teacher-notes golden pentagon
TRANSCRIPT
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+
SouthAfrica GradesRand1to3 Grades4to6 Grades7to9 Grades10to12EastAfrica NurseryandPrimary1to3 Primary4to6 Secondary1to3 Secondary4to6
USA KindergartenandG1to3 Grades4to6 Grades7to9 Grades10to12UK ReceptionandYears1to3 Years4to6 Years7to9 Years10to13