## 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 3 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 3 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 3 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 3 meet expectations for assessing grade-level content and, if applicable, content from earlier grades. The curriculum is divided into eight units and each unit contains a written End-of-Unit Assessment for individual student completion. The Unit 8 Assessment is an End-of-Course Assessment and it includes problems from across the grade. Examples from End-of-Unit Assessments include:

• Unit 1, Introducing Multiplication, End-of-Unit Assessment, Problem 4, “Elena has 5 bags. Each bag has 8 rubber bands. How many rubber bands does Elena have? Explain or show your reasoning.” (3.OA.3)

• Unit 4, Relating Multiplication to Division, End-of-Unit Assessment, Problem 4, “Lin covers her desk with 77 sticky notes. The sticky notes are in 7 equal rows. How many sticky notes are in each row? a. Write a division equation to represent the situation. Use a symbol for the unknown quantity. b. Write a multiplication equation to represent the situation. Use a symbol for the unknown quantity. c. Solve the problem. Explain or show your reasoning.” (3.OA.3, 3.OA.4, 3.OA.6)

• Unit 5, Fractions as Numbers, End-of-Unit Assessment, Problem 5, “Write two fractions that are equivalent to \frac{1}{2}.” (3.NF.3b)

• Unit 7, Two-dimensional Shapes and Perimeter, End-of-Unit Assessment, Problem 7, “Priya wants to make a rectangular playpen for her dog. She has 18 meters of fencing materials. a. Andre suggests that Priya make a playpen that is 10 meters long and 8 meters wide. Explain why Priya does not have enough fencing to make this playpen. b. What are 2 possible pairs of side lengths Priya could use for the playpen that would give different areas? Explain or show your reasoning. c. Which playpen do you think Priya should make? Explain or show your reasoning.” (3.MD.7, 3.MD.8)

• Unit 8, Putting It All Together, End-of-Course Assessment, Problem 4, “Seven bags of dog food weigh 63 kilograms. If all the bags have the same weight, how many kilograms does each bag of dog food weigh? a. Write an equation to represent the situation. Use a letter or symbol for the unknown. b. Solve the problem.” (3.MD.2, 3.OA.4)

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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 3 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 3 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 3, Wrapping Up Addition and Subtraction Within 1,000, Lesson 15 engages students with extensive work with grade-level problems for 3.NBT.1 (Use place value understanding to round whole numbers to the nearest 10 or 100). Lesson 15, Round to the Nearest Ten and Hundred, Activity 1, students round given numbers to the nearest ten and hundred and see that the result can be the same for some numbers. Student Facing, “1. Round each number to the nearest ten and the nearest hundred. Use number lines if you find them helpful. 18, 97, 312, 439, 601. 2. Kiran and Priya are rounding some numbers and are stuck when trying to round 415 and 750. Kiran said, ‘415 doesn’t have a nearest multiple of 10, so it can’t be rounded to the nearest ten.’ Priya said, ‘750 doesn’t have a nearest multiple of 100, so it can’t be rounded to the nearest hundred.’ Do you agree with Kiran and Priya? Explain your reasoning.” Activity 2, students practice rounding to the nearest ten and hundred in context. Student Facing, “The table shows the numbers of people in different parts of a school at noon during a school day. Andre and Lin are trying to estimate the number of people in the whole school. Andre plans to round the numbers to the nearest hundred. Lin plans to round them to the nearest ten. 1. Make a prediction: Whose estimate is going to be greater? Explain your reasoning. 2. Work with a partner to find Andre and Lin’s estimates. Record them in the table.” Table shows “playground 94, cafeteria 163, art room 36, library 13, classrooms 216, gymnasium 109, music room 52, total ___.” Cool-Down, Student Facing, “1. Round 237 to the nearest ten. Show or explain your reasoning. 2. Round 237 to the nearest hundred. Show or explain your reasoning.”

• Unit 5, Fraction as Numbers, Lessons 5, 6, and 7 engage students in extensive work with 3.NF.2 (Understand a fraction as a number on the number line; represent fractions on a number line diagram). Lesson 5, To the Number Line, Activity 1, students further develop the idea that fractional amounts can be represented on a number line, “Groups of 2. Distribute one set of pre-cut cards to each group of students. ‘Work with your partner to sort some number lines into categories that you choose. Make sure you have a name for each category.’ 3-5 minutes: partner work time. Select groups to share their categories and how they sorted their cards. Choose as many different types of categories as time allows. Be sure to highlight categories created based on whether the tick marks represent whole numbers or fractions. If not mentioned by students, ask, ‘Can we sort the number lines based on what the tick marks represent? Let’s look at B and E. Both are partitioned into 4 parts. What do the unlabeled tick marks in E represent?’ (1, 2, 3) ‘What do you think those in B represent?’ (... or amounts less than 1). ‘Take a minute to sort your cards by number lines where the tick marks only represent whole numbers and number lines where the tick marks represent fractions.’ 1-2 minutes: partner work time.” Lesson 6, Locate Unit Fractions on the Number line, Warm-up: Which One Doesn’t Belong, students compare four images and talk about the characteristics of the items in comparison to one another, “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.” Lesson 7, Non-unit Fractions on the Number Line, Warm-up: Choral Count, students practice counting by \frac{1}{4} and notice patterns in the count. ‘Count by \frac{1}{4}, starting at \frac{1}{4}.’ Record as students count. Record 4 fractions in each row, then start a new row. There will be 4 rows. Stop counting and recording at \frac{16}{4}.”

• Unit 4, Relating Multiplication to Division, Lessons 11 and 20; Unit 6, Measuring Length, Time, Liquid Volume, and Weight, Lesson 8; and Unit 8, Putting It All Together, Lesson 9 engage students in the extensive work with 3.OA.7 (Fluently multiply and divide within 100, using strategies such as the relationship between multiplication and division). Unit 4, Lesson 11, Multiplication Strategies on Ungridded Rectangles, Cool-Down, “1. Mark or shade this rectangle to show a strategy for finding its area. 2. Write one or more expressions that represent how you find the area.” An image of a rectangle with sides labeled 6 and 9 is provided. Unit 4, Lesson 20, Strategies for Dividing, Activity 2, students practice finding the value of division expressions using any strategy that makes sense to them. Student Facing, “Find the value of each quotient. Explain or show your reasoning. Organize it so it can be followed by others. 1. 80\div5 2. 68\div4 3. 91\div7 If you have time: Eighty-four students on a field trip are put into groups. Each group has 14 students. How many groups are there?” Unit 6, Lesson 8, Estimate and Measure Liquid Volume, Warm-up, students use strategies for dividing within 100, “Display one expression. ‘Give me a signal when you have an answer and can explain how you got it.’” Student Facing, “Find the value of each expression mentally. 30\div3, 60\div3, 63\div3, 54\div3.” Unit 8, Lesson 9, Multiplication Game Day, Activity 2, students practice multiplying within 100 by playing a game of their choice. Student Facing, “Choose a center to practice multiplying within 100. Compare, Multiply within 100: Decide which expression has the greatest value. How Close? Multiply to 100: Choose 2–3 numbers to multiply to get a product closest to 100. Rectangle Rumble, Factors 1–10: Multiply numbers to create rectangular areas to fill a grid with the most squares.”

The materials provide opportunities for all students to engage with the full intent of Grade 3 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 2, Area and Multiplication, Lessons 2, 3, 4, and 6 engage students in full intent of 3.MD.6 (Measure areas by counting unit squares (square cm, square m, square in, square ft, and improvised units). Lesson 2, How Do We Measure Area?, Activity 2, students use square tiles to measure area. Student facing, “Your teacher will give you handouts with some figures on them. Use square tiles to find the area of each figure. Record your answers here. Be prepared to explain your reasoning.” Figures have areas of 9, 18, 13, 30, 36, and 21 square units. Lesson 3, Tile Rectangles, Activity 1, students practice tiling with no gaps or overlaps. Student facing, “Your teacher will give you square tiles and a handout showing 4 rectangles and squares. 1. Describe or show how to use the square tiles to measure the area of each rectangle. You can place square tiles on the handout where squares are already shown. You can also move the tiles, if needed. 2. Describe how to use square tiles to find the area of any rectangle.” Lesson 4, Area of Rectangles, Activity 2, students use squared units to find the area of 4 different rectangles, “Find the area of each rectangle and include the units. Explain or show your reasoning.” Lesson 6, Different Square Units (Part 1), Activity 2, students estimate and then find the area of squares using square inches and centimeters. Student Facing, “Estimate how many square centimeters and inches it will take to tile this square. square inches (estimate) ___ square centimeters (estimate) ___ 1. Use the inch grid and centimeter grid to find the area of the square, square inches ___ square centimeters ___ 2. Write a multiplication expression that describes the rows and columns in the square and can tell us the area in each unit. square inches ___ square centimeters ___.”

• Unit 5, Fractions as Numbers, Lessons 10, 11, and 12 engage students in the full intent of 3.NF.3b (Recognize and generate simple equivalent fractions, e.g., \frac{1}{2}=\frac{2}{4}, \frac{4}{6}=\frac{2}{3}. Explain why the fractions are equivalent, e.g., by using a visual fraction model). Lesson 10, Equivalent Fractions, Activity 2, students use fraction strips to identify equivalent fractions and explain why they are equivalent. Student Facing, “Use your fraction strips from an earlier lesson to find as many equivalent fractions as you can that are equivalent to: 1. \frac{1}{2} 2. \frac{2}{3} 3. \frac{6}{6} 4. \frac{3}{4}.” Lesson 11, Generate Equivalent Fractions, Cool-down, students generate equivalent fractions, including for fractions greater than 1, given partially shaded diagrams. Student Facing, “1. Write two fractions that the shaded part of this diagram represents. (Bar diagram shows 3/6 shaded.) 2. Show that the shaded part of this diagram represents both \frac{5}{4} and \frac{10}{8}.” Lesson 12, Equivalent Fractions on a Number Line, Activity 3, students practice generating equivalent fractions. Student Facing, “1. Roll 6 number cubes. If you roll any fives, they count as a wild card and can be any number you’d like. 2. Can you put the numbers you rolled in the boxes to make a statement that shows equivalent fractions? Work with your partner to find out. 3. If you cannot, re-roll as many number cubes as you’d like. You can re-roll your number cubes twice. 4. If you can make equivalent fractions, record your statement and show or explain how you know the fractions are equivalent. You get 1 point for each pair of equivalent fractions you write.”

• Unit 6, Measuring Length, Time, Liquid Volume, and Weight, Lessons 9, 10, 11, and 14 engage students in the full intent of 3.MD.1 (Tell and write time to the nearest minute and measure time intervals in minutes. Solve word problems involving addition and subtraction of time intervals in minutes, e.g., by representing the problem on a number line diagram). Lesson 9, Time to the Nearest Minute, Activity 2, students tell and write time to the nearest minute. Student Facing, “1. Show the time given on each clock. (Four clock images are provided. A. 2:36 PM ,B. 3:18 PM, C. 12:17 PM, D. 9:02 PM) 2. Draw a time on this clock. Trade with a partner and tell the time on their clock.” Lesson 10, Solve Problems Involving Time (Part 1), Activity 1, students solve problems involving elapsed time in a way that makes sense to them. Student Facing, “1. Kiran arrived at the bus stop at 3:27 p.m., as shown on this clock. He waited 24 minutes for his bus to arrive. What time did his bus arrive? Show your thinking. Organize it so it can be followed by others. 2. Elena arrived at the bus stop at 3:45 p.m. She also waited 24 minutes for her bus to arrive. What time did the bus arrive? Show your thinking. Organize it so it can be followed by others.” Two clock images are provided. Lesson 11, Solve Problems Involving Time (Part 2), Activity 1, students solve problems involving addition and subtraction of time intervals when given times on a clock. Student Facing, “1. For how many minutes was Han on the bus? Explain or show your reasoning. (The times when Han got on the bus and off the bus are shown with two analog clock images displaying 5:43 and 6:36.) 2. Draw the minute hand to show that Elena waited for the bus for 32 minutes.“ 2 additional analog clocks are pictured. Lesson 14, What Makes Sense in the Problem? Cool-down, students solve problems with time intervals. Student Facing, “1. A show at the carnival starts at 2:45 p.m. and lasts 47 minutes. What time does the show end? Explain or show your reasoning. 2. Another show that is 24 minutes long ends at 5:10 p.m. Kiran says that the show starts before 4:40 p.m. Do you agree? Explain or show your reasoning.”

#### 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 3 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.

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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 3 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 8, approximately 75%.

• The number of lessons devoted to major work of the grade (including assessments and supporting work connected to the major work) is 109 out of 151, approximately 72%. The total number of lessons devoted to major work of the grade includes 101 lessons plus 8 assessments for a total of 109 lessons.

• The number of days devoted to major work of the grade (including assessments and supporting work connected to the major work) is 115 out of 159, approximately 72%.

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 72% 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 3 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, Wrapping Up Addition and Subtraction Within 1,000, Lesson 2, Cool-down connects supporting work of 3.NBT.A (Use place value understanding and properties of operations to perform multi-digit arithmetic) to the major work of 3.OA.D (Solve problems involving the four operations, and identify and explain patterns in arithmetic). Students solve multi-digit, multi-step word problems. Student Facing states, “The Statue of Liberty is 305 feet tall. The Brooklyn Bridge is 133 feet tall. How much taller is the Statue of Liberty than the Brooklyn Bridge? Explain or show your reasoning.”

• Unit 6, Measuring Length, Time, Liquid Volume, and Weight, Lesson 3, Activity 1 connects the supporting work of 3.MD.4 (Generate measurement data by measuring lengths using rulers marked with halves and fourths of an inch. Show the data by making a line plot, where the horizontal scale is marked off in appropriate units— whole numbers, halves, or quarters) to the major work of 3.NF.3 (Explain equivalence of fractions in special cases, and compare fractions by reasoning about their size). Students measure lengths using a ruler that is marked with half inches and quarter inches, students recognize that lengths that line up with a half-inch mark can be read as one-half of an inch or two-fourths of an inch. Student Facing states, “1. Kiran and Jada are discussing the length of a worm, Kiran says that the worm is 4\frac{2}{4} inches long, Jada says that the worm is 4\frac{1}{2} inches long. Use the ruler to explain how both of their measurements are correct. 2. Measure the length of the following worms.” Images of four worms of various lengths are shown.

• Unit 7, Two-Dimensional Shapes and Perimeter, Lesson 8, Activity 1 connects the supporting work of 3.MD.8 (Solve real world and mathematical problems involving perimeters of polygons, including finding the perimeter given the side lengths, finding an unknown side length, and exhibiting rectangles with the same perimeter and different areas or with the same area and different perimeters) to the major work of 3.OA.8 (Solve two-step word problems using the four operations. Represent these problems using equations with a letter standing for the unknown quantity. Assess the reasonableness of answers using mental computation and estimation strategies including rounding). Students practice finding the perimeter of shapes that have labeled side lengths. Some of the figures are not regular shapes and will require multiple steps using addition to solve. Student Facing states, “What do you notice? What do you wonder? Find the perimeter of each shape. Explain or show your reasoning.” Students find the perimeter of seven different shapes, both regular and composite shapes.

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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 3 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 2, Area and Multiplication, Lesson 8, Activity 1 connects the major work of 3.MD.C. (Geometric measurement: understand concepts of area and relate area to multiplication and to addition) to the major work of 3.OA.B (Understand properties of multiplication and the relationship between multiplication and division). Students solve an area problem with a partially tiled rectangle while using multiplication knowledge. This encourages students to multiply to solve problems involving area, but still provides some visual support to see the arrangement of the rows and columns. Student facing (students are provided two rectangles), “What do you notice? What do you wonder?, After learning about azulejos in Portugal, Elena is making her own tile artwork. This rectangle shows the project Elena is tiling. Each tile has a side length of 1 inch. How many tiles are needed to tile the whole rectangle? Explain or show your reasoning.” Activity Synthesis states, “‘How did you know how many tiles would be in each row or column?’ (The first row had 10 tiles, so I know every other row has 10 tiles because I could put more tiles to fill in the rows. It’s like an array. Each column has to have the same number of tiles, so there is 9 in each column.) ‘How did you find the total number of tiles needed?’ (I counted by ten 9 times. I multiplied 9 times 10.)”

• Unit 4, Relating Multiplication to Division, Lesson 7, Activity 1 connects the major work of 3.OA.A (Represent and solve problems involving multiplication and division) to the major work of 3.OA.B (Understand properties of multiplication and the relationship between multiplication and division). Students reason abstractly and quantitatively as they relate drawings, situations, and equations. The Launch states, “In the first box on your sheet, create a drawing that shows equal groups of objects. This drawing will be used by other students in your group to fill in the other boxes.” Student Facing states, “Your teacher will give you a sheet of paper with 4 boxes on it and instruct you to draw or write something in each box. After working on each box, pause and wait for your teacher's instructions for the next box. 1. Draw equal groups in Box 1 on your recording sheet. 2. In Box 2, write a description of a division situation that matches the drawing you just received. 3. In Box 3, write a multiplication equation that matches the drawing and division situation you just received. Use a symbol for the unknown quantity. 4. In Box 4, write a division equation that matches the drawing, division situation, and multiplication equation you just received. Use a symbol for the unknown quantity.“

• Unit 6, Measuring Length, Time, Liquid Volume, and Weight, Lesson 8, Cool-Down connects the major work of 3.MD.A (Solve problems involving measurement and estimation of intervals of time, liquid volumes, and masses of objects) to the major work of 3.NF.A (Develop understanding of fractions as numbers). Students use liters to estimate and measure liquid volumes, including fractional quantities. Student Facing states, “What is the volume of the liquid shown in each image?” An image shows two containers, containing 3 liters and 1\frac{1}{2} liters, respectively.

• Unit 7, Two-dimensional Shapes and Perimeter, Lesson 14, Cool-Down connects the supporting work of 3.MD.D (Geometric measurement: Recognize perimeter as an attribute of plane figures and distinguish between linear and area measures) to the supporting work of 3.G.A (Reason with shapes and their attributes). Students analyze an image in order to reason with shapes and their attributes. Student Facing states, “1. Describe the quadrilaterals that were used in this pattern. 2. If the image of the pattern is a rectangle with side lengths of 9 inches by 6 inches, what is the perimeter? Explain your reasoning.” Student Response states, “1. Sample responses: There are quadrilaterals in white and gray that don’t have any right angles. The black quadrilaterals are rhombuses. The grey shapes and the white shapes are quadrilaterals that have 2 equal sides. They are not rectangles, rhombuses, or squares. It looks like there are tall skinny rectangles that are shaded white and gray behind the black rhombuses. 2. 30 inches. I added 9 plus 6 to get 15, then multiplied by 2 since there would be another set of sides that were 9 inches and 6 inches.”

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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 3 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.” While future connections are all embedded within the Scope and Sequence, descriptions of prior connections are also found within the Preparation tab for specific lessons, and within the notes for specific parts of lessons.

Examples of connections to future grades include:

• Course Guide, Scope and Sequence, Unit 3, Wrapping Up Addition and Subtraction Within 1,000, Unit Learning Goals connect 3.NBT.2 (Fluently add and subtract within 1000 using strategies and algorithms based on place value, properties of operations, and/or the relationship between addition and subtraction) to addition and subtraction of multi-digit numbers using the standard algorithm in 4.NBT.4. Lesson Narrative states, “Students explore various algorithms but are not required to use a specific one. They should, however, move from strategy-based work of grade 2 to algorithm-based work to set the stage for using the standard algorithm in grade 4. If students begin the unit with knowledge of the standard algorithm, it is still important for them to make sense of the place-value basis of the algorithm.”

• Course Guide, Scope and Sequence, Unit 7, Two-dimensional Shapes and Perimeter, Unit Learning Goals connect 3.G.1 (Understand that shapes in different categories may share attributes, and that the shared attributes can define a larger category. Recognize rhombuses, rectangles, and squares as examples of quadrilaterals, and draw examples of quadrilaterals that do not belong to any of these subcategories) to work with identifying angles in Grade 4. Lesson Narrative states, “In this section, students describe, compare, and sort a variety of shapes. They have previously used terms such as square, rectangle, triangle, quadrilateral, pentagon, and hexagon to name shapes. Here, students think about ways to further categorize triangles and quadrilaterals. They see that triangles and quadrilaterals can be classified based on their sides (whether some are of equal length) and their angles (whether one or more right angles are present). Although students will not learn the formal definition of an angle until grade 4, they are introduced to the terms ‘angle in a shape’ and ‘right angle in a shape’ to describe the corners of shapes. This allows students to distinguish right triangles and to describe defining attributes of squares and rectangles.”

• Course Guide, Scope and Sequence, Unit 8, Putting It All Together, Unit Learning Goals connect 3.NF.A (Develop understanding of fractions as numbers), 3.OA.7 (Fluently multiply and divide within 100, using strategies such as the relationship between multiplication and division or properties of operations), and 3.NBT.2 (Fluently add and subtract within 1000 using strategies and algorithms based on place value, properties of operations, and/or the relationship between addition and subtraction) to major work in Grade 4, including operations with fractions and operations with multi-digit numbers. Lesson Narrative states, “The concepts and skills strengthened in this unit prepare students for major work in grade 4: comparing, adding, and subtracting fractions, multiplying and dividing within 1,000, and using the standard algorithm to add and subtract multi-digit numbers within 1 million.”

Examples of connections to prior knowledge include:

• Unit 1, Introducing Multiplication, Lesson 1, Preparation connects 3.MD.3 (Draw a scaled picture graph and a scaled bar graph to represent a data set with several categories. Solve one- and two-step “how many more” and “how many less” problems using information presented in scaled bar graphs) to the work creating and analyzing graphs from 2.MD.10. Lesson Narrative states, “In grade 2, students learned how to draw and label single-unit scale bar graphs and picture graphs and used categorical data presented in graphs to solve simple problems. In this lesson, students revisit the structure of picture graphs and bar graphs, the features of graphs that help communicate information clearly, and the information they can learn by analyzing a graph. Students learn that a key is the part of a picture graph that tells what each picture represents. Students contextualize and make sense of the data based on the title, the given values, and their own experiences (MP2).”

• Unit 3, Wrapping Up Addition and Subtraction Within 1,000, Lesson 3, Warm Up connects the work of 3.NBT.2 (Fluently add and subtract within 1000 using strategies and algorithms based on place value, properties of operations, and/or the relationship between addition and subtraction) to work with adding and subtracting within 1000 from 2.NBT.7. Narrative states, “The purpose of this Number Talk is to elicit strategies and understandings students have for adding three-digit numbers. These understandings help students develop fluency and will be helpful later in this lesson when students are to use strategies based on place value and properties of operations to add within 1,000.”

• Unit 6, Measuring Length, Time, Liquid Volume, and Weight, Lesson 4, Preparation connects work with 3.MD.4 (Generate measurement data by measuring lengths using rulers marked with halves and fourths of an inch. Show the data by making a line plot, where the horizontal scale is marked off in appropriate units–-whole numbers, halves, or quarters) to work generating measurement data from 2.MD.9. Lesson Narrative states, “In grade 2, students made line plots to show measurements to the nearest whole unit. In previous lessons, they measured objects with rulers marked with halves and fourths of an inch. In this lesson, students interpret line plots that show lengths in half inches and quarter inches and ask and answer questions about the data.”

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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 3 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 3, there are 159 days of instruction including:

• 143 lesson days

• 16 unit assessment days

There are eight units in Grade 3 and, within those units, there are between 15 and 22 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 3, 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 3 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 3 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 3 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 1, Introducing Multiplication, Lesson 9, Activity 2, students develop conceptual understanding as they represent situations involving equal groups in a way that makes sense to them. Student Facing states, “Represent each situation. 1. There are 4 people wearing shoes. Each person is wearing 2 shoes. 2. There are 2 boxes of markers. Each box has 10 markers. 3. There are 3 basketball teams. Each team has 5 players.” (3.OA.1)

• Unit 2, Area and Multiplication, Lesson 7, Warm-Up, students develop conceptual understanding of measurement units, larger square units can be useful in situations involving larger areas. Students see a picture of a girl on a playground holding a large square and Student Facing states, “What do you notice? What do you wonder?” Activity Synthesis states, “If needed, ‘What could you measure with this square?’ (You could measure the area of big areas, like the playground.) ‘Why might you want this square instead of square centimeters or square inches?’ (It takes fewer squares of this size to measure an area that is a lot larger like a playground or a room.)” (3.MD.6)

• Unit 8, Putting It All Together, Lesson 2, Warm-up, students develop conceptual understanding as they compare fractions on a number line. An image of different four number lines with fractions is provided and Student Facing states, “Which one doesn’t belong?” (3.NF.2)

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 1, Introducing Multiplication, Lesson 11, Cool-down, students demonstrate conceptual understanding as they write expressions for equal groups. Student Facing states, “There were 6 envelopes. Each envelope had 2 notes in it. Write a multiplication expression to represent the situation. Explain or show your reasoning. Create a drawing or diagram if it’s helpful.” (3.OA.1)

• Unit 2, Area and Multiplication, Lesson 8, Cool-down, students demonstrate conceptual understanding as they reason about the area of a rectangle. Students are provided a drawing of a rectangle with tick marks rather than a completed grid. Student Facing states, “The tick marks on the sides of the rectangle are 1 foot apart. What is the area of the rectangle? Explain or show your reasoning.” (3.MD.7b)

• Unit 5, Fractions as Numbers, Lesson 3, Activity 2, students demonstrate conceptual understanding of fractions as they match fractions to shaded diagrams. Student Facing states, “Your teacher will give you a set of cards for playing Fraction Match. Two cards are a match if one is a diagram and the other a number, but they have the same value. 1. To play Fraction Match: Arrange the cards face down in an array. Take turns choosing 2 cards. If the cards match, keep them and go again. If not, return them to where they were, face down. You can’t keep more than 2 matches on each turn. After all the matches have been found, the player with the most cards wins. 2. Use the cards your teacher gives you to create 4 new pairs of cards to add to the set. 3. Play another round of Fraction Match using all the cards.” (3.NF.1)

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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 3 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 1, Introducing Multiplication, Lesson 4, Activity 2, students develop procedural skill and fluency with data as they create a scaled picture graph. Activity states, “‘Represent the data that you collected in your own scaled picture graph where each picture represents 2 students.’ Circulate as students work: Encourage them to include a title, category labels, and key. Pay attention to how students are grouping by 2. Support students with questions they may have (especially around representing odd number amounts).” Student Facing states, “Represent our survey data in a scaled picture graph where each picture represents 2 students.” (3.OA.A)

• Unit 3, Wrapping Up Addition and Subtraction Within 1,000, Lesson 16, Warm-up, students develop fluency as they use strategies for finding the products of 4 and 6 as they relate to products of 5. Student Facing states, “Find the value of each expression mentally. 5\times7, 4\times7, 6\times7, 4\times8.” (3.OA.7)

• Unit 7, Two-dimensional Shapes and Perimeter, Lesson 7, Warm-up, students develop procedural skill and fluency as they use strategies they have learned to add multi-digit numbers. Student Facing states, “Decide whether each statement is true or false. Be prepared to explain your reasoning. 123+75+123+75=100+100+70+70+5+5+3+3, 123+75+123+75=(2\times123)+(2\times75), 123+75+123+75=208+208, 123+75+123+75=246+150.” (3.NBT.2)

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 3, Wrapping Up Addition and Subtraction Within 1,000, Lesson 10, Cool-Down, students demonstrate procedural skill and fluency as they use an algorithm to subtract within 1,000. Student Facing states, “Use an algorithm of your choice to find the value of 419-267.” (3.NBT.2)

• Unit 4, Relating Multiplication to Division, Lesson 9, Activity 2, students demonstrate fluency as they identify patterns in multiplication. Activity states, “‘In the right column, work independently to write down at least two multiplication facts you can figure out because you know the given multiplication fact in the left column.’ 3–5 minutes: independent work time. ‘Now, share the facts that you found with your partner. Record any facts that your partner found that you didn’t find. Be sure to explain your reasoning.’” Student Facing states, “1. In each row, write down at least two multiplication facts you can figure out because you know the given multiplication fact in the left column. Be prepared to share your reasoning. If I know…  2\times4, then I also know 4\times2, 4\times4, 2\times8.” (3.OA.7)

• Unit 8, Putting It All Together, Lesson 15, Activity 1, students demonstrate procedural skill and fluency as they reason about subtraction and write a subtraction expression. Activity states, “‘How would you find the value of each expression, without writing? For each expression, think of at least two ways. Then, share your thinking with your group.’ Reiterate to students that they are to consider how someone might reason about each difference, rather than only finding the value. 4 minutes: independent work time. 4 minutes: small-group discussion.” Student Facing states, “Here are three subtraction expressions. 600-400, 600-399, 500-399. 1. Think of at least two different ways to find the value of each difference mentally. 2. Write a fourth subtraction expression whose value can be found using one of the strategies you thought of.” (3.NBT.2)

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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 3 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 1, Introducing Multiplication, Lesson 12, Activity 1, students solve a real-world problem involving multiplication. Launch states, “Groups of 2. MLR5 Co-Craft Questions. Display only the problem stem, ‘Tyler has 3 boxes.’ without revealing the question. ‘Write a list of mathematical questions that could be asked about this situation.’ (What’s in the boxes? How many things are in the boxes? How many things does he have altogether?) 2 minutes: independent work time. 2–3 minutes: partner discussion. Invite several students to share one question with the class. Record responses. ‘What do these questions have in common? How are they different?’ Reveal the task (students open books), and invite additional connections.” Student Facing states, “Tyler has 3 boxes. He has 5 baseballs in each box. How many baseballs does he have altogether? Show your thinking using diagrams, symbols, or other representations.” (3.OA.3)

• Unit 3, Wrapping Up Addition and Subtraction within 1000, Lesson 19, Activity 2, students solve a multi-step real-world problem and then write an equation to represent the problem. Activity states, “‘Take some independent time to work on this problem. You can choose to solve the problem first or write the equation first.’ 5–7 minutes: independent work time, Monitor for different ways students: write an equation, represent the problem, such as by using a tape diagram, decide their answer makes sense, such as thinking about the situation or by rounding.” Student Facing states, “Kiran is setting up a game of mancala. He has a jar of 104 stones. From the jar, he takes 3 stones for each of the 6 pits on his side of the board. How many stones are in the jar now? 1. Write an equation to represent the situation. Use a letter for the unknown quantity. Solve the problem. Explain or show your reasoning. Explain how you know your answer makes sense.” (3.OA.8)

• Unit 7, Two-dimensional Shapes and Perimeter, Lesson 10, Cool-Down, students solve a real-world problem involving the perimeter of a rectangle. Student facing states, “Lin is building a fence around her rectangular garden. A diagram is shown. The area of the garden is 36 square feet. How many feet of fencing material will she need to enclose the whole garden?” (3.MD.8)

Examples of non-routine applications of the math include:

• Unit 4, Relating Multiplication to Division, Lesson 17, Activity 1, students develop understanding of multiplication and its relation to division to solve real-world problems. Activity states, “‘Now, work with your partner to come up with as many questions as you can about this situation.’ 3–5 minutes: partner work time. Share and record responses. Display: ‘How many guests fit at each table in Room B?’ or circle the question if mentioned by a student. ‘Now work with your partner to answer this question.’ (I found 142-94 to find out how many guests were in Room B. There were 48 guests and 6 tables. I put the same amount of guests at each table and there were 8 guests at each table.) 3–5 minutes: partner work time.”  Student Facing states, “What questions could you ask about this situation? There are 142 guests at a party. All the guests are in 2 rooms. Room A has 94 guests. Room B has 6 tables that each have the same number of guests. There are 4 pieces of silverware and 1 plate for each guest.” (3.OA.8)

• Unit 5, Fractions as Numbers, Lesson 6, Activity 1, students solve a non-routine problem as they partition a number line that extends beyond one. Launch states, “‘Today we are going to partition number lines to locate unit fractions. Take a minute to look at how Clare, Andre, and Diego have partitioned their number lines into fourths.’ 1-2 minutes: quiet think time.” Student Facing states, “Three students are partitioning a number line into fourths. Their work is shown. Whose partitioning makes the most sense to you? Explain your reasoning.” Clare’s number line partitioned into halves, Andre’s number line partitioned into fourths, and Diego’s number line partitioned into fifths are shown. (3.NF.2)

• Unit 6, Measuring Length, Time, Liquid Volume, and Weight, Lesson 15, Activity 1, students solve a real-world problem by using concepts of time, weight, and volume. Launch states, “‘We’re going to solve some problems about a day at the fair. What are some things you could do during a day at the fair?’ (go on rides, walk around, eat fair food, look at some of the animals) 30 seconds: quiet think time. Share responses. Give each group tools for creating a visual display.” Student Facing states, “You spent a day at the fair. Solve four problems about your day and create a poster to show your reasoning and solutions. 1. You arrived at the fair! Entry to the fair is \$9 a person. You went there with 6 other people. How much did it cost your group to enter the fair? 2. How did you start your day? (Choose one.) You arrived at the giant pumpkin weigh-off at 11:12 a.m. and left at 12:25 p.m. How long were you there? You spent 48 minutes at the carnival and left at 12:10 p.m. What time did you get to the carnival? 3. What was next? (Choose one.) You visited a barn with 7 sheep. The sheep drink 91 liters of water a day, each sheep drinking about the same amount. How much does each sheep drink a day? You visited a life-size sculpture of a cow made of butter. The butter cow weighs 273 kilograms, which is 277 kilograms less that the actual cow. How much does the actual cow weigh? 4. Before you went home . . . You stopped for some grilled corn on the cob. On the grill, there were 54 ears of corn arranged in 9 equal rows. How many ears of corn were in each row?” (3.MD.1, 3.MD.2, 3.OA.3)

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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 3 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 3, Wrapping Up Addition and Subtraction Within 1,000, Lesson 4, Activity 2, students develop procedural skill and fluency as they use the addition algorithm. Launch states, “Give students access to base-ten blocks. ‘Now you are going to have a chance to try the algorithms that Lin and Han used in the last activity. Take a minute to think about which algorithm you want to use for each problem.’” Student Facing states, “Try using an algorithm to find the value of each sum. Show your thinking. Organize it so it can be followed by others. 1. 475+231, 2. 136+389, 3. 670+257.” (3.NBT.2)

• Unit 7, Two-dimensional Shapes and Perimeter, Lesson 9, Cool-down, students apply their understanding of perimeter to solve a real-world problem. Student Facing states, “A rectangular swimming pool has a perimeter of 94 feet. If it is 32 feet on one side, what are the lengths of the other three sides? Explain or show your reasoning.” (3.MD.8)

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, Introducing Multiplication, Lesson 19, Cool-down, students use all three aspects of rigor, conceptual understanding, procedural skill and fluency, and application, as they use an equation to represent an array. Student Facing states, “Clare has 3 rows of baseball cards. Each row has 10 cards. How many cards does she have? 1. Write an equation with a symbol for the unknown number to represent the situation. 2. Find the number that makes the equation true. Explain or show your reasoning.” (3.OA.3)

• Unit 4, Relating Multiplication to Division, Lesson 19, Activity 2, students develop conceptual understanding alongside procedural skill and fluency as they represent division within 100. Launch states, “Give base-ten blocks to each group. Ask students to keep their materials closed. ‘Use base-ten blocks to find the value of 60\div5.’ 1–2 minutes: independent work time.” Student Facing states, “Jada and Han used base-ten blocks to represent 60\div5. 1. Make sense of Jada’s and Han’s work. a. What did they do differently? b. Where do we see the value of 60\div5 in each person’s work? 2. How would you use base-ten blocks so you could represent these expressions and find their value? Be prepared to explain your reasoning. a. 64\div4: Would you make 4 groups or groups of 4? b. 72\div6: Would you make 6 groups or groups of 6? c. 75\div15: Would you make 15 groups or groups of 15?” (3.OA.2, 3.OA.7)

• Unit 7, Two-dimensional Shapes and Perimeter, Lesson 6, Activity 1, students use procedural fluency and apply their understanding of perimeter of shapes to solve a non-routine real-world problem. Launch states, “Give each group a copy of the blackline master and 25–50 paper clips. ‘Make a prediction: Which shape do you think will take the most paper clips to build?’ 30 seconds: quiet think time. Poll the class on whether they think shape A, B, C, or D would take the most paper clips to build.” Activity states, “Work with your group to find out which shape takes the most paper clips to build. You may need to take turns with the paper clips.” Student Facing states, “Your teacher will give you four shapes on paper and some paper clips. Work with your group to find out which shape takes the most paper clips to build. Explain or show how you know. Record your findings here. Draw sketches if they are helpful.” (3.MD.8)

#### 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 3 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 3 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, Area and Multiplication, Lesson 10, Activity 1, students solve a real-world problem involving area. Student Facing states, “Noah is painting a wall in a community garden. The wall is shaped like a rectangle. A diagram of the wall is shown here. Paint is sold in 3 different sizes: A small container will cover 3 square meters. A medium container will cover 10 square meters. A large container will cover 40 square meters. What should Noah buy? Explain your reasoning.” Narrative states, “The activity includes a rectangle where the side lengths are labeled. When students solve problems with multiple solutions and have to choose and justify a solution, they make sense of problems and persevere in solving them (MP1).”

• Unit 6, Measuring Length, Time, Liquid, Volume, and Weight, Lesson 13, Cool Down, students make sense of problems involving weight and justify their reasoning. Preparation, Lesson Narrative states, “In this lesson, students solve problems involving weight in two Information Gap activities. They interpret descriptions of situations involving all four operations and in which one or more quantities are missing. Students determine the information that they need to answer the questions and then reason about the solutions.” Student facing states, “The winning pig weighed 48 kilograms when his owner decided to raise him to show at the fair. At the fair weigh-off, the pig weighed 124 kilograms. How much weight did the pig gain? Explain or show your reasoning.”

• Unit 7, Two-dimensional Shapes and Perimeter, Lesson 6, Warm-up, students make sense of perimeter concepts. Students are given an image of a shape and paper clips. Launch states, “1 minute: quiet think time.” Student Facing states, “What do you notice? What do you wonder?” Narrative states, “The purpose of this warm-up is for students to visualize the idea of perimeter and elicit observations about distances around a shape. It also familiarizes students with the context and materials they will be working with in the next activity, where they will use paper clips to form the boundary of shapes and compare or quantify their lengths.”

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 1, Introducing Multiplication, Lesson 2, Activity 2, students reason abstractly and quantitatively as they interpret data from a bar graph. Activity states, “‘Now you’re going to use your bar graph to decide if statements are true or false.’ 1–2 minutes: independent work time.” Student Facing states, “1. Decide if each statement is true or false about how our class gets home. Explain your reasoning to your partner. a. More students walk than go home any other way. b. More students ride home on a bus than in a car. c. Fewer students walk home than ride their bikes. d. More students walk or ride their bikes than ride in a van. 2. Fill in the blanks as directed by your teacher, then answer each question. a. ‘How many more students ___  than ___?’ b. ‘How many more students ___  or ___ than ___?’” Narrative states, “When students use expression, equations, or describe adding or subtracting to find how many more or less, they show they can decontextualize and recontextualize the data to make sense of and solve the problems (MP2). You will generate the questions students answer in this task from the class graph.”

• Unit 5, Fractions as Numbers, Lesson 4, Activity 2, students use diagrams to represent the fractional amount in a given situation. Narrative states, “The purpose of this activity is for students to use diagrams to represent situations that involve non-unit fractions. The synthesis focuses on how students partition and shade the diagrams and how the end of the shaded portion could represent the location of an object. When students interpret the different situations in terms of the diagrams they reason abstractly and quantitatively (MP2).” Activity states, “‘In the activity, each strip represents the length of a street where Pilolo is played. Work independently to represent each situation on a diagram.’ 3–5 minutes: independent work time. ‘With a partner, choose one of the situations and make a poster to show how you represented the situation with a fraction strip. You may want to include details such as notes, drawings, labels, and so on, to help others understand your thinking.’ Give students materials for creating a visual display. 5–7 minutes: partner work time.” Student Facing states, “Here are four situations about playing Pilolo and four diagrams. Each diagram represents the length of a street where the game is played. Represent each situation on a diagram. Be prepared to explain your reasoning. 1. A student walks \frac{4}{8} the length of the street and hides a rock. 2. A student walks \frac{2}}{3} the length of the street and hides a penny. 3. A student walks \frac{3}{4} the length of the street and hides a stick. 4. A student walks \frac{5}{6} the length of the street and hides a penny. 5. This diagram represents the location of a hidden stick. About what fraction of the length of the street did the student walk to hide it? Be prepared to explain how you know.”

• Unit 6, Measuring Length, Time, Liquid, Volume, and Weight, Lesson 2, Warm-up, students practice estimation strategies with measurements. Narrative states, “The purpose of this Estimation Exploration is to practice the skill of estimating a reasonable answer based on experience and known information. The warm-up also draws students' attention to a length between a full inch and one-half of an inch, preparing students to work with such lengths later.” Launch states, “Display the image. ‘What is an estimate that’s too high? Too low? About right?’ 1 minute: quiet, think time.” Student Facing states, “What is the length of the paper clip?” A paper clip is shown next to a ruler and the length is between the 1 and 2 inch mark on the ruler.

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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 3 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, Area and Multiplication, Lesson 10, Cool-down, students construct viable arguments as they find the area of a rectangle. Student Facing, “Kiran bought two pieces of fabric. The black fabric is 9 yards by 2 yards. The purple fabric is 4 yards by 5 yards. Which piece of fabric has the larger area? Explain or show your reasoning.”

• Unit 2, Area and Multiplication, Lesson 12, Activity 2, students find the area of a “figure” by decomposing the figure into rectangles and then critique the reasoning of others. Narrative states, “Some students may partition diagonally to split the figure into what looks like 2 symmetrical parts, or cut the figure up into more than 2 parts. These are both acceptable ways of finding the area. Ask students who partition diagonally to find the area in the way they partitioned, but then encourage them to find a second way that has partitions on one of the grid lines. As students look through each others' work, they discuss how the representations are the same and different and can defend different points of view (MP3).”  Launch states, “Groups of 2. Display the image of the gridded figure. ‘What do you notice? What do you wonder?’ (Students may notice: It looks like 2 rectangles. It looks like a big rectangle with a chunk missing. There are squares. Students may wonder: What is this shape called? Could we find the area of the shape? How would we find the area?) 1 minute: quiet think time. Share responses. ‘This isn’t a shape that we have a name for like a square or triangle. Because of this, we’ll call it a “figure” as we work with it in this activity. This word will be helpful in describing other shapes that we don’t have a name for. Talk with your partner about different ways you could find the area of this figure.’ 1 minute: partner discussion.” Student Facing states, “What do you notice? What do you wonder? Find the area of this figure. Explain or show your reasoning. Organize it so it can be followed by others.”

• Unit 5, Fractions as Numbers, Lesson 11, Activity 1, students construct a viable argument and critique the reasoning of others when they reason about fraction equivalence. Student Facing states, “1. The diagram represents 1.​​​​​​ a. What fraction does the shaded part of the diagram represent? Jada says it represents \frac{4}{8}. Tyler is not so sure. Do you agree with Jada? If so, explain or show how you would convince Tyler that Jada is correct. If not, explain or show your reasoning.” Narrative states, “In the first problem, students construct a viable argument in order to convince Tyler that \frac{4}{8} of the rectangle is shaded (MP3).”

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 1, Introducing Multiplication, Lesson 6, Activity 1, students construct a viable argument and critique the reasoning of others as they create a scaled bar graph. Student Facing states, “Here is a collection of pattern blocks. Mai, Noah, and Priya want to make a bar graph to represent the number of triangles, squares, trapezoids, and hexagons in the collection. Mai says the scale of the bar graph should be 2. Noah says the scale of the bar graph should be 5. Priya says the scale of the bar graph should be 10. 1. Who do you agree with? Explain your reasoning. 2. Use the scale that you chose to create a scaled bar graph to represent the collection of pattern blocks.” Narrative states, “They consider three students’ ideas, choose a scale of 2, 5, or 10, and create a scaled bar graph to represent the categorical data. Students must justify why they agree that a particular scale would be best. During the activity and whole-class discussion, students share their thinking and have opportunities to listen to and critique the reasoning of their peers (MP3).”

• Unit 3, Wrapping Up Addition and Subtraction within 1,000, Lesson 9, Cool-down, students solve a subtraction problem using the algorithm and then critique the work of others. Preparation, Lesson Narrative states, “Previously, students learned to record subtraction using an algorithm in which the numbers are written in expanded form. They made connections between the structure and steps of the algorithm to those of base-ten diagrams that represent the same subtraction. In this lesson, students take a closer look at the algorithm and use it to find differences. They also examine a common error in subtracting numbers when decomposition of a place value unit is required. When students discuss shown work, they construct viable arguments and critique the reasoning of others (MP3).”  Student Facing states, (Students see the thinking of a student on the problem with regrouping.)  “Andre found the value of  739-255. His work is shown. Explain how he subtracted and the value he found for 739-255.”

• Unit 4, Relating Multiplication to Division, Lesson 15, Activity 2, students construct a viable argument and critique the reasoning of others when they participate in a gallery walk and agree or disagree with other students’ work. Narrative states, “The purpose of this activity is for students to see how other students solved one of the problems that involves a factor of a teen number. While students look at each other’s work, they will leave sticky notes describing why they think the answer does or does not make sense (MP3). The synthesis will look specifically at examples of how students used the area diagram to represent the problem.” Activity states, “‘As you visit the posters with your partner, discuss what is the same and what is different about the thinking shown on each poster. Also, leave a sticky note describing why you think the solution does or does not make sense.’ 8–10 minutes: gallery walk. Monitor for different uses of the area diagram to highlight, specifically, a fully gridded area diagram with no labels and no decomposition, a gridded area diagram that was gridded, but also decomposed into parts or labeled along the sides or in the parts of the rectangle, a partitioned rectangle that was drawn with no grid, but labeled with side lengths or the area of the parts of the rectangle.” Student Facing states, “As you visit the posters with your partner, discuss what is the same and what is different about the thinking shown on each poster.”

##### Indicator {{'2g' | indicatorName}}

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 3 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 4, Relating Multiplication to Division, Lesson 22, Activity 1, students use multiplication and division to determine the arrangement of strawberry plants in a garden. Lesson Narrative states, “Students model with mathematics (MP4) as they consider constraints, make assumptions and decisions about quantities, think about how to represent the relationships among quantities, and check their solutions in terms of the situation.” Activity states, “2 minutes: independent work time. 10 minutes: partner work time. Monitor for students who: write multiplication or division expressions or equations, are able to represent the same situation with both multiplication and division.” Student Facing states, “For each situation, draw a diagram and write an equation or expression. 1. A strawberry patch has 7 rows with 8 strawberry plants in each row. a. How many strawberry plants are in the patch? b. To grow strawberries in the best way, the rows should be 4 feet apart. Each plant in the row should be 2 feet apart. How long and wide is the strawberry patch? c. You can harvest 12 strawberries per plant. How many strawberries will grow in each row? 2. With your partner, take turns explaining where you see the numbers in the expression or equation you wrote in your diagram.”

• Unit 6, Measuring Length, Time, Liquid Volume, and Weight, Lesson 14, Cool-down, students reason about situations involving elapsed time. Student Facing states, “1. A show at the carnival starts at 2:45 p.m. and lasts 47 minutes. What time does the show end? Explain or show your reasoning. 2. Another show that is 24 minutes long ends at 5:10 p.m. Kiran says that the show starts before 4:40 p.m. Do you agree? Explain or show your reasoning.” Lesson Narrative states, “In this lesson, students model with mathematics (MP4) as they determine quantities, questions, and solutions that make sense in given situations and adhere to mathematical and real-world constraints when solving problems.”

• Unit 7, Two-Dimensional Shapes and Perimeter, Lesson 13, Activity 1, students apply what they have learned about perimeter and area to design a small park. Narrative states, “The purpose of this activity is to provide students an opportunity to apply what they’ve learned about perimeter and area to design a small park. Since diagonal lines that connect the dots are not one length unit, students should use vertical and horizontal lines to design the park. When students make and describe their own choices for how they represent real-world objects, they model real-world problems with mathematics (MP4).”  Activity states, “‘Work independently to design your small park.’ 5–7 minutes: independent work time. ‘You can work with a partner or small group for the last few minutes or continue working on your own. Even if you choose to work alone, be available if your partner wants to think through something together.’ 3–5 minutes: partner, small group, or independent work time.”  Student facing states, “Your teacher will give you some dot paper for drawing. 1. The distance from 1 dot to another horizontally or vertically represents 1 yard. Connect dots on the grid horizontally or vertically to design a small park that has 5 of these features: a. basketball court  b. soccer goal  c. swings  d. a slide  e. an open area  f. picnic table  g. water play area  h. skate park  2. a feature of your choice  3. Describe the area and the perimeter of 3 features in the park.”

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 3, Wrapping Up Addition and Subtraction Within 1,000, Lesson 7, Activity 1, students use appropriate tools strategically to subtract within 1,000. Student Facing states, “Find the value of each difference in any way that makes sense to you. Explain or show your reasoning. 1. 428-213. 2. 505-398. 3. 394-127.” Lesson Narrative states, “Students may also use a variety of representations, which will be the focus of the activity synthesis. Students who choose to use base-ten blocks or number lines to represent their thinking use tools strategically (MP5).”

• Unit 4, Relating Multiplication to Division, Lesson 13, Cool-down, students use appropriate tools strategically when they multiply within 100. Student Facing states, “There are 6 bags of oranges and each bag has 11 oranges. How many oranges are in the bags? Show your thinking using objects, a drawing, or a diagram.” Activity 1 Lesson Narrative states, “This is the first time students have worked with problems with numbers in this range, so they should be encouraged to use the tools provided to them during the lesson if they choose (MP5). Students should also be encouraged to use strategies and representations from the previous section.” This outlines the goal of working with tools throughout this lesson.

• Unit 6, Measuring Length, Time, Liquid Volume, and Weight, Lesson 4, Activity 1, students analyze a line plot as a tool for representing data. Students are given the heights of seedlings in inches on a line plot. Student Facing states, “1. Write 3 statements about the measurements represented in the line plot. 2. What questions could be answered more easily with the line plot than the list? Write at least 2 questions.” Lesson Narrative states, “When students recognize how organizing data helps to read the information and to answer questions, they learn that line plots are a powerful tool to present data (MP5).”

##### Indicator {{'2h' | indicatorName}}

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 3 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 2, Area and Multiplication, Lesson 4, Activity 1, students use specific language and precision to describe the rectangle they create. Narrative states, “The purpose of this activity is for students to create and describe rectangles of a certain area. Students work in groups of 2. One partner creates a rectangle and describes it, and the other partner creates a matching rectangle based on the description. Then students compare how their rectangles are the same and different. Students should describe their rectangle to their partner without revealing the total number of squares they used, so that the focus is on understanding the rectangular structure. In the synthesis, students share language that helped them understand the rectangle their partner built. When students revise their language to be more precise in the descriptions of their rectangle, they attend to precision (MP6).”  Activity states, “‘The goal of this activity is to get both partners in a group to draw the same rectangle without looking at each other’s drawing. ‘If you are partner A, draw a rectangle and describe it to your partner. You can’t tell them how many squares you used to draw your rectangle. If you are partner B, draw the rectangle that you think your partner is describing and then compare the drawings. After you finish describing and drawing the first rectangle, switch roles and repeat.’ 10–12 minutes: partner work.”  Student Facing states, “1. Can you and your partner draw the same rectangle without looking at each other's drawing?  Partner A: Draw a rectangle on one of the grids provided. Describe it to your partner without telling them the total number of squares. Partner B: Draw the rectangle your partner describes to you. 2. Place your two rectangles next to each other. Discuss: What is the same? What is different?  3. Switch roles and repeat.”  Lesson synthesis states, “‘What language did your partner use that was most helpful for you to draw the same rectangle they drew?’ (The number of squares in each row or column and the number of rows or columns.)”

• Unit 3, Wrapping Up Addition and Subtraction Within 1,000, Lesson 8, Cool-down, students use accuracy and precision when they use base-ten diagrams to make sense of a written subtraction algorithm. Student Facing states, “Explain how the diagram matches the algorithm.” Narrative states, “As students work, encourage them to refine their descriptions of what is happening in both the diagrams and the algorithms using more precise language and mathematical terms (MP6).”

• Unit 5, Fractions as Numbers, Lesson 9, Activity 1, students attend to precision when using a number line to locate fractions. Activity states, “‘Take a few minutes to locate 1 on these number lines. Then use any of the number lines to explain how you located 1.’ 5–7 minutes: independent work time.” Student Facing states, “2. Use any of the number lines to explain how you located 1.” Lesson Narrative states, “In the second problem, they reinforce their knowledge that the denominator of a fraction tells us the number of equal parts in a whole and the size of a unit fraction, and that the numerator gives the number of those parts (MP6).” Activity Synthesis states, “‘Share your written reasoning for one of the number lines with your partner. Take turns being the speaker and the listener. If you are the speaker, share your ideas and writing so far. If you are the listener, ask questions and give feedback to help your partner improve their work.’ 3–5 minutes: structured partner discussion. Repeat with 2–3 different partners. ‘Revise your initial draft based on the feedback you got from your partners.’ 2–3 minutes: independent work time. Invite students to share their revised explanations of how they located 1 on the number lines.”

• Unit 6, Measuring Length, Time, Liquid Volume, and Weight, Lesson 1, Activity 1, students use the specialized language of mathematics as they discuss how to describe lengths using fractions of an inch. Lesson Narrative states, “In the synthesis, discuss the need for fractions of an inch to describe lengths more precisely (MP6).” Activity Synthesis states, “Display the inch ruler and an object that wasn’t exactly a whole number of inches. ‘What is the length of this object? (Between 3 and 4 inches. More than 3 but less than 4. Three-and-a-half inches.) If needed, Could we say that the length of this object is (a whole number of) inches. (No, It's between 3 inches and 4 inches.) We need a way to make our measurements more precise. We'll think about this more in the next activity.’”

• Unit 7, Two-Dimensional Shapes and Perimeter, Lesson 3, Cool-down, students attend to the specialized language of math as they describe shapes. Student Facing states, “1. Which quadrilateral is being described? Hint 1: It has 4 sides. Hint 2: All of its sides are the same length. Hint 3: It has no right angles. 2. Which hints do you need to guess the quadrilateral? Explain your reasoning.” Students see four different quadrilaterals with different features. Narrative states, “As students decide which questions to ask they think about important attributes such as side lengths and angles and have an opportunity to use language precisely (MP6, MP7).”

• Unit 8, Putting It All Together, Lesson 7, Warm-up, students attend to precision as they use a bar graph and see the importance of precise labels and titles. Narrative states, “The purpose of this warm-up is to elicit the idea that bar graphs need a title and a scale in order to be able to communicate information clearly (MP6), which will be useful when students draw a scaled bar graph in a later activity. During the synthesis, focus the discussion on the missing scale.”  Launch states, “Groups of 2. Display the graph. ‘What do you notice?  What do you wonder?’ 1 minute: quiet think time.” Activity Synthesis states, “Could each unit or each space between two lines on the graph represent 1 student? Why or why not? (No, because that would mean half of a student likes broccoli, cauliflower, and peas.) If each unit on the graph represents 2 students, how many students have broccoli as their favorite vegetable? (13) What if it represents 4 students? (26) How should you decide on a scale for your graph? (Think about how many people you surveyed and use a scale that will fit them on your graph. Use a scale that will make the bar graph easy to read.)”

##### Indicator {{'2i' | indicatorName}}

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 3 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, Introducing Multiplication, Lesson 16, Warm-up, students look for and make use of structure while they notice the arrangement of a dozen eggs in relation to arrays. Narrative states, “The purpose of this warm-up is to elicit ideas students have about objects arranged in an array, which will be useful when students arrange equal groups into arrays in a later activity. While students may notice and wonder many things about this image, ideas around arrangement and equal groups are the important discussion points. When students notice the arrangement of the eggs they look for and make use of structure (MP7).” Launch states, “Groups of 2. Display the image. ‘What do you notice? What do you wonder?’” Student Facing states, “What do you notice? What do you wonder?” An egg carton with a dozen eggs is shown. Activity Synthesis states, “‘How does having the eggs in a carton help you see equal groups?’ (I can see how they could be split into equal groups. I can see 6 eggs in each row. I can see 6 groups of 2.) The eggs are arranged in an array. An array is an arrangement of objects in rows and columns. Each column must contain the same number of objects as the other columns, and each row must have the same number of objects as the other rows.”

• Unit 3, Wrapping Up Addition and Subtraction Within 1,000, Lesson 1, Activity 1, students look for and make use of structure as they represent numbers using base-ten blocks, base-ten diagrams, expanded form, numerals, and word form. Activity states, “Work with your partner to find the expression that matches your card. Then discuss how you know the expression matches your card.” Student Facing states, “Your teacher will give you a set of cards that show numbers in different forms. Match the cards that represent the same number. Record your matches here. Be ready to explain your reasoning.” Lesson Narrative states, “As they make matches, students use their understanding of base-ten structure represented in many different ways (MP7).”

• Unit 8, Putting It All Together, Lesson 1, Cool-down, students use structure to determine if the three representations all show the same fractional value. Lesson Narrative states, “In previous lessons, students learned how to represent fractions with area diagrams, fraction strips, and number lines. In this lesson, students revisit each of these representations in an estimation context. Students have an opportunity to think about how to partition each representation to decide what fraction is shown (MP7). Additionally, if time allows and it seems of benefit to student understanding, there is an option after each activity to find the exact value of the fraction in the task statement.” Student Facing states, “Could the shaded part of the shape, the point on the number line, and the shaded part of the diagram all represent the same fraction?  Explain your reasoning.” Students see a diamond, a number line, and a diagram that do not all represent the same fraction.

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, Wrapping Up Addition and Subtraction Within 1,000, Lesson 14, Activity 2, students use repeated reasoning as they analyze the position of numbers relative to their immediate multiples of 10 and 100. Activity states, “‘Work with your partner to complete these problems.’ 5–7 minutes: partner work time. Monitor for students who use the following strategies to highlight: Reason about the midpoint between a multiple of 10 or a multiple of 100 (5 or 50) to determine which multiple is closer, such as, ‘568 is closer to 570 because 565 would be the middle point between 560 and 570’; Use place value patterns to determine which multiple is closer, such as, ‘Since the 1 in 712 is less than 5, it tells me that the number is closest to 700’. Pause for a brief discussion before students complete the last problem. Select previously identified students to share the strategies they used to find the nearest multiple of 100 and the nearest multiple of 10. ‘Now take a few minutes to complete the last problem.’ 2–3 minutes: independent work time.” Students Facing states, “1a. Is 349 closer to 300 or 400? 1b. Is 349 closer to 340 or 350? 2a. Is 712 closer to 700 or 800? 2b. Is 712 closer to 710 or 720? 3a. Is 568 closer to 500 or 600? 3b. Is 568 closer to 560 or 570? 4. Without locating a given number on a number line, how did you decide: a. the nearest multiple of 100? b. the nearest multiple of 10?” Narrative states, “When students notice and describe patterns in the relationship between the numbers and the nearest multiples of 10 or 100, they look for and express regularity in repeated reasoning (MP8).”

• Unit 5, Fractions as Numbers, Lesson 2, Cool-down, students use repeated reasoning as they partition shapes into equal parts. Student Facing states, “1. Label each part with the correct fraction. 2. Partition and shade the rectangle to show \frac{1}{4}.” Activity 1 Narrative states, “When students make halves, fourths, and eighths they observe regularity in repeated reasoning as each piece is subdivided into 2 equal pieces. They observe the same relationship between thirds and sixths (MP8).” The Cool-down provides an opportunity to demonstrate this reasoning.

• Unit 6, Measuring Length, Time, Liquid Volume, and Weight, Lesson 10, Warm-up, students use repeated reasoning as they choral count, using 15 minutes as the increment of time. Narrative states, “The purpose of this Choral Count is to invite students to practice counting times by 15 minutes and notice patterns in the count. This will be helpful later in this section when students will solve problems involving addition and subtraction of time intervals. Students have an opportunity to notice regularity through repeated reasoning (MP8) as they count by 15 minutes over a span of 3 hours.” Launch states, “‘Count by 15 minutes, starting at 12:00.’ Record as students count. Record times in the count in a single column. Stop counting and recording at 3:00.” Activity states, “‘What patterns do you see?’ 1–2 minutes: quiet think time. Record responses.” Activity Synthesis states, “‘How much time passed between 1:15 and 1:45? (30 minutes) 1:15 and 2:30?’ (75 minutes) Consider asking: ‘Who can restate the pattern in different words? Does anyone want to add an observation on why that pattern is happening here? Do you agree or disagree? Why?’”

### Usability

The materials reviewed for Kendall Hunt's Illustrative Mathematics Grade 3 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 3 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 3 meet expectations for providing teacher guidance with useful annotations and suggestions for how to enact the student materials and ancillary materials, with specific attention to engaging students in order to guide their mathematical development.

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

• IM Curriculum, How To Use These Materials, Design Principles, Coherent Progression provides an overview of the design and implementation guidance for the program, “The overarching design structure at each level is as follows: Each unit starts with an invitation to the mathematics. The first few lessons provide an accessible entry point for all students and offer teachers the opportunity to observe students’ prior understandings. Each lesson starts with a warm-up to activate prior knowledge and set up the work of the day. This is followed by instructional activities in which students are introduced to new concepts, procedures, contexts, or representations, or make connections between them. The lesson ends with a synthesis to consolidate understanding and make the learning goals of the lesson explicit, followed by a cool-down to apply what was learned. Each activity starts with a launch that gives all students access to the task. This is followed by independent work time that allows them to grapple with problems individually before working in small groups. The activity ends with a synthesis to ensure students have an opportunity to consolidate their learning by making connections between their work and the mathematical goals. In each of the activities, care has been taken to choose contexts and numbers that support the coherent sequence of learning goals in the lesson.”

• Course Guide, Scope and Sequence, provides an overview of content and expectations for the units, “The big ideas in grade 3 include: developing understanding of multiplication and division and strategies for multiplication and division within 100; developing understanding of fractions, especially unit fractions (fractions with numerator 1); developing understanding of the structure of rectangular arrays and of area; and describing and analyzing two-dimensional shapes.”

Materials include sufficient and useful annotations and suggestions that are presented within the context of the specific learning objectives. Several components focus specifically on the content of the lesson. Preparation and lesson narratives within the Warm-up, Activities, and Cool-down provide useful annotations. Examples include:

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

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

##### 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 3 meet expectations for containing adult-level explanations and examples of the more complex grade-level concepts and concepts beyond the current grade so that teachers can improve their own knowledge of the subject.

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

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

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

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

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

##### 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 3 meet expectations for including standards correlation information that explains the role of the standards in the context of the overall series.

Correlation information is present for the mathematics standards addressed throughout the grade level/series and can be found in several places, including the curriculum course guide, within unit resources, and within each lesson. Examples include:

• Grade 3, Course Guide, Lesson Standards includes a table with each grade-level lesson (in columns) and aligned grade-level standards (in rows). Teachers can search any lesson for the grade and identify the standard(s) that are addressed within.

• Grade 3, Course Guide, Lesson Standards, includes all Grade 3 standards and the units and lessons each standard appears in. Teachers can search a standard for the grade and identify the lesson(s) where it appears within materials.

• Unit 2, Resources, Teacher Guide, outlines standards, learning targets and the lesson where they appear. This is present for all units and allows teachers to identify targeted standards for any lesson.

• Unit 4, Relating Multiplication to Division, Lesson 10, the Core Standards are identified as 3.MD.C.7c and 3.OA.C.7. Lessons contain a consistent structure: a Warm-up that includes Narrative, Launch, Activity, Activity Synthesis; Activity 1, 2, or 3 that includes Narrative, Launch, Activity; an Activity Synthesis; a Lesson Synthesis; and a Cool-down that includes Responding to Student Thinking and Next Day Supports. This provides an additional place to reference standards, and language of the standard, within each lesson.

Each unit includes an overview outlining the content standards addressed within as well as a narrative describing relevant prior and future content connections. Examples include:

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

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

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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 3 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 3, Unit 2, Area and Multiplication, Family Support Materials, “In this unit, students learn about the concept of area and relate area to multiplication and addition. Section A: Concepts of Area Measurement. In this section, students make sense of the area of flat shapes. They learn that the area of a shape is the amount of space it covers, and it can be measured by the number of square units that cover it without gaps or overlaps. Students explore this idea by tiling shapes with squares and counting the number of squares. Section B: Relate Area to Multiplication. In this section, students relate the area of rectangles to multiplication. They see that rectangles can be tiled with squares in equal-size rows (or columns), so if the rectangle is 6 units by 4 units, there are 6 groups of 4 or 4 groups of 6. The number of square units is then 6\times4 or 4\times6. Students come to understand that multiplying the side lengths of a rectangle gives the same number of squares as counting them. A rectangle that is 3 units by 6 units can be tiled with 3 rows of 6 squares, so its area is 6\times3 or 18 square units. Students then use these ideas to solve real-world story problems related to area.”

• For Families, Grade 3, Unit 5, Fractions as Numbers, Family Support Materials, Try it at home!, “Near the end of the unit, ask your student to show the fractions \frac{5}{8} and \frac{6}{4} on a fraction strip and a number line. Questions that may be helpful as they work: How did you determine how many partitions needed to be made? How did you know how many parts to shade in? How did you know where to place the fraction on the number line? Which fraction is larger? How do you know?”

• For Families, Grade 3, Unit 7, Two-dimensional Shapes and Perimeter, Family Support Materials, Try it at home!, “Near the end of the unit, ask your student to find: these shapes around the house: a rhombus, a rectangle, a square, and a quadrilateral that isn't a rhombus, rectangle, or square the area and perimeter of a rectangle in the house. Questions that may be helpful as they work: What kind of quadrilateral is this? How do you know? Are you measuring area or perimeter? How do you know?”

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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 3 meet expectations for providing explanations of the instructional approaches of the program and identification of the research-based strategies.

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

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

• IM K-5 Teacher Guide, Design Principles, Coherent Progression, “Each unit starts with an invitation to the mathematics. The first few lessons provide an accessible entry point for all students and offer teachers the opportunity to observe students’ prior understandings. Each lesson starts with a warm-up to activate prior knowledge and set up the work of the day. This is followed by instructional activities in which students are introduced to new concepts, procedures, contexts, or representations, or make connections between them. The lesson ends with a synthesis to consolidate understanding and make the learning goals of the lesson explicit, followed by a cool-down to apply what was learned. Each activity starts with a launch that gives all students access to the task. This is followed by independent work time that allows them to grapple with problems individually before working in small groups. The activity ends with a synthesis to ensure students have an opportunity to consolidate their learning by making connections between their work and the mathematical goals. In each of the activities, care has been taken to choose contexts and numbers that support the coherent sequence of learning goals in the lesson.”

• IM K-5 Teacher Guide, Design Principles, Learning Mathematics by Doing Mathematics, “Students learn mathematics by doing mathematics, rather than by watching someone else do mathematics or being told what needs to be done. Doing mathematics can be defined as learning mathematical concepts and procedures while engaging in the mathematical practices—making sense of problems, reasoning abstractly and quantitatively, making arguments and critiquing the reasoning of others, modeling with mathematics, making appropriate use of tools, attending to precision in their use of language, looking for and making use of structure, and expressing regularity in repeated reasoning. By engaging in the mathematical practices with their peers, students have the opportunity to see themselves as mathematical thinkers with worthwhile ideas and perspectives. ‘Students learn mathematics as a result of solving problems. Mathematical ideas are the outcomes of the problem-solving experience rather than the elements that must be taught before problem solving’ (Hiebert et al., 1996). A problem-based instructional framework supports teachers in structuring lessons so students are the ones doing the problem solving to learn the mathematics. The activities and routines are designed to give teachers opportunities to see what students already know and what they can notice and figure out before having concepts and procedures explained to them.”

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

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

• IM K-5 Math Teacher Guide, Design Principles, Using the 5 Practices for Orchestrating Productive Discussions, “Promoting productive and meaningful conversations between students and teachers is essential to success in a problem-based classroom. The Instructional Routines section of the teacher course guide describes the framework presented in 5 Practices for Orchestrating Productive Mathematical Discussions (Smith & Stein, 2011) and points teachers to the book for further reading. In all lessons, teachers are supported in the practices of anticipating, monitoring, and selecting student work to share during whole-group discussions. In lessons in which there are opportunities for students to make connections between representations, strategies, concepts, and procedures, the lesson and activity narratives provide support for teachers to also use the practices of sequencing and connecting, and the lesson is tagged so teachers can easily identify these opportunities. Teachers have opportunities in curriculum workshops and PLCs to practice and reflect on their own enactment of the 5 Practices.”

• IM K-5 Math Teacher Guide, Key Structures in This Course, Student Journal Prompts, “Writing can be a useful catalyst in learning mathematics because it not only supplies students with an opportunity to describe their feelings, thinking, and ideas clearly, but it also serves as a means of communicating with other people (Baxter, Woodward, Olson, & Robyns, 2002; Liedtke & Sales, 2001; NCTM, 2000). NCTM (1989) suggests that writing about mathematics can help students clarify their ideas and develop a deeper understanding of the mathematics at hand.”

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

The materials reviewed for Kendall Hunt's Illustrative Mathematics Grade 3 meet expectations for providing a comprehensive list of supplies needed to support instructional activities.

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

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

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

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

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

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This is not an assessed indicator in Mathematics.

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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 3 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.

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

The materials reviewed for Kendall Hunt's Illustrative Mathematics Grade 3 meet expectations for having assessment information included in the materials to indicate which standards are assessed.

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

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

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

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

Guidance is provided within materials for assessing progress of the Mathematical Practices. According to IM K-5 Math Teacher Guide, How to Use These Materials, “Because using the mathematical practices is part of a process for engaging with mathematical content, we suggest assessing the Mathematical Practices formatively. For example, if you notice that most students do not use appropriate tools strategically (MP5), plan in future lessons to select and highlight work from students who have chosen different tools.” For each grade, there is a chart outlining a handful of lessons in each unit that showcase certain mathematical practices. There is also guidance provided for tracking progress against “I can” statements aligned to each practice, “Since the Mathematical Practices in action can take many forms, a list of learning targets for each Mathematical Practice is provided to support teachers and students in recognizing when engagement with a particular Mathematical Practice is happening. The intent of the list is not that students check off every item on the list. Rather, the ‘I can’ statements are examples of the types of actions students could do if they are engaging with a particular Mathematical Practice.” Examples include:

• IM K-5 Math Teacher Guide, How to Use These Materials, Standards for Mathematical Practices Chart, Grade 3, MP6 is found in Unit 1, Lessons 2, 10, 14, 17, and 20.

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

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

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

##### 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 3 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 5, Fractions as Numbers, End-of-Unit Assessment, Problem 4, “Which fraction is equivalent to \frac{9}{3}? A: \frac{1}{3}, B: \frac{10}{4}, C: \frac{6}{2}, D: \frac{8}{2}.” The Assessment Teacher Guide states, “Students identify a fraction that is equivalent to a whole number expressed as a fraction. While they are not directly asked to write \frac{9}{3} as a whole number, the most likely reasoning is to identify that this is 3 wholes and so is \frac{6}{2}. Students may select A if they confuse the meaning of the numerator and denominator and think of the fraction \frac{1}{3}. They may select B if they add 1 to the numerator and the denominator. They may select D if they subtract 1 from the numerator and the denominator.” This answer key aligns this problem to 3.NF.3a and 3.NF.3c.

• Unit 6, Measuring Length Time, Liquid, Volume, and Weight, End-of-Unit Assessment, Problem 6, “A young humpback whale weighs 835 kg. A young killer whale weighs 143 kg. How much heavier is the humpback whale than the killer whale? Explain or show your reasoning.” The Assessment Teacher Guide states, “Students subtract within 1,000 to answer a question about weights. Students may add 835 and 143 if they do not read the question carefully. This would be the total weight of the two whales in kilograms. Students may subtract by place value, as shown in the solution, or they may use a number line or other diagram.” This answer key aligns this problem to 3.MD.2.

• Unit 8, Putting It All Together, End-of-Course Assessment, Problem 7, “A farmer brought 525 oranges to the market. She sold 8 boxes of 12 oranges. How many oranges does she have now? Show your reasoning.” The Assessment Teacher Guide states, “Students solve a two-step arithmetic problem within 1,000. They find a product within 100 and then perform subtraction within 1,000 to solve the problem. The multiplication part of the problem may require several steps as students may not know the value of 8\times12 without calculating. They may find the difference using equations, an algorithm, or a form of compensation as in the given solution.” The answer key aligns this problem to 3.OA.7 and 3.OA.8.

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 4, Relating Multiplication to Division, Lesson 4, Cool-down, Student Facing states, “Han has 14 tops. He shares the tops equally between 2 boxes. How many tops will be in each box? Select all the ways that we could represent the situation.” Responding to Student Thinking states, “Students select responses that correspond to 7 groups of 2 instead of 2 groups of 7.” Next Day Supports states, “During the launch of the next day’s activity, have students work in partners to discuss a correct response to this cool-down.” This problem aligns to 3.OA.2.

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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 3 meet expectations for providing assessments that include opportunities for students to demonstrate the full intent of grade-level standards and practices across the series.

Formative assessment opportunities include some end of lesson cool-downs, interviews, and Checkpoint Assessments for each section. Summative assessments include End-of-Unit Assessments and the End-of-Course Assessment. Assessments regularly demonstrate the full intent of grade-level content and practice standards through a variety of item types, including multiple choice, multiple response, short answer, restricted constructed response, and extended response. Examples from summative assessments include:

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

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

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

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

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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 3 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 3 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.

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

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 3 meet expectations for providing strategies and supports for students in special populations to support their regular and active participation in learning grade-level mathematics.

Materials regularly provide strategies, supports, and resources for students in special populations to help them access grade-level mathematics as suggestions are outlined within each lesson and parts of each lesson. According to the IM K-5 Teacher Guide, Universal Design for Learning and Access for Students with Disabilities, “These materials empower all students with activities that capitalize on their existing strengths and abilities to ensure that all learners can participate meaningfully in rigorous mathematical content. Lessons support a flexible approach to instruction and provide teachers with options for additional support to address the needs of a diverse group of students, positioning all learners as competent, valued contributors. When planning to support access, teachers should consider the strengths and needs of their particular students. The following areas of cognitive functioning are integral to learning mathematics (Addressing Accessibility Project, Brodesky et al., 2002). Conceptual Processing includes perceptual reasoning, problem solving, and metacognition. Language includes auditory and visual language processing and expression. Visual-Spatial Processing includes processing visual information and understanding relation in space of visual mathematical representations and geometric concepts. Organization includes organizational skills, attention, and focus. Memory includes working memory and short-term memory. Attention includes paying attention to details, maintaining focus, and filtering out extraneous information. Social-Emotional Functioning includes interpersonal skills and the cognitive comfort and safety required in order to take risks and make mistakes. Fine-motor Skills include tasks that require small muscle movement and coordination such as manipulating objects (graphing, cutting with scissors, writing).”

Examples of supports for special populations include:

• Unit 1, Introducing Multiplication, Lesson 6, Activity 1, Narrative, Access for Students with Disabilities, “Representation: Access for Perception. Provide access to pattern blocks to model the collection of pattern blocks in the student-facing task statement. Supports accessibility for: Organization, Visual-Spatial Processing.

• Unit 3, Wrapping Up Addition and Subtraction Within 1,000, Lesson 5, Activity 1, Narrative, Access for Students with Disabilities, “Action and Expression: Develop Expression and Communication. Synthesis: Identify connections between strategies that result in the same outcomes but use differing approaches. Supports accessibility for: Conceptual Processing.”

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

• Unit 7, Two-dimensional Shapes and Perimeter, Lesson 6, Activity 2, Narrative, Access for Students with Disabilities, “Action and Expression: Develop Expression and Communication. Synthesis: Identify connections between strategies that result in the same outcomes but use differing approaches. Supports accessibility for: Visual-Spatial Processing.

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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 3 meet expectations for providing extensions and/or opportunities for students to engage with grade-level mathematics at higher levels of complexity.

While there are no instances where advanced students do more assignments than classmates, materials do provide multiple opportunities for students to investigate grade-level content at a higher level of complexity. These are found in a section titled “Exploration Problems” within lessons where appropriate. According to the IM K-5 Teacher Guide, How To Use The Materials, Exploration Problems, “Each practice problem set also includes exploration questions that provide an opportunity for differentiation for students ready for more of a challenge. There are two types of exploration questions. One type is a hands-on activity directly related to the material of the unit that students can do either in class if they have free time, or at home. The second type of exploration is more open-ended and challenging. These problems go deeper into grade-level mathematics. They are not routine or procedural, and they are not just the same thing again but with harder numbers. Exploration questions are intended to be used on an opt-in basis by students if they finish a main class activity early or want to do more mathematics on their own. It is not expected that an entire class engages in exploration problems, and it is not expected that any student works on all of them. Exploration problems may also be good fodder for a Problem of the Week or similar structure.” Examples include:

• Unit 2, Area and Multiplication, Section A: Concepts of Area Measurement, Problem 11, Exploration, “How many different rectangles can you make with 36 square tiles? Describe or draw the rectangles. How are the rectangles the same? How are they different?”

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

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

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

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

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 3 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 3, Wrapping Up Addition and Subtraction Within 1,000, Lesson 3, Activity 1, students use strategies to add within 1,000. Launch states, “Give students access to base-ten blocks. ‘Take a minute to think about how you could find the value of each sum.’” Student Facing states, “Find the value of each sum in any way that makes sense to you. Explain or show your reasoning. 1. 325+102 2. 301+52 3. 276+118 4. 298+305.”

• Unit 4, Relating Multiplication to Division, Lesson 1, Activity 1, students make posters to represent math problems. Activity states, “‘Solve these problems and show your thinking using objects, a drawing, or a diagram.’ 6–8 minutes: independent work time. As students work, consider asking: ‘How can you represent what you are thinking? Where can we see the apples in your work? Where can we see how many boxes there are in your work?’ Monitor for students who solve the first problem in the same way. Arrange them into groups of 2 to create a poster together. ‘Now you are going to create a poster to show your thinking on the first problem. You are going to work with a partner who solved the problem in the same way you did.’ Give each group tools for making a visual display. 6–8 minutes: partner work time.” Student Facing states, “Solve each problem. Show your thinking using objects, a drawing, or a diagram. If 24 apples are put into boxes with 8 apples in each box, how many boxes are there?  If 42 apples are put into boxes with 6 apples in each box, how many boxes are there?  If 32 apples are put into boxes with 4 apples in each box, how many boxes are there?”

• Unit 5, Fractions as Numbers, Lesson 16, Cool-down, students compare two fractions with the same numerator. Student Facing states, “Use the symbols > and < to make the statement true. Explain or show your reasoning. \frac{4}{3}___$$\frac{4}{6}$$.”

• Center, Mystery Number (1–4), Stage 3: Fractions with Denominators 2, 3, 4, 6, students try to guess the mystery number. Narrative states, “Students choose a mystery fraction (with a denominator of 2, 3, 4, or 6) from the gameboard. Students give clues based on the given vocabulary.”

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

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

The materials reviewed for Kendall Hunt's Illustrative Mathematics Grade 3 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 1, Introducing Multiplication, Lesson 13, Activity 1, students are grouped in pairs or groups of four as they match multiplication equations to situations and representations. Launch states, “Groups of 2 and 4. ‘Think about how you might match these equations to a situation or diagram.’ 1 minute: quiet think time.” Activity states, “‘Take turns finding a situation or diagram that matches each equation. Explain your reasoning to your partner.’ 5–7 minutes: partner discussion. Monitor for students who make direct connections between each factor representing the number in each group or the number of groups and the product representing the total number of objects to share during the synthesis. ‘Get together with another group to discuss the matches you made.’ 3-5 minutes: small-group discussion.”

• Unit 5, Fractions as Numbers, Lesson 13, Activity 2, students work with a small group as they write whole numbers as fractions. Launch states, “Groups of 3. ‘Take a moment to look at the table and the numbers in the table. How do you think the table works?’ 1 minute: quiet think time. ‘Share responses. In your group, take turns explaining why \frac{1}{4} is equivalent to 4, \frac{30}{6} is equivalent to 5, and \frac{48}{8} is equivalent to 6.’ 2 minutes: group discussion.” Activity states, ‘Work with your group to complete the table. Start by writing two fractions that are equivalent to each whole number: 4, 5, and 6.’ 2 minutes: independent work time. ‘Pass your paper to your right. On the paper you receive, write one new fraction that is equivalent to a whole number of your choice. Keep passing and writing one additional fraction for a whole number until the table is complete.’ Encourage students to ask clarifying questions before they begin. 7–8 minutes: group work time. ‘Be prepared to explain how you know what fractions to write for each whole number.’ Monitor for students who: use a number line (either draw a new one or extending those from an earlier activity), extend a pattern they noticed in fractions that are equivalent to whole numbers, use multiplication facts.”

• Unit 6, Measuring Length, Time, Liquid, Volume, and Weight, Lesson 6, Activity 1, students are organized into five groups to work with measurements using grams and kilograms. Launch states, “5 groups. ‘Grams and kilograms are some units we use to measure weight. A gram is a small metric unit of weight and is about the same weight as a paper clip. A kilogram is a larger metric unit of weight and is about the same weight as a basket of apples.’” Activity states, “‘We are going to hold some weights and get a sense of how heavy 1 gram, 10 grams, 100 grams, 1 kilogram, and 2 kilograms are. Your group will have 2 minutes with each weight. In that time, everyone in your group should hold the weight and list some things you think weigh about the same. You can share ideas with your group as you write.’ Distribute objects with the following weights, one object for each group: 1 gram, 10 grams, 100 grams, 1 kilogram, 2 kilograms. 2 minutes: small-group work time. Rotate materials. 8–10 minutes: Repeat rounds of 2 minutes with each weight until each group has had every weight. Monitor for examples of each weight that students generate. Pause for a discussion. Invite students to share 1–2 examples for each measurement. Have students add the examples that are shared to their list. ‘Now, look at the table in the second problem. Work with your group to find objects around the room that belong in each column based on their weight.’ 5–7 minutes: small-group work time.”

• Unit 8, Putting It All Together, Lesson 12, Activity 1, students work with a small group in order to design a Notice and Wonder activity. Launch states, “Groups of 3 - 4. Give each group access to picture books. ‘Work with your group to create a Notice and Wonder activity about equal groups.’ Activity states, “15 minutes: small-group work time.”

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

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 3 meet expectations for providing strategies and supports for students who read, write, and/or speak in a language other than English to regularly participate in learning grade-level mathematics.

Guidance is consistently provided for teachers to support students who read, write, and/or speak in a language other than English, providing scaffolds for them to meet or exceed grade-level standards. According to the IM K-5 Math Teacher Guide, Mathematical Language Development and Access for English Learners, “In a problem-based mathematics classroom, sense-making and language are interwoven. Mathematics classrooms are language-rich, and therefore language demanding learning environments for every student. The linguistic demands of doing mathematics include reading, writing, speaking, listening, conversing, and representing (Aguirre & Bunch, 2012). Students are expected to say or write mathematical explanations, state assumptions, make conjectures, construct mathematical arguments, and listen to and respond to the ideas of others. In an effort to advance the mathematics and language learning of all students, the materials purposefully engage students in sense-making and using language to negotiate meaning with their peers. To support students who are learning English in their development of language, this curriculum includes instruction devoted to fostering language development alongside mathematics learning, fostering language-rich environments where there is space for all students to participate.” The series provides the following principles that promote mathematical language use and development:

• “Principle 1. Support sense-making: Scaffold tasks and amplify language so students can make their own meaning.

• Principle 2. Optimize output: Strengthen opportunities for students to describe their mathematical thinking to others, orally, visually, and in writing.

• Principle 3. Cultivate conversation: Strengthen opportunities for constructive mathematical conversations.

• Principle 4. Maximize meta-awareness: Strengthen the meta-connections and distinctions between mathematical ideas, reasoning, and language.”

The series also provides Mathematical Language Routines in each lesson. According to the IM K-5 Math Teacher Guide, Mathematical Language Development and Access for English Learners, “Mathematical Language Routines (MLRs) are instructional routines that provide structured but adaptable formats for amplifying, assessing, and developing students' language. MLRs are included in select activities in each unit to provide all students with explicit opportunities to develop mathematical and academic language proficiency. These ‘embedded’ MLRs are described in the teacher notes for the lessons in which they appear.” Examples include:

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

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

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

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

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 3 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 Lin, Diego, Kiran, and Priya 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.

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

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 3 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.

##### Indicator {{'3t' | indicatorName}}

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 3 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 8, Putting It All Together, Lesson 4, Warm-up, introduces students to the idea of tiny houses. Throughout this lesson, students see different ways to live, beyond houses or apartment buildings. Narrative states, “The purpose of this warm-up is to familiarize students with tiny houses and encourage them to think about the area of a tiny house, which will be useful when students design a tiny house in a later activity. While students may notice and wonder many things about the images, the fact that the tiny houses are very small and cover small areas are the important discussion points.” Activity synthesis states, “‘These are all types of tiny houses, which are houses that are usually under 400 square feet. What might be some reasons for choosing to build or live in a tiny house?’ (It is less expensive to build. It works for those who live alone or don't want a lot of space. It is good for the environment because it takes up less space in nature.) ‘Tiny houses come in many forms or types. Some are small cabins, but some are built out of shipping containers and even school buses (as in the last image). What could be some advantages of each type? What could be some disadvantages of each type?’ (A shipping container would be strong, but it could rust and could be challenging to create windows and doors. A cabin would be kind of like building a regular house. You could have 2 stories and you could make it bigger, but some people might not like the look of a cabin. The school bus is mobile, so you wouldn’t have to stay in one spot, but there are a bunch of windows, so you would have to cover some.)”

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

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

The materials reviewed for Kendall Hunt's Illustrative Mathematics Grade 3 provide some supports for different reading levels to ensure accessibility for students.

• Unit 3, Wrapping Up Addition and Subtraction Within 1,000, Lesson 20, Activity 1, Activity states, “MLR4 Information Gap. Display the Sample Problem Card. Read the problem aloud. Listen for and clarify any questions about the context. ‘Some of the information you need to solve this problem is missing, and I have it here. What specific information do you need?’ 1–2 minutes: quiet think time. ‘With your partner, decide what information you need to solve the problem, and create a list of questions you can ask to find out.’ 2–3 minutes: partner discussion. Invite students to share 1 question at a time. Record each question on a display, and respond with: ‘Why do you need to know (restate the information requested)?’ Students should provide a justification for how they will use the information before the information is revealed. Answer questions using only information stated on the Sample Data Card (do not reveal). Record information that is shared on the display. Give students time to decide whether they have enough information to solve the problem. Repeat until students decide they have enough information to solve, then ask students to solve. 2–4 minutes: independent work time.”

• Unit 5, Fractions as Numbers, Lesson 18, Activity 1, 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, Social-Emotional Functioning.”

• Unit 6, Measuring Length, Time, Liquid Volume, and Weight, Lesson 13, Activity 1, 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 cards. 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)’?”

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

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 3 meet expectations for providing manipulatives, physical but not virtual, that are accurate representations of the mathematical objects they represent and, when appropriate, are connected to written methods.

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

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

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

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

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

#### 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 3 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 3 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 3 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 3 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 3 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

## Report for {{ report.grade.shortname }}

### Overall Summary

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