### Alignment

The materials reviewed for Grade 4 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 4 materials include all three aspects of rigor and there is a definitive balance between conceptual understanding, fluency and application. 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.

### Focus & Coherence

MEETS EXPECTATIONS

The materials reviewed for Grade 4 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 4 materials are focused and follow a coherent plan.

#### Focus

- 0
- 6 6

The instructional materials reviewed for Grade 4 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 4 standards. For example, students are assessed on understanding and fluency with multidigit multiplication, comparison of decimal fractions and using the four operations to solve problems. The instructional materials spend the majority of the time on the major clusters of the grade. This includes all clusters in 4.NBT and 4.NF and cluster A in 4.OA. Overall, the materials meet the expectations for focus.

**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 4 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 4 standards. For example, students are assessed on understanding and fluency with multidigit multiplication, comparison of decimal fractions and using the four operations to solve problems. 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 4 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 4 standards or below.
- The summative assessments focus on grade-level topics.
- Students are expected to become fluent in addition and subtraction with larger numbers within the first module.
- Students are allowed to choose a method for multiplication and encouraged to use a method based on 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 4 meet the expectations for focus by spending the majority of the time on the major clusters of the grade. This includes all clusters in 4.NBT and 4.NF and cluster A in 4.OA.

##### Indicator 1b

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

The instructional materials reviewed for Grade 4 meet the expectations for focus by spending the majority of the time on the major clusters of the grade. This includes all clusters in 4.NBT and 4.NF and cluster A in 4.OA.

- While some lessons include multiple standards, a large majority are explicitly focused on major work.
- Of seven modules, module 1 addresses major work exclusively. Modules 2, 3, 5, 6 and 7 devote a few lessons to additional and supporting work.
- Module 4 focuses on additional and supporting work.
- Of the 27 assessment days, 18 are devoted to major work.

#### Coherence

- 0
- 5
- 8 8

The instructional materials reviewed for Grade 4 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, students are solving multiplicative comparisons and conversions using their understandings of place value and operations. The materials include a full program of study that is viable content for a school year, including 180 days of lessons and assessments. This set of materials is consistent with the mathematical progression of learning set forth in the standards. All students are given extensive work on grade-level problems, even those who struggle, and this work progresses mathematically. These instructional materials are visibly shaped by the cluster headings in the standards. Module 1 includes a topic called "Place Value of Multidigit Whole Numbers," which is similar to the cluster heading "Generalize Place Value Understanding for Multidigit Whole Numbers." Connections are made between domains and clusters within the grade level. For example, fraction work is connected to data and measurement lessons. Overall, the Grade 4 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 4 meet the expectations for supporting content as a way to enhance coherence. For Grade 4, reviewers focused on the use of data, factors and multiples, measurement, and conversion of measurements as methods for supporting operations with whole numbers and fractions.

- In module 2, students are solving measurement word problems using the four operations.
- In module 2, students are solving problems using metric conversion giving them practice with multidigit arithmetic.
- In module 2, conversion strategies are suggested including strategies relying specifically on place value.
- In module 3, factors and multiples are used for multidigit arithmetic.
- In module 5, students create line plots using fractional measurements.
- In module 5, students solve measurement word problems involving fractions.
- In module 7, students are solving multiplicative comparisons and conversions using their understandings of place value and operations.

##### 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 4 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 indicated concerns for the volume of lessons.
- Some lessons may take longer than indicated.
- Days are included at the end of the year for culmination activities and preparation 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 4 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 3 or from previous Grade 4 work are included for each module.
- In later modules, standards from earlier in the school year are listed as foundational standards. For example, 4.OA.3 and 4.NBT.4 are taught in module 1 and then listed as foundational standards in module 2.
- Module 1 focuses on multidigit addition and subtraction. Module 2 includes problem solving with these strategies. Module 3 progresses to multiplication and division.
- Within the fraction progression, students first work with unit fractions, then fraction equivalence and ordering, then addition and subtraction and finally multiplication of fractions.
- 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 the English language learner (ELL) continue to reflect the high level of 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 4 foster coherence through connections at the grade level.

- Module 1 includes a topic called "Place Value of Multidigit Whole Numbers," which is similar to the cluster heading "Generalize Place Value Understanding for Multidigit Whole Numbers."
- Module 2 is called "Unit Conversions and Problem Solving with Metric Measurement," which is similar to the cluster heading "Solve Problems Involving Measurement and Conversion."
- Module 5 has a topic called "Decomposition and Fraction Equivalence," which is similar to the cluster heading "Extend Understanding of Fraction Equivalence and Ordering."
- Module 5 connects 4.NF.A to 4.NF.B and connects both to 4.MD.A and 4.MD.B.
- Module 3 connects 4.OA.A to 4.MD.A.
- Module 3 connects 4.NBT.A and 4.NBT.B to 4.OA.A.
- Module 4 connects the measurement and geometry domains.

### Rigor And Mathematical Practices

MEETS EXPECTATIONS

The materials reviewed for Grade 4 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 MPs 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 4 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 4 fluency and procedural work includes 4.NBT.B.4 which asks students to add and subtract within 1,000,000. Application problems occur in almost every lesson depending upon the focus mathematics of the lesson. This is expected to last around 3-10 minutes for each lesson.

**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 4 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 fractions, place value and operations.
- Although some sample scripts offered to the teachers show procedural methods, the methodology, instructions and guiding questions are conceptual.
- Students spend time in module 1 working on rounding using their understanding of place value.
- Module 5 spends a significant amount of time developing an understanding of fraction addition and subtraction using equivalence.
- In module 5 students are guided to use their understanding of multiplication to generate equivalent fractions.

##### 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 4 for this indicator meet the expectations by attending to fluency and procedural work within the lessons. In Grade 4 this includes 4.NBT.B.4, which focuses on adding and subtracting within 1,000,000.

- Within the distribution of instructional minutes the schedule allows for 10-15 minutes per day to practice fluency. This varies according to the timeline of the school year and the focus mathematics in the module.
- Module 1 spends a significant amount of time on fluency for addition and subtraction with whole numbers.
- As described in "How to Implement
*A Story of Units,*" "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 curriculum overview sequence.
- Lessons include mental strategies, problem sets, homework assignments and sprints.

##### 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 4 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 3-10 minutes for each lesson in Grade 4.
- If the focus standard of the lesson includes language requiring application, the application problem will become the major portion of the lesson.
- Contextual word problems are used with a variety of problem types that increase in difficulty throughout the year. These problems focus on a variety of operations.

##### 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 4 for this indicator meet the expectations by providing a balance of rigor. The three aspects are not always treated together nor are they always treated separately.

- The structure of the lessons and the distribution-of-minutes charts 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 4 partially meet this criteria. The Standards for Mathematical Practice are often identified and often used to enrich mathematics content. There are many missed opportunities for identifying MP, however, and in some instances they are misidentified. In module 1, students are directed to use a place-value chart and not given an opportunity to solve an open problem or make meaning of problem and solve. This is incorrectly identified as MP1. 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. There is little explicit reference to modeling (MP4) and lessons identifying this practice incorrectly. There are lessons where the tools are chosen for the students or the modeling expected is a simple representation. The materials reviewed for Grade 4 attend to the standards' emphasis on mathematical reasoning. For example, in module 5, students are asked to share their solution paths with their partners. Then they are asked, "Why is it necessary to decompose the total into ones and a fraction before subtracting? How does that relate to a subtraction problem such as 74- 28?" 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. Overall, the materials partially meet the criteria for practice-content connections.

**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 4 partially meet this criteria. The Standards for Mathematical Practice are often identified and often used to enrich mathematics content. There are many missed opportunities for identifying MP, however, and in some instances they are misidentified. In module 1, students are directed to use a place-value chart and not given an opportunity to solve an open problem or make meaning of problem and solve. This is incorrectly identified as MP1. 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. There is little explicit reference to modeling (MP4) and lessons identifying this practice incorrectly. There are lessons where the tools are chosen for the students or the modeling expected is a simple representation. The materials reviewed for Grade 4 attend to the standards' emphasis on mathematical reasoning. For example, in module 5, students are asked to share their solution paths with their partners. Then they are asked, "Why is it necessary to decompose the total into ones and a fraction before subtracting? How does that relate to a subtraction problem such as 74- 28?" 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. Overall, the materials partially meet the criteria for practice-content connections.

##### 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. 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 an activity tied to MP3 ("Construct Viable Arguments and Critique the Reasoning of Others") as follows: "Knowing and using the relationships between adjacent and vertical angles, students construct an argument for identifying the angle measures of all four angles generated by two intersecting lines when given the measure of one angle. Students explore the concepts of parallelism and perpendicularity on different types of grids with activities that require justifying whether or not completing specific tasks is possible on different grids."
- MPs are listed in the margins of the teacher notes, mostly in the concept development portion and the student debrief of some lessons.
- In module 5, MP7 is correctly marked in a problem in which the understanding of 6x2 as repeated addition that can be displayed on a number line is linked to what 6x1/2 must mean and how it can then equal 3x2/2.
- While reviewers appreciate that MPs are not over identified or used in contrived situations, there are missed opportunities for identifying them 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, in many lessons the tools are chosen for the students or the modeling expected is a simple representation.

- Students are using the 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 module 3, students practice MP4 to solve a real-world problem with modeling.
- Many lessons list MP without attending to the full meaning of the standard. For example, in module 1, students are directed to use a place-value chart and not given an opportunity to solve an open problem or to make meaning of a problem and solve. This is incorrectly identified as MP1. Another example in module 1 is that students are given place-value disks to use in solving the problem and this is listed as MP5, not attending to the full meaning which includes strategically choosing a tool.
- MP4 ("Model with Mathematics") 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 4 meet the requirement 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 3, students are asked to "assess the reasonableness of your answer."
- In module 5, students are asked to share their solution paths with their partners. Then they are asked, "Why is it necessary to decompose the total into ones and a fraction before subtracting? How does that relate to a subtraction problem such as 74-28?"
- In module 5, students are asked on a problem set to "Explain the reasoning you used when determining whether 11/8 or 15/12 is greater.
- In module 5, students are asked to look at the work of their classmates in order to analyze the solution paths of others.

##### 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 4 meet the requirement 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, "Why is a vertical number line a good tool to use for rounding?"
- In a module 5, the teacher is prompted to ask students to explain why they might have different answers and the reasoning they used for each.

##### Indicator 2g.iii

Materials explicitly attend to the specialized language of mathematics.

The materials reviewed for Grade 4 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."
- In module 6 students are asked to attend to precise mathematical language in their work with classifying angles.

### 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. It 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 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 a web page for parents that 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. 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 students, but 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 those 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 home language connections 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 4 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.
- "How to Implement
*A Story of Units*" 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 4 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 place value, comparing, and rounding multidigit whole numbers to addition and subtraction, and then to word problems.

##### 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 4 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 Module 6, for example, students are asked to identify decimal and decimal fractions in a variety of ways (6.A.14); construct line segments of decimal lengths (6.A.23); use an area model to represent decimal numbers (6.A.27); write decimal numbers in expanded form (6.A.42); and solve word problems (6.C.37). Other modules include written responses requiring reasoning, as in (4.B.13).

##### 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 4 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 place-value mats, number lines, folded paper, pattern blocks and fraction strips. They are connected with written methods.

##### 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 4 meet the expectations of this indicator by using a design that supports students in engaging thoughtfully with the subject.

- The visual design is clean and simple and supports students in engaging with the mathematics.
- There are no distractions on the student pages or teacher pages.
- Student pages contain only math problems and pictures/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 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 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 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" that 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 the 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 4 meet the expectations for this indicator by supporting teachers in using questions to guide mathematical development.

- Materials provide quality-suggested questions throughout the debrief section of each lesson. For example, in module 3 students are asked, "What happens to the product when one factor is doubled? Halved?"
- 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 4 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 the 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 that 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 4 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.
- "How to Implement
*A Story of Units*" includes 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 4 partially meet the expectations for this indicator. There are a few specific descriptions of the coherence of the mathematics, but 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.
- "
*A Story of Units:*A Curriculum Overview for Grades P-5" contains 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 4 do 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." It 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 4 include information and suggestions for parents.

- Eureka has a web page for parents that 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" that 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 4 contain explanation of the instructional approaches of the program.

- The Eureka web page has a section called "Reports." It details key research reports on math instruction and learning.
- There is annotation about the curriculum as it relates to these reports.
- Both "How to Implement
*A Story of Units*" and "*A Story of Units:*A Curriculum Overview for Grades P-5" contain information about instructional approaches and research connections. - The opening letter from Executive Director Lynne Munson 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 4 partially meet the expectations for this criterion. While there are no systematic ways to gather information about the prior knowledge of students, 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.

##### Indicator 3m

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

The materials reviewed for Grade 4 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 pretests.

##### Indicator 3n

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

The materials reviewed for Grade 4 meet the expectations for this indicator by including strategies to identify and address common student errors and misconceptions.

- Each summative assessment includes a chart of "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 4 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 also within the problem sets/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 4 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 4 partially meet the expectations for this indicator. The summative assessments meet the expectations, but the formative assessments do not.

- For the 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.

On the whole, materials reviewed for this indicator do not include self-monitoring for students. The one exception is 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 4 meet the criterion for differentiated instruction. The marginal notes often suggest ways to support students as a whole and subgroups of students who might need extra support or those 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 use home-language connections 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 4 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 4 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 option and materials.
- "How to Implement
*A Story of Units*" describes a variety of scaffolds and accommodation (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 4 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. These problems can be solved in a variety of ways, and students can choose their own solution strategy and/or representation.
- The embedded tasks show the students multiple representations using drawings, charts, graphs, or numbers or words to solve.

##### 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 4 meet the expectations for this indicator by including support for the English language learner 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/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 4 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*," provides teachers with suggestions for working with above-grade level students (page20).

##### Indicator 3w

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

The materials reviewed for Grade 4 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 4 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 4 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.).