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
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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
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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
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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.