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SECONDARY MATH I // MODULE 6

TRANSFORMATIONS AND SYMMETRY – 6.4

Mathematics Vision Project

Licensed under the Creative Commons Attribution CC BY 4.0

mathematicsvisionproject.org

6. 4 Leap Year

A Practice Understanding Task

CarlosandClaritaarediscussingtheirlatestbusinessventurewiththeirfriendJuanita.

Theyhavecreatedadailyplannerthatisbotheducationalandentertaining.Theplannerconsistsof

apadof365pagesboundtogether,onepageforeachdayoftheyear.Theplannerisentertaining

sinceimagesalongthebottomofthepagesformaflip-bookanimationwhenthumbedthrough

rapidly.Theplanneriseducationalsinceeachpagecontainssomeinterestingfacts.Eachmonth

hasadifferenttheme,andthefactsforthemonthhavebeenwrittentofitthetheme.Forexample,

thethemeforJanuaryisastronomy,thethemeforFebruaryismathematics,andthethemefor

Marchisancientcivilizations.CarlosandClaritahavelearnedalotfromresearchingthefactsthey

haveincluded,andtheyhaveenjoyedcreatingtheflip-bookanimation.

Thetwinsareexcitedto

sharetheprototypeof

theirplannerwithJuanita

beforesendingitto

printing.Juanita,

however,hasamajor

concern."Nextyearis

leapyear,"sheexplains,

"youneed366pages."

SonowCarlosandClarita

havethedilemmaof

needingtocreateanextra

pagetoinsertbetween

February28andMarch1.

Herearetheplanner

pagestheyhavealready

designed.

CC

BY

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in

http

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Part1

SincethethemeforthefactsforFebruaryismathematics,Claritasuggeststhattheywrite

formaldefinitionsofthethreerigid-motiontransformationstheyhavebeenusingtocreatethe

imagesfortheflip-bookanimation.

Howwouldyoucompleteeachofthefollowingdefinitions?

1.Atranslationofasetofpointsinaplane...

2.Arotationofasetofpointsinaplane...

3.Areflectionofasetofpointsinaplane...

4.Translations,rotationsandreflectionsarerigidmotiontransformationsbecause...

CarlosandClaritausedthesewordsandphrasesintheirdefinitions:perpendicular

bisector,centerofrotation,equidistant,angleofrotation,concentriccircles,parallel,image,pre-

image,preservesdistanceandanglemeasureswithintheshape.Reviseyourdefinitionssothat

theyalsousethesewordsorphrases.

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Part2

InadditiontowritingnewfactsforFebruary29,thetwinsalsoneedtoaddanotherimage

inthemiddleoftheirflip-bookanimation.TheanimationsequenceisofDorothy'shousefromthe

WizardofOzasitisbeingcarriedovertherainbowbyatornado.ThehouseintheFebruary28

drawinghasbeenrotatedtocreatethehouseintheMarch1drawing.Carlosbelievesthathecan

getfromtheFebruary28drawingtotheMarch1drawingbyreflectingtheFebruary28drawing,

andthenreflectingitagain.

VerifythattheimageCarlosinsertedbetweenthetwoimagesthatappearedonFebruary28

andMarch1worksasheintended.Forexample,

• WhatconvincesyouthattheFebruary29imageisareflectionoftheFebruary28imageaboutthegivenlineofreflection?

• WhatconvincesyouthattheMarch1imageisareflectionoftheFebruary29imageaboutthegivenlineofreflection?

• WhatconvincesyouthatthetworeflectionstogethercompletearotationbetweentheFebruary28andMarch1images?

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Images this page:

http://openclipart.org/detail/168722/simple-farm-pack-by-viscious-speed

www.clker.com/clipart.tornado-gray

21






6. 4 Leap Year – Teacher Notes A Practice Understanding Task

Purpose:Inthistaskstudentswillwriteprecisedefinitionsforthethreerigid-motiontransformations,basedontheobservationstheyhavemadeintheprevioustasksinthislearningcycle.Topreparestudentsforwritingtheirowndefinitions,theywillstudythelanguageusedinthedefinitionsgivenforcircle,angle,andangleofrotation.Theywillalsowriteadefinitionfortheworddegreebasedontheinformationgiven.Inpart2ofthistaskstudentswillusetheirdefinitionstojustifythatmultipleimageshavebeencorrectlydrawnbasedonspecifiedtransformations.Aspartofthistaskstudentswillalsoexploretheideathattwoconsecutivereflectionsproducearotationwhenthelinesofreflectionarenotparallel.CoreStandardsFocus:G.CO.4Developdefinitionsofrotations,reflections,andtranslationsintermsofangles,circles,perpendicularlines,parallellines,andlinesegments.G.CO.2Representtransformationsintheplaneusing,e.g.,transparenciesandgeometrysoftware;describetransformationsasfunctionsthattakepointsintheplaneasinputsandgiveotherpointsasoutputs.Comparetransformationsthatpreservedistanceandangletothosethatdonot(e.g.,translationversushorizontalstretch).G.CO.1Knowprecisedefinitionsofangle,circle,perpendicularline,parallelline,andlinesegment,basedontheundefinednotionsofpoint,line,distancealongaline,anddistancearoundacirculararc.G.GPE.5Provetheslopecriteriaforparallelandperpendicularlinesandusethemtosolvegeometricproblems(e.g.,findtheequationofalineparallelorperpendiculartoagivenlinethatpassesthroughagivenpoint).






RelatedStandards:G.CO.5,G.CO.6

StandardsforMathematicalPracticeoffocusinthetask:

SMP6–Attendtoprecision

SMP7–Lookforandmakeuseofstructure

AdditionalResourcesforTeachers:

Ananswerkeyforthequestionsinthetaskcanbefoundasaseparatepageattheendofthese

teachernotes.Itisrecommendedthatyouworkthroughthetaskyourselfbeforeconsultingthe

answerkeytodevelopabettersenseofhowyourstudentsmightengageinthetask.

TheTeachingCycle:

Launch(WholeClass):

ReadandclarifythedefinitionofcirclegivenontheFebruary28calendarpage.Askhowthewords

usedidentifyonlythepointsonacircleandomiteveryotherpointintheplanefrombeing

identifiedaspointsbelongingtothecircle.Askwhatwouldhappenifthewords“inaplane”were

removedfromthedefinition.

Askstudentsabouttheimagesformedintheirmindsfromthetwodefinitionsforangleandangleof

rotation.Makesuretheycanuseeitherofthesedefinitionstodescribeanangle.Thenhave

studentsreadthehistoricalnotesconsideringwhythereare360°inacircle.(Whatdoesthisreally

mean—whereareeachofthe360degreeslocated“inthecircle”?)Basedonthisdiscussion,ask

studentstowriteadefinitionfortheworddegree.Pressforsomethinglike,“Adegreeisthe

measureofanangleofrotationthatisequalto1/360ofacompleterotationaroundafixedpoint.”

Afteremphasizingtheprecisionoflanguageinaformaldefinition,havepartnerswritedefinitions

foreachofthethreerigid-motiontransformations.Letstudentsknowthatyouwillbeformalizing

thesedefinitionsasaclassbeforeworkingonpart2ofthetask.






Explore(SmallGroup),part1:

Givestudentstimetoconsiderallthreedefinitions.Listenforlanguageaboutdistanceand

directionineachdefinition.Studentswhofinishtheirthreedefinitionscanalsoworkonwritinga

definitionforrigid-motiontransformation,whichshouldincludetheideathatrigid-motion

transformationspreservedistanceandanglemeasurements.

Discuss(WholeClass),part1:

Asawholeclass,writedefinitionsthatstudentsagreewilldefineeachtransformationwith

precision.Theessentialelementsofeachdefinitionareasfollows:

• Translation:translationsmovepointsthesamedistanceanddirectionalonglinesthatareparalleltoeachother

• Rotation:rotationsmovepointsthesamedirectionalongconcentriccirclesandthroughthesameangleofrotation

• Reflection:reflectionsmovepointsacrossaspecifiedlineofreflectionsothatthelineofreflectionistheperpendicularbisectorofeachlinesegmentconnectingcorrespondingpre-imageandimagepoints

Explore(SmallGroup),part2:

Inpart2,studentsshouldjustifytheiranswerstothethreequestionsbyusingthedefinitionsthey

wroteforthethreerigid-motiontransformations.Herearesomeadditionalpromptingquestions,if

studentsarenotattendingtothedefinitions:

• Whatevidencecanyouprovidethatthefirstgivenlineisalineofreflection?Howcanwe

convinceourselvesthatthelineistheperpendicularbisectorofthelinesegments

connectingpre-imageandimagepointsofthefirsttwodrawingsofDorothy’shouse?(Hint:

YoumaywanttousethePythagoreantheoremandalsothinkaboutslopes.)

• Whatevidencecanyouprovidethatthesecondlineistheperpendicularbisectoroftheline

segmentsconnectingpre-imageandimagepointsofthelasttwodrawingsofDorothy’s

house?

• Whereisthecenterofthisrotationlocated?Whatevidencecanyouprovidethatpre-image

andimagepointsareequidistantfromthecenterofrotation?






Discuss(WholeClass),part2:

ThefocusofthisdiscussionisonusingthedefinitionstojustifythetransformationsthatCarlos

claimstohaveused.Therefore,studentswillneedtoverifythatcorrespondingvertexpointsonthe

imageandpre-imagesatisfytheconditionsdefiningtheparticulartransformation.Forexample,is

thelineofreflectiontheperpendicularbisectorofeachlinesegmentjoiningavertexpointonthe

imagewithitscorrespondingvertexpointonthepre-image?Howdidstudentsverifythis?(Note:

thequestionofdeterminingifapointisamidpointofalinesegmentmaycomeupinthis

discussion.Ifso,allowstudentstodiscusshowtheythinktheymightfindthecoordinatesofthe

midpointofasegmentwhentheyknowthecoordinatesoftheendpoints.)

Oncestudentsareconvincedthatthesetworeflectionsproducedarotationyoumightaskthemto

considerifthiswouldalwaysbethecase,andwhatmakesthemthinkthismightbeso,orunder

whatconditionsitmightnotbeso.

AlignedReady,Set,Go:TransformationsandSymmetry6.4






6.4

READY

Topic:Definingpolygonsandtheirattributes

Foreachofthegeometricwordsbelowwriteadefinitionoftheobjectthataddressestheessentialelements.

1. Quadrilateral:

2. Parallelogram:

3. Rectangle:

4. Square:

5. Rhombus:

6. Trapezoid:

SET Topic:Reflectionsandrotations,composingreflectionstocreatearotation.

7.

READY, SET, GO! Name PeriodDate

UsethecenterofrotationpointCandrotatepoint

Pclockwisearoundit900.LabeltheimageP’.

WithpointCasacenterofrotationalsorotate

pointP1800.LabelthisimageP’’.C

P

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6.4

8.

9.

10.

11.

UsethecenterofrotationpointCandrotatepointPclockwisearoundit900.LabeltheimageP’.WithpointCasacenterofrotationalsorotatepointP1800.LabelthisimageP’’.

a.Whatistheequationforthelineforreflectionthat

reflectspointPontoP’?b.Whatistheequationforthelineofreflectionsthat

reflectspointP’ontoP’’?c.CouldP’’alsobeconsideredarotationofpointP?Ifsowhatisthecenterofrotationandhowmanydegreeswas

pointProtated?



reflectspointP’ontoP’’?c.CouldP’’alsobeconsideredarotationofpointP?Ifsowhatisthecenterofrotationandhowmanydegrees

waspointProtated?



reflectspointP’ontoP’’?c.CouldP’’alsobeconsideredarotationofpointP?Ifsowhatisthecenterofrotationandhowmanydegreeswas

pointProtated?

P

C

P''

P'P

P''

P'

P

P''

P'

P

23






6.4

GO Topic:Rotationsabouttheorigin.

Plotthegivencoordinateandthenperformtheindicatedrotationinaclockwisedirectionaroundtheorigin,thepoint(0,0),andplottheimagecreated.Statethecoordinatesoftheimage.

12.PointA(4,2)rotate1800 13.PointB(-5,-3)rotate900clockwiseCoordinatesforPointA’(___,___) CoordinatesforPointB’(___,___)

14.PointC(-7,3)rotate1800 15.PointD(1,-6)rotate900clockwiseCoordinatesforPointC’(___,___) CoordinatesforPointD’(___,___)

24