### Alignment

The materials reviewed for Grade 2 are aligned to the CCSSM. The materials are focused within assessments and spend the majority of time on the major work of the grade. The materials are also coherent, following the progression of the standards and connecting the mathematics within the grade level. The Grade 2 materials include all three aspects of rigor and there is a definitive balance between conceptual understanding, fluency and application. The MPs are identified and used to enhance the mathematical content, but the materials often do not attend to the full meaning of each MP and some are misidentified. Overall the materials are aligned to the CCSSM.

### Focus & Coherence

MEETS EXPECTATIONS

The materials reviewed for Grade 2 meet the expectations for Gateway 1. These materials do not assess above-grade-level content, and they spend the majority of the time on the major clusters of each grade level. Teachers using these materials as designed will use supporting clusters to enhance the major work of the grade. These materials are consistent with the mathematical progression in the standards and students are offered extensive work with grade-level problems. Connections are made between clusters and domains where appropriate. Overall, the Grade 2 materials are focused and follow a coherent plan.

#### Focus

- 0
- 6 6

The instructional materials reviewed for Grade 2 meet the expectations for focus by not assessing any topics before the grade level in which the topic is introduced in the standards. No above-grade-level content was assessed on mid-module or end-of-module assessments in any module. All assessments, rubrics and topics relate to Grade 2 standards. For instance, students are assessed on their understanding of place value, their fluency with addition and subtraction to 20 and their ability to compare two three-digit numbers. The instructional materials reviewed spend the majority of the time on the major clusters of the grade. This includes the first two clusters of operations and algebraic thinking, all of number and operations in base ten, and the first two clusters of measurement and data.

**Criterion 1a**

- 0
- 2 2

Materials do not **assess** topics **before **the grade level in which the topic should be introduced.

The instructional materials reviewed for Grade 2 meet the expectations for this criterion by not assessing any topics before the grade level in which the topic is introduced in the standards. No above-grade-level content was assessed on mid-module or end-of-module assessments in any module. All assessments, rubrics and topics relate to Grade 2 standards. For example, students are assessed on their understanding of place value, their fluency with addition and subtraction to 20, and their ability to compare two three-digit numbers. Overall, the instructional material meets the expectations for focus within assessment.

##### Indicator 1a

The instructional material ** assesses** the grade-level content

**, content from earlier grades. Content from future grades may be introduced but students should not be held accountable on assessments for future expectations.**

__and__,__if applicable__The instructional materials reviewed for Grade 2 meet the expectations for focus within assessment. Overall, the instructional material does not assess any content from future grades within the summative assessment sections of each module.

- No above-grade-level content was assessed on mid-module or end-of-module assessments.
- All assessments, rubrics and topics relate to Grade 2 standards or below.
- The summative assessments focus on grade-level topics.
- The first five modules assess addition and subtraction and place value.
- Fractions are assessed using partitions of circles and rectangles as indicated by the standards.
- Students are expected to choose and defend a method using their understanding of the magnitude of numbers and their understanding of place value.

**Criterion 1b**

- 0
- 4 4

Students and teachers using the materials as designed devote the large majority of class time in each grade K-8 to the major work of the grade.

The instructional materials reviewed for Grade 2 meets the expectations for focus by spending the majority of the time on the major clusters of the grade. This includes the first two clusters of 2.OA, all of 2.NBT and the first two clusters of 2.MD.

##### Indicator 1b

Instructional material spends the majority of class time on the major cluster of each grade.

The instructional materials reviewed for Grade 2 meets the expectations for focus by spending the majority of the time on the major clusters of the grade. This includes the first two clusters of 2.OA, all of 2.NBT and the first two clusters of 2.MD.

- While some lessons include multiple standards, 100 out of 152 lesson days are devoted to major work.
- More than 70% of the lessons are explicitly focused on major work, with major work often included within supporting work lessons as well.
- Of eight modules, modules 1, 2, 4 and 5 address major work exclusively. Modules 3 and 7 devote a few lessons to additional and supporting work.
- Modules 6 and 8 spend the majority of the time on additional and supporting work with a few major work lessons included.
- Of the 28 assessment days, 20 are devoted to major work.

#### Coherence

- 0
- 5
- 8 8

The instructional materials reviewed for Grade 2 meet the expectations for coherence. The materials use supporting content as a way to continue work with the major work of the grade. For example, money is often used to support work in addition, subtraction and place value. The materials include a full program of study that is viable content for a school year including 180 lesson and assessment days. This set of materials is consistent with the mathematical progression of learning set forth in the standards. All students, including struggling students, are given extensive work on grade-level problems and this work progresses mathematically. These instructional materials are visibly shaped by the cluster headings in the standards including using the word "understanding" within the objectives of the place-value lesson and the word "reasoning" with regard to shapes. Connections are made between domains and clusters within the grade level. For example, materials make connections between fractional parts of a circle and telling time. Overall, the Grade 2 materials support coherence and are consistent with the progressions in the standards.

**Criterion 1c-1f**

- 0
- 5
- 8 8

Coherence: Each grade's instructional materials are coherent and consistent with the Standards.

##### Indicator 1c

Supporting content enhances focus and coherence simultaneously by engaging students in the major work of the grade.

The instructional materials reviewed for Grade 2 meet the expectations for use of supporting content as a way to enhance coherence. For Grade 2, reviewers focused on the use of data and money as methods for supporting addition and subtraction and place value.

- In module 3, students are solving word problems with money to work on their place-value understanding and their addition and subtraction strategies.
- In module 6, while setting the foundations for multiplication and division students are working on fluency with their addition within 20 and adding within 100.
- In module 6, students are skip counting by 5s, 10s and 100s as they create equal groups for multiplication.
- In module 6, students are using addition to find the sum of an array in preparation for multiplication.
- In module 7, students are fluently adding and subtracting within 100 and counting within 1,000 using data and money.

##### Indicator 1d

The amount of content designated for one grade level is viable for one school year in order to foster coherence between grades.

The instructional materials reviewed for Grade 2 meet the expectations for this indicator by providing a viable level of content for one school year.

- Materials provide for 180 days of instruction and assessment.
- Lessons are expected to be 60 minutes.
- Lessons generally include fluency practice, application problems, concept development and a student debrief.
- The materials are structured so that a teacher could make modifications if necessary.
- While a district, school or teacher would not need to make significant changes to the schedule set forth, reviewers expressed concerns about the volume of the lessons.
- Some lessons may take longer than indicated.
- Days are included at the end of the year for culmination activities and preparing for summer practice.

##### Indicator 1e

Materials are consistent with the progressions in the Standards i. Materials develop according to the grade-by-grade progressions in the Standards. If there is content from prior or future grades, that content is clearly identified and related to grade-level work ii. Materials give all students extensive work with grade-level problems iii. Materials relate grade level concepts explicitly to prior knowledge from earlier grades.

The instructional materials reviewed for Grade 2 are consistent with the mathematical progressions in the standards meeting the expectation for this indicator.

- The problem types included in the application problems show an increasing level of difficulty.
- Foundational standards from Grade 1 are included for each module.
- In later modules, standards from earlier in the school year are listed as foundational standards. For example, 2.NBT.6 is covered in module 1 and then listed as a foundational standard in module 6.
- Problem sets in each module offer students extensive work on grade-level problems.
- Within the differentiation sections teachers are given suggestions for supporting struggling students while continuing to expect that students work on grade-level problems.
- Suggestions for supporting English language learners (ELLs) continued to reflect the high expectations for these students.
- Teacher notes include suggestions for advanced students to continue working within their grade level while deepening their understanding of the content.

##### Indicator 1f

Materials foster coherence through connections at a single grade, where appropriate and required by the Standards i. Materials include learning objectives that are visibly shaped by CCSSM cluster headings. ii. 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.

The instructional materials reviewed for Grade 2 foster coherence through grade-level connections.

- Module 1 includes a topic called "Mental Strategies for Addition and Subtraction within 20," which is similar to the cluster heading "Add and subtract within 20."
- Module 2 includes a topic called "Relate Addition and Subtraction to Length," which is the exact title of a cluster heading.
- Module 3 has a topic that begins with "Understanding Place Value," the same language as a cluster heading.
- Module 6 is called "Foundations of Multiplication and Division," similar to the cluster heading "Work with Equal Groups of Objects to Gain Foundations for Multiplication."
- In Module 8, lesson objectives include "reasoning about shapes" as informed by the cluster heading "Reason with Shapes and Their Attributes."
- Modules 1 and 4 connect 2.OA and 2.NBT.
- Module 2 uses time and money for place value and work on addition and subtraction.
- Module 7 connects word problems with money to place-value understanding.
- Geometry connections are made between telling time to the nearest 5 minutes and fractional parts of a circle.

### Rigor And Mathematical Practices

MEETS EXPECTATIONS

The materials reviewed for Grade 2 meet the expectations for Gateway 2. The materials include each aspect of rigor: conceptual understanding, fluency and application. These three aspects are balanced within the lessons. The materials partially meet the expectations for the connections between the MP and the mathematical content. There are missed opportunities for identifying MP, and some instances where they are misidentified. The materials do attend to the mathematical reasoning that is embedded in the standards.

#### Rigor and Balance

- 0
- 5
- 8 8

The materials reviewed for Grade 2 meet the expectation for this criterion by providing a balance of all three aspects of rigor throughout the lessons. Within the concept development sections of each lesson the mathematical topic is developed through understanding as indicated by the standards and cluster headings. In Grade 2, fluency work includes single digit sums and differences (2.OA.B.2) and addition and subtraction (2.NBT.B.5). Application problems occur in almost every lesson depending upon the focus mathematics of the lesson. This is expected to last around 5-10 minutes for each lesson. The three aspects are balanced within the lessons and modules. Overall, the Grade 2 materials meet the criteria for rigor and balance.

**Criterion 2a-2d**

- 0
- 5
- 8 8

Rigor and Balance: Each grade's instructional materials reflect the balances in the Standards and help students meet the Standards' rigorous expectations, by helping students develop conceptual understanding, procedural skill and fluency, and application.

##### Indicator 2a

Attention to conceptual understanding: Materials develop conceptual understanding of key mathematical concepts, especially where called for in specific content standards or cluster headings.

The materials reviewed in Grade 2 for this indicator meet the expectations by attending to conceptual understanding within the lessons.

- Within the concept-development sections of each lesson the mathematical topic is developed through understanding as indicated by the standards and cluster headings.
- Significant time is spent developing understanding of addition and subtraction and place value.
- In Module 2 the teacher edition provides the following guidance: "Students should be encouraged to make connections between different solution strategies and to choose what works best for a given problem or for their way of thinking."
- In Module 4, students are instructed, "Show another model to solve the problem."
- Significant time is spent writing and comparing numbers through 1,000 in the development of place-value understanding.
- Understanding of addition and subtraction and place value is built through contextual problems.

##### Indicator 2b

Attention to Procedural Skill and Fluency: Materials give attention throughout the year to individual standards that set an expectation of procedural skill and fluency.

The materials reviewed in Grade 2 for this indicator meet the expectations by attending to fluency and procedural work within the lessons. In Grade 2 this includes work on single digit sums and differences, including sums from memory, (2.OA.B.2) and to add and subtract within 100 (2.NBT.B.5).

- Within the distribution of instructional minutes the schedule allows 10-20 minutes per day to practice fluency. This varies according to the timeline of the school year and the focus mathematics in the module.
- As described within the "how to implement" document: "Fluency is usually first-by beginning class with animated, adrenaline-rich fluency, students are more alert when presented with the Concept Development and Application Problems."
- Attention is paid to the use of the words "fluency" and "fluent" within the standards.
- Required fluencies are listed within the section called "Curriculum Overview Sequence."
- Lessons include mental strategies, sprints, dashes and flashcard activities.

##### Indicator 2c

Attention to Applications: Materials are designed so that teachers and students spend sufficient time working with engaging applications of the mathematics, without losing focus on the major work of each grade

The materials reviewed in Grade 2 for this indicator meet the expectations by attending to application within the lessons.

- Application problems occur in almost every lesson depending upon the focus mathematics of the lesson. This is expected to last around 5-10 minutes for each lesson in Grade 2.
- If the focus standard of the lesson includes language requiring application, the application problem will become the major portion of the lesson.
- Contextual two-step word problems are used with a variety of problem types that increase in difficulty throughout the year. These problems focus on addition and subtraction.

##### Indicator 2d

Balance: The three aspects of rigor are not always treated together and are not always treated separately. There is a balance of the 3 aspects of rigor within the grade.

The materials reviewed in Grade 2 for this indicator meet the expectations by providing a balance of rigor. The three aspects are not always combined together nor are they always separate.

- The distribution-of-minutes charts and the structure of the lessons show a balance of the three aspects of rigor.
- Application problems often call for fluency and procedural skills.
- Fluency work and application problems are used to develop conceptual understanding.
- Conceptual problems often involve procedures.

#### Mathematical Practice-Content Connections

- 0
- 6
- 8
- 10

The materials reviewed for Grade 2 partially meet this criteria. MPs are often identified and often used to enrich mathematics content. There are many missed opportunities for identifying MPs and some instances where they are misidentified. The materials often attend to the full meaning of each practice; however, there are instances where the students are not using the practice as written. In describing MP4 activities, the overview of module 4 says, "Students model with mathematics when they write equations to solve two-step word problems, make math drawings when solving a vertical algorithm, or when they draw place value charts and disks to represent numbers." This description does not match the full intent of MP4. For example, in Module 4, lesson 6 (pages 4.B.5-6) students are using place-value charts showing addition with regrouping, but there is no real-world context. There are lessons in which the tools are chosen for the students or the modeling expected is a simple representation. The materials reviewed for Grade 2 attend to the standards' emphasis on mathematical reasoning. Students are prompted within problem sets and application problems to explain, describe, critique and justify. Each lesson includes a debrief section with questions for the teacher to use in facilitating classroom discussion about the mathematical content. In module 3, teachers are prompted use the following in a classroom discussion: "Turn and tell your partner how counting on the place value chart is different from that on the number line." Overall, the materials partially meet the criteria for connecting practice to content.

**Criterion 2e-2g**

- 0
- 6
- 8
- 10

Practice-Content Connections: Materials meaningfully connect the Standards for Mathematical Content and the Standards for Mathematical Practice

The materials reviewed for Grade 2 partially meet this criteria. MPs are often identified and often used to enrich mathematics content. There are many missed opportunities for identifying MPs and some instances where they are misidentified. The materials often attend to the full meaning of each practice; however, there are instances where the students are not using the practice as written. In describing MP4 activities, the overview of module 4 says, "Students model with mathematics when they write equations to solve two-step word problems, make math drawings when solving a vertical algorithm, or when they draw place value charts and disks to represent numbers." This description does not match the full intent of MP4. For example, in Module 4, lesson 6 (pages 4.B.5-6) students are using place-value charts showing addition with regrouping, but there is no real-world context. There are lessons in which the tools are chosen for the students or the modeling expected is a simple representation. The materials reviewed for Grade 2 attend to the standards' emphasis on mathematical reasoning. Students are prompted within problem sets and application problems to explain, describe, critique and justify. Each lesson includes a debrief section with questions for the teacher to use in facilitating classroom discussion about the mathematical content. In module 3, teachers are prompted use the following in a classroom discussion: "Turn and tell your partner how counting on the place value chart is different from that on the number line." Overall, the materials partially meet the criteria for connecting practice to content.

##### Indicator 2e

The Standards for Mathematical Practice are identified and used to enrich mathematics content within and throughout each applicable grade.

The Standards for Mathematical Practice (MPs) are often identified and often used to enrich mathematics content. This instructional material, however, only partially meets expectations. There are many missed opportunities for identifying MPs and some instances where they are misidentified.

- MPs are listed at the beginning of each module with a description of the explicit connection to the mathematics of the module.
- Module 4 describes MP1: "Students solve two-step word problems, and are challenged to make sense of more complex relationships within situations. They flexibly solve problems with a variety of strategies at their disposal, sometimes finding many ways to solve the same problem."
- MPs are listed in the margins of the teacher notes, mostly in the "Concept Development" and the "Student Debrief" portions of some lessons.
- While reviewers appreciate that the MPsare not over identified or used in contrived situations, there are missed opportunities for identifying MP in order to enrich the content in these lessons.
- The debrief section of the lessons offers an opportunity to highlight, for both teachers and students, how they might reason abstractly and quantitatively (MP2) and construct arguments and critique the reasoning of others (MP3).
- There is little explicit reference to modeling (MP4), and some lessons identify this practice incorrectly.

##### Indicator 2f

Materials carefully attend to the full meaning of each practice standard

The materials often attend to the full meaning of each practice; however, there are instances where the students are not using the practice as written. For example, there are many lessons in which the tools are chosen for the students or the modeling expected is a simple representation. As a result, this instructional material only partially meets expectations.

- Students are using MPs when engaging with the content as designed, fully meeting Publisher's Criteria #9.
- Throughout the lessons the debrief section includes opportunities to construct viable arguments and critique the reasoning of others (MP3).
- In describing MP4 ("Model with Mathematics"), the overview of module 4 says, "Students model with mathematics when they write equations to solve two-step word problems, make math drawings when solving a vertical algorithm, or when they draw place value charts and disks to represent numbers." This description does not match the intent of MP 4. Example: Module 4, lesson 6 (pages 4.B.5-6) students are using place-value charts showing addition with regrouping, but there is no real-world context.
- Many lessons list MPs without attending to their full meaning.
- In module 4, students are offered problems to solve and specifically told how to solve each problem, not taking advantage of learning with MP1.
- In module 4, students are answering questions the teacher asks, when they could be constructing arguments and critiquing arguments of others (MP3).
- MP4 is irregularly applied. There is ambiguity over whether "model" means to draw a picture representing the problem or whether it means to create a mathematical representation in a real-world context.

##### Indicator 2g

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

##### Indicator 2g.i

Materials prompt students to construct viable arguments and analyze the arguments of others concerning key grade-level mathematics detailed in the content standards.

The materials reviewed for Grade 2 meet the expectations of this indicator by attending to the standards' emphasis on mathematical reasoning.

- Students are prompted within problem sets and application problems to explain, describe, critique and justify.
- In module 4, students are asked on an exit ticket, "Explain how problem 1(a) can help you solve problem 1(b)."
- In module 4, students are asked on an exit ticket, "Explain how Kevin's work and your work are similar."

##### Indicator 2g.ii

Materials assist teachers in engaging students in constructing viable arguments and analyzing the arguments of others concerning key grade-level mathematics detailed in the content standards.

The materials reviewed for Grade 2 meet the expectations of this indicator by attending to the standards' emphasis on mathematical reasoning.

- Each lesson includes a debrief section with questions for the teacher to use in facilitating classroom discussion about the mathematical content. For example, "What did we do today to make our counting strategy more efficient?"
- In module 3, teachers are prompted use the following in a classroom discussion: "Turn and tell your partner how counting on the place-value chart is different from that on the number line."
- In module 4, the teacher is prompted to ask, "Explain how and why you changed each number. Be sure to use place value language."
- In module 7, teachers are prompted to ask, "Can you think of other math skills we have learned where the same value can be represented in different ways?"

##### Indicator 2g.iii

Materials explicitly attend to the specialized language of mathematics.

The materials reviewed for Grade 2 meet the requirement of this indicator by attending to the standards' emphasis on mathematical reasoning.

- Each module lists terminology for the module including "new or recently introduced terms" and "familiar terms and symbols."

### Usability

MEETS EXPECTATIONS

The materials reviewed meet the expectations for usability. In reviews for use and design, the problems and exercises are developed sequentially and each activity has a mathematical purpose. Students are asked to produce a variety of assignments. Manipulatives and models are used to enhance learning and the purpose of each is explained well. The visual design is not distracting or chaotic. The visual design supports learning. The materials support teachers in learning and understanding the standards. All materials include support for teachers in using questions to guide mathematical development. Teacher editions have many annotations and examples on how to present the content. There are answer keys for all the student problem sets, exit tickets, homework and tests, including written annotations to show what student work should look like. In the teacher edition for each module, there is an overview section that has narrative information about the math content of the module. In each module, at the start of each topic, there is another section that gives a mathematical explanation of the math content in the topic. There are a few specific descriptions of the coherence of the mathematics, however there is no discussion of the grade-level content's role in Kindergarten through Grade 12. Materials do provide information on connected content standards and pacing.

Eureka has a web page for parents that contains general information about the curriculum as well as a few informational videos. There is also a section on the web page called "Eureka Math Tips for Parents," which gives information organized by grade level and module. There is information about the instructional approaches and research connection in the documents called "How to Implement *A Story of Units*" and "*A Story of Units*: A Curriculum Overview for Grades P-5." Within the assessment criterion the materials only partially met the expectations.

There are no systematic ways to gather information about the prior knowledge of their students, however the teachers are offered support in identifying and addressing common student errors and misconceptions. Materials include opportunities for ongoing review and practice. While the summative assessments include information on standards alignment and scoring rubrics, the formative assessments do not include this same information. There are no systems or suggestions for students to monitor their own progress. In reviews for differentiation the marginal notes often suggest ways to support students as a whole and subgroups of students who might need extra support or students who may be advanced. This includes support for vocabulary, representations, engagement options, and materials. Application problems, problem sets, and homework are included in almost all lessons. These problems can be solved in a variety of ways. Students can choose their own solution strategy and/or representation. Suggestions are included for supporting ELL students and other special populations in order for them to actively participate. Notes within the lessons present the teachers a variety of options for whole group, small group, partner, or individual work. Materials encourage teachers to make connections with home languages and cultural ties to facilitate learning. The materials do not include a technology component for instruction, so this criterion was not rated. Overall, the materials meet the expectations for usability.

**Criterion 3a-3e**

- 0
- 5
- 8 8

Use and design facilitate student learning: Materials are well designed and take into account effective lesson structure and pacing.

The materials meet the criterion for use and design. The problems and exercises are developed sequentially, and each activity has a mathematical purpose. Students are asked to produce a variety of assignments. Manipulatives and models are used to enhance learning and the purpose of each is explained well. The visual design is not distracting or chaotic. The visual design supports learning.

##### Indicator 3a

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.

The design of the materials reviewed for Grade 2 meet the expectations for this indicator by providing students with ongoing opportunities to practice previously learned skills alongside their learning of new content. These materials use problem sets and application problems to develop their understanding of new mathematics. These materials use homework, application problems and fluency sessions to practice previously learned concepts.

- Problems sets within the lessons include guidance on how to select and sequence the exercises.
- Fluency exercises within the lessons include guidance on the purpose of each activity allowing the teacher to determine the necessary activities for the students.
- The "how to implement" document provides information for the teacher on the purpose for each lesson section.
- "The primary goal of the problem set is for students to apply the conceptual understanding(s) learned in the lesson." (page 12)
- "The bank of fluency activities for each lesson is intentionally organized so that activities revisit previously-learned material to develop automaticity, anticipate future concepts, and strategically preview or build skills for the day's Concept Development." (page 23)
- "The homework gives students additional practice on the skills they learn in class each day. The idea is not to introduce brand-new concepts, but to build student confidence with the material learned in class." (page 13)
- "A Story of Units doesn't wait months to spiral back to a concept. Rather, once a concept is learned, it is immediately spiraled back into the daily lesson structure through fluency and applications." (page 9)

##### Indicator 3b

Design of assignments is not haphazard: exercises are given in intentional sequences.

The materials reviewed for Grade 2 meet the expectations of this indicator by using intentional sequences in the design.

- Problem sets, exit tickets and homework relate to the mathematical concept developed in the lessons each day.
- Once a concept is developed, it is spiraled back into the daily structure within the fluency and application portion.
- The sequence of topics within each module is intentional going from working with a variety of concrete and pictorial representations to more abstract work with numbers and computation.
- For example, module 1 goes from foundations for addition and subtraction within 20 to mental strategies for addition and subtraction within 20 and finally to strategies for addition and subtraction within 100.

##### Indicator 3c

There is variety in what students are asked to 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.

The materials reviewed for Grade 2 meet the expectations of this indicator by providing a variety in what students are expected to produce.

- Students are expected to produce answers and solutions throughout the fluency sections and some of the problem sets.
- Students are expected to provide arguments and explanations within the problem sets, exit tickets and homework.
- Students are asked to provide a variety of mathematical responses.
- Arguments and explanations are the basis for the debriefing section of each lesson.
- The "Read, Draw, Write" procedure requires students to represent the problem in a drawing and make connections between the drawing and the equations.
- Throughout the modules and lessons, students produce a variety of solutions, using concrete, pictorial and abstract representations.
- In the first two sections of module 4, for example, students are asked to represent subtraction in a particular method (4.A.36); to represent addition using a tape diagram (4.A.45); to solve addition and subtraction vertically (4.B.21); to provide written explanations for how two problems are related (4.B.22); to complete a "sprint" of basic addition facts (4.B.40); and to show addition using place-value methods (4.B.42).

##### Indicator 3d

Manipulatives are faithful representations of the mathematical objects they represent and when appropriate are connected to written methods.

The materials reviewed for Grade 2 meet the expectations of this indicator by using manipulatives and models as faithful representations of the mathematics they are used to represent.

- The materials use a limited set of concrete and pictorial models throughout the program.
- Each module lists suggested tools and representations that apply to the mathematics in the module.
- Students use a variety of manipulatives including rekenreks, counting objects, five and ten frames, pattern blocks, 2-D and 3-D shapes, and linking cubes. They are connected with written methods.
- In module 4, lesson 9, students draw "number disks" to represent place value, but there is a note on how to use manipulatives for below-level students.

##### Indicator 3e

The visual design (whether in print or online) is not distracting or chaotic, but supports students in engaging thoughtfully with the subject.

The materials reviewed for Grade 2 use a design that supports student engagement with the subject.

- The visual design is clean and simple and supports students in engaging with the mathematics.
- There are no distractions on student pages or teacher pages.
- Student pages contain only math problems and pictures or diagrams as part of the problems.
- The materials have very minimal pictures.

**Criterion 3f-3l**

- 0
- 5
- 8 7

Teacher Planning and Learning for Success with CCSS: Materials support teacher learning and understanding of the Standards.

The materials reviewed for this criteria meet the expectations by including materials that support teachers in learning and understanding CCSSM. All materials include support for teachers in using questions to guide mathematical development. Teacher editions have many annotations and examples on how to present content. There are answer keys for all the student problem sets, exit tickets, homework and tests, including written annotations to show what student work should look like. In the teacher edition for each module, there is an overview section that has narrative information about the mathematics content of the module. In each module, at the start of each topic, there is another section that gives a mathematical explanation of the mathematics content in the topic. There are a few specific descriptions of the coherence of the mathematics, however there is no discussion of the grade-level content's role in Kindergarten through Grade 12. Materials do provide information on connected content standards and pacing. Eureka has an parent web page that contains general information about the curriculum as well as a few informational videos. There is also a section on the web page called, "Eureka Math Tips for Parents," which gives information organized by grade level and module. There is information about the instructional approaches and research connection in the documents called "How to Implement *A Story of Units*" and "*A Story of Units:* A Curriculum Overview for Grades P-5." Overall, the materials reviewed include support for the teacher in planning and learning for success with CCSSM.

##### Indicator 3f

Materials support teachers in planning and providing effective learning experiences by providing quality questions to help guide students' mathematical development.

The materials reviewed for Grade 2 meet the expectations for this indicator by supporting teachers in using questions to guide mathematical development.

- Materials provide quality suggested questions throughout the debriefing section of each lesson. For example, in module 7, students are asked, "Which one was the easiest to find the value for? Why?"
- Quality questions are also included in the concept development portion, application problems, and problem sets of the lessons.

##### Indicator 3g

Materials contain a teacher's edition with ample and useful annotations and suggestions on how to present the content in the student edition and in the ancillary materials. Where applicable, materials include teacher guidance for the use of embedded technology to support and enhance student learning.

The materials reviewed for Grade 2 meet the expectations for this indicator by including a teacher edition with ample and useful annotations and suggestions on how to present the content.

- The concept development sections include a sample script to help the teacher understand what
__might__happen when presenting the material. These scripts can sometimes mask the mathematical concepts at hand, leading teachers to think that this script is exactly what should happen. A summary of the process and concept before the script would be useful. - Within the lessons, aside from the teacher script and wording in teacher directions, most lessons have pictures and representations with annotations, demonstrating the concepts pictorially for the teacher, to provide guidance about how to present the content.
- There are answer keys for all the student problem sets, exit tickets, homework and tests, including written annotations to show what student work should look like.
- There are also boxes in the sidebar of many lessons that annotate information about how to present content to students.
- There is a repeated process for solving word problems called the "Read, Draw, Write" approach, which the manual explains in the module overview.
- The overview of each module has several suggestions for delivering instruction.

##### Indicator 3h

Materials contain a teacher's edition (in print or clearly distinguished/accessible as a teacher's edition in digital materials) that contains full, adult-level explanations and examples of the more advanced mathematics concepts in the lessons so that teachers can improve their own knowledge of the subject, as necessary.

The materials reviewed for Grade 2 meet the expectations for this indicator by including adult-level explanations and examples of mathematical topics so that teachers can improve their own knowledge of the subject, if necessary.

- In the teacher edition for each module, there is an overview section that has narrative information about the math content of the module.
- In each module, at the start of each topic, there is another section of narrative that gives a mathematical explanation of the math content in the topic.
- These topic level explanations and overviews include mathematical coherence within and between grade levels.
- In the document called, "How to Implement
*A Story of Units,*" there are adult-level explanations of the models and representations used.

##### Indicator 3i

Materials contain a teacher's edition (in print or clearly distinguished/accessible as a teacher's edition in digital materials) that explains the role of the specific grade-level mathematics in the context of the overall mathematics curriculum for kindergarten through grade twelve.

The materials reviewed for Grade 2 partially meet the expectations for this indicator. There are a few specific descriptions of the coherence of the mathematics, however there is no discussion of the grade-level content's role in Kindergarten through Grade 12.

- There are explanations of the role previous content plays in each module. This is listed in the module overview in the Foundational Standards.
- In "
*A Story of Units:*A Curriculum Overview for Grades P-5," there is a description of the module sequence that includes the connection to the previous grade and the next future grade. No connection is made to other grade levels.

##### Indicator 3j

Materials provide a list of lessons in the teacher's edition (in print or clearly distinguished/accessible as a teacher's edition in digital materials), cross-referencing the standards covered and providing an estimated instructional time for each lesson, chapter and unit (i.e., pacing guide).

The materials reviewed for Grade 2 provide information on connected-content standards and pacing.

- Within each module overview there is a section called "Overview of the Module Topics and Lesson Objectives," which contains lessons broken down by topic, and cross-references the standards at the topic level.
- This overview also lists the number of days for each topic, as well as the total number of instructional days for the entire module, including assessments.
- Lessons include a time frame for each activity in the lesson.
- There is a yearly summary of standards and pacing.

##### Indicator 3k

Materials contain strategies for informing parents or caregivers about the mathematics program and suggestions for how they can help support student progress and achievement.

The materials reviewed for Grade 2 include information and suggestions for parents.

- Eureka has a web page for parents, which contains general information about the curriculum as well as a few informational videos.
- The web page also has a section called, "Eureka Math Tips for Parents," which gives information organized by grade level and module.

##### Indicator 3l

Materials contain explanations of the instructional approaches of the program and identification of the research-based strategies.

The materials reviewed for Grade 2 contain explanation of the instructional approaches of the program.

- The Eureka web page has a section called "Reports." It details key research reports on mathematics instruction and learning.
- There is annotation about the curriculum as it relates to these reports.
- There is information about the instructional approaches and research connections in the documents titled "How to Implement
*A Story of Units*" and "*A Story of Units:*A Curriculum Overview for Grades P-5." - The materials contain an opening letter that addresses some of the research and philosophy behind the curriculum.

**Criterion 3m-3q**

- 0
- 6 6
- 10

Assessment: Materials offer teachers resources and tools to collect ongoing data about student progress on the Standards.

The materials reviewed for Grade 2 partially meet the expectations for this criterion. While there are no systematic ways to gather information about the prior knowledge of their students, teachers are offered support in identifying and addressing common student errors and misconceptions. Materials includes opportunities for ongoing review and practice. While the summative assessments include information on standards alignment and scoring rubrics, the formative assessments do not include this same information. There are no systems or suggestions for students to monitor their own progress.

##### Indicator 3m

Materials provide strategies for gathering information about students' prior knowledge within and across grade levels.

The materials reviewed for Grade 2 do not meet the expectations for this indicator.

- Foundational standards are listed for most modules, but there are no directions for using these standards to assess prior knowledge.
- There are not systematic ways to gather information about prior knowledge.
- There are no diagnostics included other than within the rubrics for the summative assessments.
- There are no module pre-tests.

##### Indicator 3n

Materials provide strategies for teachers to identify and address common student errors and misconceptions.

The materials reviewed for Grade 2 meet the expectations for this indicator by including strategies to identify and address common student errors and misconceptions.

- Each summative assessment includes a chart titled "Progression Toward Mastery" to help teachers with the coherence towards mastery.
- On page 13, "How to Implement
*A Story of Units*" says this about addressing errors and misconception: "Distractors for such questions are written to illuminate common student errors and misconceptions." - The student debrief section of the lesson is intended to invite the students to reflect and process the lesson. Strategies include partnering to guide students in conversation to debrief the Problem Set and process the lesson.
- The marginal notes often suggest ways to support students as a whole and subgroups of students who might need support. In particular, the "Multiple Means of..." notes tend to focus on student misconceptions.

##### Indicator 3o

Materials provide opportunities for ongoing review and practice, with feedback, for students in learning both concepts and skills.

The materials reviewed for Grade 2 meet the expectations for this indicator by including ongoing review and practice.

- Ongoing review and practice is included within fluency section of lessons.
- Exit tickets can provide feedback depending upon teacher use.
- Review and practice is also found within the problem sets and homework that are included in every lesson.
- The summative assessments contain rubrics to provide feedback to the teacher and student as to a student's progression toward mastery.

##### Indicator 3p

Materials offer ongoing formative and summative assessments:

##### Indicator 3p.i

Assessments clearly denote which standards are being emphasized.

The materials reviewed for Grade 2 partially meet the expectations for this indicator. The summative assessments meet the expectations, but the formative assessments do not.

- Mid-module and end-of-module assessments align each item to specific standard(s).
- There are standards listed for each lesson; sometimes multiple standards are listed.
- There are no specific standards listed within the lesson exit tickets. These exit tickets could possibly include multiple standards.

##### Indicator 3p.ii

Assessments include aligned rubrics and scoring guidelines that provide sufficient guidance to teachers for interpreting student performance and suggestions for follow-up.

The materials reviewed for Grade 2 partially meet the expectations for this indicator. The summative assessments meet the expectations, but the formative assessments do not.

- For mid-module and end-of-module assessments, there are rubrics for scoring the items, as well as an answer key with sample answers.
- Rubrics and scoring guides are clear and helpful. Examples of student work receiving top grades on the rubric are included.
- In the "Progression Toward Mastery" section of the summative assessments there is a detailed rubric for grading student mastery from 1 to 4. If the student does not achieve total mastery (step 4), then the teacher can look at the next steps to see what or how to follow up with the student. For example, when a student's mastery is step 2, teachers can look at steps 3 and 4 to guide follow-up instruction.

##### Indicator 3q

Materials encourage students to monitor their own progress.

Materials reviewed for this indicator do not include self-monitoring for students. There is one exception within the fluency sprints. Students complete the sprint twice with a goal of increasing their score on the second round.

**Criterion 3r-3y**

- 0
- 8
- 10
- 12 12

Differentiated instruction: Materials support teachers in differentiating instruction for diverse learners within and across grades.

The materials reviewed for Grade 2 meet the criterion for differentiated instruction. The marginal notes often suggest ways to support students as a whole as well as subgroups of students who might need extra support or students who may be advanced. This includes support for vocabulary, representations, engagement options, and materials. Application problems, problem sets, and homework are included in almost all lessons. These problems can be solved in a variety of ways. Students can choose their own solution strategy and/or representation. Suggestions are included for supporting ELL students and other special populations in order for them to actively participate. Notes within the lessons present the teachers a variety of options for whole group, small group, partner, or individual work. Materials encourage teachers to make connections with home languages and cultural ties to facilitate learning.

##### Indicator 3r

Materials provide strategies to help teachers sequence or scaffold lessons so that the content is accessible to all learners.

The materials reviewed for Grade 2 meet the expectations for this indicator by including strategies to help teachers sequence and scaffold lessons.

- The lessons are sequenced to build from conceptual understanding, using concrete and pictorial representations to more abstract representations.
- The marginal notes often suggest ways to support students as a whole and subgroups of students who might need extra support. This includes support for vocabulary, representations, engagement options and materials.
- Lessons and mathematical topics are sequenced according to the CCSSM progressions of learning.
- A description of the module sequence and layout is provided.

##### Indicator 3s

Materials provide teachers with strategies for meeting the needs of a range of learners.

The materials reviewed for Grade 2 meet the expectations for this indicator by including strategies for meeting the needs of a range of learners.

- The lessons are sequenced to build from conceptual understanding, using concrete and pictorial representations to more abstract representations.
- The marginal notes often suggest ways to support students as a whole and subgroups of students who might need extra support. This includes support for vocabulary, representations, engagement options and materials.
- "How to Implement
*A Story of Units*" describes a variety of scaffolds and accommodations (page 13).

##### Indicator 3t

Materials embed tasks with multiple entry-points that can be solved using a variety of solution strategies or representations.

The materials reviewed for Grade 2 meet the expectations for this indicator by providing tasks with multiple entry points that can be solved in a variety of ways.

- Application problems, problem sets, and homework are included in almost all lessons. Students can choose their own solution strategy and/or representation from a variety of options.
- The embedded tasks show the students multiple representations using drawings, charts, graphs, or numbers or words.

##### Indicator 3u

Materials suggest support, accommodations, and modifications for English Language Learners and other special populations that will support their regular and active participation in learning mathematics (e.g., modifying vocabulary words within word problems).

The materials reviewed for Grade 2 meet the expectations for this indicator by including support for ELL students and other special populations in order for them to actively participate.

- Notes on multiple means of engagement give teachers suggestions about meeting the needs of ELL students. These margin notes include sentence starters, physical responses and vocabulary support.
- On pages 14-20 of "How to Implement
*A Story of Units*," there are suggestions for working with ELL students and students with disabilities. Page 14 states, "It is important to note that the scaffolds and accommodations integrated into*A Story of Units*might change how a learner accesses information and demonstrates learning; they do not substantially alter the instructional level, content, or performance criteria. Rather, they provide students with choices in how they access content and demonstrate their knowledge and ability."

##### Indicator 3v

Materials provide opportunities for advanced students to investigate mathematics content at greater depth.

The materials reviewed for Grade 2 meet the expectations for this indicator by offering instructional support for advanced students.

- Notes on multiple means of engagement give teachers suggestions about meeting the needs of advanced students.
- The curriculum specifies that not all pieces of each section of a lesson must be used, so advanced students could be asked to tackle problems or sections that a teacher does not use for all students.
- "How to Implement
*A Story of Units*" gives teachers suggestions for working with above-grade level students. (page 20)

##### Indicator 3w

Materials provide a balanced portrayal of various demographic and personal characteristics.

The materials reviewed for Grade 2 meet the expectations for this indicator by providing a balanced portrayal of various demographic and personal characteristics.

- The names and situations in the story problems represent a variety of cultural groups.
- The application problems include real-world situations that would appeal to a variety of cultural and gender groups.
- There is a balanced approach to the use of gender identification.

##### Indicator 3x

Materials provide opportunities for teachers to use a variety of grouping strategies.

The materials reviewed for Grade 2 include a variety of grouping strategies.

- Notes within the lessons present the teachers a variety of options for whole group, small group, partner, or individual work.
- There are opportunities for different groupings, however the fundamental model is "Modeling with Interactive Questioning; Guided Practice; and Independent Practice."
- There are also suggestions for small group work within the differentiation pages of "How to Implement
*A Story of Units*."

##### Indicator 3y

Materials encourage teachers to draw upon home language and culture to facilitate learning.

The materials reviewed for Grade 2 encourage teachers to make home language connections and cultural ties to facilitate learning.

- There are occasions (mostly with Spanish) where students are encouraged to make connections to words in their home languages.
- How to Implement
*A Story of Units*" offers teachers this guidance: "Know, use, and make the most of student cultural and home experiences. Build on the student's background knowledge."

**Criterion 3z-3ad**

Effective technology use: Materials support effective use of technology to enhance student learning. Digital materials are accessible and available in multiple platforms.

Reviews for this series were conducted using print materials, which do not include an instructional technology component. Materials were not reviewed for this indicator.

##### Indicator 3z

Materials integrate technology such as interactive tools, virtual manipulatives/objects, and/or dynamic mathematics software in ways that engage students in the Mathematical Practices.

Reviews for this series were conducted using print materials, which do not include an instructional technology component. Materials were not reviewed for this indicator.

##### Indicator 3aa

Digital materials (either included as supplementary to a textbook or as part of a digital curriculum) are web-based and compatible with multiple internet browsers (e.g., Internet Explorer, Firefox, Google Chrome, etc.). In addition, materials are "platform neutral" (i.e., are compatible with multiple operating systems such as Windows and Apple and are not proprietary to any single platform) and allow the use of tablets and mobile devices.

Reviews for this series were conducted using print materials, which do not include an instructional technology component. Materials were not reviewed for this indicator.

##### Indicator 3ab

Materials include opportunities to assess student mathematical understandings and knowledge of procedural skills using technology.

##### Indicator 3ac

Materials can be easily customized for individual learners. i. Digital materials include opportunities for teachers to personalize learning for all students, using adaptive or other technological innovations. ii. Materials can be easily customized for local use. For example, materials may provide a range of lessons to draw from on a topic.

##### Indicator 3ad

Materials include or reference technology that provides opportunities for teachers and/or students to collaborate with each other (e.g. websites, discussion groups, webinars, etc.).