Kindergarten - Gateway 3

The materials reviewed for Eureka Math² Kindergarten 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:

• Kindergarten 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.”

• Kindergarten 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.”

• Kindergarten 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.”

• Kindergarten 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 50-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.”

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 B, Lesson 8: Classify solid shapes based on the ways they can be moved, Launch Roll, Slide, or Stack, provides a teacher note with guidance for Differentiation: Support. “If students make mistakes, prompt them to check their work by asking the following question: Can you show how the ___ rolls, slides, or stacks? The difference between rolling and sliding may need clarification such as the following: Does it roll like a ball, turning as it goes? Or does it move smoothly, like when you go down a slide? You don’t turn around and around as you slide.”

• Module 4, Topic C, Lesson 12: Draw to represent put together with total unknown story problems, Learn, Duck Story, provides a teacher notes with general guidance. Teacher Note, “Some students will mimic writing equations, or number sentences. Resist the urge to correct number sentences as students experiment. Students will learn to write equations in module 5. The intent of teacher modeling in this lesson is to expose students to writing equations and to connect the three models: drawing, number bond, and equation.”

• Module 5, Topic A, Lesson 1: Represent add to with result unknown story problems by using drawings and numbers, Launch, provides a teacher note with guidance for UDL: Engagement. “Students choose the numbers for the math story. This serves as a natural source of engagement and differentiation. Numberless word problems also help students make sense of the story. A focus on the story context allows students to consider important questions. Are cookies being added or being taken away? Are there more or fewer cookies at the end of the story?”

The materials reviewed for Eureka Math² Kindergarten 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 Kindergarten 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 3: Comparison, Topic A: Compare Heights and Lengths, Topic Overview, provides teachers with an understanding of the progression of measuring as students become more precise in their terminology and rationale for a comparison of two or more objects as to their length and/or height. “Students expand their understanding of size by focusing on height and length as measurable attributes. Instead of using general terms such as bigger and smaller to describe objects, they learn to accurately compare the lengths of two objects and use specific terms, such as taller, longer, and shorter. Young children have many experiences with height and length before entering kindergarten. Visiting the doctor, building with blocks, getting new shoes, and cutting a piece of tape all involve length. Kindergarten students need support in two key areas: Developing measurement behaviors that lead to accurate comparisons, and using specific terminology to describe height and length comparisons. Students expand their understanding of size by focusing on height and length as measurable attributes. Instead of using general terms such as bigger and smaller to describe objects, they learn to accurately compare the lengths of two objects and use specific terms, such as taller, longer, and shorter. Young children have many experiences with height and length before entering kindergarten. Visiting the doctor, building with blocks, getting new shoes, and cutting a piece of tape all involve length. Kindergarten students need support in two key areas: Developing measurement behaviors that lead to accurate comparisons, and using specific terminology to describe height and length comparisons. Employing accurate measurement practices is important when two objects are close in length. Most students know whether a bookshelf or a crayon is taller upon sight. But to compare a crayon and a toy car, they need to bring the objects close together and align the endpoints to make a fair comparison. They also use these measurement behaviors when they want to create an object that is about the same length as something else. This work prepares students to measure with centimeter cubes in grade 1 and with rulers in grade 2. Using specific terminology and making complete comparison statements requires practice for many kindergarten students. Comparison statements are long: ‘The clipboard is about the same length as the shoe.’ Students must also attend to where each object falls in the sentence because the comparison word describes the first object named: ‘The shoe is longer than the pencil.’ If the objects are reversed, the sentence is no longer true: ‘The pencil is longer than the shoe.’ Labeling with comparison cards helps them form a complete sentence. In general, they find it easier to make statements about what is longer. In the latter half of the topic, students associate number and length by using cube sticks in their comparisons: ‘The sticky note is about the same length as a 4-stick.’ This relates to the nonstandard measurement students do in grade 1. Because cube sticks are made of the same units, they start to see that sticks with more cubes are longer. They explore this relationship more in topic C.”

• Module 4: Composition and Decomposition, Module Overview, Why, provides teachers with a rationale as to the importance for connecting geometry to the use of numbers to better understand value. “Why does this module combine geometry and number? Research suggests that experience with shape composition and decomposition corresponds with a student’s ability to compose and decompose numbers.¹ If students first explore the nature of composition and decomposition in a visual context, as with shapes, they can apply that understanding to new contexts such as numbers. Students benefit from concrete and pictorial experiences with both shape and number decomposition. Staples of early childhood classrooms such as unit blocks and pattern block puzzles provide playful experiences with composition and decomposition that make entry points for discussing part–whole relationships in shapes. Sorting and other familiar hands-on experiences give context for discussing part–total relationships in numbers. The study of shapes and numbers is linked by the language used to describe part–whole relationships. First students consider their everyday experiences with part–whole relationships: That is part of the whole sandwich. They use familiar language to describe composite shapes: The triangle is part of the whole square. Then they learn to use part and total as mathematical terminology when they explore relationships between numbers: 3 and 3 are the parts. 6 is the total. In both shape and number contexts, students find that there are multiple ways to decompose the whole or total.”

• Module 5: Addition and Subtraction, Module Overview, Why, “What are the levels of development as students learn to solve addition and subtraction problems? In their first years of school, students generally move through three levels of development as they solve addition and subtraction problems. Level 1: Count all; Level 2: Count on; Level 3: Make a simpler addition or subtraction problem; Many students rely on direct modeling to count all throughout the kindergarten year. To add, they represent the parts by using objects or drawings and then count all to find the total. To subtract, they first count out the total, then count to take away the known part, and finally count the remaining part. Kindergarten students often spend the full year at Level 1 because they are developing conceptual understanding of what it means to add and subtract. They are learning many different ways to represent those actions, including using concrete objects, drawings, mental images, and number sentences. They are also learning which situations call for each operation. As students build conceptual understanding of addition and subtraction through counting all, they increasingly see that parts are embedded in the total. This is foundational for counting on to add or subtract. Some students begin to use counting on to solve addition problems in kindergarten. Module 5 includes teacher notes and lessons to support these students. The lessons include examples of student strategies from Levels 1 and 2, including questions to advance student thinking from one level to the next. Comparing and connecting different student work can help them make sense of more sophisticated strategies and relate them to their own thinking. Students spend much of their first and second grade years in the third developmental level, using what they know to make simpler problems. Once they acquire several strategies, students reason about which strategy best fits the problem they are solving. The goal is to empower them to continue developing number sense and flexibility in problem solving. What is the associative property and how do kindergarten students understand it? The associative property of addition says that in an addition equation, we can choose to start by adding any two numbers that are next to each other, rather than working left to right. For example, to find 3+4+6, we can first add 4+6 to make 10, resulting in the simpler problem 3+10=13. Put more formally, the associative property states that for any numbers a, b, and c, (a+b)+c=a+(b+c). As with the commutative property, students’ early understanding of the associative property develops from their work with part–total relationships and builds on their understanding of conservation. For example, students are presented with a picture of lollipops and asked to find the total. Some students will count the lollipops from left to right. Others may notice that they can use the doubles fact 3+3=6 if they start with the lollipops on the right and then add the 2 lollipops on the left. In grade 1 students use the associative property, particularly when practicing the make 10 strategy. For example, when presented with 5+7, students may decompose 5 into 2+3, resulting in a new problem: 2+3+7. Then students add 3 and 7 first, making use of the associative property to create the simpler problem 2+10… Which word problem types, or addition and subtraction situations, must be mastered in kindergarten? See the table for explanations and examples of some problem types. 1 Darker shading in the table indicates the four kindergarten problem subtypes. Students in grades 1 and 2 work with all problem subtypes and variants. Grade 2 students master the unshaded problem types. (Image of Problem Types) Students solve all four of the kindergarten problem subtypes in module 5. Add to with result unknown: Both parts are given. An action joins the parts to form the total. Auntie had 3 apples at home. Then she went to the store and bought 5 apples. How many apples does she have now? (Module 5, Lesson 1) Take from with result unknown: The total and one part are given. An action takes away one part from the total. I bought 9 oranges. I ate 5 oranges. How many oranges do I have now? (Module 5, Lesson 9) Put together with total unknown: Both parts are given. No action joins or separates the parts. Instead, the parts are distinguished by an attribute such as type, color, size, or location. There are 6 baby ducks and 1 adult duck. How many ducks are there? (Module 4, Lesson 12) Take apart with both addends unknown: Only the total is given. Students take apart the total to find both parts. This situation is the most open ended because the parts can be any combination of numbers that make the total. There are 8 meerkats moving to a new zoo. Two trucks drive them to their new home. How could the zookeeper put the meerkats in the trucks? (Module 4, Lesson 15)  What are numberless word problems? Why are they used in kindergarten? Numberless word problems are math stories told without numbers. For example: I bake some sugar cookies. My friend brings over some chocolate chip cookies. These problems are used in two different ways in module 5. The first lesson opens with the cookies problem. Students visualize the story in their minds and then make a math drawing to show what they see. They choose the numbers for the math story. One student may see 3 of each type of cookie while another sees 8 sugar cookies and 3 chocolate chip cookies. Numberless word problems build in choice, validate emerging visualization skills, and naturally create engagement and differentiation. Once students have more experience using addition and subtraction to solve problems, numberless word problems serve a new purpose. They cause students to analyze action and relationships, which provides a scaffold as students make sense of story problems. Consider the following numberless word problem: Some students are reading in the library. Some of those students go to the computer center. The class can discuss whether students are coming or going from the library and whether there are more or fewer students in the library. They consider the relationship between quantities before they know the exact numbers. Once students make sense of the situation, numbers are inserted and they choose a solution path to solve the problem. There are 5 students. I can take away 1, 2, 3, 4, or 5 fingers, but I can’t take away 6 fingers.Sometimes problems are presented with only one number given: 5 students are reading in the library. Some of those students go to the computer center. Students can model or visualize to figure out which numbers make sense in the story. They might use 5 fingers to show the students in the library and reason that, at most, 5 students can leave because that’s how many fingers are showing. Numberless word problems focus students on reasoning about and understanding the context and relationships between quantities before they select an operation or solution strategy.”

The materials reviewed for Eureka Math² Kindergarten 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: Classify objects into two categories, Achievement Descriptors and Standards, “K.Mod1.AD10 Sort objects into categories. (K.MD.B.3)”

• Module 2, Topic B: Analyze and Name Three-Dimensional Shapes, Description, “Students continue to focus on defining attributes and extend the list to include features of three- dimensional, or solid, shapes: faces and edges. Through sorting, they discover that some attributes are common to both flat and solid shapes, such as corners. Their spatial thinking evolves as they consider how geometric attributes affect the way a solid shape can be moved or the type of imprint it leaves.” Achievement Descriptors and Standards are listed for the topic in the tab labeled, “Standards.”

• Module 4: Composition and Decomposition, Achievement Descriptors and Standards, “K.Mod4.AD1 Represent composition or decomposition of numbers with objects, fingers, mental images, drawings, sounds, acting out situations, verbal explanations, or number bonds. (K.OA.A.1)”

• Module 6: Place Value Foundations, Description, “Students compose and decompose numbers 11 to 20 as 10 ones and some more ones in various contexts. As they count to 100 by tens and ones, students explore patterns in the number sequence. This prepares them for continued work with the base ten number system.” Achievement Descriptors and Standards are listed for the module in the tab labeled, “Standards.”

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

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

• Kindergarten 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.”

• Kindergarten 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 Kindergarten, “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: Counting and Cardinality, Module Overview, Why, “What is the number core? How is it tied to counting? In this module, children have sustained interaction with four core ideas for describing the number of objects in a group. These ideas are collectively referred to as the number core. The number word list—Students say numbers in the appropriate count sequence (1, 2, 3, …). One-to-one correspondence—When counting, students pair one object with one number word, being careful not to count any objects twice or skip any objects. Cardinality—Students say a number to tell how many are in a group. They may be able to tell how many by subitizing, counting, or matching to a group of known quantity. When counting, students recognize that the last number said represents the number of objects in the group. Written numerals—Students read and write the symbols used to represent numbers. They also connect the written numeral with the number of objects in a set. Students must integrate all aspects of the number core to count and use numbers fluently. The majority of kindergarten activities should involve three or more elements of the number core in conjunction. The number core components are not learned in isolation. The number core plays a foundational role in work with number relations, operations, and place value understanding and is thus a critical start to the kindergarten year.” Works Cited include, “Carpenter, Thomas P., Young Children’s Mathematics, p. 26.”

• Module 5: Addition and Subtraction, Module Overview, Why?, “Why is it important for students to interpret number sentences in different ways? In module 4 students describe the relationships between numbers by using everyday language: and, make, take away, and is. Everyday language precedes academic language because experiences of making things and taking away are relatable to young students. Statements such as 10 take away 3 is 7 align neatly with the numbers and symbols in an equation, creating a smooth transition to the mathematical terminology of plus, minus, and equals: 10 minus 3 equals 7. In module 5 reading number sentences using everyday and mathematical terminology helps students make sense of how numbers and symbols work together in a number sentence. Another way that students read number sentences is called reading like a storyteller: The baker made 10 muffins. He sold 3 of them. There are 7 muffins left. By using story language after solving, students move from computation back to context. Rather than saying, ‘the answer is 7,’ they can more specifically say, ‘there are 7 muffins left.’ Recontextualizing the entire number sentence as a story shows that students understand the meaning of each quantity, as well as how the actions or relationships correlate to the symbols. Saying the number sentence by using mathematical and story language prepares students for the Read–Draw–Write (RDW) process. Beginning in grade 1, students write both a number sentence and a statement in the last step of the RDW process.” Works Cited include, “Common Core Standards Writing Team. Progressions for the Common Core State Standards in Mathematics, Operations and Algebraic Thinking Progression.”

The materials reviewed for Eureka Math² Kindergarten 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: Counting and Cardinality, Module Overview, Materials, "Carrots(3), Pencils(25), Chart paper, tablet(1), Personal whiteboards(25), Counting collections(25), Personal whiteboard erasers(25), Crayon sets(25), Projection Device(1), Cups(25), Puppet or stuffed animal(1), Dry-erase markers(25), Set of number gloves, left and right(1), Eureka Math²™ 5-Group™ cards(1), demonstration set1), Small resealable plastic bags(48), Eureka Math²™ Bingo boards(25), Sorting bags(25), Eureka Math²™ Hide Zero® cards, basic student set 12(24), Teach book(1), Eureka Math²™ Hide Zero® cards, demonstration set(1), Teacher computer or device(1), Eureka Math²™ Match cards, set of 12(12), Teddy bear counters, set of 96(3), Eureka Math²™ Numeral Cards(12), Two-color beans, red and white(275), Learn books(24), Unifix® Cubes, set of 1,000(1), Pad of sticky notes(1), White paper, ream(1), Paper plates(96), Please see lesson 6 for a list of organizational tools (cups, rubber bands, graph paper, etc.) suggested for counting collections.”

• Module 4, Topic C, Lesson 12: Draw to represent put together with total unknown story problems, Overview, Materials, “Teacher: 5-group™ cards, demonstration set, Personal whiteboard, Personal whiteboard eraser, Dry-erase marker, Puppet. Students: Personal whiteboard, Personal whiteboard eraser, Dry-erase marker, Student book. Lesson Preparation: None.”

• Module 5, Topic D, Lesson 25: Extend Growing Patterns, Overview, Materials, “Teacher: Plastic pattern blocks (10). Students: Circle Groups of 3 (in the student book), Quilt (in the student book), Plastic pattern blocks (10). Lesson Preparation: Consider tearing out the quilt removable and distributing it to give students a closer look at the patterns. Assemble resealable plastic bags with 10 green triangle pattern blocks per pair or triad of students plus one additional bag of 10 blocks for demonstration.”

The materials reviewed for Eureka Math² Kindergarten 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 Kindergarten Implementation Guide, “The assessment system in kindergarten helps you understand student learning by generating data from many perspectives. The system includes a recording sheet to guide your observations during lessons and Module Assessments. In kindergarten, every module has an Observational Assessment Recording Sheet. This sheet lists the module’s Achievement Descriptors, or ADs.” These formal 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 from Module Assessments include:

• Module 1, Module Assessment, Counting and Cardinality supports the full intent of K.CC.A (Know number names and the count sequence) and K.CC.B (Count to tell the number of objects). For example, “1. Give the student the bag of writing utensils. Place the number path in front of the student. ‘You can sort these any way you want.’ If needed, prompt students to sort by size. Point to the smallest group from the sort. ‘How many are in this group?’ Point to the number that tells how many. ‘How many cubes are in the group?’ If the student says none, ask for the number that shows none (0). 2. Show the picture of the flower. ‘Count the petals. Put 1 cube on each petal as you count. How many cubes are there?’ Point to the number that tells how many cubes. Scatter the cubes. ‘How many cubes are there?’ 3. Give the student the bag of 10 objects. Hold up the Hide Zero 7 card. (Hold up the 7 card.) ‘Count out this many. If you get 1 more, how many will there be? Point to the number that shows 1 more than 7.’ 4. Remove the number path. Place the numeral writing page in front of the student. ‘Write the numbers 1 through 10 in order.’ Scoring and Grading You may find it useful to score Module Assessments. Consider using the following guidelines. Give 1 point when the student shows evidence of being not yet proficient, 2 points when the student shows evidence of being partially proficient, 3 points when the student shows evidence of being proficient, and 4 points when the student shows evidence of being highly proficient. As needed, look at the ADs and proficiency indicators for examples of the type of work that corresponds to each level of proficiency. If possible, work with grade-level colleagues to standardize the number of points different types of responses earn. In conjunction with the recording sheet you completed for each student, use these scores to grade students’ overall proficiency.”  

• Module 2, Module Assessment, Two- and Three-Dimensional Shapes supports the full intent of MP7 (Look for and make use of structure) name and identify shapes regardless of their orientation or overall size. “2. Clear the work mat and remove all the shapes. Place the set of shapes shown in front of the student. Put all the triangles on the mat. Point to one of the triangles on the mat. Why is this a triangle? (Point.) Point to the rectangle. Why is this not a triangle? (Point.) Point to the open ‘triangle.’ Why is this not a triangle? (Point.) Teacher note: If students describe examples and nonexamples by using defining attributes correctly but missort a few shapes, use the defining attributes they used and ask them to look at all the shapes again and make changes if needed.”

• Module 4, Module Assessment, Composition and Decomposition, supports the full intent of K.OA.A (Understand addition as putting together and adding to, and understand subtraction as taking apart and taking from). For example, “1. Place cubes, marker, and number bond in front of the student. Show the bird scene. ‘Look at the birds. What parts do you see? Fill in the number bond to match.’ Teacher note: Students may use cubes, pictures, or numbers to complete the number bond. Point to a part in the number bond that the student has filled in. ‘What does this tell us about?’ (Point.) Teacher note: Listen for students to describe the reasoning behind their sort. ‘Birds’ is not descriptive enough. Elicit the attributes of the part. 2. The student clears or erases their number bond. ‘Look at the birds to find different parts. Fill in the number bond to match.’ Point to the total in the number bond. ‘What does this tell us about?’ (Point.) Teacher note: If a student shows the same parts in a different way (1 and 5, then 5 and 1), the student should be given credit for showing two ways. 3. Place cubes, marker, and number bond in front of the student. ‘Listen to my story problem. You can use any math tools you want. 5 brown dogs are at the park. 2 white dogs are at the park. How many dogs are at the park?’ After the student has solved the story problem, ask a follow-up question. ‘Can you say a number statement for the story?’ Teacher note: If the student provides an answer without using any math tools, ask them to tell how they know. ‘5 and 2 make 7’ and ‘7 is 5 and 2’ are both correct number sentences for this story. 4. Place the square puzzle and 5 puzzle pieces in front of the student. ‘Fill in the whole square with all the smaller parts. How many parts did you use? Write it.’ Teacher note: If a student is not on the path to success after 3 minutes or becomes discouraged, place 2 triangles together to form a rectangle on top of the puzzle. See if the student can complete the puzzle by using the remaining 3 pieces. You may find it useful to score Module Assessments. Consider using the following guidelines. Give 1 point when the student shows evidence of being not yet proficient, 2 points when the student shows evidence of being partially proficient, 3 points when the student shows evidence of being proficient, and 4 points when the student shows evidence of being highly proficient. As needed, look at the ADs and proficiency indicators for examples of the type of work that corresponds to each level of proficiency. If possible, work with grade-level colleagues to standardize the number of points different types of responses earn. In conjunction with the recording sheet you completed for each student, use these scores to grade students’ overall proficiency.”

• Module 6, Module Assessment, Place Value Foundations, supports the full intent of MP4 (Model with mathematics) as students model a situation with an appropriate representation/ strategy. “3. Place the bird picture in front of the student. Then tell this story: ‘There were 8 blue birds flying and 10 pigeons walking on the ground. How many birds are there?’ Write a number sentence that tells about all the birds. Prompt students to write a number sentence to show their thinking and explain it. Point to different parts in their number sentence and use the following questions to check for understanding. Which birds does this number tell about? Where is the total number of birds in your number sentence? Where are the parts?”

The materials reviewed for Eureka Math² Kindergarten 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 Kindergarten Implementation Guide, Page 41, “Universal Design for Learning (UDL) is a framework based on current research from cognitive neuroscience that recognizes learner variance as the norm rather than the exception. The guiding principles of the UDL framework are based on the three primary networks of the brain. Although the concept of UDL has roots in special education, UDL is for all students. When instruction is designed to meet the needs of the widest range of learners, all students benefit. Eureka Math² lessons are designed with these principles in mind. Lessons throughout the curriculum provide additional suggestions for Engagement, Representation, and Action & Expression.” Examples of supports for special populations include: 

• Module 2, Topic B, Lesson 8: Classify solid shapes based on the ways they can be moved, Fluency, Show Me Shapes, students develop fluency with analyzing and identifying three- dimensional shapes. “Language Support: The names of solid shapes can be difficult to master because some are tricky to pronounce and are not often heard or used in everyday speech. To promote command of the new terminology, consider delivering the fluency Show Me Shapes as a musical fluency, inviting students to hold up the corresponding shape when they hear it in a song. Choose from the many online options suitable for kindergarten learners. While it is well known that songs aid in memorization, they also lead students to incorporate new vocabulary into their productive language. When they hear a catchy song again and again in their mind, they have the opportunity to internally rehearse the new vocabulary.” 

• Module 3, Topic A, Lesson 5: Compare the lengths of two cube sticks, Learn, Record comparisons, students record whether their cube stick is longer, shorter, or the same as their partner’s cube stick. “UDL: Action & Expression: Drawing dots before coloring supports students in planning and remembering. Kindergarten students are accustomed to coloring a picture in its entirety. The dots remind them where to stop coloring.”

• Module 3, Topic C, Lesson 12: Relate more and fewer to length, Fluency, Beep Counting, students determine the missing number in a sequence to prepare for comparison. “Differentiation: Support: Provide a number path for students who need more support with the count sequence. Students can touch each number on their number path as you count.”

The materials reviewed for Eureka Math² Kindergarten 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 A, Lesson 3: Classify objects into two categories and count, Launch, students sort by considering attributes. “Differentiation: Challenge. Challenge students by asking them to consider other ways that the items in the bag are the same and different. In the sample shown here, most of the objects have a similar shape (long and stick-like) and can be held in the hand.”

• Module 2, Topic C, Lesson 15: Compose solid shapes to create a structure that can fit a toy inside, Learn, Pet Houses, students create a house that an animal can fit inside. “Differentiation: Challenge: Challenge students to design a multistory structure, using cardboard to serve as ceilings between each level. Invite them to use their imagination. Perhaps other pets are housed there, as in an apartment building.”

• Module 3, Topic C, Lesson 17: Count and compare sets in pictures, Learn, Recreate a Context, students use math tools to recreate the context of a video they have watched. “Differentiation: Challenge: If students easily make the comparison, then use any combination of the following suggestions to extend the activity. Find the total: How many birds altogether? Find the difference: How many more blue birds than red birds? How many fewer red birds than blue birds? ‘If–then’ scenarios: If 5 red birds fly away, then how many red birds are there? If 10 yellow birds join, then what is the new total number of birds?”

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

• Module 1, Topic D, Lesson 15: Sort the same group of objects in more than one way and count, Launch, MLL students are provided the support to participate in grade-level mathematics as described in the Language Support box. “Support language development by pointing to the bears when using words that students need to describe attributes. Do this when revoicing student ideas about how to sort. For example: This bear is blue. (Point.) This bear is green. (Point.) They are different colors. This bear is big. (Point.) This bear is small. (Point.) They are different sizes. Display the picture of 5 bears on a plate. ‘What do you notice about the bears?’ (They are blue and green. There are 5 bears on a plate. There are big bears and small bears.Invite students to think–pair–share about the following question.) ‘Think in your head: How could we sort these bears? Tell your partner. Start like this: We could sort the bears by …’ Select a few students to share their ideas, making sure that size and color are mentioned. Transition to the next segment by framing the work. ‘I wonder if the way I sort will change the number of things. Today, we will try different ways to sort and see what happens.’”

• Module 4, Topic A, Lesson 2: Decompose flat shapes and count the parts, Launch, MLL students are provided the support to participate in grade-level mathematics as described in the Language Support box. “Invite students to outline the shapes if they are still learning the language needed to describe them. Use this as an opportunity to build vocabulary related to shape, color, size, and position. Min found two gray rectangles next to each other. They make a bigger rectangle. Students study a piece of artwork and locate embedded shapes. Display the Mondrian teacher interactive featuring Piet Mondrian’s Composition with Large Red Plane, Yellow, Black, Gray, and Blue (1921). Use the painting to begin a discussion ‘This painting is by an artist named Piet Mondrian. He often used shapes to create his artwork. What shapes can you find? When you find a shape, trace it with your finger in the air.’ Allow time for students to study the art, and then invite them to share by using precise vocabulary to describe shapes, colors, and sizes. ‘Rosey found a big red square. Can you find it too? Show me.’ Invite students to trace the shape in the air as you use the interactive to show it on the painting. ‘Did anyone find a shape that is made of 2 parts?’ Continue outlining the shapes that students find, encouraging them to find shapes made of 3, 4, or 5 parts. Transition to the next segment by framing the work. ‘Today, let’s take apart shapes to see what shapes, or parts, are hidden inside.’”

• Module 6, Topic C, Lesson 14: Count by Tens, Launch, MLL students are provided the support to participate in grade-level mathematics as described in the Language Support box. “As students share, revoice their responses by using precise terminology such as digit, number, or tens. For example, if a student says, ‘The numbers in the top row go in order but the second row skips numbers,’ point to the relevant part of the chart and revoice as ‘Yes, the numbers in the top row don’t skip any number. In the bottom row we are skip-counting by tens.’ Students chorally count by ones to 10 and by tens to 100. Post a sheet of chart paper in landscape orientation. Invite students to chorally count by ones starting at 1. Guide the class to count with one unified voice. Encourage students to watch the marker carefully, without counting too quickly or slowly, as you record the count. Record up to 10 in the first row, leaving ample space around each number to record patterns and connections that students notice. Alert the class that the count pattern is about to change. Invite students to chorally count by tens starting at 10. On the left side of the paper, begin a second row with 10. Continue to record the count up to 100. Consider recounting the bottom row the Say Ten way. Invite students to share what they notice about the two counts. Use any combination of the following questions to facilitate discussion and elicit student observations: What do you notice? What changes in the count? What stays the same? What is the same and different between the two rows? As needed, give students the opportunity to come up to the chart and point to help explain what they see. Use different-color markers to record patterns and connections students notice. Each class chart will be unique based on student responses.”

The materials reviewed for Eureka Math² Grade Kindergarten 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 Kindergarten 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 1, Topic A, Lesson 5: Classify objects into three categories, count, and match to a numeral, Materials, Teacher: Unifix^® Cubes. For Fluency, Whisper-Shout Counting the Unifix^® Cubes are used to tell the number of objects with a focus on the last number name said to develop an understanding of cardinality. “Display a stick of 3 Unifix Cubes. Using a dry-erase marker, make a dot on the last cube.”

• Module 2, Topic B, Lesson 9: Match solid shapes to their two-dimensional faces, Materials, Students: bag of geometric solids. For Fluency, Show-Me Shapes, students identify a solid shape to develop fluency with analyzing and identifying three-dimensional shapes. “Spread out your shapes so you can see them all. “‘Find the sphere. Hold your shape close to keep it a secret. Stand up.’ Wait until most students stand up with the shape. ‘Show me the shape.’ (Holds up the sphere).”

• Module 3, Topic B, Lesson 9: Use a balance scale to compare an object to a group of cubes, Materials, Students: “School rocker scale (1 per student group.)” For Launch, students physically experience the concept of balance. “Display the balance scale. ‘Let’s try something. Stand still. Put your arms out. Whatever I do with this math tool, you do with your arms. Ready?’ Use a finger to tilt the scale completely to one side. Students move their arms to mimic the action of the balance scale. Anticipate that students will wobble or even fall over. This is part of the learning. Repeat with the other side, and the center, a few more times. ‘Where were your arms when your body stood up straight? Show me.’ (Stretches out arms in a T-shape) ‘Holding our arms out like this helps us keep our balance.’ Hold arms in a T-shape. Transition to the next segment by framing the work. ‘Today, we will put different objects on a scale and try to balance the sides.’”