EdReports math 6-8

Indicator 1a Materials assess the grade-level content and, if applicable, content from
earlier grades.
Scoring:
2 points
● Materials assess grade-level standards or include
above-grade assessment items that could be
removed or modified without impacting the
structure of the materials.
● Above-grade items are mathematically reasonable.
● For Grades K-5, no topics are explicitly assessed
from (i), (ii), or (iii) below.
0 points
● Materials include above-grade assessment items
that, if removed or modified, would impact the
structure of the materials.
OR
● Above-grade items are not mathematically
reasonable.
OR
● For Grades K-5, topics are explicitly assessed from
(i), (ii), or (iii) below.
About this indicator:
What is the purpose of this Indicator?
This indicator, along with indicator 1b, determines the shift of Focus. In order to maintain Focus, materials assess
grade-level standards. Assessments are determined by the publisher and are series-specific. Content from future
grades may be introduced, but students should not be held accountable on assessments for future expectations.
Indicator 1a is focused on the assessments embedded in materials for K-8 grade-level standards which define
what students should understand and be able to do in their study of mathematics .1
Research or Standards connection:
● Common Core State Standards for Mathematics (CCSSM)
● K-8 Publishers' Criteria for the CCSSM (Summer 2012)
● Student Achievement Partners (SAP) Instructional Materials Evaluation Tool for K-8 Mathematics
● Achieve EQuIP Rubric for Lessons & Units
Resources:
● Video: "The Balance Between Skills and Understanding" (The Hunt Institute)
● SAP Coherence Map
● Institute for Mathematics Education Progressions Documents
1 "Common Core State Standards for Mathematics" http://www.corestandards.org/Math/

Indicator 1a Guiding Question:
Do assessments address grade-level standards?
Evidence Collection
Look at all interim, unit, and/or summative assessments. Look at scoring rubrics, if available, to determine
acceptable responses for the items. If all questions are on grade level, provide evidence of assessment items
that are representative of the materials.
For Grades K-5, do the assessments include items addressing the following topics:
(i) Probability, including chance, likely outcomes, and/or probability models?
(ii) Statistical distributions, including center, variation, clumping, outliers, mean,
median, mode, range, and/or quartiles; and statistical association or trends,
including two-way tables, bivariate measurement data, scatter plots, trend
line, line of best fit, and/or correlation?
(iii) Similarity, transformations, and/or congruence?
Identify above-grade-level assessment items. For each item, answer the following questions:
● What above-grade-level standard does the item align to? Please explain.
● What instructional activities, including span of time, are associated with the assessment item?
● If you remove or modify the above-grade-level item and any associated activities, would this impact the
structure of the materials? Please explain.
● Is the above-grade-level item mathematically reasonable? Please explain.
Mathematically Reasonable:
For this indicator, mathematically reasonable consists of two criteria: number of grade-levels above the current
one and time of year when the item occurs. For example, assessing standards that are one grade level above
the current one could be mathematically reasonable, or assessing above-grade-level standards at the end of
the academic year could be mathematically reasonable. To determine if an above-grade-level item is
mathematically reasonable, consider the following questions:
● How many grade levels above is the assessment item?
● What time of year do students encounter the assessment item?
Cluster Meeting
Discuss evidence of grade-level assessment items that are representative of the materials.
Discuss all above-grade-level assessment items and any additional information:
● Which assessment items are mathematically reasonable? Please explain.
● Which assessment items, if removed or modified, would impact the structure of the materials? Please
explain.

Gateway 1: Focus & Coherence
Criterion 1.1 Materials assess grade-level content and give all students extensive work with
grade-level problems to meet the full intent of grade-level standards.

Indicator 1b Materials give all students extensive work with grade-level problems to meet
the full intent of grade-level standards.
Scoring:
4 points
● Materials present all students
with extensive work with
grade-level problems.
AND
● Materials present opportunities
for all students to meet the full
intent of grade-level standards.
2 points
● Materials do not present all
students with extensive work
with grade-level problems.
OR
● Materials do not present
opportunities for all students to
meet the full intent of
grade-level standards.
0 points
● Materials do not present all
students with extensive work
with grade-level problems.
AND
● Materials do not present
opportunities for all students to
meet the full intent of
grade-level standards.
About this indicator:
What is the purpose of this Indicator?
This indicator, along with indicator 1a, determines the shift of Focus. This indicator examines if the materials
provide all students with extensive work with grade-level problems, so that all students engage with the full intent
of the grade-level standards.
ALL standards in the CCSSM should be considered throughout evidence collection for Gateway 1 indicators 1b,
1c, 1d, 1e, 1f, and 1g.
Research or Standards connection:
● Common Core State Standards for Mathematics (CCSSM)
● K-8 Publishers' Criteria for the CCSSM (Summer 2012)
● Student Achievement Partners (SAP) Instructional Materials Evaluation Tool for K-8 Mathematics
● Achieve EQuIP Rubric for Lessons & Units
Resources:
● Video: "The Balance Between Skills and Understanding" (The Hunt Institute)
● SAP Coherence Map
● Institute for Mathematics Education Progressions Documents

Indicator 1b Guiding Question:
Do the materials give all students extensive work with grade-level problems to meet the full intent of
grade-level standards?
Evidence Collection
Are all grade-level standards present? If not, which ones are not present?
Do the materials engage all students in extensive work with grade-level problems? Identify where and how the
materials engage all students in extensive work with grade-level problems. Include evidence of standards
where extensive work is not present. To determine if extensive work has been given for a standard, consider
the following questions:
● Is the standard from a major cluster or a supporting cluster? (Standards from major clusters might require a
different amount of problems than standards from supporting clusters.)
● Have all aspects/parts of a standard been addressed? (Standards with more parts or aspects might require
a different amount of problems than standards with fewer parts or aspects.)
● To what aspect(s) of rigor might the standard correspond? (Standards corresponding to one aspect of rigor
might require a different amount of problems than standards that correspond to a different aspect of rigor.)
Do all students engage with the full intent of grade-level standards? Identify where and how the materials
engage all students with the full intent of grade-level standards. Include evidence of standards where the full
intent is not met.
If the materials include activities that have differentiation suggestions, are they engaging all students with
GRADE-LEVEL problems? Note any differences in the ways in which different populations engage with
grade-level problems. Also note if there is guidance for teachers to inform differentiation options.
Does the amount of off-grade-level content (even if it is mathematically reasonable) negatively impact students’
work with grade-level content to meet the full intent of the grade-level standards? If yes, include evidence that
explains which grade-level standards are affected.
Mathematically Reasonable:
For this indicator, mathematically reasonable refers to the number of grade-levels above or below the current
one. For example, content that is one grade level above or below the current grade level could be
mathematically reasonable.
Cluster Meeting
Consider the following question(s) as evidence is synthesized:
● Are there any grade-level standards absent from the materials? Evidence must be provided for all
grade-level standards absent in the materials.
● Do the materials engage all students in extensive work with grade-level problems? Please explain.
● Do all students engage with the full intent of grade-level standards? Please explain.
● Do the materials include activities that have differentiation suggestions? If so, are they engaging all
students with GRADE-LEVEL problems?
● Does the amount of off-grade-level content (even if it is mathematically reasonable) negatively impact
students’ work with grade-level content to meet the full intent of the grade-level standards? If yes, include
evidence that explains which grade-level standards are affected.

Gateway 1: Focus & Coherence
Criterion 1.2
Coherence
Each grade’s materials are coherent and consistent with the Standards.
What is the purpose of this Criterion?
In Criterion 2, reviewers consider the coherence of the materials by examining:
● how much of the materials address the major clusters of the grade,
● the connections between major and supporting standards within each grade,
● the connections between two or more clusters in a domain, or two or more domains in a grade, and
● the connections between grade-level standards to prior and future-grade-level standards.
Research Connection
● Common Core State Standards for Mathematics (CCSSM)
● K-8 Publishers' Criteria for the CCSSM (Summer 2012)
● Student Achievement Partners (SAP) Instructional Materials Evaluation Tool for High School Mathematics
● Achieve EQuIP Rubric for Lessons & Units
Scoring:
Meets Expectations
● 7-8 points
Partially Meets Expectations
● 5-6 points
Does Not Meet Expectations
● <5 points

Gateway 1: Focus & Coherence
Criterion 1.2 Each grade’s materials are coherent and consistent with the Standards.

Indicator 1c When implemented as designed, the majority of the materials address the
major clusters of each grade.
Scoring:
2 points
● The majority (at least 65%) of the materials, when
implemented as designed, address the major clusters of
the grade.
0 points
● The majority (at least 65%) of the materials,
when implemented as designed, do not
address the major clusters of the grade.
About this indicator:
What is the purpose of this Indicator?
This indicator, along with indicators 1d, 1e, 1f, and 1g, determines the shift of Coherence. This indicator addresses
coherence within the grade level by examining if the majority of the materials address the major clusters of the
grade, including connections between major and supporting clusters.
Indicator 1c examines how much of the materials addresses standards identified as major work for each grade in
the K-8 grade-level standards. This indicator also includes connections between standards identified as
supporting work and major work.
This indicator is informed by the other indicators in Coherence and should be reviewed at the end of Gateway 1.
ALL standards in the CCSSM should be considered throughout evidence collection for Gateway 1 indicators 1b,
1c, 1d, 1e, 1f, and 1g.
Research or Standards connection:
● Common Core State Standards for Mathematics (CCSSM)
● K-8 Publishers' Criteria for the CCSSM (Summer 2012)
● Student Achievement Partners (SAP) Instructional Materials Evaluation Tool for K-8 Mathematics
● Achieve EQuIP Rubric for Lessons & Units
Resources:
● Video: "The Balance Between Skills and Understanding" (The Hunt Institute)
● SAP Coherence Map
● Institute for Mathematics Education Progressions Documents
● Focus Component 2: Major Clusters of the Grade

Indicator 1c Guiding Question:
Do at least 65% of the materials, when implemented as designed, address the major clusters of the grade?
Evidence Collection
This indicator is informed by the other indicators in Coherence and should be reviewed at the end of Gateway
1. Connections between supporting work and major work should be included in the calculations for different
perspectives.
Collect evidence in the Table provided in the 1c Reviewer Evidence Collection document.
For this indicator, you should consider multiple perspectives when collecting evidence, including:
● Units/chapters (including assessments):
○ List the units/chapters that address the major work of the grade and connections between supporting
and major work.
○ Explain the specific standards/clusters being addressed.
○ What percent of units/chapters address the major work of the grade and connections between
supporting and major work?
○ State the units/chapters included in the numerator and denominator of your calculation.
● Lessons/Activities (including assessments):
○ List the lessons/activities that address the major work of the grade and connections between
supporting and major work.
○ Explain the specific standards/clusters being addressed.
○ What percent of lessons/activities address the major work of the grade and connections between
supporting and major work?
○ State the units/chapters included in the numerator and denominator of your calculation.
● Instructional Time (including assessments):
○ Time can be days, hours, or minutes. For materials where a lesson equals a day, please note above,
and use minutes as a measure for this perspective.
○ List the amounts of instructional time that address the major work of the grade and connections
between supporting and major work.
○ Explain the specific standards/clusters being addressed.
○ What percent of instructional time addresses the major work of the grade and connections between
supporting and major work?
○ State the amounts of instructional time included in the numerator and denominator of your calculation.
If it is not possible for a reviewer to capture data for any of these three perspectives, then the reviewer should
provide an explanation of why a perspective was not feasible. (For example, a set of materials is not divided
into units/chapters, so a calculation from that perspective is not possible.)
If a perspective other than the three already given is a better representation for the materials, then the reviewer
should clearly explain why the other perspective is better and include evidence and calculations to accompany
the additional perspective.
The total amount of instructional time for this indicator is calibrated to the totals used in Indicator 1g. Calibrating
the totals does not mean the totals for 1c and 1g have to be the same.
If the materials include optional chapters/lessons/activities or provide choices for teachers as to what parts of
the materials are used, please discuss calculations for 1c with EdReports’ staff.

A minimum of 65% should be viewed as a guideline for reviewers. Reviewers should discuss materials on the
borderline (e.g., 60%-64%) with EdReports staff.
Cluster Meeting
Consider the following question(s) as evidence is synthesized:
● Review the evidence collected from at least three perspectives. (Chapters/units, lessons, days, minutes,
etc.)
● Which perspective is most representative of the materials? Why?
● Do the materials spend a minimum of 65% of instructional time on major work of the grade, including
supporting work connected to major work of the grade? Please explain.
● How similar are each reviewer's calculations? If there are differences in calculations between reviewers,
those differences should be clearly evident in reviewers evidence collection documents.
● If the materials spiral concepts and skills within each grade level, please explain how you integrated the
spiraling within your calculations?
● Evidence that reflects difficulty in calculating percentages for the materials should be discussed with
EdReports staff to determine appropriate placement within the grade-level report.

Gateway 1: Focus & Coherence
Criterion 1.2 Each grade’s materials are coherent and consistent with the Standards.

Indicator 1d Supporting content enhances focus and coherence simultaneously by
engaging students in the major work of the grade.
Scoring:
2 points
● Connections between
supporting and major work
enhance focus on major work.
1 point
● Multiple connections between
supporting and major work are
entirely absent from the
grade-level materials.
0 points
● Supporting work is treated
separately and does not include
connections to major work.
About this indicator:
What is the purpose of this Indicator?
This indicator, along with indicators 1c, 1e, 1f, and 1g, determines the shift of Coherence. This indicator addresses
connections within the grade level.
In order to maintain Coherence, materials should connect mathematics within and across grade levels. Materials
should indicate how learning progresses from previous learning and toward future learning.
ALL standards in the CCSSM should be considered throughout evidence collection for Gateway 1 indicators 1b,
1c, 1d, 1e, 1f, and 1g.
Research or Standards connection:
● Common Core State Standards for Mathematics (CCSSM)
● K-8 Publishers' Criteria for the CCSSM (Summer 2012)
● Student Achievement Partners (SAP) Instructional Materials Evaluation Tool for K-8 Mathematics
● Achieve EQuIP Rubric for Lessons & Units
Resources:
● Video: "The Balance Between Skills and Understanding" (The Hunt Institute)
● SAP Coherence Map
● Institute for Mathematics Education Progressions Documents
● Focus Component 2: Major Clusters of the Grade

Indicator 1d Guiding Questions:
Is supporting work connected to the major work of the grade?
Is supporting work addressed independently?
Evidence Collection
For connections between supporting standards/clusters and major standards/clusters, please answer the
following questions:
● What connections are made?
● Where are the connections made?
● How are those connections made?
When supporting standards/clusters are not connected to the major standards/clusters of the grade, is the
separation mathematically reasonable?
Mathematically Reasonable:
For this indicator, mathematically reasonable refers to the content of the standards. Not all supporting
standards address content that should be connected to major work of the grade. For example, in Kindergarten,
K.G.5 may not be connected to major work of the grade, and that would be mathematically reasonable.
Are there connections between supporting and major standards/clusters of the grade that are entirely absent
from the materials?
Note: A missed connection between supporting and major work standards/clusters is defined as a connection
that is entirely absent from the materials. For example:
● In Grade K, if the materials do not connect classifying shapes into categories (K.MD.3) with counting the
number of shapes in the categories (K.CC.5) anywhere in the materials, then this would be a missed
connection.
● In Grade 7, if the materials do not connect scale drawings (7.G.1) with ratios and proportional relationships
(7.RP.A) anywhere in the materials, then this would be a missed connection.
Cluster Meeting
Consider the following question(s) as evidence is synthesized:
● What connections between supporting and major standards/clusters of the grade were identified?
● Where were connections of supporting and major standards/clusters of the grade found?
● How were the connections made in the materials?

Gateway 1: Focus & Coherence
Criterion 1.2 Each grade’s materials are coherent and consistent with the Standards.

Indicator 1e Materials include problems and activities that serve to connect two or more
clusters in a domain or two or more domains in a grade.
Scoring:
2 points
● There are connections from
supporting work to supporting
work and/or from major work to
major work throughout the
grade-level materials, when
appropriate.
1 point
● Multiple connections from
supporting work to supporting
work and/or from major work to
major work are entirely absent
from the grade-level materials.
0 points
● There are no connections from
supporting work to supporting
work and from major work to
major work throughout the
grade-level materials.
About this indicator:
What is the purpose of this Indicator?
This indicator, along with indicators 1c, 1d, 1f, and 1g, determines the shift of Coherence. This indicator addresses
connections within the grade level.
In order to maintain Coherence, materials should connect mathematics within and across grade levels. Materials
should indicate how learning progresses from previous learning and toward future learning.
Indicator 1e supports coherence by identifying where materials connect across clusters and/or domains. ALL
standards in the CCSSM should be considered throughout evidence collection for Gateway 1 indicators 1b, 1c, 1d,
1e, 1f, and 1g.
Research or Standards connection:
● Common Core State Standards for Mathematics (CCSSM)
● K-8 Publishers' Criteria for the CCSSM (Summer 2012)
● Student Achievement Partners (SAP) Instructional Materials Evaluation Tool for K-8 Mathematics
● Achieve EQuIP Rubric for Lessons & Units
Resources:
● Video: "The Balance Between Skills and Understanding" (The Hunt Institute)
● SAP Coherence Map
● Institute for Mathematics Education Progressions Documents
● Focus Component 2: Major Clusters of the Grade

Indicator 1e Guiding Question:
Are there connections between major domains and/or clusters?
Are there connections between supporting domains and/or clusters?
Evidence Collection
Note: Evidence collection should address connections between clusters and/or domains and not individual
standards.
For connections between supporting clusters/domains or major clusters/domains, please answer the following
questions:
● What connections are made?
● Where are the connections made?
● How are those connections made?
When major to major clusters/domains or supporting to supporting clusters/domains are not connected, is the
separation mathematically reasonable?
Mathematically Reasonable:
For this indicator, mathematically reasonable refers to the content of the clusters/domains. Not all
clusters/domains address content that should be connected to other clusters/domains. For example, in Grade 1,
1.MD.B may not be connected to other clusters/domains of the grade, and that would be mathematically
reasonable.
Are there connections between major to major clusters/domains or supporting to supporting clusters/domains
that are entirely absent from the materials?
Note: A missed connection is defined as a connection that is entirely absent from the materials. For example:
● In Grade 3, if the materials do not connect using place value understanding and properties of operations
to perform multi-digit arithmetic (3.NBT.A) with understanding properties of multiplication and the
relationship between multiplication and division (3.OA.B) anywhere in the materials, then this would be a
missed connection.
● In Grade 8, if the materials do not connect understanding and applying the Pythagorean Theorem (8.G.B)
with working with radicals and integer exponents (8.EE.A) anywhere in the materials, then this would be a
missed connection.
Note: Evidence should not include connections from supporting work to major work. Those connections are
addressed in Indicator 1d.
Cluster Meeting
Consider the following question(s) as evidence is synthesized:
● What connections from major clusters/domains to major clusters/domains of the grade were identified, and
where are those connections located?
● What connections from supporting clusters/domains to supporting clusters/domains of the grade were
identified, and where are those connections located?
● How were the connections made in the materials?

● If connections are entirely absent from the materials, explain/provide an example of what and where that
connection might have occurred.

Gateway 1: Focus & Coherence
Criterion 1.2 Each grade’s materials are coherent and consistent with the Standards.

Indicator 1f Content from future grades is identified and related to grade-level work, and
materials relate grade-level concepts explicitly to prior knowledge from earlier
grades.
Scoring:
2 points
● Content from future grades is
identified and related to
grade-level work.
AND
● Materials relate grade-level
concepts explicitly to prior
knowledge from earlier grades.
1 point
● Content from future grades is
not identified and related to
grade-level work.
OR
● Materials do not relate
grade-level concepts explicitly
to prior knowledge from earlier
grades.
0 points
● Content from future grades is
not identified and related to
grade-level work.
AND
● Materials do not relate
grade-level concepts explicitly
to prior knowledge from earlier
grades.
About this indicator:
What is the purpose of this Indicator?
This indicator, along with indicators 1c, 1d, 1e, and 1g, determines the shift of Coherence. This indicator examines
vertical coherence through identifications of future content related to grade-level work and relations of grade-level
concepts explicitly to prior knowledge from earlier grades.
ALL standards in the CCSSM should be considered throughout evidence collection for Gateway 1 indicators 1b,
1c, 1d, 1e, 1f, and 1g.
Research or Standards connection:
● Common Core State Standards for Mathematics (CCSSM)
● K-8 Publishers' Criteria for the CCSSM (Summer 2012)
● Student Achievement Partners (SAP) Instructional Materials Evaluation Tool for K-8 Mathematics
● Achieve EQuIP Rubric for Lessons & Units
Resources:
● Video: "The Balance Between Skills and Understanding" (The Hunt Institute)
● SAP Coherence Map
● Institute for Mathematics Education Progressions Documents

Indicator 1f Guiding Question:
Is content from future grades identified and related to grade-level work?
Do the materials relate grade-level concepts explicitly to prior knowledge from earlier grades?
Evidence Collection
Identify evidence at the chapter/unit level and/or lesson level where grade-level content is related to future
learning.
Identify evidence at the chapter/unit level and/or lesson level where prior learning is explicitly related to
grade-level content.
Note: The explicit relationship of grade-level concepts to prior knowledge should include the content of the
standards. If the materials include a standard code(s) with no explanation on the relationship to grade-level
content, that is not sufficient to meet the expectations for this indicator.
Cluster Meeting
Consider the following question(s) as evidence is synthesized:
● Is future learning identified and related to grade-level content? Where? How?
● Is prior learning explicitly identified and related to grade-level learning? Where? How?

Gateway 1: Focus & Coherence
Criterion 1.2 Each grade’s materials are coherent and consistent with the Standards.

Indicator 2a Materials develop conceptual understanding of key mathematical concepts,
especially where called for in specific content standards or cluster headings.
Scoring:
2 points
● Materials develop conceptual
understanding throughout the
grade level.
AND
● Materials provide opportunities
for students to independently
demonstrate conceptual
understanding throughout the
grade level.
1 point
● Materials do not develop
conceptual understanding
throughout the grade level.
OR
● Materials do not provide
opportunities for students to
independently demonstrate
conceptual understanding
throughout the grade level.
0 points
● Materials do not develop
conceptual understanding
throughout the grade level.
AND
● Materials do not provide
opportunities for students to
independently demonstrate
conceptual understanding
throughout the grade level.
About this indicator:
What is the purpose of this Indicator?
This indicator, along with 2b, 2c, and 2d, determines the shift of Rigor. Conceptual understanding of key concepts
will allow students to be able to access concepts from a number of perspectives in order to see mathematics as
more than a set of algorithmic procedures.
Research or Standards connection:
● Common Core State Standards for Mathematics (CCSSM)
● K-8 Publishers' Criteria for the CCSSM (Summer 2012)
● Student Achievement Partners (SAP) Instructional Materials Evaluation Tool for K-8 Mathematics
● Achieve EQuIP Rubric for Lessons & Units
● Achieve Framework to Evaluate Cognitive Complexity in Mathematics Assessments
Resources:
● SAP Coherence Map
● Institute for Mathematics Education Progressions Documents
● Reading: “Principles To Actions”, (NCTM) p. 42-48
● Video: "The Balance Between Skills and Understanding" (The Hunt Institute)
● Video: “Mathematics Fluency: A Balanced Approach” (The Hunt Institute)
● Video: “Building Conceptual Understanding in Mathematics” (NCTM)
● Video: “Conceptual Understanding Excerpt” (The Hunt Institute)

● Concrete Representational Abstract: Instructional Sequence for Mathematics
Indicator 2a Guiding Questions:
Do the materials develop conceptual understanding?
Do the materials provide opportunities for students to independently demonstrate conceptual understanding
throughout the grade level?
Evidence Collection
Conceptual Understanding is a flexible web of connections and relationships within and between ideas,
interpretations, and images of mathematical concepts.
Example: In Grade 4, students use their understanding of place value and properties of operations to perform
multi-digit arithmetic.
Note: The use of manipulatives does not necessarily indicate conceptual understanding. Conceptual
understanding can be developed without the use of manipulatives.
Do materials feature conceptual problems and discussion questions, including brief conceptual problems with
low computational difficulty?
● Example: 11 + 6 = __ + 2
● Example: Find a number greater than 3/5 and less than .75.
● Example: A fraction divided by a fraction is always/sometimes/never less than the original fraction.
Determine if the materials offer opportunities for students to engage with concrete and semi-concrete
representations, as well as verbalization and writing, when developing conceptual understanding. For
independent demonstration, student opportunities should not be guided or supported by teacher direction.
For example:
Do the materials develop cluster(s) and/or standard(s) that specifically relate to conceptual understanding?
(Consider the whole grade level.)
● If yes, provide evidence that explains how the materials develop conceptual understanding including unit,
lesson, and lesson part for reference for all examples.
● If no, specifically list any standards and/or clusters that attend to conceptual understanding and are not
developed by the materials. Describe how the conceptual understanding is not developed.
Examples of specific standards/clusters include, but are not limited to:
Grade 	Standards
Kindergarten 	K.CC.4

K.OA.1, 3, 4
K.NBT
Grade 1 	1.OA.B, 6
1.NBT.B
1.NBT.C
Grade 2 	2.OA.C
2.NBT.1, 7
2.MD.6
Grade 3 	3.OA.1, 2, B
3.NF
3.MD.C
Grade 4 	4.NBT.A
4.NBT.5, 6
4.NF
Grade 5 	5.NBT.A, 6, 7
5.NF.1, 3, 4, 5
5.MD.C
Grade 6 	6.RP.A
6.NS.5, 6, 7
6.EE.3-6
Grade 7 	7.NS.1, 2
7.EE.A
7.SP.1, 5
Grade 8 	8.EE.B
8.F.A
8.G.A
Cluster Meeting
Consider the following question(s) as evidence is synthesized:
● How do the materials develop conceptual understanding throughout each grade?
● How do the materials provide opportunities for students to independently demonstrate conceptual
understanding throughout each grade?
● Do the materials have a consistent lesson structure for developing conceptual understanding and where
students independently demonstrate conceptual understanding?
● Based on the student-facing materials, how do students independently demonstrate conceptual
understanding?

Gateway 2: Rigor & Mathematical Practices
Criterion 2.1 Materials reflect the balances in the Standards and help students meet the
Standards’ rigorous expectations by giving appropriate attention to:
developing students’ conceptual understanding; procedural skill and fluency;
and engaging applications.

Indicator 2b Materials give attention throughout the year to individual standards that set an
expectation for procedural skill and fluency.
Scoring:
2 points
● Materials develop procedural
skills and fluency throughout
the grade level.
AND
● Materials provide opportunities
for students to independently
demonstrate procedural skills
and fluency throughout the
grade level.
1 point
● Materials do not develop
procedural skills and fluency
throughout the grade level.
OR
● Materials do not provide
opportunities for students to
independently demonstrate
procedural skills and fluency
throughout the grade level.
0 points
● Materials do not develop
procedural skills and fluency
throughout the grade level.
AND
● Materials do not provide
opportunities for students to
independently demonstrate
procedural skills and fluency
throughout the grade level.
About this indicator:
What is the purpose of this Indicator?
This indicator, along with 2a, 2b, and 2d, determines the shift of Rigor. Students practice skills in order to access
more complex concepts and procedures.
Research or Standards connection:
● Common Core State Standards for Mathematics (CCSSM)
● K-8 Publishers' Criteria for the CCSSM (Summer 2012)
● Student Achievement Partners (SAP) Instructional Materials Evaluation Tool for K-8 Mathematics
● Achieve EQuIP Rubric for Lessons & Units
● Achieve Framework to Evaluate Cognitive Complexity in Mathematics Assessments
Resources:
● SAP Coherence Map
● Institute for Mathematics Education Progressions Documents
● Reading: “Principles To Actions”, (NCTM) p. 42-48
● Video: "The Balance Between Skills and Understanding" (The Hunt Institute)
● Video: “Mathematics Fluency: A Balanced Approach” (The Hunt Institute)
● Concrete Representational Abstract: Instructional Sequence for Mathematics

Indicator 2b Guiding Questions:
Do the materials develop procedural skill?
Do the materials provide opportunities for students to independently demonstrate procedural skills throughout
the grade level?
Evidence Collection
Procedural skill includes knowing how and why an algorithm works.
Fluency includes efficiency, accuracy, flexibility, and appropriate strategy selection.
Procedural skill and fluency may not be addressed in every unit, but should be addressed over the grade,
especially with those standards that explicitly call for fluency.
Collect evidence from each unit that shows how the materials develop procedural skill and fluency and where
students independently demonstrate procedural skill and fluency.
Identify where there are opportunities for students to progress toward fluency and procedural skill interwoven
with students’ developing conceptual understanding.
Note whether the materials include a specific section in units/chapters/lessons, etc. that are specifically
designed for procedural skill and fluency.
Examples of specific standards/clusters include, but are not limited to:
Grade 	Standards
Kindergarten 	K.OA.5
Grade 1 	1.OA.6
Grade 2 	2.OA.2
2.NBT.5
Grade 3 	3.OA.7
3.NBT.2
Grade 4 	4.NBT.4
Grade 5 	5.NBT.5
Grade 6 	6.NS.2
6.NS.3
6.EE.1
Grade 7 	7.NS.2d
7.EE.4a
7.G.5
Grade 8 	8.EE.1
8.EE.7

8.EE.8b
Cluster Meeting
Consider the following question(s) as evidence is synthesized:
● How do the materials develop procedural skill and fluency throughout each grade?
● How do the materials provide opportunities for students to independently demonstrate procedural skill and
fluency throughout each grade?
● Do the materials have a consistent lesson structure for developing procedural skill and fluency and where
students independently demonstrate procedural skill and fluency?
● Based on the student-facing materials, how do students independently demonstrate procedural skill and
fluency?

Gateway 2: Rigor & Mathematical Practices
Criterion 2.1 Materials reflect the balances in the Standards and help students meet the
Standards’ rigorous expectations by giving appropriate attention to:
developing students’ conceptual understanding; procedural skill and fluency;
and engaging applications.

Indicator 2c Materials are designed so that teachers and students spend sufficient time
working with engaging applications of the mathematics.
Scoring:
2 points
● Materials include multiple
routine and non-routine
applications of the mathematics
throughout the grade level.
AND
● Materials provide opportunities
for students to independently
demonstrate multiple routine
and non-routine applications of
the mathematics throughout the
grade level.
1 point
● Materials do not include multiple
routine and non-routine
applications of the mathematics
throughout the grade level.
OR
● Materials do not provide
opportunities for students to
independently demonstrate
multiple routine and non-routine
applications of the mathematics
throughout the grade level.
0 points
● Materials do not include multiple
routine and non-routine
applications of the mathematics
throughout the grade level.
AND
● Materials do not provide
opportunities for students to
independently demonstrate
multiple routine and non-routine
applications of the mathematics
throughout the grade level.
About this indicator:
What is the purpose of this Indicator?
This indicator, along with 2a, 2b, and 2c, determines the shift of Rigor. Materials should include activities with a
variety of contexts for both routine and non-routine applications. Students should apply mathematical knowledge
in real-world contexts to make meaning of and access the content.
Research or Standards connection:
● Common Core State Standards for Mathematics (CCSSM)
● K-8 Publishers' Criteria for the CCSSM (Summer 2012)
● Student Achievement Partners (SAP) Instructional Materials Evaluation Tool for K-8 Mathematics
● Achieve EQuIP Rubric for Lessons & Units
● Achieve Framework to Evaluate Cognitive Complexity in Mathematics Assessments
Resources:
● SAP Coherence Map
● Institute for Mathematics Education Progressions Documents
● Reading: “Principles To Actions”, (NCTM) p. 42-48
● Video: "The Balance Between Skills and Understanding" (The Hunt Institute)

● Video: “Mathematics Fluency: A Balanced Approach” (The Hunt Institute)
● Video: “Building Conceptual Understanding in Mathematics” (NCTM)
● Video: “Conceptual Understanding Excerpt” (The Hunt Institute)
● Concrete Representational Abstract: Instructional Sequence for Mathematics
● Achieve the Core: Situation Types for Operations in Word Problems
Indicator 2c Guiding Questions:
Do the materials develop students’ ability to utilize mathematical concepts and skills in engaging applications
throughout the grade level?
Do the materials provide opportunities for students to independently demonstrate application of mathematical
concepts and skills in real-world contexts throughout the grade level?
Evidence Collection
Application problems often, but not always, present a real-world scenario. Students employ a prescribed
mathematical strategy or choose their own strategy to find a solution. Example: students apply knowledge
about multiplication (Grade 3) to calculate volume (Grade 5).
Collect evidence of problems/activities that align to the standards/clusters for each grade level that address
application. Note whether the materials include a specific section in units/chapters/lessons, etc. that is
specifically designed for application.
Collect evidence of problems/activities where students independently demonstrate applying mathematics.
Evidence should include students solving problems in a variety of contexts, and students should be able to
make their own assumptions or simplifications in order to model the contexts mathematically.
Routine applications are prescriptive word problems that provide a situation and a question. They have a real
world context, but the problems generally include all necessary (and no extraneous) information. The problems
are designed to elicit particular mathematical strategies. The correct answers are usually pre-determined.
Non-Routine applications include unfamiliar/novel contexts. These problems often present a real world
scenario, and students employ their choice of mathematical strategies to find a solution. Thus, the problem is
open in the middle and closed at the beginning and end. Target answers are known, although the solutions
may be allowed to have some variation.
Collect evidence from the materials of multiple routine and non-routine applications of the mathematics
throughout the grade level.
Collect evidence of opportunities for students to independently demonstrate multiple routine and non-routine
applications of mathematics throughout the grade level.
Examples of specific standards/clusters that address application include, but are not limited to:
Grade 	Standards
Kindergarten 	K.OA.2
K.G.1

Grade 1 	1.OA.A
1.MD.4
Grade 2 	2.OA.A
2.MD.5, 8, 10
Grade 3 	3.OA.3, 8
3.MD.A, D
Grade 4 	4.OA.A
4.NF.3d, 4.NF.4c
4.MD.2
Grade 5 	5.NF.2, 6, 7c
5.MD.5b
5.G.2
Grade 6 	6.RP.3
6.NS.1
6.EE.7, 9
Grade 7 	7.RP.A
7.NS.3
7.EE.3
Grade 8 	8.EE.8c
8.F.B
8.G.7
Cluster Meeting
Consider the following question(s) as evidence is synthesized:
● How do the materials include multiple routine and non-routine applications throughout each grade?
● How do the materials provide opportunities for students to independently demonstrate multiple routine
and non-routine applications throughout each grade?
● Do the materials have a consistent lesson structure for including multiple routine and non-routine
applications?
● Based on the student-facing materials, how do students independently demonstrate multiple routine and
non-routine applications?

Gateway 2: Rigor & Mathematical Practices
Criterion 2.1 Materials reflect the balances in the Standards and help students meet the
Standards’ rigorous expectations by giving appropriate attention to:
developing students’ conceptual understanding; procedural skill and fluency;
and engaging applications.

Indicator 2d The three aspects of rigor are not always treated together and are not always
treated separately. There is a balance of the three aspects of rigor within the
grade.
Scoring:
2 points
● All three aspects of rigor are
present independently
throughout each grade level.
AND
● Multiple aspects of rigor are
engaged simultaneously to
develop students’ mathematical
understanding of a single
topic/unit of study throughout
each grade level.
1 point
● All three aspects of rigor are not
present independently
throughout each grade level.
OR
● Multiple aspects of rigor are not
engaged simultaneously to
develop students’ mathematical
understanding of a single
topic/unit of study throughout
each grade level.
0 points
● All three aspects of rigor are not
present independently
throughout each grade level.
AND
● Multiple aspects of rigor are not
engaged simultaneously to
develop students’ mathematical
understanding of a single
topic/unit of study throughout
each grade level.
About this indicator:
What is the purpose of this Indicator?
This indicator, along with 2a, 2b, and 2c, determines the shift of Rigor. Grade-level materials must include a
balance of conceptual understanding, procedural skill and fluency, and application, and this balance should be
evident in supporting students as they develop mathematical understanding.
Research or Standards connection:
● Common Core State Standards for Mathematics (CCSSM)
● K-8 Publishers' Criteria for the CCSSM (Summer 2012)
● Student Achievement Partners (SAP) Instructional Materials Evaluation Tool for K-8 Mathematics
● Achieve EQuIP Rubric for Lessons & Units
● Achieve Framework to Evaluate Cognitive Complexity in Mathematics Assessments
Resources:
● SAP Coherence Map
● Institute for Mathematics Education Progressions Documents
● Reading: “Principles To Actions”, (NCTM) p. 42-48
● Video: "The Balance Between Skills and Understanding" (The Hunt Institute)
● Video: “Mathematics Fluency: A Balanced Approach” (The Hunt Institute)

● Video: “Building Conceptual Understanding in Mathematics” (NCTM)
● Video: “Conceptual Understanding Excerpt” (The Hunt Institute)
● Concrete Representational Abstract: Instructional Sequence for Mathematics
● Achieve the Core: Situation Types for Operations in Word Problems
Indicator 2d Guiding Question:
Do the materials balance the three aspects of rigor?
Evidence Collection
Note: Consider the intent of the program to balance the three aspects of rigor, not the quality of the
materials—indicators 2a-c focus on the quality of materials. Evidence should be different than the evidence
collected for 2a, 2b, and 2c.
Collect evidence demonstrating how the three aspects of rigor are present independently throughout each
grade level.
Collect evidence demonstrating where multiple aspects of rigor are engaged in simultaneously to develop
students’ mathematical understanding of a single topic/unit of study throughout each grade level, for example:
● Consider whether materials use conceptual understanding to develop procedural skill and fluency.
● Consider if students use multiple representations (i.e. manipulatives, drawings, expressions, equations,
tables, graphs, charts, number lines, etc.) and written/oral explanations to support their work in application
problems.
Do the materials emphasize one aspect of rigor over the others in specific units? If so, do the materials work to
maintain balance throughout the grade level?
Cluster Meeting
Consider the following question(s) as evidence is synthesized:
● How are the three aspects of rigor present independently throughout each grade level?
● How are multiple aspects of rigor engaged simultaneously to develop students’ mathematical
understanding of a single topic/unit of study throughout each grade level?
● Do the materials emphasize one aspect of rigor over the others in specific units? If so, do the materials
work to maintain balance throughout the grade level?

Gateway 2: Rigor & Mathematical Practices
Criterion 2.2
Standards for Mathematical Practice
Materials meaningfully connect the Standards for Mathematical Content and the Standards for Mathematical
Practice (MPs).
What is the purpose of this Criterion?
The purpose of this criterion is to ensure the Standards for Mathematical Practice are identified and connected
to grade-level mathematical content, and the materials present opportunities for students to both learn and
independently demonstrate each of the MPs.
Research Connection
● Common Core State Standards for Mathematics (CCSSM)
Scoring:
Note: If the materials do not identify the MPs for teachers, evidence of this will be included in the
Practice-Content Connections criterion report. The lack of identification of the MPs will result in the deduction
of 1 point in the scoring for indicator 2e only.
Meets Expectations
● 9-10 points
Partially Meets Expectations
● 6-8 points
Does Not Meet Expectations
● <6 points

Gateway 2: Rigor & Mathematical Practices
Criterion 2.2 Materials meaningfully connect the Standards for Mathematical Content and
the Standards for Mathematical Practice.

Indicator 2e Materials support the intentional development of MP1: Make sense of
problems and persevere in solving them; and MP2: Reason abstractly and
quantitatively, for students, in connection to the grade-level content standards,
as expected by the mathematical practice standards.
Scoring:
2 points
● There is intentional
development of MP1 to meet its
full intent in connection to
grade-level content.
AND
● There is intentional
development of MP2 to meet its
full intent in connection to
grade-level content.
1 point
● There is intentional
development of MP1 to meet its
full intent in connection to
grade-level content.
OR
● There is intentional
development of MP2 to meet its
full intent in connection to
grade-level content.
0 points
● There is no intentional
development of MP1 to meet its
full intent in connection to
grade-level content.
AND
● There is no intentional
development of MP2 to meet its
full intent in connection to
grade-level content.
About this indicator:
What is the purpose of this Indicator?
This indicator, along with 2f, 2g, 2h, and 2i, determines the adherence to the Standards for Mathematical Practice.
This indicator specifically examines MPs 1 and 2 by examining whether the provided opportunities for student
engagement with the MPs are a) connected to the mathematical content of the grade level and b) fully developed
across the grade level.
Research or Standards connection:
● Common Core State Standards for Mathematics (CCSSM)
● K-8 Publishers' Criteria for the CCSSM (Summer 2012)
● Student Achievement Partners (SAP) Instructional Materials Evaluation Tool for K-8 Mathematics
● Achieve EQuIP Rubric for Lessons & Units
Resources:
● SAP Coherence Map
● Institute for Mathematics Education Progressions Documents
● MP1: Make sense of problems and persevere in solving them
● MP2: Reason abstractly and quantitatively
● Mathematical Practices Compilation

Indicator 2e Guiding Questions:
Across the grade level, are MP1 and MP2 identified and connected to mathematical content?
Across the grade level, is there intentional development of MP1 and MP2 that reaches the full intent of the
MPs?
Evidence Collection
Look through all teacher and student materials to ensure that MP1 and MP2 are occurring throughout the grade
level. This includes, but is not limited to: lessons, unit overviews, scope and sequence charts, and/or other
instructional guides.
Record any instances where MP1 and MP2 are misleading in the curricular materials (e.g. a lesson is marked as
aligned to an MP when only a small part of the lesson addresses the MP).
To check that MP1 and MP2 are connected to grade-level content and are developed to their full intent, look at
lessons, assessments, and any examples/descriptions of anticipated student work that require students to:
MP1 	MP2
Analyze and make sense of problems: actively
engage in solving problems by working to
understand the information in the problems and the
questions asked.
Consider units involved in a problem and attend to
the meaning of quantities.
Use a variety of strategies that make sense to solve
the problem.
Represent situations symbolically.
Monitor and evaluate their progress in solving
problems.
Explain/discuss what the numbers or symbols in an
expression/equation represent.
Determine if their answers make sense. 	Understand the relationships between problem
scenarios and mathematical representations.
Reflect on and revise their problem solving strategy.
Look at teacher directions and how teachers are guided to carry out the lessons so that students are engaged
in the MPs. In particular, look for places where teachers are expected to:
● pose rich problems (MP1),
● provide time for students to make sense of problems (MP1),
● provide opportunities for students to engage in problem solving (MP1),
● ask clarifying and probing questions (MP1, MP2), and
● ensure students make connections between mathematical representations and scenarios (MP2).
Check to see if any of the materials address only the Standards for Mathematical Practice (meaning they are
not connected to grade-level mathematical content). Record any instances where the Standards for
Mathematical Practice are not connected to the grade-level mathematics content.

Across the grade level, verify that students use the MPs to their full intent within the materials.
If MPs are only located in a specific part of the teacher manuals (e.g. the teacher-led portion of the lesson), you
will need to look at other sections (e.g. independent work, homework, assessments) to ensure that the MPs are
intentionally connected to the content. Look where the MPs are identified, but also look at places where they
are not identified.
Cluster Meeting
Consider the following question(s) as evidence is synthesized:
● When are the MPs identified and connected to grade-level mathematical content?
● In what ways do the students use the MPs to their full intent across the grade level?
● In what ways, if any, do the supports provided for teachers enable students to engage with the MPs?

Gateway 2: Rigor & Mathematical Practices
Criterion 2.2 Materials meaningfully connect the Standards for Mathematical Content and
the Standards for Mathematical Practice.

Indicator 2f Materials support the intentional development of MP3: Construct viable
arguments and critique the reasoning of others, for students, in connection to
the grade-level content standards, as expected by the mathematical practice
standards.
Scoring:
2 points
● There is intentional
development of MP3 to meet its
full intent in connection to
grade-level content.
1 point
● There is intentional
development of students
constructing viable arguments
in connection to grade-level
content.
OR
● There is intentional
development of students
critiquing the reasoning of
others in connection to
grade-level content.
0 points
● There is no intentional
development of MP3 to meet its
full intent in connection to
grade-level content.
About this indicator:
What is the purpose of this Indicator?
This indicator, along with 2e, 2g, 2h, and 2i, determines the adherence to the Standards for Mathematical
Practice. This indicator specifically examines MP3 by examining whether the provided opportunities for student
engagement with the MP are a) connected to the mathematical content of the grade level and b) fully developed
across the grade level.
Research or Standards connection:
● Common Core State Standards for Mathematics (CCSSM)
● K-8 Publishers' Criteria for the CCSSM (Summer 2012)
● Student Achievement Partners (SAP) Instructional Materials Evaluation Tool for K-8 Mathematics
● Achieve EQuIP Rubric for Lessons & Units
Resources:
● SAP Coherence Map
● Institute for Mathematics Education Progressions Documents
● MP3: Construct viable arguments and critique the reasoning of others
● Mathematical Practices Compilation

Indicator 2f Guiding Questions:
Across the grade level, is MP3 identified and connected to mathematical content?
Across the grade level, is there intentional development of MP3 that reaches the full intent of the MP?
Evidence Collection
Look through all teacher and student materials to ensure that MP3 is occurring throughout the grade level.
This includes, but is not limited to: lessons, unit overviews, scope and sequence charts, and/or other
instructional guides.
Record any instances where MP3 is misleading in the curricular materials (e.g. a lesson is marked as aligned to
an MP when only a small part of the lesson addresses the MP).
To check that MP3 is connected to grade-level content and is developed to it full intent, look at lessons,
assessments, and any examples/descriptions of anticipated student work that require students to:
Construct Viable Arguments 	Critique the Reasoning of Others
Construct mathematical arguments. 	Perform error analysis of provided student
work/solutions/arguments.
Explain/justify their strategies and thinking orally or in
writing using concrete models, drawings, actions, or
numbers.
Listen to or read the arguments of others, decide
whether they make sense, and ask useful questions
to clarify or improve the arguments.
Create their own conjectures.
Look at teacher directions and how teachers are guided to carry out the lessons so that students are engaged
in the MPs. In particular, look for places where teachers are expected to:
● provide opportunities for students to engage in discourse to construct and/or critique mathematical
arguments,
● ask clarifying and probing questions, and
● help students compare methods and strategies.
Check to see if any of the materials address only the Standard for Mathematical Practice (meaning it is not
connected to grade-level mathematical content). Record any instances where the Standard for Mathematical
Practice is not connected to the grade-level mathematics content.
Across the grade level, verify that students use the MP to its full intent within the materials.
If the MP is only located in a specific part of the teacher manuals (e.g. the teacher-led portion of the lesson),
you will need to look at other sections (e.g. independent work, homework, assessments) to ensure that the MP
is intentionally connected to the content. Look where the MP is identified, but also look at places where it is
not identified.
Cluster Meeting

During the cluster meeting:
Consider the following question(s) as evidence is synthesized:
● When is MP3 identified and connected to grade-level mathematical content?
● In what ways do the students use MP3 to its full intent across the grade level?
● In what ways, if any, do the supports provided for teachers enable students to engage with MP3?

Gateway 2: Rigor & Mathematical Practices
Criterion 2.2 Materials meaningfully connect the Standards for Mathematical Content and
the Standards for Mathematical Practice.

Indicator 2h Materials support the intentional development of MP6: Attend to precision;
and attend to the specialized language of mathematics, for students, in
connection to the grade-level content standards, as expected by the
mathematical practice standards.
Scoring:
2 points
● There is intentional
development of MP6 to meet its
full intent in connection to
grade-level content.
AND
● Materials attend to the
specialized language of
mathematics.
1 point
● There is intentional development
of MP6 to meet its full intent in
connection to grade-level
content.
OR
● Materials attend to the
specialized language of
mathematics.
0 points
● There is no intentional
development of MP6 to meet its
full intent in connection to
grade-level content.
AND
● Materials do not attend to the
specialized language of
mathematics.
About this indicator:
What is the purpose of this Indicator?
This indicator, along with 2e, 2f, 2g, and 2i, determines the adherence to the Standards for Mathematical Practice.
This indicator specifically examines MP6, including the specialized language of mathematics, by examining
whether the provided opportunities for student engagement with the MP are a) connected to the mathematical
content of the grade level and b) fully developed across the grade level.
Research or Standards connection:
● Common Core State Standards for Mathematics (CCSSM)
● K-8 Publishers' Criteria for the CCSSM (Summer 2012)
● Student Achievement Partners (SAP) Instructional Materials Evaluation Tool for K-8 Mathematics
● Achieve EQuIP Rubric for Lessons & Units
Resources:
● SAP Coherence Map
● Institute for Mathematics Education Progressions Documents
● MP6: Attend to precision
● Mathematical Practices Compilation

Indicator 2h Guiding Questions:
Across the grade level, is MP6 identified and connected to mathematical content?
Across the grade level, is there intentional development of MP6 that reaches the full intent of the MP?
Across the grade level, is the specialized language of mathematics intentionally developed?
Evidence Collection
Look through all teacher and student materials to ensure that MP6 and the specialized language of
mathematics are occurring throughout the grade level. This includes, but is not limited to: lessons, unit
overviews, scope and sequence charts, and/or other instructional guides.
Record any instances where MP6 is misleading in the curricular materials (e.g. a lesson is marked as aligned to
an MP when only a small part of the lesson addresses the MP).
Record any instances where the specialized language of mathematics is misleading and/or erroneous in the
curricular materials (e.g. the materials use made up words versus mathematical terminology, the use of
symbols is incorrect or confusing, etc.).
To check that MP6 is connected to grade-level content and is developed to it full intent, look at lessons,
assessments and any examples/descriptions of anticipated student work that require students to:
● communicate using grade-level appropriate vocabulary and conventions,
● formulate clear explanations,
● state the meaning of symbols,
● calculate accurately and efficiently,
● specify units of measure,
● use and label tables, graphs, etc. appropriately, and
● introduce and use definitions accurately.
Look at teacher directions and how teachers are guided to carry out the lessons so that students are engaged
in the MPs. In particular, look for places where teachers are expected to:
● ensure students know and use clear definitions,
● model accurate, precise mathematical language (vocabulary and conventions), and
● provide feedback to students on the accurate use of mathematical language.
Ensure that mathematical definitions and terminology are precise and accurate (e.g. “commutative property”
versus “flip-flop”; using rate/ratio/fraction/proportion precisely; using accurate geometric terminology, even at
young ages). Provide specific examples of vocabulary, symbols, numbers, etc. that are not used accurately and
precisely.
Check to see if any of the materials address only the Standard for Mathematical Practice (meaning it is not
connected to grade-level mathematical content). Record any instances where the Standard for Mathematical
Practice is not connected to the grade-level mathematics content.
Across the grade level, verify that students use the MP to its full intent within the materials.
If the MP is only located in a specific part of the teacher manuals (e.g. the teacher-led portion of the lesson),
you will need to look at other sections (e.g. independent work, homework, assessments) to ensure that the MP

is intentionally connected to the content. Look where the MP is identified, but also look at places where it is
not identified.
Cluster Meeting
Consider the following question(s) as evidence is synthesized:
● When is MP6 identified and connected to grade-level mathematical content?
● In what ways do the students use MP6 to its full intent across the grade level?
● In what ways is the specialized language of mathematics intentionally developed?
● In what ways, if any, do the supports provided for teachers enable students to engage with MP6?

Gateway 2: Rigor & Mathematical Practices
Criterion 2.2 Materials meaningfully connect the Standards for Mathematical Content and
the Standards for Mathematical Practice.

Indicator 2i Materials support the intentional development of MP7: Look for and make use
of structure; and MP8: Look for and express regularity in repeated reasoning,
for students, in connection to the grade-level content standards, as expected
by the mathematical practice standards.
Scoring:
2 points
● There is intentional
development of MP7 to meet its
full intent in connection to
grade-level content.
AND
● There is intentional
development of MP8 to meet its
full intent in connection to
grade-level content.
1 point
● There is intentional
development of MP7 to meet its
full intent in connection to
grade-level content.
OR
● There is intentional
development of MP8 to meet its
full intent in connection to
grade-level content.
0 points
● There is no intentional
development of MP7 to meet its
full intent in connection to
grade-level content.
AND
● There is no intentional
development of MP8 to meet its
full intent in connection to
grade-level content.
About this indicator:
What is the purpose of this Indicator?
This indicator, along with 2e, 2f, 2g, and 2h, determines the adherence to the Standards for Mathematical Practice.
This indicator specifically examines MPs 7 and 8 by examining whether the provided opportunities for student
engagement with the MPs are a) connected to the mathematical content of the grade level and b) fully developed
across the grade level.
Research or Standards connection:
● Common Core State Standards for Mathematics (CCSSM)
● K-8 Publishers' Criteria for the CCSSM (Summer 2012)
● Student Achievement Partners (SAP) Instructional Materials Evaluation Tool for K-8 Mathematics
● Achieve EQuIP Rubric for Lessons & Units
Resources:
● SAP Coherence Map
● Institute for Mathematics Education Progressions Documents
● MP7: Look for and make use of structure
● MP8: Look for and express regularity in repeated reasoning
● Mathematical Practices Compilation

Indicator 2i Guiding Questions:
Across the grade level, are MP7 and MP8 identified and connected to mathematical content?
Across the grade level, is there intentional development of MP7 and MP8 that reaches the full intent of the MPs?
Evidence Collection
Look through all teacher and student materials to ensure that MP7 and MP8 are occurring throughout the
grade-level. This includes, but is not limited to: lessons, unit overviews, scope and sequence charts, and/or
other instructional guides.
Record any instances where MP7 and MP8 are misleading in the curricular materials (e.g. a lesson is marked as
aligned to an MP when only a small part of the lesson addresses the MP).
To check that MP7 and MP8 are connected to grade-level content and are developed to their full intent, look at
lessons, assessments, and any examples/descriptions of anticipated student work that require students to:
MP7 	MP8
Look for patterns or structures to make
generalizations and solve problems.
Notice repeated calculations to understand
algorithms and make generalizations or create
shortcuts.
Look for and explain the structure within
mathematical representations.
Create, describe, explain a general formula, process,
method, algorithm, model, etc.
Analyze a problem and look for more than one
approach.
Evaluate the reasonableness of their answers and
thinking.
Look at and decompose “complicated” into “simpler”
things.
Look at teacher directions and how teachers are guided to carry out the lessons so that students are engaged
in the MPs. In particular, look for places where teachers are expected to:
● provide tasks/problems with patterns (MP7),
● prompt students to look for and describe structure and/or patterns (MP7),
● provide situations in which students can use a strategy to develop understanding of a concept (MP7, MP8),
● provide a variety of examples that explicitly focus on patterns and repeated reasoning (MP7, MP8), and
● prompt students to make generalizations (MP8).
Check to see if any of the materials address only the Standards for Mathematical Practice (meaning they are
not connected to grade-level mathematical content). Record any instances where the Standards for
Mathematical Practice are not connected to the grade-level mathematics content.
Across the grade level, verify that students use the MPs to their full intent within the materials.
If MPs are only located in a specific part of the teacher manuals (e.g. the teacher-led portion of the lesson), you
will need to look at other sections (e.g. independent work, homework, assessments) to ensure that the MPs are

intentionally connected to the content. Look where the MPs are identified, but also look at places where they
are not identified.
Cluster Meeting
Consider the following question(s) as evidence is synthesized:
● When are the MPs identified and connected to grade-level mathematical content?
● In what ways do the students use the MPs to their full intent across the grade level?
● In what ways, if any, do the supports provided for teachers enable students to engage with the MPs?

Gateway 3: Usability
Criterion 3.1
Teacher Supports
The program includes opportunities for teachers to effectively plan and utilize materials with integrity and to
further develop their own understanding of the content.
What is the purpose of this Criterion?
This criterion examines how the materials support teachers:
● in delivering the student and ancillary materials, especially as it relates to students’ mathematical
development.
● in understanding the instructional approaches of the program and research-based strategies.
● in improving their own knowledge of the subject beyond the grade level.
● in understanding the role of the standards in the context of the overall series.
● in planning for effective instruction that includes appropriate materials and how caregivers can support
student progress and achievement.
Scoring:
Meets Expectations
● 8-9 points
Partially Meets Expectations
● 5-7 points
Does Not Meet Expectations
● <5 points

Gateway 3: Usability
Criterion 3.1 The program includes opportunities for teachers to effectively plan and utilize
materials with integrity and to further develop their own understanding of the
content.

Indicator 3a Materials provide teacher guidance with useful annotations and suggestions
for how to enact the student materials and ancillary materials, with specific
attention to engaging students in order to guide their mathematical
development.
Scoring:
2 points
● Materials provide
comprehensive guidance that
will assist teachers in
presenting the student and
ancillary materials.
AND
● Materials include sufficient and
useful annotations and
suggestions that are presented
within the context of the specific
learning objectives.
1 point
● Materials provide
comprehensive guidance that
will assist teachers in
presenting the student and
ancillary materials.
OR
● Materials include sufficient and
useful annotations and
suggestions that are presented
within the context of the specific
learning objectives.
0 points
● Materials do not provide
comprehensive guidance that
will assist them in presenting
the student and ancillary
materials.
AND
● Materials do not include
sufficient and useful
annotations and suggestions
that are presented within the
context of the specific learning
objectives.
About this indicator:
What is the purpose of this Indicator?
This indicator examines the materials to determine whether they contain teacher guidance with sufficient and
useful annotations and suggestions for how to enact the student materials and ancillary materials.
Indicator 3a Guiding Question:
Do the materials provide teacher guidance with useful annotations and suggestions for how to enact the
student materials and ancillary materials, with specific attention to engaging students in order to guide their
mathematical development?
Evidence Collection
Review the materials, both print and digital (if available), across the series.
Look for and record evidence to:
● Describe if and how the materials include overview sections, annotations, narrative information, or other

documents that will assist the teacher in presenting the student material and/or ancillary materials.
● Describe how information and guidance provided by the materials is useful for planning instruction. Look
for suggestions about instructional strategies and guidance for presenting the content (specifically how to
support students' mathematical development), which could include identifying and addressing student
errors and misconceptions. These are often in the planning sections as well as margin notes, but could
also be in the front matter philosophy, professional development, or explanations of program components.
Cluster Meeting
Discuss and answer the following questions to support consensus scoring conversations:
● How are the materials structured to provide information that will assist the teacher in presenting the
student material or ancillary materials?
● How do the materials provide specific guidance to plan instruction and support students in the content
(specifically how to support students' mathematical development)?

Gateway 3: Usability
Criterion 3.1 The program includes opportunities for teachers to effectively plan and utilize
materials with integrity and to further develop their own understanding of the
content.

Indicator 3b Materials contain adult-level explanations and examples of the more complex
grade-level/course-level concepts and concepts beyond the current course so
that teachers can improve their own knowledge of the subject.
Scoring:
2 points
● Materials contain adult-level
explanations and examples of
the more complex
grade/course-level concepts so
that teachers can improve their
own knowledge of the subject.
AND
● Materials contain adult-level
explanations and examples of
concepts beyond the current
course so that teachers can
improve their own knowledge
of the subject.
1 point
● Materials contain adult-level
explanations and examples of
the more complex
grade/course-level concepts so
that teachers can improve their
own knowledge of the subject.
OR
● Materials contain adult-level
explanations and examples of
concepts beyond the current
course so that teachers can
improve their own knowledge
of the subject.
0 points
● Materials do not contain
adult-level explanations and
examples of the more complex
grade/course-level concepts so
that teachers can improve their
own knowledge of the subject.
AND
● Materials do not contain
adult-level explanations and
examples of concepts beyond
the current course so that
teachers can improve their own
knowledge of the subject.
About this indicator:
What is the purpose of this Indicator?
This indicator examines the materials to determine whether they deepen teacher understanding of mathematical
concepts and the mathematical practices so that teachers can improve their own knowledge of the subject.
Indicator 3b Guiding Question:
Do the materials contain adult-level explanations and examples of the more complex grade/course-level
concepts and concepts beyond the current course so that teachers can improve their own knowledge of the
subject?
Evidence Collection
Review the materials, both print and digital (if available), across the series.
Look for and record evidence to:

● Describe if and how the materials provide complete adult-level explanations and examples that support
the teacher in developing their own understanding of the content and expected student practices.
Cluster Meeting
Discuss and answer the following questions to support consensus scoring conversations:
● Where are supports provided for teachers to develop their own understanding of more advanced,
grade-level concepts?
● Where are supports provided for teachers to develop their own understanding of concepts beyond the
current course?

Gateway 3: Usability
Criterion 3.1 The program includes opportunities for teachers to effectively plan and utilize
materials with integrity and to further develop their own understanding of the
content.

Indicator 3c Materials include standards correlation information that explains the role of the
standards in the context of the overall series.
Scoring:
2 points
● Correlation information is present
for the mathematics standards
addressed throughout the grade
level/series.
AND
● Explanations of the role of the
specific grade-level/course-level
mathematics are present in the
context of the series.
1 point
● Correlation information is present
for the mathematics standards
addressed throughout the grade
level/series.
OR
● Explanations of the role of the
specific grade-level/course-level
mathematics are present in the
context of the series.
0 points
● Standards correlations are
inaccurate or missing.
OR
● There are few, if any,
explanations of the role of the
specific grade-level/course-level
mathematics in the context of the
series.
About this indicator:
What is the purpose of this Indicator?
This indicator examines whether materials provide documentation of how each lesson and unit correlate to the
Common Core State Standards for Mathematics and whether materials provide explanations of the role of the
standards at each unit/module in the context of the overall series.
Research or Standards connection:
● Common Core State Standards for Mathematics (CCSSM)
Indicator 3c Guiding Question:
Do the materials include standards correlation information that explains the role of the standards in the context
of the overall series?
Evidence Collection
Review the print and digital (if available) table of contents, pacing guides, scope and sequence, and other
teacher materials.
Look for and record evidence to:
● Describe how teacher materials provide Information to allow for coherence across multiple course levels to
allow a teacher to make prior connections and teach for connections to future content.

● Describe how individual units, lessons, or activities throughout the series are correlated to the CCSSM.
Note:
● if materials only reference the Standards for Mathematical Content but not Mathematical Practices.
● if standards correlation is inconsistent between levels (I.e. from the overview to a module or a lesson).
● if standards correlation is inaccurate.
● if standards correlation is over-identified.
Cluster Meeting
Discuss and answer the following questions to support consensus scoring conversations:
● Are the standards identified consistently and accurately?
● Do the materials identify how the current work fits into the K-12 progression of learning?

Gateway 3: Usability
Criterion 3.1 The program includes opportunities for teachers to effectively plan and utilize
materials with integrity and to further develop their own understanding of the
content.

Indicator 3f Materials provide a comprehensive list of supplies needed to support
instructional activities.
Scoring:
1 point
● Materials include a comprehensive list of supplies
needed to support the instructional activities.
0 points
● Materials do not include a comprehensive list of
supplies needed to support instructional activities.
About this indicator:
What is the purpose of this Indicator?
This indicator examines the series to determine if the materials contain a comprehensive list of materials needed
to support implementation.
Indicator 3f Guiding Question:
Do the materials provide a comprehensive list of supplies needed to support instructional activities?
Evidence Collection
Review the materials across the series.
Look for and record evidence to:
● Determine whether a comprehensive list of required materials is provided.
Cluster Meeting
During the cluster meeting:
Discuss and answer the following questions to support consensus scoring conversations:
● Does the series provide a comprehensive list of required materials? At what level(s) is the support
provided (course, unit/module, lesson, etc.)?

Gateway 3: Usability
Criterion 3.1 The program includes opportunities for teachers to effectively plan and utilize
materials with integrity and to further develop their own understanding of the
content.

Indicator 3i Assessment information is included in the materials to indicate which
standards are assessed.
Scoring:
2 points
● Materials consistently identify
the standards and practices
assessed for formal
assessments.
1 point
● Materials identify the standards
and practices assessed for
some of the formal
assessments.
OR
● Materials consistently identify
the standards and practices
assessed for formal
assessments, but do not include
all standards and practices for
the grade or course level.
0 points
● Materials do not identify the
standards and practices
assessed for the formal
assessments.
OR
● Materials identify the standards
and practices assessed for
some of the formal
assessments, but do not include
all standards and practices for
the grade or course level.
About this indicator:
What is the purpose of this Indicator?
This indicator examines the assessment materials to determine whether they identify the standards being
assessed for all formal assessment types. It is important to note that some assessments may be building toward
the standards and not intended to measure full depth of the standards; these assessments should identify which
aspects of the standards are being assessed and/or being built toward.
Formal assessments are determined by the publisher and could include all types: formative, summative, etc.
Reviewers look for a list of standards assessed for the entire assessment and/or associated with each item/task.
Reviewers look for evidence of identification only.
Indicator 3i Guiding Question:
Does assessment information included in the materials indicate which standards are assessed?
Evidence Collection
Review assessments and corresponding assessment guidance across the series. *Note: this is not an item
analysis.
Look for and record evidence to:

● Describe how and where assessments clearly identify which standards are being assessed. Include the
level at which the assessment is given (unit, lesson, etc.) and the level at which standards are identified
(assessment, task, item, etc.).
● Describe any instances where standards are listed incorrectly or are not from the appropriate grade
level/band.
● In the event that the assessments build toward grade-level or grade-band standards, describe whether the
assessment information identifies which aspects of the standards are assessed. Also, describe how and
where the materials include information that details how the assessments build toward the standards for
the grade level or band.
Cluster Meeting
Discuss and answer the following questions to support consensus scoring conversations:
● Where and how do the materials clearly identify which standards are assessed?
● Do the standards correlations or assessment guidance documents indicate if all standards for the grade
level/band are assessed by the end of the grade level/band?
● If assessments are building towards the standards, do the materials identify which aspects of the standards
are being assessed and how these contribute to building toward grade-level/grade-band standards?

Gateway 3: Usability
Criterion 3.2 The program includes a system of assessments identifying how materials
provide tools, guidance, and support for teachers to collect, interpret, and act
on data about student progress towards the standards.

Indicator 3m Materials provide strategies and supports for students in special populations to
support their regular and active participation in learning grade-level/series
mathematics.
Scoring:
2 points
● Materials regularly provide
strategies, supports, and
resources for students in
special populations to support
their regular and active
participation in grade-level
mathematics.
1 point
● Materials do not regularly
provide strategies, supports,
and resources for students in
special populations to support
their regular and active
participation in grade-level
mathematics.
0 points
● There are no strategies,
supports, or resources for
students in special populations
to support their regular and
active participation in
grade-level mathematics.
About this indicator:
What is the purpose of this Indicator?
This indicator examines whether the materials provide strategies, supports, and resources for students in special
populations to support their regular and active participation in grade-level mathematics.
Research or Standards connection:
For this indicator, special populations refers to students that must overcome barriers that may require special
consideration and attention to ensure equal opportunity for success and in an educational setting.
Resources:
● Supporting Special Populations, Office of Elementary and Secondary Education
Indicator 3m Guiding Question:
What opportunities are there for students in special populations to engage with materials to support ongoing
participation in grade-level mathematics content?
Evidence Collection
Review teacher and student materials across the series.
Look for and record evidence to:

● Describe where and how the materials provide specific strategies and supports for differentiating
instruction to meet the needs of students in special populations.
● Identify whether the materials support students in special populations in regular and active participation in
grade-level mathematics and include any instances where differentiation does not present opportunities to
engage students in the work of the grade level.
● Describe teacher guidance around using grouping strategies with students in special populations.
Note - There must be more than a statement at the beginning of the chapter or lesson that is generic or states
that the same strategy could be used with every lesson.
Cluster Meeting
Discuss and answer the following questions to support consensus scoring conversations:
● How and where do materials provide appropriate differentiated strategies and supports for students in
special populations?
● Do materials provide differentiation supports to sufficiently engage students in grade-level/course-level
mathematics?
● Do the materials include overarching guidance on strategies and accommodations for special populations?
Are these evident in lessons?

Gateway 3: Usability
Criterion 3.3 The program includes materials designed for each student’s regular and
active participation in grade-level/grade-band/series content.

Indicator 3n Materials provide extensions and/or opportunities for students to engage with
grade-level/course-level mathematics at higher levels of complexity.
Scoring:
2 points
● Materials provide multiple
opportunities for advanced
students to investigate the
grade-level mathematics at a
higher level of complexity.
AND
● No instances of advanced
students doing more problems
than their classmates.
1 point
● Materials provide some
opportunities for advanced
students to investigate the
grade-level mathematics at a
higher level of complexity.
AND
● There are few instances of
advanced students doing more
problems than their classmates.
0 points
● Materials provide few, if any,
opportunities for advanced
students to investigate the
grade-level mathematics at a
higher level of complexity.
OR
● There are many instances of
advanced students doing more
problems than their classmates.
About this indicator:
What is the purpose of this Indicator?
This indicator examines the materials to determine whether the materials provide opportunities for advanced
students to engage in grade-level mathematics at a higher level of complexity.
Indicator 3n Guiding Question:
What opportunities are present for students to engage in learning with grade-level mathematics at higher
levels of complexity?
Are the opportunities that are present purposeful investigations or extensions?
Do the opportunities extend learning of the grade-level content or topic?
Evidence Collection
Review the student materials across the series.
Look for and record evidence to:
● Describe how and where advanced students have opportunities to work at a higher level of complexity
with a standard. Note - this is not students completing additional tasks or more work, but is an extension
of their learning.

● Identify strategies and supports for advanced students exploration of grade-level content at a higher level
of complexity.
● Describe teacher guidance around using grouping strategies with advanced students.
Cluster Meeting
Discuss and answer the following questions to support consensus scoring conversations:
● Where and how do the materials present opportunities specific to extending students' learning of the
grade-level content?
● Where and how do the materials present opportunities to students to engage in grade-level content at a
higher level of complexity?
● What opportunities do students have to develop and apply higher-level thinking?
● What strategies and supports are available for students to engage in grade-level content at a higher level
of complexity?

Gateway 3: Usability
Criterion 3.3 The program includes materials designed for each student’s regular and active
participation in grade-level/grade-band/series content.