Download Fibonacci Sequence and Fractal Spirals

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FibonacciSequenceandFractalSpirals
1.First,we’regoingtofigureouttheFibonaccisequence.Fillouttheblanksbelow:
0+1=______
1+______=______
______+______=______
______+______=______
______+______=______
______+______=______
______+______=______
______+______=______
2.Listeachnumberaftertheequalsign:112______________________________
3.Now,squareeachnumber:114_______________________________
4.Addtwoadjacentnumbersfromthelistabovetogether.
1+1=_____1+4=_____4+_____=__________+_____=_____
Whatpatterndoyousee?Circlethosenumberswhereyou’veseenthembefore!
5.Howaboutwhenyouaddthesquarednumbers(from#3)sequentially?
114_________________
1+1+4=_____thenaddthenextnumberinthesequencetothat
_____+_____=______+______=______+______=______
6.Listthenumbersfromaboveaftereachequalsign(=):________________________
FibonacciSequenceandFractalSpirals
7.Howiseachnumberlistedin#6expressedasamultiplicationofnumbersinthe
Fibonaccisequence,listedafter#2?
yourfirstnumber______=_____x______yoursecondnumber______=______x______
yourthirdnumber_____=______x______yourfourthnumber______=______x______
Anotherfunandmind-blowingfact…
8.GoingbacktotheoriginalFibonaccisequence,dividethelargernumberbytheprevious
smallernumberandlet’sseewhatweget.Theoriginalsequence(#2)is:
112___________________________________
andsoon…
1÷1=_____2÷1=__________÷2=___________÷_____=___________÷_____=______
______÷_____=____________÷______=____________÷______=____________÷______=______
Goldenratio=1.618033…
9.Let’sdosomegraphingtoseemoreabouthowthisworks!
a.WhatisthefirstnumberoftheFibonaccisequence?______
Onthegraphpaperattheendofthishandout,thereissquarethatis1x1.
b.What’sthesecondnumberoftheFibonaccisequence?_______
Rightabovethesquareyoujustdrew,drawanother1x1square.
c.What’sthesecondnumberintheFibonaccisequence?______
Directlytotheleftofthetwoexistingsquares,drawina2x2square.
d.What’sthenextnumberintheFibonaccisequence?______
Rightbelowyourexistingsquares,drawa_____x_____square.
e.What’sthenextnumberintheFibonaccisequence?______
Totherightofallthatyou’vedrawn,drawa_____x_____square.
FractalsareSMART:Science,Math&Art!
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FibonacciSequenceandFractalSpirals
f.What’sthenextnumberintheFibonaccisequence?______
Aboveallthatyou’vedrawn,drawa_____x_____square.
g.What’sthenextnumber?______
Totheleftofallthatyou’vedrawn,drawa_____x_____square.
h.What’sthenextnumber?______
Belowallthatyou’vedrawn,drawa_____x_____square.
…Totherightofthatwouldbethenextsquare,butwe’verunoutofroom.
10.Nowlet’sseehowwecanmakeapatternoutofthesesquares.
Intheoriginalsquare,drawalinefromthebottomlefttothetopright.
Onthenext1x1square,continuethatlineacrossyoursquare,fromthebottomrighttothe
topleft.
Crossthe2x2squarefromthetoprighttobottomleft.
Crossthe3x3squarefromthetoplefttobottomright.
Crossthe5x5squarefrombottomlefttotopright.
Crossthe8x8squarefrombottomrighttotopleft.
Continuethelineacrossthe13x13squareandthe21x21square,wrappingupwitha
linethatwouldgothroughthe34x34square.
11.Whatpatterndoyouget?
12.Wheredowefindspiralsnaturally?
13.Countthenumberofthingsthatmakeupaspiralonapineappleorapineconeorthe
numberofpetalsonaflowerornumberofspiralsonafroccoliorseedsofasunflower.
TheyalloccurinFibonaccinumbers!Natureisfullofmathematicalpatterns!Amazing,
huh?Seewhatothercoolpatternsyoucanfigureoutinnature.
FractalsareSMART:Science,Math&Art!
www.FractalFoundation.org
Copyright2015FractalFoundation,allrightsreserved
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FibonacciSequenceandFractalSpirals
FractalsareSMART:Science,Math&Art!
www.FractalFoundation.org
Copyright2015FractalFoundation,allrightsreserved
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