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.