Upload
dinhtuong
View
213
Download
0
Embed Size (px)
Citation preview
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
GeorgeMasonUniversityCOMPLETEMath©2015
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.
GeorgeMasonUniversityCOMPLETEMath©2015
BrowniePanLessonPlan
Designedby:
JonathanThompsonGeorgeMasonUniversity,COMPLETEMath
TheTaskMr.BrownE.Panrecentlyopenedanewbusinessmakingbrowniescalled“TheBrownE.Pan.”Onhisfirstdaybaking,hestartedinhisownkitchenbyusingasinglesquarepan.Hejustfinishedhisfirstbatchofbrownies,andhewantstomakesurethatallthebrowniesarethesamesize.Hefirstcutthebrowniesverticallyasshowninthepicturebelow.
Thepanofbrowniesaboveiscutinto12congruentrectangles.Iftheperimeterofeachoftherectanglesis65cm.,whatistheareaofthepan?
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
GeorgeMasonUniversityCOMPLETEMath©2015
• Confusionbetweenpropertiesofasquarevs.propertiesofarectangle
• Using65centimetersastheareaofeachrectangleratherthantheperimeter
• Misunderstandingofthequestionthatisbeingasked.Findingonlythelengthofeachsideofthesquare,orfindingtheperimeterofthesquareinsteadofthearea.
samesizeastheactualpan?• Whatinformationareweattemptingto
find?• Whatdoyouknowaboutsquaresor
rectangles?• Whatwouldhappenwhenthewidthsof
therectanglesareputtogether?• Wouldanyrectangleworkasthepan
length,ordoesithavetobeasquare?• Howdoesasquaredifferfroma
rectangle?
GeorgeMasonUniversityCOMPLETEMath©2015
BrowniePan Name Date Mr.BrownE.Panrecentlyopenedanewbusinessmakingbrowniescalled“TheBrownE.Pan.”Onhisfirstdaybaking,hestartedinhisownkitchenbyusingasinglesquarepan.Hejustfinishedhisfirstbatchofbrownies,andhewantstomakesurethatallthebrowniesarethesamesize.Hefirstcutthebrowniesverticallyasshowninthepicturebelow.
Thepanofbrowniesaboveiscutinto12congruentrectangles.Iftheperimeterofeachoftherectanglesis65cm.,whatistheareaofthepan?
GeorgeMasonUniversityCOMPLETEMath©2016
BrowniePanAnticipationGuide
Designedby:
JonathanThompsonGeorgeMasonUniversity,COMPLETEMath
TheTask
Mr.BrownE.Panrecentlyopenedanewbusinessmakingbrowniescalled“TheBrownE.Pan.”Onhisfirstdaybaking,hestartedinhisownkitchenbyusingasinglesquarepan.Hejustfinishedhisfirstbatchofbrownies,andhewantstomakesurethatallthebrowniesarethesamesize.Hefirstcutthebrowniesverticallyasshowninthepicturebelow.
Thepanofbrowniesaboveiscutinto12congruentrectangles.Iftheperimeterofeachoftherectanglesis65cm.,whatistheareaofthepan?
GeorgeMasonUniversityCOMPLETEMath©2016
AnticipatedStrategy#1
Description:StudentsmayfindasolutionbyusingaT-chart.Theycoulddothisbyfirstfiguringoutthatthesumofthelengthandwidthofeachrectanglemustbehalfoftheperimeter,or32.5centimeters.Uponfindingthis,theycouldmakeachartwithmanydifferentpossiblecombinationsofvaluesthatcouldaddtogethertomake32.5centimeters.Theycouldfindsomecombinationsinthechart,untiltheycomeuponthatworksintheproblem.Thelastcolumninthechartwouldbethechecktoseeifitworks.Sincethereare12congruentrectangles,thesumofthewidthsofthe12rectanglesmustbeequivalenttothelengthoftherectanglebecausethefigureisstatedasasquare.Thestudentswouldneedtocontinuetodothisuntilthelengthmatchestheproductofthewidthand12.Thisworksfor30centimeters,sotheareaoftheentiresquareis900squarecentimeters.
GeorgeMasonUniversityCOMPLETEMath©2016
AnticipatedStrategy#2
Description:Thestudentsmaysolvetheproblembyusingapureguessandcheckstrategy.Sincethestudentsknowthateachrectanglehasaperimeterof65centimeters,thisistheinformationthattheymuststartwith.Upondecidingthis,theywillmostlikelydecidethattheyneedtoknowthedimensionsoftheserectangles,buttheywilldeterminethattherearedifferentpossiblecombinations.Thestudentscouldfinddifferentcombinationsuntiltheyfindonethatmakesasquare.Sincethe12widthsofthethinrectanglesareequivalenttooneofthelengthsoftherectangles,theycouldcontinuethispatternuntilmultiplyingthewidthby12isthesameasthelength.Thiswouldprovethedimensionswouldmakeasquare,whichisstatedintheproblem.Oncethestudentfindsthis,theycanmultiplythelengthbythewidth,whichshouldbethesamenumber,tofindthearea.Inthiscase,theareais900squarecentimeters.
GeorgeMasonUniversityCOMPLETEMath©2016
AnticipatedStrategy#3
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.
GeorgeMasonUniversityCOMPLETEMath©2016
BrowniePanStudentwork
Designedby:
JonathanThompsonGeorgeMasonUniversity,COMPLETEMath
Fall2016
TheTaskMr.BrownE.Panrecentlyopenedanewbusinessmakingbrowniescalled“TheBrownE.Pan.”Onhisfirstdaybaking,hestartedinhisownkitchenbyusingasinglesquarepan.Hejustfinishedhisfirstbatchofbrownies,andhewantstomakesurethatallthebrowniesarethesamesize.Hefirstcutthebrowniesverticallyasshowninthepicturebelow.
Thepanofbrowniesaboveiscutinto12congruentrectangles.Iftheperimeterofeachoftherectanglesis65cm.,whatistheareaofthepan?
GeorgeMasonUniversityCOMPLETEMath©2016
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.
GeorgeMasonUniversityCOMPLETEMath©2016
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.