4th Grade - Gateway 3

The materials reviewed for Eureka Math² Grade 4 meet expectations for providing teacher guidance with useful annotations and suggestions for how to enact the student materials and ancillary materials, with specific attention to engaging students in order to guide their mathematical development.

Materials provide comprehensive guidance that will assist teachers in presenting the student and ancillary materials. These are found in a variety of sections within the Implementation Guide, including the Overview, Why, Achievement Descriptors Overview, and Lesson Structure. Examples include:

• Grade 3-5 Implementation Guide, Inside Teach, Module-Level Components, Overview, “Your Teach book begins with the Overview, a topic-by-topic summary that shows the development of learning throughout the module. It also provides connections to work done before and after the module, helping you understand the module’s place in the overall development of learning in and across the grade levels.”

• Grade 3-5 Implementation Guide, Inside Teach, Module-Level Components, Why, “The Why section gives insight into the decisions made during the writing of the module. This insight helps you understand the underlying structure of the module, flow of the content, and coherence of the different parts of the curriculum.”

• Grade 3-5 Implementation Guide, Inside Teach, Module-Level Components, Achievement Descriptors, “The Achievement Descriptors: Overview section is a helpful guide that describes what Achievement Descriptors (ADs) are and briefly explains how to use them. It identifies specific ADs for the module, with more guidance provided in the Achievement Descriptors: Proficiency Indicators resource at the end of each Teach book.”

• Grade 3-5 Implementation Guide, Inside Teach, Lesson Structure, “Each lesson is structured in four sections: Fluency, Launch, Learn, and Land. Lessons are designed for one 60-minute instructional period. Fluency provides distributed practice with previously learned material. Launch creates an accessible entry point to the day’s learning through activities that build context and often create productive struggle that leads to a need for the learning that follows. Learn presents new learning related to the lesson objective, usually through a series of instructional segments. This lesson component takes most of the instructional time. Land helps you facilitate a brief discussion to close the lesson and provides students with an opportunity to complete the Exit Ticket.”

Materials include sufficient and useful annotations and suggestions that are presented within the context of the specific lessons. This guidance can be found for teachers within boxes called Differentiation, UDL, and Teacher Notes. The Implementation Guide states, “There are six types of instructional guidance that appear in the margin notes. These notes provide information about facilitation, differentiation, and coherence. Teacher Notes may enhance mathematical understanding, explain pedagogical choices, five background information, or help identify common misconceptions. Universal Design for Learning (UDL) suggestions offer strategies and scaffolds that address learner variance. These suggestions promote flexibility with engagement, representation, and action and expression, the three UDL principles described by CAST. These strategies and scaffolds are additional suggestions to complement the curriculum’s overall alignment with the UDL Guidelines.” Examples include:

• Module 3, Topic B, Lesson 4: Apply place value strategies to divide hundreds, tens, and ones, Learn, Draw on the Place Value Chart to Divide, UDL: Action & Expression, “Consider providing equations with blanks for students to use. This can help reduce the fine-motor demands of writing. Students draw on a place value chart and write an equation to divide hundreds, tens, and ones. Direct students to remove Place Value Chart to Hundreds from their books and insert it in their whiteboards. Write 846\div2= ____.”

• Module 4, Topic A, Lesson 6: Rename mixed numbers as fractions greater than 1, Learn, Choose a Model, UDL: Representation, “Consider activating background knowledge of different models students can use to represent whole numbers and fractions. Create a list students can refer to when they rename mixed numbers as fractions greater than 1. Area model, Tape diagram, Number bond, Number line, Equations.”

• Module 5, Topic A, Lesson 4: Write mixed numbers in decimal form with tenths, Learn, Mixed Numbers and Decimal Form on a Number Line, Differentiation: Support, “Consider using the following questions to support students as they represent the given number on a number line: What whole numbers is the number between? How does that help you draw and label a number line? What fractional unit do you need to partition each whole number into? How do you know? How can you use the unit form to help you locate the number on the number line? How can you use the number line to help you write the number as a mixed number and in decimal form?”

• Module 6, Topic A, Lesson 6: Relate geometric figures to a real-world context, Fluency, Show Me Geometric Figures: Lines and Line Segments, Teacher Notes, “Consider asking students to whisper to their partner how each gesture represents the geometric figure.” For example, after “Show me parallel lines, say, How do you think our arms show parallel lines? Whisper your idea to your partner.”

The materials reviewed for Eureka Math² Grade 4 meet expectations for containing adult-level explanations and examples of the more complex grade/course-level concepts and concepts beyond the current course so that teachers can improve their own knowledge of the subject.

Materials consistently contain adult-level explanations, examples of the more complex grade/ course-level concepts, and concepts beyond the course within Topic Overviews and/or Module Overviews. According to page 5 of the Grade 3-5  Implementation Guide, “Your Teach book begins with the Overview, a topic-by-topic summary that shows the development of learning throughout the module. It also provides connections to work done before and after the module, helping you understand the module’s place in the overall development of learning in and across the grade levels.” Page 7 outlines the purpose of the Topic Overview, “Each topic begins with a Topic Overview that is a summary of the development of learning in that topic. It typically includes information about how learning connects to previous or upcoming content.” Examples include:

• Module 1: Place Value Concepts for Addition and Subtraction, Module Overview, Why, “Why is the vertical number line used for rounding numbers? The vertical number line is used to help support conceptual understanding of rounding. In grade 3, students first see the vertical number line as an extension of reading a vertical measurement scale. Using the context of temperature, students identify the tens (i.e., benchmarks) between which a temperature falls, the halfway mark between the benchmark temperatures, and the benchmark temperature the actual temperature is closer to. Students then generalize to round numbers to the nearest ten and hundred. In grade 4, students round numbers with up to 6 digits to any place. They continue to use the vertical number line as a supportive model. Labeling the benchmark numbers and halfway tick mark in both standard form and unit form helps emphasize the unit to which a number is being rounded. This way, the place values line up vertically, helping students see the relationship between the numbers. The pictorial support of the vertical number line when rounding is eventually removed, but the conceptual understanding of place value remains as students round mentally. These experiences with the vertical number line prepare students for representing ratios with vertical double number lines and graphing pairs of values in the coordinate plane.”

• Module 2: Place Value Concepts for Multiplication and Division, Module Overview, Why, “Why is the distributive property used in reference to multiplication and the break apart and distribute strategy used in reference to division? In grade 3, students use the break apart and distribute strategy \times with multiplication and division. In grade 4, the distributive property is formally named, and students recognize that the distributive property relates multiplication and addition. For example, 7\times23=7\times(20+3)=(7\times20)+(7\times3). The distributive property is more formally the distributive property of multiplication over addition. Thus, the distributive property does not apply to division. With division, the break apart and distribute strategy continues to be referenced and describes how students approach a division problem. They break apart the total into smaller parts and then divide each part by the divisor. For example, 72\div8=(40\div8)+(32\div8). In grade 5, students learn that dividing by n is equivalent to multiplying by \frac{1}{n}, so the distributive property can be applied. For example, 

72\div8=72\times\frac{1}{8}=(40+32)\times\frac{1}{8}=(40\times\frac{1}{8})+(32\times\frac{1}{8})=(40\div8)+(32\div8).”

• Module 4: Foundations for Fraction Operations, Module Overview, Why, “Why are so many addition and subtraction strategies used? The strategies used for adding and subtracting fractions and mixed numbers in topics D and E reflect the strategies students use in grades 1, 2, and 3 to add and subtract whole numbers. These strategies reinforce the idea of fractions as numbers—we can perform operations with fractions similar to the way we perform operations with whole numbers. Because fractions are numbers, they can be composed and decomposed. Students apply the part total relationship found in addition and subtraction problems to compose and decompose the units of the parts and total. Fluency means being accurate and efficient and flexibly applying strategies to solve problems. A strategy may be efficient for solving one problem but time consuming for another. Students analyze problems and select efficient strategies, many of which develop into mental math over time. They select a model to record their work in a way that makes sense to them. Students are not expected to master all the strategies and models taught in module 4 topics D and E. Rather, they are expected to make informed decisions about which strategy to use on a problem-by-problem basis.”

The materials reviewed for Eureka Math² Grade 4 meet expectations for including standards correlation information that explains the role of the standards in the context of the overall series. 

Correlation information and explanations of standards are present for the mathematics addressed throughout the grade level. The Overview section includes Achievement Descriptors and these serve to identify, describe, and explain how to use the standards. Each module, topic, and lesson overview includes content standards and achievement descriptors addressed. Examples include:

• Module 3: Multiplication and Division of Multi-Digit Numbers, Module Overview, Achievement Descriptors, Proficiency Indicator, “4.Mod3.AD2, Multiply whole numbers of up to four digits by one-digit whole numbers. (4.NBT.B.5)” 

• Module 4, Topic B, Equivalent Fractions, Description, “Students generate equivalent fractions and equivalent mixed numbers. They decompose fractional units to find an equivalent fraction with smaller units and record their work with multiplication. They compose fractional units to find an equivalent fraction with larger units and record their work with division. Students use area models, as well as tape diagrams and number lines, to represent fractions and compose or decompose fractional units to generate equivalent fractions.” Achievement Descriptors and Standards are listed for the module in the tab labeled, “Standards."

• Module 5, Topic C, Lesson 10: Use pictorial representations to compare decimal numbers. Achievement Descriptors and Standards, “4.Mod5.AD4 Compare two decimal numbers to hundredths and justify the conclusions (4.NF.C.7).”

• Module 6, Topic C, Determine Unknown Angle Measures, Description, “Students recognize and apply the additive nature of angle measure to find the unknown measures of angles within figures without using a protractor. They use what is known and the part–total relationship to determine an unknown angle measure when right angles, straight angles, and angles of known measures are decomposed. Students extend the strategy to find the measures of multiple unknown angle measures around a point.” Achievement Descriptors and Standards are listed for the module in the tab labeled, “Standards."

The materials reviewed for Eureka Math² Grade 4 meet expectations for providing explanations of the instructional approaches of the program and identification of the research-based strategies. 

The Grade 3-5 Implementation Guide includes a variety of references to both the instructional approaches and research-based strategies. Examples include:

• Grade 3-5 Implementation Guide, What’s Included, “Eureka Math² is a comprehensive math program built on the foundational idea that math is best understood as an unfolding story where students learn by connecting new learning to prior knowledge. Consistent math models, content that engages students in productive struggle, and coherence across lessons, modules, and grades provide entry points for all learners to access grade-level mathematics.”

• Grade 3-5 Implementation Guide, Lesson Facilitation, “Eureka Math² lessons are designed to let students drive the learning through sharing their thinking and work. Varied activities and suggested styles of facilitation blend guided discovery with direct instruction. The result allows teachers to systematically develop concepts, skills, models, and discipline-specific language while maximizing student engagement.”

• Implement, Suggested Resources, Instructional Routines, “Eureka Math² features a set of instructional routines that optimize equity by increasing access, engagement, confidence, and students’ sense of belonging. The following is true about Eureka Math² instructional routines: Each routine presents a set of teachable steps so students can develop as much ownership over the routine as the teacher. The routines are flexible and may be used in additional math lessons or in other subject areas. Each routine aligns to the Stanford Language Design Principles (see Works Cited): support sense-making, optimize output, cultivate conversation, maximize linguistic and cognitive meta awareness.” Works Cited, “Smith, Margaret S., Victoria Bill, and Miriam Gamoran Sherin. The 5 Practices in Practice: Successfully Orchestrating Mathematics Discussions in Your Elementary Classroom. 2nd ed. Thousand Oaks, CA: Corwin Mathematics; Reston, VA: National Council of Teachers of Mathematics, 2018. Zwiers, Jeff, Jack Dieckmann, Sara Rutherford-Quach, Vinci Daro, Renae Skarin, Steven Weiss, and James Malamut. Principles for the Design of Mathematics Curricula: Promoting Language and Content Development. Retrieved from Stanford University, UL/SCALE website: http://ell.stanford.edu/content/mathematics-resources additional-resources, 2017.”

Each Module Overview includes an explanation of instructional approaches and reference to the research. For example, the Why section explains module writing decisions. According to the Implementation Guide for Grade 4, “The Why section gives insight into the decisions made during the writing of the module. This insight helps you understand the underlying structure of the module, flow of the content, and coherence of the different parts of the curriculum.” The Implementation Guide also states, “Works Cited, A robust knowledge base underpins the structure and content framework of Eureka Math². A listing of the key research appears in the Works Cited for each module.” Examples include:

• Module 1: Place Value Concepts for Addition and Subtraction, Module Overview, Why, “Why does the place value module begin with a topic on multiplicative comparisons? Beginning with multiplicative comparison enables students to build on their prior knowledge of multiplication from grade 3 and provides a foundation upon which students can explore the relationships between numbers and place value units. This placement also activates grade 3 knowledge of multiplication and division facts within 100 and provides students with opportunities to continue building fluency with the facts in preparation for multiplication and division in modules 2 and 3. Students are familiar with additive comparison—relating numbers in terms of how many more or how many less. Multiplicative comparison—relating numbers as times as many—is a new way to compare numbers. Students use multiplicative comparison throughout the year to relate measurement units, whole numbers, and fractions. This important relationship between factors, where one factor tells how much larger the product is compared to the other factor, is foundational to ratios and proportional relationships in later grades. Taking time to develop this understanding across the grade 4 modules sets students up for success with interpreting multiplication as scaling in grade 5 and applying or finding a scale factor in scale drawings, dilations, and similar figures.”

• Module 3: Multiplication and Division of Multi-Digit Numbers, Module Overview, Why, “Why is vertical form introduced alongside the place value chart for multiplication and division? Similar to what students experience with addition and subtraction, vertical form is introduced alongside the place value chart for multiplication and division to support conceptual understanding and the transition from a pictorial representation to a written representation. Each action represented in the place value chart (e.g., renaming units, adding or subtracting like units, distributing units, finding the total quantity of each unit) has a direct connection to a recording within vertical form. As students become proficient with recording in vertical form, they internalize the process and no longer require drawing on the place value chart to find the unknown or explain their work. Additionally, students not yet fluent with multiplication and division facts may find the place value chart helpful in keeping track of their calculations within vertical form.”

The materials reviewed for Eureka Math² Grade 4 meet expectations for providing a comprehensive list of supplies needed to support instructional activities. 

Each module includes a tab, “Materials” where directions state, “The following materials are needed to implement this module. The suggested quantities are based on a class of 24 students and one teacher.” Additionally, each lesson includes a section, “Lesson at a Glance” where supplies are listed for the teacher and students. Examples include:

• Module 2: Place Value Concepts for Multiplication and Division, Module Overview, Materials, “1 Chart paper, tablet, 25 Personal whiteboards, 25 Colored pencils (red and blue), 25 Personal whiteboard erasers, 25 Crayons (red, green, and blue), 25 Eureka Math^2™ place value disks set, ones to millions, 25 Dry-erase markers, 1 Projection device, 24 Learn books, 25 Rulers, 1 Meter stick, 1 Teach book, 27 Paper strips, 1"\times12", 1 Teacher computer or device, 25 Pencils.”

• Module 4, Topic B, Lesson 7: Write mixed numbers in decimal form with hundredths,  Materials, “Teacher: Open Number Line (in the teacher edition). Students: Open Number Line (in the student book), Deci-disks set (1 per student pair). Lesson Preparation: Consider whether to remove Open Number Line from the student books and place inside whiteboards in advance or have students prepare them during the lesson. Gather at least 5 ones disks, 6 tenths disks, and 7 hundredths disks for each student pair. Review the Math Past resource to support delivery of Land.”

• Module 6, Topic B, Lesson 8: Use a circular protractor to recognize a 1° angle as a turn through \frac{1}{360}​​ of a circle, Materials, “Teacher: Circular Protractor, Angle-maker tool. Students: Circular Protractor, Angle-maker tool. Lesson Preparation:Consider whether to remove Circular Protractor from the student books in advance or have students tear them out during the lesson. Gather the angle-maker tools created in lesson 7.”

The materials reviewed for Eureka Math² Grade 4 meet expectations for providing assessments that include opportunities for students to demonstrate the full intent of grade-level standards and practices across the series.

According to the 3-5 Implementation Guide, “The assessment system in grades 3 through 5 helps you understand student learning by generating data from many perspectives. The system includes Lesson-embedded Exit Tickets, Topic Quizzes, Module Assessments, Pre-Module Assessment in Eureka Math ² Equip, and Benchmark Assessments. These assessments use a variety of question types, such as constructed response, multiple select, multiple choice, single answer, and multi-part. Module Assessments.” These assessments consistently list grade-level content standards for each item. While Mathematical Practices are not explicitly identified on assessments, they are regularly assessed. Students have opportunities to demonstrate the full intent of the standards using a variety of modalities (e.g., oral responses, writing, modeling, etc.). Examples include:

• Module 1: Place Value Concepts for Addition and Subtraction, Topic E Quiz 2, Metric Measurement Conversion Tables, Item 3, “Decide whether each equation is true or false. 1 km = 100 m, 3 kg = 3,000 g, 4 L = 4,000 mL, 2 m = 2,000 cm.” Students engage with the full intent of 4.MD.1 (Know relative sizes of measurement units within one system of units including km, m, cm, kg, g… Within a single system of measurement, express measurements in a larger unit in terms of the smaller unit…).

• Module 3: Multiplication and Division of Multi-Digit Numbers, Topic A, Lesson 1, Divide Multiples of 100 and 1000, Land, Exit Ticket, supports the full intent of MP8 (Look for and express regularity in repeated reasoning) as students divide by using unit form and basic division facts. “Divide. Use unit form to help you. a. 800\div2=___ hundreds \div ___; =___ hundreds; =___.  b. 1200\div4=____.”

Module 5, Place Value Concepts for Decimal Fractions, Module Assessment 2, Item 2, “Add. \frac{3}{10}+\frac{25}{100}=1___, \frac{81}{100}+\frac{5}{10}=2___, 2\frac{17}{100}+1\frac{6}{10}=3___” Students engage with the full intent of 4.NF.5 (Express a fraction with denominator 10 as an equivalent fraction with denominator 100, and use this technique to add two fractions with respective denominators 10 and 100).

The materials reviewed for Eureka Math² Grade 4 meet expectations for providing strategies and supports for students in special populations to support their regular and active participation in learning grade-level mathematics.

Suggestions are outlined within Teacher Notes for each lesson. Specific recommendations are routinely provided for implementing Universal Design for Learning (UDL), Differentiation: Support, and Differentiation: Challenge, as well as supports for multilingual learners. According to the Grade 3-5 Implementation Guide, Page 46, “Universal Design for Learning (UDL) is a framework based on current research from cognitive neuroscience that recognizes learner variance as the norm rather than the exception. The guiding principles of the UDL framework are based on the three primary networks of the brain. Although the concept of UDL has roots in special education, UDL is for all students. When instruction is designed to meet the needs of the widest range of learners, all students benefit. Eureka Math² lessons are designed with these principles in mind. Lessons throughout the curriculum provide additional suggestions for Engagement, Representation, and Action & Expression.” Examples of supports for special populations include:

• Module 2, Topic A, Lesson 2: Divide two- and three-digit multiples of 10 by one-digit numbers, Launch, “Language Support: Encourage students to use precise language such as factor, unknown factor, and total as they describe the categories they identified. Rephrase student responses as necessary to include the precise language in the discussion.” Learn, Divide by Using Unit Form, “UDL: Action & Expression: Support students in expressing learning in flexible ways. Encourage students to skip-count by tens rather than use place value disks to support the use of mental math in this segment.” Learn, Multiplicative Comparison with Unknown Factors, “UDL: Representation Consider pausing and providing additional think time as students begin to represent the problem with a tape diagram. Ask questions such as the following: What information is known? What information is unknown? What letter could you use to represent the unknown information? Pausing provides time for processing of information and signifies its importance.” Land, Debrief, “UDL: Action & Expression Consider reserving time for students to reflect on their overall experiences multiplying one-digit numbers by multiples of 10 and dividing two- and three-digit multiples of 10 by a one-digit number. What strategies work well for me? Which methods do I need more practice using to become confident? What is still confusing? What can I do to help myself?”

• Module 3, Topic C, Lesson 10: Apply place value strategies to multiply four-digit numbers by one-digit numbers, Learn, Multi-Digit Multiplication on the Place Value Chart, “UDL: Action & Expression: Revisit Liz’s thinking in Launch. Ask students if, after completing the work on the place value chart, they can confirm whether Liz’s thinking was correct. After the class comes to consensus that Liz’s thinking was correct, support students in reflecting by posing the following prompts: If you agreed with Liz, what about your thinking was confirmed? If you disagreed with Liz, what about your thinking has changed?” Learn, Three Methods, “UDL: Action & Expression Consider supporting students in planning their method to multiply the four-digit number. Have students turn to a group member and summarize the steps of one method they can apply before getting started. Language Support: Encourage the use of the Ask for Reasoning section of the Talking Tool to support students with asking their group members clarifying questions.”

• Module 5, Topic D, Lesson 12: Apply fraction equivalence to add tenths and hundredths, Launch, “UDL: Action & Expression: Consider providing tools to support students with renaming tenths as hundredths. For example, students may wish to use Blank Tape Diagram, Area Models, and Number Line from lesson 9.” Learn, Choose a Strategy to Add, “UDL: Representation: Consider engaging the class in brainstorming what they already know about making equivalent fractions to activate prior knowledge.” Learn, Share, Compare, and Connect, “Language Support: Consider inviting students to refer to the Talking Tool as they share strategies and ask questions about their peers’ strategies. UDL: Engagement: Consider presenting a real-world application of adding tenths and hundredths anchored in a context that may be of interest, or familiar, to students. For example,a snake is \frac{48}{100} meters long. It grows another \frac{3}{10} meters. How long is the snake now?” Learn, Add Tenths and Hundredths, “Language Support Consider supporting students with the phrases like units, related units, and unlike units by providing examples. Like units: \frac{1}{4} and \frac{3}{4}, tenths and tenths Related units: \frac{1}{2} and \frac{1}{4}, tenths and hundredths Unlike units: \frac{1}{2} and \frac{1}{3, tenths and thirds.”

The materials reviewed for Eureka Math² Grade 4 meet expectations for providing extensions and/or opportunities for students to engage with grade-level/course-level mathematics at higher levels of complexity.

Materials do not require advanced students to do more assignments than their classmates. Instead, students have opportunities to think differently about learning with alternative questioning, or extension activities. Specific recommendations are routinely highlighted as Teacher Notes within parts of each lesson, as noted in the following examples: 

• Module 2, Topic A, Lesson 2: Divide two- and three-digit multiples of 10 by one-digit numbers, Learn, Differentiation: Challenge, “Consider providing students with more challenging numbers. Invite students to think about whether they could use a similar strategy to interpret the expression 600\div3 or 6,000\div3. What is similar and different about each of the quotients? Why?”

• Module 3, Topic C, Lesson 11: Represent multiplication by using partial products, Learn, Partial Products in Vertical Form, Differentiation: Challenge, “Consider inviting students to work independently or with a partner to find 273\times5 and  3\times4,128. Direct students to articulate the process of recording partial products with vertical form.”

• Module 5, Topic B, Lesson 6: Represent hundredths as a place value unit, Learn, Hundredths as a Fractional Unit and a Place Value Unit, Differentiation: Challenge, “Consider inviting students to shade 100 hundredths in an area model. Then ask the following questions: How can you represent the shaded amount as a fraction with a denominator of 100? How can you represent the shaded amount as a fraction with a denominator of 10? How can you represent the shaded amount as a decimal number?”

The materials reviewed for Eureka Math² Grade 4 meet expectations for providing strategies and supports for students who read, write, and/or speak in a language other than English to regularly participate in learning grade-level mathematics. 

Support for active participation in grade-level mathematics is consistently included within a Language Support Box embedded within parts of lessons. According to the Grade 3-5 Implementation Guide, “Multilingual Learner Support, Multilingual learners, or learners who speak a language other than English at home, require specific learning supports for gaining proficiency with the English needed to access the mathematics. Research suggests that best practices for these learners include opportunities and supports for student discourse and for using precise terminology. In addition to precise domain-specific terminology, high-impact academic terminology that supports learners across learning domains is explicitly introduced and used repeatedly in various contexts to build familiarity and fluency across the grade levels. Eureka Math² is designed to promote student discourse through classroom discussions, partner or group talk, and rich questions in every lesson.” According to Eureka Math² How To Support Multilingual Learners In Engaging In Math Conversations In The Classroom, “Eureka Math² supports MLLs through the instructional design, or how the plan for each lesson was created from the ground up. With the goal of supporting the clear, concise, and precise use of reading, writing, speaking, and listening in English, Eureka Math² lessons include the following embedded supports for students. 1. Activate prior knowledge  (mathematics content, terminology, contexts). 2. Provide multiple entry points to the mathematics. 3. Use clear, concise student-facing language. 4. Provide strategic active processing time. 5. Illustrate multiple modes and formats. 6. Provide opportunities for strategic review. In addition to the strong, built-in supports for all learners including MLLs outlined above, the teacher–writers of Eureka Math² also intentionally planned to support MLLs with mathematical discourse and the three tiers of terminology in every lesson. Language Support margin boxes provide these just-in-time, targeted instructional recommendations to support MLLs.” Examples include:

• Module 1, Topic E, Lesson 23: Express metric measurements of length in terms of smaller units, Learn, Relative Size of Units, MLL students are provided the support to participate in grade-level mathematics as described in the UDL: Representation, “The context video Running Meters and Kilometers is available to provide another format to illustrate the relationship between meters and kilometers. It may be used to remove language or cultural barriers and provide student engagement. Consider showing the video and facilitating a discussion about what students notice and wonder. This supports students in visualizing the situation before being asked to interpret it mathematically. Ask students how it feels to run 1 meter. Invite students who have experience with running longer distances to compare how it feels to run 1 kilometer, or 2 and 12 laps on a track. Invite students to count the number of steps they take in 1 meter and estimate how many steps they take in 1 kilometer.”

• Module 4, Topic A, Lesson 1: Decompose whole numbers into a sum of unit fractions, Fluency, Counting on the Number Line by Halves, Language Support, “Consider using strategic, flexible grouping throughout the module. Pair students who have different levels of mathematical proficiency. Pair students who have different levels of English language proficiency. Join pairs to form small groups of four. As applicable, complement any of these groupings by pairing students who speak the same native language.”

• Module 6, Topic B, Lesson 9: Identify and measure angles as turns and recognize them in various contexts, Learn, Problem Set, MLL students are provided the support to participate in grade-level mathematics as described in the Teacher Note box, “A context video for problem 5 in the Problem Set is available. It may be used to remove language or cultural barriers and provide student engagement. Before providing the problem to students, consider showing the video and facilitating a discussion about what students notice and wonder. This supports students in visualizing the situation before being asked to interpret it mathematically. Alternatively, consider showing the video during Land to facilitate students confirming their solutions. Consider inviting students to reflect on how the video helped them see how the parts of the scene are related. Invite them to describe how this helped them solve the problem and how they can apply the thinking to other situations.”

The materials reviewed for Eureka Math² Grade 4 meet expectations for providing manipulatives, both virtual and physical, that are accurate representations of the mathematical objects they represent and, when appropriate, are connected to written methods.

Each lesson includes a list of materials for the Teacher and the Students. As explained in the Grade 3-5 Implementation Guide, “Materials lists the items that you and your students need for the lesson. If not otherwise indicated, each student needs one of each listed material.” Examples include:

• Module 1, Topic B, Lesson 8: Write numbers to 1,000,000 in standard form and word form, Learn, Write Numbers in Standard and Word Forms, Materials, Teacher and Student: Place Value Chart to Millions. “Students group thousands to write numbers in word form and standard form.” Teacher Note, “To support students in writing numbers in standard form, a place value chart is used throughout the lesson. Students express a number in standard form by writing the digits on the place value chart. This scaffold helps them keep track of the place value of each digit, see where commas should be placed, and read the number. Consider removing the scaffold of the place value chart as students are ready.”

• Module 5, Topic A, Lesson 2: Decompose 1 one and express tenths in fraction form and decimal form, Learn, Tenths on a Number Line, Materials, Teacher: Meter stick. “Students decompose 1 meter and write tenths in decimal form and fraction form. Trace along the edge of a meter stick to create a number line with a length of 1 meter. Draw and label tick marks to represent 0 and 1. Let’s represent tenths another way. How can we use the meter stick to partition the number line into tenths?”

• Module 6, Topic C, Lesson 13: Decompose angles by using pattern blocks, Learn, Decompose and Angle, Materials, Student: Pattern blocks. “Students use pattern blocks to decompose an angle and find the measure.” After students use pattern blocks to decompose an obtuse angle, the teacher directs students to discuss their decompositions and equations with a partner. “Circulate and listen as they talk. Identify two or three students to share their thinking. Purposefully choose work that allows for rich discussion about connections between strategies used to find the measure of \angle{L}.”