The materials reviewed for Kendall Hunt's Illustrative Mathematics Grade 4 meet expectations for assessing grade-level content and, if applicable, content from earlier grades. The curriculum is divided into nine units and each unit contains a written End-of-Unit Assessment for individual student completion. The Unit 9 Assessment is an End-of-Course Assessment and it includes problems from across the grade. Examples from End-of-Unit Assessments include: 

• Unit 1, Factors and Multiples, End-of-Unit Assessment, Problem 1, “a. Is 27 a prime number or a composite number? Explain or show your reasoning. b. Is 29 a prime number or a composite number? Explain or show your reasoning.” (4.OA.4)

• Unit 3, Extending Operations to Fractions, End-of-Unit Assessment, Problem 4, “Jada needs 2 pounds of walnuts for a trail mix. She has 3 packages of walnuts that each weigh $\frac{3}{4}$ pound. Does Jada have enough walnuts to make the trail mix? Explain or show your reasoning.”  (4.MD.2, 4.NF.4c)

• Unit 5, Multiplicative Comparison and Measurement, End-of-Unit Assessment, Problem 1, “There are 93 students in the cafeteria. There are 3 times as many students in the cafeteria as there are students on the playground. a. Write a multiplication equation that represents the situation. b. How many students are on the playground? Explain or show your reasoning.” (4.OA.1, 4.OA.2)

• Unit 7, Angles and Angle Measurement, End-of-Unit Assessment, Problem 6, “Use a protractor to complete the following: a. Draw a ray that makes a 25 degree angle with the given ray. b. Draw a ray that makes a 60 degree angle with the given ray. c. What is the size of the angle made by the two rays you drew? Explain how you know.” One ray is provided in the problem. (4.MD.6, 4.MD.7)

• Unit 9, Putting It All Together, End-of-Course Assessment, Problem 1, “Select the number where the value of 6 is 1,000 times the value of the 6 in 463. a. 643, b. 6,118, c. 63,479, d. 627,385.” (4.NBT.1)

The materials reviewed for Kendall Hunt's Illustrative Mathematics Grade 4 meet expectations for the materials giving all students extensive work with grade-level problems to meet the full intent of grade-level standards.

The materials provide extensive work in Grade 4 as students engage with all CCSSM standards within a consistent daily lesson structure, including a Warm Up, one to three Instructional Activities, a Lesson Synthesis, and a Cool-Down. Examples of extensive work include:

• Unit 2, Fraction Equivalence and Comparison, Lessons 7, 10, and 11 engage students in extensive work with 4.NF.1 (Explain why a fraction is equivalent to a fraction by using visual fraction models, with attention to how the number and size of the parts differ even though the two fractions themselves are the same size. Use this principle to recognize and generate equivalent fractions). Lesson 7, Equivalent Fractions, Activity 2, students find equivalent fractions for fractions given numerically, “Groups of 2. ‘Work with a partner on this activity. One person is partner A and the other is B. Your task is to find two equivalent fractions for each fraction listed under A or B, and then convince your partner that your fractions are equivalent.’” Lesson 10, Use Multiples to Find Equivalent Fractions, Activity 1, students use visual representations to generate equivalent fractions, “‘Think quietly for a couple of minutes about what Elena did and how it relates to Andre’s number lines.’ 1–2 minutes: quiet think time for the first problem. 3–4 minutes: partner discussion on the first problem. Pause for a brief whole-class discussion. Invite students to share their ideas about Elena’s work and how it is related to Andre’s number lines. 4–5 minutes: independent work time for the last problem. Monitor for students who find equivalent fractions for $\frac{1}{8}$ by multiplying time a factor other than 2, 3 or 4.” Student Facing, “Elena thought of another way to find equivalent fractions. She wrote: ‘ is multiplied by $\frac{2}{2}$, $\frac{3}{3}$, $\frac{4}{4}$, $\frac{5}{5}$, and $\frac{10}{10}$.’ 1. Analyze Elena’s work. Then, discuss with a partner: a. How are Elena’s equations related to Andre’s number lines? (The equivalent fractions are displayed on a number line.) b. How might Elena find other fractions that are equivalent to $\frac{1}{5}$? Show a couple of examples. 2. Use Elena’s strategy to find five fractions that are equivalent to $\frac{1}{8}$. Use number lines to check your thinking, if they help.” Lesson 11, Use Factors to Find Equivalent Fractions, Activity 2, students generate equivalent fractions by applying the numerical strategies they learned, “‘Work on the activity independently. Then, share your responses with your partner and check each other’s work.’ 8–10 minutes: independent work time. 3–5 minutes: partner discussion.” Student Facing, “Find at least two fractions that are equivalent to each fraction. Show your reasoning. 1. $\frac{16}{8}$ 2. $\frac{40}{10}$ 3. $\frac{7}{6}$ 4. $\frac{90}{100}$ 5. $\frac{5}{4}$.“

• Unit 4, From Hundredths to Hundred-thousands, Lesson 11, 13, and 21 engage students in the extensive work with 4.NBT.2 (Read and write multi-digit whole numbers using base-ten numerals, number names, and expanded form. Compare two multi-digit numbers based on meanings of the digits in each place, using >, =, and < symbols to record the results of comparisons). Lesson 11, Large Numbers on a Number Line, Activity 1, students use their understanding of place value and the relative position of numbers within 1,000,000 to partition and place numbers on a number line. Student Facing, “1. Locate and label each number on the number line. a. 347 (a number line labeled 300 at one end and 400 at the opposite end is located under the problem), b. 3,470 (a numberline label 3000 at one end and 4000 at the opposite end is located under the problem), c. 34,700 (a numberline label 30,000 at one end and 40,000 at the opposite end is located under the problem), d. 347,000 (a numberline label 300,000 at one end and 400,000 at the opposite end is located under the problem). 2. Locate and label each number on the number line. a. 347 (a number line labeled 340 at one end and 350 at the opposite end is located under the problem), b.3,470 (a number line labeled 3400 at one end and 3500 at the opposite end is located under the problem), c. 34,700 (a number line labeled 34,000 at one end and 35,000 at the opposite end is located under the problem), d. 347,000 (a number line labeled 340,000 at one end and 350,000 at the opposite end is located under the problem). 3. What do you notice about the location of these numbers on the number lines? Make two observations and discuss them with your partner.” Activity 2, students place a set of numbers that are each ten times as much the one before it on the same number line. Student Facing, “Your teacher will assign a number for you to locate on the given number line. A. 347 B. 3470 C. 34,700 D. 347,000 1. Decide where your assigned number will fall on this number line. Explain your reasoning. 2. Work with your group to label the tick marks and agree on where each of the numbers should be placed.” Number 1 has a number line with endpoints 0 and 400,000 labeled. Number 2 has a number line with 0 and 400,000 labeled and three tick marks on the line to be labeled. Lesson 13, Order Multi-digit Numbers, Cool-Down, students use their place value understanding to order numbers. Student Facing, ”Order the following numbers from least to greatest 94,942; 9,042; 279,104; 9,420; 59,000; 500,492; 279,099.” Lesson 21, Zeros in the Standard Algorithm, Warm-up: Which One Doesn’t Belong?, students analyze and compare features of multi-digit numbers, “Groups of 2. Display numbers. ‘Pick one that doesn’t belong. Be ready to share why it doesn’t belong.’” Student Facing, “Which one doesn’t belong? A. 2,050 B. 2,055 C. 205.2 D. 20,005.”

• Unit 5, Multiplicative Comparison and Measurement, Lessons 2, 3, and 5 engage students in extensive work of 4.OA.2 (Multiply or divide to solve word problems involving multiplicative comparison, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem, distinguishing multiplicative comparison from additive comparison). Lesson 2, Interpret Representations of Multiplicative Comparisons, Activity 1, students analyze and describe how images and diagrams can show “n times as many.” Student Facing, students see pictures of connecting cubes (6 cubes and 2 cubes), “1. Jada has 4 times as many cubes as Kiran. Draw a diagram to represent the situation. 2. Diego has 5 times as many cubes as Kiran. Draw a diagram to represent the situation. 3. Lin has 6 times as many cubes as Kiran. How many cubes does Lin have? Explain or show your reasoning.” Lesson 3, Solve Multiplicative Comparison Problems, Activity 2, students make sense of and represent multiplicative comparison problems in which a factor is unknown. Student facing, “1. Clare donated 48 books. Clare donated 6 times as many books as Andre. a. Draw a diagram to represent the situation. b. How many books did Andre donate? Explain your reasoning. 2. Han says he can figure out the number of books Andre donated using division. Tyler says we have to use multiplication because it says ‘times as many’. a. Do you agree with Han or Tyler? Explain your reasoning. b. Write an equation to represent Tyler’s thinking. c. Write an equation to represent Han’s thinking. 3. Elena donated 9 times as many books as Diego. Elena donated 81 books. Use multiplication or division to find the number of books Diego donated.” Lesson 5, One- and Two-step Comparison Problems, Activity 1, students solve contextualized problems using multiplicative comparison. Student Facing, “For this year’s book fair, a school ordered 16 science books and 6 times as many picture books. Last year, the school ordered 4 times as many picture books and 4 times as many science books than they did this year. 1. How many picture books were ordered this year? 2. How many picture books were ordered last year? 3. How many more science experiment books were ordered last year than this year?”

The materials provide opportunities for all students to engage with the full intent of Grade 4 standards through a consistent lesson structure. According to the IM Teacher Guide, A Typical IM Lesson, “Every warm-up is an instructional routine. The warm-up invites all students to engage in the mathematics of the lesson. After the warm-up, lessons consist of a sequence of one to three instructional activities. The activities are the heart of the mathematical experience and make up the majority of the time spent in class. After the activities for the day, students should take time to synthesize what they have learned. This portion of class should take 5-10 minutes. The cool-down task is to be given to students at the end of the lesson and students are meant to work on the cool-down for about 5 minutes independently.” Examples of meeting the full intent include:

• Unit 5, Multiplicative Comparison and Measurement, Lesson 16 and Unit 6, Multiplying and Dividing Multi-digit Numbers, Lesson 22 engage students in the full intent of 4.MD.3 (Apply the area and perimeter formulas for rectangles in real world and mathematical problems). Unit 5, Lesson 16, Compare Perimeters of Rectangles, Cool-down, students reason about the perimeter of rectangles. Student Facing, “1. Rectangle Y has a perimeter of 20 inches. Name a possible pair of side lengths it could have. 2. Rectangle Z has a perimeter of 180 inches. Complete this statement: a. The perimeter of rectangle Z is ___ times the perimeter of rectangle Y. b. If the length of rectangle Z is 70 inches, how many inches is its width? Explain or show your reasoning. Draw a diagram if it is helpful.” Unit 6, Lesson 22, Problems About Perimeter and Area, Activity 2, students perform operations with multi-digit numbers to solve situations about perimeter and area. Student Facing, “A classroom is getting new carpet and baseboards. Tyler and a couple of friends are helping to take measurements. Here is a sketch of the classroom and the measurements they recorded. For each question, show your reasoning. 1. How many feet of baseboard will they need to replace in the classroom? How many inches is that? 2. 1,200 inches of baseboard material was delivered. Is that enough? 3. How many square feet of carpet will be needed to cover the floor area?” A composite figure is included with measurement labels for each side. 

• Unit 3, Extending Operations to Fractions, Lessons 8 and 9 engage students with the full intent of 4.NF.3a (Understand addition and subtraction of fractions as joining and separating parts referring to the same whole). Lesson 8, Addition of Fractions, Activity 2, students use number lines to represent addition of two fractions and to find the value of the sum. Student Facing, “1. Use a number line to represent each addition expression and to find its value. a. $\frac{5}{8}+\frac{2}{8}$, b. $\frac{1}{8}+\frac{9}{8}$, c. $\frac{11}{8}+\frac{9}{8}$, d. $2\frac{1}{8}+\frac{4}{8}$. 2. Priya says the sum of and $\frac{4}{5}$ is $1\frac{6}{5}$. Kiran says the sum is $\frac{11}{5}$. Tyler says it is $2\frac{1}{5}$. Do you agree with any of them? Explain or show your reasoning. Use one or more number lines if you find them helpful.” Lesson 9, Differences of Fractions, Cool-down, students use number lines to represent subtraction of a fractions with the same denominator, including mixed numbers. Student Facing, “Use a number line to represent each difference and to find its value. 1. $\frac{12}{5}-\frac{4}{5}$. 2. $2\frac{1}{5}-\frac{7}{5}$.”

• Unit 7, Angles and Angle Measurement, Lessons 7, 8, and 11 engage students in the full intent of 4.MD.5a (An angle is measured with reference to a circle with its center at the common endpoint of the rays, by considering the fraction of the circular arc between the points where the two rays intersect the circle. An angle that turns through $\frac{1}{360}$ of a circle is called a "one-degree angle," and can be used to measure angles). Lesson 7, The Size of Angles on a Clock, Activity 2, students use the clock as a tool for reasoning and for talking about “how much” of a turn. Student Facing, “1. Here are some angles formed by the two hands of a clock. In each pair of angles, which angle is larger? Explain or show your reasoning. a. 5:00, 3:00, b. 1:15, 1:20, c. 2:50, 11:20, d. 8:58, 9:35. 2. How large is this angle? Describe its size in as many ways as you can.” A clock shows 12:20. Lesson 8, The Size of Angles in Degrees, Activity 1, students compare angles on clocks and use degrees as a unit of measure. Student facing, “A ray that turns all the way around its endpoint and back to its starting place has made a full turn. We say that the ray has turned 360 degrees. 1. How many degrees has the ray turned from where it started? (part a shows a 180 degree angle, b shows a 90 degree angle and c shows a 270 degree angle) 2. Sketch two angles: a. an angle where a ray has turned $50\degree$ b. an angle where a ray has turned $130\degree$.” Lesson 11, Use a Protractor to Draw Angles, Warm Up: Estimation Exploration, students estimate the measure of an angle on a clock face using what they have learned about angles. Student Facing, “How many degrees is the angle formed by the long hand and the short hand of the clock? Make an estimate that is: too high, just right, too low.” An unlabeled clock face shows an angle that is about 3:40 for reference.

The materials reviewed for Kendall Hunt's Illustrative Mathematics Grade 4 meet expectations that, when implemented as designed, the majority of the materials address the major clusters of each grade. The instructional materials devote at least 65 percent of instructional time to the major clusters of the grade: 

• The approximate number of units devoted to major work of the grade (including assessments and supporting work connected to the major work) is 6 out of 9, approximately 67%.

• The number of lessons devoted to major work of the grade (including assessments and supporting work connected to the major work) is 121 out of 158, approximately 77%. The total number of lessons devoted to major work of the grade includes 113 lessons plus 8 assessments for a total of 121 lessons.

• The number of days devoted to major work of the grade (including assessments and supporting work connected to the major work) is 119 out of 167, approximately 71%.

A lesson-level analysis is most representative of the instructional materials as the lessons include major work, supporting work connected to major work, and the assessments embedded within each unit. As a result, approximately 77% of the instructional materials focus on major work of the grade.

The materials reviewed for Kendall Hunt's Illustrative Mathematics Grade 4 meet expectations that supporting content enhances focus and coherence simultaneously by engaging students in the major work of the grade. 

Materials are designed so supporting standards/clusters are connected to the major standards/ clusters of the grade. These connections are listed for teachers on a document titled “Pacing Guide and Dependency Diagram” found within the Course Guide tab for each unit. Examples of connections include:

• Unit 3, Extending Operations to Fractions, Lesson 13, Activity 2 connects the supporting work of 4.MD.4 (Make a line plot to display a data set of measurements in fractions of a unit [, $\frac{1}{4}$, $\frac{1}{8}$]. Solve problems involving addition and subtraction of fractions by using information presented in line plots) to the major work of 4.NF.3d (Solve word problems involving addition and subtraction of fractions referring to the same whole and having like denominators). Students create a line plot using measurements to the nearest $\frac{1}{4}$ and $\frac{1}{8}$ inch and use their understanding of fraction equivalence to plot and partition the horizontal axis. Student Facing states, “1. Andre’s class measured the length of some colored pencils to the nearest $\frac{1}{4}$ inch. The data are shown here: $1\frac{3}{4}$, $2\frac{1}{4}$, $5\frac{1}{4}$, $5\frac{1}{4}$, $4\frac{2}{4}$, $4\frac{2}{4}$, $6\frac{1}{4}$, $6\frac{3}{4}$, $6\frac{3}{4}$, $6\frac{3}{4}$ a. Plot the colored- pencil data on the line plot. b. Which colored-pencil length is the most common in the data set? c. Write 2 new questions that could be answered using the line plot data. 2. Next, Andre’s class measured their colored pencils to the nearest $\frac{1}{8}$ inch. The data are shown here: $1\frac{6}{8}$, $2\frac{2}{8}$, $5\frac{2}{8}$, $5\frac{4}{8}$, $4\frac{4}{8}$, $4\frac{4}{8}$, $6\frac{6}{8}$, $6\frac{6}{8}$, $6\frac{6}{8}$, $6\frac{4}{8}$ a. Plot the colored-pencil data on the line plot. b. Which colored-pencil length is the most common in the line plot? c. Why did some colored-pencil lengths change on this line plot? d. What is the difference between the length of the longest colored pencil and the shortest colored pencil? Show your reasoning.”

• Unit 5, Multiplicative Comparison and Measurement, Lesson 17, Activity 2 connects the supporting work of 4.MD.3 (Apply the area and perimeter formulas for rectangles in real world and mathematical problems) to the major work of 4.OA.2 (Multiply or divide to solve word problems involving multiplicative comparison, distinguishing multiplicative comparison from additive comparison). Students consolidate their learning from the past few units to solve problems about length measurements in a mathematical context. Student Facing states, “Your teacher has posted six quadrilaterals around the room. Each one has a missing side length or a missing perimeter. 1. Choose two diagrams—one with a missing length and another with a missing perimeter. Make sure that all six shapes will be visited by at least one person in your group. Find the missing values. Show your reasoning and remember to include the units. 2. Discuss your responses with your group until everyone agrees on the missing measurements for all six figures. 3. Answer one of the following questions. Explain or show your reasoning. a. The perimeter of B is how many times the perimeter of D? b. The perimeter of one figure is 1,000 times that of another figure. Which are the two figures? c. The perimeter of F is how many times the perimeter of B?”

• Unit 6, Multiplying and Dividing Multi-Digit Numbers, Lesson 4, Activity 2 connects the supporting work of 4.OA.5 (Generate a number or shape pattern that follows a given rule. Identify apparent features of the pattern that were not explicit in the rule itself) to the major work of 4.NBT.5 (Multiply a whole number of up to four digits by a one-digit whole number, and multiply two two-digit numbers, using strategies based on place value and the properties of operations. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models). In the activity, students continue to analyze patterns in numbers and use them to look at the relationship between the multiples of 99 and multiples of 100. Student Facing states, “Andre’s class did a choral count by 99. Here are the first six numbers they said. 1. Study the list of numbers. Make at least 3 observations about features of the pattern.” The list of numbers shows, “counting by 99: 99, 198, 297, 396, 495, 594.” Students then answer, “2. Extend the list with the next four multiples of 99. Be prepared to discuss how you know what numbers to write. 3. Why do you think the digits in the numbers change the way they do?”

The materials for Kendall Hunt's Illustrative Mathematics Grade 4 meet expectations for including problems and activities that serve to connect two or more clusters in a domain or two or more domains in a grade. 

Materials are coherent and consistent with the Standards. These connections can be listed for teachers in one or more of the four phases of a typical lesson: warm-up, instructional activities, lesson synthesis, or cool-down. Examples of connections include:

• Unit 3, Extending Operations to Fractions, Lesson 15, Activity 1 connects the major work of 4.NF.A (Extend understanding of fraction equivalence and ordering) to the major work of 4.NF.B (Build fractions from unit fractions). Students reason about problems that involve combining or removing fractional amounts with different denominators in the context of stacking playing bricks. Student facing states, “Priya, Kiran, and Lin are using large playing bricks to make towers. Here are the heights of their towers so far: Priya: $21\frac{1}{4}$ inches, Kiran: $32\frac{3}{8}$ inches, Lin: $55\frac{1}{2}$ inches. For each question, show your reasoning. 1. How much taller is Lin’s tower compared to: a. Priya’s tower? b. Kiran’s tower? They are playing in a room that is 109 inches tall. Priya says that if they combine their towers to make a super tall tower, it would be too tall for the room and they’ll have to remove one brick. 2. Do you agree with Priya? Explain your reasoning.”

• Unit 5, Multiplicative Comparison and Measurement, Lesson 15, Activity 1 connects the major work of 4.NF.B (Build fractions from unit fractions by applying and extending previous understandings of operations on whole numbers) to the major work of 4.OA.A (Use the four operations with whole numbers to solve problems). Students analyze length measurements listed in a chart, perform multiplication, and convert distances involving fractional amounts in order to compare them. Student Facing states, “Six students were throwing frisbees on field day. Here is some information about each person’s first throw. Elena’s frisbee went 3 times as far as Clare’s did. Andre’s frisbee went 4 times as far as Tyler’s did. 1. Complete the table with Elena and Tyler’s distances. Explain or show your reasoning. 2. Who are the top 3 throwers for that round? Find out by listing the students and their distances in feet and in order, from longest to shortest.” Values in the table show: Han 17 yards, Lin $51\frac{1}{2}$ feet, Clare $21\frac{1}{3}$ feet, Andre 22 yards 2 feet, and Elena and Tyler are blank.

• Unit 7, Angles and Angle Measurement, Lesson 5, Activity 1 connects the supporting work of 4.G.A (Draw and identify lines and angles, and classify shapes by properties of their lines and angles) to the supporting work of 4.MD.C (Geometric measurement: Understand concepts of angle and measure angles). Students work with a partner to replicate images of angles as they use the vocabulary they have learned to describe figures. Student Facing states, “Work with a partner in this activity. Choose a role: A or B. Sit back to back, or use a divider to keep one person from seeing the other person’s work. Partner A: Your teacher will give you a card. Don’t show it to your partner. Describe both images on the card—as clearly and precisely as possible—so that your partner can draw the same images. Partner B: Your partner will describe two images. Listen carefully to the descriptions. Create the drawings as described. Follow the instructions as closely as possible. 1. When done, compare the drawings to the original images. Discuss: Which parts were accurate? Which were off? How could the descriptions be improved so the drawing could be more accurate? 2. Switch roles and repeat the exercise. Compare the drawings to the original images afterwards. If you have time: Request two new cards from your teacher (one card at a time). Take turns describing and drawing the geometric figure on each card.” Activity Synthesis states, “‘How are the two drawings on each card the same?’ (They each have 2 rays. The rays start at the same point. One ray is pointing in the same direction in both drawings.) ‘How are they different?’ (The rays are pointing in different directions on some cards. The rays are farther apart in some cards.) ‘How did you describe what you saw? What terms did you use to help you describe the directions of the rays?’ (We tried to explain by describing the hands on a clock. We tried using words like north, south, east, and west. We described them in relation to vertical and horizontal.) As students share responses, update the display, by adding (or replacing) language, diagrams, or annotations. Remind students to borrow language from the display as needed. ‘Did anyone use the term “angle?” Did anyone measure something or use measurements? The figures that you drew are angles. An angle is a figure that is made up of two rays that share the same endpoint. The point where the two rays meet is called the vertex of the angle.’”

• Unit 9, Putting It All Together, Lesson 9, Activity 1 connects the major work of 4.NBT.B (Use place value understanding and properties of operations to perform multi-digit arithmetic) to the major work of 4.OA.A (Use the four operations with whole numbers to solve problems). Students analyze a situation and solutions in order to think about what questions were asked. The Teacher Launch states, “Groups of 2. ‘Have you ever gone on a long hike? What is the longest distance you ever traveled just by walking? Let’s look at the work a student did to answer questions about two men who set world records for traveling by walking.’” Student Facing states, “George Meegan walked 19,019 miles between 1977 and 1983. He finished at age 31. He wore out 12 pairs of hiking boots. Jean Beliveau walked 46,900 miles between 2000 and 2011 and finished at age 56. Here are the responses Kiran gave to answer some questions about the situation. Write the question that Kiran might be answering. In the last row, write a new question about the situation and show the answer, along with your reasoning.” A worksheet divided in half and numbered from 1-4 with the words “question” in one column and “response and reasoning” in another column is shown.

The materials reviewed for Kendall Hunt's Illustrative Mathematics Grade 4 meet expectations that content from future grades is identified and related to grade-level work, and materials relate grade-level concepts explicitly to prior knowledge from earlier grades. 

Prior and Future connections are identified within materials in the Course Guide, Section Dependency Diagrams which state, “an arrow indicates the prior section that contains content most directly designed to support or build toward the content in the current section.” Connections are further described within the Unit Learning Goals embedded in the Scope and Sequence, within the Preparation tab for specific lessons, and within the notes for specific parts of lessons. 

Examples of connections to future grades include:

• Unit 6, Multiplying and Dividing Multi-digit Numbers, Lesson 11, Preparation connects the work of 4.NBT.5 (Multiply a whole number of up to four digits by a one-digit whole number, and multiply two two-digit numbers, using strategies based on place value and the properties of operations. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models) to work with multiplying multi-digit whole numbers using the standard algorithm in 5.NBT.5. Lesson Narrative states, “This lesson extends students’ analysis to include the standard algorithm for multiplication of multi-digit numbers. In grade 4, the standards focus on understanding place value and how it is represented in different methods for finding products. The work here serves to build the groundwork for making sense of the standard algorithm in grade 5, so students are not expected to use the standard algorithm at this time.”

• Unit 8, Properties of Two-dimensional Shapes, Lesson 3, Activity 2 connects 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.) to work with classifying two-dimenationsl figures in 5.G.B. Narrative states, “Students are not expected to recognize that the attributes of one shape may make it a subset of another shape (for example, that squares are rectangles, or that rectangles are parallelograms). They may begin to question these ideas, but the work to understand the hierarchy of shapes will take place formally in grade 5. During the synthesis, highlight how sides and angles can help us define and distinguish various two-dimensional shapes.”

• Unit 9, Putting It All Together, Lesson 2, Preparation connects 4.NF.A (Extend understanding of fraction equivalence and ordering) and 4.NF.B (Build fractions from unit fractions) to work adding and subtracting fractions with unlike denominators in 5.NF.1. Lesson Narrative states, “In this lesson, students apply what they know about equivalence and addition and subtraction of fractions to solve problems. Throughout the lesson, students have opportunities to reason quantitatively and abstractly as they connect their representations, including equations, to the situations (MP2) and to compare their reasoning with others' (MP3). The work of this lesson helps prepare students for adding and subtracting with unlike denominators in grade 5. If students need additional support with the concepts in this lesson, refer back to Unit 3, Section B in the curriculum materials.”

Examples of connections to prior knowledge include:

• Unit 1, Factors and Multiples, Lesson 1, Preparation connects 4.OA.4 (Find all factor pairs for a whole number in the range 1–100. Recognize that a whole number is a multiple of each of its factors. Determine whether a given whole number in the range 1–100 is a multiple of a given one-digit number. Determine whether a given whole number in the range 1–100 is prime or composite) to work with concepts of area from 3.MD.7a. Lesson Narrative states, “In grade 3, students learned how to find the area of a rectangle by tiling and found that multiplying the side lengths yields the same result. The purpose of this lesson is for students to apply their understanding of area and multiplication to build rectangles and find their area. As students consider the areas of rectangles with a given side length, they explore the idea of multiples. Students learn that a multiple of a number is the result of multiplying that whole number by another.”

• Course Guide, Scope and Sequence, Unit 2, Fraction Equivalence and Comparison, Unit Learning Goals connects 4.NF.A (Extend understanding of fraction equivalence and ordering) to work with unit fractions from Grade 3. Lesson Narrative states, “In grade 3, students partitioned shapes into parts with equal area and expressed the area of each part as a unit fraction. They learned that any unit fraction $\frac{1}{b}$ results from a whole partitioned into b equal parts. They used unit fractions to build non-unit fractions, including fractions greater than 1, and represent them on fraction strips and tape diagrams. The denominators of these fractions were limited to 2, 3, 4, 6, and 8. Students also worked with fractions on a number line, establishing the idea of fractions as numbers and equivalent fractions as the same point on the number line.”

• Unit 6, Multiplying and Dividing Multi-digit Numbers, Lesson 5, Activity 1 connects 4.NBT.5 (Multiply a whole number of up to four digits by a one-digit whole number, and multiply two two-digit numbers, using strategies based on place value and the properties of operations. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models) to work with concepts of multiplication from Grade 3. Narrative states, “In this activity, students build on grade 3 work with arrays to consider how to find the total number in an array without counting by 1. Students are not asked to find the answer, but instead share their strategies for doing so. This allows teachers to observe how students make sense of multiplying larger numbers.”

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 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.”