AFRICAN INSTITUTE FOR MATHEMATICAL SCIENCES SCHOOLS ENRICHMENT CENTRE (AIMSSEC)

AIMING HIGH

GOLDEN PENTAGON IneachofthesediagramsofaregularpentagonfindtheratioofthelengthshowninredtothelengthinblueintermsoftheGoldenRatio𝜙 = 𝟏

𝟐(𝟏$√𝟓)

Findalltheanglesinthediagram.

LetAE=1unitandBE=xunits.

Whichtrianglesareisosceles?

Whichtrianglesaresimilar?

Usesimilartrianglestogiveanequationforxandsolvetheequation.

Prove()*)= 𝜙

Prove()(+= ()

(,= 𝜙

Prove𝑨𝑬𝑨𝑹= 𝝓

Prove𝑨𝑷𝑷𝑹= ∅

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HELP Youmayfindithelpfultomarkalltheanglesof36oinonecolour,anglesof54oinanothercolourandanglesof72oinathirdcolour.Thiswillhelpyoutoseewhichtrianglesareisosceles,andalsotoseewhichpairsoftrianglesaresimilar.Writetheratiosintermsofxand1+xandsimplifytheexpressionstogivequadraticequations.Forsomeoftheseproofsyouwillneedtomanipulatesurds.

NEXT Usingthesamediagramwritedownthevalueofcos36ointermsof√5.

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NOTES FOR TEACHERS SOLUTION Theanglesofaregularpentagonare108°soalltheanglescanbefound.

Hence x2 - x - 1 = 0 and thisquadraticequationhassolutions

𝑥 = 12(3±√5)

Asthelengthsmustbepositive𝑥 = 12(3$√5).

1. BE/AE= 𝑥 = 1263$√5789theGoldenRatio.

2. BE/BR=BE/BSbecausetriangleBRSisisosceles.

=𝒙𝟏=theGoldenRatio∅.

Alternatively,usingthefactthattrianglesAEBandSBEarecongruent:BE/BR=BE/BS=BE/AE=∅.

3. AE/EU= 𝟏𝒙<𝟏

whichwehaveprovedequalto𝒙theGoldenRatio∅.

36o 36o 36o

1 1 x -1 x -1

36o 108o 72o 72o x -1 36o 72o 36o

U

36o 36o 36o

Theedgesoftheregularpentagonare1unitandBE=xunits.

TriangleABRisisoscelessoBA=BR=1andRE=x–1units.

TrianglesABPandAREarecongruentisoscelestriangles.AR=RE=AP=x–1.

1. TrianglesABE,ABPandAREaresimilar(108o,36o,36o)henceBE/AE=AB/AP

𝑥1 =

1𝑥 − 1

B E

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4. AP/PR= 𝒙<𝟏𝟏<(𝒙<𝟏)

=𝟏𝟐(√𝟓%𝟏)𝟏𝟐(𝟑&√𝟓)

= (√𝟓%𝟏)(𝟑)√𝟓)(𝟑%√𝟓)(𝟑)√𝟓)

= (𝟐)𝟐√𝟓)𝟒

= *+,1 + √50 = 𝜙

Bysimilartriangles:AP/PR=AE/EU=∅theGoldenRatiobecausetrianglesAEUandAPRaresimilar(36o,72o,72o)

NEXT SOLUTIONcos36o=sin54o=12x =

1=(1 + √5)

so2cos36o=∅theGoldenRatio

(

DIAGNOSTICASSESSMENTThisshouldtakeabout5–10minutes.Writethequestionontheboard,saytotheclass:“Putup1fingerifyouthinktheanswerisA,2fingersforB,3fingersforCand4fingersforD”.1. Noticehowthelearners

respond.AskalearnerwhogaveanswerAtoexplainwhyheorshegavethatanswer.DONOTsaywhetheritisrightorwrongbutsimplythankthelearnerforgivingtheanswer.

2. Itisimportantforlearnerstoexplainthereasonsfortheiranswers.Puttingthoughtsintowordsmayhelpthemtogainbetterunderstandingandimprovetheircommunicationskills.

3. ThendothesameforanswersB,CandD.Trytomakesurethatlearnerslistentothesereasonsandtrytodecideiftheirownanswerwasrightorwrong.

4. Asktheclasstovotefortherightanswerbyputtingup1,2,3or4fingers.Noticeifthereisachangeandwhogaverightandwronganswers.

Thecorrectansweris:BandG–allsquaresaresimilartoeachother.

Possiblemisconceptions:Somelearnerswilljustguess.Encouragethemtolookatthesquaregridandtodecideiftheproportionsarethesame.ThisactivityisabouttheGoldenRatio.AllGoldenRectangleshaveedgesintheratioØ: 1whereØ = 1

2(3$√5)https://diagnosticquestions.com

https://diagnosticquestions.com

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Why do this activity? Thisactivityoffersablendofalgebraandgeometrysuitablefor14yearoldsandolderstudents.Thereisscaffoldingtoguidestudentstosolvetheproblemusingthesolutionstoaquadraticequation.Learning objectives Indoingthisactivitystudentswillhaveanopportunityto:• reviewanddeepentheirknowledgeandunderstandingofsimilartriangles;• reviewanddeepentheirknowledgeandunderstandingofratios;• reviewanddeepentheirknowledgeandunderstandingofhowtosolveaquadratic

equation;• reviewanddeepentheirknowledgeandunderstandingofsurds.GenericcompetencesIndoingthisactivitystudentswillhaveanopportunityto:• developproblemsolvingskills;• developvisualizationskills;• makeconnectionsbetweendifferenttopicsandusealgebratosolveaproblemin

geometry.Suggestions for teaching Startwiththediagnosticquizanddiscussthedifferentratiosbetweenthelengthsofthelongandshortedgesoftherectangles.

ThenshowthispaintingoftheLastSupper,paintedbytheSpanishartistSalvadorDaliin1955.ItisdesignedtoshowtheGoldenRectangleandtheGoldenRatio.NoticealsothepentagonalwindowsthatDaliusedbecauseoftheirgoldenproportions.

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TheyellowframeisagoldenrectanglesplitintoasquarewithChristatthelowerlefthandcorner,andanotherrectanglethatisalsoagoldenrectangle.Ifyoumeasurethelengthsoftheedgesyouwillfindthatbothrectangleshavetheproportions1.62:1(Thegoldenratiois1.618to3decimalplaces).

Usethe1–2–4–moreteachingstrategy,firstaskingstudentstoworkindividually,theninpairs,thentocomparetheirfindingsingroupsoffourstudentsandfinallyhaveaplenarywherestudentsgivepresentationsontheworkoftheirgroupoffour,tothewholeclass.Theeasiestwaytosolvetheproblemistousesimilartrianglesandtheratiosoflengthswhichleadtothequadraticequationx2-x-1=0

Key questions• haveyoufoundalltheangles?• whichtrianglesareisosceles?• haveyoulistedallthepairsofsimilartriangles?• howwouldyoudescribethesymmetriesofthepentagon/pentagramdiagram?• whathappensifyoudrawapentagraminsidethesmallinnerpentagon?Canyou

repeatthisprocessagainandagainonasmallerandsmallerscale?Tryit.FollowupElephantDreaminghttps://aiminghigh.aimssec.ac.za/years-8-12-elephant-dreaming/

OneStepTwoStepshttps://aiminghigh.aimssec.ac.za/years-7-10-one-step-two-steps/

Fibonacci’sRabbitshttps://aiminghigh.aimssec.ac.za/years-9-to-13-fibonaccis-rabbits/

SheepTalkhttps://aiminghigh.aimssec.ac.za/years-7-12-sheep-talk/

GreatPyramidhttps://aiminghigh.aimssec.ac.za/years-11-12-great-pyramid/

GototheAIMSSECAIMINGHIGHwebsiteforlessonideas,solutionsandcurriculumlinks:http://aiminghigh.aimssec.ac.zaSubscribetotheMATHSTOYSYouTubeChannelhttps://www.youtube.com/c/mathstoysDownloadthewholeAIMSSECcollectionofresourcestouseofflinewith

theAIMSSECAppseehttps://aimssec.apporfinditonGooglePlay.

Note:TheGradesorSchoolYearsspecifiedontheAIMINGHIGHWebsitecorrespondtoGrades4to12inSouthAfricaandtheUSA,toYears4to12intheUKandschoolyearsuptoSecondary5inEastAfrica.NewmaterialwillbeaddedforSecondary6.ForresourcesforteachingAlevelmathematics(Years12and13)seehttps://nrich.maths.org/12339MathematicstaughtinYear13(UK)&Secondary6(EastAfrica)isbeyondtheSACAPScurriculumforGrade12

LowerPrimaryApprox.Age5to8

UpperPrimaryAge8to11

LowerSecondaryAge11to15

UpperSecondaryAge15+

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USA KindergartenandG1to3 Grades4to6 Grades7to9 Grades10to12UK ReceptionandYears1to3 Years4to6 Years7to9 Years10to13