DilationsInvestigation-StudentActivity

Objective:Givengridpaper,acentimeterruler,aprotractor,andasheetofpattypaperthestudentswillgenerateandapplytherelationshipbetweenthescalefactorofdilationandthecoordinatesoftheimage,anglemeasures,perimeter,andareabycompletingdatatablesandbyredrawingfiguresongraphpaper.

Thestudentisexpectedtogeneratesimilarfiguresusingdilationsincludingenlargementsandreductions,graphdilations,reflections,andtranslationsonacoordinateplane,useandextendsimilaritypropertiesandtransformationstoexploreandjustifyconjecturesaboutgeometricfiguresanddescribetheresultingeffectsonperimeterandareawhendimensionsofashapearechangedproportionally.

LessonOverview:Inpartone,studentswillusepattypapertocreateadilationwithascalefactoroftwoonplanepaper.Studentswillthentransferthedilationtothecoordinategridandcreateadditionaldilationsofthesamefigurewithscalefactorsofthreeandone-half.Inparttwo,studentswillmakecoordinatedilationsofseveralfiguresandwilllookattheeffectofthescalechangeonthecoordinates,sidelengths,perimeter,andareaofeachimage.Extension:Studentscangraphthecoordinatesofthefourtrianglesinpartoneusingagraphingcalculatoranddeterminetheequationsfortheraysusedinthedilation.

Grouping:Studentsshouldworkingroupssotheycandiscusswitheachotherwhattheynoticeabouttheeffectsofthedilations.

Materials:Eachstudentwillneedacopyofthelesson,aseparatesheetofcentimetergridpaper,acentimeterruler,aprotractor,andasheetofpattypaper.Studentscantheusesamepieceofpattypaperforstepsonethrougheighttotransferthedilationstothegridpaper.Studentsshouldbeabletoworkthroughthedocumentontheirownwithteacherguidance.

Homework:Studentscancompleteparttwooftheassignmentashomework.Moststudentsshouldbeabletocompletebothpartsinaninety­minuteclassperiod.

Adaptedfromalessonfoundat:popehs.typepad,corn

Dilations-PartI

1. Onthepicturebelow,drawraysfrom0througheachvertexofMBC.

2. UsepattypapertomeasurethelengthofOA. PlaceA'onOAsothatOA'=2•OA

3. UsethesameprocesstofindB'andC'.

4. DrawM'B'C'.

5. TraceMBConapieceofpattypaper.ComparetheanglesofMBCwiththeanglesofM'B'C'.

Whatdoyounoticeabouttheanglemeasures?

6. Usearulertofindthefollowinglengthsincentimeters.

a. AB= d. A'B' =

b. BC= e.

B'C'=

c. AC= f. A'C' =

7. Whatdoyounoticeaboutthelengthsofthesidesofthetwotriangles?

8. TheScaleFactorforthisdilationis_

A

Bc

0

•

9.Useyourpattypapertotransfer MBCandM 1B1C1 tothegridpaperwithpoint0attheorigin.WritetheorderedpairsnexttoeachvertexonthegridpaperandinTable1below. Whatdoyounoticeabouttheseorderedpairs?

10. Dilate MBCusingascalefactorof3onyourgridpaper. LabelthenewtriangleM"B"C".WritetheorderedpairsnexttoeachvertexandinTable1below.

11. Dilate MBCusingascalefactorof..!_onyourgridpaper. LabelthetriangleM"'B"'C"'.Writethe

2

orderedpairsnexttoeachvertexandinTable1below.

Table1-OrderedPairs

Original
Triangle / Dilation
(ScaleFactorof2) / Dilation
(ScaleFactorof3) / Dilation
(ScaleFactorof-1)
2
A / A' / A" / A"'
B / B' / B" / B"'
c / C' / C" / C"'

12. Whatobservations/conjectures canyoumakeaboutdilationsandorderedpairsbasedonyourworksofar?

13. CompletethisconjecturebasedonTable1: Theimageof(a,b) afteradilationwithcenter0andscalefactor khascoordinates( ).

Dilations-PartII

I" RecordyourobservationsaboutthecharacteristicsofthetrianglesMBCandM'B'C'from

Dilations-PartIinthetablebelow.

Characteristics / Original MBC / M'B'C'
(ScaleFactorof2) / Observations
Coordinates / A(3,5)B(3,2),C(7,2)
AngleMeasures
LengthofSides
Perimeter
Area

2,GraphQuadrilateral JKLMonthebackofyourgridpaper(fromPart1).DilatequadrilateralJKLMwithascalefactorof3onyourgridpaper.Labeltheimage J'K'L'M'andcompletethetablebelow.

Characteristics / Original JKLM / Quad. J'K'L'M'
_(ScaleFactorof3) / Observations
Coordinates / J(-2,1)K(-2,3),
L(3,3),M(3,1)
AngleMeasures
LengthofSides
Perimeter
Area

,.,3,. GraphQuadrilateral QRSTonthesamegridpaper.Dilatequadrilateral QRSTwithascalefactorof

.!.. onyourgridpaper.Labeltheimage Q'R'S'T'andcompletethetablebelow.

2

Characteristics / Original QRST / Quad.Q'R'S'T'
(ScaleFactorof 1)
-
2 / Observations
Coordinates / Q(-2,-2)R(-2,-4),
S(-6,-4),T(-6,2)
AngleMeasures
LengthofSides
Perimeter
Area

AdiameterofacirclehasendpointsAandB.Graphthecircleonyourgridpaper.Dilatethecircle

Wl'th asea1ef:actorof2Labe1thecoordm'

atesofther.mageA'

and B'

andcomp.1etethetablebe1ow.

Characteristics / Originalcirclewith
-
diameter AB / Newcirclewith
--
diameter A'B' / Observations
Coordinates / A(4,1),B(4,-3)
Lengthofthediameter
Lengthoftheradius
Circumference
Area

JJ..Basedontheinformationinallfourtables,whatconjecturescanyoumakeabout:

e. ScaleFactorusedinthedilationandanglemeasures

f.ScaleFactorusedinthedilationandsidelengths/diameter. ------g. ScaleFactorusedinthedilationandperimeter/circumference.------

h. ScaleFactorusedinthedilationandarea.

·""Usethegraphtobelowtoanswerthefollowingdilationquestions.

a) Findthecoordinatesoftriangle A'B'C'afteradilation

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Howdoesthischangethelengthsofthesides?

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c) Ifthecoordinatesof A'are(-1.5,1)afteradilation,findthecoordinatesof B' andC'?

Howdoesthischangetheperimeterofthetriangle?------d)Ifthecoordinatesof B' are(18,-12)afteradilation,findthecoordinatesof A' and C'? __Howdoesthischangetheareaofthetriangle?

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