Problem of the Month Polly Gone © Noyce Foundation 2015. This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 Unported License (http://creativecommons.org/licenses/by-nc-nd/3.0/deed.en_US). 1 Problem of the Month Polly Gone The Problems of the Month (POM) are used in a variety of ways to promote problem solving and to foster the first standard of mathematical practice from the Common Core State Standards: “Make sense of problems and persevere in solving them.” The POM may be used by a teacher to promote problem solving and to address the differentiated needs of her students. A department or grade level may engage their students in a POM to showcase problem solving as a key aspect of doing mathematics. POMs can also be used school wide to promote a problem-solving theme at a school. The goal is for all students to have the experience of attacking and solving non-routine problems and developing their mathematical reasoning skills. Although obtaining and justifying solutions to the problems is the objective, the process of learning to problem solve is even more important. The Problem of the Month is structured to provide reasonable tasks for all students in a school. The POM is designed with a shallow floor and a high ceiling, so that all students can productively engage, struggle, and persevere. The Primary Version is designed to be accessible to all students and especially as the key challenge for grades kindergarten and one. Level A will be challenging for most second and third graders. Level B may be the limit of where fourth and fifth-grade students have success and understanding. Level C may stretch sixth and seventh-grade students. Level D may challenge most eighth and ninth-grade students, and Level E should be challenging for most high school students. These grade-level expectations are just estimates and should not be used as an absolute minimum expectation or maximum limitation for students. Problem solving is a learned skill, and students may need many experiences to develop their reasoning skills, approaches, strategies, and the perseverance to be successful. The Problem of the Month builds on sequential levels of understanding. All students should experience Level A and then move through the tasks in order to go as deeply as they can into the problem. There will be those students who will not have access into even Level A. Educators should feel free to modify the task to allow access at some level. Overview In the Problem of the Month Polly Gone, students use polygons to solve problems involving area. The mathematical topics that underlie this POM are the attributes of linear measurement, square measurement, two-dimensional geometry, perimeter, area, and geometric justification. The problem asks students to explore polygons and the relationship of their areas in various problem situations. In the first level of the POM, students are presented with 40 cubes and are asked to make all possible rectangular regions using the cubes as a border. The students are then asked to determine the area of the interior
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ProblemoftheMonth PollyGone©NoyceFoundation2015.ThisworkislicensedunderaCreativeCommonsAttribution-NonCommercial-NoDerivatives3.0UnportedLicense(http://creativecommons.org/licenses/by-nc-nd/3.0/deed.en_US).
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ProblemoftheMonth
PollyGone
TheProblemsoftheMonth(POM)areusedinavarietyofwaystopromoteproblemsolvingandtofosterthefirststandardofmathematicalpracticefromtheCommonCoreStateStandards:“Makesenseofproblemsandpersevereinsolvingthem.”ThePOMmaybeusedbyateachertopromoteproblemsolvingandtoaddressthedifferentiatedneedsofherstudents.AdepartmentorgradelevelmayengagetheirstudentsinaPOMtoshowcaseproblemsolvingasakeyaspectofdoingmathematics.POMscanalsobeusedschoolwidetopromoteaproblem-solvingthemeataschool.Thegoalisforallstudentstohavetheexperienceofattackingandsolvingnon-routineproblemsanddevelopingtheirmathematicalreasoningskills.Althoughobtainingandjustifyingsolutionstotheproblemsistheobjective,theprocessoflearningtoproblemsolveisevenmoreimportant.TheProblemoftheMonthisstructuredtoprovidereasonabletasksforallstudentsinaschool.ThePOMisdesignedwithashallowfloorandahighceiling,sothatallstudentscanproductivelyengage,struggle,andpersevere.ThePrimaryVersionisdesignedtobeaccessibletoallstudentsandespeciallyasthekeychallengeforgradeskindergartenandone.LevelAwillbechallengingformostsecondandthirdgraders.LevelBmaybethelimitofwherefourthandfifth-gradestudentshavesuccessandunderstanding.LevelCmaystretchsixthandseventh-gradestudents.LevelDmaychallengemosteighthandninth-gradestudents,andLevelEshouldbechallengingformosthighschoolstudents.Thesegrade-levelexpectationsarejustestimatesandshouldnotbeusedasanabsoluteminimumexpectationormaximumlimitationforstudents.Problemsolvingisalearnedskill,andstudentsmayneedmanyexperiencestodeveloptheirreasoningskills,approaches,strategies,andtheperseverancetobesuccessful.TheProblemoftheMonthbuildsonsequentiallevelsofunderstanding.AllstudentsshouldexperienceLevelAandthenmovethroughthetasksinordertogoasdeeplyastheycanintotheproblem.TherewillbethosestudentswhowillnothaveaccessintoevenLevelA.Educatorsshouldfeelfreetomodifythetasktoallowaccessatsomelevel.OverviewIntheProblemoftheMonthPollyGone,studentsusepolygonstosolveproblemsinvolvingarea.ThemathematicaltopicsthatunderliethisPOMaretheattributesoflinearmeasurement,squaremeasurement,two-dimensionalgeometry,perimeter,area,andgeometricjustification.Theproblemasksstudentstoexplorepolygonsandtherelationshipoftheirareasinvariousproblemsituations.InthefirstlevelofthePOM,studentsarepresentedwith40cubesandareaskedtomakeallpossiblerectangularregionsusingthecubesasaborder.Thestudentsarethenaskedtodeterminetheareaoftheinterior
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regionsandidentifytherectanglewiththelargestareatomakeapenforanimalsinazoo.InLevelB,studentsarepresentedwithatriangularshapeongridpapercomprisedoffivesmallerpolygons.Thestudentsareaskedtonameanddeterminetheareaofeachpolygon.Theyarealsoaskedtorearrangetheshapestoconstructasmanydifferentparallelogramsaspossible.InLevelC,studentsaregivenarectanglethatissubdividedintoninesmallersquares.Thestudentsaregiventheareaoftwoofthesquaresandaskedtodeterminetheareaoftheremainingsevensquares.InLevelD,thestudentsexploreconceptsformaximizingareagivenafixedperimeter.Thestudentsgrapplewithwhichpolygonwillproducethelargestareaaswellasmaintainaconstantdistancefromtheperimetertodesigntheplayingsurfaceinasportsarena.InLevelE,studentsareaskedtoconstructageometricfigurefromasquare.Anoctagonisproducedbydrawinglinesegmentsfromeachvertextoitsoppositemidpoints.Studentsareaskedtodeterminetheareaoftheoctagoninrelationshiptotheareaofthesquare.Studentsareaskedtojustifytheirsolutions.
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LevelAPollyworksinazooandneedstobuildpenswhereanimalscanliveandbesafe.Thewallsof the pens aremade out of cubes that are connected together. Polly has 40 cubes andwantstomakethelargestpenpossible,sotheanimalscanmovearoundfreelybutnotgetloose.Buildthelargestareausingall40cubes.Yourwallsmust:
• Befullyenclosed,withnodoorsorwindowssoPolly’sanimalscan’tgetout.• Haveaheightofonecube.• Bejoinedcubefacetocubeface.
HelpPollybymakingpensofseveralshapesanddeterminewhichpenprovidesthelargestareafortheanimals.Youmightwanttobuildthepenonthegridpaperfirst,sothatitwillbeeasiertodeterminethearea.Usethegridpapertoshowtheshapeofthepen.ExplaintoPollywhyyoubelieveyourpenisthelargestonethatcanbemade.
ProblemoftheMonthPollyGone
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LevelBThelargetriangleismadeupoffiveshapesandisdrawnongraphpaper.Nameeachofthefiveshapesanddeterminetheareaofeach.Rearrange theshapes to findallpossibleparallelogramsofanysizeusinganynumberofthem.Drawapictureofeachparallelogramthatyouhavefound,andthendetermineitsarea.Howdidyoufindallofthem?Howdoyouknowyoufoundthemall?
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LevelCInNovember1958,themagazine,ScientificAmerican,showedthisdiagramonitscover.Eachoftheinteriorrectanglesisasquare.IfsquareDis81squareunitsandsquareCis64squareunits,what is theareaof theother seven squares?What is theareaof theentirefigure?Whatistheperimeteroftheentirefigure?Explainyoursolutions.
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A B
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LevelDAnewarenaisgoingtobeconstructedatalocaluniversity.Astudyisbeingdonetofindthe best performance or playing area design. Since the arena will be used for manydifferentsports,aswellasshowsandconcerts,thedesignerswantaseatingarrangementthat allows spectators to be as close as possible to the action.They alsowant to seat asmany front row spectators as possible around the performance area. They have alsodecidedthattheboundaryoftheperformanceareaneedstohavestraightsides,nocurves,duetothebuildingmaterialstheyareusing.Thegoalistohavefrontrowseatsnotmorethan20metersfromthecenteroftheperformanceorplayingarea.Theywanttohireyouasaconsultanttoinvestigatethismatterandexplaintothemwhichdesign would best suit their needs. They need to see several examples of possibleperformance area designs thatwill fit their constraints. The final recommendationmustexplaintheadvantagesofthedesignintermsofthesizeoftheplayingareaandthenumberofpeopletheycanseatinthefrontrow.
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LevelECatherinesaidtoRebecca,“IneedtodrawanoctagonandIwantittobeaccurate.”Rebeccareplied,“Ihaveaneasywaytodrawanoctagon.Startwithalargesquare.Findthemidpointofeachside.Nowdrawa linesegment fromeachmidpoint to thetwooppositevertices.Inthecenterofthedrawing,anoctagonwillbeformed.”“That’sagreatmethod,Rebecca,butIwanttomakemyoctagonacertainsize.HowbigdoIneed tomake the original square in terms of area to get an octagon of a certain area?”Catherineasked.Please help Catherine and Rebecca determine these relationships. Fully explain yourreasoning.“Afteryouhavedrawnyouroctagon,youwillseethat itcomesoutasabeautifulregularoctagon,” Rebecca exclaimed. “Well, it may be beautiful, but I don’t think it is regular,”challengedCatherine.Whoisright?Determineyouranswerusingmathematics.
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PrimaryVersionLevelA Materials:20multi-linkcubesperpair,asheetof1-inchgridpapertomeasureandrecord.Discussionontherug:Theteacheraskstheclass,“Where do the animals in the zoo live?” Theteacherinvitesstudentstotellthewholeclassaboutcagesorpensatthezoo.Theteachersays, “Suppose we want to make a cage or pen where an animal can live. We want to make our pen out of these cubes.” Theteachershowsthemulti-linkcubestotheclass. “We want to make the pen as big as possible so the animal can roam around, but we have only 20 cubes we can use. What shape can we make?” Theteacheraskstheclass.Studentssharetheirideas.Thentheteachershowstwodifferentpens-onewithinteriordimensions1by7andasecondwithinteriordimensions3by5-toillustratewhatismeantbydifferent.Theteachersays, “I would like you to go back to your desk and work with your partner and make all the different possible shapes using 20 cubes.” Inpairs:Studentshavecubesandgirdpaperavailable.Teachersays, “Look at all the animal pen shapes you made. Which shape has the most room for the animal?” Students work together to find a solution. After thestudentsaredone,theteacherasksstudentstosharetheiranswersandhowtheyknow.At the end of the investigation: Students either discuss or dictate a response to thissummary question, “Show all the shapes you can make with 20 cubes. Explain which shape has the most room for the animal. How do you know?”
ProblemoftheMonthPollyGone
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ProblemoftheMonth
PollyGoneTaskDescription–LevelA
Thistaskchallengesstudentstomakeallpossiblerectanglesusing40cubesasaborder,tofindtheareaoftherectangles,andtoidentifytheonewiththelargestarea.
CommonCoreStateStandardsMath-ContentStandardsMeasurementandDataGeometricmeasurement:understandconceptsofareaandrelateareatomultiplicationandtoaddition.3.MD.5Recognizeareaasanattributeofplanefiguresandunderstandconceptsofareameasurement3.MD.6Measureareasbycountingunitsquares.3.MD.7Relateareatotheoperationsofmultiplicationandaddition.GeometrySolvereal-worldandmathematicalproblemsinvolvingarea,surfacearea,andvolume.6.G.1Findtheareaofrighttriangles,othertriangles,specialquadrilaterals,andpolygonsbycomposingintorectanglesordecomposingintotrianglesandothershapes,applythesetechniquesinthecontextofsolvingreal-worldandmathematicalproblems.
CommonCoreStateStandardsMath–StandardsofMathematicalPracticeMP.1Makesenseofproblemsandpersevereinsolvingthem.Mathematicallyproficientstudentsstartbyexplainingtothemselvesthemeaningofaproblemandlookingforentrypointstoitssolution.Theyanalyzegivens,constraints,relationships,andgoals.Theymakeconjecturesabouttheformandmeaningofthesolutionandplanasolutionpathwayratherthansimplyjumpingintoasolutionattempt.Theyconsideranalogousproblems,andtryspecialcasesandsimplerformsoftheoriginalprobleminordertogaininsightintoitssolution.Theymonitorandevaluatetheirprogressandchangecourseifnecessary.Olderstudentsmight,dependingonthecontextoftheproblem,transformalgebraicexpressionsorchangetheviewingwindowontheirgraphingcalculatortogettheinformationtheyneed.Mathematicallyproficientstudentscanexplaincorrespondencesbetweenequations,verbaldescriptions,tables,andgraphsordrawdiagramsofimportantfeaturesandrelationships,graphdata,andsearchforregularityortrends.Youngerstudentsmightrelyonusingconcreteobjectsorpicturestohelpconceptualizeandsolveaproblem.Mathematicallyproficientstudentschecktheiranswerstoproblemsusingadifferentmethod,andtheycontinuallyaskthemselves,“Doesthismakesense?”Theycanunderstandtheapproachesofotherstosolvingcomplexproblemsandidentifycorrespondencesbetweendifferentapproaches.MP.3Constructviableargumentsandcritiquethereasoningofothers.Mathematicallyproficientstudentsunderstandandusestatedassumptions,definitions,andpreviouslyestablishedresultsinconstructingarguments.Theymakeconjecturesandbuildalogicalprogressionofstatementstoexplorethetruthoftheirconjectures.Theyareabletoanalyzesituationsbybreakingthemintocases,andcanrecognizeandusecounterexamples.Theyjustifytheirconclusions,communicatethemtoothers,andrespondtotheargumentsofothers.Theyreasoninductivelyaboutdata,makingplausibleargumentsthattakeintoaccountthecontextfromwhichthedataarose.Mathematicallyproficientstudentsarealsoabletocomparetheeffectivenessoftwoplausiblearguments,distinguishcorrectlogicorreasoningfromthatwhichisflawed,and–ifthereisaflawinanargument–explainwhatitis.Elementarystudentscanconstructargumentsusingconcretereferentssuchasobjects,drawings,diagrams,andactions.Suchargumentscanmakesenseandbecorrect,eventhoughtheyarenotgeneralizedormadeformaluntillatergrades.Later,studentslearntodeterminedomainstowhichanargumentapplies.Studentsatallgradescanlistenorreadtheargumentsofothers,decidewhethertheymakesense,andaskusefulquestionstoclarifyorimprovethearguments.
CCSSMAlignment:ProblemoftheMonth PollyGone©NoyceFoundation2015.ThisworkislicensedunderaCreativeCommonsAttribution-NonCommercial-NoDerivatives3.0UnportedLicense(http://creativecommons.org/licenses/by-nc-nd/3.0/deed.en_US).
ProblemoftheMonthPollyGone
TaskDescription–LevelBThistaskchallengesstudentstoidentifyshapesdrawninsideatriangleandtofindtheirareas.Studentsarethenchallengedtocutoutshapesandfindallpossibleparallelogramsthatcanbemadefromsomecombinationoftheshapes,andtorecordtheshapesandareasofeach.Studentsmustjustifyhowtheyknowtheyhavefoundallpossibilities.
CommonCoreStateStandardsMath-ContentStandardsGeometryClassifytwo-dimensionalfiguresintocategoriesbasedontheirproperties.5.G.3Understandthatattributesbelongingtoacategoryoftwo-dimensionalfiguresalsobelongtoallsubcategoriesofthatcategory.Solvereal-worldandmathematicalproblemsinvolvingarea,surfacearea,andvolume.6.G.1Findtheareaofrighttriangles,othertriangles,specialquadrilaterals,andpolygonsbycomposingintorectanglesordecomposingintotrianglesandothershapes,applythesetechniquesinthecontextofsolvingreal-worldandmathematicalproblems.7.G.6Solvereal-worldandmathematicalproblemsinvolvingarea,volumeandsurfaceareaoftwo-andthree-dimensionalobjectscomposedoftriangles,quadrilaterals,polygons,cubes,andrightprisms.
CommonCoreStateStandardsMath–StandardsofMathematicalPracticeMP.1Makesenseofproblemsandpersevereinsolvingthem.Mathematicallyproficientstudentsstartbyexplainingtothemselvesthemeaningofaproblemandlookingforentrypointstoitssolution.Theyanalyzegivens,constraints,relationships,andgoals.Theymakeconjecturesabouttheformandmeaningofthesolutionandplanasolutionpathwayratherthansimplyjumpingintoasolutionattempt.Theyconsideranalogousproblems,andtryspecialcasesandsimplerformsoftheoriginalprobleminordertogaininsightintoitssolution.Theymonitorandevaluatetheirprogressandchangecourseifnecessary.Olderstudentsmight,dependingonthecontextoftheproblem,transformalgebraicexpressionsorchangetheviewingwindowontheirgraphingcalculatortogettheinformationtheyneed.Mathematicallyproficientstudentscanexplaincorrespondencesbetweenequations,verbaldescriptions,tables,andgraphsordrawdiagramsofimportantfeaturesandrelationships,graphdata,andsearchforregularityortrends.Youngerstudentsmightrelyonusingconcreteobjectsorpicturestohelpconceptualizeandsolveaproblem.Mathematicallyproficientstudentschecktheiranswerstoproblemsusingadifferentmethod,andtheycontinuallyaskthemselves,“Doesthismakesense?”Theycanunderstandtheapproachesofotherstosolvingcomplexproblemsandidentifycorrespondencesbetweendifferentapproaches.MP.3Constructviableargumentsandcritiquethereasoningofothers.Mathematicallyproficientstudentsunderstandandusestatedassumptions,definitions,andpreviouslyestablishedresultsinconstructingarguments.Theymakeconjecturesandbuildalogicalprogressionofstatementstoexplorethetruthoftheirconjectures.Theyareabletoanalyzesituationsbybreakingthemintocases,andcanrecognizeandusecounterexamples.Theyjustifytheirconclusions,communicatethemtoothers,andrespondtotheargumentsofothers.Theyreasoninductivelyaboutdata,makingplausibleargumentsthattakeintoaccountthecontextfromwhichthedataarose.Mathematicallyproficientstudentsarealsoabletocomparetheeffectivenessoftwoplausiblearguments,distinguishcorrectlogicorreasoningfromthatwhichisflawed,and–ifthereisaflawinanargument–explainwhatitis.Elementarystudentscanconstructargumentsusingconcretereferentssuchasobjects,drawings,diagrams,andactions.Suchargumentscanmakesenseandbecorrect,eventhroughtheyarenotgeneralizedormadeformaluntillatergrades.Later,studentslearntodeterminedomainstowhichanargumentapplies.Studentsatallgradescanlistenorreadtheargumentsofothers,decidewhethertheymakesense,andaskusefulquestionstoclarifyorimprovethearguments.
CCSSMAlignment:ProblemoftheMonth PollyGone©NoyceFoundation2015.ThisworkislicensedunderaCreativeCommonsAttribution-NonCommercial-NoDerivatives3.0UnportedLicense(http://creativecommons.org/licenses/by-nc-nd/3.0/deed.en_US).
ProblemoftheMonthPollyGone
TaskDescription–LevelCThistaskchallengesstudentstoderivemeasurementsfromacollectionofdifferentsizedsquaresarrangedinasquarewhentheareasoftwoofthesquaresareknown.Studentsmustuselogicandgeometricformulastofindtheareaandperimeterofallthesquaresincludingthelargecompositesquare.
CommonCoreStateStandardsMath-ContentStandardsMeasurementandDataGeometricmeasurement:understandconceptsofareaandrelateareatomultiplicationandtoaddition.3.MD.7Relateareatotheoperationsofmultiplicationandaddition.Geometricmeasurement:recognizeperimeterasanattributeofplanefiguresanddistinguishbetweenlinearandareameasurements.3.MD.8Solverealworldandmathematicalproblemsinvolvingperimetersofpolygons,includingfindingtheperimetergiventhesidelengths,findinganunknownsidelength,andexhibitingrectangleswiththesameperimeteranddifferentareasorwiththesameareaanddifferentperimeters.GeometrySolvereal-worldandmathematicalproblemsinvolvingarea,surfacearea,andvolume.6.G.1Findtheareaofrighttriangles,othertriangles,specialquadrilaterals,andpolygonsbycomposingintorectanglesordecomposingintotrianglesandothershapes,applythesetechniquesinthecontextofsolvingreal-worldandmathematicalproblems.7.G.6Solvereal-worldandmathematicalproblemsinvolvingarea,volumeandsurfaceareaoftwo-andthree-dimensionalobjectscomposedoftriangles,quadrilaterals,polygons,cubes,andrightprisms.ExpressionsandEquationsExpressionsandequationsworkwithradicalsandintegerexponents.8.EE.2Usesquarerootandcuberootsymbolstorepresentsolutionstoequationsintheforx2=pandx3=p.
CommonCoreStateStandardsMath–StandardsofMathematicalPracticeMP.1Makesenseofproblemsandpersevereinsolvingthem.Mathematicallyproficientstudentsstartbyexplainingtothemselvesthemeaningofaproblemandlookingforentrypointstoitssolution.Theyanalyzegivens,constraints,relationships,andgoals.Theymakeconjecturesabouttheformandmeaningofthesolutionandplanasolutionpathwayratherthansimplyjumpingintoasolutionattempt.Theyconsideranalogousproblems,andtryspecialcasesandsimplerformsoftheoriginalprobleminordertogaininsightintoitssolution.Theymonitorandevaluatetheirprogressandchangecourseifnecessary.Olderstudentsmight,dependingonthecontextoftheproblem,transformalgebraicexpressionsorchangetheviewingwindowontheirgraphingcalculatortogettheinformationtheyneed.Mathematicallyproficientstudentscanexplaincorrespondencesbetweenequations,verbaldescriptions,tables,andgraphsordrawdiagramsofimportantfeaturesandrelationships,graphdata,andsearchforregularityortrends.Youngerstudentsmightrelyonusingconcreteobjectsorpicturestohelpconceptualizeandsolveaproblem.Mathematicallyproficientstudentschecktheiranswerstoproblemsusingadifferentmethod,andtheycontinuallyaskthemselves,“Doesthismakesense?”Theycanunderstandtheapproachesofotherstosolvingcomplexproblemsandidentifycorrespondencesbetweendifferentapproaches.MP.3Constructviableargumentsandcritiquethereasoningofothers.Mathematicallyproficientstudentsunderstandandusestatedassumptions,definitions,andpreviouslyestablishedresultsinconstructingarguments.Theymakeconjecturesandbuildalogicalprogressionofstatementstoexplorethetruthoftheirconjectures.Theyareabletoanalyzesituationsbybreakingthemintocases,andcanrecognizeandusecounterexamples.Theyjustifytheirconclusions,communicatethemtoothers,andrespondtotheargumentsofothers.Theyreasoninductivelyaboutdata,makingplausibleargumentsthattakeintoaccountthecontextfromwhichthedataarose.Mathematicallyproficientstudentsarealsoabletocomparetheeffectivenessoftwoplausiblearguments,distinguishcorrectlogicorreasoningfromthatwhichisflawed,and–ifthereisaflawinanargument–explainwhatitis.Elementarystudentscanconstructargumentsusingconcretereferentssuchasobjects,drawings,diagrams,andactions.Suchargumentscanmakesenseandbecorrect,eventhoughtheyarenotgeneralizedormadeformaluntillatergrades.Later,studentslearntodeterminedomainstowhichanargumentapplies.Studentsatallgradescanlistenorreadtheargumentsofothers,decidewhethertheymakesense,andaskusefulquestionstoclarifyorimprovethearguments.
CCSSMAlignment:ProblemoftheMonth PollyGone©NoyceFoundation2015.ThisworkislicensedunderaCreativeCommonsAttribution-NonCommercial-NoDerivatives3.0UnportedLicense(http://creativecommons.org/licenses/by-nc-nd/3.0/deed.en_US).
ProblemoftheMonth
PollyGoneTaskDescription–LevelD
Thistaskchallengesstudentstoexploreconceptsformaximizingareagivenafixedperimeter.Thestudentsgrapplewithwhichpolygonwillproducethelargestareaaswellasmaintainaconstantdistancefromtheperimetertodesigntheplayingsurfaceinasportsarena.
CommonCoreStateStandardsMath-ContentStandardsMeasurementandDataGeometricmeasurement:understandconceptsofareaandrelateareatomultiplicationandtoaddition.3.MD.7Relateareatotheoperationsofmultiplicationandaddition.Geometricmeasurement:recognizeperimeterasanattributeofplanefiguresanddistinguishbetweenlinearandareameasurements.3.MD.8Solverealworldandmathematicalproblemsinvolvingperimetersofpolygons,includingfindingtheperimetergiventhesidelengths,findinganunknownsidelength,andexhibitingrectangleswiththesameperimeteranddifferentareasorwiththesameareaanddifferentperimeters.GeometrySolvereal-worldandmathematicalproblemsinvolvingarea,surfacearea,andvolume.6.G.1Findtheareaofrighttriangles,othertriangles,specialquadrilaterals,andpolygonsbycomposingintorectanglesordecomposingintotrianglesandothershapes,applythesetechniquesinthecontextofsolvingreal-worldandmathematicalproblems.7.G.4Knowtheformulasfortheareaandcircumferenceofacircleandusethemtosolveproblems;giveaninformalderivationoftherelationshipbetweenthecircumferenceandareaofacircle.7.G.6Solvereal-worldandmathematicalproblemsinvolvingarea,volumeandsurfaceareaoftwo-andthree-dimensionalobjectscomposedoftriangles,quadrilaterals,polygons,cubes,andrightprisms.
CommonCoreStateStandardsMath–StandardsofMathematicalPracticeMP.1Makesenseofproblemsandpersevereinsolvingthem.Mathematicallyproficientstudentsstartbyexplainingtothemselvesthemeaningofaproblemandlookingforentrypointstoitssolution.Theyanalyzegivens,constraints,relationships,andgoals.Theymakeconjecturesabouttheformandmeaningofthesolutionandplanasolutionpathwayratherthansimplyjumpingintoasolutionattempt.Theyconsideranalogousproblems,andtryspecialcasesandsimplerformsoftheoriginalprobleminordertogaininsightintoitssolution.Theymonitorandevaluatetheirprogressandchangecourseifnecessary.Olderstudentsmight,dependingonthecontextoftheproblem,transformalgebraicexpressionsorchangetheviewingwindowontheirgraphingcalculatortogettheinformationtheyneed.Mathematicallyproficientstudentscanexplaincorrespondencesbetweenequations,verbaldescriptions,tables,andgraphsordrawdiagramsofimportantfeaturesandrelationships,graphdata,andsearchforregularityortrends.Youngerstudentsmightrelyonusingconcreteobjectsorpicturestohelpconceptualizeandsolveaproblem.Mathematicallyproficientstudentschecktheiranswerstoproblemsusingadifferentmethod,andtheycontinuallyaskthemselves,“Doesthismakesense?”Theycanunderstandtheapproachesofotherstosolvingcomplexproblemsandidentifycorrespondencesbetweendifferentapproaches.MP.4Modelwithmathematics.Mathematicallyproficientstudentscanapplythemathematicstheyknowtosolveproblemsarisingineverydaylife,society,andtheworkplace.Inearlygradesthismightbeassimpleaswritinganadditionequationtodescribeasituation.Inmiddlegrades,astudentmightapplyproportionalreasoningtoplanaschooleventoranalyzeaprobleminthecommunity.Byhighschool,astudentmightusegeometrytosolveadesignproblemoruseafunctiontodescribehowonequantityofinterestdependsonanother.Mathematicallyproficientstudentswhocanapplywhattheyknowarecomfortablemakingassumptionsandapproximationstosimplifyacomplicatedsituation,realizingthatthesemayneedrevisionlater.Theyareabletoidentifyimportantquantitiesinapracticalsituationandmaptheirrelationshipsusingsuchtoolsasdiagrams,two-waytables,graphs,flowcharts,andformulas.Theycananalyzethoserelationshipsmathematicallytodrawconclusions.Theyroutinelyinterprettheirmathematicalresultsinthecontextofthesituationandreflectonwhethertheresultsmakesense,possiblyimprovingthemodelifithasnotserveditspurpose.
CCSSMAlignment:ProblemoftheMonth PollyGone©NoyceFoundation2015.ThisworkislicensedunderaCreativeCommonsAttribution-NonCommercial-NoDerivatives3.0UnportedLicense(http://creativecommons.org/licenses/by-nc-nd/3.0/deed.en_US).
ProblemoftheMonthPollyGone
TaskDescription–LevelEThistaskchallengesstudentstodrawanoctagonfromasquareandfindtherelationshipbetweenthetwoareas.
CommonCoreStateStandardsMath-ContentStandardsGeometrySolvereal-worldandmathematicalproblemsinvolvingarea,surfacearea,andvolume.6.G.1Findtheareaofrighttriangles,othertriangles,specialquadrilaterals,andpolygonsbycomposingintorectanglesordecomposingintotrianglesandothershapes,applythesetechniquesinthecontextofsolvingreal-worldandmathematicalproblems.7.G.4Knowtheformulasfortheareaandcircumferenceofacircleandusethemtosolveproblems;giveaninformalderivationoftherelationshipbetweenthecircumferenceandareaofacircle.Drawconstruct,anddescribegeometricalfiguresanddescribetherelationshipsbetweenthem.7.G.2Drawgeometricshapeswithgivenconditions.Focusonconstructingtrianglesfromthreemeasuresofanglesorsides,noticewhentheconditionsdetermineauniquetriangle,morethanonetriangle,ornotriangle.HighSchool–Geometry-CongruenceMakegeometricconstructions.G-CO.12Makeformalgeometricconstructionswithavarietyoftoolsandmethods.HighSchool–Geometry–Similarity,RightTriangles,andTrigonometryApplytrigonometrytogeneraltriangles.G-SRT.11UnderstandandapplytheLawofSineandtheLawofCosinestofindunknownmeasurementsinrighttrianglesandnon-righttriangles.
CommonCoreStateStandardsMath–StandardsofMathematicalPracticeMP.4Modelwithmathematics.Mathematicallyproficientstudentscanapplythemathematicstheyknowtosolveproblemsarisingineverydaylife,society,andtheworkplace.Inearlygradesthismightbeassimpleaswritinganadditionequationtodescribeasituation.Inmiddlegrades,astudentmightapplyproportionalreasoningtoplanaschooleventoranalyzeaprobleminthecommunity.Byhighschool,astudentmightusegeometrytosolveadesignproblemoruseafunctiontodescribehowonequantityofinterestdependsonanother.Mathematicallyproficientstudentswhocanapplywhattheyknowarecomfortablemakingassumptionsandapproximationstosimplifyacomplicatedsituation,realizingthatthesemayneedrevisionlater.Theyareabletoidentifyimportantquantitiesinapracticalsituationandmaptheirrelationshipsusingsuchtoolsasdiagrams,two-waytables,graphs,flowcharts,andformulas.Theycananalyzethoserelationshipsmathematicallytodrawconclusions.Theyroutinelyinterprettheirmathematicalresultsinthecontextofthesituationandreflectonwhethertheresultsmakesense,possiblyimprovingthemodelifithasnotserveditspurpose.MP.7Lookforandmakeuseofstructure.Mathematicallyproficientstudentstrytolookcloselytodiscernapatternorstructure.Youngstudents,forexample,mightnoticethatthreeandsevenmoreisthesameamountassevenandthreemore,ortheymaysortacollectionofshapesaccordingtohowmanysidestheshapeshave.Later,studentswillsee7x8equalsthewell-remembered7x5+7x3,inpreparationforlearningaboutthedistributiveproperty.Intheexpressionx2+9x+14,olderstudentscanseethe14as2x7andthe9as2+7.Theyrecognizethesignificanceofanexistinglineinageometricfigureandcanusethestrategyofdrawinganauxiliarylineforsolvingproblems.Theyalsocanstepbackforanoverviewandshiftperspective.Theycanseecomplicatedthings,suchassomealgebraicexpressions,assingleobjectsorbeingcomposedofseveralobjects.Forexample,theycansee5–3(x-y)2as5minusapositivenumbertimesasquareandusethattorealizethatitsvaluecannotbemorethan5foranyrealnumbersxandy.
CCSSMAlignment:ProblemoftheMonth PollyGone©NoyceFoundation2015.ThisworkislicensedunderaCreativeCommonsAttribution-NonCommercial-NoDerivatives3.0UnportedLicense(http://creativecommons.org/licenses/by-nc-nd/3.0/deed.en_US).
ProblemoftheMonthPollyGone
TaskDescription–PrimaryLevelThistaskchallengesstudentstoexplorealltherectanglesthatcanbemadewith20multi-linkcubes.Studentsshouldrecordtheirrectanglesongraphpaperanddeterminetheshapewiththemaximumarea.
CommonCoreStateStandardsMath-ContentStandardsMeasurementandDataDescribeandcomparemeasureableattributes.K.MD.2Directlycomparetwoobjectswithameasureableattributeincommontoseewhichobjecthas“moreof”or“lessof”theattribute,anddescribethedifference.Geometricmeasurement:understandconceptsofareaandrelateareatomultiplicationandtoaddition.3.MD.5Recognizeareaasanattributeofplanefiguresandunderstandconceptsofareameasurement3.MD.6Measureareasbycountingunitsquares.
CommonCoreStateStandardsMath–StandardsofMathematicalPracticeMP.1Makesenseofproblemsandpersevereinsolvingthem.Mathematicallyproficientstudentsstartbyexplainingtothemselvesthemeaningofaproblemandlookingforentrypointstoitssolution.Theyanalyzegivens,constraints,relationships,andgoals.Theymakeconjecturesabouttheformandmeaningofthesolutionandplanasolutionpathwayratherthansimplyjumpingintoasolutionattempt.Theyconsideranalogousproblems,andtryspecialcasesandsimplerformsoftheoriginalprobleminordertogaininsightintoitssolution.Theymonitorandevaluatetheirprogressandchangecourseifnecessary.Olderstudentsmight,dependingonthecontextoftheproblem,transformalgebraicexpressionsorchangetheviewingwindowontheirgraphingcalculatortogettheinformationtheyneed.Mathematicallyproficientstudentscanexplaincorrespondencesbetweenequations,verbaldescriptions,tables,andgraphsordrawdiagramsofimportantfeaturesandrelationships,graphdata,andsearchforregularityortrends.Youngerstudentsmightrelyonusingconcreteobjectsorpicturestohelpconceptualizeandsolveaproblem.Mathematicallyproficientstudentschecktheiranswerstoproblemsusingadifferentmethod,andtheycontinuallyaskthemselves,“Doesthismakesense?”Theycanunderstandtheapproachesofotherstosolvingcomplexproblemsandidentifycorrespondencesbetweendifferentapproaches.MP.7Lookforandmakeuseofstructure.Mathematicallyproficientstudentstrytolookcloselytodiscernapatternorstructure.Youngstudents,forexample,mightnoticethatthreeandsevenmoreisthesameamountassevenandthreemore,ortheymaysortacollectionofshapesaccordingtohowmanysidestheshapeshave.Later,studentswillsee7x8equalsthewell-remembered7x5+7x3,inpreparationforlearningaboutthedistributiveproperty.Intheexpressionx2+9x+14,olderstudentscanseethe14as2x7andthe9as2+7.Theyrecognizethesignificanceofanexistinglineinageometricfigureandcanusethestrategyofdrawinganauxiliarylineforsolvingproblems.Theyalsocanstepbackforanoverviewandshiftperspective.Theycanseecomplicatedthings,suchassomealgebraicexpressions,assingleobjectsorbeingcomposedofseveralobjects.Forexample,theycansee5–3(x-y)2as5minusapositivenumbertimesasquareandusethattorealizethatitsvaluecannotbemorethan5foranyrealnumbersxandy.
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