A Develop Understanding Task - uen.org Vision Project Licensed under the Creative Commons...
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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 A Develop Understanding Task A line that reflects a figure onto itself is called a line of symmetry. A figure that can be carried onto itself by a rotation is said to have rotational symmetry. Every four-sided polygon is a quadrilateral. Some quadrilaterals have additional properties and are given special names like squares, parallelograms and rhombuses. A diagonal of a quadrilateral is formed when opposite vertices are connected by a line segment. Some quadrilaterals are symmetric about their diagonals. Some are symmetric about other lines. In this task you will use rigid-motion transformations to explore line symmetry and rotational symmetry in various types of quadrilaterals. For each of the following quadrilaterals you are going to try to answer the question, “Is it possible to reflect or rotate this quadrilateral onto itself?” As you experiment with each quadrilateral, record your findings in the following chart. Be as specific as possible with your descriptions. Defining features of the quadrilateral Lines of symmetry that reflect the quadrilateral onto itself Center and angles of rotation that carry the quadrilateral onto itself A rectangle is a quadrilateral that contains four right angles. A parallelogram is a quadrilateral in which opposite sides are parallel. CC BY fdecomite https://flic.kr/p/gis7Cj 25
Transcript of A Develop Understanding Task - uen.org Vision Project Licensed under the Creative Commons...
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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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.
CC
BY
fdec
omite
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gis7
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SECONDARY MATH I // MODULE 6
TRANSFORMATIONS AND SYMMETRY – 6.5
Mathematics Vision Project
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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
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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
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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
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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
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Askstudentshowtheydeterminedwherethecenterofrotationislocatedinvariousclassesofquadrilaterals.Thisshouldleadtoadiscussionaboutthepointofintersectionofthediagonals.Askstudentswhichtypeofquadrilateralhasthemosttypesofsymmetry,andwhythismightbeso.AlignedReady,Set,Go:TransformationsandSymmetry6.5
SECONDARY MATH I // MODULE 6
TRANSFORMATIONS AND SYMMETRY – 6.5
Mathematics Vision Project
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6.5
READY Topic:Polygons,definitionandnames
1.Whatisapolygon?Describeinyourownwordswhatapolygonis.
2.Fillinthenamesofeachpolygonbasedonthenumberofsidesthepolygonhas.
NumberofSides NameofPolygon
3
4
5
6
7
8
9
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SET Topic:Kites,Linesofsymmetryanddiagonals.3.Onequadrilateralwithspecialattributesisakite.Findthegeometricdefinitionofakiteandwriteitbelowalongwithasketch.(Youcandothisfairlyquicklybydoingasearchonline.)4.Drawakiteanddrawallofthelinesofreflectivesymmetryandallofthediagonals.
LinesofReflectiveSymmetry Diagonals
READY, SET, GO! Name PeriodDate
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SECONDARY MATH I // MODULE 6
TRANSFORMATIONS AND SYMMETRY – 6.5
Mathematics Vision Project
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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
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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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