brownie pan - gmu complete mathcompletemath.onmason.com/files/2016/10/brownie-pan.pdfbrownie pan...

GeorgeMasonUniversityCOMPLETEMath©2015

BrowniePan

Designedby:JonathanThompson

GeorgeMasonUniversity,COMPLETEMath

TheTaskMr.BrownE.Panrecentlyopenedanewbusinessmakingbrowniescalled“TheBrownE.Pan.”Onhisfirstdaybaking,hestartedinhisownkitchenbyusingasinglesquarepan.Hejustfinishedhisfirstbatchofbrownies,andhewantstomakesurethatallthebrowniesarethesamesize.Hefirstcutthebrowniesverticallyasshowninthepicturebelow.

Thepanofbrowniesaboveiscutinto12congruentrectangles.Iftheperimeterofeachoftherectanglesis65cm.,whatistheareaofthepan?

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-Thestudentwill

StandardsofLearningforGrades6-7-8• 6.9–Thestudentwillmakeballpark

comparisonsbetweenmeasurementsintheU.S.Customarysystemandthemetricsystem.

• 6.10–Thestudentwillo Solvepracticalproblemsinvolving

areaandperimetero Describeandderminethevolumeofa

rectangularprism• 7.5–Thestudentwill

o Describevolumeofcylinderso Solvepracticalproblemsinvolvingthe

volumeofrectangularprismsand


o estimateandmeasurelengthanddescribetheresultinbothMetricandU.S.Customary

o Identifyequivalentmeasurmentsbetweenunits

• 5.8–Thestudentwillo Findperimeter,area,andvolumein

standardunitsofmeasureo Differentiatebetweenperimeter,

area,andvolumeo Identifyequivalentmeasurments

withinthemetricsystemo Chooseanappropriateunitof

measureforagivensituation

cylinderso Describehowchangingonemeasured

attributeofarectangularprismaffectsitsvolume

• 8.7–Thestudentwillo Investigateandsolvepractical

problemsinvolvingvolumeofprismsandcylinders

o Describehowchangingonemeasuredattributeofafigureaffectsthevolumeandsurfacearea

ProcessGoals• ProblemSolvingandReasoning–Studentswillapplytheirknowledgeofareaandperimetertofindthe

areaofapanofbrowniesthatisbrokendowninto12congruentrectangles,afterbeingcutvertically,giventheperimeterofoneofthese12rectangles.

• ConnectionsandRepresentations–Studentswillrelatetheperimeterofthesmallerrectanglestothedimensionsofthelargersquare.Thestudentswilldothisbyusingreasoningskillsandbyshowingtheirthinkinginwordsandinpictures.

• Communication–Studentswillusemathematicallanguagetojustifytheirfindingsnddiscusssolutionpathwayswiththeirpeers.

RelatedTask–BrownieBatterMr.BrownE.Panrecentlyopenedanewbusinessmakingbrowniescalled“TheBrownE.Pan.”Onhisfirstdaybaking,hestartedinhisownkitchenbyusingasinglerectangularpan.Hehasalreadyprepared10mixingbowlsfullofbrowniebatter.Usepictures,words,tables,graphs,and/orsymbolstofigureoutaplanshowinghowmanybrownieshecanmake.

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.


BrowniePanLessonPlan

Designedby:

JonathanThompsonGeorgeMasonUniversity,COMPLETEMath



Materials• Tasksheetforeachstudent• Tasktoprojectonboard• Stripsofrectangularpapertovisualize

pan• Graphpaper• Rulers• Presentationpaperforeachgroup

FacilitatingTask• Classwillbedistributedtheproblemsheet

whichwillbereadtogetherandclarifyingquestionswillbeanswered.

• Studentwillbesplitintogroupsof3-4students.

• Beforegroupsworktogether,eachstudentwillbegiven5-7minutesofindependenttimetobeginworkingonideasoftheirown.

• Independentworkwilltransitiontosmallgroupwork(withinthe3-4persongroups)

• Groupswillbedistributedmaterialswhenrequested

• Groupspresentfindingsbaseduponstrategieschoseninorderbyteacher.

Misconceptions• Misinterpretationofthepicture

mistakenlythinkingitisdrawnexactlytoscale

• Understandingthatthepanisasquare

SuggestedPromptsorQuestionsQuestionstoEngageStudentsThatareStuck

• Couldyoudraworlabeltheinformationthatyouaregiven?Howcouldyouvisualizewhattheproblemissaying?

• Doyouthinkthispictureistheexact


• Confusionbetweenpropertiesofasquarevs.propertiesofarectangle

• Using65centimetersastheareaofeachrectangleratherthantheperimeter

• Misunderstandingofthequestionthatisbeingasked.Findingonlythelengthofeachsideofthesquare,orfindingtheperimeterofthesquareinsteadofthearea.

samesizeastheactualpan?• Whatinformationareweattemptingto

find?• Whatdoyouknowaboutsquaresor

rectangles?• Whatwouldhappenwhenthewidthsof

therectanglesareputtogether?• Wouldanyrectangleworkasthepan

length,ordoesithavetobeasquare?• Howdoesasquaredifferfroma

rectangle?


BrowniePan Name Date Mr.BrownE.Panrecentlyopenedanewbusinessmakingbrowniescalled“TheBrownE.Pan.”Onhisfirstdaybaking,hestartedinhisownkitchenbyusingasinglesquarepan.Hejustfinishedhisfirstbatchofbrownies,andhewantstomakesurethatallthebrowniesarethesamesize.Hefirstcutthebrowniesverticallyasshowninthepicturebelow.



BrowniePanAnticipationGuide

Designedby:


TheTask

Mr.BrownE.Panrecentlyopenedanewbusinessmakingbrowniescalled“TheBrownE.Pan.”Onhisfirstdaybaking,hestartedinhisownkitchenbyusingasinglesquarepan.Hejustfinishedhisfirstbatchofbrownies,andhewantstomakesurethatallthebrowniesarethesamesize.Hefirstcutthebrowniesverticallyasshowninthepicturebelow.



AnticipatedStrategy#1

Description:StudentsmayfindasolutionbyusingaT-chart.Theycoulddothisbyfirstfiguringoutthatthesumofthelengthandwidthofeachrectanglemustbehalfoftheperimeter,or32.5centimeters.Uponfindingthis,theycouldmakeachartwithmanydifferentpossiblecombinationsofvaluesthatcouldaddtogethertomake32.5centimeters.Theycouldfindsomecombinationsinthechart,untiltheycomeuponthatworksintheproblem.Thelastcolumninthechartwouldbethechecktoseeifitworks.Sincethereare12congruentrectangles,thesumofthewidthsofthe12rectanglesmustbeequivalenttothelengthoftherectanglebecausethefigureisstatedasasquare.Thestudentswouldneedtocontinuetodothisuntilthelengthmatchestheproductofthewidthand12.Thisworksfor30centimeters,sotheareaoftheentiresquareis900squarecentimeters.



Description:Thestudentsmaysolvetheproblembyusingapureguessandcheckstrategy.Sincethestudentsknowthateachrectanglehasaperimeterof65centimeters,thisistheinformationthattheymuststartwith.Upondecidingthis,theywillmostlikelydecidethattheyneedtoknowthedimensionsoftheserectangles,buttheywilldeterminethattherearedifferentpossiblecombinations.Thestudentscouldfinddifferentcombinationsuntiltheyfindonethatmakesasquare.Sincethe12widthsofthethinrectanglesareequivalenttooneofthelengthsoftherectangles,theycouldcontinuethispatternuntilmultiplyingthewidthby12isthesameasthelength.Thiswouldprovethedimensionswouldmakeasquare,whichisstatedintheproblem.Oncethestudentfindsthis,theycanmultiplythelengthbythewidth,whichshouldbethesamenumber,tofindthearea.Inthiscase,theareais900squarecentimeters.



Description:Studentsthatarefamiliarwithvariablesorsettingupequationsmaylookatthisprobleminacompletelydifferentway.Firstofall,theyknowthattheshapeisasquare,sotheymaylookatthetwosidesofthesquareasbeingequivalent.Theycoulddothisbyassigningavariable(w)tothewidthofthethinrectanglesandthevariable(y)forthelengthoftheserectangles.Theycouldthenconcludethat12x=y,sincethereare12ofthesesmallerrectangles.Theymightthenfigureoutthat2x+2y=65,since65centimeterswasgivenastheperimeterforeachofthesesmallerrectangles.Byhavingthesetwoequations,thestudenthascreatedasystemthatcouldbesolvedinmultipleways.Onewayisbyusingelimination,whichisshownabove.Oncethestudenthasfiguredoutthey-value,theycouldmultiplythisnumberbyitselfbecausetheshapewasidentifiedasasquare.Thiswouldgiveanareaof900squarecentimeters.


BrowniePanStudentwork

Designedby:


Fall2016




Studentwork1

TeacherNotes:Thisstudentlookstohavestartedoutwiththenumber65centimeters,sincethisnumberwasgivenastheperimeterofeachofthe12rectanglesontheinsideofthesquare.Thestudentseemstohavestartedbymultiplyingthisnumberby12,mistakingitastheareaofeachoftheserectangles.Afterdoingthis,thestudentdrewoneofthesmallerrectanglesontheinside,whichisdrawnontherightofthepage,andthestudentlabeledthesidestomaketheperimeterof65centimeters.Thestudentcheckedtomakesurethiswastruebymultiplying2.5by12toprovethewidthx12wasequivalenttothelength.Thisprovedtheshapewasasquarewiththosedimensions.Thestudentfinishedbymultiplying30by30togetasolutionof900squarecentimeters.

Studentwork2

TeacherNotes:Thispieceofstudentworkillustrateswhatisprobablythemostcommonmisconceptionthatoccurswhenstudentsworkthroughthisproblem.Thestudentstartedoffbyhighlightingimportantinformation,markingthingslikesquarepan,congruentrectangles,perimeterof65centimeters,andthequestionbeingasked.Afterdoingthis,thestudentmistakenlylabeledeverysinglerectangleinsidethesquareas65centimeters.Whiletheperimeterofeachoftheserectanglesis65centimeters,thisstudentlabeledthisvalueastheareaofeachoftheserectangles.Oncethestudentmistakesthispieceofinformationforthearea,theproblemseemsextremelyeasy.Ifyouknowtheareaofonerectangle,thenallyoumustdoismultiplythisvalueby12(sincethereare12equivalentrectangles)togetthetotalareaofthewholeshape.Eventhoughthisstudenthighlightedthewordperimeter,thesetupoftheproblemandthisonemistakeledtoacompletemisconception.


Studentwork3

TeacherNotes:Thisstudentstartedoutthisproblemwiththemainpieceofinformationgiven,theperimeterofthesmallerrectanglesthatfillthelargersquare,whichisgivenas65centimeters.Oncethestudentsawthisnumberalongwiththeperimeter,thestudentchosetodividethisnumberby4.Thisisastrategythatmakessomesensebecauseeachoftheserectangleshave4sides,sodividingby4isalogicalwaytogetthevalueofeachside.Thiswouldworkinthecaseoftherectanglebeingasquare,butinthispicture,theserectanglesareclearlynot.Oncedoingthis,thestudentlabeledeachsideofthelargersquareas16.35,whichwasintendedtobe16.25if65wascorrectlydividedby4.Oncethestudenthadmistakenlylabeledthesquarewiththesesidelengthsaftermisinterpretingtheinformation,thestudentmultipliedthelengthbythewidth(thesevalueswerethesame)togetthefinalsolutiontotheproblem.Thisisacommonmistakeinthisproblem,butitisonethatcanbepreventedwhencloselyreadingtheinformationgiven.

Sequencing• Atleastfourgroupswillpresenttheirfindingsfordeterminingtheareaofthesquare

bakingpan.o Thefirstgroupwillshowastrategywheretheyvisualizedtheproblemusing

eitherrectangularpaperstripsorgraphpaper.Theymightnothaveactuallyfoundthelengthandwidthoftherectangles,buttheywilldiscussthemeaningofthegiveninformation.Theywilldiscusshowtwonumbersneedtoaddtogettheperimeterofeachofthesquares.Theywilldiscusshowthesidesoftherectanglescomparetothelargersquare.

o Thesecondgroupwilldisplayaguess-and-checkmethod.Theywilldiscusshowtheylookedattheperimeteroftherectanglesandguessedandcheckedtheirworktomakesuretheperimeteralsoworkedtomakealargersquare.Theywillshowhowtheycouldprovethatthiscouldbetheonlypossiblesetofnumberstowork.

o Thethirdgroupwillbeonethatworksbackwardsthroughtheproblem.Theywillstartwiththesquareandsetthesidelengthsofthesquareasagivennumberandtoseeifthisworksfortheirproblem.Theywillshowhowyoucouldalsoworkbackwardstofindthesolution.

o Thefourthgroupwillshowalittlemoresophisticatedstrategyifoneexists,suchasusingvariablestostandfortheunknownsintheproblemoradifferentcreativestrategy.Theywilldiscusshowtheirstrategyrelatestotheothers.

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