FractalTriangles

FractalsareSMART:Science,Math&Art!www.FractalFoundation.org

Copyright2015FractalFoundation,allrightsreserved

1

OverviewTheSierpinskiTriangleactivityillustratesthefundamentalprinciplesoffractals–howapatterncanrepeatagainandagainatdifferentscalesandhowthiscomplexshapecanbeformedbysimplerepetition.Eachstudentwillmaketheirownfractaltrianglecomposedofsmallerandsmallertriangles.Next,studentswillcutouttheirpersonalizedtrianglesandassemblethemallintoalargerfractalpatternthatreplicatesthesameshape.

Appropriatefor:gradesK–12,collegeandadultsObjectives

• Tocreateacomplexpatternbyrepeatingasimpleprocessoverandoveragain• Tocompose,describeandidentifyshapesandpatterns• Toanalyzeandcompareshapesaccordingtopropertiesoftheirlinesandangles• Touseoperationstoexplainpatternsinarithmeticandcreatemathematicalmodels• Toattendtoprecision• Toexhibitteamworkwhenusingeachstudent’sindividualfractaldesigntomakea

largerproduct• Toproduceapieceofartthatexpresseseachstudent’sindividuality• Touseappropriatetoolsstrategically

Materials • Markers,coloredpencilsorcrayons• Scissors• Ruler/straightedge(optional)• Rulerandprotractorforoldergrades• Triangleworksheet

FractalTriangles

FractalsareSMART:Science,Math&Art!www.FractalFoundation.org

Copyright2015FractalFoundation,allrightsreserved

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CommonCoreStandardsforMathematics

CommonCoreStandardsforEnglishLanguageArts

Code Standard Grade Code Standard GradeCC CountingandCardinality K G Geometry K–8NBT NumberandOperationsin

BaseTenK,1 NS NumberSystem 6,7

OA OperationsandAlgebraicThinking

K–5 RP RatiosandProportionalRelationships

6,7

MD MeasurementandData K–5 SP StatisticsandProbability 6

Code Standard GradesK–5 Grades6–8 Grades9–12RL Reading:Literature 1,4,7,10 1,4,7,10 1,4,10RI Reading:

InformationalText1,3,4,7,10 1,3,4,7,10 1,3,4,10

FS FoundationalSkills 1,2,3forgradesK–1;3and4forgrades2–5

Noneavailable Noneavailable

W Writing 2,3,8;4forgrades3–5 2,3,4 2,3,4,9SL Speakingand

Listening1,2,3,4,5,6 1,2,3,4,5,6 1,2,3,4,5,6

L Language 1,4,6;3forgrades2–5 1,3,4,6 1,3,4,6RST Scienceand

TechnicalSubjectsNoneavailable 1,3,4,6,7,10 2,3,4,6,7,10

FractalTriangles

FractalsareSMART:Science,Math&Art!www.FractalFoundation.org

Copyright2015FractalFoundation,allrightsreserved

3

InstructionsPrintoutoneactivitytemplate(page5)foreachstudent.1.)Providemarkers,coloredpencilsorcrayonsandinstructstudentstoconnectthemidpointsoftheexistingtriangletomakeanew,downward-facingtriangle.Havestudentscolorinthedownward-facingtriangleonly.Thiswillleavethreeupward-facingtrianglesremaining,eachofwhichisliketheoriginal,buthalfthewidth.2)Next,havestudentsplacedotsatthemidpointsofeachofthesidesofeachofthethreesmallertriangles.Connectthosemidpointstomakethreesmallerdownwardfacingtriangles,andthencolorinthesetriangles.3and4)Repeatthesameprocessagainforatleastthreeiterationstotal.

Rubricforevaluation:Theymustgettoatleaststep2.Doestheirtriangleexhibitself-similarity,orisitjustlotsofcopiesofonesizeoftriangle?Howaccuratearetheirmidpoints?Allowforcreativeexpression-eachstudent’sfractalisanindividualartwork-buttheymustbefractal.

1 2

3 4

FractalTriangles

FractalsareSMART:Science,Math&Art!www.FractalFoundation.org

Copyright2015FractalFoundation,allrightsreserved

4

Whenstudentsaredone,eachstudentshouldcutouttheirtriangleandthenjointheirtrianglesingroupstoformalargerversionofthesameshape!Nine,27,or81trianglesworkbest.“Trianglehandout”isthecompanionworksheetforthisactivity.

FractalTriangles

FractalsareSMART:Science,Math&Art!www.FractalFoundation.org

Copyright2015FractalFoundation,allrightsreserved

5

Youareabouttobuildtheworld’slargestfractaltriangle,madeoutof2,187individualtriangles.

Howmuchmaskingtapewillyouneed?

Theentiretriangleis96feetonaside.Eachofthesmallesttrianglesinthediagramis12feetwide.

Todoso,youmustlaydownmaskingtapeonthefloortodrawtheoutlines,soyouknowwheretoputallthetriangles.

FractalTriangles

FractalsareSMART:Science,Math&Art!www.FractalFoundation.org

Copyright2015FractalFoundation,allrightsreserved

6

Whenyoudecideyou’redone,

cutoutthebigtriangle,flipitover,andw

riteyournameand

schoolontheback.

Now

howmanyright-side-

uptrianglesdoyouhave?Fillinyourtableforoneiteration.Findthemidpointsofthethree

sidesofthesethreetrianglesandconnectthesemidpointstom

akethreesmallerupside-dow

ntriangles.

FractalTriangleTemplateInstructions

Beforedoinganythingtoyourtriangle,fillinthetablew

iththenumberofright-side-uptriangles

youhavewithzeroiterationsoftheprocess.

Joinitwithothers

tomakegroupsof

three,nine,27ormoretrianglesthat

makethesam

eshapebutlarger.

Fractalsarenever-endingpatterns,andyoucouldkeepdoingthisforalongtim

e…forever,

perhaps!

IterationMath

notationNum

berofright-side-uptriangles

030

1

31

2

32

3

33

4

34

Connectthethreedotsthatareinthemiddle

ofeachline.Thisiscalledthemidpoint.

Colorinthisupside-downtriangle.

Continuethisprocess,makingsm

allerandsmaller

trianglesandfillinthetableforeachiteration.