## Kendall Hunt’s Illustrative Mathematics

##### v1.5
###### Usability
Our Review Process

Title ISBN Edition Publisher Year
Kendal Hunt's Illustrative Mathematics Grade 1 978-1-7924-6275-7 2021 Kendall Hunt Publishing Company 2021
Kendal Hunt's Illustrative Mathematics Grade 1 978-1-7924-6289-4 2021 Kendall Hunt Publishing Company 2021
Kendal Hunt's Illustrative Mathematics Grade 4 978-1-7924-6278-8 2021 Kendall Hunt Publishing Company 2021
Kendal Hunt's Illustrative Mathematics Grade 4 978-1-7924-6292-4 2021 Kendall Hunt Publishing Company 2021
Kendal Hunt's Illustrative Mathematics Grade 3 978-1-7924-6277-1 2021 Kendall Hunt Publishing Company 2021
Kendal Hunt's Illustrative Mathematics Grade 3 978-1-7924-6291-7 2021 Kendall Hunt Publishing Company 2021
Kendal Hunt's Illustrative Mathematics Kindergarten 978-1-7924-6274-0 2021 Kendall Hunt Publishing Company 2021
Kendal Hunt's Illustrative Mathematics Kindergarten 978-1-7924-6287-0 2021 Kendall Hunt Publishing Company 2021
Kendal Hunt's Illustrative Mathematics Grade 5 978-1-7924-6279-5 2021 Kendall Hunt Publishing Company 2021
Kendal Hunt's Illustrative Mathematics Grade 5 978-1-7924-6293-1 2021 Kendall Hunt Publishing Company 2021
Kendal Hunt's Illustrative Mathematics Grade 2 978-1-7924-6276-4 2021 Kendall Hunt Publishing Company 2021
Kendal Hunt's Illustrative Mathematics Grade 2 978-1-7924-6290-0 2021 Kendall Hunt Publishing Company 2021
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### Overall Summary

The materials reviewed for Kendall Hunt's Illustrative Mathematics Grade 4 meet expectations for Alignment to the CCSSM. In Gateway 1, the materials meet expectations for focus and coherence. In Gateway 2, the materials meet expectations for rigor and meet expectations for practice-content connections.

###### Alignment
Meets Expectations
###### Usability
Meets Expectations

### Focus & Coherence

The materials reviewed for Kendall Hunt's Illustrative Mathematics Grade 4 meet expectations for focus and coherence. For focus, the materials assess grade-level content and provide all students extensive work with grade-level problems to meet the full intent of grade-level standards. For coherence, the materials are coherent and consistent with the CCSSM.

##### Gateway 1
Meets Expectations

#### Criterion 1.1: Focus

Materials assess grade-level content and give all students extensive work with grade-level problems to meet the full intent of grade-level standards.

The materials reviewed for Kendall Hunt's Illustrative Mathematics Grade 4 meet expectations for focus as they assess grade-level content and provide all students extensive work with grade-level problems to meet the full intent of grade-level standards.

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Materials assess the grade-level content and, if applicable, content from earlier grades.

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)

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Materials give all students extensive work with grade-level problems to meet the full intent of grade-level standards.

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: ‘$$\frac{1}{5}$$ 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 1\frac{2}[5} 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.

#### Criterion 1.2: Coherence

Each grade’s materials are coherent and consistent with the Standards.

The materials reviewed for Kendall Hunt's Illustrative Mathematics Grade 4 meet expectations for coherence. The materials: address the major clusters of the grade, have supporting content connected to major work, make connections between clusters and domains, and have content from prior and future grades connected to grade-level work.

##### Indicator {{'1c' | indicatorName}}

When implemented as designed, the majority of the materials address the major clusters of each grade.

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.

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Supporting content enhances focus and coherence simultaneously by engaging students in the 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 [$$\frac{1}{2}$$, \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?”

##### Indicator {{'1e' | indicatorName}}

Materials include problems and activities that serve to connect two or more clusters in a domain or two or more domains in a grade.

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.

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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.

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.”

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In order to foster coherence between grades, materials can be completed within a regular school year with little to no modification.

The materials reviewed for Kendall Hunt's Illustrative Mathematics Grade 4 foster coherence between grades and can be completed within a regular school year with little to no modification. According to the IM K-5 Teacher Guide, About These Materials, “Each grade level contains 8 or 9 units. Units contain between 8 and 28 lesson plans. Each unit, depending on the grade level, has pre-unit practice problems in the first section, checkpoints or checklists after each section, and an end-of-unit assessment. In addition to lessons and assessments, units have aligned center activities to support the unit content and ongoing procedural fluency. The time estimates in these materials refer to instructional time. Each lesson plan is designed to fit within a class period that is at least 60 minutes long. Some units contain optional lessons and some lessons contain optional activities that provide additional student practice for teachers to use at their discretion.”

In Grade 4, there are 167 days of instruction including:

• 149 lesson days

• 18 unit assessment days

There are nine units in Grade 4 and, within those units, there are between 8 and 25 lessons. According to the IM K-5 Teacher Guide, A Typical IM Lesson, “A typical lesson has four phases: 1. a warm-up 2. one or more instructional activities 3. the lesson synthesis 4. a cool-down.” There is a Preparation tab for lessons, including specific guidance and time allocations for each phase of a lesson.

In Grade 4, each lesson is composed of:

• 10 minutes Warm-up

• 10-25 minutes (each) for one to three Instructional Activities

• 10 minutes Lesson Synthesis

• 5 minutes Cool-down

### Rigor & the Mathematical Practices

The materials reviewed for Kendall Hunt's Illustrative Mathematics Grade 4 meet expectations for rigor and balance and practice-content connections. The materials help students develop procedural skills, fluency, and application. The materials also make meaningful connections between the Standards for Mathematical Content and the Standards for Mathematical Practice (MPs).

##### Gateway 2
Meets Expectations

#### Criterion 2.1: Rigor and Balance

Materials reflect the balances in the Standards and help students meet the Standards’ rigorous expectations, by giving appropriate attention to: developing students’ conceptual understanding; procedural skill and fluency; and engaging applications.

The materials reviewed for Kendall Hunt's Illustrative Mathematics Grade 4 meet expectations for rigor. The materials develop conceptual understanding of key mathematical concepts, give attention throughout the year to procedural skill and fluency, and spend sufficient time working with engaging applications of mathematics. There is a balance of the three aspects of rigor within the grade.

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Materials develop conceptual understanding of key mathematical concepts, especially where called for in specific content standards or cluster headings.

The materials reviewed for Kendall Hunt's Illustrative Mathematics Grade 4 meet expectations for developing conceptual understanding of key mathematical concepts, especially where called for in specific content standards or cluster headings.

Materials develop conceptual understanding throughout the grade level. According to IM K-5 Math Teacher Guide, Design Principles, conceptual understanding is a part of the design of the materials. Balancing Rigor states, “Opportunities to connect new representations and language to prior learning support students in building conceptual understanding. Access to new mathematics and problems prompts students to apply their conceptual understanding and procedural fluency to novel situations.” Additionally, Purposeful Representations states, “Across lessons and units, students are systematically introduced to representations and encouraged to use representations that make sense to them. As their learning progresses, students are given opportunities to make connections between different representations and the concepts and procedures they represent.” Examples include:

• Unit 2, Fraction Equivalence and Comparison, Lesson 7, Warm-up, students develop conceptual understanding as they use previous knowledge of equivalence and strategies for comparing fractions. Student Facing states, “Decide if each statement is true or false. Be prepared to explain your reasoning. \frac{4}{8}=\frac{7}{8}, \frac{3}{4}=\frac{6}{8}, \frac{2}{6}=\frac{2}{8}, \frac{6}{3}=\frac{4}{2}.” (4.NF.1)

• Unit 4, From Hundredths to Hundred-Thousands, Lesson 7, Warm-Up, students develop conceptual understanding of place value with larger numbers and notice patterns in the count. Launch states, “‘Count by 1,000, starting at 3,400.’ Record as students count. Stop counting and recording at 23,400.” Activity Synthesis states, “What parts of the numbers stay the same each time we count? (The digits in the hundreds, tens, and one place remain the same each time.) When will these digits change? (The digit in the hundreds, tens, and ones place will never change because we are counting by 1,000 each time.)” (4.NBT.2)

• Unit 6, Multiplying and Dividing Multi-Digit Numbers, Lesson 13, Activity 2, students develop conceptual understanding of dividing multi-digit numbers in the context of real-life situations. Student Facing states, “1. Priya’s mom made 85 gulab jamuns for the class to share. Priya gave 5 to each student in the class. How many students are in Priya’s class? Explain or show your reasoning. 2. Han’s uncle sent in 110 chocolate-covered breadsticks for a snack. The students in Han's class are seated at 6 tables. Han plans to give the same number of breadsticks to each table. How many breadsticks does each table get? Explain or show your reasoning.” (4.NBT.6)

According to IM K-5 Math Teacher Guide, materials were designed to include opportunities for students to independently demonstrate conceptual understanding, when appropriate. Design Principles, Coherent Progress states, “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.” A Typical IM Lesson states, “The cool-down task is to be given to students at the end of the lesson. Students are meant to work on the cool-down for about 5 minutes independently and turn it in. The cool-down serves as a brief formative assessment to determine whether students understood the lesson. Students’ responses to the cool-down can be used to make adjustments to further instruction.” Examples include:

• Unit 4, From Hundredths to Hundred-thousands, Lesson 11, Cool-down, students demonstrate conceptual understanding of place value to locate large numbers on a number line. Student Facing states, “1. Estimate the location of 28,500 on the number line and label it with a point. 2. Which point—A, B, or C—could represent a number that is 10 times as much as 28,500? Explain your reasoning.” For problem 1, an image of a number line is shown with A,B,C on the number line from 0 to 400,000. (4.NBT.1)

• Unit 5, Multiplicative Comparison and Measurement, Lesson 8, Cool-down, students demonstrate conceptual understanding while comparing and converting metric measurements. Student Facing states, “1. Kiran lives 7 kilometers from school. How many meters from school does he live? Explain or show your reasoning. 2. A classmate of Kiran’s lives 800 meters from school. Does he live closer or farther away from school than Kiran? Explain your reasoning.” Responding to Student Thinking states, “Students may say that Kiran’s classmate lives farther from school (or that 800 meters is greater than 7 kilometers) if they mistake 7 kilometers to be 700 meters instead of 7,000 meters, or if they confuse the relationship between kilometers and meters with that between meters and centimeters.” (4.MD.1, 4.MD.2)

• Unit 6, Multiplying and Dividing Multi-digit Numbers, Lesson 7, Activity 1, students demonstrate conceptual understanding as they use rectangular diagrams to represent multiplication of three-digit and one-digit numbers. Student Facing states, “1. Clare drew this diagram. a. What multiplication expression can be represented by the diagram? b. Find the value of the expression. Show your reasoning.” A rectangle that is partitioned is shown. (4.NBT.5)

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Materials give attention throughout the year to individual standards that set an expectation for procedural skill and fluency.

The materials reviewed for Kendall Hunt's Illustrative Mathematics Grade 4 meet expectations for giving attention throughout the year to individual standards that set an expectation of procedural skill and fluency.

According to IM Curriculum, Design Principles, Balancing Rigor, “Warm-ups, practice problems, centers, and other built-in routines help students develop procedural fluency, which develops over time.” Examples include:

• Unit 4, From Hundredths to Hundred-thousands, Lesson 10, Warm-up, students develop procedural skill and fluency as they use strategies and understanding of adding and subtracting multi-digit numbers. Student Facing states, “Find the value of each expression mentally. 650+75, 5,650+75, 50,650+75, 500,650+75.” (4.NBT.4)

• Unit 6, Multiplying and Dividing Multi-digit Numbers, Lesson 19, Activity 1, students develop procedural skill and fluency as they use partial quotients and interpret remainders. Activity states, “3 minutes: independent work time on the first 2 problems. Pause after problem 2 to discuss students’ responses. Display the different ways that students decompose 389 to divide it by 7. ‘Most other calculations we’ve seen so far end with a 0, but this one ends with a 4. What does the 4 tell us?’ (We cannot make a group of 7 with 4 leftover. 389 is not a multiple of 7, and there are leftovers.) ‘When we divide and end up with leftovers we call them remainders, because they represent what is remaining after we divide into equal groups.’ Display: 389=7x55+4 ‘How does this equation show that 389\div7 has a remainder?’ (It shows that 389 is not a multiple of 7. It also shows that 7 and 55 make a factor pair for 385, and 389 is 4 more than that.) 3 minutes: independent work time on the last 2 problems. As students work on the last two problems monitor for students who: start with the largest multiple of 3 and 10 within 702 that they can think of to decompose the dividend (690, 600), use the fewest steps to find the quotient. Student Facing states, “Jada used partial quotients to find out how many groups of 7 are in 389. Analyze Jada’s steps in the algorithm. (A vertical representation of partial quotients is shown.) 1. a. Look at the three numbers above 389. What do they represent? b. Look at the three subtractions below 389. What do they represent? c. What is another way you can decompose 389 to divide by 7? 2. Is 389 a multiple of 7? Explain your reasoning. 3. Use an algorithm that uses partial quotients to find out how many groups of 3 are in 702. 4. Is 702 a multiple of 3? Explain your reasoning.”

• Unit 9, Putting It All Together, Lesson 9, Warm-Up, students develop procedural skill and fluency with subtraction. Activity states, “1 minute: quiet think time. Record answers and strategy. Keep expressions and work displayed. Repeat with each expression.” Student Facing states, “Find the value of each expression mentally. 5,000-403, 5,300-473, 25,300-493, 26,000-1,493.” (4.NBT.4)

According to IM K-5 Math Teacher Guide, materials were designed to include opportunities for students to independently demonstrate procedural skill and fluency, when appropriate. Design Principles, Coherent Progress states, “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.” A Typical IM Lesson states, “The cool-down task is to be given to students at the end of the lesson. Students are meant to work on the cool-down for about 5 minutes independently and turn it in. The cool-down serves as a brief formative assessment to determine whether students understood the lesson. Students’ responses to the cool-down can be used to make adjustments to further instruction.” Examples include:

• Unit 4, From Hundredths to Hundred-thousands, Lesson 20, Cool-Down, students demonstrate procedural skill and fluency as they use the standard algorithm for subtraction. Student Facing states, “Use the standard algorithm to find the value of the difference. 173,225-114,329.” (4.NBT.4)

• Unit 7, Angles and Angle Measurement, Lesson 15, Activity 1, students demonstrate procedural skill and fluency as they find angle measurements. Activity states, “5 minutes: independent work time. 2 minutes: partner discussion. Monitor for students who: use symbols or letters to represent unknown angles, write equations to help them reason about the angle measurements.” Student Facing states, “Find the measurement of each shaded angle. Show how you know. (A. A right angle is shown, with 62 degrees and the unknown shaded part.)” (4.MD.7)

• Unit 9, Putting It All Together, Lesson 4, Activity 1, students demonstrate procedural skill and fluency as they subtract multi-digit numbers. Activity states, “6–8 minutes: independent work time.” Student Facing states, “1. Find the value of each difference. a. 700-16. b. 7,000-16. c. 70,000-16. d. 700,000-16.” (4.NBT.4)

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Materials are designed so that teachers and students spend sufficient time working with engaging applications of the mathematics.

The materials reviewed for Kendall Hunt's Illustrative Mathematics Grade 4 meet expectations for being designed so that teachers and students spend sufficient time working with engaging applications of the mathematics. According to IM Curriculum, Design Principles, Balancing Rigor, “Access to new mathematics and problems prompts students to apply their conceptual understanding and procedural fluency to novel situations.” Multiple routine and non-routine applications of the mathematics are included throughout the grade level and these single- and multi-step application problems are included within Activities or Cool-downs.

Students have the opportunity to engage with applications of math both with support from the teacher and independently. According to IM K-5 Math Teacher Guide, materials were designed to include opportunities for students to independently demonstrate application of grade-level mathematics, when appropriate. Design Principles, Coherent Progress states, “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.” A Typical IM Lesson states, “The cool-down task is to be given to students at the end of the lesson. Students are meant to work on the cool-down for about 5 minutes independently and turn it in. The cool-down serves as a brief formative assessment to determine whether students understood the lesson. Students’ responses to the cool-down can be used to make adjustments to further instruction.”

Examples of routine applications of the math include:

• Unit 3, Extending Operations to Fractions, Lesson 6, Cool-down, students apply their understanding about multiplication of a fraction by a whole number to solve real-world problems. Student Facing states, “Tyler bought 5 cartons of milk. Each carton contains \frac{3}{4} liter. How many liters of milk did Tyler buy? Explain or show your reasoning. Han bought 3 cartons of chocolate milk. Each carton contains \frac{5}{8} liter. Did Han buy the same amount of milk as Tyler? Explain or show your reasoning.” (4.NF.4c)

• Unit 6, Multiplying and Dividing Multi-digit Numbers, Lesson 23, Activity 1, students solve real-world problems as they reason about distance and use multiple operations to find a solution. Student Facing states, “Mai’s cousin is in middle school. She travels from her homeroom to math, then English, history, and science. When she finishes her science class, she takes the same path back to her homeroom. Mai’s cousin makes the same trip 5 times each week. The distances between the classes are shown. 1. How far does Mai’s cousin travel each round trip—from her homeroom to the four classes and back? Write one or more expressions or equations to show your reasoning. 2. Each week, Mai’s cousin makes 3 round trips from her homeroom to her music class. The total distance traveled on those 3 round trips is 2,364 feet. How far away is the music room from her homeroom? Show your reasoning. 3. Mai thinks her cousin travels 2 miles each week just going between classes. Do you agree? Explain or show your reasoning.” A diagram showing distances between locations, in feet, is shown. (4.NBT.4, 4.NBT.5, 4.NBT.6)

• Unit 9, Putting It All Together, Lesson 8, Activity 2, students solve a real-world problem as they interpret a situation involving equal groups and make sense of a remainder. Student Facing states, “A school is taking everyone on a field trip. It needs buses to transport 375 people. Bus Company A has small buses with 27 seats in each. Bus Company B has large buses with 48 seats in each. 1. What is the smallest number of buses that will be needed if the school goes with: Bus Company A? Show your reasoning. Bus Company B? Show your reasoning. 2. Which bus company should the school choose? Explain your reasoning. 3. Bus Company C has large buses that can take up to 72 passengers. Diego says, ‘If the school chooses Bus Company C, it will need only 6 buses, but the buses will have more empty seats.’ Do you agree? Explain your reasoning.” (4.OA.3)

Examples of non-routine applications of the math include:

• Unit 1, Factors and Multiples, Lesson 6, Activity 2, students examine factors of numbers from 1 to 20 and use them to solve problems. Launch states, “‘Let’s solve some problems about a game you read about earlier, where students take turns opening and closing lockers. Silently read and think about each question.’ 1 minute: quiet think time.” Student Facing states, “The 20 students in Tyler’s fourth-grade class are playing a game in a hallway with 20 lockers in a row. Your goal is to find out which lockers will be touched as all 20 students take their turn touching lockers. 1. Which locker numbers does the 3rd student touch? 2. Which locker numbers does the 5th student touch? 3. How many students touch locker 17? Explain or show how you know. 4. Which lockers are only touched by 2 students? Explain or show how you know. 5. Which lockers are touched by only 3 students? Explain or show how you know. 6. Which lockers are touched the most? Explain or show how you know. If you have time: Which lockers are still open at the end of the game? Explain or show how you know.” (4.OA.4)

• Unit 5, Multiplicative Comparison and Measurement, Lesson 10, Activity 2, students solve multi-step problems by using metric units of measurement and multiplicative comparison. Student Facing states, “Here are six water bottle sizes and four clues about the amount of water they each hold. One bottle holds 350 mL. A bottle in size B holds 5 times as much water as the bottle that holds 1 L. The largest bottle holds 20 times the amount of water in the smallest bottle. One bottle holds 1,500 mL, which is 3 times as much water as a bottle in size E. Use the clues to find out the amount of water, in mL, that each bottle size holds. Be prepared to explain or show your reasoning.” (4.MD.2, 4.OA.2, 4.OA.3)

• Unit 7, Angles and Angle Measurement, Lesson 16, Activity 2, students use their understanding of geometric figures and measurements to draw, describe, and identify two-dimensional figures. Activity states, “5 minutes: independent work time. 8–10 minutes: partner work time. Monitor for diagrams that reflect a variety of geometric features. Monitor for students who consider both geometric features and measurement in the description.” Student Facing states, “1. Create a two-dimensional shape that has at least 3 of the following: a. ray, b.line segment, c. right angle, d. acute angle, e. obtuse angle, f. perpendicular lines, g. parallel lines. 2. Without showing your partner, describe the figure so that your partner is able to draw it as best as possible. 3. Switch roles, and draw your partner’s shape based on their description.” (4.G.1, 4.G.2)

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The three aspects of rigor are not always treated together and are not always treated separately. There is a balance of the three aspects of rigor within the grade.

The materials reviewed for Kendall Hunt's Illustrative Mathematics Grade 4 meet expectations in that the three aspects of rigor are not always treated together and are not always treated separately. There is a balance of the three aspects of rigor within the grade.

All three aspects of rigor are present independently throughout the grade. Examples where instructional materials attend to conceptual understanding, procedural skill and fluency, or application include:

• Unit 2, Fraction Equivalence and Comparison, Lesson 2, Cool-down, students demonstrate conceptual understanding as they create visual representation of non-unit fractions. Student Facing states, “Use a blank diagram to create a representation for each fraction. Both blank diagrams represent the same quantity. 1. \frac{5}{8} 2. \frac{9}{8}” Bar models broken into two wholes are provided for each problem. (4.NF.A)

• Unit 3, Extending Operations to Fractions, Lesson 11, Activity 1, students solve routine real-world problems that involve subtracting mixed numbers where it is necessary to decompose one or both numbers. Student Facing states, “Clare, Elena, and Andre are making macramé friendship bracelets. They’d like their bracelets to be 9\frac{4}{8} inches long. For each question, explain or show your reasoning. 1. Clare started her bracelet first and has only \frac{7}{8} inch left until she finishes it. How long is her bracelet so far? 2. So far, Elena’s bracelet is 5\frac{1}{8} inches long and Andre’s is 3\frac{5}{8} inches long. How many more inches do they each need to reach 9\frac{4}{8}inches? 3. How much longer is Elena’s bracelet than Andre’s at the moment?” (4.NF.3d)

• Unit 4, From Hundredths to Hundred-thousands, Lesson 19, Activity 1, students develop procedural skill and fluency as they use the addition algorithm. Student Facing states, “1. Find the value of each sum. a. 8,299+1, b. 8,299+11, c. 8,299+111, d. 8,299+1111. 2. Use the expanded form of both 8,299 and 1,111 to check the value you found for the last sum.” (4.NBT.4)

Multiple aspects of rigor are engaged simultaneously throughout the materials in order to develop students’ mathematical understanding of a single unit of study or topic. Examples include:

• Unit 1, Factors and Multiples, Lesson 1, Cool-down, students use procedural fluency with multiples and apply their understanding of area. Student Facing states, “If a rectangle is 6 tiles wide, what could be its area? Name three possibilities. Explain or show your reasoning.” (4.OA.4)

• Unit 3, Extending Operations to Fractions, Lesson 1, Activity 1, students engage in all three aspects of rigor, conceptual understanding, procedural fluency, and application as they interpret situations involving equal groups. Activity states, “‘Take a few quiet minutes to think about the first set of problems about crackers. Then, discuss your thinking with your partner.’ 4 minutes: independent work time. 2 minutes: partner discussion. Pause for a whole-class discussion. Invite students to share their responses. If no students mention that there are equal groups, ask them to make some observations about the size of the groups in each image. Discuss the expressions students wrote: ‘What expression did you write to represent the crackers in Image A? Why? ($$6\times4$$, because there are 6 groups of 4 full crackers.) What about the crackers in Image B? Why? ($$6\times\frac{1}{4}$$, because there are 6 groups of \frac{1}{4} of a cracker.)’ Ask students to complete the remaining problems. 5 minutes: independent or partner work time. Monitor for students who reason about the quantities in terms of ‘___ groups of ___’ to help them write expressions.” Student Facing states, “Here are images of some crackers. a. How are the crackers in image A like those in B? b. How are they different? c. How many crackers are in each image? d. Write an expression to represent the crackers in each image. 2. Here are more images and descriptions of food items. For each, write a multiplication expression to represent the quantity. Then, answer the question. a. Clare has 3 baskets. She put 4 eggs into each basket. How many eggs did she put in baskets? b. Diego has 5 plates. He put \frac{1}{2} of a kiwi fruit on each plate. How many kiwis did he put on plates? c. Priya prepared 7 plates with \frac{1}{8} of a pie on each. How much pie did she put on plates? d. Noah scooped \frac{1}{3} cup of brown rice 8 times. How many cups of brown rice did he scoop?” (4.NF.4)

• Unit 7, Angles and Angle Measurement, Lesson 10, Activity 2, students develop conceptual understanding alongside procedural skill and fluency as they use a protractor to measure angles and understand perpendicular lines. Launch states, “Give each student 2 pieces of paper and colored pencils. Provide access to straightedges or rulers, in case requested. Read the opening prompts and the first question. ‘What do you think Lin did with her paper? Mark a point on a piece of paper and try folding it as Lin might have done.’ 2–3 minutes: quiet think time on the first problem. Pause for a discussion. Invite a couple of students to share how they think Lin met the challenge.” Student Facing states, “Tyler gave Lin a challenge: ‘Without using a protractor, draw four 90\degree angles. All angles have their vertex at point P.’ Lin folded the paper twice, making sure each fold goes through point P. Then, she traced the creases. 1. Your teacher will give you a sheet of paper. Draw a point on it. Then, show how Lin might have met the challenge. 2. When Lin folded the paper, the creases formed a pair of perpendicular lines. What do you think ‘perpendicular lines’ mean? 3. Use Lin’s method to create a new pair of perpendicular lines through the same point. Trace the creases with a different color. Be prepared to explain how you know the lines you created are perpendicular. 4. Which shapes have sides that are perpendicular to one another? Mark the perpendicular sides. Be prepared to explain how you know the sides are perpendicular.” A is a regular pentagon, B is a rectangle, C is a right triangle, and D is a parallelogram. (4.G.1, 4.MD.6)

#### Criterion 2.2: Math Practices

Materials meaningfully connect the Standards for Mathematical Content and Standards for Mathematical Practice (MPs).

The materials reviewed for Kendall Hunt's Illustrative Mathematics Grade 4 meet expectations for practice-content connections. The materials meaningfully connect the Standards for Mathematical Content and the Standards for Mathematical Practice (MPs).

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Materials support the intentional development of MP1: Make sense of problems and persevere in solving them; and MP2: Reason abstractly and quantitatively, for students, in connection to the grade-level content standards, as expected by the mathematical practice standards.

The materials reviewed for Kendall Hunt's Illustrative Mathematics Grade 4 meet expectations for supporting the intentional development of MP1: Make sense of problems and persevere in solving them; and MP2: Reason abstractly and quantitatively, for students, in connection to the grade-level content standards, as expected by the mathematical practice standards. Students have opportunities to engage with the Math Practices across the year and they are often explicitly identified for teachers within the Course Guide (How to Use These Materials) and within specific lessons (Instructional Routines, Lesson Preparation Narratives, and Lesson Activities’ Narratives).

MP1 is identified and connected to grade-level content, and there is intentional development of the MP to meet its full intent. Students make sense of problems and persevere in solving them as they work with support of the teacher and independently throughout the units. Examples include:

• Unit 2, Fraction Equivalence and Comparison, Lesson 14, Activity 1, students reason about fractions given descriptive clues. Preparation, Lesson Narrative states, “In the first activity, students compare sets of fractions with like and unlike denominators. They do so by using benchmarks, writing equivalent fractions, or reasoning about the numerators and denominators. In the second activity, students interpret and solve problems involving fractional measurements in context. Both activities present a new setup, structure, or context, requiring students to make sense of the given information and the problems, and to persevere in solving them (MP1).” Student Facing states, “Six friends are each given a list of 5 fractions. They each chose one fraction quietly and wrote clues about their choice. Use their clues to identify the fractions they chose.”

• Unit 6, Multiplying and Dividing Multi-Digit Numbers, Lesson 12, Cool-down, students persevere to solve and make sense of a real-world problem involving multi-digit multiplication. Preparation, Lesson Narrative states, “This lesson gives students the opportunity to apply the multiplication strategies they have learned to solve various contextual problems involving measurement. The problems vary in format and complexity—some involve a single computation and others require multiple steps to solve. The work here prompts students to make sense of problems and persevere in solving them (MP1) and to reason quantitatively and abstractly (MP2).” Student Facing states, “In a leap year, the month of February has 29 days. How many hours are in that month? Show your reasoning.”

• Unit 7, Angles and Angle Measurement, Lesson 15, Activity 2, students use their knowledge of angles to make sense of a problem and persevere in solving it. Narrative states, “In this Info Gap activity, students solve abstract multi-step problems involving an arrangement of angles with several unknown measurements. By now students have the knowledge and skills to find each unknown value, but the complexity of the diagram and the Info Gap structure demand that students carefully make sense of the visual information and look for entry points for solving the problems. They need to determine what information is necessary, ask for it, and persevere if their initial requests do not yield the information they need (MP1).” Launch states, “Groups of 2. MLR4 Information Gap Display the task statement, which shows a diagram of the Info Gap structure.1–2 minutes: quiet think time. Read the steps of the routine aloud. ‘I will give you either a problem card or a data card. Silently read your card. Do not read or show your card to your partner.’ Distribute the cards. ‘The diagram is not drawn accurately, so using a protractor to measure is not recommended.’ 1–2 minutes: quiet think time. Remind students that after the person with the problem card asks for a piece of information, the person with the data card should respond with ‘Why do you need to know (restate the information requested)?’”  Activity states, “5 minutes: partner work time. After students solve the first problem, distribute the next set of cards. Students switch roles and repeat the process with Problem Card 2 and Data Card 2.”

MP2 is identified and connected to grade-level content, and there is intentional development of the MP to meet its full intent. Students reason abstractly and quantitatively as they work with support of the teacher and independently throughout the units. Examples include:

• Unit 4, From Hundredths to Hundred-thousands, Lesson 1, Activity 2, students reason abstractly and quantitatively about decimals and their representations. Narrative states, “In this activity, students practice representing and writing decimals given another representation (fraction notation or a diagram). The idea that two decimals can be equivalent, just like two fractions can be equivalent, is made explicit here. When students make connections between quantities in word form, decimal form, and fraction form, they reason abstractly and quantitatively (MP2).” Student Facing states, “Each large square represents 1. 1. Write a fraction and a decimal that represent the shaded parts of each diagram. Then, write each amount in words. 2. Shade each diagram to represent each given fraction or decimal. a. Fraction: ___ Decimal: 0.78 b. Fraction: \frac{8}{10} Decimal: ___ c. Fraction: \frac{55}{100} Decimal: ___ d. Fraction: \frac{107}{100} Decimal: ___ e. Fraction: ___ Decimal: 1.6  3. Han and Elena disagree about what number the shaded portion represents. Han says that it represents 0.60 and Elena says it represents 0.6. Explain why both Han and Elena are correct.”

• Unit 5, Multiplicative Comparison and Measurement, Lesson 15, Activity 2, students reason abstractly and quantitatively when they convert feet and inches and solve a logic puzzle. Student Facing states, “While on an outing, a group of friends had a stone-stacking contest to see who could build the tallest stone tower. Andre’s stone tower is 3 times as tall as Diego’s, but Diego didn’t build the shortest tower. The tallest tower is 4 feet and 2 inches tall and belongs to Tyler. One person built a tower that is 39 inches tall. Tyler’s tower is 5 times as tall as the shortest tower. 1. How tall is each person’s stone tower? Be prepared to explain or show your reasoning. 2. Elena came along and built a tower that is 5 times as tall as Diego’s tower. Is Elena’s tower more than 6 feet? Show your reasoning.” Narrative states, “In this activity, students apply their knowledge of multiplicative comparison and ability to convert feet and inches to solve a logic puzzle. They use several given clues to determine the heights of four objects. As they use the clues to reason about the heights of the towers and who built them, students reason abstractly and quantitatively (MP2).”

• Unit 9, Putting It All Together, Lesson 8, Cool-down, students solve multi-step problems involving all operations. Preparation, Lesson Narrative states, “In the previous lesson, students solved word problems involving multiplicative comparison. In this lesson, they practice solving a wider variety of problems, with a focus on the relationships among multiple quantities in a situation. Students think about how to represent the relationships with one or multiple equations and using multiple operations. They also interpret their solutions and the solutions of others in context, including interpreting remainders in situations that involve division (MP2).” Student Facing states, “In one week, a train made 8 round trips between its home station and Union Station. At the end of the week, it traveled a few more miles from the home station to a repair center. That week, the train traveled a total of 1,564 miles. 1. Which statement is true for this situation? Explain or show your reasoning. a. The distance traveled for each round trip is 200 miles. The distance to the repair station is 26 miles. b. The distance traveled for each round trip is 195 miles. The distance to the repair station is 4 miles. c. The distance traveled for each round trip is 8 miles. The distance to the repair station is 1,500 miles. d. The distance traveled for each round trip is 193 miles. The distance to the repair station is 8 miles. Explain why one of the choices could not be true.”

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Materials support the intentional development of MP3: Construct viable arguments and critique the reasoning of others, for students, in connection to the grade-level content standards, as expected by the mathematical practice standards.

The materials reviewed for Kendall Hunt's Illustrative Mathematics Grade 4 meet expectations for supporting the intentional development of MP3: Construct viable arguments and critique the reasoning of others, for students, in connection to the grade-level content standards, as expected by the mathematical practice standards.

Students have opportunities to engage with MP3 across the year and it is often explicitly identified for teachers within the Course Guide (How to Use These Materials) and within specific lessons (Instructional Routines, Lesson Preparation Narratives, and Lesson Activities’ Narratives). According to the Course Guide, Instructional Routines, Other Instructional Routines, 5 Practices, “Lessons that include this routine are designed to allow students to solve problems in ways that make sense to them. During the activity, students engage in a problem in meaningful ways and teachers monitor to uncover and nurture conceptual understandings. During the activity synthesis, students collectively reveal multiple approaches to a problem and make connections between these approaches (MP3).”

Students construct viable arguments, in connection to grade-level content, as they work with support of the teacher and independently throughout the units. Examples include:

• Unit 2, Fraction Equivalence and Comparison, Lesson 10, Cool-down, students construct viable arguments as they determine if two fractions are equivalent. Student Facing states, “Problem 2, Diego wrote \frac{11}{5} and \frac{55}{10} as equivalent fractions. Are those fractions equivalent? Explain or show how you know. Use a number line, if it helps.”

• Unit 7, Angles and Angle Measurement, Lesson 3, Activity 2, students construct a viable argument as they create and reason about perpendicular and parallel lines. Narrative states, “In this activity, students are prompted to draw intersecting and parallel lines, and to explain how they know a pair of parallel lines would never intersect. Students are not expected to formally justify that two lines are parallel. They are expected to make a case that goes beyond appearance (such as ‘it looks like they would never cross’) and notice that the parallel lines maintain the same distance apart (MP3). Students are also introduced to the convention of naming lines with letters to support precision when describing and comparing lines. They are not expected to formally name lines or line segments with letters.” Launch states, “Groups of 2. Give each student access to a ruler or a straightedge. Display a field of dots. Select a student to draw a line in the field. ‘Sometimes we label lines to help communicate about different parts of a figure.’ Demonstrate labeling the line with a letter. ‘We can call this “line a” because we labeled it with an “a”.’”  Student Facing states, “Here is another field of dots. Each dot represents a point. 1. Draw a line through at least 2 points. Label it line h. 2. Draw another line that goes through at least 2 points and intersects your first line. Label it line g. 3. Can you draw a new line that you think would never intersect: a. line h?  b, line g? If so, draw the line. Be prepared to explain or show how you know the lines would never cross. If not, explain or show why it can’t be done. 4. Here is a trapezoid. Do you think its top and bottom sides are parallel? What about its left and right sides? Explain or show how you know. If you have time: Can you draw a new line that you think would never intersect either line h or line g? If so, draw the line and be prepared to explain or show how you know the lines would never cross. If not, explain why it can’t be done.”

• Unit 9, Putting It All Together, Lesson 8, Activity 1, students construct an argument and critique the reasoning of others as they interpret a problem involving equal groups. Student Facing states, “A school is taking everyone on a field trip. It needs buses to transport 375 people. Bus Company A has small buses with 27 seats in each. Bus Company B has large buses with 48 seats in each. 1. What is the smallest number of buses that will be needed if the school goes with: Bus Company A? Show your reasoning. Bus Company B? Show your reasoning. 2. Which bus company should the school choose? Explain your reasoning. 3. Bus Company C has large buses that can take up to 72 passengers. Diego says, “If the school chooses Bus Company C, it will need only 6 buses, but the buses will have more empty seats.” Do you agree? Explain your reasoning.” Activity Synthesis states, “Select students to share their responses and reasoning. Record the different representations students used to solve the problems. Consider asking: ‘How did you decide how many buses you would need from each company? Do all the buses carry the same amount of passengers? How can you see that in your representations or equations? How did you decide what operations to use to answer each question?”

Students critique the reasoning of others, in connection to grade-level content, as they work with support of the teacher and independently throughout the units. Examples include:

• Unit 3, Extending Operations to Fractions, Lesson 3, Activity 2, students construct a viable argument and critique the reasoning of others when they identify patterns using multiplication. Activity states, “3 minutes: independent work time on the first set of problems. 2 minutes: group discussion. Select students to explain how they reasoned about the missing numbers in the equations.” Student Facing states, “2. Your teacher will give you a sheet of paper. Work with your group of 3 and complete these steps on the paper. After each step, pass your paper to your right. Step 1: Write a fraction with a numerator other than 1 and a denominator no greater than 12. Step 2: Write the fraction you received as a product of a whole number and a unit fraction. Step 3: Draw a diagram to represent the expression you just received. Step 4: Collect your original paper. If you think the work is correct, explain why the expression and the diagram both represent the fraction that you wrote. If not, discuss what revisions are needed.” Narrative states, “As students discuss and justify their decisions they create viable arguments and critique one another’s reasoning (MP3).”

• Unit 5, Multiplicative Comparison and Measurement, Lesson 7, Activity 2, students convert measurement from meters to centimeters and critique student work to identify and describe errors. Narrative states, “In this activity, students analyze student work converting meters to centimeters to develop the understanding that a meter is ‘100 times as long’ as a centimeter. They correct errors in reasoning centering around place value (MP3).” Activity states, “‘Take 5 quiet minutes to spot and correct Priya’s errors and find the missing measurement. Then, share your thinking with your partner.’ 5 minutes: independent work time. 3–4 minutes: partner discussion. Monitor for students who place zeros for the measurement in centimeters and those who explicitly reason in terms of 100 times the value in meters.” Student Facing states, “Priya took some measurements in meters and recorded them in the table, but she made some errors when converting them to centimeters. She also left out one measurement.”  Students are given a table with the headings “Measurement in Meters, Measurement in Centimeters” and then the following: ”a. height of door 2 and 200 b. height of hallway 3 and 30  c. width of hallway 5 and 500  d. length of gym.18 and 180 e. length of hallway 27 and 2,700 f. length of playground 50 and ____ 1. Find and correct Priya’s conversion errors. Be prepared to explain how you know. 2. Fill in the length of the playground in centimeters. Write an equation to represent your thinking.”

• Unit 6, Multiplying and Dividing Multi-digit Numbers, Lesson 9, Activity 1, students critique the reasoning of others as they analyze an algorithm that uses partial products. Activity states, “4 minutes: independent work time on the first problem about Noah’s diagram. 4 minutes: partner discussion. 5 minutes: group work time on the rest of the activity. Monitor for students who include the place value of each digit in 124 in explaining what is happening in the algorithm.” Student Facing states, “1. Noah drew a diagram and wrote expressions to show his thinking as he multiplied two numbers. How does each expression represent Noah’s diagram? Be prepared to share your thinking with a partner. 2. Later, Noah learned another way to record the multiplication, as shown here. Make sense of each step of the calculations and record your thoughts. Be prepared to explain Noah’s steps to a partner.” An image of Noah’s work is shown along with his calculations. Narrative states, “When students interpret and make sense of Noah's work, they construct viable arguments and critique the reasoning of others (MP3).”

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Materials support the intentional development of MP4: Model with mathematics; and MP5: Choose tools strategically, for students, in connection to the grade-level content standards, as expected by the mathematical practice standards.

The materials reviewed for Kendall Hunt's Illustrative Mathematics Grade 4 meet expectations for supporting the intentional development of MP4: Model with mathematics; and MP5: Choose tools strategically, for students, in connection to the grade-level content standards, as expected by the mathematical practice standards. Students have opportunities to engage with the Math Practices across the year and they are often explicitly identified for teachers within the Course Guide (How to Use These Materials) and within specific lessons (Instructional Routines, Lesson Preparation Narratives, and Lesson Activities’ Narratives).

MP4 is identified and connected to grade-level content, and there is intentional development of the MP to meet its full intent. Students have many opportunities to solve real-world problems, model situations with appropriate representations, and describe what they do with the model and how it relates to the problem. Students model with mathematics as they work with support of the teacher and independently throughout the units. Examples include:

• Unit 1, Factors and Multiples, Lesson 8, Activity 1, students model with mathematics as they use art and concepts of area. Lesson Narrative states, “When students isolate and describe the mathematical elements in art and adhere to mathematical constraints to create art, they model with mathematics (MP4).” Student Facing states, “Create an outline for art in the Mondrian style, starting with an 18-by-24 grid. Your artwork should: be partitioned into at least 12 rectangles, include two different rectangles that have the same area, include at least one rectangle whose area is a prime number. Try at least one of these challenges. Make a design where: all but two of the rectangles have a prime number for its area, no two rectangles share a side entirely.”

• Unit 6, Multiplying and Dividing Multi-digit Numbers, Lesson 25, Activity 3, students model with mathematics as they create their own flower pattern and multi-step problem. Lesson Narrative states, “When students ask and answer questions that arise from a given situation, use mathematical features of an object to solve a problem, make choices, analyze real-world situations with mathematical ideas, interpret a mathematical answer in context, and decide if an answer makes sense in the situation, they model with mathematics (MP4).” Student Facing states, “1. Write a multi-step problem about making paper flowers. 2. Exchange the problem with your partner and solve each other’s problems.” Student Response states, “Sample response: It takes 1 sheet of tissue paper to make a big flower and \frac{1}{2} sheet to make a small flower. How much tissue paper is needed to make a garland that has 7 small and 7 large garlands? (7 sheets for big flowers and 4 sheets for small flowers, where half a sheet will not be used. 11 sheets are needed, with \frac{1}{2} a sheet left over).”

MP5 is identified and connected to grade-level content, and there is intentional development of the MP to meet its full intent. Students have multiple opportunities to identify and use a variety of tools or strategies, working with the support of the teacher and independently, throughout the units to support their understanding of grade-level math. Examples include:

• Unit 2, Fraction Equivalence and Comparison, Lesson 15, Cool-down, students use appropriate tools strategically when they compare two fractions using a strategy of their choice. Student Facing states, “In each pair of fractions, which fraction is greater? Explain or show your reasoning. 1. \frac{3}{10} or \frac{2}{6}. 2. \frac{99}{100} or \frac{9}{10}.”

• Unit 4, From Hundredths to Hundred-thousands, Lesson 16, Activity 1, students use appropriate strategies as tools when rounding numbers. Student Facing states, “Noah says that 489,231 can be rounded to 500,000. Priya says that it can be rounded to 490,000. 1. Explain or show why both Noah and Priya are correct. Use a number line if it helps. 2. Describe all the numbers that round to 500,000 when rounded to the nearest hundred-thousand. 3. Describe all the numbers that round to 490,000 when rounded to the nearest ten-thousand. 4. Name two other numbers that can also be rounded to both 500,000 and 490,000.” Lesson Narrative states, “When they find all of the numbers that round to a given number, students need to think carefully about place value and may choose to use a number line to support their reasoning (MP5).”

• Unit 8, Properties of Two-dimensional Shapes, Lesson 9, Activity 1, students use tools strategically as they identify line symmetry and solve problems. Student Facing states, “1. Mai has a piece of paper. She can get two different shapes by folding the paper along a line of symmetry. What is the shape of the paper before it was folded?” Lesson Narrative states, “The first question offers opportunities to practice choosing tools strategically (MP5). Some students may wish to trace the half-shapes on patty paper, to make cutouts of them, or to use other tools or techniques to reason about the original shape. Provide access to the materials and tools they might need.”

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Materials attend to the intentional development of MP6: Attend to precision; and attend to the specialized language of mathematics for students, in connection to the grade-level content standards, as expected by the mathematical practice standards.

The materials reviewed for Kendall Hunt's Illustrative Mathematics Grade 4 meet expectations for supporting the intentional development of MP6: Attend to precision; and attend to the specialized language of mathematics, for students, in connection to the grade-level content standards, as expected by the mathematical practice standards.

Students have many opportunities to attend to precision and the specialized language of math, in connection to grade-level content, as they work with support of the teacher and independently throughout the units. Examples include:

• Unit 1, Factors and Multiples, Lesson 7, Activity 1, students attend to the specialized language of math as they accurately describe factors and multiples. Student Facing states, “1. Complete a statement using the word “factor” and a statement using the word “multiple” for each number. 2. As you compare statements with your partner, discuss one thing you notice and one thing you wonder.” Lesson Narrative states, “The purpose of this activity is for students to find factors and multiples of a given number and make statements that use the terms “factors” and “multiples.” This work prompts students to use language precisely (MP6).”

• Unit 2, Fraction Equivalence and Comparison, Lesson 2, Warm-up, students use precise mathematical language as they describe how four shapes are partitioned and shaded. Narrative states, “This warm-up prompts students to carefully analyze and compare the features of four partitioned shapes. It allows the teacher to hear the terminologies students use to talk about fractions and fractional parts. In making comparisons, students have a reason to use language precisely (MP6).”  Launch states, “Groups of 2. Display the image. ‘Pick one that doesn’t belong. Be ready to share why it doesn’t belong.’ 1 minute: quiet think time.” Student response sample states,  A is the only one not partitioned into 3 parts. B is the only one that does not have straight edges. C is the only one not partitioned into equal parts. D is the only one whose parts are not all clear or unshaded.”  Activity Synthesis states,  “‘What does the shaded part in D represent?’ ($$\frac{1}{3}$$ or one-third of the shape). Shade one part of B and C. ‘Is each shaded part one-third of the shape as well? (Yes for B, no for C.)  Why is the shaded part not one-third of the square in C?’ (The parts aren’t equal in size.)  Shade one part of A. ‘Is it a third of the square?’ (No, it is \frac{1}{4} or one-fourth.)”

• Unit 2, Fraction Equivalence and Comparison, Lesson 9, Warm-up, students use accuracy and precision when they describe strategies in finding the value of multiplication problems. The Narrative states, “strategies of doubling and halving elicited here will be helpful later in the lesson when students generate equivalent fractions. In describing strategies, students need to be precise in their word choice and use of language (MP6).” Launch states, “Display one expression. ‘Give me a signal when you have an answer and can explain how you got it.’” Student Facing states, “Find the value of each expression mentally. 10\times6, 10\times12, 10\times24, 5\times24.”

• Unit 3, Extending Operations to Fractions, Lesson 13, Activity 1, students use accuracy and precision when they measure pencils to the nearest \frac{1}{4} and \frac{1}{8} inch. Student Facing states, “Your teacher will give your group a set of colored pencils. 1. Work with your group to measure each colored pencil to the nearest \frac{1}{4} inch. Check each other’s measurements. Record each measurement in the table. 3. Work with your group to measure each colored pencil to the nearest \frac{1}{8} inch. Check one another’s measurements. Record each measurement in the table.” The Lesson Narrative states, “Students attend to precision when they measure the pencils to the appropriate fractional unit (MP6).” The Activity Synthesis states, “Allow students to record their two sets of data on two different class line plots. (If dot stickers are available, consider using them—one sticker for each data point.) ‘How did your data and line plots change when you measured colored pencils to the nearest \frac{1}{8} inch?’ (Sample responses: We got different numbers. The marks or points on the line plots are distributed differently. The points for some of the same pencils show up as different lengths in the second line plot.) ‘What is challenging about measuring to the nearest \frac{1}{8} inch?’ (The tick marks are smaller and harder to see on the ruler.) ‘Why do you think we measure to the nearest \frac{1}{8} inch?’ (We measure to be more accurate.) “Let’s look at some other length data with measurements in halves, fourths and eighths of an inch.”

• Unit 5, Multiplicative Comparison and Measurement, Lesson 13, Cool-Down, students compare the ingredients needed to make cookies, using precision when comparing units of measure including pounds and ounces (MP6). Student Facing states, “Priya needs oats and raisins to make cookies. She needs 3 pounds of oats. That amount is 4 times as much as the amount of raisins that she needs. How many ounces of raisins does she need? Explain or show your reasoning.” Student Section Summary states, “In this section, we learned about various units for measuring length, distance, weight, capacity, and time. We saw how different units that measure the same property are related. Here are the relationships that we saw: One meter (m) is 100 times as long as 1 centimeter (cm). One kilometer (km) is 1,000 times as long as 1 meter (m). One kilogram (kg) is 1,000 times as heavy as 1 gram (g). One liter (L) is 1,000 times as much as 1 milliliter (mL). One pound (lb) is 16 times as heavy as 1 ounce (oz). One hour is 60 times as long as 1 minute. One minute is 60 times as long as 1 second. When given a measurement in one unit, we can find the value in another unit by reasoning and writing equations. Throughout the section, we used these relationships to convert measurements from one unit to another, to compare and order measurements, and to solve problems in different situations.”

• Unit 7, Angle and Angle Measurement, Lesson 1, Cool-down, students attend to precise mathematical language as they describe a drawing to a partner. Student Facing states, “Here is a drawing on a card: Write a description of the drawing that could be used by a classmate to make a copy.” Student Response sample states, “Draw two diagonal lines: one from the top left corner to the bottom right, and another from the bottom left corner to the top right. Draw a line that goes up and down through the point where the two diagonal lines cross. From the top of that line, draw a line to the bottom right corner. The bottom segment of the up-and-down line is thicker than the rest of the lines. The lines make a lot of triangles of different sizes.”

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Materials support the intentional development of MP7: Look for and make use of structure; and MP8: Look for and express regularity in repeated reasoning, for students, in connection to the grade-level content standards, as expected by the mathematical practice standards.

The materials reviewed for Kendall Hunt's Illustrative Mathematics Grade 4 meet expectations for supporting the intentional development of MP7: Look for and make use of structure; and MP8: Look for and express regularity in repeated reasoning, for students, in connection to grade-level content standards, as expected by the mathematical practice standards. Students have opportunities to engage with the Math Practices across the year and they are often explicitly identified for teachers within the Course Guide (How to Use These Materials) and within specific lessons (Instructional Routines, Lesson Preparation Narratives, and Lesson Activities’ Narratives).

MP7 is identified and connected to grade-level content, and there is intentional development of the MP to meet its full intent. Students have many opportunities to look for, describe, and make use of patterns within problem-solving as they work with support of the teacher and independently throughout the units. Examples include:

• Unit 1, Factors and Multiples, Lesson 4, Warm-up, students use structure and knowledge of math facts to find larger products. Narrative states, “The purpose of this Number Talk is to elicit strategies and understandings students have for multiplying within 100 with one factor larger than 10. These understandings help students develop fluency and will be helpful when students find factor pairs of numbers later in the lesson. In this activity, students have an opportunity to look for and make use of structure (MP7) as they use a combination of products of smaller factors to find products of larger factors.” Launch states, “Display one expression. ‘Give me a signal when you have an answer and can explain how you got it.’  1 minute: quiet think time.” Student Facing states, “Find the value of each expression mentally. 10\times6, 3\times6, 13\times6, \12\times4.” Activity Synthesis states, “How can knowing the value of the first two expressions help you find the value of the third expression? (I can multiply in parts and add the smaller parts together to find a larger product.)”

• Unit 4, From Hundredths to Hundred-thousands, Lesson 9, Activity 1, students look for and make use of structure while they compare numbers to determine value. Narrative states, “When students sort the cards, they look for how the numbers are the same and different, including their overall value or the digits that make up the numbers (MP7).” Activity states, “Give each group a set of cards from the blackline master. 5 minutes: partner and group work time on the first two problems. As students work, listen for place-value language such as: value of the digit, ten times, thousands, ten-thousands, and hundred-thousands. Record any place-value language students use to describe how they sorted the numbers and display for all to see. ‘Now work independently to write the numbers in the next problem in expanded form. Then, talk with your partner about the value of the digits.’ 3 minutes: independent work time. 5 minutes: partner work time. Monitor for students who: accurately write the numbers in expanded form, describe the relationship between the value of the digits in multiplicative terms (“ten times”).” Student Facing states, “Your teacher will give you and your partner a set of cards with multi-digit numbers on them. 1. Sort the cards in a way that makes sense to you. Be prepared to explain your reasoning. 2. Join with another group and explain how you sorted your cards. 3. Write each number in expanded form. a. 4,620 b. 46,200 c. 462,000. 4. Write the value of the 4 in each number. 5. Compare the value of the 4 in two of the numbers. Write two statements to describe what you notice about the values. 6. How is the value of the 2 in 46,200 related to the value of the 2 in 462,000?” Activity Synthesis states, “Invite students to share their expressions in expanded form and what they noticed about the value of the 4. ‘What do you notice about the value of the 6 in each number? The value of the 2?’ (The value of the 6 is different in each number. It is first 600, then 6,000, then 60,000.) Students may talk about the number of zeros in each number. Shift their focus to the place value of the 6— hundreds, thousands, ten-thousands. ‘How is the value of the 2 in 46,200 related to the value of the 2 in 462,000?’ (The value of the 2 in 462,000 is 2,000 and the same digit in 46,200 has a value of 200. 2,000 is ten times the value 200.) ‘What multiplication equation could we write to represent the relationship between the 2 in 46,200 and 462,000?’ ($$2,000=200\times10$$) ‘We can also write this equation using division: 2,000\div200=10.’”

• Unit 9, Putting It All Together, Lesson 1, Cool-down, students look for and make use of structure as they reason about sums of fractions. Lesson Narrative states, “In this lesson, students practice multiplying a fraction and a whole number and adding and subtracting fractions, including mixed numbers. They rely on their understanding of equivalence and the properties of operations to decompose fractions, whole numbers, and mixed numbers to enable comparison, addition, subtraction, and multiplication (MP7).” Student Facing states, “Here are some fractions: \frac{15}{10}, \frac{13}{10}, \frac{53}{100}, \frac{9}{10}. 1. Select two fractions that have a sum greater than 2. Explain or show your reasoning. 2. Use all four fractions to write an expression that has a value greater than 1 but less than 2.”

MP8 is identified and connected to grade-level content, and there is intentional development of the MP to meet its full intent. Students have multiple opportunities to use repeated reasoning in order to make generalizations and build a deeper understanding of grade-level math concepts as they work with support of the teacher and independently throughout the units. Examples include:

• Unit 3, Extending Operations to Fractions, Lesson 7, Activity 2, students use repeated reasoning as they decompose fractions to find sums. Narrative states, “In the previous activity, students saw that a fraction can be decomposed into a sum of fractions with the same denominator and that it can be done in more than one way. In this activity, they record such decompositions as equations. The last question prompts students to consider whether any fraction can be written as a sum of smaller fractions with the same denominator. Students see that only non-unit fractions (with a numerator greater than 1) can be decomposed that way. Students observe regularity in repeated reasoning as they decompose the numerator, 9, into different parts while the denominator in all cases is 5 (MP8).” Activity states, “‘Take a few quiet minutes to complete the activity. Then, share your responses with your partner.’ 5–6 minutes: independent work time. 3–4 minutes: partner discussion. Monitor for different explanations students offer for the last question.” Student Facing states, “1. Use different combinations of fifths to make a sum of \frac{9}{5}. a. \frac{9}{5} = ___ + ___ + ___ + ___ + ___  b.  \frac{9}{5} = ___ + ___ + ___ + ___  c. \frac{9}{5} = ___ + ___ + ___  d. \frac{9}{5}=___ + ___   2. Write different ways to use thirds to make a sum of \frac{4}{3}}. How many can you think of? Write an equation for each combination. 3. Is it possible to write any fraction with a denominator of 5 as a sum of other fifths? Explain or show your reasoning.” Activity Synthesis states, “Invite students to share their equations. Display or record them for all to see. Next, discuss students' responses to the last question. Select students with different explanations to share their reasoning. If not mentioned by students, highlight that fractions with a numerator of 1 (unit fractions) cannot be further decomposed into smaller fractions with the same denominator because it is already the smallest fractional part. Other fractions with a numerator other than 1 (non-unit fractions) can be decomposed into fractions with the same denominator.”

• Unit 8, Properties of Two-dimensional Shapes, Lesson 7, Cool-down, students use repeated reasoning as they find the perimeter of shapes and write matching expressions. Student Facing states, “Here is a rectangle with two lines of symmetry. Find its perimeter. Write an expression to show how you find it.” Activity 1 Lesson Narrative states, “In this activity, students find the perimeter of several shapes and write expressions that show their reasoning. Each side of the shape is labeled with its length, prompting students to notice repetition in some of the numbers. The perimeter of all shapes can be found by addition, but students may notice that it is efficient to reason multiplicatively rather than additively (MP8).” Students have the opportunity to demonstrate this same reasoning within the Cool-down.

• Unit 9, Putting It All Together, Lesson 4, Warm-up, students work through a number talk, using repeated reasoning to solve increasingly challenging addition problems. Narrative states, “This Number Talk encourages students to think about the base-ten structure of whole numbers and properties of operations to mentally solve subtraction problems. The reasoning elicited here will be helpful later in the lesson when students find differences of multi-digit numbers.” Activity states, “1 minute: quiet think time. Record answers and strategy. Keep expressions and work displayed. Repeat with each expression.” Student Facing states, “Find the value of each difference mentally. 87-24, 387-124, 6,387-129, 6,387-4,329.” Activity Synthesis states, “How is each expression related to the one before it? How might the first expression help us find the value of the last expression?”

### Usability

The materials reviewed for Kendall Hunt's Illustrative Mathematics Grade 4 meet expectations for Usability. The materials meet expectations for Criterion 1, Teacher Supports; partially meet expectations for Criterion 2, Assessment; and meet expectations for Criterion 3, Student Supports.

##### Gateway 3
Meets Expectations

#### Criterion 3.1: Teacher Supports

The program includes opportunities for teachers to effectively plan and utilize materials with integrity and to further develop their own understanding of the content.

The materials reviewed for Kendall Hunt's Illustrative Mathematics Grade 4 meet expectations for Teacher Supports. The materials: provide teacher guidance with useful annotations and suggestions for enacting the student and ancillary materials; contain 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; include standards correlation information that explains the role of the standards in the context of the overall series; provide explanations of the instructional approaches of the program and identification of the research-based strategies; and provide a comprehensive list of supplies needed to support instructional activities.

##### Indicator {{'3a' | indicatorName}}

Materials provide teacher guidance with useful annotations and suggestions for how to enact the student materials and ancillary materials, with specific attention to engaging students in order to guide their mathematical development.

The 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 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.)”

##### Indicator {{'3b' | indicatorName}}

Materials contain adult-level explanations and examples of the more complex grade-level/course-level concepts and concepts beyond the current course so that teachers can improve their own knowledge of the subject.

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.”

##### Indicator {{'3c' | indicatorName}}

Materials include standards correlation information that explains the role of the standards in the context of the overall series.

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.”

##### Indicator {{'3d' | indicatorName}}

Materials provide strategies for informing all stakeholders, including students, parents, or caregivers about the program and suggestions for how they can help support student progress and achievement.

The materials reviewed for Kendall Hunt's Illustrative Mathematics Grade 4 provide strategies for informing all stakeholders, including students, parents, or caregivers about the program and suggestions for how they can help support student progress and achievement.

Each unit has corresponding Family Support Materials, in English and Spanish, that provide a variety of supports for families. Each unit includes a narrative for stakeholders, describing what students will learn within each section. Additionally, Try it at home! includes suggestions for at home activities and questions families can ask, all geared towards supporting the mathematical ideas in the unit. Examples include:

• For Families, Grade 4, Unit 1, Factors and Multiples, Family Support Materials, “In this unit, students learn about factors and multiples and apply their understanding of the area of rectangles. Students determine if a number between 1 and 100 is prime or composite. Section A: Understand Factors and Multiples. In this section, students learn about the meaning of factors and multiples by relating them to the concept of area. They use square tiles to build rectangles with given length and width. Then, they find the area of the rectangles. For example, this rectangle has an area of 14 square units with side lengths of 7 and 2. We can say that 7 and 2 are a factor pair of 14, and that 7\times2=14. We can also say that 14 is a multiple of 7 and a multiple of 2. Students discover that some numbers have many factor pairs and others have only one possible factor pair. They decide if a number is prime or composite based on how many rectangles can be made with that number as the area.”

• For Families, Grade 4, Unit 4, From Hundredths to Hundred-thousands, Family Support Materials, Try it at home!, “Near the end of the unit, ask your student about the numbers 769,038 and 170,932: What is the value of the 7 in each number? Write a multiplication or division equation to show the relationship between these two values. Round each number to the nearest multiple of 1,000 and multiple of 100,000. Find the sum and difference of the two numbers. Questions that may be helpful as they work: How did you find your answer? How could you solve your problem in a different way?”

• For Families, Grade 4, Unit 6, Multiplying and Dividing Multi-digit Numbers, Family Support Materials, “In this unit, students deepen their understanding of multiplication and division and expand their ability to perform these operations on multi-digit numbers. Section A: Features of Patterns. In this section, students analyze patterns. They use ideas related to multiplication (such as factors, multiples, double, and triple) to describe and extend the patterns. If the pattern continues, could 50 represent the side length or the area of one of the rectangles? If so, which step? If not, why not? Section B: Multi-Digit Multiplication. In this section, students multiply one-digit numbers and numbers up to four digits, and pairs of two-digit numbers. They learn to use increasingly more efficient methods to multiply. Students begin by using visual representations—arrays, base-ten diagrams, and grids—to help them find products. They recall that rectangles can be used to represent multiplication, with the side lengths representing the factors and the area representing the product. Students see that it helps to decompose (break apart) the factors by place value. For example, to multiply 31 and 15, we can think of the 31 as 30+1 and the 15 as 10+5. We can then label these values on a diagram, multiply the parts separately, and add the partial products. Later, students use an algorithm that lists partial products vertically. This work prepares them to make sense of the standard algorithm for multiplication, to be studied closely in grade 5.”

##### Indicator {{'3e' | indicatorName}}

Materials provide explanations of the instructional approaches of the program and identification of the research-based strategies.

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.”

##### Indicator {{'3f' | indicatorName}}

Materials provide a comprehensive list of supplies needed to support instructional activities.

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.”

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This is not an assessed indicator in Mathematics.

##### Indicator {{'3h' | indicatorName}}

This is not an assessed indicator in Mathematics.

#### Criterion 3.2: Assessment

The program includes a system of assessments identifying how materials provide tools, guidance, and support for teachers to collect, interpret, and act on data about student progress towards the standards.

The materials reviewed for Kendall Hunt's Illustrative Mathematics Grade 4 partially meet expectations for Assessment. The materials identify the standards and the mathematical practices assessed in formal assessments. The materials provide multiple opportunities to determine students' learning and sufficient guidance to teachers for interpreting student performance but do not provide suggestions for follow-up. The materials include opportunities for students to demonstrate the full intent of grade-level standards and mathematical practices across the series.

##### Indicator {{'3i' | indicatorName}}

Assessment information is included in the materials to indicate which standards are assessed.

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.”

##### Indicator {{'3j' | indicatorName}}

Assessment system provides multiple opportunities throughout the grade, course, and/or series to determine students' learning and sufficient guidance to teachers for interpreting student performance and suggestions for follow-up.

The materials reviewed for Kendall Hunt's Illustrative Mathematics Grade 4 partially meet expectations for including an assessment system that provides multiple opportunities throughout the grade, course, and/or series to determine students’ learning and sufficient guidance to teachers for interpreting student performance and suggestions for follow-up.

Each End-of-Unit Assessment and End-of-Course Assessment provides guidance to teachers for interpreting student performance, with an answer key and standard alignment. According to the Teacher Guide, Summative Assessments, “All summative assessment problems include a complete solution and standard alignment. Multiple choice and multiple response problems often include a reason for each potential error a student might make.” Examples from the assessment system include:

• Unit 1, Factors and Multiples, End-of-Unit Assessment, Problem 2, “Select all true statements. A. 15 is a multiple of 3. B. 16 is a factor of 8. C. 80 is a multiple of 4. D. The only factor pair of 49 is 1 and 49. E. The factor pairs of 12 are 1 and 12, 2 and 6, and 3 and 4.” The Assessment Teacher Guide states, “This item assesses student understanding of the words factor and multiple. They may select B, and not select A, C, and E, if they confuse the meaning of factor and multiple. They may select D if they understand the meaning of factor but are not careful and forget the factor 7. Students may understand the meaning of factor but fail to select C if they do not see that 80=20\times4.” The answer key aligns this problem to 4.OA.4.

• Unit 3, Extending Operations to Fractions, End-of-Unit Assessment, Problem 5, “The line plot shows the lengths of some colored pencils. (There is an image of a line graph showing colored pencil lengths in inches.) a. What is the difference between the longest pencil and the shortest pencil shown in this line plot? Show your reasoning. b. How many pencils measured 4\frac{1}{2}inches or more? c. Two more colored pencils measure 2\frac{1}{4} inches and 5\frac{1}{8} inches. Plot these measurements on the line plot.” The Assessment Teacher Guide states, “Students interpret the measurement data on the line plot to answer questions and use the data to subtract fractions. For the first question, students may use the numbers on the line plot to help find the difference or they may reason more abstractly as in the provided solution.” The answer key aligns this problem to 4.MD.4.

• Unit 9, Putting It All Together, End-of-Course Assessment, Problem 4, “a. Round 73,526 to the nearest ten-thousand. Use the number line if it is helpful. b. Round 73,526 to the nearest thousand. Use the number line if it is helpful. c. Round 73,526 to the nearest hundred. Use the number line if it is helpful.” The Assessment Teacher Guide states, “Students round a number to the nearest ten-thousand, thousand, and hundred. No method is suggested so students may use their understanding of place value or they may label the number lines and use them.” The answer key aligns this problem to 4.NBT.3.

While assessments provide guidance to teachers for interpreting student performance, suggestions for following-up with students are either minimal or absent. Cool-Downs, at the end of each lesson, include some suggestions. According to IM Curriculum, Cool-Downs, “The cool-down (also known as an exit slip or exit ticket) is to be given to students at the end of the lesson. This activity serves as a brief check-in to determine whether students understood the main concepts of that lesson. Teachers can use this as a formative assessment to plan further instruction. When appropriate, guidance for unfinished learning, evidenced by the cool-down, is provided in two categories: next-day support and prior-unit support. This guidance is meant to provide teachers ways in which to continue grade-level content while also giving students the additional support they may need.” An example includes:

• Unit 8, Properties of Two-dimensional Shapes, Lesson 4, Cool-down, Student Facing states, “Which figures have more than one line of symmetry? Explain or show your reasoning.” Responding to Student Thinking states, “Students draw a line through the smiling face that splits the circle into halves but is not a line of symmetry.” Next Day Supports states, “Launch warm-up or Activity 1 by highlighting important ideas from previous lessons.” This problem aligns to 4.G.3.

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Assessments include opportunities for students to demonstrate the full intent of grade-level/course-level standards and practices across the series.

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.”

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Assessments offer accommodations that allow students to demonstrate their knowledge and skills without changing the content of the assessment.

The materials reviewed for Kendall Hunt's Illustrative Mathematics Grade 4 offer accommodations that allow students to demonstrate their knowledge and skills without changing the content of the assessment. These suggestions are provided within the Teacher Guide in a section called “Universal Design for Learning and Access for Students with Disabilities.” As such, they are included at the program level and not specific to each assessment.

Examples of accommodations include:

• IM K-5 Teacher Guide, UDL Strategies to Enhance Access states, “Present content using multiple modalities: Act it out, think aloud, use gestures, use a picture, show a video, demonstrate with objects or manipulatives. Annotate displays with specific language, different colors, shading, arrows, labels, notes, diagrams, or drawings. Provide appropriate reading accommodations. Highlight connections between representations to make patterns and properties explicit. Present problems or contexts in multiple ways, with diagrams, drawings, pictures, media, tables, graphs, or other mathematical representations. Use translations, descriptions, movement, and images to support unfamiliar words or phrases.”

• IM K-5 Teacher Guide, UDL Strategies to Enhance Access states, “It is important for teachers to understand that students with visual impairments are likely to need help accessing images in lesson activities and assessments, and prepare appropriate accommodations. Be aware that mathematical diagrams are provided as scalable vector graphics (SVG format), because this format can be magnified without loss of resolution. Accessibility experts who reviewed this curriculum recommended that students who would benefit should have access to a Braille version of the curriculum materials, because a verbal description of many of the complex mathematical diagrams would be inadequate for supporting their learning. All diagrams are provided in the SVG file type so that they can be rendered in Braille format.”

• IM K-5 Teacher Guide, UDL Strategies to Enhance Access states, “Develop Expression and Communication, Offer flexibility and choice with the ways students demonstrate and communicate their understanding. Invite students to explain their thinking verbally or nonverbally with manipulatives, drawings, diagrams. Support fluency with graduated levels of support or practice. Apply and gradually release scaffolds to support independent learning. Support discourse with sentence frames or visible language displays.”

#### Criterion 3.3: Student Supports

The program includes materials designed for each child’s regular and active participation in grade-level/grade-band/series content.

The materials reviewed for Kendall Hunt's Illustrative Mathematics Grade 4 meet expectations for Student Supports. The materials provide: strategies and supports for students in special populations and for students who read, write, and/or speak in a language other than English to support their regular and active participation in learning grade-level mathematics; multiple extensions and/or opportunities for students to engage with grade-level mathematics at higher levels of complexity; and manipulatives, both virtual and physical, that are accurate representations of the mathematical objects they represent and, when appropriate, are connected to written methods.

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Materials provide strategies and supports for students in special populations to support their regular and active participation in learning grade-level/series mathematics.

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.”

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Materials provide extensions and/or opportunities for students to engage with grade-level/course-level mathematics at higher levels of complexity.

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.”

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Materials provide varied approaches to learning tasks over time and variety in how students are expected to demonstrate their learning with opportunities for students to monitor their learning.

The materials reviewed for Kendall Hunt's Illustrative Mathematics Grade 4 provide various approaches to learning tasks over time and variety in how students are expected to demonstrate their learning, but do not provide opportunities for students to monitor their learning.

Students engage with problem-solving in a variety of ways within each lesson: Warm-up, Instructional Activities, Cool-down, and Centers. According to the IM K-5 Teacher Guide, A Typical IM 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. An activity can serve one or more of many purposes. Provide experience with a new context. Introduce a new concept and associated language. Introduce a new representation. Formalize a definition of a term for an idea previously encountered informally. Identify and resolve common mistakes and misconceptions that people make. Practice using mathematical language. Work toward mastery of a concept or procedure. Provide an opportunity to apply mathematics to a modeling or other application problem. The purpose of each activity is described in its narrative. Read more about how activities serve these different purposes in the section on design principles.” Examples of varied approaches include:

• Unit 1, Factors and Multiples, Lesson 2, Cool-down, students calculate possible side lengths for rectangles with different areas. Student Facing states, “What are all of the possible side lengths of a rectangle with an area of 21 square units? What are all of the possible side lengths of a rectangle with an area of 50 square units?”

• Unit 6, Multiplying and Dividing Multi-digit Numbers, Lesson 13, Activity 1, students reason about division contexts. Student Facing states, “Diego’s aunt is buying paletas, which are ice pops, for a class party. At the local market, paletas come in different flavors. She buys the same number of paletas of each flavor. What mathematical questions can we ask about this situation? Here is an equation: 84\div7=? In the situation about the class party, what questions could the equation represent? Find the answer to one of the questions you wrote. Show your reasoning.”

• Unit 9, Putting It All Together, Lesson 10, Warm-up, students practice their estimation skills. Student Facing states, “Here are pictures showing the exterior and interior of a parking tower in Wolfstadt, Germany. The parking is automated: each car goes up on a lift and is then placed in a parking space. How many cars can fit in the tower? Record an estimate that is: too low, about right, too high.”

• Center, Find the Number (4), Stage 1: Factors, students use factors to play a game. Narrative states, “Students find all the factors for a given number. One player chooses a number on the gameboard (1–36) and they get that many points. The other player covers all the factors of that number. Their score is the sum of all the covered factors. Students take turns choosing the starting number. If a player chooses a number that doesn’t have any uncovered factors, they lose their next turn. When there are no numbers remaining with uncovered factors, the player with the most points wins.”

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Materials provide opportunities for teachers to use a variety of grouping strategies.

The materials reviewed for Kendall Hunt's Illustrative Mathematics Grade 4 provide opportunities for teachers to use a variety of grouping strategies. Suggestions are consistently provided for teachers within the facilitation notes of lesson activities and include guidance for a variety of groupings, including whole group, small group, pairs, or individual. Examples include:

• Unit 4, From Hundredths to Hundred-thousands, Lesson 3, Activity 1, students work with a small group in order to reason about the relative size of decimals. Launch states, “Groups of 3–4.” Activity states, “If creating a giant number line, lead the activity as outlined in the Activity Narrative. Otherwise, ask students to work with their group on the first two problems. Pause and discuss: how students knew where to put each decimal, how the number line could help us see the least and greatest. ‘Take a few quiet minutes to complete the rest of the activity.’ 5–6 minutes: independent work time.”

• Unit 6, Multiplying and Dividing Multi-Digit Numbers, Lesson 25, Activity 1, students work in groups of four to make paper flowers and describe patterns within. Launch states, “Groups of 4. Give students strips of tissue paper and rubber bands or pipe cleaners. ‘Today we are going to make paper flowers. Has anybody made this type of flower or seen them used for decorations before?’ Demonstrate how to make a paper flower. ‘In your group, make some small and large paper flowers. Make as many paper flowers as you can in 10 minutes. As you’re working, try to keep in mind the different things you have to think about to make the flowers.’” Activity states, “10 minutes: small-group work time.”

• Unit 8, Properties of Two-dimensional Shapes, Lesson 2, Activity 1, students work with a partner to describe and sort triangles based on the length of their sides and the size of their angles. Launch states, “Groups of 2. Give each group a set of cards from the first lesson. Provide access to rulers and protractors. Make available the chart with vocabulary from the previous lesson for reference during this lesson. ‘Use only the cards with triangles for this activity. Complete the triangle hunt with a partner. When your group is done, compare responses with another group.’” Activity states, “5 minutes: group work time. 3 minutes: discussion with another group. 2 minutes: individual work time on the last question.”

• Unit 9, Putting It All Together, Lesson 3, Activity 1, students work in pairs to practice solving addition and subtraction problems involving decimal fractions. Launch states, “Groups of 2.” Activity states, “1–2 minutes: independent work time. ‘Compare your strategies with your partner’s.’ 5 minutes: partner discussion. Monitor for expressions, strategies, and representations students use to determine connections between strategies and evidence of reasoning about equivalence.”

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Materials provide 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.

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.

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Materials provide a balance of images or information about people, representing various demographic and physical characteristics.

The materials reviewed for Kendall Hunt's Illustrative Mathematics Grade 4 provide a balance of images or information about people, representing various demographic and physical characteristics.

Images of characters are included in the student facing materials when they connect to the problem tasks. These images represent different races and portray people from many ethnicities in a positive, respectful manner, with no demographic bias for who achieves success based on the grade-level mathematics and problem circumstances. Names include multi-cultural references such as Han, Lin, Andre, and Mai and problem settings vary from rural, to urban, and international locations. Additionally, lessons include a variety of problem contexts to interest students of various demographic and personal characteristics.

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Materials provide guidance to encourage teachers to draw upon student home language to facilitate learning.

The materials reviewed for Kendall Hunt's Illustrative Mathematics Grade 4 partially provide guidance to encourage teachers to draw upon student home language to facilitate learning.

The student materials are available in Spanish. Directions for teachers are in English with prompts for students available in Spanish. The student materials including warm ups, activities, cool-downs, centers, and assessments are in Spanish for students.

The IM K-5 Teacher Guide includes a section titled “Mathematical Language Development and Access for English Learners” which outlines the program’s approach towards language development in conjunction with the problem-based approach to learning mathematics, which includes the regular use of Mathematical Language Routines, “The MLRs included in this curriculum were selected because they simultaneously support students’ learning of mathematical practices, content, and language. They are particularly well-suited to meet the needs of linguistically and culturally diverse students who are learning mathematics while simultaneously acquiring English.” While Mathematical Language Routines (MLRs) are regularly embedded within lessons and support mathematical language development, they do not include specific suggestions for drawing on a student’s home language.

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Materials provide guidance to encourage teachers to draw upon student cultural and social backgrounds to facilitate learning.

The materials reviewed for Kendall Hunt's Illustrative Mathematics Grade 4 provide some guidance to encourage teachers to draw upon student cultural and social backgrounds to facilitate learning.

Spanish materials are consistently accessible for a variety of stakeholders within the Family Support Materials for each unit. Within lessons, cultural connections are included within the context of problem solving, picture book centers, or games. Examples include:

• Unit 1, Factors and Multiples, Lesson 8, Warm-up, students examine the artwork of Piet Mondrian and analyze the art through the mathematical lens. Narrative states, “The purpose of this task is to introduce students to the artwork of Piet Mondrian. Students may notice that his paintings are composed of rectangles of various sizes. Students will create their own versions of Mondrian art in the first activity. To show students additional artwork by Mondrian, consider visiting a virtual installation of Piet Mondrian's work on the website of Museum of Modern Art (MoMA) or visiting the website of the Tate Gallery.”

• Unit 6, Multiplying and Dividing Multi-Digit Numbers, Lesson 25, Activity 2, students describe and use patterns to make paper flowers. In the activity, students are working on a problem related to preparing for a Quinceanera. Student facing states, “Priya and Jada are making paper flower garlands for their friend’s quinceañera. Each garland uses 12 flowers. 1. Priya wants 2 big flowers, followed by 2 small flowers. Jada wants 1 big flower, followed by 2 small flowers. Use their patterns to draw the garlands. 2. Priya and Jada make 25 garlands of each type. How many large and small flowers will they need altogether? 3. Diego and Kiran also made flowers. They made a total of 155 flowers for garlands that require 16 flowers each. How many garlands can they make?  4. It takes 1 minute to cut the strips for a flower and 2 minutes to finish it. How long did it take Diego and Kiran to make the 155 flowers, if they each make about the same number of flowers?”

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Materials provide supports for different reading levels to ensure accessibility for students.

The materials reviewed for Kendall Hunt's Illustrative Mathematics Grade 4 provide some supports for different reading levels to ensure accessibility for students.

• Unit 3, Extending Operations to Fractions, Lesson 19, Activity 2, Launch states, “MLR6 Three Reads. Display only the problem stem, without revealing the question(s). ‘We are going to read this problem 3 times.’ 1st Read: ‘Jada and Noah’s class are hiking at a park. Here is a map of the trails. The length of each trail is shown. What is this situation about?’ 1 minute: partner discussion. Listen for and clarify any questions about the context. 2nd Read: ‘Jada and Noah’s class are hiking at a park. Here is a map of the trails. The length of each trail is shown. (Display the trail map). Name the quantities. What can we count or measure in this situation?’ 30 seconds: quiet think time. 2 minutes: partner discussion. Share and record all quantities. Reveal the question(s). 3rd Read: Read the entire problem, including question(s) aloud. ‘What are some strategies we can use to solve this problem?’ 30 seconds: quiet think time. 1–2 minutes: partner discussion.”

• Unit 5, Fractions as Numbers, Lesson 18, Activity 2, Access for Students with Disabilities, “Representation: Access for Perception. Provide appropriate reading accommodations and supports to ensure student access to written directions, word problems, and other text-based content. Supports accessibility for: Language, Visual-Spatial Processing, Social-Emotional Functioning.”

• Unit 6, Multiplying and Dividing Multi-digit Numbers, Lesson 24, Activity 1, Access for Students with Disabilities, “Representation: Access for Perception. Read the problems aloud. Students who both listen to and read the information will benefit from extra processing time. Supports accessibility for: Language, Memory.

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Manipulatives, both virtual and physical, are accurate representations of the mathematical objects they represent and, when appropriate, are connected to written methods.

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.”

#### Criterion 3.4: Intentional Design

The program includes a visual design that is engaging and references or integrates digital technology, when applicable, with guidance for teachers.

The materials reviewed for Kendall Hunt's Illustrative Mathematics Grade 4 do not integrate technology such as interactive tools, virtual manipulatives/objects, and/or dynamic mathematics software in ways that engage students in the grade-level standards. The materials do not include or reference digital technology that provides opportunities for teachers and/or students to collaborate with each other. The materials have a visual design that supports students in engaging thoughtfully with the subject, and is neither distracting nor chaotic. The materials do not provide teacher guidance for the use of embedded technology to support and enhance student learning.

##### Indicator {{'3w' | indicatorName}}

Materials integrate technology such as interactive tools, virtual manipulatives/objects, and/or dynamic mathematics software in ways that engage students in the grade-level/series standards, when applicable.

The materials reviewed for Kendall Hunt's Illustrative Mathematics Grade 4 do not integrate technology such as interactive tools, virtual manipulatives/objects, and/or dynamic mathematics software in ways that engage students in the grade-level/series standards, when applicable.

According to the IM K-5 Teacher Guide, About These Materials, “Teachers can access the teacher materials either in print or in a browser as a digital PDF. When possible, lesson materials should be projected so all students can see them.” While this format is provided, the materials are not interactive.

According to the IM K-5 Teacher Guide, Key Structures in This Course, “Across lessons and units, students are systematically introduced to representations and encouraged to use representations that make sense to them. As their learning progresses, students are given opportunities to make connections between different representations and the concepts and procedures they represent. Over time, they will see and understand more efficient methods of representing and solving problems, which support the development of procedural fluency. In general, more concrete representations are introduced before those that are more abstract.” While physical manipulatives are referenced throughout lessons and across the materials, they are not virtual or interactive.

##### Indicator {{'3x' | indicatorName}}

Materials include or reference digital technology that provides opportunities for teachers and/or students to collaborate with each other, when applicable.

The materials reviewed for Kendall Hunt's Illustrative Mathematics Grade 4 do not include or reference digital technology that provides opportunities for teachers and/or students to collaborate with each other, when applicable.

According to IM K-5 Teacher Guide, Key Structures in this Course, “Classroom environments that foster a sense of community that allows students to express their mathematical ideas—together with norms that expect students to communicate their mathematical thinking to their peers and teacher, both orally and in writing, using the language of mathematics—positively affect participation and engagement among all students(Principles to Action, NCTM).” While the materials embed opportunities for mathematical community building through student task structures, discourse opportunities, and journal and reflection prompts, these opportunities do not reference digital technology.

##### Indicator {{'3y' | indicatorName}}

The visual design (whether in print or digital) supports students in engaging thoughtfully with the subject, and is neither distracting nor chaotic.

The materials reviewed for Kendall Hunt's Illustrative Mathematics Grade 4 have a visual design (whether in print or digital) that supports students in engaging thoughtfully with the subject, and is neither distracting nor chaotic.

There is a consistent design within units and lessons that supports student understanding of the mathematics. According to the IM K-5 Teacher Guide, Design Principles, “Each unit, lesson, and activity has the same overarching design structure: the learning begins with an invitation to the mathematics, is followed by a deep study of concepts and procedures, and concludes with an opportunity to consolidate understanding of mathematical ideas.” Examples from materials include:

• Each lesson follows a common format with the following components: Warm-up, one to three Activities, Lesson Synthesis, and Cool-Down, when included in lessons. The consistent structure includes a layout that is user-friendly as each component is included in order from top to bottom on the page.

• Student materials, in printed consumable format, include appropriate font size, amount and placement of directions, and space on the page for students to show their mathematical thinking.

• Teacher digital format is easy to navigate and engaging. There is ample space in the printable Student Task Statements, Assessment PDFs, and workbooks for students to capture calculations and write answers.

##### Indicator {{'3z' | indicatorName}}

Materials provide teacher guidance for the use of embedded technology to support and enhance student learning, when applicable.

The materials reviewed for Kendall Hunt's Illustrative Mathematics Grade 4 do not provide teacher guidance for the use of embedded technology to support and enhance student learning, when applicable.

While the IM K-5 Teacher Guide provides guidance for teachers about using the “Launch, Work Synthesize” structure of each lesson, including guidance for Warm-ups, Activities, and Cool-Downs, there is no embedded technology.

## Report Overview

### Summary of Alignment & Usability for Kendall Hunt’s Illustrative Mathematics | Math

#### Math K-2

The materials reviewed for Illustrative Mathematics Kendall Hunt Grades K-2 meet expectations for Alignment to the CCSSM. In Gateway 1, the materials meet expectations for focus and coherence. In Gateway 2, the materials meet expectations for rigor and practice-content connections. In Gateway 3, the materials meet expectations for usability including Teacher Supports and Student Supports; the materials partially meet expectations for Assessment.

##### Kindergarten
###### Alignment
Meets Expectations
###### Usability
Meets Expectations
###### Alignment
Meets Expectations
###### Usability
Meets Expectations
###### Alignment
Meets Expectations
###### Usability
Meets Expectations

#### Math 3-5

The materials reviewed for Illustrative Mathematics Kendall Hunt Grades 3-5 meet expectations for Alignment to the CCSSM. In Gateway 1, the materials meet expectations for focus and coherence. In Gateway 2, the materials meet expectations for rigor and practice-content connections. In Gateway 3, the materials meet expectations for usability including Teacher Supports and Student Supports; the materials partially meet expectations for Assessment.

###### Alignment
Meets Expectations
###### Usability
Meets Expectations
###### Alignment
Meets Expectations
###### Usability
Meets Expectations
###### Alignment
Meets Expectations
###### Usability
Meets Expectations

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### Overall Summary

###### Alignment
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###### Usability
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##### Gateway {{ gateway.number }}
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