Implementing a Common Core Curriculum

Boston, MassachusettsUpper Saddle River, New Jersey

Implementing a Common Core Curriculum

CMP2_OIG_FM_i-iv.qxp 3/3/11 12:24 PM Page i

13-digit ISBN 978-0-13-318385-6

10-digit ISBN 0-13-318385-8

1 2 3 4 5 6 7 8 9 10 V036 15 14 13 12 11

Copyright © by Pearson Education, Inc., publishing as Pearson Prentice Hall, Boston, Massachusetts 02116. All rightsreserved. Printed in the United States of America. This publication is protected by copyright, and permission should beobtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. The publisher hereby grants permissionto reproduce these pages, in part or in whole, for classroom use only, the number not to exceed the number of students ineach class. Notice of copyright must appear on all copies. For information regarding permission(s), write to: Rights and Permissions Department, One Lake Street, Upper Saddle River, New Jersey 07458.

Common Core State Standards: © Copyright 2010. National Governors Association Center for Best Practices and Council ofChief State School Officers. All rights reserved.

Connected Mathematics™ is a trademark of Michigan State University.

Pearson Prentice Hall™ is a registered trademark of Pearson Education, Inc.

Pearson® is a registered trademark of Pearson plc.

Prentice Hall® is a registered trademark of Pearson Education, Inc.

ExamView® is a registered trademark of eInstruction Corporation.

TeacherExpress™ is a trademark of Pearson Education, Inc.

CMP2_OIG_FM_i-iv.qxp 3/3/11 12:24 PM Page ii

Implementing a Common Core Curriculum with CMP2Grades 6–8

Introduction From the CMP2 Authors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

The Standards for Mathematical Content . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

Areas of Emphasis by Grade. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

Domain Progressions Across Grades . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

What’s Different? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

The Standards for Mathematical Practice. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

Mathematical Reflection Preparing for an Investigation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

After an Investigation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

CCSS Assessments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

Partnership for Assessment of Readiness for College and Career . . . . . . . . . . . . . . . . . . . . . 19

SMARTER Balanced Assessment Consortium. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

Parent Letters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

Correlations of Grade 6-8 Content Standards to CMP2 Units . . . . . . . . . . . 25

Ratios and Proportional Relationships . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

The Number System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

Expressions and Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

Statistics and Probability. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

CMP2_OIG_FM_i-iv.qxp 3/3/11 12:24 PM Page iii

CMP2_OIG_FM_i-iv.qxp 3/3/11 12:24 PM Page iv

From the CMP2 Authors 1

The Connected Mathematics Project (CMP) and The Common Core State Standards for Mathematics (CCSSM)

CMP was developed to help both students and teachers develop mathematical knowledge, understanding andskill, as well as an awareness of and appreciation for the rich connections among mathematical strands andbetween mathematics and other disciplines. The development of the program was supported by two grantsfrom the National Science Foundation. These grants allowed for the time and support necessary to trial ideasin classrooms across the nation. The result is a curriculum that has provided powerful mathematicalexperiences for students since its release in 1996.

From the beginning, our overarching mathematical goal for students has been that all students be able toreason and communicate proficiently in mathematics. They should have knowledge of and skill in the use ofthe vocabulary, forms of representation, materials, tools, techniques, and intellectual methods of the disciplineof mathematics. This knowledge should include the ability to define and solve problems with reason, insight,inventiveness, and technical proficiency.

Connected Mathematics differs from more conventional US curricula in that it is problem-centered. Theproblem situations that students encounter shape the perceptions they have about the discipline. The study ofmathematics begins with undefined terms, axioms, and definitions, and deduces important conclusions logicallyfrom those starting points. However, mathematics itself is produced and used in a much more complexcombination of exploration, experience-based intuition, and reflection. To be able to develop the skills to solve“new” problems, CMP students spend time solving problems that require thinking, planning, reasoning,computing, connecting, proving, and evaluating. Such a problem-centered, connected curriculum helpsstudents to not only make sense of the mathematics, but also learn the mathematics in retrievable ways.

In 2010, the Common Core State Standards Initiative released the Common Core State Standards forMathematics (CCSSM). The CCSSM, which have been adopted by more than 40 states, consists of Standardsfor Mathematical Content and Standards for Mathematical Practice, They define what mathematicalunderstandings and skills students should have and be able to do. The writers of the CCSSM note thefollowing:

One hallmark of understanding is the ability to justify, in a way appropriate to the student’s mathematicalmaturity, whether a particular mathematical statement is true or where a mathematical rule comes from . . .Mathematical understanding and procedural skill are equally important and both are assessable usingmathematical tasks of sufficient richness. (CCSS for Mathematics. 2010. p.4)

The eight Standards for Mathematical Practice are: Make sense of problems and persevere in solving them;Reason abstractly and quantitatively; Construct viable arguments and critique the reasoning of others; Modelwith mathematics; Use appropriate tools strategically; Attend to precision; Look for and make sense of structure;Look for and express regularity in repeated reasoning. The first practice, Make sense of problems and perseverein solving them, can be thought of as the key practice with the remaining practices as elaborations of thispractice. These practices permeate the CMP curriculum. In addition, CMP reflects an additional essentialpractice: Build on and connect to prior knowledge to develop deeper understandings and new insights.

CMP2_OIG_CON_001-007.qxp 3/3/11 12:22 PM Page 1

2 Implementing a Common Core Curriculum with CMP2

The Common Core State Standards for Mathematics represent the collaborative efforts ofmathematicians, researchers, educators, and state education officials from 48 states and the District ofColumbia to develop a single set of rigorous and internationally-benchmarked standards for K-12mathematics. These standards set clear expectations for what students are to know and be able to doat each grade level from Kindergarten through High School.

Two guiding principles framed the development of the Standards for Mathematical Content: thatthey be based on evidence and research and build on the strengths of current state standards andthat at each grade level, the standards be fewer in number, but clearer and more rigorous. Thereduced number of standards at each grade level challenged the writing team to articulate aprogression of concepts and from that progression, to identify critical areas for each grade level.

Main Areas of Emphasis in Grade 6

• ratio and unit rate

• division of fractions and positive rational (ordecimal) numbers

• variables, expressions, and equations

• statistical thinking

• area, surface area, and volume

Grade 6 students draw from their knowledge ofwhole-number multiplication and division as theybegin their study of ratios and rates. They seeequivalent ratios as deriving from multiplicationtables, and analyze drawings that illustrate relativesize of quantities to expand their understanding ofmultiplication and division to ratios.

Students extend their work with fraction operationsto dividing fractions and mixed numbers. They drawon their knowledge of multiplication and divisionand of fractions to explain both the meaning of andprocess for dividing fractions. They extend theirstudy of numbers to include all positive rationalnumbers. They locate and order integers on a numberline, and reason about the order and absolute valueof rational numbers. They apply spatial orientationskills as they determine location of points in thecoordinate plane (all four quadrants).

In Grade 6, students begin their formal study ofalgebraic expressions. They make use of theirunderstandings of unknowns in expressions andequations to interpret expressions, to writeexpressions that match given situations, and toevaluate expressions. They draw on their knowledgeof properties of operations and of equality to solveone-step equations. They also represent and analyzethe relationships between quantities using graphs,tables, and equations.

Students in Grade 6 undertake a study of statisticsfor the first time. They develop their understandingof statistical variability by examining measures ofcenter (mean and median) and measures ofvariability (interquartile range and mean absolutedeviation). They create and analyze different datadisplays (e.g., box plots, histograms, and dot plots) todescribe and summarize data distributions.

Students build on their geometric concepts to reasonabout relationships among shapes to determine area,surface area, and volume. Through the manipulationof two- and three-dimensional shapes, studentsreason about and justify the formulas for area,surface area, and volume. They apply theirknowledge of operations with fractions to find thevolume of prisms with fractional side lengths. Theydraw polygons in the coordinate plane as afoundation for scale drawings introduced in Grade 7.


Standards for Mathematical Content 3


• proportional relationships and applying thoserelationships to solve problems

• operations with rational numbers; expressionsand linear equations

• scale drawings and informal geometricconstructions; attributes of circles

• drawing inferences about populations based onsamples; concepts of chance

In Grade 7, students extend their study of ratios andrates to develop an understanding of proportionality.They solve problems related to proportionalrelationships, including a range of percent problems(e.g., discounts, interest, taxes, tips, percent ofincrease/decrease). They apply the concept ofproportionality to scale drawings and solve problemsof scale. They graph proportional relationships anddevelop an informal understanding of slope. Theydifferentiate proportional relationships from otherrelationships.

Students develop a holistic understanding of numberas they express rational numbers in different formsof representations (e.g., fractions, decimals, and

percents) and interpret negative numbers ineveryday context (e.g., temperature change).Students extend their knowledge of operations andof properties of operations to solve problems withany rational numbers, both positive and negativerational numbers. They write expressions andequations in one variable to solve problems.

Students expand their reasoning about shapes toinclude circles. They build on their knowledge ofgeometric attributes and formulas to determine thecircumference and area of a circle and the surfacearea and volume of any polyhedron. They reasonabout relationships among two-dimensional figuresusing scale drawings and informal geometricconstructions. They solve problems involving anglesformed by intersecting lines, and they examine cross-sections of three-dimensional figures.

Students continue their study of statistical thinkingby exploring the concept of random sampling. Bycomparing two data distributions and describingdifferences between the populations, students beginto understand the importance of representativesamples for drawing valid inferences aboutpopulations.


• expressions and equations, including systems oflinear equations

• functions to describe quantitative relationships

• analysis of two- and three-dimensional spaceand figures and the Pythagorean Theorem

In Grade 8, students engage in formal study ofalgebraic expressions and linear equations. Theywrite and use linear equations, linear functions, andsystems of linear equations to represent, analyze, andsolve problems. Students recognize that linearequations in the form y = mx represent proportionalrelationships, and can explain slope as a constant ofproportionality. Students use a linear model todescribe the relationship between two quantities in abivariate data set (e.g., arm span to height) andinformally describe the fit of the model to the data.

Students choose efficient and generalizablemethods to solve linear equations in one variable,applying properties of operations and of equality.They solve systems of two linear equations in two

variables and relate the systems to pairs of lines inthe plane.

Students begin their study of functions in Grade 8.They explain the concept and uses of functions anddescribe functions in different representations(graphic, tabular, algebraic). Further, they analyzefunctions to describe how elements of the functionare shown in each representation.

Students’ study of geometry formalizes concepts ofcongruence and similarity through the study oftransformations. They explore the behavior of shapesunder translations, rotations, reflections, anddilations, and relate these behaviors to concepts ofcongruence and similarity. Students study anglerelationships in shapes (triangles) and on parallellines cut by a transversal. Students are able toexplain the Pythagorean Theorem and its converse,and why it holds. They apply the PythagoreanTheorem to solve problems. Students complete theirstudy of volume by finding the volumes of cones,cylinders, and spheres.



Domain Progressions Across Grades

The Common Core State Standards promote a more conceptual and analytical approach to thestudy of mathematics. In the elementary years, the focus is on interpreting operations to helpstudents develop strategic competence. This helps them make sense of operations conceptually andapply this conceptual understanding in different contexts with different forms of numbers. In themiddle school years, the focus is on strengthening students’ multiplicative reasoning skills and onbuilding a solid foundation of numeric and algebraic concepts and skills.

As students begin their study of mathematics in the middle years, they have completed their study ofthe four arithmetic operations with whole numbers and decimals and will complete their study of thefour operations with fractions in Grade 6. The focus now turns to the development of algebraicconcepts and skills, which grows from students’ understanding of arithmetic operations. Students willapply their knowledge of place value, properties of operations, and the inverse relationships betweenoperations (addition and subtraction; multiplication and division) to explore and evaluate algebraicexpressions. As they did with arithmetic expressions, students will interpret and analyze parts of theexpression and explore the relationships among the parts of the expressions to build not justprocedural fluency, but conceptual understanding and strategic competence. This analytic focus helpsstudents to look more fully at the equations and expressions so that they begin to see patterns in thestructure. These patterns will be useful when students explore more complex algebraic concepts inhigh school.

Ratios and Proportional Relationships

Students begin their study of ratios and rates inGrade 6. Using different ratio notations, studentsexplain the concept of a ratio and describe ratiorelationships between two quantities. They displaythese ratio relationships in tables and by plottingpoints on the coordinate plane. They solve real-world and mathematical problems using models andratio and rate reasoning. These problems includeconverting measurements and finding unit rates,percents of a quantity, and the whole given a partand a percent.

In Grade 7, students build on their knowledge ofratios as they study proportional relationships. Theyanalyze proportional relationships using variousstrategies and models and write equations torepresent these relationships. They look to identifythe constant of proportionality and use proportionalrelationships to solve a real-life problems, includinga wide variety of percent problems (e.g., discounts,interest, taxes, tips, and percent of increase ordecrease). They also graph proportionalrelationships and understand unit rate as a measureof the steepness (slope) of the related line.

In Grade 8, students compare two differentproportional relationships and formally study theslope of linear equations.

Functions

Students begin study of functions in Grade 8. Theydraw from their understanding of ratios andproportional relationships to examine functionspresented in different ways (algebraically,numerically in tables, and graphically). Theycompare the properties of different functionspresented in different ways. Students describe thefunctional relationship between two quantities andrepresent the relationship graphically. This formalintroduction to functions provides students with animportant foundation for the study of high schoolmathematics.



The Number System

In Grade 6, students extend their understanding ofnumbers to the system of rational numbers,including positive and negative rational numbers.They describe the uses of these quantities in real-world and mathematical contexts and extend thenumber line to represent and order negative valuesand to find the absolute value of integers. Theyexpand their work with factors and multiples to findgreatest common factors and least commonmultiples. Students also extend their study of thecoordinate plane to include all four quadrants.

In Grade 7, students gain an understanding ofterminating and repeating decimals as well ascomplex fractions. They perform operations with anyrational numbers, including negative rationalnumbers (e.g., integers). They can explain theadditive inverse and its value when subtractingnegative rational numbers. As they carry outoperations, students interpret the solutions bydescribing real-world contexts.

In Grade 8, students augment the rational numberswith the irrational numbers to form the realnumbers. In high school, students will augment thereal numbers with the imaginary numbers to formthe complex numbers.

Expressions and Equations

Students begin a formal study of algebraicexpressions in Grade 6.They start by drawing fromtheir understanding of arithmetic expressions andproperties of operations to write and evaluatealgebraic expressions. They manipulate the parts ofthe expressions, describing each part inmathematical terms and the relationship of eachpart within the expression. Students analyzeexpressions and identify equivalent expressions.Students also evaluate expressions, performingarithmetic operations following the Order ofOperations. Finally, students explore quantities thatchange in relationship to one another and analyzeand represent the relationship between the twoquantities. They come to see one of the quantities asthe dependent variable and the other as theindependent variable. Students explore inequalitiesfor the first time, writing inequalities to represent acondition or constraint in a real-world ormathematical problem. Representing solutions toinequalities on the number line, students realize thatinequalities have an infinite number of solutions.

In Grade 7, students extend their analysis ofexpressions to explain the relationships among thequantities that are revealed when an expression isrewritten. They expand their understanding ofexpressions to solving multi-step algebraicequations. Students continue their study ofinequalities, solving word problems that lead toinequalities and graphing the solutions on numberlines.

In Grade 8, students solve linear equations in onevariable with more than two steps as well as pairs ofsimultaneous linear equations. They draw from theirunderstanding of proportional relationships andtheir exploration of unit rate as a measure ofsteepness to define more formally the slope of linearequations.



Geometry

Grade 6 students build on their knowledge ofgeometric concepts to reason about relationshipsamong shapes and determine area, surface area, andvolume. Students reason about and justify theformulas for area, surface area, and volume andapply these formulas to solve real-world problems.They draw nets of three-dimensional shapes and usethem to find surface area. They apply theirknowledge of operations with fractions to find thevolume of prisms with fractional side lengths. Grade6 students plot polygons in the coordinate plane anduse coordinates to find side lengths.

In Grade 7, students synthesize their understandingof geometric attributes and properties as they creategeometric shapes with given attributes or conditions.They extend previous work with perimeter and areato include circumference and area of circles. Theyalso extend their study of volume and surface areato other polyhedra. They solve problems involvingscale drawings while exploring concepts of similarityand studying angle relationships.

In Grade 8, one focus of study in geometry is oncongruence and similarity through transformations.Students verify the properties of transformations(reflections, translations, rotations, dilations) anddescribe the effect of each on two-dimensionalshapes using coordinates. They extend their study ofangles and angle relationships in congruent andsimilar figures. Students explain that shapes arecongruent or similar based on a sequence oftransformations, and conversely they describe aseries of transformations that will illustrate that twoshapes are congruent. A second focus in geometry isthe Pythagorean Theorem. Students explain thetheorem and its converse, and apply the theorem tosolve real-world and mathematical problems. Inaddition, students continue their work with volumeof three-dimensional solids (cones, cylinders, andspheres).

Statistics and Probability

Statistics is introduced in Grade 6, with an emphasison statistical variability and on summarizing anddescribing distributions. Students look at measuresof center (mean, median, mode) and measures ofvariability (interquartile range and mean absolutedeviation) to describe a data set. They work withdifferent data displays to summarize and describedata sets.

Grade 7 students work with samples to drawinferences about populations and to compare twopopulations. They analyze data in light of the samplepopulation from which the data were collected.Students consider the effect of the sampling on thedata collected and understand the importance ofrandom sampling when drawing inferences ormaking generalizations about a population based onthe data collected.

The focus of study in Grade 8 is on bivariate data.Students investigate patterns with bivariatedata.They look at graphical representations of thedata (scatter plots) to describe the pattern shown (ifany).They use the equation of a linear model tosolve problems related to bivariate data, interpretingthe slope and intercept.

Students undertake a comprehensive study ofprobability in Grade 7, students’ primary encounterwith probability concepts in their elementary andmiddle years to date. Students develop probabilitymodels from which they find theoretical probabilityof events. They compare the theoretical probabilitiesto observed frequencies (experimental probabilities)and investigate both simple and compound eventsand represent theoretical probabilities usingdifferent models.



What’s Different?

The Common Core State Standards identify a limited number of topics at each grade level, allowing enoughtime for students to achieve fluency, if not mastery of these concepts. The subsequent year of study builds onthe concepts of the previous year. While review of topics from earlier grades is appropriate and encouraged,the CCSS writers argue that re-teaching of these topics should not be the focus. The focus should be to preparestudents to take the next steps in a mathematical development trajectory that deepens and builds on earliergrade topics and explorations.

The Common Core State Standards in Grades K–7 are developed to ensure that students have a strongfoundation to begin the study of algebra in Grade 8. Much of the work with number, which begins withcounting and cardinality at the kindergarten level, is completed in Grade 7. Students are expected to achievefluency with operations with most rational numbers by the end of Grade 7.

The Common Core State Standards for Grade 8 are algebra-focused. Students work with linear equations,functions and applications involving functions, and geometric concepts that include the Pythagorean theoremand transformations. At the end of Grade 8, students are well prepared for a high school algebra course.

Topics, such as operations with fractions and decimals, traditionally taught in Grade 6, have been shifted todifferent grades. In Grade 7, certain topics that have often been part of the curriculum are not included in theCCSS. Among the most noticeable are a pared-down set of geometry, and a different focus in the data analysisstandards. Certain topics that have often been part of the Grade 8 curriculum are not included in the CCSS. Ingeneral, these topics are not included because students are expected to have achieved fluency with them inearlier grades. These topics include rational numbers and operations with rational numbers; ratios, rates, andproportional relationships; properties and attributes of two- and three-dimensional shapes; and measures ofcenter and variability.

An overview of the changes by topic is listed below.

• Number and Operations Students are expected to have achieved fluency with operations involving wholenumbers, decimals, and with addition, subtraction, and multiplication of fractions, so these are not part ofthe Grade 6–8 standards. Operations with integers are part of the Grade 7 standards.

• Measurement The study of circles begins in Grade 7.

• Geometry The CCSS introduce the study of congruence and transformations in Grade 8.

• Probability Students first encounter probability concepts in Grade 7.



The Common Core State Standards (CCSS) articulate Standards for Mathematical Practice thathave been central to the development of the Connected Mathematics Project (CMP) materials fromtheir inception. The investigations in CMP are set in mathematical situations that give studentsopportunities to develop mathematical proficiency in these mathematical practices. Even more, theCMP materials help students build deep understanding of important math concepts and to makeconnections among these important concepts.

The essays that follow explain how CMP supports the development of mathematical proficiency instudents, citing some examples of how each Standard for Mathematical Practice is embedded in theCMP materials.

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

Mathematically proficient students start by explaining to themselves the meaning of a problem and looking forentry points to its solution. They analyze givens, constraints, relationships, and goals. They make conjecturesabout the form and meaning of the solution and plan a solution pathway rather than simply jumping into asolution attempt. They consider analogous problems, and try special cases and simpler forms of the originalproblem in order to gain insight into its solution. They monitor and evaluate their progress and change courseif necessary. Older students might, depending on the context of the problem, transform algebraic expressionsor change the viewing window on their graphing calculator to get the information they need. Mathematicallyproficient students can explain correspondences between equations, verbal descriptions, tables, and graphs ordraw diagrams of important features and relationships, graph data, and search for regularity or trends. Youngerstudents might rely on using concrete objects or pictures to help conceptualize and solve a problem.Mathematically proficient students check their answers to problems using a different method, and theycontinually ask themselves, “Does this make sense?” They can understand the approaches of others to solvingcomplex problems and identify correspondences between different approaches.

This goal is fundamental to CMP, a problem-centered curriculum. To be effective, problems mustnot just embody critical concepts and skills, but mustalso have the potential to engage students in makingsense of problem situations and mathematics.

A growing body of evidence from the cognitivesciences suggests that students make sense ofmathematics when concepts and skills are embeddedwithin a context or problem. This research is one ofthe cornerstones for developing the problemsituations in the CMP2 program. These student-centered problem situations engage students inarticulating the knowns and determining a logicalsolution pathway. The student-student and student-teacher dialogues, another hallmark of the program,help students to not just make sense of the problems,but also persevere in finding appropriate strategiesto solve them. The suggested questions in theTeacher Guides can provide the metacognitivescaffolding to help students monitor and refine theirproblem-solving strategies.

The Applications, Connections, and Extensionsproblems assure that all students make sense of newproblem situations and persevere in finding theappropriate solution.

For examples of this Standard for MathematicalPractice, see: Bits and Pieces I (Inv. 3); Bits andPieces II (Inv. 3); Bits and Pieces III (Inv. 4); PrimeTime (Inv. 2); Data About Us (Inv. 1); How Likely Is It? (Inv. 4); Covering and Surrounding (Inv. 2);Shapes and Designs (Inv. 1); Stretching and Shrinking(Inv. 1, 5); What Do You Expect? (Inv. 3, 4); DataDistributions (Inv. 2); Accentuate The Negative(Inv. 1, 3);Variables and Patterns (Inv. 1, 2); MovingStraight Ahead (Inv. 1); Comparing and Scaling(Inv. 4); Filing and Wrapping (Inv. 1, 3); Looking For Pythagoras (Inv. 2); Say It With Symbols (Inv. 3);Kaleidoscopes, Hubcaps, and Mirrors (Inv. 3, 4);Thinking With Mathematical Models (Inv. 1, 2);Growing, Growing, Growing (Inv. 4); The Shapes ofAlgebra (Inv. 4); Samples and Populations (Inv. 1);Frogs, Fleas, and Painted Cubes (Inv. 1)

CMP2_OIG_MATH_008-015.qxp 3/3/11 12:24 PM Page 8

Standards for Mathematical Practice 9

2. Reason abstractly and quantitatively.

Mathematically proficient students make sense of quantities and their relationships in problem situations.They bring two complementary abilities to bear on problems involving quantitative relationships: the ability todecontextualize—to abstract a given situation and represent it symbolically and manipulate the representingsymbols as if they have a life of their own, without necessarily attending to their referents—and the ability tocontextualize, to pause as needed during the manipulation process in order to probe into the referents for thesymbols involved. Quantitative reasoning entails habits of creating a coherent representation of the problemat hand; considering the units involved; attending to the meaning of quantities, not just how to compute them;and knowing and flexibly using different properties of operations and objects.

In the CMP classroom, students are supported intheir acquisition of mathematical language andmathematical ways of reasoning, both of which areunderpinnings of abstract and quantitativereasoning. The problem situations in CMP2 aredesigned to support the development of students’mathematical reasoning abilities. As studentsexplore a set of connected problems within aninvestigation, they look to understand the quantitiesin the problem and the relationship among thesequantities. Students are frequently expected totranslate a problem situation in an expression orequation and to then manipulate the equation to finda solution, or in other words, to “decontextualize.”

Conversely, throughout the problem-solving process,students are encouraged to translate from anequation or expression back to the problem situationto verify that the equation accurately represents thesituation, with particular attention to the units andquantities of the problems.

For examples of this Standard for MathematicalPractice, see: Prime Time (Inv. 2 p. 25); Bits andPieces I (Inv. 3 pp 40–41); Bits and Pieces II (Inv. 3pp. 36–38); Bits and Pieces III (Inv. 3 p. 46); DataAbout Us (Inv. 3 pp. 54–55); How Likely Is It? (Inv. 2pp. 22–23); Covering and Surrounding (Inv. 5 p. 88);Shapes and Designs (Inv. 3 p. 69); Stretching andShrinking (Inv. 3 pp. 40–41); What Do You Expect?(Inv. 2 p. 37); Data Distributions (Inv. 1, 2, 3, 4);Accentuate The Negative (Inv. 3, 4);Variables andPatterns (Inv. 1, 2 p. 48); Moving Straight Ahead(Inv. 2, 3); Comparing and Scaling (Inv. 1, 3, 4); Filingand Wrapping (Inv. 3 p. 47); Looking For Pythagoras(Inv. 3); Say It With Symbols (Inv. 5); Kaleidoscopes,Hubcaps, and Mirrors (Inv. 3, 4); Thinking WithMathematical Models (Inv. 1); Growing, Growing,Growing (Inv. 2); The Shapes of Algebra (Inv. 5);Samples and Populations (Inv. 2); Frogs, Fleas, andPainted Cubes (Inv. 4)



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

Mathematically proficient students understand and use stated assumptions, definitions, and previouslyestablished results in constructing arguments. They make conjectures and build a logical progression ofstatements to explore the truth of their conjectures. They are able to analyze situations by breaking them intocases, and can recognize and use counterexamples. They justify their conclusions, communicate them to others,and respond to the arguments of others. They reason inductively about data, making plausible arguments thattake into account the context from which the data arose. Mathematically proficient students are also able tocompare the effectiveness of two plausible arguments, distinguish correct logic or reasoning from that which isflawed, and—if there is a flaw in an argument—explain what it is. Elementary students can constructarguments using concrete referents such as objects, drawings, diagrams, and actions. Such arguments can makesense and be correct, even though they are not generalized or made formal until later grades. Later, studentslearn to determine domains to which an argument applies. Students at all grades can listen or read thearguments of others, decide whether they make sense, and ask useful questions to clarify or improve thearguments.

Reasoning and justification are central to theprogram at all levels. In the CMP classroom, studentsroutinely participate in mathematics discourse asthey explain their thinking about a problem situationand their reasoning for a solution pathway. Theproblems that students encounter in the programoffer opportunities to construct mathematicalarguments and to critique other students’ solutionsand strategies.

Students are called on to defend their solutions andtheir strategies for solving the problems, and toexplain their reasoning that led to the solution theyput forth. The Teacher Guides offer questions thatsupport the development of a classroom culture thatfocuses on argument and critique as a part of solvingmathematical problems.

For examples of this Standard for MathematicalPractice, see: Bits and Pieces I (Inv. 1 pp. 7–8, 10,Inv. 2 pp. 24–25, Inv. 4 pp. 56–58, 60); Bits andPieces II (Inv. 4 p. 54); Bits and Pieces III (Inv. 3p. 39); Prime Time (Inv. 1 p. 10); Data About Us(Inv. 2 p. 33); How Likely Is It? (Inv. 3); Covering andSurrounding (Inv. 1, pp. 6–8); Shapes and Designs(Inv. 2 pp. 36–37); Stretching and Shrinking (Inv. 1ACE 20); What Do You Expect? (Unit Project p. 62);Data Distributions (Inv. 2, 3); Accentuate TheNegative (Inv. 4);Variables and Patterns (Inv. 1, 2, 4);Moving Straight Ahead (Inv. 1, 2, 3); Comparing andScaling (Inv. 2, 3); Filing and Wrapping (Inv. 3);Looking For Pythagoras (Inv. 2 p. 30); Say It WithSymbols (Inv. 4 p. 71); Kaleidoscopes, Hubcaps, andMirrors (Inv. 2, p. 47); Thinking With MathematicalModels (Inv. 1); Growing, Growing, Growing (Inv. 5p. 58); The Shapes of Algebra (Inv. 3); Samples andPopulations (Inv. 2 p. 46); Frogs, Fleas, and PaintedCubes (Inv. 3, 4)



Mathematical modeling begins in Grade 6, andcontinues to grow in sophistication throughout theprogram. CMP engages students in learning toconstruct, make inferences from, and interpretconcrete, symbolic, graphic, verbal, and algorithmicmodels of mathematical relationships in problemsituations. Students are also asked to translateinformation from one model to another.

Throughout the program, students apply conceptsand skills to particular problem situations providedas well as to problem situations of their own creationor design. The regular and dynamic application ofmathematical concepts to solve real-world problemsis yet another hallmark of the ConnectedMathematics Project. Students are well-equipped torespond to these challenges as they regularlyconstruct and analyze a range of visual, graphicaland algebraic models. As students develop fluencywith these models, they realize the applicability ofthese models in different problem situations.

For examples of this Standard for MathematicalPractice, see: Bits and Pieces I (Inv. 3 pp. 36–38,40–41); Bits and Pieces II (Inv. 3, pp. 34–35); Bits andPieces III (Inv. 4 p. 58) Prime Time (Inv. 2 p. 25); DataAbout Us (Inv. 1 pp. 18–20) How Likely Is It? (Inv. 4pp. 57–58); Covering and Surrounding (Inv. 2 p. 30);Shapes and Designs (Inv. 1 pp. 14–15); Stretching andShrinking (Inv. 1, 2, 3); What Do You Expect? (Inv. 2);Data Distributions (Inv. 1); Accentuate The Negative(Inv. 1); Variables and Patterns (Inv. 4); MovingStraight Ahead (Inv. 2); Comparing and Scaling(Inv. 4); Filing and Wrapping (Inv. 3); Looking ForPythagoras (Inv. 4); Say It With Symbols (Inv. 1);Kaleidoscopes, Hubcaps, and Mirrors (Inv. 1, 3);Thinking With Mathematical Models (Inv. 1, 3);Growing, Growing, Growing (Inv. 2); The Shapes ofAlgebra (Inv. 1); Samples and Populations (Inv. 3);Frogs, Fleas, and Painted Cubes (Inv. 4)

4. Model with mathematics.

Mathematically proficient students can apply the mathematics they know to solve problems arising ineveryday life, society, and the workplace. In early grades, this might be as simple as writing an additionequation to describe a situation. In middle grades, a student might apply proportional reasoning to plan aschool event or analyze a problem in the community. By high school, a student might use geometry to solve adesign problem or use a function to describe how one quantity of interest depends on another. Mathematicallyproficient students who can apply what they know are comfortable making assumptions and approximationsto simplify a complicated situation, realizing that these may need revision later. They are able to identifyimportant quantities in a practical situation and map their relationships using such tools as diagrams, two-waytables, graphs, flowcharts and formulas. They can analyze those relationships mathematically to drawconclusions. They routinely interpret their mathematical results in the context of the situation and reflect onwhether the results make sense, possibly improving the model if it has not served its purpose.



In the CMP program, students work with a small setof tools as the primary vehicles for exploringproblem situations. Once students gain familiaritywith these tools, they make decisions about whichtools are most appropriate for a given problemsituation. Students can describe uses of differenttools; for example, they realize that calculators canbe used to compute, to check their thinking, toexplore possibilities, to see whether an approachmakes sense, and to use the graphing capability toexamine functions to see how they behave. Studentsbecome facile with graphing tools as a way “see intoa problem situation” and to find solutions to problems.

Students come to recognize that tools such aspolystrips can be used to explore concepts such asthe rigidity of triangle forms and the lack of rigidityof square forms and plastic two-dimensional shapesto explore the question of what shape has thegreatest area when built from a given number ofsquares.

For examples of this Standard for MathematicalPractice, see: Shapes and Designs (Inv. 2, 4); Bits andPieces I (Inv. 1); Bits and Pieces II (Inv. 3); PrimeTime (Inv. 1, 2); How Likely Is It? (Inv. 1, 2, 3, 4);Stretching and Shrinking (Inv. 1, 2, 5);What Do YouExpect? (Inv. 1, 2, 3, 4); Data Distributions (Inv. 2, 3);Accentuate The Negative (Inv. 1, 2, 3); Variables andPatterns (Inv. 1, 2, 4); Moving Straight Ahead (Inv. 1);Comparing and Scaling (Inv. 3, 4); Filing and Wrapping(Inv. 1, 2, 3, 4); Thinking With Mathematical Models(Inv. 1, 2, 3); The Shapes of Algebra (Inv. 2, 3);Growing, Growing, Growing (Inv. 1, 5); Say It WithSymbols (Unit Project pp. 85–86); Looking ForPythagoras (Inv. 1, 3)

5. Use appropriate tools strategically.

Mathematically proficient students consider the available tools when solving a mathematical problem. Thesetools might include pencil and paper, concrete models, a ruler, a protractor, a calculator, a spreadsheet, acomputer algebra system, a statistical package, or dynamic geometry software. Proficient students aresufficiently familiar with tools appropriate for their grade or course to make sound decisions about when eachof these tools might be helpful, recognizing both the insight to be gained and their limitations. For example,mathematically proficient high school students analyze graphs of functions and solutions generated using agraphing calculator. They detect possible errors by strategically using estimation and other mathematicalknowledge. When making mathematical models, they know that technology can enable them to visualize theresults of varying assumptions, explore consequences, and compare predictions with data. Mathematicallyproficient students at various grade levels are able to identify relevant external mathematical resources, suchas digital content located on a website, and use them to pose or solve problems. They are able to usetechnological tools to explore and deepen their understanding of concepts.



6. Attend to precision.

Mathematically proficient students try to communicate precisely to others. They try to use clear definitions indiscussion with others and in their own reasoning. They state the meaning of the symbols they choose,including using the equal sign consistently and appropriately. They are careful about specifying units ofmeasure, and labeling axes to clarify the correspondence with quantities in a problem. They calculateaccurately and efficiently, express numerical answers with a degree of precision appropriate for the problemcontext. In the elementary grades, students give carefully formulated explanations to each other. By the timethey reach high school they have learned to examine claims and make explicit use of definitions.

A key goal of CMP is helping students to learn to“talk” mathematics using precise terms anddefinitions, arguing that the clarity of a student’sthinking is reflected in the student’s use of precisemathematical language. The key mathematical goalsin each unit identify the important mathematicalterms, definitions, and ways of thinking andreasoning.

Student books include mathematical definitions thatare student-friendly while being mathematicallyaccurate. The goal of presenting definitions instudent-friendly language is to develop students’facility in talking mathematics at an appropriatelevel of mathematical maturity.

In addition to supporting the development of preciseuse of mathematical language, CMP supportsstudents in developing precision in theirpresentation of arguments. The series of questions ina problem push students to articulate more clearlytheir solutions and the processes by which they havereached these solutions.

A regular feature of the CMP student materials, theMathematical Reflections (MR) pages that occur atthe end of each investigation, also helps studentsdevelop precision in their thinking andcommunicating of mathematical ideas. The MRpages consist of a set of questions that help studentssynthesize and organize their understandings ofimportant concepts and strategies.

For more examples of this Standard for MathematicalPractice, see: Shapes and Designs (Inv. 2); Filling andWrapping (Inv. 3, 4); Stretching and Shrinking (Inv. 1,2, 3, 4, 5); Samples and Populations (Inv. 2, 3);Looking For Pythagoras (Inv. 2, 4); Kaleidoscopes,Hubcaps, and Mirrors (Inv. 3)



7. Look for and make use of structure.

Mathematically proficient students look closely to discern a pattern or structure. Young students, for example,might notice that three and seven more is the same amount as seven and three more, or they may sort acollection of shapes according to how many sides the shapes have. Later, students will see 7 × 8 equals the well remembered 7 × 5 + 7 × 3, in preparation for learning about the distributive property. In the expression x2 + 9x + 14, older students can see the 14 as 2 × 7 and the 9 as 2 + 7. They recognize the significance of anexisting line in a geometric figure and can use the strategy of drawing an auxiliary line for solving problems.They also can step back for an overview and shift perspective. They can see complicated things, such as somealgebraic expressions, as single objects or as being composed of several objects. For example, they can see 5 − 3(x − y)2 as 5 minus a positive number times a square and use that to realize that its value cannot be morethan 5 for any real numbers x and y.

The CMP materials were designed to help studentsbuild mathematical understandings in ways thatilluminate and make use of mathematical structure.In Grade 6, for example, students look for patterns inthe data presented in data tables, and they analyzenumbers to determine their prime structure.

As Grade 7 students examine proportionalreasoning situations of various kinds, they begin tosee patterns in proportions from which they drawgeneralizations about strategies for solvingproportions. They examine the structure of algebraicexpressions to understand patterns in algebraicoperations.

Grade 8 students examine graphical representationsof linear, exponential, and quadratic relationships sothey can begin to see the structure of theserelationships and functions. Although it is unusualfor students to examine quadratic and exponentialfunctions in middle school, the benefit is that theybegin to see the attributes of these differentrelationships, the structure of functions, in particular,what is revealed about the function through itsstructure.

In all grades, students see structure in measurement.They examine formulas, create algorithms forcomputation with rational numbers, and comparealgorithms for scope of use and efficiency.

For examples of this Standard for MathematicalPractice, see: Bits and Pieces I (Inv. 1, 2, 3); Bits andPieces II (Inv. 2, 3, 4); Bits and Pieces III (Inv. 1, 2, 5);Prime Time (Inv. 1, 2, 4); Covering and Surrounding(Inv. 1, 2, 3, 4, 5); How Likely Is It? (Inv. 2, 4); DataAbout Us (Inv. 1, 2, 3); Shapes and Designs (Inv. 1,2, 3, 4); What Do You Expect? (Inv. 1, 2); DataDistributions (Inv. 1, 2, 3, 4); Accentuate The Negative(Inv. 1, 2, 3); Variables and Patterns (Inv. 1, 2, 3, 4);Moving Straight Ahead (Inv. 1, 2, 3, 4); Looking ForPythagoras (Inv. 3); Say It With Symbols (Inv. 1);Kaleidoscopes, Hubcaps, and Mirrors (Inv. 1, 2, 3,4, 5); Growing, Growing, Growing (Inv. 1, 2, 3, 4, 5);Samples and Populations (Inv. 1, 2, 3, 4); Frogs, Fleas,and Painted Cubes (Inv. 1, 3, 4)



8. Look for and express regularity in repeated reasoning.

Mathematically proficient students notice if calculations are repeated, and look both for general methods andfor shortcuts. Upper elementary students might notice when dividing 25 by 11 that they are repeating the samecalculations over and over again, and conclude they have a repeating decimal. By paying attention to thecalculation of slope as they repeatedly check whether points are on the line through (1, 2) with slope 3, middleschool students might abstract the equation (y − 2)/(x − 1) = 3. Noticing the regularity in the way terms cancelwhen expanding (x − 1)(x + 1), (x − 1)(x2 + x + 1), and (x − 1)(x3 + x2 + x + 1) might lead them to the generalformula for the sum of a geometric series. As they work to solve a problem, mathematically proficient studentsmaintain oversight of the process, while attending to the details. They continually evaluate the reasonablenessof their intermediate results.

One of the guiding principles of the CMP2curriculum is helping students see regularity inmathematics. As students investigate problems ateach grade level, they are encouraged to look forconnections to previous problems and previoussolution strategies. Students are aided in seeingopportunities to use strategies previously used tosolve a problem in order to solve a new problem thatlooks on the surface to be very different. This kind ofthinking and reasoning about solving problemspromotes a view of mathematics as connected inmany different ways, rather than as an endless set ofproblems to be solved and forgotten.

The CMP classroom promotes student-to-studentdiscourse around mathematics. The problems arewritten to be engaging to students in the middlegrades and to encourage the development ofmathematical thinking and reasoning. Even the titlesof the materials express the importance the authorsplace on making connections–all kinds ofmathematics connections. Noting such connections isfundamental in seeing mathematics as a connectedwhole rather than an endless string of algorithms orprocesses to be learned.

For examples of this Standard for MathematicalPractice, see: Prime Time (Inv. 2, 3); Bits and Pieces I(Inv. 3); Bits and Pieces II (Inv. 2, 3); Shapes andDesigns (Inv. 1, 3); Covering and Surrounding (Inv. 1,2, 4); Moving Straight Ahead (Inv. 1, 2, 3, 4);Variablesand Patterns (Inv. 1, 2, 3, 4); Stretching and Shrinking(Inv 4); Kaleidoscopes, Hubcaps, and Mirrors (Inv. 1,2, 3, 4, 5)


Before an Investigation

Before each CMP2 Investigation, think about the behaviors you expect yourstudents will demonstrate and the Mathematical Practices they will haveopportunities to use/develop. Take a few minutes to write down answers to thefollowing questions.

Unit:

Investigation:

1. a. What are your mathematical goals for this investigation?

b. What evidence do you expect to observe during each lesson that will helpyou know that students are meeting these goals?

2. What prior knowledge might students use to build their understanding ofthese goals? How will you facilitate these connections?

3. What part of each problem might provide an opportunity to observe studentsdeveloping proficiency with the mathematical practices?

© Pearson Education, Inc., publishing as Pearson Prentice Hall. All rights reserved.


CMP2_OIG_REF_016-017.qxp 3/3/11 12:26 PM Page 16

After an Investigation

After each CMP2 Investigation, think about the behaviors your studentsdemonstrated. Take a few minutes to write down answers to the followingquestions.

Unit:

Investigation:

1. a. What were your mathematical goals for this Investigation?

b. What evidence do you have that students have accomplished these goals?

2. What evidence do you have of students’ use of the mathematical practices?What evidence do you have that students are growing in their sophisticationof their use of the practices?

3. How will you use this evidence to plan the next investigation?

© Pe

arson

Educ

ation

, Inc

., pu

blish

ing as

Pears

on Pr

entic

e Hall

. All r

ights

reserv

ed.

Mathematical Reflections 17

CMP2_OIG_REF_016-017.qxp 3/3/11 12:26 PM Page 17


The adoption and implementation of the Common Core State Standards for Mathematics is animportant and critical step to improving students’ math achievement in the United States. A second,equally important step is creating assessments grounded in these standards to measure students’progress against these new standards. These common assessments can also ensure that all studentshave access to these new standards.

The Race to the Top Assessment Program, funded by the American Recovery and Reinvestment Actof 2009 (ARRA) awarded funding to two state consortia to develop next generation assessment andaccountability systems. These valid and reliable assessments will be used to measure students’progress against the Common Core State Standards, provide a common measure of college andcareer readiness, and make use of new technologies in assessment and reporting so that parents andteachers have timely information about student performance. These next generation assessmentsystems, which are to be operational by 2014-2015, are to meet the dual needs of accountability andinstructional improvement. With these common assessments, state and local school officials can getan accurate view of how their students’ performances compare to those of students in other districtsor states. They can also reduce challenges associated with student mobility. Students in over 40 stateswill be expected to learn the same content and will take the same or similar assessments.

These new assessments will focus on assessing the critical areas that the Common Core StateStandards identify for each grade from Kindergarten through Grade 8. These assessments will alsoinclude tasks to measure students’ mathematical proficiency as described in the Standards forMathematical Practice.

The two state consortia are the Partnership for Assessment of Readiness for College and Careers(PARCC) and the SMARTER Balanced Assessment Consortium (SBAC).

On the following pages are brief descriptions of the assessments plans for PARCC and SBAC.

CMP2_OIG_ASSESS_018-020.qxp 3/3/11 12:22 PM Page 18

Assessing the Common Core State Standards 19

Partnership for Assessment of Readiness for College and Career

The Partnership for Assessment of Readiness for College and Careers (PARCC) Consortium ismade up of 24 states and the District of Columbia. The fiscal agent for the consortium is Florida.The consortium is working with Achieve, an independent, bipartisan, non-profit education reformorganization, and more than 200 colleges or universities to develop its next generation assessmentsystem.

The PARCC assessment system will be made up of a series of summative assessments giventhroughout the school year, an aligned formative assessment that teachers can use in the classroom.The summative assessment will have three Through-Course Assessments and one End-of-YearComprehensive Assessment.

• The Through-Course Assessments will be given towards the end of the first, second, and thirdquarters and will focus on the critical areas for each grade. Assessing these areas closer to the timeof instruction allows for mid-year corrections as needed.

• The first and second Through-Course Assessment will require one class period to complete. Thethird Through-Course Assessment may require more than one class period to complete.

• Students will take these assessments primarily on computers or other digital devices. They willencounter a range of item types, including performance tasks and computer-enhanced items.

• The scoring for these assessments will be a combination of computer-scored and human-scored.The results will be reported within a few weeks of administration.

• The End-of-Year Assessment will assess all of the standards at the grade level. It will measurestudents’ conceptual understanding, procedural fluency, and problem solving.

• It will be taken online during the last 4 to 6 weeks of the school year.

• It will have 40 to 65 items, with a range of item types (i.e., selected-response, constructed-response, performance tasks) and cognitive demand.

• It will be entirely computer-scored.

A student’s score will be based on his or her scores on the three Through-Course Assessments andthe End-of-Year Assessment. This score will be used for the purposes of accountability.

PARCC will also develop the Partnership Resource Center, a web-based platform where teacherswill find:

• Model curricula and exemplar lesson plans

• Released test items and performance tasks

• Formative assessment items and tasks

• Professional development materials regarding test administration, scoring, and use of data

• Practice tests

• Tools and resources developed by Partner states

• Optional performance tasks for children in Kindergarten through Grade 2



SMARTER Balanced Assessment Consortium

The SMARTER Balanced Assessment Consortium (SBAC) Consortium is made up of 31 states. Thefiscal agent for the consortium is Washington. The consortium is working with WestEd, a nonprofitpublic research and development agency, and more than 170 colleges and universities to develop itsnext generation assessment system.

The SBAC assessment system will consist of performance tasks and an End-of-Year AdaptiveAssessment. It will also make available optional computer-adaptive interim assessments.

• Performance Tasks Students will complete two performance tasks during the last 12 weeks of theschool year. These tasks will measure students’ abilities to integrate knowledge and skills from theCommon Core State Standards for Mathematics.

• Students will take these assessments primarily on computers or other digital devices.

• Each task will take one or two class periods to complete.

• The tasks will involve student-initiating planning, interaction with other materials, andproduction of an extended response.

• The tasks will be scored by both computers and humans.

• Results will be available as soon as possible.

• The End-of-Year Assessment will use computer adaptive delivery so that the scores accuratelyreflect student achievement.

• It will be administered during the last 12 weeks of the school year.

• It will be made up of 40 to 65 items, with a range of item types (i.e., selected-response,constructed-response, technology-enhanced).

• Some items will be computer-scored while others will be human-scored.

• Some students may be approved for a re-take with a new set of items.

A student’s score will be based on his or her scores on the Performance tasks and the End-of-YearAdaptive Assessment. This score will be used for the purposes of accountability.

SBAC will also make available an optional system of computer adaptive interim assessments whoseuses can be customized by local districts or schools. These assessments will have items similar tothose on the End-of-Year Assessment. The report of student results will also identify appropriatestudent resources based on student performance.

The SMARTER Balanced Assessment System will be built around a comprehensive electronicplatform with a collection of resources for teachers:

• The Educator Dashboard will be a secure site where teachers can find resources related toassessment, including model curricula, research-based instructional strategies, sample performancetasks, and longitudinal data regarding assessments

• Formative Tools, Processes, and Practices Clearinghouse that will include formative assessmenttools and strategies, teacher-made assessments, and research-based instructional tools andprocesses.


Parents’ Letter (English) 21

Dear Parents,

Recently, more than 40 states in the Unites States have developed and adopted a common set ofacademic standards in mathematics. These standards, called the Common Core State Standards weredeveloped in collaboration with teachers, school administrators, and mathematics and educationexperts under the auspices of the bipartisan National Governors’ Association and the Council forChief State School Officers (CCSSO).

These standards will serve as important benchmarks to ensure that all students are receiving highquality education and are well prepared for success in post-secondary education and the workforce.Students will be assessed on a regular basis throughout their school career to monitor their progresstowards meeting these benchmarks.

As individual states have adopted these new standards, they have committed to a shared grade-by-grade sequence of topics to be taught. For many states, this requires a shift in the instructionalmaterials used, to match both the content skills and the mathematical understandings contained inthe Common Core State Standards.

Your son or daughter is using Connected Mathematics Project (CMP2) in his/her math course. Thisprogram provides comprehensive coverage of the Common Core State Standards for Mathematics(CCSSM) at the middle years (Grades 6–8) so that students completing the three years of the CMP2 program are well-prepared for study of high school mathematics. At individual grade level,however, the CCSSM recommends the teaching of some content in grades different from CMP2. Toaddress these content shifts, Pearson has worked with the CMP authors to develop supplementalinvestigations to ensure that CMP2 students have studied all of the grade-level content for theirend-of-year state assessment.

All of the problems and exercises in the program support students in deepening their contentknowledge and in developing the skills and habits of mind that are embodied in the Standards forMathematical Practices, one set of the standards that make up the Common Core State Standards.You can help your son or daughter develop these abilities by asking the questions found on the backof this letter as they work on homework assignments.

Pearson is committed to providing quality instructional materials that can help all students achievemastery of the Common Core State Standards and be well prepared for success after high school.We hope that your child has a successful and rewarding year in the study of mathematics!

Sincerely,

CMP2_OIG_PARENT_021-024.qxp 3/3/11 12:25 PM Page 21


A Parent’s Guide to the Standards for Mathematical Practice

The Standards for Mathematical Practice elaborate criteria for problem solving and sense making. Thepractices are:

1. Make sense of problems and persevere in solving them.2. Reason abstractly and quantitatively.3. Construct viable arguments and critique the reasoning of others.4. Model with mathematics.5. Use appropriate tools strategically.6. Attend to precision.7. Look for and make use of structure.8. Look for and express regularity in repeated reasoning.

As your son or daughter works through homework exercises, you can help him/her develop skill with thesestandards by asking some of these questions:

Getting Ready

• What is the problem that you are solving for?

• How is this problem similar to others you have solved? How might this help you solve the problem?

• How will you go about solving the problem? (I.e., What’s your plan?)

• What tools (graphing calculator, geometry software, ruler, manipulatives, . . .) might be helpful? Why?

• What patterns do you see? Can you make a generalization?

• What relationships do you see in the problem?

Carry Out the Plan

• Can you verify your conjecture?

• What representation, such as a picture, concrete model, table, graph or equation, can you use torepresent this relationship?

• What information does the representation provide about the problem situation?

• What information do the numbers and symbols in your equation represent in the problem situation?

• Explain to me what this term (from the lesson) means?

• Are you progressing towards a solution? How do you know you are on the right path? Should you try adifferent solution plan?

• What might be a good estimate for the answer?

Evaluating the Solution

• Is your answer reasonable? Does it make sense?

• Suppose I said this was the answer (a wrong answer), how would you explain to me why I am wrong?

• What if I changed this number or this condition, how would it affect your answer?

Reflecting

• What do you notice about this problem and its answer? How is it related to previous problems in thisinvestigation? Previous investigations? Previous units?

• How does this add to your understanding of (mention the concept being studied)?


Parents’ Letter (Spanish) 23

Estimados padres:

Recientemente, más de 40 estados de los Estados Unidos han desarrollado y adoptado un conjuntode estándares académicos de Matemáticas común. Estos estándares, llamados los Common CoreState Standards (Estándares estatales comunes), fueron desarrollados en colaboración con maestros, administradores de escuelas y expertos en Matemáticas y Educación bajo los auspicios de organizaciones bipartidarias: la National Governors’ Association (Asociación Nacional deGobernadores) y el Council for Chief State School Officers (Consejo de Oficiales Jefes Estatales de Escuelas; CCSSO, por sus siglas en inglés).

Estos estándares servirán como importantes puntos de referencia o metas para asegurar que todos los estudiantes estén recibiendo una educación de alta calidad y estén preparados para teneréxito en los estudios postsecundarios y en la fuerza laboral. Los estudiantes serán evaluados conregularidad durante toda su carrera escolar para observar su progreso en el cumplimiento de estas metas .

A medida que los diferentes estados han adoptado estos estándares nuevos, se han comprometido acompartir una secuencia de temas, grado por grado, que se debe enseñar. Para muchos estados, estorequiere un cambio en los materiales de enseñanza que se utilizan, para lograr que concuerden tantolas destrezas de contenido como la comprensión matemática en los Common Core State Standards;CCSS, por sus siglas en inglés.

Su hijo/hija están utilizando Connected Mathematics Project (Proyecto de Matemáticas Conectadas 2;CMP2, por sus siglas en inglés) en su curso de Matemáticas. Este programa proporciona extensacobertura de los Common Core State Standards for Mathematics (Estándares estatales comunes de Matemáticas; CCSSM, por sus siglas en inglés) en la escuela intermedia (Grados 6–8) para que los estudiantes que completen los tres años del programa CMP2 estén bien preparados para el estudio de Matemáticas en la escuela secundaria. Sin embargo, al nivel de cada grado, los CCSSM recomiendan que se enseñe algún contenido de otros grados que no se cubre en CMP2.Para abordar estos cambios de contenido, Pearson ha trabajado con los autores de CMP eninvestigaciones suplementarias para asegurar que los estudiantes de CMP2 hayan estudiado todo el contenido de su grado para su evaluación estatal al final del año.

Todos los problemas y ejercicios del programa ayudan a los estudiantes a profundizar suconocimiento del contenido y a desarrollar las destrezas y hábitos mentales que son parte de losStandards for Mathematical Practices (Estándares para las prácticas matemáticas), que son parte de los Common Core Standards. Usted puede ayudar a su hijo o hija a desarrollar estos hábitoshaciendo las preguntas que se encuentran al dorso de esta carta mientras ellos hacen su tarea.

Pearson está comprometido a proporcionar materiales de enseñanza de alta calidad que puedenayudar a todos los estudiantes a lograr el dominio de los Common Core State Standards y aprepararse para tener éxito después de terminar la escuela secundaria. ¡Esperamos que su hijo o hijatengan un año de mucho éxito y gratificación en el estudio de las Matemáticas!

Atentamente,



Una guía para los padres sobre los Estándares de prácticasmatemáticas

Los Estándares de prácticas matemáticas elaboran los criterios necesarios para resolver problemas yencontrar sentido en las Matemáticas. Éstas son:

1. Entender los problemas y perseverar hasta resolverlos.2. Razonar de manera abstracta y cuantitativa.3. Construir argumentos válidos y evaluar el razonamiento de otros.4. Representación de modelos en Matemáticas.5. Utilizar estratégicamente herramientas apropiadas.6. Poner atención a la precisión.7. Buscar y utilizar la estructura.8. Buscar y expresar regularidad en el razonamiento repetido.

Mientras su hijo o hija hagan su tarea, ustedes pueden ayudarlos a desarrollar sus destrezas en estosestándares haciéndoles algunas de las preguntas siguientes:

Preparación

• ¿Cuál es el problema que estás resolviendo?

• ¿Puedes pensar en un problema que resolviste recientemente que se parezca a éste? ¿Cómo te puedeayudar eso en al resolución de este problema?

• ¿Qué vas a hacer para resolver el problema? (Es decir, ¿cuál es tu plan?)

• ¿Qué herramientas (calculadora gráfica, software de Geometría, regla, materiales manipulables, ...)podrían serte útiles? ¿Por qué?

• ¿Qué patrones puedes ver? ¿Puedes hacer una generalización?

• ¿Qué relaciones ves en el problema?

Implementación del plan

• ¿Puedes verificar tu conjetura?

• ¿Qué representación, ya sea un dibujo, modelo concreto, tabla, gráfica o ecuación puedes usar pararepresentar esta relación?

• ¿Qué información te da esta representación sobre el problema que hay que resolver?

• ¿Qué información representan los números y símbolos de tu ecuación en el problema que hay queresolver?

• Explícame qué es [término de la lección].

• ¿Estás acercándote a la solución? ¿Cómo lo sabes? ¿Debes probar otra manera diferente de resolverlo?

• ¿Cuál sería una buena estimación para la respuesta?

Evaluación de la solución

• ¿Es razonable tu respuesta? ¿Tiene sentido?

• Supón que te dije que yo pienso que la respuesta es [una respuesta equivocada], ¿cómo me explicaríaspor qué no tengo razón?

• Si cambiara este número o condición, ¿cómo afectaría esto tu respuesta?

Reflexión

• ¿Qué puedes observar sobre este problema y su respuesta? ¿Se relaciona con problemas anteriores de lainvestigación? ¿con investigaciones hechas previamente? ¿con unidades previas?

• ¿Cómo ayuda a tu comprensión del concepto de [mencione el concepto que se está estudiando]?


COMMON CORE STATE STANDARDS FOR MATHEMATICS CMP2 UNIT

CCSSM Correlations to all CMP2 Units 25

DOMAIN: RATIOS AND PROPORTIONAL RELATIONSHIPS

6.RP.1Understand the concept of a ratio and use ratio language to describe a ratiorelationship between two quantities.

6.RP.2Understand the concept of a unit rate a/b associated with a ratio a:b with b ≠ 0, anduse rate language in the context of a ratio relationship.

6.RP.3Use ratio and rate reasoning to solve real-world and mathematical problems, e.g., byreasoning about tables of equivalent ratios, tape diagrams, double number linediagrams, or equations.

6.RP.3.aMake tables of equivalent ratios relating quantities with whole numbermeasurements, find missing values in the tables, and plot the pairs of values on thecoordinate plane. Use tables to compare ratios.

6.RP.3.bSolve unit rate problems including those involving unit pricing and constant speed.

6.RP.3.cFind a percent of a quantity as a rate per 100 (e.g., 30% of a quantity means 30/100times the quantity); solve problems involving finding the whole, given a part and thepercent.

6.RP.3.dUse ratio reasoning to convert measurement units; manipulate and transform unitsappropriately when multiplying or dividing quantities.

Bits and Pieces I Inv. 4Comparing and Scaling Inv. 1–3

Variables and Patterns Inv. 1 Comparing and Scaling Inv. 3

Bits and Pieces I Inv. 3–4Shapes and Designs Inv. 2 How Likely Is It? Inv. 1–4Comparing and Scaling Inv. 1–3Moving Straight Ahead Inv. 1–4

Bits and Pieces I Inv. 4Variables and Patterns Inv. 3Moving Straight Ahead Inv. 1–2Growing, Growing, Growing Inv. 4

Bits and Pieces I Inv. 4Variables and Patterns Inv. 1–3Comparing and Scaling Inv. 3Moving Straight Ahead Inv. 1–2

Bits and Pieces III Inv. 4–5Comparing and Scaling Inv. 2

Comparing and Scaling Inv. 3–4Covering and Surrounding Inv. 1

(ACE)Stretching and Shrinking Inv. 2, 4,

5 (ACE)Filling and Wrapping Inv. 5 (ACE)

The following is a correlation of the Common Core State Standards for Mathematics (Grades 6–8) toall of the Connected Mathematics 2 (CMP2) ©2009 units

© Copyright 2010. National Governors Association Center for Best Practices and Council of Chief State School Officers. All rights reserved.

CMP2_OIG_CORR_025-038.qxp 3/3/11 12:23 PM Page 25



7.RP.1Compute unit rates associated with ratios of fractions, including ratios of lengths,areas and other quantities measured in like or different units.

7.RP.2Recognize and represent proportional relationships between quantities.

7.RP.2.aDecide whether two quantities are in a proportional relationship, e.g., by testing forequivalent ratios in a table or graphing on a coordinate plane and observing whetherthe graph is a straight line through the origin.

7.RP.2.bIdentify the constant of proportionality (unit rate) in tables, graphs, equations,diagrams, and verbal descriptions of proportional relationships.

7.RP.2.cRepresent proportional relationships by equations.

7.RP.2.dExplain what a point (x, y) on the graph of a proportional relationship means in termsof the situation, with special attention to the points (0, 0) and (1, r) where r is theunit rate.

7.RP.3Use proportional relationships to solve multistep ratio and percent problems.Examples: simple interest, tax, markups and markdowns, gratuities and commissions,fees, percent increase and decrease, percent error.

Bits and Pieces I Inv. 3–4Comparing and Scaling Inv. 3Stretching and Shrinking Inv. 1–5

Comparing and Scaling Inv. 1–4Stretching and Shrinking Inv. 1–4Filling and Wrapping Inv. 5How Likely Is It? Inv. 1–4What Do You Expect? Inv. 1–4

Comparing and Scaling Inv. 4Stretching and Shrinking Inv. 4Moving Straight Ahead Inv. 1–4

Comparing and Scaling Inv. 3–4Moving Straight Ahead Inv. 1–4Thinking With Mathematical

Models Inv. 1Say It With Symbols Inv. 1–4

Variables and Patterns Inv. 1–4Comparing and Scaling Inv. 4Stretching and Shrinking Inv. 4–5Moving Straight Ahead Inv. 1–4

Variables and Patterns Inv. 2Comparing and Scaling Inv. 3Moving Straight Ahead Inv. 1–4

Bits and Pieces III Inv. 4–5Variables and Patterns Inv. 4Comparing and Scaling Inv. 1–4Stretching and Shrinking Inv. 4–5Growing, Growing, Growing Inv. 4




DOMAIN: THE NUMBER SYSTEM

6.NS.1Interpret and compute quotients of fractions, and solve word problems involvingdivision of fractions by fractions, e.g., by using visual fraction models and equationsto represent the problem.

6.NS.2Fluently divide multi-digit numbers using the standard algorithm.

6.NS.3Fluently add, subtract, multiply, and divide multi-digit decimals using the standardalgorithm for each operation.

6.NS.4Find the greatest common factor of two whole numbers less than or equal to 100 andthe least common multiple of two whole numbers less than or equal to 12. Use thedistributive property to express a sum of two whole numbers 1–100 with a commonfactor as a multiple of a sum of two whole numbers with no common factor.

6.NS.5Understand that positive and negative numbers are used together to describequantities having opposite directions or values (e.g., temperature above/below zero,elevation above/below sea level, credits/debits, positive/negative electric charge); usepositive and negative numbers to represent quantities in real-world contexts,explaining the meaning of 0 in each situation.

6.NS.6Understand a rational number as a point on the number line. Extend number linediagrams and coordinate axes familiar from previous grades to represent points onthe line and in the plane with negative number coordinates.

6.NS.6.aRecognize opposite signs of numbers as indicating locations on opposite sides of 0 onthe number line; recognize that the opposite of the opposite of a number is thenumber itself, e.g., –(–3) = 3, and that 0 is its own opposite.

6.NS.6.bUnderstand signs of numbers in ordered pairs as indicating locations in quadrants ofthe coordinate plane; recognize that when two ordered pairs differ only by signs, thelocations of the points are related by reflections across one or both axes.

6.NS.6.cFind and position integers and other rational numbers on a horizontal or verticalnumber line diagram; find and position pairs of integers and other rational numberson a coordinate plane.

Bits and Pieces II Inv. 4

Bits and Pieces I Inv. 3Bits and Pieces III Inv. 3

Bits and Pieces III Inv. 1–3

Prime Time Inv. 2–3

Bits and Pieces II Inv. 2Accentuate the Negative Inv. 1

Bits and Pieces I Inv. 1–4Bits and Pieces II Inv. 1–4Bits and Pieces III Inv. 1–4Accentuate the Negative Inv. 1–4Looking for Pythagoras Inv. 2

Bits and Pieces II Inv. 2Accentuate the Negative Inv. 1

Accentuate the Negative Inv. 2Variables and Patterns Inv. 1–2, 4Moving Straight Ahead Inv. 1–4Thinking With Mathematical

Models Inv. 1–3Looking for Pythagoras Inv. 1Kaleidoscopes, Hubcaps, and

Mirrors Inv. 5Frogs, Fleas, and Painted Cubes

Inv. 1–4Say It With Symbols Inv. 1–4The Shapes of Algebra Inv. 1

Variables and Patterns Inv. 1–2Stretching and Shrinking Inv. 2Accentuate the Negative Inv. 1–2Moving Straight Ahead Inv. 1–2Thinking With Mathematical

Models Inv. 1–3Looking for Pythagoras Inv. 1Growing, Growing, Growing

Inv. 2, 4Frogs, Fleas, and Painted Cubes

Inv. 1–4Kaleidoscopes, Hubcaps, and

Mirrors Inv. 5Say It With Symbols Inv. 3The Shapes of Algebra Inv. 1, 3, 5Samples and Populations Inv. 4




6.NS.7Understand ordering and absolute value of rational numbers.

6.NS.7.aInterpret statements of inequality as statements about the relative position of twonumbers on a number line diagram.

6.NS.7.bWrite, interpret, and explain statements of order for rational numbers in real-worldcontexts.

6.NS.7.cUnderstand the absolute value of a rational number as its distance from 0 on thenumber line; interpret absolute value as magnitude for a positive or negativequantity in a real-world situation.

6.NS.7.dDistinguish comparisons of absolute value from statements about order.

6.NS.8Solve real-world and mathematical problems by graphing points in all four quadrantsof the coordinate plane. Include use of coordinates and absolute value to finddistances between points with the same first coordinate or the same secondcoordinate.

7.NS.1Apply and extend previous understandings of addition and subtraction to add andsubtract rational numbers; represent addition and subtraction on a horizontal orvertical number line diagram.

7.NS.1.aDescribe situations in which opposite quantities combine to make 0.

7.NS.1.bUnderstand p + q as the number located a distance |q| from p, in the positive ornegative direction depending on whether q is positive or negative. Show that anumber and its opposite have a sum of 0 (are additive inverses). Interpret sums ofrational numbers by describing real-world contexts.

7.NS.1.cUnderstand subtraction of rational numbers as adding the additive inverse, p – q = p + (–q). Show that the distance between two rational numbers on thenumber line is the absolute value of their difference, and apply this principle in real-world contexts.

7.NS.1.dApply properties of operations as strategies to add and subtract rational numbers.

Bits and Pieces I Inv. 1–3Bits and Pieces II Inv. 1Accentuate the Negative Inv. 1–2Looking for Pythagoras Inv. 2

Bits and Pieces I Inv. 1–4Bits and Pieces II Inv. 2Accentuate the Negative Inv. 1

Bits and Pieces II Inv. 2Bits and Pieces III Inv. 1Accentuate the Negative Inv. 1

Accentuate the Negative Inv. 2


Covering and Surrounding Inv. 2Data About Us Inv. 2Accentuate the Negative Inv. 2Thinking With Mathematical


Inv. 2, 4Frogs, Fleas, and Painted Cubes

Inv. 1–4Kaleidoscopes, Hubcaps, and

Mirrors Inv. 5Say It With Symbols Inv. 3The Shapes of Algebra Inv. 1, 3, 5Samples and Populations Inv. 4

Accentuate the Negative Inv. 2, 4


Accentuate the Negative Inv. 1–2


Bits and Pieces II Inv. 1–2Bits and Pieces III Inv. 1Accentuate the Negative Inv. 2, 4




7.NS.2Fluently divide multi-digit numbers using the standard algorithm.

7.NS.2.aUnderstand that multiplication is extended from fractions to rational numbers byrequiring that operations continue to satisfy the properties of operations, particularlythe distributive property, leading to products such as (–1)(–1) = 1 and the rules formultiplying signed numbers. Interpret products of rational numbers by describingreal-world contexts.

7.NS.2.bUnderstand that integers can be divided, provided that the divisor is not zero, andevery quotient of integers (with non-zero divisor) is a rational number. If p and q areintegers, then –(p/q) = (–p)/q = p/(–q). Interpret quotients of rational numbers bydescribing real-world contexts.

7.NS.2.cApply properties of operations as strategies to multiply and divide rational numbers.

7.NS.2 .dConvert a rational number to a decimal using long division; know that the decimalform of a rational number terminates in 0s or eventually repeats

7.NS.3Solve real-world and mathematical problems involving the four operations withrational numbers.NOTE: Computations with rational numbers extend the rules for manipulatingfractions to complex fractions.

8.NS.1Understand informally that every number has a decimal expansion; the rationalnumbers are those with decimal expansions that terminate in 0s or eventually repeat.Know that other numbers are called irrational.

8.NS.2Use rational approximations of irrational numbers to compare the size of irrationalnumbers, locate them approximately on a number line diagram, and estimate thevalue of expressions (e.g., π2).


Bits and Pieces II Inv. 3Bits and Pieces III Inv. 2Accentuate the Negative Inv. 3–4



Comparing and Scaling Inv. 3Accentuate the Negative Inv. 3


Looking For Pythagoras Inv. 4





DOMAIN: EXPRESSIONS AND EQUATIONS

6.EE.1Write and evaluate numerical expressions involving whole-number exponents.

6.EE.2Write, read, and evaluate expressions in which letters stand for numbers.

6.EE.2.aWrite expressions that record operations with numbers and with letters standing fornumbers.

6.EE.2.bIdentify parts of an expression using mathematical terms (sum, term, product, factor,quotient, coefficient); view one or more parts of an expression as a single entity.

6.EE.2.cEvaluate expressions at specific values of their variables. Include expressions that arisefrom formulas used in real-world problems.Perform arithmetic operations, including those involving whole number exponents, inthe conventional order when there are no parentheses to specify a particular order(Order of Operations).

6.EE.3Apply the properties of operations to generate equivalent expressions.

Prime Time Inv. 4

Bits and Pieces II Inv. 2–4Bits and Pieces III Inv. 1–4Variables and Patterns Inv. 1–3Stretching and Shrinking Inv. 2Comparing and Scaling Inv. 3Moving Straight Ahead Inv. 1–4Thinking With Mathematical


Inv. 1–5Frogs, Fleas, and Painted Cubes

Inv. 1–4Say It With Symbols Inv. 1–5The Shapes of Algebra Inv. 1–5Samples and Populations Inv. 4

Bits and Pieces II Inv. 2–4Bits and Pieces III Inv. 1–3Variables and Patterns Inv. 1–3Comparing and Scaling Inv. 3Moving Straight Ahead Inv. 1–4Thinking With Mathematical



Inv. 1–4Say It With Symbols Inv. 1–5The Shapes of Algebra Inv. 1–2, 5Samples and Populations Inv. 4

Prime Time Inv. 1, 3–5Bits and Pieces II Inv. 2–4Bits and Pieces III Inv. 1–3Variables and Patterns Inv. 3Thinking With Mathematical

Models Inv. 1–3Frogs, Fleas, and Painted Cubes

Inv. 2Say It With Symbols Inv. 1–4

Covering and Surrounding Inv. 1–5Variables and Patterns Inv. 3Stretching and Shrinking Inv. 2Moving Straight Ahead Inv. 1–2Filling and Wrapping Inv. 1–4Thinking With Mathematical




Accentuate the Negative Inv. 2–4Moving Straight Ahead Inv. 3Frogs, Fleas, and Painted Cubes

Inv. 2Say It With Symbols Inv. 1–4The Shapes of Algebra Inv. 3–4




6.EE.4Identify when two expressions are equivalent (i.e., when the two expressions namethe same number regardless of which value is substituted into them).

6.EE.5Understand solving an equation or inequality as a process of answering a question:which values from a specified set, if any, make the equation or inequality true? Usesubstitution to determine whether a given number in a specified set makes anequation or inequality true.

6.EE.6Use variables to represent numbers and write expressions when solving a real-worldor mathematical problem; understand that a variable can represent an unknownnumber, or, depending on the purpose at hand, any number in a specified set.

6.EE.7Solve real-world and mathematical problems by writing and solving equations of theform x + p = q and px = q for cases in which p, q and x are all nonnegative rationalnumbers.

6.EE.8Write an inequality of the form x > c or x < c to represent a constraint or condition ina real-world or mathematical problem. Recognize that inequalities of the form x > cor x < c have infinitely many solutions; represent solutions of such inequalities onnumber line diagrams.

6.EE.9Use variables to represent two quantities in a real-world problem that change inrelationship to one another; write an equation to express one quantity, thought of asthe dependent variable, in terms of the other quantity, thought of as theindependent variable. Analyze the relationship between the dependent andindependent variables using graphs and tables, and relate these to the equation.

7.EE.1Apply properties of operations as strategies to add, subtract, factor, and expandlinear expressions with rational coefficients.

Accentuate the Negative Inv. 2, 4Frogs, Fleas, and Painted Cubes

Inv. 2Say It With Symbols Inv. 1–4The Shapes of Algebra Inv. 3

Bits and Pieces II Inv. 2–4Bits and Pieces III Inv. 1–3Shapes and Designs Inv. 2 Variables and Patterns Inv. 3Moving Straight Ahead Inv. 1–4Thinking With Mathematical

Models Inv. 1–3Growing, Growing, Growing


Inv. 1–4Say It With Symbols Inv. 1–4Shapes of Algebra Inv. 2–5

Shapes and Designs Inv. 3–4Covering and Surrounding Inv. 5Variables and Patterns Inv. 1–3Comparing and Scaling Inv. 3Moving Straight Ahead Inv. 1–4Thinking With Mathematical




Shapes and Designs Inv. 3–4Covering and Surrounding Inv. 5Moving Straight Ahead Inv. 3

The Shapes of Algebra Inv. 2, 5

Covering and Surrounding Inv. 2Data About Us Inv. 2Variables and Patterns Inv. 1–3Moving Straight Ahead Inv. 1–4Thinking With Mathematical




Moving Straight Ahead Inv. 3–4Thinking With Mathematical

Models Inv. 2–3The Shapes of Algebra Inv. 1–5Say It With Symbols Inv. 1–4




7.EE.2Understand that rewriting an expression in different forms in a problem context canshed light on the problem and how the quantities in it are related.

7.EE.3Solve multi-step real-life and mathematical problems posed with positive andnegative rational numbers in any form (whole numbers, fractions, and decimals),using tools strategically. Apply properties of operations to calculate with numbers inany form; convert between forms as appropriate; and assess the reasonableness ofanswers using mental computation and estimation strategies.

7.EE.4Use variables to represent quantities in a real-world or mathematical problem, andconstruct simple equations and inequalities to solve problems by reasoning about thequantities.

7.EE.4.aSolve word problems leading to equations of the form px + q = r and p(x + q) = r,where p, q, and r are specific rational numbers. Solve equations of these formsfluently. Compare an algebraic solution to an arithmetic solution, identifying thesequence of the operations used in each approach

7.EE.4.bSolve word problems leading to inequalities of the form px + q > r or px + q <r,where p, q, and r are specific rational numbers. Graph the solution set of theinequality and interpret it in the context of the problem.

8.EE.1Know and apply the properties of integer exponents to generate equivalentnumerical expressions.

8.EE.2Use square root and cube root symbols to represent solutions to equations of theform x2 = p and x3 = p, where p is a positive rational number. Evaluate square rootsof small perfect squares and cube roots of small perfect cubes. Know that √2 isirrational.

8.EE.3Use numbers expressed in the form of a single digit times an integer power of 10 toestimate very large or very small quantities, and to express how many times as muchone is than the other.

8.EE.4Perform operations with numbers expressed in scientific notation, including problemswhere both decimal and scientific notation are used. Use scientific notation andchoose units of appropriate size for measurements of very large or very smallquantities (e.g., use millimeters per year for seafloor spreading). Interpret scientificnotation that has been generated by technology.

Thinking With MathematicalModels Inv. 2–3

Growing, Growing, Growing Inv. 1Frogs, Fleas, and Painted Cubes

Inv. 2Say It With Symbols Inv. 1–4Shapes of Algebra Inv. 2–5

Variables and Patterns Inv. 2–4Accentuate the Negative Inv. 1–4Moving Straight Ahead Inv. 1–4Thinking With Mathematical

Models Inv. 1–3Looking for Pythagoras Inv. 3–4Growing, Growing, Growing


Inv. 1–4The Shapes of Algebra Inv. 2–5Say It With Symbols Inv. 1–5

Variables and Patterns Inv. 1–3Moving Straight Ahead Inv. 1–4Thinking With Mathematical

Models Inv. 1–3Looking for Pythagoras Inv. 3–4Growing, Growing, Growing


Inv. 1–4Growing, Growing, Growing

Inv. 1–4The Shapes of Algebra Inv. 2–5Say It With Symbols Inv. 1–5

Variables and Patterns Inv. 1–3Moving Straight Ahead Inv. 1–4Say It With Symbols Inv. 1–5Thinking With Mathematical

Models Inv. 1–3

Moving Straight Ahead Inv. 2 The Shapes of Algebra Inv. 2

Growing, Growing, Growing Inv. 5

Looking for Pythagoras Inv. 2–4

Growing, Growing, Growing Inv.1–2, 4–5

Growing, Growing, Growing Inv. 5




8.EE.5Graph proportional relationships, interpreting the unit rate as the slope of the graph.Compare two different proportional relationships represented in different ways.

8.EE.6Use similar triangles to explain why the slope m is the same between any two distinctpoints on a non-vertical line in the coordinate plane; derive the equation y = mx for aline through the origin and the equation y = mx + b for a line intercepting thevertical axis at b.

8.EE.7Solve linear equations in one variable.

8.EE.7.aGive examples of linear equations in one variable with one solution, infinitely manysolutions, or no solutions. Show which of these possibilities is the case by successivelytransforming the given equation into simpler forms, until an equivalent equation ofthe form x = a, a = a, or a = b results (where a and b are different numbers).

8.EE.7.bSolve linear equations with rational number coefficients, including equations whosesolutions require expanding expressions using the distributive property and collectinglike terms.

8.EE.8Analyze and solve pairs of simultaneous linear equations.

8.EE.8.aUnderstand that solutions to a system of two linear equations in two variablescorrespond to points of intersection of their graphs, because points of intersectionsatisfy both equations simultaneously.

8.EE.8.bSolve systems of two linear equations in two variables algebraically, and estimatesolutions by graphing the equations. Solve simple cases by inspection.

8.EE.8.cSolve real-world and mathematical problems leading to two linear equations in twovariables.

Moving Straight Ahead Inv. 4Thinking With Mathematical

Models Inv. 2


Models Inv. 2




Models Inv. 1–3Say It With Symbols Inv. 5The Shapes of Algebra Inv. 3-4



The Shapes of Algebra Inv. 2–4

The Shapes of Algebra Inv. 2–4

Moving Straight Ahead Inv. 2–4The Shapes of Algebra Inv. 1–4

Moving Straight Ahead Inv. 2–4The Shapes of Algebra Inv. 2–4




DOMAIN: FUNCTIONS

8.F.1Understand that a function is a rule that assigns to each input exactly one output.The graph of a function is the set of ordered pairs consisting of an input and thecorresponding output.NOTE Function notation is not required in Grade 8.

8.F.2Compare properties of two functions each represented in a different way(algebraically, graphically, numerically in tables, or by verbal descriptions).

8.F.3Interpret the equation y = mx + b as defining a linear function, whose graph is astraight line; give examples of functions that are not linear.

8.F.4Construct a function to model a linear relationship between two quantities.Determine the rate of change and initial value of the function from a description of arelationship or from two (x, y) values, including reading these from a table or from agraph. Interpret the rate of change and initial value of a linear function in terms ofthe situation it models, and in terms of its graph or a table of values.

8.F.5Describe qualitatively the functional relationship between two quantities byanalyzing a graph e.g., where the function is increasing or decreasing, linear ornonlinear). Sketch a graph that exhibits the qualitative features of a function that hasbeen described verbally.


Models Inv. 1–3Growing, Growing, Growing


Inv. 1–4Say It With Symbols Inv. 1–4


Models Inv. 1Growing, Growing, Growing Inv. 1 Frogs, Fleas, and Painted Cubes

Inv. 2–4Say It With Symbols Inv. 2


Models Inv. 2–3, 5Growing, Growing, Growing Inv. 3Frogs, Fleas, and painted Cubes

Inv. 1–4The Shapes of Algebra Inv. 4Say It With Symbols Inv. 4


Models Inv. 1–3The Shapes of Algebra Inv. 4Say It With Symbols Inv. 4

Moving Straight Ahead Inv. 1–2, 4Thinking With Mathematical

Models Inv. 2Growing, Growing, Growing


Inv. 1–4Say It With Symbols Inv. 4




DOMAIN: GEOMETRY

6.G.1Find the area of right triangles, other triangles, special quadrilaterals, and polygonsby composing into rectangles or decomposing into triangles and other shapes; applythese techniques in the context of solving real-world and mathematical problems.

6.G.2Find the volume of a right rectangular prism with fractional edge lengths by packingit with unit cubes of the appropriate unit fraction edge lengths, and show that thevolume is the same as would be found by multiplying the edge lengths of the prism.Apply the formulas V = l w h and V = b h to find volumes of right rectangular prismswith fractional edge lengths in the context of solving real-world and mathematicalproblems.

6.G.3Draw polygons in the coordinate plane given coordinates for the vertices; usecoordinates to find the length of a side joining points with the same first coordinateor the same second coordinate. Apply these techniques in the context of solving real-world and mathematical problems.

6.G.4Represent three-dimensional figures using nets made up of rectangles and triangles,and use the nets to find the surface area of these figures. Apply these techniques inthe context of solving real-world and mathematical problems.

7.G.1Solve problems involving scale drawings of geometric figures, including computingactual lengths and areas from a scale drawing and reproducing a scale drawing at adifferent scale.

7.G.2Draw (freehand, with ruler and protractor, and with technology) geometric shapeswith given conditions. Focus on constructing triangles from three measures of anglesor sides, noticing when the conditions determine a unique triangle, more than onetriangle, or no triangle.

7.G.3Describe the two-dimensional figures that result from slicing three-dimensionalfigures, as in plane sections of right rectangular prisms and right rectangularpyramids.

7.G.4Know the formulas for the area and circumference of a circle and use them to solveproblems; give an informal derivation of the relationship between the circumferenceand area of a circle.

7.G.5Use facts about supplementary, complementary, vertical, and adjacent angles in amulti-step problem to write and solve simple equations for an unknown angle in afigure.

7.G.6Solve real-world and mathematical problems involving area, volume and surface areaof two- and three-dimensional objects composed of triangles, quadrilaterals,polygons, cubes, and right prisms.

Covering and Surrounding Inv. 1–5

Filling and Wrapping Inv. 2

Shapes and Designs Inv. 2Stretching and Shrinking Inv. 2Kaleidoscopes, Hubcaps, and

Mirrors Inv. 5The Shapes of Algebra Inv. 1

Covering and Surrounding Inv. 3 Filling and Wrapping Inv. 1, 3

Stretching and Shrinking Inv. 1–5Comparing and Scaling Inv. 4Filling and Wrapping Inv. 4

Shapes and Designs Inv. 4Filling and Wrapping Inv. 1–4

Covering and Surrounding Inv. 5Filling and Wrapping Inv. 2–5

Stretching and Shrinking Inv. 3 Filling and Wrapping Inv. 1

Covering and Surrounding Inv. 1–4Stretching and Shrinking Inv. 2–3Filling and Wrapping Inv. 1–5




8.G.1Verify experimentally the properties of rotations, reflections, and translations:

8.G.1.aLines are taken to lines, and line segments to line segments of the same length.

8.G.1.bAngles are taken to angles of the same measure.

8.G.1.cParallel lines are taken to parallel lines.

8.G.2Understand that a two-dimensional figure is congruent to another if the second canbe obtained from the first by a sequence of rotations, reflections, and translations;given two congruent figures, describe a sequence that exhibits the congruencebetween them.

8.G.3Describe the effect of dilations, translations, rotations, and reflections on two-dimensional figures using coordinates.

8.G.4Understand that a two-dimensional figure is similar to another if the second can beobtained from the first by a sequence of rotations, reflections, translations, anddilations; given two similar two-dimensional figures, describe a sequence that exhibitsthe similarity between them.

8.G.5Use informal arguments to establish facts about the angle sum and exterior angle oftriangles, about the angles created when parallel lines are cut by a transversal, andthe angle-angle criterion for similarity of triangles.

8.G.6Explain a proof of the Pythagorean Theorem and its converse.

8.G.7Apply the Pythagorean Theorem to determine unknown side lengths in righttriangles in real-world and mathematical problems in two and three dimensions.

8.G.8Apply the Pythagorean Theorem to find the distance between two points in acoordinate system.

8.G.9Know the formulas for the volumes of cones, cylinders, and spheres and use them tosolve real-world and mathematical problems.

Kaleidoscopes, Hubcaps, andMirrors Inv. 1–5

Stretching and Shrinking Inv. 2Kaleidoscopes, Hubcaps, and

Mirrors Inv. 1–5


Mirrors Inv. 1–5


Mirrors Inv. 1–5

Kaleidoscopes, Hubcaps, andMirrors Inv. 3


Mirrors Inv. 2, 5


Mirrors Inv. 2

Shapes and Designs Inv. 3


Looking For Pythagoras Inv. 3–4

Looking For Pythagoras Inv. 2–3

Filling and Wrapping Inv. 3–5Kaleidoscopes, Hubcaps, and

Mirrors Inv. 1–3Looking For Pythagoras Inv. 3–4Say It With Symbols Inv. 1, 3–4




6.SP.1Recognize a statistical question as one that anticipates variability in the data relatedto the question and accounts for it in the answers

6.SP.2Understand that a set of data collected to answer a statistical question has adistribution which can be described by its center, spread, and overall shape.

6.SP.3Recognize that a measure of center for a numerical data set summarizes all of itsvalues with a single number, while a measure of variation describes how its valuesvary with a single number.

6.SP.4Display numerical data in plots on a number line, including dot plots, histograms, andbox plots.

6.SP.5Summarize numerical data sets in relation to their context, such as by:

6.SP.5.aReporting the number of observations.

6.SP.5.bDescribing the nature of the attribute under investigation, including how it wasmeasured and its units of measurement.

6.SP.5.cGiving quantitative measures of center (median and/or mean) and variability(interquartile range and/or mean absolute deviation), as well as describing any overallpattern and any striking deviations from the overall pattern with reference to thecontext in which the data were gathered.

6.SP.5.dRelating the choice of measures of center and variability to the shape of the datadistribution and the context in which the data were gathered.

7.SP.1Understand that statistics can be used to gain information about a population byexamining a sample of the population; generalizations about a population from asample are valid only if the sample is representative of that population. Understandthat random sampling tends to produce representative samples and support validinferences.

7.SP.2Use data from a random sample to draw inferences about a population with anunknown characteristic of interest. Generate multiple samples (or simulated samples)of the same size to gauge the variation in estimates or predictions.

7.SP.3Informally assess the degree of visual overlap of two numerical data distributionswith similar variabilities, measuring the difference between the centers by expressingit as a multiple of a measure of variability.

7.SP.4Use measures of center and measures of variability for numerical data from randomsamples to draw informal comparative inferences about two populations.

Data About Us Inv. 1–3, UnitProject

Data About Us Inv. 1–3

Data About Us Inv. 1–3

Data About Us Inv. 1, 3Data Distributions Inv. 1–4Samples and Populations Inv. 1–4

Data About Us Inv. 1–3Data Distributions Inv. 1–4Samples and Populations Inv. 1–4

How Likely Is It? Inv. 1–4Data Distributions Inv. 1–4Samples and Populations Inv. 1–4

Data About Us Inv. 1–2Data Distributions Inv. 1–4Samples and Populations Inv. 1–4

Data About Us Inv. 3Data Distributions Inv. 1–3Samples and Populations Inv. 1–4

Data About Us Inv. 3Data Distributions Inv 1–3Samples and Populations Inv. 1–4

Samples and Populations Inv. 2

Samples and Populations Inv. 2–3

Data Distributions Inv. 2

Data Distributions Inv. 3–4

DOMAIN: STATISTICS AND PROBABILITY




7.SP.5Understand that the probability of a chance event is a number between 0 and 1 thatexpresses the likelihood of the event occurring. Larger numbers indicate greaterlikelihood. A probability near 0 indicates an unlikely event, a probability around 1/2indicates an event that is neither unlikely nor likely, and a probability near 1 indicatesa likely event.

7.SP.6Approximate the probability of a chance event by collecting data on the chanceprocess that produces it and observing its long-run relative frequency, and predict theapproximate relative frequency given the probability.

7.SP.7Develop a probability model and use it to find probabilities of events. Compareprobabilities from a model to observed frequencies; if the agreement is not good,explain possible sources of the discrepancy.

7.SP.7.aDevelop a uniform probability model by assigning equal probability to all outcomes,and use the model to determine probabilities of events.

7.SP.7.bDevelop a probability model (which may not be uniform) by observing frequencies indata generated from a chance process.

7.SP.8Find probabilities of compound events using organized lists, tables, tree diagrams,and simulation.

7.SP.8.aUnderstand that, just as with simple events, the probability of a compound event isthe fraction of outcomes in the sample space for which the compound event occurs.

7.SP.8.bRepresent sample spaces for compound events using methods such as organized lists,tables and tree diagrams. For an event described in everyday language (e.g., “rollingdouble sixes”), identify the outcomes in the sample space which compose the event.

7.SP.8.cDesign and use a simulation to generate frequencies for compound events.

8.SP.1Construct and interpret scatter plots for bivariate measurement data to investigatepatterns of association between two quantities. Describe patterns such as clustering,outliers, positive or negative association, linear association, and nonlinear association.

8.SP.2Know that straight lines are widely used to model relationships between twoquantitative variables. For scatter plots that suggest a linear association, informally fita straight line, and informally assess the model fit by judging the closeness of thedata points to the line.

8.SP.3Use the equation of a linear model to solve problems in the context ofbivariate measurement data, interpreting the slope and intercept.

8.SP.4Understand that patterns of association can also be seen in bivariate categorical databy displaying frequencies and relative frequencies in a two-way table. Construct andinterpret a two-way table summarizing data on two categorical variables collectedfrom the same subjects. Use relative frequencies calculated for rows or columns todescribe possible association between the two variables.

How Likely Is It? Inv. 1–4What Do You Expect? Inv. 1

How Likely Is It? Inv. 4What Do You Expect? Inv. 1–4

How Likely Is It? Inv. 1–4What Do You Expect? Inv. 1–4

What Do You Expect? Inv. 1–4

How Likely Is It? Inv. 1–4What Do You Expect? Inv. 1–4





Samples and Populations Inv. 4


Models Inv. 2Samples and Populations Inv. 4


Models Inv. 2–3The Shapes of Algebra Inv. 2–3

Data About Us Inv. 2


Implementing a Common Core Curriculum

Documents

Transcript of Implementing a Common Core Curriculum