UCL SCRATCHMATHS CURRICULUM
MODULE 5:Exploring
Mathematical Relationships
TEACHER MATERIALS
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
Investigation1PolygonFireworks
NightSkyline
Investigation2Mathematically
SimilarRectangles
Investigation3GridWorld:For
ExploringSimilarity
Investigation4Exploring
Proportionality
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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.
Explorehowtousetheaskandanswerblockstodrawdifferenttypesofregularpolygons.Explainwhatisthesameandwhatisdifferentbetweenregularandirregularpolygons.
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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
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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.
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ADDITIONAL SUPPORT
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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).
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ADDITIONAL SUPPORT CONTINUEDINVESTIGATION 1
Activity 5.1.1
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ADDITIONAL SUPPORT CONTINUED
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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.
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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
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Explorehowtousevariableswithinascripttostoredifferentvaluesatthesametime.Explainwhywenowneedvariablestodrawmultipleregularpolygonsofdifferentsizes.
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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.
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ADDITIONAL SUPPORT
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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:
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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…
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
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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.
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INVESTIGATION 1Activity 5.1.4
Inthedefinitionofthesquare blockweusethesidelengthvariable,thevalueofwhichwillbesetinthewhenthisspriteclicked scriptbyask andanswer.
InthescriptfortheBeetleeachsquarecanhavethesamecolour ormayhavedifferentpenshadesordifferentpencolours,pupilscanchoose.
Pupilsshouldstartusingthevocabulary:Thespriteasksfor…thensavesorkeepstheanswer(oransweredvalue)inavariable…
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INVESTIGATION 1Activity 5.1.4
Alternativesolutionwithnumberoffloorsandsidelengthsetatrandom,withoutanyasking:Carefullychooseanappropriateminimumandmaximumforeachvalue,includingtherangesforxpositionandypositionforrandomjumping.
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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.
Explorerepeatedlychangingavariablebyasetamounttocreateasequenceofincreasingsquares.Explainthedifferencebetweenthechange<variable>byandset<variable>toblocks.
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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
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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
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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
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❽
Threeslightlydifferentalternativesolutions,allarecorrect.Whichonedoyoufindeasiesttoread?
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INVESTIGATION 2Activity 5.2.1
Theideasbuiltearlierinthisactivitymightbeusedandfurtherdevelopedinmanydirections.Seesomeofthembellowandaskpupilstobuildscriptstodrawtheseorsimilaroutcomes.
TheBeetledrawsseveralsquaresofthegrowingsize.Pupilscanexperimentwithdifferentanglesintheturn blocktoworkoutgoodoutcome.
EXTENSION IDEAS
ACTIVITY INSTRUCTIONS MATHEMATICS CONNECTIONS
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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
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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
ACTIVITY INSTRUCTIONS MATHEMATICS CONNECTIONS
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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
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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.
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h b
90 30
100 20
95 45
120 30
60 60
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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.
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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
SEQUENCE OF RECTANGLESG SEQUENCE OF RECTANGLESH
SEQUENCE OF RECTANGLESE SEQUENCE OF RECTANGLES F
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INVESTIGATION 2Activity 5.2.3NAME
WHAT TO DOForeachtableE toH alwaysusebothscripts,decidefortheinitialvaluesofheight andbase,thenchoosethevaluestochangebyandwriteintheemptyholes.BuildinScratch,settheinitialvalues,runthesecondscriptrepeatedlyandnotedownthevaluedisplayedeachtime.
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.
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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
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INVESTIGATION 2Ext. Activity 5.2.4NAME
WHAT TO DOInthetopjumblealltherectanglesdrawnbytheBeetleareproportional,theheightisalways3timesthebase.
Inthebottomjumbletherearethreetypesofrectangles– theheightiseither2timesthebase,or3timesthebase,or4timesthebase.Sorttherectanglesinthebottomjumbleintothreedifferentgroups,writingeither2to1,or3to1,or4to1oneach.
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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.
Envisagethesizeofagridtileusingcoordinates.Explorehowtomovehorizontallyandverticallybyaspecifiednumberof“tiles”.
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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
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Pleasenotetheblue numbersontheleftlinktothenumberedstepsintheactivityinstructions
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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
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EXTENSION
Explorehowtodrawarectanglethatisdefinedbythepositionofthedotsprite.Explainhowtocontrolthespriteusingthearrowkeys.
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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.
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
❸
❷
❶
❹
❺
Pleasenotetheblue numbersontheleftlinktothenumberedstepsintheactivityinstructions
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
❻
❼
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
❽
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.
55
INVESTIGATION 3Ext. Activity 5.3.3
Thekeyparthereisthis:
TheBeetle firstpointstowardstheDot,thenslowlystartsmovingbyrepeatingmove5steps,untilittouchestheedgeofthestage.Thenitjumpsbackhome.
ADDITIONAL SUPPORT
❸
❷
❶
Pleasenotetheblue numbersontheleftlinktothenumberedstepsintheactivityinstructions
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
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)
❷
❸ Ifwerunthisscript:
a) HowmanytimeswilltheBeetleask“whatsidelength?”?
b) DescribewhattheBeetlewilldraw.
c) IftheperimeterofthepolygonthattheBeetledrawsis160,whatnumberwastypedin?
58
ASSESSMENT TASKS CONTINUEDINVESTIGATION 3
Activity 5.3.4
Yes NoWhy?
Yes NoWhy?
Yes NoWhy?
Yes NoWhy?
Foreachofthefollowingdrawingsdecidewhetheritcanbeanoutcomeofthescriptontherightornot.CircleortickYes orNo andexplainwhy.
❹
❺
Circlethescriptthatwillproducethedrawingbelow.
Explainwhyyoupickedthatscript:
59
ASSESSMENT TASKS CONTINUEDINVESTIGATION 3
Activity 5.3.4
❼
❻ IfwerunthisscriptfortheBeetle whatkindofpolygonisdrawn?
Answerandexplainyourthinking:
ForeachofthefollowingscriptsoftheBeetlewritedownwhatsidelengththedrawnsquarewillhave:
sidelengthofsquare=
sidelengthofsquare=
sidelengthofsquare=
sidelengthofsquare=
sidelengthofsquare=
sidelengthofsquare=
60
ASSESSMENT TASKS CONTINUEDINVESTIGATION 3
Activity 5.3.4
❿
❾ Matchthescripttothepicturewhichitcouldproduce.
❽
Ifwerunthisscript…
HowmanysquareswilltheBeetledraw?
Whatwillbethesidelengthofthefirstone?
Whatwillbethesidelengthofthelastone?
Whatwillbethevalueofthesidelengthvariableafterthescriptisrun?
[Extension]Ifwerunscript(a),theBeetlewilldrawthebelowrectangle.Thinkoftwomorepairsofvalues oftheheight andbase variablesthatwouldoutputamathematicallysimilarbiggerandsmaller rectangle(i.e.fitonthemagicline)andwritethenumbersintheemptyholesontheright(markedby?).
proportionallybigger
proportionallysmaller
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.
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
❶
63
Transferallyourpreviousanswersintothetable,andcompletethecalculations.
INVESTIGATION 4Activity 5.4.1NAME
WHAT TO DO
WORKSHEET 1
A B A÷ B
3
6
21
36
10
1
2
❶
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
❷
INVESTIGATION 4Activity 5.4.1
65
Transferallyourpreviousanswersintothetable,andcompletethecalculations.
INVESTIGATION 4Activity 5.4.1NAME
WHAT TO DO
WORKSHEET 2
A B A÷ B
3
6
18
30
10
2
4
❷
66
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.
❷
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 .
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.
68
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
69
INVESTIGATION 4Activity 5.4.2NAME
WHAT TO DOUsetheGridWorldtosolvetheproblems,drawdiagramstoexplainyourworking.
WORKSHEET
Thetworectanglesareproportional(mathematicallysimilar)tooneanother.Findlengthx andgiveareasonforyouranswer.
❶
❷ Twomatchstickshavethesamelengthasthreebottletops.Howmanybottletopswillhavethesamelengthas12matchsticks?
Answerandexplainyourthinking
❸ HereisapictureofMr.Short. MrShortis4buttonsor6paperclipsinheight.MrTallmeasures6buttons,howhighisMrTallinpaperclips?
Answerandexplainyourthinking
70
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
❹