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SECONDARY MATH I // MODULE 8
CONNECTING ALGEBRA & GEOMETRY – 8.3
8.3 Prove It!
CCBY
A Practice Understanding Task
Inthistaskyouneedtouseallthethingsyouknowabout
quadrilaterals,distance,andslopetoprovethatthe
shapesareparallelograms,rectangles,rhombi,or
squares.Besystematicandbesurethatyougiveallthe
evidencenecessarytoverifyyourclaim.
1.
a.IsABCDaparallelogram?Explainhowyouknow.
b.IsEFGHaparallelogram?Explainhowyouknow.
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SECONDARY MATH I // MODULE 8
CONNECTING ALGEBRA & GEOMETRY – 8.3
2.
a.IsABCDarectangle?Explainhowyouknow.
b.IsEFGHarectangle?Explainhowyouknow.
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SECONDARY MATH I // MODULE 8
CONNECTING ALGEBRA & GEOMETRY – 8.3
3.
a.IsABCDarhombus?Explainhowyouknow.
b.IsEFGHarhombus?Explainhowyouknow.
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SECONDARY MATH I // MODULE 8
CONNECTING ALGEBRA & GEOMETRY – 8.3
4.
a.IsABCDasquare?Explainhowyouknow.
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SECONDARY MATH I // MODULE 8
CONNECTING ALGEBRA & GEOMETRY – 8.3
8.3 Prove It! – Teacher Notes
A Practice Understanding Task
Purpose:Thepurposeofthistaskistosolidifystudentunderstandingofquadrilateralsandto
connecttheirunderstandingofgeometryandalgebra.Inthetasktheywilluseslopesanddistance
toshowthatparticularquadrilateralsareparallelograms,rectangles,rhombiorsquares.Thistask
willalsostrengthenstudentunderstandingofjustificationandproof,andtheneedtoputfortha
completeargumentbaseduponsoundmathematicalreasoning.
CoreStandardsFocus:
G.GPE.4Usecoordinatestoprovesimplegeometrictheoremsalgebraically.Forexample,proveor
disprovethatafiguredefinedbyfourgivenpointsinthecoordinateplaneisarectangle.
RelatedStandards:G.GPE.5,G.GPE.7
StandardsforMathematicalPracticeofFocusintheTask:
SMP3–Constructviableargumentsandcritiquethereasoningofothers
SMP6–Attendtoprecision
TheTeachingCycle:
Launch(WholeClass):
Launchthetaskwithadiscussionofwhatstudentsknowaboutthepropertiesofquadrilaterals,for
instancethatarhombushastwopairsofparallelsides(makingitaparallelogram),congruentsides,
andperpendiculardiagonals.Discusswhatyouwouldneedtoshowtoproveaclaimthatafigureis
aparticularquadrilateral.Forinstance,itisnotenoughtoshowthatashapeisarhombusby
showingthatthetwopairsofsidesareparallel,butitwouldbeenoughtoshowthatthediagonals
areperpendicular.Why?
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SECONDARY MATH I // MODULE 8
CONNECTING ALGEBRA & GEOMETRY – 8.3
Explore(SmallGroup):
Monitorstudentsastheywork.Itmaybehelpfultorecognizethateachsetofproblemsissetupso
thereisasimplecaseandamorecomplicatedcase.Thesimplecaseisdesignedtohelpstudents
getideasforhowtoprovethemorecomplicatedcase.Keeptrackofthevariousapproachesthat
studentsusetoverifytheirclaimsandpressthemtoorganizetheirworksothatitcommunicatesto
anoutsideobserver.Studentsshouldbeshowingsidesordiagonalsareparallelorperpendicular
usingtheslopeproperties,andthedistanceformulatoshowthatsidesordiagonalsarecongruent.
Youmayalsoseestudentstrytoshowonefiguretobeaparticularquadrilateralandthenuse
transformationstoshowthatthesecondfigureisthesametype.Selectonegroupofstudentsthat
havearticulatedaclearargumentforeachtypeofquadrilateral.Besuretoalsoselectavarietyof
approachessothatstudentshaveopportunitytomakeconnectionsandbecomemorefluent.
Discuss(WholeClass):
Thediscussionshouldproceedinthesameorderasthetask,withdifferentgroupsdemonstrating
theirstrategiesforoneparallelogram,onerectangle,onerhombus,andonesquare.Selectthe
shapethatdoesnothavesidesthatareonagridline,sothatstudentsaredemonstratingthemore
challengingcases.Onerecommendedsequenceforthediscussionwouldbe:
1b.QuadrilateralEFGHisnotaparallelogramdemonstratedbyshowingthatonepairofopposite
sidesarenotparallelusingslopes.
2b.RectangleEFGHdemonstratedbyusingthedistanceformulatoshowthatthediagonalsare
congruent.
3a.ShowingquadrilateralABCDisarhombusbecausethediagonalsareperpendicularandthe
sidesarecongruent(orthatthesidesarecongruentandtheoppositesidesareparallel).
3b.ShowingthatquadrilateralEFGHisnotarhombusbecausethesidesarenotcongruent.Askif
thefigureisaparallelogram?Howdoweknow?
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SECONDARY MATH I // MODULE 8
CONNECTING ALGEBRA & GEOMETRY – 8.3
4a.ShowingthatquadrilateralABCDisasquarebecauseadjacentsidesareperpendicularand
sidesarecongruent.Askifitsufficienttoshowthatthesidesarecongruent?Howdoweknowthat
theoppositesidesareparallel?
AlignedReady,Set,Go:ConnectingAlgebraandGeometry8.3
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SECONDARY MATH I // MODULE 8
8.3
CONNECTING ALGEBRA & GEOMETRY – 8.3
READY, SET, GO!
Name
PeriodDate
READY
Topic:Interpretingtablesofvalueasorderedpairs.
Findthevalueof! ! forthegivendomain.Write!and! ! asanorderedpair.
1.! ! = 3! − 2
3.! ! = 5 ! 2.! ! = ! ! ! ! ! !
! ! !
! ! !, ! ! !, ! ! -2 -2
-2
-1 -1
-1
0
0
0
1
1
1
2
2
2
!, ! ! SET
Topic:Identifyingspecificquadrilaterals
4.a)Isthefigureattherightarectangle?Justifyyouranswer.
b)Isthefigureattherightarhombus?Justifyyouranswer.
c)Isthefigureattherightasquare?Justifyyouranswer.
GO
Topic:Calculatingperimetersofgeometricshapes
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SECONDARY MATH I // MODULE 8
CONNECTING ALGEBRA & GEOMETRY – 8.3
Findtheperimeterofeachfigurebelow.Roundanswerstothenearesthundredth.
5.
6.
7.
8.
9.
10.
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8.3