symmetries of regular polygons - utah education network
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SECONDARY MATH I // MODULE 6
TRANSFORMATIONS AND SYMMETRY – 6.6
Mathematics Vision Project
Licensed under the Creative Commons Attribution CC BY 4.0
mathematicsvisionproject.org
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
Licensed under the Creative Commons Attribution CC BY 4.0
mathematicsvisionproject.org
5. Aregularoctagon
6. Aregularnonagon
Whatpatternsdoyounoticeintermsofthenumberandcharacteristicsofthelinesofsymmetryina
regularpolygon?
Whatpatternsdoyounoticeintermsoftheanglesofrotationwhendescribingtherotational
symmetryinaregularpolygon?
31
SECONDARY MATH I // MODULE 6
TRANSFORMATIONS AND SYMMETRY – 6.6
Mathematics Vision Project
Licensed under the Creative Commons Attribution CC BY 4.0
mathematicsvisionproject.org
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
Mathematics Vision Project
Licensed under the Creative Commons Attribution CC BY 4.0
mathematicsvisionproject.org
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
Mathematics Vision Project
Licensed under the Creative Commons Attribution CC BY 4.0
mathematicsvisionproject.org
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
Mathematics Vision Project
Licensed under the Creative Commons Attribution CC BY 4.0
mathematicsvisionproject.org
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
Licensed under the Creative Commons Attribution CC BY 4.0
mathematicsvisionproject.org
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
Mathematics Vision Project
Licensed under the Creative Commons Attribution CC BY 4.0
mathematicsvisionproject.org
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
Mathematics Vision Project
Licensed under the Creative Commons Attribution CC BY 4.0
mathematicsvisionproject.org
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
SECONDARY MATH I // MODULE 6
TRANSFORMATIONS AND SYMMETRY – 6.6
Mathematics Vision Project
Licensed under the Creative Commons Attribution CC BY 4.0
mathematicsvisionproject.org
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
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