6. 1 Leaping Lizards! t- for Mathematical Practice of focus in the task: SMP 1 – Make sense of...

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

CC

0 Sh

are

d b

y T

racy

04-

140

-201

1

http

://w

ww

.clk

er.c

om/c

lipar

t-gr

een-

geck

o.ht

ml

1






LazyLizard

Translatetheoriginallizardsothepointatthetipofitsnoseislocatedat(24,20),makingthe

lizardappearstobesunbathingontherock.

LungingLizard

Rotatethelizard90°aboutpointA(12,7)soitlookslikethelizardisdivingintothepuddle

ofmud.

LeapingLizard

Reflectthelizardaboutgivenline soitlookslikethelizardisdoingabackflip

overthecactus.

€

y = 12 x +16

2






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






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






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






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






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






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:



• Next,haveastudentpresentwhousedaprotractortomeasure90°andarulertomeasure

distancesfromthecenterofrotation.Drawinthelinesegmentsbetween(12,7)andthe

correspondingimageandpre-imagepoints,usingadifferentcolorforeachimage/pre-

imagepair.Thiswillhighlightthe90°angleofrotation,centeredat(12,7).






• Next,haveastudentpresentwhodrewconcentriccircles(orarcs)toshowthatpairsof

image/pre-imagepointsarethesamedistancefrom(12,7)becausetheylieonthesame

circle.

• Finally,haveastudentpresentwhoshowedthatimage/pre-imagepointsarethesame

distancefrom(12,7)byusingthePythagoreanTheorem,orsomestrategythatisintuitively

equivalent.

Debriefingthereflection:



• Next,haveastudentpresentwhousedarulertomeasuredistancesfromthelineof

reflection.

• Ifavailable,haveastudentdescribehowtheydeterminedthesedistancesfromthelineof

reflectionusingthePythagoreanTheorem,orsomestrategythatisintuitivelyequivalent.

• Next,haveastudentpresentwhonoticedthatthesegmentsconnectingpairsofimage/pre-

imagepointsareparallel,perhapsbypointingoutthattheyhavethesameslope.

• Finally,haveastudentpresentwhomightarguethatthesegmentsconnectingpairsof

image/pre-imagepointsareperpendiculartothelineofreflection.

AlignedReady,Set,Go:TransformationandSymmetry6.1






3


TRANSFORMATIONS AND SYMMETRY - 6.1




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






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






6.1

GO Topic:Graphinglinearequations.

Grapheachfunctiononthecoordinategridprovided.Extendthelineasfarasthegridwillallow.

9.!(!) = 2! − 3 10.!(!) = −2! − 3

11.Whatsimilaritiesanddifferences

aretherebetweenthefunctionsf(x)

andg(x)?

12.ℎ(!) = !! ! + 1 13.!(!) = − !

! ! + 1


aretherebetweentheequationsh(x)

andk(x)?

15.!(!) = ! + 1 16.!(!) = ! − 3


aretherebetweentheequationsa(x)

andb(x)?

-5

-5

5

5

-5

-5

5

5

-5

-5

5

5

-5

-5

5

5

-5

-5

5

5

-5

-5

5

5

6

6. 1 Leaping Lizards! t- for Mathematical Practice of focus in the task: SMP 1 – Make sense of...

Documents

Transcript of 6. 1 Leaping Lizards! t- for Mathematical Practice of focus in the task: SMP 1 – Make sense of...