Elementary Math Agenda Primary: January 2012 Norms
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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
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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.
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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.
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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.
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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. ●
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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. ●
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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?