New Instructional Materials Evaluation Tool Connected Mathematics...
Transcript of New Instructional Materials Evaluation Tool Connected Mathematics...
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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
Major Total: 116 65% Non-Major Total: 68 35%
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
Major Total: 123 83% Non-Major Total: 25 17%
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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SECTION I (continued):
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
SECTION I (continued):
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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Standard CMP3 Unit and Investigation
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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Standard CMP3 Unit and Investigation
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.
Thinking With Mathematical
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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Standard CMP3 Unit and Investigation
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.
Thinking With Mathematical
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?
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 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?
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 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?
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 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.