2nd Grade - Gateway 3

The materials reviewed for Eureka Math² Grade 2 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 1-2 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 1-2 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 1-2 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 1-2 Implementation Guide, Inside Teach, Module-Level Components, 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. It is designed to prepare students for new learning by activating prior knowledge and bridging small learning gaps. 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. Every launch ends with a transition statement that sets the goal for the day’s learning. 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. Suggested facilitation styles vary and may include direct instruction, guided instruction, group work, partner activities, interactive video, and digital elements. The Problem Set, an opportunity for independent practice, is included in Learn. Land helps you facilitate a brief discussion to close the lesson and provides students with an opportunity to complete the Exit Ticket. In the Debrief portion of Land, suggested questions, including key questions related to the objective, help students synthesize the day’s learning. The Exit Ticket provides a window into what students understand so that you can make instructional decisions.”

Materials include sufficient and useful annotations and suggestions that are presented within the context of 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 margins. These notes provide information about facilitation, differentiation, and coherence. Teacher Notes communicate information that helps with implementing the lesson. 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 2, Topic C, Lesson 15: Use compensation to subtract within 100, Launch, provides a teacher note for Differentiation: Support. “While measuring tapes are provided in the next segment, consider making them available now for students who can benefit from the concrete support of a number line.”

• Module 3, Topic A, Lesson 1: Determine the defining attributes of a polygon, Launch, provides a teacher note with general guidance, Teacher Note. “Much of the terminology in this topic (e.g., closed, side, corner, and square corner) is familiar from previous grades. However, given the specificity of the word use and the time that has passed, it may be appropriate to introduce the terminology as if it were new. Consider creating an anchor chart with the terms and visuals for students to reference as needed throughout the topic.”

• Module 4, Topic C, Lesson 12: Take from a ten or a hundred to subtract, Learn, Reason about Efficiency, provides a teacher note for UDL: Representation. “Consider providing students with a copy of the student work samples to use as a reference as they participate in the Five Framing Questions routine.”

The materials reviewed for Eureka Math² Grade 2 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 7 of the Grade 1-2 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 9 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 2: Addition and Subtraction within 200, Topic D: Topics for Decomposing a Ten and a Hundred to Subtract, Topic Overview, connects the work being done in Grade 2 to the development of student understanding as they work towards using the standard algorithm in Grade 4. “In topic D, students build on their understanding of place value strategies and taking from a unit of ten or a hundred. Consistent use of place value disks and place value drawings helps students systematically model the steps they take when they decompose a larger value unit. When students relate models to the unit form, it primes them to use the vertical form in module 4. As with addition, students are not expected to master the standard subtraction algorithm until grade 4. Throughout the topic, students move through concrete and pictorial representations to develop a conceptual understanding of subtraction. First, students use place value disks to represent the total and subtract numbers concretely. Then they use place value drawings to represent subtraction problems. Students see they can decompose a unit of ten, and later a hundred, when they need more ones in the ones place or tens in the tens place to subtract. Problems gradually increase in complexity as students decompose once, and then twice, to subtract. As it was in topic C, the language is intentionally consistent and repetitive. This secures familiarity with the representations and anchors students’ understanding as they move toward work with more abstract numbers. Students relate place value drawings to more abstract recordings that show the minuend and subtrahend in unit form. While students are not expected to write unit form recordings independently, they make connections that deepen their place value understanding. For example, when they unbundle 1 ten into 10 ones in a drawing, students see how 1 hundred 2 tens 6 ones can be renamed as 1 hundred 1 ten 16 ones. The fact that the value of the minuend does not change when it is renamed is a key understanding in grade 2.”

• Module 4: Addition and Subtraction Within 1000, Module Overview, Why, “Why do you show new groups below when adding in vertical form? The decision to show newly composed units on the line below the addends, as opposed to above, has several advantages that support conceptual understanding with the standard algorithm: 1. The digits are written in close proximity to each other, so students do not see them as unrelated. The close proximity reduces the likelihood of students reversing the order of the numbers when recording the regrouping. 2. When composing a new unit, students write the teen number in order, for example as 1 new unit of ten on the line first and then the additional ones next to it below the line. It is natural for students to write numbers in their usual order (e.g., 1 then 6), rather than the reverse. 3. Since students typically add digits from the top down in a given column, the additional 1 can be easily counted on to a larger sum at the end. Why is there an entire topic devoted to simplifying strategies for subtraction but not for addition?”

• Module 6: Multiplication and Division Foundations, Topic C: Rectangular Arrays as a Foundation for Multiplication and Division, Topic Overview, relates the use of arrays to the more complex strategy of using the area module in Grade 3. “Topic C naturally follows topic B, where students compose and manipulate the rows and columns of an array. This topic is designed to deepen students’ understanding of spatial relationships and structure as they build and partition rectangles with rows and columns of same-size squares. To begin, students build a rectangle by making a tile array without gaps or overlaps. Then they build square arrays and recognize that only one repeated addition equation can be used to represent the array because the number of rows is equal to the number of columns. Next, students use square tiles to draw an array. They reason about how they can compose a larger rectangle by using smaller units. Much like students iterated a length unit in module 1 to create a centimeter ruler, now students iterate the square unit to construct a row or column and, ultimately, a rectangle. As they draw, they realize the structure of an array is a collection of same-size squares arranged in rows and columns. Students begin to see a row or column in two ways: as a composition of multiple units (e.g., 3 tiles) and as a single unit (1 row of 3). This supports the transition from repeated addition to multiplication and the area model in grade 3.”

The materials reviewed for Eureka Math² Grade 2 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 1, Topic A, Lesson 2: Draw and label a bar graph to represent data, Achievement Descriptors and Standards, “2.Mod1.AD8 Draw and label picture and bar graphs to represent a data set with up to four categories. (2.MD.D.10)”

• Module 3: Shapes and Time with Fraction Concepts, Achievement Descriptors and Standards, “2.Mod3.AD3 Label a given daily event as taking place in the a.m. or p.m. (2.MD.C.7)”

• Module 4, Topic C: Simplifying Strategies for Subtracting Within 1000, Description, “Students expand upon their toolbox of simplifying strategies for subtraction by using familiar models and recording methods from module 2. Students apply the take from a ten or a hundred strategy to subtract within 1,000. They also explore three types of compensation as shown in the table.” Achievement Descriptors and Standards are listed for the topic in the tab labeled, “Standards.”

• Module 6: Multiplication and Division Foundations, “Students count and solve problems with equal groups of objects. Students organize equal groups into rows and columns to create rectangular arrays. As they compose and decompose arrays, students gain foundations for multiplication.” Achievement Descriptors and Standards are listed for the module in the tab labeled, “Standards.”

The materials reviewed for Eureka Math² Grade 2 meet expectations for providing explanations of the instructional approaches of the program and identification of the research-based strategies. 

The Grade 1-2 Implementation Guide includes a variety of references to both the instructional approaches and research-based strategies. Examples include:

• Grade 1-2 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 1-2 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 Grade 1-2 Implementation Guide, “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 from Module Overviews Include:

• Module 1: Place Value Concepts Through Metric Measurement and Data: Place Value, Counting, and Comparing Within 1,000, Module Overview, Why?, “Part 1: Place Value Concepts Through Metric Measurement and Data, Why does the year start with categorical data? During the first week of school, teachers and students spend time establishing a classroom community. By launching with categorical data, teachers can leverage getting-to-know-you activities to generate student data, create graphs, and answer questions. Bar graphs provide students with a concrete and visual experience of comparison. Comparing categories on a bar graph sets up students for solving compare word problems by using a more abstract model, the tape diagram. Labeling the categories on a bar graph supports the practice of labeling tape diagrams, where students must visualize the amount or length.” Works Cited include, “Common Core Standards Writing Team, Progressions for the Common Core (draft), Grades K–5, Counting and Cardinality & Operations and Algebraic Thinking, 9. These word problem types come from Progressions for the Common Core State Standards in Mathematics, Operations and Algebraic Thinking Progression, and an explanation and example of some types are included here. See the table for examples. Darker shading indicates the four Kindergarten problem subtypes. Grade 1 and 2 students work with all subtypes and variants. Unshaded (white) problems are the four difficult subtypes or variants that students should work with in Grade 1 but need not master until Grade 2.”

• Module 2: Addition and Subtraction Within 200, Module Overview, Why?, “Why are two topics devoted to simplifying strategies for addition and subtraction?By the end of grade 2, students are expected to add and subtract fluently within 100 by using strategies based on place value, properties of operations, and the relationship between addition and subtraction. Fluency means being able to operate with numbers flexibly, efficiently, and accurately. Because students are not expected to work fluently with the standard addition and subtraction algorithms until grade 4, topics A and C are intentionally devoted to Level 3 addition and subtraction methods, in which students use simplifying strategies to make simpler problems. This gives students time to work through and to make connections between various strategies. As students apply place value understanding from module 1 and leverage familiar tools, they develop confidence and flexibility. While students are not expected to master all of the Level 3 strategies, they are expected to reason about the numbers in a problem and to consider efficient solution paths by using tools and written recordings. This builds their capacity toward mental math.” Works Cited include, “Common Core Standards Writing Team, Progressions for the Common Core (draft), Grades K–5, Counting and Cardinality & Operations and Algebraic Thinking, 9.”

The materials reviewed for Eureka Math² Grade 2 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 1, Topic D, Lesson 16: Use a measuring tape as a number line to subtract efficiently, Overview, Materials, “Teacher: Double-sided meter sticks(2), Chart paper. Students: Measuring tape (1 per student pair).”

• Module 4: Addition and Subtraction Within 1000, Module Overview, Materials, “Chart paper, tablet(1), Eureka Math2™ place value disks set, ones to thousands(25), Computer with internet access(1), Pencils(25), Dot dice, set of 12(1), Personal whiteboards(25), Dry-erase markers(25), Personal whiteboard erasers(25), Index cards(12), Projection device(1), Learn books(24), Sticky notes, pad(5), Markers(25), Teach book(1), Eureka Math2™ measuring tape(24), Unifix® Cubes, set of 300(1), Please see lessons 1 and 24 for a list of organizational tools (cups, bowls, plates, trays, or rubber bands) suggested for the counting collection.”

• Module 6, Topic C, Lesson 12: Reason about how equal arrays can be composed differently, Overview, Materials, “Teacher: Arrays (digital download), Scissors. Students: Eureka Math² Numeral Cards (1 set per student pair), Hidden Addends Mat (1 per student pair), Arrays (in the student book), Color tiles, plastic, 1”(25), Scissors. Lesson Preparation: Gather the Hidden Addends Mats used in the previous lesson. Tear out the Arrays page from the student books. Consider whether to prepare this material in advance or have students remove it during the lesson. Print one copy of the Arrays page to use for demonstration.”

The materials reviewed for Eureka Math² Grade 2 meet expectations for providing assessments that include opportunities for students to demonstrate the full intent of grade-level standards and practices across the series.

Assessments consistently list grade-level content standards for each item. While Mathematical Practices are not explicitly identified on assessments, they are 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 from Module Assessments, Topic Tickets, and Exit Tickets include:

• Module 1, Module Assessment, Place Value Concepts Through Metric Measurement and Data, Place Value, Counting, and Comparing Within 1000, Problem 5, supports the full intent of 2.NBT.4 (Compare two three-digit numbers based on meanings of the hundreds, tens, and ones digits, using >, =, and < symbols to record the results of comparisons). “Show the numbers on the place value chart. Then write >, =, or <. 20 tens 9 ones ___ 290, Write >, =, or <. This assessment has 10 items. Use the scoring guide and the answer key from the Assessment resource in the Teach book when scoring each student’s assessment. As needed, use the Achievement Descriptors and the proficiency indicators to help interpret student work and assign points. 2.Mod1.AD11, Students can earn up to 2 points for showing work in the place value charts. 2.Mod1.AD16, Students can earn 1 point for correctly completing the comparison statement.”

• Module 3, Topic B, Lesson 9: Interpret equal shares in composite shapes as halves, thirds, and fourths, Land, Topic Ticket, Problem 4, students develop the full intent of  2.G.3 (Partition circles and rectangles into two, three, or four equal shares, describe the shares using the words halves, thirds, half of, a third of, etc., and describe the whole as two halves, three thirds, four fourths. Recognize that equal shares of identical wholes need not have the same shape). “Partition the rectangle into thirds. Then shade 1 third.” From the scoring guide, the teacher is given the following guidance, “Module 3 Topic B Ticket, This assessment has four items. Use the scoring guide and the answer key from the Assessment resource in the Teach book when scoring each student’s assessment. As needed, use the Achievement Descriptors and the proficiency indicators to help interpret student work and assign points. 2. Mod3.AD6, Students can earn 1 point for partitioning the rectangle. Students can earn 1 point for shading 1 third.” 

• Module 4, Topic E, Lesson 22: Solve compare with smaller unknown word problems, Land, Exit Ticket, support the full intent of MP1 (Make sense of problems and persevere in solving them). as students understand and solve word problems. “Read. Matt shoots the basketball 63 times. Matt shoots the basketball 18 more times than Nate. How many times does Nate shoot the basketball? Draw. Write.”

• Module 4: Addition and Subtraction Within 1,000, Module Assessment, Problem 13, supports the full intent of MP3 (Construct viable arguments and critique the reasoning of others) as students reason about problems with multiple addends. Students see a T-Chart split into tens and ones where dots represent a 10 or a 1 on the chart. This is Tim’s Way. “Tim finds 35+23+17+20. Look at Tim’s work. Show a different way. Tim’s Way 35+23+17+20=95, Your Way.”

• Module 6, Topic B, Lesson 7: Distinguish between rows and columns and use math drawings to represent arrays, Land, Exit Ticket, supports the full intent of 2.OA.3 (Determine whether a group of objects (up to 20) has an odd or even number of members) and 2.OA.4 (Use addition to find the total number of objects arranged in rectangular arrays with up to 5 rows and up to 5 columns; write an equation to express the total as a sum of equal addends). “Provide up to 5 minutes for students to complete the Exit Ticket. It is possible to gather formative data even if some students do not complete every problem. Draw an array with 5 rows of 4. Draw a line between each row. 5 rows of 4 is equal to ___. Draw 1 more row. Write a repeated addition equation to match the new array.”

The materials reviewed for Eureka Math² Grade 2 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 1-2 Implementation Guide, Page 47, “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 1, Topic A, Lesson 4: Use information presented in a bar graph to solve compare problems, Learn, Use a Bar Graph to Solve Compare Problems, students use different strategies to compare data on a bar graph. “Differentiation: Support: A common student error is to color the wrong row. Have students put a finger between rows before coloring. Or, suggest that they put their finger on the label and move it across the row as they color. Another common student error is to color more than the total number of boxes for each category. Suggest that students put their finger on the total on the scale and slide it up to the appropriate row. They can put a mark in the box to signal where to stop coloring.” 

• Module 4, Topic B, Lesson 5: Use the associative property to make a benchmark number to add within 1,000, Learn, Add to Make a Ten or Make a Hundred, students decompose an addend to make the next ten or next hundred to add. “UDL: Action & Expression: Consider posting questions for students to think about as they strategize. This scaffold can be gradually released as individuals are ready to apply the strategy independently. Are the addends close to a benchmark number? What is the next ten or next hundred? Is it easier to make a ten or a hundred? What new addition problem can you write?”

• Module 6, Topic B, Lesson 6: Decompose arrays into rows and columns and relate them to repeated addition, Learn, Decompose Arrays into Rows or Columns, students decompose an array into rows or columns and write repeated addition equations. “Language Support: Consider making a chart with key terms from the module with an example for each term. Include the following terms: Group, Row, Column, Array.” The Teacher Note includes a sample chart with illustrations and words.

The materials reviewed for Eureka Math² Grade 2 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 1, Topic B, Lesson 10: Reason about the relationship between the size of the unit and the number of units needed to measure, Learn, Compare Length Units, students analyze various length units to understand the relationship among them. “Differentiation: Challenge: Invite students to apply knowledge of mental benchmarks. Consider asking the following questions. Make a thoughtful guess: How many digits are in a cubit? What did you use as your mental benchmark?” 

• Module 4, Topic B, Lesson 10: Choose and defend efficient solution strategies for addition, Learn, Share and Defend Strategy Choices, students use place value understanding to defend the efficiency of their solution strategies. “Differentiation: Challenge: Encourage students to show multiple solution strategies, beginning with the one they feel is most efficient. As time allows, consider inviting students to explain their alternate strategies, or ask the class to reason about what steps they think the student took to solve. For example, for 399+499, a student might solve the problem by thinking 400+500-2=900-2=898 When students share their flexible thinking, it provides multiple access points for all members of the class.”

• Module 6, Topic B, Lesson 8: Use square tiles to create arrays with gaps, Learn, Develop Contexts to Match Arrays, students construct scenarios to match a given array. “Differentiation: Challenge: Challenge students to develop a context in which the total and the number of rows are known, but the number of objects in each row is unknown.”

The materials reviewed for Eureka Math² Grade 2 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 1-2 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 2, Topic A, Lesson 7: Solve word problems by using simplifying strategies for addition, Learn, Share and Defend Solution Strategies, MLL students are provided the support to participate in grade-level mathematics as described in the Language Support box. “As the small group discussion unfolds, consider providing students with sentence frames and the key terminology as a scaffold for peer conversations. For example, provide a smaller version of the addition strategies chart for students to reference as they defend their strategy choice. Also include the following sentence frames: I chose the ___strategy because ___. My strategy ___ works because ___. When I used ___, it made an easier problem because ___. Students explain their reasoning for selecting a particular solution strategy. Introduce the Take a Stand routine to the class. Draw students’ attention to the signs hanging in the classroom. Take a moment to read aloud the strategy on each of the signs. (Signs show the different strategies students may have used to solve the problem.)  Invite students to stand beside the sign that shows the strategy they used for this problem. When all students are standing near a sign, allow 2 minutes for groups to discuss the reasons why they chose that strategy. Then call on each group to defend their strategy choice by sharing reasons for their selection. Invite students who change their minds during the discussion to join a different group. Show student work as students share their reasoning.”

• Module 2, Topic D, Lesson 20: Reason about when to unbundle a ten to subtract, Land, Debrief, MLL students are provided the support to participate in grade-level mathematics as described in the Language Support box. “Consider providing students with sentence frames that include key terminology to help students share their thinking with a partner. For example, post the following sentence frames: I think you ___(do, do not) need to unbundle because ___. I think I can rename 52 as ___. Objective: Reason about when to unbundle a ten to subtract. Gather students with their Problem Sets and invite them to think–pair–share about the following questions. ‘Tell your partner which problems you can answer without unbundling a ten. How do you know when you need to unbundle a ten to subtract?’ (In problem 1, I didn’t need to unbundle because I had enough ones in the ones place to subtract ones from ones. In problem 4, I needed to unbundle a ten because I couldn’t subtract 7 ones from 2 ones, so I had to unbundle a ten to get more ones. In problem 5, I didn’t need to unbundle a ten because I had enough ones. I took 4 ones from 8 ones. In problem 6, I needed to unbundle a ten because I couldn’t take 4 ones away from 3 ones.) Direct students to problem 3. ‘How did you rename 52 to subtract 6?’ (I unbundled one of the tens and renamed it as 10 ones, so then I had 4 tens and 12 ones. I didn’t have enough ones to subtract 6 ones, so I crossed off a ten and drew 10 ones. Then I had 40 and 12.)”

• Module 4, Topic B, Lesson 8: Use place value drawings to represent addition and relate them to written recordings, part 1, Launch, MLL students are provided the support to participate in grade-level mathematics as described in the Language Support box. “The place value recordings are intentionally labeled with letters rather than the name of each recording. This reduces the cognitive demand of students struggling with language and allows them to focus on making mathematical connections. While students are not expected to master the names of each recording, they have been introduced to those names. Consider supporting students by creating a chart that shows the various recordings alongside a sample of each. Students reason about similarities and differences between different representations of addition. Display the picture of the place value drawing, the totals below recordings, and the new groups below recording. ‘What do you notice? What do you wonder?’ (I notice they all show the same addition problem and have the same total. I notice they all show how we add like units. I notice the new ten in A and D. In B & C, they found the totals for each place value unit. I wonder whether D will work for all numbers. It looks like it has the fewest steps! I wonder why you can add in any order in some of the recordings, but in others you always start in the ones place.) Invite students to turn and talk about how the recordings are alike and different. Then facilitate a class discussion. Invite students to share their thinking with the whole group. As students discuss, highlight thinking that shows the role of place value understanding in each recording.”

The materials reviewed for Eureka Math² Grade 2 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 1-2 Implementation Guide, page 11, “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 of manipulatives include: 

• Module 1, Topic C, Lesson 13: Estimate and measure height to model metric relationships, Materials, Students: measuring tape. For Fluency, Add on the Measuring Tape, students add 10 cm to a measurement to build an understanding of length units. “Invite students to roll out the measuring tape to 2 m and lay it in front of them. Consider displaying your own measuring tape as a model. After asking each question, provide think time and then signal for students to respond. ‘Wait for my signal to say the answer to each question. Put your finger on 30 cm. What is 10 more centimeters?’ (40 cm) ‘Slide your finger up to 40 cm while you say the addition equation, starting with 30 cm. Ready?’ (30cm +10cm =40cm Slides finger from 30 cm to 40 cm.)”

• Module 4, Topic C, Lesson 14: Use compensation to keep a constant difference by adding the same amount to both numbers, Materials, Teacher: Unifix^® Cubes. For Learn, Show Compensation with a Tape Diagram, the teacher uses Unifix^® Cubes to support instruction about Tape Diagram models. “Let’s look at another way to show why this strategy works.’ Show two rows of 5 Unifix Cubes in one color. Add 3 cubes of another color to the right end of the top row ‘There are 5 cubes in the bottom row. How many cubes are in the top row?’ (8) ‘What is the difference between 8 and 5?’ (3) Write 8-5=3. Then add 1 cube of a third color to the left end of each row. ‘Did the difference change?’ (No.) ‘What new number sentence can we write to represent the difference now?’ (9-6=3.)  Write 9-6=3. Then draw a tape diagram to represent the two rows of cubes. ‘I started with 8 and 5. When I added 1 more to each row, I changed the amounts to 9 and 6, but the difference stayed the same. We can say 8-5=9-6 because they both equal 3.’ Write 8-5=9-6=3.”

• Module 6, Topic C, Lesson 9: Determine the attributes of a square array, Materials, Students: “Color tiles, plastic, 1″ (25).” For Learn, Compose Rectangular Arrays students use tiles to compose arrays. “Direct students to place 12 tiles into 2 equal groups. ‘How many groups are there?’ (2) ‘How many tiles are in each group?’ (6) Direct students to arrange the 2 equal groups into 2 rows with no gaps or overlaps. ‘How many rows are there? (2) ‘How many tiles are in each row?’ (6) Direct students to outline the array to show its shape. ‘What shape is the array?’ (It’s a rectangle.) Invite students to think–pair–share about how they know the array is a rectangle. (I know it’s a rectangle because it has 4 sides. It has 4 right angles. It has 2 opposite pairs of parallel sides.) ‘What repeated addition equation matches the rows?’ (6+6=12) What repeated addition equation matches the columns?’ (2+2+2+2+2+2=12) The array can be represented by two different repeated addition equations depending on whether we think of the rows or columns as groups.) Direct students to rearrange their tiles to show 4 equal groups.”