module 6: transformations & symmetry

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

Mathematics Vision Project

Licensed under the Creative Commons Attribution CC BY 4.0

mathematicsvisionproject.org

MODULE 6 - TABLE OF CONTENTS


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)


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)


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)


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)


SECONDARY MATH 1 // MODULE 6





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)


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)


SECONDARY MATH I // MODULE 6

TRANSFORMATION AND SYMMETRY – 6.1




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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2






3


TRANSFORMATIONS AND SYMMETRY - 6.1




6.1

READY

Topic:PythagoreanTheorem

Foreachofthefollowingrighttrianglesdeterminethemeasureofthemissingside.Leavethemeasuresinexactformifirrational.

1. 2. 3.

4. 5. 6.

READY, SET, GO! Name PeriodDate

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4

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5

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3

√10

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√174

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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)?

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5

5

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-5

5

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TRANSFORMATIONS AND SYMMETRY – 6.2




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


andB(7,5).

8






Experiment4


andB(0,6).

Basedonexperiments1-4,stateanobservationabouttheslopesofperpendicularlines.

Whilethisobservationisbasedonafewspecificexamples,canyoucreateanargumentor

justificationforwhythisisalwaystrue?(Note:Youwillexamineaformalproofofthisobservation

inafuturemodule.)

9






6.2

READY Topic:FindingDistanceusingPythagoreanTheoremUsethecoordinategridtofindthelengthofeachsideofthetrianglesprovided.Giveanswersin

exactformandwherenecessaryroundedtothenearesthundredth.

1. 2.

3. 4.

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-5

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6.2

SET Topic:Slopesofparallelandperpendicularlines.

5.Graphalineparalleltothegivenline.



Equationforgivenline:Equationfornewline:



8.Graphalineperpendiculartothegivenline.






11






6.2

GO Topic:Solvethefollowingequations.Solveeachequationfortheindicatedvariable.

11.3 ! − 2 = 5! + 8;Solveforx. 12.−3 + ! = 6! + 22;Solveforn.

13.! − 5 = !(! − 2);Solveforx. 14.!" + !" = !;Solvefory.

12






6. 3 Leap Frog


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

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image4

image1

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images this page:

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6.3

READY Topic:RotationsandReflectionsoffigures.Ineachproblemtherewillbeapre-imageandseveralimagesbasedonthegivepre-image.Determinewhichoftheimagesarerotationsofthegivenpre-imageandwhichofthemarereflectionsofthepre-image.Ifanimageappearstobecreatedastheresultofarotationandareflectionthenstateboth.(Compareallimagestothepre-image.)1. 2.


ImageA ImageB ImageC ImageD

ImageA

ImageB

ImageC

ImageD

Pre-Image

Pre-Image

15






6.3

SET Topic:Reflectingandrotatingpoints.Oneachofthecoordinategridsthereisalabeledpointandline.Usethelineasalineofreflectiontoreflectthegivenpointandcreateitsreflectedimageoverthelineofreflection.(Hint:pointsreflectalongpathsperpendiculartothelineofreflection.Useperpendicularslope!)3. 4.

5. 6.

Foreachpairofpoint,PandP’drawinthelineofreflectionthatwouldneedtobeusedtoreflectPontoP’.Thenfindtheequationofthelineofreflection.

7. 8.

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C

ReflectpointAoverlinemandlabeltheimageA’

ReflectpointBoverlinekandlabeltheimageB’

ReflectpointCoverlinelandlabeltheimageC’ ReflectpointDoverlinemandlabeltheimageD’

k

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5

5

B

m

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5

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5

P'

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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

17






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

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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.

19






Part2

InadditiontowritingnewfactsforFebruary29,thetwinsalsoneedtoaddanotherimage

inthemiddleoftheirflip-bookanimation.TheanimationsequenceisofDorothy'shousefromthe

WizardofOzasitisbeingcarriedovertherainbowbyatornado.ThehouseintheFebruary28

drawinghasbeenrotatedtocreatethehouseintheMarch1drawing.Carlosbelievesthathecan

getfromtheFebruary28drawingtotheMarch1drawingbyreflectingtheFebruary28drawing,

andthenreflectingitagain.

VerifythattheimageCarlosinsertedbetweenthetwoimagesthatappearedonFebruary28

andMarch1worksasheintended.Forexample,

• WhatconvincesyouthattheFebruary29imageisareflectionoftheFebruary28imageaboutthegivenlineofreflection?

• WhatconvincesyouthattheMarch1imageisareflectionoftheFebruary29imageaboutthegivenlineofreflection?

• WhatconvincesyouthatthetworeflectionstogethercompletearotationbetweentheFebruary28andMarch1images?

20






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.


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






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.

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Atrapezoidisaquadrilateralwithonepairofoppositesidesparallel.Isitpossibletoreflectorrotateatrapezoidontoitself?Drawatrapezoidbasedonthisdefinition.Thenseeifyoucanfind:

• anylinesofsymmetry,or• anycentersofrotationalsymmetry,

thatwillcarrythetrapezoidyoudrewontoitself.Ifyouwereunabletofindalineofsymmetryoracenterofrotationalsymmetryforyourtrapezoid,seeifyoucansketchadifferenttrapezoidthatmightpossesssometypeofsymmetry.

Arhombusisaquadrilateralinwhichallsidesarecongruent.

Asquareisbotharectangleandarhombus.

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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


27






6.5

5.Listalloftherotationalsymmetryforakite.

6.Arelinesofsymmetryalsodiagonalsinapolygon?Explain.

6.Arealldiagonalsalsolinesofsymmetryinapolygon?Explain.

7.Whichquadrilateralshavediagonalsthatarenotlinesofsymmetry?Namesomeanddrawthem.

8.Doparallelogramshavediagonalsthatarelinesofsymmetry?Ifso,drawandexplain.Ifnotdrawand

explain.

28






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):

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-5

5

5

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6.6 Symmetries of Regular

Polygons


Alinethatreflectsafigureontoitselfiscalledalineofsymmetry.Afigurethatcanbecarriedonto

itselfbyarotationissaidtohaverotationalsymmetry.Adiagonalofapolygonisanyline

segmentthatconnectsnon-consecutiveverticesofthepolygon.

Foreachofthefollowingregularpolygons,describetherotationsandreflectionsthatcarryitonto

itself:(beasspecificaspossibleinyourdescriptions,suchasspecifyingtheangleofrotation)

1. Anequilateraltriangle

2. Asquare

3. Aregularpentagon

4. Aregularhexagon

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5. Aregularoctagon

6. Aregularnonagon

Whatpatternsdoyounoticeintermsofthenumberandcharacteristicsofthelinesofsymmetryina

regularpolygon?

Whatpatternsdoyounoticeintermsoftheanglesofrotationwhendescribingtherotational

symmetryinaregularpolygon?

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6.6

READY

Topic:Rotationalsymmetry,connectedtofractionsofaturnanddegrees.

1.Whatfractionofaturndoesthewagonwheel

belowneedtoturninordertoappearthevery

sameasitdoesrightnow?Howmanydegreesof

rotationwouldthatbe?

2.Whatfractionofaturndoesthepropeller

belowneedtoturninordertoappearthevery

sameasitdoesrightnow?Howmanydegreesof


3.WhatfractionofaturndoesthemodelofaFerriswheelbelowneedtoturninordertoappearthe

verysameasitdoesrightnow?Howmanydegreesof



32






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?

34






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






6.7 Quadrilaterals—Beyond

Definition

A Practice Understanding Task

Wehavefoundthatmanydifferentquadrilateralspossesslinesofsymmetryand/orrotational

symmetry.Inthefollowingchart,writethenamesofthequadrilateralsthatarebeingdescribedin

termsoftheirsymmetries.

Whatdoyounoticeabouttherelationshipsbetweenquadrilateralsbasedontheirsymmetriesand

highlightedinthestructureoftheabovechart?

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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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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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Inthefollowingchart,writethenamesofthequadrilateralsthatarebeingdescribedintermsof

theirfeaturesandproperties,andthenrecordanyadditionalfeaturesorpropertiesofthattypeof

quadrilateralyoumayhaveobserved.Bepreparedtosharereasonsforyourobservations.

Whatdoyounoticeabouttherelationshipsbetweenquadrilateralsbasedontheircharacteristics

andthestructureoftheabovechart?

Howarethechartsatthebeginningandendofthistaskrelated?Whatdotheysuggest?

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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.


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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

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module 6: transformations & symmetry

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