common core state standards standards for mathematical...

Common Core State Standards Standards for Mathematical PracticeThe Standards for Mathematical Practice are an important part of the Common Core State Standards. They describe varieties of proficiency that teachers should focus on to develop their students. These practices draw from the NCTM Process Standards of problem solving, reasoning and proof, communication, representation, connections, and the strands of mathematical proficiency specified in the National Research Council’s report Adding It Up: adaptive reasoning, strategic competence, conceptual understanding, procedural fluency, and productive disposition.

This document includes an explanation of the features and elements of Pearson’s Prentice Hall Middle Grades Mathematics program that help students develop mathematical proficiency for each of the Standards for Mathematical Practice.

Common Core Edition ©2013

Understanding and Solving Problems

The structure of Prentice Hall Middle Grades Mathematics supports students as they make sense of problems and persevere in solving them. The program was designed around a four-step problem-solving approach. Students are reminded of the problem-solving plan in the Problem Solving Handbook found at the beginning of the Student Edition.

Apply Problem-Solving Strategies

In every lesson, students make sense of problems with two Guided Problem Solving exercises found within the Homework Exercises. One has an accompanying student workbook page that models the questions students should ask themselves to analyze the givens of the problems, determine a solution plan, and persevere to a solution. Students are also encouraged to check if the answers they found make sense within the context of the problem. Using Real-World

Math Applications

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Standard for Mathematical Practice

Mathematically proficient students start by explaining to themselves the meaning of a problem and looking for entry points to its solution.

Theyanalyzegivens,constraints,relationships,andgoals.Theymakeconjecturesabouttheformandmeaningofthesolutionandplanasolutionpathwayratherthansimplyjumpingintoasolutionattempt.Theyconsideranalogousproblems,andtryspecialcasesandsimplerformsoftheoriginalprobleminordertogaininsightintoitssolution.Theymonitorandevaluatetheirprogressandchangecourseifnecessary.Olderstudentsmight,dependingonthecontextoftheproblem,transformalgebraicexpressionsorchangetheviewingwindowontheirgraphingcalculatortogettheinformationtheyneed.

Mathematicallyproficientstudentscanexplaincorrespondencesbetweenequations,verbaldescriptions,tables,andgraphsordrawdiagramsofimportantfeaturesandrelationships,graphdata,andsearchforregularityortrends.Youngerstudentsmightrelyonusingconcreteobjectsorpicturestohelpconceptualizeandsolveaproblem.Mathematicallyproficientstudentschecktheiranswerstoproblemsusingadifferentmethod,andtheycontinuallyaskthemselves,“Doesthismakesense?”Theycanunderstandtheapproachesofotherstosolvingcomplexproblemsandidentifycorrespondencesbetweendifferentapproaches.

1. Make sense of problems and persevere in solving them.

How Pearson Develops Mathematical Proficiency

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Each chapter includes a Guided Problem Solving activity which requires that students apply their sense-making and perseverance skills to solve real-world problems. As students progress through the program, scaffolding becomes less structured and students analyze a problem situation and formulate solutions with greater autonomy and proficiency.

Symbolic Representation

Reasoning is another important theme of Prentice Hall Middle Grades Mathematics. Many of the examples in lessons are Application Examples in which students are guided to represent the situation symbolically, either numerically or algebraically.

Pause and Explain Thinking

Each lesson ends with a Check Your Understanding

feature, in which students explain their thinking related

to the concepts studied in the lesson.


2. Reason abstractly and quantitatively.

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Mathematically proficient students make sense of quantities and their relationships in problem situations.

Theybringtwocomplementaryabilitiestobearonproblemsinvolvingquantitativerelationships:theabilitytodecontextualize—toabstractagivensituationandrepresentitsymbolicallyandmanipulatetherepresentingsymbolsasiftheyhavealifeoftheirown,withoutnecessarilyattendingtotheirreferents—andtheabilitytocontextualize,topauseasneededduringthemanipulationprocessinordertoprobeintothereferentsforthesymbolsinvolved.Quantitativereasoningentailshabitsofcreatingacoherentrepresentationoftheproblemathand;consideringtheunitsinvolved;attendingtothemeaningofquantities,notjusthowtocomputethem;andknowingandflexiblyusingdifferentpropertiesofoperationsandobjects.

Contextualize Your Response

Through the solving process, as they manipulate expressions, students are reminded to check back to the problem situation with the Check for Reasonableness.

Focus on the Meaning

Throughout the exercise sets, Reasoning Exercises focus students’ attention on the structure or meaning of an operation rather than the solution.


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Focus on Critical Reasoning

Consistent with a focus on reasoning and sense-making is a focus on critical reasoning/ argumentation and critique of arguments. In Prentice Hall Middle Grades Mathematics, students are frequently asked to explain their solutions and the thinking that led to them. The many Reasoning Exercises found throughout the program call for students to justify or explain their solutions. In the More Than One Way features, students analyze and critique the solution plans and reasoning of two students, each of whom presents a different solution for the same problem.

Analyze the Solution

The Error Analysis exercises provide students additional opportunities to analyze and critique the solution presented to a problem.

Apply the Math

The Chapter Projects, in the back of the Student Edition, provide students with opportunities to apply the mathematics they are learning to solve meaningful, real-life situations.

Building Mathematical Models

In Prentice Hall Middle Grades Mathematics students are guided to build mathematical models using equations, graphs, tables, and technology. Application Examples show students how the mathematical concept under study can be applied as a model for a real-world problem situation. The Stepped-Out Process shows students the thinking that can help them apply models to the problem situations presented.

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Mathematically proficient students understand and use stated assumptions, definitions, and previously established results in constructing arguments.

Theymakeconjecturesandbuildalogicalprogressionofstatementstoexplorethetruthoftheirconjectures.Theyareabletoanalyzesituationsbybreakingthemintocases,andcanrecognizeandusecounterexamples.Theyjustifytheirconclusions,communicatethemtoothers,andrespondtotheargumentsofothers.Theyreasoninductivelyaboutdata,makingplausibleargumentsthattakeintoaccountthecontextfromwhichthedataarose.Mathematicallyproficientstudentsarealsoabletocomparetheeffectivenessoftwoplausiblearguments,distinguishcorrectlogicorreasoningfromthatwhichisflawed,and—ifthereisaflawinanargument—explainwhatitis.Elementarystudentscanconstructargumentsusingconcretereferentssuchasobjects,drawings,diagrams,andactions.Suchargumentscanmakesenseandbecorrect,eventhoughtheyarenotgeneralizedormadeformaluntillatergrades.Later,studentslearntodeterminedomainstowhichanargumentapplies.Studentsatallgradescanlistenorreadtheargumentsofothers,decidewhethertheymakesense,andaskusefulquestionstoclarifyorimprovethearguments.

3. Construct viable arguments and critique the reasoning of others.



Mathematically proficient students can apply the mathematics they know to solve problems arising in everyday life, society, and the workplace.

Inearlygrades,thismightbeassimpleaswritinganadditionequationtodescribeasituation.Inmiddlegrades,astudentmightapplyproportionalreasoningtoplanaschooleventoranalyzeaprobleminthecommunity.Byhighschool,astudentmightusegeometrytosolveadesignproblemoruseafunctiontodescribehowonequantityofinterestdependsonanother.Mathematicallyproficientstudentswhocanapplywhattheyknowarecomfortablemakingassumptionsandapproximationstosimplifyacomplicatedsituation,realizingthatthesemayneedrevisionlater.Theyareabletoidentifyimportantquantitiesinapracticalsituationandmaptheirrelationshipsusingsuchtoolsasdiagrams,two-waytables,graphs,flowchartsandformulas.Theycananalyzethoserelationshipsmathematicallytodrawconclusions.Theyroutinelyinterprettheirmathematicalresultsinthecontextofthesituationandreflectonwhethertheresultsmakesense,possiblyimprovingthemodelifithasnotserveditspurpose.

4. Model with mathematics.


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Working with Math Tools

Students become fluent in the use of an assortment of tools ranging from physical devices, such as rulers, protractors, and pencil and paper, to technological tools, such as calculators, graphing calculators,and computers. They use various manipulatives and technology tools in the Activity Labs. By developing fluency in the use of different tools, students are able to select the appropriate tool(s) to solve a given problem.

Applying Digital Technology

Technology and technology tools, such as graphing calculators, are an integral part of Prentice Hall Middle Grades Mathematics and are used in these ways: to develop understanding of mathematicalconcepts, to solve problems that would beunapproachable without the use of technology, and to build models based on real-world data.

Explain the Process

For the Writing in Math exercises, students are once again expected to provide clear, concise explanations of terms, concepts, or processes or to use specific terminology accurately and precisely. Students are reminded to use appropriate units of measure when working through solutions and accurate labels on axes when making graphs to represent solutions.

Focus on Precise Communication

Students are expected to use mathematical terms and symbols with precision. Key terms are highlighted in each lesson and important concepts explained in the Key Concepts features. In the Check Your Understanding feature, students revisit these key terms and provide explicit definitions or explanations of the terms.


Mathematically proficient students consider the available tools when solving a mathematical problem.

Thesetoolsmightincludepencilandpaper,concretemodels,aruler,aprotractor,acalculator,aspreadsheet,acomputeralgebrasystem,astatisticalpackage,ordynamicgeometrysoftware.Proficientstudentsaresufficientlyfamiliarwithtoolsappropriatefortheirgradeorcoursetomakesounddecisionsaboutwheneachofthesetoolsmightbehelpful,recognizingboththeinsighttobegainedandtheirlimitations.Forexample,mathematicallyproficienthighschoolstudentsanalyzegraphsoffunctionsandsolutionsgeneratedusingagraphingcalculator.Theydetectpossibleerrorsbystrategicallyusingestimationandothermathematicalknowledge.Whenmakingmathematicalmodels,theyknowthattechnologycanenablethemtovisualizetheresultsofvaryingassumptions,exploreconsequences,andcomparepredictionswithdata.Mathematicallyproficientstudentsatvariousgradelevelsareabletoidentifyrelevantexternalmathematicalresources,suchasdigitalcontentlocatedonaWebsite,andusethemtoposeorsolveproblems.Theyareabletousetechnologicaltoolstoexploreanddeepentheirunderstandingofconcepts.

5. Use appropriate tools strategically.



Mathematically proficient students try to communicate precisely to others.

Theytrytousecleardefinitionsindiscussionwithothersandintheirownreasoning.Theystatethemeaningofthesymbolstheychoose,includingusingtheequalsignconsistentlyandappropriately.Theyarecarefulaboutspecifyingunitsofmeasure,andlabelingaxestoclarifythecorrespondencewithquantitiesinaproblem.Theycalculateaccuratelyandefficiently,expressnumericalanswerswithadegreeofprecisionappropriatefortheproblemcontext.Intheelementarygrades,studentsgivecarefullyformulatedexplanationstoeachother.

6. Attend to precision.


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Solve Problems More Efficiently

Once again, throughout each course and the program as a whole, students are prompted to look for repetition in calculations to devise general methods or shortcuts that can make the problem-solving process more efficient. Students are prompted to look for similar problems they have previously encountered or to generalize results to other problem situations. The Online Active Math activities offer students opportunities to notice regularity in the way operations or functions behave by easily inputting different values.


Mathematically proficient students look closely to discern a pattern or structure.

Youngstudents,forexample,mightnoticethatthreeandsevenmoreisthesameamountassevenandthreemore,ortheymaysortacollectionofshapesaccordingtohowmanysidestheshapeshave.Later,studentswillsee7×8equalsthewellremembered7×5+7×3,inpreparationforlearningaboutthedistributiveproperty.Intheexpressionx2+9x+14,olderstudentscanseethe14as2×7andthe9as2+7.Theyrecognizethesignificanceofanexistinglineinageometricfigureandcanusethestrategyofdrawinganauxiliarylineforsolvingproblems.Theyalsocanstepbackforanoverviewandshiftperspective.Theycanseecomplicatedthings,suchassomealgebraicexpressions,assingleobjectsorasbeingcomposedofseveralobjects.Forexample,theycansee5−3(x−y)2as5minusapositivenumbertimesasquareandusethattorealizethatitsvaluecannotbemorethan5foranyrealnumbersxandy.

7. Look for and make use of structure.



Mathematically proficient students notice if calculations are repeated, and look both for general methods and for shortcuts.

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,mathematicallyproficientstudentsmaintainoversightoftheprocess,whileattendingtothedetails.Theycontinuallyevaluatethereasonablenessoftheirintermediateresults.

8. Look for and express regularity in repeated reasoning.

Identify Patterns and Structure

Throughout the program, students are encouraged to discern patterns and structure as they look to formulate solution pathways. The Pattern/Look for a Pattern exercises explicitly ask students to find patterns in operations or graphic displays.


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