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Geometry

Transform it! Dilations

Grades

7-10 Aligned to the Common Core State Standards

8.G.A.3

Teacher’sGuide

CCSS8.G.A.3Describetheeffectsofdilations,translations,rotations,andre<lectionsontwo-dimensional<iguresusingcoordinates.Objectives•  Studentswillbeabletodescribetheeffectsofdilationsonpolygonsusing

coordinates.

Materials•  “Transformit!GraphsandCoordinatesofDilations”Handout•  “Transformit!DilationsSummaryandExtension”Handout•  RulersProcedure•  Allowstudentstoworkthroughthehandoutindividuallyorinpairs.

Intermittentlycheckforunderstandingasalargegroup.ThisproductispartoftheDiscovery-BasedWorksheetSeries.Discovery-BasedWorksheetshavebeenspeciallydesignedtoengagestudentsinlearningthatmovesbeyondtraditionalskillspractice.Studentswilldevelopadeeperunderstandingofthebigideaandwillmakeconnectionsbetweenconcepts.Theseworksheetsmakeagreatintroductiontoanewtopic

orsummaryattheendofalessonorunit.Enjoy!

Name:______________________________________________________________Class:__________Date:____________________

Transformit!GraphsandCoordinatesofDilations

Ageometricdilationcanbecomparedtotheprocessofhavingyoureyesdilated.Justasad-EYE-lationchangesthesizeofpupilsattheoptometrists’of<ice,italsochangesthesizeofageometric<igure.Withadilation,theoriginalshapechangessizebasedonascalefactorandwillbecenteredatadesignatedpoint,oftentheorigin.Theresulting<igureisasimilarimageoftheoriginalshape.Inthisactivity,youwillexplorethepatternsthatexistwithdilations.Activity1a)  Plotthefollowingpointsonthecoordinateplane:{(-1,1),(-4,0),(-5,5),(-2,3)}LabelthemA,B,C,andDrespectively.Connectthepoints.b)  Dilatethe<igurebydoublingeveryverticalandhorizontaldistancefromtheorigin.Forexample,(-1,1)islocated1unitleftand1unitupfrom(0,0).Afterthedilation,itshouldbelocated2unitsleftand2unitsup.LabelthenewcoordinatesA’,B’,C’,andD’.Connectthepoints.c)  WritedownthecoordinatesofA’B’C’D’.d)Howdothecoordinatesof<igureA’B’C’D’comparetothoseof<igureABCD?e)  Thistypeofdilationiscenteredattheoriginbecausethedistancewasmeasuredfromthatpoint

eachtime.Whatdoyousupposewasthescalefactor?

f)  Hypothesizewhatyoubelievetherelationshipbetweenthecoordinatesandthescalefactormight

be.WewilltestthishypothesiswithanotherexampleinActivity2.

© Free to Discover (Amanda Nix) – 2015

Activity2a)  Plotthefollowingpointsonthecoordinateplane:{(-3,0),(-6,-9),(-6,-3)}LabelthemE,F,andGrespectively.Connectthem.b)  Dilatethe<igurebytakingone-thirdofeveryverticalandhorizontaldistancefromtheorigin.Forexample,(-3,0)islocated3unitsleftand0unitsupfrom(0,0).Afterthedilation,itshouldbelocated1unitleftand0unitsup.LabelthenewcoordinatesE’,F’,andG’.Connectthepoints.c)  WritedownthecoordinatesofE’F’G’.

d)Howdothecoordinatesof<igureE’F’G’comparetothoseof<igureEFG?

e)Whatwasthescalefactor?Doesthisresultalignwithyouroriginalhypothesis?Explain.

f)Letrrepresentthescalefactorofadilationcenteredattheorigin.Recordthegeneralrulehere:Activity3a)  Let’spracticedilatinga<igurecenteredatapointotherthantheorigin.Plotthefollowingpointsonthecoordinateplane:{(2,-2),(4,-1),(3,-4)}LabelthemH,I,andJrespectively.Connectthepoints.b)  Dilatethe<igurewithascalefactorof2centeredat(1,1).Forexample,H(2,-2)is1unitrightand3unitsdownfrom(1,1).H’willbe2unitsrightand6unitsdownfrom(1,1).LabelthenewcoordinatesH’I’J’.Connectthepoints.

Dilation Centered at the Origin: (x, y) à ( , ) !

© Free to Discover (Amanda Nix) – 2015

Name:______________________________________________________________Class:__________Date:____________________

Transformit!GraphsandCoordinatesofDilations–AnswerKey

Ageometricdilationcanbecomparedtotheprocessofhavingyoureyesdilated.Justasad-EYE-lationchangesthesizeofpupilsattheoptometrists’of<ice,italsochangesthesizeofageometric<igure.Withadilation,theoriginalshapechangessizebasedonascalefactorandwillbecenteredatadesignatedpoint,oftentheorigin.Theresulting<igureisasimilarimageoftheoriginalshape.Inthisactivity,youwillexplorethepatternsthatexistwithdilations.Activity1a)  Plotthefollowingpointsonthecoordinateplane:{(-1,1),(-4,0),(-5,5),(-2,3)}LabelthemA,B,C,andDrespectively.Connectthepoints.b)  Dilatethe<igurebydoublingeveryverticalandhorizontaldistancefromtheorigin.Forexample,(-1,1)islocated1unitleftand1unitupfrom(0,0).Afterthedilation,itshouldbelocated2unitsleftand2unitsup.LabelthenewcoordinatesA’,B’,C’,andD’.Connectthepoints.c)  WritedownthecoordinatesofA’B’C’D’.

{(-2,2),(-8,0),(-10,10),(-4,6)}d)Howdothecoordinatesof<igureA’B’C’D’comparetothoseof<igureABCD?

Boththex-andy-valuesweredoubled.e)  Thistypeofdilationiscenteredattheoriginbecausethedistancewasmeasuredfromthatpoint

eachtime.Whatdoyousupposewasthescalefactor?Thescalefactorwas2.

f)  Hypothesizewhatyoubelievetherelationshipbetweenthecoordinatesandthescalefactormight

be.WewilltestthishypothesiswithanotherexampleinActivity2.SampleAnswer:Thescalefactoristhenumbermultipliedtoboththex-andy-valuesofeach

coordinate.

A!B!

C

D

D’!

C’!

B’!

A’!

© Free to Discover (Amanda Nix) – 2015

Activity2a)  Plotthefollowingpointsonthecoordinateplane:{(-3,0),(-6,-9),(-6,-3)}LabelthemE,F,andGrespectively.Connectthem.b)  Dilatethe<igurebytakingone-thirdofeveryverticalandhorizontaldistancefromtheorigin.Forexample,(-3,0)islocated3unitsleftand0unitsupfrom(0,0).Afterthedilation,itshouldbelocated1unitleftand0unitsup.LabelthenewcoordinatesE’,F’,andG’.Connectthepoints.c)  WritedownthecoordinatesofE’F’G’.

{(-1,0),(-2,-3),(-2,-1)}

d)  Howdothecoordinatesof<igureE’F’G’comparetothoseof<igureEFG?Thex-andy-valuesofE’F’G’areone-thirdthevalueofthoseinEFG.

e)Whatwasthescalefactor?Doesthisresultalignwithyouroriginalhypothesis?Explain.

Thescalefactorwas.Sampleanswer:Yes,thescalefactorwasmultipliedtotheoriginalcoordinatestocreatethenewcoordinates.

f)Letrrepresentthescalefactorofadilationcenteredattheorigin.Recordthegeneralrulehere:Activity3a)  Let’spracticedilatinga<igurecenteredatapointotherthantheorigin.Plotthefollowingpointsonthecoordinateplane:{(2,-2),(4,-1),(3,-4)}LabelthemH,I,andJrespectively.Connectthepoints.b)  Dilatethe<igurewithascalefactorof2centeredat(1,1).Forexample,H(2,-2)is1unitrightand3unitsdownfrom(1,1).H’willbe2unitsrightand6unitsdownfrom(1,1).LabelthenewcoordinatesH’I’J’.Connectthepoints.

Dilation Centered at the Origin: (x, y) à (rx, ry)!

E !

F!

G !

E’!

F’!

G’!

13

H! I !

J!H’!

I’!

J’!

© Free to Discover (Amanda Nix) – 2015

Name:______________________________________________________________Class:__________Date:____________________

Transformit!DilationsSummaryandExtension

Summarize a)  List three words or phrases that sum up geometric dilations.!

!!!b)  Based on your observations of the graphs in this activity, does it seem that dilations ! result in figures that are congruent, similar, both, or neither? !

!!!c)  Based on your observations of the graphs in this activity, circle all attributes that appear

to change given a dilation of a geometric figure.!!

size shape direction location !

Extend a)  Create your own dilation.!

q  Draw a figure using 6-12 coordinates. !q  Label each vertex with a letter (A, B, C, etc).!q  Determine the scale factor that you will apply. State it here: ______________!q  Dilate the figure centered at the origin based on the scale factor you selected.!q  Label the new vertices (A’, B’, C’).!

!

!!

!!!b) Create a hashtag that cleverly sums up the lesson.!

!

© Free to Discover (Amanda Nix) – 2015

Name:______________________________________________________________Class:__________Date:____________________

Transformit!DilationsSummaryandExtension–AnswerKey

Summarize a)  List three words or phrases that sum up geometric dilations.!

Sample answer: similar, size change, scale factor !!!b)  Based on your observations of the graphs in this activity, does it seem that dilations ! result in figures that are congruent, similar, both, or neither? !

similar !!!c)  Based on your observations of the graphs in this activity, circle all attributes that appear

to change given a dilation of a geometric figure.!!

size shape direction location !

Extend a)  Create your own dilation. – Answers will vary.!

q  Draw a figure using 6-12 coordinates. !q  Label each vertex with a letter (A, B, C, etc).!q  Determine the scale factor that you will apply. State it here: ______________!q  Dilate the figure centered at the origin based on the scale factor you selected.!q  Label the new vertices (A’, B’, C’).!

!

!!

!!!b) Create a hashtag that cleverly sums up the lesson.!

Sample Answer: #dEYElated !

© Free to Discover (Amanda Nix) – 2015

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