Problem of the Month: On Balance - Inside Mathematics of the Month: On Balance The Problems of the Month (POM) are used in a variety of ways to promote problem solving and to foster

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

1

ProblemoftheMonth:OnBalance

TheProblemsoftheMonth(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.ThePOMisstructuredwithashallowfloorandahighceiling,sothatallstudentscanproductivelyengage, struggle,andpersevere.ThePrimaryVersion isdesigned to be accessible to all students and especially as the key challenge forgradesK–1.LevelAwillbechallengingformostsecondandthirdgraders.LevelBmay be the limit of where fourth and fifth-grade students have success andunderstanding.LevelCmaystretchsixthandseventh-gradestudents.LevelDmaychallengemosteighthandninth-gradestudents,andLevelEshouldbechallengingformosthighschoolstudents.Thesegrade-levelexpectationsarejustestimatesandshouldnotbeusedasanabsoluteminimumexpectationormaximumlimitationforstudents. Problem solving is a learned skill, and students may need manyexperiences to develop their reasoning skills, approaches, strategies, and theperseverancetobesuccessful.TheProblemoftheMonthbuildsonsequentiallevelsofunderstanding.AllstudentsshouldexperienceLevelAandthenmovethroughthetasks in order to go as deeply as they can into the problem. Therewill be thosestudentswhowillnothaveaccess intoevenLevelA.Educatorsshould feel freetomodifythetasktoallowaccessatsomelevel.OverviewIntheProblemoftheMonthOnBalance,studentsareengagedintasksandpuzzlesthat involve equality, inequalities, equations, and simultaneous constraints. Themathematical topics that underlie this POM aremeasurement, number sentences,equality,inequality,variables,inverseoperations,andsimultaneoussystems.In the first level of the POM, students are presented with a puzzle involving theweightofthreeapplesandabalancescale.Thetaskistofindwhichappleislighterusingonlyoneweighing.Theyneedtouselogicandtheirknowledgeofequalityandinequality.InLevelB,studentsarepresentedwithfiveapples,oneofwhichisbad.Theyknowthebadappleisadifferentweightthantheotherfour,butarenotsure

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

2

whetherthebadoneisheavierorlighter.Studentsneedtousethebalancescaletodeterminethebadapple,usingtheleastamountofweighings.InLevelC,studentsarepresentedwithadrawingofbalancescalesshowinghowdifferentkindsoffruitweighinrelationshiptooneanother.Thestudentsusetheirknowledgeofequalityandequationstodeterminehowthestrawberriescomparetothelimesinweight.InLevelD,studentsaregivenasimilardrawingtotheonetheyhadinLevelB,butthistimetheyhavenineapplestoweighinordertodeterminethebadapple.Thegoalisto find thebadapple in the leastnumberofweighings.Thestudentsareasked tomakeageneralizationbasedonthenumberofapplesbeingconsidered.InthefinalLevelE,studentsareaskedtoconsiderfivesimultaneousequationsrepresentedinadrawing of fruit on a balance scale. The task is to compare the weights of the 6differentkindsof fruit and to find theweightof each, given that the strawberriesweigh3oz.MathematicalConceptsAlgebraisthecornerstoneofsecondarymathematics.Algebraicthinkingistaughtinprimarygradeswiththefoundationsofalgebrataughtusuallybytheendofmiddleschool. Even though the term algebraic thinking is routinely used, it cannot besimply defined. The underpinnings of algebra involve abstractions and language.Thereareseveralresourcesthatdefinethemostimportantconceptsinalgebra.OneresourceisFosteringAlgebraicThinking.

ProblemoftheMonth OnBalance 1©NoyceFoundation2015.ThisworkislicensedunderaCreativeCommonsAttribution-NonCommercial-NoDerivatives3.0UnportedLicense(http://creativecommons.org/licenses/by-nc-nd/3.0/deed.en_US).

ProblemoftheMonthOnBalance

LevelA

Makingonlyoneweighing,canyoudetermineawaytoknowforsurewhichappleisbad?Explain.

Debbie works at an apple orchard. She knows that every so often there is a bad or rotten apple in the baskets of apples she picks. She knows that bad apples weigh less than good ones. There are a lot of apples in her basket so she wants to spend as little time as possible checking apples. She thinks she knows a fast way to check. She has three apples and one is bad. How can she weigh apples just one time and still find out which one is bad?


LevelBDebbiehasa lotmoreapplestocheck. Nowshehas fiveapples. Fourapplesaregood,allweighingthesame.Butthebadoneeitherweighsmoreorweighslessthantheothers.Listtheprocessstepsanddecisionyouneedtomakeinordertodeterminewhichappleisbad,andwhetheritisheavierorlighterthanthegoodones. You know one of the apples is either heavier or lighter than all the others.What are theminimumnumberofweighingsneeded todeterminewhichapple isbad,andwhether it isheavierorlighterthantheothers?Showandjustifyyoursolution.

A B E D C


LevelCEach type of fruit has a uniqueweight that is consistent. Use the scales to determine therelationshipbetweenthedifferenttypesoffruit. Howdotheweightsofstrawberriescomparewithlimes?


LevelDSupposeyouwerepresentedwithnineapples(A-I).Eightapplesarethesameweight,andthe ninth either weighs more or weighs less than the others. List the process steps anddecisionsyouneedtomakeinordertodeterminewhichappleisdifferent,andwhetheritisheavierorlighterthantheothers.Youaregivenanynumberofapplesbetween3and15andyouknowthatoneoftheapplesiseitherheavieror lighter than theothers,whichallweigh thesame.Foreachsetofapples,whataretheminimumnumberoftimesyouneedtoweightheitemstoinsurethatyoufindtheonethatisadifferentweightthantheothers?Showtherelationshipbetweenthenumberofapplesandtheminimumnumberofweighingsneeded.

E

A B D C

F G I H


LevelEEachtypeoffruithasauniqueweightthatisconsistent.Usethefivescalestodeterminetherelationshipbetweenthedifferenttypesoffruits.Howdotheweightsofeachfruitcomparewith each other?Whichweighs themost?Whichweighs the least? Suppose a strawberryweighs3ounces.Howmanyofonekindoffruitisequaltoanotherkindoffruit?Explain.


ProblemoftheMonthOnBalancePrimaryVersion

Materials:Ascaleandthreeobjectsofdifferentweights(maybeanapple,pearandorange),paper,andpencil. Discussionon the rug: Teacher shows abalance scale. “What do we use this object for?” Studentsvolunteerideas.Ifstudentsdon’tknow,thentheteacherputstwodifferentobjects on the scale and asks the question, “What do you think it tells us now?” Students respond with ideas. The teacher asks, “Okay, which object is heavier?” Studentsrespond.Theteacherasks, “How do we know? Which one weighs the most? Which one weighs the least?” Insmallgroups:Eachstudentpairhasascaleandthreedifferentobjects–maybeafruit,apiece of paper and a pencil. The teacher explains that they can use the scale to find theanswertothesequestions:

Which object weighs the most? Which object weighs the least? What do we know about the third object (fruit)? Put the objects in order from heaviest to lightest. At the end of the investigation have students either discuss or dictate a response to thissummaryquestion: How did you figure out which object (fruit) weighed the most? Tell me how you figured it out and how you know for sure.

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

ProblemoftheMonth

OnBalanceTaskDescription–LevelA

This taskpresentsstudentswithapuzzle involving theweightof threeapplesandabalancescale.Studentsarechallengedtofindwhichappleisthelighterafteronlyoneweighing.Studentswillneedtouselogicandtheirknowledgeofequalityandinequalitytohelpthem.

CommonCoreStateStandardsMath-ContentStandardsMeasurementandDataDescribeandcomparemeasurableattributes.K.MD.1 Describe measurable attributes of objects, such as length or weight. Describe severalmeasurableattributesofasingleobject.K.MD.2Directlycomparetwoobjectswithameasurableattributeincommon,toseewhichobjecthas“more of”/“less of” the attribute, and describe the difference. For example, directly compare theheightsoftwochildrenanddescribeonechildastaller/shorter.

CommonCoreStateStandardsMath–StandardsofMathematicalPracticeMP.3 Constructviableargumentsandcritiquethereasoningofothers.Mathematicallyproficientstudentsunderstandandusestatedassumptions,definitions,andpreviouslyestablishedresultsinconstructing arguments. Theymake conjectures and build a logical progression of statements to explore the truth of theirconjectures. They are able to analyze situations by breaking them into cases, and can recognize and use counterexamples.Theyjustifytheirconclusions,communicatethemtoothers,andrespondtotheargumentsofothers.Theyreasoninductivelyabout data, making plausible arguments that take into account the context from which the data arose. Mathematicallyproficient students are also able to compare the effectiveness of two plausible arguments, distinguish correct logic orreasoning from thatwhich is flawed, and—if there is a flaw in an argument—explainwhat it is. Elementary students canconstruct arguments using concrete referents such as objects, drawings, diagrams, and actions. Such arguments canmakesense and be correct, even though they are not generalized or made formal until later grades. Later, students learn todeterminedomains towhichanargumentapplies.Studentsatallgradescan listenorread theargumentsofothers,decidewhethertheymakesense,andaskusefulquestionstoclarifyorimprovethearguments.MP.6 Attendtoprecision.Mathematicallyproficientstudentstrytocommunicatepreciselytoothers.Theytrytousecleardefinitionsindiscussionwithothers and in their own reasoning. They state the meaning of the symbols they choose, including using the equal signconsistently and appropriately. They are careful about specifying units of measure, and labeling axes to clarify thecorrespondencewith quantities in a problem. They calculate accurately and efficiently, express numerical answerswith adegree of precision appropriate for the problem context. In the elementary grades, students give carefully formulatedexplanationstoeachother.Bythetimetheyreachhighschooltheyhavelearnedtoexamineclaimsandmakeexplicituseofdefinitions.MP.7Lookforandmakeuseofstructure.Mathematicallyproficientstudents lookclosely todiscernapatternorstructure.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.MP.8 Lookforandexpressregularityinrepeatedreasoning.Mathematicallyproficientstudentsnotice ifcalculationsarerepeated,and lookboth forgeneralmethodsand forshortcuts.Upperelementarystudentsmightnoticewhendividing25by11thattheyarerepeatingthesamecalculationsoverandoveragain,andconclude theyhavea repeatingdecimal.Bypayingattention to thecalculationof slopeas theyrepeatedlycheckwhetherpointsareonthelinethrough(1,2)withslope3,middleschoolstudentsmightabstracttheequation(y–2)/(x–1)=3.Noticing the regularity in theway termscancelwhenexpanding (x–1)(x+1),(x–1)(x2+x+1),and(x–1)(x3+x2+x+1)might lead them to the general formula for the sum of a geometric series. As they work to solve a problem,mathematicallyproficientstudentsmaintainoversightoftheprocess,whileattendingtothedetails.Theycontinuallyevaluatethereasonablenessoftheirintermediateresults.


ProblemoftheMonth

OnBalanceTaskDescription–LevelB

Thistaskchallengesstudentstofindwhichappleoutoffiveapplesisthebadone.Studentrealizesthatthebadapplemaybeheavierorlighterbuttheyaren’tsurewhich.Studentswillneedtousethebalancescaletodeterminethebadappleusingtheleastnumberofweighings.

CommonCoreStateStandardsMath-ContentStandardsStatisticsandProbabilityInvestigatechanceprocessesanddevelop,use,andevaluateprobabilitymodels.7.SP.8 Find probabilities of compound events using organized lists, tables, tree diagrams, andsimulation.7.SP.8a. Understand that, just as with simple events, the probability of a compound event is thefractionofoutcomesinthesamplespaceforwhichthecompoundeventoccurs.7.SP.8b.Representsamplespacesforcompoundeventsusingmethodssuchasorganizedlists,tablesandtreediagrams.Foraneventdescribedineverydaylanguage(e.g.,“rollingdoublesixes”),identifytheoutcomesinthesamplespace,whichcomposetheevent.



ProblemoftheMonth

OnBalanceTaskDescription–LevelC

Whenpresentedwithadrawingofbalancescalesshowingdifferentkindsoffruitandtheirweightsinrelationtoeachother,studentsarechallengedtousethisinformationandtheirunderstandingofqualityandequationstodeterminehowthestrawberriescomparetothelimesinweight.




ProblemoftheMonth:

OnBalanceTaskDescription–LevelD

Thistaskchallengesstudentstodeterminewhichappleoutofnineapplesisthebadone(i.e.,whichapplehasadifferentweightthantheothereightapples)usingtheleastamountofweighingsonthescales.ThisproblemissimilartoLevelB.Studentsareaskedtomakeageneralizationbasedonthenumberofapplesbeingconsidered.

CommonCoreStateStandardsMath-ContentStandardsMeasurementandDataDescribeandcomparemeasurableattributes.K.MD.1Describemeasurableattributesofobjects,suchaslengthorweight.Describeseveralmeasurableattributesofasingleobject.K.MD.2Directlycomparetwoobjectswithameasurableattributeincommon,toseewhichobjecthas“moreof”/“lessof”theattribute, anddescribe thedifference. For example, directly compare theheightsof two childrenanddescribeone child astaller/shorter.StatisticsandProbabilityInvestigatechanceprocessesanddevelop,use,andevaluateprobabilitymodels.7.SP.8Findprobabilitiesofcompoundeventsusingorganizedlists,tables,treediagrams,andsimulation.7.SP.8a.Understandthat, justaswithsimpleevents, theprobabilityofacompoundevent is the fractionofoutcomes in thesamplespaceforwhichthecompoundeventoccurs.7.SP.8b.Representsamplespacesforcompoundeventsusingmethodssuchasorganizedlists,tablesandtreediagrams.Foraneventdescribedineverydaylanguage(e.g.,“rollingdoublesixes”),identifytheoutcomesinthesamplespace,whichcomposetheevent.

CommonCoreStateStandardsMath–StandardsofMathematicalPracticeMP.3Constructviableargumentsandcritiquethereasoningofothers.Mathematicallyproficientstudentsunderstandandusestatedassumptions,definitions,andpreviouslyestablishedresultsinconstructing arguments. Theymake conjectures and build a logical progression of statements to explore the truth of theirconjectures. They are able to analyze situations by breaking them into cases, and can recognize and use counterexamples.Theyjustifytheirconclusions,communicatethemtoothers,andrespondtotheargumentsofothers.Theyreasoninductivelyabout data, making plausible arguments that take into account the context from which the data arose. Mathematicallyproficient students are also able to compare the effectiveness of two plausible arguments, distinguish correct logic orreasoning from thatwhich is flawed, and—if there is a flaw in an argument—explainwhat it is. Elementary students canconstruct arguments using concrete referents such as objects, drawings, diagrams, and actions. Such arguments canmakesense and be correct, even though they are not generalized or made formal until later grades. Later, students learn todeterminedomains towhichanargumentapplies.Studentsatallgradescan listenorread theargumentsofothers,decidewhethertheymakesense,andaskusefulquestionstoclarifyorimprovethearguments.MP.6Attendtoprecision.Mathematicallyproficientstudentstrytocommunicatepreciselytoothers.Theytrytousecleardefinitionsindiscussionwithothers and in their own reasoning. They state the meaning of the symbols they choose, including using the equal signconsistently and appropriately. They are careful about specifying units of measure, and labeling axes to clarify thecorrespondencewith quantities in a problem. They calculate accurately and efficiently, express numerical answerswith adegree of precision appropriate for the problem context. In the elementary grades, students give carefully formulatedexplanationstoeachother.Bythetimetheyreachhighschooltheyhavelearnedtoexamineclaimsandmakeexplicituseofdefinitions.MP.7Lookforandmakeuseofstructure.Mathematicallyproficientstudents lookclosely todiscernapatternorstructure.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.MP.8Lookforandexpressregularityinrepeatedreasoning.Mathematicallyproficientstudentsnotice ifcalculationsarerepeated,and lookboth forgeneralmethodsand forshortcuts.Upperelementarystudentsmightnoticewhendividing25by11thattheyarerepeatingthesamecalculationsoverandoveragain,andconclude theyhavea repeatingdecimal.Bypayingattention to thecalculationof slopeas theyrepeatedlycheckwhetherpointsareonthelinethrough(1,2)withslope3,middleschoolstudentsmightabstracttheequation(y–2)/(x–1)=3.Noticingtheregularityinthewaytermscancelwhenexpanding(x–1)(x+1),(x–1)(x2+x+1),and(x–1)(x3+x2+x+1)mightleadthem to thegeneral formula for the sumof a geometric series.As theywork to solveaproblem,mathematicallyproficientstudentsmaintain oversight of the process,while attending to the details. They continually evaluate the reasonableness oftheirintermediateresults.


ProblemoftheMonth

OnBalanceTaskDescription–LevelE

This task challenges students to consider five simultaneousequations represented inadrawingoffruit on a balance scale. Students are to find the weight of the six different fruits given that thestrawberriesweigh3oz.

CommonCoreStateStandardsMath-ContentStandardsExpressionsandEquationsAnalyzeandsolvelinearequationsandpairsofsimultaneouslinearequations.8.EE.8Analyzeandsolvepairsofsimultaneouslinearequations.8.EE8c. Solvereal-worldandmathematicalproblemsleadingtotwolinearequationsintwovariables.HighSchool-Algebra-CreatingEquationsCreateequationsthatdescribenumbersorrelationships.A-CED.2Createequationsintwoormorevariablestorepresentrelationshipsbetweenquantities…

CommonCoreStateStandardsMath–StandardsofMathematicalPracticeMP.3Constructviableargumentsandcritiquethereasoningofothers.Mathematicallyproficientstudentsunderstandandusestatedassumptions,definitions,andpreviouslyestablishedresultsinconstructing arguments. Theymake conjectures and build a logical progression of statements to explore the truth of theirconjectures. They are able to analyze situations by breaking them into cases, and can recognize and use counterexamples.Theyjustifytheirconclusions,communicatethemtoothers,andrespondtotheargumentsofothers.Theyreasoninductivelyabout data, making plausible arguments that take into account the context from which the data arose. Mathematicallyproficient students are also able to compare the effectiveness of two plausible arguments, distinguish correct logic orreasoning from thatwhich is flawed, and—if there is a flaw in an argument—explainwhat it is. Elementary students canconstruct arguments using concrete referents such as objects, drawings, diagrams, and actions. Such arguments canmakesense and be correct, even though they are not generalized or made formal until later grades. Later, students learn todeterminedomains towhichanargumentapplies.Studentsatallgradescan listenorread theargumentsofothers,decidewhethertheymakesense,andaskusefulquestionstoclarifyorimprovethearguments.MP.6Attendtoprecision.Mathematicallyproficientstudentstrytocommunicatepreciselytoothers.Theytrytousecleardefinitionsindiscussionwithothers and in their own reasoning. They state the meaning of the symbols they choose, including using the equal signconsistently and appropriately. They are careful about specifying units of measure, and labeling axes to clarify thecorrespondencewith quantities in a problem. They calculate accurately and efficiently, express numerical answerswith adegree of precision appropriate for the problem context. In the elementary grades, students give carefully formulatedexplanationstoeachother.Bythetimetheyreachhighschooltheyhavelearnedtoexamineclaimsandmakeexplicituseofdefinitions.MP.7Lookforandmakeuseofstructure.Mathematicallyproficientstudents lookclosely todiscernapatternorstructure.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.MP.8Lookforandexpressregularityinrepeatedreasoning.Mathematicallyproficientstudentsnotice ifcalculationsarerepeated,and lookboth forgeneralmethodsand forshortcuts.Upperelementarystudentsmightnoticewhendividing25by11thattheyarerepeatingthesamecalculationsoverandoveragain,andconclude theyhavea repeatingdecimal.Bypayingattention to thecalculationof slopeas theyrepeatedlycheckwhetherpointsareonthelinethrough(1,2)withslope3,middleschoolstudentsmightabstracttheequation(y–2)/(x–1)=3.Noticing the regularity in theway termscancelwhenexpanding (x–1)(x+1),(x–1)(x2+x+1),and(x–1)(x3+x2+x+1)might lead them to the general formula for the sum of a geometric series. As they work to solve a problem,mathematicallyproficientstudentsmaintainoversightoftheprocess,whileattendingtothedetails.Theycontinuallyevaluatethereasonablenessoftheirintermediateresults.


ProblemoftheMonth

OnBalanceTaskDescription–PrimaryLevel

Thistaskchallengesstudentstoworkwithascaleandthreedifferentfruitsofdifferentweights.Theteacher holds a discussion about a balance scale – asking students what they know about it ingeneral,andspecificallywhentwodifferentobjectsareplacedonthescale.Theteacheraskswhichobject is heavier and how they know. Then, in small groups, students have 3 objects,which theyweigh to decide which one weighs the most and the least; then they order the objects from theheaviest to the lightest. At the end of the investigation a student will dictate a response to thequestion,“Howdoyoufigureoutwhichobjectweighsthemost?”


CommonCoreStateStandardsMath–StandardsofMathematicalPracticeMP.3Constructviableargumentsandcritiquethereasoningofothers.Mathematicallyproficientstudentsunderstandandusestatedassumptions,definitions,andpreviouslyestablishedresultsinconstructing arguments. Theymake conjectures and build a logical progression of statements to explore the truth of theirconjectures. They are able to analyze situations by breaking them into cases, and can recognize and use counterexamples.Theyjustifytheirconclusions,communicatethemtoothers,andrespondtotheargumentsofothers.Theyreasoninductivelyabout data, making plausible arguments that take into account the context from which the data arose. Mathematicallyproficient students are also able to compare the effectiveness of two plausible arguments, distinguish correct logic orreasoning from thatwhich is flawed, and—if there is a flaw in an argument—explainwhat it is. Elementary students canconstruct arguments using concrete referents such as objects, drawings, diagrams, and actions. Such arguments canmakesense and be correct, even though they are not generalized or made formal until later grades. Later, students learn todeterminedomains towhichanargumentapplies.Studentsatallgradescan listenorread theargumentsofothers,decidewhethertheymakesense,andaskusefulquestionstoclarifyorimprovethearguments.MP.6Attendtoprecision.Mathematicallyproficientstudentstrytocommunicatepreciselytoothers.Theytrytousecleardefinitionsindiscussionwithothers and in their own reasoning. They state the meaning of the symbols they choose, including using the equal signconsistently and appropriately. They are careful about specifying units of measure, and labeling axes to clarify thecorrespondencewith quantities in a problem. They calculate accurately and efficiently, express numerical answerswith adegree of precision appropriate for the problem context. In the elementary grades, students give carefully formulatedexplanationstoeachother.Bythetimetheyreachhighschooltheyhavelearnedtoexamineclaimsandmakeexplicituseofdefinitions.MP.7Lookforandmakeuseofstructure.Mathematicallyproficientstudents lookclosely todiscernapatternorstructure.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.MP.8Lookforandexpressregularityinrepeatedreasoning.Mathematicallyproficientstudentsnotice ifcalculationsarerepeated,and lookboth forgeneralmethodsand forshortcuts.Upperelementarystudentsmightnoticewhendividing25by11thattheyarerepeatingthesamecalculationsoverandoveragain,andconclude theyhavea repeatingdecimal.Bypayingattention to thecalculationof slopeas theyrepeatedlycheckwhetherpointsareonthelinethrough(1,2)withslope3,middleschoolstudentsmightabstracttheequation(y–2)/(x–1)=3.Noticingtheregularityinthewaytermscancelwhenexpanding(x–1)(x+1)(x–1)(x2+x+1),and(x–1)(x3+x2+x+1)mightleadthem to thegeneral formula for the sumof a geometric series.As theywork to solveaproblem,mathematicallyproficientstudentsmaintain oversight of the process,while attending to the details. They continually evaluate the reasonableness oftheirintermediateresults.

Documents

Problem of the Month: On Balance - Inside Mathematics of the Month: On Balance The Problems of the Month (POM) are used in a variety of ways to promote problem solving and to foster