1st Grade - Gateway 3

The materials reviewed for Eureka Math² Grade 1 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 1, Topic B, Lesson 10: Count on from 5 within a set, Learn, Find 5 and Count On, provides a teacher note with guidance for UDL: Representation. “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.” A picture is shown of a ten frame, number bond, and number list, all representing counting on 2 more from 5.

• Module 3, Topic B, Lesson 9: Make ten with either addend, Learn, Make 10 Bingo, provides a note with guidance for Differentiation: Support. “Students may refer to or use the Add to 6 or 7 removable and the Add to 8 or 9 removable from lesson 8 (inserted in the other side of the whiteboard) for support.”

• Module 5, Topic C, Lesson 12: Decompose an addend to make the next ten, Land, Debrief, provides a Teacher Note with general guidance. “Level 3 strategies such as make the next ten require time and practice to learn. At first students directly model with a drawing or cubes. They progress to independently using number bonds and number sentences. Expect variety in their representations. If students use cubes, encourage them to show what they did with a drawing. When working independently, it is not necessary for students to draw a picture and use number bonds; they may choose one or the other. Some students may choose to use the number path and record their hops with arrows.” 

The materials reviewed for Eureka Math² Grade 1 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 1: Counting, Comparison, and Addition, Topic B: Count on From a Visible Part, Topic Overview, explains the change the focus of instruction to subitizing and being able to count on from any number for addition. “This topic opens by inviting students to discover and verbalize that when finding a total of two parts, such as a box of 6 markers and 3 additional markers, using the Level 2 strategy of counting on from a known part is more efficient than counting all. As topic B progresses, students use the Level 1 strategy of counting all less frequently. They learn to trust the cardinality of a part, seeing it as a unit from which they can count on. They select parts that they ‘just know,’ or subitize. Subitizing a part and counting on the other part to find the total requires practice over time. Students may find it difficult to represent the second part in number bonds or number sentences. For example, students might ask, ‘When we count on 2 more from 5, why do we write 2 when we counted 6 and 7?’ Several representations help students make sense of these part–total relationships: numerical recordings of the count sequence (5, 6, 7), number bonds, number sentences that relate counting on to addition, As they better understand part–total relationships, students count on from both parts and notice that the total remains the same. Students take note of special part–part– total relationships and doubles, and they continue to practice these in fluency activities throughout module 1. Mid-topic, a subtle shift advances student learning. Students conceptually subitize, or isolate one part, within a visible set, such as those represented on dot cards, and count on from that part. Familiar 5+n facts support the shift. Students move toward counting on from any known part to find the total. Their sense of efficiency and flexibility grows as they realize that some parts are more helpful to see and count on from, and that totals can be found in many ways. To prepare for addition expressions, students count on from 10. Because the second part is now shown as a numeral rather than a set of objects, students use fingers to track. Students learn that the term unknown refers to what needs to be figured out, and they use it to describe the total. Although students worked with 10 + n facts in kindergarten, fluency with these facts is essential to their success with Level 3 strategies.” 

• Module 3: Property of Operations to Make Easier Problems, Module Overview, Why, provides a synopsis of how strategies become more complex both in Grade 1 as well as how current strategies will impact learning in later grades. “What are Level 3 strategies for addition and subtraction? Module 3 marks a critical moment in grade 1 when students transition away from finding totals by counting all (Level 1 strategy) or counting on (Level 2 strategy) to finding totals by making an equivalent yet easier problem (Level 3 strategy). The following tables show different ways to make addition or subtraction problems easier. These strategies often rely on knowledge of the properties of operations, such as the commutative and associative properties of addition, and leverage ten as a benchmark number. Level 3 strategies take time and practice before they are truly easier for students than Level 2 strategies. Students learn the following Level 3 strategies, though they may not master them all. After learning them, students may self-select strategies that are most efficient for them personally or that are most appropriate for the problem. It is important for students to internalize the concepts embedded within these strategies to improve their number sense. Ultimately, students who have a command of these concepts use numbers flexibly to solve problems efficiently. These strategies extend to the ranges of numbers that students work with in later grades. In grades 1–3, students apply these strategies for making an easier problem to larger units, such as hundreds and thousands. In grades 4–6, they apply these strategies to smaller units, such as decimals and fractions.”

• Module 5: Place Value Concepts to Compare, Add, and Subtract, Topic E: Addition of Two-Digit Numbers, Topic Overview, describes the reasoning for the importance of providing students with different strategies to solve addition problems. These options will allow students to select a strategy that works best for them. “Now that students have experienced adding two-digit numbers to one-digit numbers and adding a multiple of 10 to a two-digit number, they are ready to add 2 two-digit numbers. Students leverage place value understanding to make problems easier. They use a variety of concrete, pictorial, and abstract tools to model addends as tens and ones. Students record their reasoning by using a written method and then explain their strategy. The goal of topic E is to build number sense that allows students to flexibly manipulate two-digit addends. At first, students self-select ways to combine groups of cubes that represent 2 two-digit numbers. They share how they decomposed each group and combined the resulting parts. Subsequent lessons present the three following ways to add 2 two-digit numbers: Add like units: Decompose both addends into tens and ones, combine tens with tens and ones with ones, and then put tens and ones together. Add tens first: Decompose one addend into tens and ones, combine the tens with the other addend, and then add the ones. Make the next ten: Decompose one addend into tens and ones, combine some (or all) of the ones with the other addend (in many cases to make the next ten), and then add the remaining parts. These three strategies present different ways to add 2 two-digit numbers, primarily to promote flexible thinking; mastering each of the strategies is less important than attaining flexible thinking. When students compare their various recordings, it helps them to identify equivalent expressions that make a problem easier. For example, 35+15 is equivalent to 30+5+10+5, but the second expression makes it easy to add 3 tens +1 ten and 5+5. This type of discussion leads students to the general understanding that different ways of thinking about a problem result in the same total. Using Level 3 strategies, such as those presented in this topic, takes time and practice. Students may self-select the strategies and tools they use to solve the problems, as long as they are able to record and explain their solution pathways.”

The materials reviewed for Eureka Math² Grade 1 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 4: Find the total number of data points and compare categories in a picture graph, Achievement Descriptors and Standards, “1.Mod1.AD8 Compare category totals in graphs by using the symbols >, =, and <. (1.MD.C.4, 1.NBT.B.3)”

• Module 3: Properties of Operations to Make Easier Problems, Achievement Descriptors and Standards, “1.Mod3.AD3 Add within 20 by using strategies such as applying the commutative and associative properties to make 10 or by counting on to 10. (1.OA.B.3, 1.OA.C.6)”

• Module 4: Comparison and Composition of Length Measurements, Description, “In module 4, students explore units within the context of measurement. After comparing lengths indirectly, students iterate length units, such as centimeter cubes and 10-centimeter sticks, to describe and compare lengths.” Achievement Descriptors and Standards are listed for the module in the tab labeled, “Standards.”

• Module 6, Topic B: Composition of Shapes, Description, “Students decompose and compose flat and solid composite shapes in increasingly complex ways: They identify shapes within a composite shape. They name composite shapes using defining attributes. They create composite shapes by combining shapes. Geometric composition is an important concept because it deepens understanding of part–whole relationships in other areas, such as composing 10 ones to make 1 ten, decomposing 8 into 2 and 6, partitioning a whole into halves, or recognizing that a clock is partitioned into hours and minutes. After students compose a shape in a variety of ways, they compare the number of shapes they used. They realize that the smaller the shapes they use to make a composed shape, the more shapes they need to make the composition.” Achievement Descriptors and Standards are listed for the topic in the tab labeled, “Standards.”

The materials reviewed for Eureka Math² Grade 1 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 2: Addition and Subtraction Relationships, Module Overview, Why?, “How does the story of subtraction unfold in this module? Module 2 presents three interpretations of subtraction. In-depth experience is necessary with all three interpretations to avoid the misconception that subtraction always means take away. Take from: Move or remove objects from a set, resulting in a smaller set, Part–whole: Find an unknown part when given a part and the total,Comparison: Compare two different sets and identify the difference between the two, Subtraction can be more challenging than addition because the given (or known) part is embedded within the total. Students represent the total and then isolate the known part to find the unknown part.” 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.”

• Module 6: Attributes of Shapes - Advancing Place Value, Addition, and Subtraction, Module Overview, Why?, “Part 1: Attributes of Shapes: How do attributes of shapes help students name and describe them?Grade 1 students expand their knowledge of defining attributes, or the mathematical characteristics of a shape, to describe flat shapes with increasing precision. They use attributes, such as the number of straight sides and whether the shape has equal-length sides, parallel sides, or square corners, to sort a variety of shapes into different categories. They find that the fewer attributes a shape category has, the more shapes that fit into that category. In contrast, the more attributes a category has, the fewer shapes that fit into that category. Students see that the same shape can have more than one name or fit into more than one category, depending on the attributes they are considering. This concept connects to students’ experience of naming and representing numbers in various ways.” Works cited include, “These word problem types come from the document Grades K–5, Counting and Cardinality & Operations and Algebraic Thinking, one of the Progressions for the Common Core State Standards in Mathematics. 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 grade 2 students work with all subtypes and variants. Students master the types that are not shaded in grade 2.”

The materials reviewed for Eureka Math² Grade 1 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: Addition and Subtraction Relationships, Module Overview, Materials, “Chart paper, pad(1), Computer with internet access(1), Craft sticks(240), Crayons, set of 3 colors(24), Dry-erase markers(25), Eureka Math²™ Addition expression cards, 13 decks(1), Eureka Math²™ Centimeter number paths(24), Eureka Math²™ Hide Zero^® cards, basic student set of 12(2), Eureka Math²™ Hide Zero^® cards, demonstration set(1), Farm animal counters, set of 72(2), Learn books(24), Markers(7), Pencils(25), Pennies(132), Personal whiteboards(24), Personal whiteboard erasers(24), Projection device(1), Sticky notes (pad)(2), Teach book(1), Unifix^® Cubes, set of 1,000(1).”

• Module 5, Topic E, Lesson 22: Decompose both addends and add like units, Overview, Materials, “Teacher: 100-bead rekenrek, Make 50 cards (digital download). Students: Make 50 cards (1 set per student pair, in the student book), Unifix^® Cubes. Lesson Preparation: The Make 50 cards must be torn out of student books and cut apart. Consider whether to prepare these materials in advance or to have students assemble them during the lesson, or to use the ones prepared in lesson 21. Copy or print the Make 50 cards for demonstration, or use the ones prepared in lesson 21.”

• Module 6, Topic F Lesson 26: Make a total in more than one way, Overview, Materials, “Teacher: Chart paper, Centimeter cubes(5), Base 10 rods(6). Students: Rectangles removable (in the student book), Match: Make 65 Recording Sheet (in the student book), Match: Make 65 cards (1 set of 14 cards per student pair, in the student book), Centimeter cubes(5), Base 10 rods(6). Lesson Preparation: The Rectangles removables and the Match: Make 65 Recording Sheets must both be torn out of student books and placed in personal whiteboards. Consider whether to prepare these materials in advance or to have students prepare them during the lesson. The Match: Make 65 cards need to be torn out of student books and cut apart. Prepare this material before the lesson. Consider creating resealable plastic bags with 6 base 10 rods and 5 centimeter cubes for easy distribution during the lesson. Save these for use in the next lesson.”

The materials reviewed for Eureka Math² Grade 1 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, Topic D, Lesson 20: Find all two-part expressions equal to 6., Land, Exit Ticket, students develop the full intent of MP6 (Show attention to precision) as they color three different ways to make six from a row of circles. Then they write a number sentence for each. “Color three ways to make 6. Fill in each number bond. Write each number sentence.”

• Module 3, Topic A: Make Easier Problems with Three Addends, Topic Ticket, Problem 2, supports the full intent of MP2 (Reason abstractly and quantitatively) as students create an equation for a story problem and then solve it. They use symbols and numbers correctly to create the equation. “Read. Lan has 8 red apples. Meg has 2 yellow apples. Ned has 4 green apples. How many apples do they have? Draw. Write. They have ____ apples.”

• Module 4, Topic A, Lesson 2: Reason to order and compare heights, Land, Exit Ticket, supports the full intent of 1.MD.1 (Order three objects by length; compare the lengths of two objects indirectly by using a third object). “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 and write the animals shortest to tallest.” Students see a picture of a pig, cow, and hen. The cow is taller than the pig. The hen is shorter than the pig. Students indicate with a check mark taller or shorter.

• Module 5, Module Assessment, Place Value Concepts to Compare, Add, and Subtract, Problem 1, supports the full intent of 1.NBT.2 (Understand that the two digits of a two-digit number represent amounts of tens and ones). Students see pictures of fish in groups of tens, some ones and an empty number bond. “Circle tens. Write how many tens and ones. Fill in the number bond. ___tens ___ones is ___.” From the scoring guide, the teacher is given the following guidance, “Module 5 Module Assessment, This assessment has six 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. Item number 1, 1.Mod5.AD1, 1.Mod5.AD2, 1.Mod5.AD3, Students can earn 1 point for circling tens in each part. Students can earn up to 3 points for completing the number bond and writing the number of tens and ones in each part.”

• Module 6, Module Assessment, Attributes of Shapes, Advancing Place Value, Addition, and Subtraction, Problem 6, supports the full intent of MP4 (Model with mathematics). Students model their thinking when solving four 2-digit addition problems. “Add. Show how you know. 43+53=, 38+52=, 62+9=, 54+27=.“

The materials reviewed for Eureka Math² Grade 1 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 3, Topic A, Lesson 2: Make ten with three addends, Learn, Lollipops Problem, students solve a word problem with three addends by using the Read–Draw–Write (RDW) process. “Differentiation: Support: If drawing poses a challenge for students, have them represent the problem with cubes on a whiteboard. Encourage students to draw where they see 10. Students may choose to move the parts that make ten next to each other.” 

• Module 4, Topic A, Lesson 1: Compare and order objects by length, Launch, students reason about the comparative terms shorter, taller, shortest, and tallest. “Language Support: Consider helping students understand the suffixes -er and -est by making an anchor chart like the one shown. Explain that -er is used to compare two objects and -est is used when comparing three or more objects.” The Language Support Teacher Note includes a sketch of a possible anchor chart showing pictures and words with suffixes highlighted. 

• Module 5, Topic B, Lesson 9: Compare two quantities and make them equal, Learn, Make It Equal, students compare two quantities and add to the lesser amount to make the totals equal. “UDL: Representation: If coins still prove difficult for some students to work with, consider providing the information in another format. Provide students with manipulatives they can use to represent the values of the dimes and pennies. Consider having students use Unifix cubes to represent dimes by stacking 10 cubes.”

The materials reviewed for Eureka Math² Grade 1 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 3, Topic C, Lesson 12: Represent and compare related situation equations, Part 2, Launch, students discuss the number sentences and solution pathways for a pair of related addition word problems. “Differentiation: Challenge: Extend the work by asking students to find the total number of ants on paths A and B.”

• Module 4, Topic C, Lesson 10: Compare to find how much longer, Learn, How much longer?, students measure to show how many more cubes are needed to measure the longer caterpillar. “Differentiation: Challenge: If some students find the difference mentally, provide a challenge by changing caterpillar B’s length to 5 or 7 centimeters.”

• Module 5, Topic A, Lesson 2: Count a collection and record the total in units of tens and ones, Learn, Share, Compare, Connect, students share and discuss counting collections. “Differentiation: Challenge: At another time, invite pairs who count collections with more than 100 objects to share their work. Facilitate discussion by using the following questions: Do you all agree this recording shows a total of 103 cubes? Why? How many tens and ones do you see? How many is 10 tens 3 ones?”

The materials reviewed for Eureka Math² Grade 1 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 3, Topic A, Lesson 1: Group to make ten when there are three parts, 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 two pairs to form small groups of four. As applicable, complement any of these groupings by pairing students who speak the same native language. As students share, support student-to-student dialogue by inviting the class to agree or disagree, ask a question, give a compliment, make a suggestion, or restate an idea in their own words. Students share ways to organize a set and find the total. Gather students and display the cupcake. Use the Math Chat routine to engage students in mathematical discourse. ‘How many cupcakes are there? How do you know?’ Give students a moment of silent think time to find the total. Prompt students to give a silent signal to indicate that they are finished. Invite students to discuss their thinking with a partner. Circulate and listen as they talk. Have a few students share their thinking. Purposely choose work that allows for rich discussion about connections between strategies. Then facilitate a class discussion. Invite students to share their thinking with the whole group, and record their reasoning. (I saw 4 on top and 4 under that. I know that’s 8. Then I counted on 2 more so I know there are 10. I saw the 6 cupcakes on the bottom first. Then I saw 2 and 2. I know 6 plus 2 is 8 and 8 plus 2 is 10. I saw 4, 4, and 2. I know 4 plus 4 equals 8 and 2 more is 10.) Some students see the cupcakes in two parts. Others see three or more parts. We can combine two, three, or even four parts to find a total. Transition to the next segment by framing the work. ‘Today, we will discover a helpful way to add three parts.’”

• Module 5, Topic B, Lesson 9: Compare two quantities and make them equal, Fluency, Choral Response: 5-Groups to 30 with Pennies and Dimes, MLL students are provided the support to participate in grade-level mathematics as described in the Language Support box. “Later in this lesson, students see circles as representations of coins without the visual cue of the actual coin. Consider supporting students by posting a coin anchor chart to reference as they work. Students recognize the value of a group of coins and tell how many more to make the next ten to prepare for comparing coin combinations. After asking each question, wait until most students raise their hands, and then signal for students to respond. ‘Raise your hand when you know the answer to each question. Wait for my signal to say the answer.’ Display 8 pennies. ‘How many cents?’ (8 cents) ‘How many more cents to make the next ten?’ (2 cents) ‘When I give the signal, say the addition sentence starting with 8 cents.’ (8 cents+2 cents=10) Display the addition sentence and the additional pennies. ‘What can we exchange 10 pennies for?’ (1 dime) Display the 10 pennies exchanged for a dime.”

• Module 6, Topic A, Lesson 2: Sort and name two-dimensional shapes based on attributes, Learn, Parallel Sides, MLL students are provided the support to participate in grade-level mathematics as described in the Language Support box. “To help students internalize the term parallel, have them hold their arms as shown in the photo. Have students say the term. Repeat the process, with students holding their arms vertically. Explain that, like a pair of mittens or a pair of socks, parallel sides come in pairs. Students identify parallel sides of shapes as a defining attribute. Ask students to keep their Square Corners and Parallel Sides removable and then distribute craft sticks. Use the rectangle to model and explain how to test for parallel lines as students follow along. ‘Let’s put our craft sticks along the top and bottom of the rectangle. Those sides are across from each other. (Point to the space between the sticks.) Notice that our sticks do not touch. Imagine that these sticks stretch out very far on both ends. Would they touch then?’ (No.) ‘When two sides that are across from each other never touch, we call them parallel. What do we call sides across from each other that never touch?’ (Parallel) Guide students to test the vertical sides of the rectangle. ‘Are these sides parallel? How do you know?’ (Yes, the sticks do not touch.) ‘A rectangle has 2 pairs of parallel sides.’ (Point to the vertical and horizontal parallel lines.) Have students use their craft sticks to test the square for parallel lines. ‘What do you notice about the parallel sides on a square?’ (A square has 2 pairs of parallel sides, just like a rectangle.) ‘Both squares and rectangles have 4 sides, 4 square corners, and 2 pairs of parallel sides. What is different about these shapes?’ (The rectangle is longer. All 4 sides of the square are the same length, but the sides of the rectangle are not all the same.) Have students test the rhombus, trapezoid, triangles, and hexagons for parallel sides. Discuss the results. Students should find the following: The rhombus has 2 pairs of parallel sides. The trapezoid only has 1 pair of parallel sides. The triangles do not have parallel sides. One hexagon has 3 pairs of parallel sides and the other has 2 pairs.”

The materials reviewed for Eureka Math² Grade 1 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 include: 

• Module 3, Topic E, Lesson 22: Take from ten to subtract from a teen number, Part 2, Materials, Teacher and Student: “Unifix^® Cubes (20.)” For Learn, Take from Ten, students use Unifix Cubes to show and discuss the take from ten strategy. “Distribute sticks of cubes to students. Write 16-7. ‘Show the total, 16, with your cubes.’ Model making ten in one color and 6 in another color. ‘Say 16 the Say Ten way.’ (Ten 6) Point to the ten. ‘Let’s take 7 all at once from ten.’ Model snapping off and setting aside 7 cubes as students do the same. ‘How many do we have left?’ (9)”

• Module 4, Topic C, Lesson 13: Find the unknown shorter length, Materials, Students: “Set of base 10 rods and centimeter cubes (5 base 10 rods and 20 centimeter cubes.)” For Launch students work with partners to build two equal lengths and then take away from one length. “Make sure partners have their set of 10-centimeter sticks and centimeter cubes and one whiteboard (red side up) placed between them. ‘Use cubes to show 10 centimeters. Partner A, make your cubes the top row. Partner B, make your cubes the bottom row. Line up the endpoints.’ Roll the die and show students the result (for example, 7). ‘Both partners add 7 to your length. Class, what length did we build?’ (17 centimeters) ‘Now, only partner B, take 7 centimeters away from your length. What can we say about partner B’s length compared to partner A’s length’ (It is shorter. It is missing the 7 cubes.) ‘Partner B, what is your length? How do you know?’ (10 centimeters. I had 17. Then I took away 7. Now there are 10.) ‘Partner A, how much shorter is your partner’s length? How do you know?’ (It’s 7 centimeters shorter because they took away 7 cubes.)”

• Module 5, Topic C, Lesson 13: Reason about related problems that make the next ten, Materials, Students: “Number Path to 40.” Lesson Preparation Notes, “The Number Path to 40 must be torn out of student books and placed in personal whiteboards. Consider whether to prepare these materials in advance or to have students prepare them during the lesson.” For Fluency, Number Path Hop: Hop to the Next Ten, students represent addition within 40 on the number path in preparation for later learning in the lesson involving a number path to 120. “Make sure students have a personal whiteboard with a Number Path to 40 removable inside. After each prompt for a written response, give students time to work. When most students are ready, signal for them to show their whiteboards. Provide immediate and specific feedback. If students need to revise, briefly return to validate their corrections. Display the expression 18+3 ‘Write the expression 18+3. Circle 18 on your number path.’ Display the number 18 circled. ‘Hop to the next ten on your number path. Label your hop.’ Display the labeled hop. ‘How many more do we need to hop to add 3 altogether?’ (1) ‘Hop 1 more on your number path. Label your hop.’ Display the labeled hop.”