Elementary Math Agenda December/January 2011 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. I understand that there is a new ODE Math Scoring Guide and have explored its dimensions. Agenda • • • • • Introductions to Common Core State Standards Digging into Mathematical Practices Content Knowledge and Sheltered Instruction: Decimal Models, Fractions and Division Problem Solving Math Scoring Guide 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 ____________. The way I see it is ____________. Another idea is ____________. Another way to do it is ____________. I tried something different, I ____________. That’s how I see it too because ____________. I did it that same way. I ___________. When You Have an Idea: When You Want to Clarify: I have an idea ____________. Can you explain why ____________. Let’s try ____________. I don’t quite understand ____________. Maybe we could ____________. Can you model _________ with manipulatives? Would it work if ____________. _____ is greater than _____. _____ is less than _____. _____ is equal to _____. Both _____ and _____ are greater than _____. Both _____ and Both _____ and _____ are less _____ are equal than _____. to _____. _____ is greater than _____. _____ is less than _____. 2011-2012 Mathematics Problem Solving Official Scoring Guide 2011-2012 Apply mathematics in a variety of settings. Build new mathematical knowledge through problem solving. Solve problems that arise in mathematics and in other contexts. Apply and adapt a variety of appropriate strategies to solve problems. Monitor and reflect on the process of mathematical problem solving. Process Dimensions Making Sense of the Task Interpret the concepts of the task and translate them into mathematics. Representing and Solving the Task Use models, pictures, diagrams, and/or symbols to represent and solve the task situation and select an effective strategy to solve the task. Communicating Reasoning Coherently communicate mathematical reasoning and clearly use mathematical language. Accuracy Support the solution/outcome. Reflecting and Evaluating State the solution/outcome in the context of the task. Defend the process, evaluate and interpret the reasonableness of the solution/outcome. **6/ 5 4 3 *2 / 1 The interpretation and/or translation of the task are thoroughly developed and/or enhanced through connections and/or extensions to other mathematical ideas or other contexts. The strategy and representations used are elegant (insightful), complex, enhanced through comparisons to other representations and/or generalizations. The interpretation and translation of the task are adequately developed and adequately displayed. The interpretation and/or translation of the task are partially developed, and/or partially displayed. The strategy that has been selected and applied and the representations used are effective and complete. The strategy that has been selected and applied and the representations used are partially effective and/or partially complete. The interpretation and/or translation of the task are underdeveloped, sketchy, using inappropriate concepts, minimal, and/or not evident. The strategy selected and representations used are underdeveloped, sketchy, not useful, minimal, not evident, and/or in conflict with the solution/outcome. The use of mathematical language and communication of the reasoning are elegant (insightful) and/or enhanced with graphics or examples to allow the reader to move easily from one thought to another. The solution/outcome is correct and enhanced by extensions, connections, generalizations, and/or asking new questions leading to new problems. Justifying the solution/outcome completely, the student reflection also includes reworking the task using a different method, evaluating the relative effectiveness and/or efficiency of different approaches taken, and/or providing evidence of considering other possible solution/outcomes and/or interpretations. The use of mathematical language and communication of the reasoning follow a clear and coherent path throughout the entire work sample and lead to a clearly identified solution/outcome. The use of mathematical language and communication of the reasoning are partially displayed with significant gaps and/or do not clearly lead to a solution/outcome. The solution/outcome given is correct, mathematically justified, and supported by the work. The solution/outcome given is incorrect due to minor error(s), or a correct answer but work contains minor error(s) partially complete, and/or partially correct The solution/outcome is stated within the context of the task, and the reflection justifies the solution/outcome completely by reviewing the interpretation of the task concepts, strategies, calculations, and reasonableness. The solution/outcome is not stated clearly within the context of the task, and/or the reflection only partially justifies the solution/outcome by reviewing the task situation, concepts, strategies, calculations, and/or reasonableness. The use of mathematical language and communication of the reasoning are underdeveloped, sketchy, inappropriate, minimal, and/or not evident. The solution/outcome given is incorrect and/or incomplete, or correct, but o conflicts with the work, or o not supported by the work. The solution/outcome is not clearly identified and/or the justification is underdeveloped, sketchy, ineffective, minimal, not evident, and/or inappropriate. **6 for a given dimension would have most attributes in the list; 5 would have some of those attributes. *2 for a given dimension would be underdeveloped or sketchy, while a 1 would be minimal or nonexistent. For use beginning with 2011-2012 Assessments Oregon Department of Education Office of Assessment and Evaluation Adopted May 19, 2011 2011-­‐12 Mathematics Problem Solving Scoring Guide: Plain Language Student Version **6/5 Process Dimensions Making Sense of the Task Understand the ideas and change them into a math task WHAT? Representing and Solving the Task Choose the strategy that works best for this problem. HOW? 4 3 2011-­‐12 2/1* ·The problem is changed into ·The problem is changed into math ·Parts of the problem are changed into ·Only a small amount of the problem is complete ideas that work. ideas that work. math ideas that can work or parts of the problem are understood. understood OR ·The ideas are connected to other math ·No understanding is shown. ideas. ·A complete plan is used that contains ·A plan using pictures, charts, words, ·The plan could solve some parts of the pictures, charts, words, graphs or numbers and may contain more than one step. graphs or numbers is used to solve the problem. problem or the plan has a few missing parts. ·The steps to complete the work are ·The path through the work can be ·The path is not clear OR doesn’t show very clear. followed to a clearly identified solution. much of the work The plan ·has many missing parts, ·cannot work OR ·No work is shown ·A superior strategy is used to solve the problem. Communicating Reasoning Use the language of math (words, equations, graphs, charts) to make your ideas clear to others. WHY? Accuracy The answer is… The steps to complete the work are ·just started OR ·There are no steps shown. ·An explanation for each part is given. ·Some attempt is made to explain why each step was used. ·The solution is correct and may be ·The answer given is correct. extended or shown another way. ·The answer given may have a small error but the important parts work fine. The answer given · is not correct, · is not finished OR · doesn’t match the work. IS IT RIGHT? Reflecting and Evaluating Check your answer and explain why it makes sense. CHECK? ·A second look has been taken to ·The problem is solved a second time ·Some, but not all of the work is checked. completely check the work. to check the work and method. · The explanation cleary shows why the solution makes sense. ·A new way may be used to check the · The student makes an attempt to explain why the answer makes sense. work. **6 for a given dimension would have most attributes in the list; 5 would have some of those attributes. *2 for a given dimension would be inadequate in some of the attributes; while a 1 would be inadequate in most of the attributes The check ·doesn’t work, ·is barely started OR · is not there at all. Name _____________________________ Date ______________________ Making sense of the task Rater Representing and Solving Communicating Reasoning Reflecting and Evaluating Accuracy Zoo Dilemma Grade 4 (Standard 4.1.6) Todd and Kay read in a newspaper that the zoo was going to have a special admission rate of two for $5.00 to celebrate the birth of a new baby gorilla. They didn’t have the money, so they decided to wash windows to earn money for their tickets. Kay washed two windows, each of which had a dozen windowpanes. Todd washed three windows, each of which had eight windowpanes. If they charged 10 cents for each windowpane they washed, and combined their money, will they have enough to go to the zoo? Explain your answer(s) and your thinking. NCSD Math Problem Solving Prompt: Grade 4, Standard 4.1.6 6 November 17, 2011 Name _____________________________ Date ______________________ Making sense of the task Rater Representing and Solving Communicating Reasoning Reflecting and Evaluating Accuracy Marty’s Activities Grade 4 (Standard 4.1.6) Marty spent $10.00 at the amusement park. He did two activities and bought one snack food. What activities did Marty do, and what food did Marty buy? Explain your thinking in the space below. Activities Cost Snacks Cost Putt Putt Golf $3.00 French Fries $1.75 Skating $2.25 Hamburger $3.75 Go-­‐Kart Rides $4.50 Ice Cream $3.25 Bounce House $1.75 Snow Cone $1.00 Super Coaster $3.75 NCSD Math Problem Solving Prompt: Grade 4, Standard 4.1.6 November 17, 2011 Name _____________________________ Date ______________________ Making sense of the task Rater Representing and Solving Communicating Reasoning Reflecting and Evaluating Accuracy Tiling the Counter Grade 4: (Standards 4.3.5 & 4.3.7) Suzanne is going to tile her bathroom counter, which measures 3 ft by 6ft. She has a choice between several kinds of tiles, but she can’t spend more than $20. Which tile/s can she afford? Tile Color Number of tiles needed to cover 1 square ft. Tile Cost White 1 $1.03 Gray 2 $0.56 Blue 4 $0.23 NCSD Math Problem Solving Prompt: Grade 4, Core Standard 3 November 28, 2011 Name _____________________________ Date ______________________ Making sense of the task Rater Representing and Solving Communicating Reasoning Reflecting and Evaluating Accuracy School Playground Grade 4: Standard 4.3.7 Franklin School has a rectangular playground that is 40 meters by 10 meters. The playground will be covered with recycled shoe rubber. Each bag of shredded shoe rubber covers 50 square meters and costs $30. How much will it cost Franklin School to cover the playground? NCSD Math Problem Solving Prompt: Grade 4, Core Standard 3 November 28, 2011 Name _____________________________ Date ______________________ Making sense of the task Rater Representing and Solving Communicating Reasoning Reflecting and Evaluating Accuracy Lemurs and Sloths Grade 4: Core Standard 2 A lemur sleeps 16 hours each day. A sloth sleeps 4 more hours each day than a lemur. How many total hours of sleep do a lemur and sloth get during a week? Sloth Lemur NCSD Math Problem Solving Prompt: Grade 4, Core Standard 2 November 17, 2011 Name _____________________________ Date ______________________ Making sense of the task Rater Representing and Solving Communicating Reasoning Reflecting and Evaluating Accuracy Math Homework Grade 4: Core Standard 2 Allison completed 30 problems for her math homework each night. She has math homework 5 nights a week. Write a real world problem using this information. Then solve and justify your thinking. Problem I am going to solve: Solution: NCSD Math Problem Solving Prompt: Grade 4, Core Standard 2 November 17, 2011