Brownie Batter - onMason

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


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?


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?


BrownieBatter Name Date Mr.BrownE.Panrecentlyopenedanewbusinessmakingbrowniescalled“TheBrownE.Pan.”Onhisfirstdaybaking,hestartedinhisownkitchenbyusingasinglerectangularpan.Hehasalreadyprepared10mixingbowlsfullofbrowniebatter.



BrownieBatterAnticipationGuide

Designedby:


TheTask



AnticipatedStrategy#1

Description:Studentsmightsolvetheproblembyfirststartingoffwithsomepriorknowledgeofthesizeofamixingbowl.Baseduponpreviousbakingormeasuringexperience,theymayknowarelativeestimatedsizeofhowbigthebowlcouldbe.Thiscouldalsocomefrommixingbowlsthataremadeavailabletothestudents.Bymultiplyingthisnumberby10,theywouldhavethetotalamountofbatter.Aftertheyhavethis,theycouldestimateormeasurethesizeofthepanandconverttofigureouthowmuchbattertheywouldneedforonepan.Tofinish,theywoulddividethetotalamountofbatterbytheamountneededperpantogetthetotalnumberofpansthatcouldbebaked.Fromthispoint,theywouldonlyneedtofigureouthowmanybrowniesareineachpantogetatotalnumber.



Description:Studentsmayrelatethemixingbowlstothepansina1to1ratio.Throughpreviousbakingexperience,theyaremostlikelyusedtousingamixingbowltomakeonlyonepanofbrownies.Duetothis,theymightthinkthateachmixingbowlisrestrictedtocreatingonepan.Ifthisistrue,then10batchesofbrowniesarepossible.Thestudentwouldthenhavetofigureouthowmanybrowniescouldfitintoonesinglepan.Thiscouldbeanestimatebaseduponapicture,ortheycouldactuallyusearealpanandmeasuretogetamoreaccurateestimate.Uponknowingthis,theycouldmultiplytheamountofbrowniesineachpanbythetotalnumberofpanstogetthetotalnumberofbrownies.



Description:Studentsmightstartoffbythinkingaboutthesizeofonesinglebrownie.Thiscouldbedrawnonapieceofpaperorcutouttoplaceinsideapantomakesureofthesize.Oncethissizeisdetermined,theycouldfindthevolumeofeachbrowniethatwouldbebaked.Afterdoingthis,theycouldthinkaboutthevolumeofapan.Theycoulddothisbyestimating,byusingpriorknowledge,ortheycouldlookatthepanashalfofasphereandfindthevolume.Thisnumberwouldneedtobemultipliedby10toaccommodateeachmixingbowl.Thetotalvolumeofall10mixingbowlswouldbedividedbythevolumeofeachbrownietoarriveatatotalnumberofbrownies.Thisanswercouldvarywidelybaseduponhowbigthestudentsthinkabrowniecouldbe.


BrownieBatterStudentwork

Designedby:


TheTask


Usepictures,words,tables,graphs,and/orsymbolstofigureoutaplanshowinghowmanybrownieshecanmake.Studentwork1

TeacherNotes:Thisstudentstartedoffwithaknownpieceofinformationthatamixingbowlwouldbeabletohold3quartsofbatter.Knowingthatthereare10mixingbowlsintheproblemscenario,thestudentmultiplied10by3toarriveattheconclusionthattheremustbeatotalof30quartsofbatterintotal.Thestudentestimatedthatonequartwouldmakeapan,sotheremustbe30pans.Afterfiguringthisout,thestudentwrotedownthatthepansizeforbakingthebrownieswouldbe9x13.Thiswasmostlikelybaseduponpriorknowledgeorfrommeasuringthepanprovided.Ithinkthestudenthadtroublerelatingthepansizetoaspecificnumberofbrowniesbecausethestudentwrotethateachbrowniewouldbe2inchesx2incheswithoutreallyshowingareason.Thestudentthenwrotethatonly4ofthesebrownieswouldfitintoeachpan.Thestudentmultiplied30pansby4browniesineachpantogetatotalof120brownies.


Studentwork2

TeacherNotes:Thisstudentstartedoffbyfindingthesizeofthepan,mostlikelybymeasuringorbyusingpriorknowledge,tobe13inchesx9inchesx2inches.Thestudentfoundthateachpanhasatotalvolumeof234cubicinches.Afterfiguringthisout,thestudentthentriedtofigureouthowmuchbatterwouldfitintoeachmixingbowl.Todothis,thestudentusedthevolumeofasphereanddivideditby2togetamixingbowl.Thestudentestimatedtheradiusofthebowltobe5inches,whichgaveeachbowlatotalvolumeof2612/3cubicinches.Thestudentmultipliedthisnumberby10toaccountforall10bowlsandcameupwithatotalbatteramountof2620cubicinches.Thestudenttookthetotalamountofbatteranddividedbytheamountofbatterineachpantofigureoutthattherewouldbeabout11totalpans.Bydeterminingthesizeofeachbrownietobe3inchesx3inchesx2inches,thestudentfiguredoutthat12brownieswouldfitintoeachpanforatotalof132brownies.Thisstudentdidalsoindicateinhis/herworkthattherewouldbesomebrowniebatterthatwouldbeleftoverorpossiblywasted.


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


butstillcomesupwithananswerthatgivesusabenchmarktostartthediscussion.

o Thesecondgroupwillbeonethatusedthetoolsgiven:themeasuringcupsandthepans.Theywilldiscusshowtheydeterminedhowmuchwouldfitintoasinglepan,andtheywilldiscusshowthishelpedthemtofigureouthowmanytotalpanscouldbemade.

o Thethirdgroupwillmakeaconnectiontolinearvolumeofarectangularprism.Theywilldiscusshowtheycanusearulertomeasurethevolumeofbatterthatwouldbeneeded.Theywillthendiscusshowthiswouldcomparetocapacitywithstandardandmetricunits.Thisgroupwilldiscusshowtheyusedthisideatodeterminehowmanypanscouldbefilled.

o Afourthgroupwillbegiventimetodiscussastrategythatisuniqueanddifferentfrommostoftheothergroups.Timeshouldbegiventoagroupthatsolvedtheprobleminawaythatusesthestrategiessimilartothoseabove,butatthesametime,usesauniqueidea.Thisgroupwillbeaskedquestionstocomparehowtheirideasaresimilartotheothergroups.

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Brownie Batter - onMason