitscontext.a) Interpretpartsofanexpression,suchasterms,factors,andcoefficients.

A-SSE.6 Usethestructureofanexpressiontoidentifywaystorewriteit. A-CED.2 Createequationsintwoormorevariablestorepresentrelationshipsbetween

quantities;graphequationsoncoordinateaxeswithlabelsandscales.F-LE.5 Interprettheparametersinalinearorexponentialfunctionintermsofacontext.StandardsforMathematicalPracticeofFocusintheTask:

SMP2–Reasonabstractlyandquantitatively.

SMP7–Lookforandexpressregularityinrepeatedreasoning.

TheTeachingCycle

Launch(WholeClass):

Inprevioustasksstudentshaveworkedwithvisualpatternssuchastheoneinthistask.StartthelessonbytellingstudentsthatZacandSionehaveworkedtheproblemandcomeupwithtwodifferentanswers,whichtheyaretryingtoresolvewithsoundreasoning.Studentsneedtofigure

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

outhowZacandSionehavearrivedatdifferentequationsandwhoisrightthrougheachofthescenariosinthetask.

Explore(SmallGroup):

Monitorstudentsastheyworkthroughthetasktoseethattheyunderstandeachscenario.Inproblem#1,watchforstudentsthathavelabeledthefigurestomatchtheequations;eitherstartingwithn=0orn=1.Forproblems2-5,watchtoseethatstudentsarenoticingpatternsinhowthenumbersareusedintheequationandmakingsenseofthetables.

Discuss(WholeClass):

Bepreparedforthewholegroupdiscussionbyhavinglargeversionsofthefigurein#1readytobeused.AskastudenttoexplainthedifferencebetweenZacandSione’sequationsandwhytheybothmakesenseasmodelsforthefigures.Askastudenttoshowwhetherornotthetwoequationsareequivalent.

Movetothenextscenario,askingforverbaldescriptionsofthepatterntheynoticedin#3.Askforastudenttogivesomeexamplesofequationsthattheywrotefor#4usingthepattern.Ask,“Aretheequationsequivalent?Howdoyouknow?”Askforstudentstogivetheiranswerfor#4.Iftherearedifferencesinequationsamongthegroups,discussthedifferences.Finally,askstudentsforreasonswhythisrelationshipshouldholdforanylinearfunction.Afterdiscussingtheirreasons,offerthatthispatternisoftenusedasaformulaforwritingequationsandgraphinglinesandiscalledpoint/slopeformoftheequationofaline.Youmaywishtoshowthemthatthisformcanbederivedfromtheslopeformula:

! = !!!!!!!!

Withalittlerearranging:

! ! − !! = ! − !!

! = ! ! − !! + !!

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

Thefocusofthistaskisontheconnectionsbetweenrepresentationsandhowanypointcanbeusedtocreateanequation.Thisisthetaskforstudentstothinkaboutthisconcept.Thederivationisimportantbutmaycomelater.Foreachproblem,demonstratetheconnectionbetweenthestrategiesthatstudentsusedandtheslope/interceptformula.Afterthesethreeproblems,solicitanswersforquestion#10;whatisitaboutlinesthatcausethisconnection?Answersshouldincludetheideathattheconstantrateofchangemakesitpossibletostartatanypointonthelineandfindanotherpoint.Inthepast,studentshaveusedthey-interceptasthestartingpoint,butanyknownpointwillworkaswelltowriteanequationortographtheline.

AlignedReady,Set,Go:LinearandExponentialFunctions2.5

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

READY Topic:Writingequationsoflines.

Writetheequationofalineinslope-interceptform:y=mx+b,usingthegiveninformation.

1.m=-7,b=4 2.m=3/8,b=-3 3.m=16,b=-1/5

Writetheequationofthelineinpoint-slopeform:y=m(x–x1)+y1,usingthegiveninformation.

4.m=9,(0.-7) 5.m=2/3,(-6,1) 6.m=-5,(4,11)

7.(2,-5)(-3,10) 8.(0,-9)(3,0) 9.(-4,8)(3,1)

READY, SET, GO! Name PeriodDate

32

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

SET

Topic:Graphinglinearandexponentialfunctions

Makeagraphofthefunctionbasedonthefollowinginformation.Addyouraxes.Choosean

appropriatescaleandlabelyourgraph.Thenwritetheequationofthefunction.

10.Thebeginningvalueis5anditsvalueis3

unitssmallerateachstage.

Equation:

11.Thebeginningvalueis16anditsvalueis¼

smallerateachstage.

Equation:

12.Thebeginningvalueis1anditsvalueis10

timesasbigateachstage.

Equation:

13.Thebeginningvalueis-8anditsvalueis2

unitslargerateachstage.

Equation:

33

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

GO Topic:Equivalentequations

Provethatthetwoequationsareequivalentbysimplifyingtheequationontherightsideofthe

equalsign.Thejustificationintheexampleistohelpyouunderstandthestepsforsimplifying.

YoudoNOTneedtojustifyyoursteps.

Example: Justification 2! − 4 = 8 + ! − 5! + 6 ! − 2 Add! − 5!anddistributethe6over ! − 2 = 8 − 4! + 6! − 12 Combineliketerms. = −4 + 2! 2! − 4 = 2! − 4 Commutativepropertyofaddition

14.! − 5 = 5! − 7 + 2 3! + 1 − 10!

15.6 − 13! = 24 − 10 2! + 8 + 62 + 7!

16.14! + 2 = 2! − 3 −4! − 5 − 13

17. ! + 3 = 6 ! + 3 − 5 ! + 3

18.4 = 7 2! + 1 − 5! − 3 3! + 1

19.! = 12 + 8! − 3 ! + 4 − 4!

20.Writeanexpressionthatequals ! − 13 . Itmusthaveatleasttwosetsofparenthesesandoneminussign.Verifythatitisequalto ! − 13 .

34