8.3 prove it! - utah education network · 2017-10-26 · 8.3 prove it! a practice understanding...
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SECONDARY MATH I // MODULE 8
CONNECTING ALGEBRA & GEOMETRY – 8.3
Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org
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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SECONDARY MATH I // MODULE 8
CONNECTING ALGEBRA & GEOMETRY – 8.3
Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org
2.
a.IsABCDarectangle?Explainhowyouknow.
b.IsEFGHarectangle?Explainhowyouknow.
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SECONDARY MATH I // MODULE 8
CONNECTING ALGEBRA & GEOMETRY – 8.3
Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org
3.
a.IsABCDarhombus?Explainhowyouknow.
b.IsEFGHarhombus?Explainhowyouknow.
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SECONDARY MATH I // MODULE 8
CONNECTING ALGEBRA & GEOMETRY – 8.3
Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org
4.
a.IsABCDasquare?Explainhowyouknow.
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SECONDARY MATH I // MODULE 8
CONNECTING ALGEBRA & GEOMETRY – 8.3
Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org
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
Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org
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
Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org
4a.ShowingthatquadrilateralABCDisasquarebecauseadjacentsidesareperpendicularandsidesarecongruent.Askifitsufficienttoshowthatthesidesarecongruent?Howdoweknowthattheoppositesidesareparallel?
AlignedReady,Set,Go:ConnectingAlgebraandGeometry8.3
SECONDARY MATH I // MODULE 8
CONNECTING ALGEBRA & GEOMETRY – 8.3
Mathematics Vision Project
Licensed under the Creative Commons Attribution CC BY 4.0
mathematicsvisionproject.org
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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SECONDARY MATH I // MODULE 8
CONNECTING ALGEBRA & GEOMETRY – 8.3
Mathematics Vision Project
Licensed under the Creative Commons Attribution CC BY 4.0
mathematicsvisionproject.org
8.3
Findtheperimeterofeachfigurebelow.Roundanswerstothenearesthundredth.5.
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7.
8.
9.
10.
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