3rd Grade - Gateway 3

The materials reviewed for Kendall Hunt's Illustrative Mathematics Grade 3 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 3 include: developing understanding of multiplication and division and strategies for multiplication and division within 100; developing understanding of fractions, especially unit fractions (fractions with numerator 1); developing understanding of the structure of rectangular arrays and of area; and describing and analyzing two-dimensional shapes.”

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 2, Area and Multiplication, Lesson 13, Activity 1, teachers are provided context about finding the area of figures. Narrative states, “Partially gridded figures help to prepare students to find the area of figures with only side length measurements. Students should be encouraged to find side lengths and multiply, rather than rely on counting, as the grids disappear. If students continue to draw in the squares, ask them if there is another way to find the area.” Launch states, “Groups of 2. ‘Sketch or display a rotated L-shape figure as shown. What do you notice? What do you wonder? (Students may notice: The figure is not a rectangle. It could be split into smaller rectangles. Students may wonder: Why are there no squares inside? How can I find out how many squares will cover that shape?)’ 1 minute: quiet think time. Share and record responses. ‘What information would help you find the area of this figure? (The side lengths. Being able to see the squares inside the figure.)’ 1 minute: quiet think time. Share responses. Display image from the first problem. ‘What information is given in this figure that could help you find the area? (Grid lines. The side lengths. Some of the squares.)’ Share responses.” Activity states, “‘Now work with your partner to find the area of this figure.’ 5 minutes: partner work time. Monitor for strategies for finding the side lengths and decomposing into rectangles. ‘Let's look at the first figure.’ Have students share strategies for finding the side lengths and area of figures with a partial grid. ‘Take a look at the next figure. Think about how you could find the area of this figure.’ 1 minute: quiet think time. ‘Work with your partner to find the area of this figure.’ 5 minutes: partner work time. Monitor for strategies for finding the side lengths.”

• Unit 8, Putting It All Together, Lesson 10, Lesson Synthesis provides teachers guidance for closing the lesson with representations of multiplication and division, “Today we created posters that showed ways to represent division. How does an area diagram show us the relationship between multiplication and division? (It shows that multiplying is like finding the area of a rectangle when the two side lengths are known, and dividing is like finding a side length when we know the area and the other side length.) How does a tape diagram or equal-groups diagram show multiplication and division? (Both show multiplying as a way to find the total when we know the number of groups and how many in each group, and dividing as a way to find either the number of groups or the size of each group when the total is known.) What were some aspects of the posters you saw that helped make the math your classmates used clear for you? (Clear labels on diagrams that helped me understand their thinking. Units on their answers. When other students wrote their explanations, it helped me understand their thinking.)”

The materials reviewed for Kendall Hunt's Illustrative Mathematics Grade 3 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:

• IM K-5 Math Teacher Guide, About These Materials, Unit 1, “Ratio Tables are not Elementary. In this blog post, McCallum discusses the difference between multiplication tables and tables of equivalent ratios, highlighting how K–5 arithmetic work prepares students to make sense of these tables.”

• IM K-5 Math Teacher Guide, About These Materials, Unit 3, “To learn more about the order of operations, see: A world without order (of operations). In this blog post, McCallum describes a world with only parentheses to guide the order of operations and discusses why the conventional order of operations is useful.”

• Unit 3, Wrapping Up Addition and Subtraction Within 1000, Lesson 17, Preparation, Lesson Narrative states, “Previously, students extended their understanding of addition and subtraction within 1,000 and learned how to round to the nearest ten and hundred. In this lesson, students work with two-step word problems and decide if a given answer for a two-step problem is reasonable. Students estimate answers to two-step problems and determine if each other's solutions make sense after they solve two-step word problems in a way that makes sense to them.”

• Unit 8, Putting It All Together, Lesson 8, Preparation, Lesson Narrative states, “Throughout the course, students have worked to develop fluency with multiplication and division within 100. In this lesson, they reflect on their progress and ways to improve their fluency with products within 100. Students sort multiplication facts into groups based on whether they know them right away, can find them quickly, or don’t know them yet. They then consider strategies for finding the value of unfamiliar products efficiently and practice applying those strategies. At the end of the year, grade 3 students are expected to fluently multiply and divide within 100 and to know from memory all products of two single-digit numbers. If students need additional support with the concepts in this lesson, refer back to Unit 1, Section B in the curriculum materials.”

The materials reviewed for Kendall Hunt's Illustrative Mathematics Grade 3 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 3, 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 3, Course Guide, Lesson Standards, includes all Grade 3 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 2, 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, Relating Multiplication to Division, Lesson 10, the Core Standards are identified as 3.MD.C.7c and 3.OA.C.7. 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 3, Course Guide, Scope and Sequence, Unit 1: Introducing Multiplication, Unit Learning Goals, includes an overview of how the math of this module builds from previous work in math, “In this unit, students interpret and represent data on scaled picture graphs and scaled bar graphs. Then, they learn the concept of multiplication. This is the first of four units that focus on multiplication. In this unit, students explore scaled picture graphs and bar graphs as an entry point for learning about equal-size groups and multiplication. In grade 2, students analyzed picture graphs in which one picture represented one object and bar graphs that were scaled by single units. Here, students encounter picture graphs in which each picture represents more than one object and bar graphs that were scaled by 2 or 5 units. The idea that one picture can represent multiple objects helps to introduce the idea of equal-size groups. Students learn that multiplication can mean finding the total number of objects in groups of objects each, and can be represented by a\times b. They then relate the idea of equal groups and the expression \frac{1}{5} to the rows and columns of an array. In working with arrays, students begin to notice the commutative property of multiplication. In all cases, students make sense of the meaning of multiplication expressions before finding their value, and before writing equations that relate two factors and a product. Later in the unit, students see situations in which the total number of objects is known but either the number of groups or the size of each group is not known. Problems with a missing factor offer students a preview to division. Throughout the unit, provide access to connecting cubes or counters, as students may choose to use them to represent and solve problems.”

• Grade 3, Course Guide, Scope and Sequence, Unit 2: Area and Multiplication, Unit Learning Goals, includes an overview of how the math of this module builds from previous work in math, “In this unit, students encounter the concept of area, relate the area of rectangles to multiplication, and solve problems involving area. In grade 2, students explored attributes of shapes, such as number of sides, number of vertices, and length of sides. They measured and compared lengths (including side lengths of shapes). In this unit, students make sense of another attribute of shapes: a measure of how much a shape covers. They begin informally, by comparing two shapes and deciding which one covers more space. Later, they compare more precisely by tiling shapes with pattern blocks and square tiles. Students learn that the area of a flat figure is the number of square units that cover it without gaps or overlaps. Students then focus on the area of rectangles. They notice that a rectangle tiled with squares forms an array, with the rows and columns as equal-size groups. This observation allows them to connect the area of rectangles to multiplication—as a product of the number of rows and number of squares per row. To transition from counting to multiplying side lengths, students reason about area using increasingly more abstract representations. They begin with tiled or gridded rectangles, move to partially gridded rectangles or those with marked sides, and end with rectangles labeled with their side lengths. 6\times3=18 (Tiled rectangles are shown.) Students also learn some standard units of area—square inches, square centimeters, square feet, and square meters—and solve real-world problems involving area of rectangles. Later in the unit, students find the area and missing side lengths of figures composed of non-overlapping rectangles. This work includes cases with two non-overlapping rectangles sharing one side of the same length, which lays the groundwork for understanding the distributive property of multiplication in a later unit.”

The materials reviewed for Kendall Hunt's Illustrative Mathematics Grade 3 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 K-5 Math Teacher Guide, About These Materials, 3–5, Fraction Division Parts 1–4, “In this four-part blog post, McCallum and Umland discuss fraction division. They consider connections between whole-number division and fraction division and how the two interpretations of division play out with fractions with an emphasis on diagrams, including a justification for the rule to invert and multiply. In Part 4, they discuss the limitations of diagrams for solving fraction division problems. Fraction Division Part 1: How do you know when it is division? Fraction Division Part 2: Two interpretations of division Fraction Division Part 3: Why invert and multiply? Fraction Division Part 4: Our final post on this subject (for now). Untangling fractions, ratios, and quotients. In this blog post, McCallum discusses connections and differences between fractions, quotients, and ratios.“

• 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 3 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:

• Unit 1, Introducing Multiplication, Lesson 18, Activity 1, Required Materials, “Connecting cubes or counters.” Launch states, “Give students access to connecting cubes or counters. Take a minute to represent this situation with an array. You can use drawings or objects.” Activity states, “Work with your partner to represent the next three situations with an array. Be prepared to share how you see equal groups in your array. Have students share an array for problems 2–4. Try to show both drawings and arrays made of objects.”

• Course Guide, Required Materials for Grade 3, Materials Needed for Unit 2, Lesson 6, teachers need, “Patty paper, Rulers (whole units), Scissors. Same Rectangle, Different Units (groups of 2).” 

• Course Guide, Required Materials for Grade 3, Materials Needed for Unit 5, Lesson 4, teachers need, “Colored pencils, Folders, Materials for creating a visual display. Secret Fractions Stage 1 Gameboard (groups of 2), Secret Fractions Stage 1 Cards (groups of 2).” 

• Unit 8, Putting It All Together, Lesson 14, Activity 1, Required Materials, “Picture books, ruleres.” Launch states, “Give each group picture books and a ruler. Work with your group to create an Estimation Exploration activity about measuring objects to the nearest half or fourth of an inch.”

The materials reviewed for Kendall Hunt's Illustrative Mathematics Grade 3 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 3, Wrapping Up Addition and Subtraction Within 1,000, End-of-Unit Assessment, Assessment Teacher Guide denotes standards addressed for each problem. Problem 2, 3.NBT.2, “Find the value of each sum. Explain or show your reasoning. a. 256+123 b. 389+415.”

• Unit 5, Fractions as Numbers, End-of-Unit Assessment, Assessment Teacher Guide denotes standards addressed for each problem. Problem 3, 3.NF.3, “What fraction of the large rectangle is shaded? Select all that apply.”

• Unit 8, Putting it All Together, End-of-Course Assessment, Assessment Teacher Guide denotes standards addressed for each problem. Problem 17, 3.OA.2, “a. Write a division equation for each situation. Use ‘?’ for the unknown quantity. i. There are 35 students in the room. They are seated at 7 tables, with the same number of students at each table. How many students are at each table? ii. There are 35 students in the room. There are 7 students seated at each table. How many tables of students are there? b. How are the situations the same? How are they different?”

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 3, MP6 is found in Unit 1, Lessons 2, 10, 14, 17, and 20.

• IM K-5 Math Teacher Guide, How to Use These Materials, Standard for Mathematical Practices Chart, Grade 3, MP7 is found in Unit 5, Lessons 3, 5, 11, 13, and 16. 

• IM K-5 Math Teacher Guide, How to Use These Materials, Standards for Mathematical Practice Student Facing Learning Targets, “MP3 I Can Construct Viable Arguments and Critique the Reasoning of Others. I can explain or show my reasoning in a way that makes sense to others. I can listen to and read the work of others and offer feedback to help clarify or improve the work. I can come up with an idea and explain whether that idea is true.”

• IM K-5 Math Teacher Guide, How to Use These Materials, Standards for Mathematical Practice Student Facing Learning Targets, “MP8 I Can Look for and Express Regularity in Repeated Reasoning. I can identify and describe patterns and things that repeat. I can notice what changes and what stays the same when working with shapes, diagrams, or finding the value of expressions. I can use patterns to come up with a general rule.”

The materials reviewed for Kendall Hunt's Illustrative Mathematics Grade 3 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 1, Introducing Multiplication, End-of-Unit Assessment supports the full intent MP2 (Reason abstractly and quantitatively) as students interpret an array and relate the quantities to an equation. For example, Problem 6 states, “Kiran has 18 cards. He arranges the cards in 3 rows. Each row has the same number of cards. a. Explain how the 3\times?=18 equation relates to Kiran’s cards. b. How many cards are in each row? Explain how you know.”

• Unit 4, Relating Multiplication to Division, End-of-Unit Assessment develops the full intent of 3.OA.8 (Solve two-step word problems using the four operations. Represent these problems using equations with a letter standing for the unknown quantity. Assess the reasonableness of answers using mental computation and estimation strategies including rounding. This standard is limited to problems posed with whole numbers and having whole-number answers; students should know how to perform operations in the conventional order when there are no parentheses to specify a particular order [Order of Operations]). For example, Problem 7 states, “There are 278 students at the school. 197 go home by foot or by car. The rest take a school bus. a. How many students take the bus home? Explain or show your reasoning. b. Each bus holds 35 students. Explain why the students who take the bus cannot all fit in 2 buses. C. There are 3 buses and each bus carries the same number of students. How many students are in each bus? Explain or show your reasoning.“

• Unit 6, Measuring Length, Time, Liquid Volume, and Weight, End-of-Unit Assessment supports the full intent of MP6 (Attend to precision) as students choose objects that weigh about one kilogram. For example, Problem 2 states, “Select 3 items that weigh about 1 kilogram. A. pencil, B. laptop computer, C. pineapple, D. paper clip, E. car, F. dictionary.”

• Unit 7, Two-dimensional Shapes and Perimeter, End-of-Unit Assessment develops the full intent of 3.MD.8 (Solve real world and mathematical problems involving perimeters of polygons, including finding the perimeter given the side lengths, finding an unknown side length, and exhibiting rectangles with the same perimeter and different areas or with the same area and different perimeters). For example, Problem 3 states, “Find the perimeter of the rectangle. Explain or show your reasoning.” A picture of a rectangular shape with sides of 6 inches and 10 inches is shown.

The materials reviewed for Kendall Hunt's Illustrative Mathematics Grade 3 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, Introducing Multiplication, Lesson 6, Activity 1, Narrative, Access for Students with Disabilities, “Representation: Access for Perception. Provide access to pattern blocks to model the collection of pattern blocks in the student-facing task statement. Supports accessibility for: Organization, Visual-Spatial Processing.”

• Unit 3, Wrapping Up Addition and Subtraction Within 1,000, Lesson 5, Activity 1, 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.”

• Unit 4, Relating Multiplication to Division, Lesson 3, Activity 3, Narrative, Access for Students with Disabilities, “Engagement: Provide Access by Recruiting Interest. Leverage choice around perceived challenge. Invite students to select at least 3 of the 6 problems to complete. Supports accessibility for: Organization, Attention, Social-emotional skills.”

• Unit 7, Two-dimensional Shapes and Perimeter, 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: Visual-Spatial Processing.”

The materials reviewed for Kendall Hunt's Illustrative Mathematics Grade 3 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 2, Area and Multiplication, Section A: Concepts of Area Measurement, Problem 11, Exploration, “How many different rectangles can you make with 36 square tiles? Describe or draw the rectangles. How are the rectangles the same? How are they different?”

• Unit 3, Wrapping Up Addition and Subtraction Within 1,000, Section A: Add Within 1,000, Problem 14, Exploration, “Write an addition problem with 3-digit numbers that you think is well suited for each of the following methods. Then, find the value of the sum using that method: mental strategies, base-ten blocks, an algorithm.”

• Unit 5, Fractions as Numbers, Section C: Equivalent Fractions, Problem 6, Exploration, “If you continue to fold fraction strips, how many parts can you fold them into? Can you fold them into 100 equal parts?”

• Unit 7, Two-dimensional Shapes and Perimeter, Section C: Expanding on Perimeter, Problem 4, Exploration, “Clare draws a rectangle. She tells you that the perimeter is 36. What rectangle could Clare have drawn? Then she tells you that her rectangle has the biggest area possible. What rectangle could Clare have drawn?”

The materials reviewed for Kendall Hunt's Illustrative Mathematics Grade 3 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 2, Area and Multiplication, Lesson 15, Activity 1, Teaching Notes, Access for English Learners, “MLR5 Co-Craft Questions. Display the image of the floor plan, and invite students to write a list of possible mathematical questions they could ask about the situation. Invite students to compare their questions, ‘What do these questions have in common? How are they different?’ Amplify questions related to comparison and areas of rectangles. Advances: Reading, Writing.”

• Unit 3, Wrapping Up Addition and Subtraction Within 1,000, Lesson 4, Activity 1, Teaching notes, Access for English Learners, “MLR7 Compare and Connect. Synthesis: Invite groups to prepare a visual display that shows the strategy they used to find the value of the sums. Encourage students to include details that will help others interpret their thinking. For example, specific language, using different colors, shading, arrows, labels, notes, diagrams or drawings. Give students time to investigate each others’ work. During the whole-class discussion, ask students, ‘What did the representations have in common?’, ‘How were they different?’, ‘How did the total sum show up in each method?’ Advances: Representing, Conversing.”

• Unit 8, Putting It All Together, Lesson 14, Activity 2, Teaching Notes, Access for English Learners, “MLR8 Discussion Supports. Synthesis: Display sentence frames to support whole-class discussion: “I learned . . . “ “The next time I create an estimation exploration, I will . . . “  Advances: Speaking, Representing.”

The materials reviewed for Kendall Hunt's Illustrative Mathematics Grade 3 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, Wrapping Up Addition and Subtraction Within 1,000, Lesson 16, Activity 2, students use index cards to play a rounding game. Launch states, “‘We’re going to play a game in which you have to guess a mystery number that someone in your group writes down.’ Choose a mystery number and give the class three clues. Play a round of the game with the class and discuss the clues. Consider using 275 and these clues: My mystery number is odd. My mystery number rounds to 300. My mystery number is between 270 and 278. ‘You’ll give your group three clues by finishing three sentences. The first clue should tell whether the number is even or odd.’ Take a couple minutes to choose a mystery number and write down your three clues.”

• Unit 4, Relating Multiplication to Division, Lesson 10, Activity 2, identifies colored pencils, crayons or markers as strategies/tools for students to represent expressions. Launch states, “Give students access to colored pencils, crayons, or markers.” Activity states, “Mark or shade each diagram to represent how each student found the area.”

• Unit 7, Two-dimensional Shapes and Perimeter, Lesson 11, Activity 2, references dot paper, scissors, and tape to help students draw rectangles and reason about perimeter and area. Launch states, “Groups of 2. Display the visual display labeled with each of the four perimeters in the first problem. Give each group 2 sheets of dot paper, scissors, and access to tape.” Activity states, “‘Work with your partner to complete the first problem.’ 6–8 minutes: partner work time. ‘Choose which rectangles you want to share and put them on the appropriate poster. Try to look for rectangles that are different from what other groups have already placed.’ 3–5 minutes: partner work time. Monitor to make sure each visual display has a variety of rectangles. When all students have put their rectangles on the posters, ask students to visit the posters with their partner and discuss one thing they notice and one thing they wonder about the rectangles. 5 minutes: gallery walk.”

• Unit 8, Putting It All Together, Lesson 11, Activity 1, references dice to play a game called Race to 1, where students practice dividing. Launch states, “‘Let’s look at a sample game. Jada rolled a 3 on her first turn, then rolled 2 a few times afterwards. Talk with your partner about what her next move should be if she rolls 2 on her next turn.’ (She should divide 4 or 6 by 2 because those moves get her really close to one.) Give each group a number cube.” Activity states, “Play Race to 1 with your partner.”