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SECONDARY MATH I // MODULE 6

TRANSFORMATIONS AND SYMMETRY – 6.5

Mathematics Vision Project

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6. 5 Symmetries of Quadrilaterals

A Develop Understanding Task

Alinethatreflectsafigureontoitselfiscalledalineofsymmetry.Afigurethatcanbe

carriedontoitselfbyarotationissaidtohaverotationalsymmetry.

Everyfour-sidedpolygonisaquadrilateral.Somequadrilateralshaveadditionalproperties

andaregivenspecialnameslikesquares,parallelogramsandrhombuses.Adiagonalofa

quadrilateralisformedwhenoppositeverticesareconnectedbyalinesegment.Somequadrilaterals

aresymmetricabouttheirdiagonals.Somearesymmetricaboutotherlines.Inthistaskyouwilluse

rigid-motiontransformationstoexplorelinesymmetryandrotationalsymmetryinvarioustypesof

quadrilaterals.

Foreachofthefollowingquadrilateralsyouaregoingtotrytoanswerthequestion,“Isit

possibletoreflectorrotatethisquadrilateralontoitself?”Asyouexperimentwitheachquadrilateral,

recordyourfindingsinthefollowingchart.Beasspecificaspossiblewithyourdescriptions.

Definingfeaturesofthequadrilateral

Linesofsymmetrythatreflectthequadrilateral

ontoitself

Centerandanglesofrotationthatcarrythequadrilateral

ontoitselfArectangleisaquadrilateralthatcontainsfourrightangles.

Aparallelogramisaquadrilateralinwhichoppositesidesareparallel.

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SECONDARY MATH I // MODULE 6

TRANSFORMATIONS AND SYMMETRY – 6.5

Mathematics Vision Project

Licensed under the Creative Commons Attribution CC BY 4.0

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Atrapezoidisaquadrilateralwithonepairofoppositesidesparallel.Isitpossibletoreflectorrotateatrapezoidontoitself?Drawatrapezoidbasedonthisdefinition.Thenseeifyoucanfind:

• anylinesofsymmetry,or• anycentersofrotationalsymmetry,

thatwillcarrythetrapezoidyoudrewontoitself.Ifyouwereunabletofindalineofsymmetryoracenterofrotationalsymmetryforyourtrapezoid,seeifyoucansketchadifferenttrapezoidthatmightpossesssometypeofsymmetry.

Arhombusisaquadrilateralinwhichallsidesarecongruent.

Asquareisbotharectangleandarhombus.

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SECONDARY MATH I // MODULE 6

TRANSFORMATIONS AND SYMMETRY – 6.5

Mathematics Vision Project

Licensed under the Creative Commons Attribution CC BY 4.0

mathematicsvisionproject.org

6. 5 Symmetries of Quadrilaterals – Teacher Notes A Develop Understanding Task

Purpose:Inthislearningcycle,studentsfocusonclassesofgeometricfiguresthatcanbecarried

ontothemselvesbyatransformation—figuresthatpossessalineofsymmetryorrotational

symmetry.Inthistasktheideaof“symmetry”issurfacedrelativetofindinglinesthatreflecta

figureontoitself,ordeterminingifafigurehasrotationalsymmetrybyfindingacenterofrotation

aboutwhichafigurecanberotatedontoitself.Thisworkisintendedtobeexperimental(e.g.,

foldingpaper,usingtransparencies,usingtechnology,measuringwithrulerandprotractor,etc.),

withthedefinitionsofreflectionandrotationbeingcalledupontosupportstudents’claimsthata

figurepossessessometypeofsymmetry.Theparticularclassesofgeometricfiguresconsideredin

thistaskarevarioustypesofquadrilaterals.

CoreStandardsFocus:

G.CO.3Givenarectangle,parallelogram,trapezoid,orregularpolygon,describetherotationsand

reflectionsthatcarryitontoitself.

G.CO.6Usegeometricdescriptionsofrigidmotionstotransformfiguresandtopredicttheeffectof

agivenrigidmotiononagivenfigure.

RelatedStandards:G.CO.4,G.CO.5

StandardsforMathematicalPractice:

SMP7–Lookforandmakeuseofstructure

AdditionalResourcesforTeachers:

Acopyofthechartfromthetaskcanbefoundattheendofthissetofteachernotes.Thischartcan

beprintedforusewithstudentswhomaybeaccessingthetaskonacomputerortablet.In

addition,eachofthequadrilateralshasbeenprovidedonamastercopythatcanbereproducedand

SECONDARY MATH I // MODULE 6

TRANSFORMATIONS AND SYMMETRY – 6.5

Mathematics Vision Project

Licensed under the Creative Commons Attribution CC BY 4.0

mathematicsvisionproject.org

distributedtostudents.Studentscancutouttheindividualquadrilateralsinordertomanipulatethem,suchasfoldingthemalongadiagonalorrotatingthemaboutapoint.TheTeachingCycle:

Launch(WholeClass):

Discusstheconceptofsymmetryintermsoffindingarigid-motiontransformationthatcarriesageometricfigureontoitself.Helpstudentsrecognizethattwosuchtypesofsymmetriesexist:alineofsymmetrymightexistthatreflectsafigureontoitself,oracenterofrotationmightexistaboutwhichafiguremightberotatedontoitself.Alsoremindstudentsofthedefinitionsof“quadrilateral”and“diagonal”givingintheintroductionofthetask.Studentsaretoexperimentwiththevarioustypesofquadrilateralslistedinthetasktodetermineiftheycanfindanylinesofsymmetryorcentersandanglesofrotationthatwillcarrythegivenquadrilateralontoitself.Youwillneedtodecidewhattoolstomakeavailableforthisinvestigation.Forexample,youcouldprovidecut-outsofeachofthefigureswhichwouldallowstudentstofindlinesofsymmetrybyfoldingthefiguresontothemselves(notethatahandoutofthesetoffiguresisprovidedattheendoftheteachernotes).Thiswouldalsobeagoodtasktosupportusingdynamicgeometrysoftwareprograms,suchasGeometer’sSketchpadorGeogebra.Ifyouusetechnology,studentswillneedtobeprovidedwithasetofwell-constructedquadrilaterals,sotheycanfocusonsearchingforlinesofsymmetryandcentersofrotation,ratherthanontheconstructionofthegeometricfiguresthemselves.Explore(SmallGroup):

Sincestudentsaredealingwithclassesofquadrilaterals,ratherthanindividualquadrilaterals,inadditiontofindingthelineofsymmetryorthecenterandangleofarotation,theyshouldalsoprovidesometypeofjustificationastohowtheyknowthatthissymmetryexistsforallmembersoftheclass.Thegivendefinitionsforeachquadrilateralshouldsupportmakingsuchanargument.Forexample,ifstudentssaythatthediagonalofasquareisalineofsymmetry,theymightnotethatdistanceandanglearepreservedbythisreflectionsinceadjacentsidesofasquarearecongruent

SECONDARY MATH I // MODULE 6

TRANSFORMATIONS AND SYMMETRY – 6.5

Mathematics Vision Project

Licensed under the Creative Commons Attribution CC BY 4.0

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andoppositeanglesofasquarearebothrightangles.Trytopressstudentstomoveawayfrom

basingtheirdecisionsaboutlinesofsymmetryorcentersofrotationsimplyonintuitionand“it

lookslikeitworks”typeofjustification,andtowardsargumentsbasedonthedefinitionsofthe

rigid-motiontransformationsandthedefiningpropertiesofthegeometricfigures.

Lookforstudentswhofindallofthelinesofsymmetryordescribeallofthepossiblerotationsthat

mightexistforeachtypeofquadrilateral.Forexample,asquarehastwodifferenttypesoflinesof

symmetry:thediagonals,andthelinespassingthroughthemidpointsofoppositesides.Hence,

therearefourlinesofsymmetryinasquare.Thepointwherethetwodiagonalsofasquare

intersectlocatesthecenterofrotationfordescribingtherotationalsymmetryofasquare.Asquare

canberotated90°,180°,270°or360°aboutthiscenterofrotation.Eachreflectionorrotation

carriesasegmentontoanothersegmentofthesamelength,orarightangleontoanotherright

angle,duetothedefiningpropertiesofasquare.

Watchformisconceptionsthatmightarise,suchasthediagonalsofaparallelogrambeingidentified

aslinesofreflection.Experimentationwithtechnologyorpaperfoldingwilldisprovethis

conjecture,butitisimportanttohavestudentsdescribewhytheyinitiallythoughtitwastrue,and

howtheymightconvincethemselvesthatthisconjectureisn’ttruebasedonthedefinitionofa

reflection.

Discuss(WholeClass):

Startthediscussionbyaskingifthediagonalsofaparallelogramarelinesofsymmetryforthe

parallelogram.Askstudentshowtheyknowadiagonalisnotalineofsymmetry.Iftheironly

argumentsareexperimentalinnature(e.g.,“ifyoufolditonthediagonal,oppositeverticesdon’t

matchup”or“whenIusedthediagonalasamirrorlineinGSPitdidn’twork”),pressforan

explanationbasedontheessentialideasofareflection(e.g.,“sinceadjacentsidesofaparallelogram

aren’tnecessarilycongruent,wecan’tfindalineofreflectionthatwillreflectasideofa

parallelogramontoanadjacentsidesothatdistanceispreserved”).

SECONDARY MATH I // MODULE 6

TRANSFORMATIONS AND SYMMETRY – 6.5

Mathematics Vision Project

Licensed under the Creative Commons Attribution CC BY 4.0

mathematicsvisionproject.org

Askstudentshowtheydeterminedwherethecenterofrotationislocatedinvariousclassesofquadrilaterals.Thisshouldleadtoadiscussionaboutthepointofintersectionofthediagonals.Askstudentswhichtypeofquadrilateralhasthemosttypesofsymmetry,andwhythismightbeso.AlignedReady,Set,Go:TransformationsandSymmetry6.5

SECONDARY MATH I // MODULE 6

TRANSFORMATIONS AND SYMMETRY – 6.5

Mathematics Vision Project

Licensed under the Creative Commons Attribution CC BY 4.0

mathematicsvisionproject.org

6.5

READY Topic:Polygons,definitionandnames

1.Whatisapolygon?Describeinyourownwordswhatapolygonis.

2.Fillinthenamesofeachpolygonbasedonthenumberofsidesthepolygonhas.

NumberofSides NameofPolygon

3

4

5

6

7

8

9

10

SET Topic:Kites,Linesofsymmetryanddiagonals.3.Onequadrilateralwithspecialattributesisakite.Findthegeometricdefinitionofakiteandwriteitbelowalongwithasketch.(Youcandothisfairlyquicklybydoingasearchonline.)4.Drawakiteanddrawallofthelinesofreflectivesymmetryandallofthediagonals.

LinesofReflectiveSymmetry Diagonals

READY, SET, GO! Name PeriodDate

27

SECONDARY MATH I // MODULE 6

TRANSFORMATIONS AND SYMMETRY – 6.5

Mathematics Vision Project

Licensed under the Creative Commons Attribution CC BY 4.0

mathematicsvisionproject.org

6.5

5.Listalloftherotationalsymmetryforakite.

6.Arelinesofsymmetryalsodiagonalsinapolygon?Explain.

6.Arealldiagonalsalsolinesofsymmetryinapolygon?Explain.

7.Whichquadrilateralshavediagonalsthatarenotlinesofsymmetry?Namesomeanddrawthem.

8.Doparallelogramshavediagonalsthatarelinesofsymmetry?Ifso,drawandexplain.Ifnotdrawand

explain.

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SECONDARY MATH I // MODULE 6

TRANSFORMATIONS AND SYMMETRY – 6.5

Mathematics Vision Project

Licensed under the Creative Commons Attribution CC BY 4.0

mathematicsvisionproject.org

6.5

GO Topic:Equationsforparallelandperpendicularlines.

FindtheequationofalinePARALLELtothegiveninfoandthroughtheindicated

y-intercept.

FindtheequationofalinePERPENDICULARtothe

givenlineandthroughtheindicatedy-intercept.

9.Equationofaline:! = 4! + 1.

a.Parallellinethroughpoint(0,-7):

b.Perpendiculartothelinelinethroughpoint(0,-7):

10.Tableofaline:

x y3 -84 -105 -126 -14

a.Parallellinethroughpoint(0,8):

b.Perpendiculartothelinethroughpoint(0,8):

11.Graphofaline:

a.Parallellinethroughpoint(0,-9):

b.Perpendiculartothelinethroughpoint(0,-9):

-5

-5

5

5

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