SECONDARY MATH I // MODULE 6

TRANSFORMATIONS AND SYMMETRY – 6.6

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6.6 Symmetries of Regular

Polygons

A Solidify Understanding Task

Alinethatreflectsafigureontoitselfiscalledalineofsymmetry.Afigurethatcanbecarriedonto

itselfbyarotationissaidtohaverotationalsymmetry.Adiagonalofapolygonisanyline

segmentthatconnectsnon-consecutiveverticesofthepolygon.

Foreachofthefollowingregularpolygons,describetherotationsandreflectionsthatcarryitonto

itself:(beasspecificaspossibleinyourdescriptions,suchasspecifyingtheangleofrotation)

1. Anequilateraltriangle

2. Asquare

3. Aregularpentagon

4. Aregularhexagon

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SECONDARY MATH I // MODULE 6

TRANSFORMATIONS AND SYMMETRY – 6.6

Mathematics Vision Project

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5. Aregularoctagon

6. Aregularnonagon

Whatpatternsdoyounoticeintermsofthenumberandcharacteristicsofthelinesofsymmetryina

regularpolygon?

Whatpatternsdoyounoticeintermsoftheanglesofrotationwhendescribingtherotational

symmetryinaregularpolygon?

31

SECONDARY MATH I // MODULE 6

TRANSFORMATIONS AND SYMMETRY – 6.6

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6.6 Symmetries of Regular Polygons – Teacher Notes A Solidify Understanding Task

Purpose:Inthistask,studentscontinuetofocusonclassesofgeometricfiguresthatcanbecarriedontothemselvesbyatransformation—figuresthatpossessalineofsymmetryorrotationalsymmetry.Studentssolidifytheideaof“symmetry”relativetofindinglinesthatreflectafigureontoitself,ordeterminingifafigurehasrotationalsymmetrybyfindingacenterofrotationaboutwhichafigurecanberotatedontoitself.Theyalsolookforanddescribethestructurethatdeterminesifafigurepossessessometypeofsymmetry.Thisworkcanbeexperimental(e.g.,foldingpaper,usingtransparencies,usingtechnology,measuringwithrulerandprotractor,etc.),ortheoretical,withthedefinitionsofreflectionandrotationbeingcalledupontosupportstudents’claimsthatafigurepossessessometypeofsymmetry.Theparticularclassesofgeometricfiguresconsideredinthistaskarevarioustypesofregularpolygons,andstudentswilllookforpatternsinthetypesandnumbersoflinesofsymmetryaregularpolygonwithanoddnumberofsidespossesses,versusthosewithanevennumberofsides.Theyshouldalsonoteapatternbetweenthesmallestangleofrotationthatcarriesaregularpolygonontoitselfandthenumberofsidesofthepolygon.CoreStandardsFocus:G.CO.3Givenarectangle,parallelogram,trapezoid,orregularpolygon,describetherotationsandreflectionsthatcarryitontoitself.G.CO.6Usegeometricdescriptionsofrigidmotionstotransformfiguresandtopredicttheeffectofagivenrigidmotiononagivenfigure.RelatedStandards:G.CO.4,G.CO.5

SECONDARY MATH I // MODULE 6

TRANSFORMATIONS AND SYMMETRY – 6.6

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StandardsforMathematicalPracticeoffocusinthetask:

SMP7–Lookforandmakeuseofstructure

SMP6–Attendtoprecision

SMP8–Lookforandexpressregularityinrepeatedreasoning

AdditionalResourcesforTeachers:

Imagesofeachoftheregularpolygonshavebeenprovidedonamastercopythatcanbereproduced

anddistributedtostudents.Studentscancutouttheindividualpolygonsinordertomanipulate

them,ortheycantracethemontracingpaperinordertofoldthemalongadiagonalorrotatethem

aboutapoint.

TheTeachingCycle:

Launch(WholeClass):

Basedonthelevelofstudentthinkingthatexistsinyourclass,decideifyouwanttheworkofthis

tasktobeexperimental(e.g.,usingtools,cut-outsand/ordynamicgeometricsoftware),or

theoretical,makingdecisionsbasedonthedefinitionsofreflectionsandrotations.Introducethe

taskasbeingsimilarinnaturetotheprevioustask.Ifyouhavedecidedtoprovideexperimental

tools,outlinewhatisavailableforstudents(notethatahandoutofthefiguresisprovidedattheend

oftheteachernotes,ifneeded.)Youmightalsochoosetojusthandoutthetaskandletstudents

decideiftheywanttodrawuponreasoningortoolstosupporttheirwork.Or,youmightwantto

pressstudentstoanalyzethefiguresusingreasoningbasedonthedefinitionsofreflectionand

rotation.

Beforehavingstudentsstartonthetask,readthelasttwoquestionstogether,andpointoutthatthis

isthegoalofthetask:tolookforpatternsinthenumberandcharacteristicsofthelinesofsymmetry

inaregularn-gon,andtolookforpatternsthatdescribethenatureoftherotationalsymmetryina

regularn-gon.Encouragestudentstokeepthesegoalsinmindastheyexplore.

Explore(SmallGroup):

Listenforhowstudentsaredeterminingthetypesofsymmetrythatexistsforeachregularpolygon,

andmakesuretheyareidentifyingbothtypesofsymmetry—linesofsymmetryandrotational

SECONDARY MATH I // MODULE 6

TRANSFORMATIONS AND SYMMETRY – 6.6

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symmetry.Forrotationalsymmetry,makesuretheyareidentifyingallpossibleanglesofrotation.

Asyouobservestudentswork,youmaywanttosuggestadditionaltoolsforexploration,orsettools

asideifyoufeelstudentsarecapableofnotingthesymmetriesbyreasoningwiththedefinitionsof

rotationandreflectionandthepropertiesofregularpolygons.

Itisimportantforstudentstoexaminethelasttwoquestionsinsmallgroups,beforemovingtoa

wholeclassdiscussion.Keepremindingstudentsofthesegoals,evenbeforetheyhaveexaminedall

ofthelistedregularpolygons.Thatway,studentswillbeabletoattendtothepresentationsduring

thediscussion,eveniftheyhaven’texaminedthecompletesetofpolygons.

Discuss(WholeClass):

Thediscussionshouldfocusonthelasttwoquestions,andstudentsmightdrawuponspecific

examplesofregularpolygonstosupporttheirconjectures.

Askstudentstostateaconjectureastothenumberoflinesofsymmetryinaregularn-gon,andto

providesomejustificationoftheirconjecture.Ifstudentshavenotnoticedanypatternsinthe

numberoflinesofsymmetry,makeatableontheboardconsistingof“numberofsides”astheinput

and“numberoflinesofsymmetry”astheoutput.Askwhattheymayhavenoticedaboutthetypes

oflinesofsymmetryinaregularpolygonwithanevennumberofsidesversusanoddnumberof

sides.Whataccountsforthedifferencesinthewaystheylocatedthelinesofsymmetry?

Studentsshouldnoticethatinaregularpolygonwithanoddnumberofsidestheycanonlydraw

linesofsymmetry(orlocateacreaselinethatfoldsthepolygonontoitself)byusingthelinesthat

passthroughavertexpointandthemidpointoftheoppositeside.Sincesuchalinecanbedrawn

througheachvertex,aregularn-gonwithanoddnumberofsideswillpossessnlinesofsymmetry.

Inaregularpolygonwithanevennumberofsidesyoucandraw(orfold)alineofreflectionthrough

oppositevertices.Sinceonlyonelineofsymmetryexistsforeachpairofoppositevertices,thereare

n/2suchlinesofsymmetry.Youcanalsodrawlinesofsymmetrythroughthemidpointsofopposite

sidesofthepolygon—thesidesthatareparalleltoeachother.Sinceonlyonelineofsymmetry

SECONDARY MATH I // MODULE 6

TRANSFORMATIONS AND SYMMETRY – 6.6

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existsforeachpairofoppositesides,therearealson/2suchlinesofsymmetry.Consequently,a

regularpolygonwithanevennumberofsidesalsohasn/2+n/2=nlinesofsymmetry,butfor

differentreasonsthaninthecaseofregularpolygonswithanoddnumberofsides.Makesurethat

theargumentsfornlinesofsymmetryinanyregularpolygonarebasedonthestructureofthe

geometricfiguresthemselves,andnotjustonthepatternobservedinthetable.

Turnthefocusofthediscussiontothesecondquestion:Whatpatternsdoyounoticeintermsofthe

anglesofrotationwhendescribingtherotationalsymmetryinaregularpolygon?Againitmaybe

helpfultocreateatablewiththeinputrepresenting“numberofsides”andoutputrepresenting“the

smallestangleofrotation”.Pointoutthateveryregularpolygoncanberotatedontoitselfby

rotating360°aboutthepointofintersectionofthediagonalsofthepolygon.Howmightthesmallest

angleofrotationberelatedtothis360°rotation?Youmightdrawthelinesegmentsbetweenapair

ofconsecutiveverticesandthecenterofrotationandask,“Whatisthemeasureofthisangle,and

howdoyouknow?”Studentsshouldnoticethatthesmallestangleofrotationinaregularn-gonis

360°/nandtheyshouldbeabletojustifywhythisisso.Theyshouldalsonotethatanywhole-

numbermultipleofthissmallestangleofrotationisalsoanangleofrotationforthepolygon.

AlignedReady,Set,Go:TransformationsandSymmetry6.6

SECONDARY MATH I // MODULE 6

TRANSFORMATIONS AND SYMMETRY – 6.6

Mathematics Vision Project

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6.6

READY

Topic:Rotationalsymmetry,connectedtofractionsofaturnanddegrees.

1.Whatfractionofaturndoesthewagonwheel

belowneedtoturninordertoappearthevery

sameasitdoesrightnow?Howmanydegreesof

rotationwouldthatbe?

2.Whatfractionofaturndoesthepropeller

belowneedtoturninordertoappearthevery

sameasitdoesrightnow?Howmanydegreesof

rotationwouldthatbe?

3.WhatfractionofaturndoesthemodelofaFerriswheelbelowneedtoturninordertoappearthe

verysameasitdoesrightnow?Howmanydegreesof

rotationwouldthatbe?

READY, SET, GO! Name PeriodDate

32

SECONDARY MATH I // MODULE 6

TRANSFORMATIONS AND SYMMETRY – 6.6

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6.6

SET Topic:Findinganglesofrotationalsymmetryforregularpolygons,linesofsymmetryanddiagonals

4.Drawthelinesofsymmetryforeachregularpolygon,fillinthetableincludinganexpressionforthe

numberoflinesofsymmetryinan-sidedpolygon.

5.Drawallofthediagonalsineachregularpolygon.Fillinthetableandfindapattern,isitlinear,

exponentialorneither?Howdoyouknow?Attempttofindanexpressionforthenumberofdiagonalsin

an-sidedpolygon.

Numberof

Sides

Numberoflines

ofsymmetry

3

4

5

6

7

8

n

Numberof

Sides

Numberof

diagonals

3

4

5

6

7

8

n

33

SECONDARY MATH I // MODULE 6

TRANSFORMATIONS AND SYMMETRY – 6.6

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6.6

6.Findtheangle(s)ofrotationthatwillcarrythe12sidedpolygonbelowontoitself.

7.Whataretheanglesofrotationfora20-gon?Howmanylinesofsymmetry(linesofreflection)willit

have?

8.Whataretheanglesofrotationfora15-gon?Howmanylineofsymmetry(linesofreflection)willit

have?

9.Howmanysidesdoesaregularpolygonhavethathasanangleofrotationequalto180?Explain.

10.Howmanysidesdoesaregularpolygonhavethathasanangleofrotationequalto200?Howmany

linesofsymmetrywillithave?

34

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TRANSFORMATIONS AND SYMMETRY – 6.6

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6.6

GO Topic:Reflectingandrotatingpointsonthecoordinateplane.

(Thecoordinategrid,compass,rulerandothertoolsmaybehelpfulindoingthiswork.)

9.ReflectpointAoverthelineofreflectionandlabeltheimageA’.

10.ReflectpointAoverthelineofreflectionandlabeltheimageA’.

11.ReflecttriangleABCoverthelineofreflectionandlabeltheimageA’B’C’.

12.ReflectparallelogramABCDoverthelineofreflectionandlabeltheimageA’B’C’D’.

13.GiventriangleXYZanditsimageX’Y’Z’drawthelineofreflectionthatwasused.

14GivenparallelogramQRSTanditsimageQ’R’S’T’drawthelineofreflectionthatwasused.

Z'

Y'

X'

ZY

X

R'

Q'T'T

Q

S S'R

line of reflection

A

line of reflection

A

line of reflection

AC

B

line of reflectionA

CB

D

35