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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
Te
d K
irw
in
http
s://f
lic.k
r/p/
bmsd
Ub
18
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
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.
19
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
Part2
InadditiontowritingnewfactsforFebruary29,thetwinsalsoneedtoaddanotherimage
inthemiddleoftheirflip-bookanimation.TheanimationsequenceisofDorothy'shousefromthe
WizardofOzasitisbeingcarriedovertherainbowbyatornado.ThehouseintheFebruary28
drawinghasbeenrotatedtocreatethehouseintheMarch1drawing.Carlosbelievesthathecan
getfromtheFebruary28drawingtotheMarch1drawingbyreflectingtheFebruary28drawing,
andthenreflectingitagain.
VerifythattheimageCarlosinsertedbetweenthetwoimagesthatappearedonFebruary28
andMarch1worksasheintended.Forexample,
• WhatconvincesyouthattheFebruary29imageisareflectionoftheFebruary28imageaboutthegivenlineofreflection?
• WhatconvincesyouthattheMarch1imageisareflectionoftheFebruary29imageaboutthegivenlineofreflection?
• WhatconvincesyouthatthetworeflectionstogethercompletearotationbetweentheFebruary28andMarch1images?
20
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
Images this page:
http://openclipart.org/detail/168722/simple-farm-pack-by-viscious-speed
www.clker.com/clipart.tornado-gray
21
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 – 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).
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
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.
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
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?
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
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
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
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
22
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
8.
9.
10.
11.
UsethecenterofrotationpointCandrotatepointPclockwisearoundit900.LabeltheimageP’.WithpointCasacenterofrotationalsorotatepointP1800.LabelthisimageP’’.
a.Whatistheequationforthelineforreflectionthat
reflectspointPontoP’?b.Whatistheequationforthelineofreflectionsthat
reflectspointP’ontoP’’?c.CouldP’’alsobeconsideredarotationofpointP?Ifsowhatisthecenterofrotationandhowmanydegreeswas
pointProtated?
a.Whatistheequationforthelineforreflectionthat
reflectspointPontoP’?b.Whatistheequationforthelineofreflectionsthat
reflectspointP’ontoP’’?c.CouldP’’alsobeconsideredarotationofpointP?Ifsowhatisthecenterofrotationandhowmanydegrees
waspointProtated?
a.Whatistheequationforthelineforreflectionthat
reflectspointPontoP’?b.Whatistheequationforthelineofreflectionsthat
reflectspointP’ontoP’’?c.CouldP’’alsobeconsideredarotationofpointP?Ifsowhatisthecenterofrotationandhowmanydegreeswas
pointProtated?
P
C
P''
P'P
P''
P'
P
P''
P'
P
23
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
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