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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
WCPSS Math 2 Unit 1: MVP MODULE 6
Transformations & Symmetry
SECONDARY
MATH ONE
An Integrated Approach
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SECONDARY MATH 1 // MODULE 6
TRANSFORMATIONS AND SYMMETRY
Mathematics Vision Project
Licensed under the Creative Commons Attribution CC BY 4.0
mathematicsvisionproject.org
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 (NC.M2.G-CO.4, NC.M2.G-CO.5, NC.M2.F-IF.1, NC.M2.F-IF.2)
READY, SET, GO Homework: Transformations and Symmetry 6.1
6.3 Leap Frog – A Solidify Understanding Task
Determining which rigid-motion transformations will carry one image onto another congruent
image (NC.M2.G.CO.4, NC.M2.G.CO.5, NC.M2.F-IF.1, NC.M2.F-IF.2)
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 (NC.M2.G.CO.2, NC.M2.G.CO.4)
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 (NC.M2.G.CO.3)
READY, SET, GO Homework: Transformations and Symmetry 6.5
6.6 Symmetries of Regular Polygons – A Solidify Understanding Task
Examining characteristics of regular polygons that emerge from rotational symmetry and lines
of symmetry (NC.M2.G.CO.3)
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 (NC.M2.G.CO.3, NC.M2.G.CO.4)
READY, SET, GO Homework: Transformations and Symmetry 6.7
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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
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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
CC
0 Sh
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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
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LazyLizard
Translatetheoriginallizardsothepointatthetipofitsnoseislocatedat(24,20),makingthe
lizardappearstobesunbathingontherock.
LungingLizard
Rotatethelizard90°aboutpointA(12,7)soitlookslikethelizardisdivingintothepuddle
ofmud.
LeapingLizard
Reflectthelizardaboutgivenline soitlookslikethelizardisdoingabackflip
overthecactus.
€
y = 12 x +16
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SECONDARY MATH I // MODULE 6
TRANSFORMATION AND SYMMETRY – 6.1
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SECONDARY MATH I // MODULE 6
TRANSFORMATIONS AND SYMMETRY - 6.1
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6.1
READY
Topic:PythagoreanTheorem
Foreachofthefollowingrighttrianglesdeterminethemeasureofthemissingside.Leavethemeasuresinexactformifirrational.
1. 2. 3.
4. 5. 6.
READY, SET, GO! Name Period Date
3
4
?
?
5
12
?4
1
?
3
√10
?
√174
?
2
√13
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SECONDARY MATH I // MODULE 6
TRANSFORMATIONS AND SYMMETRY - 6.1
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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’’’
-5
-5
5
5
A
y = x-5
-5
5
5
B
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SECONDARY MATH I // MODULE 6
TRANSFORMATIONS AND SYMMETRY - 6.1
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6.1
GO Topic:Graphinglinearequations.
Grapheachfunctiononthecoordinategridprovided.Extendthelineasfarasthegridwillallow.
9. !(!) = 2! − 3 10. !(!) = −2! − 311. Whatsimilaritiesanddifferences
aretherebetweenthefunctionsf(x)
andg(x)?
12. ℎ(!) = !! ! + 1 13. !(!) = − !
! ! + 1
14. Whatsimilaritiesanddifferences
aretherebetweentheequationsh(x)
andk(x)?
15. !(!) = ! + 1 16. !(!) = ! − 317. Whatsimilaritiesanddifferences
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
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SECONDARY MATH I // MODULE 6
TRANSFORMATIONS AND SYMMETRY – 6.3
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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 image2
image2 image3
image3 image4
image1 image5
image2 image4
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0 F
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SECONDARY MATH I // MODULE 6
TRANSFORMATIONS AND SYMMETRY – 6.3
Mathematics Vision Project
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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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SECONDARY MATH I // MODULE 6
TRANSFORMATIONS AND SYMMETRY – 6.3
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6.3
READY
Topic:RotationsandReflectionsoffigures.
Ineachproblemtherewillbeapre-imageandseveralimagesbasedonthegivepre-image.Determinewhichoftheimagesarerotationsofthegivenpre-imageandwhichofthemarereflectionsofthepre-image.Ifanimageappearstobecreatedastheresultofarotationandareflectionthenstateboth.(Compareallimagestothepre-image.)
1.
2.
READY, SET, GO! Name Period Date
ImageA ImageB ImageC ImageD
ImageA
ImageB
ImageC
ImageD
Pre-Image
Pre-Image
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SECONDARY MATH I // MODULE 6
TRANSFORMATIONS AND SYMMETRY – 6.3
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6.3
SET Topic:Reflectingandrotatingpoints.Oneachofthecoordinategridsthereisalabeledpointandline.Usethelineasalineofreflectiontoreflectthegivenpointandcreateitsreflectedimageoverthelineofreflection.(Hint:pointsreflectalongpathsperpendiculartothelineofreflection.Useperpendicularslope!)3. 4.
5. 6.
Foreachpairofpoint,PandP’drawinthelineofreflectionthatwouldneedtobeusedtoreflectPontoP’.Thenfindtheequationofthelineofreflection.
7. 8.
m-5
-5
5
5A
l
-5
-5
5
5
C
ReflectpointAoverlinemandlabeltheimageA’ ReflectpointBoverlinekandlabeltheimageB’
ReflectpointCoverlinelandlabeltheimageC’ ReflectpointDoverlinemandlabeltheimageD’
k
-5
-5
5
5
B
m
-5
-5
5
5 D
-5
-5
5
5
P'
P-5
-5
5
5
P'
P
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SECONDARY MATH I // MODULE 6
TRANSFORMATIONS AND SYMMETRY – 6.3
Mathematics Vision Project
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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! = 8
Foreachlinearequationwritetheslopeofalineperpendiculartothegivenline.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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SECONDARY MATH I // MODULE 6
TRANSFORMATIONS AND SYMMETRY – 6.4
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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.
CC
BY
Te
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http
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bmsd
Ub
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TRANSFORMATIONS AND SYMMETRY – 6.4
Mathematics Vision Project
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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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TRANSFORMATIONS AND SYMMETRY – 6.4
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Part2
InadditiontowritingnewfactsforFebruary29,thetwinsalsoneedtoaddanotherimage
inthemiddleoftheirflip-bookanimation.TheanimationsequenceisofDorothy'shousefromthe
WizardofOzasitisbeingcarriedovertherainbowbyatornado.ThehouseintheFebruary28
drawinghasbeenrotatedtocreatethehouseintheMarch1drawing.Carlosbelievesthathecan
getfromtheFebruary28drawingtotheMarch1drawingbyreflectingtheFebruary28drawing,
andthenreflectingitagain.
VerifythattheimageCarlosinsertedbetweenthetwoimagesthatappearedonFebruary28
andMarch1worksasheintended.Forexample,
• WhatconvincesyouthattheFebruary29imageisareflectionoftheFebruary28imageaboutthegivenlineofreflection?
• WhatconvincesyouthattheMarch1imageisareflectionoftheFebruary29imageaboutthegivenlineofreflection?
• WhatconvincesyouthatthetworeflectionstogethercompletearotationbetweentheFebruary28andMarch1images?
Page 15
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TRANSFORMATIONS AND SYMMETRY – 6.4
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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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TRANSFORMATIONS AND SYMMETRY – 6.4
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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 Period Date
UsethecenterofrotationpointCandrotatepoint
Pclockwisearoundit900.LabeltheimageP’.
WithpointCasacenterofrotationalsorotate
pointP1800.LabelthisimageP’’.C
P
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TRANSFORMATIONS AND SYMMETRY – 6.4
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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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SECONDARY MATH I // MODULE 6
TRANSFORMATIONS AND SYMMETRY – 6.4
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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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TRANSFORMATIONS AND SYMMETRY – 6.5
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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
ontoitself
Arectangleisaquadrilateralthatcontainsfourrightangles.
Aparallelogramisaquadrilateralinwhichoppositesidesareparallel.
CC
BY
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http
s://f
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gis7
Cj
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TRANSFORMATIONS AND SYMMETRY – 6.5
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Atrapezoidisaquadrilateralwithonepairofoppositesidesparallel.Isitpossibletoreflectorrotateatrapezoidontoitself?Drawatrapezoidbasedonthisdefinition.Thenseeifyoucanfind:
• anylinesofsymmetry,or• anycentersofrotationalsymmetry,
thatwillcarrythetrapezoidyoudrewontoitself.
Ifyouwereunabletofindalineofsymmetryoracenterofrotationalsymmetryforyourtrapezoid,seeifyoucansketchadifferenttrapezoidthatmightpossesssometypeofsymmetry.
Arhombusisaquadrilateralinwhichallsidesarecongruent.
Asquareisbotharectangleandarhombus.
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TRANSFORMATIONS AND SYMMETRY – 6.5
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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 Period Date
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TRANSFORMATIONS AND SYMMETRY – 6.5
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6.5
5. Listalloftherotationalsymmetryforakite.
6. Arelinesofsymmetryalsodiagonalsinapolygon?Explain.
6. Arealldiagonalsalsolinesofsymmetryinapolygon?Explain.
7. Whichquadrilateralshavediagonalsthatarenotlinesofsymmetry?Namesomeanddrawthem.
8. Doparallelogramshavediagonalsthatarelinesofsymmetry?Ifso,drawandexplain.Ifnotdrawand
explain.
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TRANSFORMATIONS AND SYMMETRY – 6.5
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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
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TRANSFORMATIONS AND SYMMETRY – 6.6
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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)
2. Asquare
3. AregularpentagonC
C B
Y Jo
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Jara
mill
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http
s://f
lic.k
r/p/
rbd9
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1. Anequilateraltriangle
4. Aregularhexagon
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TRANSFORMATIONS AND SYMMETRY – 6.6
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5. Aregularoctagon 6. Aregularnonagon
Whatpatternsdoyounoticeintermsofthenumberandcharacteristicsofthelinesofsymmetryina
regularpolygon?
Whatpatternsdoyounoticeintermsoftheanglesofrotationwhendescribingtherotational
symmetryinaregularpolygon?
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TRANSFORMATIONS AND SYMMETRY – 6.6
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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 Period Date
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TRANSFORMATIONS AND SYMMETRY – 6.6
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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
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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?
Page 31
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TRANSFORMATIONS AND SYMMETRY – 6.6
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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
Page 32
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
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TRANSFORMATIONS AND SYMMETRY – 6.7
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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.
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TRANSFORMATIONS AND SYMMETRY – 6.7
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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.
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TRANSFORMATIONS AND SYMMETRY – 6.7
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Inthefollowingchart,writethenamesofthequadrilateralsthatarebeingdescribedintermsof
theirfeaturesandproperties,andthenrecordanyadditionalfeaturesorpropertiesofthattypeof
quadrilateralyoumayhaveobserved.Bepreparedtosharereasonsforyourobservations.
Whatdoyounoticeabouttherelationshipsbetweenquadrilateralsbasedontheircharacteristics
andthestructureoftheabovechart?
Howarethechartsatthebeginningandendofthistaskrelated?Whatdotheysuggest?
Page 36
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TRANSFORMATIONS AND SYMMETRY – 6.7
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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. Hastwolinesofsymmetrythatarenotdiagonals.
9. Hascongruentdiagonals. 10. Hasdiagonalsthatbisecteachother.
11. Hasdiagonalsthatareperpendicular. 12. Hascongruentangles.
READY, SET, GO! Name Period Date
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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
Page 38