Fractal Triangles - Fractal Foundation › fractivities › Triangle.pdf · Fractal Triangles...
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Fractal Triangles Fractals are SMART: Science, Math & Art! www.FractalFoundation.org Copyright 2015 Fractal Foundation, all rights reserved 1 Overview The Sierpinski Triangle activity illustrates the fundamental principles of fractals – how a pattern can repeat again and again at different scales and how this complex shape can be formed by simple repetition. Each student will make their own fractal triangle composed of smaller and smaller triangles. Next, students will cut out their personalized triangles and assemble them all into a larger fractal pattern that replicates the same shape. Appropriate for: grades K – 12, college and adults Objectives • To create a complex pattern by repeating a simple process over and over again • To compose, describe and identify shapes and patterns • To analyze and compare shapes according to properties of their lines and angles • To use operations to explain patterns in arithmetic and create mathematical models • To attend to precision • To exhibit teamwork when using each student’s individual fractal design to make a larger product • To produce a piece of art that expresses each student’s individuality • To use appropriate tools strategically Materials • Markers, colored pencils or crayons • Scissors • Ruler/straight edge (optional) • Ruler and protractor for older grades • Triangle worksheet
Transcript of Fractal Triangles - Fractal Foundation › fractivities › Triangle.pdf · Fractal Triangles...
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
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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.
