6.7 quadrilaterals—beyond definition e n€¦ · we have found that many different quadrilaterals...
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SECONDARY MATH I // MODULE 6
TRANSFORMATIONS AND SYMMETRY – 6.7
Mathematics Vision Project
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6.7 Quadrilaterals—Beyond
Definition
A Practice Understanding Task
Wehavefoundthatmanydifferentquadrilateralspossesslinesofsymmetryand/orrotational
symmetry.Inthefollowingchart,writethenamesofthequadrilateralsthatarebeingdescribedin
termsoftheirsymmetries.
Whatdoyounoticeabouttherelationshipsbetweenquadrilateralsbasedontheirsymmetriesand
highlightedinthestructureoftheabovechart?
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TRANSFORMATIONS AND SYMMETRY – 6.7
Mathematics Vision Project
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Basedonthesymmetrieswehaveobservedinvarioustypesofquadrilaterals,wecanmakeclaims
aboutotherfeaturesandpropertiesthatthequadrilateralsmaypossess.
1.Arectangleisaquadrilateralthatcontainsfourrightangles.
Basedonwhatyouknowabouttransformations,whatelsecanwesayaboutrectanglesbesidesthe
definingpropertythat“allfouranglesarerightangles?”Makealistofadditionalpropertiesof
rectanglesthatseemtobetruebasedonthetransformation(s)oftherectangleontoitself.Youwill
wanttoconsiderpropertiesofthesides,theangles,andthediagonals.Thenjustifywhythe
propertieswouldbetrueusingthetransformationalsymmetry.
2.Aparallelogramisaquadrilateralinwhichoppositesidesareparallel.
Basedonwhatyouknowabouttransformations,whatelsecanwesayaboutparallelogramsbesides
thedefiningpropertythat“oppositesidesofaparallelogramareparallel?”Makealistofadditional
propertiesofparallelogramsthatseemtobetruebasedonthetransformation(s)ofthe
parallelogramontoitself.Youwillwanttoconsiderpropertiesofthesides,anglesandthediagonals.
Thenjustifywhythepropertieswouldbetrueusingthetransformationalsymmetry.
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3.Arhombusisaquadrilateralinwhichallfoursidesarecongruent.
Basedonwhatyouknowabouttransformations,whatelsecanwesayaboutarhombusbesidesthe
definingpropertythat“allsidesarecongruent?”Makealistofadditionalpropertiesofrhombuses
thatseemtobetruebasedonthetransformation(s)oftherhombusontoitself.Youwillwantto
considerpropertiesofthesides,anglesandthediagonals.Thenjustifywhythepropertieswouldbe
trueusingthetransformationalsymmetry.
4.Asquareisbotharectangleandarhombus.
Basedonwhatyouknowabouttransformations,whatcanwesayaboutasquare?Makealistof
propertiesofsquaresthatseemtobetruebasedonthetransformation(s)ofthesquaresontoitself.
Youwillwanttoconsiderpropertiesofthesides,anglesandthediagonals.Thenjustifywhythe
propertieswouldbetrueusingthetransformationalsymmetry.
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Inthefollowingchart,writethenamesofthequadrilateralsthatarebeingdescribedintermsof
theirfeaturesandproperties,andthenrecordanyadditionalfeaturesorpropertiesofthattypeof
quadrilateralyoumayhaveobserved.Bepreparedtosharereasonsforyourobservations.
Whatdoyounoticeabouttherelationshipsbetweenquadrilateralsbasedontheircharacteristics
andthestructureoftheabovechart?
Howarethechartsatthebeginningandendofthistaskrelated?Whatdotheysuggest?
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6.7 Quadrilaterals—Beyond Definition –
Teacher Notes A Practice Understanding Task
Purpose:Thistaskallowsstudentstoextendtheirworkwithsymmetriesofquadrilateralsandpracticemakingconjecturesaboutgeometricfiguresthatarebasedonreasoningwiththedefinitionsofreflectionandrotation.TheworkofthistaskwillberevisitedinMathematicsII,wherestudentswillbeaskedtocreateformalproofsfortheconjecturestheyaremakinginthistaskaboutthepropertiesofdifferenttypesofquadrilaterals.Therefore,whilethisisclassifiedasapracticeunderstandingtask,themathematicsstudentsshouldbepracticingismakingandjustifyingconjecturesaboutgeometricfiguresbasedonthedefinitionsofrigid-motiontransformations,ratherthanpracticingknowledgeaboutthespecificpropertiesofdifferenttypesofquadrilaterals.Whateverpropertiesaboutsides,anglesanddiagonalsofquadrilateralsstudentssurfaceissufficientforthistask.
CoreStandardsFocus:G.CO.3Givenarectangle,parallelogram,trapezoid,orregularpolygon,describetherotationsandreflectionsthatcarryitontoitself.G.CO.4Developdefinitionsofrotations,reflections,andtranslationsintermsofangles,circles,perpendicularlines,parallellines,andlinesegments.G.CO.6Usegeometricdescriptionsofrigidmotionstotransformfiguresandtopredicttheeffectofagivenrigidmotiononagivenfigure.RelatedStandards:G.CO.11
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StandardsforMathematicalPracticeoffocusinthetask:
SMP3–Constructviableargumentsandcritiquethereasoningofothers
SMP7–Lookandmakeuseofstructure
AdditionalResourcesforTeachers:
Cutoutsortrackingpaperforthequadrilateralsfrom6.5maybeusedagaininthistask.Ananswer
keyforthequestionsinthetaskcanalsobefoundasaseparatepageattheendoftheseteacher
notes.Itisrecommendedthatyouworkthroughthetaskyourselfbeforeconsultingtheanswerkey
todevelopabettersenseofhowyourstudentsmightengageinthetask.
TheTeachingCycle:
Launch(WholeClass):
Givestudentsafewminutestoexaminethechartonthefirstpageofthetask.Theyshould
summarizetheirworkwithsymmetriesofquadrilateralsbyidentifyingthetypesofquadrilaterals
thatpossessthesymmetriesbeingdescribedineachportionofthechart.Discussthequestion:
Whatdoyounoticeabouttherelationshipsbetweenquadrilateralsbasedontheirsymmetriesand
highlightedinthestructureoftheabovechart?Helpstudentsnoticethatspecialquadrilaterals
inheritthesymmetriesofallcategoriesofquadrilateralstowhichtheybelong.
Remindstudentsthatintheprevioustasktheywereabletomakesomeconjecturesabout
propertiesofregularpolygonsbasedonfeaturesthatrevealedthemselveswhentheylookedatthe
symmetriesofthepolygons.Inthistasktheywillreturntoquadrilateralsandseewhatconjectures
theymightmakeaboutrelationshipsbetweensides,anglesanddiagonalsofdifferenttypesof
quadrilateralsbasedonthesymmetriesofthequadrilateral.Havestudentspracticemaking
conjecturesbyworkingonproblem1:whatelsemightbetrueaboutarectangle—inadditionto
beingaquadrilateralwithfourrightangles—andhowcantheyjustifytheseobservationsbasedon
thedefinitionsofreflectionandrotation.
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Afterafewminutessummarizewhatgroupshaveobservedandsharetheargumentstheymight
maketosupporttheirclaims.Don’tbringuppropertiesthatstudentshavenotobservedontheir
own,butyoumightpromptfurtherdiscussionbyaskingifstudentsnoticedanythingaboutthe
diagonalsofarectanglethatcouldbejustifiedbasedonlinesofsymmetryorrotationalsymmetry.If
theyhaven’talreadynoticedanythingaboutthediagonalssuggestthattheycouldaddthinking
aboutthediagonalstotheirlistofthingstopayattentiontoduringtheirexploration.
Nowthatstudentshaveasenseoftheworkthatisexpectedofthemonthistask,assignthemto
workonmakingconjecturesabouttheremainingquadrilaterals.
Explore(SmallGroup):
Listenforthetypesofconjecturesstudentsaremakingabouteachquadrilateral.Pressthemtolook
formoreconjecturesbyaskingquestionslike,“Isthereanythingyoucansayaboutoppositesides?
Oppositeangles?Adjacentsides?Adjacentangles?Thediagonalsandthewaytheyinteractwith
thesidesandanglesandwitheachother?”
Wheneverstudentsstateaconjectureaskthemwhytheythinkthatconjectureistrue—isitbased
onintuitiveguessing,experimentationwithtools,oronreasoningwiththedefinitionsofreflection
androtation?Pressforjustificationsthatarebasedonreasoningwithtransformations.
Discuss(WholeClass):
Sincethepurposeofthistaskistopracticemakingconjecturesbasedonsymmetry,thereareno
specificconjecturesthatneedtobehighlighted.Selectstudentstoshareconjecturesforwhichthey
havesomejustificationbasedontransformations.Asconjecturesaresharedanddiscussed,have
studentslistthoseconjecturesintheappropriateplacesonthechartattheendofthetask.(They
willfirstneedtolabeltheportionsofthechartbasedonthedefiningpropertiesofthedifferent
typesofquadrilaterals.Asthechartevolves,youwillwanttorelatethechartatthebeginningofthis
tasktothechartattheendasawayofacknowledgingtherolethatsymmetryplaysintheinherited
propertiesthatquadrilateralspossessbasedonthedifferentcategoriesofquadrilateralstowhich
theybelong.)
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Possiblelistsofpropertiesofquadrilateralsthatmaysurfacearesummarizedinthefollowingchart.However,notallofthesepropertiesneedtobediscussed.
AlignedReady,Set,Go:TransformationsandSymmetry6.7
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6.7
READY Topic:Definingcongruenceandsimilarity.
1.Whatdoyouknowabouttwofiguresiftheyarecongruent?2.Whatdoyouneedtoknowabouttwofigurestobeconvincedthatthetwofiguresarecongruent?3.Whatdoyouknowabouttwofiguresiftheyaresimilar?4.Whatdoyouneedtoknowabouttwofigurestobeconvincedthatthetwofiguresaresimilar? SET Topic:Classifyingquadrilateralsbasedontheirproperties.Usingtheinformationgivendeterminethemostaccurateclassificationofthequadrilateral.5.Has1800rotationalsymmetry. 6.Has900rotationalsymmetry.7.Hastwolinesofsymmetrythatarediagonals. 8.Hastwolinesofsymmetrythatarenot diagonals.9.Hascongruentdiagonals. 10.Hasdiagonalsthatbisecteachother.11.Hasdiagonalsthatareperpendicular. 12.Hascongruentangles.
READY, SET, GO! Name PeriodDate
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6.7
GO Topic:Slopeanddistance.Findtheslopebetweeneachpairofpoints.Then,usingthePythagoreanTheorem,findthedistancebetweeneachpairofpoints.Distancesshouldbeprovidedinthemostexactform.13.(-3,-2),(0,0) a.Slope:b.Distance:
14.(7,-1),(11,7) a.Slope:b.Distance:
15.(-10,13),(-5,1)a.Slope:b.Distance:
16.(-6,-3),(3,1) a.Slope:b.Distance:
17.(5,22),(17,28)a.Slope:b.Distance:
18.(1,-7),(6,5) a.Slope:b.Distance:
S
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