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CONNECTING ALGEBRA & GEOMETRY – 8.3

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8.3 Prove It!

A Practice Understanding Task

Inthistaskyouneedtouseallthethingsyouknowaboutquadrilaterals,distance,andslopetoprovethattheshapesareparallelograms,rectangles,rhombi,orsquares.Besystematicandbesurethatyougivealltheevidencenecessarytoverifyyourclaim.

1.

a.IsABCDaparallelogram?Explainhowyouknow.

b.IsEFGHaparallelogram?Explainhowyouknow.

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CONNECTING ALGEBRA & GEOMETRY – 8.3

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2.

a.IsABCDarectangle?Explainhowyouknow.

b.IsEFGHarectangle?Explainhowyouknow.

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CONNECTING ALGEBRA & GEOMETRY – 8.3

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3.

a.IsABCDarhombus?Explainhowyouknow.

b.IsEFGHarhombus?Explainhowyouknow.

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CONNECTING ALGEBRA & GEOMETRY – 8.3

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4.

a.IsABCDasquare?Explainhowyouknow.

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CONNECTING ALGEBRA & GEOMETRY – 8.3

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8.3 Prove It! – Teacher Notes

A Practice Understanding Task

Purpose:Thepurposeofthistaskistosolidifystudentunderstandingofquadrilateralsandtoconnecttheirunderstandingofgeometryandalgebra.Inthetasktheywilluseslopesanddistancetoshowthatparticularquadrilateralsareparallelograms,rectangles,rhombiorsquares.Thistaskwillalsostrengthenstudentunderstandingofjustificationandproof,andtheneedtoputforthacompleteargumentbaseduponsoundmathematicalreasoning.CoreStandardsFocus:

G.GPE.4Usecoordinatestoprovesimplegeometrictheoremsalgebraically.Forexample,proveordisprovethatafiguredefinedbyfourgivenpointsinthecoordinateplaneisarectangle.

RelatedStandards:G.GPE.5,G.GPE.7

StandardsforMathematicalPracticeofFocusintheTask:

SMP3–Constructviableargumentsandcritiquethereasoningofothers

SMP6–Attendtoprecision

TheTeachingCycle:

Launch(WholeClass):

Launchthetaskwithadiscussionofwhatstudentsknowaboutthepropertiesofquadrilaterals,forinstancethatarhombushastwopairsofparallelsides(makingitaparallelogram),congruentsides,andperpendiculardiagonals.Discusswhatyouwouldneedtoshowtoproveaclaimthatafigureisaparticularquadrilateral.Forinstance,itisnotenoughtoshowthatashapeisarhombusbyshowingthatthetwopairsofsidesareparallel,butitwouldbeenoughtoshowthatthediagonalsareperpendicular.Why?

SECONDARY MATH I // MODULE 8

CONNECTING ALGEBRA & GEOMETRY – 8.3

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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?

SECONDARY MATH I // MODULE 8

CONNECTING ALGEBRA & GEOMETRY – 8.3

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4a.ShowingthatquadrilateralABCDisasquarebecauseadjacentsidesareperpendicularandsidesarecongruent.Askifitsufficienttoshowthatthesidesarecongruent?Howdoweknowthattheoppositesidesareparallel?

AlignedReady,Set,Go:ConnectingAlgebraandGeometry8.3

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CONNECTING ALGEBRA & GEOMETRY – 8.3

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8.3

READY Topic:Interpretingtablesofvalueasorderedpairs.

Findthevalueof! ! forthegivendomain.Write!and! ! asanorderedpair.1.! ! = 3! − 2 2.! ! = !! 3.! ! = 5!

! ! ! !, ! !

-2

-1

0

1

2

! ! ! !, ! !

-2

-1

0

1

2

! ! ! !, ! !

-2

-1

0

1

2

SET Topic:Identifyingspecificquadrilaterals

4.a)Isthefigureattherightarectangle?Justifyyouranswer.b)Isthefigureattherightarhombus?Justifyyouranswer.c)Isthefigureattherightasquare?Justifyyouranswer.

GO

Topic:Calculatingperimetersofgeometricshapes

READY, SET, GO! Name PeriodDate

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CONNECTING ALGEBRA & GEOMETRY – 8.3

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8.3

Findtheperimeterofeachfigurebelow.Roundanswerstothenearesthundredth.5.

6.

7.

8.

9.

10.

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