3rd Grade - Gateway 3

The materials reviewed for Eureka Math² Grade 3 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, 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. 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 1, Topic A, Lesson 5: Represent and solve multiplication word problems by using drawings and equations, Fluency, Choral Response: Relating Multiplication Models, Teacher Note, “Arrays in the sequence of pictures with more than 5 rows are shaded to support students in quickly determining the number of rows in each array without having to count each one.” Learn, Equal Groups Word Problem, Teacher Note, “This is the first use of a context video. It is shown before a related word problem to build familiarity and engagement with the context. It also allows students to visualize and discuss the situation before being asked to interpret it mathematically.”

• Module 2, Topic A, Lesson 7: Solve one-step word problems using metric units, Learn, Solve a One-Step Bigger Unknown Comparison Word Problem UDL: Action & Expression, “Consider providing an opportunity for students to self-monitor their thinking. What helped you make sense of the problem? Which tape diagram helped you better understand the problem? How can you use what you learned in today’s lesson to solve other problems? Can you use what you learned to make your own comparison problem?”

• Module 4, Topic B, Lesson 6: Tile rectangles with squares to make arrays and relate the side lengths to area, Launch, Differentiation: Support, “Students reason about how to find an exact measurement for area. Open and display the Trying to Cover a Rectangle digital interactive. Does the rectangle take up space? Does it have area? We need to find the area of the rectangle. Invite students to think–pair–share about how they might use the circles to find the area of the rectangle. Cover the rectangle with circles, extending some over the edge of the rectangle. Overlap circles to cover all the white space. Does the total area of the circles represent the area of the rectangle? Will using the circles this way help us measure the area of the rectangle? Why or why not? Let’s try using the circles with no overlaps. Demonstrate making an array of circles inside the rectangle. Ask students how many circles are in the array. Invite students to think–pair–share about whether the area of the circles is an accurate representation of the area of the rectangle. How could we use the squares to find the area of the rectangle?”

The materials reviewed for Eureka Math² Grade 3 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 2: Place Value Concept Through Metric Measurement, Module Overview, Why, “Why relate metric units to the place value system? One of the advantages of the metric system of measurement is its base-ten structure. In grade 2, students connect the base-ten system and the metric system for measuring length by using centimeters and meters. Relating place value concepts to measurement provides a natural application to strengthen understanding and highlight connections. The composition and decomposition of 1 thousand as 10 hundreds, 100 tens, and 1,000 ones parallels the composition and decomposition of 1 kilogram as 10 hundred grams, 100 ten grams, and 1,000 grams and 1 liter as 10 hundred milliliters, 100 ten milliliters, and 1,000 milliliters. Tens and hundreds are used to show the progression from 1 to 1,000, but emphasis is placed on creating the new units of 1 liter and 1 kilogram from milliliters and grams.”

• Module 3: Multiplication and Division with Units of 0, 1, 6, 7, 8, and 9, Module Overview, Why, “How do students continue to develop fluency with multiplication facts after module 3? In module 4, students use their multiplication skills and strategies to find the areas of rectangles and rectangular arrays. In modules 4, 5, and 6, fluency activities reinforce multiplication and division concepts and skills through counting the math way, relating multiplication and division, and finding unknown factors. Sprints also provide practice with multiplication and division. A lesson at the end of module 6 includes activities that reinforce fluency. These activities can be implemented at any time after module 3 and can be repeated to aid in developing fact fluency.”

• Module 5: Fraction as Numbers, Module Overview, Why, “Why is distance from zero used to compare fractions, even though in later grades that strategy won’t work for negative numbers?Students conceptually understand a fraction on a number line in two ways: as a position or location on the number line, and as a distance from zero. So, it is natural for them to use the same two ways of thinking when comparing fractions: the fraction located to the right is greater, and the fraction further from zero is greater. The first way of thinking  will continue to work once negative numbers are introduced in grade 6. The second way of thinking will need to be amended because, for example, −5 is further from zero than −3, but −5 is less than −3. When comparing two negative numbers, the second way of thinking is the exact opposite: the number further from zero is less, not greater. This opposite relationship is consistent with the way students will learn about negative numbers themselves—as the (additive) opposites of positive numbers. The introduction of negative numbers in grade 6 will challenge students’ understanding of the number system in general, which is what makes the use of this comparison strategy acceptable for grade 3.” An Image of two number lines is displayed.

The materials reviewed for Eureka Math² Grade 3 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: Multiplication and Division with Units of 2,3, 4, 5, and 10, Description, “In module 1, students relate repeated addition, equal groups, and arrays to multiplication and division. With a focus on units of 2, 3, 4, 5, and 10, students use the commutative and distributive properties as strategies to multiply, and they write expressions with three factors as a foundation of the associative property. Students express division as both unknown factor problems and division equations and break apart and distribute the total to divide. They use their understanding of multiplication and division concepts to reason about and solve one- and two-step word problems.” Achievement Descriptors and Standards are listed for the module in the tab labeled, “Standards."

• Module 2, Topic D, Lesson 20: Add measurements using the standard algorithm to compose larger units once. Achievement Descriptors and Standards, “3.Mod2.AD2 Add and subtract within 1,000 fluently using strategies based on place value, properties of operations, or the relationship between addition and subtraction (3.NBT.A.2).”

• Module 5: Fraction as Numbers, Description, “In module 5, students develop an understanding of fractions as numbers. They partition a whole into equal parts and recognize 1 of a fractional unit as a unit fraction. Students compose non-unit fractions from unit fractions and use visual fraction models and written fractions to represent parts of a whole. Students use fractions to represent numbers equal to, less than, and greater than 1. They compare fractions by using visual fraction models and by reasoning about the size of fractions that have the same numerator or denominator. Students identify equivalent fractions, and they apply fraction concepts by using rulers to measure to the nearest quarter inch and by plotting fractional length data on line plots.” Achievement Descriptors and Standards are listed for the module in the tab labeled, “Standards."

• Module 6, Topic C, Lesson 14: Measure side lengths in whole-number units to determine the perimeters of polygons. Achievement Descriptors and Standards, ”3.Mod6.AD5 Solve real-world and mathematical problems involving perimeters of polygons, Students analyze two multiplication strategies for finding the perimeter of a quadrilateral (3.MD.D.8).”

The materials reviewed for Eureka Math² Grade 3 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 3-5, “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 4: Multiplication and Area, Module Overview, Why, “Why is a trapezoid defined as a quadrilateral with at least 1 pair of parallel sides instead of a quadrilateral with exactly 1 pair of parallel sides? The term trapezoid can have two different meanings. Exclusive definition: A trapezoid is a quadrilateral with exactly 1 pair of parallel sides. Inclusive definition: A trapezoid is a quadrilateral with at least 1 pair of parallel sides. Both definitions are legitimate, and at grade 3 there is not a significant advantage to either. The inclusive definition is chosen primarily for the conveniences it allows at later grades and for consistency with most geometry textbooks for college-bound students. One nice consequence of using the inclusive definition is that it means trapezoids and parallelograms have a similar relationship to the relationship between rectangles and squares: In each case, the latter is always also the former.” Images of a trapezoid and a parallelogram are shown. 

• Module 6: Geometry, Measurement, and Data, Module Overview, Why, “Why do lessons 24 and 25 introduce place value units to 1 million?Lessons 24 and 25 intentionally extend student thinking around place value units in preparation for the major work of grade 4. By organizing, counting, and representing a collection with a total value greater than 1,000 in lesson 24, students study patterns to identify and broaden their understanding of the place value system. Students make sense of relationships and patterns in the place value system as they repeatedly bundle ten smaller units to compose one of the next larger unit, only to realize that much larger units are needed. Lesson 25 is an optional lesson where students repeatedly count and bundle bills to one million. This lesson builds the foundation for students seeing that the value of a digit is 10 times as much as the value of the same digit in the place to its right, a key concept for expanding whole-number relationships to decimals. Consider including lesson 25 to provide students with experience with numbers larger than 1,000 before grade 4.”

The materials reviewed for Eureka Math² Grade 3 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 C, Lesson 12: Demonstrate the distributive property using a unit of 4,  Materials, “Teacher: Interlocking cubes, 1 cm (40). Students: Interlocking cubes, 1 cm (40). Lesson Preparation: Prepare 40 interlocking cubes, 20 in one color and 20 in another color, per student and teacher.”

• Module 2: Place Value Concepts Through Metric Measurement, Module Overview, Materials, “9 2-liter containers, 24 Learn books, 1 Bottle of liquid food coloring, blue, 1 Pad of chart paper, 30 Clear plastic cups, about 150 mL (5 oz), 1 Pad of sticky notes, 1 Container, about 1 liter, 25 Pencils, 1 Container, greater than 1 liter, 9 Permanent markers, 1 Container, less than 1 liter, 25 Personal whiteboards, 1 Demonstration thermometer, 25 Personal whiteboard erasers, 6 Digital compact scale with 5-cup bowl, 9 Plastic pitchers, 1.5 L or larger, 25 Dry-erase markers, 6 Platform scales, 14 Envelopes, 1 Projection device, 25 Eureka Math^® place value disks set, ones to thousands, 1 Resealable bag, gallon size, 1 Eureka Math^® tape measure, 150 cm, 24 Resealable bags, sandwich size, 1 Eureka Math^® whole number place value cards,1 Roll of painter's tape, 1 Graduated cylinder, 100 mL, 1 Syringe, 10 ml, 1 Graduated cylinder, 1,000 mL, 1 Teach book, 11 Index cards, 1 Teacher computer or device, 2,443 Interlocking cubes, 1 cm.”

• Module 4, Topic C, Lesson 12: Find all possible side lengths of rectangles with a given area, Materials, “Teacher: Interlocking cubes, 1 cm (24). Students: Multiply and Divide by 6 Sprint (in the student book), Interlocking cubes, 1 cm (24), Centimeter Grid (in the student book), Chart paper, sheet (1 per student group), Marker set (1 per student group), Scissors (1 per student group), Sticky note (1 per student group). Lesson Preparation: Consider tearing out the Sprint pages in advance of the lesson. Consider whether to remove Centimeter Grid from the student books and place inside personal whiteboards in advance or to have students prepare them during the lesson. Prepare one sheet of chart paper, pair of scissors, marker set, and sticky note per group of three students.”

The materials reviewed for Eureka Math² Grade 3 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: Multiplication and Division with Units of 2, 3, 4, 5, and 10, Topic E Quiz 1, Application of Multiplication and Division Concepts, Items 3 and 4, Item 3, “A bookcase has 6 shelves. There are 5 books on each shelf. How many books are in the bookcase?” Item 4, “Deepa buys a pack of 12 juice boxes. The pack has an equal number of juice boxes in lime, grape, and peach flavors. Deepa drinks all the lime-flavored juice boxes. How many juice boxes does Deepa have left?” Students engage with the full intent of 3.OA.3 (Use multiplication and division within 100 to solve word problems in situations involving equal groups, arrays, and measurement quantities).

• Module 2: Place Value Concepts through Metric Measurement, Module Assessment 1, Items 5 and 7, Item 5, “Add. 328+239=____.” Item 7, “Subtract. 428-169=____.” Students engage with the full intent of 3.NBT.2 (Fluently add and subtract within 1000 using strategies and algorithms based on place value, properties of operations, and/or the relationship between addition and subtraction).

• Module 3: Multiplication and Division with Units of 0, 1, 6, 7, 8, and 9, Topic A, Lesson 3, Count by units of 𝟖 to multiply and divide by using arrays, Land, Exit Ticket, supports the full intent of MP4 (Model with mathematics) as students draw a model to skip-count. “Draw a model and skip-count to find 6\times8. Write a related division equation.”

The materials reviewed for Eureka Math² Grade 3 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 1, Topic D, Lesson 15: Model division as an unknown factor problem, Learn, Represent Measurement Division with a Tape Diagram, “Language Support: Make an anchor chart for the terms sum, difference, product, and quotient to help students associate the terms with the correct operations. Discussion of these terms should include how the parts and the whole are related in the operations. When we add and multiply, the sum and product represent the whole. But when we subtract, the difference is a part of the whole. The quotient represents either the number of groups or the size of the group. UDL: Action & Expression: Consider modeling a think aloud for drawing the tape diagram from problem 2 to guide students through drawing the tape diagram for additional problems. Students may need continued support drawing a tape diagram when the number of groups is unknown. In addition, encourage students to share their thought process by asking them to think aloud and address misconceptions as needed. UDL: Action & Expression: Consider including opportunities for students to self-reflect on their process by displaying the following sentence frames for students to refer to either independently or during partner work: After reading the word problem I ask myself ____. I look for ____. If I get stuck I can ____. It is important to ____.”

• Module 3, Topic C, Lesson 15: Reason about and explain patterns of multiplication and division with units of 1 and 0, Launch, “Differentiation: Support: Assign each set of partners a number with which the students have some proficiency. Partners need to focus on making sense of the situation and noticing the patterns, not struggling to find the products or quotients. Ensure that each number, 2 through 9, is assigned to at least one pair of students. Language Support: The statements throughout this lesson are very similar. Students must correctly interpret each statement and make distinctions between the statements to generalize the patterns. Consider highlighting the parts of the statement in different colors to support students in interpreting the statements and describing the patterns.” Learn, Multiply and Divide with 0, “UDL: Representation: Consider presenting the information in another format. Provide students with 10 cubes to act out the following situations concretely before moving to the problems in the book: Show me 1 group of 0. How many? Show me 2 groups of 0. How many? Show me 0 groups of 1. How many? Show me 0 groups of 9. How many?”

• Module 6, Topic C, Lesson 15: Recognize perimeter as an attribute of shapes and solve problems with unknown measurements, Launch, “Differentiation: Support:Students may need support in understanding why two of the shorter side lengths are not included in the perimeter of the larger rectangle. Consider displaying two index cards, not touching, and the larger rectangle side by side. Ask students to trace the border of each of the index cards and then the border of the larger rectangle. Then ask them to reason why the two smaller sides are not included in the perimeter of the larger rectangle.” Learn, Partner Practice, “Language Support: As partners compare solutions, consider asking students to use the Agree or Disagree section of the Talking Tool to support respectful and productive conversation. UDL: EngagementConsider facilitating personal coping skills and strategies as students participate in the Partner Practice activity. Ask students to recall that if they are struggling to find the perimeter, they can choose a different approach or ask their partner a clarifying question.”

The materials reviewed for Eureka Math² Grade 3 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 D, Lesson 21: Add measurements using the standard algorithm to compose larger units twice, Learn, Renaming Twice to Add, Differentiation: Challenge, “As time allows, challenge students to find the sums by using a different strategy. Experience with multiple strategies will allow for richer discussion.”

• Module 3, Topic B, Lesson 10: Use parentheses in expressions with different operations, Learn, Same Problem, Different Solutions, Differentiation: Challenge, “Ask students to extend their thinking to write equivalent expressions by using parentheses. A student may write (3\times5)−2=13 as (3\times5)−2=(7\times2)−1.”

• Module 6, Topic B, Lesson 8: Compare and classify quadrilaterals, Learn, Decompose Quadrilaterals into Two Triangles, Differentiation: Challenge, “Challenge students to determine what happens to other polygons when a diagonal line is drawn inside of them. Is there a pattern?”

The materials reviewed for Eureka Math² Grade 3 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 2 ,Topic C, Lesson 16: Use compensation to add, Learn, Error Analysis, MLL students are provided the support to participate in grade-level mathematics as described in the Language Support box, “Consider strategically pairing partners with different levels of English proficiency to complete the error analysis. Students may be able to identify the error, but need support with how to explain it or why it might have happened. Designate a student in each pair to write down their explanation of the error to prepare for sharing their thinking with the class.”

• Module 4, Topic, Lesson 10: Compose large rectangles and reason about their areas. Launch, MLL students are provided the support to participate in grade-level mathematics as described in the Teacher Note box, “A context video for this word problem is available. It may be used to remove language or cultural barriers and encourage 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.”

• Module 5, Topic A, Lesson 1: Partition a whole into equal parts and name the fractional unit, Launch, MLL students are provided the support to participate in grade-level mathematics as described in the Language Support box, “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.”

The materials reviewed for Eureka Math² Grade 3 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 A, Lesson 1: Organize, count, and represent a collection of objects, Fluency, Counting on the Rekenrek by Tens, Materials, Teacher: Rekenrek. “Students count by tens in unit and standard form to develop an understanding of multiplication. Show students the rekenrek. Start with all the beads to the right side. Say how many beads there are as I slide them over. The unit is 10. In unit form, we say 1 ten. Say 10 in unit form. Slide the second row of beads all at once to the left side. How many beads are there now? Say it in unit form. Continue sliding over each row of beads all at once as students count. Slide all the beads back to the right side. Now let’s practice counting by tens in standard form. Say how many beads there are as I slide them over. Let’s start at 0. Ready? Slide over each row of beads all at once as students count.”

• Module 2, Topic B, Lesson 9: Round two-digit numbers to the nearest ten on the vertical number line, Whiteboard Exchange: Halfway on the Number Line, “Students identify the number halfway between consecutive units of ten on a number line to prepare for rounding to the nearest 10. Display the vertical number line. Draw the vertical number line. What number is halfway between 0 and 10? Label it on your number line. Give students time to work. When most students are ready, signal for students to show their whiteboards. Provide immediate and specific feedback. If students need to revise, briefly return to validate their corrections.” Learn, Round on a Vertical Number Line, “Students represent measurements on a vertical number line and use the vertical number line to help them round the measurements to the nearest ten. Show a graduated cylinder containing 73 milliliters of water. This graduated cylinder has 73 milliliters of water in it. Let’s use a vertical number line to help us round that measurement. Display a vertical number line. Direct students to the first vertical number line in problem 1 in their books. How many tens are in 73? What is 1 more ten than 7 tens? Our number line needs to show the interval from 7 tens, or 70, to 8 tens, or 80. Invite students to label the number line as you model labeling the lowest tick mark as  70 = 7 tens and the highest tick mark as  80 = 8 tens. What number is halfway between 7 tens and 8 tens? Let’s label the halfway mark. Label the halfway tick mark as  tens and 5 ones. Is 73 more than or less than halfway between the two tens? Watch as I plot and label 73 on the number line. Say “Stop!” when my finger points to where 73 should be. Move your finger up the number line from 70 toward 75. Stop when the students say to stop. Put your pencil where 73 should be on your number line. Label the spot as 73 = 7 tens and 3 ones . Now that we know where 73 is, we can see how to round 73 milliliters to the nearest 10 milliliters. Which ten is 73 closer to? How do you know? So what ten does 73 round to? Display the sentence frame: ___ rounded to the nearest ten is ___. Point to the sentence frame as you say: Complete the sentence. 73 milliliters rounded to the nearest ten milliliters is about how many milliliters?” Image of a vertical number line is shown.

• Module 5, Topic E, Lesson 26: Create a ruler with 𝟏-inch, half-inch, and quarter-inch intervals, Launch, Materials, Teacher: Paper strip. “Students summarize the characteristics of various fraction models. Display the picture of the models students have used to represent fractions. Invite students to think–pair–share about what is important about each model and how the models are alike and different. As students share, consider placing special emphasis on the ruler in preparation for Learn and highlight the following concepts: The ruler’s partitions must be equally spaced. For the model to be understandable, there needs to be enough information, such as the location of whole numbers, labeled. However, it is not necessary to label every mark on the number line or ruler. The wholes must be the same size to be used to compare fractions or find equivalent fractions. Consider inviting students to share what makes some models easier for them to use than others. Show the paper strip. I want to create a ruler with this strip of paper. My ruler needs to precisely measure to the nearest quarter inch. Invite students to think about strategies they have used to partition and suggest ways to create the ruler.” Images of various fraction models are shown.