2nd Grade - Gateway 3

The materials reviewed for Kendall Hunt's Illustrative Mathematics 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.

Materials provide comprehensive guidance that will assist teachers in presenting the student and ancillary materials. This is located within IM Curriculum, How to Use These Materials, and the Course Guide, Scope and Sequence. Examples include:

• IM Curriculum, How To Use These Materials, Design Principles, Coherent Progression provides an overview of the design and implementation guidance for the program, “The overarching design structure at each level is as follows: 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.”

• Course Guide, Scope and Sequence, provides an overview of content and expectations for the units, “The big ideas in 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. In these materials, particularly in units that focus on addition and subtraction, teachers will find terms that refer to 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.”

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 within the Warm-up, Activities, and Cool-down provide useful annotations. Examples include:

• Unit 1, Adding, Subtracting, and Working with Data, Lesson 3, Activity 1, teachers are provided context to support students finding numbers within 20 to make an equation true. Narrative states, “In this activity, students learn stage 3 of the What’s Behind My Back center. In this new stage, called 20 cubes, students work with 20 cubes, organized into two towers of 10 cubes. One partner snaps the tower and puts one part behind their back and shows the other part to their partner. The other partner figures out how many cubes are behind their partner’s back. Students record an addition equation with a blank to represent the missing cubes. Students may write equations with the blank as the first or second addend. Ask students to explain what each number and blank in the equation represents in the context of the center activity (MP2).” Launch states, “Groups of 2. Give each group 20 connecting cubes and a recording sheet. ‘We are going to play What’s Behind My Back, this time with 20 cubes. How did you figure out how many connecting cubes were behind your partner’s back last time? (I thought about an addition expression that would make 10. I subtracted what they showed me from 10.) Let’s play a round with 20.’ Show students 2 towers of 10 cubes. Put the towers behind your back. Break off and display 8 of the cubes. ‘This time when you play, you are going to record an addition equation with a blank to represent the missing cubes, before you figure out how many are behind your partner’s back. What equation should we record? ($$8+$$___$$=20$$).’ 30 seconds: quiet think time. Share responses. ‘How many cubes are behind my back? How do you know? (12 because 2 more makes 10 and then here’s another tower of 10.)’ 30 seconds: quiet think time. 30 seconds: partner discussion. ‘Play with your partner. Don’t forget to record an equation each round.’” Activity Synthesis states, “Display 9 cubes. ‘What’s an addition equation I can write to represent the number of cubes you know and the number of cubes you need to figure out? ($$9+$$___$$=20$$). Tell your partner how you can figure out how many cubes are missing.’ Monitor for students who talk about making a 10 and knowing there is one more 10. Share responses.”

• Unit 5, Numbers to 1000, Lesson 2, Warm-Up, provides information to the teacher about the importance of students being able to count by 10, as a precursor to counting by larger numbers. Narrative states, “The purpose of this Choral Count is for students to practice counting by 10 beyond 120 and notice patterns in the count. These understandings help students develop fluency and will be helpful later in this lesson when students will need to be able to recognize multiples of 100 written as numerals and make connections between groups of 10 tens and hundreds.”

The materials reviewed for Kendall Hunt's Illustrative Mathematics 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.

Within the Teacher’s Guide, IM Curriculum, About These Materials, 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 grade, so that teachers can improve their own understanding of the content. Professional articles support teachers with learning opportunities about topics such as ensuring mathematical success for all, early understanding of equality, and repeating patterns. Additionally, each lesson provides teachers with a lesson narrative, including adult-level explanations and examples of the more complex grade/course-level concepts. Examples include:

• Unit 1, Adding, Subtracting, and Working with Data, Lesson 16, Preparation, 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.”

• IM K-5 Math Teacher Guide, About These Materials, Unit 4, “To learn more about the essential nature of the number line (which is introduced in this unit) in mathematics beyond grade 2, see: The Nuances of Understanding a Fraction as a Number. In this blog post, Gray discusses the role the number line plays in student understanding of fractions as numbers. Why is 3–5=3+(-5)? In this blog post, McCallum discusses the use of the number line in introducing negative numbers.” 

• Unit 7, Adding and Subtracting Within 1000, Lesson 2, Preparation, Lesson Narrative states, “In grade 1, students added and subtracted multiples of 10 within 100. In a previous unit, students represented three-digit numbers with base-ten blocks, drawings, and words. Students used equations to represent three-digit numbers as sums of the value of hundreds, tens, and ones using the number and name of each unit (235 = 2 hundreds + 3 tens + 5 ones) and using expanded form (235=200+30+5). In this lesson, students add and subtract three-digit numbers and multiples of 10 and 100 using what they know about tens and hundreds. Students compare representations such as base-ten blocks, base-ten diagrams, and equations to understand that when adding or subtracting multiples of 10, the tens place changes and when adding or subtracting multiples of 100 the hundreds place changes (MP7, MP8).”

• IM K-5 Math Teacher Guide, About These Materials, Unit 8, “What is Multiplication? In this blog post, McCallum discusses multiplication beyond repeated addition—as equal groups. The foundation of this understanding is laid in this unit of grade 2.”

The materials reviewed for Kendall Hunt's Illustrative Mathematics Grade 2 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 2, Course Guide, Lesson Standards includes a table with each grade-level lesson (in columns) and aligned grade-level standards (in rows). Teachers can search any lesson for the grade and identify the standard(s) that are addressed within.

• Grade 2, Course Guide, Lesson Standards, includes all Grade 2 standards and the units and lessons each standard appears in. Teachers can search a standard for the grade and identify the lesson(s) where it appears within materials.

• Unit 5, 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 7, Adding and Subtracting within 1,000, Lesson 1, the Core Standards are identified as 2.NBT.A.2, 2.NBT.B.7, and 2.NBT.B.3. Lessons contain a consistent structure: a Warm-up that includes Narrative, Launch, Activity, Activity Synthesis; Activity 1, 2, or 3 that includes Narrative, Launch, Activity; an Activity Synthesis; a Lesson Synthesis; and 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 outlining the content standards addressed within as well as a narrative describing relevant prior and future content connections. Examples include:

• Grade 2, Course Guide, Scope and Sequence, Unit 3: Measuring Length, Unit Learning Goals, includes an overview of how the math of this module builds from previous work in math, “This unit introduces students to standard units of lengths in the metric and customary systems. In grade 1, students expressed the lengths of objects in terms of a whole number of copies of a shorter object laid without gaps or overlaps. The length of the shorter object serves as the unit of measurement. Here, students learn about standard units of length: centimeters, meters, inches, and feet. They examine how different measuring tools represent length units, learn how to use the tools, and gain experience in measuring and estimating the lengths of objects. Along the way, students notice that the length of the same object can be described with different measurements and relate this to differences in the size of the unit used to measure.”

• Grade 2, Course Guide, Scope and Sequence, Unit 6: Geometry, Time, and Money, Unit Learning Goals, includes an overview of how the math of this module builds from previous work in math, “In this unit, students transition from place value and numbers to geometry, time, and money. In grade 1, students distinguished between defining and non-defining attributes of shapes, including triangles, rectangles, trapezoids, and circles. Here, they continue to look at attributes of a variety of shapes and see that shapes can be identified by the number of sides and vertices (corners). Students then study three-dimensional (solid) shapes, and identify the two-dimensional (flat) shapes that make up the faces of these solid shapes. Next, students look at ways to partition shapes and create equal shares. They extend their knowledge of halves and fourths (or quarters) from grade 1 to now include thirds. Students compose larger shapes from smaller equal-size shapes and partition shapes into two, three, and four equal pieces. As they develop the language of fractions, students also recognize that a whole can be described as 2 halves, 3 thirds, or 4 fourths, and that equal-size pieces of the same whole need not have the same shape. Which circles are not examples of circles partitioned into halves, thirds, or fourths? Later, students use their understanding of halves and fourths (or quarters) to tell time. In grade 1, they learned to tell time to the half hour. Here, they relate a quarter of a circle to the features of an analog clock. They use ‘quarter past’ and ‘quarter till’ to describe time, and skip-count to tell time in 5-minute intervals. They also learn to associate the notation ‘a.m.’ and ‘p.m.’ with their daily activities. To continue to build fluency with addition and subtraction within 100, students conclude the unit with a money context. They skip-count, count on from the largest value, and group like coins, and then add or subtract to find the value of a set of coins. Students also solve one- and two-step story problems involving sets of dollars and different coins, and use the symbols $ and ¢.”

The materials reviewed for Kendall Hunt's Illustrative Mathematics Grade 2 meet expectations for providing explanations of the instructional approaches of the program and identification of the research-based strategies. 

The IM K-5 Math Teacher Guide, Design Principles, outlines the instructional approaches of the program, “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 mathematics through a coherent progression, and provides teachers the opportunity to deepen their knowledge of mathematics, student thinking, and their own teaching practice. In order to design curriculum and professional learning materials that support student and teacher learning, we need to be explicit about the principles that guide our understanding of mathematics teaching and learning. This document outlines how the components of the curriculum are designed to support teaching and learning aligning with this belief.” Examples of the design principles include:

• IM K-5 Math Teacher Guide, Design Principles, All Students are Capable Learners of Mathematics, “All students, each with unique knowledge and needs, enter the mathematics learning community as capable learners of meaningful mathematics. Mathematics instruction that supports students in viewing themselves as capable and competent must leverage and build upon the funds of knowledge they bring to the classroom. In order to do this, instruction must be grounded in equitable structures and practices that provide all students with access to grade-level content and provide teachers with necessary guidance to listen to, learn from, and support each student. The curriculum materials include classroom structures that support students in taking risks, engaging in mathematical discourse, productively struggling through problems, and participating in ways that make their ideas visible. It is through these classroom structures that teachers will have daily opportunities to learn about and leverage their students’ understandings and experiences and how to position each student as a capable learner of mathematics.”

• IM K-5 Teacher Guide, Design Principles, 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 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.” 

• IM K-5 Teacher Guide, Design Principles, Learning Mathematics by Doing Mathematics, “Students learn mathematics by doing mathematics, rather than by watching someone else do mathematics or being told what needs to be done. Doing mathematics can be defined as learning mathematical concepts and procedures while engaging in the mathematical practices—making sense of problems, reasoning abstractly and quantitatively, making arguments and critiquing the reasoning of others, modeling with mathematics, making appropriate use of tools, attending to precision in their use of language, looking for and making use of structure, and expressing regularity in repeated reasoning. By engaging in the mathematical practices with their peers, students have the opportunity to see themselves as mathematical thinkers with worthwhile ideas and perspectives. ‘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 ones doing the problem solving to learn the mathematics. The activities and routines are designed to give teachers opportunities to see what students already know and what they can notice and figure out before having concepts and procedures explained to them.”

Research-based strategies are cited and described within the IM Curriculum and can be found in various sections of the IM K-5 Math Teacher Guide. Examples of research-based strategies include:

• IM Certified, Blog, Making Sense of Story Problems, Deborah Peart, How can we support “sense-making” of stories in math class?, “The mission of Illustrative Mathematics is to create a world where learners know, use, and enjoy mathematics. By using stories to help students see math in the world around them and recognize the ways in which using math is a part of their daily lives, word problems can become an enjoyable part of math learning. This starts with calling word problems ‘story problems’ in the early grades. From there, other supports embedded in the curriculum include: providing relevant contexts and images with which students can engage, supporting reading comprehension with routines and instructional practices, like Act it Out and Three Reads, encouraging students to use visual representations to support sense-making, inviting students to write their own math stories and ask questions that can be answered by them.”

• IM K-5 Math Teacher Guide, 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.”

• IM K-5 Math Teacher Guide, Key Structures in This Course, 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.”

The materials reviewed for Kendall Hunt's Illustrative Mathematics Grade 2 meet expectations for providing a comprehensive list of supplies needed to support instructional activities. 

The Course Guide includes a section titled “Required Materials” that includes a breakdown of materials needed for each unit and for each lesson. Additionally, specific lessons outline materials to support the instructional activities and these can be found on the “Preparation” tab in a section called “Required Materials.” Examples include:

• Course Guide, Required Materials for Grade 2, Materials Needed for Unit 1, Lesson 4, teachers need, “Connecting cubes, Number cards 0–10, How Close? Stage 1 Recording Sheet (groups of 1).” 

• Unit 2, Adding and Subtracting within 100, Lesson 7, Activity 1, Required Materials, “Base-ten blocks, Connecting cubes.” Launch states, “Groups of 2. Give students access to connecting cubes and base-ten blocks.” Activity states, “‘Find the value of each difference and share your method and solution with your partner.’ 7 minutes: independent work time. MLR8 Discussion Supports ‘After your partner shares their method, repeat back what they told you.’ Display the sentence frames: I heard you say . . . . Our methods are alike because . . . . Our methods are different because . . . . 5 minutes: partner discussion. Monitor for students who use base-ten blocks to show decomposing a ten.”

• Unit 4, Addition and Subtraction on the Number Line, Lesson 5, Activity 2, Required Materials, “Chart paper, Markers. Materials to Copy: Order Numbers on the Number Line Cards.” Launch states, “Groups of 3. Give each group chart paper, markers, and a set of number cards.” Activity states, “‘You will be working with your group to arrange the number cards on the number line. Take turns picking a card and placing it near its spot on the number line. Explain how you decided where to place your card. If you think you need to rearrange other cards, explain why. When you agree that you have placed all the numbers in the right spots, mark each of the numbers on your cards with a point on the number line. Label each point with the number it represents.’ 10 minutes: small-group work time.”

• Course Guide, Required Materials for Grade 2, Materials Needed for Unit 8, Lesson 6, teachers need, “Dry erase markers, Materials from previous centers, Sheet protectors, Write the Number Stage 4 Gameboard (groups of 2).”

The materials reviewed for Kendall Hunt's Illustrative Mathematics Grade 2 meet expectations for having assessment information included in the materials to indicate which standards are assessed. 

End-of-Unit Assessments and the End-of-Course Assessments consistently and accurately identify grade-level content standards. Content standards can be found in each Unit Assessment Teacher Guide. Examples from formal assessments include:

• Unit 6, Geometry, Time, and Money, End-of-Unit Assessment, Assessment Teacher Guide denotes standards addressed for each problem. Problem 5, 2.MD.7, “a. Jada gets up in the morning at 6:45. Show the time on the clock face. Then circle a.m. or p.m. b. Jada goes to bed at the time on the clock. Write the time and circle a.m. or p.m.”

• Unit 8, Equal Groups, End-of-Unit Assessment, Assessment Teacher Guide denotes standards addressed for each problem. Problem 5, 2.OA.3, “For each number, decide whether the number is even or odd. Write each even number as the sum of 2 equal addends. a. 6   b. 11  c. 14.”

• Unit 9, Putting it All Together, End-of-Course Assessment, Assessment Teacher Guide denotes standards addressed for each problem. Problem 9, 2.NBT.7, “Find the value of each expression. Show your thinking. a. 347+583. b. 612-174.”

Guidance is provided within materials for assessing progress of the Mathematical Practices. According to IM K-5 Math Teacher Guide, How to Use These Materials, “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 Math Teacher Guide, How to Use These Materials, Standards for Mathematical Practices Chart, Grade 2, MP3 is found in Unit 3, Lessons 4, 8, 9, 12, and 16.

• IM K-5 Math Teacher Guide, How to Use These Materials, Standard for Mathematical Practices Chart, Grade 2, MP8 is found in Unit 6, Lessons 9, 12, and 16. 

• IM K-5 Math Teacher Guide, How to Use These Materials, Standards for Mathematical Practice Student Facing Learning Targets, “MP6 I Can Attend to Precision. I can use units or labels appropriately. I can communicate my reasoning using mathematical vocabulary and symbols. I can explain carefully so that others understand my thinking. I can decide if an answer makes sense for a problem.”

• IM K-5 Math Teacher Guide, How to Use These Materials, 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 Kendall Hunt's Illustrative Mathematics Grade 2 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 some end of lesson cool-downs, interviews, and Checkpoint Assessments for each section. 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 from summative assessments include:

• Unit 5, Numbers to 1,000, End-of-Unit Assessment develops the full intent of 2.NBT.3 (Read and write numbers to 1000 using base-ten numerals, number names, and expanded form). For example, Problem 3 states, “Select 2 ways to represent the number 518. A. 500+10+8 B. 5+1+8 C. 5 hundreds and 18 tens D. 51 tens and 8 ones E. 4 hundreds and 11 tens.”

• Unit 6, Geometry, Time, and Money, End-of-Unit Assessment develops the full intent of 2.G.1 (Recognize and draw shapes having specified attributes, such as a given number of angles or a given number of equal faces. Sizes are compared directly or visually, not compared by measuring. Identify triangles, quadrilaterals, pentagons, hexagons, and cubes). For example, Problem 1 states, “Draw a quadrilateral with one square corner and two equal sides.”

• Unit 8, Equal Groups, End-of-Unit Assessment supports the full intent of MP3 (Construct viable arguments and critique the reasoning of others) as students reason about a situation involving sharing. For example, Problem 6 states, “​​Here are some pattern blocks that Jada and Diego want to share. (There are images of some pattern blocks shown.) a. Explain why there are an even number of trapezoids. b. Jada says that she and Diego can share the pattern blocks so they each have 9 pattern blocks. Explain why Jada is correct. c. Can Jada and Diego share all of the pattern blocks so that they each have the same set of pattern block shapes? Explain or show your reasoning.”

• Unit 9, Putting It All Together, End-of-Course Assessment supports the full intent of MP2 (Reason abstractly and quantitatively) as students compare numbers within 1,000. For example, Problem 2 states, “Fill in each blank with <, =, or > to make the statements true. a. 675 ___ 576 b. 98 ___ 205 c. 500+40+3___$$543$$ d. 675___$$600+70+1$$.”

The materials reviewed for Kendall Hunt's Illustrative Mathematics Grade 2 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 as suggestions are outlined within each lesson and parts of each lesson. According to the IM K-5 Teacher Guide, Universal Design for Learning and 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 of supports for special populations include: 

• Unit 1, Adding, Subtracting, and Working with Data, Lesson 3, Activity 1, Narrative, Access for Students with Disabilities, “Representation: Internalize Comprehension. Synthesis: Invite students to identify which details were important or most useful to pay attention to. Display the sentence frame, ‘To figure out how many cubes are behind my partner’s back, I can . . . .’ Supports accessibility for: Visual-Spatial Processing.”

• Unit 3, Measuring Length, Lesson 5, Activity 2, Narrative, Access for Students with Disabilities, “Engagement: Provide Access by Recruiting Interest. Synthesis: Optimize meaning and value. Invite students to share people they know or specific jobs they recognize that may use the measuring tools they have been exposed to. ‘When and why might someone use these measuring tools?’ Supports accessibility for: Conceptual Processing.”

• Unit 6, Geometry, Time, and Money, Lesson 8, Activity 2, Narrative, Access for Students with Disabilities, “Representation: Develop Language and Symbols. Synthesis: Maintain a visible display to record images of ways to make thirds (also add fourths and halves) to reiterate that fractions have equal parts and can be made in certain ways. Invite students to suggest details (words or pictures) that will help them remember the meaning of the fractions. Supports accessibility for: Memory, Language, Organization.”

• Unit 7, Adding and Subtracting within 1,000, Lesson 6, Activity 2, Narrative, Access for Students with Disabilities, “Action and Expression: Develop Expression and Communication. Synthesis. Identify connections between strategies that result in the same outcomes but use differing approaches. Supports accessibility for: Conceptual Processing.”

The materials reviewed for Kendall Hunt's Illustrative Mathematics Grade 2 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 To Use The Materials, 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 directly related to the material of the unit that students can do 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 1, Adding, Subtracting, and Working with Data, Section A: Add and Subtract Within 20, Problem 10, Exploration, “Clare has a set of cards numbered 1, 2, 3, 4, 5, 6, 7, 8, 9. She picks out seven of the cards. Clare was NOT able to make 20 with 3 of her 7 cards. Which cards do you think she picked out if she was NOT able to make 20?”

• Unit 2, Adding and Subtracting within 100, Section A: Add and Subtract, Problem 7, Exploration, “Jada added 3 different numbers between 1 and 9 and got 20. What could Jada’s numbers be? Give three different examples. If Jada used 6, what are the other two numbers? Explain your reasoning.”

• Unit 4, Addition and Subtraction on the Number Line, Section B: Add and Subtract on a Number Line, Problem 8, Exploration, “Using addition or subtraction, how many equations can you make with these three numbers: 20, 13, 7? Draw number lines to match each of the equations you wrote. How are the number lines the same? How are they different?”

• Unit 8, Equal Groups, Section B: Rectangular Arrays, Problem 10, Exploration, “What are some things in the classroom that you know there are an even number of without counting them? Explain your reasoning.”

The materials reviewed for Kendall Hunt's Illustrative Mathematics Grade 2 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 IM K-5 Math Teacher Guide, Mathematical Language Development and Access for English Learners, “In a problem-based mathematics classroom, sense-making and language are interwoven. Mathematics classrooms are language-rich, and therefore language demanding learning environments for every student. The linguistic demands of doing mathematics include reading, writing, speaking, listening, conversing, and representing (Aguirre & Bunch, 2012). Students are expected to say or write mathematical explanations, state assumptions, make conjectures, construct mathematical arguments, and listen to and respond to the ideas of others. In an effort to advance the mathematics and language learning of all students, the materials purposefully engage students in sense-making and using language to negotiate meaning with their peers. To support students who are learning English in their development of language, this curriculum includes instruction devoted to fostering language development alongside mathematics learning, fostering language-rich environments where there is space for all students to participate.” The series provides the following 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 in each lesson. According to the IM K-5 Math Teacher Guide, Mathematical Language Development and Access for English Learners, “Mathematical Language Routines (MLRs) are instructional routines that provide structured but adaptable formats for amplifying, assessing, and developing students' language. MLRs are included in select activities in each unit to provide all students with explicit opportunities to develop mathematical and academic language proficiency. These “embedded” MLRs are described in the teacher notes for the lessons in which they appear.” Examples include:

• Unit 1, Adding, Subtracting, and Working with Data, Lesson 2, Activity 1, Teaching notes, Access for English Learners, “MLR8 Discussion Supports. Provide all students with an opportunity for verbal output. Invite students to read each expression they create to their partner. Amplify words and phrases such as: add, addition, sum, take away, difference, value, and expression. Advances: Speaking, Listening, Representing.”

• Unit 2, Adding and Subtracting within 100, Lesson 12, Activity 1, Activity, “MLR6 Three Reads, Display only the problem stem for the first problem, without revealing the question. ‘We are going to read this problem 3 times.’ 1st Read: ‘Clare captured 54 seeds. Han captured 16 fewer seeds. What is this story about?’ 1 minute: partner discussion. Listen for and clarify any questions about the context. 2nd Read: ‘Clare captured 54 seeds. Han captured 16 fewer seeds. What are all the things we can count in this story?’ (Clare’s seeds. Han’s seeds. The difference between their seeds.) 30 seconds: quiet think time. 1 minute: partner discussion. Share and record all quantities. Reveal the question. 3rd Read: Read the entire problem, including the question aloud. Ask students to open their books. ‘Which of the diagrams shows a way we could represent this problem?’ (See Student Responses for the first problem). 30 seconds: quiet think time. 1–2 minutes: partner discussion. Share responses. ‘Read each story with your partner. Then choose a diagram that matches on your own. When you have both selected a match, compare your choices and explain why the diagram matches the story or why other diagrams do not match the story.’ 5 minutes: partner work time.”

• Unit 7, Adding and Subtracting within 1,000, Lesson 13, Activity 1, Teaching Notes, Access for English Learners, “MLR5 Co-Craft Questions. Keep books or devices closed. Display only the images, without revealing the question, and ask students to write down possible mathematical questions that could be asked about the situation. Invite students to compare their questions before revealing the task. Ask, ‘What do these questions have in common? How are they different?’ Reveal the intended questions for this task and invite additional connections. Advances: Reading, Writing.”

The materials reviewed for Kendall Hunt's Illustrative Mathematics Grade 2 meet expectations for providing manipulatives, physical but not virtual, that are accurate representations of the mathematical objects they represent and, when appropriate, are connected to written methods.

Suggestions and/or links to manipulatives are consistently included within materials, often in the Launch portion of lessons, to support the understanding of grade-level math concepts. Examples include: 

• Unit 3, Measuring Length, Lesson 9, Activity 1, references inch tiles, rulers, and tape to introduce students to measurement units and tools. Launch states, “Give each student an inch ruler and access to inch tiles. ‘You have measured the length of objects and the sides of shapes using inches. If you are measuring longer objects, like the fish in the warm-up, you might want to use a different unit. A foot is a longer length unit in the U.S. Customary Measurement System. A foot is the same length as 12 inches. When we measure a length that starts at 0 on the ruler and ends at the 12, we can say the length is 12 inches or we can say the length is 1 foot. What are some things you see around the classroom that are about 1 foot long?’"

• Unit 5, Numbers to 1,000, Lesson 11, Activity 2, identifies number cards 0-10 to support students’ reasoning about place value and the greatest possible digit. Launch states, “Give each group a set of number cards and each student a recording sheet. ‘Now you will be playing the Greatest of Them All center with your partner. You will try to make the greatest three-digit number you can.’ Display spinner and recording sheet. Demonstrate spinning. ‘If I spin a (2), I need to decide whether I want to put it in the hundreds, tens, or ones place to make the largest three-digit number. Where do you think I should put it? (I think it should go in the ones place because it is a low number. In the hundreds place, it would only be 200.) At the same time, my partner is spinning and building a number, too. Take turns using the spinner and writing each digit in a space. Read your comparison aloud to your partner.’”

• Unit 7, Adding and Subtracting within 1,000, Lesson 2, Activity 1, base-ten blocks and numbers cubes are identified to support adding and subtracting three-digit numbers. Launch states, “Groups of 2. Give each group base-ten blocks and a number cube.” Activity states, “‘Work with your partner to show each number with base-ten blocks. Take turns rolling the number cube to see how many tens or hundreds to add or subtract.’ 10 minutes: partner work time. Monitor for students to share their equations for the number of hundreds they subtract from 805.”

• Unit 9, Putting It All Together, Lesson 8, Activity 1, references number cards to play a game called Heads Up, practicing addition and subtraction within 100. Launch states, “Give students number cards.” Activity states, “‘We are going to play a game called Heads Up.’ Demonstrate with 2 students. ‘Players A and B pick a card and put it on their foreheads without looking at it. I am Player C. My job is to find the value of the sum and tell my group. Players A and B use the other player’s number and the value of the sum to determine what number is on their head. Finally, each player writes the equation that represents what they did.’ Demonstrate writing an equation for each of the players. After each round switch roles and play again.”