5th Grade Math Agenda November 2011 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.  I understand that there is a new ODE Math Scoring Guide and have explored its dimensions. Agenda
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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.
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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 ____________.
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 Fruit Scales Julie went to the store and bought apples, oranges, and bananas. The total weight of the fruit was 496.725grams. The oranges weighed 121.256grams. The apples weighed 152.145grams. How much did the bananas weigh? Total Weight: 496.725g NCSD Math Problem Solving Prompt: Grade 5, Standard 5.1.5 November 9, 2011 Name _____________________________ Date ______________________ Making sense of the task Rater Representing and Solving Communicating Reasoning Reflecting and Evaluating Accuracy Bike Ride Grade 5 (Standard 5.1.5) Amelia rode her bike 18 miles each week. She rode 1.6 more miles on Thursday than on Tuesday. Find the number of miles she rode on Tuesday and Thursday. Monday Tuesday Wednesday Thursday Friday Total 1.2 mi ? 3.7 mi ? 5.5 mi 18 mi NCSD Math Problem Solving Prompt: Grade 5, Standard 5.1.5 November 9, 2011