problem of the month: between the lines - inside mathematics

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ProblemoftheMonth:BetweentheLines

TheProblemsoftheMonth(POM)areusedinavarietyofwaystopromoteproblemsolvingandtofosterthefirststandardofmathematicalpracticefromtheCommonCoreStateStandards:“Makesenseofproblemsandpersevereinsolvingthem.”ThePOMmaybeusedbyateachertopromoteproblemsolvingandtoaddressthedifferentiatedneedsofherstudents.AdepartmentorgradelevelmayengagetheirstudentsinaPOMtoshowcaseproblemsolvingasakeyaspectofdoingmathematics.POMscanalsobeusedschoolwidetopromoteaproblem-solvingthemeataschool.Thegoalisforallstudentstohavetheexperienceofattackingandsolvingnon-routineproblemsanddevelopingtheirmathematicalreasoningskills.Althoughobtainingandjustifyingsolutionstotheproblemsistheobjective,theprocessoflearningtoproblem-solveisevenmoreimportant.TheProblemoftheMonthisstructuredtoprovidereasonabletasksforallstudentsinaschool.ThePOMisdesignedwithashallowfloorandahighceiling,sothatallstudentscanproductivelyengage,struggle,andpersevere.ThePrimaryVersionisdesignedtobeaccessibletoallstudentsandespeciallyasthekeychallengeforgradesK–1.LevelAwillbechallengingformostsecondandthirdgraders.LevelBmaybethelimitofwherefourthandfifth-gradestudentshavesuccessandunderstanding.LevelCmaystretchsixthandseventh-gradestudents.LevelDmaychallengemosteighthandninth-gradestudents,andLevelEshouldbechallengingformosthighschoolstudents.Thesegrade-levelexpectationsarejustestimatesandshouldnotbeusedasanabsoluteminimumexpectationormaximumlimitationforstudents.Problemsolvingisalearnedskill,andstudentsmayneedmanyexperiencestodeveloptheirreasoningskills,approaches,strategies,andtheperseverancetobesuccessful.TheProblemoftheMonthbuildsonsequentiallevelsofunderstanding.AllstudentsshouldexperienceLevelAandthenmovethroughthetasksinordertogoasdeeplyastheycanintotheproblem.TherewillbethosestudentswhowillnothaveaccessintoevenLevelA.Educatorsshouldfeelfreetomodifythetasktoallowaccessatsomelevel.OverviewIntheProblemoftheMonthBetweentheLines,studentsusepolygonstosolveproblemsinvolvingarea.ThemathematicaltopicsthatunderliethisPOMaretheattributesoflinearmeasurement,squaremeasurement,two-dimensionalgeometry,area,thePythagoreanTheorem,andgeometricjustification.Theproblemasksstudentstoexplorepolygonsandtherelationshipoftheirareasinvariousproblemsituations.InthefirstlevelofthePOM,studentsarepresentedwithanoutlineofanirregularly-shapedanimalandpatternblocks.Thestudentsarethenaskedtocover(tile)theinteriorregionswithpatternblocks.Thesecondpartofthetaskinvolvesthesamefigure,onlyflippedhorizontally.Theyareaskedtocoverthe

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regionagainusingadifferentarrangement.InLevelB,studentsarepresentedwithasetofisoscelestrapezoidsthatarescaledlargerbyalinearfactor.Thestudentsareaskedtocovertheareaofeachtrapezoidwithunittrapezoids.Theyarealsoaskedtoexaminethegrowthinthesizeoftheareaandtofindapattern.InLevelC,studentsaregivenagroupoftrianglesdrawnondotpaper.Thestudentsareaskedtofindtheareaofeachofthetriangles.InLevelD,thestudentsexplorealogodesignthatinvolvesthreesquaresofdifferingsizespositionedonthepaper.Thestudentsgrapplewithfindingarelationshipamongthethreesquares.InLevelE,studentsareshownhowthreetrianglesareconstructedinastepwiseprocess.Theyareaskedtodeterminearelationshipbetweenthethreetriangles.Studentsareaskedtojustifytheirsolutions.MathematicalConceptsInthisPOM,studentsexploretherelationshipoftheareaofpolygons.Studentsusepolygonsinspatialvisualization,tessellationandareaproblems.LaterinthePOM,studentswilluseknowledgeofarea,geometricpropertiesandthePythagoreanTheorem,aswellasreasoningandjustification,todetermineandverifyproblemsinvolvinggeometryandmeasurement.

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ProblemoftheMonth

BetweentheLines LevelABrianhascreatedanewanimalusingpatternblocks.Hetracedtheoutsideofhisanimal.Below is the outline of his animal.Whichpatternblocksdidheuse tomakehis animal?Showhowhemadehisanimal.

Canyoumakethesameanimalagainusingsomedifferentpatternblocks?Nowitisfacingintheotherdirection,show a different way to make theanimalwithpatternblocks.

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LevelBStartwithanisoscelestrapezoid,thesamesizeasthepatternblock.1.Useonlythepatternblocktrapezoidstotileandcoverthetrapezoiddrawnbelow.Howmanypatternblocktrapezoidsdoesit taketotiletheabovetrapezoid?Howaretheblocksarranged?Istheremorethanonearrangementyoucanmake?2.Useonlythepatternblocktrapezoidstotileandcoverthetrapezoiddrawnbelow.Howmanypatternblocktrapezoidsdoesittaketotiletheabovetrapezoid?Howaretheblocksarranged?Istheremorethanonearrangementyoucanmake?

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3.Usethepatternblocktrapezoidstotileandcoverthetrapezoiddrawnbelow:Howmanypatternblocktrapezoidsdoesittaketotiletheabovetrapezoid?Howaretheblocksarranged?Istheremorethanonearrangementyoucanmake?4.Usethepatternblocktrapezoidstotileandcoverthetrapezoiddrawnbelow.Howmanypatternblocktrapezoidsdoesittaketotiletheabovetrapezoid?Howaretheblocksarranged?Istheremorethanonearrangementyoucanmake?Examine the tiling arrangements youmade in figures 1 – 4.What patterns can you see?Howistheareagrowing?Explainwhythepatternsmakesense.

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LevelCFindtheareaofthesetriangles.

Explainyourreasoning.

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LevelDAhightechcompanyhasdesignedanewcompanylogousingthreesquares.Whataretherelationshipsbetweenthesizesofthethreesquares?Explainyourfindings.

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LevelEGivenanytwosquaresAandB-theycanbearrangedtoshareonevertexthatformsananglebetween0ºand180º.A third square C can be foundwhich has a side length equal to the distance between adifferentvertexofsquareAandadifferentvertexofsquareB(shownbelow),thusformingatriangularregionbetweenthethreesquares(coloredblack).

A B

A

B

A

B

C

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A line segment can be drawn between the vertices of the squares, forming three moretriangles(eachshadedwhite).What are the relationships between the threewhite triangles? What is the relationshipbetween the black triangle and the threewhite triangles? Explain and justifywhat youknow.

A

B

C

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ProblemoftheMonth

BetweentheLines

PrimaryVersionLevelAMaterials:Asetofpatternblocksforeachpairorapageoftheprintedpatternblockscutout,acopyoftheoutlineoftheanimalfacingbothdirections,gluestick,pencilandpaper.Discussion on the rug: Teacher shows the pattern blocks. “Here are the pattern blocks. What do you notice about them?” Teachercontinuestoaskchildrentonoticethat theyaredifferentcolors, shapes, sizesanddifferent length. The teacherencouragesthestudentstoplaywiththemandmakedifferentthings. Insmallgroups: Eachgrouphasasetofpatternblocks/ 1. Which of these blocks do we know? What is its name? How many sides does it have? How many corners does it have? Introducethenameofblocksoncethestudentsdemonstrateknowledgeoftheirattributes. 2. A boy named Brian made a picture of a new animal using pattern blocks. Here is the outline of the animal. Can you make Brian’s animal using pattern blocks? Show me how you made it. 3. There is a second picture of the animal. It is looking in a different direction. Can you make Brian’s animal using pattern blocks? Show me how you made it. At the end of the investigation have students draw either a picture, paste a picture ordictatearesponsetorepresenttheirsolution.

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Level A (Actual Pattern Block Template)

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Level A , Part 2 (Actual Pattern Block Template)

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Level B (Actual Pattern Block Template) Part 1 Part 2

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Level B (Actual Pattern Block Template) Part 3

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Level B (Actual Pattern Block Template) Part 4

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ProblemoftheMonthBetweentheLines

TaskDescription–LevelAThistaskchallengesstudentstovisualizeafigurewithdifferentconfigurationsfromasetofshapes(triangles,squares,rhombi,trapezoids,and/orhexagons).Thisdevelopmentofspatialsensewillsupportthedevelopingunderstandingsofattributesinplanegeometryandtheuseofbasictransformations.

CommonCoreStateStandardsMath-ContentStandardsGeometryIdentifyanddescribeshapes(squares,circles,triangles,rectangles,hexagons,cubes,cones,cylinders,andspheres).K.G.1Describeobjectsintheenvironmentusingnamesofshapes,anddescribetherelativepositionsoftheseobjectsusingtermssuchasabove,below,beside,infrontof,behind,andnextto.K.G.2Correctlynameshapesregardlessoftheirorientationsoroverallsize.Analyze,compare,create,andcomposeshapes.K.G.6Composesimpleshapestoformlargershapes.Reasonwithshapesandtheirattributes.1.G.1Distinguishbetweendefiningattributes(e.g.,trianglesareclosedandthreesided)versusnon-definingattributes(e.g.,color,orientation,overallsize);buildanddrawshapestopossessdefiningattributes.1.G.2Composetwo-dimensionalshapes(rectangles,squares,trapezoids,triangles,half-circles,andquarter-circles)orthree-dimensionalshapes(cubes,rightrectangularprisms,rightcircularcones,andrightcircularcylinders)tocreateacompositeshape,andcomposenewshapesfromthecompositeshape.*

*Studentsdonotneedtolearnformalnamessuchas“rightrectangularprism.”Reasonwithshapesandtheirattributes.2.G.1Recognizeanddrawshapeshavingspecifiedattributes,suchasagivennumberofanglesoragivennumberofequalfaces.**Identifytriangles,quadrilaterals,pentagons,hexagons,andcubes.

**Sizesarecompareddirectlyorvisually,notcomparedbymeasuring.CommonCoreStateStandardsMath–StandardsofMathematicalPractice

MP.6 Attendtoprecision.Mathematicallyproficientstudentstrytocommunicatepreciselytoothers.Theytrytousecleardefinitionsindiscussionwithothersandintheirownreasoning.Theystatethemeaningofthesymbolstheychoose,includingusingtheequalsignconsistentlyandappropriately.Theyarecarefulaboutspecifyingunitsofmeasure,andlabelingaxestoclarifythecorrespondencewithquantitiesinaproblem.Theycalculateaccuratelyandefficiently,expressnumericalanswerswithadegreeofprecisionappropriatefortheproblemcontext.Intheelementarygrades,studentsgivecarefullyformulatedexplanationstoeachother.Bythetimetheyreachhighschooltheyhavelearnedtoexamineclaimsandmakeexplicituseofdefinitions.MP.7Lookforandmakeuseofstructure.Mathematicallyproficientstudentslookcloselytodiscernapatternorstructure.Youngstudents,forexample,mightnoticethatthreeandsevenmoreisthesameamountassevenandthreemore,ortheymaysortacollectionofshapesaccordingtohowmanysidestheshapeshave.Later,studentswillsee7x8equalsthewell-remembered7x5+7x3,inpreparationforlearningaboutthedistributiveproperty.Intheexpressionx2+9x+14,olderstudentscanseethe14as2x7andthe9as2+7.Theyrecognizethesignificanceofanexistinglineinageometricfigureandcanusethestrategyofdrawinganauxiliarylineforsolvingproblems.Theyalsocanstepbackforanoverviewandshiftperspective.Theycanseecomplicatedthings,suchassomealgebraicexpressions,assingleobjectsorasbeingcomposedofseveralobjects.Forexample,theycansee5–3(x–y)2as5minusapositivenumbertimesasquareandusethattorealizethatitsvaluecannotbemorethan5foranyrealnumbersxandy.

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ProblemoftheMonthBetweentheLines

TaskDescription–LevelBThistaskchallengesstudentstovisualizedifferentwaystoconfigurefourdifferentsizedtrapezoids,usinganisoscelestrapezoidastheunitwhole.Thistaskthenchallengesstudentstonoticepatternsthatcanbeseenintheconstructionofthesefourfigures-suchastherelationshipbetweenlineardimensionsofthefourisoscelestrapezoidsandtheirareas.Thisdevelopmentofspatialsensewillsupportthedevelopingunderstandingsofattributesinplanegeometryandtheuseofbasictransformations,andthisexplorationofpatternswillsupportthedevelopingunderstandingoffunctionalrelationships.

CommonCoreStateStandardsMath-ContentStandardsGeometryReasonwithshapesandtheirattributes.1.G.1Distinguishbetweendefiningattributes(e.g.,trianglesareclosedandthreesided)versusnon-definingattributes(e.g.,color,orientation,overallsize);buildanddrawshapestopossessdefiningattributes.1.G.2Composetwo-dimensionalshapes(rectangles,squares,trapezoids,triangles,half-circles,andquarter-circles)orthree-dimensionalshapes(cubes,rightrectangularprisms,rightcircularcones,andrightcircularcylinders)tocreateacompositeshape,andcomposenewshapesfromthecompositeshape.*

*Studentsdonotneedtolearnformalnamessuchas“rightrectangularprism.”Reasonwithshapesandtheirattributes.2.G.1Recognizeanddrawshapeshavingspecifiedattributes,suchasagivennumberofanglesoragivennumberofequalfaces.**Identifytriangles,quadrilaterals,pentagons,hexagons,andcubes.

**Sizesarecompareddirectlyorvisually,notcomparedbymeasuring.Reasonwithshapesandtheirattributes.3.G.2Partitionshapesintopartswithequalareas.Expresstheareaofeachpartasaunitfractionofthewhole.Forexample,partitionashapeinto4partswithequalarea,anddescribetheareaofeachpartas¼oftheareaoftheshape.OperationsandAlgebraicThinkingGenerateandanalyzepatterns.4.OA.5Generateanumberorshapepatternthatfollowsagivenrule.Identifyapparentfeaturesofthepatternthatwerenotexplicitintheruleitself.FunctionsDefine,evaluate,andcomparefunctions.8.F.1Understandthatafunctionisarulethatassignstoeachinputexactlyoneoutput.Thegraphofafunctionisthesetoforderedpairsconsistingofaninputandthecorrespondingoutput.HighSchool–Functions-BuildingFunctionsBuildafunctionthatmodelsarelationshipbetweentwoquantities.F-BF.1Writeafunctionthatdescribesarelationshipbetweentwoquantities.F-BF.1a.Determineanexplicitexpression,arecursiveprocess,orstepsforcalculationfromacontext.

CommonCoreStateStandardsMath–StandardsofMathematicalPracticeMP.6 Attendtoprecision.Mathematicallyproficientstudentstrytocommunicatepreciselytoothers.Theytrytousecleardefinitionsindiscussionwithothersandintheirownreasoning.Theystatethemeaningofthesymbolstheychoose,includingusingtheequalsignconsistentlyandappropriately.Theyarecarefulaboutspecifyingunitsofmeasure,andlabelingaxestoclarifythecorrespondencewithquantitiesinaproblem.Theycalculateaccuratelyandefficiently,expressnumericalanswerswithadegreeofprecisionappropriatefortheproblemcontext.Intheelementarygrades,studentsgivecarefullyformulatedexplanationstoeachother.Bythetimetheyreachhighschooltheyhavelearnedtoexamineclaimsandmakeexplicituseofdefinitions.MP.7Lookforandmakeuseofstructure.Mathematicallyproficientstudentslookcloselytodiscernapatternorstructure.Youngstudents,forexample,mightnoticethatthreeandsevenmoreisthesameamountassevenandthreemore,ortheymaysortacollectionofshapesaccordingtohowmanysidestheshapeshave.Later,studentswillsee7x8equalsthewell-remembered7x5+7x3,inpreparationforlearningaboutthedistributiveproperty.Intheexpressionx2+9x+14,olderstudentscanseethe14as2x7andthe9as2+7.Theyrecognizethesignificanceofanexistinglineinageometricfigureandcanusethestrategyofdrawinganauxiliarylineforsolvingproblems.Theyalsocanstepbackforanoverviewandshiftperspective.Theycanseecomplicatedthings,suchassomealgebraicexpressions,assingleobjectsorasbeingcomposedofseveralobjects.Forexample,theycansee5–3(x–y)2as5minusapositivenumbertimesasquareandusethattorealizethatitsvaluecannotbemorethan5foranyrealnumbersxandy.

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ProblemoftheMonthBetweentheLines

TaskDescription–LevelCThistaskchallengesstudentstofindtheareasofdifferenttypesoftrianglesongeoboardgridpaper.Anunderstandingoftherelationshipsbetweenrectanglesandtrianglesisoneavenuefordeterminingtheseareas.

CommonCoreStateStandardsMath-ContentStandardsMeasurementandDataGeometricmeasurement:understandconceptsofareaandrelateareatomultiplicationandtoaddition.3.MD.5Recognizeareaasanattributeofplanefiguresandunderstandconceptsofareameasurement.3.MD.5a.Asquarewithsidelength1unit,called“aunitsquare,”issaidtohave“onesquareunit”ofarea,andcanbeusedtomeasurearea.3.MD.5b.Aplanefigurewhichcanbecoveredwithoutgapsoroverlapsbynunitsquaresissaidtohaveanareaofnsquareunits.3.MD.6Measureareasbycountingunitsquares(squarecm,squarem,squarein,squareft,andimprovisedunits).Drawandidentifylinesandangles,andclassifyshapesbypropertiesoftheirlinesandangles.4.G.1Classifytwo-dimensionalfiguresbasedonthepresenceorabsenceofparallelorperpendicularlines,orthepresenceorabsenceofanglesofaspecifiedsize.Recognizerighttrianglesasacategory,andidentifyrighttriangles.Solvereal-worldandmathematicalproblemsinvolvingarea,surfacearea,andvolume.6.G.1Findtheareaofrighttriangles,othertriangles,specialquadrilaterals,andpolygonsbycomposingintorectanglesordecomposingintotrianglesandothershapes;applythesetechniquesinthecontextofreal-worldandmathematicalproblems.

CommonCoreStateStandardsMath–StandardsofMathematicalPracticeMP.6Attendtoprecision.Mathematicallyproficientstudentstrytocommunicatepreciselytoothers.Theytrytousecleardefinitionsindiscussionwithothersandintheirownreasoning.Theystatethemeaningofthesymbolstheychoose,includingusingtheequalsignconsistentlyandappropriately.Theyarecarefulaboutspecifyingunitsofmeasure,andlabelingaxestoclarifythecorrespondencewithquantitiesinaproblem.Theycalculateaccuratelyandefficiently,expressnumericalanswerswithadegreeofprecisionappropriatefortheproblemcontext.Intheelementarygrades,studentsgivecarefullyformulatedexplanationstoeachother.Bythetimetheyreachhighschooltheyhavelearnedtoexamineclaimsandmakeexplicituseofdefinitions.MP.7Lookforandmakeuseofstructure.Mathematicallyproficientstudentslookcloselytodiscernapatternorstructure.Youngstudents,forexample,mightnoticethatthreeandsevenmoreisthesameamountassevenandthreemore,ortheymaysortacollectionofshapesaccordingtohowmanysidestheshapeshave.Later,studentswillsee7x8equalsthewell-remembered7x5+7x3,inpreparationforlearningaboutthedistributiveproperty.Intheexpressionx2+9x+14,olderstudentscanseethe14as2x7andthe9as2+7.Theyrecognizethesignificanceofanexistinglineinageometricfigureandcanusethestrategyofdrawinganauxiliarylineforsolvingproblems.Theyalsocanstepbackforanoverviewandshiftperspective.Theycanseecomplicatedthings,suchassomealgebraicexpressions,assingleobjectsorasbeingcomposedofseveralobjects.Forexample,theycansee5–3(x–y)2as5minusapositivenumbertimesasquareandusethattorealizethatitsvaluecannotbemorethan5foranyrealnumbersxandy.

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ProblemoftheMonth:BetweentheLines

TaskDescription–LevelDThistaskchallengesstudentstofindtherelationshipbetweenthesizesofthreegivenoverlappingsquares.Anunderstandingofanglerelationshipsanddefinitions,congruency,andthePythagoreanTheoremdeepenstudents’understandingandaccesstothemathematicsinthislevel.

CommonCoreStateStandardsMath-ContentStandardsGeometryUnderstandcongruenceandsimilarityusingphysicalmodels,transparencies,orgeometrysoftware.8.G.2Understandthatatwo-dimensionalfigureiscongruenttoanotherifthesecondcanbeobtainedfromthefirstbyasequenceofrotations,reflections,andtranslations;giventwocongruentfigures,describeasequencethatexhibitsthecongruencebetweenthem.UnderstandandapplythePythagoreanTheorem.8.G.7ApplythePythagoreanTheoremtodetermineunknownsidelengthsinrighttrianglesinreal-worldandmathematicalproblemsintwoandthreedimensions.HighSchool–Geometry-CongruenceUnderstandcongruenceintermsofrigidmotionsG-CO.7Usethedefinitionofcongruenceintermsofrigidmotionstoshowthattwotrianglesarecongruentifandonlyifcorrespondingpairsofsidesandcorrespondingpairsofanglesarecongruent.ProvegeometrictheoremsG-CO.9Provetheoremsaboutlinesandangles.Theoremsinclude:verticalanglesarecongruent;whenatransversalcrossesparallellines,alternateinterioranglesarecongruentandcorrespondinganglesarecongruent;pointsonaperpendicularbisectorofalinesegmentareexactlythoseequidistantfromthesegment’sendpoints.G-CO.10Provetheoremsabouttriangles.Theoremsinclude:measuresofinterioranglesofatrianglesumto180!;baseanglesofisoscelestrianglesarecongruent;thesegmentjoiningmidpointsoftwosidesofatriangleisparalleltothethirdsideandhalfthelength;themediansofatrianglemeetatapoint.

CommonCoreStateStandardsMath–StandardsofMathematicalPracticeMP.6Attendtoprecision.Mathematicallyproficientstudentstrytocommunicatepreciselytoothers.Theytrytousecleardefinitionsindiscussionwithothersandintheirownreasoning.Theystatethemeaningofthesymbolstheychoose,includingusingtheequalsignconsistentlyandappropriately.Theyarecarefulaboutspecifyingunitsofmeasure,andlabelingaxestoclarifythecorrespondencewithquantitiesinaproblem.Theycalculateaccuratelyandefficiently,expressnumericalanswerswithadegreeofprecisionappropriatefortheproblemcontext.Intheelementarygrades,studentsgivecarefullyformulatedexplanationstoeachother.Bythetimetheyreachhighschooltheyhavelearnedtoexamineclaimsandmakeexplicituseofdefinitions.MP.7Lookforandmakeuseofstructure.Mathematicallyproficientstudentslookcloselytodiscernapatternorstructure.Youngstudents,forexample,mightnoticethatthreeandsevenmoreisthesameamountassevenandthreemore,ortheymaysortacollectionofshapesaccordingtohowmanysidestheshapeshave.Later,studentswillsee7x8equalsthewell-remembered7x5+7x3,inpreparationforlearningaboutthedistributiveproperty.Intheexpressionx2+9x+14,olderstudentscanseethe14as2x7andthe9as2+7.Theyrecognizethesignificanceofanexistinglineinageometricfigureandcanusethestrategyofdrawinganauxiliarylineforsolvingproblems.Theyalsocanstepbackforanoverviewandshiftperspective.Theycanseecomplicatedthings,suchassomealgebraicexpressions,assingleobjectsorasbeingcomposedofseveralobjects.Forexample,theycansee5–3(x–y)2as5minusapositivenumbertimesasquareandusethattorealizethatitsvaluecannotbemorethan5foranyrealnumbersxandy.

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ProblemoftheMonthBetweentheLines

TaskDescription–LevelEThistaskchallengesstudentstohaveastrongunderstandingoftransformationalgeometryandpropertiesoftriangles.

CommonCoreStateStandardsMath-ContentStandardsHighSchool–Geometry–CongruenceExperimentwithtransformationsintheplaneG-CO.2Representtransformationsintheplaneusing,e.g.,transparenciesandgeometrysoftware;describetransformationsasfunctionsthattakepointsintheplaneasinputsandgiveotherpointsasoutputs.Comparetransformationsthatpreservedistanceandangletothosethatdonot(e.g.,translationversushorizontalstretch).G-CO.3Givenarectangle,parallelogram,trapezoidorregularpolygon,describetherotationsandreflectionsthatcarryitontoitself.G-CO.4Developdefinitionsofrotations,reflections,andtranslationsintermsofangles,circles,perpendicularlines,parallellines,andlinesegments.UnderstandcongruenceintermsofrigidmotionsG-CO.7Usethedefinitionofcongruenceintermsofrigidmotionstoshowthattwotrianglesarecongruentifandonlyifcorrespondingpairsofsidesandcorrespondingpairsofanglesarecongruent.

CommonCoreStateStandardsMath–StandardsofMathematicalPracticeMP.6Attendtoprecision.Mathematicallyproficientstudentstrytocommunicatepreciselytoothers.Theytrytousecleardefinitionsindiscussionwithothersandintheirownreasoning.Theystatethemeaningofthesymbolstheychoose,includingusingtheequalsignconsistentlyandappropriately.Theyarecarefulaboutspecifyingunitsofmeasure,andlabelingaxestoclarifythecorrespondencewithquantitiesinaproblem.Theycalculateaccuratelyandefficiently,expressnumericalanswerswithadegreeofprecisionappropriatefortheproblemcontext.Intheelementarygrades,studentsgivecarefullyformulatedexplanationstoeachother.Bythetimetheyreachhighschooltheyhavelearnedtoexamineclaimsandmakeexplicituseofdefinitions.MP.7Lookforandmakeuseofstructure.Mathematicallyproficientstudentslookcloselytodiscernapatternorstructure.Youngstudents,forexample,mightnoticethatthreeandsevenmoreisthesameamountassevenandthreemore,ortheymaysortacollectionofshapesaccordingtohowmanysidestheshapeshave.Later,studentswillsee7x8equalsthewell-remembered7x5+7x3,inpreparationforlearningaboutthedistributiveproperty.Intheexpressionx2+9x+14,olderstudentscanseethe14as2x7andthe9as2+7.Theyrecognizethesignificanceofanexistinglineinageometricfigureandcanusethestrategyofdrawinganauxiliarylineforsolvingproblems.Theyalsocanstepbackforanoverviewandshiftperspective.Theycanseecomplicatedthings,suchassomealgebraicexpressions,assingleobjectsorasbeingcomposedofseveralobjects.Forexample,theycansee5–3(x–y)2as5minusapositivenumbertimesasquareandusethattorealizethatitsvaluecannotbemorethan5foranyrealnumbersxandy.

CCSSMAlignment:ProblemoftheMonth BetweentheLines©NoyceFoundation2015.ThisworkislicensedunderaCreativeCommonsAttribution-NonCommercial-NoDerivatives3.0UnportedLicense(http://creativecommons.org/licenses/by-nc-nd/3.0/deed.en_US).

ProblemoftheMonthBetweentheLines

TaskDescription–PrimaryLevelThistaskchallengesstudentstonameanddefineattributesoftwo-dimensionalshapes(triangles,squares,rhombi,trapezoids,and/orhexagons).Thistaskalsochallengesstudentstorecognizeafigurewithtwodifferentorientationsfromasetofshapes(triangles,squares,rhombi,trapezoids,and/orhexagons).Thisdevelopmentofspatialsensewillsupportthedevelopingunderstandingofattributesinplanegeometryandtheuseofbasictransformations.

CommonCoreStateStandardsMath-ContentStandardsGeometryIdentifyanddescribeshapes(squares,circles,triangles,rectangles,hexagons,cubes,cones,cylinders,andspheres.)K.G.1Describeobjectsintheenvironmentusingnamesofshapes,anddescribetherelativepositionsoftheseobjectsusingtermssuchasabove,below,beside,infrontof,behind,andnextto.K.G.2Correctlynameshapesregardlessoftheirorientationsoroverallsize.Analyze,compare,create,andcomposeshapes.K.G.6Composesimpleshapestoformlargershapes.

CommonCoreStateStandardsMath–StandardsofMathematicalPracticeMP.6Attendtoprecision.Mathematicallyproficientstudentstrytocommunicatepreciselytoothers.Theytrytousecleardefinitionsindiscussionwithothersandintheirownreasoning.Theystatethemeaningofthesymbolstheychoose,includingusingtheequalsignconsistentlyandappropriately.Theyarecarefulaboutspecifyingunitsofmeasure,andlabelingaxestoclarifythecorrespondencewithquantitiesinaproblem.Theycalculateaccuratelyandefficiently,expressnumericalanswerswithadegreeofprecisionappropriatefortheproblemcontext.Intheelementarygrades,studentsgivecarefullyformulatedexplanationstoeachother.Bythetimetheyreachhighschooltheyhavelearnedtoexamineclaimsandmakeexplicituseofdefinitions.MP.7Lookforandmakeuseofstructure.Mathematicallyproficientstudentslookcloselytodiscernapatternorstructure.Youngstudents,forexample,mightnoticethatthreeandsevenmoreisthesameamountassevenandthreemore,ortheymaysortacollectionofshapesaccordingtohowmanysidestheshapeshave.Later,studentswillsee7x8equalsthewell-remembered7x5+7x3,inpreparationforlearningaboutthedistributiveproperty.Intheexpressionx2+9x+14,olderstudentscanseethe14as2x7andthe9as2+7.Theyrecognizethesignificanceofanexistinglineinageometricfigureandcanusethestrategyofdrawinganauxiliarylineforsolvingproblems.Theyalsocanstepbackforanoverviewandshiftperspective.Theycanseecomplicatedthings,suchassomealgebraicexpressions,assingleobjectsorasbeingcomposedofseveralobjects.Forexample,theycansee5–3(x–y)2as5minusapositivenumbertimesasquareandusethattorealizethatitsvaluecannotbemorethan5foranyrealnumbersxandy.