ErinAndrusKatherineGrasso

LessonPlanStandards:CCSS.MATH.CONTENT.HSA.REI.C.6Solvesystemsoflinearequationsexactlyandapproximately(e.g.,withgraphs),focusingonpairsoflinearequationsintwovariables.

LearningGoal/Objective:Studentswillbeabletomodelrealworldsituationswithlinearrelationshipsandidentify/usethreestrategies(elimination,substitution,graphically)forsolving.

TypeofTask:ProcedureswithConnections

PrerequisiteKnowledge:Studentsshouldhaveabasicunderstandingofhowtosolvesystemsoflinearequations.Studentshavebeenintroducedtosystemsoflinearequationsbefore(AlgebraIandAlgebraII).Thepurposeofthislessonistohelpactivatepriorknowledgesothatstudentswillbemoresuccessfulandpreparedduringupcominglessons.Studentsshouldalsoknowhowtographlinearequationseitherbyhandorbyusingagraphingcalculator.

ToolsandGroupings:Studentsmaychoosetousegraphingcalculators.Aswedonotknowthestudentswell,wewillgroupstudentsbycountingoff.

IntroductionsandLessonPlan12Minutes Introductions LessonPurpose:Reviewhowtosolvesystemsoflinearequationsandmakeexplicitthe

connectionsbetweensolutionsfoundalgebraicallyandthegraphsofthesystem.Thisknowledgewillbeneededforupcomingsectionsinthetextbook.

GooverscheduleSmallGroupWork:LinearEquations1520minutes

Worksheet:Solvethreesystemsoflinearequations Worksheet:Makingconnectionsbetweenmultiplemathematicalrepresentations Purpose:Activatepriorknowledge,formativeassessment,makeconnectionsbetweenmultiple

mathematicalrepresentations,prepareforupcominglessons Questions:

Whatareyoutryingtosolvefor? Whatwouldtheequationslooklikegraphically? Istherearelationshipbetweenthealgebraicequationsandthegraphical

representations?Explainthisrelationshipinwords. Istherewayyoucancombineliketermsbetweentheequations? Howcanyoucheckyouranswer? Whatmethoddidyouusetosolvethesystemofequations? Canyouthinkofanotherapproach?Doesitmakemoreorlesssensetosolveusingthis

approach? Explain/writeinwordswhatthesolutionmeans. Whatdoorderedpairsrepresent?

BreakforLunchClassDiscussion:Shareoutresults1015Minutes

Purpose: Reviewmethodsforsolvingsystemsoflinearequations(substitution,elimination,

graphing) Explainwhatitiswearefinding/solvingforandthesignificanceofthethreedifferent

typesofsolutions(1,many,none) Makeexplicittheconnectionbetweensolutionsfoundalgebraicallyandthegraphsof

thesystem. Questions:

WhenwesaySolvethesystemofequations,whatarewetryingtoaccomplish? Whatdowemeanbysolve? Whatdoesthesolutiontypetellusaboutthealgebraicrepresentation?Whataboutthe

graphicalrepresentation? Whatdoesthegraphtellusaboutthesolutiontype?Whataboutthesystemof

equations? Giventhesolutiontype,whatdoweknowaboutthesystemofequations?Whatwould

thegraphlooklike?IntroduceTask12minutes

LaunchHasanyoneattendedWaukeeWinterfestbefore? Readtaskaloud Explainschedulefortherestoftheperiod

IndividualReflection:PossibleStrategies23minutes

Instructstudentstorereadthroughtheproblemontheirownandwritedownpossiblestrategiestheycouldusetosolvetheproblem.Remindthematthispointtheyshouldnotbecomingwithasolution.Rather,theyshouldbethinkingabouttheprocess.

ClassDiscussion:PossibleStrategies23minutes

AskstudentstoshareoutthestrategiestheycameupwithandwritethemontheboardSmallGroupWork:CashBoxTask2025minutes

Dividestudentsintogroupsof34.Explaintothemthattheyshouldworkthroughthetasktogether.OncegroupsthinktheyhavecomeupwithasolutionANDeveryoneinthegroupunderstandstheprocess,studentsshouldtryadifferentmethodandseeiftheyarriveatthesamesolution.

SeesolutionsforquestionsandpossiblesolutionsClassDiscussion:Shareoutresults1520minutes

PurposeofDiscussion: Shareoutthedifferentstrategiesusedontheboard Emphasizeprocess(whatworked,whatdidntwork,commonroadblocks,howdidyou

knowwhattotryfirst?) Makeexplicittheconnectionsbetweenthetable,algebraicexpressions,andgraphical

representation Sequencing:

Chart Howdidthecharthelpyoucreatethealgebraicexpressions Whatrelationshipsdidyounoticebetweenthecolumnsinthechart?

Algebra Didyouusethetabletohelpcreateyourequations? Howdidyoucomeupwithyoursystemofequations? Whydidyousetyourequationequalto200/201? Whatdoxandyrepresent? Whatmethoddidyouusetosolveyoursystemofequations?Why?

Graph Whatistherelationshipbetweenthegraphandthechart? Whatdoesthepointofintersectiontellyou? Howdothecolumnsinthetablerelatetothepointsonthegraph?

Name:______________________Solvethesystemsofequationsbelow.Explainyourresultsusingcompletesentences.1. 3=2xy,9=5xyAlgebraicSolution: GraphicalRepresentation:

InterpretationsofResults:

2.8=6x+y,24=18x+3y

AlgebraicSolution: GraphicalRepresentation:

InterpretationsofResults:

3.y=2(3x+4),y=6x5AlgebraicSolution: GraphicalRepresentation:

InterpretationsofResults:

Name:_______________________

1)Usingtheeliminationmethodwhensolvingasystemofequations,youcomeupwith10=2.Whatdoyouknowaboutthefollowing?Justifyyouranswer.a.Graphicalrepresentation?b.Solutiontype?c.Algebraicequations?2)Giventhefollowinggraphingwindow,whatpredictionscanyoumakeaboutthefollowing?Justifyyouranswer.

a.Solutiontype?b.Algebraicequations?3)Considerthefollowingsystemoflinearequations.y=12x+9y=3(4x+3)Withoutsolvingorgraphing,whatpredictionscanyoumakeaboutthefollowing?Justifyyouranswer.a.Graphicalrepresentation?b.Solutiontype?

Name:_______________________

NolawassellingticketsfortheWaukeeWinterfestcarriagerides.Attheendoftheevening,shepickedupthecashboxandnoticedadollarlyingonthefloornexttoit.Shesaid,

Iwonderwhetherthedollarbelongsinsidethecashboxornot.

Thepriceofticketsforthecarriageridewas1ticketfor$5(forindividuals)or2ticketsfor$8(forcouples).Shelookedinsidethecashboxandfound$200andticketstubsforthe47studentsinattendance.Doesthedollarbelonginsidethecashboxornot?Makeapredictionastowhetherornotthedollarbelongsinsidethecashboxandexplainwhyorwhynot.

Name:_______________________

Completethefollowingquestionsasyouworktosolvethecashboxproblem.Answereachquestionusingcompletesentences.1)Usethecharttoproveordisproveyourprediction.2)Whatpatternsdoyounoticeinthetable?3)Howcanyouusethepatternsinthetabletohelpyourepresentthisinformationalgebraically?ExpresseachcolumnofthechartintermsofXandY.4) Whatdothevariablesrepresent?5) Solvetheproblemalgebraically.6) Considerthesolution(s)youfoundwhensolvingforxandy.Whatdothesevaluestellyou?

7) Comparethesolution(s)foundin#6totheinformationyougatheredfromthechart.Isitthesame?Different?Explainwhy.8) Generateagraphicalrepresentationoftheproblemandsketchbelow.9) Lookingatthevaluesyoucameupwithinthetable,howdotheserelatetothelinesinthegraph?10)Howdoesthegraphicalrepresentationshowthesamesolution(s)asyouralgebraicapproach?

#ofsinglesticketssold

#ofcouplesticketssold

Total#ofticketssold

Singlestickets

totalcost

Couplestickets

totalcost

Totalcost

PossibleSolutionsSolutionbyTable

Singleticketssoldx

Coupleticketssold(numberofcouples)y

Totalticketssold(x+yORx+2y)

Singleticketcost(5x)

Coupleticketscost(4yOR8y)

Totalincashbox(5x+4yOR5x+8y)

47 0 47 235 0 235

45 2(1) 47 225 8 233

43 4(2) 47 215 16 231

... ... ... ... ... ...

13 34(17) 47 65 136 201

12 35(17.5) 47 60 140 200

Questions:Howcanyoucheckyouranswer?Whatpatternsdoyounoticeinthetable?Howcanyouusethepatternsinthetabletohelpyourepresentthisinformationalgebraically?Howwouldyourepresentintermsofxandyeachofthecolumnsinyourtable?Whatwouldthislooklikealgebraically/graphically?Doyougetasimilaranswerwhenyousolvealgebraically/graphically?Misconceptions:NumberofcoupleticketssoldcanbyoddSomestudentsmightmixupthenumberofcouplesandthenumberofcouplesticketssoldElimination/Substitutionx=individualticket y=coupleticket5x+4y=200 5x+4y=201x+y=47 x+y=47x=12,y=35 x=13,y=34x=numberofindividuals y=numberofcouples5x+8y=200 5x+8y=201x+2y=47 x+2y=47x=12,y=15.5 x=13,y=17

Questions:Whatinformationdidyouusetocreateyourequations?Whatdothevariablesrepresent?Howdidyoucomeupwithyourcoefficients?Whydidyousetyourequationequalto200/201?Usewordstoexplainthesignificanceofthealgebraicexpressions.Considerthesolutionsyoufoundwhensolvingforxandy.Whatdothesevaluestellyou?Whatmethoddidyouusetosolvethesystemofequations?Canyouthinkofanotherapproach?Doesitmakemoreorlesssensetosolveusingthisapproach?Doyougetasimilarsolutionifyousolvegraphically?Misconceptions:Studentsmayhavetroubledefiningvariablesandasaresultcreateaincorrectalgebraicrepresentations.

coupleticketssoldvnumberofcouples whatthefinalvalueforyrepresents

Studentsmayfindasolutionwhenx+y=200andthinktheyarefinished.Askthemtointerprettheirresultsandaskifthesolutionmakessense.Studentsmayassumethathavinganoddnumberofcouplesmeanstherewereanoddnumberofcoupleticketssold.SolutionbyGraphing

x=numberofindividualticketssold y=coupleticketssold5x+4y=201 5x+4y=200x+y=47 x+y=47

x=numberofindividuals y=numberofcouples5x+8y=201 5x+8y=200x+2y=47 x+2y=47Questions:Howdoesthisrelatethesolutionyoufoundusingalgebra/table/reasoning?Lookingatthevaluesyoucameupwithinthetable,howdotheserelatetothelinesinthegraph?Misconceptions:Studentsmayhavetroubledefiningvariablesandasaresultcreateincorrectalgebraicrepresentations.

coupleticketssoldvnumberofcouples whatthefinalvalueforyrepresents

YoucannothaveanoddnumberofcouplesticketssoldbutyoucanhaveanoddnumberofcouplesSolutionbyReasoning

Anevennumberofcoupleticketsweresoldat$4/ticket.Thus,thetotalamountforthecostofthoseticketsiseven(evenxeven=even)

Sinceanoddnumberofticketsweresold,weknowtherewereanoddnumberofsingleticketssold(even+odd=odd)andthatthetotalamountforthecostofsingleticketsisodd(oddxodd=odd)

Ifweaddthetotalamountfromcoupleticketsandthetotalamountfromsingletickets,weknowtheamountofmoneyinthecashboxmustbeodd(even+odd=odd).

Isthereawaytocheckthisusingthenumbersgiven?Whatwouldthislooklikealgebraically/graphically?Doyougetasimilarsolutionifyousolvealgebraically/graphically?Misconceptions:

Studentsmayassumethathavinganoddnumberofcouplesmeanstherewereanoddnumberofcoupleticketssold.