problem of the month: circular reasoning · circular reasoning level a janet and lydia want to...

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ProblemoftheMonth:CircularReasoningTheProblemsoftheMonth(POM)areusedinavarietyofwaystopromoteproblemsolvingandtofosterthefirststandardofmathematicalpracticefromtheCommonCoreStateStandards:“Makesenseofproblemsandpersevereinsolvingthem.”ThePOM may be used by a teacher to promote problem solving and to address thedifferentiatedneedsofherstudents.Adepartmentorgradelevelmayengagetheirstudents in a POM to showcase problem solving as a key aspect of doingmathematics. POMs can also be used school wide to promote a problem-solvingthemeataschool.Thegoalisforallstudentstohavetheexperienceofattackingandsolving non-routine problems anddeveloping theirmathematical reasoning skills.Although obtaining and justifying solutions to the problems is the objective, theprocessoflearningtoproblemsolveisevenmoreimportant.TheProblemoftheMonthisstructuredtoprovidereasonabletasksforallstudentsinaschool.ThePOMisdesignedwithashallowfloorandahighceiling,sothatallstudentscanproductivelyengage, struggle,andpersevere.ThePrimaryVersion isdesigned to be accessible to all students and especially as the key challenge forgradeskindergartenandone.LevelAwillbechallengingformostsecondandthirdgraders. Level B may be the limit of where fourth and fifth-grade students havesuccessandunderstanding.LevelCmaystretchsixthandseventh-gradestudents.LevelDmaychallengemosteighthandninth-gradestudents,andLevelEshouldbechallenging formosthigh school students.These grade-level expectations are justestimatesandshouldnotbeusedasanabsoluteminimumexpectationormaximumlimitation for students. Problem solving is a learned skill, and studentsmay needmanyexperiencestodeveloptheirreasoningskills,approaches,strategies,andtheperseverancetobesuccessful.TheProblemoftheMonthbuildsonsequentiallevelsofunderstanding.AllstudentsshouldexperienceLevelAandthenmovethroughthetasks in order to go as deeply as they can into the problem. Therewill be thosestudentswhowillnothaveaccessintoevenLevelA. Educatorsshouldfeelfreetomodifythetasktoallowaccessatsomelevel.OverviewIntheProblemoftheMonthCircularReasoning,studentsusegeometricreasoningtosolveproblemsinvolvingcirclesandthenumberπ.Themathematicaltopicsthatunderlie this POMare the attributes of polygons, circles, the irrational numberπ,spatialvisualization,andanglemeasurement.InthefirstlevelofthePOM,studentsarepresentedwiththetaskofexaminingtherelationshipbetweenthemeasureofthediameterofacircleanditscircumference.Their task involves determining the number of times bigger the circumference isthan the diameter (approximation of π). Level B requires students to continue toinvestigate the number π by investigating the relationship between themeasurements of a tennis ball can. They find the height of the can and the

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circumferenceofthebaseofthecan.Theyreasonastowhythetwomeasurementsareclosetothesamesizeandtheysearchforapatterntodeterminearationale.InLevelC,studentsinvestigatedifferentsizedpizzasmadeintheformofacircle.Thepizzasaredividedintofractionalslicesshapedlikeasectorofacircle.Thestudentsinvestigate the relationships between the sizes of the pizzas, including thedimensions of the slices (radii and degrees), the costs, andwhether the costs areproportionaltotheirsizes.InLevelD,thestudentsinvestigatetheringsthatsupportwoodenbarrels. In thisproblem, the ringsaremadea foot longer than theyweresupposedtobe.Since thebarrelscome inarangeofsizes fromvery large toverysmall, the foot-long error must be explored. The goal is to determine therelationshipbetweentheerrorintheringsandtheoriginalsize,andhowtheextralength affects a redesigned barrel. In Level E, students investigate Archimedes’approachtocalculatinganapproximationforπ.Studentsareaskedtoduplicatethedrawings and calculations of Archimedes and determine the accuracy of hisapproximation.Theyarealsoaskedtojustifytheformulafortheareaofacircle.MathematicalConceptsGeometry and measurement have important real-world applications. Whetherdesigning the most complex structures created by designers, architects, andconstruction workers or arranging the furniture in a room, geometry andmeasurementareessential.InthisPOM,studentsexplorevariousaspectsofcirculargeometry. This includes understanding the attributes of circles. In particular,students study regular polygons, diameters, circumference, and angles. Theattributes studied include the diagonals of polygons as well as angles and theirmeasurements.Studentsexploretheirrationalnumberπ.InLevelE,studentsuseanhistoricalapproachtocalculateanapproximationofπ.Inadditiontothegeometricandmeasurement aspects of this POM, the students are seeking to find patterns,developfunctionalrelationships,makegeneralizations,andjustifytheirconclusions.Themathematicsinvolveshigher-levelcognitiveskills.

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ProblemoftheMonth

CircularReasoningLevelAJanetandLydiawant to learnmoreaboutcircles.Theydecidetomeasuredifferentsizedcircles that they can find. They measure the circles in two ways. One way is across acenterlineof the circle.The centerline is called thediameter.Anotherway is around theoutsideofthecircle.Thedistancearoundtheoutsideofacircleiscalledthecircumference.Measurevarioussizedcirclesusingeitherphysicalobjectsortheincludedpageofcircles.Measureboththecircumferenceanddiameterofeachcirclewithatapemeasure(paperorclothtape).Createatablethatcomparesthelengthofthediameterwiththecircumferenceofthecircle.Measureatleastfivedifferentsizedcircles.

Diameter Circumference

Comparethenumbersbetweenthetwocolumns.Whatpatternsdoyousee?Explainanyrelationshipyouseebetweenthelengthofthediameterandthecircumference.

π π


LevelBMeasureatennisballcan.Firstmeasuretheheightofthecan,fromthetopofthecantothebottom.Recordthemeasurement.

Next,measurethecircumferencearoundthecan.Beaccuratewithyourmeasurements.

Comparethemeasurements.Explaintherelationshipbetweenthetwomeasurements.Howdoesthenumberofballsinthecanrelatetothemeasurementsandtheirrelationship?Whatotherpartsofthecanortennisballscanyoumeasureandcompare?Describetherelationshipofthosemeasurements?Howaretheyrelated?


LevelCYouworkforapizzaparlor.Youcurrentlyhavethreesizesofpizza:regular(area25squareinches);large(area36squareinches);andgiant(area64squareinches).Thestoreownerhasaskedyoutodeterminesomeinformationabouttheamountofpizzathatismade.Howmuchbiggeristhegiantpizzathanthelargepizzaintermsofarea?Howmuchbiggeristhelargepizzathantheregularpizzaintermsofarea?Theownersellsaregularcheesepizzafor$8.25.Hewantstoknowwhattochargeforthelarge and giant sized pizzas. He wants to cover the cost of the ingredients andmake aproportionalamountofprofitontheothertwopizzas.Whatshouldhechargefora largeandgiantcheesepizza?Theownersellspizzabytheslice.Hehasbeensellingalargepizzabytheslicefor$1.35.Eachpizzaiscutintoeightequalslices.Theownerwantstoknowwhetherhehaspriceditright tomakemoremoneysellingbytheslice thanbysellingthewholepizza.Explaintohim,usingmathematics,howhispricesfortheslicesandthewholepizzacompare.Nowtheownerwantstocuteachpizzaintonineslicesinsteadofeight.Howmuchsmallerwill the slices be? He is interested in comparing both the width (angle in numbers ofdegrees)andtheareaofeachslice.Theownerwantstosellthenineslicesatthesamepriceashedidwhenthepizzawascutintoeight slices.Hewants toknowhowmuchmorehewillmakecutting thepizzas intosmallerslices.Hewants tocompare thepricesof selling thenineslices individuallywiththepriceofsellinganentirelargepizza.Heasksyoutocalculateandexplainthedifferenceinpricesandamounts.Theowneralsothinkshecansavemoremoneyifhesellsslicesfromregularsizedpizzasinsteadof large sizedpizzas. If youcutboth the regularand largepizzas intonineequalslices,howmuchlargerisasingleslicefromalargepizzathanthatofaslicefromaregularpizza?Explainyouranswerintermsofdegreesoftheangleandareaoftheslice.

Regular 25 sq. in. Large

36 sq. in. Giant 64 sq. in.


Level D You are a quality controller for a barrel manufacturing company. Your companymakesbarrels of all different sizes. The companymakes very large storage vats for vineyards.They evenmake very small barrels as earrings that are sold in the gift shopnext to thefactory.Withoutexception,thecompanymakeseverybarrelthesameway.Theyusewoodsupported with metal rings that go around the barrels. A computer controls the ring-makingmachine.Thecomputermalfunctionedandmadeeveryring,fromthesmallestfortheearringstothelargestforthewinevats,largerthannormalbyexactlyonefoot.Inotherwords,everyringyourfactoryhasturnedoutrecentlyhasacircumferencethatistwelveincheslongerthantherightsize.Yourbosshasaskedyoutoinvestigatethismatter.

Howdothealteredringscompareto theoriginalrings?Howmuchbiggerdothealteredrings look incomparison to theringson thebarrels that theyweredesigned to fit?Howwill thealteredringscompare insize to theringson thesmallbarrels, like theearrings?Howwillthealteredringscomparetotheringsonthelarge�barrels,likethewinevats?Whatshouldthecompanydowithallthosealteredrings?Itwouldbeverycostlytodisposeofthem.Yourbosshasaskedyoutotellhimwhatsizeofbarrelsneedtobemadetousethoserings.Arethebarrelsusable?Whatsizewouldyouhavetomakethenewbarrelsinrelationshiptotheoriginalsizedbarrels? What conclusion can you draw from your findings? Can you explain your findings usingmathematics?Whyaretheresultswhattheyare?


LevelEArchimedes isconsideredoneof thegreatestmathematiciansofall time. He lived in theGreek cityof Syracuse, Sicily,between287BCand212BCand isknown forhiswork inmanyareasofmathematicsandscience.Archimedes’mathematicalwork includes inventingmethods for calculating the area andvolume of geometric objects such as circles, parabolas, cylinders, cones, spheres,hyperbolasandellipses.Healsospenttimeinvestigatingarationalnumberapproximationof π. Themethod he used to approximateπ involved examining the size ofmany-sidedpolygons.Archimedes,asthestoryistold,drewalargecircleonthefloorofhisapartment.Hemeasuredthediameterofthecircleandcalledthatdistanceoneunitinlength.Nexthebegan to inscribe and circumscribe different sized polygons, tangent to the circle.Archimedes would measure the perimeters of the polygons and average the two toapproximateπ.Asthenumbersofsidesof thepolygonswere increased, theshapeof thepolygonlookedmorelikethecircle.Thepolygonsgotcloserandclosertobeingthesizeofthecircleandtheperimetersbecameeverclosertothevalueofπ.Usingthatmethod,hewasabletocalculatetheapproximationofπ,correcttoavaluebetween31/7and310/71.How precise was his approximation of π? Give your answer in terms of the number ofcorrectdecimalplaces.Estimate howmany sides his final polygonneeded to be to get a number that accurate?Explainyourestimate.IllustratethemethodArchimedesusedbydrawinganinteriorregularpolygoninscribedinacircleandanexteriorregularpolygoncircumscribedabout thesamecircle. Determinetheperimeterofbothpolygonsandshowhowπmightbeapproximated.Findtheareaofbothpolygonsandcomparetheareaswiththatofthecircle.Explainhowtheareaformulaisderivedfromtheareaformulaofthepolygon.


Problem of the Month Circular Reasoning

Primary Version Level A Materials: Apicture of a circle, circular objects (i.e. counters, coins), theπ-day picture,paperandpencil. Discussionontherug:Theteacherholdsupapictureofacircle. “What is the name of this shape?” Students respond to the question. The teacher hands out circular objects. “These objects are shaped like a circle. Tell us about the shapes?” Studentsexploretheobjectsanddescribethecircles.Theteacherholdsupacircle. “A circle is a special shape. Long ago people discovered the circle and tried to measure it. They found a special number called pi. Some people who like math celebrate circles on a special day, March 14. It is called pi day. Here is a picture of people having a picnic on pi day.” Theteacherhandsoutthepictureoftheπ-daypicnic.Insmallgroups:Eachstudenthasthepictureoftheπ-daypicnic. “Look at the picture of the picnic. Find all the things in the picture that are shaped like a circle. Circle each shape you find.” Studentsanswerthefollowingquestions:How many things did you find that are shaped like a circle? Name the things in the picture that are shaped like a circle. At theendof the investigation,havestudentseitherdiscussordictatea response to thissummaryquestionbelow:Explain how you decided which ones to circle.

π π


π day picnic


Diameter Circumference

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ProblemoftheMonthCircularReasoningTaskDescription–LevelA

Thistaskchallengesstudentstoexplorevariousaspectsofcirculargeometry.Studentsareaskedtoexamine the relationship between the measure of the diameter of a circle and its circumference.Theirtaskinvolvesdeterminingthenumberoftimesbiggerthecircumferenceisthanthediameter(approximationofπ).

CommonCoreStateStandardsMath-ContentStandardsGeometryIdentifyanddescribeshapes(squares,circles,triangles,rectangles,hexagons,cubes,cones,cylinders,andspheres.)K.G.1Describeobjects in theenvironmentusingnamesof shapes,anddescribe therelativepositionsof theseobjectsusingtermssuchasabove,below,beside,infrontof,behind,andnextto.K.G.2Correctlynameshapesregardlessoftheirorientationsoroverallsize.MeasurementandDataDescribeandcomparemeasurableattributes.K.MD.1Describemeasurableattributesofobjects,suchaslengthorweight.Describemeasurableattributesofasingleobject.Measureandestimatelengthsinstandardunits.2.MD.1Measurethelengthofanobjectbyselectingandusingappropriatetoolssuchasrulers,yardsticks,metersticks,andmeasuringtapes.2.MD.4Measure to determine howmuch longer one object is than another, expressing the length difference in terms of astandardlengthunit.OperationsandAlgebraicThinkingGenerateandanalyzepatterns.4.OA.5Generateanumberorshapepatternthatfollowsagivenrule.…ExpressionsandEquationsSolvereal-lifeandmathematicalproblemsusingnumericalandalgebraicexpressionsandequations.7.EE.4Use variables to represent quantities in a real-world ormathematical problem, and construct simple equations andinequalitiestosolveproblemsbyreasoningaboutthequantities.FunctionsDefine,evaluate,andcomparefunctions.8.F.1Understandthatafunctionisarulethatassignstoeachinputexactlyoneoutput.…Usefunctionstomodelrelationshipsbetweenquantities.8.F.4Constructafunctiontomodelalinearrelationshipbetweentwoquantities.…HighSchool–Functions-BuildingFunctionsBuildafunctionthatmodelsarelationshipbetweentwoquantities.F-BF.1Writeafunctionthatdescribesarelationshipbetweentwoquantities.F-BF.1a.Determineanexplicitexpression,arecursiveprocess,orstepsforcalculationfromacontext.

CommonCoreStateStandardsMath–StandardsofMathematicalPracticeMP.2Reasonabstractlyandquantitatively.Mathematically proficient studentsmake sense of quantities and their relationships in problem situations. They bring twocomplementaryabilitiestobearonproblemsinvolvingquantitativerelationships:theabilitytodecontextualize—toabstractagiven situation and represent it symbolically andmanipulate the representing symbols as if they have a life of their own,withoutnecessarilyattendingtotheirreferents—andtheabilitytocontextualize,topauseasneededduringthemanipulationprocess in order to probe into the referents for the symbols involved. Quantitative reasoning entails habits of creating acoherentrepresentationof theproblemathand;consideringtheunits involved;attendingto themeaningofquantities,notjusthowtocomputethem;andknowingandflexiblyusingdifferentpropertiesofoperationsandobjects.MP.4Modelwithmathematics.Mathematicallyproficientstudentscanapplythemathematicstheyknowtosolveproblemsarisingineverydaylife,society,andtheworkplace.Inearlygrades,thismightbeassimpleaswritinganadditionequationtodescribeasituation.Inmiddlegrades,astudentmightapplyproportionalreasoningtoplanaschooleventoranalyzeaprobleminthecommunity.Byhighschool, a studentmightusegeometry to solveadesignproblemorusea function todescribehowonequantityof interestdependsonanother.Mathematicallyproficientstudentswhocanapplywhattheyknowarecomfortablemakingassumptionsandapproximationstosimplifyacomplicatedsituation,realizingthatthesemayneedrevisionlater.Theyareabletoidentifyimportantquantitiesinapracticalsituationandmaptheirrelationshipsusingsuchtoolsasdiagrams,two-waytables,graphs,flowchartsandformulas.Theycananalyzethoserelationshipsmathematicallytodrawconclusions.Theyroutinelyinterprettheirmathematicalresults inthecontextofthesituationandreflectonwhethertheresultsmakesense,possiblyimprovingthemodelifithasnotserveditspurpose.


ProblemoftheMonthCircularReasoningTaskDescription–LevelB

Thistaskchallengesstudentstoexplorevariousaspectsofcirculargeometry.Studentsarerequiredtocontinuetoinvestigatethenumberπbyinvestigatingtherelationshipbetweenthemeasurementsofatennisballcan-suchastheheightofthecanandthecircumferenceofthebaseofthecan.Theyare toreasonwhythe twomeasurementsareclose to thesamesizeand tosearch forapattern todeterminearationaleforthisrelationship.Additionally,studentsareaskedtoexploreanddescribetherelationshipbetweenthenumberofballsinacanandtheheightofacan.

CommonCoreStateStandardsMath-ContentStandardsGeometryIdentifyanddescribeshapes(squares,circles,triangles,rectangles,hexagons,cubes,cones,cylinders,andspheres).K.G.1Describeobjects in theenvironmentusingnamesofshapes,anddescribe therelativepositionsof theseobjectsusingtermssuchasabove,below,beside,infrontof,behind,andnextto.K.G.2Correctlynameshapesregardlessoftheirorientationsoroverallsize.MeasurementandDataDescribeandcomparemeasurableattributes.K.MD.1Describemeasurableattributesofobjects,suchaslengthorweight.Describemeasurableattributesofasingleobject.Measureandestimatelengthsinstandardunits.2.MD.1Measurethelengthofanobjectbyselectingandusingappropriatetoolssuchasrulers,yardsticks,metersticks,andmeasuringtapes.2.MD.4Measure to determine howmuch longer one object is than another, expressing the length difference in terms of astandardlengthunit.OperationsandAlgebraicThinkingGenerateandanalyzepatterns.4.OA.5Generateanumberorshapepatternthatfollowsagivenrule.…ExpressionsandEquationsSolvereal-lifeandmathematicalproblemsusingnumericalandalgebraicexpressionsandequations.7.EE.4Use variables to represent quantities in a real-world ormathematical problem, and construct simple equations andinequalitiestosolveproblemsbyreasoningaboutthequantities.FunctionsDefine,evaluate,andcomparefunctions.8.F.1Understandthatafunctionisarulethatassignstoeachinputexactlyoneoutput.…Usefunctionstomodelrelationshipsbetweenquantities.8.F.4Constructafunctiontomodelalinearrelationshipbetweentwoquantities.…HighSchool–Functions-BuildingFunctionsBuildafunctionthatmodelsarelationshipbetweentwoquantities.F-BF.1Writeafunctionthatdescribesarelationshipbetweentwoquantities.F-BF.1a.Determineanexplicitexpression,arecursiveprocess,orstepsforcalculationfromacontext.

CommonCoreStateStandardsMath–StandardsofMathematicalPracticeMP.2Reasonabstractlyandquantitatively.Mathematically proficient studentsmake sense of quantities and their relationships in problem situations. They bring twocomplementaryabilitiestobearonproblemsinvolvingquantitativerelationships:theabilitytodecontextualize—toabstractagiven situation and represent it symbolically andmanipulate the representing symbols as if they have a life of their own,withoutnecessarilyattendingtotheirreferents—andtheabilitytocontextualize,topauseasneededduringthemanipulationprocess in order to probe into the referents for the symbols involved. Quantitative reasoning entails habits of creating acoherentrepresentationof theproblemathand;consideringtheunits involved;attendingto themeaningofquantities,notjusthowtocomputethem;andknowingandflexiblyusingdifferentpropertiesofoperationsandobjects.MP.4Modelwithmathematics.Mathematicallyproficientstudentscanapplythemathematicstheyknowtosolveproblemsarisingineverydaylife,society,andtheworkplace.Inearlygrades,thismightbeassimpleaswritinganadditionequationtodescribeasituation.Inmiddlegrades,astudentmightapplyproportionalreasoningtoplanaschooleventoranalyzeaprobleminthecommunity.Byhighschool, a studentmightusegeometry to solveadesignproblemorusea function todescribehowonequantityof interestdependsonanother.Mathematicallyproficientstudentswhocanapplywhattheyknowarecomfortablemakingassumptionsandapproximationstosimplifyacomplicatedsituation,realizingthatthesemayneedrevisionlater.Theyareabletoidentifyimportantquantitiesinapracticalsituationandmaptheirrelationshipsusingsuchtoolsasdiagrams,two-waytables,graphs,flowchartsandformulas.Theycananalyzethoserelationshipsmathematicallytodrawconclusions.Theyroutinelyinterprettheirmathematicalresults inthecontextofthesituationandreflectonwhethertheresultsmakesense,possiblyimprovingthemodelifithasnotserveditspurpose.


ProblemoftheMonthCircularReasoningTaskDescription–LevelC

Thistaskchallengesstudentstoexplorevariousaspectsofcirculargeometry.Studentsareaskedtoinvestigatedifferentsizedpizzasmadeintheformofacircle.Thepizzasaredividedintofractionalslicesshapedlikeasectorofacircle.Thestudentsinvestigatetherelationshipsbetweenthesizesofthepizzas,includingthedimensionsoftheslices(radiianddegrees),thecosts,andwhetherthecostsareproportionaltotheirsizes.

CommonCoreStateStandardsMath-ContentStandardsOperationsandAlgebraicThinkingRepresentandsolveproblemsinvolvingadditionandsubtraction.2.OA.1Useadditionandsubtractionwithin100tosolveone-andtwo-stepwordproblemsinvolvingsituationsofaddingto,takingfrom,puttingtogether,takingapart,andcomparing,…RatioandProportionalRelationshipsUnderstandratioconceptsanduseratioreasoningtosolveproblems.6.RP.3c.Findapercentofaquantityasarateper100(e.g.,30%ofaquantitymeans30/100timesthequantity);solveproblemsinvolvingfindingthewhole,givenapartandthepercent.Analyze proportional relationships and use them to solve real-world and mathematicalproblems.7.RP.1Recognizeandrepresentproportionalrelationshipsbetweenquantities.7.RP.3Useproportionalrelationshipstosolvemultistepratioandpercentproblems.…GeometrySolve real-life andmathematical problems involving anglemeasure, area, surface area, andvolume.7.G.4Knowtheformulasfortheareaandcircumferenceofacircleandusethemtosolveproblems…7.G.5 Use facts about…angles in amulti-step problem towrite and solve simple equations for anunknownangleinafigure.

CommonCoreStateStandardsMath–StandardsofMathematicalPracticeMP.2Reasonabstractlyandquantitatively.Mathematicallyproficientstudentsmakesenseofquantitiesandtheirrelationshipsinproblemsituations.Theybring two complementary abilities to bear on problems involving quantitative relationships: the ability todecontextualize—to abstract a given situation and represent it symbolically andmanipulate the representingsymbolsas if theyhavea lifeof theirown,withoutnecessarilyattendingtotheirreferents—andtheabilitytocontextualize,topauseasneededduringthemanipulationprocessinordertoprobeintothereferentsforthesymbolsinvolved.Quantitativereasoningentailshabitsofcreatingacoherentrepresentationoftheproblemathand;consideringtheunitsinvolved;attendingtothemeaningofquantities,notjusthowtocomputethem;andknowingandflexiblyusingdifferentpropertiesofoperationsandobjects.MP.4Modelwithmathematics.Mathematicallyproficientstudentscanapplythemathematicstheyknowtosolveproblemsarisingineverydaylife, society, and the workplace. In early grades, this might be as simple as writing an addition equation todescribeasituation. Inmiddlegrades,astudentmightapplyproportionalreasoningtoplanaschooleventoranalyzeaprobleminthecommunity.Byhighschool,astudentmightusegeometrytosolveadesignproblemoruseafunctiontodescribehowonequantityofinterestdependsonanother.Mathematicallyproficientstudentswho can apply what they know are comfortable making assumptions and approximations to simplify acomplicatedsituation,realizingthatthesemayneedrevisionlater.Theyareabletoidentifyimportantquantitiesin a practical situation and map their relationships using such tools as diagrams, two-way tables, graphs,flowcharts and formulas. They can analyze those relationships mathematically to draw conclusions. Theyroutinelyinterprettheirmathematicalresultsinthecontextofthesituationandreflectonwhethertheresultsmakesense,possiblyimprovingthemodelifithasnotserveditspurpose.


ProblemoftheMonth:CircularReasoningTaskDescription–LevelD

Thistaskchallengesstudentstoexplorevariousaspectsofcirculargeometry.Studentsareaskedtoinvestigatetheringsthatsupportwoodenbarrels.Inthisproblem,theringsaremadeafootlongerthan theywere supposed tobe. Since thebarrels come in a rangeof sizes fromvery large toverysmall, the foot-longerrormustbeexplored.Thegoal is todetermine the relationshipbetween theerrorintheringsandtheoriginalsize,andhowtheextralengthaffectsaredesignedbarrel.

CommonCoreStateStandardsMath-ContentStandardsGeometrySolve real-life andmathematical problems involving anglemeasure, area, surface area, andvolume.7.G.4Knowtheformulasfortheareaandcircumferenceofacircleandusethemtosolveproblems…7.G.6Solvereal-worldandmathematicalproblems involvingarea,volumeandsurfaceareaof two-andthree-dimensionalobjects…RatioandProportionalRelationshipsAnalyze proportional relationships and use them to solve real-world and mathematicalproblems.7.RP.2Recognizeandrepresentproportionalrelationshipsbetweenquantities.7.RP.3Useproportionalrelationshipstosolvemultistepratioandpercentproblems.…ExpressionsandEquationsSolve real-life and mathematical problems using numerical and algebraic expressions andequations.7.EE.4Usevariablestorepresentquantitiesinareal-worldormathematicalproblem,andconstructsimpleequationsandinequalitiestosolveproblemsbyreasoningaboutthequantities.7.EE.4a.Solvewordproblemsleadingtoequationsoftheformpx+q=randp(x+q)=r,wherep,q,andarespecificrationalnumbers.Solveequationsoftheseformsfluently.Compareanalgebraicsolutionto an arithmetic solution, identifying the sequence of the operations used in each approach. Forexample,theperimeterofarectangleis54cm.Itslengthis6cm.Whatisitswidth?

CommonCoreStateStandardsMath–StandardsofMathematicalPracticeMP.2Reasonabstractlyandquantitatively.Mathematicallyproficientstudentsmakesenseofquantitiesandtheirrelationshipsinproblemsituations.Theybring two complementary abilities to bear on problems involving quantitative relationships: the ability todecontextualize—to abstract a given situation and represent it symbolically andmanipulate the representingsymbolsas if theyhavea lifeof theirown,withoutnecessarilyattendingtotheirreferents—andtheabilitytocontextualize,topauseasneededduringthemanipulationprocessinordertoprobeintothereferentsforthesymbolsinvolved.Quantitativereasoningentailshabitsofcreatingacoherentrepresentationoftheproblemathand;consideringtheunitsinvolved;attendingtothemeaningofquantities,notjusthowtocomputethem;andknowingandflexiblyusingdifferentpropertiesofoperationsandobjects.MP.4Modelwithmathematics.Mathematicallyproficientstudentscanapplythemathematicstheyknowtosolveproblemsarisingineverydaylife, society, and the workplace. In early grades, this might be as simple as writing an addition equation todescribeasituation. Inmiddlegrades,astudentmightapplyproportionalreasoningtoplanaschooleventoranalyzeaprobleminthecommunity.Byhighschool,astudentmightusegeometrytosolveadesignproblemoruseafunctiontodescribehowonequantityofinterestdependsonanother.Mathematicallyproficientstudentswho can apply what they know are comfortable making assumptions and approximations to simplify acomplicatedsituation,realizingthatthesemayneedrevisionlater.Theyareabletoidentifyimportantquantitiesin a practical situation and map their relationships using such tools as diagrams, two-way tables, graphs,flowcharts and formulas. They can analyze those relationships mathematically to draw conclusions. Theyroutinelyinterprettheirmathematicalresultsinthecontextofthesituationandreflectonwhethertheresultsmakesense,possiblyimprovingthemodelifithasnotserveditspurpose.


ProblemoftheMonthCircularReasoningTaskDescription–LevelE

Thistaskchallengesstudentstoexplorevariousaspectsofcirculargeometry.Studentsareaskedtoinvestigate Archimedes’ approach to calculating an approximation for π. Students are asked toduplicate the drawings and calculations of Archimedes and determine the accuracy of hisapproximation.Theyarealsoaskedtojustifytheformulafortheareaofacircle.

CommonCoreStateStandardsMath-ContentStandardsGeometrySolve real-life andmathematical problems involving anglemeasure, area, surface area, andvolume.7.G.4Knowtheformulasfortheareaandcircumferenceofacircleandusethemtosolveproblems…7.G.6Solvereal-worldandmathematicalproblems involvingarea,volumeandsurfaceareaof two-andthree-dimensionalobjects…HighSchool–Geometry-CongruenceMakegeometricconstructions.G-CO.12Make formal geometric constructions with a variety of tools andmethods (compass andstraightedge, string, reflective devices, paper folding, dynamic geometric software, etc.). Copying asegment;copyinganangle;bisectingasegment;bisectinganangle;constructingperpendicularlines,includingtheperpendicularbisectorofalinesegment;andconstructingalineparalleltoagivenlinethroughapointnotontheline.G-CO.13Constructanequilateraltriangle,asquare,andaregularhexagoninscribedinacircle.HighSchool–Geometry-Similarity,RightTriangles,andTrigonometryApplytrigonometrytogeneraltriangles.G-SRT.11 (+) Understand and apply the Law of Sines and the Law of Cosines to find unknownmeasurementsinrightandnon-righttriangles(e.g.,surveyingproblems,resultantforces).

CommonCoreStateStandardsMath–StandardsofMathematicalPracticeMP.2Reasonabstractlyandquantitatively.Mathematically proficient students make sense of quantities and their relationships in problemsituations. They bring two complementary abilities to bear on problems involving quantitativerelationships: the ability to decontextualize—to abstract a given situation and represent itsymbolically andmanipulate the representing symbols as if they have a life of their own,withoutnecessarilyattendingtotheirreferents—andtheabilitytocontextualize,topauseasneededduringthemanipulationprocessinordertoprobeintothereferentsforthesymbolsinvolved.Quantitativereasoningentailshabitsof creatinga coherent representationof theproblemathand; consideringthe units involved; attending to the meaning of quantities, not just how to compute them; andknowingandflexiblyusingdifferentpropertiesofoperationsandobjects.MP.4Modelwithmathematics.Mathematicallyproficientstudentscanapplythemathematicstheyknowtosolveproblemsarisingin everyday life, society, and theworkplace. In early grades, thismight be as simple aswriting anaddition equation to describe a situation. In middle grades, a student might apply proportionalreasoningtoplanaschooleventoranalyzeaprobleminthecommunity.Byhighschool,astudentmight use geometry to solve a design problem or use a function to describe how one quantity ofinterestdependsonanother.Mathematicallyproficientstudentswhocanapplywhattheyknowarecomfortablemakingassumptionsandapproximations to simplifya complicatedsituation, realizingthat these may need revision later. They are able to identify important quantities in a practicalsituation and map their relationships using such tools as diagrams, two-way tables, graphs,flowchartsandformulas.Theycananalyzethoserelationshipsmathematicallytodrawconclusions.They routinely interpret their mathematical results in the context of the situation and reflect onwhethertheresultsmakesense,possiblyimprovingthemodelifithasnotserveditspurpose.


ProblemoftheMonthCircularReasoning

TaskDescription–PrimaryLevelThis task challenges students to explore various aspects of circular geometry. This level asksstudents to describe and identify circular shapes in real world objects such as coins, counters,containers,etc.Thenstudentsareaskedtocircleand identifyallshapes inagivenpicturethatareshapedlikeacircle.

CommonCoreStateStandardsMath-ContentStandardsGeometryIdentify anddescribe shapes (squares, circles, triangles, rectangles, hexagons, cubes, cones,cylinders,andspheres).K.G.1Describeobjectsintheenvironmentusingnamesofshapes,anddescribetherelativepositionsoftheseobjectsusingtermssuchasabove,below,beside,infrontof,behind,andnextto.K.G.2Correctlynameshapesregardlessoftheirorientationsoroverallsize.

CommonCoreStateStandardsMath–StandardsofMathematicalPracticeMP.2Reasonabstractlyandquantitatively.Mathematically proficient students make sense of quantities and their relationships in problemsituations. They bring two complementary abilities to bear on problems involving quantitativerelationships: the ability to decontextualize—to abstract a given situation and represent itsymbolically andmanipulate the representing symbols as if they have a life of their own,withoutnecessarilyattendingtotheirreferents—andtheabilitytocontextualize,topauseasneededduringthemanipulationprocessinordertoprobeintothereferentsforthesymbolsinvolved.Quantitativereasoningentailshabitsof creatinga coherent representationof theproblemathand; consideringthe units involved; attending to the meaning of quantities, not just how to compute them; andknowingandflexiblyusingdifferentpropertiesofoperationsandobjects.MP.4Modelwithmathematics.Mathematicallyproficientstudentscanapplythemathematicstheyknowtosolveproblemsarisingin everyday life, society, and theworkplace. In early grades, thismight be as simple aswriting anaddition equation to describe a situation. In middle grades, a student might apply proportionalreasoningtoplanaschooleventoranalyzeaprobleminthecommunity.Byhighschool,astudentmight use geometry to solve a design problem or use a function to describe how one quantity ofinterestdependsonanother.Mathematicallyproficientstudentswhocanapplywhattheyknowarecomfortablemakingassumptionsandapproximations to simplifya complicatedsituation, realizingthat these may need revision later. They are able to identify important quantities in a practicalsituation and map their relationships using such tools as diagrams, two-way tables, graphs,flowchartsandformulas.Theycananalyzethoserelationshipsmathematicallytodrawconclusions.They routinely interpret their mathematical results in the context of the situation and reflect onwhethertheresultsmakesense,possiblyimprovingthemodelifithasnotserveditspurpose.

problem of the month: circular reasoning · circular reasoning level a janet and lydia want to...

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