SECONDARY MATH I // MODULE 5

SYSTEMS OF EQUATIONS AND INEQUALITIES – 5.7

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5. 7 Get to the Point

A Solidify Understanding Task

CarlosandClaritaneedtocleanthestorageshedwheretheyplantoboardthepets.Theyhavedecidedtohireacompanytocleanthewindows.Aftercollectingthefollowinginformation,theyhavecometoyouforhelpdecidingwhichwindowcleaningcompanytheyshouldhire.

• SunshineExpressWindowCleanerscharges$50foreachservicecall,plus$10perwindow.

• “Pane”lessWindowCleanerscharges$25foreachservicecall,plus$15perwindow.

1. Whichcompanywouldyourecommend,andwhy?PrepareanargumenttoconvinceCarlosandClaritathatyourrecommendationisreasonable.(Itisalwaysmoreconvincingifyoucansupportyourclaiminmultipleways.Howmightyousupportyourrecommendationusingatable?Agraph?Algebra?)

YourpresentationtoCarlosremindshimofsomethinghehasbeenthinkingabout—howto

findthecoordinatesofthepointswheretheboundarylinesinthe“PetSitter”constraintsintersect.Hewouldliketodothisalgebraicallysincehethinksguessingthecoordinatesfromagraphmightbelessaccurate.

2. Writeequationsforthefollowingtwoconstraints.

• Space• Start-upCostsFindwherethetwolinesintersectalgebraically.Recordenoughstepssothatsomeoneelsecanfollowyourstrategy.

3. Nowfindthepointofintersectionforthetwotimeconstraints.• FeedingTime

• PamperingTime

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SECONDARY MATH I // MODULE 5

SYSTEMS OF EQUATIONS AND INEQUALITIES – 5.7

Mathematics Vision Project

Licensed under the Creative Commons Attribution CC BY 4.0

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5. 7 Get to the Point – Teacher Notes A Solidify Understanding Task

Purpose:Thistaskisdesignedtosolidifygraphical,numericalandalgebraicstrategiesforsolvingasystemoftwolinearequations.Whilethepointofintersectiononagraphrepresentsthesolutiontothesystem,itcanbedifficulttoidentifytheexactcoordinatesofthispointofintersection.Atablecanprovideanefficient“guessandcheck”strategyforclosinginonthecoordinatesofapointofintersectionwhenthecoordinatesarenotintegers.Atablemightalsosuggestanalgebraicstrategy:sincewearelookingforaninputvalueforwhichbothoutputvaluesarethesame,wecansetthetwoequationsequaltoeachothertofindthex-valueforwhichthetwoy-valuesareequal.Oncethis“settheequationsequaltoeachother”strategyisestablished,thenotionofsubstitutingoneoftheexpressionsforyintotheotherequationcanbesuggested,sincesettingbothequationsequaltoeachotherisequivalenttosubstitutingoneexpressionforyintotheotherequation.Solvingasystemoflinearequationsbysubstitutionorbygraphingarebothpartofthegrade8CCSSMstandards,andtherefore,thistaskshouldhelpremindstudentsoftheworktheyhavedonepreviously.Thistasksetsthestageforsolvingsystemsoflinearequationsbyelimination,whichwillbethetopicofthenextsequenceoftasks.CoreStandardsFocus:A.REI.6Solvesystemsoflinearequationsexactlyandapproximately(e.g.,withgraphs),focusingonpairsoflinearequationsintwovariables.RelatedStandards:N.Q.2,A.CED.2,A.CED.2,A.REI.10

StandardsforMathematicalPracticeoffocusinthetask:

SMP2–Reasonabstractlyandquantitatively

SMP4–Modelwithmathematics

SMP8–Lookforandexpressregularityinrepeatedreasoning

SECONDARY MATH I // MODULE 5

SYSTEMS OF EQUATIONS AND INEQUALITIES – 5.7

Mathematics Vision Project

Licensed under the Creative Commons Attribution CC BY 4.0

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TheTeachingCycle:

Launch(WholeClass):Q1

Readtheinitialwindow-cleaningcontextwiththeclass,andpointoutthestatementthatthe

studentsshouldprovidemultiplerepresentationstojustifytheirrecommendationastowhich

windowcleaningcompanyCarlosandClaritashoulduse.

Explore(SmallGroup):

Yourstudentsshouldalreadybefamiliarwithstrategiesforfindingthepointofintersectionofthe

twolinesrepresentingthewindowcleaningcompanycostsbasedontheworkwithsolvingsystems

oflinearequationsingrade8.Thecontextpointsoutanimportantissue—whichcompanythey

shouldselectdependsonthenumberofwindowsinthestorageshed.Sincewedon’tknowthis

number,therecommendationshouldbemadeintermsofthisunknownamount.Inthiscase,both

companiescharge$100towash5windows.“Pane”lessWindowCleanersischeaperifthetwins

havefewerthan5windowstobecleaned,andSunshineExpressischeaperiftheyhavemore.

Pressstudentstoexploregraphical,numericalandalgebraicsolutionsandtobeabletodescribe

theconnectionsbetweeneach(seepurposestatementabove).

Discuss(WholeClass):

Beginbyhavingagraphofthescenariopresentedinordertoidentifythatthepointofintersection

determinesthenumberofwindowsforwhichbothcompanieswouldcostthesame,andthatthe

graphcanbeusedtodeterminewhichcompanyischeaperoneithersideofthepointof

intersection.

Nextexamineatableshowinganinputcolumnforthenumberofwindows,andtwocolumnsforthe

amountchargedbyeachcompanytocleanthatnumberofwindows.Connecttherowwherethe

outputsarethesametothepointofintersectiononthegraph.Addarowtothistabletorepresent

thegeneralcase,asfollows.Note,also,they-interceptofthelinesrepresentstheinitialchargefor

eachcompany,ifnowindowsarecleaned.

SECONDARY MATH I // MODULE 5

SYSTEMS OF EQUATIONS AND INEQUALITIES – 5.7

Mathematics Vision Project

Licensed under the Creative Commons Attribution CC BY 4.0

mathematicsvisionproject.org

NumberofWindows SunshineExpressCost “Pane”lessCost

0 50 25

1 60 40

2 70 55

3 80 70

4 90 85

5 100 100

6 110 115

7 120 130

N 50+10N 25+15N

Usethistabletodiscusshowsettingthetwoexpressions50+10Nand25+15Nequaltoeachother

wouldbeequivalenttofindingtherowwherebothcompanieschargethesameamount.Thisisalso

asubstitutionmethodiftheexpressionsaretreatedaspartsoftheequationsC=50+10Nand

C=25+15N.Thentheexpression50+10NcanbesubstitutedintotheequationC=25+15NforC.

Aspossible,usestudentworktodiscusseachalgebraicstrategy.

Launch(WholeClass):Q2&Q3

Havestudentsturntheirattentiontosolvingforpointswheretwoofthe“PetSitter”constraints

intersect.Whiletherearemanypointsofintersectionbetweenvariousconstraints,wewill

considertwosuchpointsinthistask.

Explore(SmallGroup):

Thecoordinatesofthepointofintersectionforthespaceandstart-upcostconstraintsarenotwhole

numbers.Thisshouldmotivateanalgebraicsolutionstrategy.Studentswhoareworkingwiththe

constraintswritteninstandardformmayfindasubstitutionstrategymoreefficientthansolving

bothequationsforavariableandsettingthemequaltoeachother.Watchforbothalgebraic

strategies.

SECONDARY MATH I // MODULE 5

SYSTEMS OF EQUATIONS AND INEQUALITIES – 5.7

Mathematics Vision Project

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Discuss(WholeClass):

Makesurethatbothalgebraicstrategiesforsolvingsystemsoftwolinearequations(i.e.,

substitutionandsettingexpressionsequal)getpresentedanddiscussed.Discusstheissueofthe

solutiontothespaceandstart-upcostssystemofequationsnothavingwholenumbercoordinates.

Whilewecan’thave131/3dogsand62/3catsasareasonablesolutiontothe“PetSitters”

scenario,thisisthesolutiontothesystemofequations.Aswehavenoticed,thesolutiontothe

contextualizedsituationmaybeapointintheinterioroftheregion.However,itisimportantto

notethatthisisanexampleofthemodelingstandard—wehavedecontextualizedthesituationto

findamathematicalmodelthatwillhelpusreasonaboutthe“PetSitters”context.Eventually,any

conclusionswemakeusingthemathematicalmodelwillhavetobeinterpretedintermsofthe

originalcontext.

AlignedReady,Set,Go:SystemsofEquationsandInequalities5.7

SECONDARY MATH I // MODULE 5

SYSTEMS – 5.7

Mathematics Vision Project

Licensed under the Creative Commons Attribution CC BY 4.0

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5.7

READY Topic:PythagoreantheoremAneasywaytocheckifatrianglecontainsa90°angle(alsocalledarighttriangle)istousethePythagoreantheorem.Youmayrememberthetheoremas!! + !! = !!,where! isthelengthofthelongestside(thehypotenuse)and! and !arethelengthsofthetwoshortersides.Identifywhichlengthsmakearighttriangle.Example:Given5,12,13Replace!, !, and !withthenumbers 5! + 12! = 13! → 25 + 144 = 169 → 169 = 169 Since169=169,atrianglewithsidelengthsof5,12,and13mustbearighttriangle.Dothesenumbersrepresentthesidesofarighttriangle?WriteYESintheboxesthatapply.1.9,40,41

2.3,4,5 3.6,7,8 4.20,21,29

5.9,12,15

6.10,11,15 7.6,8,10 8.8,15,17

SET Topic:Solvingsystemsofequationsusingsubstitution.Solveeachsystemofequationsusingsubstitution.Checkyoursolutioninbothequations.Inthisproblem,substitute ! + 1 inplaceof! inthesecondequation.9. ! = ! + 1

! + 2! = 8

Inthisproblem,substitute 3 + ! inplaceof! inthefirstequation.10. ! + 2! = 7

! = 3 + !

11.! = 9 + 2!3! + 5! = 20

12.! = 2! − 4

3! + 21! = 15

READY, SET, GO! Name PeriodDate

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SECONDARY MATH I // MODULE 5

SYSTEMS – 5.7

Mathematics Vision Project

Licensed under the Creative Commons Attribution CC BY 4.0

mathematicsvisionproject.org

5.7

13.! = −1 − 2!3! + 5! = −1

14.! = 2! − 3! + ! = −5

15.Ticketstoaconcertcost$10inadvanceand$15atthedoor.If120ticketsweresoldforatotalof$1390,howmanyoftheticketswerepurchasedinadvance?

GO Topic:Solvingonevariableinequalities

Solvethefollowinginequalities.Writethesolutionsetinintervalnotationandgraphthe

solutionsetonanumberline.

16.4x+10<2x+14

17.2x+6>55–5x

18.2( !! + 3) > 6(! − 1)

19.9! + 4 ≤ −2(! + !! )

Solveeachinequality.Givethesolutioninsetbuildernotation(e.g. ! ∈ ℝ ! < ! ).

20.− !! > − !"

! 21.5! > 8! + 27

22.!! >!!

23.3! − 7 ≥ 3(! − 7)

24.2! < 7! − 36

25.5 − ! < 9 + !

–10 –5 5 100

–10 –5 5 100

–10 –5 5 100

–10 –5 5 100

29