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Elementary Math Agenda Primary: January 2012 Norms • • • • Participant actively. Be respectful of people's time, ideas, and needs. Maintain a positive tone. Be solution oriented. Learning Targets I can describe to parents and colleagues the North Clackamas Common Core Standard implementation plan for math. I can implement two of the eight Common Core Mathematical Practices into my classroom instruction. I can teach grade level content through a variety of models that progress from concrete to abstract. (Build-­‐Sketch-­‐Record) I can describe characteristics of Sheltered Instruction by talking to my peers about the techniques. Agenda • • • • • Introductions to Common Core State Standards Digging into Mathematical Practices Content Knowledge and Sheltered Instruction: Number Sense Break Out Sessions Exit Card Exit Card: Using the scale below, rate how comfortable you are incorporating the four Sheltered Instruction Strategies modeled today into your math instruction. 1 Not ready 2 I need support in this area (what support is needed?) 3 Ready to implement Frayer Model Language Objectives Pictorial Input Chart Sentence Frames In what ways can we further support your work in math this year? 4 Currently in practice. What are the next steps? Common Core State StandardS for matHematICS mathematics | Standards for mathematical Practice TheStandardsforMathematicalPracticedescribevarietiesofexpertisethat mathematicseducatorsatalllevelsshouldseektodevelopintheirstudents. Thesepracticesrestonimportant“processesandproficiencies”withlongstanding importanceinmathematicseducation.ThefirstofthesearetheNCTMprocess standardsofproblemsolving,reasoningandproof,communication,representation, andconnections.Thesecondarethestrandsofmathematicalproficiencyspecified intheNationalResearchCouncil’sreportAdding It Up:adaptivereasoning,strategic competence,conceptualunderstanding(comprehensionofmathematicalconcepts, operationsandrelations),proceduralfluency(skillincarryingoutprocedures flexibly,accurately,efficientlyandappropriately),andproductivedisposition (habitualinclinationtoseemathematicsassensible,useful,andworthwhile,coupled withabeliefindiligenceandone’sownefficacy). 1 Make sense of problems and persevere in solving them. Mathematicallyproficientstudentsstartbyexplainingtothemselvesthemeaning ofaproblemandlookingforentrypointstoitssolution.Theyanalyzegivens, constraints,relationships,andgoals.Theymakeconjecturesabouttheformand meaningofthesolutionandplanasolutionpathwayratherthansimplyjumpinginto asolutionattempt.Theyconsideranalogousproblems,andtryspecialcasesand simplerformsoftheoriginalprobleminordertogaininsightintoitssolution.They monitorandevaluatetheirprogressandchangecourseifnecessary.Olderstudents might,dependingonthecontextoftheproblem,transformalgebraicexpressionsor changetheviewingwindowontheirgraphingcalculatortogettheinformationthey need.Mathematicallyproficientstudentscanexplaincorrespondencesbetween equations,verbaldescriptions,tables,andgraphsordrawdiagramsofimportant featuresandrelationships,graphdata,and searchforregularityortrends.Younger studentsmightrelyonusing concreteobjectsorpicturestohelpconceptualize andsolveaproblem.Mathematicallyproficientstudentschecktheiranswersto problemsusingadifferentmethod,andtheycontinuallyaskthemselves,“Doesthis makesense?”Theycanunderstandtheapproachesofotherstosolvingcomplex problemsandidentifycorrespondencesbetweendifferentapproaches. 2 Reason abstractly and quantitatively. 3 Construct viable arguments and critique the reasoning of others. Mathematicallyproficientstudentsunderstandandusestatedassumptions, definitions,andpreviouslyestablishedresultsinconstructingarguments.They makeconjecturesandbuildalogicalprogressionofstatementstoexplorethe truthoftheirconjectures.Theyareabletoanalyzesituationsbybreakingtheminto cases,andcanrecognizeandusecounterexamples.Theyjustifytheirconclusions, StandardS for matHematICal praCtICe | Mathematicallyproficientstudentsmakesenseofquantitiesandtheirrelationships inproblemsituations.Theybringtwocomplementaryabilitiestobearonproblems involvingquantitativerelationships:theabilitytodecontextualize—toabstract agivensituationandrepresentitsymbolicallyandmanipulatetherepresenting symbolsasiftheyhavealifeoftheirown,withoutnecessarilyattendingto theirreferents—andtheabilitytocontextualize,topauseasneededduringthe manipulationprocessinordertoprobeintothereferentsforthesymbolsinvolved. Quantitativereasoningentailshabitsofcreatingacoherentrepresentationof theproblemathand;consideringtheunitsinvolved;attendingtothemeaningof quantities,notjusthowtocomputethem;andknowingandflexiblyusingdifferent propertiesofoperationsandobjects. 6 Common Core State StandardS for matHematICS communicatethemtoothers,andrespondtotheargumentsofothers.Theyreason inductivelyaboutdata,makingplausibleargumentsthattakeintoaccountthe contextfromwhichthedata arose.Mathematicallyproficientstudentsarealsoable tocomparetheeffectivenessoftwoplausiblearguments,distinguishcorrectlogicor reasoningfromthatwhichisflawed,and—ifthereisaflawinanargument—explain whatitis.Elementarystudentscanconstructargumentsusingconcretereferents suchasobjects,drawings,diagrams,andactions.Suchargumentscanmakesense andbecorrect,eventhoughtheyarenotgeneralizedormadeformaluntillater grades.Later,studentslearntodeterminedomainstowhichanargumentapplies. Studentsatallgradescanlistenorreadtheargumentsofothers,decidewhether theymakesense,andaskusefulquestionstoclarifyorimprovethearguments. 4 Model with mathematics. Mathematicallyproficientstudentscanapplythemathematicstheyknowtosolve problemsarisingineverydaylife,society,andtheworkplace.Inearlygrades,thismight beassimpleaswritinganadditionequationtodescribeasituation.Inmiddlegrades, astudentmightapplyproportionalreasoningtoplanaschooleventoranalyzea probleminthecommunity.Byhighschool,astudentmightusegeometrytosolvea designproblemoruseafunctiontodescribehowonequantityofinterestdepends onanother.Mathematicallyproficientstudentswhocanapplywhattheyknoware comfortablemakingassumptionsandapproximationstosimplifyacomplicated situation,realizingthatthesemayneedrevisionlater.Theyareabletoidentify importantquantitiesinapracticalsituationandmaptheirrelationshipsusingsuch toolsasdiagrams,two-waytables,graphs,flowchartsandformulas.Theycananalyze thoserelationshipsmathematically todrawconclusions.Theyroutinelyinterprettheir mathematicalresultsinthecontextofthesituationandreflectonwhethertheresults makesense,possiblyimprovingthemodelifithasnotserveditspurpose. 5 Use appropriate tools strategically. 6 Attend to precision. Mathematicallyproficientstudentstrytocommunicatepreciselytoothers.They trytousecleardefinitionsindiscussionwithothersandintheirownreasoning. Theystatethemeaningofthesymbolstheychoose,includingusingtheequalsign consistentlyandappropriately.Theyarecarefulaboutspecifyingunitsofmeasure, andlabelingaxestoclarifythecorrespondencewithquantitiesinaproblem.They calculateaccuratelyandefficiently,expressnumericalanswerswithadegreeof precisionappropriatefortheproblemcontext.Intheelementarygrades,students givecarefullyformulatedexplanationstoeachother.Bythetimetheyreachhigh schooltheyhavelearnedtoexamineclaimsandmakeexplicituseofdefinitions. StandardS for matHematICal praCtICe | Mathematicallyproficientstudentsconsidertheavailabletoolswhensolvinga mathematicalproblem.Thesetoolsmightincludepencilandpaper,concrete models,aruler,aprotractor, acalculator,aspreadsheet,acomputeralgebrasystem, astatisticalpackage,ordynamicgeometrysoftware.Proficientstudentsare sufficientlyfamiliarwithtoolsappropriatefortheirgradeorcoursetomakesound decisionsaboutwheneachofthesetoolsmightbehelpful,recognizingboththe insighttobegainedandtheirlimitations.Forexample,mathematicallyproficient highschoolstudentsanalyzegraphsoffunctionsandsolutionsgeneratedusinga graphingcalculator.Theydetectpossible errorsbystrategicallyusingestimation andothermathematicalknowledge.Whenmakingmathematicalmodels,theyknow thattechnologycanenablethemtovisualizetheresultsofvaryingassumptions, exploreconsequences,andcomparepredictionswithdata.Mathematically proficientstudentsatvariousgradelevelsareabletoidentifyrelevantexternal mathematicalresources,suchasdigitalcontentlocatedonawebsite,andusethem toposeorsolveproblems.Theyareabletousetechnologicaltoolstoexploreand deepentheirunderstandingofconcepts. 7 Common Core State StandardS for matHematICS 7 Look for and make use of structure. Mathematicallyproficientstudentslookcloselytodiscernapatternorstructure. Youngstudents,forexample,mightnoticethatthreeandsevenmoreisthesame amountassevenandthreemore,ortheymaysortacollectionofshapesaccording tohowmanysidestheshapeshave.Later,studentswillsee7×8equalsthe wellremembered7×5+ 7×3,inpreparationforlearningaboutthedistributive property.Intheexpressionx2+9x+14,olderstudentscanseethe14as2×7and the9as2+7.Theyrecognizethesignificanceofanexistinglineinageometric figureandcanusethestrategyofdrawinganauxiliarylineforsolvingproblems. Theyalsocanstepbackforanoverviewandshiftperspective.Theycansee complicatedthings,suchassomealgebraicexpressions,assingleobjectsoras beingcomposedofseveralobjects.Forexample,theycansee5–3(x–y)2as5 minusapositivenumbertimesasquareandusethattorealizethatitsvaluecannot bemorethan5foranyrealnumbersxandy. 8 Look for and express regularity in repeated reasoning. Mathematicallyproficientstudentsnoticeifcalculationsarerepeated,andlook bothforgeneralmethodsandforshortcuts. Upperelementarystudentsmight noticewhendividing25by11thattheyarerepeatingthesamecalculationsover andoveragain,andconcludetheyhavearepeatingdecimal.Bypayingattention tothecalculationofslopeastheyrepeatedlycheckwhetherpointsareontheline through(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)mightleadthemtothe generalformulaforthesumofageometricseries.Astheyworktosolveaproblem, mathematicallyproficientstudentsmaintainoversightoftheprocess,while attendingtothedetails.Theycontinuallyevaluatethereasonablenessoftheir intermediateresults. Connecting the Standards for Mathematical Practice to the Standards for Mathematical Content TheStandardsforMathematicalPracticedescribewaysinwhichdevelopingstudent practitionersofthedisciplineofmathematicsincreasinglyoughttoengagewith thesubjectmatterastheygrowinmathematicalmaturityandexpertisethroughout theelementary,middleandhighschoolyears.Designersofcurricula,assessments, andprofessionaldevelopmentshouldallattendtotheneedtoconnectthe mathematicalpracticestomathematicalcontentinmathematicsinstruction. Inthisrespect,thosecontentstandardswhichsetanexpectationofunderstanding arepotential“pointsofintersection”betweentheStandardsforMathematical ContentandtheStandardsforMathematicalPractice.Thesepointsofintersection areintendedtobeweightedtowardcentralandgenerativeconceptsinthe schoolmathematicscurriculumthatmostmeritthetime,resources,innovative energies,andfocusnecessarytoqualitativelyimprovethecurriculum,instruction, assessment,professionaldevelopment,andstudentachievementinmathematics. StandardS for matHematICal praCtICe | TheStandardsforMathematicalContentareabalancedcombinationofprocedure andunderstanding.Expectationsthatbeginwiththeword“understand”areoften especiallygoodopportunitiestoconnectthepracticestothecontent.Students wholackunderstandingofatopicmayrelyonprocedurestooheavily.Without aflexiblebasefromwhichtowork,theymaybelesslikelytoconsideranalogous problems,representproblemscoherently,justifyconclusions,applythemathematics topracticalsituations,usetechnologymindfullytoworkwiththemathematics, explainthemathematicsaccuratelytootherstudents,stepbackforanoverview,or deviatefromaknownproceduretofindashortcut.Inshort,alackofunderstanding effectivelypreventsastudentfromengaginginthemathematicalpractices. 8 Standards for Mathematical Practice Look-­for Tool Mathematical Practice 1. Make sense of problems and persevere in solving them. Mathematically Proficient Students: Examples of what students would say: 3. Construct viable arguments and critique the reasoning of others. Teacher Actions to engage students in Practices: Examples of what the teacher would say: Examples of what students would say: Examples of what the teacher would say: Revised for North Clackamas School District #12 November 9, 2011 Construct viable arguments and critique the reasoning of others I can make conjectures and critique the mathematical thinking of others. When You Disagree With Someone’s Thinking: When You Agree with Someone’s Thinking: I disagree ____________. I agree because ____________. What about ____________? This makes sense because ____________. That’s not how I see it because ____________. That’s how I see it too because ____________. The way I see it is ____________. I did it that same way. I ____________. Another idea is ____________. Another way to do it is ____________. I tried something different ____________. When You Have an Idea: When You Want to Clarify: Can you explain why ____________. I have an idea ____________. I don’t quite understand ____________. Let’s try ____________. Can you model _________ with manipulatives? Maybe we could ____________. Would it work if ____________. Pictorial Input Chart Draw a picture of your story problem. Identify the key math vocabulary, including prepositions, and determine how you will teach them. ● Create a physical gesture to support the key words. ● Determine what manipulatives will be used to model the solution to the problem. ● ● Picture Gestures: Manipulatives: Key Words Pictorial Input Chart Draw a picture of your story problem. Identify the key math vocabulary, including prepositions, and determine how you will teach them. ● Create a physical gesture to support the key words. ● Determine what manipulatives will be used to model the solution to the problem. ● ● Picture Gestures: Manipulatives: Key Words 2nd Grade Word Problems 1. Jake had 36 stickers. He gave 10 to his sister and 10 to a friend. How many stickers does Jake have left? 2. Jack has the lead role in the school play. He has to learn 55 lines before opening night. Jack learns 11 lines each day. How many days will it take Jack to learn all his lines? 3. Lauren solved 36 math problems. Griffin solved 23 more problems than Lauren. How many problems did Griffin solve? 4. Tracy caught 9 fish in the morning. She threw 5 of them back because they were too small. She caught 8 more in the afternoon. How many fish did Tracy have then? 5. Kira and Sally were playing Cover Up with 52 counters. Kira hid some of the counters, and then 29 were showing. How many counters did Kira hide? 6. Franco had 66 car stickers. Jake gave him 56 car stickers. How many car stickers does Franco have now?