african institute for mathematical sciences ... - aimssec
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AFRICAN INSTITUTE FOR MATHEMATICAL SCIENCES SCHOOLS ENRICHMENT CENTRE (AIMSSEC) AIMING HIGH
This INCLUSION AND HOME LEARNING GUIDE suggests related learning activities for all ages from 4 to 18
on the theme of QUADRILATERALS The original BENDY QUAD ACTIVITY was designed for Years 11 to 12
but this document has versions for all ages. Choose what seems suitable for the age or attainment level of your learners.
BENDY QUADS SeetheBendyQuadsvideohttps://bit.ly/BendyQuadsVideo
Fourrodsarehingedattheirendstoformaconvexquadrilateralwithsidesoflength3,4,5and6(inthatorder).Investigatethedifferentshapesthatthequadrilateralcantakeifthepolygonisalwaysconvex.
Howdotheangleschangeasthebendyquadchangesshape?
Cananyoftheanglesreducetozerodegrees?
Cananyoftheanglesincreaseto180degrees?
CalculatethesizeofangleCwhentherodsformatriangleasshown.Ifthepolygonremainsconvex,canangleCgetanysmallerthanshowninthisdiagram?WhatisthesmallestsizeofangleCandwhatisthelargest?
Findthesmallestandlargestvaluesthattheotheranglescantakeinasimilarway.
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HELP Youmightmakeamodelthatyoucanmanipulateandexperimentwith,changingtheangles.Youcoulduse4papersticksoflengths3,4,5and6unitschoosingyourownscale.Forexample,yourstickscouldbe6cm,8cm,10cmand14cm(linearscalefactor2).Thespecialquadsinthetwopictures,withedgelengths2,3,2and5,canbothformasymmetrictrapezium.
Forthestiffquadmodelcut4stripsofcardandjointhemtoformaquadrilateralofthegivendimensionsusingsplitpinstolinkthestripsofcard.Thefinalcalculationsonlyrequiretheuseofcosineandsinerules.
NEXT Youcouldinvestigatenon-convexquadrilaterals.Youcouldinvestigatetheareaofthequadrilateralandhowthischanges.Canyoumakeallthetypesofquadrilateralwith4rods,forexampleatrapeziumoracyclicquadrilateral?Tryaquadrilateralwithedgesoflengths:3,5,8and6.Whatisspecialaboutthisquadrilateral?
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INCLUSION AND HOME LEARNING GUIDETHEME: QUADRILATERALS
Early Years Makealargesupplyofpapersticksofdifferentlengths.
Seethevideohttp://bit.ly/HowToMakePaperSticksVideo
Maketrianglesbytyingtheendsof3stickstogether.Makeagameoftryingtofindwhocanbethefirsttomakeatrianglethatisdifferentfromanyyouhavefoundbefore.
Talkaboutwhatisthesameandwhatisdifferent.
Makequadrilateralsbytyingtheendsof4stickstogether.Makeagameoftryingtofindwhocanbethefirsttomakeaquadrilateralthatisdifferentfromanyyouhavefoundbefore.
Talkaboutwhatisthesameandwhatisdifferent.
Lower Primary Theactivityissimilartothatdescribedforearlyyearsbutnowintroducesomeofthenamesofthedifferentshapes.
Makealargesupplyofpapersticksofdifferentlengths.
Seethevideohttp://bit.ly/HowToMakePaperSticksVideo
Maketrianglesbytyingtheendsof3stickstogether.Makeagameoftryingtofindwhocanbethefirsttomakeatrianglethatisdifferentfromanyyouhavefoundbefore.
Talkaboutwhatisthesameandwhatisdifferent.
Makequadrilateralsbytyingtheendsof4stickstogether.Makeagameoftryingtofindwhocanbethefirsttomakeaquadrilateralthatisdifferentfromanyyouhavefoundbefore.
Talkaboutwhatisthesameandwhatisdifferent.
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Upper Primary TheactivityissimilartothatdescribedforearlyyearsandLowerPrimarybutnowmaketrianglesandquadrilateralsofallthedifferentshapespossibleandintroducethenamesofthedifferentshapes.
Makealargesupplyofpapersticksofdifferentlengths. Seethevideohttp://bit.ly/HowToMakePaperSticksVideo
Maketrianglesbytyingtheendsof3stickstogether.Makeagameoftryingtofindwhocanbethefirsttomakeatrianglethatisdifferentfromanyyouhavefoundbefore.
Talkaboutwhatisthesameandwhatisdifferent.Makeaposteroftrianglesinwhichyoustickthepaperstickedgesontoabackingsheetandwritethenamesandpropertiesbesidethemodels.
Makequadrilateralsbytyingtheendsof4stickstogether.
Makeagameoftryingtofindwhocanbethefirsttomakeaquadrilateralthatisdifferentfromanyyouhavefoundbefore.
Talkaboutwhatisthesameandwhatisdifferent.
Makeaquadrilateralsposterinwhichyoustickthepaperstickedgesontoabackingsheetandwriteallthenamesandpropertiesbesidethemodels.
ByKrishnavedala-Ownwork,CCBY-SA4.0,https://commons.wikimedia.org/w/index.php?curid=37238992
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Lower Secondary Solution by modelling, scale drawing and measurement Youwillneedalargesupplyofscrappaper,stringandsomescissors,orasupplyofready-madepapersticksoflengths12cm,16cm,20cmand24cm.
Organiseyourclasssothatlearnersworkinpairsorgroupsoffour.Startyourlessonbyreviewingwhattheleanersknowaboutscaleandenlargement.Inthislearningactivitythelengthsoftheedgesofthequadrilateralare3,4,5and6units.Sticksoflengths3cm,4cm,5cmand6cmsticksaretoosmalltomakeaccuratelyandeventohandle.
Thepercentageerrorinmeasurementofangleswillbelessforsimilarquadrilateralsmadewithlongersticks.Rememberthatanglesremainthesamewhenobjectsareenlarged.
Ifyoudon’thaveaready-madesupplyofpapersticksthengiveoutscrappaperandstringtothelearners,demonstratehowtomakeapaperstickandthelearnersshouldallmakeastick.Seethevideohttp://bit.ly/HowToMakePaperSticksVideo
Useascaleof1unit=4cmsothesticksinyourmodelshavelengths12cm,16cm,20cmand24cm.Iflearnersworkingroupsof4eachlearnercanmakeastickofoneofthefourlengthssotheyarereadytostarttheinvestigationwiththeir4sticks.
Usingyourmodel,explaintotheclassbrieflythattheymustexploreallthedifferentlyshapedquadrilateralsthatcanbemadeby‘bending’thequadrilateral.
Discusswhatitmeansforaquadrilateraltobeconvexsoithasinterioranglesalllessthan180o.Explainthat,forsimplicity,theyaregoingtoworkwithconvexquadrilaterals(andnotarrowordartshapedquadrilaterals).
Askthemtoexplorehowtheangleschange.Givesometimeforthelearnerstothinkabouttherangeofpossibilitiesandtodrawsketches.
Youmightmakeademonstrationmodelbycutting4stripsofcardandjoiningthemtoformaquadrilateralofthegivendimensionsusingsplitpinstolinkthestripsofcard.
Allowtimeforlearnerstoexplorethedifferentshapesthatthequadrilateralcantake,usingamodelasdescribedordynamicgeometrysoftwaresuchasGeogebra.Thiswill
1 unit = 1 cm 1 unit = 2 cm 1 unit = 3 cm Edge lengths are: Edge lengths are: 6 cm, 8 cm, 10 cm, 12 cm 9 cm, 12 cm, 15 cm and 18 cm
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helpthemtoidentifywhatcanbevariedandhowmuchvariationispossible.Askthemtoanswerthequestions(asonpage1).Theyshouldfirstdecideontheshapesthatgivethegreatestandsmallestanglespossible,thendrawaccuratescaledrawingsandmeasuretheangles.
Youcanmakeitmorechallengingbysimplyshowingthefirstdiagramonpage1andleavingittothelearnerstodiscoverhowsomeanglescanreducetozeroorincreaseto180o,andhowthetriangleformsthelimitingshapeifthequadrilateralremainsconvex(sothatthequadrilateralisnotanarrowhead).
GuidetheworkbyaskingKeyQuestions.
YoumightalsomakeiteasierforthelearnersbysuggestingthattheyconsidertheconfigurationswhereABCandADCbecomestraightlinesorwhereDABbecomesastraightline.
Whenlearnershavehadtimetodoallthishaveageneraldiscussioninwhichthelearnerssharetheirdiscoveries.Foreachangletheyhavefound,writethelearners’answersontheboardandfindtheaveragetogetanapproximationanswer.
Key questions • Ifyouflexthequadrilateralcantheanglesbeanysize?• Cananyoftheanglesreduceto0o?• Cananyoftheanglesincreaseto180o?• Cantherodsformatriangle?
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Upper Secondary Solution by modelling and calculation using trigonometry Youwillneedasupplyofscrappaper,stringandsomescissors,orasupplyofready-madepapersticksoflengths12cm,16cm,20cmand24cm.
Followingtheinstructionsinthevideohttp://bit.ly/HowToMakePaperSticksVideoifyouhavenotmadethembefore,makepapersticksoflengths12cm,16cm,20cmand24cm.Makeascalemodelofthe3-4-5-6quadrilateralusingascaleof1unit=4cm,because3cm,4cm,5cmand6cmsticksaretoosmalltohandle,andthepercentageerrorinmeasurementofangleswillbelessforsimilarquadrilateralsmadewithlongersticks.
Startbendingyourquadrilateraltodiscoverhowtheangleschangeandtofindthesmallestandlargestpossibleangles.ExplorethedifferentshapesthatthequadrilateralcantakeusingamodelordynamicgeometrysoftwaresuchasGeogebra.Thiswillhelpyoutoidentifywhatcanbevariedandhowmuchvariationispossible.Thenmeasurethesmallestandlargestpossibleangles.Thinkabouttherangeofpossibilitiesandtomakesomenotes.
Ifyouwanttobeabletomeasuretheanglesmoreaccuratelythenmakeamodelbycutting4stripsofcardandjoiningthemtoformaquadrilateralofthegivendimensionsusingsplitpinstolinkthestripsofcard.
Aconvexquadrilateralhasinterioranglesalllessthan180o(orequalto180oontheextremecasewhentwoedgesformastraightline).
Non-convexquadrilateralsliketheoneshownherearecalledarrowheadsordarts.Forthisinvestigationonlyworkonconvexquadrilateralsatfirst.Ifyouwanttoextendyourinvestigationtonon-convexquadrilaterals,thendothatasa‘follow-up’.
Togetaccurateanswersyouneedtocalculatetheangles.Therearedifferentmethodsfordoingthis.Youmightworkoutyourownsolution,andthendiscusswhatyouhavedonewithotherstudentsandexplainyourmethodstoeachother.Writeasummaryofyourworkandexplainhowbothmethodsapply.
Key questions • Ifyouflexthequadrilateralcantheanglesbeanysize?• Cananyoftheanglesreduceto0o?• Cananyoftheanglesincreaseto180o?• Cantherodsformatriangle?
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SOLUTION AsAB+BC=CD+DA=9
weseethatanglesAandCcanreduceto0o.
SoangleAcanchangeinsizefrom0oto180o
However,ifwejustconsiderconvexpolygons,thenangleCcannotgetsmallerthanshowninthisdiagramwheretherodsformatriangle.
ThesmallestpossiblesizesofanglesC,BandDarefoundfromthisdiagram.
Bythecosinerule:72=62+52–60cosC
SocosC=12/60=1/5andangleC=78.5otothenearesttenthofadegree.
AngleCcanchangefrom0oto78.5o.
Usingthesinerule:
sinB=6/7(sinC)=5/7(sinD)soangleB=57.1oandangleD=44.4o
AngleBchangesfrom57.1oto180o.
AngleDchangesfrom44.4oto1800.
Why do this activity? Thisactivityinvolvestheinterpretationofaverysimpleconcretestructure,alinkageof4rodswiththejointsbetweentherodsattheverticestotallyflexible.Experimentandinvestigationleadtoideasabouttheanglesthatcanbeformedinthesebendyquadrilaterals.Differentcasescanbeconsidered,includingconvexandnon-convexbendyquadsin2Dandevenin3D.Theconjecturesneedjustificationandproofbyformingconvincingarguments.
Tofindtheconstraintsontheanglesinthegeneralcaserequiresanargumentusinginequalities.
Solutionscanbefoundbymathematicalthinkingandscaledrawing.Accuratevaluesoftheanglescanbecalculatedusingthecosineandsinerules.
Learning objectives Indoingthisactivitystudentswillhaveanopportunityto:• investigatearangeofgeometricalpossibilitiesforaquadrilateral;• findsolutionsbyscaledrawing;• practiseapplyingthesineandcosinerules.
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Generic competences Indoingthisactivitystudentswillhaveanopportunityto:• thinkflexibly,becreativeandinnovativeandapplyknowledgeandskills;• visualizeanddeveloptheskillofinterpretingandcreatingvisualimagesto
representconceptsandsituations.
Diagnostic Assessment Thisshouldtakeabout5–10minutes.Writethequestionontheboard,saytotheclass:
“Putup1fingerifyouthinktheanswerisA,2fingersforB,3fingersforCand4forD”.1. Noticehowthelearners
responded.AskalearnerwhogaveanswerAtoexplainwhyheorshegavethatanswerandDONOTsaywhetheritisrightorwrongbutsimplythankthelearnerforgivingtheanswer.
2. Itisimportantforlearnerstoexplainthereasonfortheiranswersothattheydeveloptheircommunicationskillsanddeepentheirunderstandingbyputtingtheirthoughtsintowords.
3. ThendothesameforanswersB,CandD.Trytomakesurethatlearnerslistentothesereasonsandtrytodecideiftheirownanswerwasrightorwrong.
4. Asktheclassagaintovotefortherightanswerbyputtingup1,2,3or4fingers.Noticeifthereisachangeandwhogaverightandwronganswers.
5. Theconceptisneededforthelessontofollow,soexplaintherightanswerorgivearemedialtask.
ThecorrectanswerisAusingthecosinerule.StudentsgivinganswersBandCareincorrectlytryingtousethesinerule.StudentsgivinganswerDaremisusingthecosinerulegettingthatsignswrong.https://diagnosticquestions.com
Follow up Achallengingquestionthatrequiresthesettingupandsolutionofaquadraticequation:https://aiminghigh.aimssec.ac.za/years-11-12-solve-the-triangle/
GototheAIMSSECAIMINGHIGHwebsiteforlessonideas,solutionsandcurriculumlinks:http://aiminghigh.aimssec.ac.zaSubscribetotheMATHSTOYSYouTubeChannelhttps://www.youtube.com/c/mathstoysDownloadthewholeAIMSSECcollectionofresourcestouseofflinewith
theAIMSSECAppseehttps://aimssec.apporfinditonGooglePlay.