Instructional Materials Evaluation Tool Connected...

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Instructional Materials Evaluation Tool Connected Mathematics Project (CMP3)

ALIGNMENT TO THE COMMON CORE STATE STANDARDS

Evaluators of materials should understand that at the heart of the Common Core State Standards is a substantial shift in

mathematics instruction that demands the following:

1) Focus strongly where the Standards focus

2) Coherence: Think across grades and link to major topics within grade

3) Rigor: In major topics, pursue conceptual understanding, procedural skill and fluency, and application with equal

intensity.

Evaluators of materials must be well versed in the Standards for the grade level of the materials in question, including

understanding the major work of the grade1 vs. the supporting and additional work, how the content fits into the

progressions in the Standards, and the expectations of the Standards with respect to conceptual understanding, fluency,

and application. It is also recommended that evaluators refer to the Spring 2013 K–8 Publishers' Criteria for

Mathematics while using this tool

ORGANIZATION SECTION I: NON-NEGOTIABLE ALIGNMENT CRITERIA

All submissions must meet all of the non-negotiable criteria at each grade level to be aligned to CCSS and before

passing on to Section II.

SECTION II: ADDITIONAL ALIGNMENT CRITERIA AND INDICATORS OF QUALITY

Section II includes additional criteria for alignment to the standards as well as indicators of quality. Indicators of quality

are scored differently from the other criteria; a higher score in Section II indicates that materials are more closely

aligned.

Together, the non-negotiable criteria and the additional alignment criteria reflect the 10 criteria from the K–8 Publishers’

Criteria for Mathematics. The indicators of quality are taken from the K–8 Publishers’ Criteria as well. For more

information on these elements, see achievethecore.org/publisherscriteria.

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SECTION I:

Non-Negotiable 1. FOCUS ON MAJOR WORK: Students and teachers using the materials as designed

devote the large majority of time in each grade K–8 to the major work of the grade.

Grade Major Clusters Days Spent on Cluster

% of Total Time Spent on Cluster

Additional or Supporting Clusters or

Other

Days Spent on Cluster

% of Total Time Spent on

Cluster

1G. Grade 6 6.NS: B 22 14%

6.RP: A 27 17% 6.G: A 12 7%

6.NS: A, C 19 12% 6.SP: A, B 23 14%

6.EE: A, B, C 59 36% OTHER

Major Total: 105 65% Non-Major Total: 57 35%

1H. Grade 7 7.RP: A 39 22% 7.G: A, B 33 18%

7.NS: A 32 18% 7.SP: A, B, C 38 30%

7.EE: A, B 45 25% OTHER


1I. Grade 8 8.NS: A 5 3%

8.EE: A, B, C 43 29% 8.G: C 6 4%

8.F: A, B 38 26% 8.SP: A 14 10 %

8.G: A, B 42 28% OTHER


To be aligned to the CCSSM, materials should devote at least 65% and up to approximately 85% of class time to the major work of each grade with Grades K–2 nearer the upper end of that range, i.e., 85%. Each grade must meet the criterion; do not average across two or more grades.

Meet? (Y/N) Yes

Justification/Notes

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SECTION I (continued):

Non-Negotiable 2. FOCUS IN K–8: Materials do not assess any of the following topics before the grade level

indicated.

Topic Grade level introduced

Materials assess only at, or after, the

indicated grade level

Evidence

2A. Probability, including chance,

likely outcomes, probability

models.

7 T F Students are first introduced to probability

in the Grade 7 Unit What Do You Expect?

2B. Statistical distributions, including

center, variation, clumping,

outliers, mean, median, mode,

range, quartiles; and statistical

association or trends, including

two-‐way tables, bivariate

measurement data, scatter

plots, trend line, line of best fit,

correlation.

6 T F Students are first introduced to statistics

in the Grade 6 Unit Data About Us.

2C. Similarity, congruence, or geometric transformations.

8 T F Students explore the concepts of similarity

through scale drawings in the Grade 7

Unit Stretching and Shrinking. Formalized

study of congruence and transformations

is found in the Grade 8 Unit Butterflies,

Pinwheels, and Wallpaper.

2D. Symmetry of shapes, including

line/reflection symmetry,

rotational symmetry.

4 T F Students are assumed to have proficient

understanding of symmetry, which they

apply throughout different Units that

involve geometry concepts, such as

Covering and Surrounding (Grade 6),

Shapes and Designs, Stretching and

Shrinking, and Filling and Wrapping

(Grade 7), and Butterflies, Pinwheels, and

Wallpaper (Grade 8).

To be aligned to the CCSSM, materials cannot assess above-named topics before they are introduced in the CCSSM. All four of the T/F items above must be marked ‘true’ (T).

Meet? (Y/N) Yes

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Justification/Notes

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Non-Negotiable 3. RIGOR and BALANCE: The instructional materials for each grade reflect the balances in the

Standards and help students meet the rigorous expectations set by the standards, by helping students

develop conceptual understanding, procedural skill and fluency, and application.

Aspects of Rigor True/False Evidence

3A. Attention to Conceptual Understanding:

Materials develop conceptual

understanding of key

mathematical concepts, especially

where called for in specific

content standards or cluster

headings.

T F CMP3 emphasizes developing conceptual understanding

through its three-part lessons. In the Launch phase, teachers

present the Problem to the class, which they position within

students’ prior learning. Teachers work with students to

clarify the task presented in the Problem. In the Explore

phase, students work together to solve the problem

presented, and plan for the presentation of their solution.

The Summarize phase has students engage in discourse

about the solutions shared.

Table 1 indicates the clusters in the CCSSM that call explicitly

for the initial conceptual development of key concepts and

the CMP3 Units where that development takes place.

3B. Attention to Procedural Skill and Fluency:

Materials give attention

throughout the year to individual

standards that set an expectation

of procedural skill and fluency.

T F The nature of the CMP3 Problems requires students to

practice and apply concepts and skills learned in previous

Units or grades, allowing for frequent opportunities to

develop procedural fluency. Table 2 identifies the fluency

expectations called for in the Standards and the CMP3 Units

that are designed to help students towards fluency.

3C. Attention to Applications: Materials are

designed so that teachers and

students spend sufficient time

working with engaging

applications, without losing focus

on the major work of each grade

T F CMP3 provides a wealth of opportunities for students to

work with engaging applications. The structure of the

program presents students with a rich problem in the

Launch phase of the daily lesson, and students learn

appropriate concepts and skills that they can then integrate

into their problem solution pathway.

3D. Balance: The three aspects of rigor

are not always treated together,

and are not always treated

separately

T F In CMP3, conceptual understanding underpins fluency work;

fluency is developed in the context of application through

the ACE exercises. Within each Investigation, students think

about and apply concepts in ways that cultivate and deepen

their conceptual understanding. Through this deeper

conceptual understanding, students are able to achieve

fluency when appropriate.

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Aspects of Rigor True/False Evidence

To be aligned to the CCSSM, materials for each grade must attend to each element of rigor and must represent the balance reflected in the Standards. All four of the T/F items above must be marked ‘true’ (T).’

Meet? (Y/N) Yes

Justification/Notes


Non-Negotiable 4. PRACTICE-CONTENT CONNECTIONS: Materials meaningfully connect the Standards for

Mathematical Content and the Standards for Mathematical Practice.

Practice-Content Connections True/False Evidence

4A. The materials connect the Standards for

Mathematical Practice and the Standards for

Mathematical Content.

T F CMP3 embraces the essence of the Common Core

State Standards at a deep and organic level,

starting with its CCSS-aligned tables of contents

through its instructional philosophy with an

emphasis on inquiry, problem-solving strategies

and applications. CMP3 students learn to

communicate their reasoning by constructing

viable arguments, offering proofs, and using

representations. These approaches, which are

aligned with the Standards for Mathematical

Practice, are explicitly woven within the content of

the curriculum and connected to the Common

Core Content Standards.

4B. The developer provides a description or

analysis, aimed at evaluators, which shows

how materials meaningfully connect the

Standards for Mathematical Practice to the

Standards for Mathematical Content within

each applicable grade.

T F The Guide to Connected Mathematics 3:

Understanding, Implementing,

and Teaching reference book offers an overview of

how the program helps students develop

proficiency with the Standards for Mathematical

Practice. This information is summarized in

Appendix A.

To be aligned to the CCSSM, materials must connect the practice standards and content standards and the developer must provide a narrative that describes how the two sets of standards are meaningfully connected within the set of materials for each grade. Both of the T/F items above must be marked ‘true’ (T).

Meet? (Y/N)

Yes

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Justification/Notes

Materials must meet all four non-negotiable criteria listed above to be aligned to the CCSS and to continue to the evaluation in Section II.

# Met All of the Non-Negotiable criteria

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SECTION II ADDITIONAL ALIGNMENT CRITERIA AND INDICATORS OF QUALITY Materials must meet all four non-negotiable criteria listed above to be aligned to the CCSS and to continue to the

evaluation in Section II.

Section II includes additional criteria for alignment to the Standards as well as indicators of quality. Indicators of

quality are scored differently from the other criteria: a higher score in Section II indicates that materials are more

closely aligned. Instructional materials evaluated against the criteria in Section II will be rated on the following scale:

• 2 – (meets criteria): A score of 2 means that the materials meet the full intention of the criterion in all grades.

• 1 – (partially meets criteria): A score of 1 means that the materials meet the full intention of the criterion for some

grades or meets the criterion in many aspects but not the full intent of the criterion.

• 0 – (does not meet criteria): A score of 0 means that the materials do not meet many aspects of the criterion.

For Section II parts A, B, and C, districts should determine the minimum number of points required for approval.

Before evaluation, please review sections A – C, decide the minimum score according to the needs of your district,

and write in the number for each section.

II(A) Alignment Criteria for Standards for Mathematical Content Score Justification/Notes

1. Supporting content enhances

focus and coherence

simultaneously by engaging

students in the major work of

the grade.

2 1 0 The richness of the Problems presented in CMP3 offers

robust opportunities for students to engage in both major

and supporting content. This interweaving of content from

different domains and clusters is a hallmark of the program.

As Table 3 shows, even when students are working in a

geometry Unit, they are drawing on and using algebraic

concepts.

2. Materials are consistent with the progressions in the Standards.

2A. Materials base content

progressions on the grade-by-

grade progressions in the

Standards.

2 1 0 The content progressions in CMP3 align to the progressions

in the Standards. Students make tangible progress each year

with minimum review. In CMP3, grade level work begins

during the first two to four weeks.

2B. Materials give all students

extensive work with grade-level

problems.

2 1 0 The structure of the CMP3 program provides students with

extensive work with grade-level problems. Rather than asking

students to spend important classroom time with a series of

skills practice exercises, CMP3 asks them to ponder solutions

to rich problems that require them to draw on concepts and

skills that are grade-level appropriate.

2C. Materials relate grade level

concepts explicitly to prior

knowledge from earlier grades.

2 1 0 The organization and sequencing of Units in CMP3 are based

on many years of research and field-testing. The program

embodies a thoughtful development of math concepts in the

different domains. Within the Units, students and teachers

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II(A) Alignment Criteria for Standards for Mathematical Content Score Justification/Notes

will frequently find references to Units or Investigations

studied in previous years. As an example, Table 4 shows the

progression of proportionality concept development across

Grades 6–8.

3. Materials foster coherence through connections at a single grade, where appropriate and where required by the

Standards.

3A. Materials include learning

objectives that are visibly

shaped by CCSSM cluster

headings.

2 1 0 The learning objectives of the CMP3 program are clearly

shaped by the goals of the CCSSM: helping students become

proficient math thinkers. The Investigations and Units fully

embody this goal. Students interact with mathematics in

meaningful and relevant situations that help them connect

concepts, enhancing their conceptual understanding and

proficiency with both fluency and application.

3B. Materials include problems and

activities that serve to connect

two or more clusters in a

domain, or two or more

domains in a grade, in cases

where these connections are

natural and important.

2 1 0 As was noted earlier, the Problems presented in CMP3

provide natural connections between domains and clusters.

Math concepts are not taught in isolation, but within a rich

problem-solving setting. Students find natural connections

among domains and clusters as they work out solutions to

the Problems presented.

3C. Materials preserve the focus,

coherence, and rigor of the

Standards even when targeting

specific objectives.

2 1 0 The organization and sequencing of Units are intentionally

structured to promote focus, coherence, and rigor.

MUST HAVE ----- POINTS IN SECTION II(A) FOR APPROVAL Score: 14

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SECTION II ADDITIONAL ALIGNMENT CRITERIA AND INDICATORS OF QUALITY II(B) Alignment Criteria for Standards for

Mathematical Practice Score Justification/Notes

4. Focus and Coherence via Practice

Standards: Materials promote

focus and coherence by

connecting practice

standards with content that

is emphasized in the

Standards.

2 1 0 CMP3 makes the connection of the Practice Standards with the

Content Standards explicit in every lesson. For example, in the

first Investigation of the Grade 8 Unit Say It With Symbols,

students spend time understanding algebraic expressions. As

they explore different ways of representing tile borders and

areas of pools with algebraic expressions, students are brought

to see the structure of such expressions. They make use of that

structure in their work with expressions. Students revisit the

Distributive Property in Problem 1.4. In the Problem

Introduction, they are challenged with writing two different but

equivalent expressions to represent the area of a swimming

pool and explaining how the diagrams and expressions

illustrate the Distributive Property. These kinds of prompts help

focus students on the structure of expressions, leading them to

recognize and write equivalent expressions. Students continue

to make use of this structure later in the Unit, when they use

the Distributive Property in the process of solving equations.

5. Careful Attention to Each Practice

Standard: Materials attend to

the full meaning of each

practice standard.

2 1 0 The heart and soul of the Mathematical Practices have been

the foundation of the CMP3 classroom from its beginning,

especially the practice “make sense of problems and persevere

in solving them.” A description of how CMP3 attends to each

practice standard can be found in Appendix A. In CMP, one

additional practice has been critical in helping students

develop new and deeper understandings and strategies. New

knowledge is developed by connecting and building on prior

knowledge. In the process, understanding of prior knowledge

is extended and deepened.

6. Emphasis on Mathematical Reasoning: Materials support the Standards' emphasis on mathematical reasoning by:

6A. Materials prompt students to

construct viable arguments

and critique the arguments

of other concerning key

grade-level mathematics that

is detailed in the content

2 1 0 The inquiry-based learning approach of CMP3 affords students

opportunities to share with classmates their thinking about

problems, their solution methods, and their reasoning about

the solutions. Many Problems found throughout the program

specifically call for students to justify or explain their solutions,

or critique a sample explanation. These Problems are not

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II(B) Alignment Criteria for Standards for Mathematical Practice Score Justification/Notes

standards (cf. MP.3). optional or avoidable sections of the material but rather major

parts of the core of CMP3. In addition, the Mathematical

Reflections at the end of each Investigation help students

develop foundational critical reasoning skills by having them

describe processes they have developed throughout the

Investigation, and construct explanations for why these

processes work. The ability to articulate a clear explanation for

a process is a stepping stone to critical analysis and reasoning

of both the student’s own processes and those of others.

6B. Materials engage students in

problem solving as a form of

argument.

2 1 0 CMP3 is built on a foundation of inquiry-based instruction that

has sense-making and problem-solving at its heart. As was

noted above, the structure of the program encourages

students to engage in mathematical discourse as they present

and defend their solutions.

6C. Materials explicitly attend to

the specialized language of

mathematics.

2 1 0 Essential vocabulary terms are set in bold and blue in order to

call students’ attention to them. The teacher support provides

teachers with many suggestions for supporting students’

vocabulary development. Students are encouraged to create

their own mathematical dictionaries to use as a reference from

year to year.

MUST HAVE ----- POINTS IN SECTION II(B) FOR APPROVAL Score: 10

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SECTION II - ADDITIONAL ALIGNMENT CRITERIA AND INDICATORS OF QUALITY II(C) Indicators of Quality Score Justification/Notes

7. The underlying design of the materials

distinguishes between problems and exercises.

In essence the difference is that in solving

problems, students learn new mathematics,

whereas in working exercises, students apply

what they have already learned to build

mastery. Each problem or exercise has a

purpose.

2 1 0 The design of CMP3 clearly distinguishes

between problems and exercises. The Launch,

Explore, and Summarize phases of each

Investigation are designed as learning

experiences while the ACE Exercises are

practice exercises.

8. Design of assignments is not haphazard:

exercises are given in intentional sequences.

2 1 0 The sequencing of Units and Investigations is

based on the years of research and field

training carried out by the CMP3 authors. The

Problems and supporting activities have been

carefully designed based on the research and

field-testing.

9. There is variety in the pacing and grain size of

content coverage.

2 1 0 The number of days devoted to each cluster

varies depending on the grain size and

content coverage of the cluster. See the table

in Section I (on page 2) of this evaluation tool.

10. There is variety in what students produce. For

example, students are asked to produce

answers and solutions, but also, in a grade-

appropriate way, arguments and explanations,

diagrams, mathematical models, etc.

2 1 0 The tasks and assignments in CMP3 ask

students to produce a range of solutions, from

calculations and computations to models and

written explanations. For example, each

Investigation ends with a Mathematical

Reflection in which students record their

thinking about their work from the

Investigation.

11. Lessons are thoughtfully structured and

support the teacher in leading the class

through the learning paths at hand, with

active participation by all students in their

own learning and in the learning of their

classmates.

2 1 0 The teacher support for CPM3, delivered in

both print and digital formats, provides

extensive information about both math

content and math pedagogy. Within the

pedagogy are guiding questions that teachers

are encouraged to ask to maximize student

learning.

12. There are separate teacher materials that

support and reward teacher study including,

2 1 0 As was noted above, the teacher support for

CMP3 provides extensive teacher notes about

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II(C) Indicators of Quality Score Justification/Notes

but not limited to: discussion of the

mathematics of the units and the

mathematical point of each lesson as it relates

to the organizing concepts of the unit,

discussion on student ways of thinking and

anticipating a variety of students responses,

guidance on lesson flow, guidance on

questions that prompt students thinking, and

discussion of desired mathematical behaviors

being elicited among students.

the math content students are learning. There

is also information about the progression of

topics and tasks within a grade and across

grades.

Perhaps the most essential feature of the

teacher support materials are the guiding

questions that teachers can ask students as

they engage in problem solving and as they

present their solutions. These questions have

been carefully crafted to create an

environment of mathematical discourse in the

classroom.

13. Manipulatives are faithful representations of

the mathematical objects they represent.

2 1 0 CMP3 has a suite of 12 digital math tools that

students can access online as they work

through problem solutions. There is also a

Problem (lesson)-based manipulatives kit for

every grade. Many labsheets provide

templates for creating paper manipulatives

specific to a Problem, such as the Shapes Set

and the Polystrips used in the Unit Shapes

and Designs.

14. Manipulatives are connected to written

methods.

2 1 0 Many of the digital math tools include a

feature that translates the content of the

workmat to a symbolic representation. For

example, the integer chips tool will model the

equation represented on the workmat.

15. Materials are carefully reviewed by qualified

individuals, whose names are listed, in an

effort to ensure freedom from mathematical

errors and grade-level appropriateness.

2 1 0 The CMP authors from Michigan State

University are well-respected math educators

and active members of the NCTM community

who have called on colleagues in the

community to review the program at regular

intervals. Because the program is so widely

field-tested and researched, it is often

reviewed and critiqued for mathematical

accuracy.

16. The visual design isn't distracting or chaotic,

but supports students in engaging

2 1 0 The clean design of CMP3 supports students

in their learning by creating a non-distracting

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II(C) Indicators of Quality Score Justification/Notes

thoughtfully with the subject. yet engaging visual environment.

17. Support for English Language Learners and

other special populations is thoughtful and

helps those students meet the same standards

as all other students. The language in which

problems are posed is carefully considered.

2 1 0 The teacher support includes a section called

"Providing for Individual Needs,” which gives

teachers ideas for supporting individual

student needs. CMP3 Launch videos often

help introduce vocabulary and contexts

difficult for ELL students to grasp. The Guide

to Connected Mathematics 3:

Understanding, Implementing,

and Teaching reference book provides an

overview of strategies to support English

Language Learners in the CMP3 classroom.

The support also includes guidance to help

ELLs adapt to the social environment of the

classroom, and suggested ways of simplifying

language for ELLS. (See Guide to Connected

Mathematics 3 pages 99–111.)

MUST HAVE ----- POINTS IN SECTION II(C) FOR APPROVAL Score: 22

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FINAL EVALUATION SECTION PASS/FAIL Final Justification/Notes

Section 1 - Non-Negotiable Criteria 1–4 P

Section II(A) - Alignment Criteria for Standards for Mathematical Content

P

Section II(B) - Alignment Criteria for Standards for Mathematical Practice

P

Section II(C) - Indicators of Quality P

FINAL DECISION FOR THIS MATERIAL PURCHASE? Y/N

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Table 1 Conceptual Development Standards

Standard CMP3 Unit and Investigation

Grade 6

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

quantities.

Comparing Bits an Pieces: Inv. 1–

4;

D i l O I 16.RP.A.2 Understand the concept of a unit rate a/b associated with a ratio a : b with b ≠ 0, and use rate

language in the context of a ratio relationship.

Comparing Bits an Pieces: Inv. 1–

4;

D i l O I 16.NS.C.5 Understand that positive and negative numbers are used together to describe quantities having

opposite directions or values … .

Comparing Bits an Pieces: Inv. 3

6.NS.C.6 Understand a rational number as a point on the number line... .

b. Understand signs of numbers in ordered pairs as indicating locations in quadrants of the

di t l

Variables and Patterns: Inv. 2

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

b. Understand the absolute value of a rational number as its distance from 0 on the number

li

Comparing Bits and Pieces: Inv. 3

6.EE.B.5 Understand solving an equation or inequality as a process of answering a question: which values from

a specified set, if any, make the equation or inequality true? ... .

Decimal Ops: Inv. 2;

Variables and Patterns: Inv. 4

6.EE.B.6 . . . understand that a variable can represent an unknown number, or, depending on the purpose at

hand, any number in a specified set.

Let's Be Rational: Inv. 4;

Covering and Surrounding: Inv.

1–4;

Decimal Ops: Inv. 2, 4;

V i bl d P tt I 3 4

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6.SP.A.2 Understand that a set of data collected to answer a statistical question has a distribution, which can

be described by its center, spread, and overall shape.

Data About Us: Inv. 1, 2, 3, 4

Grade 7

7.NS.A.1b Understand p + q as the number located a distance | q | from p, in the positive or negative direction

depending on whether q is positive or negative.

Accentuate the Negative: Inv. 1,

2;

7.NS.A.1c Understand subtraction of rational numbers as adding the additive inverse, p – q = p + (–q). Accentuate the Negative: Inv. 1,

2

7.NS.A.2a Understand that multiplication is extended from fractions to rational numbers by requiring that

operations continue to satisfy the properties of operations, particularly the distributive property, leading

Accentuate the Negative: Inv. 3,

4

7.NS.A.2b Understand that integers can be divided, provided that the divisor is not zero, and every quotient of

integers (with non-zero divisor) is a rational number.

Accentuate the Negative: Inv. 3

7.EE.A.2 Understand that rewriting an expression in different forms in a problem context can shed light on the

problem and how the quantities in it are related.

Shapes and Designs: Inv. 2;

Moving Straight Ahead: Inv. 3, 4;

Filling and Wrapping: Inv. 1, 3

7.SP.A.1 Understand that statistics can be used to gain information about a population by examining a sample of

the population; generalizations about a population from a sample are valid only if the sample is

representative of that population. Understand that random sampling tends to produce representative

samples and support valid inferences.

Samples and Populations: Inv.

2

7.SP.C.5 Understand that the probability of a chance event is a number between 0 and 1 that expresses the

likelihood of the event occurring.

What Do You Expect?: Inv. 2–5;

Samples and Populations: Inv.

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7.SP.C.8a Understand that, just as with simple events, the probability of a compound event is the fraction of

outcomes in the sample space for which the compound event occurs.

What Do You Expect?: Inv. 2–5

Grade 8

8.NS.A.1 Understand Informally that every number has a decimal expansion; for rational numbers show that the

decimal expansion repeats eventually, and convert a decimal expansion, which repeats eventually into a

rational number.

Looking for Pythagoras: Inv. 4

8.EE.C.8a Understand that solutions to a system of two linear equations in two variables correspond to points of

intersection of their graphs, because points of intersection satisfy both equations simultaneously.

Thinking With Mathematical

Models: Inv. 2;

Say It With Symbols: Inv. 3;

It’s In the System: Inv. 1, 2, 3

8.F.A.1 Understand that a function is a rule that assigns to each input exactly one output. The graph of a

function is the set of ordered pairs consisting of an input and the corresponding output.


Models: Inv. 1, 2, 3

Growing, Growing, Growing:

Inv. 1–4;

8.G.A.2 Understand that a two-dimensional figure is congruent to another if the second can be obtained from

the first by a sequence of rotations, reflections, and translations; given two congruent figures, describe a

sequence that exhibits the congruence between them.

Butterflies, Pinwheels, and

Wallpaper: Inv. 2, 3

8.G.A.4 Understand that a two-dimensional figure is similar to another if the second can be obtained from the

first by a sequence of rotations, reflections, translations, and dilations; given two similar two-dimensional

figures, describe a sequence that exhibits the similarity between them.

Butterflies, Pinwheels, and

Wallpaper: Inv. 4

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8.SP.A.4 Understand that patterns of association can also be seen in bivariate categorical data by displaying

frequencies and relative frequencies in a two-way table.


Models: Inv. 5

Table 2 Fluency Development

Standard CMP3 Unit

Grade 6

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

6.NS.B.3 Fluently add, subtract, multiply, and divide multi-digit decimals using the standard algorithm for

each operation.

Let’s Be Rational, Decimal Ops

Grade 7

7.EE.B.4a Solve word problems leading to equations of the form px + q = r and p(x + q) = r, where p, q, and

r are specific rational numbers. Solve equations of these forms fluently.

Moving Straight Ahead

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Table 3 Supporting Clusters and Major Cluster Connections

Supporting Cluster and Standards CMP3 Units Major Cluster Connection Comment

Grade 6

Solve real-world and mathematical

problems involving area, surface

area, and volume. (6.G.1, 6.G.2, 6.G.3,

6.G.4)

Covering and

Surrounding

Apply and extend previous understandings of

numbers to the system of rational numbers.

Apply and extend previous understandings of

arithmetic to algebraic expressions.

Students solve measurement problems

involving computation with rational

numbers. Also, as they work with formulas,

students apply their work with expressions.

Grade 7

Use random sampling to draw

inferences about a population.

(7.SP.A.1, 7.SP.A.2)

Samples and

Populations

Analyze proportional relationships and use them

to solve real-world and mathematical problems

Solve real-life and mathematical problems using

numerical and algebraic expressions and

equations.

Students use numeric and algebraic

expressions and equations to analyze data

from random samples of a population and

draw conclusions about that sample. They

then use proportional reasoning to apply

such a conclusion to the whole population.

Investigate chance processes and

develop, use, and evaluate

probability models. (7.SP.C.5,

7.SP.C.6, 7.SP.C.7a, 7.SP.C.7b,

7.SP.C.8a, 7.SP.C.8b, 7.SP.C.8c)

What Do You

Expect?

Solve real-life and mathematical problems using

numerical and algebraic expressions and

equations.

Students develop probability models and

use numeric and algebraic expressions and

equations to evaluate them.

Grade 8

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Supporting Cluster and Standards CMP3 Units Major Cluster Connection Comment

Know that there are numbers that

are not rational, and approximate

them by rational numbers. (8.NS.A.1,

8.NS.A.2)

Looking for

Pythagoras

Work with radicals and integer exponents.

Understand and apply the Pythagorean Theorem.

While working with radicals and the

Pythagorean Theorem, students encounter

square roots of numbers that are not

perfect squares. In this context, they explore

irrational numbers.

Investigate patterns of association in

bivariate data. (8.SP.A.1, 8.SP.A.2,

8.SP.A.3, 8.SP.A.4)

Thinking with

Mathematical

Models

Understand the connections between

proportional relationships, lines and linear

equations.

Analyze and solve linear equations and pairs of

simultaneous linear equations.

Define, evaluate and compare functions.

Use functions to model relationships between

quantities.

Using their understanding of functions,

students analyze bivariate data and use

linear models to represent the relationship

between the two quantities.

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Table 4 Grade-to-Grade Concept Development -- Proportionality

CMP3 Units Development of Proportionality

Grade 6

Comparing Bits and Pieces

Decimal Ops

Variables and Patterns

Students’ study of proportionality begins with a study of ratios and rates. Students draw on their understanding of the

relationship between units of measure to formalize ratios. They-think of fractions, decimals, and percents as ratios and

they explore ratio equivalence.

Grade 7

Stretching and Shrinking

Comparing and Scaling

Moving Straight Ahead

What Do you Expect?

In Grade 7, students build on the foundation of ratios and rates from Grade 6 to formalize their understanding of

proportionality. They study the constant of proportionality as they create models for situations with proportionality. They

express proportionality in geometry and in real-world settings, such as banking (simple and compound interest) and

consumerism (mark-ups and mark-downs).

Grade 8

Thinking with Mathematical

Models

Students spend some time graphing proportional relationships as an introduction to linear equations and functions.

They study slope as an expression of proportionality.

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Table 5 - Coherent Connections Across Domains and Clusters

Unit and Investigation Focus Question Domain and Cluster Standard Connections

Decimal Ops,

Problem 4.1

What’s the Tax on

This Item?

How do you find the tax and

total cost of an item from a

given selling price and tax

rate?

How do you find the base

price from a given tax rate

and amount?

6.RP.A – Understand ratio concepts

and use ratio reasoning to solve

problems.

6.NS.B – Compute fluently with

multi=digit numbers and find

common factors and multiples.

6.RP.A.3

6.NS.B.3

In order to calculate tax, total price, and

base price, students will solve problems

involving percent. In order to solve these

percent problems, students will need to

compute fluently with multi-digit

decimals.

Decimal Ops,

Problem 4.2

Computing Tips

How do you find the tip and

total cost of a restaurant meal

from a given meal price and

tip rate?

How do you find the meal

price from a given tip percent

and amount?



problems.




6.RP.A.3

6.NS.B.3

In order to calculate tip, total price, and

meal price, students will solve problems

involving percent. In order to solve these

percent problems, students will need to

compute fluently with multi-digit

decimals.

Decimal Ops,

Problem 4.3

Percent Discounts

How do you find the discount

and total cost of an item from

a given selling price and

discount rate?

How do you express change in

a given amount as a percent

change?



problems.




6.RP.A.3

6.NS.B.3

In order to calculate discount price, total

price, and base price, students will solve

problems involving percent. In order to

solve these percent problems, students

will need to compute fluently with multi-

digit decimals.

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Unit and Investigation Focus Question Domain and Cluster Standard Connections

Decimal Ops,

Problem 4.4

Putting

Operations

Together

How do you decide which

operations to perform when a

problem involves decimals and

percents?



problems.




6.RP.A.3

6.NS.B.3

In order to solve these percent problems,

students will need to compute fluently

with multi-digit decimals.

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Appendix A

Overview of the Standards for Mathematical Practice in CMP3

The Common Core Standards for Mathematical Practice come alive in the CMP3 classroom as students and teachers

interact around a sequence of rich tasks to discuss, conjecture, validate, generalize, extend, connect, and

communicate. As a result, students develop deep understanding of concepts and the inclination and ability to

reason and make sense of new situations.

The heart and soul of the Mathematical Practices have been the foundation of the CMP3 classroom from its

beginning, especially the practice “make sense of problems and persevere in solving them.” In CMP, one additional

practice has been critical in helping students develop new and deeper understandings and strategies. New

knowledge is developed by connecting and building on prior knowledge. In the process, understanding of prior

knowledge is extended and deepened. Thus, CMP3’s additional practice is:

Build on and connect to prior knowledge in order to build deeper understandings and new insights.

Students use many of the Mathematical Practices each day in class. To enhance students’ metacognition of the role

of the Mathematical Practices in developing their understanding and reasoning, examples of student reasoning that

reflect several of the Mathematical Practices are given at the end of each Investigation in the Student Edition. The

teacher support offers additional examples of student reasoning. Below is an explanation of how CMP addresses

mathematical practices throughout the student editions:

MP1: Make sense of problems and persevere in solving them:

This mathematical practice comes alive in the Connected Mathematics classroom as students and teachers interact

around a sequence of rich problems, to conjecture, validate, generalize, extend, connect, and communicate.

MP2: Reason abstractly and quantitatively:

As students observe, experiment with, analyze, induce, deduce, extend, generalize, relate and manipulate information

from problems, they develop the disposition to inquire, investigate, conjecture and communicate with others around

mathematical ideas.

MP3: Construct viable arguments and critique the reasoning of others:

The student and teacher materials support a pedagogy that focuses on explaining thinking and understanding the

reasoning of others.

MP4: Model with mathematics:

The student materials provide opportunities to construct, make inferences from, and interpret concrete, symbolic,

graphic, verbal, and algorithmic models of quantitative, statistical, probabilistic and algebraic relationships.

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MP5: Use appropriate tools strategically:

Problem settings encourage the selection and intelligent use of calculators, computers, drawing and measuring tools,

and physical models to measure attributes and to represent, simulate, and manipulate relationships.

MP6: Attend to precision:

Students are encouraged to decide whether an estimate or an exact answer for a calculation is called for, to

compare estimates to computed answers, and to choose an appropriate measure or scale depending on the degree

of accuracy needed.

MP7: Look for and make use of structure:

Problems are deliberately designed and sequenced to prompt students to look for interrelated ideas, and take

advantage of patterns that show how data points, numbers, shapes or algebraic expressions are related to each

other.

MP8: Look for and express regularity in repeated reasoning:

Students are encouraged to observe and explain patterns in computations or symbolic reasoning that lead to further

insights and fluency with efficient algorithms.

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