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GeorgeMasonUniversityCOMPLETEMath©2015
BrownieBatter
Designedby:JonathanThompson
GeorgeMasonUniversity,COMPLETEMath
TheTask
Mr.BrownE.Panrecentlyopenedanewbusinessmakingbrowniescalled“TheBrownE.Pan.”Onhisfirstdaybaking,hestartedinhisownkitchenbyusingasinglerectangularpan.Hehasalreadyprepared10mixingbowlsfullofbrowniebatter.Usepictures,words,tables,graphs,and/orsymbolstofigureoutaplanshowinghowmanybrownieshecanmake.
BigIdeas
• Usingperimeter,area,andvolume• Findingrelationshipsbetweenperimeter,area,andvolume• Measurement
StandardsofLearningforGrades3-4-5• 3.9–ThestudentwillestimateanduseU.S.
CustomaryandMetricUnitstomeasure:o Lengtho LiquidVolumeo Weight/Masso AreaandPerimeter
• 4.6-Thestudentwillo Estimateandmeasureweight/mass
anddescribetheresultsinU.S.CustomaryandMetricUnits
o Identifyequivalentmeasurmentsbetweenunits
• 4.7-Thestudentwillo estimateandmeasurelengthand
describetheresultinbothMetricandU.S.Customary
o Identifyequivalentmeasurmentsbetweenunits
• 5.8–Thestudentwillo Findperimeter,area,andvolumein
standardunitsofmeasureo Differentiatebetweenperimeter,
area,andvolumeo Identifyequivalentmeasurments
withinthemetricsystemo Chooseanappropriateunitof
measureforagivensituation
StandardsofLearningforGrades6-7-8• 6.9–Thestudentwillmakeballpark
comparisonsbetweenmeasurementsintheU.S.Customarysystemandthemetricsystem.
• 6.10–Thestudentwillo Solvepracticalproblemsinvolving
areaandperimetero Describeandderminethevolumeofa
rectangularprism• 7.5–Thestudentwill
o Describevolumeofcylinderso Solvepracticalproblemsinvolvingthe
volumeofrectangularprismsandcylinders
o Describehowchangingonemeasuredattributeofarectangularprismaffectsitsvolume
• 8.7–Thestudentwillo Investigateandsolvepractical
problemsinvolvingvolumeofprismsandcylinders
o Describehowchangingonemeasuredattributeofafigureaffectsthevolumeandsurfacearea
GeorgeMasonUniversityCOMPLETEMath©2015
ProcessGoals• ProblemSolvingandReasoning–Studentswillusemathematicalmodelingtoreasonthroughhow
manypansofbrownies(andanactualnumberofbrownies)couldbemadegiven10mixingbowlsofbrowniebatter.
• ConnectionsandRepresentations–Studentswillusevarioustoolstogatherinformationonhowmanypansofbrowniecouldbemadefromagallonofbrowniebatter,andtheywillgeneralizetheirthinkingtofigureouthowmanybrowniescouldbemadefrom10mixingbowlsofbrowniebatter.Theywillrepresenttheirthinkinginpictures,tables,andintheactualmodelingitself.
• Communication–Studentswillusemathematicallanguagetojustifytheirfindingsnddiscusssolutionpathwayswiththeirpeers.
RelatedTask–BiteSizeBrownies
Mr.BrownE.Panrecentlyopenedanewbusinessmakingbrowniescalled“TheBrownE.Pan.”Onhisfirstdaybaking,hestartedinhisownkitchenbyusingarectangularpan.
1. Drawapicturetoshowhowmanybrownieswouldfillthepan. 2. Sincehewantedtomakethemostofhistime,hewondered,“WhatifImakethebrowniessmaller
andaddanotherrowofbrownies?"Howmanybrownieswouldnowfitintothepan?Whatifheaddedanotherrow?
3. Continuethepatterntofindthexstage.Findawaytorecordyourresults.4. Ifthepanheld120brownies,howmanytimesdidMr.BrownE.Panaddanewrow?5. Ifhecontinuesthispattern,woulditbepossibleforapantohold500brownies?Explainhowyou
know.RelatedTask–BrowniePan
Mr.BrownE.Panrecentlyopenedanewbusinessmakingbrowniescalled“TheBrownE.Pan.”Onhisfirstdaybaking,hestartedinhisownkitchenbyusingasinglesquarepan.Hejustfinishedhisfirstbatchofbrownies,andhewantstomakesurethatallthebrowniesarethesamesize.Hefirstcutthebrowniesverticallyasshowninthepicturebelow.
Thepanofbrowniesaboveiscutinto12congruentrectangles.Iftheperimeterofeachoftherectanglesis65cm.,whatistheareaofthepan?
GeorgeMasonUniversityCOMPLETEMath©2015
BrownieBatterLessonPlan
Designedby:
JonathanThompsonGeorgeMasonUniversity,COMPLETEMath
TheTask
Mr.BrownE.Panrecentlyopenedanewbusinessmakingbrowniescalled“TheBrownE.Pan.”Onhisfirstdaybaking,hestartedinhisownkitchenbyusingasinglerectangularpan.Hehasalreadyprepared10mixingbowlsfullofbrowniebatter.
Usepictures,words,tables,graphs,and/orsymbolstofigureoutaplanshowinghowmanybrownieshecanmake.
Materials• Tasksheetforeachstudent• Tasktoprojectonboard• Differentsizerectangularbakingpans• Differentsizemixingbowls• GraduatedCylinders• Water(usedasbrowniebatter
substitute)• Graphpaper• Rulers• Presentationpaperforeachgroup
FacilitatingTask• Classwillbedistributedtheproblemsheet
whichwillbereadtogetherandclarifyingquestionswillbeanswered.
• Studentwillbesplitintogroupsof3-4students.
• Beforegroupsworktogether,eachstudentwillbegiven5-7minutesofindependenttimetobeginworkingonideasoftheirown.
• Independentworkwilltransitiontosmallgroupwork(withinthe3-4persongroups)
• Groupswillbedistributedmaterialswhenrequested
• Groupspresentfindingsbaseduponstrategieschoseninorderbyteacher.
Misconceptions• Thereisnotonesolutiontothe
problem• Determiningtherelationshipsbetween
differenttypeofcapacitymeasurementsbothstandardandmetric
• Asinglemixingbowldoesnotnecessarilyhavetomakeawholenumberofpansofbrownies
• Browniesshouldbethesamesize• Relationshipbetweenlinearvolume(lx
wxh)tocapacitymeasurements.
SuggestedPromptsorQuestions• Howcouldyouusethetoolsthatare
availabletoyou?• Howcouldyoufigureoutthevolume
thatwouldbeneededforapanofbrownies?Couldyoumeasurewitharuler?Howcouldyouusethis?
• Howcouldyousplitupasinglemixingbowl?Howmanypansofbrownieswouldthatmake?
• Howbigshouldthebrowniesbe?Whywouldthismatter?Woulditmatteriftheyarethesamesizeornot?
• Howcanwemeasurethepantodeterminethevolumeofbatteritwouldneed?Ifwegetavolumeincubicunits,howdoesthiscomparetogallonsorothercapacitymeasurements?
GeorgeMasonUniversityCOMPLETEMath©2015
BrownieBatter Name Date Mr.BrownE.Panrecentlyopenedanewbusinessmakingbrowniescalled“TheBrownE.Pan.”Onhisfirstdaybaking,hestartedinhisownkitchenbyusingasinglerectangularpan.Hehasalreadyprepared10mixingbowlsfullofbrowniebatter.
Usepictures,words,tables,graphs,and/orsymbolstofigureoutaplanshowinghowmanybrownieshecanmake.
GeorgeMasonUniversityCOMPLETEMath©2016
BrownieBatterAnticipationGuide
Designedby:
JonathanThompsonGeorgeMasonUniversity,COMPLETEMath
TheTask
Mr.BrownE.Panrecentlyopenedanewbusinessmakingbrowniescalled“TheBrownE.Pan.”Onhisfirstdaybaking,hestartedinhisownkitchenbyusingasinglerectangularpan.Hehasalreadyprepared10mixingbowlsfullofbrowniebatter.
Usepictures,words,tables,graphs,and/orsymbolstofigureoutaplanshowinghowmanybrownieshecanmake.
AnticipatedStrategy#1
Description:Studentsmightsolvetheproblembyfirststartingoffwithsomepriorknowledgeofthesizeofamixingbowl.Baseduponpreviousbakingormeasuringexperience,theymayknowarelativeestimatedsizeofhowbigthebowlcouldbe.Thiscouldalsocomefrommixingbowlsthataremadeavailabletothestudents.Bymultiplyingthisnumberby10,theywouldhavethetotalamountofbatter.Aftertheyhavethis,theycouldestimateormeasurethesizeofthepanandconverttofigureouthowmuchbattertheywouldneedforonepan.Tofinish,theywoulddividethetotalamountofbatterbytheamountneededperpantogetthetotalnumberofpansthatcouldbebaked.Fromthispoint,theywouldonlyneedtofigureouthowmanybrowniesareineachpantogetatotalnumber.
GeorgeMasonUniversityCOMPLETEMath©2016
AnticipatedStrategy#2
Description:Studentsmayrelatethemixingbowlstothepansina1to1ratio.Throughpreviousbakingexperience,theyaremostlikelyusedtousingamixingbowltomakeonlyonepanofbrownies.Duetothis,theymightthinkthateachmixingbowlisrestrictedtocreatingonepan.Ifthisistrue,then10batchesofbrowniesarepossible.Thestudentwouldthenhavetofigureouthowmanybrowniescouldfitintoonesinglepan.Thiscouldbeanestimatebaseduponapicture,ortheycouldactuallyusearealpanandmeasuretogetamoreaccurateestimate.Uponknowingthis,theycouldmultiplytheamountofbrowniesineachpanbythetotalnumberofpanstogetthetotalnumberofbrownies.
GeorgeMasonUniversityCOMPLETEMath©2016
AnticipatedStrategy#3
Description:Studentsmightstartoffbythinkingaboutthesizeofonesinglebrownie.Thiscouldbedrawnonapieceofpaperorcutouttoplaceinsideapantomakesureofthesize.Oncethissizeisdetermined,theycouldfindthevolumeofeachbrowniethatwouldbebaked.Afterdoingthis,theycouldthinkaboutthevolumeofapan.Theycoulddothisbyestimating,byusingpriorknowledge,ortheycouldlookatthepanashalfofasphereandfindthevolume.Thisnumberwouldneedtobemultipliedby10toaccommodateeachmixingbowl.Thetotalvolumeofall10mixingbowlswouldbedividedbythevolumeofeachbrownietoarriveatatotalnumberofbrownies.Thisanswercouldvarywidelybaseduponhowbigthestudentsthinkabrowniecouldbe.
GeorgeMasonUniversityCOMPLETEMath©2016
BrownieBatterStudentwork
Designedby:
JonathanThompsonGeorgeMasonUniversity,COMPLETEMath
TheTask
Mr.BrownE.Panrecentlyopenedanewbusinessmakingbrowniescalled“TheBrownE.Pan.”Onhisfirstdaybaking,hestartedinhisownkitchenbyusingasinglerectangularpan.Hehasalreadyprepared10mixingbowlsfullofbrowniebatter.
Usepictures,words,tables,graphs,and/orsymbolstofigureoutaplanshowinghowmanybrownieshecanmake.Studentwork1
TeacherNotes:Thisstudentstartedoffwithaknownpieceofinformationthatamixingbowlwouldbeabletohold3quartsofbatter.Knowingthatthereare10mixingbowlsintheproblemscenario,thestudentmultiplied10by3toarriveattheconclusionthattheremustbeatotalof30quartsofbatterintotal.Thestudentestimatedthatonequartwouldmakeapan,sotheremustbe30pans.Afterfiguringthisout,thestudentwrotedownthatthepansizeforbakingthebrownieswouldbe9x13.Thiswasmostlikelybaseduponpriorknowledgeorfrommeasuringthepanprovided.Ithinkthestudenthadtroublerelatingthepansizetoaspecificnumberofbrowniesbecausethestudentwrotethateachbrowniewouldbe2inchesx2incheswithoutreallyshowingareason.Thestudentthenwrotethatonly4ofthesebrownieswouldfitintoeachpan.Thestudentmultiplied30pansby4browniesineachpantogetatotalof120brownies.
GeorgeMasonUniversityCOMPLETEMath©2016
Studentwork2
TeacherNotes:Thisstudentstartedoffbyfindingthesizeofthepan,mostlikelybymeasuringorbyusingpriorknowledge,tobe13inchesx9inchesx2inches.Thestudentfoundthateachpanhasatotalvolumeof234cubicinches.Afterfiguringthisout,thestudentthentriedtofigureouthowmuchbatterwouldfitintoeachmixingbowl.Todothis,thestudentusedthevolumeofasphereanddivideditby2togetamixingbowl.Thestudentestimatedtheradiusofthebowltobe5inches,whichgaveeachbowlatotalvolumeof2612/3cubicinches.Thestudentmultipliedthisnumberby10toaccountforall10bowlsandcameupwithatotalbatteramountof2620cubicinches.Thestudenttookthetotalamountofbatteranddividedbytheamountofbatterineachpantofigureoutthattherewouldbeabout11totalpans.Bydeterminingthesizeofeachbrownietobe3inchesx3inchesx2inches,thestudentfiguredoutthat12brownieswouldfitintoeachpanforatotalof132brownies.Thisstudentdidalsoindicateinhis/herworkthattherewouldbesomebrowniebatterthatwouldbeleftoverorpossiblywasted.
GeorgeMasonUniversityCOMPLETEMath©2016
Studentwork3
TeacherNotes:Thisstudentstartedoffbydrawingeachofthe10mixingbowls.Althoughitdoesn’tseemtohaveinfluencedthesolution,itisinterestingtonotethatthestudentdreweachoftheseascylinders.Thestudentindicatedonhis/herpaperthateachmixingbowlhasenoughbattertomakeonepanofbrownies.Thestudentthendrewanillustrationofthepotentialbakingpantoshowthat18brownieswouldfitineverypanthatisbaked.Thisdoesn’tseemtobebaseduponanymeasurementsshownonthepaper,soitismostlikelyanestimate.Oncethiswasfiguredout,thestudentonlyhadtomultiplythenumberofbrowniesineachpan(18)bythetotalnumberofbrowniepansbaked(10)togetatotalnumberofbrownies(180).Thissolutionwasfiguredoutwithoutusinganyspecificmeasurements.
Sequencing• Theactivityfocusesontwomainpieces:Howmanybrowniesshouldfitinonepan,and
howmanypanscanbemadewith10mixingbowlsofbrowniebatter.o Thefirstportionofstudentresponseswillfocusondetermininghowmany
browniesshouldbeineachpan.o Thesecondportionwouldfocusonhowwecandeterminehowmanypanscanbe
madewith10mixingbowlsofbrowniebatter.• Threegroupswouldpresenttheirunderstandingsofhowmanybrownieswouldfitina
singlepan.o Onegroupwillshowadrawingordiagramdrawntoshowhowthebrownies
wouldfittogetherinapan.Thegroupwilltalkabouthowtheydeterminedthatthisamountofbrownieswouldmakesensebecauseofthesizeofthebrownies.
o Asecondgroupwillshowastrategywheretheywillonlydeterminethelengthandwidthofbrowniesinthepantofindtheareaofhowmanywouldfitinasinglepan.Theywillalsodiscusshowsizewouldaffecthowtheymadetheirdecision.
o Athirdgroupwillbechosenthatdidsomethingdifferentfromthefirsttwogroups.Thiswillbeagroupthatchosetomakethebrowniesmuchsmallerthantheothers.Wewilldiscusshowthisisdifferentfromtheotherstudents,andwewilldiscusshowthischangesthesituation.
• Atleastfourgroupswillpresenttheirfindingsfordetermininghowmanypanscouldbefilledgiventhe35gallonsofbrowniebatter.
o Thefirstgroupwillbeonethatsolvedtheprobleminthemostgenericway.Theywillbeagroupthatusedmoreofaestimationapproachtofigureouthowmuchvolumewouldfillasinglepanaswellashowmanypansofbrowniescouldbemadewithanentiregallon.Theywillbeagroupthatusesthetoolsonlysparingly
GeorgeMasonUniversityCOMPLETEMath©2016
butstillcomesupwithananswerthatgivesusabenchmarktostartthediscussion.
o Thesecondgroupwillbeonethatusedthetoolsgiven:themeasuringcupsandthepans.Theywilldiscusshowtheydeterminedhowmuchwouldfitintoasinglepan,andtheywilldiscusshowthishelpedthemtofigureouthowmanytotalpanscouldbemade.
o Thethirdgroupwillmakeaconnectiontolinearvolumeofarectangularprism.Theywilldiscusshowtheycanusearulertomeasurethevolumeofbatterthatwouldbeneeded.Theywillthendiscusshowthiswouldcomparetocapacitywithstandardandmetricunits.Thisgroupwilldiscusshowtheyusedthisideatodeterminehowmanypanscouldbefilled.
o Afourthgroupwillbegiventimetodiscussastrategythatisuniqueanddifferentfrommostoftheothergroups.Timeshouldbegiventoagroupthatsolvedtheprobleminawaythatusesthestrategiessimilartothoseabove,butatthesametime,usesauniqueidea.Thisgroupwillbeaskedquestionstocomparehowtheirideasaresimilartotheothergroups.
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