SECONDARY MATH I // MODULE 5

SYSTEMS OF EQUATIONS AND INEQUALITIES – 5.9

Mathematics Vision Project

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5.9 Can You Get to the Point,

Too?

A Solidify Understanding Task

Part1In“ShoppingforCatsandDogs,”Carlosfoundawaytofindthecostofindividualitems

whengiventhepurchasepriceoftwodifferentcombinationsofthoseitems.Hewouldliketomakehisstrategymoreefficientbywritingitoutusingsymbolsandalgebra.Helphimformalizehisstrategybydoingthefollowing:

• Foreachscenarioin“ShoppingforCatsandDogs”writeasystemofequationsto

representthetwopurchases.

• Showhowyourstrategiesforfindingthecostofindividualitemscouldberepresentedbymanipulatingtheequationsinthesystem.Writeoutintermediatestepssymbolically,sothatsomeoneelsecouldfollowyourwork.

• Onceyoufindthepriceofoneoftheitemsinthecombination,showhowyouwouldfindthepriceoftheotheritem.

Part2WritingouteachsystemofequationsremindedCarlosofhisworkwithsolvingsystemsof

equationsgraphically.Showhowthefollowingscenariofrom“ShoppingforCatsandDogs”canberepresentedgraphically,andhowthecostofeachitemshowsupinthegraphs.

Carlospurchased6dogleashesand6catbrushesfor$45.00forClaritatousewhilepamperingthepets.Laterinthesummerhepurchased3additionaldogleashesand2catbrushesfor$19.00.Basedonthisinformation,figureoutthepriceofeachitem.

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SECONDARY MATH I // MODULE 5

SYSTEMS OF EQUATIONS AND INEQUALITIES – 5.9

Mathematics Vision Project

Licensed under the Creative Commons Attribution CC BY 4.0

mathematicsvisionproject.org

5.9 Can You Get to the Point, Too? – Teacher Notes A Solidify Understanding Task

Purpose:Thistasksolidifiesthestrategiesforsolvingsystemsofequationsthatsurfacedduringtheprevioustask.Studentswillbeginbywritingasystemofequationstorepresenttheshoppingscenarios.Studentswillrecognizethatwecanobtainanequivalentsystemofequationsbyreplacingoneorbothequationsinthesystemusingoneofthefollowingsteps:• Replaceanequationinthesystemwithaconstantmultipleofthatequation• Replaceanequationinthesystemwiththesumordifferenceofthetwoequations• Replaceanequationwiththesumofthatequationandamultipleoftheother

Thegoalofthesestepsistoobtainasystemofequationsinwhichthecoefficientofoneofthevariablesisthesameinbothequations.Then,whenwesubtractoneoftheequationsfromtheother,wewillobtainanequationthatcontainsonlyonevariable.Thisequationcanbesolvedforitsvariableandtheresultcanbesubstitutedbackintooneoftheoriginalequationstoobtainanequationthatcanbesolvedfortheothervariable.CoreStandardsFocus:A.REI.5Provethat,givenasystemoftwoequationsintwovariables,replacingoneequationbythesumofthatequationandamultipleoftheotherproducesasystemwiththesamesolutions.A.REI.6Solvesystemsoflinearequationsexactlyandapproximately(e.g.,withgraphs),focusingonpairsoflinearequationsintwovariables.RelatedStandards:N.Q.1,A.SSE.1a,A.CED.2,A.CED.3

SECONDARY MATH I // MODULE 5

SYSTEMS OF EQUATIONS AND INEQUALITIES – 5.9

Mathematics Vision Project

Licensed under the Creative Commons Attribution CC BY 4.0

mathematicsvisionproject.org

StandardsforMathematicalPracticeoffocusinthetask:

SMP7–Lookforandmakeuseofstructure

SMP8–Lookforandexpressregularityinrepeatedreasoning

AdditionalResourcesforTeachers:

TheStandardsforMathematicalPracticePromptCardsavailablefromMathematicsVisionProject

canbeusedwiththistask.Havestudentsfocusonthepromptsandsentencestemsfromcards7or

8astheywork,andusethesepromptstosupporttheirlanguageandexplanations.

TheTeachingCycle:

Launch(WholeClass):

Provideamodelofhowstudentsmightworkwithsystemsofequationsusingtheintuitive

reasoningtheydevelopedintheprevioustaskbyworkingthroughscenario1from“Shoppingfor

CatsandDogs”together.Writeoutthesystemusingequationsinstandardform:

SincethecoefficientsofTarethesameinbothequations,wewillsubtractequation#1from

equation#2toget2F=11.00.WecansolvethisequationforFbydividingbothsidesofthe

equationby2togetF=5.50,whichmustbethepriceofabagFigaroFlakes.Wecansubstitutethis

amountintoeitherequationtosolveforthepriceofTabithaTidbits.Forexample,substituting5.50

intothefirstequationforFyields3T+22.00=43.00.Therefore,3T=21.00,orT=7.00.

Explore(SmallGroup):

Watchandlistenforthewaysstudentswriteandsolvethesystemsofequationsrepresentedin

eachoftheotherscenarios.Encouragethemtoconnecttheirintuitivereasoningwiththeshopping

scenariostothesymbolicreasoningwithvariables.Part2ofthetaskgivesstudentsanopportunity

toconnectthisworktosolvingasystemoflinearequationsgraphically.

3T + 4F = 43.003T + 6F = 54.00

⎧ ⎨ ⎩

SECONDARY MATH I // MODULE 5

SYSTEMS OF EQUATIONS AND INEQUALITIES – 5.9

Mathematics Vision Project

Licensed under the Creative Commons Attribution CC BY 4.0

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Discuss(WholeClass):

Invitestudentstoarticulateageneralstrategyforsolvingsystemsofequationsbyeliminatinga

variable.Recordthewayswecanobtainnew,equivalentsystemsofequationsbyusingthe

procedureslistedinthepurposestatementabove.Helpstudentsidentifythatthegoalofwriting

equivalentsystemsistoobtainasystemofequationsinwhichthecoefficientofoneofthevariables

isthesameinbothequations.Pointoutthatoncewehavedeterminedthevalueofoneoftheitems

wecansolveforthevalueoftheotheritembysubstitution.

Giventime,itmightbebeneficialtohavestudentsdemonstratethisstrategywithoneofthemore

challengingsystemsfromthe“PetSitters”context,suchasthefollowingsystemthatinvolvesthe

spaceconstraintandthepamperingtimeconstraint.

�

24x + 6y = 36013x +

415

y = 8

⎧ ⎨ ⎪

⎩ ⎪

Onepossiblestrategyforsolvingthissystemwouldbetomultiplythebottomequationby15to

obtainwholenumbercoefficients.

�

24x + 6y = 3605x + 4y = 120⎧ ⎨ ⎩

Thenmultiplythetopequationby4andthebottomequationby6togetthey-coefficientthesame

inbothequations.

�

96x + 24y = 144030x + 24y = 720⎧ ⎨ ⎩

Subtractingthebottomequationfromthetopyieldsthesinglevariableequation

66x = 720.

Solvingthisequationforxgives

x = 72066

=101011.Thecompletesolutionis

10 1011 ,16 411( ).

Fortunately,thisisnotoneoftheimportantpointsofintersectioninthe“PetSitters”context,since

itliesoutsidethefeasibleregion.

AlignedReady,Set,Go:SystemsofEquationsandInequalities5.9

SECONDARY MATH I // MODULE 5

SYSTEMS – 5.9

Mathematics Vision Project

Licensed under the Creative Commons Attribution CC BY 4.0

mathematicsvisionproject.org

5.9

READY Topic:Matchingdefinitionsofgeometricfigures.

Matchthenameofthefigurewithitsgeometricdefinition.

a.isoscelestriangle b.equilateraltriangle c.scalenetriangle d.righttrianglee.rectangle f.rhombus g.square h.trapezoid

1.__________ Aquadrilateralwithonlyonepairofparallelsides. 2.__________ Allofthesidesofthistrianglearethesamelength.3.__________ Allofthesidesofthisquadrilateralarethesamelength.4.__________ Thistrianglehasexactlyonerightangle.5.__________ Thisquadrilateralhasfourrightangles.6.__________ Noneofthesidesofthistrianglearethesamelength.7.__________ Thisquadrilateralisboth#3and#5.8.__________ Onlytwosidesofthistrianglearethesamelength.

SET Topic:SolvingsystemsofequationsbyeliminationSolveeachsystemofequationsusingeliminationofavariable.Checkyoursolution.

9. 2! + ! = 32! + 2! = 2

10. 2! + 5! = 3 ! + 5! = 6

11. 2! + 0.5! = 3! + 2! = 8.5 12. 3! + 5! = −1

! + 2! = −1

READY, SET, GO! Name PeriodDate

35

SECONDARY MATH I // MODULE 5

SYSTEMS – 5.9

Mathematics Vision Project

Licensed under the Creative Commons Attribution CC BY 4.0

mathematicsvisionproject.org

5.9

13.3! + 5! = −3! + 2! = − !

!

14.A150-yardpipeiscuttoprovidedrainagefortwofields.Ifthelengthofonepiece(a)isthreeyardslessthantwicethelengthofthesecondpiece(b),whatarethelengthsofthetwopieces?

GO

Topic:Identifyingfunctions

Foreachgraphdetermineiftherelationshiprepresentsafunction.Ifitisafunction,writeyes.Ifitisnotafunction,explainwhyitisnot.

15.

16.

17.

18.

19.

20.

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