K-2nd Grade - Gateway 3

The materials reviewed for Illustrative Mathematics® v.360 Kindergarten through Grade 2 meet expectations for providing teacher guidance with useful annotations and suggestions for how to enact the student materials and ancillary materials, with specific attention to engaging students in order to guide their mathematical development. 

The Course Guide contains sections titled What’s in an IM Lesson, Key Structures in This Course, and Scope and Sequence, along with a Pacing Guide and Dependency Diagram, which provide instructional guidance related to the use of student and ancillary materials. Examples include:

• Course Guide, Key Structures in This Course, Coherent Progression, “Each unit starts with an invitation to the mathematics. The first few lessons provide an accessible entry point for all students and offer the opportunity to observe students’ prior understandings. Each lesson starts with a Warm-up to activate students’ prior knowledge and to 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 Synthesis at the end consolidates understanding and makes the learning goals of the lesson explicit. In the Cool-down that follows, students apply what they learned. Each activity starts with a Launch that gives all students access to the task. Independent work time follows, allowing them to grapple with problems individually before working in small groups. In the Activity Synthesis at the end, students consolidate their learning by making connections between their work and the mathematical goals. Each activity includes carefully chosen contexts and numbers that support the coherent sequence of learning goals in the lesson.” 

• Course Guide, Key Structures in This Course, Principles of IM Curriculum Design, Productive Discussions states, "Promoting productive and meaningful conversations between students and teachers is essential to success in a problem-based classroom. To facilitate these conversations, the IM curriculum incorporates the framework presented in 5 Practices for Orchestrating Productive Mathematical Discussions (Smith & Stein, 2011). The 5 Practices are: anticipating, monitoring, selecting, sequencing, and making connections between students’ responses. All IM lessons support the practices of anticipating, monitoring, and selecting students’ work to share during whole-group discussions. In lessons in which students make connections between representations, strategies, concepts, and procedures, the Lesson Narrative and the Activity Narrative support the practices of sequencing and connecting as well, and the lesson is tagged so that these opportunities are easily identifiable. For additional opportunities to connect students’ work, look for activities tagged with MLR7 Compare and Connect. Similar to the 5 Practices routine, MLR7 supports the practices of monitoring, selecting, and making connections. In curriculum workshops and PLCs, rehearse and reflect on enacting the 5 Practices."

• Course Guide, What’s in an IM Lesson, Narratives Tell The Story states, “The story of each grade is told in eight or nine units. Each unit has a narrative that describes the mathematical work that will unfold in that unit. Each lesson and each activity in the unit also has a narrative. The Lesson Narrative explains: The mathematical content of the lesson and its place in the learning sequence. The meaning of any new terms introduced in the lesson. How the mathematical practices come into play, as appropriate. The Activity Narrative explains: The mathematical purpose of the activity and its place in the learning sequence. What students are doing during the activity. What to look for, while students are working on an activity, to orchestrate an effective Activity Synthesis. Connections to the mathematical practices, when appropriate.”

• Course Guide, Scope and Sequence, lists each of the units, explains connections to prior learning, describes the progression of learning throughout the unit, and integration of new terminology/vocabulary throughout each unit.

• The Course Guide, Pacing Guide, and Dependency Diagram show the interconnectedness between lessons and units within each grade and across all grades.

• The glossary provides a visual glossary for teachers that includes both definitions and illustrations.

Materials include sufficient annotations and suggestions presented within the context of the specific learning objectives. An example includes:

• In Kindergarten, Unit 5, Composing and Decomposing Numbers to 10, Lesson 4, Warm-up, Choral Count Instructional Routine states, “While Choral Counting offers students the opportunity to practice verbal counting, the recorded count is the primary focus of the routine. As students reflect on the recorded count, they make observations, predict upcoming numbers in the count, and justify their reasoning.” Activity Narrative states, “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.”

• In Grade 1, Unit 7, Geometry and Time, Lesson 3, Lesson Synthesis, provides teachers guidance on ways to sort and describe two-dimensional shapes. “Display Card A. ‘Today we described flat shapes in different ways in order to sort them. How might you describe this shape? (3 sides are the same. There are 3 corners. It is a triangle.) Display Card Q. ‘How might you describe this shape?’ (There are 4 sides. 3 sides are the same length and the bottom side is long.) Continue with shape cards U and C as time allows.”

• In Grade 2, Unit 7, Adding and Subtracting within 1,000, Lesson 12, Lesson Synthesis, provides teachers guidance on ways to decompose 10 to subtract within 1,000. “‘Today we saw that we can subtract by place with greater numbers, and sometimes a ten is decomposed.’ ‘How did you know when a ten would be decomposed when you subtracted three-digit numbers?’ (I could tell when I looked at the ones place and saw I didn't have enough ones to subtract ones from ones.) ‘How was this the same as when you subtracted two-digit numbers? How was it different?’ (It was just like when we subtracted two-digit numbers. It’s different because one of the numbers has hundreds.)”

The materials reviewed for Illustrative Mathematics® v.360 Kindergarten through Grade 2 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. 

The Further Reading section, located within the Course Guide, connects research to pedagogy. It includes explanations and examples of both grade-level and above-grade-level content to support teacher understanding. Additionally, Unit Overviews, Lesson Narratives, and Activity Synthesis sections throughout the lessons provide similar support, offering explanations and examples of how grade-level concepts connect to content in other grade levels.

 Examples include:

• Kindergarten, Course Guide, 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 in IM Kindergarten.”

• Kindergarten, Unit 1, Math in Our World, Lesson 6, Lesson Narrative states, “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. In the Lesson Synthesis, students practice saying the verbal count sequence to 10 in preparation for counting objects in an upcoming section. Add variety to the choral count by providing movement. For example, students can count as they clap, stomp their feet, or jump.”

• Grade 1, Course Guide, Further Reading, “Unit 2, Representing Subtraction of Signed Numbers: Can You Spot the Difference?” supports teachers with context for work beyond the grade. “In this blog post, Anderson and Drawdy discuss how counting on to find the difference plays a foundational role in understanding subtraction with negative numbers, on the number line, in middle school.”

• Grade 1, Unit 5, Adding Within 100, Lesson 14, Lesson Narrative states, “This lesson is optional because it does not address any new mathematical content standards. This lesson does provide students with an opportunity to apply precursor skills of mathematical modeling (MP4). In previous lessons, students found the value of sums within 100 using methods based on place value and the properties of operations, including adding tens and tens and ones and ones, and adding on by place. In this lesson, students apply these methods to make sense of and solve real-world problems within 100. Students may use base-ten representations or equations to represent their thinking. In the Warm-up, they are introduced to a food drive context which will be used throughout the lesson as students use data about the number of cans collected by each grade to determine how the cans should be packed in boxes. Students combine quantities of cans in different ways. Students make choices about which numbers to combine based on their values and the constraints of the problem. Then students are asked to think about the different ways the cans can be packed and choose a way they think makes the most sense.”

• Grade 2, Course Guide, Further Reading, “Unit 4, The Nuances of Understanding a Fraction as a Number”, supports teachers with context for work beyond the grade. “In this blog post, Gray discusses the role the number line plays in students' understanding of fractions as numbers.”

• Grade 2, Unit 1, Adding, Subtracting, and Working with Data, Lesson 16, Solve All Kinds of Compare Problems, Lesson Narrative states, “The number choices in the Compare problems in this lesson encourage students to use methods based on place value to find the unknown value. Students may look for ways to compose a ten or subtract multiples of ten when finding unknown values within 100. Students will subtract numbers other than multiples of ten within 100 in future lessons. Encourage students to use a tape diagram to make sense of the problem if it is helpful.”

The materials reviewed for Illustrative Mathematics® v.360 Kindergarten through Grade 2 meet expectations for including a year-long scope and sequence with standards correlation information.

The Course Guide includes multiple components that support planning and understanding of the program’s structure and standards alignment. 

Examples in Kindergarten include:

• The Scope and Sequence section narratively outlines unit content, prior knowledge, future learning, and terminology. The materials state, “The big ideas in IM Kindergarten include: representing and comparing whole numbers, initially with sets of objects; understanding and applying addition and subtraction; and describing shapes and space. In IM Kindergarten, more time is devoted to numbers than to other topics. The materials, particularly units that focus on addition and subtraction, include problem types, such as Add To, Take From, Put Together or Take Apart, Compare, Result Unknown, and so on. These problem types are based on common addition and subtraction situations, as outlined in Table 1 of the ‘Mathematics Glossary’ section of the Common Core State Standards (NGA & CCSSO).”

• Lessons by Standard, which provides a table that shows each content standard for the grade level and the lessons in which it appears. For example, K.CC.A.2 is addressed in Unit 4, Lessons 14 and 18; Unit 5, Lessons 4 and 9; Unit 6, Lessons 1 and 11; and Unit 8, Lessons 1, 3, 4, 5, and 14.

• Standards by Lesson provides a table listing the standards covered within each lesson. For example, Unit 1, Lesson 12 includes K.CC.A.1, K.CC.B.4, K.CC.B.4.a, and K.G.B.

• Standards for Mathematical Practice, mapping practice standards (MPs) to lessons. For example, Unit 3, Lesson 12 integrates MP1 and MP3.

Examples in Grade 1 include:

• The Scope and Sequence section narratively outlines unit content, prior knowledge, future learning, and terminology. The materials state, “The big ideas in IM Grade 1 include: developing understanding of addition and subtraction, and strategies for addition and subtraction within 20; developing understanding of whole-number relationships and place value, including grouping tens and ones; developing understanding of linear measurement and measuring lengths as iterating length units; and reasoning about attributes of, and composing and decomposing, geometric shapes. The materials, particularly units that focus on addition and subtraction, include problem types, such as Add To, Take From, Put Together or Take Apart, Compare, Result Unknown, and so on. These problem types are based on common addition and subtraction situations, as outlined in Table 1 of the ‘Mathematics Glossary’ section of the Common Core State Standards (NGA & CCSSO, 2010).”

• Lessons by Standard, which provides a table that shows each content standard for the grade level and the lessons in which it appears. For example, 1.OA.A.2 is addressed in Unit 3, Lessons 15, 20, and 28; Unit 6, Lesson 11; and Unit 8, Lesson 6.

• Standards by Lesson provides a table listing the standards covered within each lesson. For example, Unit 5, Lesson 13 addresses 1.NBT.A.1; 1.NBT.B.3; 1.NBT.C.4; 1.OA.C.5; 1.OA.C.6; and 1.OA.D.8.

• Standards for Mathematical Practice, mapping practice standards (MPs) to lessons. For example, Unit 3, Lesson 6 integrates MP1, MP2, and MP3.

Examples in Grade 2 include:

• The Scope and Sequence section narratively outlines unit content, prior knowledge, future learning, and terminology. The materials state, “The big ideas in IM Grade 2 include: extending understanding of the base-ten number system;, building fluency with addition and subtraction; using standard units of measure; and describing and analyzing shapes. The materials, particularly units that focus on addition and subtraction, include problem types such as Add To, Take From, Put Together or Take Apart, Compare, Result Unknown, and so on. These problem types are based on common addition and subtraction situations, as outlined in Table 1 of the ‘Mathematics Glossary’ section of the Common Core State Standards (NGA & CCSSO).”

• Lessons by Standard, which provides a table that shows each content standard for the grade level and the lessons in which it appears. For example, 2.NBT.A.3 is addressed in Unit 5, Lessons 4, 5, 6, and 11; Unit 6, Lesson 5; and Unit 9, Lesson 6.

• Standards by Lesson provides a table listing the standards covered within each lesson. For example, Unit 5, Lesson 6 addresses 2.NBT.A.3.

• Standards for Mathematical Practice, mapping practice standards (MPs) to lessons. For example, Unit 1, Lesson 15 integrates MP1, MP2, and MP7.

In addition, the Pacing Guide and Dependency Diagram within the Course Guide outline the number of lessons and suggested teaching days per unit, supporting year-long planning and implementation. Each lesson includes references to the standards addressed and often notes how the lesson builds on prior learning.

The materials reviewed for Illustrative Mathematics® v.360 Kindergarten through Grade 2 meet expectations for explaining the program’s instructional approaches, identifying research-based strategies, and explaining the role of the standards. Examples include:

• Course Guide, Problem-Based Teaching and Learning, “Illustrative Mathematics is a problem-based curriculum that fosters the development of mathematics learning communities in classrooms, gives students access to the mathematics through a coherent progression, and offers teachers the opportunity to deepen their knowledge of mathematics, students’ thinking, and their own teaching practice. The curriculum and the professional-learning materials support students’ and teachers’ learning, respectively. This document defines the principles that guide IM’s approach to mathematics teaching and learning. It then outlines how each component of the curriculum supports teaching and learning, based on these principles.”

• Course Guide, Problem-Based Teaching and Learning, Learning Mathematics By Doing Mathematics, “Students learn mathematics as a result of solving problems. Mathematical ideas are the outcomes of the problem-solving experience rather than the elements that must be taught before problem solving” (Hiebert et al., 1996). A problem-based instructional framework supports teachers in structuring lessons so students are the problem solvers learning the mathematics.”

• Course Guide, Key Structures in This Course, Coherent Progression, “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. Activities and lessons are parts 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.”

• Course Guide, What’s in an IM Lesson, Instructional Routines, Instructional routines (IRs) in the materials are designed to promote student engagement in mathematical conversations through predictable, discourse-driven structures. Instructional Routines state, “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). A small, intentionally selected set of IRs is used throughout the curriculum to support consistent implementation and reduce cognitive load for teachers. Each routine is aligned to specific unit, lesson, or activity learning goals and is intended to support student access to mathematics by requiring them to think and communicate mathematically. Routines are identified by name within activities, and professional learning includes classroom videos and opportunities for educators to observe, practice, and reflect on their use.

The materials reviewed for Illustrative Mathematics® v.360 Kindergarten through Grade 2 meet expectations for providing a comprehensive list of supplies needed to support instructional activities. 

The Course Guide includes a comprehensive Required Materials list detailing all materials needed for the grade-level content. At the lesson level, the Required Materials section specifies what is needed for that specific lesson; if no materials are required, this section will indicate that by being left blank or stating "none."

Examples include: 

• Kindergarten, Unit 6, Numbers 0 to 20, Lesson 9, Required Materials, “Activity 3, Connecting cubes, Materials from previous centers,Two-color counters.” Required Preparation, “Activity 1, Create a set of cards from the blackline master for each group of 4. Activity 3, Gather materials from: Bingo, Stages 1‒4, Number Race, Stages 1 and 2, Grab and Count, Stage 1, Tower Build, Stages 1 and 2.”

• Grade 1, Unit 5, Adding Within 100, Lesson 2, Activity 3, Materials To Copy (from Blackline Masters), “Five in a Row Addition and Subtraction, Stage 5 Gameboards. To Gather: Paper clips, Two-color counters.”

• Grade 2, Unit 3, Measuring Length, Lesson 8, Required Materials, “Activity 1, Inch tiles, Objects of various lengths, Rulers (inches), Activity 2, Inch tiles, Rulers (inches)” Required Preparation, “Activity 1, Each group of 4 needs access to several objects between 1–11 inches long. Consider using classroom objects such as markers, colored pencils, 11 connecting cubes, books, or any other object with a length shorter than 12 inches.”

The materials reviewed for Illustrative Mathematics® v.360 Kindergarten through Grade 2 meet expectations for providing consistent opportunities to determine student learning throughout the school year. The assessment system provides sufficient teacher guidance for evaluating student performance and determining instructional next steps. 

Each End-of-Unit Assessment and the End-of-Course Assessment includes answer keys and standards alignment to support teachers in interpreting student understanding. According to the Course Guide and Assessment Guidance, “All summative-assessment problems include a complete solution and standards alignment. Multiple-choice and multiple-response problems often include a reason for each potential error that a student might make. Unlike formative assessments, problems on summative assessments generally do not prescribe a method of solution.” Examples include: 

• Kindergarten, Unit 8, Putting It All Together, End-of-Course Assessment and Resources, Problem 7 states, “There are 8 crabs on the beach. Then 5 of the crabs go into the ocean. How many crabs are on the beach now? Show your thinking, using drawings, numbers, or words.” The Narrative for Problem 7 states, “Students solve a Take Away, Result Unknown story problem. They may draw a picture, as in the provided solution, or they may write an equation or explain their reasoning in words.” The answer key aligns the item to standards K.OA.1 and K.OA.2, and includes sample representations to support teacher interpretation of student strategies.

• Grade 1, Unit 3, Adding and Subtracting Within 20, End-of-Unit Assessment, Problem 7 states, “Find the value of each expression. 1. 5+3, 2. 11+6, 3. 9-7, 4. 18-5, 5. 10+3, 6. 15-10.” The Narrative for Problem 7 states, “Students find the value of sums and differences within 20. No explanation is expected. The problems cover several important skills: Fluency within 10 (problems 1 and 3), Understanding teen numbers as a ten and some more (problems 2, 4, 5, and 6).” The answer key aligns the item to standard 1.OA.6, and includes sample representations to support teacher interpretation of student strategies.

• Grade 2, Unit 5, Numbers to 1,000, End-of-Unit Assessment, Problem 1 states, “Label the tick marks on the number line.” A number line is shown with 15 tick marks, labeled from 0 to 150, counting by 10s. The Narrative for Problem 1 states, “Students label the tick marks on a number line starting at 0 where each unit interval represents a length of 10. This is a version of skip counting by 10 where the students record the count as labels on the number line. This provides the opportunity for students to show how to skip count by 10 and appropriately label the tenth tick mark as 100. If students make mistakes when labeling the tick marks, invite them to orally count by 10 starting at 0 to differentiate their ability to count and their ability to label numbers on the number line.” The answer key aligns the item to standards 2.NBT.2 and 2.NBT.3, and includes sample representations to support teacher interpretation of student strategies.

The materials also include guidance for determining next instructional steps, integrated into both formative and summative assessment opportunities. Most lessons conclude with a Cool-down task designed to assess student thinking in relation to the lesson’s learning goal. The Course Guide section titled Key Structures in This Course and Authentic Use of Contexts and Suggested Launch Adaptations, Response to Student Thinking includes the following description:

• “The materials offer guidance to support students in meeting the learning goals. This guidance falls into one of two categories, Next-Day Support or Prior-Unit Support, based on anticipated student responses. This guidance offers ways to continue teaching grade-level content, with appropriate and aligned practice and support for students. These suggestions range from providing students with more concrete representations in the next day’s lesson to recommending a section from a prior unit, with activities that directly connect to the concepts in the lesson.”

In addition to this formative support, the materials provide teachers with structured guidance following summative assessments. The End-of-Unit Assessment Guidance describes how teachers might observe patterns of student understanding and offers suggestions for addressing unfinished learning alongside upcoming grade-level instruction: “The End-of-Unit Assessment Guidance includes example observations of students’ unfinished learning and strategies for support in the Next-Unit Support. The guidance is organized around evidence for understanding and mastery of the grade-level content standards. Rather than provide item-by-item analysis, the observations encourage analyzing multiple items (when appropriate) to look for evidence of what students understand about the standards. The Next-Unit Support offers ideas for how to address any unfinished learning alongside upcoming grade-level work or before the concept is needed for upcoming grade-level work. These supports include suggestions for questions to ask during activities, representations to use, centers to encourage, and ways to incorporate the End-of-Unit Assessment as an additional learning opportunity. When needed, supports also include ways to revisit activities (for example, a Card Sort) in new ways to build on what students already know and focus on both unfinished and new learning.” For example: 

• Kindergarten, Unit 6, Numbers 0–20, End-of-Unit Assessment, Problem 2, Responding To Student Thinking states, “Next Unit Support Observation: Students count to answer ‘how many?’ questions, but they do not yet write numbers greater than 10. Response: In the next unit, invite students to play Number Race, Stage 2 during Center Choice Time. Ask them to say the number they land on before they write it. Next Unit Support Observation: Students show they may count a number of objects between 11‒20 when there is a noticeable group of 10 (2.a, 2.c), but they may not yet count accurately when a group of 10 is not obvious (2.b). Response: In the next unit, invite students to revisit the optional activity from Grade K Unit 6 Section C where students count images in organized arrangements. Begin by focusing on the images not arranged by 10 and some more. Discuss different strategies students use to count the images. Then look at the images arranged on 10-frames. Discuss how students can count these and how the 10-frame is helpful for keeping track of and counting images. Next Unit Support Observation: Students do not yet accurately represent a given number of objects from 10–20. Response: While students play Pattern Blocks, Stage 7 in the next unit, encourage them to use a 10-frame to help keep track of how many pattern blocks they are counting out. Ask, ‘How do you know you have ___ pattern blocks?’ Next Day Support Observation: Students show they may be counting all to determine the number of objects in a 10-frame. Response: Encourage students to play Counting Collections, Stage 1 during Center Choice Time in the next unit. Ask students to organize their objects on a 10-frame and place the remaining objects below the frame. If students begin to count all as they place objects in the 10-frame, ask them how many objects there would be if the 10-frame was full. Invite them to fill the 10-frame without counting and ask them how many there are. If students count all to determine how many total objects are in the collection, ask them again how many are in the 10-frame. Ask them to start at 10 and continue counting the remaining objects. Next Unit Support Observation: Students count all to figure out how many objects are in a group organized into 10 ones and some more ones, or to find the value of a 10 + n  expression, rather than using this structure to recognize or represent teen numbers. Response: Encourage students to play Counting Collections, Stage 1 during Center Choice Time in the next unit. Ask students to organize their objects on a 10-frame and place the remaining objects below the frame. Point to the 10-frame and ask, ‘How many objects are here?’ Point to the objects below the frame. ‘How many objects are here?’ Then ask, ‘How many objects are there all together?’ After students figure out how many there are all together, say: ‘There are 10 here and ___ here, so there are ___ all together.’ Encourage students to write each number on their recording sheet. For example, they may draw a picture and label the group of 10, label the other group, and write the teen number that represents the total.”

• Grade 1, Unit 7, Geometry and Time, End-of-Unit Assessment, Problem 1, Responding To Student Thinking states, “Next Unit Support Observation: Students show they consider some defining attributes, but they rely on some non-defining attributes when identifying shapes. For example, students show that they understand triangles have 3 straight sides, but do not yet recognize triangles with less familiar angles. Response: During center time in the next unit, invite students to play Stage 1 of How are They The Same? Monitor for students selecting or drawing shapes that may look less familiar to them. Display the headings triangle, rectangle, and square. Invite students to share a shape card, or a shape they drew, and explain in which category the shape belongs and their reasoning why.  If needed, choose a shape card, or draw a shape, and ask, ‘Is this a _______? How do you know?’ Ask students to describe how shapes in the same category are alike and how they are different. Next Unit Support Observation: Students show they know some, but not all, attributes of triangles, squares, and rectangles. For example, students show they understand triangles have 3 sides, but draw or select a triangle with curved sides. Response: During center time in the next unit, invite students to play Stage 1 of Can You Draw It? Monitor for students drawing rectangles, squares, and triangles. Ask: ‘How do you know this is a ____?’ As needed, create a poster with examples and non examples of triangles, squares, and rectangles, and invite students to reference these examples as they play.”

• Grade 2, Unit 3, Measuring Length, End-of-Unit Assessment, Problem 3, Responding To Student Thinking states, “Next Unit Support Observation: Students show they may not yet understand how to label the scale of a line plot. For example, they only label the measurements given on the line plot or start the scale with 0 on the left side. Students do not yet represent the data accurately on the line plot. Response: Throughout the next unit, encourage students to play Creating Line Plots, Stage 1. Invite students to share how they decided what numbers to include on their scale and how they know their scale matches the way numbers are represented on a ruler. Invite partners to check each other's work and give feedback on how the line plots clearly and accurately reflect the data collected.”

The materials reviewed for Illustrative Mathematics® v.360 Kindergarten through Grade 2 meet expectations for providing strategies and support for students in special populations to work with grade-level content and meet or exceed grade-level standards, which support their regular and active participation in learning. 

Examples include:

• Course Guide, Advancing Mathematical Language and Access For English Learners, states, “To support students who are learning English in their development of language, this curriculum includes instruction devoted to advancing language development alongside mathematics learning, and fostering language-rich environments in which there is space for all students to participate.” Mathematical Language Routines states, “Mathematical Language Routines (MLRs) are instructional routines that provide structured but adaptable formats for amplifying, assessing, and developing students' language. The MLRs included in this curriculum were selected because they simultaneously support students’ learning of mathematical practices, content, and language. They are particularly well-suited to meet the needs of linguistically and culturally diverse students who are learning mathematics while concurrently acquiring English.” MLRs are included in select activities of each unit and are described in the Teacher Guide for the lessons in which they appear within the Activity Narrative, Supporting English Learners.

• Course Guide, Universal Design for Learning and Access for Students with Disabilities, Access for Students with Disabilities states, “Supplemental instructional strategies, included in Access for Students with Disabilities of each lesson, increase access, reduce barriers and maximize learning. Each support is aligned to the Universal Design for Learning Guidelines (udlguidelines.cast.org), and based on one of the three principles of UDL, providing alternative means of engagement, representation, or action and expression. These supports offer additional ways to adjust the learning environment so that students can access activities, engage in content, and communicate their understanding. Supports are tagged, with the areas of cognitive functioning they are designed to address, to help identify and select appropriate supports for students. Designed to facilitate access to Tier 1 instruction by capitalizing on students’ strengths to address obstacles related to cognitive functions or challenges, these strategies and supports are appropriate for any student who needs additional support to access rigorous, grade-level content. Use these lesson-specific supports, as needed, to help students succeed with a specific activity, without reducing the mathematical demands of the task. Phase them out as students gain understanding and fluency. Use a UDL approach and students’ IEPs, their strengths, and their challenges to ensure access. When students may benefit from alternative means of access or support, draw on ideas from the tables below or visit udlguidelines.cast.org for more information.”

• Course Guide, Universal Design for Learning and Access for Students with Disabilities, Accessibility For Students With Visual Impairments states, “For students with visual impairments, accessibility features are built into the materials: 1. A palette of colors distinguishable to people with the most common types of color blindness. 2. Tasks and problems are designed so that success does not depend on the ability to distinguish between colors. 3. Mathematical diagrams, presented in scalable vector graphic (SVG) format, can be magnified, without loss of resolution, and rendered in Braille. 4. Where possible, text associated with images is not part of the image file, but rather included as an image caption accessible to screen readers. 5. Alt text on all images makes interpretation easier for users accessing the materials, with a screen reader. All images in the curriculum have alt text: a very short indication of the image’s contents, so that the screen reader doesn’t skip over as if nothing is there. Some images have a longer description to help students’ with visual impairments recreate the image in their mind. Understand that students with visual impairments likely will need help accessing images in lesson activities and assessments. Prepare appropriate accommodations. Accessibility experts, who reviewed this curriculum, recommended that eligible students have access to a Braille version of the curriculum materials, because a verbal description of many of the complex mathematical diagrams is inadequate to support their learning.”

• Kindergarten, Unit 4, Understanding Addition and Subtraction, Lesson 2, Activity 2, 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.”

• Grade 1, Unit 6, Length Measurements Within 120 Units, Lesson 15, Activity 2, Access for Students with Disabilities, “Action and Expression: Internalize Executive Functions. Check for understanding by inviting students to rephrase directions in their own words. Keep a display of directions visible throughout the activity. Supports accessibility for: Memory, Organization.”

• Grade 2, Unit 9, Putting It All Together, Lesson 12, Activity 2, Access for Students with Disabilities, “Action and Expression: Develop Expression and Communication. Provide students with alternatives to writing on paper. Students can share their learning by drawing or creating a picture of their story problem, or they can share verbally and visually by creating a video that tells their story. Supports accessibility for: Attention, Organization, Language.”

The materials reviewed for Illustrative Mathematics® v.360 Kindergarten through Grade 2 meet expectations for regularly providing extensions and/or opportunities for advanced students to engage with grade-level mathematics at greater depth. Examples include:

• Course Guide, What's in an IM Lesson?, Practice Problems, Exploration Problems states, “Each section has two or more exploration practice problems that offer differentiation for students ready for a greater challenge. There are two types of exploration problems. One type is a hands-on activity directly related to the material of the unit that students complete either in class if they have free time, or at home. The second type of exploration problem is more open ended and challenging. These problems go deeper into grade-level mathematics. They are not routine or procedural, and they are not ‘the same thing again but with harder numbers.’” While there are no instances where advanced students do more assignments than their classmates, materials do provide multiple opportunities for students to investigate grade-level content at greater depth.

• Course Guide, Key Structures in This Course, Authentic Use of Contexts and Suggested Launch Adaptations, Advancing Student Thinking states, “This section offers look-fors and questions to support students as they engage in an activity. Effective teaching requires supporting students as they work on challenging tasks, without taking over the process of thinking for them (Stein, Smith, Henningsen, & Silver, 2000). As teachers monitor during the course of an activity, they gain insight into what students know and are able to do. Based on these insights, the Advancing Student Thinking section provides questions that advance students’ understanding of mathematical concepts, strategies, or connections between representations.” Respond to Student Thinking states, “Most lessons end with a Cool-down to formatively assess students’ thinking in relation to the learning goal of the day. The materials offer guidance to support students in meeting the learning goals. This guidance falls into one of two categories, Next-Day Support or Prior-Unit Support, based on anticipated student responses. This guidance offers ways to continue teaching grade-level content, with appropriate and aligned practice and support for students. These suggestions range from providing students with more concrete representations in the next day’s lesson to recommending a section from a prior unit, with activities that directly connect to the concepts in the lesson.”

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

• Grade 1, Unit 7, Geometry and Time, Section A: Flat and Solid Shapes, Section A Practice Problems, Problem 12 Exploration, “1. What is the least number of pattern blocks you can use to fill in the puzzle? 2. What is the greatest number of pattern blocks you can use to fill in the puzzle? 3. Can you fill in the puzzle, using exactly 12 pattern blocks?”

• Grade 2, Unit 4, Addition and Subtraction on the Number Line, Section A: The Structure of the Number Line, Section A Practice Problems, Problem 11 Exploration, “1. Here is a picture of a thermometer. How are the thermometer and a number line alike? How are they different? 2. Here is a picture of a rain gauge. How are the rain gauge and a number line alike? How are they different?”

The materials reviewed for Illustrative Mathematics® v.360 Kindergarten through Grade 2 meet expectations for providing manipulatives, both virtual and physical, that are accurate representations of the mathematical objects they support and, when appropriate, are connected to written methods.

Course Guide, Key Structures in This Course, Purposeful Representations states, “Curriculum representations, and the grade levels at which they are used, are determined by their usefulness for particular mathematical learning goals. More concrete representations are introduced before those that are more abstract. For example, in IM Kindergarten, students begin by counting and moving objects before they represent these objects in 5- and 10-frames—to lay the foundation for understanding the base-ten system. In later grades, these familiar representations are extended so that as students encounter greater numbers, they use place-value diagrams and more symbolic methods, such as equations, to represent their understanding. When appropriate, the reasoning behind the selection of certain representations in the materials is made explicit.” Manipulatives are referenced within lessons as appropriate to support concept development. Examples include:

• Kindergarten, Unit 4, Understanding Addition and Subtraction, Lesson 6, Activity 3, students build a tower, roll to determine how many cubes to subtract, and count to determine how many cubes are left. Launch states, “Groups of 2. 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? I rolled a 3. 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 between 5–10 cubes to build a tower. 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.’”

• Grade 1, Unit 4, Numbers to 99, Lesson 20, Activity 1, students use connecting cubes in towers of 10 and singles to make given numbers with different combinations of tens and ones. Launch states, “Groups of 2. Give each group access to connecting cubes in towers of 10 and singles. Activity states, ‘Today’s challenge is to find as many ways as you can to make 94 using tens and ones. You can use cubes if they will help you. Each way you make 94 should have a different number of tens.’ 10 minutes: independent work time. 4 minutes: partner discussion. Monitor for students who: Use connecting cubes to physically break apart a ten at a time to move between representations. Use tens and ones notation. Use addition expressions.”

• Grade 2, Unit 8, Equal Groups, Lesson 7, Activity 1, students use counters to create arrays. Launch states, “Groups of 2. Give each group 3 sets of counters with 6, 7, and 9. Display Image A from the Warm-up or arrange counters to show: (Image of 4 rows: first and third row have 4 counters, second and fourth row have 2 counters.) ‘The red counters are arranged in rows, but it is not an array. How could we rearrange the counters to make an array like image B?’ (We could move the bottom 2 counters to the middle row. We could move 1 from the top row to the next row and 1 from the third row to the bottom row.) If needed, also display Image B from the Warm-Up. 1 minute: quiet think time, 1 minute: partner discussion. Share responses.” Activity states, “‘Arrange each of your sets of counters into an array. Your arrays should have the same number of counters in each row with no extra counters. Be prepared to explain how you made an array out of each set.’ ‘If you have time, try to figure out a different way to make an array out of each set of counters.’ 12 minutes: partner work time.”