Kindergarten - Gateway 3

The materials reviewed for Imagine Learning Illustrative Mathematics 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. Examples include:

• IM Curriculum, Why is the curriculum designed this way?, Design Principles, Coherent Progression, “To support students in making connections to prior understandings and upcoming grade-level work, it is important for teachers to understand the progressions in the materials. Grade level, unit, lesson, and activity narratives describe decisions about the organization of mathematical ideas, connections to prior and upcoming grade-level work, and the purpose of each lesson and activity. When appropriate, the narratives explain whether a decision about the scope and sequence is required by the standards or a choice made by the authors. The basic architecture of the materials supports all learners through a coherent progression of the mathematics based both on the standards and on research-based learning trajectories. Each activity and lesson is part of a mathematical story that spans units and grade levels. This coherence allows students to view mathematics as a connected set of ideas that makes sense. Each unit, lesson, and activity has the same overarching design structure: the learning begins with an invitation to the mathematics, is followed by a deep study of concepts and procedures, and concludes with an opportunity to consolidate understanding of mathematical ideas. The invitation to the mathematics is particularly important because it offers students access to the mathematics. It builds on prior knowledge and encourages students to use their own language to make sense of ideas before formal language is introduced, both of which are consistent with the principles of Universal Design for Learning.”

• IM Curriculum, Scope and sequence information, provides an overview of content and expectations for the units. “The big ideas in kindergarten include: representing and comparing whole numbers, initially with sets of objects; understanding and applying addition and subtraction; and describing shapes and space. More time in kindergarten is devoted to numbers than to other topics.”

• Unit 3, Flat Shapes All Around Us, Section B, Making Shapes, Section Overview, Section Narrative, provides an overview of the content and expectations for the section. For example, “In this section, students develop spatial reasoning by manipulating shapes and solving geometric puzzles while using geometric language from earlier work. Students use pattern blocks to compose geometric figures, explore shapes in different orientations, find shapes that match exactly, and complete puzzles that require reorienting shapes. Throughout the section, students use their own language to describe how the shapes they are working with are alike and different, including descriptions of the side lengths of shapes in their comparison.”

Materials include sufficient and useful annotations and suggestions that are presented within the context of the specific learning objectives. Several components focus specifically on the content of the lesson. Preparation and Lesson Narratives, Warm-up, Activities, and Cool-down Narratives all provide useful annotations. IM Curriculum, Why is the curriculum designed this way?, Design Principles, Coherent Progressions, “Each unit starts with an invitation to the mathematics. The first few lessons provide an accessible entry point for all students and offer teachers the opportunity to observe students’ prior understandings. Each lesson starts with a warm-up to activate prior knowledge and set up the work of the day. This is followed by instructional activities in which students are introduced to new concepts, procedures, contexts, or representations, or make connections between them. The lesson ends with a synthesis to consolidate understanding and make the learning goals of the lesson explicit, followed by a cool-down to apply what was learned. Each activity starts with a launch that gives all students access to the task. This is followed by independent work time that allows them to grapple with problems individually before working in small groups. The activity ends with a synthesis to ensure students have an opportunity to consolidate their learning by making connections between their work and the mathematical goals. In each of the activities, care has been taken to choose contexts and numbers that support the coherent sequence of learning goals in the lesson.” Examples include:

• Unit 2, Numbers 1–10, Lesson 2, Activity 1, teachers are provided context as they help students recognize the arrangement of groups does not change the number in each group. Narrative, “Students grab a handful of connecting cubes and count to see how many they have. They then rearrange the connecting cubes using a 5-frame and discover that although the connecting cubes are arranged differently, the number of connecting cubes stays the same. This understanding develops over time with repeated experience working with quantities in many different arrangements. Students may continue to recount the objects in this and future lessons until they understand and are confident that the number of objects remains the same when they are rearranged.” Launch, “Groups of 2. Give each group of students connecting cubes. ‘We are going to play a game with our connecting cubes and 5-frame. One person will grab a handful of connecting cubes and figure out and tell their partner how many there are. Then the other partner will organize the connecting cubes using the 5-frame, and figure out and tell their partner how many there are. Take turns playing with your partner.’” Activity, “5 minutes: partner work time. Monitor for students who notice that the number of objects is the same after they are rearranged.”

• Unit 5, Composing and Decomposing Numbers to 10, Lesson 4, Find All the Ways, Warm-up, Lesson Plan, Teaching Notes provide information to the teacher for teaching specific parts of the lesson. For example, “Pacing: 10 minutes for warm-up activity and synthesis; About the warm-up: Warm-ups help students get ready for the day's lesson, or give students an opportunity to strengthen their number sense or procedural fluency. Activity Narrative: The purpose of this warm-up is to count on from a given number. As students count, point to the numbers posted so that students can follow along.”

The materials reviewed for Imagine Learning Illustrative Mathematics Kindergarten meet expectations for containing adult-level explanations and examples of the more complex grade-level concepts and concepts beyond the current grade so that teachers can improve their own knowledge of the subject. 

Within the Teacher’s Guide, IM Curriculum, Why is the curriculum designed this way?, there are sections entitled “Further Reading” that consistently link research to pedagogy. There are adult-level explanations including examples of the more complex grade-level concepts and concepts beyond the current grade so that teachers can improve their own knowledge of the subject. Professional articles support teachers with learning opportunities about topics such as ensuring mathematical success for all, early understanding of equality, and repeating patterns. Examples include:

• Why is the curriculum designed this way? Further Reading, Unit 1, When is a number line not a number line?, supports teachers with context for work beyond the grade. “In this blog post, McCallum shares why the number line is introduced in grade 2 in IM K–5 Math, emphasizing the importance of foundational counting skills.”

• Why is the curriculum designed this way? Further Reading, Unit 7, What is a Measurable Attribute?, “In this blog post, Umland wonders about what counts as a measurable attribute and discusses how this interesting and important mathematical idea begins to develop in kindergarten.”

• Unit 1, Math in Our World, Lesson 6, Look for Small Groups, About this Lesson, “This skill (subitizing) is essential to students’ number work. Students communicate how many there are by showing quantities on their fingers and saying number words (MP6). Although some students may count to determine how many, the focus of this lesson is on recognizing groups of objects without counting. Students learn two new routines that will be used throughout the year to develop counting concepts. These routines will continue to be developed throughout the section and will be used across the year Throughout the section, observe students for the look-fors on the Unit 1, Sections A-D Checkpoint. In the Lesson Synthesis, students practice saying the verbal count sequence to 10 in About this Lesson for counting objects in an upcoming section. Add variety to the counting by adding movement. For example, students can count as they clap, stomp their feet, or jump.”

• Unit 5, Composing and Decomposing Numbers to 10, Lesson 15, About this Lesson, “This lesson does provide students with an opportunity to apply precursor skills of mathematical modeling. In previous lessons, students represented and solved Put Together/Take Apart, Both Addends Unknown story problems. This lesson builds on students’ experience in the Math Stories center. In this lesson, students use familiar contexts to generate and solve Put Together/Take Apart, Both Addends Unknown story problems. In the second activity, students are encouraged to find all possible solutions and use reasoning based on patterns explored in previous lessons (MP8). When students attend to the mathematical features of a situation, adhere to mathematical constraints, make choices, and translate a mathematical answer back into the context they model with mathematics (MP4).”

The materials reviewed for Imagine Learning Illustrative Mathematics Kindergarten meet expectations for including standards correlation information that explains the role of the standards in the context of the overall series. 

Correlation information is present for the mathematics standards addressed throughout the grade level/series and can be found in several places, including the Curriculum Course Guide, within unit resources, and within each lesson. Examples include:

• Grade-level resources, Kindergarten standards breakdown, standards are addressed by lesson. Teachers can search for a standard in the grade and identify the lesson(s) where it appears within materials.

• Course Guide, Lesson Standards, includes all Kindergarten standards and the units and lessons each standard appears in. 

• Unit 3, Resources, Teacher Guide, outlines standards, learning targets and the lesson where they appear. This is present for all units and allows teachers to identify targeted standards for any lesson.

• Unit 4, Understanding Addition and Subtraction, Lesson 18, the Core Standards are identified as K.CC.1, K.CC.2, K.OA.1, K.OA.2. Lessons contain a consistent structure that includes a Warm-up with a Narrative, Launch, Activity, Activity Synthesis. An Activity 1, 2, or 3 that includes Narrative, Launch, Activity, Activity Synthesis, Lesson Synthesis. A Cool-down that includes Responding to Student Thinking and Next Day Supports. This provides an additional place to reference standards, and language of the standard, within each lesson.

Each unit includes an overview identifying the content standards addressed within the unit, as well as a narrative outlining relevant prior and future content connections. Examples include: 

• Unit 2, Numbers 1-10, Unit Overview, Unit Learning Goals, full Unit Narrative, includes an overview of how the math of this module builds from previous work in math. “In this unit, students continue to develop counting concepts and skills, including comparing, while learning to write numbers. Previously, students answered “how many” and “are there enough” questions and counted groups of up to 10 objects. They also learned the structures and routines for activities and centers. Here, students rely on familiar activity structures to build their counting skills and concepts. First, they count and compare the number of objects, and then do the same with groups of images. The images are given in different arrangements—in lines, arrays, number cube patterns, on 5-frames—to help students connect different representations to the same number.”

• Unit 6, Numbers 0-20, Unit Overview, Unit Learning Goals, full Unit Narrative, includes an overview of how the math of this module builds from previous work in math. “In this unit, students count and represent collections of objects and images within 20. They apply previously developed counting concepts—such as one-to-one correspondence, keeping track of what has been counted, and conservation of numbers—to larger numbers. Previously, students have counted, composed, and decomposed numbers up to 10, using tools such as counters, connecting cubes, 5-frames, 10-frames, drawings, and their fingers. They wrote expressions to record compositions and decompositions. Here, students use the 10-frame to organize groups of 11-19 objects and images. This tool encourages students to see teen numbers as 10 ones and some more ones, emphasizing the 10 + n structure of the numbers 11–19. They use this structure as they represent teen numbers with their fingers, objects, drawings, expressions, and equations. Students see equations with the addend written first, such as 10 + 6 = 16.”

The materials reviewed for Imagine Learning Illustrative Mathematics Kindergarten meet expectations for providing explanations of the instructional approaches of the program and identification of the research-based strategies.

Instructional approaches of the program are described within the Curriculum Guide, Why is the curriculum designed this way? Design Principles. “It is our intent to create a problem-based curriculum that fosters the development of mathematics learning communities in classrooms, gives students access to the materials through coherent progression, and provides teachers the opportunity to deepen their knowledge of mathematics, student thinking, and their own teaching practice. Examples from the Design principles include:

• Curriculum Guide, Why is the curriculum designed this way?, Design principles, includes information about the 11 principles that informed the design of the materials. Balancing Rigor, “There are three aspects of rigor essential to mathematics: conceptual understanding, procedural fluency, and the ability to apply these concepts and skills to mathematical problems with and without real-world contexts. These aspects are developed together and are therefore interconnected in the materials in ways that support student understanding.  Opportunities to connect new representations and language to prior learning support students in building conceptual understanding. Access to new mathematics and problems prompts students to apply their conceptual understanding and procedural fluency to novel situations. Warm-ups, practice problems, centers, and other built-in routines help students develop procedural fluency, which develops over time.”

• Curriculum Guide, Why is the curriculum designed this way?, Design Principles, Task Complexity, “Mathematical tasks can be complex in different ways, with the source of complexity varying based on students’ prior understandings, backgrounds, and experiences. In the curriculum, careful attention is given to the complexity of contexts, numbers, and required computation, as well as to students’ potential familiarity with given contexts and representations. To help students navigate possible complexities without losing the intended mathematics, teachers can look to warm-ups and activity launches for built-in preparation, and to teacher-facing narratives for further guidance. In addition to tasks that provide access to the mathematics for all students, the materials provide guidance for teachers on how to ensure that during the tasks, all students are provided the opportunity to engage in the mathematical practices. More details are given below about teacher reflection questions, and other fields in the lesson plans help teachers assure that all students not only have access to the mathematics, but the opportunity to truly engage in the mathematics.”

Research-based strategies within the program are cited and described within the Curriculum Guide, within Why is the curriculum designed this way?. There are four sections in this part of the Curriculum Guide including Design Principles, Key Structures, Mathematical Representations, and Further Reading. Examples of research-based strategies include: 

• Curriculum Guide, Why is the curriculum designed this way?, Further Reading, Entire Series, The Number Line: Unifying the Evolving Definition of Number in K–12 Mathematics. “In this article, the authors (Lahme, McLeman, Nakamaye, and Umland) focus their attention on the selection of definitions, notation, and graphical conventions surrounding the development of the real numbers from kindergarten to grade 12, and address the work that students might do in later years.“

• Curriculum Guide, Why is the curriculum designed this way?, Design Principles, Instructional Routines, “Instructional routines provide opportunities for all students to engage and contribute to mathematical conversations. Instructional routines are invitational, promote discourse, and are predictable in nature.” They are “enacted in classrooms to structure the relationship between the teacher and the students around content in ways that consistently maintain high expectations of student learning while adapting to the contingencies of particular instructional interactions.” (Kazemi, Franke, & Lampert, 2009)

• Curriculum Guide, Why is the curriculum designed this way?, Key Structures in these materials, Student Journal Prompts, “Writing can be a useful catalyst in learning mathematics because it not only supplies students with an opportunity to describe their feelings, thinking, and ideas clearly, but it also serves as a means of communicating with other people (Baxter, Woodward, Olson, & Robyns, 2002; Liedtke & Sales, 2001; NCTM, 2000). NCTM (1989) suggests that writing about mathematics can help students clarify their ideas and develop a deeper understanding of the mathematics at hand.”

• Curriculum Guide, Why is the curriculum designed this way?, Design Principles, Using the 5 Practices for Orchestrating Productive Discussions, “Promoting productive and meaningful conversations between students and teachers is essential to success in a problem-based classroom. The Instructional Routines section of the teacher course guide describes the framework presented in 5 Practices for Orchestrating Productive Mathematical Discussions (Smith & Stein, 2011) and points teachers to the book for further reading. In all lessons, teachers are supported in the practices of anticipating, monitoring, and selecting student work to share during whole-group discussions. In lessons in which there are opportunities for students to make connections between representations, strategies, concepts, and procedures, the lesson and activity narratives provide support for teachers to also use the practices of sequencing and connecting, and the lesson is tagged so teachers can easily identify these opportunities. Teachers have opportunities in curriculum workshops and PLCs to practice and reflect on their own enactment of the 5 Practices.”

The materials reviewed for Imagine Learning Illustrative Mathematics Kindergarten meet expectations for providing a comprehensive list of supplies needed to support instructional activities. 

The Course Overview, Grade-level resources, provides a Materials List intended for teachers to gather materials for each grade level. Additionally, specific lessons include a Teaching Notes section and a Materials List, which include specific lists of instructional materials for lessons. Examples include:

• Course Overview, Grade Level Resources, Grade K Materials List, contains a comprehensive chart of all materials needed for the curriculum. It includes the materials used throughout the curriculum, whether they are reusable or consumable, quantity needed, lessons the materials are used in, and suitable substitutes for the materials. Each lesson listed in the chart and any additional virtual materials noted for a lesson are digitally linked in the materials for quick access. Geoblocks are a reusable material used in lessons K.1.4, K.1.5, K.1.6, K.1.7, K.1.8, K.1.9, K.1.10, K.1.11, K.1.12, K.1.13, K.1.14, K.1.15, K.1.16, K.1.17, K.2.14, K.3.10, K.3.11, K.3.12, K.3.13, K.3.14, K.7.1, K.7.2, K.7.3, K.7.4, K.7.5,… 15 sets with at least 10 shapes in each set are needed per 30 students. Block set or cardboard boxes are suitable substitutes. Brown paper bags are a consumable material used in lessons K.2.3, K.2.6, K.2.12, and K.2.16. 45 brown bags are needed per 30 students. No suitable substitutes for the material are listed. Play dough or modeling clay is a reusable material used in lessons K.3.7, K.7.7, K.7.8, K.7.9, K.7.10, K.7.11, K.7.12, and  K.7.13. 15 are needed per 30 students. No suitable substitutes are listed.

• Course Overview, Grade Level Resources, Grade K Picture Books, contains a “list of suggested picture books to read throughout the curriculum.” Unit 2, The Little Red Hen (Makes a Pizza) by Philomen Sturges is suggested. Unit 3, Stitchin’ and Pullin’: A Gee’s Bend Quilt by Patricia McKissack is suggested. Unit 7, The Seesaw by Judith Koppens is suggested.

• Unit 8, Putting It All Together, Lesson 14, Activity 1, Teaching Notes, Materials to gather, “Colored pencils, crayons, or markers.” Launch, “Display the student book. This code tells us which color to use. If the group of dots or expression shows 5, you are going to color that section brown. This section says 2+0. What color should I color this section? How do you know? (You should color it green. 2+0 is 2.) Figure out the total number of dots in each image. Find the value of each expression. Check the key to determine which color to use in this section. If the expression is 2+1, you would color that section red, because in the key it says that ‘3’ should be colored red.”

The materials reviewed for Imagine Learning Illustrative Mathematics Kindergarten meet expectations for having assessment information included in the materials to indicate which standards are assessed. The materials consistently identify the content standards assessed for formal assessments, and the materials provide guidance, including the identification of specific lessons, as to how the mathematical practices can be assessed across the series.

End-of-Unit Assessments and End-of-Course Assessments consistently and accurately identify grade-level content standards within each End-of-Unit Assessment answer key. Examples from formal assessments include:

• Unit 3, Flat Shapes All Around Us, End-of-Unit Assessment answer key, denotes standards addressed for each problem. Problem 3, K.MD.2, “a. Circle the rectangle that is longer. b. Circle the rectangle that is shorter.” 3a has images of red and blue horizontal rectangles. 3b has images of red and blue vertical rectangles.

• Unit 6, Numbers 0-20, End-of-Unit Assessment answer key, denotes standards addressed for each problem. Problem 1, K.NBT.1, “Draw 17 dots. Use the 10-frame if it helps you.”

• Unit 7, Solid Shapes All Around Us, End-of-Unit Assessment answer key, denotes standards addressed for each problem. Problem 2, K.MD.2, “a. Circle the object that is heavier. An image shows an apple and a pencil. b. Circle the object that is lighter. An image shows a desk and paper clip.”

Guidance is provided within materials for assessing progress of the Mathematical Practices. According to IM K-5 Curriculum Guide, How do you assess progress?, Standards For Mathematical Practice, “Because using the mathematical practices is part of a process for engaging with mathematical content, we suggest assessing the Mathematical Practices formatively. For example, if you notice that most students do not use appropriate tools strategically (MP5), plan in future lessons to select and highlight work from students who have chosen different tools.” For each grade, there is a chart outlining a handful of lessons in each unit that showcase certain mathematical practices. There is also guidance provided for tracking progress against “I can” statements aligned to each practice. “Since the Mathematical Practices in action can take many forms, a list of learning targets for each Mathematical Practice is provided to support teachers and students in recognizing when engagement with a particular Mathematical Practice is happening. The intent of the list is not that students check off every item on the list. Rather, the “I can” statements are examples of the types of actions students could do if they are engaging with a particular Mathematical Practice.” Examples include:

• IM K-5 Curriculum Guide, How do you assess progress?, Standards For Mathematical Practice, Standards for Mathematical Practices Chart, Grade K, MP1 is found in Unit 2, Lessons 4, 19, 20, and 21. 

• IM K-5 Curriculum Guide, How do you assess progress?, Standards For Mathematical Practice, Standards for Mathematical Practices Chart, Grade K, MP4 is found in Unit 3, Lessons 2, 9, 14, and 15. 

• IM K-5 Curriculum Guide, How do you assess progress?, Standards For Mathematical Practice, Standards for Mathematical Practice Student Facing Learning Targets, “MP5 I Can Use Appropriate Tools Strategically. I can choose a tool that will help me make sense of a problem. These tools might include counters, base-ten blocks, tiles, a protractor, ruler, patty paper, graph, table, or external resources. I can use tools to help explain my thinking. I know how to use a variety of math tools to solve a problem.”

• IM K-5 Curriculum Guide, How do you assess progress?, Standards For Mathematical Practice, Standards for Mathematical Practice Student Facing Learning Targets, “MP7 I Can Look for and Make Use of Structure. I can identify connections between problems I have already solved and new problems. I can compose and decompose numbers, expressions, and figures to make sense of the parts and of the whole. I can make connections between multiple mathematical representations. I can make use of patterns to help me solve a problem.”

The materials reviewed for Imagine Learning Illustrative Mathematics 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.

Formative assessment opportunities include instructional tasks, practice problems, and checklists in each section of each unit. Summative assessments include End-of-Unit Assessments and the End-of-Course Assessment. Assessments regularly demonstrate the full intent of grade-level content and practice standards through a variety of item types including multiple choice, multiple response, short answer, restricted constructed response, and extended response. Examples of summative assessment items include:

• Unit 2, Number 1-10, End-of-Unit Assessment develops the full intent of K.CC.6, identify whether the number of objects in one group is greater than, less than, or equal to the number of objects in another group, e.g., by using matching and counting strategies. Problem 3, “a. Circle the group that has more things. (2 images: a 5 frame with dots in each box and 1 added dot outside of the frame and an image of two hands with 8 fingers raised.) b. Circle the group that has fewer things. (2 images: a straight row of 6 black dots and a circular configuration of 9 black dots).”

• Unit 4, Understanding Addition and Subtraction, End-of-Unit Assessment problems support the full intent of MP4, model with mathematics, as students show their thinking using drawings, numbers, words or objects to solve a subtraction problem. Problem 3, “There are 6 kids playing in the park. 2 of the kids leave the park to go home. How many kids are playing in the park now? Show your thinking using drawings, numbers, words, or objects.”

• Unit 5, Composing and Decomposing Numbers to 10, End-of-Unit Assessment develops the full intent of K.OA.3, decompose numbers less than or equal to 10 into pairs in more than one way, e.g., by using objects or drawings, and record each decomposition by a drawing or equation (e.g., 5=2+3 and 5=4+1). Problem 3, “Mai has a train of 7 connecting cubes. (image of 7 connecting cubes shown) She snaps the train into two pieces. Show 1 way to snap the cubes. Show a different way to snap the cubes.”

• Unit 8, Putting It All Together, End-of-Course Assessment supports the full intent of MP6, attend to precision, as students build shapes from geoblocks. Problem 12, “Build a shape with pattern blocks or with geoblocks. Describe your shape. How many blocks did you use to build your shape? Write and solve a story problem about your shape.”

The materials reviewed for Imagine Learning Illustrative Mathematics 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.

Materials regularly provide strategies, supports, and resources for students in special populations to help them access grade-level mathematics. In the Curriculum Guide, How do the materials support all learners? Access for Students with Disabilities, “These materials empower all students with activities that capitalize on their existing strengths and abilities to ensure that all learners can participate meaningfully in rigorous mathematical content. Lessons support a flexible approach to instruction and provide teachers with options for additional support to address the needs of a diverse group of students, positioning all learners as competent, valued contributors. When planning to support access, teachers should consider the strengths and needs of their particular students. The following areas of cognitive functioning are integral to learning mathematics (Addressing Accessibility Project, Brodesky et al., 2002). Conceptual Processing includes perceptual reasoning, problem solving, and metacognition. Language includes auditory and visual language processing and expression. Visual-Spatial Processing includes processing visual information and understanding relation in space of visual mathematical representations and geometric concepts. Organization includes organizational skills, attention, and focus. Memory includes working memory and short-term memory. Attention includes paying attention to details, maintaining focus, and filtering out extraneous information. Social-Emotional Functioning includes interpersonal skills and the cognitive comfort and safety required in order to take risks and make mistakes. Fine-motor Skills include tasks that require small muscle movement and coordination such as manipulating objects (graphing, cutting with scissors, writing).” 

Examples include: 

• Unit 4, Understanding Addition and Subtraction, Lesson 2, Activity 2, Narrative, Access for Students with Disabilities, “Action and Expression: Develop Expression and Communication. Some students may benefit from using 5-frames to help count the number of green and red apples. Give students access to 5-frames and counters to represent the apples in each problem. Invite students to use the 5-frames to figure out how many apples there are altogether. Supports accessibility for: Organization, Conceptual Processing.”

• Unit 5, Composing and Decomposing Numbers to 10, Lesson 4, Activity 3, Narrative, Access for Students with Disabilities, “Engagement: Provide Access by Recruiting Interest. Use visible timers or audible alerts to help learners anticipate and prepare to transition between activities. Supports accessibility for: Social-Emotional Functioning, Organization.”

• Unit 7, Solid Shapes All Around Us, Lesson 9, Activity 2, Narrative, Access for Students with Disabilities, “Engagement: Develop Effort and Persistence. Invite students to generate a list of shared expectations for group work. Ask students to share explicit examples of what those expectations would look like in this activity. Supports accessibility for: Social-Emotional Functioning.”

The materials reviewed for Imagine Learning Illustrative Mathematics Kindergarten meet expectations for providing extensions and/or opportunities for students to engage with grade-level mathematics at higher levels of complexity.

While there are no instances where advanced students do more assignments than classmates, materials do provide multiple opportunities for students to investigate grade-level content at a higher level of complexity. These are found in a section titled, “Exploration Problems” within lessons where appropriate. According to the IM K-5 Teacher Guide, How do you use the materials?, Practice Problems, Exploration Problems, “Each practice problem set also includes exploration questions that provide an opportunity for differentiation for students ready for more of a challenge. There are two types of exploration questions. One type is a hands-on activity that students can do directly related to the material of the unit, either in class if they have free time, or at home. The second type of exploration is more open-ended and challenging. These problems go deeper into grade-level mathematics. They are not routine or procedural, and they are not just “the same thing again but with harder numbers. Exploration questions are intended to be used on an opt-in basis by students if they finish a main class activity early or want to do more mathematics on their own. It is not expected that an entire class engages in exploration problems, and it is not expected that any student works on all of them. Exploration problems may also be good fodder for a Problem of the Week or similar structure.”

Examples include:

• Unit 2, Numbers 1–10, Section B: Count and Compare Groups of Images, Problem 7, Exploration, “‘Are there fewer students than chairs? Explain how you know.​​​​​’ An image is provided of a classroom with students and chairs.”

• Unit 4, Understanding Addition and Subtraction, Section B: Represent and Solve Story Problems, Problem 8, Exploration, “There are 6 dolphins swimming around the boat. Complete the story in two different ways. Solve your problems or share with a partner and solve your partner's problems.”

• Unit 6, Numbers 0–20, Section A: Count Groups of 11-20 Objects, Problem 1, Exploration, “​​​How many shapes do you see on the soccer ball?”

The materials reviewed for Imagine Learning Illustrative Mathematics 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.

Guidance is consistently provided for teachers to support students who read, write, and/or speak in a language other than English, providing scaffolds for them to meet or exceed grade-level standards. According to the Curriculum Guide, How do the materials support all learners? Mathematical language development, “Embedded within the curriculum are instructional routines and supports to help teachers address the specialized academic language demands when planning and delivering lessons, including the demands of reading, writing, speaking, listening, conversing, and representing in math (Aguirre & Bunch, 2012). While these instructional routines and supports can and should be used to support all students learning mathematics, they are particularly well-suited to meet the needs of linguistically and culturally diverse students who are learning mathematics while simultaneously acquiring English. Mathematical Language Routines (MLR) are also included in each lesson’s Support for English learners, to provide teachers with additional language strategies to meet the individual needs of their students. Teachers can use the suggested MLRs as appropriate to provide students with access to an activity without reducing the mathematical demand of the task. When selecting from these supports, teachers should take into account the language demands of the specific activity and the language needed to engage the content more broadly, in relation to their students’ current ways of using language to communicate ideas as well as their students’ English language proficiency. Using these supports can help maintain student engagement in mathematical discourse and ensure that struggle remains productive. All of the supports are designed to be used as needed, and use should fade out as students develop understanding and fluency with the English language.” The series provides principles that promote mathematical language use and development: 

• Principle 1. Support sense-making: Scaffold tasks and amplify language so students can make their own meaning. 

• Principle 2. Optimize output: Strengthen opportunities for students to describe their mathematical thinking to others, orally, visually, and in writing. 

• Principle 3. Cultivate conversation: Strengthen opportunities for constructive mathematical conversations. 

• Principle 4. Maximize meta-awareness: Strengthen the meta-connections and distinctions between mathematical ideas, reasoning, and language. 

The series also provides Mathematical Language Routines (MLR) in each lesson. Curriculum Guide, How do the materials support all learners? Mathematical language development, “A mathematical language routine is a structured but adaptable format for amplifying, assessing, and developing students' language. The MLRs were selected because they are effective and practical for simultaneously learning mathematical practices, content, and language. These routines can be adapted and incorporated across lessons in each unit to fit the mathematical work wherever there are productive opportunities to support students in using and improving their English and disciplinary language use. These routines facilitate attention to student language in ways that support in-the-moment teacher, peer, and self-assessment. The feedback enabled by these routines will help students revise and refine not only the way they organize and communicate their own ideas, but also ask questions to clarify their understanding of others’ ideas.” Examples include:

• Unit 2, Numbers 1-10, Lesson 9, Activity 1, Teaching Notes, Access for English Learners, “MLR8 Discussion Supports. Make sure students can explain how they know which card has more. Invite groups to rehearse what they will say when they share with the whole class. Advances: Speaking, Conversing.”

• Unit 6, Numbers 0-20, Lesson 13, Activity 2, Teaching Notes, Access for English Learners, “MLR8 Discussion Supports. Pair gestures with verbal directions to clarify the meaning of any unfamiliar terms. Students may benefit from discussing possible strategies they can use to determine order before they begin. Advances: Listening, Representing.”

• Unit 8, Putting It All Together, Lesson 5, Activity 1, Teaching Notes, Access for English Learners, “MLR8 Discussion Supports. Invite each group to chorally read numbers 1–20 in order once the group agrees on the order. Listen for and clarify questions. Advances: Speaking, Conversing.”

The materials reviewed for Imagine Learning Illustrative Mathematics 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.

The materials consistently include suggestions and/or links, within the lesson notes, for virtual and physical manipulatives that support the understanding of grade-level math concepts. Examples include: 

• Unit 4, Understanding Addition and Subtraction, Lesson 6, Activity 3, students use connecting cubes, and a number mat to support understanding of subtraction. Launch, “Give each group of students 10 connecting cubes and a number mat. ‘We are going to learn a center called Subtraction Towers.’ Display a connecting cube tower with 7 cubes. ‘How many cubes are in the tower? If I have to subtract, or take away, 3 cubes from my tower, what should I do?’ (Break off 3 cubes, take off 1 cube at a time as you count.) ‘One partner uses up 5-10 cubes to build a tower. Then the other partner rolls to figure out how many cubes to take away, or subtract, from the tower. Then work together to figure out how many cubes are left in the tower. Take turns building the tower.’”

• Unit 5, Composing and Decomposing Numbers to 10, Lesson 7, Activity 1, students use connecting cubes or two color counters to support solving word problems. Launch, “Give students access to connecting cubes or two-color counters. ‘Many families and cultures make special desserts. Are there desserts that you make with your family?’ 30 seconds: quiet think time 1 minute: partner discussion Share responses. Display the image. ‘Paletas are a type of ice pop popular in Mexico. They are usually made with fruit. Read and display the task statement. Tell your partner what happened in the story. What are we trying to figure out? (How many of the paletas had lime and how many had coconut.) Show your thinking using drawings, numbers, words, or objects.’”

• Unit 8, Putting It All Together, Lesson 13, Activity 2, students use domino cards and sorting cards to compare groups. Launch, “Invite each student to make a pile with half of the domino cards. ‘We are going to play a comparing game with our dominoes. You and your partner will both flip over one card. One partner will compare the number of dots using 'fewer' or 'the same number' and explain how they know. The other partner will compare the number of dots using 'more' or 'the same number. Let’s play one round together.’ Display 2 domino cards. Choose who will go first. Compare the number of dots using 'fewer' or 'the same number' and explain how you know. ‘The domino with 2 dots and 1 dot has fewer dots than the domino with 2 dots and 2 dots. 3 is less than 4. If you did not go first, compare the number of dots using 'more' or 'the same number' and explain how you know.’”