7.10"Sand"Castles" A"Solidify"Understanding"Task"

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1 SandCastles ASolidifyUnderstandingTask Benji,ChauandKassandraplantoenterasandcastle buildingcontestbeingsponsoredbyalocalradiostation. Thewinningteamgetsaprivatebeachpartyatalocal resortforalloftheirfriends.tobeselectedforthe competition,theteamhastosubmitadrawingoftheir castleandverificationthatthedesignfitswithintherules. 2013www.flickr.com/photos/klbm Thethreefriendsactuallyplantobuildthreeidenticalcastles,eachonetwiceasbigastheprevious one.theyhopethatreplicatingthesamedesignthreetimes whilepayingattentiontothetiniest littledetails willimpressthejudgeswiththeircreativityandsandsculptingskill. Benjiispuzzlingoveracoupleofquestionsontheapplication.Theysoundlikemathquestions,and hewantschauandkassandratomakesurethatheanswersthemcorrectly. Please&provide&the&following&information&about&your&sand&sculpture:& &&What&is&the&total&area&of&the&footprint&of&your&planned&sand&sculpture?& [Thisinformationwillallowtheplanningcommitteetolocatesandsculpturessotheviewingpublicwillhave easyaccesstoallsculptures.rememberthatthetotalareaoccupiedbyyoursculpturecannotexceed50sq.ft.] &What&is&the&total&volume&of&sand&required&to&build&your&sand&sculpture?& [Wewillprovideclean,siftedsandforeachteamsowewillnotbeliableforanydebrisorharmfulsubstances thatcanbepresentinbeachsand.] I&certify&that&the&above&information&is&correct.&& Signature&of&team&leader:&& date:& & Thefriendshaveonlydesignedoneofthecastles,sincetheotherswillbescaledupversionsofthis one,eachonebeing twiceasbig. AfterstudyingthediagramBenjisaid, Icalculatedtheareaofthefootprintofthesmallestcastleto be2.5sq.ft.,sothenextonewilloccupy5sq.ft.,andthelargest10sq.ft.that satotalof17.5sq.ft. Wellwithinthelimits. 1. WhatdoyouthinkofBenji scomment?designacoupleofpossible footprints forasand castlethatwilloccupy2.5squareunitsofarea.thenscaleeachdesignupsoitis twiceas big,andcalculatethearea.whatdoyounotice? 2013MathematicsVisionProject MVP InpartnershipwiththeUtahStateOfficeofEducation LicensedundertheCreativeCommonsAttribution4NonCommercial4ShareAlike3.0Unportedlicense.

2 53 2. Imaginestackingcubesonyoursandcastle footprints tocreateasimple3zdsculpture. Thenscaleupeachdesignsoitis twiceasbig andcalculatethevolume.whatdoyou notice? 3. Howdidyouinterpretthephrase twiceasbig inyourworkonquestions1and2?isyour interpretationthesameasbenji s? 4. Toavoidconfusion,itwouldbemoreappropriateforBenjiandhisfriendstosaytheyare goingto scaleup theirinitialsandcastlebyafactorof2.ifthe footprint ofasandcastle occupies2.5sq.ft.,isitpossibletocalculatetheareaoccupiedbyasandcastlethathasbeen enlargedbyascalefactorof2,oristheareaoftheenlargedshapedependentuponthe shapeoftheoriginalfigure?thatis,dotriangles,parallelograms,pentagons,etc.allscaleup inthesameway?writeaconvincingargumentexplainingwhyorwhynot? 5. Whathappenstotheperimeterofthe footprint ofyoursandcastlewhenitisscaledzupby afactorof2? 6. Supposeyoursandcastle footprint wascutoutofapieceofstyrofoamthatisonezinch thick.whathappenstothevolumewhenthis 3ZDfootprint isscaledupbyafactorof2? 7. Whatwouldhappentotheperimeterofanyface,thetotalsurfacearea,andthevolumeofa triangularprismifitisscaledzupbyafactorof3?buildmodelsorsketchdiagramsto illustratethereasoningbehindyouranswer. WhileBenjihasbeenworkingoncorrectingtheapplication,ChauandKassandrahavebeenlooking onlineforinformationtohelpthemcalculatethevolumeofsandrequired,sincetheirplansinclude prisms,pyramids,cylindersandcones. Chaufoundthisinformation:ThevolumeofarightprismorrightcircularcylinderisgivenbyV=Bh, wherebistheareaofthesurfacethatformsthebase,andhistheheightoftheprism orcylinder.thevolumeofapyramidorconeis1/3ofthevolumeofaprismor cylinderwiththesamebaseandheight. Kassandrafoundthisinformation:ThevolumeofaprismorconeisgivenbyV=Bh,whereBisthe areaofthecongruentcrosssectionsparalleltothebase,andhistheheightofthe prism.thevolumeofapyramidorconeis1/3ofthevolumeofaprismorcylinder withthesamebaseandheight. 8. Howdothesetwostatementsdifferandwhatdothosedifferencesimply? 2013MathematicsVisionProject MVP InpartnershipwiththeUtahStateOfficeofEducation LicensedundertheCreativeCommonsAttribution4NonCommercial4ShareAlike3.0Unportedlicense.

3 54 ChauandKassandraarecuriousaboutthedifferencesinthesetwodefinitionsandsotheycontinue searchingonlineformoreinformation.chauisparticularlyconcernedaboutwhyapyramidor conealwayshas1/3ofthevolumeoftherelatedprismorcylinderwiththesamebaseandheight. Sheremembersateacherillustratingthisideabyfillingafewplasticpyramidswithsandand pouringthemintoprismswiththesameheightandbase.eachtimeittookthree pyramid sfull of sandtoexactlyfillthecorrespondingprism.butnowthatshehastakengeometry,chauismore skeptical.shewonders, Justbecauseitworksforthosemanufacturedexamples,doesitworkin everycase? 9. DoanonlinesearchforthefollowingtopicsthatChauandKassandrafoundinterestingand relevanttotheirquestions.bepreparedtoreporttoyourclassmateswhatyoulearned. Keeptrackofparticularlyusefulwebsitese.g.,easytounderstand,greatillustrations, dynamicimages,handszonactivities,etc.)tosharewithyourpeers. a. Cavalieri sprinciple b. Disectingaprismintothreepyramidsofequalvolume c. OthernonZcalculus)proofsoftherelationshipbetweenthevolumesofpyramidsand prisms,cylindersandcones 10. ChauandKassandra splansforthesmallestsandcastleincludescolumnsintheshapeof hexagonalprismswiththebasebeingahexagonthatcouldbeinscribedinacirclewitha radiusof1inch.theheightofthecolumnis12inches. a. Whatisthevolumeofsandrequiredtomakeeachofthesecolumnsinthesmallestsand castle? b. WhatisthevolumeofsandrequiredtomakethiscolumninthemiddleZsizedsand castle? c. Whatisthevolumeofsandrequiredtomakethiscolumninthelargestsandcastle? d. Whatisthecircumferenceofthecirclethatwillcircumscribethebaseofthiscolumnin thelargestsandcastle? 11. Theplansforthesmallestsandcastleincludeaconethatis5incheshighandhasacircular basewitharadiusof2inches. a. Whatisthevolumeofsandrequiredtomakethisconeinthesmallestsandcastle? b. WhatisthevolumeofsandrequiredtomakethisconeinthemiddleZsizedsandcastle? c. Whatisthevolumeofsandrequiredtomakethisconeinthelargestsandcastle? d. Whatisthecircumferenceofthecirclethatformsthebaseofthisconeinthelargest sandcastle? 2013MathematicsVisionProject MVP InpartnershipwiththeUtahStateOfficeofEducation LicensedundertheCreativeCommonsAttribution4NonCommercial4ShareAlike3.0Unportedlicense.

4 7.10SandCastles TeacherNotes ASolidifyUnderstandingTask Purpose:Thepurposeofthistaskistodeepenstudentsunderstandingofthevolumeformulasfor prisms,pyramids,cylindersandcones,particularlywhenappliedtoobliquenonzright)examples ofthesesolids.studentshaveusedtheseformulasinpreviousmathcoursesascomputational tools.inthistasktheyexamineinformal,nonzcalculusargumentssupportingthesevolume formulasinthesamewaytheyexaminedtheformulasforcircumferenceandareaofacirclebased oninformallimitargumentsseetask7.4and7.5).inthistaskstudentsalsoexaminethe proportionalityrelationshipsoflengths,areas,andvolumeswhengeometricfiguresarescaledup. CoreStandardsFocus: G.GMD.1Giveaninformalargumentfortheformulasforthecircumferenceofacircle,areaofa circle,volumeofacylinder,pyramid,andcone.usedissectionarguments,cavalieri sprinciple,and informallimitarguments. G.GMD.3Usevolumeformulasforcylinders,pyramids,cones,andspherestosolveproblems. NoteforMathematicsII:Informalargumentsforareaandvolumeformulascanmakeuseofthe wayinwhichareaandvolumescaleundersimilaritytransformations:whenonefigureintheplane resultsfromanotherbyapplyingasimilaritytransformationwithscalefactork,itsareaisk 2 timesthe areaofthefirst.similarly,volumesofsolidfiguresscalebyk 3 underasimilaritytransformationwith scalefactork. RelatedStandards: LaunchWholeClass): Introducethecontextofthistaskbyreadingthefirstfewparagraphs,includingtheapplicationfor thesandcastlebuildingcompetition.remindstudentsthattheyhavepreviouslyusedformulasfor findingperimetersandareasofregularpolygonsandcircles,andinothermathcourses)formulas forfindingvolumesofprisms,pyramids,cylindersandcones.pointoutquestions10and11where theywillbeaskedtousetheseformulastosolvesomeproblemsrelativetothesandcastlecontext. Pointoutthe scalingup ideathatispresentinthesequestionsandthenreturntotheparagraphs followingtheapplicationandreadthroughbenji scommentandquestion1ofthetask.thenset studentstoworkon1z8.iftheyfinishearly,theyaretoreadthroughquestion9,whichwillbe assignedashomework,andthenworkonquestions10and11. ExploreSmallGroup): Watchforstudentswhointerpret twiceasbig inthesamewaybenjididdoublethearea),aswell asstudentswhointerpretitasdoublingthelineardimensionsofthefigure.allowboth perspectivestocoexistuntilquestion4,whenallstudentsshouldbeworkingfromthe scalefactor 2013MathematicsVisionProject MVP InpartnershipwiththeUtahStateOfficeofEducation LicensedundertheCreativeCommonsAttribution4NonCommercial4ShareAlike3.0Unportedlicense.

5 perspectiveofdoublingallofthelineardimensions.studentsshouldhaveatleasttwodifferent footprints oftheirowndesigntoexperimentwithastheyconsiderthesequestions.watchfor unusualdesignse.g.,nonzrectangular).youmaywanttopromptsomegroupstotrysomething unusual,likeatrianglewithanareaof2.5sq.in.,oranlzshapedfigure,orsomethingelse.listen forargumentsthatindicatewhetherornotstudentsunderstandthatthedesignofthebasedoesn t matterincalculatingtheareaofthescaledupfigure,sincelineardimensionswillallgetdoubledas thefiguregetsscaledup. Havefastfinishersreadthroughtherestofthetaskandworkonproblems10and11. DiscussWholeClass): Thewholeclassdiscussionshouldfocusonthewayareaandvolumescaleundersimilarity transformations:whenonefigureintheplaneresultsfromanotherbyapplyingasimilarity transformationwithscalefactork,itsareaisk 2 timestheareaofthefirst.similarly,volumesof solidfiguresscalebyk 3 underasimilaritytransformationwithscalefactork.emphasizetheideaof scalingup asasimilaritytransformation.usestudentworkonquestions5z7tohighlightthis concept. Asaclass,readthroughthetwostatementsthatprecedequestion8,andthendiscussquestion8. Onebigideathatshouldcomeoutofthisdiscussionisthataslongasthecorresponding slices or crosssectionsparalleltothebaseofaprism,pyramid,cylinderorconehavethesamearea,the resultingsolidhasthesamevolume.asimplisticillustrationofthiswouldbeadeckofcards skewedsoitisnolongerarightprism,orastackofpenniesskewedsoitisnolongeraright cylinderperhapsstackedsoitisn t cylindrical atall).thesearesimpleexamplesofcavalieri s principle,buttheprinciplesaysmore:thecorresponding slices don tneedtobecongruent shapes,wejustneedtoverifythateachslicehasthesamearea.studentsmayencounterexamples ofthisduringtheironlineresearch. Bynow,studentsshouldrecognizeChau sconcernabouttheformulaforthevolumeofapyramid basedonafewexamplesasalegitimateone.thismaynotbefullyresolvedbylookingatsome onlinearguments,butatleaststudentswillhaveacknowledgedthatthisissueisattheheartof mathematicalproof. Assignproblem9,doingsomeonlineresearchaboutthesetopics,ashomework.Therearesome greatvisualresourcesonline.hereareafewyoumightsuggest: %20a%20cube [Thissiteprovidesatemplatefornetsthatcanbefoldedintopyramidssothatthreesuch pyramidsformacube.thenetsareprovidedintwosizes,sotheissueofscalinguparea andvolumeisalsoaccessiblewiththesenets] 2013MathematicsVisionProject MVP InpartnershipwiththeUtahStateOfficeofEducation LicensedundertheCreativeCommonsAttribution4NonCommercial4ShareAlike3.0Unportedlicense.

6 thezvolumezofzazcylinder [Thissiteincludesadynamicanimationofthethreepyramidsthatformacube.Italso illustrateshowaconecanhavethesamevolumeasapyramid thusillustratingcavalieri s principlefornonzcongruentcrosssections] [ThissiteincludesadynamicanimationofCavalieri sprinciple.italsoillustrateshowto decomposeatriangularprismintothreepyramids,andthengeneralizesthevisual proof ofthepyramidformulatogeneralpyramidsbyshowingthatallprismscanbedecomposed intotriangularprisms.] Ifthereistimeremaining,havestudentsworkonproblems10and11.Assignthesequestionsas homeworkifstudentsdon tfinishtheminclass. AlignedReady,Set,Go:Circles:aGeometricPerspective MathematicsVisionProject MVP InpartnershipwiththeUtahStateOfficeofEducation LicensedundertheCreativeCommonsAttribution4NonCommercial4ShareAlike3.0Unportedlicense.

7 Name: Ready,Set,Go Circles:)a)Geometric)Perspective)) 7.10) www.flickr.com/photos/klbm Ready Topic:Findingthecenterofacircle. Previouslyyouhaveworkedwithfindingthecenterorrotationbetweenpre=imageand imagepoints.usingaverysimilarstrategyfindthecenterofthecirclesbelow. Usechordsofthecircletopinpointthecenter.) Justifyyourworkforfindingthecenterofthecirclesabove.Whydoesitwork?Whydoesit pinpointthecenterofthecircle? 2013MATHEMATICSVISIONPROJECT MV P InpartnershipwiththeUtahStateOfficeofEducation LicensedundertheCreativeCommonsAttribution4NonCommercial4ShareAlike3.0Unportedlicense

8 Set Topic:Findingsurfaceareaandvolume. Findthevolumeandsurfaceareaofeachpyramid. 4. Circles:)a)Geometric)Perspective)) 7.10) 5.Apyramidthatissimilartothepyramidin number4butscaledupbyafactorof Apyramidthatissimilartothepyramidin number6butscaledupbyafactorof5. Findthevolumeandsurfaceareaofeachrectangularprisms Aprismsimilartotheoneontheleft thathasbeenenlargedbyafactorof MATHEMATICSVISIONPROJECT MV P InpartnershipwiththeUtahStateOfficeofEducation LicensedundertheCreativeCommonsAttribution4NonCommercial4ShareAlike3.0Unportedlicense

9 Findthevolumeforeachcylinderandcone Radius=5ft Height=15ft Circles:)a)Geometric)Perspective)) 7.10) 11.Acylinderthatissimilartotheoneat theleftthathasbeenscaledupbyafactor of2. 13.Aconethatissimilartotheoneatthe leftthathasbeenscaledupbyafactorof4. 57 Go Topic:Radiansanddegreeconversions,sectorsofcircles Findthemeasurethatismissing,eitherdegreesorradiansgiventheothermeasure =Radians =Radians =Radians 17. Radians=Degrees Radians=Degrees 19. Radians=Degrees =Radians =Radians =Radians Findtheareaofeachsector Findthemeasureofthelengthofeacharcindicatedbelow MATHEMATICSVISIONPROJECT MV P InpartnershipwiththeUtahStateOfficeofEducation LicensedundertheCreativeCommonsAttribution4NonCommercial4ShareAlike3.0Unportedlicense

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