Problem of the Month Tri-Triangles - inside · PDF fileProblem of the Month Tri-Triangles ......

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ProblemoftheMonth

Tri-Triangles

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. A POM can also be used schoolwide 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. ThePrimaryVersionisdesigned 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 thosestudentswhowillnothaveaccess intoevenLevelA.Educatorsshould feel freetomodifythetasktoallowaccessatsomelevel.OverviewIntheProblemoftheMonthTri-Triangles,studentsusealgebraicthinkingtosolveproblems involvingpatterns,sequences,generalizations,and linearandnon-linearfunctions.ThemathematicaltopicsthatunderliethisPOMarefindingandextendingpatterns, creatinggeneralizations, finding functions,developing inverseprocesses,exploringnon-linearfunctions,andjustifyingsolutions.

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In the first level of the POM, students view a pattern of triangles composed oftoothpicks.Theirtaskistodeterminethenumberoftoothpicksthatmakeupeachpattern.Thefunctionisproportional.InLevelB,studentsexaminealinearpatternthat involves a constant. The task involves a set of triangular tables that arearrangedadjacentlyinarow.Thetaskasksstudentstodeterminetherelationshipbetween the number of tables and the number of peoplewho can sit around thetables.Studentsneedtoextendthepattern.Theyalsofindtheinverserelationship,i.e.,findthepatternnumberwhengiventhetotalnumberofpeopleseated.LevelCrequiresstudentstodeterminehowapatterngrows.Studentsneedtoseethatthepatterngrowsbysquarenumbers.Theyidentifytherelationshipandthenexplainavalidprocessforfindingthesevalues.InLevelD,studentsareaskedtogeneralizearule for findingavalue in the triangularnumbersequence.Theyarealsoaskedtoexplaintheprocessforfindinganinversevalueforthetriangularnumbersequencebyfindingthetermwhengiventhetotal. Inthe finalLevelE,studentsgenerateaclosed expression for a sequence that grows as the sum of two exponentialfunctions.Inaddition,thestudentsmustjustifytheirfindings.

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Problem of the Month

Tri - Triangles Level A Lisa is making triangle patterns out of toothpicks of all the same length. A triangle is made from three toothpicks. Her first pattern is a single triangle. Her second pattern is shown below. How many toothpicks are needed for her second pattern? Her third pattern is shown below. How many toothpicks are needed for her third pattern? If she continued the same pattern, how many toothpicks are needed for the fifth pattern? How many toothpicks are needed for the tenth pattern? If you had 81 toothpicks, what pattern number could you make? Explain how you know.


Level B Your classroom has triangular shaped tables. Three students can sit around one table. Two tables can be pushed together so that two sides are adjacent. How many students can sit around the tables in this arrangement? Tables can be added to the arrangement by pushing together tables so that each additional table is adjacent to one side of the row of tables. The arrangement may grow to be a long row of tables. How many students can sit around three tables in this arrangement? How many students can sit around five tables in a row arrangement? Without drawing the arrangement, determine how many students can sit around twelve desks in a row. Explain how you figured it out. How many tables in a row are needed to seat 105 students? Explain your answer.


Level C Anne uses triangles to make the following patterns: The pattern continues in the same geometric design. Draw Pattern 4. How many triangles are needed? How many triangles are needed to construct Pattern 7? How many triangles are needed to construct Pattern 16? Explain how you determined your rule. Write a rule to find the number of triangles needed for the nth pattern? Explain your rule. Suppose a pattern had 2,025 triangles. What is the pattern number? Explain.

Pattern 1 Pattern 2 Pattern 3


Level D Jo constructs triangular patterns using dots. The pattern continues in the same geometric design. How many dots are needed to make Pattern 5? How many dots are needed for the nth pattern? Explain your rule. Jo was born in 1953 and she was wondering if she could make a triangular pattern out of exactly 1,953 dots. If she could, what would the pattern number be? Explain your answer.



Level E Craig constructs the designs above from equal line segments. The design in Pattern 1 is made up of three line segments. Pattern 2 is made up of nine line segments. Pattern 3 is made up of thirty line segments, and so on. How many line segments are needed to make Pattern 8? How many line segments are needed to make Pattern 16? Determine a function for finding the number of line segments needed to make the pattern for any number n. Justify why your function works. You have 6,294,528 equal line segments. Can you construct a design that belongs in this sequence using just those line segments? If so, what pattern number would that be? If not, how many more line segments might you need to construct a design that fits the sequence?



Problem of the Month

Tri - Triangles

Primary Version Level A

Materials: A picture of the three patterns, paper, a pencil and toothpicks for students to make the different patterns. Discussion on the rug: Teacher holds up the pictures of the triangle patterns. “Here are different patterns. How many toothpicks would it take to make the first pattern?” Students may build and count. “How many toothpicks do you need to build the second pattern?” Students may build and count. Teacher asks the children to think about how the number of toothpicks changed from the first pattern to the second. In small groups: Each student has access to toothpicks, paper, a pencil and the picture of the first three patterns. The teacher explains that they may either build or draw the pattern to help find the answers. The teacher asks the following questions, only going on to the next question if students have success. How many toothpicks do you need to build the first pattern? How many toothpicks do you need to build the second pattern? How many toothpicks do you need to build the third pattern? How many toothpicks do you need to build the fourth pattern? How many toothpicks do you need to build the sixth pattern? At the end of the investigation have students either discuss or dictate a response to this summary question: “Tell me how you know.”


Problem of the Month Tri - Triangles

First Pattern

Second Pattern

Third Pattern

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ProblemoftheMonth

Tri-TrianglesTaskDescription–LevelA

Thistaskchallengesstudentstoviewapatternoftrianglescomposedoftoothpicks.Studentsmustdeterminethenumberoftoothpicksthatmakeupeachpattern.Thisfunctionisproportional.

CommonCoreStateStandardsMath-ContentStandardsOperationsandAlgebraicThinkingRepresentandsolveproblemsinvolvingmultiplicationanddivision.3.OA.1.Interpretproductsofwholenumbers,e.g.interpret5x7asthetotalnumberofobjectsin5groupsof7objectseach.3.OA.2Interpretwholenumberquotientsofwholenumbers,e.g.,interpret56÷8asthenumberofobjectsineachsharewhen56objectsarepartitionedequallyinto8shares,,orasanumberofobjectsineachsharewhen56objectsarepartitionedintoequalsharesof8objectseach.3.OA.3Usemultiplicationanddivisionwithin100tosolvewordproblemsinsituationsinvolvingequalgroups,arrays,andmathematicalquantities,e.g.,byusingdrawingsandequationswithasymbolfortheunknownnumbertorepresenttheproblems.Understandpropertiesofmultiplicationandtherelationshipbetweenmultiplicationanddivision.3.OA.5Applypropertiesofoperationsasstrategiestomultiplyanddivide.3.OA.6Understanddivisionasanunknown-factorproblem.Solveproblemsinvolvingthefouroperations,andidentifyandexplainpatternsinarithmetic.3.OA.8Solvetwo-stepwordproblemsusingthefouroperations.Representtheseproblemsusingequationswithaletterstandingfortheunknownquantity.Assthereasonablenessofanswersusingmentalcomputationandestimationstrategiesincludingrounding.Generateandanalyzepatterns.4.OA.5Generateanumberorshapepatternthatfollowsagivenrule.Identifyapparentfeaturesofthepatternthatwerenotexplicitintheruleitself.

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.


ProblemoftheMonth

Tri-TrianglesTaskDescription–LevelB

Thistaskchallengesstudentstoexaminealinearpatternthatinvolvesaconstant.Thetaskinvolvesaset of triangular tables that are arranged adjacently in a row. Students determine the relationshipbetweenthenumberoftablesandthenumberofpeoplewhocansitaroundthetables.Studentsneedtoextendthepatternandfindtheinverserelationship,i.e.,findthepatternnumberwhengiventhetotalnumberofpeopleseated.

CommonCoreStateStandardsMath-ContentStandardsOperationsandAlgebraicThinkingUnderstandpropertiesofmultiplicationandtherelationshipbetweenmultiplicationanddivision.3.OA.5Applypropertiesofoperationsasstrategiestomultiplyanddivide.3.OA.6Understanddivisionasanunknown-factorproblem.Solveproblemsinvolvingthefouroperations,andidentifyandexplainpatternsinarithmetic.3.OA.8 Solve two-step word problems using the four operations. Represent these problems using equations with a letterstanding for theunknownquantity.Ass thereasonablenessofanswersusingmentalcomputationandestimationstrategiesincludingrounding.Generateandanalyzepatterns.4.OA.5Generateanumberorshapepatternthatfollowsagivenrule.Identifyapparentfeaturesofthepatternthatwerenotexplicitintheruleitself.Writeandinterpretnumericalexpressions.5.OA.2 Write simple expressions that record calculations with numbers, and interpret numerical expressions withoutevaluationsthem.Analyzepatternsandrelationships.5.OA.3Generatetwonumericalpatternsusingtwogivenrules.Identifyapparentrelationshipsbetweencorrespondingterms.Formorderedpairs consistingof corresponding terms from the twopatterns andgraph theorderedpairsona coordinateplane.ExpressionsandEquationsRepresentandanalyzequantitativerelationshipsbetweendependentandindependentvariables.6.EE.9Usevariablestorepresenttwoquantitiesinareal-worldproblemthatchangeinrelationshiptooneanother;writeanequation to express one quantity, thought of as the dependent variable, in terms of the other quantity, thought of as theindependentvariable.Analyze the relationshipbetweenthedependentand independentvariableusinggraphsandtables,andrelatethesetotheequations.Solvereal-lifeandmathematicalproblemsusingnumericalandalgebraicexpressionsandequations.7.EE.4 Use variables to represent quantities in a real-world or mathematical problem, and construct simpleequationsandinequalitiestosolveproblemsbyreasoningaboutthequantities.

CommonCoreStateStandardsMath–StandardsofMathematicalPracticeMP.3Constructviableargumentsandcritiquethereasoningofothers.Mathematicallyproficientstudentsunderstandandusestatedassumptions,definitions,andpreviouslyestablishedresultsinconstructingarguments. Theymake conjecturesandbuilda logicalprogressionof statements toexplore the truthof theirconjectures. Theyareable toanalyze situationsbybreaking them into cases, and can recognizeanduse 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 inanargument–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 todeterminedomainstowhichanargumentapplies. Studentsatallgradescanlistenorreadtheargumentsofothers,decidewhethertheymakesense,andaskusefulquestionstoclarifyorimprovethearguments.MP.4Modelwithmathematics.Mathematicallyproficientstudentscanapplythemathematicstheyknowtosolveproblemsarisingineverydaylife,society,andtheworkplace.Inearlygradesthismightbeassimpleaswritinganadditionequationtodescribeasituation.Inmiddlegrades,astudentmightapplyproportionalreasoningtoplanaschooleventoranalyzeaprobleminthecommunity.Byhighschool, a studentmightusegeometry to solveadesignproblemorusea function todescribehowonequantityof interestdependsonanother.Mathematicallyproficientstudentswhocanapplywhattheyknowarecomfortablemakingassumptionsandapproximationstosimplifyacomplicatedsituation,realizingthatthesemayneedrevisionlater.Theyareabletoidentifyimportantquantitiesinapracticalsituationandmaptheirrelationshipsusingsuchtoolsasdiagrams,two-waytables,graphs,flowcharts,andformulas.Theycananalyzethoserelationshipsmathematicallytodrawconclusions.Theyroutinelyinterprettheirmathematicalresults inthecontextofthesituationandreflectonwhethertheresultsmakesense,possiblyimprovingthemodelifithasnotserveditspurpose.


ProblemoftheMonth

Tri-TrianglesTaskDescription–LevelC

Thistaskchallengesstudentstodeterminevaluesthatgrowinasquarearearelationshipandthentoexplainavalidprocess for finding thesevaluesandextending thepattern.Theymustalso find theinverserelationshipforspecificvalues.

CommonCoreStateStandardsMath-ContentStandardsOperationsandAlgebraicThinkingGenerateandanalyzepatterns.4.OA.5Generateanumberorshapepattern that followsagivenrule. Identifyapparent featuresofthepatternthatwerenotexplicitintheruleitself.Writeandinterpretnumericalexpressions.5.OA.2 Write simple expressions that record calculations with numbers, and interpret numericalexpressionswithoutevaluationsthem.Analyzepatternsandrelationships.5.OA.3 Generate two numerical patterns using two given rules. Identify apparent relationshipsbetweencorrespondingterms.Formorderedpairsconsistingofcorrespondingtermsfromthetwopatternsandgraphtheorderedpairsonacoordinateplane.ExpressionsandEquationsApplyandextendpreviousunderstandingsofarithmetictoalgebraicexpressions.6.EE.1Writeandevaluatenumericalexpressionsinvolvingwhole-numberexponents.Represent and analyze quantitative relationships between dependent and independentvariables.6.EE.9Usevariablestorepresenttwoquantitiesinareal-worldproblemthatchangeinrelationshiptooneanother;writeanequationtoexpressonequantity,thoughtofasthedependentvariable, intermsoftheotherquantity,thoughtofastheindependentvariable.Analyzetherelationshipbetweenthedependentandindependentvariableusinggraphsandtables,andrelatethesetotheequations.Solve real-life and mathematical problems using numerical and algebraic expressions andequations.7.EE.4Usevariablestorepresentquantitiesinareal-worldormathematicalproblem,andconstructsimpleequationsandinequalitiestosolveproblemsbyreasoningaboutthequantities.

CommonCoreStateStandardsMath–StandardsofMathematicalPracticeMP.3Constructviableargumentsandcritiquethereasoningofothers.Mathematicallyproficientstudentsunderstandandusestatedassumptions,definitions,andpreviouslyestablishedresultsinconstructingarguments. Theymake conjecturesandbuilda logicalprogressionof statements toexplore the truthof theirconjectures. Theyareable toanalyze situationsbybreaking them into cases, and can recognizeanduse 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 inanargument–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 todeterminedomainstowhichanargumentapplies. Studentsatallgradescanlistenorreadtheargumentsofothers,decidewhethertheymakesense,andaskusefulquestionstoclarifyorimprovethearguments.MP.4Modelwithmathematics.Mathematicallyproficientstudentscanapplythemathematicstheyknowtosolveproblemsarisingineverydaylife,society,andtheworkplace.Inearlygradesthismightbeassimpleaswritinganadditionequationtodescribeasituation.Inmiddlegrades,astudentmightapplyproportionalreasoningtoplanaschooleventoranalyzeaprobleminthecommunity.Byhighschool, a studentmightusegeometry to solveadesignproblemorusea function todescribehowonequantityof interestdependsonanother.Mathematicallyproficientstudentswhocanapplywhattheyknowarecomfortablemakingassumptionsandapproximationstosimplifyacomplicatedsituation,realizingthatthesemayneedrevisionlater.Theyareabletoidentifyimportantquantitiesinapracticalsituationandmaptheirrelationshipsusingsuchtoolsasdiagrams,two-waytables,graphs,flowcharts,andformulas.Theycananalyzethoserelationshipsmathematicallytodrawconclusions.Theyroutinelyinterprettheirmathematicalresults inthecontextofthesituationandreflectonwhethertheresultsmakesense,possiblyimprovingthemodelifithasnotserveditspurpose.


ProblemoftheMonth

Tri-TrianglesTaskDescription–LevelD

This task challenges students to generalize a rule for finding a value in the triangular numbersequence. Studentsalsoexplain theprocess for findingan inversevalue for the triangularnumbersequencebyfindingtheterm,giventhetotal.

CommonCoreStateStandardsMath-ContentStandardsGeometrySolvereal-worldandmathematicalproblemsinvolvingarea,surfacearea,andvolume.6.G.1 Find the area of right triangles, other triangles, special quadrilaterals and polygons bycomposingintorectanglesordecomposingintotrianglesandothershapes;applythesetechniquesinthecontextofsolvingreal-worldandmathematicalproblems.ExpressionsandEquationsApplyandextendpreviousunderstandingsofarithmetictoalgebraicexpressions.6.EE.1Writeandevaluatenumericalexpressionsinvolvingwhole-numberexponents.Represent and analyze quantitative relationships between dependent and independentvariables.6.EE.9Usevariablestorepresenttwoquantitiesinareal-worldproblemthatchangeinrelationshiptooneanother;writeanequationtoexpressonequantity,thoughtofasthedependentvariable, intermsoftheotherquantity,thoughtofastheindependentvariable.Analyzetherelationshipbetweenthedependentandindependentvariableusinggraphsandtables,andrelatethesetotheequations.Solve real-life and mathematical problems using numerical and algebraic expressions andequations.7.EE.4Usevariablestorepresentquantitiesinareal-worldormathematicalproblem,andconstructsimpleequationsandinequalitiestosolveproblemsbyreasoningaboutthequantities.

CommonCoreStateStandardsMath–StandardsofMathematicalPracticeMP.3Constructviableargumentsandcritiquethereasoningofothers.Mathematically proficient students understand and use stated assumptions, definitions, and previouslyestablished results in constructing arguments. They make conjectures and build a logical progression ofstatementstoexplorethetruthoftheirconjectures.Theyareabletoanalyzesituationsbybreakingthemintocases,andcanrecognizeandusecounterexamples.Theyjustifytheirconclusions,communicatethemtoothers,andrespondtotheargumentsofothers.Theyreasoninductivelyaboutdata,makingplausibleargumentsthattake into account the context fromwhich thedata arose. Mathematicallyproficient students are also able tocomparetheeffectivenessoftwoplausiblearguments,distinguishcorrectlogicorreasoningfromthatwhichisflawed,and–ifthereisaflawinanargument–explainwhatitis.Elementarystudentscanconstructargumentsusingconcretereferentssuchasobjects,drawings,diagrams,andactions.Suchargumentscanmakesenseandbe correct, even though they are not generalized ormade formal until later grades. Later, students learn todeterminedomains towhichanargumentapplies. Studentsatall gradescan listenor read theargumentsofothers,decidewhethertheymakesense,andaskusefulquestionstoclarifyorimprovethearguments.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 acomplicated situation, realizing that these may need revision later. They are able to identify importantquantities in a practical situation andmap their relationships using such tools as diagrams, two-way tables,graphs, flowcharts, and formulas. They can analyze those relationshipsmathematically to draw conclusions.They routinely interpret theirmathematical results in the contextof the situationand reflectonwhether theresultsmakesense,possiblyimprovingthemodelifithasnotserveditspurpose.


ProblemoftheMonth

Tri-TrianglesTaskDescription–LevelE

Thistaskchallengesstudentstogenerateaclosedexpressionforasequencethatgrowsasthesumoftwoexponentialfunctions.Inaddition,thestudentsmustjustifytheirfindings.

CommonCoreStateStandardsMath-ContentStandardsExpressionsandEquationsExpressionsandequationsworkwithradicalsandintegerexponents.8.EE.1 Know and apply the properties of inter exponents to generate equivalent numericalexpressions.HighSchool–Algebra–CreatingEquationsCreateequationsthatdescribenumbersorrelationships.A-CED.1Createequationsand inequalities inonevariableandusethemtosolveproblems. Includeequationsarisingfromlinearandquadraticfunctionsandsimplerationalandexponentialfunctions.HighSchool–Functions–BuildingFunctionsBuildafunctionthatmodelsrelationshipbetweentwoquantities.F-BF.1Writeafunctionthatdescribesarelationshipbetweentwoquantities.F-BF.2Writearithmeticandgeometricsequencesbothrecursivelyandwithanexplicitformula,usethemtomodelsituationsandtranslatebetweenthetwoforms.Buildnewfunctionsfromexistingfunctions.F-BF.4Findinversefunctions.HighSchool–Functions–Linear,Quadratic,andExponentialModelsInterpretexpressionsforfunctionsintermsofthesituationtheymodel.F-LE.5Interprettheparametersinalinearorexponentialfunctionintermsofacontext.

CommonCoreStateStandardsMath–StandardsofMathematicalPracticeMP.4Modelwithmathematics.Mathematicallyproficientstudentscanapplythemathematicstheyknowtosolveproblemsarisingineverydaylife,society,andtheworkplace.Inearlygradesthismightbeassimpleaswritinganadditionequationtodescribeasituation.Inmiddlegrades,astudentmightapplyproportionalreasoningtoplanaschooleventoranalyzeaprobleminthecommunity.Byhighschool, a studentmightusegeometry to solveadesignproblemorusea function todescribehowonequantityof interestdependsonanother.Mathematicallyproficientstudentswhocanapplywhattheyknowarecomfortablemakingassumptionsandapproximationstosimplifyacomplicatedsituation,realizingthatthesemayneedrevisionlater.Theyareabletoidentifyimportantquantitiesinapracticalsituationandmaptheirrelationshipsusingsuchtoolsasdiagrams,two-waytables,graphs,flowcharts,andformulas.Theycananalyzethoserelationshipsmathematicallytodrawconclusions.Theyroutinelyinterprettheirmathematicalresults inthecontextofthesituationandreflectonwhethertheresultsmakesense,possiblyimprovingthemodelifithasnotserveditspurpose.MP.8Lookforandexpressregularityinrepeatedreasoning.Mathematicallyproficientstudentsnotice ifcalculationsarerepeated,and lookboth forgeneralmethodsand forshortcuts.Upperelementarystudentsmightnoticewhendividing25by11thattheyarerepeatingthesamecalculationsoverandoveragain,andconcludetheyhavearepeatingdecimal. Bypayingattentiontothecalculationofslopeastheyrepeatedlycheckwhetherpointsareonthelinethrough(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)mightleadthemtothegeneralformulaforthesumofageometricseries.Astheyworktosolveaproblem,mathematicallyproficientstudentsmaintainoversightof theprocess,whileattending to thedetails. Theycontinuallyevaluate the reasonablenessoftheirintermediateresults.


ProblemoftheMonth

Tri-TrianglesTaskDescription–PrimaryLevel

Thistaskchallengesstudentstobuildatrianglepatternwithtoothpicksandcountthetoothpicks.Studentsalsorecordthefindingsusingdrawingsandnumbers.Studentsextendthepatternbyidentifyingattributesofthepatternandbuilding.

CommonCoreStateStandardsMath-ContentStandardsCountingandCardinalityCounttotellthenumberofobjects.K.CC.5 Count to answer “how many?” questions about as many as 20 things arranged in a line, arectangular array, or a circle, or asmany as 10 things in a scattered configuration; given a numberfrom1-20,countoutthatmanyobjects.OperationsandAlgebraicThinkingRepresentandsolveproblemsinvolvingadditionandsubtraction.1.OA.2Solvewordproblemsthatcall foradditionof threewholenumberswhosesumis lessthatorequalto20,e.g.byusingobjects,drawings,andequationswithasymbolfortheunknownnumbertorepresenttheproblem.2.OA.1Useadditionandsubtractionwithin100tosolveone-andtwo-stepwordproblemsinvolvingsituationsofaddingto, takingfrom,puttingtogether, takingapart,andcomparing,withunknownsnall positions, e.g., by using drawings and equations with a symbol for the unknown number torepresenttheproblem.Workwithequalgroupsofobjectstogainfoundationsformultiplication.2.OA.4Writeanequationtoexpressthetotalasasumofequaladdends.GeometryAnalyze,compare,create,andcomposeshapes.K.G.4 Analyze and compare two- and three-dimensional shapes, in different sizes and orientations,using informal language to describe their similarities, differences, parts (e.g. number of sides andvertices/cornersandotherattributes.K.G.5Modelshapesintheworldbybuildingshapesfromcomponents(e.g.,sticksandclayballs)anddrawingshapes.

CommonCoreStateStandardsMath–StandardsofMathematicalPracticeMP.3Constructviableargumentsandcritiquethereasoningofothers.Mathematically proficient students understand anduse stated assumptions, definitions, andpreviously established results inconstructing arguments. Theymake conjectures and build a logical progression of statements to explore the truth of theirconjectures.Theyareabletoanalyzesituationsbybreakingthemintocases,andcanrecognizeandusecounterexamples.Theyjustifytheirconclusions,communicatethemtoothers,andrespondtotheargumentsofothers.Theyreasoninductivelyaboutdata, making plausible arguments that take into account the context fromwhich the data arose. Mathematically proficientstudentsarealsoabletocomparetheeffectivenessoftwoplausiblearguments,distinguishcorrectlogicorreasoningfromthatwhichisflawed,and–ifthereisaflawinanargument–explainwhatitis.Elementarystudentscanconstructargumentsusingconcrete referents such as objects, drawings, diagrams, and actions. Such arguments canmake sense and be correct, eventhough they are not generalized ormade formal until later grades. Later, students learn to determine domains towhich anargumentapplies. Studentsatallgradescanlistenorreadtheargumentsofothers,decidewhethertheymakesense,andaskusefulquestionstoclarifyorimprovethearguments.MP.4Modelwithmathematics.Mathematicallyproficientstudentscanapplythemathematicstheyknowtosolveproblemsarisingineverydaylife,society,andtheworkplace.Inearlygradesthismightbeassimpleaswritinganadditionequationtodescribeasituation.Inmiddlegrades,astudentmightapplyproportionalreasoningtoplanaschooleventoranalyzeaprobleminthecommunity. Byhighschool,astudentmightusegeometrytosolveadesignproblemorusea functiontodescribehowonequantityof interestdependsonanother. Mathematically proficient students who can apply what they know are comfortable making assumptions andapproximations to simplify a complicated situation, realizing that these may need revision later. They are able to identifyimportantquantitiesinapracticalsituationandmaptheirrelationshipsusingsuchtoolsasdiagrams,two-waytables,graphs,flowcharts,andformulas. Theycananalyzethoserelationshipsmathematicallytodrawconclusions. Theyroutinelyinterprettheirmathematicalresultsinthecontextofthesituationandreflectonwhethertheresultsmakesense,possiblyimprovingthemodelifithasnotserveditspurpose.

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