4th Grade - Gateway 3

The materials reviewed for Kendall Hunt's Illustrative Mathematics Grade 4 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 4 include: developing understanding and fluency with multi-digit multiplication, and developing understanding of dividing to find quotients involving multi-digit dividends; developing an understanding of fraction equivalence, addition and subtraction of fractions with like denominators, and multiplication of fractions by whole numbers; understanding that geometric figures can be analyzed and classified based on their properties, such as having parallel sides, perpendicular sides, particular angle measures, and symmetry.”

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, Fraction Equivalence and Comparison, Lesson 4, Activity 1, teachers are provided context to help students reason about fractions and locate them on a number line. Narrative states, “This activity serves two main goals: to revisit the idea of equivalence from grade 3, and to represent non-unit fractions with denominator 10 and 12. Students use diagrams of fraction strips, which allow them to see and reason about fractions that are the same size. In the next activity, students will apply a similar process of partitioning to represent these fractional parts on number lines.” Launch states, “Groups of 2. Give each student a straightedge. ‘Here is a diagram of fraction strips you saw before, with two new rows added. How can we show tenths and twelfths in the two rows? Think quietly for a minute.’ 1 minute: quiet think time.” Activity states, “‘Work on the first two questions on your own. Afterward, discuss your responses with your partner. Use a straightedge when drawing your diagram.’ 5–6 minutes: independent work time. 2–3 minutes: partner discussion. Monitor for students who found the size of tenths and twelfths as noted in student responses. Pause for a brief discussion. Select students who used different strategies to find tenths and twelfths to share. After each person shares, ask if others in the class did it the same way or if they had anything to add to the explanation. ‘Look at your completed diagram. What can you say about the relationship between \frac{1}{5} and \frac{1}{10}? (There are two \frac{1}{10}s in every \frac{1}{5}. One fifth is twice one tenth. One fifth is the same size as 2 tenths.) What can you say about the relationship between \frac{1}{6} and \frac{1}{12}? (There are two \frac{1}{12}s in every \frac{1}{6}. One sixth is twice one twelfth. One sixth is the same size as 2 twelfths.) Take 2 minutes to answer the last question.’ 2 minutes: independent or group work time.”

• Unit 5, Multiplicative Comparison and Measurement, Lesson 7, Warm-up, provides teachers guidance about how to support students when working with length measurements. Launch states, “Groups of 2. Display the image. ‘What do you notice? What do you wonder?’ 1 minute: quiet think time.” Activity states, “‘Discuss your thinking with your partner.’ 1 minute: partner discussion. Share and record responses.” Activity Synthesis states, “Consider sharing that the large insect is a stick insect. (The longest species ever found measured more than 60 cm.) The small insect is a green potato bug. ‘If each unit in the ruler is 1 centimeter, about how long is the potato bug? (1 cm) What about the stick insect?’ (About 16 cm with the antennae, about 12 cm otherwise.)”

The materials reviewed for Kendall Hunt's Illustrative Mathematics Grade 4 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, Factors and Multiples, Lesson 2, Preparation, Lesson Narrative states, “In grade 3, students learned that a factor is a number being multiplied by another number. For instance, when we multiply 3 and 5 to find the total in 3 groups of 5, or to find the area of a rectangle that is 3 units by 5 units, the 3 and 5 are factors. In this lesson, students learn that a factor pair of a number n is a pair of whole numbers that multiply to result in n. For example, 3 and 5 a factor pair of 15. Previously, students made sense of multiples of a number in the context of area: they built and drew rectangles with given a side length and reasoned about their area. Here, they use the same context to make sense of factor pairs. Students build and draw rectangles with a given area and reason about their side lengths. Students then analyze the rectangles that the class has drawn in a gallery walk. They make observations about the side lengths of the rectangles and consider whether all possible rectangles have been drawn for each area. In these activities, a rectangle with 3 rows and 2 columns is considered the same as a rectangle with 2 rows and 3 columns.”

• IM K-5 Math Teacher Guide, About These Materials, Unit 2, “Fractions: Units and Equivalence. In this blog post, McCallum discusses equivalent fractions as the same numbers in different units.” 

• Unit 5, Multiplicative Comparison and Measurement, Lesson 6, Preparation, Lesson Narrative states, “In this lesson, students apply place value understanding, where they look for and make use of structure, to what they have learned about representing and solving multiplicative comparison problems (MP7). They use tape diagrams and equations to represent multiplicative comparisons that are ‘10 times as many’. Students will build on this understanding in the next section as they convert measurements from larger metric units into smaller ones (for instance, from meters to centimeters, kilograms to grams, or liters to milliliters).”

• IM K-5 Math Teacher Guide, About These Materials, Unit 7, ”Making Peace with the Basics of Trigonometry. In this blog post, Phillips highlights how student exploration in trigonometry allows them to see that trigonometric ratios come from measuring real triangles, fostering conceptual understanding. This blog is included in this unit as an example of how concepts of angle come into play in mathematics beyond elementary school.”

The materials reviewed for Kendall Hunt's Illustrative Mathematics Grade 4 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 4, 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 4, Course Guide, Lesson Standards, includes all Grade 4 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 1, 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 8, Properties of Two-dimensional Shapes, Lesson 3, the Core Standards are identified as 4.G.A.1, 4.G.A.2, and 4.MD.C. 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 4, Course Guide, Scope and Sequence, Unit 4: From Hundredths to Hundred Thousands, Unit Learning Goals, includes an overview of how the math of this module builds from previous work in math, “In this unit, students learn to express both small and large numbers in base ten, extending their understanding to include numbers from hundredths to hundred-thousands. In previous units, students compared, added, subtracted, and wrote equivalent fractions for tenths and hundredths. Here, they take a closer look at the relationship between tenths and hundredths and learn to express them in decimal notation. Students analyze and represent fractions on square grids of 100 where the entire grid represents 1. They reason about the size of tenths and hundredths written as decimals, locate decimals on a number line, and compare and order them. Students then explore large numbers. They begin by using base-ten blocks and diagrams to build, read, write, and represent whole numbers beyond 1,000. Students see that ten-thousands are related to thousands in the same way that thousands are related to hundreds, and hundreds are to tens, and tens are to ones. As they make sense of this structure (MP7), students see that the value of the digit in a place represents ten times the value of the same digit in the place to its right. Students then reason about the size of multi-digit numbers and locate them on number lines. To do so, they need to consider the value of the digits. They also compare, round, and order numbers through 1,000,000. They also use place-value reasoning to add and subtract numbers within 1,000,000 using the standard algorithm. Throughout the unit, students relate these concepts to real-world contexts and use what they have learned to determine the reasonableness of their responses.”

• Grade 4, Course Guide, Scope and Sequence, Unit 8: Properties of Two-dimensional Shapes, Unit Learning Goals, includes an overview of how the math of this module builds from previous work in math, “In this unit, students deepen their understanding of the attributes and measurement of two-dimensional shapes. Prior to this unit, students learned about some building blocks of geometry—points, lines, rays, segments, and angles. They identified parallel and intersecting lines, measured angles, and classified angles based on their measurement. Here, they apply those insights to describe and reason about characteristics of shapes. In the first half of the unit, students analyze and categorize two-dimensional shapes—triangles and quadrilaterals—by their attributes. They classify two-dimensional shapes based on the presence or absence of parallel or perpendicular lines, or the presence or absence of angles of a specified size. Students also learn about symmetry. They identify line-symmetric figures and draw lines of symmetry. Quadrilaterals N, U, and Z are parallelograms. Quadrilaterals AA, EE, and JJ are rhombuses. Write 4–5 statements about the sides and angles of the quadrilaterals in each set. Each statement must be true for all the shapes in the set. The second half of the unit gives students opportunities to apply their understanding of geometric attributes to solve problems about measurements (side lengths, perimeters, and angles). Included in this unit are three optional lessons that offer opportunities for students to strengthen and extend their understanding of symmetry and other attributes of two-dimensional shapes.”

The materials reviewed for Kendall Hunt's Illustrative Mathematics Grade 4 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 4 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 2, Fraction Equivalence and Comparison, Lesson 2, Activity 1. Required Materials, “Straightedges.” Launch states, “Groups of 2. Give each student a straightedge. Record and display the fraction \frac{1}{4}. ‘Describe to your partner what the diagram would look like for this fraction.’ 30 seconds: partner discussion. Record and display the fraction \frac{2}{4}. ‘Describe what the diagram would look like for this fraction.’ 30 seconds: partner discussion. Share responses. ‘In an earlier lesson, we looked at fractions with 1 for the numerator. Now let’s look at fractions with other numbers for the numerator.’ As a class, read aloud the word name of each fraction in the task.”

• Course Guide, Required Materials for Grade 4, Materials Needed for Unit 5, Lesson 16, teachers need, “Pipe cleaners, Rulers (inches), Rulers or straightedges, Tape.” 

• Unit 7, Angles and Angle Measurement, Lesson 6, Activity 2, Required Materials, “Materials from a previous activity, Patty paper.” Launch states, “Groups of 2, Make sure each group has the angle cards from the previous activity. Make patty paper available, if requested.”

• Course Guide, Required Materials for Grade 4, Materials Needed for Unit 8, Lesson 10, teachers need, “Paper Patty paper, Protractors, Rulers, Scissors.”

The materials reviewed for Kendall Hunt's Illustrative Mathematics Grade 4 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 2, Fraction Equivalence and Comparison, End-of-Unit Assessment, Assessment Teacher Guide denotes standards addressed for each problem. Problem 4, 4.NF.1, “List three different fractions that are equivalent to \frac{4}{5}. Explain or show your reasoning.”

• Unit 8, Properties of Two-dimensional Shapes, End-of-Unit Assessment, Assessment Teacher Guide denotes standards addressed for each problem. Problem 2, 4.G.2 and 4.G.3, “Which statement is true?  A. A right triangle never has a line of symmetry. B. A right triangle sometimes has a line of symmetry. C. A right triangle always has a line of symmetry. D. If a triangle has a line of symmetry then it is a right triangle.”

• Unit 9, Putting it All Together, End-of-Course Assessment, Assessment Teacher Guide denotes standards addressed for each problem. Problem 5, 4.NBT.5 and 4.OA.2, “Andre ran 1,270 meters. Clare ran 3 times as far as Andre. How many meters did Clare run? Explain or show your reasoning.”

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 4, MP3 is found in Unit 3, Lessons 3, 6, 9, and 11.

• IM K-5 Math Teacher Guide, How to Use These Materials, Standard for Mathematical Practices Chart, Grade 4, MP7 is found in Unit 5, Lessons 6-8, 10, 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, “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 4 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, Factors and Multiples, End-of-Unit Assessment supports the full intent of MP6 (Attend to precision) as students examine multiples of different numbers. For example, Problem 4 states, “Han is playing a card game with friends. The number of cards never changes, but the number of players does. a. With 5 players, the cards can be divided equally between the players. Could there be 50 cards? Explain or show your reasoning. b. With 3 players, the cards can be divided equally between the players. Could there be 50 cards? Explain or show your reasoning. c. With 4 players, the cards can be divided equally between the players. How many cards could there be? Explain or show your reasoning.”

• Unit 4, From Hundredths to Hundred-thousands, End-of-Unit Assessment develops the full intent of 4.NBT.4 (Fluently add and subtract multi-digit whole numbers using the standard algorithm). For example, Problem 8 states, “A school district in Los Angeles reported 633,621 students in 2016. A school district in New York City reported 984,462 students in the same year. a. Which school district had more students? Explain your reasoning. b. How many more students? Explain or show your reasoning. c. How many more students does the school district in New York need to have 1,000,000 students? Explain or show your reasoning.”

• Unit 8, Properties of Two-dimensional Shapes, End-of-Unit Assessment develops the full intent of 4.G.2 (Classify two-dimensional figures based on the presence or absence of parallel or perpendicular lines, or the presence or absence of angles of a specified size. Recognize right triangles as a category, and identify right triangles). For example, Problem 4 states, “Select all true statements. A. All rhombuses have a right angle. B. All rectangles have a right angle. C. Lines containing opposite sides of rectangles are parallel. D. Some rhombuses have an obtuse angle. E. Some rectangles have an obtuse angle.”

• Unit 9, Putting It All Together, End-of-Course Assessment supports the full intent of MP4 (Model with mathematics) as students reason about operations with fractions. For example, Problem 13 states, “a. Mai’s house is \frac{5}{8} mile from school. She walked to school all 5 days of the week. How many miles did Mai walk altogether from home to school? Explain or show your reasoning. b. Mai wants to walk 6 miles total for the week. How much farther does she need to walk? Explain or show your reasoning.”

The materials reviewed for Kendall Hunt's Illustrative Mathematics Grade 4 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, Factors and Multiples, Lesson 7, Activity 2, Narrative, Access for Students with Disabilities, “Engagement: Develop Effort and Persistence. Invite students to generate a list of shared expectations and possible language to use during group work, especially when playing a game that has a winner. Encourage students to discuss how they might support their partner’s learning or collaborate to find solutions, even though they are on opposing teams. Record responses on a display and keep visible during the activity. Supports accessibility for: Language, Social-Emotional Functioning.”

• Unit 3, Extending Operations to Fractions, Lesson 1, Activity 1, Narrative, Access for Students with Disabilities, “Representation: Access for Perception. Use pictures (or actual crackers, if possible) to represent the situation. Ask students to identify correspondences between this concrete representation and the diagrams they create or see. Supports accessibility for: Conceptual Processing, Visual-Spatial Processing.”

• Unit 6, Multiplying and Dividing Multi-digit Numbers, Lesson 2, Activity 3, Narrative, 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 verbally. Supports accessibility for: Language, Conceptual Processing.”

• Unit 9, Putting It All Together, Lesson 6, Activity 1, Narrative, Access for Students with Disabilities, “Engagement: Provide Access by Recruiting Interest. Provide choice. Tell students they will be finding the value of 7,465\div5, and that there are four unfinished strategies to look at. Invite students to choose whether they want to solve it in their own way or look at the unfinished strategies first. Supports accessibility for: Organization, Attention, Social-Emotional Functioning.”

The materials reviewed for Kendall Hunt's Illustrative Mathematics Grade 4 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, Factors and Multiples, Section A: Understand Factors and Multiples, Problem 7, Exploration, “1. You want to arrange all of the students in your class in equal rows. a. How many rows can you have? How many students would be in each row? b. What if you add the teacher to the arrangement? How would your rows change? 2. Find some objects at home (such as silverware, stuffed animals, cards from a game) and decide how many rows you can arrange them in and how many objects are in each row.”

• Unit 2, Fraction Equivalence and Comparison, Section B: Equivalent Fractions, Problem 6, Exploration, “Jada is thinking of a fraction. She gives several clues to help you guess her fraction. Try to guess Jada’s fraction after each clue. 1. My fraction is equivalent to \frac{2}{3}. 2. The numerator of my fraction is greater than 10. 3. 8 is a factor of my numerator. 4. 8 and 5 are a factor pair of my numerator.

• Unit 6, Multiplying and Dividing Multi-Digit Numbers, Section C: Multi-digit Division, Problem 10, Exploration, “Mai has a special way to see that 531 is a multiple of 9. She says, ‘Each hundred is 11 nines and 1 more and each ten is one nine and 1 more, so 531 is 58 nines and 9 more.’ 1. Make sense of and explain Mai’s reasoning. Is 531 a multiple of 9? 2. Use Mai's reasoning to decide if 648 is a multiple of 9.”

• Unit 8, Properties of Two-Dimensional Shapes, Section B: Reason About Attributes to Solve Problems, Problem 5, Exploration, “Make a shape or design with one or more lines of symmetry. Trade shapes with a partner and find all of the lines of symmetry of your partner's shape. You may find pattern blocks helpful to make your shape or design.”

The materials reviewed for Kendall Hunt's Illustrative Mathematics Grade 4 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, Fraction Equivalence and Comparison, Lesson 16, Activity 1, Teaching notes, Access for English Learners, “MLR8 Discussion Supports. Students should take turns explaining their reasoning to their partner. Display the following sentence frames for all to see: ‘___ is greater than ___ because  . . .’, and ‘___ and ___ are equivalent because . . . .’ Encourage students to challenge each other when they disagree. Advances: Speaking, Conversing.”

• Unit 4, From Hundredths to Hundred-thousands, Lesson 1, Activity 2, Teaching notes, Access for English Learners, “MLR2 Collect and Display. Synthesis: Direct attention to words collected and displayed from the previous activity. Invite students to borrow language from the display as needed, and update it throughout the lesson. Advances: Conversing, Reading.”

• Unit 9, Putting It All Together, Lesson 4, Activity 1, Teaching notes, Access for English Learners, “MLR7 Compare and Connect. Synthesis: After all strategies have been presented, lead a discussion comparing, contrasting, and connecting the different approaches. Ask, ‘What did the approaches have in common?’, ‘How were they different?’, or ‘Did anyone solve the problem the same way, but would explain it differently?’ Advances: Representing, Conversing.”

The materials reviewed for Kendall Hunt's Illustrative Mathematics Grade 4 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 2, Fraction Equivalence and Comparison, Lesson 6, Activity 2, students sort a set of fraction cards from a blackline master copy called “Where Do They Belong.” Launch states, “Give each group one set of fraction cards.” Activity states, “Work with your group to sort the fraction cards into three groups: less than \frac{1}{2}, equal to \frac{1}{2}, or greater than \frac{1}{2}. Be prepared to explain how you know. When you are done, compare your sorting results with another group. If the two groups disagree about where a fraction belongs, discuss your thinking until you reach an agreement.”

• Unit 4, From Hundredths to Hundred-thousands, Lesson 5, Activity 2, references grid paper, to help students understand algorithms for multiplying a multi-digit number and a single-digit number. Launch states, “Groups of 2. Provide access to grid paper, in case it is needed to align digits when multiplying.” Activity states, “3–4 minutes: independent work time on the first two problems. 1–2 minutes: partner discussion. Monitor for students who can explain the numbers in the standard algorithm. 5 minutes: independent work time on the remaining questions.”

• Unit 6, Multiplying and Dividing Multi-digit Numbers, Lesson 17, Activity 1, outlines base-ten diagrams to help students compute and understand quotients of two-digit dividends and single-digit divisors. Student Facing states, “1. Priya draws a base-ten diagram to find the value of 64\div4. A rectangle represents 10. A small square represents 1. Use the diagram (or actual blocks) to help Priya complete the division. Explain or show your reasoning. (Image of base ten blocks are provided.) 2. Use this base-ten diagram (or actual blocks) to find the value of 117\div3.” Launch states, “Groups of 4. Give students access to base-ten blocks. Display the first diagram. Make sure students can explain why it represents 64. 1 minute: quiet think time.” Activity states, “5 minutes: quiet think time. 2 minutes: group discussion. Monitor for students who see that a larger piece can be decomposed into 10 of the next smaller piece to help with distribution.”

• Unit 8, Putting It All Together, Lesson 10, Activity 1, references paper, patty paper, protractors, rulers, and scissors as students reason about angle measurements. Launch states, “Read the opening paragraph of the activity statement as a class. Display the four images. Clarify the context as needed before students begin the activity. Ask each group member to start with the drawing for a different student (one member starts with Noah’s, another with Clare’s, and so on), but try to complete at least 2 of the 4 drawings. Give a protractor and a ruler to each student. Provide access to patty paper, scrap paper, and scissors.” Student Facing states, “Noah, Clare, Andre, and Elena each have a sheet of paper with one line of symmetry. When they folded their paper along the line of symmetry, they all produced the same shape. The dashed line represents the folding line. 1. Draw the shape of the unfolded paper that each student received. Be as precise as possible. 2. Without measuring, find the measurement of all angles within the shape (of the unfolded paper) that you drew.”