module 6: transformations & symmetry
TRANSCRIPT
The Mathematics Vision Project Scott Hendrickson, Joleigh Honey, Barbara Kuehl, Travis Lemon, Janet Sutorius
© 2016 Mathematics Vision Project Original work © 2013 in partnership with the Utah State Off ice of Education
This work is licensed under the Creative Commons Attribution CC BY 4.0
MODULE 6
Transformations & Symmetry
SECONDARY
MATH ONE
An Integrated Approach
SECONDARY MATH 1 // MODULE 6
TRANSFORMATIONS AND SYMMETRY
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MODULE 6 - TABLE OF CONTENTS
TRANSFORMATIONS AND SYMMETRY
6.1 Leaping Lizards! – A Develop Understanding Task
Developing the definitions of the rigid-motion transformations: translations, reflections and rotations
(G.CO.1, G.CO.4, G.CO.5)
READY, SET, GO Homework: Transformations and Symmetry 6.1
6.2 Is It Right? – A Solidify Understanding Task
Examining the slope of perpendicular lines (G.CO.1, G.GPE.5)
READY, SET, GO Homework: Transformations and Symmetry 6.2
6.3 Leap Frog – A Solidify Understanding Task
Determining which rigid-motion transformations will carry one image onto another congruent image
(G.CO.4, G.CO.5)
READY, SET, GO Homework: Transformations and Symmetry 6.3
6.4 Leap Year – A Practice Understanding Task
Writing and applying formal definitions of the rigid-motion transformations: translations, reflections and
rotations (G.CO.1, G.CO.2, G.CO.4, G.GPE.5)
READY, SET, GO Homework: Transformations and Symmetry 6.4
6.5 Symmetries of Quadrilaterals – A Develop Understanding Task
Finding rotational symmetry and lines of symmetry in special types of quadrilaterals (G.CO.3, G.CO.6)
READY, SET, GO Homework: Transformations and Symmetry 6.5
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6.6 Symmetries of Regular Polygons – A Solidify Understanding Task
Examining characteristics of regular polygons that emerge from rotational symmetry and lines of
symmetry (G.CO.3, G.CO.6)
READY, SET, GO Homework: Transformations and Symmetry 6.6
6.7 Quadrilaterals—Beyond Definition – A Practice Understanding Task
Making and justifying properties of quadrilaterals using symmetry transformations
(G.CO.3, G.CO.4, G.CO.6)
READY, SET, GO Homework: Transformations and Symmetry 6.7
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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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LazyLizard
Translatetheoriginallizardsothepointatthetipofitsnoseislocatedat(24,20),makingthe
lizardappearstobesunbathingontherock.
LungingLizard
Rotatethelizard90°aboutpointA(12,7)soitlookslikethelizardisdivingintothepuddle
ofmud.
LeapingLizard
Reflectthelizardaboutgivenline soitlookslikethelizardisdoingabackflip
overthecactus.
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6.1
READY
Topic:PythagoreanTheorem
Foreachofthefollowingrighttrianglesdeterminethemeasureofthemissingside.Leavethemeasuresinexactformifirrational.
1. 2. 3.
4. 5. 6.
READY, SET, GO! Name PeriodDate
3
4
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?
5
12
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3
√10
?
√174
?
2
√13
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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’’’
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y = x-5
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B
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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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TRANSFORMATIONS AND SYMMETRY – 6.2
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6. 2 Is It Right?
A Solidify Understanding Task
InLeapingLizardsyouprobablythoughtalotaboutperpendicularlines,particularlywhen
rotatingthelizardaboutagivencentera90°angleorreflectingthelizardacrossaline.
Inprevioustasks,wehavemadetheobservationthatparallellineshavethesameslope.In
thistaskwewillmakeobservationsabouttheslopesofperpendicularlines.PerhapsinLeaping
Lizardsyouusedaprotractororsomeothertoolorstrategytohelpyoumakearightangle.Inthis
taskweconsiderhowtocreatearightanglebyattendingtoslopesonthecoordinategrid.
Webeginbystatingafundamentalideaforourwork:
Horizontalandverticallinesareperpendicular.Forexample,ona
coordinategrid,thehorizontalliney=2andtheverticalline
x=3intersecttoformfourrightangles.
Butwhatifalineorlinesegmentisnothorizontalorvertical?Howdowedeterminethe
slopeofalineorlinesegmentthatwillbeperpendiculartoit?
Experiment1
1. ConsiderthepointsA(2,3)andB(4,7)andthelinesegment,
�
AB ,betweenthem.Whatistheslopeofthislinesegment?
2. LocateathirdpointC(x,y)onthecoordinategrid,sothepointsA(2,3),B(4,7)andC(x,y)formtheverticesofarighttriangle,with
�
AB asitshypotenuse.
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3. Explainhowyouknowthatthetriangleyouformedcontainsarightangle?
4. Nowrotatethisrighttriangle90°aboutthevertexpoint(2,3).Explainhowyouknowthatyouhaverotatedthetriangle90°.
5. Comparetheslopeofthehypotenuseofthisrotatedrighttrianglewiththeslopeofthehypotenuseofthepre-image.Whatdoyounotice?
Experiment2
Repeatsteps1-5aboveforthepointsA(2,3)
andB(5,4).
Experiment3
Repeatsteps1-5aboveforthepointsA(2,3)
andB(7,5).
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Experiment4
Repeatsteps1-5aboveforthepointsA(2,3)
andB(0,6).
Basedonexperiments1-4,stateanobservationabouttheslopesofperpendicularlines.
Whilethisobservationisbasedonafewspecificexamples,canyoucreateanargumentor
justificationforwhythisisalwaystrue?(Note:Youwillexamineaformalproofofthisobservation
inafuturemodule.)
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6.2
READY Topic:FindingDistanceusingPythagoreanTheoremUsethecoordinategridtofindthelengthofeachsideofthetrianglesprovided.Giveanswersin
exactformandwherenecessaryroundedtothenearesthundredth.
1. 2.
3. 4.
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READY, SET, GO! Name PeriodDate
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6.2
SET Topic:Slopesofparallelandperpendicularlines.
5.Graphalineparalleltothegivenline.
6.Graphalineparalleltothegivenline.
7.Graphalineparalleltothegivenline.
Equationforgivenline:Equationfornewline:
Equationforgivenline:Equationfornewline:
Equationforgivenline:Equationfornewline:
8.Graphalineperpendiculartothegivenline.
9.Graphalineperpendiculartothegivenline.
10.Graphalineperpendiculartothegivenline.
Equationforgivenline:Equationfornewline:
Equationforgivenline:Equationfornewline:
Equationforgivenline:Equationfornewline:
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6.2
GO Topic:Solvethefollowingequations.Solveeachequationfortheindicatedvariable.
11.3 ! − 2 = 5! + 8;Solveforx. 12.−3 + ! = 6! + 22;Solveforn.
13.! − 5 = !(! − 2);Solveforx. 14.!" + !" = !;Solvefory.
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6. 3 Leap Frog
A Solidify Understanding Task
JoshisanimatingasceneinwhichatroupeoffrogsisauditioningfortheAnimalChannel
realityshow,"TheBayou'sGotTalent".Inthisscenethefrogsaredemonstratingtheir"leapfrog"
acrobaticsact.Joshhascompletedafewkeyimagesinthissegment,andnowneedstodescribethe
transformationsthatconnectvariousimagesinthescene.
Foreachpre-image/imagecombinationlistedbelow,describethetransformationthat
movesthepre-imagetothefinalimage.
• Ifyoudecidethetransformationisarotation,youwillneedtogivethecenterofrotation,thedirectionoftherotation(clockwiseorcounterclockwise),andthemeasureoftheangleofrotation.
• Ifyoudecidethetransformationisareflection,youwillneedtogivetheequationofthelineofreflection.
• Ifyoudecidethetransformationisatranslationyouwillneedtodescribethe"rise"and
"run"betweenpre-imagepointsandtheircorrespondingimagepoints.
• Ifyoudecideittakesacombinationoftransformationstogetfromthepre-imagetothefinalimage,describeeachtransformationintheordertheywouldbecompleted.
Pre-image FinalImage Description
image1
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images this page:
CC0 http://openclipart.org/detail/33781/architetto
CC0 http://openclipart.org/detail/33979/architetto
CC0 http://openclipart.org/detail/33985/architetto
CC0 http://openclipart.org/detail/170806/hatalar205
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6.3
READY Topic:RotationsandReflectionsoffigures.Ineachproblemtherewillbeapre-imageandseveralimagesbasedonthegivepre-image.Determinewhichoftheimagesarerotationsofthegivenpre-imageandwhichofthemarereflectionsofthepre-image.Ifanimageappearstobecreatedastheresultofarotationandareflectionthenstateboth.(Compareallimagestothepre-image.)1. 2.
READY, SET, GO! Name PeriodDate
ImageA ImageB ImageC ImageD
ImageA
ImageB
ImageC
ImageD
Pre-Image
Pre-Image
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6.3
SET Topic:Reflectingandrotatingpoints.Oneachofthecoordinategridsthereisalabeledpointandline.Usethelineasalineofreflectiontoreflectthegivenpointandcreateitsreflectedimageoverthelineofreflection.(Hint:pointsreflectalongpathsperpendiculartothelineofreflection.Useperpendicularslope!)3. 4.
5. 6.
Foreachpairofpoint,PandP’drawinthelineofreflectionthatwouldneedtobeusedtoreflectPontoP’.Thenfindtheequationofthelineofreflection.
7. 8.
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ReflectpointAoverlinemandlabeltheimageA’
ReflectpointBoverlinekandlabeltheimageB’
ReflectpointCoverlinelandlabeltheimageC’ ReflectpointDoverlinemandlabeltheimageD’
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6.3
Foreachpairofpoint,AandA’drawinthelineofreflectionthatwouldneedtobeusedtoreflectAontoA’.Thenfindtheequationofthelineofreflection.9. 10.
GO Topic:Slopesofparallelandperpendicularlinesandfindingslopeanddistancebetweentwopoints.Foreachlinearequationwritetheslopeofalineparalleltothegivenline.11.! = −3! + 5 12.! = 7! – 3 13.3! − 2! = 8Foreachlinearequationwritetheslopeofalineperpendiculartothegivenline.14.! = − !
! ! + 5 15.! = !! ! − 4 16.3! + 5! = −15
Findtheslopebetweeneachpairofpoints.Then,usingthePythagoreanTheorem,findthedistancebetweeneachpairofpoints.Youmayusethegraphtohelpyouasneeded.17.(-2,-3)(1,1) a.Slope:b.Distance:18.(-7,5)(-2,-7) a.Slope:b.Distance:
A'
A
A'
A
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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.
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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
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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?
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
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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’(___,___)
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6. 5 Symmetries of Quadrilaterals
A Develop Understanding Task
Alinethatreflectsafigureontoitselfiscalledalineofsymmetry.Afigurethatcanbe
carriedontoitselfbyarotationissaidtohaverotationalsymmetry.
Everyfour-sidedpolygonisaquadrilateral.Somequadrilateralshaveadditionalproperties
andaregivenspecialnameslikesquares,parallelogramsandrhombuses.Adiagonalofa
quadrilateralisformedwhenoppositeverticesareconnectedbyalinesegment.Somequadrilaterals
aresymmetricabouttheirdiagonals.Somearesymmetricaboutotherlines.Inthistaskyouwilluse
rigid-motiontransformationstoexplorelinesymmetryandrotationalsymmetryinvarioustypesof
quadrilaterals.
Foreachofthefollowingquadrilateralsyouaregoingtotrytoanswerthequestion,“Isit
possibletoreflectorrotatethisquadrilateralontoitself?”Asyouexperimentwitheachquadrilateral,
recordyourfindingsinthefollowingchart.Beasspecificaspossiblewithyourdescriptions.
Definingfeaturesofthequadrilateral
Linesofsymmetrythatreflectthequadrilateral
ontoitself
Centerandanglesofrotationthatcarrythequadrilateral
ontoitselfArectangleisaquadrilateralthatcontainsfourrightangles.
Aparallelogramisaquadrilateralinwhichoppositesidesareparallel.
CC
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Atrapezoidisaquadrilateralwithonepairofoppositesidesparallel.Isitpossibletoreflectorrotateatrapezoidontoitself?Drawatrapezoidbasedonthisdefinition.Thenseeifyoucanfind:
• anylinesofsymmetry,or• anycentersofrotationalsymmetry,
thatwillcarrythetrapezoidyoudrewontoitself.Ifyouwereunabletofindalineofsymmetryoracenterofrotationalsymmetryforyourtrapezoid,seeifyoucansketchadifferenttrapezoidthatmightpossesssometypeofsymmetry.
Arhombusisaquadrilateralinwhichallsidesarecongruent.
Asquareisbotharectangleandarhombus.
26
SECONDARY MATH I // MODULE 6
TRANSFORMATIONS AND SYMMETRY – 6.5
Mathematics Vision Project
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6.5
READY Topic:Polygons,definitionandnames
1.Whatisapolygon?Describeinyourownwordswhatapolygonis.
2.Fillinthenamesofeachpolygonbasedonthenumberofsidesthepolygonhas.
NumberofSides NameofPolygon
3
4
5
6
7
8
9
10
SET Topic:Kites,Linesofsymmetryanddiagonals.3.Onequadrilateralwithspecialattributesisakite.Findthegeometricdefinitionofakiteandwriteitbelowalongwithasketch.(Youcandothisfairlyquicklybydoingasearchonline.)4.Drawakiteanddrawallofthelinesofreflectivesymmetryandallofthediagonals.
LinesofReflectiveSymmetry Diagonals
READY, SET, GO! Name PeriodDate
27
SECONDARY MATH I // MODULE 6
TRANSFORMATIONS AND SYMMETRY – 6.5
Mathematics Vision Project
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6.5
5.Listalloftherotationalsymmetryforakite.
6.Arelinesofsymmetryalsodiagonalsinapolygon?Explain.
6.Arealldiagonalsalsolinesofsymmetryinapolygon?Explain.
7.Whichquadrilateralshavediagonalsthatarenotlinesofsymmetry?Namesomeanddrawthem.
8.Doparallelogramshavediagonalsthatarelinesofsymmetry?Ifso,drawandexplain.Ifnotdrawand
explain.
28
SECONDARY MATH I // MODULE 6
TRANSFORMATIONS AND SYMMETRY – 6.5
Mathematics Vision Project
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6.5
GO Topic:Equationsforparallelandperpendicularlines.
FindtheequationofalinePARALLELtothegiveninfoandthroughtheindicated
y-intercept.
FindtheequationofalinePERPENDICULARtothe
givenlineandthroughtheindicatedy-intercept.
9.Equationofaline:! = 4! + 1.
a.Parallellinethroughpoint(0,-7):
b.Perpendiculartothelinelinethroughpoint(0,-7):
10.Tableofaline:
x y3 -84 -105 -126 -14
a.Parallellinethroughpoint(0,8):
b.Perpendiculartothelinethroughpoint(0,8):
11.Graphofaline:
a.Parallellinethroughpoint(0,-9):
b.Perpendiculartothelinethroughpoint(0,-9):
-5
-5
5
5
29
SECONDARY MATH I // MODULE 6
TRANSFORMATIONS AND SYMMETRY – 6.6
Mathematics Vision Project
Licensed under the Creative Commons Attribution CC BY 4.0
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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
CC
BY
Jorg
e Ja
ram
illo
http
s://f
lic.k
r/p/
rbd9
zs
30
SECONDARY MATH I // MODULE 6
TRANSFORMATIONS AND SYMMETRY – 6.6
Mathematics Vision Project
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5. Aregularoctagon
6. Aregularnonagon
Whatpatternsdoyounoticeintermsofthenumberandcharacteristicsofthelinesofsymmetryina
regularpolygon?
Whatpatternsdoyounoticeintermsoftheanglesofrotationwhendescribingtherotational
symmetryinaregularpolygon?
31
SECONDARY MATH I // MODULE 6
TRANSFORMATIONS AND SYMMETRY – 6.6
Mathematics Vision Project
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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
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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
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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
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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
SECONDARY MATH I // MODULE 6
TRANSFORMATIONS AND SYMMETRY – 6.7
Mathematics Vision Project
Licensed under the Creative Commons Attribution CC BY 4.0
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6.7 Quadrilaterals—Beyond
Definition
A Practice Understanding Task
Wehavefoundthatmanydifferentquadrilateralspossesslinesofsymmetryand/orrotational
symmetry.Inthefollowingchart,writethenamesofthequadrilateralsthatarebeingdescribedin
termsoftheirsymmetries.
Whatdoyounoticeabouttherelationshipsbetweenquadrilateralsbasedontheirsymmetriesand
highlightedinthestructureoftheabovechart?
CC
BY
Gab
riel
le
http
s://f
lic.k
r/p/
9tK
TT
n
36
SECONDARY MATH I // MODULE 6
TRANSFORMATIONS AND SYMMETRY – 6.7
Mathematics Vision Project
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Basedonthesymmetrieswehaveobservedinvarioustypesofquadrilaterals,wecanmakeclaims
aboutotherfeaturesandpropertiesthatthequadrilateralsmaypossess.
1.Arectangleisaquadrilateralthatcontainsfourrightangles.
Basedonwhatyouknowabouttransformations,whatelsecanwesayaboutrectanglesbesidesthe
definingpropertythat“allfouranglesarerightangles?”Makealistofadditionalpropertiesof
rectanglesthatseemtobetruebasedonthetransformation(s)oftherectangleontoitself.Youwill
wanttoconsiderpropertiesofthesides,theangles,andthediagonals.Thenjustifywhythe
propertieswouldbetrueusingthetransformationalsymmetry.
2.Aparallelogramisaquadrilateralinwhichoppositesidesareparallel.
Basedonwhatyouknowabouttransformations,whatelsecanwesayaboutparallelogramsbesides
thedefiningpropertythat“oppositesidesofaparallelogramareparallel?”Makealistofadditional
propertiesofparallelogramsthatseemtobetruebasedonthetransformation(s)ofthe
parallelogramontoitself.Youwillwanttoconsiderpropertiesofthesides,anglesandthediagonals.
Thenjustifywhythepropertieswouldbetrueusingthetransformationalsymmetry.
37
SECONDARY MATH I // MODULE 6
TRANSFORMATIONS AND SYMMETRY – 6.7
Mathematics Vision Project
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3.Arhombusisaquadrilateralinwhichallfoursidesarecongruent.
Basedonwhatyouknowabouttransformations,whatelsecanwesayaboutarhombusbesidesthe
definingpropertythat“allsidesarecongruent?”Makealistofadditionalpropertiesofrhombuses
thatseemtobetruebasedonthetransformation(s)oftherhombusontoitself.Youwillwantto
considerpropertiesofthesides,anglesandthediagonals.Thenjustifywhythepropertieswouldbe
trueusingthetransformationalsymmetry.
4.Asquareisbotharectangleandarhombus.
Basedonwhatyouknowabouttransformations,whatcanwesayaboutasquare?Makealistof
propertiesofsquaresthatseemtobetruebasedonthetransformation(s)ofthesquaresontoitself.
Youwillwanttoconsiderpropertiesofthesides,anglesandthediagonals.Thenjustifywhythe
propertieswouldbetrueusingthetransformationalsymmetry.
38
SECONDARY MATH I // MODULE 6
TRANSFORMATIONS AND SYMMETRY – 6.7
Mathematics Vision Project
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Inthefollowingchart,writethenamesofthequadrilateralsthatarebeingdescribedintermsof
theirfeaturesandproperties,andthenrecordanyadditionalfeaturesorpropertiesofthattypeof
quadrilateralyoumayhaveobserved.Bepreparedtosharereasonsforyourobservations.
Whatdoyounoticeabouttherelationshipsbetweenquadrilateralsbasedontheircharacteristics
andthestructureoftheabovechart?
Howarethechartsatthebeginningandendofthistaskrelated?Whatdotheysuggest?
39
SECONDARY MATH I // MODULE 6
TRANSFORMATIONS AND SYMMETRY – 6.7
Mathematics Vision Project
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6.7
READY Topic:Definingcongruenceandsimilarity.
1.Whatdoyouknowabouttwofiguresiftheyarecongruent?2.Whatdoyouneedtoknowabouttwofigurestobeconvincedthatthetwofiguresarecongruent?3.Whatdoyouknowabouttwofiguresiftheyaresimilar?4.Whatdoyouneedtoknowabouttwofigurestobeconvincedthatthetwofiguresaresimilar? SET Topic:Classifyingquadrilateralsbasedontheirproperties.Usingtheinformationgivendeterminethemostaccurateclassificationofthequadrilateral.5.Has1800rotationalsymmetry. 6.Has900rotationalsymmetry.7.Hastwolinesofsymmetrythatarediagonals. 8.Hastwolinesofsymmetrythatarenot diagonals.9.Hascongruentdiagonals. 10.Hasdiagonalsthatbisecteachother.11.Hasdiagonalsthatareperpendicular. 12.Hascongruentangles.
READY, SET, GO! Name PeriodDate
40
SECONDARY MATH I // MODULE 6
TRANSFORMATIONS AND SYMMETRY – 6.7
Mathematics Vision Project
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6.7
GO Topic:Slopeanddistance.Findtheslopebetweeneachpairofpoints.Then,usingthePythagoreanTheorem,findthedistancebetweeneachpairofpoints.Distancesshouldbeprovidedinthemostexactform.13.(-3,-2),(0,0) a.Slope:b.Distance:
14.(7,-1),(11,7) a.Slope:b.Distance:
15.(-10,13),(-5,1)a.Slope:b.Distance:
16.(-6,-3),(3,1) a.Slope:b.Distance:
17.(5,22),(17,28)a.Slope:b.Distance:
18.(1,-7),(6,5) a.Slope:b.Distance:
S
41