Overarching Goal for Students and Teachers of CMP · CMP: A Curriculum for Students and Teachers The CMP materials reﬂect the understanding that teaching and learning are not distinct—“what

2 Implementing and Teaching Guide

CMP: A Curriculum for Students and TeachersThe CMP materials reflect the understanding thatteaching and learning are not distinct—“what to teach” and “how to teach it” are inextricablylinked. The circumstances in which students learnaffect what is learned. The needs of both studentsand teachers are considered in the developmentof the CMP curriculum materials. This curriculumhelps teachers and those who work to supportteachers examine their expectations for studentsand analyze the extent to which classroommathematics tasks and teaching practices alignwith their goals and expectations.

Overarching Goal of CMPThe overarching goal of Connected Mathematics isto help students and teachers develop mathematicalknowledge, understanding, and skill along with an awareness of and appreciation for the richconnections among mathematical strands andbetween mathematics and other disciplines. All the CMP curriculum development has been guidedby a single mathematical standard.

All students should be able to reason andcommunicate proficiently in mathematics.They should have knowledge of and skill in the use of the vocabulary, forms ofrepresentation, materials, tools, techniques,and intellectual methods of the discipline ofmathematics, including the ability to defineand solve problems with reason, insight,inventiveness, and technical proficiency.

The Connected Mathematics Project (CMP) was funded by the National ScienceFoundation between 1991 and 1997 to develop a mathematics curriculum for

grades 6, 7, and 8. The result was Connected Mathematics, a complete mathematicscurriculum that helps students develop understanding of important concepts, skills,procedures, and ways of thinking and reasoning in number, geometry, measurement,algebra, probability, and statistics.

In 2000, the National Science Foundation funded a revision of the Connected Mathematicsmaterials, CMP2, to take advantage of what we learned in the six years that the first edition of CMP has been used in schools. This Implementation Guide elaboratesthe goals of CMP2, the process we used for the revision, the scope of the curriculum,and a process for implementation that will support student and teacher learning.

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CMP2 at a GlanceBelow are some key features of ConnectedMathematics 2:

Problem CenteredImportant mathematical concepts are embeddedin engaging problems. Students developunderstanding and skill as they explore theproblems, individually, in a group, or with the class.

Practice With Concepts and Related SkillsThe in-class development problems and thehomework exercises give students practicedistributed over time with important concepts,related skills, and algorithms.

Complete CurriculumThe twenty-four Connected Mathematics 2 units—eight units for each grade—form a completemiddle school curriculum that developsmathematical skills and conceptual understandingacross mathematical strands. (Three units fromthe first edition of CMP—Ruins of Montarek,Data Around Us, and Clever Counting—willcontinue to be available to help schools reachindividual state mathematics expectations.) In addition, the program provides a completeassessment package, including quizzes, tests,and projects.

For Teachers as well as StudentsThe Connected Mathematics materials were writtento support teacher learning of both unfamiliarcontent and pedagogical strategies. The Teacher’sGuides include extensive help with mathematics,pedagogy, and assessment.

Research BasedEach Connected Mathematics unit was field tested,evaluated, and revised over a five-year period.Approximately 200 teachers and 45,000 students indiverse school settings across the United Statesparticipated in the development of the curriculum.

It WorksIt works. Research results consistently show CMP students outperform non-CMP students on tests of problem-solving ability, conceptualunderstanding, and proportional reasoning.And CMP students do as well as, or better than,non-CMP students on tests of basic skills.

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Influence of Theory andResearch on CMP2 CurriculumThe curriculum, teacher support, and assessmentmaterials that comprise the ConnectedMathematics program reflect influence from avariety of sources:

• knowledge of theory and research;

• authors’ imagination and personal teachingand learning experiences;

• advice from teachers, mathematicians, teachereducators, curriculum developers, andmathematics education researchers;

• advice from teachers and students who usedpilot and field-test versions of the materials.

The fundamental features of the CMPprogram—focus on big ideas of middle gradesmathematics, teaching through student-centeredexploration of mathematically rich problems, andcontinual assessment to inform instruction—reflect the distillation of advice and experiencefrom those varied sources.

Our work was influenced in significant ways bywhat we knew of existing theory and research inmathematics education. Here we mention andexplain briefly the key themes in thetheory/research basis for our work.

Research From the Cognitive Sciences

1. Social Constructivism We are in generalagreement with constructivist explanations ofthe ways that knowledge is developed,especially the social constructivist ideas aboutinfluence of discourse on learning. Thisposition is reflected in the authors’ decision towrite materials that would support student-centered investigation of mathematicalproblems and in our attempt to designproblem content and formats that wouldencourage student-student and student-teacher dialogue about the work.

2. Conceptual and Procedural KnowledgeWe have been influenced by theory andresearch indicating that mathematicalunderstanding is fundamentally a web oflogical and psychological connections amongideas. Furthermore, we have interpretedresearch on the interplay of conceptual andprocedural knowledge to say that soundconceptual understanding is an importantfoundation for procedural skill, not anincidental and delayed consequence ofrepeated rote procedural practice.

3. Multiple Representations An importantindication of students’ connectedmathematical knowledge is their ability torepresent ideas in a variety of ways. We have interpreted this theory to imply thatcurriculum materials should frequentlyprovide and ask for knowledge representationusing graphs, number patterns, writtenexplanations, and symbolic expressions.

4. Cooperative Learning There is a consistentand growing body of research indicating thatwhen students engage in cooperative work on appropriate problem-solving tasks, theirmathematical and social learning will beenhanced. We have interpreted this line oftheory and research to imply that we shoulddesign student and teacher materials that are suitable for use in cooperative learninginstructional formats as well as individuallearning formats—the mathematical tasksdictate the format.

Research From Mathematics Education

5. Rational Numbers/Proportional ReasoningThe extensive psychological literature ondevelopment of rational numbers andproportional reasoning has guided ourdevelopment of curriculum materialsaddressing this important middle school topic.Furthermore, the implementation of CMPmaterials in real classrooms has allowed us tocontribute to that literature with research

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publications that show the effects of newteaching approaches to traditionally difficulttopics.

6. Probability and Statistical Reasoning The interesting research literature concerningdevelopment of and cognitive obstacles tostudent learning of statistical concepts, such asmean and graphic displays, and probabilityconcepts, such as the law of large numbers andconditional probability, has been used as wedeveloped the statistics and probability unitsof CMP materials.

7. Algebraic Reasoning The differentconceptualizations of algebra described andresearched in the literature contributed to thetreatment of algebra in CMP. Various scholarsdescribe algebra as a study of modeling,functions, generalized arithmetic, and/or as a problem-solving tool. CMP has aspects ofeach of these descriptions of algebra, butfocuses more directly on functions and on theeffects of rates of change on representations.The research literature illuminates some of thecognitive complexities inherent in algebraicreasoning and offers suggestions on helpingstudents overcome difficulties. Researchconcerning concepts, such as equivalence,functions, the equal sign, algebraic variables,graphical representations, multiplerepresentations, and the role of technology,were used as we developed the algebra unitsof the CMP materials.

8. Geometric/Measurement Reasoning Results from national assessments and researchfindings show that student achievement ingeometry and measurement is weak. Researchon student understanding and learning ofgeometric/measurement concepts, such as angle,area, perimeter, volume, and processes such asvisualization, contributed to the development ofgeometry/measurement units in CMP materials.As a result of research shifting from a focus onshape and form to the related ideas ofcongruence, similarity, and symmetrytransformations, CMP geometry units weredesigned to focus on these important ideas.

Research From Education Policy and Organization

9. Motivation One of the fundamental challengesin mathematics teaching is convincing studentsthat serious effort in study of the subject will berewarding and that learning of mathematics canalso be an enjoyable experience. We have paidcareful attention to literature on extrinsic andintrinsic motivation, and we have done someinformal developmental research of our own todiscover aspects of mathematics and teachingthat are most effective in engaging studentattention and interest.

10. Teacher and School Change The mostattractive school mathematics curriculummaterials will be of little long-term value oreffect if they are not put into use in schools. Inthe process of helping teachers throughprofessional development, we have paid closeattention to what is known about effectiveteacher professional development and theschool strategies that seem to be mosteffective.

A good reference book to read for more insightinto what research says in these areas is Kilpatrick,J., Martin, W.G., & Schifter, D. (Eds.) (2003) A Research Companion to Principles andStandards for School Mathematics, Reston,VA:NCTM. (ISBN 0-87353-537-5).

While each of these ten points indicatesinfluence of theory and research on design anddevelopment of the CMP curriculum, teacher, andassessment materials, it would be misleading tosuggest that the influence is direct and controllingin all decisions. As the authors have read theresearch literature reporting empirical andtheoretical work, research findings and new ideashave been absorbed and factored into thecreative, deliberative, and experimental processthat leads to a comprehensive mathematicsprogram for schools.


The authors were guided by the followingprinciples in the development of the

Connected Mathematics materials. Thesestatements reflect both research and policystances in mathematics education about whatworks to support students’ learning of importantmathematics.

• The “big” or key mathematical ideas aroundwhich the curriculum is built are identified.

• The underlying concepts, skills, or proceduressupporting the development of a key idea areidentified and included in an appropriatedevelopment sequence.

• An effective curriculum has coherence—itbuilds and connects from investigation toinvestigation, unit-to-unit, and grade-to-grade.

• Classroom instruction focuses on inquiry andinvestigation of mathematical ideas embeddedin rich problem situations.

• Mathematical tasks for students in class andin homework are the primary vehicle forstudent engagement with the mathematicalconcepts to be learned. The key mathematicalgoals are elaborated, exemplified, andconnected through the problems in aninvestigation.

• Ideas are explored through these tasks in thedepth necessary to allow students to makesense of them. Superficial treatment of anidea produces shallow and short-livedunderstanding and does not support makingconnections among ideas.

• The curriculum helps students grow in theirability to reason effectively with informationrepresented in graphic, numeric, symbolic,and verbal forms and to move flexibly amongthese representations.

• The curriculum reflects the information-processing capabilities of calculators andcomputers and the fundamental changes suchtools are making in the way people learnmathematics and apply their knowledge ofproblem-solving tasks.

Connected Mathematics is different from manymore familiar curricula in that it is problemcentered. The following section elaborates what

we mean by this and what the value added is forstudents of such a curriculum.

Rationale for a Problem-Centered CurriculumStudents’ perceptions about a discipline comefrom the tasks or problems with which they areasked to engage. For example, if students in ageometry course are asked to memorizedefinitions, they think geometry is aboutmemorizing definitions. If students spend amajority of their mathematics time practicingpaper-and-pencil computations, they come tobelieve that mathematics is about calculatinganswers to arithmetic problems as quickly aspossible. They may become faster at performingspecific types of computations, but they may notbe able to apply these skills to other situations orto recognize problems that call for these skills.

Formal mathematics begins with undefinedterms, axioms, and definitions and deducesimportant conclusions logically from thosestarting points. However, mathematics is producedand used in a much more complex combination ofexploration, experience-based intuition, andreflection. If the purpose of studying mathematicsis to be able to solve a variety of problems, thenstudents need to spend significant portions oftheir mathematics time solving problems thatrequire thinking, planning, reasoning, computing,and evaluating.

A growing body of evidence from the cognitivesciences supports the theory that students canmake sense of mathematics if the concepts andskills are embedded within a context or problem.If time is spent exploring interesting mathematicssituations, reflecting on solution methods,examining why the methods work, comparingmethods, and relating methods to those used inprevious situations, then students are likely tobuild more robust understanding of mathematicalconcepts and related procedures. This method isquite different from the assumption that studentslearn by observing a teacher as he or shedemonstrates how to solve a problem and thenpractices that method on similar problems.

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A problem-centered curriculum not only helpsstudents to make sense of the mathematics, it alsohelps them to process the mathematics in aretrievable way.

Teachers of CMP report that students insucceeding grades remember and refer to a concept,technique, or problem-solving strategy by the nameof the problem in which they encountered the ideas.For example, the Basketball Problem from WhatDo You Expect? in Grade Seven becomes a triggerfor remembering the processes of findingcompound probabilities and expected values.

Results from the cognitive sciences also suggestthat learning is enhanced if it is connected to priorknowledge and is more likely to be retained andapplied to future learning. Critically examining,refining, and extending conjectures and strategiesare also important aspects of becoming reflectivelearners.

In CMP, important mathematical ideas areembedded in the context of interesting problems.As students explore a series of connectedproblems, they develop understanding of theembedded ideas and, with the aid of the teacher,abstract powerful mathematical ideas, problem-solving strategies, and ways of thinking. They learnmathematics and learn how to learn mathematics.

Characteristics of Good Problems To be effective, problems must embody criticalconcepts and skills and have the potential toengage students in making sense of mathematics.And, since students build understanding byreflecting, connecting, and communicating, theproblems need to encourage them to use theseprocesses.

Each problem in Connected Mathematicssatisfies the following criteria:

• The problem must have important, usefulmathematics embedded in it.

• Investigation of the problem shouldcontribute to students’ conceptualdevelopment of important mathematical ideas.

• Work on the problem should promote skillfuluse of mathematics and opportunities topractice important skills.

• The problem should create opportunities forteachers to assess what students are learningand where they are experiencing difficulty.

In addition each problem satisfies some or allof the following criteria:

• The problem should engage students andencourage classroom discourse.

• The problem should allow various solutionstrategies or lead to alternative decisions thatcan be taken and defended.

• Solution of the problem should requirehigher-level thinking and problem solving.

• The mathematical content of the problemshould connect to other importantmathematical ideas.

Practice With Concepts,Related Skills, and Algorithms Students need to practice mathematical concepts,ideas, and procedures to reach a level of fluencythat allows them to “think” with the ideas in newsituations. To accomplish this we were guided bythe following principles related to skills practice.

• Immediate practice should be related to thesituations in which the ideas have beendeveloped and learned.

• Continued practice should use skills andprocedures in situations that connect to ideasthat students have already encountered.

• Students need opportunities to use the ideasand skills in situations that extend beyondfamiliar situations. These opportunities allowstudents to use skills and concepts in newcombinations to solve new kinds of problems.

• Students need practice distributed over timeto allow high ideas, concepts and proceduresto reach a level of fluency of use in familiarand unfamiliar situations and to buildconnections to other concepts and procedures.

• Students need guidance in reflecting on whatthey are learning, how the ideas fit together,and how to make judgments about what ishelpful in which kinds of situations.

• Throughout the Number and Algebra Strandsdevelopment, students need to learn how tomake judgments about what operation orcombination of operations or representationsis useful in a given situation, as well as, how tobecome skillful at carrying out the neededcomputation(s). Knowing how to, but notwhen to, is insufficient.


Rationale for Depth versus SpiralingThe concept of a “spiraling” curriculum isphilosophically appealing; but, too often, notenough time is spent initially with a new conceptto build on it at the next stage of the spiral. Thisleads to teachers spending a great deal of time re-teaching the same ideas over and over again.Without a deeper understanding of concepts andhow they are connected, students come to viewmathematics as a collection of differenttechniques and algorithms to be memorized.

Problem solving based on such learningbecomes a search for the correct algorithm ratherthan seeking to make sense of the situation,considering the nature and size of a solution,putting together a solution path that makes sense,and examining the solution in light of the originalquestion. Taking time to allow the ideas studied tobe more carefully developed means that whenthese ideas are met in future units, students have a solid foundation on which to build. Rather thanbeing caught in a cycle of relearning the sameideas superficially which are quickly forgotten,students are able to connect new ideas topreviously learned ideas and make substantiveadvances in knowledge.

With any important mathematical concept,there are many related ideas, procedures, andskills. At each grade level, a small, select set ofimportant mathematical concepts, ideas, andrelated procedures are studied in depth ratherthan skimming through a larger set of ideas in ashallow manner. This means that time is allocatedto develop understanding of key ideas in contrastto “covering” a book. The Teacher’s Guidesaccompanying CMP materials were developed tosupport teachers in planning for and teaching aproblem-centered curriculum. Practice on relatedskills and algorithms are provided in a distributedfashion so that students not only practice theseskills and algorithms to reach facility in carryingout computations, but they also learn to put theirgrowing body of skills together to solve newproblems.

Developing Depth ofUnderstanding and UseThrough the field trials process we were able todevelop units that result in student understandingof key ideas in depth. An example is illustrated inthe way that Connected Mathematics treatsproportional reasoning—a fundamentallyimportant topic for middle school mathematicsand beyond. Conventional treatments of thiscentral topic are often limited to a brief expositorypresentation of the ideas of ratio and proportion,followed by training in techniques for solvingproportions. In contrast, the CMP curriculummaterials develop core elements of proportionalreasoning in a seventh grade unit, Comparing andScaling, with the groundwork for this unit havingbeen developed in four prior units. Five succeedingunits build on and connect to students’understanding of proportional reasoning. Theseunits and their connections are summarized asfollows:

Grade 6 Bits and Pieces I and II introducestudents to fractions and their various meaningsand uses. Models for making sense of fractionmeanings and of operating with fractions areintroduced and used. These early experiencesinclude fractions as ratios. The extensive workwith equivalent forms of fractions builds the skillsneeded to work with ratio and proportionproblems. These ideas are developed further inthe probability unit How Likely Is It? in whichratio comparisons are informally used to compareprobabilities. For example, is the probability ofdrawing a green block from a bag the same if wehave 10 green and 15 red or 20 green and 30 red?

Grade 7 Stretching and Shrinking introducesproportionality concepts in the context ofgeometric problems involving similarity. Studentsconnect visual ideas of enlarging and reducingfigures, numerical ideas of scale factors and ratios,and applications of similarity through work withproblems focused around the question: “Whatwould it mean to say two figures are similar?”

The next unit in grade seven is the coreproportional reasoning unit, Comparing andScaling, which connects fractions, percents, andratios through investigation of various situationsin which the central question is: “What strategiesmake sense in describing how much greater onequantity is than another?” Through a series of

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problem-based investigations, students explore the meaning of ratio comparison and develop,in a progression from intuition to articulateprocedures, a variety of techniques for dealingwith such questions.

A seventh grade unit that follows, MovingStraight Ahead, is a unit on linear relationshipsand equations. Proportional thinking is connectedand extended to the core ideas of linearity—constant rate of change and slope. Then in theprobability unit What Do You Expect?, studentsagain use ratios to make comparisons ofprobabilities.

Grade 8 Thinking With Mathematical Models;Looking For Pythagoras; Growing, Growing,Growing, and Frogs, Fleas, and Painted Cubesextend the understanding of proportionalrelationships by investigating the contrast betweenlinear relationships and inverse, exponential, andquadratic relationships. Also in Grade Eight,Samples and Populations uses proportionalreasoning in comparing data situations and inchoosing samples from populations.

These unit descriptions show two things aboutConnected Mathematics—the in-depthdevelopment of fundamental ideas and theconnected use of these important ideasthroughout the rest of the units.

Support for Classroom TeachersWhen mathematical ideas are embedded inproblem-based investigations of rich context, theteacher has a critical responsibility for ensuringthat students abstract and generalize theimportant mathematical concepts and proceduresfrom the experiences with the problems. In aproblem-centered classroom, teachers take onnew roles—moving from always being the onewho does the mathematics to being the one whoguides, interrogates, and facilitates the learner indoing and making sense of the mathematics.

The Teacher’s Guides and AssessmentResources developed for Connected Mathematicsprovide these kinds of help for the teacher:

• The Teacher’s Guide for each unit engagesteachers in a conversation about what ispossible in the classroom around a particularlesson. Goals for each lesson are articulated.Suggestions are made about how to engage thestudents in the mathematics task, how to

promote student thinking and reasoningduring the exploration of the problem, andhow to summarize with the students theimportant mathematics embedded in theproblem. Support for this Launch—Explore—Summarize sequence occurs for each problemin the CMP curriculum.

• An overview and elaboration of themathematics of the unit is located at thebeginning of each Teacher’s Guide, along with examples and a rationale for the modelsand procedures used. This mathematical essayhelps a teacher stand above the unit and see the mathematics from a perspective thatincludes the particular unit, connects to earlierunits, and projects to where the mathematicsgoes in subsequent units and years.

• Actual classroom scenarios are included tohelp stimulate teachers’ imaginations aboutwhat is possible.

• Questions to ask students at all stages of thelesson are included to help teachers supportstudent learning.

• Reflections questions are provided at the endof each investigation to help teachers assesswhat sense students are making of the ‘big”ideas and to help students abstract, generalize,and record the mathematical ideas andtechniques developed in the Investigation.

• Diverse kinds of assessments are included in the student units and the AssessmentResources that mirror classroom practices as well as highlight important concepts, skills,techniques, and problem solving strategies.

• Multiple kinds of assessment are included tohelp teachers see assessment and evaluationas a way to inform students of their progress,apprise parents of students’ progress, andguide the decisions a teacher makes aboutlesson plans and classroom interactions.

See pages 73–77 for more details about teachersupport materials.


Research, Field Testing,and EvaluationBefore starting the design phase of the materials,we commissioned individual reviews of the firstedition of CMP units from 84 individuals in 17states and comprehensive reviews from more than 20 schools in 14 states.

Individual Reviews These reviews focused onparticular strands over all three grades (such asnumber, algebra, or statistics) on particular sub-populations (such as students with special needsor students who are commonly underserved), oron topical concerns (such as language use andreadability).

Comprehensive Reviews These reviews wereconducted in groups that included teachers,administrators, curriculum supervisors,mathematicians, experts in special education,language, and reading level analyses, Englishlanguage learners, issues of equity, and others.Each group reviewed an entire grade level of thecurriculum. All responses were coded and enteredinto a database that allowed reports to be printedfor any issue or combination of issues that wouldbe helpful to an author or staff person indesigning a unit.

In addition, CMP issued a call to schools toserve as pilot schools for the development ofCMP2. We received 50 applications from districtsfor piloting. From these applications we chose 15that included 49 school sites in 12 states and theDistrict of Columbia. We received evaluationfeedback from these sites over the five-year cycleof development.

Based on the commissioned reviews, whatthe authors had learned from CMP schoolsover a 6-year period, and input from our AdvisoryBoard, the authors started with grades 6 and 7and systematically revised and restructured theunits and their sequence for each grade-level tocreate a first draft of the revision. These were sentto our pilot schools to be taught during the secondyear of the project. These initial grade level unitdrafts were the basis for substantial feedbackfrom our trial teachers.

Examples of the kinds of questions we asked

the trial teachers following each iteration of a unitor grade level are given below.

“BIG PICTURE” UNIT FEEDBACK

1. Is the mathematics of the unit important forstudents at this grade level? Explain.

2. Are the mathematical goals of the unit clear to you?

3. Overall, what are the strengths and weaknessesin this unit?

4. Please comment on your students’ achievementof mathematics understanding at the end of thisunit. What concepts/skills did they “nail”?Which concepts/skills are still developing?Which concepts/skills need a great deal morereinforcement?

5. Is there a flow to the sequencing of theInvestigations? Does the mathematics developsmoothly throughout the unit? Are there anybig leaps where another problem is needed to help students understand a big idea in anInvestigation? What adjustments did you makein these rough spots?

PROBLEM-BY-PROBLEM FEEDBACK

1. Are the mathematical goals of each problem/investigation clear to you?

2. Is the language and wording of each problemunderstandable to students?

3. Are there any grammatical or mathematicalerrors in the problems? (Please be as specificas possible.)

4. Are there any problems that you think can bedeleted?

5. Are there any problems that needed seriousrevision?

APPLICATIONS•CONNECTIONS•EXTENSIONSFEEDBACK

1. Does the format of the ACE exercises workfor you and your students? Why or why not?

2. Which ACE exercises work well, which wouldyou change, and why?

3. What needs to be added to or deleted from theACE exercises? Is there enough practice for

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students? How do you supplement and why?

4. Are there sufficient ACE exercises thatchallenge your more interested and capablestudents? If not, what needs to be added and why?

5. Are there sufficient ACE exercises that areaccessible to and helpful to students that needmore scaffolding for the mathematical ideas?

MATHEMATICAL REFLECTIONS ANDLOOKING BACK, LOOKING AHEAD FEEDBACK

1. Are these reflections useful to you and yourstudents in identifying and making moreexplicit the “big” mathematical ideas in theunit? If not, how could they be improved?

ASSESSMENT MATERIALS FEEDBACK

1. Are the check-ups, quizzes, unit tests, andprojects useful to you? If not, how can they beimproved? What should be deleted and whatshould be added? (Please give specifics.)

2. How do you use the assessment materials? Do you supplement the materials? If so, howand why?

TEACHER’S GUIDE FEEDBACK

1. Is the Teacher’s Guide useful to you? If not,what changes do you suggest and why?

2. Which parts of the Teacher’s Guide help youand which do you ignore or seldom use?

3. What would be helpful to add or expand in the Teacher’s Guide?

YEAR-END GRADE LEVEL FEEDBACK

1. Are the mathematical concepts, skills andprocesses in the units appropriate for thegrade level?

2. Is the grade level placement of units optimalfor your school district? Why or why not?

3. Does the mathematics flow smoothly for thestudents over the year?

4. Once an idea is learned, is there sufficientreinforcement and use in succeeding units?

5. Are connections made between units withinthe grade level?

6. Does the grade level sequence of units seemappropriate? If not, what changes would you

make and why?

7. Overall, what are the strengths andweaknesses in the units for the year? (Pleasebe as specific as possible.)

BIG PICTURE QUESTION

1. What three to five things would you have usseriously improve, change, or drop at eachgrade level? Please be specific about exactlywhat you suggest and why you would like tosee this change.

Development SummaryCMP development followed the very rigorousdesign, field-test, evaluate loop pictured in the

diagram below.The units for each grade level went through at

least three cycles of field trials–data feedback–revision. If needed, units had four rounds of fieldtrials. This process of (1) commissioning reviewsfrom experts, (2) using the field trials– feedbackloops for the materials, (3) conducting keyclassroom observations by the CMP staff of units being taught, and (4) monitoring studentperformance on state and local tests by trialschools comprises research-based development of curriculum. This process takes five years toproduce the final drafts of units that are sent tothe publisher. Another 6 to 18 months is neededfor editing, design, and layout for the publishedunits. This process produces materials that arecohesive and effectively sequenced.

unit DESIGN

FIELD trials

REVISION

data FEEDBACK

FINAL Pre-Production Unit

DATAFEEDBACK

FIELDTRIALS


An Example of EffectiveSequencing of ProblemsTo be effective, problems must be carefullysequenced to help students develop appropriateunderstanding and skill. The following set ofproblems from the Grade 6 unit, Covering andSurrounding, develops methods for finding thecircumference and area of a circle.

The first problem uses the context ofirregularly–shaped lakes to explore possiblerelationships between the perimeter of a curvedfigure and its area. Using a square grid to estimateperimeter and area helps students to understandthe meaning of perimeter and area before usingformulas.

In the second problem, students measure thediameter and circumference of several circularobjects. They create a table and a graph of theirdata and look for a pattern relating the twomeasurements. Students should discover that thecircumference is the diameter times “a little bitmore than 3.” With the help of the teacher,students are introduced to the idea of pi or pand find a closer approximation of its value.

The third problem asks students to estimate thearea of a circle. Students are encouraged to thinkof several different methods and to explain theirthinking. This problem is intended to motivate theneed for a shortcut for calculating the area. Toanswer Parts C and D, students must consider therelationships between each of the measurements—radius, diameter, circumference, and area—and thepossible price of each pizza.

Problem 5.1 Estimating Perimeter and Area

Scale pictures for Loon Lake and Ghost Lake are on the grid.

A. Estimate the area and perimeter of Loon Lake and Ghost Lake.

B. Which lake is larger? Explain your reasoning.

C. Use your estimates to answer the questions. Explain your answers.

1. Naturalists claim that water birds need long shorelines for nestingand fishing. Which lake will better support water birds?

2. Sailboaters and waterskiers want a lake with room to cruise.Which lake works better for boating and skiing?

3. Which lake has more space for lakeside campsites?

4. Which lake is better for swimming, boating, and fishing? Which lake is better for the nature preserve?

D. 1. Is your estimate of the area of each lake more or less than the actual area of that lake? Explain.

2. How could you get a more accurate estimate?

Homework starts on page 78.

Loon Lake

Ghost Lake

� 100 m

Covering and Surrounding • page 71

Problem 5.2 Finding Circumference

When you want to find out if measurements are related, looking at patterns from many examples will help.

A. Use a tape measure or string to measure the circumference anddiameter of several different circular objects. Record your results in atable with columns for the object, diameter, and circumference.

B. Study your table. Look for patterns and relationships between thecircumference and the diameter. Test your ideas on some other circular objects.

1. Can you find the circumference of a circle if you know its diameter? If so, how?

2. Can you find the diameter of a circle if you know its circumference? If so, how?



Problem 5.3 Exploring Area and Circumference

A. Find as many different ways as you can to estimate the area of thepizzas. For each method, give your estimate for the area and describehow you found it.

B. Copy the table and record each pizza’s size, diameter, radius,circumference, and area in a table.

C. Examine the data in the table and your strategies for finding area.Describe any shortcuts that you found for finding the area of a circle.

D. In your opinion, should the owner of the pizzeria base the cost of apizza on area or on circumference? Explain.


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In the fourth problem, students estimate thenumber of “radius squares” (squares with sidelength equal to the radius of the circle) it takes tocover a circle. This problem helps studentsdiscover the formula for the area of a circle and tounderstand why it makes sense. Students shouldfind that the area of a circle is “a little bit morethan 3” radius squares. With the help of theteachers, students relate “a little bit more than 3”to the number p, and develop the area formula A = p ? r 2. Mental images such as the squareembedded in a circle trigger a way for students torecall the formula for the area of a circle and toremember why the formula makes sense.

The sequencing of this set of problems and itseffectiveness is reflective of the interactionsbetween the authors and the teachers andstudents at our trial sites.

CMP: A Curriculum Co-Developed With Teachers and Students Developing a curriculum with a complex set ofinterrelated goals takes time and input from manypeople. As authors, our work was based on a setof deep commitments we had about what wouldconstitute a more powerful way to engagestudents in making sense of mathematics. OurAdvisory Board took an active role in reading andcritiquing units in their various iterations. In orderto enact our development principles, we foundthat three full years of field trials in schools wasessential for each year of the materials. Thisfeedback from teachers and students across thecountry is the key element in the success of theConnected Mathematics Project materials. Thefinal materials comprised the ideas that stood the test of time in classrooms across the country.Nearly 200 teachers in 15 trial sites around thecountry (and their thousands of students) are asignificant part of the team of professionals thatmade these materials happen. The scenarios of teacher and student interactions with thematerials became the most compelling parts of the Teacher’s Guides.

Without the bravery of these teachers in usingmaterials that were never perfect in their firstversions, CMP would have been a set of ideas thatlived in the brains and imaginations of the fiveauthors. Instead, they are materials with classroomheart because our trial teachers and students madethem so. We believe that such materials have thepotential to dramatically change what studentsknow and are able to do in mathematical situations.The emphasis on thinking and reasoning, on makingsense of ideas, on making students responsible for both having and explaining their ideas, ondeveloping sound mathematics habits gives studentsopportunities to learn in ways that can change howthey think of themselves as learners of mathematics.

From the authors’ perspectives, our hope is todevelop materials that play out deeply held beliefsand firmly grounded theories about whatmathematics is important for students to learnand how they should learn it. We hope that wehave been a part of helping to challenge andchange the curriculum landscape of our country.Our students are worth the effort.

Problem 5.4

5.4 “Squaring” a Circle

Earlier you developed formulas for the area of triangles and parallelograms by comparing them to rectangles. Now you can find outmore about the area of circles by comparing them to squares.

Finding Area

A portion of each circle is covered by a shaded square. The length of a side of the shaded square is the same length as the radius of the circle. Wecall such a square a “radius square.”

A. How many radius squares does it take to cover the circle? (You can cut out radius squares, cover the circle and see how many it takes tocover.)

Record your data in a table with columns for circle number, radius,area of the radius square, area of the circle, and number of radiussquares needed.

B. Describe any patterns and relationships you see in your table that willallow you to predict the area of the circle from its radius square. Testyour ideas on some other circular objects.

C. How can you find the area of a circle if you know the radius?

D. How can you find the radius of a circle if you know the area?


Circle 1Circle 3

Circle 2



The mathematical content developed in Connected Mathematics covers number,geometry, measurement, statistics, probability, and algebra appropriate for the

middle grades.

Connected Mathematics 2 provides 24 units (8 at each grade level). Three additionalunits continue to be available from the first edition of Connected Mathematics to meetspecific state or local needs (1 at each grade level). Every unit develops a “bigmathematical idea,” that is, an important cluster of related concepts, skills, procedures,and ways of thinking. The following table gives an overview of the curriculum at eachgrade level.

Mathematics Content of CMP2 15

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Contents in Brief, by Unit

Grade 8

Thinking With Mathematical ModelsLinear and Inverse Variationintroduction to functions andmodeling; finding the equation of aline; inverse functions; inequalities

Looking for PythagorasThe Pythagorean Theoremsquare roots; the PythagoreanTheorem; connections amongcoordinates, slope, distance, andarea; distances in the plane

Growing, Growing, GrowingExponential Relationshipsrecognize and represent exponentialgrowth and decay in tables, graphs,words, and symbols; rules ofexponents; scientific notation

Frogs, Fleas and Painted CubesQuadratic Relationshipsrecognize and represent quadraticfunctions in tables, graphs, wordsand symbols; factor simple quadraticexpressions

Kaleidoscopes, Hubcaps and MirrorsSymmetry and Transformationssymmetries of designs, symmetrytransformations, congruence,congruence rules for triangles

Say It With SymbolsMaking Sense of Symbolsequivalent expressions, substituteand combine expressions, solvequadratic equations, the quadraticformula

Shapes of AlgebraLinear Systems and Inequalitiescoordinate geometry, solveinequalities, standard form of linearequations, solve systems of linearequations and linear inequalities

Samples and PopulationsData and Statisticsuse samples to reason aboutpopulations and make predictions,compare samples and sampledistributions, relationships amongattributes in data sets

Grade 7

Variables and PatternsIntroducing Algebravariables; representations ofrelationships, including tables,graphs, words, and symbols

Stretching and ShrinkingSimilaritysimilar figures; scale factors; sidelength ratios; basic similaritytransformations and their algebraicrules

Comparing and ScalingRatio, Proportion, and Percentrates and ratios; making comparisons;proportional reasoning; solvingproportions

Accentuate the NegativePositive and Negative Numbersunderstanding and modelingpositive and negative integers andrational numbers; operations; orderof operations; distributive property;four-quadrant graphing

Moving Straight AheadLinear Relationshipsrecognize and represent linearrelationships in tables, graphs, words,and symbols; solve linear equations;slope

Filling and WrappingThree-Dimensional Measurementspatial visualization, volume andsurface area of various solids, volumeand surface area relationship

What Do You Expect?Probability and Expected Valueexpected value, probabilities of two-stage outcomes

Data DistributionsDescribing Variability andComparing Groupsmeasures of center, variability indata, comparing distributions ofequal and unequal sizes

Grade 6

Prime TimeFactors and Multiplesnumber theory, including factors,multiples, primes, composites, primefactorization

Bits and Pieces IUnderstanding Rational Numbersmove among fractions, decimals, andpercents; compare and order rationalnumbers; equivalence

Shapes and DesignsTwo-Dimensional Geometryregular and non-regular polygons,special properties of triangles andquadrilaterals, angle measure, anglesums, tiling, the triangle inequality

Bits and Pieces IIUnderstanding Fraction Operationsunderstanding and skill withaddition, subtraction, multiplication,and division of fractions

Covering and SurroundingTwo-Dimensional Measurementarea and perimeter relationships,including minima and maxima; areaand perimeter of polygons andcircles, including formulas

Bits and Pieces IIIComputing With Decimals and Percentsunderstanding and skill withaddition, subtraction, multiplication,and division of decimals, solvingpercent problems

How Likely Is It?Probabilityreason about uncertainty, calculateexperimental and theoreticalprobabilities, equally-likely and non-equally-likely outcomes

Data About UsStatisticsformulate questions; gather,organize, represent, and analyzedata; interpret results from data;measures of center and range


Connected Mathematics develops fourmathematical strands: Number and

Operation, Geometry and Measurement, DataAnalysis and Probability, and Algebra. Themathematical learning goals below signify whatstudents should be able to do by the end of eighthgrade in each strand. Beside each bulleted goal isa reference to the grade level (6, 7 or 8) when thespecific content is covered. It is important to notethat many of the goals are revisited in later unitsmany times, either within classroom Problems or in the Connections Problems in the ACEhomework assignments. For example, the bulletedgoal under Number Sense of “Express rationalnumbers in equivalent forms” is labeled a Grade 6goal because the unit Bits and Pieces I includesthis goal as a “big idea”. However, practice withthis goal occurs throughout the curriculum.

Goals by Mathematics Strand

NUMBER AND OPERATION GOALS

Number Sense

• Use numbers in various forms to solveproblems (6, 7, 8)

• Understand and use large numbers, includingin exponential and scientific notation (6, 7, 8)

• Reason proportionally in a variety of contextsusing geometric and numerical reasoning,including scaling and solving proportions (6, 7, 8)

• Compare numbers in a variety of ways,including differences, rates, ratios, and percentsand choose when each comparison isappropriate (6, 7, 8)

• Order positive and/or negative rationalnumbers (6, 7, 8)

• Express rational numbers in equivalent forms(6)

• Make estimates and use benchmarks (6, 7, 8)

Operations and Algorithms

• Develop understanding and skill with all fourarithmetic operations on fractions anddecimals (6)

• Develop understanding and skill in solving avariety of percent problems (6)

• Use the order of operations to write, evaluate,and simplify numerical expressions (7, 8)

• Develop fluency with paper and pencilcomputation, calculator use, mental calculation,and estimation; and choose among these whensolving problems (6, 7)

Properties

• Understand the multiplicative structure ofnumbers, including the concepts of prime andcomposite numbers, evens, odds, and primefactorizations (6)

• Use the commutative and distributiveproperties to write equivalent numericalexpressions (7, 8)

DATA AND PROBABILITY GOALS

Formulating Questions

• Formulate questions that can be answeredthrough data collection and analysis (6, 7, 8)

• Design data collection strategies to gather datato answer these questions (6, 7, 8)

• Design experiments and simulations to testhypotheses about probability situations (8)

Data Collection

• Carry out data collection strategies to answerquestions (6, 7, 8)

• Distinguish between samples and populations(8)

• Characterize samples as representative or non-representative, as random (8)

• Use these characterizations to evaluate thequality of the collected data (8)


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Data Analysis

• Organize, analyze, and interpret data to makepredictions, construct arguments, and makedecisions (6, 7)

• Use measures of center and spread to describeand to compare data sets (6, 7)

• Be able to read, create, and choose datarepresentations, including bar graphs, line plots,coordinate graphs, box and whisker plots,histograms, and stem and leaf plots (6, 7)

• Informally evaluate the significance ofdifferences between sets of data (7, 8)

• Use information from samples to drawconclusions about populations (8)

Probability

• Distinguish between theoretical andexperimental probabilities and understand therelationship between them (6)

• Use probability concepts to make decisions (6)

• Find and interpret expected value (7)

• Compute and compare the chances of variousoutcomes, including two-stage outcomes (7)

GEOMETRY AND MEASUREMENT GOALS

Shapes and Their Properties

• Generate important examples of angles, lines,and two- and three-dimensional shapes (6)

• Categorize, define, and relate figures in avariety of representations (6, 7)

• Understand principles governing theconstruction of shapes with reasons whycertain shapes serve special purposes (e.g. triangles for trusses) (6)

• Build and visualize three-dimensional figuresfrom various two-dimensional representationsand vice versa (7)

• Recognize and use shapes and their propertiesto make mathematical arguments and to solveproblems (6, 7, 8)

• Use the Pythagorean Theorem and propertiesof special triangles (e.g. isosceles righttriangles) to solve problems (8)

• Use a coordinate grid to describe andinvestigate relationships among shapes (7, 8)

• Recognize and use standard, essentialgeometric vocabulary (6, 7, 8)

Transformations—Symmetry, Similarity, and Congruence

• Recognize line, rotational, and translationalsymmetries and use them to solve problems (6, 8)

• Use scale factor and ratios to create similarfigures or determine whether two or moreshapes are similar or congruent (7)

• Predict ways that similarity and congruencetransformations affect lengths, angle measures,perimeters, areas, volume, and orientation (7, 8)

• Investigate the effects of combining one ormore transformations of a shape (8)

• Identify and use congruent triangles and/orquadrilaterals to solve problems about shapesand measurement (6, 8)

• Use properties of similar figures to solveproblems about shapes and measurement (7)

• Use a coordinate grid to explore and verifysimilarity and congruence relationships (7, 8)

Measurement

• Understand what it means to measure anattribute of a figure or a phenomenon (6)

• Estimate and measure angles, line segments,areas, and volumes using tools and formulas (6, 7)

• Relate angle measure and side lengths to theshape of a polygon (6)

• Find area and perimeter of rectangles,parallelograms, triangles, circles, and irregularfigures (7)

• Find surface area and volume of rectangularsolids, cylinders, prisms, cones, and pyramidsand the volume of spheres (7)

• Relate units within and between the customaryand metric systems (6, 7)

• Use ratios and proportions to derive indirectmeasurements (7)

• Use measurement concepts to solve problems(6, 7, 8)

Geometric Connections

• Use geometric concepts to build understandingof concepts in other areas of mathematics (6, 7, 8)

• Connect geometric concepts to concepts inother areas of mathematics (6, 7, 8)


ALGEBRA GOALS

Patterns of Change—Functions

• Identify and use variables to describerelationships between quantitative variables in order to solve problems or make decisions(7, 8)

• Recognize and distinguish among patterns of change associated with linear, inverse,exponential and quadratic functions (7, 8)

Representation

• Construct tables, graphs, symbolic expressionsand verbal descriptions and use them todescribe and predict patterns of change invariables (7, 8)

• Move easily among tables, graphs, symbolicexpressions, and verbal descriptions (7, 8)

• Describe the advantages and disadvantages ofeach representation and use these descriptionsto make choices when solving problems (7, 8)

• Use linear, inverse, exponential, and quadraticequations and inequalities as mathematicalmodels of situations involving variables (7, 8)

Symbolic Reasoning

• Connect equations to problem situations (7, 8)

• Connect solving equations in one variable tofinding specific values of functions (8)

• Solve linear equations and inequalities andsimple quadratic equations using symbolicmethods (7, 8)

• Find equivalent forms of many kinds ofequations, including factoring simple quadraticequations (7, 8)

• Use the distributive and commutativeproperties to write equivalent expressions andequations (8)

• Solve systems of linear equations (8)

• Solve systems of linear inequalities bygraphing (8)

Content Goals in Each UnitConnected Mathematics 2 provides eight studentunits for each grade level. Each unit is organizedaround an important mathematical idea or clusterof related ideas as described in the table on page15. Each unit covers material in a particularstrand of mathematics. This classification by strandis meant to highlight the strand that is the primary

focus of the unit. However, there are problems inevery unit that connect to the other three strands.For example, the unit Shapes of Algebra isclassified under Algebra. Even though this unit’sfocus is primarily on algebraic ideas, there aremany connections to geometry, as the unit’s nameimplies.

UNITS ORGANIZED BY STRAND

Number

Prime Time (Grade 6)

Bits and Pieces I (Grade 6)

Bits and Pieces II (Grade 6)

Bits and Pieces III (Grade 6)

Comparing and Scaling (Grade 7)

Accentuate the Negative (Grade 7)

Geometry

Shapes and Designs (Grade 6)

Covering and Surrounding (Grade 6)

Stretching and Shrinking (Grade 7)

Filling and Wrapping (Grade 7)

Looking of Pythagoras (Grade 8)

Kaleidoscopes, Hubcaps, and Mirrors (Grade 8)

Algebra

Variables and Patterns (Grade 7)

Moving Straight Ahead (Grade 7)

Thinking With Mathematical Models (Grade 8)

Growing, Growing, Growing (Grade 8)

Frogs, Fleas, and Painted Cubes (Grade 8)

Say It With Symbols (Grade 8)

Shapes of Algebra (Grade 8)

Data Analysis and Probability

How Likely Is It? (Grade 6)

Data About Us (Grade 6)

What Do You Expect? (Grade 7)

Data Distributions (Grade 7)

Samples and Populations (Grade 8)


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In order to have a clearer idea of the particulargoals for each unit, the goals are listed below byunit name. The units are sequenced in the orderthey are intended to be taught.

The ninth unit at each grade level, availablefrom the first edition of CMP, can be used asa stand-alone unit for various purposes. Forexample, the unit Ruins of Montarek has beentaught at Grade 5, as well as in art classes andsocial studies classes. Depending on your state or local standards, parts or all of these threeadditional units can be supplemented into thecurriculum. The goals of each supplemental unitare listed below, after the CMP2 units.

Some questions to ask yourself as you examinethe list of unit goals that follow:

• How does a particular strand play out? For example, how are the number unitssequenced? What units in each grade are in thenumber strand? How do the number systemswith which students work grow as thecurriculum progresses (ie, whole numbers,fractions, decimals, irrational numbers)

• When following a key goal for a unit: Does alater unit further develop this same goal and ifso how?

• What goals have my students already met fromprior units? What prior knowledge do theyhave that I can draw on?

• How does a concept grow? For example, whichunits are setting the groundwork for linearfunctions? What units cover this topic and howdoes this idea grow in complexity?

• Why are the units from different strandsinterspersed? What connections are madebetween the strands, between the units within agrade level, between the units in different gradelevels?

GRADE SIX GOALS

Prime Time (Number)

• Understand relationships among factors,multiples, divisors, and products

• Recognize and use properties of prime andcomposite numbers, even and odd numbers,and square numbers

• Use rectangles to represent the factor pairs ofnumbers

• Develop strategies for finding factors andmultiples, least common multiples, and greatestcommon factors

• Recognize and use the fact that every wholenumber greater than 1 can be written inexactly one way as a product of prime numbers

• Use factors and multiples to solve problemsand to explain some numerical facts ofeveryday life

• Develop a variety of strategies for solvingproblems—building models, making lists andtables, drawing diagrams, and solving simplerproblems

Bits and Pieces I (Number)

• Build an understanding of fractions, decimals,and percents and the relationships betweenand among these concepts and theirrepresentations

• Develop ways to model situations involvingfractions, decimals, and percents

• Understand and use equivalent fractions toreason about situations

• Compare and order fractions

• Move flexibly among fraction, decimal, andpercent representations

• Use benchmarks such as 0, 1/2 and 1 to helpestimate the size of a number or sum

• Develop and use benchmarks that relatedifferent forms of representations of rationalnumbers (for example, 50% can be representedas 0.5)

• Use physical models and drawings to helpreason about a situation

• Look for patterns and describe how tocontinue the pattern

• Use context to help reason about a situation

• Use estimation to understand a situation

Shapes and Designs (Geometry)

• Understand some important properties ofpolygons and recognize polygonal shapes bothin and out of the classroom

• Investigate the symmetries of a shape—rotational or reflectional

• Estimate the size of any angle using referenceto a right angle and other benchmark angles

• Use an angle ruler for making more accurateangle measurements

• Explore parallel lines and angles created bylines intersecting parallel lines


• Find patterns that help determine angle sumsof polygons

• Determine which polygons fit together to covera flat surface and why

• Explain the property of triangles that makesthem useful as a stable structure for building

• Find that the sum of any two side lengths of atriangle is greater than the third side length

• Find that the sum of any three side lengths of aquadrilateral is greater than the fourth sidelength

• Reason about and solve problems involvingshapes

Bits and Pieces II (Number)

• Use benchmarks and other strategies toestimate the reasonableness of results ofoperations with fractions

• Develop ways to model sums, differences,products, and quotients with areas, strips, andnumber lines

• Use estimates and exact solutions to makedecisions

• Look for and generalize patterns in numbers

• Use knowledge of fractions and equivalence offractions to develop algorithms for adding,subtracting, multiplying and dividing fractions

• Recognize when addition, subtraction,multiplication, or division is the appropriateoperation to solve a problem

• Write fact families to show the inverserelationship between addition and subtraction,and between multiplication and division

• Solve problems using arithmetic operations onfractions

Covering and Surrounding (Geometry)

• Understand area and relate area to covering a figure

• Understand perimeter and relate perimeter tosurrounding a figure

• Develop strategies for finding areas andperimeters of rectangular shapes and non-rectangular shapes

• Discover relationships between perimeter andarea. including that each can vary while theother stays fixed

• Understand how the areas of simple geometricfigures relate to each other (e.g. the area of a

parallelogram is twice the area of a trianglewith the same base and height)

• Develop formulas and procedures—stated inwords and/or symbols—for finding areas andperimeters of rectangles, parallelograms,triangles, and circles

• Develop techniques for estimating the areaand perimeter of an irregular figure

• Recognize situations in which measuringperimeter or area will help answer practicalquestions

Bits and Pieces III (Number)

• Build on knowledge of operations on fractionsand whole numbers

• Develop and use benchmarks and otherstrategies to estimate the answers tocomputations with decimals

• Develop meaning of and algorithms foroperations with decimals

• Use the relationship between decimals andfractions to develop and understand whydecimal algorithms work

• Use the place value interpretation of decimalsto make sense of short-cut algorithms foroperations

• Generalize number patterns to help makesense of decimal operations

• Choose between addition, subtraction,multiplication or division as an appropriateoperation to use to solve a problem

• Understand that decimals are often associatedwith measurements in real world situations

• Solve problems using operations on decimals

• Use understanding of operations and themeaning of percents to solve percent problemsof the form a% of b equals c for any one of thevariables a, b, or c

• Create and interpret circle graphs

How Likely Is It? (Probability)

• Understand that probabilities are useful forpredicting what will happen over the long run

• Understand the concepts of equally likely andnot-equally likely

• Understand that fairness implies equally likelyoutcomes


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• Understand that there are two ways to buildprobability models: by gathering data fromexperiments (experimental probability) and byanalyzing the possible equally likely outcomes(theoretical probability)

• Understand that experimental probabilities arebetter estimates of theoretical probabilitieswhen they are based on larger numbers of trials

• Develop strategies for finding bothexperimental and theoretical probabilities

• Critically interpret statements of probability tomake decisions or answer questions

Data About Us (Data Analysis)

• Understand and use the process of datainvestigation by posing questions, collectingdata, analyzing data distributions, and makinginterpretations to answer questions

• Represent data distributions using line plots,bar graphs, stem-and-leaf plots, and coordinategraphs

• Compute the mean, median, or mode and therange of the data

• Distinguish between categorical data andnumerical data and identify which graphs andstatistics may be used to represent each kind ofdata

• Make informed decisions about which graph orgraphs and which of the measures of center(mean, median, or mode) and range may beused to describe a data distribution

• Develop strategies for comparing datadistributions

Ruins of Montarek (Geometry)Available in first edition ©2004.

• Read and make two-dimensionalrepresentations of three-dimensional cubebuildings

• Observe that the back view of a cube building isthe mirror image of the front view and that theleft view is the mirror image of the right view

• Explain how drawings of the base outline, frontview, and right view describe a building

• Construct cube buildings that fit two-dimensional building plans

• Develop a way to describe all buildings thatcan be made from a set of plans

• Understand that a set of plans can have morethan one minimal building but only onemaximal building

• Explain how a cube can be represented onisometric dot paper, how the angles on the cubeare represented with angles on the dot paper,and how the representations fit what the eyesees when viewing the corner of a cube building

• Make isometric drawings of cube buildings

• Visualize transformations of cube buildingsand make isometric drawings of thetransformed buildings

• Reason about spatial relationships

• Use models and representations to solveproblems

GRADE SEVEN GOALS

Variables and Patterns (Algebra)

• Recognize problem situations in which two ormore quantitative variables are related to eachother

• Identify quantitative variables in situations

• Describe patterns of change between twovariables that are shown in words, tables andgraphs of data

• Construct tables and graphs to display relationsamong variables

• Observe relationships between two variables as shown in a table, graph, or equation anddescribe how the relationship can be seen ineach of the other forms of representation

• Use algebraic symbols to write equationsrelating variables

• Use tables, graphs, and equations to solveproblems

• Use graphing calculators to construct tablesand graphs of relations between variables andto answer questions about these relations

Stretching and Shrinking (Geometry)

• Identify similar figures by comparingcorresponding parts

• Use scale factors and ratios to describerelationships among the side lengths of similarfigures

• Construct similar polygons

• Draw shapes on coordinate grids and then usecoordinate rules to stretch and shrink thoseshapes

• Predict the ways that stretching or shrinking afigure affect lengths, angle measures, perimeters,and areas


• Use the properties of similarity to calculatedistances and heights that can’t be directlymeasured

Comparing and Scaling (Number)

• Analyze comparison statements made aboutquantitative data

• Use ratios, fractions, differences, and percentsto form comparison statements in a givensituation, such as

“What is the ratio of boys to girls in our class?”

“What fraction of the class is going to the spring picnic?”

“What percent of the girls play basketball?”

“Which model of car has the best fuel economy?”

“Which long-distance telephone company is more popular?”

• Judge whether comparison statements makesense and are useful

• See how forms of comparison statements arerelated, for example, a percent and a fractioncomparison

• Make judgments about which statements aremost informative or best reflect a particularpoint of view

• Decide when the most informative comparisonis to find the difference between two quantitiesand when it is to form ratios between pairs ofquantities

• Scale a ratio, rate, or fraction up or down tomake a larger or smaller object or populationwith the same relative characteristics as theoriginal

• Represent related data in tables

• Look for patterns in tables that will allowpredictions to be made beyond the tables

• Write an equation to represent the pattern in atable of related variables

• Apply proportional reasoning to solve for theunknown part when one part of two equalratios is unknown

• Set up and solve proportions that arise inapplications

• Recognize that constant growth in a table isrelated to proportional situations

• Connect unit rates with the equationdescribing a situation

Accentuate the Negative (Number)

• Use appropriate notation to indicate positiveand negative numbers

• Locate rational numbers (positive and negativefractions and decimals and zero) on a numberline

• Compare and order rational numbers

• Understand the relationship between a positiveor negative number and its opposite (additiveinverse)

• Develop algorithms for adding, subtracting,multiplying, and dividing positive and negativenumbers and write mathematics sentences toshow relationships

• Write and use related fact families foraddition/subtraction and multiplication/divisionto solve simple equations with missing facts

• Use parentheses and order of operations tomake computational sequences clear

• Understand and use the CommutativeProperty for addition and multiplication ofnegative and positive numbers

• Apply the Distributive Property with positiveand negative numbers to simplify expressionsand solve problems

• Use positive and negative numbers to graph infour quadrants, model and answer questionsabout applied settings

Moving Straight Ahead (Algebra)

• Recognize problem situations in which two ormore variables have a linear relationship toeach other

• Construct tables, graphs, and symbolicequations that express linear relationships

• Translate information about linear relationsgiven in a table, a graph, or an equation to oneof the other forms

• Understand the connections between linearequations and patterns in the tables and graphsof those relations—rate of change, slope, and y-intercept

• Solve linear equations

• Solve problems and make decisions aboutlinear relationships using information given intables, graphs, and symbolic expressions

• Use tables, graphs, and equations of linearrelations to answer interesting questions


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Filling and Wrapping (Geometry)

• Understand volume as a measure of filling anobject and surface area as a measure ofwrapping or covering an object

• Use flat patterns to visualize and calculatesurface areas of prisms and cylinders

• Develop formulas for the volumes of prisms,cylinders, cones, pyramids, and spheres eitherdirectly or by comparison with known volumes

• Understand that three-dimensional figures mayhave the same volume but quite differentshapes and surface areas or that they may havethe same surface area but different shapes andvolumes

• Use surface area and volume to solve a varietyof real-world problems

• Understand how changes in one or moredimensions of a rectangular prism or cylinderaffects the prism’s volume

• Extend students’ understanding of similarityand scale factors to three-dimensional figures

• Understand the effect on surface area andvolume of applying a scale factor to arectangular prism

Data Distributions (Data Analysis)

• Apply the process of statistical investigation topose questions, identify ways data arecollected, determine strategies for analyzingdata and interpreting the analysis in order toanswer the questions posed

• Compare the distributions of data using theirrelated centers, variability, and shapes

• Use the shape of a distribution to estimate themean and median

• Recognize that variability occurs whenever dataare collected and use properties of distributionsto describe the variability in a given data set

• Identify sources of variability, including naturalvariability and variability that results fromerrors in measurement

• Decide if a difference among data valuesand/or summary measures matters

• Understand and decide when to use the meanand median to describe a distribution

• Make effective use of a variety ofrepresentations to display distributions,including tables, value bar graphs, dot or lineplots, and bar graphs

• Understand and use counts or percents toreport frequencies of occurrence of data

• Develop and use strategies for comparing equal-size and unequal-size data sets to solve problems

What Do You Expect? (Probability)

• Interpret experimental and theoreticalprobabilities and the relationship betweenthem

• Distinguish between equally likely and non-equally likely outcomes

• Review strategies for identifying possibleoutcomes and analyzing probabilities, such asusing lists or counting trees

• Understand that fairness implies equally likelyoutcomes

• Analyze situations that involve two-stages (or two actions)

• Use area models to analyze situations thatinvolve two stages

• Determine the expected value of a probabilitysituation

• Analyze binomial situations

• Use probability and expected value to makedecisions

Numbers Around Us (Number)Available in first edition ©2004.

• Choose sensible units for measuring

• Understand that a measurement has twocomponents, a unit of measure and a count

• Build a repertoire of benchmarks to relate the measures of unfamiliar objects or events to the measures of objects or events that arepersonally meaningful

• Review the concept of place value as it relatesto reading, writing, and using large numbers

• Read, write, and interpret the large numbersthat occur in real-life measurements usingstandard, scientific, and calculator notation

• Review and extend the use of exponents

• Use estimates and rounded values fordescribing and comparing objects and events

• Develop strategies for operating with largenumbers

• Choose sensible ways of comparing counts andmeasurements, including using differences,rates, and ratios

• Draw sensible conclusions from giveninformation.


GRADE EIGHT GOALS

Thinking With Mathematical Models(Algebra)

• Recognize linear and non-linear patterns incontexts, tables and graphs and describe thosepatterns using words and symbolic expressions

• Write equations to express linear patternsappearing in tables, graphs, and verbal contexts

• Write linear equations when specificinformation such as two points or a point and aslope, is given for a line

• Approximate linear data patterns with graphand equation models

• Solve linear equations

• Interpret inequalities

• Write equations describing inverse variation

• Use linear and inverse variation equations tosolve problems and to make predictions anddecisions

Looking For Pythagoras (Algebra)

• Relate the area of a square to the length of aside of the square

• Estimate square roots

• Develop strategies for finding the distancebetween two points on a coordinate grid

• Understand and apply the PythagoreanTheorem

• Use the Pythagorean Theorem to solve avariety of problems

Growing, Growing, Growing (Algebra)

• Recognize situations where one variable is anexponential function of another variable

• Recognize the connections between exponentialequations and growth patterns in tables andgraphs of those relations

• Construct equations to express exponentialpatterns that appear in data tables, graphs, andproblem conditions

• Understand and apply the rules for operatingon numerical expressions with exponents

• Solve problems about exponential growth anddecay in a variety of situations such as scienceor business

• Compare exponential and linear relationships

Frogs, Fleas, and Painted Cubes(Algebra)

• Recognize the patterns of change for quadraticrelationships in a table, graph, equation, andproblem situation

• Construct equations to express quadraticrelationships that appear in tables, graphs andproblem situations

• Recognize the connections between quadraticequations and patterns in tables and graphs ofthose relationships

• Use tables, graphs, and equations of quadraticrelationships to locate maximum and minimumvalues of a dependent variable and the x- andy-intercepts and other important features ofparabolas

• Recognize equivalent symbolic expressions forthe dependent variable in quadraticrelationships

• Use the distributive property to writeequivalent quadratic expressions in factoredform or expanded form

• Use tables, graphs, and equations of quadraticrelations to solve problems in a variety ofsituations from geometry, science, and business

• Compare properties of quadratic, linear, andexponential relationships

Kaleidoscopes, Hubcaps, and Mirrors(Geometry)

• Understand important properties of symmetry

• Recognize and describe symmetries of figures

• Use tools to examine symmetries andtransformations

• Make figures with specified symmetries

• Identify basic design elements that can be usedto replicate a given design

• Perform symmetry transformations of figures,including reflections, translations, and rotations

• Examine and describe the symmetries of adesign made from a figure and its image(s)under a symmetry transformation

• Give precise mathematical directions forperforming reflections, rotations, andtranslations

• Draw conclusions about a figure, such asmeasures of sides and angles, lengths ofdiagonals, or intersection points of diagonals,based on symmetries of the figure


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• Understand that figures with the same shapeand size are congruent

• Use symmetry transformations to explorewhether two figures are congruent

• Give examples of minimum sets of measures ofangles and sides that will guarantee that twotriangles are congruent

• Use congruence of triangles to explorecongruence of two quadrilaterals

• Use symmetry and congruence to deduceproperties of figures

• Write coordinate rules for specifying the imageof a general point (x, y) under particulartransformations

• Use transformational geometry to describemotions, patterns, designs, and properties ofshapes in the real world

Say It With Symbols (Algebra)

• Model situations with symbolic statements

• Write equivalent expressions

• Determine if different symbolic expressionsare mathematically equivalent

• Interpret the information equivalentexpressions represent in a given context

• Determine which equivalent expression to useto answer particular questions;

• Solve linear equations involving parentheses

• Solve quadratic equations by factoring

• Use equations to make predictions anddecisions

• Analyze equations to determine the patterns ofchange in the tables and graphs that theequation represents

• Understand how and when symbols should beused to display relationships, generalizations,and proofs

The Shapes of Algebra (Algebra)

• Write and use equations of circles

• Determine lines are parallel or perpendicularby looking at patterns in their graphs,coordinates, and equations

• Find coordinates of points that divide linesegments in various ratios

• Write inequalities that satisfy given situations

• Find solutions to inequalities represented by agraph or an equation

• Solve systems of linear equations by graphing,combining equations, and by substitution

• Write linear inequalities in two variables tomatch constraints in problem conditions

• Graph linear inequalities and systems ofinequalities and use the results to solveproblems

Samples and Populations (Data Analysis)

• Revisit and use the process of statisticalinvestigation to explore problems

• Distinguish between samples and populationsand use information drawn from samples todraw conclusions about populations

• Explore the influence of sample size and ofrandom or nonrandom sample selection

• Apply concepts from probability to selectrandom samples from populations

• Compare sample distributions using measuresof center (mean or median), measures ofdispersion (range or percentiles), and datadisplays that group data (histograms and box-and-whisker plots)

• Explore relationships between paired values ofnumerical attributes

Clever Counting (Number)Available in first edition ©2004.

• Recognize situations in which countingtechniques apply

• Construct organized lists of outcomes forcomplex processes and uncover patterns thathelp in counting the outcomes of thoseprocesses

• Use diagrams, tables, and symbolic expressionsto organize examples in listing and countingtasks

• Analyze the usefulness of counting trees anduse counting trees

• Use mental arithmetic to make estimates inmultiplication and division calculations

• Invent strategies for solving problems thatinvolve counting

• Analyze counting problems involving choicesin various contexts

• Differentiate among situations in which orderdoes and does not matter and in which repeatsare and are not allowed


• Analyze the number of paths through anetwork

• Compare the structures of networks withproblems involving combinations

• Create networks that satisfy given constraints

• Apply thinking and reasoning skills to anopen-ended situation in which assumptionsmust be made and create a persuasiveargument to support a conjecture

Mathematics Process GoalsIn setting mathematical goals for a schoolcurriculum, the choice of content topics must beaccompanied by an analysis of the kinds ofthinking students will be able to demonstrateupon completion of the curriculum. The textbelow describes the eleven key mathematicalprocesses developed in all the main contentstrands of Connected Mathematics.

CountingDetermining the number of elements in finite datasets, trees, graphs, or combinations by applicationof mental computation, estimation, countingprinciples, calculators and computers, and formalalgorithms.

VisualizingRecognizing and describing shape, size, andposition of one-, two-, and three-dimensionalobjects and their images under transformations;interpreting graphical representations of data,functions, relations, and symbolic expressions.

ComparingDescribing relationships among quantities andshapes using concepts such as equality andinequality, order of magnitude, proportion,congruence, similarity, parallelism, perpendicularity,symmetry, and rates of growth or change.

EstimatingDetermining reasonableness of answers. Using“benchmarks” to estimate measures. Usingvarious strategies to approximate a calculationand to compare estimates.

MeasuringAssigning numbers as measures of geometricobjects and probabilities of events. Choosingappropriate measures in a decision-makingproblem. Choosing appropriate units or scales and making approximate measurements orapplying formal rules to find measures.

ModelingConstructing, making inferences from, andinterpreting concrete, symbolic, graphic, verbal,and algorithmic models of quantitative, visual,statistical, probabilistic, and algebraic relationshipsin problem situations. Translating information fromone model to another.

ReasoningBringing to any problem situation the dispositionand ability to observe, experiment, analyze,abstract, induce, deduce, extend, generalize, relate,and manipulate in order to find solutions or proveconjectures involving interesting and importantpatterns.

ConnectingIdentifying ways in which problems, situations,and mathematical ideas are interrelated andapplying knowledge gained in solving oneproblem to other problems.

RepresentingMoving flexibly among graphic, numeric, symbolic,and verbal representations and recognizing theimportance of having various representations ofinformation in a situation.

Using ToolsSelecting and intelligently using calculators,computers, drawing tools, and physical models torepresent, simulate, and manipulate patterns andrelationships in problem settings.

Becoming MathematiciansHaving the disposition and imagination to inquire,investigate, tinker, dream, conjecture, invent, andcommunicate with others about mathematical ideas.


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Alignment with the NCTMPrinciples and Standards 2000In 1989, the National Council of Teachers ofMathematics (NCTM) released its Curriculumand Evaluation Standards for SchoolMathematics. This document provided guidancefor developing and implementing a vision ofmathematics and instruction that serves allstudents. In 2000, NCTM expanded and elaboratedon the 1989 standards to create the Principles andStandards for School Mathematics. This documentreflects the research on teaching, learning, andtechnology that has evolved over the past ten years.These standards have served as a guide indeveloping Connected Mathematics.

The following chart shows the alignment ofConnected Mathematics with NCTM Principlesand Standards 2000.

CONTENT STANDARDS

Number and Operations

Prime Time (Grade 6)

Bits and Pieces I (Grade 6)

Bits and Pieces II (Grade 6)

Bits and Pieces III (Grade 6)

Comparing and Scaling (Grade 7)

Accentuate the Negative (Grade 7)

Looking for Pythagoras (Grade 8)

Algebra

Variables and Patterns (Grade 7)

Moving Straight Ahead (Grade 7)

Thinking With Mathematical Models (Grade 8)


Growing, Growing, Growing (Grade 8)

Frogs, Fleas, and Painted Cubes (Grade 8)

Say It With Symbols (Grade 8)

Shapes of Algebra (Grade 8)

Geometry


Ruins of Montarek (Grade 6)




Kaleidoscopes, Hubcaps, and Mirrors (Grade 8)

Measurement


Covering and Surrounding (Grade 6)



Data Around Us (Grade 7)



Data About Us (Grade 6)

How Likely Is It? (Grade 6)

What Do You Expect? (Grade 7)

Data Distributions (Grade 7)

Samples and Populations (Grade 8)

PROCESS STANDARDS

PROCESS STANDARDS

Problem Solving

All unitsBecause Connected Mathematics is a problem-centered curriculum, problem solving is animportant part of every unit.

Reasoning and Proof

All unitsThroughout the curriculum, students areencouraged to look for patterns, makeconjectures, provide evidence for theirconjectures, refine their conjectures andstrategies, connect their knowledge, and extendtheir findings. Informal reasoning evolves intomore deductive arguments as students proceedfrom Grade 6 through Grade 8.


Communication

All unitsAs students work on the problems, they mustcommunicate ideas with others. Emphasis isplaced on students’ discussing problems in class, talking through their solutions, formalizingtheir conjectures and strategies, and learning to communicate their ideas to a more generalaudience. Students learn to express their ideas, solutions, and strategies using writtenexplanations, graphs, tables, and equations.

Connections

All unitsIn all units, the mathematical content is connectedto other units, to other areas of mathematics, toother school subjects, and to applications in thereal world. Connecting and building on priorknowledge is important for building and retainingnew knowledge.

Representation

All unitsThroughout the units, students organize, record,and communicate information and ideas usingwords, pictures, graphs, tables, and symbols.They learn to choose appropriate representationsfor given situations and to translate amongrepresentations. Students also learn to interpretinformation presented in various forms.

Alignment with State FrameworksConnected Mathematics addresses all contenttopics that might be required at middle schoollevel. Because topics are covered in depth inindividual units, districts may choose to use aparticular unit at a grade level above or below itsposition in the teaching sequence. The chart onpage 18 shows the recommended teaching orderwithin each mathematics strand. If units aremoved out of sequence to be taught before therecommended location and grade level, thedistrict should carefully check to see that requisiteconnected units have been taught. Obviousexamples are Bits and Pieces I, II, and III. Theseshould not be taught in a different order. Anotheris the sequence of units in grade 7, Variables andPatterns, Stretching and Shrinking, Comparingand Scaling, Accentuate the Negative, and MovingStraight Ahead. These build on each other andshould be taught in the recommended order.

Number and Operations

Grade 6 Grade 7 Grade 8

Whole Numbers

divisors, factors, greatest common factor

divisibility rules

multiples, least commonmultiple

even, odd numbers

prime numbers

composite numbers

squares

square roots

prime factorization

place value

comparing

Prime Time IMBits and Pieces I RShapes and Designs RCovering and

Surrounding RData About Us R

Prime Time IM

Prime Time IMBits and Pieces I RBits and Pieces III RData About Us R

Prime Time IM

Prime Time IM

Prime Time IM

Prime Time IMShapes and Designs R

Prime Time I

Prime Time IMShapes and Designs R

Prime Time RBits and Pieces I R

Prime Time R

Variables and Patterns RComparing and Scaling RAccentuate the Negative RFilling and Wrapping R

Comparing and Scaling R

Variables and Patterns R

Filling and Wrapping R

Stretching and Shrinking R

Stretching and Shrinking I

Data Around Us RData Distributions R

Variables and Patterns RStretching and Shrinking RComparing and Scaling RData Distributions R

Thinking WithMathematical Models R

Growing, Growing,Growing R

Frogs, Fleas, and PaintedCubes R

Say It with Symbols R



Say It With Symbols R



Looking for Pythagoras RFrogs, Fleas, and Painted

Cubes RSay It with Symbols R

Looking for Pythagoras IM



Shapes of Algebra RSamples and Populations R


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Scope and Sequence for CMP2Deep understanding of the concepts and skills aredeveloped in the units listed. In some cases, thetopics are introduced in one unit and more fullydeveloped in a later unit. In other cases, the topics are revisited in the same or other units inConnections questions, or are used to developunderstanding of new concepts. The development

of a concept includes understanding relationshipsamong and between concepts, as well as developingskills, procedures, and algorithms.

As a problem solving curriculum, every unithelps students develop a variety of strategies forsolving problems, such as building models, makinglists and tables, drawing diagrams, and solvingsimpler problems.

Key: I = introduced M =mastered R = reinforced; applied

Number and Operations (cont.)


exponential form(notation)

laws of exponents

Decimals

place value

models

on a number line

comparing and ordering

related to fractions andpercents

terminating and repeatingdecimals

estimating/benchmarks

rounding

scientific notation

operations with

Prime Time IMBits and Pieces III R

Bits and Pieces I IMBits and Pieces III R

Bits and Pieces I IMBits and Pieces III R

Bits and Pieces I IMBits and Pieces II RBits and Pieces III RData About Us R

Bits and Pieces I IMBits and Pieces II RCovering and

Surrounding RBits and Pieces III RHow Likely Is It? R

Bits and Pieces I IMBits and Pieces II RBits and Pieces III RHow Likely Is It? RData About Us R

Bits and Pieces III IM

Bits and Pieces I IMBits and Pieces II RBits and Pieces III R

Bits and Pieces I IMBits and Pieces III RHow Likely Is It? R

Bits and Pieces III IMHow Likely Is It? RData About Us R

Variables and Patterns RStretching and Shrinking RAccentuate the Negative R

Data Distributions R

Comparing and Scaling RStretching and Shrinking R

Variables and Patterns RComparing and Scaling RWhat Do You Expect? RData Distributions R

Variables and Patterns RComparing and Scaling RAccentuate the Negative RWhat Do You Expect? RData Distributions R

Stretching and Shrinking RComparing and Scaling RFilling and Wrapping RData Distributions R

Comparing and Scaling RFilling and Wrapping RData Distributions R

Filling and Wrapping RData Distributions R

Stretching and Shrinking RAccentuate the Negative RMoving Straight Ahead RFilling and Wrapping RWhat Do You Expect? RData Distributions R


Say It with Symbols RShapes of Algebra R

Growing, Growing,Growing IM


Looking for Pythagoras RGrowing, Growing,

Growing R

Looking for Pythagoras R





Growing RSamples and Populations R



Growing R




Growing R









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Fractions


related to decimals andpercents

equivalent


models

reciprocals

operations with

Ratio and Proportion

ratios, rates, unit rates

Bits and Pieces I IMShapes and Designs RBits and Pieces II RCovering and

Surrounding RBits and Pieces III RHow Likely Is It? R


Bits and Pieces I IMShapes and Designs RBits and Pieces II RBits and Pieces III RHow Likely Is It? R

Bits and Pieces I IMBits and Pieces II RBits and Pieces III RHow Likely Is It? R

Bits and Pieces I IMShapes and Designs RBits and Pieces II RBits and Pieces III RHow Likely Is It? R

Bits and Pieces II IM

Bits and Pieces I IShapes and Designs IBits and Pieces II IMCovering and

Surrounding RBits and Pieces III RHow Likely Is It? RData About Us R

Bits and Pieces I IShapes and Designs IBits and Pieces II IBits and Pieces III IHow Likely Is It? IData About Us I

Variables and Patterns RComparing and Scaling RAccentuate the Negative RWhat Do You Expect? RData Distributions R

Stretching and Shrinking RComparing and Scaling RWhat Do You Expect? RData Distributions R

Stretching and Shrinking RComparing and Scaling RWhat Do You Expect? R

Comparing and Scaling RWhat Do You Expect? RData Distributions R

Comparing and Scaling RFilling and Wrapping RWhat Do You Expect? R

Moving Straight Ahead R

Variables and Patterns RStretching and Shrinking RAccentuate the Negative RMoving Straight Ahead RFilling and Wrapping RWhat Do You Expect? RData Distributions R

Variables and Patterns IStretching and Shrinking IComparing and Scaling IMMoving Straight Ahead RFilling and Wrapping RData Distributions R



Growing RShapes of Algebra RSamples and Populations R



Growing R


Samples and Populations R







Growing RSay It with Symbols R



Growing RShapes of Algebra RSamples and Populations R




equivalent ratios

proportions

comparing proportionaland nonproportionalrelationships

scaling/scale factors

scale factors with similar 3-dimensional figures

estimating

proportional reasoning

Percents

related to fractions anddecimals

Bits and Pieces I IShapes and Designs IBits and Pieces II IBits and Pieces III IHow Likely Is It? I

Bits and Pieces I IBits and Pieces II IBits and Pieces III IHow Likely Is It? I

Bits and Pieces I IBits and Pieces III IHow Likely Is It? IData About Us I

Bits and Pieces I IBits and Pieces III IHow Likely Is It? I

Bits and Pieces I IBits and Pieces III IHow Likely Is It? IData About Us I

Bits and Pieces I IBits and Pieces II IBits and Pieces III IHow Likely Is It? I


Stretching and Shrinking IMComparing and Scaling RMoving Straight Ahead RWhat Do You Expect? RData Distributions R

Stretching and Shrinking IComparing and Scaling IMMoving Straight Ahead R

Variables and Patterns IStretching and Shrinking IComparing and Scaling IMMoving Straight Ahead RWhat Do You Expect? RData Distributions R

Stretching and Shrinking IMComparing and Scaling RWhat Do You Expect? R

Filling and Wrapping IM

Stretching and Shrinking IComparing and Scaling IMFilling and Wrapping RWhat Do You Expect? RData Distributions R

Variables and Patterns IStretching and Shrinking IComparing and Scaling IMMoving Straight Ahead RFilling and Wrapping RWhat Do You Expect? RData Distributions R








Growing RFrogs, Fleas, and Painted

Cubes RKaleidoscopes, Hubcaps,

and Mirrors RSay It with Symbols RShapes of Algebra R




Growing RKaleidoscopes, Hubcaps,




Say It with Symbols RSamples and Populations R




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models


finding

percent of a number

solving problems with

Integers

models

opposites/inverseoperations

absolute value


on a number line

operations with


Bits and Pieces I IMBits and Pieces III RHow Likely Is It? R


Bits and Pieces I IBits and Pieces III IMHow Likely Is It? RData About Us R

Bits and Pieces I IBits and Pieces III IMHow Likely Is It? RData About Us R

Bits and Pieces III IMHow Likely Is It? R

Bits and Pieces II I




What Do You Expect? RData Distributions R

Variables and Patterns RStretching and Shrinking RComparing and Scaling RMoving Straight Ahead RWhat Do You Expect? RData Distributions R

Stretching and Shrinking RComparing and Scaling RMoving Straight Ahead RWhat Do You Expect? RData Distributions R


Accentuate the Negative IMData Distributions R

Accentuate the Negative IMMoving Straight Ahead R

Accentuate the Negative IM

Accentuate the Negative IMData Distributions R

Accentuate the Negative IMWhat Do You Expect? R

Accentuate the Negative IMMoving Straight Ahead RWhat Do You Expect? R














Shapes of Algebra R






Data InvestigationNote: Opportunities for students to question, collect, analyze, and interpret data occur in almost every unit.





Irrational Numbers

models

pi

Pythagorean Theorem

square roots

estimating

Real Numbers

defined

Order of Operations

Properties

distributive

commutative

associative

collecting data

Covering and Surrounding I

How Likely Is It? I

Covering and Surrounding IM

Bits and Pieces III RHow Likely Is It? R

Prime Time I

Bits and Pieces II ICovering and

Surrounding IBits and Pieces III I

Prime Time I

How Likely Is It? IData About Us IM

Filling and Wrapping I

Variables and Patterns RFilling and Wrapping R




Accentuate the Negative I

Variables and Patterns RMoving Straight Ahead RFilling and Wrapping RWhat Do You Expect? RData Distributions R



Looking for Pythagoras IMShapes of Algebra R










Say It with Symbols IM

Thinking withMathematical Models R



Data RepresentationNote: Opportunities for students to create or use tables occur in almost every unit.

Data Analysis and Probability (cont.)



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analyze data

interpret data

samples

randomness

draw conclusions/makepredictions

compare data

conduct surveys

evaluate methods ofsampling

line plots

single, double, stacked bargraphs

stem-and-leaf plots

Bits and Pieces III IHow Likely Is It? IData About Us IM


How Likely Is It? IData About Us I

How Likely Is It? IM






Bits and Pieces I IBits and Pieces III IHow Likely Is It? IData About Us IM

Data About Us IM

Variables and Patterns RComparing and Scaling RMoving Straight Ahead RFilling and Wrapping RWhat Do You Expect? RData Distributions R

Variables and Patterns RComparing and Scaling RMoving Straight Ahead RFilling and Wrapping RWhat Do You Expect? RData Distributions R

What Do You Expect? I

What Do You Expect? R

What Do You Expect? RComparing and Scaling RMoving Straight Ahead RFilling and Wrapping RData Distributions R

Variables and Patterns IComparing and Scaling IMoving Straight Ahead IFilling and Wrapping RWhat Do You Expect? IData Distributions IM

Data Distributions I

What Do You Expect? IData Distributions I

Variables and Patterns RWhat Do You Expect? RData Distributions R

Comparing and Scaling RData Distributions R








Samples and Populations IM








Samples and PopulationsIM








coordinate graphs

tables

frequency tables

circle graphs (pie charts)

histograms

box-and-whisker plots(box plots)

scatter plots

analyze trends/trend lines

decide on appropriatenessand effectiveness

Describing Data

mode

median

mean (average)

range


Data About Us I

Shapes and Designs RCovering and

Surrounding IData About Us IM

Data About Us IMHow Likely Is It? R

Bits and Pieces III IMData About Us R

Data About Us I

Data About Us I


Data About Us IM

Data About Us IM

Bits and Pieces III IData About Us IM

Data About Us IM

Variables and Patterns IMMoving Straight Ahead RFilling and Wrapping RData Distributions R

Variables and Patterns IMComparing and Scaling RMoving Straight Ahead RFilling and Wrapping RData Distributions R



Data Distributions IM

Variables and Patterns IComparing and Scaling IMoving Straight Ahead IData Distributions I



Variables and Patterns RAccentuate the Negative RData Distributions R

Variables and Patterns RAccentuate the Negative RData Distributions R

Accentuate the Negative RData Distributions R




Say It With Symbols RShapes of Algebra RSamples and Populations R




Cubes RSay It With Symbols RSamples and Populations R






Thinking withMathematical Models IM





Thinking withMathematical Models IM






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outliers

choose the best datadescriptor

shape of data

data distribution

quartiles, interquartilerange (IQR)

maximum, minimum

Probability

predicting, computing

equally and unequallylikely outcomes

certain, possible,impossible events

experimental

theoretical

dependent andindependent events

expected value

fair and unfair games

lists, charts, tree diagrams,area models

counting techniques

simulations/experiments

Data About Us I

Data About Us IM

Data About Us I

Data About Us I







How Likely Is It? I


How Likely Is It? I

How Likely Is It? I


Data Distributions I





Variables and Patterns RStretching and Shrinking RComparing and Scaling RWhat Do You Expect? RData Distributions R



Variables and Patterns RComparing and Scaling RWhat Do You Expect? R


What Do You Expect? IM



Variables and Patterns IStretching and Shrinking IComparing and Scaling IWhat Do You Expect? IM


Variables and Patterns RMoving Straight Ahead RWhat Do You Expect? R










Shapes of Algebra R





Measurement



Angles

estimating

measuring

of similar polygons

triangle, special right

Perimeter

polygons

circles (circumference)

irregular polygons

constant perimeter,changing area

relationships of perimetersof similar figures

Shapes and Designs IMBits and Pieces III RHow Likely Is It? R

Shapes and Designs IMBits and Pieces III RHow Likely Is It? R

Shapes and Designs ICovering and

Surrounding IMBits and Pieces III R


Bits and Pieces III R




Variables and Patterns RStretching and Shrinking RComparing and Scaling RData Distributions R

Stretching and Shrinking RComparing and Scaling R

Stretching and Shrinking IM


Variables and Patterns RStretching and Shrinking RMoving Straight Ahead RFilling and Wrapping R


Stretching and Shrinking RFilling and Wrapping R

Variables and Patterns RMoving Straight Ahead R



Looking for Pythagoras RKaleidoscopes, Hubcaps,

and Mirrors R


and Mirrors R


and Mirrors R








Kaleidoscopes, Hubcaps,and Mirrors R



and Mirrors RSay It with Symbols R







Shapes of Algebra R

Measurement (cont.)



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Area

rectangles

triangles

parallelograms

circles

irregular polygons

trapezoids

constant area, changingperimeter

Prime Time IBits and Pieces I IShapes and Designs RBits and Pieces II ICovering and








Bits and Pieces I IBits and Pieces II ICovering and




Variables and Patterns RStretching and Shrinking RComparing and Scaling RMoving Straight Ahead RFilling and Wrapping RWhat Do You Expect? R


Variables and Patterns RStretching and Shrinking RFilling and Wrapping R


Stretching and Shrinking RFilling and Wrapping R

Variables and Patterns R


















Frogs, Fleas, and PaintedCubes IM




Measurement (cont.)



relationships of areas ofsimilar figures

Volume

models

cubes

prisms

cylinders

cones

pyramids

spheres

irregular figures

similar figures and scalefactors

Data About Us R


Comparing and Scaling RMoving Straight Ahead RFilling and Wrapping RWhat Do You Expect? R

Filling and Wrapping IMWhat Do You Expect? R













and Mirrors RShapes of Algebra R




Cubes R



Cubes R






Looking for Pythagoras RSay It with Symbols R

Looking for Pythagoras RSay It with Symbols R



Measurement (cont.)



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effects when thedimensions of a solidare changedproportionally

Surface Area

flat patterns (nets) forsolid figures

models

cubes

prisms

cylinders

pyramids

irregular figures

formulas

Finding Missing Lengths

similar figures using ratiosor scale factor

Covering and Surrounding R

How Likely Is It? R

How Likely Is It? I










Comparing and Scaling RMoving Straight Ahead R















Cubes RSay It with Symbols RShapes of Algebra R




Looking for Pythagoras I

Say It with Symbols I



and Mirrors R

Geometry

Measurement (cont.)



on a coordinate grid

using the PythagoreanTheorem

Indirect

similar triangles


Units of Measure

converting within thesame measurementsystem

converting amongcustomary and metric

Line

parallel lines

perpendicular lines

transversals

midpoints

Angles

classifying

congruent

Shapes and Designs R

Shapes and Designs IM










Moving Straight Ahead RFilling and Wrapping R

Comparing and Scaling RMoving Straight Ahead RData About Us R

Stretching and Shrinking RMoving Straight Ahead R






Looking for Pythagoras IMGrowing, Growing,




and Mirrors R


and Mirrors R

Looking for Pythagoras RShapes of Algebra R






and Mirrors R

Geometry (cont.)



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complementary andsupplementary

of a polygon

n-gon angle sum

on a circular grid

Polygons

properties of

regular

tilings/tessellations

diagonals

triangles, classifying

quadrilaterals, classifying

similar

congruent


Shapes and Designs IMBits and Pieces III R









How Likely Is It? I



Variables and Patterns RStretching and Shrinking R





Stretching and Shrinking RMoving Straight Ahead RFilling and Wrapping R

Stretching and Shrinking RMoving Straight Ahead R


Moving Straight Ahead RFilling and Wrapping R



and Mirrors R


Shapes of Algebra R







and Mirrors R


and Mirrors R






Looking for Pythagoras IKaleidoscopes, Hubcaps,

and Mirrors IM

Geometry (cont.)



enlarging and shrinking(dilations)

drawing on coordinategrid

Pythagorean Theorem

Circles

Relationship betweenradius/diameter/circumference

Three-Dimensional Figures

cubes

prisms

cylinders/spheres/cones

pyramids

base plans/top, side, andfront views

spatial visualization



How Likely Is It? I

















Looking for Pythagoras IMSay It with Symbols R






Cubes R



and Mirrors R







Algebra

Geometry (cont.)



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Transformations

reflections

rotations

translations

combinations oftransformations

symmetry

constructing symmetricfigures

dilations

algebraic rules/propertiesfor

on a coordinate plane

Patterns

look for and describe

Shapes and Designs I





Data About Us I

Accentuate the Negative I

Stretching and Shrinking IAccentuate the Negative I


Accentuate the Negative R


Stretching and Shrinking IAccentuate the Negative I

Variables and Patterns IMComparing and Scaling RAccentuate the Negative RMoving Straight Ahead RFilling and Wrapping RData Distributions R

Frogs, Fleas, and PaintedCubes I

Kaleidoscopes, Hubcaps,and Mirrors IM

Shapes of Algebra R




Shapes of Algebra R




Shapes of Algebra R


Shapes of Algebra R


Shapes of Algebra R


Looking for Pythagoras IKaleidoscopes, Hubcaps,

and Mirrors IMShapes of Algebra R




Cubes RSay It with Symbols RSamples and Populations R

Algebra (cont.)



numerical

geometric

rates of change

rules

analyzing and makingpredictions from

functions

Variables/Expressions

dependent, independent


Data About Us I


Shapes and Designs ICovering and

Surrounding I




Variables and Patterns RAccentuate the Negative RFilling and Wrapping R

Variables and Patterns IComparing and Scaling IMAccentuate the Negative RMoving Straight Ahead RFilling and Wrapping R

Variables and Patterns IMComparing and Scaling RMoving Straight Ahead RFilling and Wrapping R


Variables and Patterns IMComparing and Scaling RMoving Straight Ahead RFilling and Wrapping R

Variables and Patterns IMMoving Straight Ahead R

























Algebra (cont.)



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coefficients

like, constant, linear terms

evaluating

equivalent

factored form/ expandedform

Relationships

continuous/ discrete

linear

nonlinear

Data About Us I


Data About Us I

Variables and Patterns IComparing and Scaling IMoving Straight Ahead IM


Variables and Patterns IMoving Straight Ahead IM

Variables and Patterns IMoving Straight Ahead I



Variables and Patterns IMAccentuate the Negative R

Variables and Patterns IComparing and Scaling IAccentuate the Negative IMoving Straight Ahead IMFilling and Wrapping RData Distributions R

Variables and Patterns IMoving Straight Ahead IFilling and Wrapping IData Distributions I



Cubes RSay It with Symbols RShapes of Algebra R





Growing, Growing,Growing I


Say It with Symbols IMShapes of Algebra R







Say It with Symbols RShapes of Algebra RSamples and Populations R

Thinking WithMathematical Models IM




Algebra (cont.)



inverse

exponential growth/exponential decay

quadratic

slope

slopes of perpendicularlines/parallel lines

Equations, Linear

tables for

graphs for

fitting to a graph





Moving Straight Ahead IM


Variables and Patterns IComparing and Scaling IAccentuate the Negative IMoving Straight Ahead IM

Moving Straight Ahead IMData Distributions R











Growing RShapes of Algebra R





Shapes of Algebra R









Algebra (cont.)



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Slope-intercept formy � mx � b

Standard formax � by � c

writing

solving with tables

solving by graphing

solving symbolically

solving with graphingcalculator

Variables and Patterns IMoving Straight Ahead IMData Distributions R

Variables and Patterns IMComparing and Scaling RMoving Straight Ahead R








Shapes of Algebra IM





















Algebra (cont.)



solving systems of

formulate given a problemsituation (and viceversa)

Equations, Quadratic

writing

graphs for

solving

finding roots

inequalities

Equations, Nonlinear

models

cubic






Thinking WithMathematical Models I


Say It with Symbols IShapes of Algebra IM













Shapes of Algebra I






Say It with Symbols I

Algebra (cont.)



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exponential

inverse

of circles

Graphing

explore shapes of graphs

ordered pairs

polar coordinates

equations

inequalities

systems of linearinequalities

Data About Us I

Data About Us I


Variables and Patterns I



Variables and Patterns IMStretching and Shrinking RComparing and Scaling RAccentuate the Negative RMoving Straight Ahead R

Variables and Patterns IComparing and Scaling IMoving Straight Ahead IMData Distributions R















and Mirrors RShapes of Algebra RSamples and Populations R






Say It with Symbols IShapes of Algebra IM


Algebra (cont.)



using a table

with a graphing calculator

slope

x-intercept

y-intercept

maximum and minimum

systems of equations


Data About Us I





























Shapes of Algebra R

Problem Solving Strategies As a problem solving curriculum, every unit helps students develop a variety ofstrategies for solving problems such as; building models, making lists and tables,drawing diagrams, and solving simpler problems.

Problem Solving Skills



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drawing a diagram

looking for a pattern

making a graph

making a table

simulating a problem

try, check, revise

Bits and Pieces I RBits and Pieces II RBits and Pieces III RHow Likely Is It? R

Bits and Pieces I RShapes and Designs RBits and Pieces II RCovering and



Data About Us R


Surrounding RBits and Pieces III RData About Us R

Prime Time IMCovering and

Surrounding RHow Likely Is It? R

Prime Time IMBits and Pieces I RShapes and Designs RBits and Pieces II RBits and Pieces III RHow Likely Is It? RData About Us R

Stretching and Shrinking RAccentuate the Negative RFilling and Wrapping RWhat Do You Expect? R

Variables and Patterns RStretching and Shrinking RComparing and Scaling RMoving Straight Ahead RFilling and Wrapping RData Distributions R

Variables and Patterns RStretching and Shrinking RComparing and Scaling RMoving Straight Ahead RData Distributions R

Variables and Patterns RComparing and Scaling RMoving Straight Ahead RFilling and Wrapping RData Distributions R

Variables and Patterns RMoving Straight Ahead RWhat Do You Expect? RData Distributions R

Stretching and Shrinking RComparing and Scaling RAccentuate the Negative RMoving Straight Ahead R




and Mirrors R





















Problem Solving Skills (cont.)


Communication Student explanations are requested throughout in Problems, in the ACE, and in teacherquestioning from the teacher’s guides.


write an equation

Reasonableness

justify answers

make and test conjectures

reason from graphs

recognize patterns

validate conclusions usingmathematical properties


Surrounding R

Prime Time IMBits and Pieces I RBits and Pieces II RBits and Pieces III RHow Likely Is It? RData About Us R

Prime Time IMBits and Pieces I RShapes and Designs RBits and Pieces II RBits and Pieces III RHow Likely Is It? RData About Us R


Data About Us R

Bits and Pieces I RShapes and Designs RBits and Pieces II RBits and Pieces III RHow Likely Is It? R

Prime Time IMBits and Pieces I RShapes and Designs RBits and Pieces II RCovering and


Variables and Patterns RComparing and Scaling RAccentuate the Negative RMoving Straight Ahead RFilling and Wrapping R

Variables and Patterns RStretching and Shrinking RComparing and Scaling RAccentuate the Negative RMoving Straight Ahead RFilling and Wrapping RWhat Do You Expect? R

Stretching and Shrinking RComparing and Scaling RAccentuate the Negative RMoving Straight Ahead RFilling and Wrapping RWhat Do You Expect? RData Distributions R

Variables and Patterns RStretching and Shrinking RComparing and Scaling RAccentuate the Negative RMoving Straight Ahead RData Distributions R

Variables and Patterns RStretching and Shrinking RComparing and Scaling RAccentuate the Negative RMoving Straight Ahead RFilling and Wrapping RData Distributions R

Variables and Patterns RStretching and Shrinking RComparing and Scaling RAccentuate the Negative RMoving Straight Ahead RFilling and Wrapping RWhat Do You Expect? RData Distributions R











and Mirrors RShapes of Algebra RSamples and Populations R














and Mirrors RSay It with Symbols RShapes of Algebra RSamples and Populations R


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Connected Mathematics is committed to thedevelopment of mathematical skills—skills that

are much more than just quickness with paper-and-pencil algorithms. The overarching goal of CMPdiscussed on page 2 makes this commitment to skill:

All students should be able to reason andcommunicate proficiently in mathematics.They should have knowledge of and skill in the use of the vocabulary, forms ofrepresentation, materials, tools, techniques,and intellectual methods of the discipline ofmathematics, including the ability to defineand solve problems with reason, insight,inventiveness, and technical proficiency.

In Connected Mathematics, students developunderstanding of algorithms and strategies forcomputing and estimating in a variety of ways.They learn to recognize when an algorithm orstrategy applies to a new context and when theycan build on the skills and strategies they know inorder to develop new strategies. In theseprocesses, students practice skills as an ongoingactivity throughout the curriculum.

Students need to know how and when to usepaper-and pencil algorithms, mental computation,calculator procedures, and estimation strategies.They need to recognize when an exact answer isrequired and when an approximate answer issufficient, and they need a variety of methods forfinding an answer. In some situations anapproximate answer is sufficient and in thesesituations a paper-and-pencil algorithm may notbe the most efficient (or practical) method. In Bitsand Pieces III Problem 4.2, Question D(1)students estimate to find a 20% tip for $7.93. It ismore efficient for most students to estimate that10% is about 80 cents so 20% is $1.60 than tomultiply 0.20 times 7.93 to obtain $1.57.

Students also need to know methods forjudging the reasonableness of an answer. Forexample, to estimate or judge the reasonableness

of the answer to the sum and , students might

argue that is close to but less than , and

is more than but less than . Thus, the answer is

more than but less than 1, or about .

Skills with the four basic operations onfractions are developed and maintainedthroughout the curriculum. Students should beable to add two simple fractions quickly byfinding a common denominator, but they shouldalso understand why this algorithm works.Connected Mathematics helps students build a strong foundation for the development ofaddition, subtraction, multiplication, and divisionof fractions. In this phase, the essential buildingblocks of equivalent fractions, meaning offractions, models and representation of fractions are developed and used. For example,in Bits and Pieces I, students develop anunderstanding of equivalences. In Problem 2.2students represent fractions on a number lineand use the number line to develop a method offinding equivalent fractions. A portion of thisproblem is shown on the next page.

34

12

12

14

13

12

25

13

25

C. The sales tax in Kadisha’s state is 5%. Kadisha says she computes a 15% tip by multiplying the tax shown on herbill by three. For a bill with a tax charge of $0.38,Kadisha’s tip is $0.38 3 3 = $1.14.

1. Why does Kadisha’s method work?

2. Use a similar method to compute a 20% tip. Explain.

D. When people leave a 15% or 20% tip, they often round up to the nearest multiple of 5 or 10 cents. For example,in Question C, Kadisha might leave a tip of $1.15 rather than $1.14.

1. If Kadisha always rounds up, what is a 20% tip on her bill?

ITEM

Food

5% Tax

TOTAL

AMOUNT

$7.55

.38

$7.93

Bits and Pieces III • page 53

Students continue to use equivalent fractions todevelop understanding of probability, linearfunctions, and proportional reasoning and todevelop algorithms for the operations of fractions.

In Bits and Pieces II, students develop and usealgorithms to add, subtract, multiply, and dividefractions. After completing this unit, students aregiven numerous opportunities to use their fractionknowledge and skills in operating with fractions tosolve problems in number, geometry, measurement,data analysis probability, and algebra. Below are two examples which illustrate how studentscontinue to use fraction multiplication.

Example 1 Using Fraction Multiplication SkillsIn Bits and Pieces III, the algorithm of multiplyingdecimals is connected to multiplying fractions. Herestudents use their fraction multiplication skills as astrategy to multiply decimals.

Example 2 Using Fraction Multiplication SkillsStudents continue to use their understanding ofequivalent fractions to build and consolidate theirunderstanding of new ideas in new contexts. Thesequence of the units is carefully chosen with this in mind. For example, in Stretching and Shrinking,students use fractional scale factors to find andidentify similar figures.

A portion of an Application question and asolution are given below to illustrate continueduse of fraction multiplication.

Possible Solution One pair of similar rectanglesis Rectangles M and Q. The scale factor from

Rectangle Q to Rectangle M is . This means

that scaling the dimensions of Rectangle Q

by results in the dimensions of Rectangle M.

Use fraction multiplication to verify that this is

the correct scale factor: 3 cm � = 2 cm.

As illustrated with rational numbers above,a similar development is given to integers inAccentuate the Negative and irrational numbers in Looking for Pythagoras. As students movethrough the curriculum, they expand their workwith the real number system and continue topractice operating with real numbers in a varietyof situations.

23

23

23

Problem 2.2 Finding Equivalent Fractions

A. 1. On a number line like the one below, carefully label marks thatshow where and are located.

2. Use the same number line. Mark the point that is halfway between0 and and the point that is halfway between and 1.

3. Label these new marks with appropriate fraction names.

4. What are additional ways to label and Explain.

5. Use the same number line. Mark halfway between each of themarks that were already made.

6. Label the new marks on your number line. Add additional names tothe marks that were already named.

7. Write three number sentences that show equivalent fractions onyour number line. Here is an example: =

8. Write two number sentences to show fractions that are equivalentto

B. 1. On your number line, the distance between the mark and the

1 mark is of a unit. The distance between the 0 mark and the

mark on your number line is of a unit. Name two other

fractions that are of a unit apart on your number line.

2. What is the distance between the and marks on your numberline? How do you know?

3. Name at least two other fraction pairs that are the same distanceapart as and

4. Describe the distance between and in two ways.

C. 1. Here is another number line with a mark for and for What isthe distance between these two marks? On a copy of the numberline, show how you know.

2. Suppose a number line is marked with tenths. Which marks can alsobe labeled with fifths?

0 135

710

35 .

710

56

23

12 .

13

12

13

13

13

13

12

12

912 .

36 .R

12Q

23 ?

12 ,

13 ,

23

13

0 112

23

13

Bits and Pieces I • page 22

For Exercises 23 and 24 on page 87, use the rectangles below. The rectanglesare not shown at actual size.

6 cm

L4 cm

7 cm

P1 cm

8 cm

R2 cm3 cm

Q 3 cm

2 cm

2 cm

M3 cm

2 cm

N

Stretching and Shrinking • page 86

23. Multiple Choice Which pair of rectangles is similar?

A. L and M B. L and Q C. L and N D. P and R

24. a. Find at least one more pair of similar rectangles.

b. For each similar pair, find both the scale factor relating the largerrectangle to the smaller rectangle and the scale factor relating thesmaller rectangle to the larger rectangle.

c. For each similar pair, find the ratio of the area of the largerrectangle to the area of the smaller rectangle.

Stretching and Shrinking • page 87

To find the product of 0.3 3 2.3, you can use equivalent fractions.

0.3 = and 2.3 = or , so 0.3 3 2.3 = 3

• What is the product written as a fraction?

• What is the product written as a decimal?

• How can knowing the product as a fraction help you write the productin decimal form?

2310

310

23102 3

10310

Getting Ready for Problem 2.1




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Proportional reasoning skills are essential to a student’s mathematical development. Manyproblems in K–12 mathematics and beyond call forstudents to utilize proportional reasoning skills. It is the core idea in being able to write equivalentfractions, in all operations with fractions, in makingsense of scales, in similarity, in size transformations,and in solving some linear equations. Students can learn to mimic these skills by learning a newstrategy for every skill, but their learning will bemuch more powerful if they can see an underlyingidea and its connections. In the following problemfrom Comparing and Scaling, students aredeveloping proportional reasoning skills. As they solve Problem 2.1, they build on their priorknowledge of fractions and rational numbers.

The students of Team 1 used a part-to-wholestrategy. They showed that they knew how to findand use percents to make comparisons.

The students on Team 3 used a part-to-partstrategy. They gave the ratio in fraction form andthen made the numerators the same to makecomparisons. They recognized that the smallestdenominator shows the mix that is the most“orangey,” because it uses the least water per canof concentrate. The work shows that students haveflexibility in using fractions, decimals, and percents

to make their comparisons of ratios.

Problem 2.1

2.1 Mixing Juice

Julia and Mariah attend summer camp. Everyone at the camp helps withthe cooking and cleanup at meal times.

One morning, Julia and Mariah make orange juice for all the campers.They plan to make the juice by mixing water and frozen orange-juiceconcentrate. To find the mix that tastes best, they decide to test some mixes.

Developing Comparison Strategies

A. Which mix will make juice that is the most “orangey”? Explain.

B. Which mix will make juice that is the least “orangey”? Explain.

C. Which comparison statement is correct? Explain.

of Mix B is concentrate. of Mix B is concentrate.

D. Assume that each camper will get cup of juice.

1. For each mix, how many batches are needed to make juice for240 campers?

2. For each mix, how much concentrate and how much water areneeded to make juice for 240 campers?

E. For each mix, how much concentrate and how much water are neededto make 1 cup of juice?


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Mix A2 cups

concentrate3 cups

cold water

Mix B5 cups

concentrate9 cups

cold water

Mix C1 cup

concentrate2 cups

cold water

Mix D3 cups

concentrate5 cups

cold water

Comparing and Scaling • page 19

Team 1 Student Work

Team 3 Student Work


Knowing when to use a particular operation isalso a skill. Bits and Pieces III is designed toprovide experiences in building algorithms for thefour basic operations with decimals, as well asopportunities for students to consider when suchoperations are useful in solving problems. Forexample, what features of a problem indicate to thestudent that division will help solve it? Buildingthis kind of thinking and reasoning supports thedevelopment of skill with the algorithms. InProblem 3.1 B. of Bits and Pieces III, students needto interpret the problem situation and determinewhat decimal operation will solve the problem. Thisskill is practiced as students begin to develop anduse algorithms for decimal operations.

Connected Mathematics recognizes that studentsmust have an opportunity to practice skills in avariety of situations throughout the course of theirmathematical career. The Connections feature ofthe ACE Exercises (discussed more on page 43)offers a way for student to continue practicing skillslearned in previous units. As a problem-centeredcurriculum, Connected Mathematics providesstudents the opportunity to use their skills in a widerange of situations that promote higher-orderthinking and help students develop problem-solvingskills essential to their mathematical future inschool and in life. Students explore both problemsituations that are purely mathematical and othersthat are real world. They learn to use what theyknow to solve contextualized situations and to docomputation in “naked” number situations.

B. Examine each situation. Decide what operation to use and thenestimate the size of the answer.

1. Ashley eats five 5.25-ounce slices of watermelon in a contest at thepicnic. How many ounces of watermelon does she eat?

2. Stacey needs $39.99 for a pair of sneakers. She has $22.53 in hersavings and a $15 check from babysitting. Can she buy the shoes?

3. Li Ming’s allowance for transportation is $12.45. How many timescan she ride the bus if it costs $0.75 a trip?




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Traditionally, an Algebra 1 course focuses onrules or specific strategies for solving

standard types of symbolic manipulationproblems—usually to simplify or combineexpressions or solve equations. For many students,symbolic rules for manipulation are memorizedwith little attempt to make sense of why theywork. They retain the ideas for only a short time.There is little evidence that traditional experienceswith algebra help students develop the ability to“read” information from symbolic expression orequations, to write symbolic statements torepresent their thinking about relationships in aproblem, or to meaningfully manipulate symbolicexpressions to solve problems.

In the United States, algebra is generally taughtas a stand-alone course rather than as a strandintegrated and supported by other strands. Thispractice is contrary to curriculum practices inmost of the rest of the world. Today, there is agrowing body of research that leads many UnitedStates educators to believe that the developmentof algebraic ideas can and should take place overa long period of time and well before the first yearof high school. Developing algebra across thegrades and integrating it with other strands helpsstudents become proficient with algebraicreasoning in a variety of contexts and gives thema sense of the coherence of mathematics.

Developing Algebraic Reasoningin Connected MathematicsThe Connected Mathematics program aims toexpand student views of algebra beyond symbolicmanipulation and to offer opportunities for studentsto apply algebraic reasoning to problems in manydifferent contexts throughout the course of thecurriculum. The development of algebra inConnected Mathematics is consistent with therecommendations in the NCTM Principles andStandards for School Mathematics 2000 and moststate frameworks.

Algebra in Connected Mathematics focuseson the overriding objective of developingstudents’ ability to represent and analyzerelationships among quantitative variables.From this perspective, variables are notletters that stand for unknown numbers.Rather they are quantitative attributes ofobjects, patterns, or situations that changein response to change in other quantities.The most important goals of mathematicalanalysis in such situations are understandingand predicting patterns of change invariables. The letters, symbolic equations,and inequalities of algebra are tools forrepresenting what we know or what wewant to figure out about a relationshipbetween variables. Algebraic procedures for manipulating symbolic expressions intoalternative equivalent forms are also meansto the goal of insight into relationshipsbetween variables. To help students acquirequantitative reasoning skills, we have foundthat almost all of the important tasks towhich algebra is usually applied can develop naturally as aspects of thisendeavor. (Fey, Phillips 2005)


There are eight units which focus formally onalgebra. Titles and descriptions of the mathematicalcontent for these units are:

Variables and PatternsIntroducing AlgebraRepresenting and analyzing relationships betweenvariables, including tables, graphs, words, andsymbols

Moving Straight AheadLinear Relationships Examining the pattern of change associated withlinear relationships; recognizing, representing, andanalyzing linear relationships in tables, graphs,words and symbols; solving linear equations;writing equations for linear relationships

Thinking With Mathematical ModelsLinear and Inverse VariationIntroducing functions and modeling; finding theequation of a line; representing and analyzinginverse functions

Looking for PythagorasThe Pythagorean Theorem Exploring square roots; exploring and using thePythagorean Theorem, making connections in thecoordinate plane among coordinates, slope,and distance

Growing, Growing, GrowingExponential Relationships Examining the pattern of change associated withexponential relationships; comparing linear and exponential patterns of growths; recognizing,representing, and analyzing exponential growthand decay in tables, graphs, words and symbols;developing rules of exponents

Frogs, Fleas, and Painted CubesQuadratic Relationships Examining the pattern of change associated withquadratic relationships and comparing thesepatterns to linear and exponential patterns,recognizing, representing, and analyzing quadraticfunctions in tables graphs, words, and symbols;determining and predicting important features ofthe graph of a quadratic functions, such as themaximum/minimum point, line of symmetry, andthe x-and y-intercepts; factoring simple quadraticexpressions

Say It With SymbolsMaking Sense of SymbolsWriting and interpreting equivalent expressions;combining expressions; looking at the pattern of

change associated with an expression; solvinglinear and quadratic equations

Shapes of AlgebraLinear Systems and Inequalities Exploring coordinate geometry; solvinginequalities; solving systems of linear equationsand linear inequalities

Early Experiences WithAlgebraic ReasoningEven though the first primarily algebra unitoccurs at the start of seventh grade, students studyrelationships among variables in grade 6.

There also are opportunities in 6th and in 7thgrade for students to begin to examine andformalize patterns and relationships in words,graphs, tables, and with symbols.

• In Shapes and Designs (Grade 6), studentsexplore the relationship between the numberof sides of a polygon and the sum of theinterior angles of the polygon. They develop a rule for calculating the sum of the interiorangle measures of a polygon with N sides.

• In Covering and Surrounding (Grade 6),students estimate the area of three different-size pizzas and then relate the area to the price.This problem requires students to consider tworelationships: one between the price of a pizzaand its area and the other between the area ofa pizza and its radius. Students also developformulas and procedures—stated in words andsymbols—for finding areas and perimeters ofrectangles, parallelograms, triangles, and circles.

• In Bits and Pieces I, II and III (Grade 6),students learn, through fact families, thataddition and subtraction are inverseoperations and that multiplication anddivision are inverse operations. This is afundamental idea in equation solving. Theyuse these ideas to find a missing factor oraddend in a number sentence.

• In Data About Us (Grade 6), students repre-sent and interpret graphs for the relationshipbetween variables, such as the relationshipbetween length of an arm span and height ofa person, using words, tables, and graphs.

• In Accentuate the Negative (Grade 7),students explore properties of real numbers,including the commutative, distributive, andinverse properties. They use these propertiesto find a missing addend or factor in anumber sentence.


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• In Filling and Wrapping (Grade 7), studentsdevelop formulas and procedures—stated inwords and symbols—for finding surface areaand volume of rectangular prisms, cylinders,cones, and spheres.

Developing FunctionsIn a problem-centered curriculum, quantities(variables) and the relationships betweenvariables naturally arise. Representing andreasoning about patterns of change becomes away to organize and think about algebra. Lookingat specific patterns of change and how this changeis represented in tables, graphs, and symbols leadsto the study of linear, exponential, and quadraticrelationships (functions).

Linear FunctionsIn Moving Straight Ahead, students investigatelinear relationships. They learn to recognize linearrelationships from patterns in verbal, tabular,graphical, or symbolic representations. They alsolearn to represent linear relationships in a variety ofways and to solve equations and make predictionsinvolving linear equations and functions.

Problem 1.3 illustrates the kinds of questionsstudents are asked when they meet a new type ofrelationship or function—in this case, a linearrelationship. In this problem students are lookingat three pledge plans that students suggest for awalkathon.

Whereas many algebra texts choose to focusalmost exclusively on linear relationships, inConnected Mathematics students build on theirknowledge of linear functions to investigate otherpatterns of change. In particular, students exploreinverse variation relationships in Thinking WithMathematical Models, exponential relationships in Growing, Growing, Growing, and quadraticrelationships in Frogs, Fleas, and Painted Cubes.

Examples are given below which illustrate the different types of functions studentsinvestigate and some of the questions they areasked about these functions. By contrasting linearrelationships with exponential and otherrelationships, students develop deeperunderstanding of linear relationships.

Inverse FunctionsIn Thinking With Mathematical Models, studentsare introduced to inverse functions.

Exponential FunctionsIn Growing, Growing, Growing, students are giventhe context of a reward figured by placing coinscalled rubas on a chessboard in a particular patternwhich is exponential. The coins are placed on thechessboard as follows.

Place 1 ruba on the first square of achessboard, 2 rubas on the second square,4 on the third square, 8 on the fourth square, and so on, until you have covered all 64 squares. Each square should havetwice as many rubas as the previous square.

Problem 1.3

• Leanne’s sponsors will pay $10 regardless of how far she walks.

• Gilberto’s sponsors will pay $2 per kilometer (km).

• Alana’s sponsors will make a $5 donation plus 50¢ per kilometer.

Using Linear Relationships

A. 1. Make a table for each student’s pledge plan, showing the amount ofmoney each of his or her sponsors would owe if he or she walkeddistances from 0 to 6 kilometers. What are the dependent andindependent variables?

2. Graph the three pledge plans on the same coordinate axes. Use adifferent color for each plan.

3. Write an equation for each pledge plan. Explain what informationeach number and variable in your equation represents.

4. a. What pattern of change for each pledge plan do you observe inthe table?

b. How does this pattern appear in the graph? In the equation?

B. 1. Suppose each student walks 8 kilometers in the walkathon. Howmuch does each sponsor owe?

2. Suppose each student receives $10 from a sponsor. How manykilometers does each student walk?

3. On which graph does the point (12, 11) lie? What information doesthis point represent?

4. In Alana’s plan, how is the fixed $5 donation represented in

a. the table? b. the graph? c. the equation?

Moving Straight Ahead • page 9

Problem 2.4 Intersecting Linear Models

A. Use the table to find a linear equation relating the probability of rain p to

1. Saturday attendance AB at Big Fun.

2. Saturday attendance AG at Get Reel.

B. Use your equations from Question A to answer these questions. Showyour calculations and explain your reasoning.

1. Suppose there is a 50% probability of rain this Saturday. What is theexpected attendance at each attraction?

2. Suppose 400 people visited Big Fun one Saturday. Estimate theprobability of rain on that day.

3. What probability of rain would give a predicted Saturdayattendance of at least 360 people at Get Reel?

4. Is there a probability of rain for which the predicted attendance isthe same at both attractions? Explain.

Thinking with Mathematical Models • page 32


In this problem students use tables, graphs, andequations to examine exponential relationships anddescribe the pattern of change for this relationship.

Quadratic FunctionsIn Problem 1.3 from Frogs, Fleas and PaintedCubes, students use tables, graphs, and equationsto examine quadratic relationships and describethe pattern of change for this relationship.

As students explore a new type of relationship,whether it is linear, quadratic, inverse, orexponential, they are asked questions like these:

• What are the variables? Describe the pattern ofchange between the two variables.

• Describe how the pattern of change can beseen in the table, graph, and equation.

• Decide which representation is the mosthelpful for answering a particular question.(see Question D in Problem 1.3 in the firstcolumn).

• Describe the relationships between thedifferent representations (table, graph, andequation).

• Compare the patterns of change for differentrelationships. For example, compare thepatterns of change for two linear relationships,or for a linear and an exponential relationship.

Developing Symbolic ReasoningAfter students have explored importantrelationships and their associated patterns ofchange and ways to represent these relationships,the emphasis shifts to symbolic reasoning.

Equivalent ExpressionsStudents use the properties of real numbers tolook at equivalent expressions and the informationeach expression represents in a given context and to interpret the underlying patterns that asymbolic statement or equation represents. Theyexamine the graph and table of an expression aswell as the context the expression or statementrepresents. The properties of real numbers areused extensively to write equivalent expressions,combine expressions to form new expressions,predict patterns of change, and to solve equations.Say It With Symbols pulls together the symbolicreasoning skills students have developed through a focus on equivalent expressions. It also continuesto explore relationships and patterns of change.

Problem 1.1 in Say It With Symbols introducesstudents to equivalent expressions.

Problem 1.2 Representing Exponential Relationships

A. 1. Make a table showing the number of rubas the king will place onsquares 1 through 10 of the chessboard.

2. How does the number of rubas change from one square to thenext?

B. Graph the (number of the square, number of rubas) data for squares 1to 10.

C. Write an equation for the relationship between the number of thesquare n and the number of rubas r.

D. How does the pattern of change you observed in the table show up inthe graph? How does it show up in the equation?

E. Which square will have 230 rubas? Explain.

F. What is the first square on which the king will place at least one millionrubas? How many rubas will be on this square?

Growing, Growing, Growing • page 7

Problem 1.3 Writing an Equation

A. Consider rectangles with a perimeter of 60 meters.

1. Sketch a rectangle to represent this situation. Label one side O.Label the other sides in terms of O.

2. Write an equation for the area A in terms of O.

3. Use a calculator to make a table for your equation. Use your table to estimate the maximum area. What dimensions correspondto this area?

4. Use a calculator or data from your table to help you sketch a graphof the relationship between length and area.

5. How can you use your graph to find the maximum area possible?How does your graph show the length that corresponds to themaximum area?

B. The equation for the areas of rectangles with a certain fixed perimeteris A = O(35 - O), where O is the length in meters.

1. Draw a rectangle to represent this situation. Label one side O.Label the other sides in terms of O.

2. Make a table showing the length, width, and area for lengths of 0, 5, 10, 15, 20, 25, 30, and 35 meters. What patterns do you see?

3. Describe the graph of this equation.

4. What is the maximum area? What dimensions correspond to thismaximum area? Explain.

5. Describe two ways you could find the fixed perimeter. What is theperimeter?

C. Suppose you know the perimeter of a rectangle. How can you write anequation for the area in terms of the length of a side?

D. Study the graphs, tables, and equations for areas of rectangles withfixed perimeters. Which representation is most useful for finding themaximum area? Which is most useful for finding the fixed perimeter?


Frogs, Fleas and Painted Cubes • page 10

Problem 1.1 Writing Equivalent Expressions

In order to calculate the number of tiles needed for a project, the CustomPool manager wants an equation relating the number of border tiles to thesize of the pool.

A. 1. Write an expression for the number of border tiles N based on the side length s of a square pool.

2. Write a different but equivalent expression for the number of tiles N needed to surround such a square pool.

3. Explain why your two expressions for the number of border tiles are equivalent.

B. 1. Use each expression in Question A to write an equation for thenumber of border tiles N. Make a table and a graph for eachequation.

2. Based on your table and graph, are the two expressions for thenumber of border tiles in Question A equivalent? Explain.

s

s

1 ft

1 ft

border tile

Say It With Symbols • page 6


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In Problem 2.1 students revisit Problem 1.3from Moving Straight Ahead (see page 35) tocombine expressions. They also use the newexpression to find information and to predict the underlying pattern of change associated with the expression.

Solving EquationsEquivalence is an important idea in algebra. Asolid understanding of equivalence is necessary forunderstanding how to solve algebraic equations.Through experiences with different functionalrelationships, students attach meaning to thesymbols. This meaning helps student when they aredeveloping the equation-solving strategies integralto success with algebra.

In CMP, solving linear equation is an algebraidea that is developed across all three gradelevels, with increasing abstraction and complexity.In grade six, students write fact families to showthe inverse relationships between addition andsubtraction and between multiplication anddivision. The inverse relationships betweenoperations are the fundamental basis for equationsolving. Students are exposed early in sixth gradeto missing number problems where they use factfamilies. Below is a description of fact familiesand a few examples of problems where studentsuse fact families to solve algebraic equations ingrades 6 and 7. These experiences precede formalwork on equation solving.

In Bits and Pieces II (Grade 6), Bits and Pieces III (Grade 6), and Accentuate the Negative(Grade 7), students use fact families to findmissing addends and factors.

Problem 2.1 Adding Expressions

A. 1. Write equations to represent the money M that each student will raise for walking x kilometers.

a. MLeanne = 7

b. MGilberto = 7

c. MAlana = 7

2. Write an equation for the total money Mtotal raised by thethree-person team for walking x kilometers.

B. 1. Write an expression that is equivalent to the expression for thetotal amount in Question A, part (2). Explain why it is equivalent.

2. What information does this new expression represent about thesituation?

3. Suppose each person walks 10 kilometers. Explain whichexpression(s) you would use to calculate the total amount of moneyraised.

C. Are the relationships between kilometers walked and money raisedlinear, exponential, quadratic, or none of these? Explain.


Say It With Symbols • page 24Problem 2.3 Fact Families

A. For each number sentence, write its complete fact family.

1. + = 2. - =

B. For each mathematical sentence, find the value of N. Then write eachcomplete fact family.

1. + = N 2. - = N

3. + N = 4. N - =

C. After writing several fact families, Rochelle claims that subtraction undoes addition. Do you agree or disagree? Explain your reasoning.

D. In the mathematical sentence below, find values for M and N that make the sum exactly 3. Write your answer as a sum that equals 3.

+ + + M + N = 323

14

58

38

12

1712

34

123316123335

110

25

510

119

59

23

Bits and Pieces II • page 22

C. Find the value of N that makes the mathematical sentence correct.Fact families might help.

1. 63.2 + 21.075 = N 2. 44.32 - 4.02 = N

3. N + 2.3 = 6.55 4. N - 6.88 = 7.21


C. 1. Write a related sentence for each.

a. n - +5 = +35 b. n - -5 = +35 c. n + +5 = +35

2. Do your related sentences make it easier to find the value for n?Why or why not?

D. 1. Write a related sentence for each.

a. +4 + n = +43 b. -4 + n = +43 c. -4 + n = -43

2. Do your related sentences make it easier to find the value for n?Why or why not?

Accentuate the Negative • page 30

Find the value of N.

17. 3.2 3 N = 0.96 18. 0.7 3 N = 0.042 19. N 3 3.21 = 9.63



In Variables and Patterns (Grade 7), studentssolve linear equations using a variety of methodsincluding graph and tables. As students movethrough the curriculum, these informal equation-solving experiences prepare them for the formalsymbolic methods which are developed in MovingStraight Ahead (Grade 7), and revisitedthroughout the five remaining algebra units ineighth grade.

Say It With Symbols (Grade 8), pulls together thesymbolic reasoning skills students have developedthrough a focus on equivalent expressions and onsolving linear and quadratic equations.

Shapes of Algebra (Grade 8), explores solvinglinear inequalities and systems of linear equationsand inequalities.

By the end of Grade 8, students in CMP shouldbe able to analyze situations involving relatedquantitative variables in the following ways:

• identify variables

• identify significant patterns in therelationships among the variables

• represent the variables and the patternsrelating these variables using tables, graphs,symbolic expressions, and verbal descriptions

• translate information among these forms ofrepresentation

Students should be adept at identifying thequestions that are important or interesting to askin a situation for which algebraic analysis iseffective at providing answers. They shoulddevelop the skill and inclination to representinformation mathematically, to transform thatinformation using mathematical techniques tosolve equations, create and compare graphs andtables of functions, and make judgments about thereasonableness of answers, accuracy, andcompleteness of the analysis.3.4 Solving Quadratic Equations

In the last problem, you explored ways to write a quadratic expression infactored form. In this problem, you will use the factored form to findsolutions to a quadratic equation.

If you know that the product of two numbers is zero,what can you say about the numbers?

• How can you solve the equation 0 = x2+ 8x + 12 by factoring?

First write x2+ 8x + 12 in factored form to get (x + 2)(x + 6). This

expression is the product of two linear factors.

• When 0 = (x + 2)(x + 6), what must be true about one of the linearfactors?

• How can this information help you find the solutions to 0 = (x + 2)(x + 6)?

• How can this information help you find the x-intercepts of y = x2

+ 8x + 12?

Solving Quadratic Equations

A. 1. Write x2+ 10x + 24 in factored form.

2. How can you use the factored form to solve x2+ 10x + 24 = 0

for x?

3. Explain how the solutions to 0 = x2+ 10x + 24 relate to the graph

of y = x2+ 10x + 24.

B. Solve each equation for x without making a table or graph.

1. 0 = (x + 1)(2x + 7) 2. 0 = (5 - x)(x - 2)

3. 0 = x2+ 6x + 9 4. 0 = x2

- 16

5. 0 = x2+ 10x + 16 6. 0 = 2x2

+ 7x + 6

7. How can you check your solutions without using a table or graph?


Problem 3.4

Say It With Symbols • page 42

37. Solve each equation and check your answers.

a. 2x + 3 = 9 b. x + 3 = 9 c. x + 3 =

d. x + = 9 e. = 9

38. Use properties of equality and numbers to solve each equation for x.Check your answers.

a. 3 + 6x = 4x + 9 b. 6x + 3 = 4x + 9

c. 6x - 3 = 4x + 9 d. 3 - 6x = 4x + 9

x + 3_____2

1_2

9_2

1_2

Moving Straight Ahead • Investigation 4 ACEpage 85

Components of CMP2 65

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In addition to 24 student units and accompanying Teacher’s Guides, CMP2includes Additional Practice and Skills Workbooks, Assessment Resources,

Teaching Transparencies, Manipulatives Kits, a Special Needs Handbook forTeachers, and a Parent Guide.

The student units, the Additional Practice and Skills Workbooks, and the AssessmentResources are available in Spanish.

Technology components include ExamView® CD-ROM (a test generator that includesEnglish and Spanish practice and assessment items), Teacher ExpressTM CD-ROM (alesson planning tool with electronic versions of all the print resources), and a StudentActivities CD-ROM (interactivities to support conceptual understanding and practice).

In the following pages you will find a detailed discussion of the structure of thestudent units and the accompanying support found in the Teacher’s Guides and other components.


Organization of the Student UnitsConnected Mathematics 2 provides eight studentunits for each grade. One additional unit is offeredfor each grade from the first edition, ©2004.This allows some flexibility in meeting individualstate expectations by allowing the choice of an additional unit when needed. Each unit isorganized around an important mathematical idea or cluster of related ideas, such as area andperimeter, operations on fractions, ratio andproportion, linear relationships, or quadraticrelationships. The format of the student bookspromotes student engagement with an explorationof important mathematical concepts and relatedskills and procedures. Since students developstrategies and conceptual understanding by solvingproblems and discussing their solutions in class,the books do not contain worked-out examples.Instead the students record their work andexplanations as well as their growing understandingof definitions and rules in their notebooks.

The organization and features of each studentunit are described below.

Unit OpenerEach unit opens with a set of three focusingquestions that reflect the major mathematicalgoal(s) of the unit. These questions are intendedto draw students into the unit, pique theircuriosity, and point to the kinds of ideas they willinvestigate. As the students move through the unitthey will encounter these questions either as aproblem to explore in class or as homework.

Suppose a piece of ropewraps around Earth. Rope isadded to make the entire rope3 feet longer. The new ropecircles Earth exactly the samedistance away from the surfaceat all points. How far is thenew rope from Earth’s surface?

Suppose you need to makesails shaped as triangles andparallelograms for aschooner (SKOON ur). Whatmeasurements must youmake to find how muchcloth you need for the sails?

Suppose you are buildinga playhouse. How muchcarpeting do you need tocover the floor? How muchmolding (used to protectthe bases of walls) do youneed around the edges ofthe floor?

Two-Dimensional Measurement

Unit Opener • Covering and Surrounding


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Mathematical HighlightsNext, the unit provides a set of goals, orMathematical Highlights, that preview theimportant ideas of the unit. The highlights helpstudents track their progress through the unit andprovide parents and guardians with an overviewof the mathematical concepts, processes, and waysof thinking developed in the unit.

InvestigationsThe Investigations form the core of a ConnectedMathematics unit. It is by working through theInvestigations that the students develop conceptualunderstanding, reasoning, and procedural skill. EachInvestigation builds toward the mathematical goals.Each unit includes three to five Investigations withthe following key elements:

Problem An Investigation includes two to fivecarefully sequenced Problems. Each Problem islaunched by the teacher; then the students explorethe Problem individually, in groups, or as a wholeclass. As students solve the Problems, theyuncover important mathematical relationships and develop problem-solving strategies and skills.A summary occurs at the end of each Problem.The teacher pulls the class together and helpsstudents explicitly describe the mathematics of the Problem, ideas, patterns, relationships, andstrategies they found and used.

Getting Ready This feature occurs occasionallybefore a problem. It is intended to be used as part of the launch for the problem. It reviews orintroduces the mathematical ideas needed in the problem.

Two-Dimensional Measurement

In Covering and Surrounding, you will explore areas and perimeters offigures, especially quadrilaterals, triangles, and circles.

You will learn how to

• Use area and relate area to covering a figure

• Use perimeter and relate perimeter to surrounding a figure

• Analyze what it means to measure area and perimeter

• Develop strategies for finding areas and perimeters of rectangular andnon-rectangular shapes

• Discover relationships between perimeter and area, including that onecan vary while the other stays fixed

• Analyze how the area of a triangle and the area of a parallelogram arerelated to the area of a rectangle

• Develop formulas and procedures, stated in words or symbols, for findingareas and perimeters of rectangles, parallelograms, triangles, and circles

• Develop techniques for estimating the area and perimeter of an irregular figure

• Recognize situations in which measuring perimeter or area will helpanswer practical questions

As you work on the problems in this unit, ask yourself questions about situations that involve area and perimeter.

How do I know whether area or perimeter are involved?

What attributes of a shape are important to measure?

What am I finding when I find area and when I find perimeter?

What relationships involving area or perimeter, or both, will help solvethe problem?

How can I find the area and perimeter of a regular or irregular shape? Is an exact answer required?

Mathematical Highlights • Covering and Surrounding

5.1 Measuring Lakes

Geographers must know the scale of the picture to estimate the area andperimeter of a lake from a picture.

To estimate perimeter, they can

• Lay a string around the lake’s shoreline in the picture of the lake.

• Measure the length of the string.

• Scale the answer.

To estimate area, they can

• Put a transparent grid over the picture of the lake.

• Count the number of unit squares needed to cover the picture.

• Use the scale of the picture to tell what the count means.

The state Parks and Recreations Division bought a property containingLoon Lake and Ghost Lake. Park planners will develop one lake forswimming, fishing, and boating. The other lake will be used as a naturepreserve for hiking, camping, and canoeing. Planners have to think aboutmany things when deciding how to use a lake. The perimeter, area, andshape of the lake influence their decisions.

Problem 5.1 • Covering and Surrounding

A student is tired of counting the individual rail sections around the outside of each bumper-car track. She starts to think of them as one long rail. She wraps a string around the outside of Design B, as shown.

What do you think she does next? How does this help her to find the perimeter of the figure? How could she determine the area?

bumper-cartile

1 m

1 m


A Getting Ready • Covering and Surrounding

Did You Know? This feature occasionally occursto present interesting facts related to the contextof an investigation.

Applications—Connections—Extensions (ACE)The Problems in each Investigation are followedby a set of exercises meant to be used ashomework at the end of each Problem. Studentsare asked to compare, visualize, model, measure,count, reason, connect, and/or communicate theirideas in writing. To truly own an idea, strategy, orconcept, a student must apply it, connect it towhat he or she already knows or has experienced,and seek ways to extend or generalize it.

Applications These exercises help students solidifytheir understanding by providing practice with ideas and strategies that were in the Investigation.Applications contain contexts both similar to anddifferent from those in the Investigation.

In the upper Midwest of the United States, there is concern that the levelof water in the Great Lakes is decreasing. The lakes get smaller as a result. The United States Great Lakes Shipping Association reports thatfor every inch of lost clearance due to low water, a vessel loses from 90 to115 metric tons of cargo-carrying capacity.

In the year 2000, the water level in the Great Lakes decreased. Carriersthat transported iron ore, coal, and other raw cargoes had to reduce theircarrying load by 5 to 8 percent. Prices for these items increased as a result.

For: Information about the Great Lakes

Web Code: ame-9031

Did You Know? • Covering and Surrounding

11. Trace this circle and draw three different radii (RAY dee eye, theplural for radius).

a. What is the measure, in centimeters, of each radius?

b. What can you say about the measure of the radii in the same circle?

c. Estimate the circumference of this circle using the radiusmeasurements you found.

12. Terrell says that when you know the radius of a circle, you can find the diameter by doubling the radius. Do you agree? Why or why not?

13. Enrique says that when you know the diameter of a circle you can find the radius. How does he find the measure of a radius if he knowsthe measure of the diameter? Give an example in your explanation.

14. Multiple Choice A soft-drink can is about 2.25 inches in diameter.What is its circumference?

A. 3.53 in. B. 3.97 in.2 C. 7.065 in. D. 14.13 in.

15. Best Crust Pizzeria sells three different sizes of pizza. The small sizehas a radius of 4 inches, the medium size has a radius of 5 inches, andthe large size has a radius of 6 inches.

a. Make a table with these headings. Fill in the table. Explain how youfound the area of the pizzas.

b. Jamar claims the area of a pizza is about 0.75 3 (diameter)2. Is hecorrect? Explain.

Radius (in.)

Best Crust Pizzeria

Diameter (in.)

Small

Medium

Large

�

�

�

�

�

�

Area (in.2)

�

�

�

Circumference (in.)

�

�

�

Pizza Size

Applications • Covering and Surrounding



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Connections A powerful learning strategy is toconnect new knowledge to prior learning. TheConnections section of the homework providesthis opportunity. This section also providescontinued review of concepts and skills across thegrades. For example, the Connections in Coveringand Surrounding, a unit on measurement, containpractice with operations on decimals and fractions.Connections can also connect to “real-worldproblems.” Often these are problems that containoriginal data sets. For example, in Moving StraightAhead, a unit on linear relationships, there areconnections to sports records.

Extensions These exercises may provide achallenge for students to think beyond what is covered in the Problems in class, provide an interesting excursion “side ways” that looks at related mathematical ideas, foreshadowmathematics in future units or pursue aninteresting application.

Investigation 2 Changing Area, Changing Perimeter 31

Connections16. a. The floor area of a rectangular storm shelter is 65 square meters,

and its length is meters. What is the width of the storm shelter?

b. What is its perimeter?

c. A one-meter wall panel costs $129.99. Use benchmarks to estimatethe total cost of the wall panels for this four-sided shelter.

17. Multiple Choice The area of a storm shelter is 24 square meters. The

length is meters. What is the width of the storm shelter in meters?

F. G. H. J.

18. These sketches show rectangles without measurements or gridbackground. Use a centimeter ruler to make any measurements youneed to find the perimeter and area of each figure.

a.

b. c.

415414413412

513

612

Connections • Covering and Surrounding

49. a. Suppose a piece of rope wraps around Earth. Then the rope is cut,and rope is added to make the entire rope 3 feet longer. Suppose the new rope circles the earth exactly the same distance away from the surface at all points. How far is the new rope from Earth’s surface?

b. A piece of rope is wrapped around a person’s waist. Then rope isadded to make it 3 inches longer. How far from the waist is the ropeif the distance is the same all around?

c. Compare the results in parts (a) and (b).

Extensions • Covering and Surrounding

Extensions

Mathematical ReflectionsAt the end of each Investigation, students areasked to reflect on what they have learned. Aset of questions helps students organize theirthoughts and summarize important concepts andstrategies. After thinking about the questions andsketching their own ideas, students discuss thequestions with their teacher and their classmatesand then write a summary of their findings.

Unit Project At least four units at each grade level includeprojects. Projects are typically introduced at thebeginning of a unit and formally assigned at theend. A list of projects is given on page 53. Projectsare open-ended tasks that provide opportunitiesfor students to engage in independent work and todemonstrate their broad understanding of themathematics of the unit.

In this investigation, you discovered strategies for finding the area andcircumference of a circle. You examined relationships between thecircumference and diameter of a circle and between the area and radius of a circle. You also used grids to find accurate estimates of the area andperimeter of irregular shapes. These questions will help you to summarizewhat you have learned.

Think about your answers to these questions. Discuss your ideas with otherstudents and your teacher. Then write a summary of your findings in yournotebook.

1. Describe how you can find the circumference of a circle by measuringits radius or its diameter.

2. Describe how you can find the area of a circle by measuring its radiusor its diameter.

3. Describe how you can, with reasonable accuracy, find the area andperimeter of an irregular shape such as a lake or an island.

4. What does it mean to measure the area of a shape? What kinds of unitsare appropriate for measuring area? Why?

5. What does it mean to measure the perimeter or circumference of ashape? What kinds of units are appropriate for measuring perimeter or circumference? Why?

Mathematical Reflections • Covering and Surrounding

Plan a Park

- rectangular (120 yd by 100 yd)- picnic area and playground- circular flower garden in picnic area- another garden- trees- family interest

The City Council of Roseville is planning to build a park for families inthe community. Your job is to design a park to submit to the City Councilfor consideration. You will need to make an argument for why your designshould be chosen. Use what you know about parks and what you learnedfrom this unit to prepare your final design.

Part 1: The Design

Your design should satisfy the following constraints:

• The park is rectangular with dimensions 120 yards by 100 yards.

• About half of the park consists of a picnic area and a playground.These sections need not be located together.

• The picnic area contains a circular flower garden. There also is a garden in at least one other place in the park.

• There are trees in several places in the park. Young trees will be planted, so your design should show room for the trees to grow.

• The park must appeal to families. There should be more than just a picnic area and a playground.

Unit Project • Covering and Surrounding

Part 2: Write a Report

Your design package should be neat, clear, and easy to follow. Draw andlabel your design in black and white. In addition to a scale drawing of your design for the park, your project should include a report that gives:

1. the size (dimensions) of each item (include gardens, trees, picnic tables, playground equipment, and any other item in your design).

2. the amount of land needed for each item and the calculations you used to determine the amount of land needed

3. the materials needed (include the amount of each item needed and the calculations you did to determine the amounts)

• each piece of playground equipment

• fencing

• picnic tables

• trash containers

• the amount of land covered by concrete or blacktop (so thedevelopers can determine how much cement or blacktop will beneeded)

• other items

Extension Question

Write a letter to the City Council. Explain why they should choose yourdesign for the park. Justify the choices you made about the size andquantity of items in your park.

Unit Project (continued) • Covering and Surrounding



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Looking Back and Looking AheadThis feature provides a review of the “big” ideasand connections in the unit. It includes problemsthat allow students to demonstrate theirunderstanding, explain their reasoning,summarizing and connecting what they havelearned within and across units.

GlossaryAlthough students are encouraged to developtheir own definitions and examples for key terms,a glossary in English and Spanish is provided atthe back of each student book. Glossaries canserve as a guide for the student, the teacher, andparents as students develop understanding of keyideas and strategies.

Unit Review

Working on problems in this unit helped you to understand area andperimeter. You learned

• efficient strategies for estimating and calculating the area andperimeter of figures such as triangles, rectangles, parallelograms,and circles

• to investigate the relationship between area and perimeter of simplepolygons

Use Your Understanding: Area and Perimeter

Test your understanding and skill in working with area and perimeter onthese problems.

1. The diagram shows a hexagon drawn on a centimeter grid.

a. Find the area of the hexagon.

b. Describe two different strategies for finding the area.

For: Vocabulary Review Puzzle

Web Code: amj-5051

Looking Back and Looking Ahead Covering and Surrounding

2. The Nevins’ living room floor is a square 20 feet by 20 feet. It is covered with wood.They have carpeted a quarter-circle region as shown.

a. What is the area of the uncovered wood, to the nearest square foot?

b. A 1-quart can of floor wax covers 125 square feet of wood flooring. How many cans of floor wax are needed to wax the uncovered wood?

c. A special finishing trim was placed along the curved edge of the carpet.How much trim, to the nearest tenth of a foot, was needed?

Explain Your Reasoning

3. Give a rule for finding the area and perimeter of each figure.

a. rectangle b. triangle

c. parallelogram d. circle

e. irregular figure

4. Describe why the rules you wrote for Problem 3 work.

Look Ahead

Area and perimeter are among the most useful concepts for measuring thesize of geometric figures. You will use strategies for estimating andcalculating the size of geometric figures in many future units of ConnectedMathematics. The units will include surface area and volume of solid figures, similarity, and the Pythagorean theorem. You will also find that area and volume estimates and calculations are used in a variety of practicaland technical problems.

20 ft

20 ft

carpet

wood floor

Looking Back and Looking Ahead (continued)Covering and Surrounding

area The measure of the amount of surfaceenclosed by the boundary of a figure. To find thearea of a figure, you can count how many unitsquares it takes to cover the figure. You can find thearea of a rectangle by multiplying the length by thewidth. This is a shortcut method for finding thenumber of unit squares it takes to cover therectangle. If a figure has curved or irregular sides,you can estimate the area. Cover the surface with agrid and count whole grid squares and parts of gridsquares. When you find the area of a shape, write theunits, such as square centimeters (cm2), to indicatethe unit square that was used to find the area.

área La medida de la cantidad de superficieencerrada por los límites de una figura. Para hallarel área de una figura, puedes contar cuántasunidades cuadradas se requieren para cubrir lafigura. Puedes hallar el área de un rectángulomultiplicando el largo por el ancho. Esto es unmétodo más corto para hallar el número deunidades cuadradas requeridas para cubrir elrectángulo. Si una figura tiene lados curvos oirregulares, puedes estimar el área. Para ello, cubrela superficie con una cuadrícula y cuenta loscuadrados enteros y las partes de cuadrados en lacuadrícula. Cuando halles el área de una figura,escribe las unidades, como centímetros cuadrados(cm2), para indicar la unidad cuadrada que se usópara hallar el área. El área del cuadradorepresentado a continuación es de 9 unidadescuadradas y el área del rectángulo es de 8 unidadescuadradas.

A

A � 9 square units A � 8 square units

Glossary • Covering and Surrounding

Technology for StudentsConnected Mathematics was developed with thebelief that calculators should be available tostudents, and that students should know when andhow to use them. In grade 6, students needstandard, four-function calculators. Students usefour-function calculators to simplify complicatedcalculations and explore patterns in computations.

For some units in grades 7 and 8, students needaccess to graphing calculators with table andstatistical-display capabilities. Graphing calculatorsare used to investigate functions and as a tool forsolving problems. Students use graphing calculatorsto explore the shape and features of graphs oflinear, exponential, and quadratic functions aswell as the patterns of change in the tables of suchfunctions. And in addition to using symbolicsolution methods, students use graphing-calculatortables and graphs to solve equations.

Although computers are not required for anyof the investigations, applets are provided formany units. Some applets are designed to be usedduring the Launch, Explore, Summarize sequenceand some can be used at various stages of theinstruction, including additional practice with theideas in the unit. The applets are provided on theStudent Activities CD-ROM.

ManipulativesIn Connected Mathematics, manipulatives are used only when they can help students developunderstanding of mathematical ideas. For example,in Filling and Wrapping, students find all thedifferent rectangular arrangements possible for agiven number of cubes. They find the surface areaof each arrangement by creating a net (covering)for the arrangement that exactly fits, with nooverlap or underlap. They then identify thearrangements that require the least and the mostmaterial to wrap. This activity sets the stage fordeveloping the ideas of surface area and volume of rectangular prisms. Most of the manipulativesused in Connected Mathematics are commonlyavailable, and many schools may already havethem. Included are rulers, protractors, angle rulers,cubes, square tiles, counters, spinners, and dice.

The two manipulatives described below areunique to Connected Mathematics.

Polystrips are plastic strips that can be piecedtogether with brass fasteners to form polygons.These manipulatives are used in grade 6 toinvestigate the relationship among the sidelengths of triangles and quadrilaterals. They alsoare useful in the eighth grade geometry unit,Kaleidoscopes, Hubcaps, and Mirrors.

The CMP Shapes Set® is a set of polygons used ingrade 6 to explore sides, angles, and tilings.

Blackline masters are provided for teachers whodo not have Polystrips or the CMP Shapes Set.

A list of materials for each unit is found in theUnit Introduction of each Teacher’s Guide.



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In Connected Mathematics, teachers are anintegral part of the learning process. From the

beginning, the authors have viewed ConnectedMathematics as a curriculum for both students andteachers. Connected Mathematics provides teacherswith ways to think about and enact problem-centered teaching that address the followingaspects of instruction.

The CMP Instructional ModelProblem-centered teaching opens the mathematicsclassroom to exploring, conjecturing, reasoning,and communicating. The Connected Mathematicsteacher materials are organized around aninstructional model that supports this kind ofteaching. This model is very different from the“transmission” model in which teachers tellstudents facts and demonstrate procedures andthen students memorize the facts and practice theprocedures. The CMP model looks at instructionin three phases: launching, exploring, andsummarizing. The following text describes thethree instructional phases and provides thegeneral kinds of questions that are asked. Specificnotes and questions for each problem areprovided in the Teacher’s Guides.

In the first phase, the teacher launches the problemwith the whole class. This involves helping studentsunderstand the problem setting, the mathematicalcontext, and the challenge. The following questionscan help the teacher prepare for the launch:

• What are students expected to do?

• What do the students need to know tounderstand the context of the story and thechallenge of the problem?

• What difficulties can I foresee for students?

• How can I keep from giving away too much of the problem solution?

The launch phase is also the time when theteacher introduces new ideas, clarifies definitions,reviews old concepts, and connects the problem to

past experiences of the students. It is critical that,while giving students a clear picture of what isexpected, the teacher leaves the potential of thetask intact. He or she must be careful to not telltoo much and consequently lower the challenge of the task to something routine, or to cut off therich array of strategies that may evolve from amore open launch of the problem.

The nature of the problem suggests whetherstudents work individually, in pairs, in smallgroups, or occasionally as a whole class to solvethe problem during the explore phase. TheTeacher’s Guide suggests an appropriate grouping.As students work, they gather data, share ideas,look for patterns, make conjectures, and developproblem-solving strategies.

It is inevitable that students will exhibitvariation in their progress. The teacher’s roleduring this phase is to move about the classroom,observing individual performance and encouragingon-task behavior. The teacher helps studentspersevere in their work by asking appropriatequestions and providing confirmation andredirection where needed. For students who areinterested in and capable of deeper investigation,the teacher may provide extra questions related tothe problem. These questions are called GoingFurther and are provided in the explore discussionin the Teacher’s Guide. Suggestions for helpingstudents who may be struggling are also providedin the Teacher’s Guide. The explore part of theinstruction is an appropriate place to attend todifferentiated learning.

The following questions can help the teacherprepare for the explore phase:

• How will I organize the students to explorethis problem? (Individuals? Pairs? Groups?Whole class?)

• What materials will students need?

• How should students record and report theirwork?

• What different strategies can I anticipate theymight use?

Explore 1.11.1

Launch 1.11.1

• What questions can I ask to encourage studentconversation, thinking, and learning?

• What questions can I ask to focus theirthinking if they become frustrated or off-task?

• What questions can I ask to challenge studentsif the initial question is “answered”?

As the teacher moves about the classroomduring the explore, she or he should attend to thefollowing questions:

• What difficulties are students having?

• How can I help without giving away thesolution?

• What strategies are students using? Are theycorrect?

• How will I use these strategies during thesummary?

It is during the summary that the teacher guidesthe students to reach the mathematical goals of theproblem and to connect their new understanding toprior mathematical goals and problems in the unit.The summarize phase of instruction begins whenmost students have gathered sufficient data ormade sufficient progress toward solving theproblem. In this phase, students present and discusstheir solutions as well as the strategies they used toapproach the problem, organize the data, and findthe solution. During the discussion, the teacherhelps students enhance their conceptualunderstanding of the mathematics in the problemand guides them in refining their strategies intoefficient, effective, generalizable problem-solvingtechniques or algorithms.

Although the summary discussion is led by theteacher, students play a significant role. Ideally,they should pose conjectures, question each other,offer alternatives, provide reasons, refine theirstrategies and conjectures, and make connections.As a result of the discussion, students shouldbecome more skillful at using the ideas andtechniques that come out of the experience withthe problem.

If it is appropriate, the summary can end byposing a problem or two that checks students’understanding of the mathematical goal(s) thathave been developed at this point in time. Check

For Understanding questions occur occasionally in the summary in the Teacher’s Guide. Thesequestions help the teacher to assess the degree towhich students are developing their mathematicalknowledge. The following questions can help theteacher prepare for the summary:

• How can I help the students make sense of andappreciate the variety of methods that may beused?

• How can I orchestrate the discussion so thatstudents summarize their thinking about theproblem?

• What questions can guide the discussion?

• What concepts or strategies need to beemphasized?

• What ideas do not need closure at this time?

• What definitions or strategies do we need togeneralize?

• What connections and extensions can be made?

• What new questions might arise and how do Ihandle them?

• What can I do to follow up, practice, or applythe ideas after the summary?

Organization of the Teacher’s GuideThe extensive field-testing of ConnectedMathematics has helped produce teacher materials that are rich with field teachers’successful strategies, classroom dialogues andquestions, and examples of student solutions and reasoning (see page 31). The Teacher’s Guide for each unit contains a discussion of the mathematics underlying the Investigations,mathematical and problem-solving goals for each Investigation, connections to other units,in-depth teaching notes, and an extensiveassessment package.

The teacher materials are designed as a resourceto facilitate teaching Connected Mathematics. Thefeatures and organization of the Teacher’s Guideare described on the next page.

Summarize 1.11.1



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Unit IntroductionTeachers can use the material in the UnitIntroduction to prepare for teaching the unit.The following features are included in this section:

Unit Introduction• Goals of the Unit• Developing Students’ Mathematical Habits

The Mathematics in the Unit• Overview• Summary of the Investigations• Mathematics Background A detailed

description designed to assist teachers in understanding the content

Content Connections to Other Units A chart highlighting how the big ideas of the unit connect to ideas from previous and future units

Planning for the Unit• Pacing Suggestions and Materials• Vocabulary

Program Resources• Components• Technology

Assessment Summary (see page 54)• Ongoing Informal Assessment• Formal Assessment• Correlations to Standardized Tests

Launching the Unit• Using the Unit Opener• Using the Mathematical Highlights• Introducing the Unit Project

Teaching NotesDetailed teaching notes are included for eachInvestigation. These include the following:

Mathematical Goals for the Investigation

For each problem:

• Specific Mathematical and Problem-SolvingGoals

• Detailed Teaching Notes Includes problem-by-problem discussions with examples of theinstructional role of the teacher during thethree phases of problem instruction, as wellas samples of student responses to questions.

• Going Further The explore sections includeoccasional “Going Further” questions forstudents who finish early or need anotherchallenge.

• Check for Understanding The summarysection, when appropriate, may end withquestions for the teacher to use to checkstudents’ understanding.

Lesson At a Glance This is a two-sided one-pagelesson guide for each problem. (A blank At aGlance template is included in each Teacher’sGuide to facilitate a teacher’s personalization ofthe lesson plan.) At a Glance contains:

• The mathematical goal for the Problem

• Materials needed for the Problem

• Definitions that need to be addressed

• Key questions for the Launch, Explore, andSummarize phases of the instruction

• Answers to the Problem

• Homework assignment guide

Back to the Bees!3.3

Launch

Explore

Summarize

Mathematical Goal

• Decide which regular polygons will tile by themselves or in combinationsusing information about interior angles

Materials

• Overhead Shapes Set(Transparency 1.1E)

Materials

• Shapes Sets(1 per group)

Materials

• Student notebooks

At a Glance

Students should have an idea about which regular polygons will tile andwhich will not. If needed, remind them about the discussions fromInvestigation 1.

• Which of the regular polygon shapes (Shapes A–F) did we learn wouldtile a surface—fit together so that there are no gaps or overlaps—bythemselves?

Students should recall that the triangle, square, and hexagon all tile.

• How can we tell for sure that a shape, like these hexagons, fits exactlyaround each point in a tiling? We know the fit looks good, but how canwe use mathematics to tell for sure?

Have students work together in groups of 2–4.

As you move from group to group, ask questions about angles to helpstudents focus on them as a consideration in forming a tiling.

• How much turn must there be to completely surround a vertex point?

• How many degrees are in the angles of the polygon you areinvestigating?

• What would we expect the angle sum around a point to be?

When students have completed the exploration of the regular polygons,they should move to the combinations of regular polygons.

Going Further

Ask students to explore whether any quadrilateral will tile.

Students should be able to explain why there are only three regularpolygons that tile using angle measure as part of their argument. Theyshould also be able to explain why certain combinations of regular polygonswork using angle measures.

• Why do you think your design forms a tiling with no gaps or overlaps?

PACING 1 day

continued on next page

At a Glance • Shapes and Designs

Blackline Masters of Labsheets and other thingsare provided. Students use these blackline mastersas they work on the problem sets.

Descriptive Glossary/Index Key concepts aresummarized, often with illustrations or examples,in both English and Spanish.

Support Materials for TeachersIn addition to the Teacher’s Guide for each unit,there are several resources that are designed toassist teachers. They are:

Teaching Transparencies support problems fromthe student books. All the Getting Ready featuresare available on transparencies.

Parent materials for Connected Mathematicsinclude a parent letter for each unit with the goalsof the unit and examples of worked problems,as well as a website for parents to help withhomework for each unit.

Special Needs Handbook for Teachers includessuggestions for adapting instruction, examples ofmodified problems and ACE exercises from thestudent books, and assessment items for each unit.

Assessment Resources include blackline mastersfor Partner Quizzes, Check-Ups, Unit Tests,multiple- choice items, Question Bank, NotebookCheck and Self-Assessment for each grade level.They are also available in Spanish and on a CD-ROM.

Additional Practice and Skills Workbook for eachgrade level provides practice exercises for eachinvestigation as well as additional skills practice toreinforce student learning.

Technology A Student Activities CD-ROMprovides activities to enhance and supportclassroom learning in the Problems/Investigationsof the Student books.

Teacher ExpressTM CD-ROM includes lessonplanning software, the Teacher’s Guide pages, andall the teaching resources.

Exam View® Test Generator CD-ROM includesall the items from the Assessment Resources andthe Additional Practice and Skills Workbook inboth English and Spanish. Items can be editedelectronically and saved. Many items are dynamic,and can be used to create multiple versions ofpractice sheets.


Summarize

There are eight combinations of regular polygons that will tile so thateach vertex has exactly the same pattern of polygons. (note the numbersin parentheses refer to the polygon by side number):

2 octagons and 1 square (8-8-4)1 square, 1 hexagon, and 1 dodecagon (4-6-12)4 triangles and 1 hexagon (3-3-3-3-6)3 triangles and 2 squares (4-3-4-3-3)1 triangle, 2 squares, and 1 hexagon (4-3-4-6)1 triangle and 2 dodecagons (3-12-12)3 triangles and 2 squares (4-3-3-3-4)2 triangles and 2 hexagons (3-6-3-6)

Note there are two arrangements with triangles and squares, butdepending on the arrangement they produce different patterns.

ACE Assignment Guidefor Problem 3.3Core 11, 12Other Connections 18, 19; Extensions 25;unassigned choices from previous problems

Adapted For suggestions about adapting ACEexercises, see the CMP Special Needs Handbook.

Answers to Problem 3.3

A. 1.

2. Equilateral triangles, squares, and hexagonstile because the measure of an interiorangle for each (608, 908, 1208) divides evenlyinto 3608.

B. A regular polygon will form a tiling only if itsangle measurement is a factor of 3608, and theangle measurements of pentagons, heptagons,and octagons are not factors of 3608.

C. 1. Answers and sketches will vary. Example:

There are eight tilings using combinationsof regular polygons. See Summarize above.

2. The sum of the angles that meet at a pointin the tiling is always 3608.

continued

At a Glance (continued) • Shapes and Designs


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Assessment DimensionsAssessment in Connected Mathematics is anextension of the learning process, as well as anopportunity to check what students can do. Forthis reason, the assessment is multidimensional,giving students many ways to demonstrate howthey are making sense of the mathematics.

The Curriculum and Evaluation Standards forSchool Mathematics (NCTM, 1989), the AssessmentStandards for School Mathematics (NCTM, 1995),and the Principles and Standards for SchoolMathematics (NCTM 2000) provide guidelines thatdescribe mathematics education in schools, not onlyin terms of mathematical objectives, but in terms ofthe methods of instruction, the processes used bystudents in learning and doing mathematics, and the students’ disposition towards mathematics.Assessment in Connected Mathematics is designedto collect data concerning these three dimensions of student learning:

Content knowledgeAssessing content knowledge involvesdetermining what students know and what theyare able to do.

Mathematical dispositionA student’s mathematical disposition is healthywhen he or she responds well to mathematicalchallenges and sees himself or herself as a learnerand inventor of mathematics. Disposition alsoincludes confidence, expectations, andmetacognition (reflecting on and monitoring one’sown learning).

Work habitsA student’s work habits are good when he or sheis willing to persevere, contribute to group tasks,and follow tasks to completion. These valuableskills are used in nearly every career. To assesswork habits, it is important to ask questions, suchas “Are the students able to organize andsummarize their work?” and “Are the studentsprogressing in becoming independent learners?”

The NCTM Principles and Standards 2000reinforces the CMP philosophy on assessment.Its Assessment Principle states:

Assessment should support the learning ofimportant mathematics and furnish usefulinformation to both teachers and students.

Assessment ToolsConnected Mathematics provides a variety of toolsfor student assessment. These assessments fall intothree broad categories:

CheckpointsSome of the assessment tools—such as ACEassignments, notebooks, Mathematical Reflections,and the Unit Review—give teachers and studentsan opportunity to check student understanding atkey points in the unit. Checkpoints help studentssolidify their understanding, determine the areasthat need further attention, and help teachersmake decisions about whether students are readyto move on. The “Check for Understanding”feature of some summaries gives students andteachers an additional checkpoint on studentsprogress.

Surveys of KnowledgeCheck-ups, quizzes, unit tests, and projects provideteachers with a broad view of student knowledgeboth during a unit and at the end of a unit.

ObservationsThe curriculum provides teachers with numerousopportunities to assess student understanding byobserving students during group work and classdiscussions. This form of assessment is important,since some students are better able to showunderstanding in verbal situations than in formal,written assignments. Teachers may also receivefeedback from parents—who may comment ontheir child’s enthusiasm or involvement with aparticular problem—and from students who mayobserve that another student’s method is moreefficient or useful, or who may offer an importantobservation, conjecture or extension. Moreinformation about each assessment tool is givenbelow.

CheckpointsACE By assigning ACE exercises as homework,teachers can assess each student’s developingknowledge of concepts and skills.

Notebooks and Notebook Checklist Manyteachers require their students to keep organizednotebooks, which include homework, notes from class, vocabulary, and assessments. Each unit includes a checklist to help students organizetheir notebooks before they turn them in forteacher feedback. Teachers can also assess student understanding during their study of the unit by examining their work or summariesfor particular problems.

Mathematical Reflections A set of summarizingquestions, called Mathematical Reflections, occursat the end of each investigation. These questionscan help teachers assess students’ developingconceptual knowledge and skills in the investigation.(See page 70.)

Looking Back and Looking Ahead This UnitReview feature includes two to four problems that ask students to explain their reasoning.Collectively, the pieces have students summarizeand connect what they have learned within andacross units. This component can be used as areview, helping students to stand back and look at the “big” ideas and connections in the unit.(See page 71.)

Surveys of Knowledge Check-ups Check-ups are short, individualassessment instruments. Check-up questions tendto be less complex and more skill-oriented thanquestions on quizzes and unit tests. These questionsprovide insight into student understanding of thebaseline mathematical concepts and skills of theunit. Student responses to Check-ups can helpteachers plan further instruction for the unit.

Partner Quizzes Each unit has at least onepartner quiz. Quiz questions are richer and morechallenging than checkup questions. Many quizquestions are extensions of ideas studentsexplored in class. These questions provide insightinto how students apply the ideas from the unit tonew situations. The quizzes were created with thefollowing assumptions:

• Students work in pairs.

• Students are permitted to use their notebooksand any other appropriate materials, such ascalculators.

• Pairs have an opportunity to submit a draft of the quiz for teacher input. They may thenrevise their work and turn in the finishedproduct for assessment.

Unit Tests Each unit includes a test that is intendedto be an individual assessment. The test informsteachers about a student’s ability to apply, refine,modify, and possibly extend the mathematicalknowledge and skills acquired in the unit. Some of the questions draw on ideas from the entire unit, while others are smaller, focusing on aparticular idea or concept. Some of the questionsare skill oriented, while others require students todemonstrate problem-solving abilities and more in-depth knowledge of the unit concepts. Teacherscan use holistic scoring techniques and rubrics thattake into account the many dimensions addressedby the test.

Self-Assessment After every unit, studentscomplete a self-assessment, summarizing themathematics they learned in the unit and theideas with which they are still struggling. The self-assessment also asks students to provide examplesof what they did in class to add to the learning ofthe mathematics. The goal of this activity is tohave students reflect on their learning. For manystudents, self-assessment is a new experience, andthey may struggle with this at first. However, byreceiving feedback from teachers and using otherstudents’ work as models, students can learn toreflect on their own progress in making sense of mathematics.

Project At least four units in each grade includeprojects that can be used to replace or supplementthe unit test. Projects give teachers an opportunityto assign tasks that are more product/performance-based than those on traditional tests. Project tasksare typically open-ended and allow students toengage in independent work and to demonstratebroad understanding of ideas in the unit. Throughstudents’ work on the projects, teachers can gatherinformation about their disposition towardmathematics. Project guidelines, student examples,and scoring rubrics appear in the AssessmentResources section for the unit. The table on thenext page gives locations and descriptions ofprojects by grade level.


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Question Bank A bank of questions is provided for each unit.Teachers may use these questions ashomework problems, as class investigation problems,or as replacements for quiz and check-up questions.

Some of these questions give students an additionalopportunity to work on problems similar to those inthe unit, while others extend the ideas of the unit.

Unit Projects by Grade

Grade 8

Growing, Growing, GrowingPROJECT: Half-LifeStudents use cubes to simulate theradioactive decay of a substance andestimate its half-life. They then createa new situation involving radioactivedecay and design and carry out theirown simulation.

Kaleidoscopes, Hubcaps and MirrorsPROJECT: Making TessellationsStudents analyze the symmetries ofvarious tessellations and create theirown tessellations.

PROJECT: Making a Wreath and a PinwheelStudents make an origami wreathand transform it into a pinwheel.They investigate and describe thesymmetries of their creation atvarious stages.

Say It With SymbolsPROJECT: Finding the Surface Areaof Red StacksStudents find the volume and surfacearea of stacks and rods. They look forpatterns and then apply what theyhave learned about writing algebraicexpressions to describe patterns that they observe and verify theequivalence of those expressions.

Samples and Populations

PROJECT: Estimating a DeerPopulationStudents simulate a capture-recapturemethod for estimating deerpopulations, conduct some research,and write a report.

Grade 7

Stretching and ShrinkingPROJECT: Shrinking or EnlargingPicturesStudents shrink or enlarge a drawingor photograph by hand. They thenanalyze the relationships among the lengths, areas, and anglemeasurements of the original andthose of the new drawing.

PROJECT: All-Similar ShapesStudents analyze a variety of shapesto determine which shapes are alwaysmathematically similar to other shapesof the same kind.

Comparing and ScalingPROJECT: Paper PoolStudents look at several simplifiedpool tables to determine the numberof hits a ball will make before itgoes into a pocket and the pocket inwhich it lands. They use their resultsto make predictions for other tables.

Filling and WrappingPROJECT: Package Design ContestStudents design packages for table-tennis balls, calculate the costs of theirpackages, and justify the designs oftheir packages.

What Do You Expect?PROJECT: The Carnival GameStudents design carnival games andanalyze the probabilities of winningand the expected values. They thenwrite a report explaining why theirgames should be included in theschool carnival.

Accentuate the NegativePROJECT: Dealing DownStudents apply what they havelearned to a game. They then write areport explaining their strategies andtheir use of mathematics.

Moving Straight AheadPROJECT: Conducting anExperimentStudents collect data about drippingwater or rebounding balls and makepredictions based on their data.

Grade 6

Prime TimePROJECT: My Special NumberStudents choose a “special number”and use all they have learned in the unit to describe mathematicalproperties and real-world applicationsor occurrences of their numbers.

Shapes and DesignsPROJECT: What I Know AboutShapes and DesignsStudents create representations ofwhat they have learned about variouspolygons, the relationships of theirsides and angles, and where theseshapes can be found in their world.

Covering and SurroundingPROJECT: Plan a ParkStudents create scale drawings for apark that meet given constraints andsubmit a written proposal highlightingthe features of their parks.

Bits and Pieces IIIPROJECT: Ordering From a CatalogStudents select items from a catalogand fill out an order form, calculatingshipping, tax and discounts.

Data About UsPROJECT: Is Anyone Typical?Students apply what they havelearned in the unit to gather,organize, analyze, interpret, anddisplay information about the“typical” middle school student.


ObservationsGroup Work Many problems provide theopportunity to observe students as they “domathematics,” applying their knowledge, exhibitingtheir mathematical disposition, and displayingtheir work habits as they contribute to group tasks.

Class Discussions The summary portion of eachproblem and the Mathematical Reflections at the end of each Investigation provide ongoingopportunities to assess students’ understandingthrough class discussions.

Students and Parents Through Self Assessments,Partner Quizes, group work, and class discussions,students have the opportunity to observe and

assess their own content knowledge, mathematicaldisposition, and work habits. Parents may alsoobserve their child’s progress, disposition, and workhabits and share them with the teacher.

Summary of AssessmentDimensions and ToolsFinally, this chart summarizes the assessment toolsin Connected Mathematics and the dimensionsaddressed by each assessment item.


Mathematical Disposition

✔

✔

✔

✔

✔

✔

✔

✔

✔

✔

Checkpoints

ACE

Notebooks

Mathematical Reflections

Looking Back and Looking Ahead

Surveys of Knowledge

Check-ups

Partner Quizzes

Unit Tests

Self-Assessment

Project

Question Bank

Observations

Group Work

Class Discussions

Students and Parents

Content Knowledge

✔

✔

✔

✔

✔

✔

✔

✔

✔

✔

✔

✔

✔

Work Habits

✔

✔

✔

✔

✔

✔

✔

✔

✔

✔

Assessment Tool

Assessment Dimension

Implementing CMP2 81

Because a problem-centered curriculum (see page 6 this document) may be quite different from that experienced by teachers, administrators, and

parents/guardians in the community, a shift to such a curriculum is likely to generateexcitement as well as some discomfort and uncertainty.

Some concerns are shared by all members of the community. For example, in theearly stages of the adoption process, members of the community will want to knowwhat data already exists that describes the effectiveness of Connected Mathematics.As the implementation proceeds, members of the community have an interest in theresults of a well-planned, ongoing evaluation of student learning in their schooldistrict. Some issues are of greater concern to administrators and teachers than toparents/guardians. For example, administrators will need to plan for multi-facetedprofessional development to support teachers through different stages of thecurriculum, from novice first-year implementers to confident, successful teachers of CMP. Teacher concerns include the need for ongoing support, not just in the initial implementation stage, but also in the development of a thriving culture ofprofessionals working together to make decisions about improving the teaching andlearning of mathematics in the building.

Parent/guardian concerns are likely to focus on two issues: how to help their studentsbe successful in a curriculum that looks unfamiliar to them, and how CMP studentswill make the transition to the expectations of high school mathematics classes.Below are some suggestions for informing, engaging, and preparing all members of the community as they tackle the work necessary—from early evidence gathering and planning to well-supported, successful implementation of CMP.

Curriculum by itself is not enough, but a good curriculum in the hands of a goodteacher, with support of the administration, the local community, and long-termprofessional development, can provide the kinds of mathematical experiences thatsupport higher levels of mathematical performance for all students.

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Initial StagesConnected Mathematics is standards based andproblem centered. Adopting and implementing astandards-based mathematics curriculum can be a major step for school districts.

To replace a curriculum often described as “a mile wide and an inch deep,” standards-based mathematics curricula focus attentionon a core agenda of important and broadlyuseful mathematical ideas. To replaceinstruction that asks students to watchpassively and imitate teacher demonstrationsof routine computational techniques, thesecurricula engage students in challengingmathematical investigations that help themconstruct solid understanding of key ideas and confident ability to solve toughmathematical problems. (James Fey,Montgomery Gazette, Montgomery County,September 10, 1999)

Standards-based curricula such as CMP areorganized differently than traditional mathematicstextbooks. Parents and teachers are accustomed totextbooks that have examples followed by pages of skill exercises. Many teachers are accustomed to instruction characterized by working a fewexamples for the whole class, then assigning a set ofpractice problems to be done independently by thestudents. Although most teachers supplement thispattern by occasionally having students solve non-routine problems, such activity is not at the core oftheir curriculum or central to their teaching.

Not only do the standards-based textbooks lookdifferent but they also ask teachers, students, andparents to play different roles. In the traditionalcurriculum teachers actively demonstrate solutions,students somewhat passively observe, parents andguardians support and supervise practice, and thetextbook provides examples. In CMP classrooms,students actively investigate problems and developsolutions, and teachers question, challenge, andorchestrate explicit summaries of the mathematicsbeing learned. Parents and guardians will benefitfrom information about the rationale for theorganization and sequencing of the problems, therole of ACE questions and Looking Back/Looking

Ahead activity in each unit, and the way thatstudent notebooks are designed to capture astudent’s evolving knowledge of a topic.Standards-based curricula like CMP lookdifferent and make different demands on allmembers of the community.

Connected Mathematics teaches mathematicsthrough a sequence of connected problems in aninquiry-based classroom. This a major shift awayfrom a focus on developing skills and proceduresto a focus on mathematics as a set of relationshipsbetween a specialized symbolic language, concepts,facts, ways of thinking, and procedures. Schooldistricts can make this shift and smooth theimplementation of CMP by following guidelines fororchestrating the process. Bay et al. (1999) list tenimportant factors related to successful selection andimplementation of standards-based mathematicscurricula. These include: administrative support;opportunities to study and pilot the curricula; timefor daily planning and interaction with colleagues;knowledge of appropriate assessment techniquesand tools; ongoing communication with parents;and articulation with colleagues at the elementaryand secondary level.

Before adopting Connected Mathematics,school personnel should take time to:

Gather Evidence

• Seek reviews of research that might answerthe question, “Why do we want to change?What more do we want for our students thatthe current texts do not provide?”

• Seek reviews of research that might guideevaluation of textbooks, rather than relysolely on the data provided by the publisher.“What are we seeking in a new text, and howcan we find helpful and reliable information?”

• Know how middle school students in yourdistrict are performing on state or localassessments. Ask teachers to provide input intowhere they see strengths and/or weaknesses inthe current curriculum materials. Involve aparent group in reviewing the evidence andestablishing goals. “Are there particular areas ofstrength or weakness?”


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• Seek, or request from the publisher, datadescribing the effectiveness of CMP or anyother curricula under consideration. “Was thecurriculum used in a district like ours?”

• Create an evaluation plan for measuringstudent achievement. Plan to collect baselinedata the year before implementation.“What is our goal for student achievement?What realistically can we expect? How will we evaluate student’s achievement over the long run?”

• Seek information about the preparation andconfidence level of teachers in the district.“What support is there for teachers either inthe textbook itself or from other sources?”

Garner Administrative and Community Support

• Superintendents, principals, and otheradministrators as well as school boards andparents must have access to clear informationabout the CMP materials. The administrationand staff need a well-developed strategy forproviding the mechanisms through which suchinformation is made available to the schoolboard and to parents and kept updated.

• Familiarize principals and other administratorswith CMP. They should know the rationale forthe change in curricular emphasis and howCMP will better meet the needs of students.

• Work with parents to gain support. Somedistricts have found that a parent/communityadvisory group is helpful. Involving parentsduring the conceptualization of theimplementation can avoid misunderstandingslater.

• Work with teachers to gain support. Requestand respect input from all teachers.

Address Important Issues and Questions

• How does CMP “fit” with district and stateframeworks? Correlate curriculum goals withlocal/state requirements and assessmentinstruments.

• How does CMP handle basic skills? Theanswer to this question, as well as theevidence of the impact of the curriculum on students’ basic skills, is readily available.The results consistently show that CMP

students do as well as, or better than, non-CMPstudents on tests of basic skills. And CMPstudents outperform non-CMP students ontests of problem solving ability, conceptualunderstanding, and proportional reasoning.(See www.math.msu.edu/CMP: Research and Reports.)

• Will CMP be used in all grades (6–8), with alllevels of students? Will students with learningdifficulties or reading difficulties find it toodifficult? (See pages 87–99.)

• If algebra is offered as a separate course inmiddle school, will CMP be used in thiscourse? If so, how does CMP support thedevelopment of algebra concepts and skills?(See www.msu.edu/CMP: Curriculum.)Experiences of other schools that aresuccessfully using CMP in 6–8, including foran 8th grade algebra class, can be a powerfulresource. (See www.math.msu.edu/CMP:Research and Reports: State and DistrictData.)

• How will students make the transition fromCMP to high school? Who should be involvedin making a transition plan?

• Are students coming out of our K–5 ready forCMP? Should we involve elementary teachersin making a transition plan from upperelementary to CMP?

Enable Teacher Buy-InThe size, composition of staff, and past experiencesof the school district and staff will determine howthe following actions are handled.

• Consider piloting units before adopting theentire curriculum (if feasible).

• Develop an implementation plan. Will allstudents (and teachers) begin using thematerials at the same time? Will they be phasedin over the course of two to three years?

• Establish plans for long-term professionaldevelopment that coincides with theimplementation schedule (more on this in the next section).

• Designate teacher or building leadersresponsible for scheduling/planningprofessional development.

Districts need to take early action on thepreceding items. The best professional developmentplans have gone astray because schools did not


take the time to share with the key players in theirdistricts the rationale for a new curriculum focusand to develop a wide base of support. If a districtcommittee, representing all the key players, hascollected evidence about CMP as suggested above,then any questions from parents or teachers can behonestly and rationally answered. If the adoptionprocess has been transparent, then the communitywill have had an opportunity to ask questions andseek reassurance.

The Principal’s RoleThe successful principal of a school has manyroles to play as he/she interacts with all membersof the school’s community. Broadly speaking, withdistrict level administrators the interaction isabout the business of managing a school, withteachers the interaction is about the teaching andlearning of a curriculum, and with parents theinteraction is about the success of individualchildren. When CMP is adopted, the principal whotakes the time to become knowledgeable aboutthe curriculum will be better able to supportteachers and answer parent questions. Becomingknowledgeable is the first step.

Becoming Knowledgeable About the Curriculum

The District Textbook Committee In the initialstages of selection and adoption, the districtcommittee will have sought out evidence ofreasons for adopting CMP, of current achievementlevels of students, of the success of CMP in otherdistricts, and of the preparation of teachers toimplement CMP. The district committee will alsohave sought answers to questions about howstudents with disabilities or gifted and talentedstudents succeed in CMP, about the place ofAlgebra in CMP, and about the issues to beconsidered in helping students make successfultransitions from elementary school to middleschool, and from middle school to high school.This evidence and current thinking about theseissues should be shared with principals so thatthey feel confident that a thoughtful decision hasbeen made, and so that they can relateachievement goals and teacher preparation totheir own buildings. The building principal needsto be knowledgeable so that he/she can speakconfidently and supportively about the curriculumand related issues to teachers and to parents.

Since CMP is standards based and problemcentered, the principal will need to find time tounderstand what these two terms mean (see pages3, 6–7, 27–28 of this document), and how this kindof curriculum looks in a classroom when it is beingsuccessfully implemented. In this role the principalis acting like the principal teacher in the building,asking questions such as, “What research backs this kind of approach? What does this kind ofcurriculum mean for student groupings? What roledoes the teacher have? What help will the teacherneed in organizing this kind of classroom? Whatcan I do to help with classroom issues?” If theprincipal is knowledgeable about the curriculumand teacher needs, then more teachers are apt to“buy-in” to the curriculum.

After becoming knowledgeable about thecurriculum, the principal will also have to considerhis/her role in professional development activities.There is only so much a principal can do to learnabout a curriculum by reading about it. Theprincipal who actively participates in professionaldevelopment will be much more knowledgeable,and will be perceived as much more supportive by teachers.

Being Supportive of TeachersWhen teachers attend professional developmentactivities to help themselves successfully implementCMP, they learn first hand about the mathematicsin the units, about the connections among units,and about how problems are sequenced to develop mathematical ideas. They also learn aboutpedagogical aspects of the curriculum: why thecurriculum is organized the way it is, what theteacher role is in the launch phase of a lesson,in the student exploration of a problem, and in the crucial summary phase. The principal who ispresent at professional development activities withteachers is likely to be attending less to the actualmathematics under study than to the perceivedneeds of teachers. Just as teachers learn to askquestions about what their students have learnedand how they can be more supportive of studentlearning, so the principal can ask, “Do some of mybuilding’s teachers seem to need more help with themathematics than others? How can I get them thishelp? Who seems fearful or resistant? Why is that?What can I do to increase the confidence of theteachers in my building that they can implementCMP? Do they need more information? Moreencouragement?”


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As the building leader, the principal is the pointperson when questions come from districtadministrators and from parents/guardians. Andas such the principal can be an advocate forteachers. When professional support for teachersneeds district approval, the principal can make acase for what is needed; or when concernedparents/guardians have questions about ateacher’s unfamiliar classroom practices, theprincipal can knowledgeably reassure the parentand support the teacher.

It is important to be supportive of teachers inthe evaluation process. When both teachers andprincipals understand the goals of CMP and howthese goals are achieved, then the process ofevaluating teachers has integrity and validity thatwould be lacking if the principal had one picturein mind and the teacher had another. Evaluationbased on mutually valued goals and practicesfosters professionalism. The opportunity for aprincipal to observe and evaluate how CMP isbeing enacted in a particular classroom, can be an opportunity to reinforce the common goals and aspirations that the building personnel share.

Communicating with Parents/GuardiansThe building principal knows better than any otherdistrict administrator what his part of the districtlooks like, which parents he/she can expect to beinvolved in building activities, and which he/she canexpect to have questions or concerns. The districtcan make general plans and suggestions to involveparents/ guardians, but the principal is uniquelyplaced to know best which plans are good fits forhis/her school. From the list of suggestions below,or from others, each principal must make a wiseselection so that parents and guardians feelincluded, and can be more supportive of theirstudents’ success in learning mathematics.

Involving Parents/GuardiansSome parents and guardians may have beeninvolved in the initial stages described above. Butmost will only become aware that a new curriculumhas been adopted after the fact. They may havequestions about why CMP was adopted, why itlooks different from traditional textbooks, andwhat evidence there is of student success. Asmentioned above, the unfamiliarity of the problem-centered approach may be an obstacle that parentsneed help in overcoming. These are questions thata district committee that has done its work well inthe initial stages can answer, or can arrange to haveanswered in a variety of ways.

Conscientious parents have always beenconcerned about their children’s middle schooleducation. Their concerns usually have twodistinct foci:

• What is my role in helping my child besuccessful now?

• How well does this class prepare my child forhigh school mathematics and for post-secondaryeducation?

Keeping Parents and Guardians InformedParents/guardians need to understand the goals of the program. Administrators and teachers canhelp them do this by keeping them informed,early and often, about both long-term and unitgoals. They should know that the primary goal of CMP is to have students make sense ofmathematical concepts, become proficient withbasic skills, and communicate their reasoning andunderstanding clearly. The concepts and topicsthat students study should be familiar toparents/guardians, but the problem-centeredtextbooks may not make the particular topic orskill as explicit as the associated student work and reflections will. Parents and guardians needadvice and help in making good use of theirstudents’ classwork as a resource.

The emphasis in reasoning and communicationmay be less familiar. Curriculum leaders andteachers can help parent/guardians understandwhy reasoning and communication are valued andthat the program provides many opportunities todemonstrate students’ progress in these areas.There are many specific ways that a district cangain the support of parents and guardians, andkeep them informed:


Form a Community Advisory Group The groupshould be composed of knowledgeable and strongadvocates for the program. The committee shouldconsist of parents and guardians, teachers,university people (if there is a university in thearea), business people (particularly those whoappreciate the need for critical thinking) andadministrators. This group will play a crucial rolein the early stages of implementation, and less ofa role as the success of CMP speaks for itself.

Present Information to the Community Asimplementation matures, a district might create apamphlet for parents and guardians, including theresults of district evaluation studies showing howwell CMP students did on state tests.

Conduct Parent Workshops These can be helpfulat the beginning of the school year, and atdifferent stages of the implementation during theyear. Topics to be discussed might include:overarching goals, evidence of effectiveness ofCMP, specific mathematical information, theinstructional model, mathematical expectationsfor students by the end of the year and the end of the program, the use of calculators, andtransition to high school. An effective strategy forconducting these workshops is to engageparents/guardians in a problem from one of thestudent units, so they can experience first handhow understanding and skill are developed inCMP. These workshops might be tailored to fitspecific concerns such as use of calculators andother technology, and how this affects learning orthe particular mathematical goals of a unit that isabout to start.

Send an Introductory Letter An introductoryletter complements the Parent Workshopsoutlined above. A sample letter is included in theParent Guide for CMP2.

Send a Parent/Guardian Letter As students begin a new unit, the teacher can send a letter toparents/guardians stating the goals of the unit andsuggesting questions that parents/ guardians can asktheir children. The Parent Guide for CMP2 containsa sample parent/guardian letter for each unit.

Send Home Parent/Guardian Handbooks Adistrict can create and send home handbooksaddressing the mathematics in units and suggestingways that parents/guardians can help their children.

Send Home Newsletters A newsletter is anexcellent way to highlight the mathematicsstudents are studying. A newsletter might includestudent work, stories about student insights,summaries of rich class discussions, or otherevidence of achievement. If your district alreadyhas a community newsletter, then it may bepossible to include news from the mathematicsclassroom in the newsletter.

Inform Parents/Guardians of Resources CMPprovides a parent Web site offering bothbackground information and specific mathematicalhelp to parents seeking to assist their students withhomework. See www.math.msu.edu/CMP/parentsand PHSchool.com.

Tutoring Labs Conduct a tutoring lab after schoolto reassure parents that additional help is availableto students for homework. In one CMP district,a mathematics lab is held two days a week afterschool. Students sign up with their mathematicsteacher to attend, and must bring with them workto do, such as homework, redoing a past assignment,organizing their notebooks, working on vocabularylists or projects, or studying for a test or quiz.Copies of student units and Teacher’s Guides andother materials and tools typically found in theclassroom are available in the lab.

Teachers’ Guides Make a copy of the Teacher’sGuides, with answers removed, available in theschool library for checking out.

Parent and Guardian Role: A Supportive Parent CMP Web siteParents/guardians are an invaluable resource tothe district if their knowledge, good intentions,and caring can be channeled to be compatiblewith the problem-centered approach of CMP.

“The first teachers are the parents, both byexample and conversation.”

– Lamar Alexander.


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In helping children learn, a parent/guardian’sfirst goal should be to assist children in figuringout as much as they can for themselves. They canhelp by asking questions that guide, withouttelling what to do. Good questions and goodlistening will help children make sense ofmathematics, build self-confidence, and encouragemathematical thinking and communication. Agood question opens up a problem and supportsdifferent ways of thinking about it. A list of suchquestions is available at the CMP Parent Web site(www.math.msu.edu/CMP/parents), along withbackground information about the curriculum. Inaddition to general information and advice, theWeb site also offers specific mathematicalinformation. A vocabulary list, with examples toilluminate meaning and use of new vocabulary, andexamples of solutions of homework exercises arejust two of the aids that parents/guardians will findat this site. This site can help parents/guardians tohave meaningful mathematical conversations withtheir children.

Parents/guardians are some of theknowledgeable experts in their child’s universe.Their expertise may be in the mathematicalideas, or in the learning process itself. They can help with homework by learning how toscaffold a problem for a child, without takingaway all the gains to be made from the student’sindividual struggle.

Selecting and ImplementingConnected Mathematics Unitsin Grades 6, 7, and 8

The units in Connected Mathematics wereidentified, developed, and carefully sequenced tohelp students build deep understanding and skillwith important mathematical ideas. Once a unithas been taught, then the understandings andprocedural skills developed in the unit are used in

succeeding units to build unerstandings of newconcepts and skills. Therefore the order in whichthe units are listed for each grade is therecommended order for teaching the units. (Seepage 15 for the units in the recommendedteaching order.)

Two typical implementation plans are:

• A three-year plan–one grade level at atime, and

• All three grades in the first year.

If the school chooses to implement all threegrades in the first year, it is important for teachersat each grade level to know what mathematics istaught the previous year and the succeeding year.Most of the material that is taught in 6th gradeoccurs in most 6th grade textbooks. However, ifsomething is taught in CMP that was not taughtthe previous year in your school, then the 7thgrade teachers may need to add an extra lesson ortwo to cover this topic. The same proceduresshould be done by 6th and 8th grade teachers.After the first year, there should be very little, ifany, adjustments needed to the curriculum that isprescribed for each grade.

How many units a school teaches per gradelevel is difficult to predict. Length of classperiods, number of days spent on instruction,district obectives, and the background ofincoming 6th grade students are just some of thevariables. Class periods vary from 40 to 90minutes across the country. The difference of tenminutes can mean as much as one extra unitbeing taught.


Planning for the transition to high schoolshould be part of any successful

implementation plan. As soon as the district hasmade the decision to adopt CMP in middleschool, teachers and administrators at the highschool level must take time to get to know theCMP curriculum and what students can beexpected to do, long before CMP students startarriving in 9th grade. Likewise, parents andguardians of middle school students, especiallythose going into 8th grade, should be included inthe information loop about this transition.

Presently a trend throughout the country is tomake the study of algebra a goal for all eighthgrade students. If successful, these students may go on to take calculus, or other advancedmathematics classes, in their senior year of highschool. Certainly, it is a worthy goal for allstudents to become more proficient in algebra andto include more algebra in the curriculum prior tohigh school. Indeed, the Connected MathematicsProject was funded by the National ScienceFoundation and designed by its authors with thisas one of its goals.

One strategy, tried by some schools, is to movethe traditional Algebra 1 course to 8th grade.However, experience has shown that many eighthgrade students fail a traditional Algebra 1 course,and must repeat it in high school. A morepromising strategy, recommended by the NCTMPrinciples and Standards 2000, is the developmentof algebraic ideas over a longer period of time,well before the first year of high school, to betterprepare students to deal with abstraction andsymbols. This philosophy is consistent with theway that algebra is taught in other countries. TheNCTM Principles and Standards guided thedevelopment of the algebra strand in theConnected Mathematics Project.

Algebra Goals in CMAlgebra is developed in all three grades ofConnected Mathematics. By the end of Grade 8,CMP students have studied an impressive array of algebraic ideas and skills. Most students shouldbe able to meet the following goals.

Patterns of Change—Functions

• Identify and use variables to describerelationships between quantitative variablesin order to solve problems or make decisions.

• Recognize and distinguish among patterns ofchange associated with linear, inverse,exponential, and quadratic functions.

Representation

• Construct tables, graphs, symbolic expressions,and verbal descriptions and use them todescribe and predict patterns of change invariables.

• Move easily among tables, graphs, symbolicexpressions, and verbal descriptions.

• Describe the advantages and disadvantages ofeach representation and use these descriptionsto make choices when solving problems.

• Use linear and inverse equations andinequalities as mathematical models ofsituations involving variables.

Symbolic Reasoning

• Connect equations to problem situations.

• Connect solving equations in one variable tofinding specific values of functions.

• Solve linear equations and inequalities andsimple quadratic equations using symbolicmethods.

• Find equivalent forms of many kinds ofequations, including factoring simple quadraticequations.

• Use the distributive and commutativeproperties to write equivalent expressionsand equations.

• Solve systems of linear equations.


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High School Math Courses for CMP Students

If the high school offers a standards-basedmathematics curriculum, then the approach willbe compatible with CMP. High school and middleschool teachers need to communicate with eachother about what CMP students can do comingout of Grade 8 in order to make sure that there isno unintended duplication or unexpected gap. Itmay well be the case that students who have beensuccessful in CMP in 8th grade can skip the firstyear of the high school program. Obviously thisdecision can only be made based on knowledge ofboth programs, and the best guides are theteachers involved.

If the high school in the district is still offeringa traditional Algebra 1, Geometry, Algebra 2sequence, then, based on what courses areavailable at 9th grade, and on how successful aparticular student has been in 8th grade CMP,there are several options for the district toconsider. Two options are outlined below. Inneither of these options is it necessary for astudent who has been successful in the algebraunits in CMP to spend a valuable year of highschool in a traditional Algebra 1 class.

Students who have been successful in the CMPalgebra units will have met and mastered many ofthe ideas and skills that are part of a traditionalAlgebra 1. But, they also will have done verymuch more than this in their study of algebra inCMP. Their experience will have been a coherent

functions approach to important mathematicalrelationships, especially linear, exponential,inverse proportion, and quadratic,—includingsolving linear, exponential, and quadraticequations, and inverse and direct proportions.Therefore, CMP algebra units are an excellentpreparation for a traditional functions-basedapproach in Algebra 2. Because of this extensiveand thorough study of algebraic ideas in CMP,many students entering a high school with atraditional curriculum in place may successfullyproceed to Algebra 2.

If, on examining what is expected of studentscoming out of 8th grade CMP, teachers in a highschool offering the traditional curriculum seeskills which they believe are integral to Algebra 1and which CMP students have not met, then theymay create a short “patch” which can be added to the 8th grade CMP units. However, Algebra 2textbooks typically include a lot of review ofAlgebra 1, and, therefore, would review andsupplement what students know from CMP.

In summary, many students who complete all 8 algebra units of CMP2 and meet other districtcriteria may successfully proceed to a traditionalGeometry and/or Algebra 2 course. Whateveroptions are offered to students entering 9th gradeafter a successful CMP experience in 8th grade,they should be based on teacher input, knowledgeof the CMP curriculum and the high schoolcurriculum, data about student achievement—particularly on algebraic topics—and input from all the professionals involved.


Successful Long-TermProfessional DevelopmentExperience and research suggest that effectiveprofessional development models have somecommon characteristics. Effective, professionaldevelopment

• Begins prior to curriculum implementationand continues through several years ofimplementation.

• Is centered on the particular curriculum thatwill be/has been adopted, in this case, CMP.

• Develops teachers’ knowledge of mathematicsand pedagogy.

• Models and reflects good mathematicalpedagogy.

• Addresses teacher concerns about change.

• Involves teachers in reflecting and planningfor improvement.

• Creates strong leadership.

• Includes a plan for training new teachers asthey join the district.

• Reflects strong support from administrationand parents.

• Establishes an expectation among teachers ofworking together to learn from and with eachother during and after the formal professionaldevelopment has ceased.

Change in itself can be problematic and thechanges for some teachers associated with usingany standards-based curricula are significant. TheConcerns-Based Adoption Model (CBAM) (Hall& Hord, 1987, Hord, et al. 1987, Loucks-Horsley,1989, Friel and Gann, 1993) offers help inaddressing these issues. Teachers may need helpmoving through levels of concern, from non-awareness to ownership of new ideas, from a focuson themselves and their own needs to a focus ontheir students’ learning needs. The stages ofconcern can be described as

• Self-concerns—What is this new change andhow will it affect me?

• Task-oriented concerns—How do I implementthis change? What do I need to do to make thischange happen with my students?

• Impact-oriented concerns—How are mystudents learning? Are they learning more andare they learning better? How do I work withothers who are also implementing these newideas?

Progressing through these stages of concernwhile one is implementing CMP takes time—twoto three years is a good target.

The change process is ongoing with differentneeds surfacing during the period of professionaldevelopment and implementation. Early in theprofessional development component, time isneeded to address teachers’ concerns aboutimplementing CMP. In the beginning theseconcerns may tend to focus on management,grading issues, special needs students, tracking,skills, transitions to high school, etc. While theseissues are important and should be addressed,they can divert attention from content andinstruction. These concerns can be addressedgradually during the first phase of professionaldevelopment. Let teachers have time to get theirconcerns on the table early in the process and beassured that these concerns will be addressed.Many of the concerns become less urgent as theteachers engage in studying the mathematics andsharing their knowledge with colleagues. Theseexperiences help teachers integrate previousteaching practice with new expectations.

Good professional development to support astandards-based curricula like CMP weavesmathematics, pedagogy, and assessment together.To make significant changes, professionaldevelopment must address teachers’ stages ofconcerns and concurrently provide opportunitiesfor growth. Growth should focus on

• Developing a deeper understanding andbroader view of mathematics (mathematicalknowledge).

• Strengthening teachers’ pedagogicalknowledge (teaching & learning).

• Exploring assessment aligned with inquiry-based instructional strategies (assessment).


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Professional development must be based onsound criteria and principles that have evolvedfrom research and been verified in experience.The research discussed above as well as otherresearch described by Loucks-Horsley et. al(1996), the extensive experiences of the CMPauthors and staff, and the Professional Standardsfor Teaching Mathematics (NCTM 1991) serve as important references for our professionaldevelopment design. Examples of professionaldevelopment plans are available on the CMP Website. See www.math,msu.edu/CMP: ProfessionalDevelopment.

Three components—content, teaching andlearning, and assessment—are core areas of theprofessional development model and each comesto the foreground at critical times during theprofessional development. Below is a rationale forthe components that need to be included, and anorder of inclusion that has been successful.

Mathematical Knowledge An effective professional development modelassociated with preparing for and implementingCMP begins with an emphasis on mathematicalcontent, with supporting pedagogy being modeledby the professional development leader. Teachersneed to be comfortable with the mathematicsembedded in the problems in order to begin toexamine how the materials can be taught to reachtheir full potential.

Teachers benefit from examining the completepicture of how mathematical ideas build onprevious ideas and how those ideas in turn providethe foundation for the mathematics in later unitsand in subsequent grades. Even teachers who have taught mathematics for some time will find that ideas that they have accepted withoutquestioning are presented in a new light, one thatilluminates both meaning and connections to othermathematics and to other uses of the mathematics.There is value in trying to see mathematical ideasas they are first encountered by a student, ratherthan reproducing what has been stored in memory.Asking, “How do I know this? Why does it makesense?” are not questions that teachers typicallytake the time to ask about familiar mathematics,yet they are at the core of understanding howstudents learn new material.

Good instructional decisions and practice relyon deep understanding of the mathematics that

is embedded in the problems. We suggest thatdeveloping the mathematics of the units in early professional development be given primary focus, with any pedagogical discussionsfocusing only on how to help students learn themathematics. Discussions of management andassessment are more effective if they occurtoward the end of the early professionaldevelopment. First developing mathematical andpedagogical content knowledge keeps theprofessional development from becoming miredin discussions of issues that as yet have no realbasis for a substantive conversation.

Teaching and LearningHaving developed a better understanding of themathematics within CMP, the focus of professionaldevelopment can shift teaching and learning to theforeground. Teachers need to experience inquiry-based pedagogy in their professional developmentso that it will serve as a model for their ownteaching. They also need to be involved in sufficient conversations about teaching problem-centered materials to feel comfortable duringimplementation.

More in-depth work on instruction afterteachers have experience in teaching units isneeded and is very effective in improving teacherpractice. The idea that teachers should beencouraged to reflect upon, revise, and refine theirinitial understandings of the mathematics in a unit,and of ways to teach, after practical experienceparallels the learning process that is expected ofstudents. It is worth noting that when the focusshifts to pedagogy, teachers continue to developtheir own understanding of the mathematicsthrough conversations that analyze student workand assess student understanding.

While the teacher and student books serve animportant role in helping the teacher implementthe curriculum within their classroom, teachersalso need time away from their classroom to talkwith peers and to fully investigate the potential ofthe curriculum.

AssessmentOnce teachers have begun using inquiry-basedinstruction in CMP, it becomes clear thattraditional forms of assessment that focus only onskills may be insufficient to gauge the depth andbreadth of student learning. CMP offers a variety


of forms of assessment to support teachers,including embedded assessment, which may beunfamiliar to teachers.

Orchestrating different types of assessmentrequires new skills for teachers and should beincluded in the professional developmentprogram. Figuring out how to assess and how tograde assessments is a concern of teachers thattends to arise as implementation progresses.Therefore the time for assessment to be anemphasis in professional development is afterteachers have experienced some of thecurriculum. Equally important is the influence ofassessment as evidence to promote teacherreflection and decision making. For example, withsupport and experience, teachers begin to seeassessment as data to drive instructional decisions.

Contexts for Professional GrowthThe activities of a strong professional developmentprogram for teachers implementing CMPemphasize four areas.

• experiencing• planning• teaching• reflecting

ExperiencingFirst and foremost, to implement a problem-centered curriculum like CMP, teachers need deepunderstandings of the key mathematical ideas andways of reasoning that are embedded in solving theproblems within a unit. In addition, they need to seehow understanding of these ideas develops overtime and connects to content in other units. Thus,during the workshops (or whenever teachers arebeing introduced to a new unit), teachers shouldexperience the curriculum in a way that is similar towhat their students will experience. This does notmean they need as much time for each problem,nor that they must do every problem. Problems forthe workshops should be chosen to highlight thedevelopment of key mathematical ideas. Thesupporting problems can be more quickly examinedso that the flow of development is clear, but themain focus is on the key idea.

Professional development leaders should modelgood teaching; they should set a context for teacherlearning, encourage teachers to investigate, and

help teachers make their conclusions explicit. Thisallows teachers to focus on making sense of themathematics needed to solve the problems posed.By setting the context as, “How do you think your students might solve this problem?”, theworkshop leader can shift the focus to students’understanding. Teachers should be encouraged tomake a good faith effort not to superimpose theirown store of remembered knowledge on to eachproblem. The goal is not so much to find answers to the problem as it is to ask, “What would mystudents bring to this problem? What solutionstrategies might they try? Which seem productiveand rich in mathematical ideas? What are some ofthe misunderstandings that students might evince,and how can I best use discussions around thesemisunderstandings to help everyone learn more?”

Some teachers may think that the problems, orthe mathematical ideas, are too hard. A powerfulstrategy for helping teachers with the mathematicsand showing what students can learn is to useexamples of student work. This alleviates theanxiety of teachers who have never learned orunderstood the mathematics in the problem,or have no confidence in their ability to do themathematics. It allows teachers to ask questionsthey might be reluctant to ask. In such anenvironment teachers can and do learn themathematics of a unit. Positioning the mathematicsand the teaching through the lens of the studenthelps provide a comfortable environment fordiscussions of teaching and learning of themathematics.

Through follow-up discussions of the problems,the mathematical potential of the problems, thereasoning that students employ, and the connectionsthat can be made become more explicit. Throughsuch interactions teachers begin to value suchquestions as, “What is the mathematics? At whatstage are we in the development of understanding ofthe key idea? What do students need to bring to theproblem? To what do these ideas connect in astudent’s future study of mathematics?”

While effective teaching strategies are modeledand occasionally discussed during the study of aunit by the participants, it is most effective ifattention to teaching becomes explicit in theprofessional development. Teachers will need helpwith the teaching model. Knowing how to launcha problem, how to assist and guide all studentsduring the exploration, and how to summarizestudent understandings and strategies are very


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crucial to the development of the mathematics.A good stimulus for discussions on teaching isobservation of good teaching, either live in aclassroom or on video. Analyzing students’strategies can lead to conversations on how theclassroom environment/discussion may haveaffected learning. This is also a time for teachersto experience collaborating to make sense of whatevidence there is of good teaching. Developingthe habit of asking, “What aspects of the launchwere effective? What aspects of the summary wereeffective? How would I address that studentquestion?” prepares teachers to make ofthemselves the same demands.

PlanningPlanning is key to success with any problem-centered curriculum such as CMP. Professionaldevelopment activities should include opportunitiesto plan collaboratively. Occasionally teachers shouldbe asked to plan together to teach a problem,asking, “What is the mathematics? What difficultieswill students encounter? What mathematicaldiscoveries might they make? How will I launchthis? (See page 12.) It is crucial that administratorsrecognize that, while the planning load reducessomewhat after the initial implementation stage,there will always be a need for teachers to planlessons and reflect on what students learned fromthe lessons, and for administrators to help find time for these planning and reflecting activities.This is a way to optimize and continue professionaldevelopment.

For each class session it is important for teachersto identify the mathematical concepts or strategies,their stages of development, and the time neededto develop these understandings. The power ofCMP does not lie in any one activity or any oneunit. Important ideas are studied in depth within aunit and further developed and used in subsequentunits. It is both the depth of understanding withinunits and the careful building and connecting of theunits that allow students to develop to their fullestmathematical potential.

Initial planning can occur in the first summer,prior to the implementation of the CMP. However,teachers also need time during the year to plan,particularly with their colleagues. Planning sessionsallow teachers to share problems they haveexperienced, learn new ideas from their colleagues,probe the mathematics more deeply, look forconnections, and plan upcoming class sessions.

Once teachers are comfortable with themathematics and inquiry-based instructional model,they are ready to look more closely at assessmentand how to use assessment to evaluate students’knowledge and to inform their teaching. Fullerdiscussions on assessment are appropriate duringthe second year of teaching CMP and continuedprofessional development. However, examiningstudent work with a colleague is valuable. Askingquestions about what the students’ work shows notonly deepens teachers’ knowledge, but it can alsoserve as a guide to planning effective teachingstrategies. Planning also allows teachers time todiscuss and share management and gradingstrategies as well as ways to address the needs ofdiverse student populations.

TeachingTeachers need to think critically about creating aclassroom environment that fosters students’expectation that they will work together to solveproblems, reason about possibilities, justify theirideas, and solutions, and look for connections.Posing problems that provide a challenge for thestudents, allowing students to explore the problemand guiding class discussion on the solution of theproblems requires the teacher to play many rolesat the same time.

Teachers need help in learning how to askeffective questions that can guide and probestudents’ understanding, and at the same time theyneed to learn to listen carefully to their students.These are not skills that teachers, even those with many years of experience, have traditionallypracticed. District administrators who take thetime to become knowledgeable about inquiry-based learning are better able to support teachersdirectly. They can help set the expectation thatteachers will collaborate and learn from and witheach other as the curriculum is enacted.

Setting and achieving high expectations forunderstanding, problem solving, representing, andcommunicating for all students is a task thatconfronts teachers on a daily basis. Reflecting onone’s practice with a lens on student understandingis important for teachers to make progress.Establishing the kind of environment in a schoolwhere administrators support and expect teachersto collaborate with each other can change thewhole school’s daily focus to such teacherquestions as, “What evidence do I have that mystudents learned something? What did they learn?”


Reflecting on LessonsProfessional development activities should modelreflective practices. It is through reflection ontheir teaching and their students’ understandingthat teachers continue to grow in their capacity tobuild powerful mathematical experiences for allstudents. Planning with a colleague, peer coaching,observing a peer, or sharing with colleagues aresome ways to encourage reflection.

Videotapes of lessons can serve as a catalyst forreflections. Caution must be exercised if videosare used. If a teacher within a building agrees tobe video taped, then the focus should be students’learning rather than critique of the teacher.Centering conversations on student learning is away to help the teacher think about his or herpractice. “What is there about my students’ ways ofapproaching the problem that I like?” Why do Ithink this is effective? What should I do toencourage more of this? What aspects of mystudents’ actions are not productive? Why is this?What can I do to redirect my students?” Findingthe fine line between trying to help the studentsbe successful with the problem and allowing thestudents freedom to explore a more open problemwill take reflection and growth over time.

Similarly, using rich collections of student workin professional development activities can focusteachers’ attention on the role and importance ofthe summary phase of the lesson. When teachersstudy a collection of student work, some say thatall of the solutions are acceptable, or they correctthose that are not, and go on to the next lesson.But it is the analysis and comparison of thecollection of student work that can bring theimportant mathematics to the forefront. Studentwork can also be the way to center discussionsand reflections on students’ understandings.

A variety of assessment tools can be usedincluding mathematical reflections, quizzes, unittests, projects, and district wide instruments. To beeffective, discussions on student learning should gohand in hand with discussions on teaching. A focuson student learning leads naturally to looking atthe development of ideas over time. Talking andplanning with colleagues in different grade levelsprovides the opportunity for teachers to build andshare a coherent curriculum vision. Collaborationand reflection are key elements in creating acommunity of teachers and administrators withinthe school that can support improvement inteaching and learning over time.

Teacher’s Guides andProfessional DevelopmentThe teacher materials that accompany each unit of CMP offer help with the same components thatare included in a good professional developmentprogram. Part of the professional developmentshould be to model how these materials might beused. But these materials should not be considereda replacement for the professional developmentnecessary to get teachers started and for thedistrict’s support in keeping teachers enthused andlearning after the initial implementation phase.Each unit contains:

• Help with the mathematics—an in-depth lookat the mathematical ideas and how these aredeveloped.

• Suggestions for planning effective launches,asking good questions, and leading powerfulsummaries.

• Suggestions for good assessments and how tomanage them.

• Help in working with diverse or special needspopulations.

• Connections to prior and future units andassistance in tracking where students shouldbe in their development of key ideas.

If we believe it is a worthy goal to establish asafe and healthy environment in each classroomto enable students to learn together, then itbecomes equally important that we acknowledgethat teachers need time and opportunities to workwith each other. No matter how informative theteacher support materials are, teachers will getmore from them if they can plan or reflect with a colleague.


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Encouraging teachers to share their successesis one way for schools and districts to

promote teacher ownership of the curriculum.Sharing can be done within the school throughmentoring new teachers or through sharedplanning times with other teachers, through districtnewsletters, through online discussion groups, orby volunteering to speak at local or state meetings.Networking, going to professional meetings, andjoining mathematics teachers’ association are all ways to continue to grow in mathematicalknowledge and in pedagogical strategies.

Often overlooked is the problem of teacherturnover that occurs in virtually all middleschools. It is critical to develop a plan to provideprofessional development for new hires. It isequally important, and not simple, to developcollaborative relationships with experiencedmathematics teachers. Such relationships aremutually beneficial to the new and experiencedteachers, and in many instances result in loweringthe rate of attrition of new teachers.

As teachers become comfortable with CMP, itbecomes a natural part of the fabric of the school.A sense of complacency, a “We’ve done it!”feeling, sets in. This is a time for taking a moreexacting look at the potential of the curriculum.These more advanced professional developmentexperiences can re-energize teachers and result in improved student learning. Moving to this next level of implementation is a crucial step andone that is very often overlooked. Professionaldevelopment is not a one-time nor a briefexperience, but the essence of having teachersstaying fresh, enthusiastic, and highly effective.

The Strength of CollaborationIt is through collaboration that progress is madeand continued. It is through these collaborativeprofessional exchanges of sharing ideas, planning,examining student work, looking for gaps, andfinding ways to make even bigger gains in studentunderstanding, reasoning, and communication thatwe continue to move forward. In the early stagesof implementation, the community may includethe entire staff of mathematics teachers, but asimplementation continues, it is likely that teacherswill rely heavily on their grade-level colleaguesfor support, ideas, and guidance. Professionaldevelopment opportunities are needed to ensurethat these collaborations are able to continuethroughout the implementation, even after thecurriculum appears to be institutionalized at the school.

Some issues that collaboration might focus onare: student understanding, perceived weaknessesas evidenced on local and state testing, teacherstrategies, reports from teachers who haveattended state or national conferences, preparingpresentations for administrators or parents or stateor national meetings, effective use of technology.

In SummaryWorking as a community ahead of time to carefullyselect materials, become knowledgeable about thestrengths of the materials and the reasons for thechoice, examine the transitions points for studentscoming into CMP as well as the transition into highschool, prepare the parents, administrators, schoolboard members, and plan the implementation andprofessional development for teachers, will make asmooth and powerful implementation of CMP. Allthe constituent groups in the system are important in the education of children and, as such, need tobe informed so that they can offer the most helppossible to teachers and students.


This section provides information about planning to teach CMP and suggestionsfor management. The discussion and suggestions come from our work with

experienced CMP teachers. This material is intended to be read and used by teachers,so the text addresses teachers directly.

Connected Mathematics may be very different from curricula with which you arefamiliar. Because important concepts are embedded within problems rather thanexplicitly stated and demonstrated in the student text, you play a critical role inhelping students develop appropriate understanding, strategies, and skills. It is yourthoughtful engagement with the curriculum and your reflections on student learningthat will create a productive classroom environment.

In planning to teach a unit, the first thing you need to do is become familiar with the content and the way the concepts, reasoning, and skills are developed. As youprepare, you will want to try to anticipate your students’ learning and assess wheredifficulties might occur. The following section provides suggestions you can use as a guide as you plan to teach a unit.

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Planning—Getting to Know a CMP UnitThe first stage in planning to teach a unit isbecoming familiar with the key concepts and theway the unit develops concepts, reasoning, andskills. In general, the unit subtitle gives a broadview of the important ideas that will be developedin the unit. For example, the Moving Straight Aheadunit has the subtitle “Linear Relationships,” whichidentifies linear relationships and functions as thecentral idea. What the title does not reveal arewhat aspects of linear relationships are developedand how understanding is enhanced. The followingsuggestions can serve as a guide for getting toknow a unit at this more detailed level.

The Mathematical Ideas in a Unit

• In the Teacher’s Guide, read the introductorymaterial including the Goals of the Unit,Developing Students’ Mathematical Habits,Overview, Mathematics Background andContent Connections to Other Units. These willgive you a broad view of the mathematicalgoals and connections to prior and future units.

• Read the Summary of Investigations near thebeginning of the Teacher’s Guide and theMathematical Reflections in the studentbooks at the end of each Investigation. Theseoutline the development of the mathematicsin the unit.

• Look over the Assessment Resources for theUnit. They give you an idea of what studentsare expected to know at various points in theunit, and the level and type of understandingstudents are expected to develop.

The Development of the Ideas in a UnitTo help you investigate the details of concept andskill development and guide you as you teacheach investigation, read the student unit and all ofthe problems and ACE questions. Then read theMathematical and Problem-Solving Goals and theSummary of Problems that are given at the startof each Investigation. Ask yourself questions suchas the following:

• What part of the main mathematical goal ofthe unit is being developed? How does eachproblem in the Investigation contribute to thedevelopment of the mathematics? What levelof sophistication do I expect my students toachieve in answering the problems in theInvestigation?

• How will student responses show developmentin understanding the big ideas of the unit?

• What mathematical ideas will need emphasis?

• What connections can be made among theproblems in this Investigation, to otherInvestigations in this unit, and to other units?

• How can I structure the writing assignment forthe Mathematical Reflections so students getthe most from it?

• What ACE questions are appropriate for mystudents to do after each problem?

• How long should this Investigation take?

• What can I do to assure the amount of timespent in class is appropriate for the problemsand the goals of the Investigation?

Guidance in answering these questions can befound throughout the Teacher’s Guide in PacingCharts, Assignment Guides, and sample answersfor ACE exercises and Mathematical Reflections.

Teaching a Student UnitThe role of the teacher in a problem-centeredcurriculum is different from the curriculum inwhich the teacher explains ideas clearly anddemonstrates procedures so students can quicklyand accurately duplicate these procedures. Aproblem-centered curriculum such as ConnectedMathematics is best suited to an inquiry model of instruction. As the teacher and studentsinvestigate a series of problems, it is throughdiscussion of methods of solutions, embeddedmathematics, and appropriate generalizations thatstudents grow in their ability to become reflectivelearners. Teachers have a critical role to play inestablishing the norms and expectations fordiscussion in the classroom and for orchestratingdiscourse on a daily basis. It is through theinteractions in the classroom that students learn


to recognize acceptable mathematical practices,and those needing explanations or justifications.

The CMP materials are designed in ways thathelp students and teachers build a differentpattern of interaction in the classroom. The CMPmaterials are written to build a community ofmutually supportive learners working together tomake sense of the mathematics. This is donethrough the problems themselves, the justificationstudents are asked to provide on a regular basis,student opportunities to discuss and write abouttheir ideas, and the help provided to the teacherthrough the assessment package and theembedded problem-centered instructional model.In addition, the following are useful:

• To help teachers think about their teaching,the three-phase instructional model containsa launch of the problem, an exploration ofthe problem, and a summary of the problem.(See a detailed discussion of the instructionalmodel for teaching CMP on pages 73– 74).

• The teacher is provided with detailed help—Investigation by Investigation, and Problemby Problem. The Teacher’s Guide contains a discussion of the Launch, Explore, andSummarize phases for each Problem. Thesediscussions contain specific help on the focusfor each Problem, how to build on previousProblem(s) or Investigation(s), what strategiesor misconceptions students might have, andconnections to other mathematical concepts.Also included are suggestions for specificquestions to ask during each phase ofinstruction. Before you engage your class in a Problem, you will find it helpful to read thedetailed teaching notes for it.

• The discussion on Organizing the Classroom(see page 101) contains helpful suggestionsfor organizing the classroom and encouragingstudent participation.

Reflecting as You TeachThe following questions are all part of teacherreflections on the effectiveness of the classroomenvironment:

• Do the tasks engage the students, and are theyeffective in helping them learn mathematics?

• Do the activities stimulate the richness ofdiscussion that helps students to developmathematical power?

• Does classroom discussion encourage learnerindependence? Curiosity? Mathematicalthinking? Confidence? Disposition to domathematics?

• Does the classroom environment reach everystudent and support his or her mathematicaldevelopment?

• What do my students know? What is theevidence? How does this shape what I plan fortomorrow?

It is through reflection that teachers continueto grow and to develop the kind of classroomenvironment that encourages all students tobecome independent, confident, and reflectivelearners. The suggestions below are adapted fromthose submitted by CMP teachers:

Using Feedback From ClassIn their Teacher’s Guide or in a separate notebook,many teachers write brief notes or comments onimportant ideas or suggestions for what worked and what to do differently the next time they teachthe unit.

• Use the classroom discussions, homework, orMathematical Reflections as benchmarks foryour students’ understanding.

• Re-evaluate where you and your students areeach day.

• Reflect on each student’s understanding. Whatdo you know about this student? Is this studentparticipating in class discussions? Is he or shecompleting homework?

Finally, at the end of each day, each Investigation,or each unit ask yourself:

• What evidence do I have of what my studentslearned?

• How should this affect my instructionaldecisions?

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Collaborating With ColleaguesMany teachers have found it valuable to plan with a colleague before, during, and after teaching theunit. Very often, student work is a focus for theirdiscussions, as it provides a platform for discussingthe mathematics in the Unit, Investigation, orProblem. Discussion can also cover effectiveteaching strategies and other issues related toteaching. The following sets of summary questionscan be useful for working either alone or withcolleagues. The Teacher’s Guides also contain awealth of information to help you plan your lessons.

A Quick Guide to Planning

GETTING TO KNOW THE UNIT

• It is important to understand the mathematicsand how it is being developed. Read the Goalsof the Unit, Mathematics of the Unit, andContent Connections to Other Units.

• Read the Mathematical Reflections in theStudent Unit—they tell the story of themathematics that is being developed in the unit.

• Look over the Assessment Resources.

• Work all of the Problems and ACE for eachInvestigation.

• Make use of the help provided in the studentand teacher books for teaching.

• Use the Launch-Explore-Summarize (LES) asa guide for teaching each Problem.

• Keep notes on important ideas or suggestionsfor the next time you teach the unit.

• Use the Mathematical Reflections asbenchmarks for your students’ understanding.

• Reevaluate where you and your students areeach day—teacher reflections are an importantpart in becoming a more effective teacher.

• Use the following questions as you plan toteach the Unit, each Investigation, or eachProblem.

Questions to Think AboutUNIT BY UNIT

• What are the big mathematical ideas of thisunit?

• What do I want students to know when this unit is finished?

• What mathematical vocabulary does this unitbring out?

• What might be conceptually difficult?

• What are important connections to other units?

Questions to Think AboutINVESTIGATION BY INVESTIGATION

• What part of the mathematical goal is beingdeveloped?

• How does each Problem in the Investigationcontribute to the development?

• What level of sophistication do I expect mystudents to achieve in answering the questions?

• Will their responses show the development intheir understanding the goals of the unit?

• What ideas will need emphasis?

• What are the connections among the Problems,Investigations, and with other Units?

• How can I structure the writing assignment forthe Mathematical Reflections to get the mostfrom them?

• What ACE questions are appropriate for mystudents to do after the 1st problem, the 2ndproblem, etc. in this Investigation?

• How long will this Investigation take?

• What can I do to assure the time spent in classmatches the size of the problems and the goalsof the Investigation?

Questions to Think AboutPROBLEM BY PROBLEM

Launch

• How will I launch this Problem?

• What prior knowledge do my students need tocall upon?

• What do the students need to know tounderstand the story and the challenge of theProblem?

• What advantages or difficulties can I foresee?

• How can I keep from giving away too much ofthe Problem?

• How can I make it personal to them?

Explore

• How will I organize the students to explore thisProblem? (Individual? Pair? Group? Wholeclass?)

• What materials will students need?

• What are different strategies I anticipate themusing?

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• What kinds of questions can I ask:—to prompt their thinking if the level of

frustration is high?—to make them probe further into the

Problem if the initial question is “answered”?—to encourage student-to-student

conversation, thinking, learning, etc.?

Summarize

• How can I help the students make sense of andappreciate the variety of methods that may occur?

• How can I orchestrate the discussion so studentssummarize their thinking in the Problem?

• What mathematics and processes need to bedrawn out?

• What needs to be emphasized?

• What ideas do not need closure at this time?

• What do we need to generalize?

• How can we go beyond? What new questionsmight arise?

• What will I do to follow-up, practice, or applythe ideas after the summary?

Teacher’s ReflectionsAt the end of each day, Investigation, or Unit,ask yourself:

• What evidence do I have of what my studentslearned?

• How does this affect my instructional decisions?

Finally, it is important to remember that “Romewas not built in one day.” It takes time andpatience to become the teachers we all aspire to be.

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Organizing the ClassroomHelping students become independent learners is an important goal for Connected Mathematics.The Teacher’s Guides point out opportunities forhelping students reach this goal.

Classroom SetupThe way your classroom is set up can have asignificant impact on learning. Here are somesuggestions for creating an effective learningenvironment for students:

• Post the Mathematical and Problem-SolvingGoals for the unit given in the Teacher’sGuide and student books and check them offas the class meets them.

• Post a vocabulary list in the room so studentsknow what words should be in theirglossaries. Add words to the list for each unitas you proceed through the unit.

• Keep a list of assignments in the room forstudents who have missed class.

• Post upcoming assessments and due dates sostudents can anticipate your expectations

• Make tools (rulers, grid paper, angle rulers,and so on) accessible so students can decidewhich tools are appropriate for solving aproblem.

• On the board, keep a list of unresolvedquestions for future discussion.

• Have textbooks, a mathematics dictionary, andother reference materials (almanacs, atlases,and so on) available for students to use.

• Have materials on hand that allow students to display their work and share their resultswith the class. Save student work for future reference by the class or for parent meetings. Record student work on overheadtransparencies for class discussion and thenmake copies for absent students or save forfuture reference by the class.

Homework in CMP2In Connected Mathematics, homework takes a roledifferent from that in other curricula. Homeworkin Connected Mathematics is intended as anopportunity for students to think further aboutthe ideas in a lesson. The lesson, rather than thehomework assignment, is the primary unit ofinstruction, with homework as a vehicle forteachers to help students to process, practice,connect, and extend the ideas from the lesson.On a typical day in a Connected Mathematicsclassroom, far less time is spent assigning, doingand checking homework than may have been thecase with other programs. The following sectionscontain some approaches Connected Mathematicsteachers have taken in order to maximize theeffectiveness of their time and students’ timespent on homework.

Assigning HomeworkYou can use the Assignment Guide feature in theTeacher’s Guide to help you assign homework.This feature appears on the At a Glance page foreach Problem and indicates the ACE exerciseswhich students should be able to answer aftercompleting the Problem. The Assignment Guidefor a Problem typically includes questions fromeach of the three ACE sections.

Teachers have generally found that the ACEexercises in Connected Mathematics are moresubstantial than the homework assigned in othercurricula. So they often think differently abouthomework assignments. In particular, teachersbegin to make more careful choices about whichquestions to assign and how to assign and gradethem.

In general, the Assignment Guide in theTeacher’s Guide for an Investigation will includeall ACE exercises in that Investigation. In thespirit of Connected Mathematics materials, manymore ACE exercises are provided than canreasonably be assigned as homework. However,this gives the teacher choices, so the materials canbe tailored to the needs of a particular classroomof students. In addition, different communitieshave different expectations about homework, andclasses meet for different lengths of time. These


and other factors influence the pace and amountof homework assigned.

By answering the ACE exercises yourself beforeyou assign them, you can anticipate difficulties andestimate the time it will take students to completethe assignment. Some teachers read and brieflydiscuss the ACE exercises in class before assigningthem so that students understand what they are to do.

Students should attempt to answer all theassigned ACE exercises, but they may strugglewith some. You might suggest that, if a studentcannot solve a problem, he or she write a questionabout it, such as, “What are ‘increments of 5campers’?” or “Which variable should go on the x-axis?” Questions such as these focus the studenton the area of difficulty, let you know the student’sthoughts about the problem, and give you insightinto the difficulty the student may be having. SomeConnected Mathematics teachers begin class byallowing students to ask questions about theprevious night’s homework. The students are thengiven the opportunity to revise their work beforeturning it in.

Responding to and Grading HomeworkHow you respond to student work will depend on the reason you assigned the work. ConnectedMathematics teachers have listed, among manyothers, the following reasons for assigninghomework:

• To provide additional explanation and practiceof the key mathematical ideas in the lesson

• To grade students’ work

• To assess what students do and do not knowin order to plan instruction

• To connect learning experiences on twoconsecutive days

• To instill good study habits

• To accomplish more mathematical studyoutside the time limits of the classroom

Because ACE exercises are rich, they mayelicit a variety of answers and strategies fromstudents. Dealing with this variety of responsescan be time-consuming for teachers. ManyConnected Mathematics teachers adapt how theyrespond to student homework based on thereason for a particular assignment. Clearly, if anassignment is given to assess understanding toplan instruction, the work will need more careful

attention from the teacher than if it is assigned toinstill good study habits.

Some methods used by Connected Mathematicsteachers to respond to students’ homework arelisted here.

• Prepare an answer key that covers the mainelements of each exercise. (Detailed answersfor all Problems and ACE exercises are in the Teacher’s Guides.) Assign points to eachProblem, and have students correct their workand total the points for correct answers.

• Write the exercise numbers for the previousnight’s assignment on the board. As studentscome into class, have them make a checkmarknext to the numbers for which they havequestions. Discuss only those problems.

• Choose a few exercises to read carefully andgrade; discuss the rest in class.

• Go over the answers in class and havestudents check and revise their work. Then,each Friday choose a small set of exercisesfrom the week’s homework for students toturn in (this is sometimes referred to as a“homework quiz”) for a grade.

• Collect the homework papers and check eachexercise.

• To prevent losing some class time each daywhile checking homework, assign a fewexercises over the course of the week andgrade them all on one day.

• Assign a set of exercises at the beginning ofthe Investigation, informing students which of these exercises they should attempt eachday. Spend a few minutes each day takingstudent questions about the previous night’swork. Collect all of the exercises at the end ofthe Investigation.

• Give complete credit for satisfactorycompletion of the assignment. Give partialcredit, as warranted, based on the number ofexercises completed satisfactorily.

CMP Student NotebooksIt is helpful for students to keep their work in an organized notebook. The notebook can includenotes, vocabulary, solutions to investigationproblems, homework, and responses tomathematical reflections. By reviewing yourstudents’ notebooks, you can get a clearer pictureof their mathematical development.

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Because the Connected Mathematics units arethree-hole punched, students can keep theirbooks, along with their important work, in a three-ring binder. The binder can easily be divided intosections for a journal or notes, homework,vocabulary lists, quizzes, and tests. The binder caninclude work and notes written on loose-leafpaper or in spiral notebooks, which can beremoved when the teacher wants to check somepart of the notebook.

Some teachers have students designate a sectionof the notebook as a journal. In their journals,students record solutions to the Investigationproblems, responses to Mathematical Reflections,

and respond to queries generated by the classdiscussion, the teacher, or other students. Journalsshould be seen as an aid for students as they tryout their thinking and develop complete responsesand thoughtful conjectures.

Some teachers combine the journal and notes.Students record all journal entries on the left-hand side and the notes on the right-hand side.With this arrangement, students and teachers canseparate the experimentation ideas from thesummary of classroom ideas.

The following example shows the notebookguidelines one Connected Mathematics teachergave to her students.

1 These suggestions are adapted from Jan Palkowski, a middle school teacher in Traverse City, Michigan.This teacher also has the students keep daily logs of their participation in classroom activities and dialogue.

Example of a Notebook Organization1

Section 1: FORMS

In this section, keep assignment sheets,participation logs, and classroom rules andprocedures.

Section 2: JOURNAL

This section should include:

• Any and all work you do for in-class problems;this includes your work on InvestigationProblems and follow-ups and any handouts.Include words, charts, pictures, or anything elseto show your thinking.

• Any notes you take; write anything that willhelp you remember your thinking. You shouldalso record notes about the class summary ofthe ideas in each Investigation. These notes arefor your reference as you solve in-classproblems, answer homework questions, workon quizzes, and prepare for tests.

• Your Mathematical Reflections from eachInvestigation.

Section 3: VOCABULARY

In this section, you will create mathematicaldescriptions with examples of words you need toknow. Use lined loose-leaf notebook paper forthis section.

Section 4: HOMEWORK ASSIGNMENTS

This section should include your work on theACE assignments. Your homework should bewritten on lined loose-leaf notebook paper orgraph paper and clearly identified.

Section 5: ASSESSMENT

This section will include all check-ups, partnerquizzes, tests, projects, and self-assessment.

Section 6: YOUR BOOK

Keep your unit inside your binder at all times.Please do the following to help you organizeyour work and to make it easier for me to reviewyour notebook:

1. Date every entry and identify problems withproblem numbers and the unit name.

2. Always revise what you have written bycrossing it out, rather than by erasing. Thissaves you time and helps me to follow yourthinking. It does not count against you to crossout your old work.

I will check your notebooks at unannouncedtimes, and homework grades will be given.

After you complete a unit, clean out all thesections of your notebook except the “Forms”and “Vocabulary” sections. I will file your workfor future reference and portfolio selections.


It is recommended that you check notebooksoften during the first few weeks of the semester.It is important to give students feedback early to make sure notebooks are being used correctlyand to address any problems. Many teachers walkaround the room while students are working andgive comments or suggestions on maintainingnotebooks.

Since keeping notes in mathematics class isnew for many students, it is helpful to keepmodels of outstanding notes. This helps studentsunderstand your expectations. You can photocopygood examples to share with students. It might behelpful to have students evaluate their notes,journal entries, or vocabulary according to themodels. At the end of the year, ask a student ifyou may keep his or her notes for the next yearto have a complete example of how a notebookshould look.

The Notebook Checklist can be used toevaluate students’ notes periodically throughout a unit or at the end of a unit. In the HomeworkAssignments, list the items you would like toassess. Having students assess their notebooksbefore turning them in allows them to criticallyreview their entries and organization.

Many teachers grade the journal, notes, orvocabulary sections of students’ notebooks as wellas the overall organization. Rubrics lend themselvesnicely to the grading of notebooks, as you aregenerally looking for the completeness of ideas,notes, and vocabulary descriptions and records ofthe discussions from class. Some teachers give“Credit,” “Partial Credit,” or “No Credit” as a gradefor notebooks.

There are a variety of methods for checkingstudent notebooks. Here are some ideas youmight try:

• Read and respond to a few students’ journalentries each day.

• Collect papers from students at the end ofeach Investigation. Grade or respond tostudent work and then return the papers forstudents to replace in their notebooks.

• Collect notes at the end of a unit and gradethem.

• Spot-check notebooks while students work onan assessment resource.

• Check notebooks at random.

• Give notebook quizzes. That is, periodicallyhave students copy information from theirnotes on a sheet of paper; then grade just that information. (For example, What were the three strategies we discussed for solvingProblem 3.1? What was the answer to Problem 3.3 Part B?)

VocabularyVocabulary lists appear near the front of eachTeacher’s Guide. These lists are generally dividedinto three categories:

1. Essential terms developed in the unit 2. Terms developed in previous units 3. Useful terms developed in the unit

These lists indicate the mathematical termsdeveloped in the unit. Based on your students’ orschool’s needs, you may add to the lists or shiftwords from the useful to the essential column.You may choose to hand out the list of vocabularyterms from the Teacher’s Guide when you beginthe unit, or you may prefer to have studentsgenerate their own list as they encounter theterms in the unit.

Although there is a glossary in the back of eachstudent book, we suggest that you have yourstudents develop their own lists of definitions andexamples. For important mathematicalvocabulary, students need to have descriptionsthat carry meaning at their level of verbalsophistication. Encourage students to view theirlists as working glossaries that they can add to andrefine as they gain new insight and encounter newexamples. Revising and updating descriptions canhelp students improve their working knowledge of the vocabulary. You might find it helpful tohave students occasionally work in a group or as a whole class to discuss the various descriptionsthey have written.

The vocabulary lists and the definitionsstudents generate can become quite involved andpersonal. Many students like to keep their listsfrom each unit to use as reference tools in laterunits. In some schools, students are required tosave the lists they generate during the year to usethe following year. This helps them make sense ofnew ideas by giving them previous references onwhich to build. Some schools give extra credit tostudents who begin the year with their vocabularyfrom the previous year.

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To keep the vocabulary section organized,students are directed by some teachers to beginthe school year with 26 sheets of paper in thevocabulary section of their notebooks. Each pieceof paper is then labeled with a different letter ofthe alphabet. Students write the words,descriptions, and examples under the appropriateletter. Although the words are not in alphabeticalorder on the page, there is enough organizationfor students to locate specific words.

PacingWhen using Connected Mathematics, teachersshould try to maintain a steady pace that willallow them to get through as much of the materialas possible. Because ideas are developed overseveral problems, it is important for teachers notto spend too much time on any one problem. Insome districts, district coordinators set timelineschedules to help teachers establish a sense ofpacing. Each unit contains pacing schedules for50-minute periods and block scheduling that werebased on field testing. Depending on your districtneeds and schedule, it should be possible to do 6 to 8 units for each grade.

In the first year of implementation, someteachers may feel the need to supplement thematerials with drill and practice. This will take timeaway from Connected Mathematics and slow thepace. Over time, teachers will learn the curriculumand understand that drill is incorporated into the lessons.

Although the primary focus of professionaldevelopment is on the mathematics and pedagogy,teachers who are new to Connected Mathematicsoften have concerns about pacing, homework,grading, basic skills, and collaborative learning.These concerns may affect how a teacher sets thepacing of a unit. These issues should be addressedduring professional development.

AbsenteeismBeing absent in a CMP class is different from being absent in a traditional class. For example,students miss the experience of developing theirunderstanding by working on a Problem anddiscussing key concepts and strategies. InConnected Mathematics, key concepts and skillsare developed over several classroom

Investigations. If students are absent for only a dayor two, they have not missed the entire discussionon a key idea. The following suggestions comefrom CMP teachers.

When Students Are Absent

• Keep assignments and activities posted in theclassroom so students know what they missed.

• Have group members collect any materialsthat are passed out for absent group members.

• Establish note-taking buddies so studentshave someone to provide the notes.

• Keep a master copy of the classroom notesfor students’ reference.

• Have group or class members summarizewhat was done the previous day.

When Teachers Are AbsentMany teachers have found that, without someprofessional development, it is difficult for asubstitute teacher to teach a Problem or lesson.Suggestions of activities that can be done when asubstitute teaches the class are given here.

• Partner Quizzes Pairs can work on quizzesfairly independently, using their notes andbooks as resources. When it is possible,assigning partners the day before can reduceconfusion as the class begins.

• Review Compile worksheets, using theadditional practice problems from the unitsyou have completed. If you teach seventh oreighth grade, you can use the problems fromprevious years.

• ACE Exercises Assign a set of ACE exercisesto be done in class. You can provide incentivesfor students. For example, if work is donediligently the first part of the period, you mightallow students to work with a partner for thesecond half of the period.

Note that all three suggestions work well if theteacher knows in advance that he or she will beabsent. The last two suggestions also work forunexpected absences.


Student Configurations and Classroom ParticipationConnected Mathematics provides opportunities for students to tackle mathematical problemsindividually, in pairs, in small groups, and as awhole class. Each of these arrangements ofstudents enhances learning. The way you groupyour students will depend on the size, nature, anddifficulty of the task. For a particular Problem,students might do individual work as part of theLaunch phase or the launch may be a whole-classdiscussion. In the Explore phase, students maywork individually, in pairs, or in small groups. Theymay take part in a whole-class discussion of aproblem during the Summarize phase. Therationale for each of these grouping decisions is the nature of the problem and the goals of thelesson. The Teacher’s Guide for each unit offersspecific suggestions for grouping.

Whole-Class WorkThe Launch of a CMP lesson is typically done as a whole class, yet during this launch phase ofinstruction, students are sometimes asked to thinkabout a question individually before discussingtheir ideas as a whole class.

However, it is during the Summarize phase,when individuals and groups share their results,that substantive whole-class discussion most oftenoccurs. Led by the teacher’s questions, thestudents investigate ideas and strategies anddiscuss their thoughts. Whole-class discussionallows a variety of ideas to be presented and themathematical validity of solutions to be tested.Questioning by other students and the teacherchallenges students’ ideas, allowing importantconcepts to be developed more fully. Workingtogether, the students synthesize information,look for generalities, and extract the strategiesand skills involved in solving the Problem. Sincethe goal of the summarize phase is to make the mathematics in the problem more explicit,teachers often pose, toward the end of thesummary, a quick problem or two to be doneindividually as a check on how the students areprogressing. Moving flexibly between whole-class

and individual work keeps the whole classfocused, but allows each student to test his or herunderstanding of the ideas being discussed.

Individual WorkThe teacher’s notes often suggest that studentsspend some time working on a question individuallybefore working with their partner or group. Askingstudents to first think about and try a question ontheir own gives them time to sort out their ownideas and assess what makes sense to them andwhat causes them difficulty.

For an occasional question, it is suggested thatstudents work entirely on their own. Suchquestions may be less demanding than questionsfor which group work is suggested, or they mayprovide an opportunity for teachers to assess eachstudent’s understanding or skill at an importantstage in the development of key mathematicalideas in the unit.

The ACE exercises at the end of an Investigationare intended to be solved individually, outside ofclass. These exercises give students a chance topractice and make sense of ideas developed in class.These exercises are narrower in scope and demandthan are the Problems in the Investigations.

Pairs and Small-Group WorkWorking collaboratively allows students to tacklemore complicated and more conceptually difficultproblems. Carefully managed, collaborative learningcan be a powerful tool for teachers to use duringclassroom instruction. Connected Mathematicssuggests two types of collaborative-learninggroupings: partner work and small-group work.

Many of the problems in ConnectedMathematics are mathematically demanding,requiring students to gather data, consider ideas,look for patterns, make conjectures, and useproblem-solving strategies to reach a solution.For this reason, the Teacher’s Guide oftensuggests that students work on the exploration ofa problem collaboratively. Group work supportsthe generation of a variety of ideas and strategiesto be discussed and considered, and it enhancesthe perseverance of students in tackling morecomplicated multi-step and multipart problems.

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It is appropriate to ask students to think about aproblem individually before moving into groups,allowing them to formulate their own ideas andquestions to bring to the group. These multipleperspectives often lead to interesting and diversestrategies for solving a problem.

Group work is also suggested for some of theUnit Projects. These projects tend to be large,complicated tasks. Working in a group allowsstudents to consider a variety of ideas and helpsthem complete the task in a reasonable amount of time.

You will want to determine group configurationsin an efficient manner so class time is not wasted.You may find it easiest to decide before class how students will be grouped. There are variousmethods you can use to establish groups, such asassigning students to a group for a whole unit ofstudy or randomly drawing for group assignmentson a more frequent basis. You might also want toarrange the seating in the room to minimizemovement during the transition from individual to group to whole-class settings.

Guidelines for Working in Pairs or Small GroupsIt is important that you clearly communicate yourexpectations about group work to your studentsand then hold them to those expectations. Youmay want to hand out or post a set of guidelinesso students understand their responsibilities.Below is a suggested set of guidelines.

Student Guidelines for Group Work

• Move into your groups quickly and get rightto work.

• Read the instructions aloud or recap what the teacher has challenged you to find out.Be sure every group member knows what thechallenge is.

• Part of group work is learning to listen to eachother. Don’t interrupt your classmates. Makesure each person’s ideas are heard and thatthe group answers each person’s questions.

• If you are confused, ask your group toexplain. If no one in the group can answer thequestion, and it is an important question, raiseyour hand for the teacher.

• If someone in your group uses a word or an idea you do not understand, ask for anexplanation. You are responsible for learningall you can from your group. You are alsoresponsible for contributing to the work ofyour group. Your attempts to explain to otherswill help you to understand even better.

• Give everyone in the group a chance to talkabout his or her ideas. Talking out loud aboutyour thinking will help you learn to expressyour arguments and clarify your ideas.

• If your group gets stuck, go over what theproblem is asking and what you know so far.If this does not give you a new idea, raiseyour hand for the teacher.

• Be prepared to share your group’s ideas,solutions, and strategies and to explain whyyou think you are correct. Make sure youlook back at the original problem and checkthat your solutions make sense.

• You are responsible for recording your group’sideas and solutions in your notes.

Suggestions for Encouraging ParticipationWhen students work in groups, there is always apossibility that some students will dominate, whileothers will not participate. Making sure the size ofthe group is appropriate for the size of the taskcan help ensure that all students play a role. Youcan also facilitate participation by requiring thateach group member be given the opportunity toshare his or her thoughts and ideas before thegroup discussion begins. It is also helpful to givestudents some time to think about or work on theProblem individually before discussing it withtheir groups.

In the Summarize phase of instruction, groupsshare their findings with the class. It is importantthat all students have an opportunity to participatein this phase. To make sure all group members areprepared, you can randomly choose the presenterfrom each group or employ questioning techniquesthat involve all group members. Teachers havefound these strategies to be useful:

• Have students assign numbers to each studentin their group. Then, have them roll a numbercube or draw to determine who will presentthe group’s findings.


• Write each student’s name on a craft stick,store the sticks in a cup at the front of theroom, and choose one stick at random todetermine who will present.

• Have the students choose the presenter fortheir group, but ask each of the other studentsa question related to the work.

The classroom conversation that occurs duringthe Summarize phase provides an importantopportunity to push students’ mathematicalthinking. By examining and testing ideas, studentscan learn mathematical skills and strategies andmake connections and generalizations. You mightuse the following suggestions to increase interactionand participation.

• Encourage students to respond to anothergroup’s or student’s presentation, conjectures,strategies, or questions.

• Have students summarize the essence of agroup’s or student’s presentation.

• After a group or student presents, have othersin the class ask questions to challenge thegroup’s or student’s thinking.

• Ask a student to create and post an incorrectsolution to stimulate the thinking of the classand generate a conversation.

• If you have a student who struggles, findopportunities for him or her to present whenyou know he or she has a correct answer.

• If there is repetition among strategies, havestudents discuss the similarities or contributenew thoughts, rather than just repeat ideas.

• Encourage students to look for common ideasin their strategies and representations.

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Grading in CMP2The multidimensional assessment in ConnectedMathematics provides opportunities to collect broadand rich information about students’ knowledge.Teachers face the challenge of converting some ofthis information into a grade to communicate alevel of achievement to both students and parents.

The following assessment items offer teachers an opportunity to assign grades: ACE exercises,Check-Ups, Quizzes, Mathematical Reflections,Looking Back and Looking Ahead (Unit Reviews)Unit Tests, Projects, notebooks, and Self-Assessments. The use of these assessments forgrading and the value assigned to them vary from teacher to teacher. While most teachers viewthe problems as the time to learn and practicemathematical concepts and skills, some teacherswill occasionally assign a grade to a problem. Someteachers also choose to grade class participation.

Two teachers’ grading schemes for their CMPmathematics classes follow. These are given asexamples of possible grading schemes. Note thateach of these teachers has made independent

decisions about how best to use the assessmenttools in CMP for grading purposes.

Example 1: Ms. Jones’ Grading System

I try to take several things into account whengrading students in mathematics class. I work tobuild a learning community where everyone feelsfree to voice his or her thoughts so that we canmake sense of the mathematics together. I tryvery hard to assess and grade only those thingsthat we value in the classroom.

ParticipationBecause participating in discussions and activitiesis so important in helping the students make senseof the mathematics, this is one part of thestudents’ grades. They rate themselves at the endof each week on how well they participatedthroughout the week. Below is a sample of thegrading sheet they fill out. The participation gradecounts as 15% of their total mathematics grade.

Participation Grading Sheet Name _____________________________________________

Week of __________________________________________

We have completed almost a full week of math class. Think about how well you participated in class this week.

1. Answer the following questions, as they will help you give yourself a fair participation grade for this week.

2. Now count your “yes” responses.

If you answered “yes” to ALL of them, HOORAY for you! You are doing a great job. Give yourself a 5.

If you answered “yes” to most of them, give yourself a 4.

If you answered “yes” to a couple of them, give yourself a 3.

If you answered “no” to several of these, give yourself a 2,and rethink your role in this class or talk to your teacher.

3. I grade myself a _____ for this week. Signature _________________________________________

❑ Did you participate in the discussions?

❑ Did you come prepared to class, havingdone your homework, so that youcould ask questions?

❑ Did you ask questions when you didn’tunderstand?

❑ Did you LISTEN carefully to others?


JournalIdeas become clear when we talk about them and when we write about them. Because I feel it is very important to be able to communicatemathematically in writing, students’ journals alsofigure into their grade. We use the journals forproblem solving, communicating what they do and do not understand, and reflecting on eachInvestigation to summarize the ideas. I try tocollect them at least once every two weeks so I remain in constant communication with eachstudent. The journal grade counts as 15% of theirtotal grade. I use the rubric shown at the bottomof the page to grade journals.

HomeworkThe curriculum is problem centered. This meansthat the students will investigate mathematical ideaswithin the context of a realistic problem, as opposedto looking only at numbers. Students spend much ofeach class period working with a partner or in asmall group trying to make sense of a problem. Wethen summarize the investigation with a whole classdiscussion. The ACE exercises assigned offerstudents an opportunity to practice those ideasalone and to think about them in more depth.Homework assignments are very important! Theyprovide students the opportunity to assess their ownunderstanding. They then can bring their insightand/or questions with them to class the next day.We usually start each class period going over the

exercises that caused difficulty or that students justwanted to discuss. Keeping up with the homework(given about 3 or 4 times a week) helps students tostay on top of their learning. It also allows me to seewhat students are struggling with and making senseof. Homework assignment grades count as 20% oftheir total grade.

Partner QuizzesAll of the quizzes from CMP are done with apartner. Because a lot of what we do in class is donewith others, I want to assess students “putting theirheads together,” as well. Again, I try to grade what Ivalue, which is working together. Quiz grades countas 20% of their total grade.

Final AssessmentAt the end of each unit an individual assessment isgiven. Sometimes it is a written test, sometimes aproject, and sometimes a writing assignment. Theseserve as an opportunity for students to show whatthey, as individuals, have learned from the wholeunit. Test/project grades count as 30% of their totalgrade, as they are a culmination of the whole unit.

Grading Summary:

• Participation . . . . . . 15%• Journals . . . . . . . . . . . 15%• Homework . . . . . . . . 20%• Partner Quizzes . . . . 20%• Tests/Projects . . . . . . 30%

Journal Grading SheetYou will earn a 5, if:

• You effectively communicate your thoughts. • You use appropriate vocabulary. • You use a variety of strategies to solve

problems.• You write as if you are talking about

mathematics.• Your journal is well organized, and entries

are labeled and dated.

You will earn a 4, if:

• You are effective in communicating yourthoughts most of the time.

• You use some appropriate vocabulary. • You use some different strategies when

solving problems. • Your journal is fairly well organized, and

most entries are labeled and dated.

You will earn a 3, if:

• You attempt to communicate your thoughtsbut your entries are hard to follow at times;be sure to write ALL that you know.

• You use some appropriate vocabulary butneed to use more.

• You need to work on using a variety ofstrategies to solve problems.

• Your journal is not organized with theentries labeled and dated.

If you earn a 2:

• Please see me.

Date Graded Grade Received

_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _

_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _

_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _

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Journals (Part of the Notebook)Collect student journals once a week.

Scoring Rubric for JOURNALS

5 Work for all Investigation problems (done inclass, to date) and Reflections (well labeledand easy to find/follow)

4 Most class work and Reflections (well labeledand easy to find/follow)

3 Some missing class work or Reflections (notwell labeled or easy to find/follow)

Below a 3 is not acceptable. Students have tocome in at lunch or after school and meet with meand work on their journal until it is at least level 3.

ParticipationParticipation means questioning, listening, andoffering ideas. Students are given a participationgrading sheet every Monday, to be handed in onFriday. Students fill these out throughout theweek, giving evidence of their participation in theclass. On the sheets they are to note when andhow they contribute to class discussion and whenthey use an idea from class discussion to revisetheir work or their thinking.

Scoring Rubric for PARTICIPATION

5 Student has made an extra effort to participateand help others in the class to understand themathematics. Student gave evidence ofparticipating all 5 days of the week.

4 Student made an effort to participate, givingevidence of at least 4 days of participation forthe week.

3 Student made some effort to participate, givingevidence of at least 3 days of classparticipation for the week.

Below a 3 is not acceptable. I talk with studentabout his or her lack of effort. If no improvementis seen in the next week, a parent or guardian iscalled and informed of the problem.

Homework (selected ACE exercises)In class, before homework is checked or collected,students are given the opportunity to ask questionsabout the assignment. I do not give answers or tellhow to solve the exercise but, with the class’s help,work with students to help them understand whatthe exercise is asking. Students have the right torevise any of their work while this conversation is

going on and not be marked down. Grading is stricton this work because students have the opportunityto take care of it themselves and get help.

Scoring Rubric for HOMEWORK

✔ + Close to perfect✔ All problems attempted, most work

done correctly✔ – Most problems attempted, some given

answers wrong or incomplete✔– – Not much work, most work wrong

or incomplete0 No work

ProjectsA 6-point holistic rubric is used for all projects.

Scoring Rubric for PROJECTS

5 Project is complete, mathematics is correct,work is neat and easy to follow.

4 Project is mostly complete, most of themathematics is correct, work is neat and easyenough to follow.

3 Project has some missing pieces, some of themathematics is correct, work takes some effortby the teacher to follow.

2 Project is missing some major parts, there areseveral problems with the mathematics, it takesextra effort for the teacher to follow the work.

1 Project shows little to no significant work.0 No project is submitted.

Check-Ups, Partner Quizzes, and Unit TestsWith partner quizzes, only the revised paper (theone turned in the second time) is scored for a grade.

Scoring Rubric for Check-Ups, Partner Quizzes, and Unit TestsEach assessment has its own point-marking schemedevised by me. Points are determined by the amountof work asked for to solve each problem. Not allproblems are awarded the same number of points.

Assigning grades to numbers and checks5’s and ✔+ = A4’s and ✔ = B3’s and ✔– = C2’s and ✔–– = D1’s and 0’s = E

Example 2: Mr. Smith’s Grading Scheme


Comments on Partner QuizzesThe quizzes provided in the Connected Mathematicsassessment package are a feature unique to thecurriculum. The assumptions under which thequizzes were created present a unique managementand grading situation for teachers.

• Students work in pairs.

• Students are permitted to use their notebooks,calculators, and any other appropriatematerials.

• Pairs submit a draft of the quiz for teacherinput, revise their work, and turn in thefinished product for assessment.

Partner quizzes are designed to be completed bystudents working in pairs. There are several ways tochoose student pairs for a quiz. Most teachers useone or more of the following:

• Students choose their own partners.

• Partners are chosen in some random way.

• The teacher picks the pairs to work together.

• Seating assignment determines partners.

Many teachers keep track of who works withwhom and have a rule that you cannot have thesame partner twice until you have been pairedwith everyone in the class at least once.

It is assumed that each pair of students willhave one opportunity to revise their work on thequiz based on teacher feedback before submittingit for a grade. When a pair has completed the quiz,they can submit separate papers or one paperwith both names on it.

Giving feedback generally involves tellingstudents which questions they have answeredincorrectly or how many of the possible points

they would receive for a question. It should beseen as an opportunity to let students know ifthey are on track or if they need to rethink aproblem. Giving feedback should not meanreteaching or leading students to the correctsolution. Here are some methods ConnectedMathematics teachers have used for givingfeedback to students.

• Check the quizzes and write the number ofpoints achieved next to each question. Then,allow the pair to revise all the questions.

• Check the quizzes and write the number ofpoints achieved next to each question. Then,allow the pair to revise one question of theirchoice. (If they write in a different color, youneed to check only the new information.)

• While students take the quiz, allow each pairto confer with you once about one problem.

Allowing students to revise their work is a newconcept for many mathematics teachers. If youhave never done this before, you might ask one ofthe language-arts teachers in your school how he orshe orchestrates revision work for student writing,since this is a common practice in that discipline.

Quiz questions are richer and more challengingthan Check-up questions. Many quiz questions areextensions of ideas students explored in class.These questions provide insight into how studentsapply the ideas from the unit to new situations.The nature of the partner quizzes provides agrading situation in which rubrics can assist in theevaluation of the students’ knowledge. You maywant to refer to the teacher suggestions, gradingrubrics, and samples of student work in theTeacher’s Guides.

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This section contains specific suggestions of ways to present the ConnectedMathematics curriculum to English language learners, to special needs

students, and to gifted students.

In addition to the material presented here, please see the separate publicationConnected Mathematics 2 Special Needs Handbook for Teachers, which contains manysamples of ways to adapt CMP materials for special needs students.


Mathematics and English Language LearnersEnglish language learners (ELL) come into ourclassrooms from a variety of countries with adiverse set of experiences. They face the dauntingtasks of adjusting to a new home and culturalenvironment, learning a new language, makingnew friends, and making sense of the rules,appropriate behaviors, and mechanics of a newschool. Simultaneously, ELL students areexperiencing many losses and trying to “fit in”with their new surroundings.

For teachers, working successfully with ELLstudents requires more than just teaching thecontent of courses. For language learners toachieve academic success, we must also supportlanguage goals and general learning strategies inthe mathematics classroom (Richard-Amato &Snow, 2005). In addition, it is critical to create afriendly, supportive, and predictable classroomcommunity. Some general suggestions are:

• Learn about your students’ home countries,languages, and previous educationalexperiences.

• Value students’ differences as resources.

• Stay connected to families.

• Clearly communicate school and classroomnorms and expectations and be willing tocheck your assumptions at the door.

Teaching and learning with English languagelearners is a “lifelong process of learning,discovering, accepting, and trying” (Carger, 1997,p.45).

Classroom Environment and Teacher TalkEnglish language learners are often anxious aboutbeing in a classroom when they cannot speakEnglish. Efforts to create a friendly environmentthat is respectful of students’ diverse experiencesand sets high expectations for learning will greatlysupport ELL students’ opportunities for success.Part of establishing this kind of learningenvironment includes modifying the ways inwhich you talk with students. Patterns of speech,

intonation, or pace can often interfere withstudents’ understanding of your expectations andtherefore impact their abilities to engage in themathematics lesson. Many of the suggestions belowwork for all students, including ELL students.

Classroom Community Create a classroomcommunity that recognizes and values students’diverse backgrounds and experiences. Every childis born into a culture that socializes them to thinkin specific ways about many things we take forgranted as common sense. When left unexamined,some cultural beliefs and practices can interferewith students’ success in our classrooms. Find outwho your students are, where they come from, andwhich languages they speak.

Expectations Keep expectations high andconsistent. Provide effective feedback. Too oftenELL students receive “feedback that relates topersonality variables or the neatness of their workrather than to academic quality” (Jackson, 1993,p. 55). If we want our students to learn andimprove their work and understanding, it is crucialto be specific, focus our comments on the academiccomponents of students’ work, and clearlycommunicate to students how to improve theoverall quality of the work they do (Jackson, 1993).

Speak Slowly Slow down the rate at which youspeak and simplify the language you use. Consideryour intonation; avoid using slang, idioms,extraneous words, and long, complex sentences.Repeat key points. Rephrase to promote clarityand understanding. Summarize frequently. Useclear transition markers such as first, next, and inconclusion. Ask clear, succinct, high-levelquestions. (Carrasquillo & Rodriguez, 2002,Jameson, 1998)

Visual Communication Pair your instructionaltalk with visual communication cues such aspictures, graphs, objects, and gestures (Peregoy &Boyle, 1997).

Seating Up Front Seat students toward themiddle or front of the classroom, in a place whereyou can observe them closely and where they canobserve the classroom interactions of otherstudents (Peregoy & Boyle, 1997).


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Predictable Routine Even though your contentwill vary, follow a predictable routine and a stableschedule. Predictability in routine creates a senseof security for students who are experiencing a lotof change in their lives (Peregoy & Boyle, 1997).

Dictionaries Have dictionaries and other learningtools available and easily accessible to students.

Teaching Students the Norms of SchoolStudents come from a variety of places and theirconstructs of school and purposes for educationoften greatly differ. We cannot expect that any ofour students will tacitly understand the ways inwhich we “do” school. We must be explicit aboutour expectations. Problems between teachers andELL students often occur because of languagedifferences and unidentified assumptions aboutthe social aspects of schooling. Therefore, it isimportant for teachers to help students learn whatis expected of them in the school building andinside your classroom.

Create and consistently reinforce classroomnorms to support students’ understanding of whatis expected socially and academically in theclassroom.

Post homework assignments in a public place inthe classrooms so students can be responsible forchecking their assignments and keeping track ofwhether or not they submitted them.

Provide each student with a daily agenda. As aclass, write the day’s objectives, activities, and thehomework assignment. Students keep theseagendas in their notebooks for personal reference.It is also helpful to provide space on the agenda forstudents to check off the homework assignmentonce it has been completed and turned in. (Seepage 123 for an example of a daily agenda link.)

Pedagogical Strategies in the Mathematics ClassroomEnglish language learners benefit from a variety of instructional strategies that lowertheir anxiety and help make content morecomprehensible. Mathematical objectives shouldbe cognitively demanding and grade appropriate.Language-related adjustments and modificationsshould be made, including how you modify yourinstructional delivery, but the cognitive demandof the mathematics should not be changed. The

learning and teaching philosophy of ConnectedMathematics support many, if not all, of thefollowing strategies for ELL. These strategies are also just good teaching strategies to be usedwith all students.

Strategies Inherent in Connected Mathematics

Effective Questioning Use effective questioningtechniques. Research highlights the fact thatteachers “frequently use few higher orderquestions to all students, especially to those forwhom they had low expectations” (Jackson, 1993,p. 55). Higher-order questions promote analyticaland evaluative thinking, affirm students’ self-perceptions as learners, and support students tothink of themselves as knowledge producersrather than knowledge consumers (Jackson, 1993).(See pages 6–7 for a discussion of inquiry-basedinstruction.)

Cooperative Groups Use cooperative group work(see pages 106–108 for a discussion of cooperativegroup work). Research evidence demonstrates thatcooperative group work can have a “strong positiveimpact on language and literacy development andon achievement in content areas” (Richard-Amato& Snow, 2005, p. 190).

Active Participation Create opportunities forstudents to participate with you, each other, andthe mathematical content. Active participationprovides students with opportunities to learn bothmathematics and English. Encourage yourstudents to ask questions of each other. (See pages73–74 for a discussion of classroom interactionsduring the Launch–Explore–Summarize phases ofa lesson.)

Brainstorming Use class brainstorms, predictions,quick writes, and outlines as ways to accessstudents’ prior knowledge. It is also helpful towrite students’ ideas on the chalkboard so theycan see them written correctly in English.

Prior Knowledge Consider the context of theproblem. Context is meant to support students’entry into a problem by connecting to their priorknowledge and preparing them for what liesahead. If students are unfamiliar with names,places, or objects, it will be difficult for them toaccess the mathematics. Sometimes it is possible tochange the context of a problem without affecting


the mathematics of the problem or the objectivesof the lesson. Incorporate names and places fromstudents’ home countries or situate actions withincultural practices with which students are familiar.This is also a great opportunity for students to learncommon English words used in daily life. Includewords in the math problems that students need toknow and avoid using slang, idioms, or extraneouslanguage. (See pages 6–7 for a discussion of contextin a problem-centered curriculum.)

Expression of Ideas Provide many opportunitiesduring class for students to explain and justifytheir ideas.

Journals Use journals and quick writes to provide students with opportunities to write in the mathematics classroom. Use the followingsuggestions as meaningful writing activities:

• Restate the problem in your own words.

• Explain how you solved the problem.

• How do you know your answer is right?(Richard-Amato & Snow, 2005)

• What do you know so far about …?

(See pages 102–103 for a discussion of studentjournals.)

Model Behaviors Model what you want yourstudents to do. Students may not understand what you say, and actions will support theirunderstanding. For example, use visual promptssuch as hand movements, facial expressions, orother body movement that suggests meaning for a word or phrase.

Support Vocabulary Development

Highlight Mathematical Vocabulary Studentsmust understand mathematical terminology andkey words to gain access to any math problem.Isolate important vocabulary and phrases bycircling or underlining them in the text.

Bilingual Vocabulary Chart Create and maintaina Word Cluster or Vocabulary Chart in theclassroom and in students’ notebooks where new terms and their definitions are written inboth English and the student’s first language.Pictures are also useful additions. (See page 121 for examples of graphic organizers and 104–105for a discussion of the development of vocabularyin Connected Mathematics.)

Practice Out Loud Practice speaking hard-to-pronounce words verbally as a class. It isbeneficial for students to practice reading andpronouncing words correctly.

(See page 122 for five guidelines for simplifyinglanguage.)

Graphic OrganizersUse graphic organizers to scaffold your learningactivities and provide ELL students access to themathematical content. (See page 121 for examplesof graphic organizers.) Graphic organizers include:

• Venn diagrams • concept webs• timelines• lists• outlines• tree diagrams • charts

Reading, Writing, and Waiting

Time to Prepare Give students time to readsilently before asking them to discuss their ideaswith a small or large group. It is alsorecommended to provide time for students towrite their ideas on paper before they share thempublicly. This will give students time to sortthrough their ideas before they are asked toperform in front of teachers and peers.

Write and Speak Directions Post task directionson the overhead or chalkboard while yousimultaneously read the directions and havestudents follow along. This affords ELL studentsthe opportunity to read the English text silentlywhile they hear it spoken correctly.

Write in English Encourage ELL students towrite in English even if the spelling and grammaris incorrect. It is also helpful for students to use acombination of English and their first languagewhen they write in their notebooks.

Give them Time Use extra wait time so ELLstudents will have an opportunity to hear thequestion, translate the work, understand itscontent, formulate a response, and then speak.


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Assessment of English Language LearnersStudents’ lack of English proficiency will affect testperformance when tests are given only in English.It is also necessary to consider how students’cultural backgrounds and previous experiencesmight affect their ability or willingness toparticipate in an assessment activity. “Becauseschooling practices tend to conform more or less tomiddle-class European-American experiences andvalues, students from other cultural backgroundsmay be misassessed by virtue of cultural and otherexperiential differences.” (Peregoy & Boyle, 1997,page 93) Therefore, your assessment practicesshould allow students to show what they know in avariety of ways.

Diversity When creating assessments, considerthe diversity of students’ cultural, linguistic andspecial needs (Peregoy & Boyle, 1997).

Variety Use a variety of assessments in a varietyof formats including small-group work, individualactivities, drawing pictures, creating posters,engaging in interviews, constructing portfolios,journal writing, projects, and self-assessment.(See pages 77–80 for a discussion of assessment inConnected Mathematics.)

Rubrics Be clear and consistent with your gradingsystem and standards. Rubrics are an excellent toolfor itemizing the criteria on which students will beassessed and helping students understand what youare looking for (Richard-Amato & Snow, 2005).

Working Together Peer editing is an opportunityfor students to read, edit, and comment on eachother’s work while gaining reading and writingexperience.

Time Allow sufficient time for all students tocomplete the assessment.

Fewer Exercises Consider the number of exercisesyou assign students for homework. It will take ELLstudents much longer to read and make sense of the exercises than native-English speakers. OftenELL students get so bogged down in the readingcomprehension that they never get to themathematics. It will be much more meaningful andproductive for both you and the students if youassign 5 or 6 well-designed exercises (and they’ll bemore motivated to try them), rather than a page ortwo of 10 to 20 exercises.

Rebus Techniques The following suggestions follow guidelinesknown as rebus techniques for English languagelearners. Rebus is a general term referring to theuse of pictures or other visual images to representwords or symbols. Some of these techniques aresimilar to those in the preceding sections.

Original Rebus TechniqueOn a sheet of paper, students copy the text fromall or part of a page before it is discussed. Duringdiscussion, students then generate their ownrebuses for words they did not understand as thewords are made comprehensible through pictures,objects, or demonstrations.

This strategy ensures that English learnersbenefit from written communications in the sameway as their English-proficient peers. Whilewritten text summarizes key concepts, includesbackground information, and provides directionsfor completing tasks, English learners often do notbenefit from such communication.

In the past, English learners have beentraditionally paired with English-proficientstudents who are asked to read aloud written text.However, this approach does not provide Englishlearners with access to written communication.For example, English learners are asked to rely onmemory when trying to recall the writteninformation—something not required of theirpeers. Furthermore, simply reading informationaloud does not ensure that the words are madecomprehensible to the English learner. Therefore,the Original Rebus technique offers a strategythat makes written communication meaningful toEnglish learners, without depending on peercooperation or memory.

1. Teachers identify text perceived to be difficultfor English learners to comprehend. Examplesof such text may be questions appearing inMathematical Reflections, Applications, andConnections sections of the program.

2. English learners receive a copy of the rewrittentext when the corresponding page is introducedto the class. As the information from the studentbook is read aloud, teachers make key wordsunderstandable. For example, a teacher maydemonstrate the word “snapshot” by showing aphoto of a pet.


3. After students comprehend the word, theteacher writes it on the board so Englishlearners can connect the written word with aspecific meaning. At this time, English learnerscreate an original rebus over that key word ontheir sheet of paper. This rebus will then helpthe English learners recall the meaning of theword when referring back to the text duringindependent work.

Note: It is essential that English learners draw their own rebuses. This ensures that whateversymbol they choose to draw has meaning to them.The problem with providing professional orteacher-drawn rebuses is that simple drawings,by themselves, do not often convey a universalunderstanding of the words. For example, manyEnglish-proficient students were not able tocorrectly identify a rebus when the word below wascovered, yet could do so when they were able toview both the word and rebus. This suggests that thewritten word, not the rebus, conveyed the meaningin such situations. Moreover, if English learners arerequired to create their own rebuses, they thenchoose which words need to be coded. Dependingon the level of English proficiency, the number ofcoded words can vary greatly among students.

Diagram Code TechniqueStudents use a minimal number of words,drawings, diagrams, or symbols to respond toquestions requiring writing. Learning to organizeand express mathematical concepts in writing is a skill students develop over time. If Englishlearners are not given this same opportunity,they miss an important component of the mathcurriculum. This strategy provides alternate waysfor students not yet proficient in writing Englishto express mathematical thinking on paper. Whiletheir responses will not be in the same format asthose of their English-proficient peers, Englishlearners still have the same challenge: they mustrecord and communicate mathematical ideas sothat someone else can understand their thinking.

1. At the beginning of the program, teachersmodel and encourage English learners to usethis approach when writing answers toquestions presented in the program.

2. To introduce this approach, the teacher writesseveral questions requiring written responses

on the board. These questions should besimple with obvious answers.

3. The teacher then shows the English learnershow to answer each question without writingcomplete sentences and paragraphs. At theend of this session, the teacher should havemodeled answering questions by using and/orcombining minimal words, drawings, diagrams,or symbols.

Note: This approach can be used for any writtenresponse in the program, but it is especially usefulfor responding to questions found inMathematical Reflections. Since this part of theprogram provides a vehicle for assessing how wellstudents have understood key concepts of theunit, this approach enables teachers to evaluatetheir English learners’ progress as well.

Chart Summary TechniqueThis technique involves presenting information bycondensing it into a pictorial chart with minimalwords. This extension of the Diagram Codetechnique offers English learners another way toorganize and express mathematical thinking witha minimal amount of writing.

1. At the beginning of the program, the teachershows various charts on any subject. The chartsneed to be simple, include pictures, and have aminimal number of words.

2. The teacher then creates and writes a questionon the board that relates to each chart. Forexample, the teacher might show a chart of thelife cycle of a plant divided into four sections.For this chart, the teacher could ask thisquestion: What are the growth stages of a plant?

3. The teacher continues by showing how thechart answers this question by pointing to thedrawings in each section, showing the seeds,roots, stem, and flower. The teacher also pointsout how each section has been labeled.

4. At the end of this session, English learnersshould be able to respond to a question bycreating a chart with pictures and minimalwords.

Note: This approach may be an alternative forEnglish learners when responding to some of theUnit Projects requiring detailed writing.


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Rebus Scenario TechniqueTeachers make use of rebuses on the chalkboardduring discussions and when presentinginformation. While modifications for primarymathematical concepts may be perceived asnecessary for English learners, there may be atendency to omit such techniques for“enrichment” information, such as text appearingunder “Did You Know?” However, if programsoffer English-proficient students suchinformation, then English learners should alsohave an opportunity to acquire the sameknowledge. Therefore, the Rebus Scenario offersteachers a simple way to ensure that all studentshave access to both the core and enrichmentaspects of the Connected Mathematics program.

The teacher assesses what key words may notbe understood by the English learners. As each ofthose words is presented, the teachersimultaneously draws a rebus on the board.

Note: If there are English-proficient “artists” inthe classroom, teachers may opt to implement thisapproach in a slightly different way. Prior to thelesson, a teacher can ask an artistic student tocome to the chalkboard to draw rebuses fortargeted words. When using this approach, theteacher can then just point to the appropriatedrawings during the lesson. If there is no timeprior to a lesson, the artistic student can be askedto draw the rebuses as key words are presented.With this latter approach, it is important that theartist knows which words to represent as rebusesand to draw quickly.

Enactment TechniqueStudents act out mini-scenes and use props to makeinformation accessible. This technique ensures thatall students comprehend hypothetical scenariospresented throughout CMP. With this technique,English learners are not excluded from lessonsinvolving situations reflective of real-life scenarios.

1. Teachers decide which simple props, if any, willenhance the enactment. These props aregathered prior to teaching the lesson.

2. At the time of the lesson, students are selectedto assume the roles of characters mentioned ina CMP problem or scenario.

3. These students then pantomime and/orimprovise speaking parts as they enact thewritten scenario presented in CMP.

Note: There may be a tendency to select onlyEnglish-proficient students for mini-scene roles;however, many parts can also be given to Englishlearners. For example, roles such as pantomimingshooting baskets or pretending to ride a bicycle canbe easily enacted by English learners, as these kindsof parts do not require students to speak English.

Visual Enhancement TechniqueThe Visual Enhancement technique uses maps,photographs, pictures in books, and objects tomake information understandable by providingnonverbal input. This technique is most helpful for conveying information that is unlikely to beunderstood through enactment or creating rebuses.When pictures or real objects are added to lessons,English learners have the opportunity to receivethe same information presented to their English-proficient peers, who are able to understand thewritten text without visual aids. This approachensures that English learners equally acquire and benefit from descriptive and/or backgroundinformation sections of the program.

1. Teachers decide if information on a page isunlikely to be understood with a rebus or byhaving students create an enactment. Forexample, maps are often used with thistechnique to help students understand whatpart of the world an informative section orinvestigation is centered around. In contrast, amere rebus “outline” of the same countrywould not be likely to be understood byanyone. Likewise, topics such as video games,different kinds of housing, and newspaperadvertisements are more easily comprehendedby merely showing examples than by trying todraw something representative of the topic.

2. When teachers decide visual aids are the bestapproach for making information accessible,examples are sought prior to teaching thelesson.

3. Teachers then show the visual aid at theappropriate time during the lesson.

Note: In the first year of implementation,English-proficient students can earn extra creditby finding appropriate visual aids for targetedlessons. Teachers can then keep the pictures,objects (if possible), and book names (with pagenumber) on file for use in subsequent years.


SummaryThe six techniques (Original Rebus, Diagram Code,Chart Summary, Rebus Scenario, Enactment, andVisual Enhancement) ensure that English learnerswill receive the same mathematics curriculum as their English-proficient peers. Although thetechniques differ in implementation, they all offer

ways for English learners to acquire and expressthe mathematical ideas presented in CMP.

Although these approaches have been createdspecifically for English learners, they can beequally effective for many special-educationstudents.

Original Rebus Technique On a sheet of paper,students copy the text from all or part of a pagebefore it is discussed. During the discussion,students generate their own rebuses for wordsthey did not understand. This technique offers astrategy that makes written communicationmeaningful to students with language difficulties.First the teacher identifies text which containsimportant ideas and may be difficult for studentswith language difficulties. As this part of the textis discussed, the teacher tries to make key wordsunderstandable through pictures, objects, ordemonstrations. Students create their ownpictorial rebus for each of the key words.

Diagram Code Technique Students use a minimalnumber of words, complemented by diagrams ordrawings to organize and respond to questions.The teacher should introduce and demonstratehow to express mathematical thinking withouthaving to write in complete sentences. Theultimate goal is, of course, to have studentsprogress towards being able to communicatemathematical thinking in writing as well as inthese diagram codes.

Chart Summary Technique This is an extensionof the Diagram Code. The technique involvespresenting information by condensing it into apictorial chart. As before, the teacher must modelthis technique so students see what is expected.

Rebus Scenario Technique To make enrichmentinformation available to English learners, theteacher supplies quickly drawn rebuses on thechalkboard for key words in material like a “DidYou Know?” passage. An artistic student mayalso be asked to sketch key word rebuses.

Enactment Technique Students act out mini-scenes and use props to make informationaccessible. This technique ensures that allstudents comprehend hypothetical real-lifescenarios presented throughout CMP.

Visual Enhancement Technique The teacherdecides if information in the text is unlikely to beunderstood with a rebus, as above, and providesvisual aids to make information accessible.

SUMMARY OF REBUS TECHNIQUES


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Examples of Classroom Materials

Graphic Organizers Graphic organizers can be used by the teacher topresent information or by the students to organizeinformation and to compare and contrast conceptsand ideas. Graphic organizers such as word clusters,rebuses, and vocabulary charts can be used tosupport vocabulary development. Venn diagrams,concept maps, and other techniques can helpstudents organize information.

Word Clusters Write mathematical terms onsentence strips and group them together to showhow they are connected. Hang the sentence stripsfrom the ceiling or on a wall for quick reference.

Rebuses Create rebus pictures or symbols forwords the students need help to understand.

• Students can draw the symbols directly over the words.

• A sheet of these symbols can be kept in thevocabulary section of students’ binders.

Vocabulary Charts The use of word cognates(linguistically related words) help students connectwords in English to words from their own languagethat are familiar. Not all words have cognates.However, all terms are put on this chart even ifthey don’t have cognates.

Venn Diagrams Use Venn diagrams as a way tocompare and contrast information. The examplebelow is from CMP2, Data About Us.

Concept Maps Concept maps are used to organizetopics or categories and to visually representconnections between concepts and ideas.

Tree Diagrams/Hierarchy Use tree diagrams toorganize ideas from the general to the specific andto support understanding of the relationshipsbetween concepts.

Charts, Lists, and Timelines Other ways toorganize information include making charts, lists,and timelines.

figure

polygon

triangle

isosceles scalene

equilateral

line

y-interceptLinearEquation/Graph

slope

y=mx + b

Similarities

Table Line plot

Bar Graph

Add + Subtract – Triangle

Addition ±

Addend

Plus

Add

Sum

Divide 4

Quotient

Dividend

Division

Multiply 3

Mutiples

Factor

Product

Term Description Example Cognate

Factor One of two or more 2 x 3 = 6 Factorwhole numbers

that are multiplied to get a product.

Prime Number with 3 1 x 3 Númeronumber only 2 factors, 1 1 x 7 Primo

1 and itself


Simplified Version

Sam and Adam have two candy bars.Sam eats half of one candy bar. Samgives Adam half of the other candy bar.Adam is mad. He thinks Sam has morecandy. What is the problem?

Jorge made two graphs. Jorge forgotto label the graphs.

Problem 1.1A. How are the graphs alike?

How are the graphs different?

B. How can you use the graph to findthe total number of letters in all ofthe names?

C. Collect the data for your class anduse graphs to represent the datadistribution.

Look at the graphs you made.A. Find a graph where y increases and x

increases.

B. Find a graph where y decreases andx increases.

C. Use words to describe each graph.

Bank Earns Spends

Maria $25.00 $5.00/wk $3.00/wkSuzanna $15.00 $2.00/day $4.00/wk

Original Text

Samuel is getting a snack for himselfand his little brother, Adam. There are two candy bars in the refrigerator.Samuel takes half of one candy bar for himself and half of the other candybar for Adam. Adam complains thatSamuel got more. Samuel says this isn’ttrue, since he got half and Adam gothalf. What might be the problem?

Jorge made two graphs but he forgotto label them.

Problem 1.1A. How are the graphs alike?

How are the graphs different?

B. How can you use the graph to findthe total number of letters in all ofthe names?

C. Collect the data for your class anduse graphs to represent the datadistribution.

Look back at the graphs that you havemade in this unit. Find several graphsthat show relationships in which y bothincreases and decreases as x increases.Describe each graph in words.

Maria has $25.00 in the bank. She mowsthe lawn once each week and earns$5.00 each time. Suzanna only has$15.00 in the bank. She baby-sits herlittle brother for $2.00 each weekday.

Maria spends $3.00 each week to go tothe basketball game with her friends.Suzanna spends $4.00 each week to goto the movies.

Guideline

1. Use short sentencesand eliminateextraneous material.

2. Change pronounsto nouns.

3. Underline key pointsor vocabulary.

4. Turn narratives intolists.

5. Use charts anddiagrams.

Five Guidelines for Simplifying Language


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Daily Agenda • 6th Grade

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Special Education Within CMP Connected Mathematics can be and has beensuccessfully implemented in classrooms thatinclude special education students. We believe that Connected Mathematics provides all students,even those with special needs, with opportunitiesto engage in cooperative learning, to takeleadership roles, and to enhance self-esteem andself-acceptance.

Making Accommodations Some may claim CMP offers more of a challengefor special education students due to its language-based curriculum; however, many suggestions byresearchers in the field of special education forassisting in making mathematics accessible tospecial needs students are already incorporatedwithin the CMP curriculum. The CMP SpecialNeeds Handbook for Teachers contains a wealth of samples of accommodated materials. Furtheraccommodations will most likely still need to be made for special education students. Thoseaccommodations should come from each students’Individual Education Program (IEP) and additionalaccommodations that you, as a teacher, feel arebeneficial to the students you serve.

Please keep in mind that the guidelines in theSpecial Needs Handbook are suggestions. Not allsuggestions are applicable for every student, norwill every suggestion work for all students. It isimportant to have good communication with yourstudents’ special education teacher and otherproviders, as well as his/her parents to enable that the maximum benefit from learning is beingcarried through.

Embedded Special Needs StrategiesThe curriculum of the Connected MathematicsProject is already embedded with many of thestrategies that researchers and practitionersindicate as beneficial to special educationstudents. The conceptual framework upon whichCMP is built involves sound teaching principlesand practices for students, which is essentially the same foundation for working with special

education students. To begin with, CMP wasdeveloped with the belief that calculators shouldbe made available to students, which aligns withmany accommodations that special educationstudents are given in terms of calculator use.Furthermore, the CMP curriculum involvesmanipulatives. While it is stressed within theframework of CMP that manipulatives are to beused only when they can help students develop an understanding of mathematical ideas, it shouldbe clear that special needs students may need to use the manipulatives to help develop theirunderstanding more often than general educationstudents.

CMP uses real-life problems, a pedagogicaltechnique repeatedly stressed in reaching specialeducation students in mathematics classrooms.An emphasis on significant connections which are meaningful to students, among variousmathematical topics and between mathematicsand problems in other disciplines, assisted inguiding the development of CMP. Maccini andGagnon (2000) demonstrated that the embeddingof problems within real world contexts improvesthe motivation, participation, and generalizationfor special education students.

Other practices that help to facilitate teachingmathematics to students with special needsalready within the framework of the ConnectedMathematics Project include: repetition andreview, keeping expectations high, teachingconceptual knowledge, and cooperative or groupactivities. The student materials of CMP enablerepetition and review. The ACE section at the endof every Investigation allows students to tackleadditional exercises from the unit as well as towork on problems that are connected to earlierunits. Furthermore, the end of each book includesa Looking Back and Looking Ahead sectionwhich summarizes through problems the learningstudents have completed in the particular Unit,and also connects it to earlier units.

The Connected Mathematics Project holds highexpectations for its students—all of its students.This belief is reflected in the ideology and theoverarching standard of the curriculum:

All students should be able to reason andcommunicate proficiently in mathematics.They should have knowledge of and skill in the use of the vocabulary, forms ofrepresentation, materials, tools, techniques,and intellectual methods of the discipline ofmathematics. This knowledge should includethe ability to define and solve problems withreason, insight, inventiveness, and technicalproficiency.

CMP teaches conceptual knowledge and skill.As in the above definition, skill means not onlyproficiency, but also the ability to use mathematicsto make sense of situations. CMP helps students tounderstand the methods, algorithms, and strategiesthey use.

Cooperative Learning Groups ConnectedMathematics provides opportunities for students to work in small groups and pairs, as well as wholeclass, or individually. Research in the field suggeststhat cooperative groups can be beneficial to specialeducation students; however, some attention shouldbe paid to the groupings to ensure that studentswith special needs are able to actively participate.Merely placing a student within a group does notresult in that student becoming a part of the group.While studies have shown that cooperative learninghas positive benefits on students’ motivation,self-esteem, cognitive development, and academicachievement, the very dynamic of these learningmethods may exclude special education studentsdue to their disparities in skills, such skills ascontent area, communication, and social skills(Brinton, Fujiki, & Montague, 2000). In discussingthe structure of cooperative groups, researchersstress the importance of providing opportunitiesfor children with special needs (or any diverselearners) to actively participate.

Examples of Classroom MaterialsFor each Investigation in CMP2, the Special NeedsHandbook contains a sample modification of an ACE exercise. Taken together, these nearly 100 samples illustrate the wide variety ofaccommodation techniques that may be applied to individual students as needed.

The Special Needs Handbook also contains asample modification of one assessment tool foreach CMP2 Unit.

Name ____________________________________________ Date ____________ Class _______________

1.ACE Exercise 6Covering and Surrounding

6. Use the following design to answer these questions:

a. If possible, draw a figure with the same area, but with a perimeter of 20 units. If this is not possible, explain why.

i. Hint: What is the area of the design above?

ii. Hint: What is the perimeter of the design above?

iii. Is it possible to draw a figure with the same area as the designabove but a perimeter of 20 units? If so, draw the figure.

iv. If it is not possible, explain why.

b. If possible, draw a figure with the same area, but with a perimeter of 28 units. If this is not possible, explain why.

Investigation 1

,,p

gg

ACE Accommodation

gg

Name ____________________________________________ Date ____________ Class _______________

Check-UpCovering and Surrounding

Investigation 1for use after

2. Find the area and perimeter of this rectangle. Explain how you foundyour answers.

i. Area =

ii. Explain how you found your answer for the area.

iii. Perimeter =

iv. Explain how you found your answer for the perimeter.

5 cm

4 cm 4 cm

5 cm

Check-Up Modification

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Connected Mathematics is a curriculumdeveloped to provide challenges appropriate

for every student including the mathematicallygifted child. The curriculum gives all students theopportunity to learn key mathematical concepts indepth and to make valuable connections that willbenefit them in future mathematics classes.

Components of an EffectiveProgram for Gifted Students

The National Council of Teachers ofMathematics made suggestions on how to provideopportunities for the mathematically gifted in the publication Providing Opportunities for theMathematically Gifted K-12 (NCTM, 1987). Theyproposed 16 essential components for programsfor the mathematically gifted. A subset of thesesixteen components, that directly relate to themathematics curriculum are listed below. Theprogram should:

• Contain good, high quality mathematicswhich is challenging, broad, and deep

• Nurture higher-order thinking processes andopen-ended investigations

• Prompt students to communicate effectivelyby reading, writing, listening, speaking, andthinking mathematically

• Have problem solving as a major focus and include applications of mathematics to real situations

• Encourage students to experiment, explore,conjecture, and even guess

• Provide opportunities to use learningresources (texts, calculators and computers,concrete manipulatives)

• Relate mathematics to other content areas.

Connected Mathematics possesses all of thecomponents described above, while maintaining agoal of mathematical proficiency for all students.

Modifications for Gifted StudentsIn order to provide a curriculum appropriate forgifted students, modifications in both the contentand learning environment may be necessary.Maker and Nielson (1995) describe content andprocess modifications that should be made.

Modifications in Content:

• Students need a variety of problems to work.

• The content of the curriculum needs to beorganized around key concepts or abstractideas, rather some other organization (asnoted by Bruner, 1960).

• Problems should be complex and studentsshould be pushed to abstraction. (Additionalopportunities for abstraction are described inTeacher’s Guides of Connected Mathematics,particular in Going Further features describedbelow.)

Modifications in Process:

• Promote higher levels of thinking by stressinguse rather than acquisition of information.(Students continue to use information fromprevious units in the current unit they arestudying in the Connected Mathematics series.)

• Provide open-ended questions in order tostimulate divergent thinking and to “contributeto the development of an interaction pattern in which learning, not the teacher, is focus”(see page 5).

• Teachers should guide student discovery ofcontent and encourage questions of why andhow things work. (Connected MathematicsProblems often ask students to think aboutthe questions of why and how.)

• Students must be given opportunities toexpress their reasoning; (Students areconstantly asked to explain or justify theirreasoning in Connected Mathematics.)

• Group interaction should be a regular part ofthe curriculum for gifted students to enablethem to develop social and leadership skills.


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Connected Mathematics is designed so that manyof the modifications described by Maker andNielson are embedded in the curriculum. Othersimple modifications are possible in order supportgifted students and still maintain the integrity ofthe curriculum.

Other Modifications Renzoulli and Reis (2003)discuss the Schoolwide Enrichment Model (SEM),which can be used to promote challenging andhigh-end learning in schools. The SEM modelaccommodates the needs of the gifted student and offers suggestions on how to adjust the level,depth, and enrichment opportunities provided by a curriculum.

Connected Mathematics offers students richexperiences with a variety of mathematicalcontent. Students are introduced to importantareas of mathematics, such as combinatorics,graph theory, probability, statistics, andtransformational and Euclidean Geometry early

in their career so that they can see the vast terrainof mathematics. The algebra strand in ConnectedMathematics is organized around functions, whichare the cornerstone of calculus, and the structureof the real numbers, which brings coherence tothe exploration of algebraic ideas.

There are particular features of ConnectedMathematics which support the mathematicallygifted child. In the Teacher’s Guides, there arequestions in the Launch–Explore–Summarizesequence labeled Going Further that teachers can ask students who are ready to go furthermathematically. In the homework ACEassignments, Extensions questions often go beyondwhat was done in class. Extension questions can be used as additional exercises to push students’thinking. These features, in conjunction with therich, deep problems offered in this curriculum,provide mathematically gifted students challengingproblems to explore each day in class.

The articles, units, and books in this sectioninfluenced or supported the development of the CMP philosophy of teaching and learningmathematics.

NCTM (2000). Principles and Standards for SchoolMathematics. Reston, VA.

Phillips, E., G. Lappan, and Y. Grant. (2000).Implementing Standards-Based Mathematics:Preparing the Community, the District, and Teacherswww.showmecenter.missouri.edu orwww.math.msu.edu/cmp

Adams, L. M., K. K. Tung, V. M. Warfield, K. Knaub,B. Mudavanhu, and D. Yong. (2002). Middle SchoolMathematics Comparisons for SingaporeMathematics, Connected Mathematics Program, andMathematics in Context. Report submitted to theNational Science Foundation by the Department ofApplied Mathematics, University of Washington.

American Association for the Advancement ofScience: Project 2061. (1999). Middle gradesmathematics textbooks: A benchmarks-basedevaluation: Evaluation report prepared by theAmerican Association for the Advancement ofScience. Washington, D.C.

Anderson, J. R., J. G. Greeno, L. M. Reder, and H. A. Simon. (2000). “Perspectives on Learning,Thinking, and Activity.” Educational Researcher29(4):11–13.

Battista, M. T. (February 1999).“The MathematicalMiseducation of America’s Youth: IgnoringResearch and Scientific Study in Education.”Phi Delta Kappan 80(6):424–433.

Bay, J. M., B. J. Reys, and R. E. Reys. (March 1999).“The Top 10 Elements that Must Be in Place toImplement Standards-Based MathematicsCurricula.” Phi Delta Kappan 80(7):503–506.

Ben-Chaim, D., J. Fey, W. Fitzgerald, C. Benedetto,and J. Miller. (1997). A study of proportionalreasoning among seventh and eighth grade students.Paper presented at the annual meeting of theAmerican Educational Research Association, Chicago.

Ben-Chaim, D., J. Fey, W. Fitzgerald, C. Benedetto,and J. Miller. (1998). Proportional reasoning amongseventh grade students with different curriculaexperiences. Educational Studies in Mathematics,36:247–73. Kluwer Academic Publishers. TheNetherlands.

Ben-Chaim, D., G. Lappan, and R. T. Houang.(1988). “Spatial Visualization: An Intervention Study.”American Educational Research Journal 25(1):51–57

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