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SECONDARY MATH I // MODULE 6
TRANSFORMATION AND SYMMETRY – 6.1
Mathematics Vision Project
Licensed under the Creative Commons Attribution CC BY 4.0
mathematicsvisionproject.org
6. 1 Leaping Lizards!
A Develop Understanding Task
Animatedfilmsandcartoonsarenowusuallyproducedusingcomputertechnology,
ratherthanthehand-drawnimagesofthepast.Computeranimationrequiresbothartistic
talentandmathematicalknowledge.
Sometimesanimatorswanttomoveanimagearoundthecomputerscreenwithout
distortingthesizeandshapeoftheimageinanyway.Thisisdoneusinggeometric
transformationssuchastranslations(slides),reflections(flips),androtations(turns),or
perhapssomecombinationofthese.Thesetransformationsneedtobepreciselydefined,so
thereisnodoubtaboutwherethefinalimagewillenduponthescreen.
Sowheredoyouthinkthelizardshownonthegridonthefollowingpagewillendup
usingthefollowingtransformations?(Theoriginallizardwascreatedbyplottingthe
followinganchorpointsonthecoordinategrid,andthenlettingacomputerprogramdrawthe
lizard.Theanchorpointsarealwayslistedinthisorder:tipofnose,centerofleftfrontfoot,
belly,centerofleftrearfoot,pointoftail,centerofrearrightfoot,back,centeroffrontright
foot.)
Originallizardanchorpoints:
{(12,12),(15,12),(17,12),(19,10),(19,14),(20,13),(17,15),(14,16)}
Eachstatementbelowdescribesatransformationoftheoriginallizard.Dothe
followingforeachofthestatements:
• plottheanchorpointsforthelizardinitsnewlocation
• connectthepre-imageandimageanchorpointswithlinesegments,orcirculararcs,
whicheverbestillustratestherelationshipbetweenthem
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SECONDARY MATH I // MODULE 6
TRANSFORMATION AND SYMMETRY – 6.1
Mathematics Vision Project
Licensed under the Creative Commons Attribution CC BY 4.0
mathematicsvisionproject.org
LazyLizard
Translatetheoriginallizardsothepointatthetipofitsnoseislocatedat(24,20),makingthe
lizardappearstobesunbathingontherock.
LungingLizard
Rotatethelizard90°aboutpointA(12,7)soitlookslikethelizardisdivingintothepuddle
ofmud.
LeapingLizard
Reflectthelizardaboutgivenline soitlookslikethelizardisdoingabackflip
overthecactus.
€
y = 12 x +16
2
SECONDARY MATH I // MODULE 6
TRANSFORMATION AND SYMMETRY – 6.1
Mathematics Vision Project
Licensed under the Creative Commons Attribution CC BY 4.0
mathematicsvisionproject.org
6. 1 Leaping Lizards! – Teacher Notes A Develop Understanding Task
Purpose:Thistaskprovidesanopportunityforformativeassessmentofwhatstudentsalready
knowaboutthethreerigid-motiontransformations:translations,reflections,androtations.As
studentsengageinthetasktheyshouldrecognizeaneedforprecisedefinitionsofeachofthese
transformationssothatthefinalimageundereachtransformationisauniquefigure,ratherthanan
ill-definedsketch.Theexplorationandsubsequentdiscussiondescribedbelowshouldallow
studentstobegintoidentifytheessentialelementsinaprecisedefinitionoftherigid-motion
transformations,e.g.,translationsmovepointsaspecifieddistancealongparallellines;rotations
movepointsalongacirculararcwithaspecifiedcenterandangle,andreflectionsmovepointsacross
aspecifiedlineofreflectionsothatthelineofreflectionistheperpendicularbisectorofeachline
segmentconnectingcorrespondingpre-imageandimagepoints.
Inadditiontotheworkwiththerigid-motiontransformations,thistaskalsosurfacesthinking
abouttheslopecriteriafordeterminingwhenlinesareparallelorperpendicular.Inatranslation,
thelinesegmentsconnectingpre-imageandimagepointsareparallel,havingthesameslope.Ina
90°rotation,thelinesegmentsconnectingpre-imageandimagepointsareperpendicular,having
oppositereciprocalslopes.Likewise,inareflection,thelinesegmentsconnectingpre-imageand
imagepointsareperpendiculartothelineofreflection.
Finally,thistaskremindsstudentsthatrigid-motiontransformationspreservesdistanceandangle
measureswithinashape—implyingthatthefiguresformingthepre-imageandimageare
congruent.Studentswillbeattendingtotwodifferentcategoriesofdistances—thelengthsofline
segmentsthatareusedinthedefinitionsofthetransformations,andthelengthsofthecongruent
linesegmentsthatarecontainedwithinthepre-imageandimagefiguresthemselves.Studentsmay
determinethattheselengthsarepreservedbycountingunitsof“rise”and“run”,orbyusingthe
SECONDARY MATH I // MODULE 6
TRANSFORMATION AND SYMMETRY – 6.1
Mathematics Vision Project
Licensed under the Creative Commons Attribution CC BY 4.0
mathematicsvisionproject.org
PythagoreanTheorem.Ultimately,thisworkwillleadtothedevelopmentofthedistanceformula
infuturetasks.
CoreStandardsFocus:
G.CO.4Developdefinitionsofrotations,reflections,andtranslationsintermsofangles,circles,
perpendicularlines,parallellines,andlinesegments.
G.CO.5Givenageometricfigureandarotation,reflection,ortranslation,drawthetransformed
figureusing,e.g.,graphpaper,tracingpaper,orgeometrysoftware.Specifyasequenceof
transformationsthatwillcarryagivenfigureontoanother.
G.CO.1Knowprecisedefinitionsofangle,circle,perpendicularline,parallelline,andlinesegment,
basedontheundefinednotionsofpoint,line,distancealongaline,anddistancearoundacircular
arc.
RelatedStandards:G.CO.2,G.CO.6,G.GPE.5
TeacherNote:Students’previousexperienceswithrigidmotionsmayhavesurfacedintuitive
waysofthinkingaboutthesetransformations,butsuchinformaldefinitionswillnotsupport
studentsinprovinggeometricpropertiesbasedonatransformationalapproach.Experienceswith
sliding,flippingandturningrigidobjectswillhaveprovidedexperimentalevidencethatrigid-
motiontransformationspreservedistanceandanglewithinashape,suchthat,
• Linesaretakentolines,andlinesegmentstolinesegmentsofthesamelength.
• Anglesaretakentoanglesofthesamemeasure.
• Parallellinesaretakentoparallellines.
Studentswhohaveusedtechnologytotranslate,rotateorreflectobjectsmaynothaveattendedto
theessentialfeaturesthatdefinesuchtransformations.Forexample,astudentcanmarkamirror
lineandclickonabuttontoreflectanobjectacrossthemirrorlinewithoutnotingtherelationship
betweenthepre-imageandimagepointsrelativetothelineofreflection.Consequently,research
SECONDARY MATH I // MODULE 6
TRANSFORMATION AND SYMMETRY – 6.1
Mathematics Vision Project
Licensed under the Creative Commons Attribution CC BY 4.0
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showsthatstudentsharbormanymisconceptionsabouttheplacementofanimageafteratransformation—erroneousassumptionssuchas:
• oneofthesidesofareflectedimagemustcoincidewiththelineofreflection• thecenterofarotationmustbelocatedatapointonthepre-image(e.g.,avertexpoint)or
attheorigin• apre-imagepointandcorrespondingimagepointdonotneedtobethesamedistanceaway
fromthecenteroftherotationWatchforthesemisconceptionsasstudentsengageinthistask.
StandardsforMathematicalPracticeoffocusinthetask:
SMP1–Makesenseofproblemsandpersevereinsolvingthem
SMP5–Useappropriatetoolsstrategically
SMP7–Lookforandmakeuseofstructure
AdditionalResourcesforTeachers:
Anenlargedcopyoftheimageonthesecondpageofthetaskcanbefoundattheendofthissetofteachernotes.Thisimagecanbeprintedforusewithstudentswhomaybeaccessingthetaskonacomputerortablet.TheTeachingCycle:
Launch(WholeClass):
Setthestagefortheworkofthislearningcyclebydiscussingtheideasofcomputeranimationasoutlinedinthefirstfewparagraphsofthistask.Aspartofthelaunchaskstudentswhytheythinkweneedonlykeeptrackofafewanchorpoints,sincetheimageofthelizardconsistsofinfinitelymanypoints,inadditiontotheeightpointsthatarelisted.Theissuetoberaisedhereisthatrigid-motiontransformationspreservedistanceandangle(propertiesthathavebeenestablishedinMath8).Thereforeasoftwareanimationprogramcoulddrawfeaturesofthelizard,suchasthetoesoneachofthefeet,bystartingatananchorpointandusingpredeterminedangleanddistancemeasurestolocateotherpointsonthetoes.Makesurestudentspayattentiontotheorderinwhich
SECONDARY MATH I // MODULE 6
TRANSFORMATION AND SYMMETRY – 6.1
Mathematics Vision Project
Licensed under the Creative Commons Attribution CC BY 4.0
mathematicsvisionproject.org
eachoftheanchorpointsshouldbelistedaftercompletingeachofthetransformations.Thiswill
helpstudentspayattentiontoindividualpairsofpre-imageandimagepoints.
Providemultipletoolsforstudentstodothiswork,suchastransparenciesortracingpaper,
protractors,rulers,andcompasses.Thecoordinategridonwhichtheimagesaredrawnisalsoa
toolfordoingthiswork,butinitiallystudentsmaynotrecognizetheusefulnessofthegridasaway
ofcarryingoutthetransformations,butratherjustasawayofdesignatingthelocationofthepoints
afterthetransformationiscomplete.Technologytoolsmayobscuretheideasbeingsurfacedinthe
task,soitisbesttousethetoolsdescribed,whichwillallowstudentstopayattentiontothedetails
oftheirwork.
Itisintendedthatstudentsshouldworkonthetransformationsintheorderlistedinthetask.
Explore(SmallGroup):
Thistaskprovidesagreatopportunitytopre-assesswhatstudentsknowabouteachoftherigid-
motiontransformations,sodon’tworryifnotallstudentsarelocatingthefinalimagescorrectly.
Payattentiontothemisconceptionsthatmayarise(seeteachernote).
Ifstudentsusetransparencies(ortracingpaper)tocopytheoriginallizardandthenlocatethe
imagebysliding,turningorflippingthetransparency,youwillwanttomakesuretheyalsothink
aboutthesemovementsrelativetothecoordinategrid.Ask,“Howcouldyouhaveusedthe
coordinategridtolocatethissamesetofpoints?”Focusingstudents’attentiononthecoordinate
gridwillfacilitateconnectingthedetailsthatneedtobearticulatedinthedefinitionsoftherigid-
motiontransformationstocoordinategeometryideas,suchasusingslopetodetermineiflinesare
parallelorperpendicular.Inthistask,theseideasaresurfacedandinformallyexplored.In
subsequenttaskstheseideasaremademoreexplicitandeventuallyjustified.
Studentsshouldbefairlysuccessfultranslating“LazyLizard”,sincethepointatthetipofthenose
movesup8unitsandright12units,everyanchorpointmustmovethesame.Watchfortwo
differentstrategiestoemerge:somestudentsmaymoveeachpointup8,right12;othersmaymove
SECONDARY MATH I // MODULE 6
TRANSFORMATION AND SYMMETRY – 6.1
Mathematics Vision Project
Licensed under the Creative Commons Attribution CC BY 4.0
mathematicsvisionproject.org
onepointtothecorrectlocation,andthenduplicatetherelativepositionsofthepointsinthepre-imagetolocatepointsintheimage—therebypreservingdistanceandanglebetweenthepointsinthepre-imageandthosesamepointsintheimage.Togetstartedon“LungingLizard”,youmaywanttodirectstudents’attentiontothepointatthetipofthelizard’snose,whichliesonaverticalline,5unitsabovethecenterofrotation.Askstudentswherethispointwouldendupafterrotating90°counterclockwise.Watchforstudentswhoareattendingtothe90°angleofrotationbydrawinglinesegmentsfromthecenterofrotationtotheimageandcorrespondingpre-imagepoints.Alsowatchforhowstudentsdeterminethatanimagepointisthesamedistanceawayfromthecenterofrotationasitscorrespondingpre-imagepoint:dotheymeasurewitharuler,dotheydrawconcentriccirclescenteredat(12,7),dotheycounttheriseandrunfrom(12,7)toapointonthelizardandthenusearelatedwayofcountingriseandruntolocatetheimagepoint—intuitivelyusingthePythagoreanTheoremtokeepthesamedistance,ordotheyignoredistancealtogether?For“LeapingLizard”,watchforstudentswhomayhavenoticedthatanimagepointanditscorrespondingpre-imagepointareequidistantfromthelineofreflection.Listenforhowtheyjustifythatthesedistancesarethesame:dotheymeasurewitharuler,dotheyfoldthepaperalongthelineofreflection,dotheycounttheriseandrunfromthepre-imagetothelineofreflectionandthenfromthelineofreflectiontotheimagepoint—intuitivelyusingthePythagoreanTheoremtokeepthesamedistance.Alsowatchforstudentswhonoticethatthelinesegmentsconnectingtheimagepointstotheircorrespondingpre-imagepointsareallparalleltoeachother—perhapsevennoticingthatalloftheselinesegmentshaveaslopeof-2.
Discuss(WholeClass):
Ifstudentshavenotalllocatedthesamesetofpointsfortheimagesofthetransformations,havestudentsdiscusswhetherthisisreasonableornot.Informstudents,“Thattransformationsarelikefunctions—anysetofpointsthatformapre-imageshouldhaveauniquesetofpointsthatformtheimagethatistheresultofthetransformation.Ifwehavenotobtaineduniqueimages,thenwehavenotrecognizedtheprecisenatureofthesetransformations.Thatisthegoalofourworktoday,to
SECONDARY MATH I // MODULE 6
TRANSFORMATION AND SYMMETRY – 6.1
Mathematics Vision Project
Licensed under the Creative Commons Attribution CC BY 4.0
mathematicsvisionproject.org
noticewhatisimportantabouteachtransformationsotheimagesproducedbythetransformation
arepreciselydefined.”
Discussstrategiesforlocatingtheimagesoftheanchorpointsforeachtransformation.Hereisa
suggestedlistofasequenceofideastobepresented,ifavailable.Whilewewillnotbewriting
precisedefinitionsforthetransformationsuntilthetaskLeapYear,itisimportantthattheideasof
distanceanddirection(e.g.,alongaparallelline,perpendiculartoaline,oralongacircle)emerge
duringthisdiscussion.Ifnotallofthesuggestedstrategiesareavailableinthestudentwork,at
leastmakesurethedebriefofeachtransformationdoesfocusonbothdistanceanddirection.If
eitherideaismissing,askadditionalquestionstopromptforit.Forexample,“Howdidyouknow
howfarawayfromthecenterpoint(orthereflectingline)thisimagepointshouldbe?”Also,be
awareofthetasksthatfollowinthislearningcycle—noteverythingneedstobeneatlywrappedup
inthisdiscussion.
Debriefingthetranslation:
• Haveastudentpresentwhousedatransparencyortracingpapertogetasetofimage
pointsthatthewholeclasscanagreeupon.
• Next,haveastudentpresentwhomovedeachanchorpointup8,right12units.
• Finally,haveastudentpresentwhomovedoneanchorpointup8,right12unitsandthen
usedtherelativepositionsofthepointsintheoriginalfiguretolocaterelatedpointsinthe
imagefigure.Discussthatthisispossiblebecausetranslationspreservedistance,angleand
parallelism.
Debriefingtherotation:
• Haveastudentpresentwhousedatransparencyortracingpapertogetasetofimage
pointsthatthewholeclasscanagreeupon.
• Next,haveastudentpresentwhousedaprotractortomeasure90°andarulertomeasure
distancesfromthecenterofrotation.Drawinthelinesegmentsbetween(12,7)andthe
correspondingimageandpre-imagepoints,usingadifferentcolorforeachimage/pre-
imagepair.Thiswillhighlightthe90°angleofrotation,centeredat(12,7).
SECONDARY MATH I // MODULE 6
TRANSFORMATION AND SYMMETRY – 6.1
Mathematics Vision Project
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• Next,haveastudentpresentwhodrewconcentriccircles(orarcs)toshowthatpairsof
image/pre-imagepointsarethesamedistancefrom(12,7)becausetheylieonthesame
circle.
• Finally,haveastudentpresentwhoshowedthatimage/pre-imagepointsarethesame
distancefrom(12,7)byusingthePythagoreanTheorem,orsomestrategythatisintuitively
equivalent.
Debriefingthereflection:
• Haveastudentpresentwhousedatransparencyortracingpapertogetasetofimage
pointsthatthewholeclasscanagreeupon.
• Next,haveastudentpresentwhousedarulertomeasuredistancesfromthelineof
reflection.
• Ifavailable,haveastudentdescribehowtheydeterminedthesedistancesfromthelineof
reflectionusingthePythagoreanTheorem,orsomestrategythatisintuitivelyequivalent.
• Next,haveastudentpresentwhonoticedthatthesegmentsconnectingpairsofimage/pre-
imagepointsareparallel,perhapsbypointingoutthattheyhavethesameslope.
• Finally,haveastudentpresentwhomightarguethatthesegmentsconnectingpairsof
image/pre-imagepointsareperpendiculartothelineofreflection.
AlignedReady,Set,Go:TransformationandSymmetry6.1
SECONDARY MATH I // MODULE 6
TRANSFORMATION AND SYMMETRY – 6.1
Mathematics Vision Project
Licensed under the Creative Commons Attribution CC BY 4.0
mathematicsvisionproject.org
3
SECONDARY MATH I // MODULE 6
TRANSFORMATIONS AND SYMMETRY - 6.1
Mathematics Vision Project
Licensed under the Creative Commons Attribution CC BY 4.0
mathematicsvisionproject.org
6.1
READY
Topic:PythagoreanTheorem
Foreachofthefollowingrighttrianglesdeterminethemeasureofthemissingside.Leavethemeasuresinexactformifirrational.
1. 2. 3.
4. 5. 6.
READY, SET, GO! Name PeriodDate
3
4
?
?
5
12
?4
1
?
3
√10
?
√174
?
2
√13
4
SECONDARY MATH I // MODULE 6
TRANSFORMATIONS AND SYMMETRY - 6.1
Mathematics Vision Project
Licensed under the Creative Commons Attribution CC BY 4.0
mathematicsvisionproject.org
6.1
SET Topic:Transformations.
Transformpointsasindicatedineachexercisebelow.
7a.RotatepointAaroundtheorigin90oclockwise,labelasA’
b.ReflectpointAoverx-axis,labelasA’’
c.Applytherule(! − 2 , ! − 5),topointAandlabelA’’’
8a.ReflectpointBovertheline! = !,labelasB’b.RotatepointB180oabouttheorigin,labelasB’’
c.TranslatepointBthepointup3andright7units,
labelasB’’’
-5
-5
5
5
A
y = x-5
-5
5
5
B
5
SECONDARY MATH I // MODULE 6
TRANSFORMATIONS AND SYMMETRY - 6.1
Mathematics Vision Project
Licensed under the Creative Commons Attribution CC BY 4.0
mathematicsvisionproject.org
6.1
GO Topic:Graphinglinearequations.
Grapheachfunctiononthecoordinategridprovided.Extendthelineasfarasthegridwillallow.
9.!(!) = 2! − 3 10.!(!) = −2! − 3
11.Whatsimilaritiesanddifferences
aretherebetweenthefunctionsf(x)
andg(x)?
12.ℎ(!) = !! ! + 1 13.!(!) = − !
! ! + 1
14.Whatsimilaritiesanddifferences
aretherebetweentheequationsh(x)
andk(x)?
15.!(!) = ! + 1 16.!(!) = ! − 3
17.Whatsimilaritiesanddifferences
aretherebetweentheequationsa(x)
andb(x)?
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