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UCL SCRATCHMATHS CURRICULUM MODULE 5: Exploring Mathematical Relationships TEACHER MATERIALS
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Page 1: Exploring Mathematical Relationships - UCL · 2018-07-26 · Module 5 is focused around exploring different types of mathematical relationships including proportionality and ratio

UCL SCRATCHMATHS CURRICULUM

MODULE 5:Exploring

Mathematical Relationships

TEACHER MATERIALS

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DevelopedbytheScratchMathsteamattheUCLKnowledgeLab,London,England

Imagecredits(pg.3):Topleft:andreas (https://www.flickr.com/photos/andreas/2951113717/)[CCBY2.0(http://creativecommons.org/licenses/by/2.0)],viaWikimediaCommons� Topright:SeanMacintosh[CCBY-SA4.0(http://creativecommons.org/licenses/by-sa/4.0)],viaWikimediaCommons � Bottomleft:FranciscoAnzola

(Dohaskylineinthemorning)[CCBY2.0(http://creativecommons.org/licenses/by/2.0)],viaWikimediaCommons �Bottomright:WilsonHuifromCalgary,Canada- ShanghaiSkylinefromtheBund,CCBY2.0,

https://commons.wikimedia.org/w/index.php?curid=47927655

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Investigation1PolygonFireworks

NightSkyline

Investigation2Mathematically

SimilarRectangles

Investigation3GridWorld:For

ExploringSimilarity

Investigation4Exploring

Proportionality

3

MODULE 5:EXPLORING MATHEMATICAL RELATIONSHIPS

INTRODUCTION TO MODULE 5

Module5isfocusedaroundexploringdifferenttypesofmathematicalrelationshipsincludingproportionalityandratioaswellasintroducingtheconceptofvariable.ThismodulecouldpotentiallybelinkedwithseveraldifferentareasoftheKeyStage2curriculumbeyondmathematicsandcomputingsuchasgeographyaswellasartanddesign.

GEOGRAPHY:LOCATIONAL KNOWLEDGE

Thefirstinvestigationculminateswithpupilsbuildingpolygonfireworkssetagainstthebackdropofacityskyline,whichcouldbeadaptedtolinkwithspecificplaces/regionsthatpupilsarelearningaboutwithinthegeographycurriculum.

ART AND DESIGN:ARCHITECTUREPupilsarealsoaskedtobuilduppolygonskyscraperswhichcouldbeconnectedwithlearningaboutspecificarchitecturalmovementssuchasbrutalist,aspartoftheartanddesigncurriculum.

SCRATCH COMPUTING MATHEMATICS

► Askandanswerblocks► Joinblock► …*…,…/…blocks► <variablename>block► Set<variablename>to…

block►When…keypressed

► Variables► Algorithm► Repetition► Expressions► Definitions► Sequence► Debugging► LogicalReasoning► Decomposition

► Division,multiplication► Angles► Regular andirregularpolygons► Perimeter► Randomness► Coordinates► Algebraicexpressions► Factorpairs► Ratioandproportion

KEY VOCABULARY AND CONCEPTS COVERED BY MODULE 5

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MAP OF MODULE 5

Thered dashedlineindicatesthecoreactivitieswhichareimportanttocompletebeforemovingontothenextmodule.

ForactivitieswhichrequirepupilstocontinuewithaprojectfromapreviouslessonyoucanalternativelyusethesuggestedINTorFINALprojectforthosepupilswhodonothaveaprojecttocontinuewithorifyouwishallpupilstobeginfromthesamepoint.

Activity1

Activity4

Investigation1PolygonFireworks,

NightSkyline

Investigation2Mathematically

SimilarRectangles

Investigation3GridWorld:For

ExploringSimilarity

Investigation4Exploring

Proportionality

AskandAnswer

Starterproject:5-PolygonFirework

Unplugged:Polygon

Predictions

NamingValues

Continuewith:5-PolygonFirework

orstartwith:5-Polygon

FireworkINT1

TheSkyatNight

Continuewith:5-PolygonFirework

orstartwith:5-Polygon

FireworkINT2

Activity1 Activity2 Activity3 Activity4

SequencesofSquares

Starterproject:5-AlteringPolygons

AlteringRectangles

Continuewith:5-AlteringPolygons

ExploringMathematicalSimilarity

Continuewith:5-AlteringPolygons

Unplugged:RectangleJungle

EntertheGridWorld

Starterproject:5-GridWorld

ConnectingCorners

Continuewith:5-GridWorldorstartwith:5-GridWorld

INT1

MeettheMagicLine

Continuewith:5-GridWorldorstartwith:5-GridWorld

INT2

Unplugged:Module5Assessment

UsingtheGridWorld

Continuewith:5-GridWorldorstartwith:5-GridWorld

FINAL

BridgEing andSolvingProblems

Continuewith:5-GridWorldorstartwith:5-GridWorld

FINAL

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CONNECTIONS TO KS2 COMPUTING CURRICULUM

CURRICULUM OBJECTIVES LINK WITH SCRATCHMATHS

Design,writeanddebugprogramsthataccomplishspecificgoals

Solveproblemsbydecomposingthemintosmallerparts

Usesequenceandrepetitioninprograms

Workwithvariables

Usinglogicalreasoningtoexplainhowsomesimplealgorithmsworkandtodetectandcorrecterrorsinalgorithmsandprograms

► Inthefirstinvestigationpupilsarerequiredtodesign,buildanddebugaprogramwithinScratchwhichwillcreateanightskylinethatcanchangebasedonuserinput.

► Pupilsareshownanexampleofthefinalnightskylineproject(withoutseeingthescripts)andaskedtodecomposetheprogramintosmallerparts,thinkingaboutthescriptsthatwouldneedtobebuilttoreplicateeachoftheseparts.

► Pupilsarerequiredtoconsiderthesequenceofblockswhenusingaskandanswertoenvisagewhatthevalueofanswerwillbeatdifferentpointsinthescript.

► Pupilsusedifferentvariablestocontrolthenumberoftimestheirscriptsarerepeated.Repetitionisalsousedtocreateaseriesofrectanglestohelpexploretheconceptofratioandproportion.

► Pupilstaketheirfirststeptowardstheuseofvariableatthestartofthismodulethroughusingaskandanswer.Thenlaterinthemodulepupilsarerequiredtousevariablestosetandchangethevaluesofthesidelengthsofpolygons,thenumberofsidesandalsothenumberofpolygonstobedrawn.Theuseofvariablesenablestheexplorationofratioandproportion.Theimportanceofnamingconventionsforvariablesisalsoaddressedduringthismodule.

► Pupilsarerequiredtouselogicalreasoningwhenenvisagingtheoutcomeoftheirscriptsbasedondifferentuserinputsordifferentinitialvariablevalues.

► Withinanextensionactivitypupilsareaskedtoidentifyandsuggestfixesforbugsinanumberofscriptswhichdrawregularpolygons.

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MODULE 5: INVESTIGATION 1Polygon Fireworks, Night Skyline

Thisinvestigationfocusesondevelopingandbuildingpupils’understandingofvariablethroughthecreationofpolygonfireworkpatterns.TheinitialinvestigationrecallsthepolygonpatternsfromYear5placingagreaterfocusontheuseofanunknownthrough“askandanswer”tovarydifferentattributesofthepolygonpatterns.

Fromaconceptdevelopmentperspective,theanswer blockisdistancedfromitscompanionask blockwithinthescriptbypromptingthepupiltovaryaspectsofthepolygon.Variabledevelopmentcontinuesaspupilsrealisethelimitationsof“askandanswer”andgiveanametoavaluebydefiningauservariablefortheskylinetowers.

uActivity5.1.1 – AskandAnsweruActivity5.1.2– Unplugged:PolygonPredictionsuActivity5.1.3– NamingValuesuActivity5.1.4– TheSkyatNight

Scratchstarterproject 5-PolygonFirework5-PolygonFireworkINT15-PolygonFireworkINT25-PolygonFireworkFINAL

LINKS TO PRIMARY NATIONAL CURRICULUM

CURRICULUM OBJECTIVES LINK WITH SCRATCHMATHS

MathematicsSolve problems,includingmissingnumberproblems,usingmultiplicationanddivisionUsesimple formulae

Pupilscalculatetheperimeterofrectanglesandrelatedcompositeshapes

FindallfactorpairsofanumberDraw2DshapesusinggivendimensionsandanglesDistinguishbetweenregularandirregularpolygons(KS3)Workwithexperimentsthatinvolverandomnumbers

Describepositionsonthefullcoordinategrid

► Pupilsarerequiredtodiscussandbuildsimpleformulawhichincorporatesmultiplevariablesandinvolvesmultiplicationanddivisiontocreatetheirpolygonfireworks.

► [Extension] Asanextensionpupilsareaskedtobuildascriptfortheirspritetocalculateandsaytheperimeterofthepolygonithasdrawn.

► Pupilsarepromptedtorecallfactorpairsof360°.► Pupilsarerequiredtousevariablestospecifytheside

lengthandangleofageneralised polygon.► Pupilsareaskedtodiscusswhatisthesameandwhatis

differentbetweenregularandirregularpolygons.► [Extension]Pupilsbuildscriptsthatrandomlyposition

polygonfireworksandtowersofsquareswithinasetarea.

► Pupilsarerequiredtousetheirknowledgeofthefullcoordinategridtopositiontheirpolygonsandtowers.

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Explorehowtousetheaskandanswerblockstodrawdifferenttypesofregularpolygons.Explainwhatisthesameandwhatisdifferentbetweenregularandirregularpolygons.

7

MODULE 5 ● INVESTIGATION 1 ● ACTIVITY 5.1.1Ask and Answer

❶ Pupilsexploretheask andanswer blocks:theykeepthemisolated,clicktheaskblockandtypeintheanswer.Whereisthetextoftheanswer(thevalue)stored?Pupilsclicktheanswer blocktofindout.Theyalsoclickthecheckbox nexttotheanswer blocktoseeitssmallmonitorwindow inthestage.

❷ Pupilsbuildascript:WhentheBeetleisclicked,itwillaskWhatisyourname?WhentheyanswerandpressEnter,theBeetlewillgreetthembythename,usingthesay blockwiththeanswer.Theyexplorethejoin word1 word2 blocktobuildasentencefornicergreeting.

❸ Pupilsmodifythescript:WhentheBeetleisclicked,itaskswhatpensize itshoulduse,thensetsitanddrawsaline,asquare,aregularpolygon…

❹ Pupilsmodifythescript:Whenclicked,theBeetleaskswhatsizethesideofthesquare(apolygon)shouldbe.

continuesonthenextpage

Defineregularpolygon[allsidesequal,allanglesequal].Showexampleofanirregularpolygon,e.g.ahouseshape.Whatisthiscalled? [anirregularpentagon] Useexamplesasking:Isthisirregular,isthisregular? Supportpupilstodefinetheregularpolygonconceptbyasking:Whatitis? andWhatitisnot?

TheBeetleturnsthroughtheexteriorangleofthepolygon,itcanbehelpfultodrawasadiagramontheboard.Notetheconnectionbetweentheinteriorandtheexteriorangle[interiorangle+exteriorangle=180°]

Notethatthroughtheanswer blockwearemakingthenextsteptowardstheconceptofavariable.Ifweaskforavalueandtypeine.g.55,wecanthenusetheanswer blockasavariableinanalgebraicexpression,seeaboveontheright.

ACTIVITY INSTRUCTIONS

LEARNING OBJECTIVES

MATHEMATICS CONNECTIONS

Pupilsopenproject5-PolygonFirework,Saveasacopy (online)orSaveas (offline)andrename.Thefinal versionofthisprojectattheendofActivity5.1.1willbe 5-PolygonFireworkINT1.

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❺ [Extension] Afterdrawingthepolygon,theBeetlewillsaywhattheperimeterofthatsquare(polygon)is.

❻ Pupilscontinuemodifyingtheirscript:TheBeetlefirstasksforthepensize andsetsit,thenaskswhatlengththesideofthesquare(polygon)shouldbe.

❼ TheBeetleaskshowmanysidesthepolygonshouldhave,thendrawssuchpolygonwithafixedsidelength,e.g.30.

❽ Pupilsswitchthebackdroptothenightskyline andgeneralizethepreviousscript:theBeetleasksforthenumberofsides,thendrawsmanysmallpolygonsofthattypescatteredaroundthesky– byjumpingto randompositions.Theymayusethesetrandompenshade blockorotherrandomblocks.Theymayrunthesamescriptseveraltimes– givingdifferentanswerstotheaskblock.

❾ Encouragepupilstosimplifytheirscripts(makingthemmorereadable)bymakinganewblockpolygon withtheanswerblockinitanduseitintheirscriptsasashortcut.

Askpupilstodrawaregularhexagonwithaperimeterof180:Whatisthelengthoftheside?Drawaregularpentagonwithaperimeterof95:Whatisthelengthoftheside?Drawanequilateraltrianglewithaperimeterof100.Writeasimpleformulawhichconnectsperimeterofhexagonandsidelength[e.g.perimeterofhexagon=6×sidelength]

Recallfactorpairsof360° fromyear5,e.g.90°and4(square);60° and6(hexagon);72° and5(pentagon);120° and3(equilateraltriangle).Whyis360° important? [TheBeetleturnsthrough360° asitmovesaroundthepolygon.]Anirregularpolygonwillalsoturnthrough360° degrees.Thisfactisusefulwhensolvinggeometryproblemswhereanangleisunknown.Anotherwayofseeingthis:Foranyregularpolygontheexteriorangleofaregularpolygonisthesameastheanglethatacircleisdividedinto.

INVESTIGATION 1Activity 5.1.1…continued

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INVESTIGATION 1Activity 5.1.1

TheapproachweapplyherehasbeenintroducedinModule1andusedthroughoutallY5modules:Alwaysbuildscripts“frominsideout”,i.e.makesureyouunderstandwhateach‘bit’does,onlythenstartcombiningthem.Thefollowingpictureisanexamplesequenceofsuchsteps:

CONNECTIONS TO Y5SCRATCHMATHS

❽ InModule2,Activity2.3.4weappliedanotherstrategy:pupilswereprovidedwithseveralnewsetrandom… blocks,usedtheblocksintheirscripts,andonlythenexploredtheirdefinitionsbydecomposingandmodifying.Thustheybecomefamiliarwiththepickrandom…to… block.

❸ InModule2,Activity2.1.1westartedusingapentoolofasprite,withsomeofitsattributes,namelypencolour andpensize.WestartedusingthefollowingPen blocks:

Pupilslearnedhowtousethecolour pickerofthesetpencolorto… oralternativelytousethesetpencolortonumber_of_colour block.IntheadditionalmaterialsforModule2thereisaposterwith40colours andtheirnumbercodes.Alsothereareseveralotherpostersandsheetswithchallenges,oneofthemexploringthepensize,howtosetit,useitandchangeit.

NotethatinChallenge3:Explorethepensize oftheextensionmaterialsforModule2pupilsareencouragedtouseapairofblockssetpensizeto…andchangepensizeby…,whichenablesustosetacertainvalue andthenchangeit.Thisisexactlywhatwewilldolaterinthismodulewithvariables.

Pleasenotetheblue numbersontheleftlinktothenumberedstepsintheactivityinstructions

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ADDITIONAL SUPPORT

10

INVESTIGATION 1Activity 5.1.1

Ifweclicktheask block(1a),thespriteasksthequestion(1b)andaneditline(1c)appearsinthestage.WetypeinourresponseandpressEnterorclickthecheckmark.Theansweristhen“stored”intheanswer reportblock(1d)andcanbeusedinourscript(s).Clicktheisolatedanswer blocktoseethevalue(1e).

Toviewthevalueinthemonitor ofanswer,wecanalsoclickthecheckboxnexttotheanswer blockintheScriptstab(1f).Thekeydifferencebetweentheanswerblockandthemonitoristhattheblockcanbeusedinanotherblockasitsinput(see2below)whilethemonitorisjustvisualinformationforustoread. Theanswer reporterblockcanbeusedase.g.

aninputforthesay block,see(2a).SowhentheBeetleisclickeditwillaskthequestion,thenusetheanswer inthesay blocktogreetus,see(2b).Explorethejoin block(inOperators)tojointogetherHello andthevalueoftheanswer.NotethatweaddedaspaceattheendofHellosothatthetwowordsareseparatedbyaspace,see(2c).

Pleasenotetheblue numbersontheleftlinktothenumberedstepsintheactivityinstructions

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ADDITIONAL SUPPORT CONTINUEDINVESTIGATION 1

Activity 5.1.1

8

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ADDITIONAL SUPPORT CONTINUED

12

INVESTIGATION 1Activity 5.1.1

Pupilswillmaketheirownblockjumptorandomposition,thinkingaboutappropriatevaluesforthepickrandom…to… Itmightbereasonablenottousenumbers-240and240butreducethemabitsothattheBeetledoesnothittheedgewhendrawingapolygon.

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ACTIVITY INSTRUCTIONS

LEARNING OBJECTIVES

SOLUTION

Envisagethebehaviourofascriptwhichusestheask andanswer blocksindifferentways.Explainhowthecorrespondingoutcomedrawingwaschangedbytheanswer.

Printanddistributethepupilworksheet5.1.2ordotheactivityasaclass.Askthepupilstoexplainhowtheask andanswer blocksarebeingusedinthescripts,whatthescriptswillproduceandwhetherthescriptscanbesimplifiedorimproved.

13

MODULE 5 ● INVESTIGATION 1 ● ACTIVITY 5.1.2Unplugged: Polygon Predictions

TheBeetleasksforthepensize,selectsarandomcolouranddrawsasquareusingtheanswer asthepensize.Thepensize,however,isunnecessarilysetfourtimes– insidetherepeat block.Itonlyneedstobesetoncebeforetherepeat block.

TheBeetleasksforthepensizeandusestheanswer inthesetpensize…block.TheBeetlethendrawsasquaresettingarandomcolourforeachside.

TheBeetlesetsarandompencolourthendrawsasquare.Foreachsideitasksforthepensizeandusesittodrawthatside.

TheBeetleasksforthepensizebutdoesnotusetheanswer anywhere.Itdrawsasquareusingrandompencolourforeachside,settingpensizeto10insiderepeat againandagain,insteadofsettingitjustonceatthebeginning.

TheBeetleasksforthepensizeandusestheanswer inthesetpensize…block.TheBeetlethendrawsasquareandincreasesthepensizebytheanswer repeatedlyafterdrawingeachside.

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INVESTIGATION 1Activity 5.1.2

Readthescriptsbelow.Foreachofthemdrawthepictureitwillcreateandexplaininwordswhateachscriptwilldointheboxontheright.

NAME

WHAT TO DO

ASK ANSWER SCRIPTS

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INVESTIGATION 1Activity 5.1.2

Dothefollowingasaclass:Eachofthescriptsbelowwasintendedtodrawaregularpolygon.However,ineachscriptthereisabug.Envisagetheoriginalintention,explainthebugandsuggestafix.

Inthisscripttheanswer isnotusedintheturn blockatall.Thereforeinsteadofdrawingapolygonoftheanswer sides,theBeetledrawsonlyanswer sidesofanoctagonofthefixedsize,see(a)below.

Inthisscripttheangletoturnbyiswrong,theBeetlemustturnby360/answer,thatistwiceasmuchasitturnsnow,see(b).

Inthisscriptthequestionisaskedfourtimes,astheask blockisinsiderepeat.Itmeansthatifwedonotanswerthesamevalueeachtime,theBeetlewillnotdrawaregularpolygon,see(c).

Inthisscripttwoquestionsareaskedbuttheanswer tothefirstoneisneverusedforanythingbutoverwrittenbythesecondanswer immediately.

EXTENSION ACTIVITY INSTRUCTIONS

ADDITIONAL SUPPORT

❶ ❷

❹❸

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Explorehowtousevariableswithinascripttostoredifferentvaluesatthesametime.Explainwhywenowneedvariablestodrawmultipleregularpolygonsofdifferentsizes.

16

MODULE 5 ● INVESTIGATION 1 ● ACTIVITY 5.1.3Naming Values

❶ PupilscombinetwoquestionsintheirBeetlescript:theBeetleshouldfirstask aboutthesidelengthofthepolygontobedrawn,thenaboutthenumberofitssides.Howeverthisisnotpossibleusingonlythetoolswehavealreadyused.Observingthemonitoroftheanswer block,gothroughthescriptstepbystepsothatpupilsdiscoverthisproblemthemselves.

❷ Toremember theanswerofthequestionasked,wehavetogivethatvalueaname– tostorethevalueinavariable.Pupilsmakeavariablenamedsidelength.Theydragtwoisolatedblocksinthescriptsarea:setsidelengthto… andthereporterblocksidelength,keepthemisolatedandexplore,observingalsothesmallreporterwindow.Theysetdifferentvaluestothevariable.Similarly,theycreatethesecondvariablenumberofsides.

❸ Pupilssnaptwoblocks:aquestionWhatsidelength?intheask blockandsetsidelengthto…theanswer,runitandexplorethevalueofthesidelength variableinitssmallmonitor.

❹ Pupilsbuildthewholescriptfromstep1 again,askingtwoquestionsandsettingeachvariabletothecorrespondinganswer.Thentheymodifythepolygonblocksothatitusesthesetwovariablesinsteadoftheanswer block.

❺ Pupilsmakethethirdvariablenumberofpolygonsandaddanotherquestioninthescript:Howmanypolygons?Whenclicked,theBeetlewillaskthreequestionsanddrawthatmanypolygonsofthesizeandtypeasansweredbythepupils.

Notethatthereisonlyonesetvariable to…blockwithadropdownlistofallthevariables.

ACTIVITY INSTRUCTIONS

LEARNING OBJECTIVES

MATHEMATICS CONNECTIONSPupilscontinueintheirownversionofproject5-PolygonFirework,oropenthe5-PolygonFirework INT1,Saveasacopy (online)orSaveas (offline)andrename.ThefinalversionofthisprojectattheendofActivity5.1.3willbe5-PolygonFirework INT2.

Notethattheactualsettingofavariablehappensonlyafteryouruntheblock– byclickingorrunningascriptcontainingthatblock.

Youmayprefertodomostofthisactivity(uptopoint4,including)usingtheemptyplain backdrop.

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INVESTIGATION 1Activity 5.1.3

Hereisanattempttosolvethetaskbutitdoesnotworkproperly.Theanswer blockappearsthreetimesinthescript,(1a)and(1c)refertothesecondanswerand(1b)referstothefirstanswer.However,assoonasweanswerthesecondquestion,thefirstvalueofanswer islost andreplacedbythesecondanswer,see(1d).ThatiswhytheBeetleusesvalue8for(1a),(1b),and(1c)anddraws(1e)insteadofintended(1f).

TomakeavariablewegototheData groupandclicktheMakeaVariable button(see2a).AfterwetypeinthenameofthenewvariableandclickOKbutton(1b),severalnewblocksappearintheDatagroup.Inthisactivityweusethereporterblocksidelengthandthesetsidelengthblock.

ADDITIONAL SUPPORT

Pleasenotetheblue numbersontheleftlinktothenumberedstepsintheactivityinstructions

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INVESTIGATION 1Activity 5.1.3

TheBeetleaskstwoquestionsandkeepstheanswersinvariablessidelength andnumberofsides.Bothvariablesarethenusedtodrawapolygon,numberofsidesisusedtwice.(4b)isanalternativesolutionusingourownblockpolygon.

Variablenumberofpolygonsisusedastherepeat value,bothsidelengthandnumberofsidesvariablesareusedinsidethepolygon blockdefinition.Encouragepupilstomakeandusethepolygon blocksothatthewhenthisspriteclickedscriptisshorterandmorecomprehensible.Alternatively,bothpenupandpendownblocksmightbemovedinsidethejumptorandompositiondefinition.

ADDITIONAL SUPPORT CONTINUED

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ExplorethefollowingSurprisingpolygons:

19

INVESTIGATION 1Activity 5.1.3

Starpolygonsaredrawnbyconnectingonevertexofaregularpolygontoanother(non-adjacentone)andrepeatinguntilyoureturntothestart(thefirstoneintherowabove).Todemonstratewhatishappening,trywalkingaroundafive-pointedstar,payingcarefulattentiontoyourturning.Youwillseethefourwallsoftheroomtwice,notonceasyouwouldforaregularpolygon.Youhaveturnedatotalof360° twice,or720°.Allofthestarpolygonsherearefoundbyusingmultiplesof360°.

EXTENSION IDEAS

ADDITIONAL SUPPORT CONTINUED

ChoosingnamesforvariablesAlthoughpupilsareencouraged– andsupportedbyScratch– togiveanynametovariablesastheywish,anamecaneasilybecomeconfusing,insteadofhelpful.Totheright,youseearealexamplefromaschool:apupilusedthetextofthequestionasthenameofavariable,thevalue.Theconfusionmayoccurwhenthevariableisthenusedinotherblocks,see(b)and(c).

Thenameofavariableshouldreflectwhatthe‘answer’represents.Inthiscaseitcouldbee.g.length orsidelength…

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Explorehowtodrawtowersofsquaresofdifferentheightsandinrandompositions.bridgE tomathematicalquantitiesandformulastocalculatesidelengthorheightofatower.

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MODULE 5 ● INVESTIGATION 1 ● ACTIVITY 5.1.4The Sky at Night

❶ Pupilsmaketheirownblocksquare usingthesidelength variabletodrawit.Theybuildascript:whentheBeetleisclicked,itwillaskWhatsidelength? thendrawatowerof10smallidenticalsquaresatopeachother.

❷ Itisnotnecessarytohaveonly10floortowers.Pupilsmakeanewvariablenumberoffloors andbuildamorepowerfulblocktower whichwilldrawatowerofidenticalsquares– definedbythenumberoffloorsvariable.

❸ PupilsmodifytheirscriptfortheBeetletofirstaskforthenumberoffloorsandsavetheanswerinvariablenumberoffloors.Thenitwillaskforthesidelengthandsavetheanswerinvariablesidelength anddrawacorrespondingtower.

❹[Extension] Pupilsgeneralisetheirsolutionsothatthescriptdrawsanightskylineofmanytowersofdifferentnumbersoffloorsanddifferentsidelengths.Thescriptwillrepeatthetower part,askingeachtimefortheinputvalue– or,alternatively,settingthematrandomwithanappropriateminimumandmaximum.Alltowerswillbescatteredatrandom.

❺[Extension] Pupilscreateaskyfullofpolygonfireworks,thenaskylineoftowers,combiningallprevioussteps.Notethatforfireworkpartandfortheskylinepartitmightbeusefultohavetwodifferentjumptorandompositionblocks,sothatthewholescenecouldbecreatedinoneclick.

Drawoutthestructureofthetowersonthewhiteboard,indicatethestartingandendingpointoftheBeetledrawingit.Wheredoweneedtomovetostartthenextfloor? Whatisthealgorithm? [firstdrawasquarethenmoveupwardsthesidelength]

ConnecttheBeetleoutputwithmathematicalquantitiesandformula.E.g.Howtallisthetower?Writeasasimpleformulae.[heightoftower=sidelength*numberoffloors]Posequestions:Ifatoweris120tall,andsidelengthofthesquareis15,howmanyfloorsdoesithave?

ACTIVITY INSTRUCTIONS

LEARNING OBJECTIVES

MATHEMATICS CONNECTIONS

Pupilscontinueintheirownversionofproject5-PolygonFirework,oropenthe5-PolygonFirework INT2,Saveasacopy (online)orSaveas (offline)andrename.ThefinalversionofthisprojectattheendofActivity5.1.4willbe 5-PolygonFirework FINAL.

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INVESTIGATION 1Activity 5.1.4

InModule2,Activity2.2.1pupilsdrewasquareandaequilateraltriangle.InActivity2.2.3theywereencouragedtogiveanametotheirsquarescript,makingtheirownsquareblock.InActivity2.2.4theymadeanothernewblock– triangle andwereaskedtousethesenewblockstodrawatoweroftwosquaresandalsoahouse.

CONNECTIONS TO Y5SCRATCHMATHS

Intheadditionalsupportofthatactivitywesuggestedtoencouragepupilstobuildascriptandrunitstepbystepthinkingaboutthequestionsbelow:

■ WherewillmyBeetlefinishafterdrawingthefirstsquare?

■ Whichdirectionwillitpointin?■ WhereexactlydoIwantittodraw

thesecondsquare?WhichblockwillmaketheBeetlegetthere?

■ Willitthenpointinthecorrectdirection?

■ Wherewillitfinishafterdrawingthesecondsquare?

Nowinournewsquare blockfromActivity5.1.4wemakeuseofavariablesidelength whichissetbyusingtheask andanswerblocks.SomepupilsmaycomewiththefollowingsolutionbasedonageneralizationoftheY5task:

Whileitlookslikeacorrectsolutionforthesituationwhenwesetthevalueofsidelengthto10(example(a)above),itiseasytodemonstratetheproblemifwesetthevalueofsidelengthtobee.g.20(example(b)above).TheBeetleneedsmove exactlybysidelength fromonesquaretothenextone,whatevervalueitis.

Pleasenotetheblue numbersontheleftlinktothenumberedstepsintheactivityinstructions

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INVESTIGATION 1Activity 5.1.4

Inthedefinitionofthesquare blockweusethesidelengthvariable,thevalueofwhichwillbesetinthewhenthisspriteclicked scriptbyask andanswer.

InthescriptfortheBeetleeachsquarecanhavethesamecolour ormayhavedifferentpenshadesordifferentpencolours,pupilscanchoose.

Pupilsshouldstartusingthevocabulary:Thespriteasksfor…thensavesorkeepstheanswer(oransweredvalue)inavariable…

ADDITIONAL SUPPORT

Pleasenotetheblue numbersontheleftlinktothenumberedstepsintheactivityinstructions

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INVESTIGATION 1Activity 5.1.4

Alternativesolutionwithnumberoffloorsandsidelengthsetatrandom,withoutanyasking:Carefullychooseanappropriateminimumandmaximumforeachvalue,includingtherangesforxpositionandypositionforrandomjumping.

ADDITIONAL SUPPORT CONTINUED

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MODULE 5: INVESTIGATION 2Mathematically Similar Rectangles

Thisinvestigationexploresthemathematicalpropertyofsimilarity.Thematerialsdeliberatelyusethewords“mathematicallysimilar”todrawattentiontothespecialmathematicalpropertiesofthetermsimilar:havingcorrespondingsidelengthsproportional(inthesameratio)andcorrespondinganglesequal.

Pupilsstartbyexploringandbuildingsequencesofgrowingsquareswhichuseonevariableandthechange<variable> byblocktochangethevalueofthevariable.Theydeveloptheirscripttoexploreandbuildpatternsofsequencesofgrowingrectangleswhichrequiretwovariables.Theideaofa“magicline”– whichconnectseachtoprightcornerandbottomleftcornerofthesequences- isintroducedasavisualtooltotestifthesequenceofrectanglesaremathematicallysimilar.ThisideaisdevelopedwhentheBeetleisprogrammedtosaytheresult(ratio)ofheight/base.PupilsrecognizethatwhentheBeetlesaysthesamenumber,thesequencesofrectanglesaremathematicallysimilar,e.g.therectanglesareproportional.

uActivity5.2.1–SequenceofSquaresuActivity5.2.2– AlteringRectanglesuActivity5.2.3– ExploringMathematicalSimilarityu [Extension]Activity5.2.4– Unplugged:RectangleJumble

Scratchstarterproject 5-AlteringPolygons

LINKS TO PRIMARY NATIONAL CURRICULUMCURRICULUM OBJECTIVES LINK WITH SCRATCHMATHS

MathematicsKnowthatrectanglesarenotalways similartoeachother.Solvesproblemsinvolvingsimilarshapeswherethescalefactorisknownorcanbefound.

Pupilsrecognise incontextswhentherelationsbetweenquantitiesarethesameratio(forexamplesimilarshapes).

Pupilsshouldconsolidatetheirunderstandingofratiowhencomparingquantities,sizesandscaledrawingsbysolvingavarietyofproblems.

► Pupilsarerequiredtoidentify fromscaledrawingsaswellasfromscripts(inScratch)rectangleswhichareproportionaltooneanotherandtoexplaintheirchoices.

► Pupilsareaskedtoidentifywhichsequencesofsquaresandrectanglesareproportionallysimilarandtousetheideaofa“magic”linewhichcanbedrawnthroughtheupperrightcornerstohelpwiththis.

► Pupilsarerequired toexploreratioandproportioninanumberofways,throughexploringdrawingrectanglesofdifferentratioswithinScratchusingvariables,throughidentifyingandexplainingproportionalscaledrawingsandalsothroughenvisagingtheoutcomesofexistingscripts.

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Explorerepeatedlychangingavariablebyasetamounttocreateasequenceofincreasingsquares.Explainthedifferencebetweenthechange<variable>byandset<variable>toblocks.

25

MODULE 5 ● INVESTIGATION 2 ● ACTIVITY 5.2.1Sequence of Squares

Askpupils:Describethesidelengthsofeachofyoursquares.Whatisthefinalsidelengthdrawn?Whatisthevalueofthesidelengthvariableattheend,explainthisdifference?[Startingwith20,repeat5andchangesidelengthby20,willdraw5squares,20,40,60,80,100.However,thesidelengthvariablewillreport120attheendduetothepositionofthechangesidelengthblock.Writethevalueofthesidelengthvariableonthewhiteboardasyoustepthroughthecodeblockbyblock.] Seeadditionalsupport❺ formoreexamples.

❶ Pupilswillusethesquare block– identicaltotheonefromthepreviousinvestigation.UsingthisblocktheBeetleshoulddrawasquarethesideofwhichisdefinedbythesidelengthvariable.Pupilsbuildthefollowingshortscriptanduseitrepeatedlytodrawseveralsquareswithdifferentvaluesofthesidelength.

❷ Pupilsclearthestageandusethesamescripttodrawasquareofside20,then40,then60,80,100…

❸ Pupilsdraginthechangesidelengthby…blockandkeepitisolated.Theyexplorethedifferencebetweensetsidelengthto…andchangesidelengthby…

❹ Pupilsclearthestageandbuildascripttodrawthewholepatternofgrowingsquaresinonego,usingthechangesidelength by… block.

❺ Pupilsexperimentwithchangesidelength by… insidetherepeat blocktomakethesamescriptshort.

❻ Pupilsbuildascripttodrawarowofseveralattachedsquares:(a)ofthesameside,then(b)ofgrowingside,thefirstone20andthenincreasingby10.

❼[Extension] Pupilsmodifythepreviousscript(b)sothatthesquareshaveagapof10betweenthem.

❽[Extension] Pupilsmodifythepreviousscriptsothateachsquarehassidesbetween30and60,chosenatrandom.

ACTIVITY INSTRUCTIONS

LEARNING OBJECTIVES

MATHEMATICS CONNECTIONS

Pupilsopenproject5-AlteringPolygons,Saveasacopy (online)orSaveas (offline)andrename.

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INVESTIGATION 2Activity 5.2.1

Withinthe5.1.1ConnectionstoY5ScratchMaths itdescribedusingapairofblockstosetpensizeto…andchangepensizeby… withine.g.Module2:Challenge3.

CONNECTIONS TO Y5SCRATCHMATHS

ForexampleinModule3,Activity3.1.3weusedchangeyby…blocktomakeTera jumphighthenslowlyfloatback:

Notehoweverthereareotherpairsofblocksworkinginasimilarway:thefirstonesetsacertainvalue (ofthexposition,yposition,orthepenshadeetc.)andtheotheronechangesthatvaluebyacertainamount.

Nowweareintroducinganothersimilarpair– ablocktosetavalueofavariableandablocktochangethatvaluebyacertainincrement(apositivenumber)oradecrement(anegativenumber).

Pleasenotetheblue numbersontheleftlinktothenumberedstepsintheactivityinstructions

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INVESTIGATION 2Activity 5.2.1

Clickscript(a)againandagain.Observethevalueofsidelength.Thenclickscript(b)severaltimesandobserve.

ADDITIONAL SUPPORT

❹ ❺

Belowseealternativesolutionofthesametask.Discussitwiththepupils.

Pleasenotetheblue numbersontheleftlinktothenumberedstepsintheactivityinstructions

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INVESTIGATION 2Activity 5.2.1

Discussasaclass,envisagethedifferentoridenticaloutputsofthefollowingscripts,makesurepupilsunderstandthesesubtledifferences.Howmanysquareswillbedrawn?Whatwillbethesidelengthofthefirstdrawnsquareineachscript?Ofthesecond?Ofthethird?…Ofthebiggestone?

Whatistheinitialvalueofthesidelengthvariableineachscriptandwhatisitsfinalvalueafterfinishingtherepeat loop?

Twoalternativesolutionsof6(b):thefirstonemakestheBeetle“move” totheright,thesecondonemakestheBeetle“jump” totheright.BotharecorrectbutnotethatthesecondsolutionwillnotworkproperlyiftheBeetledoesnotpointindirection0(up)atthebeginning.

ADDITIONAL SUPPORT CONTINUED

❻ Twoalternativesolutionsof6(a).

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INVESTIGATION 2Activity 5.2.1ADDITIONAL SUPPORT CONTINUED

Threeslightlydifferentalternativesolutions,allarecorrect.Whichonedoyoufindeasiesttoread?

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INVESTIGATION 2Activity 5.2.1

Theideasbuiltearlierinthisactivitymightbeusedandfurtherdevelopedinmanydirections.Seesomeofthembellowandaskpupilstobuildscriptstodrawtheseorsimilaroutcomes.

TheBeetledrawsseveralsquaresofthegrowingsize.Pupilscanexperimentwithdifferentanglesintheturn blocktoworkoutgoodoutcome.

EXTENSION IDEAS

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ACTIVITY INSTRUCTIONS MATHEMATICS CONNECTIONS

31

MODULE 5 ● INVESTIGATION 2 ● ACTIVITY 5.2.2Altering Rectangles

❶ Pupilsbuildascripttodrawarectangle,e.g.oftheheight100andbase30.Theywillthenmakeanewblockrectangle.

❷ Pupilsmakeanewvariableheight anduseitinthedefinitionoftherectangle block.Theydrawdifferentrectanglesbysettingthevalueofheight andrunningtherectangleblock.

❸ Pupilsusechangeheight by…blockinsiderepeat todrawvariouspatternsofgrowingrectangles.

❹ Pupilsmakeanothervariable– base andgeneralisetherectanglescriptsothatitdrawsarectanglespecifiedbytwovariables:height andbase.Theysetbothofthemanddrawdifferentrectangles.TheyexperimentwiththeBeetlepointingindifferentdirectionsanddrawingthesamerectangle.

❺ Therearefourscriptsandfouroutcomesshowninthepresentation.Pupilswillreadthescripts,envisageandmatcheachofthemagainstoneofthepossibleoutcomes.

❻ Discussasaclasswhathappensifweconnectallupperrightcornersinapatternofrectanglesbyaline.

LEARNING OBJECTIVES

Explorehowtodrawasequenceofrectanglesofincreasingsizeusingtwovariables.Envisagetheoutcomesofscriptschangingtheheightand/orbasebydifferentamounts.Explainwhatthemagiclineis.

Pupilscontinueintheirownversionofproject5-AlteringPolygons,orstartagainwiththeinitialstarterproject5-AlteringPolygons.

Connectthestructureoftheabovescriptwiththesymmetryoftherectangle.[Arectanglehastwolinesofsymmetry,ithastwopairsofoppositeandequalsides]

Pupilsshouldcreatetwosequencesofrectanglesontheirscreenwheretheheightand thebasechange.Usetheedgeofapieceofpaperoverlaidonthescreentosupportvisualizingalineconnectingtheupperrightcornersoftherectangles.Askpupilstocomparetheirlines.Whatisthesameandwhatisdifferent?

Encourage pupilstocomparetheshapeoftheinitialrectangle,tothefinalrectangleineachsequence.Aretheythesameshape?[Whenthelineis“magic”therectanglesaremathematicallysimilar.]

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INVESTIGATION 2Activity 5.2.2

Encouragepupilstospottheregularityindrawingtherectangleandensurethescriptreflectsthatregularity.

Inthisstepthedefinitionoftherectangle blockwillbegeneralisedtodrawrectanglesofafixedbaseof25butofaheightspecifiedbythevariableheight.

ADDITIONAL SUPPORT

Pleasenotetheblue numbersontheleftlinktothenumberedstepsintheactivityinstructions

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INVESTIGATION 2Activity 5.2.2ADDITIONAL SUPPORT CONTINUED

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INVESTIGATION 2Activity 5.2.2ADDITIONAL SUPPORT CONTINUED

❻Whenthelineconnectingthetoprightcornerofeachrectanglegoesthroughthebottomleftcorneritis“magic”.Themagiclineindicatesthatthesequenceofrectanglesaremathematically similar.

Correspondingsidesareinthesameratio,andcorrespondinganglesareequal.Wecansaythattherectanglesareproportionaltoeachanother.

Anotherconnectionistousethelanguageofenlargement:WhenthelineIsmagic,onerectangleisanenlargementoftheother.It’simportanttonoticethatwhentherectanglesareanenlargementormathematicallysimilar,theirshapeismaintained.Encouragepupilstocomparethefirstrectanglewiththefinalrectangleineachsequence.

Inthissequence,thereisnomagiclinesinceitdoesnotgothroughthebottomleftcorner.Ifwedescribethefirstrectanglewemightsaythatitistallandthin,whereasthefinalrectanglelookslikeasquare,theshapehaschanged.Therectanglesinthissequencearenotmathematicallysimilar,theyarenotinproportiontooneanother.

Pupilsmaybegintonoticeanddefinetherelationshipbetweenthesidesofeachrectangle.Inthisexamplethebaseofthefirstrectangleistwicetheheight,ineachrectanglethisrelationshipismaintained.Theserectanglesaremathematicallysimilar,theratioofthesidelengthsareequal.

Allsquaresaremathematicallysimilarsincethebaseisequaltotheheight,eachsquareinthesequenceisanenlargementofanothersquare.

Inthenextactivitypupilsexplorecalculatingheight/baseasanumericalresulttosupportwhatcanbeseenvisually.Theybegintousethisrelationshipinproblemsolvingsituations.

Aslidefromthepupilpresentationforactivity5.2.2.

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INVESTIGATION 2Activity 5.2.2

Challengepupilstocreateinterestingoutcomesusingtherectangle blockinsidearepeat.

EXTENSION IDEAS

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ACTIVITY INSTRUCTIONS MATHEMATICS CONNECTIONS

36

MODULE 5 ● INVESTIGATION 2 ● ACTIVITY 5.2.3Exploring Mathematical Similarity

InActivity5.1.1pupilsusedthesay blocktoshowavalue(ananswertoaquestion,thenalsointheextensiontoshowtheperimeterofapolygon).Nowwewillusesay tocalculateandshowthevalueofheight dividedbybase.

❶ Pupilsbuildascriptthatwillsetthevariablesheightandbase to60and30,thendrawacorrespondingrectangle.TheBeetlewillthenusethesay…blocktocalculateandshowthevalueofheight dividedbybase.

❷ Pupilsnowattachtwoblockssettingtheinitialvaluesoftheheight andbase tothesetupscript.

Thentheybuildascriptwhichdrawsarectangle(withoutsettingfirstvaluesofheight andbase,astherearebeingsetinthesetupscriptnow),saysthevalueofheight dividedbybase andfinallychangesheight byanumber andbase byanumber.Theyrunthescriptseveraltimesandobservetheresultingsequenceofvaluesandpatternofincreasingrectangles.

❸ Distributetheprintedpupilworksheetsforthisactivity(2pages)andletthemrecordthesequenceofvaluesforscriptsintablesA toD (firstpage).Discussthemasaclass.

Letpupilschoosetheinitialnumbersandthenumberstochangeheight byandbase byinthescripts,tablesE toH (secondpage).

LEARNING OBJECTIVES

Explorehowtodisplaythevalueoftheheightdividedbythebaseofarectangle.Explainwhyforcertainsequencesofrectanglesthevalueofheight/basestaysthesame.

Pupilscontinueintheirownversionofproject5-AlteringPolygons,orstartagainwiththeinitialstarterproject5-AlteringPolygons.

Intheaboveexample,Whatdoesthe2represent? [Thelengthoftheheightis2timesthelengthofthebase].Demonstrateacalculationforadifferentstartingheightandbase,appropriatetotheclass.

Iftheheightandbasewereswapped,whatwouldtheBeetlesay? [0.5sincetheheightis0.5oforhalfofthebase].

ExplorewhattheBeetlewillsayforsquaresandexplaintheresult.[TheBeetlewillalwayssay1forsquaressincesquareshaveequallengthsides,andanythingdividedbyitselfis1].

Whenthereisa“magic”line,whatwilltheBeetlesay?[TheBeetlewillrepeatthesamenumberforeachrectangledrawn,sincethebaseandheightareinthesameproportion.Therelationshipoftheheighttothebaseisthesame,thesidesareinthesameratio.Therectanglesaremathematicallysimilar].

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INVESTIGATION 2Activity 5.2.3

Pupilsbuildthesay blockstepbystep,frominsideout,tryingthepartsbyclickingtheminthescriptsarea.Notethatifthesay block(withoutanyfor…secs)isused,togetridofthebubbleinthestageweeitherusesay againorclicktheredStop sign.

Itmayhappenthatsomepupilsgetthefollowingoutputsinthesayblock.NaNmeansNotaNumber.Thesetblocksmighthavebeenusedwith“nothing”orwithatextinsteadofanumber.TheBeetlewillsay Infinity ifthedivisorequals0.

Clickingtheright-handscriptagainandagainissimilartorunningitinarepeatloop.Ifrepeatedmanually,pupilshavetimetowritedownthevalueofdivision.

Notethatourrectangle blockworksevenwithoneorbothnegativeinputs.Gothroughitsdefinitionwithpupilsstepbysteptoseewhy.

ADDITIONAL SUPPORT

Pleasenotetheblue numbersontheleftlinktothenumberedstepsintheactivityinstructions

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INVESTIGATION 2Activity 5.2.3ADDITIONAL SUPPORT CONTINUED

Thepurposeofthesecondactivityisforpupilstofindvalueswhichproducea“magic”linee.g.thesequenceofrectanglesaremathematicallysimilar.

Provideamorestructuredtaskifneeded.E.g.startwithwritingontheboardaheightof90,abaseof30andchangeheightby10.PupilsfindthevalueofchangebasebytokeeptheBeetlesaying3.

AnotherapproachistoprovidevaluestosetstartingheightsandbasesasinTable1.Encouragepupilstodiscovertherelationshipbetweenthevaluesformathematicallysimilarrectangles.[Thevalueofheight/basemustbethesameasthechangeheightby/changebaseby.Wecanalsosaythattheratiooftheheighttothebaseisthesameastheratioofthechangeheightbytothechangebaseby.Addingonquantitiesinthesameratiowillcreateamathematicallysimilarrectangle.]

Playa‘beattheteacher’game.Askpupilstochoosevaluesforbaseandforheight[multiplesof10tostartwith]andyouwillcalculatethevaluesforthechangeheightbyandchangebasebytoproducemathematicallysimilarrectangles.TrythevaluesinthescripttoensurethattheBeetlesaysthesamenumberandthereisamagicline.

h b

90 30

100 20

95 45

120 30

60 60

25 100

15 120

WorksheetSolutions

DiscussthesequencesofnumbersgeneratedbytheBeetle.Agoodstartingquestionmightbe,Whatisthesameandwhatisdifferent?Whatwillhappentothenumbersifwekeepincreasingthepattern?

SequenceA– Thevalueofheight/base isdecreasing.Theshapeoftherectanglelooksmoreandmoresquarelikeaswecontinuethepattern.Wecanenvisagethatthesequencewouldgetverycloseto1.Itcan’tbeexactly1sincetheheightwillalwaysbe20morethanthebase.

SequenceB– Thevalueofheight/base isincreasing.Thepatternofnumberswillcontinuetoalternate.E.g.,2,2.5,3,3.5…

SequenceC– Thevalueofheight/base isdecreasing.Theshapeoftherectangleisbecomingverylongandthinsincetheheightisalways40.Wecanenvisagethatthesequencewouldgetcloseto0,butnotexactly,since40dividedbyalargenumberisalmost0.

SequenceD– Thevalueofheight/base isalways2.Thereisa“magic”line.Thestartingheightistwicethebase,thechangeinheightistwicethechangeinbase.Addingonvaluesinthesameratiowillalwayscreateanothermathematicallysimilar(proportional)rectangle.

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ForeachtableA toD alwaysbuildbothscripts.Besureyoufirstsetheight to40andbase to20.Clickthesecondscripttoseetheoutcomeandnotedownthevalueshown,againandagain.

1 22345678

1 22345678

1 22345678

1 22345678

INVESTIGATION 2Activity 5.2.3NAME

WHAT TO DO

SEQUENCE OF RECTANGLESA SEQUENCE OF RECTANGLESB

SEQUENCE OF RECTANGLESC SEQUENCE OF RECTANGLESD

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Modelanexampleonthewhiteboardbeforedistributingtheworksheet.Useheight60,base30andchangeheightandbaseby10.Itisimportantthatpupilscomparetheirresultswiththeirclassmatestoensurethaterrorsarespotted.TheBeetleshouldsay2 asthefirstvalueineachofthesequencesA-D.SequenceDwillproducerectangleswithamagicline,rectanglesthataremathematicallysimilarorinproportiontooneanother.

1 22 1.673 1.504 1.405 1.336 1.297 1.258 1.22

1 22 2.503 34 3.505 46 4.507 58 5.50

1 22 1.333 14 0.805 0.676 0.577 0.508 0.44

INVESTIGATION 2Activity 5.2.3

1 22 23 24 25 26 27 28 2

WORKSHEET SOLUTIONS

SEQUENCE OF RECTANGLESA SEQUENCE OF RECTANGLESB

SEQUENCE OF RECTANGLESC SEQUENCE OF RECTANGLESD

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SEQUENCE OF RECTANGLESG SEQUENCE OF RECTANGLESH

SEQUENCE OF RECTANGLESE SEQUENCE OF RECTANGLES F

41

12345678

12345678

12345678

12345678

INVESTIGATION 2Activity 5.2.3NAME

WHAT TO DOForeachtableE toH alwaysusebothscripts,decidefortheinitialvaluesofheight andbase,thenchoosethevaluestochangebyandwriteintheemptyholes.BuildinScratch,settheinitialvalues,runthesecondscriptrepeatedlyandnotedownthevaluedisplayedeachtime.

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ACTIVITY INSTRUCTIONS

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MODULE 5 ● INVESTIGATION 2 ● ACTIVITY 5.2.4[Extension] Unplugged: Rectangle Jumble

Activity5.2.4consistsofthreeworksheets.Printanddistributethemorworkasaclass.❶ Inthefirstonepupilsareaskedtoidentifypossibleplansfortheschoolswimmingpool60

metres by15metres.❷ Inthesecondworksheetpupilsareaskedtosortajumbleofrectanglesintothreedifferent

groupsofthosehavingtheheighteither2timesthebase,or3times,or4times.❸ Pupilsareaskedtosort9scriptsintothreegroups:scriptsineachgroupdrawproportional

rectangles.Lookfortherelationshipwithinarectangleandalsobetweentworectangles.

Allrectanglesproportionaltotheswimmingpool60metresby15metres arehighlighted.

Therearethreegroupsofproportionalrectangles

Forthethirdworksheet

thesolutionis

1– 4,2– 5,3– 1,4– 2,5– 3.c

LEARNING OBJECTIVES

WORKSHEETS SOLUTIONS

Envisagewhichrectanglesareproportionaltooneanother.Explainwhytworectanglesareproportional.bridgE toknowledgeofratioandproportion.

❷ ❸

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INVESTIGATION 2Ext. Activity 5.2.4NAME

WHAT TO DOAschoolhasaswimmingpool60metres by15metres.Whichoftheseplanscouldbeascaledrawingofthepool?Marktheonesyouthinkarecorrect.

SWIMMING POOL PLANS

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RECTANGLE JUMBLES

44

INVESTIGATION 2Ext. Activity 5.2.4NAME

WHAT TO DOInthetopjumblealltherectanglesdrawnbytheBeetleareproportional,theheightisalways3timesthebase.

Inthebottomjumbletherearethreetypesofrectangles– theheightiseither2timesthebase,or3timesthebase,or4timesthebase.Sorttherectanglesinthebottomjumbleintothreedifferentgroups,writingeither2to1,or3to1,or4to1oneach.

Sorttherectanglesintothreegroups(2to1,3to1and4to1)thejumblebelow.

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Readthescriptsbelow.Foreachscriptintheleftcolumnfindoneintherightcolumnwhichwoulddrawarectanglethatisproportional toit.

INVESTIGATION 2Ext. Activity 5.2.4NAME

WHAT TO DO

MATCH SCRIPTS BETWEEN COLUMNS

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MODULE 5: INVESTIGATION 3Grid World: For Exploring Similarity

Thisinvestigationbringstogetherideasfromtheprevioustwoinvestigations:constructingdifferentsizedrectanglesusingtwouserdefinedvariablesandtheconceptofa“magicline”indicatorwhenrectangles(orquantities)areinproportion.

Pupilscodeatoolwhichwillallowsthemtoexploreproportionalrelationshipsandtosolveproportionalproblems.UsingfourarrowkeyspupilsmovethedottothetoprightcornerofarectangleandclicktheBeetle.TheBeetlethenconstructstherectangleandthemagiclinefromthe‘origin’throughthediagonallyoppositecorneroftherectangleanddisplaystheratioofonequantity(sideA)totheother(sideB).Theprocesscanberepeatedinordertomakecomparisons.

uActivity5.3.1– EnterTheGridWorldu [Extension]Activity5.3.2– ConnectingCornersu [Extension]Activity5.3.3– MeettheMagicLineu [Extension]Activity5.3.4 – Unplugged:Module5Assessment

Scratchstarterproject 5-GridWorld5-GridWorldINT15-GridWorldINT2

LINKS TO PRIMARY NATIONAL CURRICULUM

CURRICULUM OBJECTIVES LINK WITH SCRATCHMATHS

MathematicsSolvesproblemsinvolvingsimilarshapeswherethescalefactorisknownorcanbefound.

Pupilsrecognise incontextswhentherelationsbetweenquantitiesarethesameratio(forexamplesimilarshapes).

Pupilsshouldconsolidatetheirunderstandingofratiowhencomparingquantities,sizesandscaledrawingsbysolvingavarietyofproblems.

Describepositionsonthefullcoordinategrid;Usesimpleformula

► Pupils useScratchtodisplaythepositionofthebeetleonthescreenusingcoordinates.Pupilsusescaletotransformcoordinatesforusewithdifferentgridtilesizes.

► Pupilsarerequiredtobuildascripttocalculatetheratioofonequantity(side)totheanother.TheGridWorldprovidestheopportunityforthemtoexplorethisrelationshipthroughexperimentingwithdifferentquantitiesandcomparing.

► BuildingtheGridWorldprovidesopportunitiesforpupilstoexploreandusecoordinatesincontext.

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Envisagethesizeofagridtileusingcoordinates.Explorehowtomovehorizontallyandverticallybyaspecifiednumberof“tiles”.

47

MODULE 5 ● INVESTIGATION 3 ● ACTIVITY 5.3.1Enter The Grid World

IntheGridWorldprojecttheBeetlealwaysturnsleft90whendrawingarectangle.Itstartsfromthebottomleftcornerofthescreenandpointsright.

TheScratchstageusesacoordinategrid-240to240pixelshorizontallyand-180to180pixelsvertically.1Beetlestepcorrespondsto1pixel.https://wiki.scratch.mit.edu/wiki/Pixels

Envisageascripttodrawarectanglewhichis4stepswideand3stepshighusingapensizeof1.Whatexactlywouldbedrawnonthescreen?Howbigwoulditbe?Explainyouranswer.[Therectanglewouldbedrawnverysmallcomparedtothebeetle:

TheScratchstageisarectangle480by360steps,soa4by3rectangleisverysmallincomparison.]

Theprojectusesbackgroundsofdifferentgridtilesizestoscaletheoutputonthestage.E.g.Usinggrid=50,foreachgridtiletheBeetlemoves50steps.HowmanystepswouldtheBeetlemoveifitmoved4,50gridtiles?[TheBeetlewouldmove200steps.]

❶ Pupilsreadthesetupscriptandrunit.InthisactivitytheBeetle willdrawrectangles,movingonlyalongthegridlines.Byreadingthemousecoordinates inthestagepupilsfindouthowbigagridtileisi.e.50.

❷ Pupilsverifytheirconjecturebydraggingintwoisolatedblocks– i.e.move50stepsandturnleft90.ByclickingthemrepeatedlyinthecorrectordertheymaketheBeetle drawarectangle,e.g.4tileswideand3tileshigh.TheBeetle shouldendinthesameposition,pointinginthesamedirectionasbefore.

❸ Pupilsmaketheirownblockmove1tilewhichwillmaketheBeetle move forwardby1tile– eitherinone‘jump’ormoreslowlyandsmoothly.

❹ Rectanglestobedrawnwillbesetbytwovariables– AtodefinehowmanytilesitwillhavehorizontallyandBtodefinehowmanytilesitwillhavevertically.Pupilsmakethesevariablesandbuildascript:whentheBeetle isclicked,A andB willbesettocertainvalues(e.g.4and2)andtheBeetlewilldrawtherectangle.Theymaymaketheirscriptshortandmorereadablebymakingtwomorenewblocks–movehorizontally(byA tiles)andmovevertically(byB tiles).

❺ Pupilsexploreotherbackdrops,findingoutwhattheirnamesareandwhattheirgridtilesizesare.Theychangebackdroptogrid20 andmodifytheirscriptssothattheBeetle drawsrectangleA byB tilescorrectly.

❻ Pupilsmakeanewvariablegrid,setitto20anduseitinthemove1tiledefinition.

❼[Extension] Pupilsexperimentwiththeirmove1tiledefinitiontomaketheBeetlemovequicklyorslowly…

ACTIVITY INSTRUCTIONS

LEARNING OBJECTIVES

MATHEMATICS CONNECTIONSOpentheproject5-GridWorldFINAL withinScratchontheIWBtodemonstratehowthefinalexplorationtoolworks.

Pupilsthenopenproject5-GridWorld,Saveasacopy (online)orSaveas (offline)andrename.ThefinalversionofthisprojectattheendofActivity5.3.1willbe 5-GridWorldINT1.

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INVESTIGATION 3Activity 5.3.1

Theactualcoordinatesofthemousecursorwithinthestagearedisplayedunderthelowerrightcornerofthestage.

TheBeetlemaymoveby1tileeitherinonego(onemove)orbyrepeatingsmallerstepseveraltimes.

NotethatiftheBeetle isclickedagainwhilestilldrawingtherectangle,twodrawingswillsomehowruninparallelandbothmightbespoilt.

ADDITIONAL SUPPORT

❷❸

Pleasenotetheblue numbersontheleftlinktothenumberedstepsintheactivityinstructions

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49

INVESTIGATION 3Activity 5.3.1

Thenameofthesecondbackdropisgrid20 and,naturally,thesizeofthegridtileis20by20.TheonlydetailinthescriptwhichmustbemodifiedsothattheBeetle againdrawsrectanglessetbythenumbersofthetilesA andB,isthemove1tile definition.Ontherighttherearetwoalternativesforhowtodoit.

Itmightbehelpfultohavethreesmallscriptstoswitchbetweendifferentgridseasily.Allotherscriptswillworkproperly,ifweusethegrid variabletodefinethemove1tileblock.

Herearedifferentdefinitionsofthemove1tile block.EnvisagewhichofthemwillmaketheBeetlemoveveryquick,veryslow,slow…

ADDITIONAL SUPPORT CONTINUED

EXTENSION

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Explorehowtodrawarectanglethatisdefinedbythepositionofthedotsprite.Explainhowtocontrolthespriteusingthearrowkeys.

50

MODULE 5 ● INVESTIGATION 3 ● ACTIVITY 5.3.2[Extension] Connecting Corners

BecarefultosettheinitialpositionoftheDot initssetupscript tomatchtheinitialvaluesofA andB.Ifgridis20andwesetA to3andB to2,thentheinitialpositionoftheDotmustbe3×20totherightand2× 20upwardsfromtheBeetle’sinitialposition.

Pupilswillbuildanewwaytosetthesidesoftherectangle:theDot spritewillreacttopressingthearrowkeys.Itwillmoveup,down,leftorright,alwaysfromonegridpointtoanother.ThesemovementswillincreaseordecreasevariablesA andB by1.Then,whentheBeetle isclickeditwilldrawarectanglewiththeDot sittinginitsdiagonallyoppositecorner.

❶ PupilsmodifythesetupscriptoftheDot:theydeletethehide blockandmaketheDot visible.

❷ AstheDot willnowberesponsibleforsettingandchangingvariablesA andB,pupilsmovethesetA…andsetB…blocksfromtheBeetle’s whenthisspriteclickedscriptintothesetupscript oftheDot andsetthemto3and2.

❸ Pupilsbuildthewhenrightarrowkeypressed scriptfortheDot:thespritewillpoint totheright,move1tile inthatdirectionandchange thevalueofA by1.

❹ InasimilarwaypupilsbuildthreemorescriptsfortheDot toreactcorrespondinglytotheotherthreearrowkeys.

❺ BuildonemorescriptfortheDot:itwillforever say theactualvaluesofA andB,e.g.3,2 or9,1…

❻ Pupilsswitchthebackdroptogrid10andmodifyallnecessarybitsofscriptssothatthewholeprojectworkscorrectlyagain.

❼[AdditionalExtension] Pupilstrytofullyautomatizeswitchingfromonegridtoanother.Whatchanges?

❽[AdditionalExtension] Pupilsmodifythewhen…arrowkeypressed scriptssothatwhentheDotreachesthegriddotsclosesttotheedgeofthestage,itwouldnottrytomoveanyfurther.

ACTIVITY INSTRUCTIONS

LEARNING OBJECTIVES

MATHEMATICS CONNECTIONS

Pupilscontinueintheirownversionofproject5-GridWorld,oropenthe5-GridWorldINT1,Saveasacopy (online)orSaveas (offline)andrename.ThefinalversionofthisprojectattheendofActivity5.3.2willbe5-GridWorldINT2.

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51

INVESTIGATION 3Ext. Activity 5.3.2

WecancheckwhatistheinitialpositionoftheBeetle andfortheDot increasethosexandypositionsby3× gridtilesize and2× grid tilesize respectively.

Notealsothatifwedeletethehideblockandclicktheshow blockonce,weinfactdonotneeditinthesetupscript atall.

TheBeetle isgoingtousevariablestodrawarectangleA byB gridtilesbutnotsetting(initializing)themnorchangingthem.Thereforewemustdeletethoselinesfromthescript.

Wecancopythemove1tileblockfromtheBeetle astheDotmaymoveinexactlythesameway.

MovingtheDot spritetotherightfromtheBeetlemeansincreasingvariableA by1.

DiscusswithyourclasswhicharrowsaffectvariableAandhowandwhicharrowsaffectvariableB.

ADDITIONAL SUPPORT

Pleasenotetheblue numbersontheleftlinktothenumberedstepsintheactivityinstructions

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52

INVESTIGATION 3Ext. Activity 5.3.2

Wheneverthebackdropisswitched,thegridsize,i.e.thegrid variablechangesitsvalue.Whatelseshouldhappenasanautomaticconsequenceofthatchange?

• Thebackdrophasswitched,buttheremightbesomelinesdrawnfromourpreviousexplorations,soitshouldbecleared.• Bothspritesshouldbeinitializedbyrunningtheirinitialscripts.

Bothissuesabovecaneasilybesolvedbybroadcastingamessageafterresettingthegridvariable.Thatbroadcastwillrunthesamesetupscriptsasclickingthegreenflag.

NotethattheinitialpositionoftheBeetle isthesameforgrid50,grid20 orgrid10…(thebackdropsweredesignedthisway).However,theDot shouldbemovedtoapositionderivedfromtheactualvalueofthegrid variable– forexample,alwayssettinginitialvaluesofA to3andofB to2andthenmovingtheDot AtilestotherightfromtheBeetle andB tilesupwards.

NotethatifwelaterdecidetousedifferentinitialvaluesforA andB than3 and2,wesimplymodifytheset blocksinthisscriptonly.

Infact,theonlydetailtomodifyistheinitialpositionoftheDot.Sowecaneasilyfindoutcorrectnewcoordinatesbyadding3× 10 and2× 10 respectively.

ADDITIONAL SUPPORT CONTINUED

ADDITIONAL EXTENSION SUPPORT

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53

INVESTIGATION 3Ext. Activity 5.3.2

Letusanalysewhatshouldhappenwhenrightarrowkeypressed(thescriptsforotherthreearrowkeyswouldneedsimilaranalysisandmodifications).

WhendowewanttheDot toreacttotherightarrowpressed?Onlyinthecasewhencertainconditionistrue– ingeneral,onlyiftheDot isstillnot“toofartotheright”.Sothegeneralstructureofthescriptmightbe:

Ifthegrid sizeis50,theconditioncouldbeassimpleas(a)above:iftheDot hasalreadymoved8gridtilesawayfromtheBeetle (inthehorizontaldirection),itshouldnotmoveanyfurther.

However,ifthegrid sizechanges,thisnumberwillbedifferent.Forexample,withthegrid20itshouldbe21,seecondition(b).

Thegeneralsolution(inthesenseofadvancedextension7)coulde.g.checkwhethertheDot isstillatleastagrid sizeawayfromtheedge,see(c)above.

ADDITIONAL EXTENSION SUPPORT CONTINUED

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Explorehowtodrawthe“magicline”.

54

MODULE 5 ● INVESTIGATION 3 ● ACTIVITY 5.3.3[Extension] Meet the Magic Line

Theblueandredpencolouredlinesareausefulvisualaidfortherelationshipbetweenthecoordinatesofthetoprightcornerandthesidelengthsoftherectangle.Thelengthoftheredlinecorrespondstothehorizontaldistanceinthecoordinate,andthebluelinetheverticaldistance.

Ifthedotisatcoordinate(12,8),theBeetlewilldrawfromtheorigin(0,0)arectanglewhichhasbase12andheight8.

Inthisfinalsteppupilswilladddrawingthemagiclineitself,connectingtheBeetle’sandDot’spositions.

❶ Pupilsextendthebehavioursofbothsprites:WhenevertheBeetle drawsitsrectangle,theDotstampsitssecond(turquoise)costumeatitsposition,i.e.atthediagonallyoppositecornerofthatrectangleandswitchthecostumebacktoblue.

❷ Insteadofdrawingthewholerectangleinonecolour,pupilshavetheBeetle drawhorizontallinesinadifferentpencolourandpensizethantheverticallines.Pupilsmodifythemovehorizontally andmovevertically blocks.

❸ PupilsbuildonemorescriptfortheBeetle:whenspacekeypressed,drawa“magicline”connectingtheBeetle’s andDot’s positions.

❹ Pupilsfinalisethewholeproject sothatitproperlyworksinthegrid10 backdrop.

ACTIVITY INSTRUCTIONS

LEARNING OBJECTIVES

MATHEMATICS CONNECTIONS

Pupilscontinueintheirownversionofproject5-GridWorld,oropenproject5-GridWorldINT2,Saveasacopy (online)orSaveas (offline)andrename.ThefinalversionofthisprojectattheendofActivity5.3.3willbe 5-GridWorldFINAL.

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55

INVESTIGATION 3Ext. Activity 5.3.3

Thekeyparthereisthis:

TheBeetle firstpointstowardstheDot,thenslowlystartsmovingbyrepeatingmove5steps,untilittouchestheedgeofthestage.Thenitjumpsbackhome.

ADDITIONAL SUPPORT

Pleasenotetheblue numbersontheleftlinktothenumberedstepsintheactivityinstructions

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bridgEtoknowledgeofmultiplication,regularpolygons,angles,perimeter,ratio/proportion.Envisagetheoutcomeofdifferentscripts.Explainwhyascriptwouldhaveaparticularoutcomeandhowtocompleteascripttogenerateaspecifiedoutcome.

uPrintanddistributetheunpluggedpupilworksheet5.3.4.

uAskpupilstoworkindividuallytocheckwhattheyhavelearnedduringModule5.

uTheanswerstotheworksheetarebelow:

1. Foranswer20theBeetlewillsay40,for1200itwillsay2400,for45itwillsay90.

2. Seconddrawing.

3. TheBeetlewillaskonlyonce,itwilldrawaregularoctagon.Ifitsperimeteris160,wemusthaveansweredthequestionbytypingin20.

4. Correctscriptis(b).Script(a)willdrawgapsbetweenthesquaresbecauseittakesthepenupeachtimeitmovestothenextsquareandscript(c)doesnotincreasethesidelengthatallsoallthesquareswouldbethesamesize.

5. FirstdrawingYes,ifweanswer60.Eachofthreesideshasthesamelength.SeconddrawingNo,asthesideshavedifferentlengths.ThirddrawingYes,ifweanswer90.FourthdrawingNo,asithasfourlines.

6. Theoutcomecanberegulartriangle,square,pentagonorhexagon.

7. 20;40;50;80;35;60

8. Itwilldraw5squares,thefirstoneof20,thelastoneof60sidelength.Thevalueofthevariableattheendwillbe70.

9. Script(a)producesthepictureontheright.Script(b)producesthepictureinthemiddle.Script(c)producesthepictureontheleft.

10. [Extension]Proportionallybiggerrectanglescouldbe25and100;30and120;40and16Proportionallysmallerrectanglescouldbe10and40;5and20andmanyothers.Theseareallrectangleswithsides1:4.

56

MODULE 5 ● INVESTIGATION 3 ● ACTIVITY 5.3.4Unplugged: Module 5 Assessment

LEARNING OBJECTIVES

ACTIVITY INSTRUCTIONS

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Readthetasksbelowandanswerthequestions.Note- Thepentoolisalwaysdownatthebeginningofeachtask.

57

INVESTIGATION 3Activity 5.3.4NAME

WHAT TO DO

ASSESSMENT TASKS

❶ Whenwerunthisscript,Beetle willask foranumber.NotethenumberthattheBeetlewillsay ifwegivethefollowinganswers:(notethat*inScratchmeanstomultiply)

Ifweanswer20 theBeetlewillsay

Ifweanswer1200 theBeetlewillsay

Ifweanswer45 theBeetlewillsay

Whatwillhappenifwerunthisscriptandanswerthequestion“Whatpensizenow?” bytypingin20?Circlethecorrectdrawingbelow.(notethestartingpointismarkedbytheredarrow)

❸ Ifwerunthisscript:

a) HowmanytimeswilltheBeetleask“whatsidelength?”?

b) DescribewhattheBeetlewilldraw.

c) IftheperimeterofthepolygonthattheBeetledrawsis160,whatnumberwastypedin?

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58

ASSESSMENT TASKS CONTINUEDINVESTIGATION 3

Activity 5.3.4

Yes NoWhy?

Yes NoWhy?

Yes NoWhy?

Yes NoWhy?

Foreachofthefollowingdrawingsdecidewhetheritcanbeanoutcomeofthescriptontherightornot.CircleortickYes orNo andexplainwhy.

Circlethescriptthatwillproducethedrawingbelow.

Explainwhyyoupickedthatscript:

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59

ASSESSMENT TASKS CONTINUEDINVESTIGATION 3

Activity 5.3.4

❻ IfwerunthisscriptfortheBeetle whatkindofpolygonisdrawn?

Answerandexplainyourthinking:

ForeachofthefollowingscriptsoftheBeetlewritedownwhatsidelengththedrawnsquarewillhave:

sidelengthofsquare=

sidelengthofsquare=

sidelengthofsquare=

sidelengthofsquare=

sidelengthofsquare=

sidelengthofsquare=

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60

ASSESSMENT TASKS CONTINUEDINVESTIGATION 3

Activity 5.3.4

❾ Matchthescripttothepicturewhichitcouldproduce.

Ifwerunthisscript…

HowmanysquareswilltheBeetledraw?

Whatwillbethesidelengthofthefirstone?

Whatwillbethesidelengthofthelastone?

Whatwillbethevalueofthesidelengthvariableafterthescriptisrun?

[Extension]Ifwerunscript(a),theBeetlewilldrawthebelowrectangle.Thinkoftwomorepairsofvalues oftheheight andbase variablesthatwouldoutputamathematicallysimilarbiggerandsmaller rectangle(i.e.fitonthemagicline)andwritethenumbersintheemptyholesontheright(markedby?).

proportionallybigger

proportionallysmaller

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61

MODULE 5: INVESTIGATION 4Using the Grid World

ThisinvestigationusestheGridWorldbuiltinInvestigation3toexploreproportionalrelationshipsandtosolveproblemswhichinvolvethemathematicalideasofratio,proportion,similarityandenlargement.

Pupilsusethetooltorepresentthequantitieswithintheproblemasthetwosidesofarectangle,drawamagiclineandthenreadofforcalculatethequantitiesrequiredintheproblem.Exploringpairsofvalueswhichfitonthemagiclinewithinthetoolprovidespupilswithanopportunitytodeveloptheirownconnectionwithinandbetweenmathematicallysimilarrectangles.Itisenvisagedthatthegraphicalrepresentationbeusedalongsideotherrepresentationswithintheclassroom.

uActivity5.4.1– UsingtheGridWorlduActivity5.4.2– BridgEing AndSolvingProblems

Scratchstarterproject 5-GridWorld5-GridWorldFINAL

LINKS TO PRIMARY NATIONAL CURRICULUMCURRICULUM OBJECTIVES LINK WITH SCRATCHMATHS

MathematicsPupilsrecogniseproportionalityincontextswhentherelationsbetweenquantitiesareinthesameratio(forexample,similarshapesandrecipes).

Pupilsconsolidatetheirunderstandingofratiowhencomparingquantities,sizesandscaledrawingsbysolvingavarietyofproblems.Theymightusethenotationa:btorecordtheirwork.

Pupilssolveproblemsinvolvingunequalquantities(forexample,‘foreveryeggyouneedthreespoonfuls offlour’)

► Pupilsuse theGridWorldtoexploreandbuildmathematicallysimilarrectangles.Theyexplaintherelationshipsbetweenthelengthsofsidesinthesameratio,withinandbetweenrectangles.

► Pupilssolvevariousproblemsincontextswhichrequireproportionality.Theyworkonandoffthecomputertoexplaintheirsolutions.

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Explorehowtobuildmathematicallysimilarrectangles.Explainmathematicalrelationships.

62

MODULE 5 ● INVESTIGATION 4 ● ACTIVITY 5.4.1Using the Grid World

Pupilsusetheirfinal5-GridWorld project,ortheprovided5-GridWorldFINAL project.Theyusethegrid20 backdrop.AisthebaseoftherectangleandBistheheight.

❶ Usingtheproject,pupilsconstructarectanglewhereA=3andB=1.❷ PupilsconstructarectanglewhereA=6andB=2.❸ Thentheyaddthemagicline.❹ Pupilslookforandconstructtwomorerectangleswhichfitonthemagicline.❺ DoestherectanglewhereA=15andB=5fitontheline?Canyouexplainyouranswer?❻ IfA=21,whatisthevalueofB?Explainyouranswer.❼ IfB=10,whatisthevalueofA?Explainyouranswer.❽ Pupilstransfertheiranswersintothetableontheprintableworksheet1onthenextpage,

thentheycompletethecalculations(solutionsbelow).

ACTIVITY INSTRUCTIONS

LEARNING OBJECTIVES

A B A÷ B

3

6

21

36

10

1

2

3

3

3

3

3

3

3

12

9

30

12

4

3

7

WORKSHEET SOLUTIONS

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63

Transferallyourpreviousanswersintothetable,andcompletethecalculations.

INVESTIGATION 4Activity 5.4.1NAME

WHAT TO DO

WORKSHEET 1

A B A÷ B

3

6

21

36

10

1

2

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Explorehowtobuildmathematicallysimilarrectangles.Explainmathematicalrelationships.

64

Pupilsusetheirfinal5-GridWorld project,ortheprovided5-GridWorldFINAL project.Theyusethegrid20 backdrop.AisthebaseoftherectangleandBistheheight.

❶ Usingtheproject,pupilsconstructarectanglewhereA=3andB=2.❷ PupilsconstructarectanglewhereA=6andB=4.❸ Thentheyaddthemagicline.❹ Pupilslookforandconstructtwomorerectangleswhichfitonthemagicline.❺ DoestherectanglewhereA=12andB=6fitontheline?Canyouexplainyouranswer?❻ IfA=18,whatisthevalueofB?Explainyouranswer.❼ IfB=10,whatisthevalueofA?Explainyouranswer.❽ Pupilstransfertheiranswersintothetableontheprintableworksheet2onthenextpage,

thentheycompletethecalculations(solutionsbelow).

ACTIVITY INSTRUCTIONS

LEARNING OBJECTIVES

A B A÷ B

3

6

18

30

10

2

4

1.5

1.5

1.5

1.5

1.5

1.5

1.5

12

9

15

20

8

6

12

WORKSHEET SOLUTIONS

INVESTIGATION 4Activity 5.4.1

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65

Transferallyourpreviousanswersintothetable,andcompletethecalculations.

INVESTIGATION 4Activity 5.4.1NAME

WHAT TO DO

WORKSHEET 2

A B A÷ B

3

6

18

30

10

2

4

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INVESTIGATION 4Activity 5.4.1MATHEMATICS CONNECTIONS

Wecanexploretheconstant relationshipsofmathematicallysimilarrectanglesintwoways:

• therelationshipofthesidelengthswithinarectangle

• therelationshipofonesidelengthtoacorrespondingsidelengthbetween tworectangles

❶ Worksheet

Thesecondworksheetlooksverysimilartothefirstbutisdesignedtoencouragethedevelopmentoftherelationshipbetween rectangles.

Therelationshipwithineachrectanglehasaconstantratio,base÷height =3/2=1.5.Thebaseis1.5timesbiggerthentheheight.Therelationshipisnotaseasytoseeortocalculatewithasforwholenumberratios.However,performingcalculationsbetween rectanglesiseasier.

Question6:IfA=18,whatisthevalueofB?Explainyouranswer.

ItcanbedifficulttocalculateBsinceAis1.5timesbiggerthanB.However,ifweknowthatamathematicallysimilarrectangleisA=3andB=2,thenbetweentherectanglesthereisa6timesbiggerrelationship,soifA=18thenB=2*6=12.

Question7:IfB=10,whatisthevalueofA?Explainyouranswer.

Between thetworectanglesthereisa5timesbiggerrelationship,soA=3*5=15.

NOTE:TheGridWorlddealswithwholenumbersonly,butwecanobservethatthe‘magicline’isaninfinitelinewhichpassesthroughfractionalanddecimalpartsofthegrid.Encouragepupilstocalculate(theycannotchecktheirworkeasily)withdecimalvaluesforAandBineachoftheaboveworksheetstodevelopfluency.

Encouragepupilstochoosethemethod(within orbetween)whichmakescalculationseasier.Askthemtoselectwhichmethodtheyusewhensolvingproblems.

Worksheet

Thisworksheetisdesignedtoencouragethedevelopmentoftherelationshipwithin arectangle,i.e.themultiplicativerelationshipbetweenbase(A)andtheheight(B).

Theratiocanbecalculated,base÷height =3.Itisimportanttorecognizethattheinverseratioheight÷base isalsoaconstant 1/3.

Wecanwrite:

• base=heightx3,or

• A =B x3,or

• B =A/3B =1/3 ofA

Question6issmallenoughtofitonthegrid,howeverpupilsshouldbeencouragedtocalculatefirstandthentesttheirsolutionintheGridWorld.Question7cannotbetestedongrid20 butcanbecheckedbyswitchingtogrid10 .

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ACTIVITY INSTRUCTIONS

67

MODULE 5 ● INVESTIGATION 4 ● ACTIVITY 5.4.2BridgEing And Solving Problems

Activity5.4.2consistsoftwoworksheets.Printanddistributethemorworkasaclass.Pupilsusetheirfinal5-GridWorld project,ortheprovided5-GridWorldFINAL projecttosolvetheproblems.Encouragepupilstosketchrectangles.Discusssolutionsandapproachesasaclass.Additionalnotesinsolutions.

1. Thebaseofthefirstrectangleistwiceabigastheheight.Thereforeforamathematicallysimilarrectanglewhichhasaheightof3,thebasemustbe3x2=6[Thisisanexampleofawithin relationship]

2. Therelationshipofthematchestobottletopscanbethoughtofascomparingthesidesofarectangle.Drawadiagramofa2x3rectangletohelppupils.

Itiseasiertouseabetween relationship.

12matchsticksis6timesbiggerthan2matchsticks,soweneed3x6=18bottletops.

3. Oneapproachtosolvethisproblemistocombinetwosimilarrectangles.We’veseeninthegrowingrectanglesequencethatifyouchangetheheightandbaseinthesameproportionyoumakeasimilarrectangle.

MrShortis4buttonsand6paperclips,wewant6buttons.

Wecouldfindtherelationshipwithin (2/3)orbetween (3/2)butbothofthesearedifficulttocalculatewith.

Aneasierapproachistohalfboth,so2buttonsand3paperclips.

Wecannowaddtogethersinceweareaddinginthesameratio(proportion)

So6buttonstallisthesameas6+3=9paperclipstall.MrTallis9paperclipstall.

LEARNING OBJECTIVES

WORKSHEET SOLUTIONS

bridgE toproblemswhichinvolvethemathematicalideasofratio,proportionandsimilarity.

Page 68: Exploring Mathematical Relationships - UCL · 2018-07-26 · Module 5 is focused around exploring different types of mathematical relationships including proportionality and ratio

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MODULE 5 ● INVESTIGATION 4 ● ACTIVITY 5.4.2BridgEing And Solving Problems

4. Thisproblemisadaptedfromataskwithinthehttp://iccams-maths.org/ project.

a)Althoughthisarecipeproblem,wecanstilluseGridWorldtohelpuscomparetwoquantitiesandtofindquantitieswhichhavethesamerelationshipbyusingthe“magicline”.

Thetwosidesoftherectanglecanrepresentanytwoquantities,inthisproblemthebaserepresentsthequantityoftabascosauceandtheheightrepresentsthenumberofpeople.

Pupilsdraw:

b)Pupilsdrawarectanglewhichhasaheightof6,thebasewillbe18.Thiswillfitonthemagicline.

c)Therearemanysolutions,theamountofTabascoisalways3xthenumberofpeople.

Thegriddefaultgridwordwillshow(3,1),(6,2),(12,4),(15,5).

d)TheamountofTabasco=3xthenumberofpeople.

e)For30people,youwillneed3x30=90mlofTabasco.

f)57/3=19

19peoplecouldbeserved.

WORKSHEET SOLUTION CONTINUED

Page 69: Exploring Mathematical Relationships - UCL · 2018-07-26 · Module 5 is focused around exploring different types of mathematical relationships including proportionality and ratio

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INVESTIGATION 4Activity 5.4.2NAME

WHAT TO DOUsetheGridWorldtosolvetheproblems,drawdiagramstoexplainyourworking.

WORKSHEET

Thetworectanglesareproportional(mathematicallysimilar)tooneanother.Findlengthx andgiveareasonforyouranswer.

❷ Twomatchstickshavethesamelengthasthreebottletops.Howmanybottletopswillhavethesamelengthas12matchsticks?

Answerandexplainyourthinking

❸ HereisapictureofMr.Short. MrShortis4buttonsor6paperclipsinheight.MrTallmeasures6buttons,howhighisMrTallinpaperclips?

Answerandexplainyourthinking

Page 70: Exploring Mathematical Relationships - UCL · 2018-07-26 · Module 5 is focused around exploring different types of mathematical relationships including proportionality and ratio

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INVESTIGATION 4Activity 5.4.2NAME

WHAT TO DOUsetheGridWorldtosolvetherecipeproblem.

WORKSHEET

Adamismakingaspicysoupfor3people.Heuses9mloftabascosauce.

a)CreatearectangleintheGridWorldwhichhasabase of9andaheight of3,drawthemagicline.

b)Davinaismakingthesamesoupfor6people.Howmuchtabascosauceshouldsheuse?[Clue:Drawarectanglewhichfitsonthemagiclineandhasaheightof6.]

c)Findanothersolutionthatworks(i.e.fitsonthemagicline).

d)Whatistherelationshipbetweenpairsofnumbers?

e)IfMattismakingthesamesoupforalargepartyof30,howmuchtabascowouldheneed?

f)Tabascoissoldin57mlbottles.Howmanypeoplecouldbeservedthesamespicinessofsoupusingonebottle?

Answerandexplainyourthinking


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