## Imagine Learning Illustrative Mathematics K-5 Math

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### Overall Summary

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

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

### Focus & Coherence

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

##### Gateway 1
Meets Expectations

#### Criterion 1.1: Focus

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

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

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

The materials reviewed for Imagine Learning Illustrative Mathematics Grade 4 meet expectations for assessing grade-level content and if applicable, content from earlier grades. The materials for Grade 4 are divided into nine units, and each unit contains a written End-of-Unit Assessment. Additionally, the Unit 9 Assessment is an End-of-Course Assessment, and it includes problems from the entire grade level. Examples of End-of-Unit Assessments include:

• Unit 2, Fraction Equivalence and Comparison, End-of-Unit Assessment, Problem 4, “List three different fractions that are equivalent to \frac{4}{5}. Explain or show your reasoning.” (4.NF.1)

• Unit 3, Extending Operations to Fractions, End of Unit Assessment, Problem 7, “Find the value of each expression. 1. \frac{7}{100}+\frac{8}{100}+\frac{1}{10} 2. \frac{3}{10}+\frac{17}{10}+\frac{2}{10} 3. \frac{14}{100}+\frac{5}{10}+\frac{26}{100}.” (4.NF.5)

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

• Unit 7, Angles and Angle Measurement, End-of-Unit Assessment, Problem 2, “Select all correct statements. A.There are 360 one-degree angles in a circle. B.There are 180 one-degree angles in a circle. C.There are 90 one-degree angles in a right angle. D.There are 180 one-degree angles in a right angle. E.There are 4 right angles in a circle.” (4.MD.5, 4.MD.7)

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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 Imagine Learning Illustrative Mathematics Grade 4 meet expectations for giving all students extensive work with grade-level problems to meet the full intent of grade-level standards. The materials provide extensive work with and opportunities for students to engage in the full intent of Grade 4 standards by including in every lesson a Warm Up, one to three instructional activities, and Lesson Synthesis. Within Grade 4, students engage with all CCSS standards.

Examples of extensive work include:

• Unit 2, Fraction Equivalence and Comparison, Lesson 12, Ways to Compare Fractions, engages students with extensive work with 4.NF.2 (Extend understanding of fraction equivalence and ordering). Activity 1: The Greatest of Them All, students compare two fractions that share the same denominator or the same numerator. Student Task Statements, “Here are 25 fractions in a table. For each question, be prepared to explain your reasoning. 1. Identify the greatest fraction in each column (A, B, C, D, and E). 2. Identify the greatest fraction in each row (1, 2, 3, 4, and 5). 3. Which fraction is the greatest fraction in the entire table?’ The table includes the following fractions: \frac{2}{3}, \frac{2}{5}, \frac{2}{10}, \frac{2}{12}, \frac{2}{100}, \frac{4}{3}, \frac{4}{5}, \frac{4}{10}, \frac{4}{12}, \frac{4}{100}, \frac{7}{3}, \frac{7}{5}, \frac{7}{10}, \frac{7}{12}, \frac{7}{100}, \frac{11}{3}, \frac{11}{5}, \frac{11}{10}, \frac{11}{12}, \frac{11}{100}, \frac{26}{3}, \frac{26}{5}, \frac{26}{10}, \frac{26}{12}, \frac{26}{100}.” Cool-down: Pick the Greater Fraction, Student Task Statements, students use strategies to compare and order common fractions. “In each pair of fractions, which fraction is greater? Explain or show your reasoning. \frac{5}{12} and \frac{5}{8}, \frac{11}{10} and \frac{18}{100}, \frac{6}{10} and \frac{7}{12}.”

• Unit 5, Multiplicative Comparison and Measurement, Lessons 1, 2, and 3 engage students with extensive work with grade-level problems from 4.OA.2 (Multiply or divide to solve word problems involving multiplicative comparison, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem, distinguishing multiplicative comparison from additive comparison). Lesson 1, Times as Many, Activity 2, Times As Many, students extend the idea of representing “twice as many” to represent “4, 6, and 8 times as many. “Draw a picture to show the number of cubes the students have in each situation. Be prepared to explain your reasoning.’ Problem 1, ‘Andre has the following cubes and Han has 4 times as many.’ (image of 5 linking cubes with the name “Andre” next to it, and the name “Han” with empty space next to it.)” Lesson 2, Interpret Representations of Multiplicative Comparison, Warm-Up: How Many Do You See: Times As Many, students use grouping strategies to describe the images they see. “‘How many do you see and how do you see them?’ Flash the image. (Images- 3 connected blue rectangles horizontal; 6 connected rectangles horizontal, 3 blue, 3 orange; 12 connected rectangles horizontal, 3 blue, 3 orange, 3 blue, 3 orange).” Lesson 3, Solve Multiplicative Comparison Problems, Activity 1: A Book Drive, students rely on the relationship between multiplication and division to solve multiplicative comparison problems. “‘This diagram shows the books Lin and Diego donated for the school book drive (diagram shows a discrete tape diagram to represent Lin’s books (16) and Diego’s books (4)).’ Problem 1, ‘Lin donated 16 books. Diego donated 4 books. How many times as many books did Lin donate as Diego did? Explain or show your reasoning. Use the diagram if it is helpful.’”

• Unit 6, Multiplying and Dividing Multi-digit Numbers, Lesson 20, Interpret Remainders in Division Situations, engages students with extensive work with grade-level problems with 4.OA.3 (Use the four operations with whole numbers to solve problems). Activity 2: Save for a Garden, students solve word problems with remainders. Student Task Statements, “1. A school needs $1,270 to build a garden. After saving the same amount each month for 8 months, the school is still short by$6. How much did they save each month? Explain or show your reasoning. 2. Choose one of the following division expressions. 711\div3, 3128\div8. a. Write a situation to represent the expression. b. Find the value of the quotient. Show your reasoning. c. What does the value of the quotient represent in your situation?”

Examples of full intent include:

• Unit 3, Extending Operations to Fractions, Lessons 8, Addition of Fractions, and Lesson 9, Differences of Fractions, engage students with the full intent of 4.NF.3a (Understand addition and subtraction of fractions as joining and separating parts referring to the same whole). In Lesson 8, Addition of Fractions, Activity 2, What is the Sum?, Student Task Statements, students use number lines to represent addition of two fractions and to find the value of the sum. “Use a number line to represent each addition expression and to find its value. a. \frac{5}{8}+\frac{2}{8} b. \frac{1}{8}+\frac{9}{8} c. \frac{11}{8}+\frac{9}{8} d. 2\frac{1}{8}+\frac{4}{8}.” In Lesson 9, Differences of Fractions, Cool-down: Differences of Fifths, students use number lines to represent subtraction of a fraction by another fraction with the same denominator including a mixed number and by a whole number. “Use a number line to represent each difference and to find its value. 1. \frac{12}{5}-\frac{4}{5}. 2.  2\frac{1}{5}-\frac{7}{5}.”

• Unit 4, From Hundredths to Hundred-thousands, Lessons 16, Round Numbers, and Lesson 17, Apply Rounding,  provides the opportunity for students to engage with the full intent of standard 4.NBT.3 (Use place value understanding to round multi-digit whole numbers to any place). Lesson 16, Round Numbers, Activity 1: Round to What, students connect the idea of “nearest multiple” to rounding. “Noah says that 489,231 can be rounded to 500,000. Priya says that it can be rounded to 490,000. 1. Explain or show why both Noah and Priya are correct. Use a number line if it helps. 2. Describe all the numbers that round to 500,000 when rounded to the nearest hundred-thousand. 3. Describe all the numbers that round to 490,000 when rounded to the nearest ten-thousand. 4. Name two other numbers that can also be rounded to both 500,000 and 490,000.” Lesson 17, Apply Rounding, Activity 1: Apart in the Air, students make sense of a situation and decide how to round the quantities. Problem 2, “Planes flying over the same area need to stay at least 1,000 feet apart in altitude. Mai said that one way to tell if planes are too close is to round each plane's altitude to the nearest thousand. Do you agree that this is a reliable strategy? In the last column, round each altitude to the nearest thousand. Use the rounded values to explain why or why not. A 2 column table included with labels “plane” in column 1 and “altitude (feet)” in column 2. The following altitudes are listed: WN11-35,625; SK51-28,999; VT35-15,450; BQ 64-36,000; AL16-31,000; AB25-35,175; CL48-16,600; WN90-30,775; NM44-30,245”

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

#### Criterion 1.2: Coherence

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

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

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

The materials reviewed for Imagine Learning Illustrative Mathematics Grade 4 meet expectations that, when implemented as designed, the majority of the materials address the major clusters of the 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 the major work of the grade (including assessments and supporting work connected to major work) is 6 out of 9, approximately 67%.

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

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

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

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Supporting content enhances focus and coherence simultaneously by engaging students in the major work of the grade.

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

Materials are designed with supporting standards/clusters 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 on the Course Guide tab for each Unit. Teacher Notes also provide the explicit standards listed within the lessons. Examples of connections include:

• Unit 3, Extending Operations to Fractions, Lesson 14, Problems About Fractional Measurement Data, Activity 2, Larger Shoes, Anyone? connects the supporting work of 4.MD.4 (Make a line plot to display a data set of measurements in fractions of a unit ($$\frac{1}{2}, \frac{1}{4}, \frac{1}{8}$$). Solve problems involving addition and subtraction of fractions by using information presented in line plots) with the major work of 4.NF.3 (Understand a fraction \frac{1}{b} with a>1 as a sum of fractions \frac{1}{b}). Student Task Statement Problem 2, “If Han’s shoe length now is 9\frac{1}{8} inches, what was his shoe length in third grade?”

• Unit 5, Multiplicative Comparison and Measurement, Lesson 10, Multi-step Measurement Problems, Cool-down: Hydration Here and There, connects the supporting work of 4.MD.2, (Use the four operations to solve word problems involving distances, intervals of time, liquid volumes, masses of objects, and money, including problems involving simple fractions or decimals, adn problems that require expression measurements given in a larger unit in terms of a smaller unit), to the major work of 4.OA.3, (Solve multistep word problems posed with whole numbers and having whole-number answers using the four operations, including problems in which remainders must be interpreted). Students use conversions to solve multi-step problems. Student Task Statements, “Halfway through a soccer game, Han drank 210 mL of water. At the end of the game, he drank 4 times as much as he did at halftime. Did Han drink more or less than 1 L of water in total? Explain or show your reasoning.”

• Unit 6, Multiplying and Dividing Multi-digit Numbers, Lesson 12, Solve Problems Involving Multiplication, Cool-down, Leap Year, connects the supporting work of 4.MD.2, (Use the four operations to solve word problems involving distances, intervals of time, liquid volumes, masses of objects, and money, including problems involving simple fractions or decimals, and problems that require expression measurements given in a larger unit in terms of a smaller unit), to the major work of 4.OA.3, (Solve multistep word problems posed with whole numbers and having whole-number answers using the four operations, including problems in which remainders must be interpreted). Students use multiplication strategies to solve problems. Student Task Statements, “In a leap year, the month of February has 29 days. How many hours are in that month? Show your reasoning.”

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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 reviewed for Imagine Learning Illustrative Mathematics Grade 4 meet expectations for including problems and activities that serve to connect two or more clusters in a domain or two or more domains in a grade.

Materials are coherent and consistent with the Standards. Examples of connections between major work to major work and/or supporting work to supporting work throughout the materials, when appropriate, include:

• Unit 3, Extending Operations to Fractions, Lesson 15, An Assortment of Fractions, Cool-down, connects the major work of 4.NF.A (Extend Understanding Of Fraction Equivalence And Ordering) to the major work of 4.NF.B (Build Fractions From Unit Fractions By Applying And Extending Previous Understandings Of Operations On Whole Numbers). Students add two sets of fractions and decide which set is greatest: “Which stack of foam blocks is taller: Two \frac{1}{3}-foot blocks and one \frac{1}{6}-foot block, or One \frac{1}{2}-foot block and two \frac{1}{6}-foot blocks?”

• Unit 6, Multiplying and Dividing Multi-Digit Numbers, Lesson 24, Assess the Reasonableness of Solutions connects the major work of 4.OA.A (Use the four operations with whole numbers to solve problems) to the major work of 4.NBT.B (Use place value understanding and properties of operations to perform multi-digit arithmetic). In Activity 2, Languages in Philadelphia and Chicago, students use place value understanding with the four operations to solve problems. Problem 2, “What is the difference between the number of people who speak only English and those who speak another language? Show how you know (table has two columns; first column- language and second column- number of people in Philadelphia: English only-1,224,539; Spanish-127,352; Other Indo-European-6,750; Asian-364).”

• Unit 7, Angles and Angle Measurements, Lesson 5, What is an Angle?, Activity 3: Discover Angles connects the supporting work of 4.G.A (Draw And Identify Lines And Angles, And Classify Shapes By Properties Of Their Lines And Angles) to the supporting work of 4.MD.C (Geometric Measurement: Understand Concepts Of Angle And Measure Angles) as students identify and sketch angles. Student Task Statements, “Here are two figures. 1. Find 2–3 angles in each figure. Draw pairs of rays to show the angles. 2. Sketch a part of your classroom that has 2–3 angles. Draw pairs of rays to show the angles.” An image shows the number 7 and letter K.

• Unit 9, Putting It All Together, Lesson 12, Number Talk, Activity 1: Related Numbers, Related Expressions connects the major work 4.NBT.A (Generalize place value understanding for multi-digit whole numbers) to the major work of 4.NBT.B (Use place value understanding and properties of operations to perform multi-digit arithmetic). Students decompose, rearrange, and regroup numbers or make use of structure to find the value of expressions. Student Task Statements, “1. Here are two addition expressions. Think of at least two different ways to find the value of each sum mentally. a. 15+29 b. 30+58. 2. Here are three subtraction expressions. Think of at least two different ways to find the value of each difference mentally. a. 91-11. b 91-16.  c. 391-86. 3. Can you write a fourth subtraction expression that uses the same strategy you used to find the value of the other differences?”

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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 Imagine Learning Illustrative Mathematics Grade 4 meet expectations that content from future grades is identified and related to grade-level work and materials relate grade-level concepts explicitly to prior knowledge from earlier grades.

The Section Dependency Chart explores the Unit sections relating to future grades. The Section Dependency Chart states, “arrow indicates the prior section that contains content most directly designed to support or build toward the content in the current section.”

Examples of connections to future grades include:

• Unit 5, Multiplicative Comparison and Measurement, Lesson 11, Pounds and Ounces, Activity 2, Party Prep, connects 4.MD.2 (Use the four operations to solve word problems involving distances, intervals of time, liquid volumes, masses of objects, and money, including problems involving simple fractions or decimals, and problems that require expressing measurements given in a larger unit in terms of a smaller unit…) and 4.OA.3 (Solve multistep word problems posed with whole numbers and having whole-number answers using the four operations, including problems in which remainders must be interpreted. 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) to the work in grade 5. “In the first activity, students learned that one pound is 16 times as heavy as 1 ounce. Here they apply this knowledge to convert quantities into ounces and to solve multi-step problems. The quantities include a fractional number of pounds and one expressed in a combination of pounds and ounces. As in earlier lessons in which they encountered a fractional amount of a unit of measurement, students are not expected to find the number of ounces in \frac{1}{2} pound by writing \frac{1}{2}\times16. Instead, they can reason about half of a quantity using their understanding of fractions and by dividing an amount by 2. Those who do write \frac{1}{2}\times16 represent the situation correctly, but this reasoning and the related operation will be developed in grade 5.”

• Unit 8, Properties of Two-dimensional Shapes, Lesson 3, Lesson Preparation connects 4.G.1 (Draw points, lines, line segments, rays, angles (right, acute, obtuse), and perpendicular and parallel lines. Identify these in two-dimensional figures), 4.G.2 ( Classify two-dimensional figures based on the presence or absence of parallel or perpendicular lines, or the presence or absence of angles of a specified size. Recognize right triangles as a category, and identify right triangles) to work in grade 5. About this lesson, “In this lesson, students identify and sort quadrilaterals based on their angles and sides, including whether their sides are parallel. Students are introduced to the term parallelogram to describe quadrilaterals with two pairs of parallel sides, but they are not expected to use this term throughout the unit. In grade 5, students will continue the work of classifying polygons using these categories.”

• Unit 9, Putting It All Together, Lesson 2, Lesson Preparation connects 4.NF.A (Extend Understanding Of Fraction Equivalence And Ordering) to work in grade 5. About this lesson, “In this lesson, students apply what they know about equivalence and addition and subtraction of fractions to solve problems. Throughout the lesson, students have opportunities to reason quantitatively and abstractly as they connect their representations, including equations, to the situations (MP2) and to compare their reasoning with others' (MP3). The work of this lesson helps prepare students for adding and subtracting with unlike denominators in grade 5.”

Examples of connections to prior knowledge include:

• Unit 1, Factors and Multiples, Lesson 1, Multiples of a Number, Activity 1: Build Rectangles and Find Area connects 4.OA.4 (Find all factor pairs for a whole number in the range 1–100. Recognize that a whole number is a multiple of each of its factors…) to the work in grade 3, “In grade 3, students learned how to find the area of a rectangle by tiling and found that multiplying the side lengths yields the same result (3.MD.7).”

• Unit 3, Fraction Operations to Fractions, Lesson 1, About this lesson, connects 4.NF.4 (Apply and extend previous understandings of multiplication to multiply a fraction by a whole number.) with work done in grade 3. “In grade 3, students represented multiplication of whole numbers with arrays, equal-group drawings, area diagrams, and expressions. In an earlier unit, students used diagrams to represent and compare fractions. In this unit, they extend their understanding of multiplication to include equal groups of unit fractions while using familiar representations to support their thinking.”

• Unit 4, From Hundredths to Hundred-thousands, Lesson 16, Round Numbers, About this lesson, connects 4.NBT.3 (Use place value understanding to round multi-digit whole numbers to any place.) to work done in grade 3. “In grade 3, students rounded whole numbers to the nearest 10 and 100. In previous lessons, they worked to find the closest multiples of powers of 10. Here, students build on this work to round whole numbers to the nearest 1,000, 10,000, and 100,000. Students revisit the convention of rounding up when a number is exactly halfway between two consecutive multiples of a power of 10.”

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

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

• 149 lesson days

• 18 days of assessments

There are 9 units with each unit containing 8 to 25 lessons which contain a mixture of four components: Warm-Up (approx. 10 minutes), Activities (20-45 minutes), Lesson Synthesis (no time specified), and Cool Down (no time specified). In the Curriculum Guide, Quick Facts, teachers are instructed “that each lesson plan is designed to fit within a class period that is at least 60 minutes long.”  Also, “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.” Since no minutes are allotted for the last two components (Lesson Synthesis and Cool Down), this can impact the total number of minutes per lesson.

### Rigor & the Mathematical Practices

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

##### Gateway 2
Meets Expectations

#### Criterion 2.1: Rigor and Balance

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

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

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

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

Materials develop conceptual understanding throughout the grade level. According to IM Curriculum, Design Principles, Purposeful Representations, “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.” Each lesson begins with a Warm-up, designed to highlight key learning aligned to the objective and to support the development of conceptual understanding through student discourse and reflection. Examples include:

• Unit 3, Fraction Operations to Fractions, Lesson 16, Activity 2, students develop conceptual understanding as they represent equations with tenths and hundredths using number lines. Activity, “Take a few quiet minutes to work on the first two problems. Then, share your responses with your partner. 5 minutes: independent work time. 2 minutes: partner discussion. Monitor for the ways students think about the total distance Noah has walked (third problem) given a fraction in tenths and one in hundredths. Now try finding the values of the sums in the last problem. 5 minutes: independent or partner work time.” Student Task Statements, “Noah walks \frac{2}{10} kilometer (km), stops for a drink of water, walks \frac{5}{100} kilometer, and stops for another sip. 1. Which number line diagram represents the distance Noah has walked? Explain how you know. 2. The diagram that you didn’t choose represents Jada’s walk. Write an equation to represent: a. the total distance Jada has walked b. the total distance Noah has walked 3. Find the value of each of the following sums. Show your reasoning. Use number lines if you find them helpful., a. \frac{5}{10}+\frac{1}{10}, b. \frac{50}{100}+\frac{10}{100}, c. \frac{5}{10}+\frac{30}{100}, d. \frac{15}{100}+\frac{4}{10}.” The number line diagrams shown in problem 1 are from 0 to 1 km, and divided into tenths. The first number line shows a jump from 0 to 2 tenths, then 5 more tenths ending on 7 tenths. The second number line shows a jump from 0 to 2 tenths, then a half tenth jump ending on 25 hundredths. The number lines in problem 3 are from 0 to 1 and are broken into tenths but are otherwise blank. Activity synthesis, “Invite students to share how they know which diagram represents Noah’s walk and their equations for the distances Noah and Jada walked. Given the number line diagram for support, students are likely to write \frac{2}{10}+\frac{50}{100}=\frac{25}{100}. Discuss why this is true. How do you know that the sum of \frac{2}{10} and \frac{5}{100} is \frac{25}{100}? Highlight that \frac{2}{10} is equivalent to \frac{20}{100} and another \frac{5}{100} makes \frac{25}{100}. Consider displaying a number line that is partitioned into tenths and hundredths and shows \frac{25}{100} as \frac{20}{100}+\frac{5}{100}.” (4.NF.5)

• Unit 4, From Hundredths to Hundred-thousands, Lesson 9, Warm-up, students develop conceptual understanding as they use commutative and associative properties of addition to compose numbers and determine equivalent sums. Launch, “Display one statement. Give me a signal when you know whether the statement is true and can explain how you know. 1 minute: quiet, think time.” Student Task Statements, “Decide if each statement is true or false. Be prepared to explain your reasoning. 4,000+600+70,000=70,460. 900,000+20,000+3,000=920,000+3,000. 80,000+800+8,000=800,000+80+8.” Activity Synthesis, “Focus question: How can you explain your answer without finding the value of both sides? We can write numbers in different forms. What form is used to represent the numbers in this True or False? (expanded form).” (4.NBT.2)

• Unit 6, Multiplying and Dividing Multi-digit Numbers, Lesson 7, Activity 1, students develop conceptual understanding as they use rectangular diagrams to represent multiplication of three-digit and one-digit numbers. Activity, “Work with your partner on the first problem. Then, try the rest of the activity independently. 2 minutes: group work time on the first problem. 5–7 minutes: independent work time on the rest of the activity. 3 minutes: partner discussion. Monitor for students who: write expressions that show the multi-digit factor decomposed by place value, draw diagrams that partition the multi-digit factor by place value.” Student Task Statement, “1. Clare drew this diagram. a. What multiplication expression can be represented by the diagram? b. Find the value of the expression. Show your reasoning.” A rectangle that is partitioned is shown. Activity synthesis, “Invite several students to share diagrams they drew to represent 6\times252. As each student shares, consider asking the class: Where do you see the parts of the factor that was decomposed? What expressions are represented in the diagram? Record expressions as students share.” (4.NBT.5)

Materials provide opportunities for students to independently demonstrate conceptual understanding throughout the grade level. Design Principles, Coherent Progress, “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.” The Cool-down part of the lesson includes independent work.  Curriculum Guide, How Do You Use the Materials, A Typical Lesson, Four Phases of a Lesson, Cool-down, “the cool-down task is to be given to students at the end of a lesson.  Students are meant to work on the Cool-down for about 5 minutes independently and turn it in.” Independent work could include practice problems, problem sets, and time to work alone within groups. Examples include:

• Unit 2, Fraction Equivalence and Comparison, Lesson 7, Cool-down, students demonstrate conceptual understanding as they generate and explain equivalent fractions. Student Task Statement, “Name two fractions that are equivalent to \frac{5}{3}. Explain or show your reasoning.” (4.NF.1)

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

• Unit 6, Multiplying and Dividing Multi-digit Numbers, Lesson 15, Situations Involving Area, Cool Down, Sticky Notes on the Door, students demonstrate conceptual understanding when they find whole number quotients by determining the number of rows of sticky notes needed to cover a door. “Jada’s class is decorating their door with square sticky notes for their teacher. Each sticky note has a drawing or a message from a student. The class used 234 square sticky notes to cover their classroom door completely, leaving no gaps or overlaps between the notes. It takes 9 square notes to cover the width of the door. How many square notes does it take to cover the full height of the door? Show how you know.” (4.NBT.6)

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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 Imagine Learning Illustrative Mathematics Grade 4 meet expectations for giving attention throughout the year to individual standards that set an expectation of procedural skill and fluency.

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

• Unit 4, From Hundredths to Hundred-thousands, Lesson 21, Activity 2, students use subtraction to calculate characters' ages. Student Task Statements, “Jada recorded the birth year of each of her maternal grandparents for a family history project. As of this year, what is the age of each family member? Show your reasoning. Use the standard algorithm at least once.” The problem includes a table with different characters’ ages listed in two columns with the headings “family member” and “birth year.” (4.NBT.4)

• Unit 7, Angles and Angle Measurement, Lesson 2, Warm-up, students mentally subtract numbers. Student Task Statements, “Find the value of each expression mentally. 90-45, 270-45, 270-135, 360-135.” (4.NBT.4)

• Unit 9, Putting It All Together, Lesson 9, Activity 2, students create problems with a given answer using established constraints. Student Task Statements, “Elena, Noah, and Han each created a problem with an answer of 1,564. Elena used multiplication. Noah used multi-digit numbers and addition only. Han used multiplication and subtraction. Write a problem that each student could have written. Show that the answer to the question is 1,564.” (4.NBT.4, 4.NBT.5)

The instructional materials provide opportunities for students to independently demonstrate procedural skills and fluency throughout the grade level. Activities can be completed during a lesson. Cool-downs or end of lesson checks for understanding are designed for independent completion. Examples include:

• Unit 4, From Hundredths to Hundred-thousands, Lesson 18, Centers, Number Puzzles, Standard Algorithm to Add and Subtract, 19, Compose and Decompose to Add and Subtract, 20, Add and Subtract Within 1,000,000. “Students use the digits 0–9 to make each addition or subtraction equation true. Stage 6: Beyond 1,000, Stage Narrative Students use the digits 0–9 to make addition equations true. They work with sums and differences beyond 1,000.” (4.NBT.4)

• Unit 4, From Hundredths to Hundred-thousands, Lesson 20, Cool-down, students use the standard algorithm to solve a subtraction problem. Student Task Statements, “Use the standard algorithm to find the value of the difference. 173,225-114,329.” (4.NBT.4)

• Unit 9, Putting It All Together, Lesson 4, Cool-down, students use the standard algorithm for subtraction. Student Task Statements, “Find the value of each difference. Show your reasoning. 1. 8,050-213. 2. 60,000-1,984.” (4.NBT.4)

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

The materials reviewed for Imagine Learning Illustrative Mathematics Grade 4 meet expectations for being designed so that teachers and students spend sufficient time working with engaging applications of the mathematics.

Students have the opportunity to engage with applications of math both with support from the teacher, and independently. According to the K-5 Curriculum Guide, a typical lesson has four phases including Warm-up and one or more instructional Activities which include engaging single and multi-step application problems. Lesson Synthesis and Cool-downs provide opportunities for students to demonstrate multiple routine and non-routine applications of the mathematics throughout the grade level. Cool-downs or end of lesson checks for understanding are designed for independent completion.

Examples of routine applications include:

• Unit 3, Extending Operations to Fractions, Lesson 6, Cool-down, students multiply whole numbers by a fraction. (4.NF.4) Student Task Statements, Problem 2, “Han bought 4 cartons of chocolate milk. Each carton contains \frac{5}{8} liter. Did Han buy the same amount of milk as Tyler? Explain or show your reasoning.”

• Unit 3, Extending Operations to Fractions, Lesson 11, Activity 1, students solve a real-world problem involving subtraction of fractions referring to the same whole (4.NF.3d). Student Task Statements, “Clare, Elena, and Andre are making macramé friendship bracelets. They’d like their bracelets to be 9\frac{4}{8} inches long. For each question, explain or show your reasoning. 2. So far, Elena’s bracelet is 5\frac{1}{8} inches long and Andre’s is 3\frac{5}{8} inches long. How many more inches do they each need to reach 9\frac{4}{8} inches?”

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

Examples of non-routine applications include:

• Unit 6, Multiplying and Dividing Multi-digit Numbers, Lesson 22, Cool-down, students solve a measurement problem using multiplication (4.MD.2, 4.MD.3, 4.NBT.5). Student Task Statements, “Han has a rectangular piece of paper that is 96 inches by 36 inches. He is using it to create a banner for Awards Day. Last year the banner measured 2,304 square inches. 1. Will the new banner fit in the same area that the old banner was? Show your reasoning. 2. What is the difference in square inches between the area of last year's banner and this year's banner?”

• Unit 7, Angles and Angle Measurement, Lesson 16, Activity 2, students use their understanding of geometric figures and measurements to draw, describe, and identify two-dimensional figures to solve a real world problem (4.G.1 and 4.G.2). Activity, “5 minutes: independent work time.” Student Task Statements, “1. Create a two-dimensional shape that has at least 3 of the following: ray, line segment, right angle, acute angle, obtuse angle, perpendicular lines, parallel lines.”

• Unit 9, Putting It All Together, Lesson 10, Activity 2, students use estimation to solve a real-world problem (4.OA.3). Student Task Statements, “It’s your turn to create an estimation problem. 1. Think of situations or look around for images that would make interesting estimation problems. Write down 4–5 ideas or possible topics. 2. Choose your favorite idea. Then, write an estimation question that would encourage others to use multiplication of multi-digit numbers to answer. Record an estimate that is ___.”

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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 Imagine Learning Illustrative Mathematics Grade 4 meet expectations in that the three aspects of rigor are not always treated together and are not always treated separately. There is a balance of the three aspects of rigor within the grade.

In the K-5 Curriculum Guide, Why is the curriculum designed this way?, Design Principles, Balancing Rigor, “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. Warm-ups, practice problems, centers, and other built-in routines help students develop procedural fluency, which develops over time.”

All three aspects of rigor are present independently throughout Grade 4. Examples where instructional materials attend to conceptual understanding, procedural skill and fluency, or application include:

• Unit 1, Factors and Multiples, Lesson 2, Cool-down, students apply their understanding of factor pairs to find possible sides of a rectangle. Activity, “What are all of the possible side lengths of a rectangle with an area of 21 square units? What are all of the possible side lengths of a rectangle with an area of 50 square units?” (4.OA.4)

• Unit 2, Fraction Equivalence and Comparison, Lesson 2, Activity 2, students extend their conceptual understanding as they create representations of fractions. Launch, “Give each student a straightedge and access to their fraction strips from a previous lesson. How can you show \frac{3}{4} with fraction strips? (Find the strip showing fourths, highlight 3 parts of fourths.) How can you show \frac{8}{4}? If students say that they don’t have enough strips to show 8 fourths, ask them to combine their strips with another group’s. Invite groups to share their representations of \frac{8}{4}. Students may use different fractional parts (fourths and halves, or fourths and eighths).” Student Task Statements, Problem 2, “Here are four fractions and four blank diagrams. Partition each diagram and shade the parts to represent the fraction. a. \frac{2}{2}, b. \frac{4}{2}, c. \frac{5}{4}, d. \frac{10}{8}.” (4.NF.1)

• Unit 4, From Hundredths to Hundred-thousands, Lesson 20, Activity 1, students develop procedural skill and fluency as they use the standard addition algorithm. Student Task Statements, “1. Use the standard algorithm to find the value of each sum and difference. If you get stuck, try writing the numbers in expanded form. a. 7,106+2,835, b. 8,179+3,599, c. 142,571+10,909, d. 268,322+72,145.” (4.NBT.4)

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

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

• Unit 5, Multiplicative Comparison and Measurement, Lesson 4, Cool-down, students develop conceptual understanding alongside application as they solve for an unknown factor using multiplicative comparisons. Student Task Statements, “Priya read some pages on Monday. Jada read 63 pages, which is 7 times as many pages as Priya read. 1. Write an equation to show the comparison. Use a symbol for the unknown. 2. How many pages did Priya read?” (4.OA.2)

• Unit 9, Putting It All Together, Lesson 9, Cool-down, students use procedural fluency and apply their understanding as they write their own mathematical questions and solve them. Student Task Statements, “The school band wants to raise $1,700 for a music festival. They have raised$175 each week for the past 6 weeks. Write a question that could be asked about this situation and answer it. Show your reasoning.” (4.NBT.4, 4.NBT.5, 4.NBT.6, 4.OA.3)

#### Criterion 2.2: Math Practices

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

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

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

The materials reviewed for Imagine Learning Illustrative Mathematics Grade 4 meet expectations for supporting the intentional development of MP1: Make sense of problems and persevere in solving them; and MP2: Reason abstractly and quantitatively, for students, in connection to the grade-level content standards, as expected by the mathematical practice standards. Students have opportunities to engage with the Math Practices across the year, and they are often explicitly identified for teachers in several places including the Instructional Routines (Warm-up Routines and Other Instructional Routines), Activity Narratives, and About this lesson.

MP1 is identified and connected to grade-level content, and there is intentional development of MP1 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 3, Extending Operations to Fractions, Lesson 18, Activity 1: Stack Centavos and Pesos, students use a variety of strategies to solve problems involving tenths and hundredths. Student Task Statements, “Diego and Lin each have a small collection of Mexican coins. The table shows the thickness of different coins in centimeters (cm) and how many of each Diego and Lin have. 1. If Diego and Lin each stack their centavo coins, whose stack would be taller? Show your reasoning. 2. If they each stack their peso coins, whose stack would be taller? Show your reasoning. 3. If they each stack all their coins, whose stack would be taller? Show your reasoning. 4. If they combine their coins to make a single stack, would it be more than 2 centimeters tall? Show your reasoning.” Activity Narrative, “Though the mathematics here is not new, the context and given information may be novel to students. Students have a wide variety of approaches available for these problems with no solution approach suggested (MP1).”

• Unit 4, From Hundredths to Hundred-thousands, Lesson 6, Activity 1: Build Numbers, students represent multi-digits in different ways. Student Task Statements, “1. Use two cards to make a two-digit number. Name it and build the number with base-ten blocks. 2. Use a third card to make a three-digit number. Name it and build it with base-ten blocks. 3. Use a fourth card to make a four-digit number. Name it and build it. If you don’t have enough blocks, describe what you would need to build the number. 4. Your teacher will give you one more digit card. Use the last card from your teacher to make a five-digit number. Make the card the first digit. Name it and build it. If you don’t have enough blocks, describe what blocks you would need to build the number.” Activity Narrative, “As students build numbers to the ten-thousands place, they may struggle to name the number. As they make sense of the value of the number, they should realize a need for more base-ten blocks, but should be given space to represent the number in a way that makes sense to them. It is not critical to name the number correctly or accurately describe how to build it. The idea is to create a bit of struggle to motivate another way to make sense of the number (MP1).”

• Unit 5, Multiplicative Comparison and Measurement, Lesson 13, Activities 1 and 2, students make sense of and persevere in solving problems using the info gap instructional routine. Activity 1 narrative, “The purpose of this activity is to introduce students to the structure of the MLR4 Information Gap routine. This routine facilitates meaningful interactions by positioning some students as holders of information that is needed by other students. Tell students that first, a demonstration will be conducted with the whole class, in which they are playing the role of the person with the problem card. Explain to students that it is the job of the person with the problem card (in this case, the whole class) to think about what information they need to answer the question. For each question that is asked, students are expected to explain what they will do with the information, by responding to the question, ‘Why do you need to know (that piece of information)?’ If the problem card person asks for information that is not on the data card (including the answer!), then the data card person must respond with, ‘I don’t have that information.’ Once the students have enough information to solve the problem, they solve the problem independently. The info gap routine requires students to make sense of problems by determining what information is necessary and then ask for information they need to solve them. This may take several rounds of discussion if their first requests do not yield the information they need (MP1). Data Card, On each school day, Noah spends \frac{1}{4} hour getting to school., Noah’s homework and reading time is 5 times as long as he spends getting to school., Noah spends \frac1}{2} hour on his bedtime routine.”  Student Task Statement, “Problem Card, On a school day, Noah usually spends 40 minutes on his morning routine and 75 minutes on his sports practice. Which takes more time: 1. Noahs’ morning routine or his bedtime routine? 2. Noah’s sports practice or his homework and reading time?”

MP2 is identified and connected to grade-level content, and there is intentional development of MP2 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 6, Multiplying and Dividing Multi-digit Numbers, Lesson 5, Activity 2, students consider units and strategies in the context of a real-world problem. The Activity Narrative states, “In this activity, students use strategies and representations that make sense to them to find products beyond 100. As before, the context of stickers lends itself to be represented with an array. The factors are large enough, however, that doing so would be inconvenient, motivating other representations or strategies (MP2). Look for the ways that students extend or generalize previously learned ideas or representations to find multiples of larger two-digit numbers. While many of the student responses are written with expressions, students are not expected to represent their reasoning using equations and expressions at this time. Teachers may choose to represent student reasoning using equations and expressions so students can start connecting representations.” The Student Task Statement, “1. Elena has another sheet of stickers that has 9 rows and 21 stickers in each row. How many stickers does Elena have? Explain or show your reasoning. 2. Noah’s sticker sheet has 3 rows with 48 stickers in each row. Andre’s sticker sheet has 7 rows with 23 stickers in each row. Who has more stickers? Explain or show your reasoning.”

• Unit 9, Putting It All Together, Lesson 8, About this lesson, “In the previous lesson, students solved word problems involving multiplicative comparison. In this lesson, they practice solving a wider variety of problems, with a focus on the relationships among multiple quantities in a situation. Students think about how to represent the relationships with one or multiple equations and using multiple operations. They also interpret their solutions and the solutions of others in context, including interpreting remainders in situations that involve division (MP2).”  Cool-down, “In one week, a train made 8 round trips between its home station and Union Station. At the end of the week, it traveled a few more miles from the home station to a repair center. That week, the train traveled a total of 1,564 miles. 1. Which statement is true for this situation? Explain or show your reasoning. a. The distance traveled for each round trip is 200 miles. The distance to the repair station is 26 miles. b. The distance traveled for each round trip is 195 miles. The distance to the repair station is 4 miles. c. The distance traveled for each round trip is 8 miles. The distance to the repair station is 1,500 miles. d. The distance traveled for each round trip is 193 miles. The distance to the repair station is 8 miles.”

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

Students have opportunities to meet the full intent of MP3 over the course of the year. The Mathematical Practices are explicitly identified for teachers in several places in the materials including Instructional routines, Activity Narratives, and the About this Lesson section. Students engage with MP3 in connection to grade level content as they work with support of the teacher and independently throughout the units.

Examples of constructing viable arguments include:

• Unit 1, Factors and Multiples, Lesson 5, Activity 2, students construct viable arguments as they find multiples. Student Task Statements, “Each package of hot dogs has 10 hot dogs. Each package of hot dog buns has 8 buns. 1. Lin expects to need 50 hot dogs for a class picnic. a. How many packages of hot dogs should Lin get? Explain or show your reasoning. b. Can Lin get exactly 50 hot dog buns? How many packages of hot dog buns should Lin get? Explain or show your reasoning. 2. Diego expects to need 72 hot dogs for a class picnic. a. How many packages of hot dogs should Diego get? Explain or show your reasoning. b. How many packages of hot dog buns should Diego get? Explain or show your reasoning. 3. Is it possible to buy exactly the same number of hot dogs and buns? If you think so, what would that number be? If not, explain your reasoning.” The Activity Narrative, “As multiple answers can be expected, the focus is on explaining why the solutions make sense (MP3).”

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

• Unit 5, Multiplicative Comparison and Measurement, Lesson 3, Activity 2, students construct a viable argument and critique the reasoning of others as they represent missing amounts in multiplication situations. Activity Narrative, “Students use the relationship between multiplication and division to write equations to represent multiplicative comparisons. These problems have larger numbers than in previous lessons in order to elicit the need for using more abstract diagrams, which are the focus of upcoming lessons. When students analyze Han's and Tyler's claims they construct viable arguments (MP3).” Student Task Statements, “1. Clare donated 48 books. Clare donated 6 times as many books as Andre. a. Draw a diagram to represent the situation. b. How many books did Andre donate? Explain your reasoning. 2. Han says he can figure out the number of books Andre donated using division. Tyler says we have to use multiplication because it says ‘times as many’. a. Do you agree with Han or Tyler? Explain your reasoning. b. Write an equation to represent Tyler’s thinking. c. Write an equation to represent Han’s thinking. 3. Elena donated 9 times as many books as Diego. Elena donated 81 books. Use multiplication or division to find the number of books Diego donated.”

Examples of critiquing the reasoning of others include:

• Unit 4, From Hundredths to Hundred-thousands, Lesson 2, Activity 2, students critique the reasoning of others as they determine if fractions are equivalent to decimals. The Activity Narrative states, “The last question refers to decimals on a number line and sets the stage for the next lesson where the primary representation is the number line. As students discuss and justify their decisions about the claim in the last question, they critically analyze student reasoning (MP3).” Student Task Statements, “1. Decide whether each statement is true or false. For each statement that is false, replace one of the numbers to make it true. (The numbers on the two sides of the equal sign should not be identical.) Be prepared to share your thinking. \frac{50}{100}=0.50, 0.05=0.5, 0.3=\frac{3}{10}, 0.3=\frac{30}{100}, 0.3=0.30, 1.1=1.10, 3.06=3.60, 2.70=0.27. 2. Jada says that if we locate the numbers 0.05, 0.5, and 0.50 on the number line, we would end up with only two points. Do you agree? Explain or show your reasoning.” Problem 2 is followed by a number line labeled 0 on the left and 1 on the right.

• Unit 8, Properties of Two-dimensional Shapes, Lesson 5, Activity 1, students critique the reasoning of others as they identify symmetry in a figure. Student Task Statements, “Each shaded triangle is half of a whole figure that has a line of symmetry shown by the dashed line. Clare drew in some segments to show the missing half of each figure. Do you agree that the dashed line is a line of symmetry for each figure Clare completed? Explain your reasoning. If you disagree with Clare's work, show a way to complete the drawing so the dashed line is a line of symmetry.” Activity Narrative, “This activity highlights that having two identical halves on each side of a line doesn’t necessarily make a figure symmetrical. It encourages students to use their understanding of symmetry and the line of symmetry to articulate why this is so as they critique supplied reasoning (MP3).”

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

The materials reviewed for Imagine Learning Illustrative Mathematics Grade 4 meet expectations for supporting the intentional development of MP4: Model with mathematics; and MP5: Use appropriate 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 in several places including the Instructional Routines (Warm-up Routines and Other Instructional Routines), Activity Narratives, and About this Lesson.

MP4 is identified and connected to grade-level content, and there is intentional development of MP4 to meet its full intent. Students use mathematical modeling with support of the teacher and independently throughout the units. Examples include:

• Unit 4, From Hundredths to Hundred-thousands, Lesson 17, Activity 2, students model with mathematics as they consider aspects of rounding numbers. Activity Narrative, “In this activity, students continue to consider rounding in the same context as in the first activity. Students think about why rounding the altitudes to the nearest 1,000 may make it appear that two planes are a safe distance apart while the exact altitudes may show otherwise. As they consider different ways and consequences of rounding in this situation, students practice reasoning quantitatively and abstractly (MP2) and engage in aspects of mathematical modeling (MP4).” Student Task Statements, “Use the altitude data table from earlier for the following problems. 1. Look at the column showing exact altitudes. a. Find two or more numbers that are within 1,000 feet of one another. Mark them with a circle or a color. b. Find another set of numbers that are within 1,000 feet of one another. Mark them with a square or a different color. c. Based on what you just did, which planes are too close to one another? 2. Repeat what you just did with the rounded numbers in the last column. If we look there, which planes are too close to one another? 3. Which set of altitude data should air traffic controllers use to keep airplanes safe while in the air? Explain your reasoning. 4. Are there better ways to round these altitudes, or should we not round at all? Explain or show your reasoning.”

• Unit 9, Putting It All Together, Lesson 9, Activity 1, students model with mathematics as they consider mathematical questions that could be asked given certain information and answers. The Activity Narrative states, “In this activity, students analyze a situation and given solutions and think about what questions were asked. They also write their own question that can be answered with the quantities in the situation. As they connect equations to a context, students reason quantitatively and abstractly (MP2) and engage in aspects of modeling (MP4).” Student Task Statements, “George Meegan walked 19,019 miles between 1977 and 1983. He finished at age 31. He wore out 12 pairs of hiking boots. Jean Beliveau walked 46,900 miles between 2000 and 2011 and finished at age 56. Here are the responses Kiran gave to answer some questions about the situation. Write the question that Kiran might be answering. In the last row, write a new question about the situation and show the answer, along with your reasoning.” There is a table showing questions 1-3 along with their response and reasoning, and a blank space for question 4. “1. 1983-1977=6, 12\div6=2, 2 pairs or hiking boots; 2. 56-31=23, 23 years; 3. 2011-2000=11, 11 years, 11\times___=46,900, $$11\times4,000=44,000$$, 11\times200=2,200, 11\times70=770, 44,000+2,200+770=46,970, 4,000+200+70=4,270, 4,270 miles., 4._____.”

MP5 is identified and connected to grade-level content, and there is intentional development of MP5 to meet its full intent. Students choose appropriate tools strategically with support of the teacher and independently throughout the units. Examples include:

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

• Unit 8, Properties of Two-Dimensional Shapes, Lesson 4, Activity 2, students choose tools strategically to help them determine symmetry in shapes. Activity Narrative, “In this activity, students practice identifying two-dimensional figures with line symmetry. They sort a set of figures based on the number of lines of symmetry that the figures have. Continue to provide access to patty paper, rulers, and protractors. Students who use these tools to show that a shape has or does not have a line of symmetry use tools strategically (MP5). Consider allowing students to fold the cards, if needed.” Student Task Statements, “Your teacher will give your group a set of cards. 1. Sort the figures on the cards by the number of lines of symmetry they have. 0 lines of symmetry, 1 line of symmetry, 2 lines of symmetry, 3 lines of symmetry 2. Find another group that has the same set of cards. Compare how you sorted the figures. Did you agree with how their figures are sorted? If not, discuss any disagreement.”

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

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

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

Students have many opportunities to attend to precision and to attend to the specialized language of mathematics in connection to grade-level work. This occurs with the support of the teacher as well as independent work throughout the materials. Examples include:

• Unit 1, Factors and Multiples, Lesson 8, Activity 2, students use precise language to identify shapes that meet certain conditions. The Activity Narrative states, “In this activity, students use their understanding of factor pairs, prime, and composite numbers to analyze their peers’ artwork. They look for rectangles that have the same area and those with a prime number or a composite number for their area. Students practice communicating with precision as they identify rectangles and how they know the rectangles meet these conditions (MP6).” Student Task Statement, “Trade artwork with your partner. Using your partner’s artwork, look for and describe each of the following: 1. Rectangles that have the same area 2. Rectangles with an area that is a prime number 3. Rectangles with an area that is a composite number 4. Which challenge they completed.”

• Unit 2, Fraction Equivalence and Comparison, Lesson 7, Activity 2, students use precise language to discuss fractions. Activity Narrative, “In this activity, students find equivalent fractions for fractions given numerically. They also work to clearly convey their thinking to a partner, which involves choosing and using words, numbers, or other representations with care. In doing so, students practice attending to precision (MP6) as they communicate about mathematics.” Student task statement, “For each fraction, find two equivalent fractions. Partner A 1. \frac{3}{2}, 2. \frac{10}{6}, Partner B 1. \frac{4}{3}, 2. \frac{14}{10}. Next, show or explain to your partner how you know that the fractions you wrote are equivalent to the original. Use any representation that you think is helpful.” Students repeat this independently during the Cool-down. Student Task Statement, “Name two fractions that are equivalent to \frac{5}{3}. Explain or show your reasoning.”

• Unit 4, From Hundredths to Hundred-thousands, Lesson 12, Cool-down, students use precise language to compare numbers and discuss place value. Lesson Narrative, “In this lesson, students use their understanding of place value to compare numbers and articulate how they reason about the size of the numbers. In doing so, they reinforce their understanding of place value and the base-ten number system (MP7). In communicating their thinking, they also practice attending to precision (MP6).” Cool-down Task Statement, “Here are two numbers, each with the same digit missing in different places. 17,__42,1___{_,}742 If the missing digit in both numbers is 1, which number will be greater: the first or the second? 2. Name all the digits from 0 to 9 that will make the second number greater. Explain how you know.”

• Unit 8, Properties of Two-dimensional Shapes, Lesson 1, Activity 1, students use the specialized language of mathematics as they sort shapes into categories. The Activity Narrative, “Students will likely use informal language in describing their categories (for example, “they have all the same sides” or “the sides are slanted the same way”). Encourage students to use more precise mathematical language (MP6) and support them in doing so by revoicing their ideas (for example, “all sides have the same length” for “all sides are the same”).” Student Task Statements, “1. Sort the shapes from your teacher into 3–5 categories. For each category, write a title on a sticky note. 2. Share your categories with another group. Take turns listening to each other’s explanations. Do your categories make sense to them? Do their categories make sense to you? Any suggestions or corrections?” The Activity Synthesis, “‘What were some of the categories you used to sort the shapes?’ As students share responses, update the display by adding (or revising) language, diagrams, or annotations. Point out to students that some categories had to do with the sides of the shapes and others had to do with the angles. Remind students to borrow language from the display in the next activity.”

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

The materials reviewed for Imagine Learning Illustrative Mathematics Grade 4 meet expectations for supporting the intentional development of MP7: Look for and make use of structure; and MP8: Look for and express regularity in repeated reasoning, for students, in connection to grade-level content standards, as expected by the mathematical practice standards. Students have opportunities to engage with the Math Practices throughout the year.

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 throughout the units to look for, describe, and make use of patterns within problem-solving as they work with support of the teacher and independently. Examples include:

• Unit 4, From Hundredths to Hundred-thousands, Lesson 7, Cool-down, students use the structure of the base ten number system to answer questions about a number. Lesson Narrative, “In this lesson, students count to read and write multi-digit numbers up to the ten-thousands place. They also count to develop a sense of the magnitude of 10,000. In the previous lesson, students counted by thousands and created 10 groups of 1,000 to make 10,000. This continues to build awareness of the structure of our number system with the base of ten (MP7). In this lesson, students practice writing numbers up to 100,000, which sets the stage for 100,000 as a new unit in base-ten in the lessons that follow.” Cool-down Task Statement, “Consider the number 57,000. 1. How many thousands are in it? 2. How many ten-thousands are in it? 3. Write the number in words.”

• Unit 5, Multiplicative Comparison and Measurement, Lesson 6, Activity 1, students look for and make use of structure as they represent and solve multiplicative comparison problems involving multiples of 10. Student Task Statements, “Here is a diagram that represents two quantities, A and B. 1. What are some possible values of A and B? 2. Select the equations that could be represented by the diagram. A. 15\times10=150. B. 16\times100=1,600. C. 30\div3=10. D. 5,000\div5=1,000. E. 80\times10=800. F. 12,000\div10=1,200.” Lesson Narrative, “The activity also reinforces what students previously learned about the product of a number and 10—namely, that it ends in zero and each digit in the original number is shifted one place to the left because its value is ten times as much (MP7).” Activity Synthesis, “Display possible values for A and corresponding values for B in a table such as this: ‘What do you notice about the values of each set? How do the values compare? (The values for B are all multiples of 10. Each value for B is ten times the corresponding value for A.)’ Select students to share their responses and reasoning.  For students who used the diagram to reason about the equations, consider asking, ‘How might you label the diagram to show that it represents the equations you selected? (Sample response: For the equation 30\div3=10 I would label A “3” and B “30” I know that 30 is ten times as much as 3, so the diagram represents the equation.)’ For students who used their understanding of numerical patterns to support their reasoning, consider asking, ‘How did you know that the equation could be represented as a comparison involving ten times as many? (Sample response: I know that when we multiply a number by 10, the product will be ten times the value. I also know that division is the inverse of multiplication, so I looked for equations that were multiplying or dividing by 10 or had ten as a quotient.)’”

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 throughout the materials, with support of the teacher or during independent practice, to use repeated reasoning in order to make generalizations and build a deeper understanding of grade-level math concepts. Examples include:

• Unit 1, Factors and Multiples, Lesson 2, Warm-up, students use repeated reasoning to complete related multiplication problems mentally. Activity Narrative, “The purpose of this Number Talk is to elicit strategies and understandings students have for multiplying single-digit numbers. These understandings help students develop fluency and will be helpful later in this lesson when students find factor pairs of numbers. As students use earlier problems to find the new products, they look for and make use of structure (MP7) and use repeated reasoning (MP8).” Student Task Statement, “Find the value of each expression mentally. 2\times7, 4\times7, 3\times7, 7\times7.” Activity Synthesis, “How did the first three expressions help you find 7\times7? (The 7 breaks apart into 3 and 4, so I could multiply in parts and add them.)’ Consider asking: ‘Who can restate _____’s reasoning in a different way? Did anyone have the same strategy but would explain it differently? Did anyone approach the expression in a different way? Does anyone want to add on to _____’s strategy?’”

• Unit 2, Fraction Equivalence and Comparison, Lesson 3, Activity 1, students reason about the relative sizes of two fractions with the same numerator. Students Facing, “1. This diagram shows a set of fraction strips. Label each rectangle with the fraction it represents. 2. Circle the greater fraction in each of the following pairs. If helpful, use the diagram of fraction strips. a. \frac{3}{4} or \frac{5}{4}. b. \frac{3}{5} or \frac{5}{5}. c. \frac{3}{6} or \frac{6}{6}. d. \frac{3}{8} or \frac{5}{8}. e. \frac{3}{10} or \frac{5}{10}. 3. What pattern do you notice about the circled fractions? How can you explain the pattern?” Lesson Narrative, “When students observe that 5 equal parts are greater than 3 of the same equal part, regardless of the size of those parts, they see regularity in repeated reasoning (MP8).”  Activity Synthesis, “What do you notice about each pair of fractions in the second question? (They all have 3 and 5 for the numerators, and they have the same denominator.) What does it mean when two fractions, say \frac{3}{8} and \frac{5}{8}, have the same denominator? (They are made up of the same fractional part—eighths in this case.) How can we tell which fraction is greater? (Because the fractional parts are the same size, we can compare the numerators. The fraction with the greater numerator is greater.)”

• Unit 8, Properties of Two-dimensional Shapes, Lesson 7, Activity 2, students use repeated reasoning as they find the perimeter of shapes and write a matching expression. Student Task Statements, “1. Find the perimeter of each shape. Write an expression that shows how you find the perimeter. 2. Compare your expressions with your partners’ expressions. Make 1–2 observations.” Lesson Narrative, “In this activity, students find the perimeter of several shapes and write expressions that show their reasoning. Each side of the shape is labeled with its length, prompting students to notice repetition in some of the numbers. The perimeter of all shapes can be found by addition, but students may notice that it is efficient to reason multiplicatively rather than additively (MP8).” Activity Synthesis, “Read Mai’s reasoning aloud. “What do you think Mai means? What is unclear? Are there any mistakes? 1 minute: quiet think time, 2 minute: partner discussion. With your partner, work together to write a revised explanation for Mai. Display and review the following criteria: We can’t find the perimeter of a quadrilateral if one or more side lengths are missing and _____.’ 3–5 minute: partner work time Select 1–2 groups to share their revised explanation with the class. Record responses as students share. ‘How did you know that the perimeter of C can be found by combining twice 4 cm and twice 7\frac{1}{2} cm or (2\times4)+(2\times7\frac{1}{2})? (We saw earlier that when quadrilaterals have two pairs of parallel sides, opposite sides have the same length. If the sides are parallel, they are the same distance apart.) How did you know the unlabeled side in Shape D is also 7\frac{1}{2} cm long? (The figure has a line of symmetry through the midpoint of the top and bottom segment.) What expression can we write for the perimeter of D? (4+5\frac{3}{4}+7\frac{1}{2}+7\frac{1}{2}) or 4+5\frac{3}{4}+(2\times7\frac{1}{2}) or equivalent).’”

### Usability

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

##### Gateway 3
Meets Expectations

#### Criterion 3.1: Teacher Supports

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

The materials reviewed for Imagine Learning Illustrative Mathematics Grade 4 meet expectations for Teacher Supports. The materials: provide teacher guidance with useful annotations and suggestions for enacting the student and ancillary materials; contain adult-level explanations and examples of the more complex grade-level concepts and concepts beyond the current grade so that teachers can improve their own knowledge of the subject; include standards correlation information that explains the role of the standards in the context of the overall series; provide explanations of the instructional approaches of the program and identification of the research-based strategies; and provide a comprehensive list of supplies needed to support instructional activities.

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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 Imagine Learning Illustrative Mathematics Grade 4 meet expectations for providing teacher guidance with useful annotations and suggestions for how to enact the student materials and ancillary materials, with specific attention to engaging students in order to guide their mathematical development.

Materials provide comprehensive guidance that will assist teachers in presenting the student and ancillary materials. Examples include:

• IM Curriculum, Why is the curriculum designed this way?, Design Principles, Coherent Progression, “To support students in making connections to prior understandings and upcoming grade-level work, it is important for teachers to understand the progressions in the materials. Grade level, unit, lesson, and activity narratives describe decisions about the organization of mathematical ideas, connections to prior and upcoming grade-level work, and the purpose of each lesson and activity. When appropriate, the narratives explain whether a decision about the scope and sequence is required by the standards or a choice made by the authors. The basic architecture of the materials supports all learners through a coherent progression of the mathematics based both on the standards and on research-based learning trajectories. Each activity and lesson is part of a mathematical story that spans units and grade levels. This coherence allows students to view mathematics as a connected set of ideas that makes sense. 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. The invitation to the mathematics is particularly important because it offers students access to the mathematics. It builds on prior knowledge and encourages students to use their own language to make sense of ideas before formal language is introduced, both of which are consistent with the principles of Universal Design for Learning.”

• IM Curriculum, Scope and sequence information, provides an overview of content and expectations for the units. “The big ideas in grade 4 include: developing understanding and fluency with multi-digit multiplication, and developing understanding of dividing to find quotients involving multi-digit dividends; developing an understanding of fraction equivalence, addition and subtraction of fractions with like denominators, and multiplication of fractions by whole numbers; understanding that geometric figures can be analyzed and classified based on their properties, such as having parallel sides, perpendicular sides, particular angle measures, and symmetry.”

• Unit 3, Extending Operations to Fractions, Section B, Addition and Subtraction of Fractions, Section Overview, Section Narrative, provides an overview of the content and expectations for the section. “In this section, students learn to add and subtract fractions by decomposing them into sums of smaller fractions, writing equivalent fractions, and using number lines to support their reasoning. Students begin by thinking about a fraction as a sum of unit fractions with the same denominator and then as a sum of other smaller fractions. They represent different ways to decompose a fraction by drawing “jumps” on number lines and writing different equations. Working with number lines helps students see that a fraction greater than 1 can be decomposed into a whole number and a fraction, and then be expressed as a mixed number. This can in turn help us add and subtract fractions with the same denominator. For example, to find the value of 3−\frac{2}{5}, it helps to first decompose the 3 into 2+\frac{5}{5}, and then subtract \frac{2}{5} from the \frac{5}{5}. Later in the section, students organize fractional length measurements ($$\frac{1}{2}$$, \frac{1}{4}, and \frac{1}{8} inch) on line plots. They apply their ability to interpret line plots and to add and subtract fractions to solve problems about measurement data.”

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, Warm-up, Activities, and Cool-down narratives all provide useful annotations. IM Curriculum, Why is the curriculum designed this way?, Design Principles, Coherent Progressions, “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.” Examples include:

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

• Unit 7, Angles and Angle Measurement, Lesson 13, Lesson Synthesis provides teachers guidance on how to help students find unknown measurements by composing or decomposing known measurements. “‘Today we used different operations to find the measurement of different angles.’ Display: ‘Here are some angles whose measurements we tried to find: anglep, angle s, and some angles composed of smaller angles. We used different operations to find the unknown measurements. Which of these angles can we find by using division? (Angle p: If we know that 2 copies ofpmake a right angle, which is 90\degree, then dividing 90\degree by 2 gives us the measure of .) Which unknown angle can we find by multiplication?’ (The angle made up of four 30° angles has a measurement of 4\times30.) ‘Which unknown angle can we find by subtracting one angle from another? (Angle s: We can subtract 30\degree from 180\degree and divide by 2 to find the measure of s, which is 75\degree.) Which unknown angle can we find by adding known angles? (Once we know the measure of angle s, we can find the last angle: 15+75+15, which is 150\degree.)’”

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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 Imagine Learning Illustrative Mathematics Grade 4 meet expectations for containing adult-level explanations and examples of the more complex grade-level concepts and concepts beyond the current grade so that teachers can improve their own knowledge of the subject.

Within the Teacher’s Guide, IM Curriculum, Why is the curriculum designed this way?, 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 current grade so that teachers can improve their own knowledge of the subject. Professional articles support teachers with learning opportunities about topics such as ensuring mathematical success for all, early understanding of equality, and repeating patterns. Examples include:

• Why is the curriculum designed this way? Further Reading, Unit 2, Fractions: Units and Equivalence, supports teachers with context for work beyond the grade. “In this blog post, McCallum discusses equivalent fractions as the same numbers in different units.”

• Why is the curriculum designed this way? Further Reading, Unit 7, Making Peace with the Basics of Trigonometry, “In this blog post, Phillips highlights how student exploration in trigonometry allows them to see that trigonometric ratios come from measuring real triangles, fostering conceptual understanding. This blog is included in this unit as an example of how concepts of angle come into play in mathematics beyond elementary school.”

• Unit 5, Multiplicative Comparison and Measurement, Lesson 18, About this Lesson, “This lesson does provide students with an opportunity to apply precursor skills of mathematical modeling. In this lesson, students write true and false statements involving multiplicative comparisons and unit conversions. They have an opportunity to choose the animals to compare and which facts to use. Then, students determine which of their classmates’ statements are true and which are false. At the end of this activity, they have an opportunity to revise their earlier statements to make them clearer or stronger. As students discuss and justify their decisions, they share a mathematical claim and the thinking behind it (MP3).”

• Unit 6, Multiplying and Dividing Multi-digit Numbers, Lesson 25, Paper Flower Decorations, About this Lesson, “This lesson does provide students with an opportunity to apply precursor skills of mathematical modeling. In this lesson, students build on their prior understanding and experiences with creating and analyzing patterns to solve multi-step problems in a real-world context. In the first activity, students make different types of paper flowers. In the second activity, they consider patterns and solve problems involving paper flower garlands. In the third activity, students think of their own pattern and multi-step problems inspired by their process of making paper flowers. When students ask and answer questions that arise from a given situation, use mathematical features of an object to solve a problem, make choices, analyze real-world situations with mathematical ideas, interpret a mathematical answer in context, and decide if an answer makes sense in the situation, they model with mathematics (MP4).”

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

The materials reviewed for Imagine Learning Illustrative Mathematics Grade 4 meet expectations for including standards correlation information that explains the role of the standards in the context of the overall series.

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

• Grade-level resources, Grade 4 standards breakdown, standards are addressed by lesson. Teachers can search for a standard in the grade and identify the lesson(s) where it appears within materials.

• Course Guide, Lesson Standards, includes all Grade 4 standards and the units and lessons each standard appears in.

• Unit 1, Resources, Teacher Guide, outlines standards, learning targets and the lesson where they appear. This is present for all units and allows teachers to identify targeted standards for any lesson.

• Unit 5, From Hundredths to Hundred-thousands, Lesson 5, the Core Standards are identified as 4.NBT.A.2. Lessons contain a consistent structure that includes a Warm-up with a Narrative, Launch, Activity, Activity Synthesis. An Activity 1, 2, or 3 that includes Narrative, Launch, Activity, Activity Synthesis, Lesson Synthesis. 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 identifying the content standards addressed within the unit, as well as a narrative outlining relevant prior and future content connections. Examples include:

• Unit 2, Fraction Equivalence and Comparison, Unit Overview, Unit Learning Goals, full Unit Narrative, includes an overview of how the math of this module builds from previous work in math. “In grade 3, students partitioned shapes into parts with equal area and expressed the area of each part as a unit fraction. They learned that any unit fraction \frac{1}{b} results from a whole partitioned into b equal parts. They used unit fractions to build non-unit fractions, including fractions greater than 1, and represent them on fraction strips and tape diagrams. The denominators of these fractions were limited to 2, 3, 4, 6, and 8. Students also worked with fractions on a number line, establishing the idea of fractions as numbers and equivalent fractions as the same point on the number line.  Here, students follow a similar progression of representations. They use fraction strips, tape diagrams, and number lines to make sense of the size of fractions, generate equivalent fractions, and compare and order fractions with denominators 2, 3, 4, 5, 6, 8, 10, 12, and 100.”

• Unit 7, Angles and Angle Measurement, Unit Overview, Unit Learning Goals, full Unit Narrative, includes an overview of how the math of this module builds from previous work in math. “In earlier grades, students learned about two-dimensional shapes and their attributes, which they described informally early on but with increasing precision over time. Here, students formalize their intuitive knowledge about geometric features and draw them. They identify and define some building blocks of geometry (points, lines, rays, and line segments), and develop concepts and language to more precisely describe and reason about other geometric figures. Students analyze cases where lines intersect and where they don’t, as in the case of parallel lines. They learn that an angle as a figure composed of two rays that share an endpoint. Later, students compare the size of angles and consider ways to quantify it. They learn that angles can be measured in terms of the amount of turn one ray makes relative to another ray that shares the same vertex.”

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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 Imagine Learning Illustrative Mathematics Grade 4 provide strategies for informing all stakeholders, including students, parents, or caregivers about the program and suggestions for how they can help support student progress and achievement.

Each unit has corresponding Family Support Materials (also in Spanish) that provide a variety of supports for families, including the core focus for each section in each unit, and Try It At Home!. Examples include:

• Course Overview, Unit 1, Factors and Multiples, Additional Resources, Home School Connection, Family Support Material, “Print or share this guide to support families support their students with the key concepts and ideas in Grade 4 Unit 1. In this unit, students learn about factors and multiples and apply their understanding of the area of rectangles. Students determine if a number between 1 and 100 is prime or composite. Section A: Understand Factors and Multiples, Section B: Find Factor Pairs and Multiples.” The guide also includes a Spanish language version.

• Course Overview, Unit 2, Fraction Equivalence and Comparison, Additional Resources, Home School Connection, Family Support Material, Try It At Home! section, “Near the end of the unit, ask your student to compare \frac{3}{5} and \frac{3}{7}. Questions that may be helpful as they work: How are the two fractions alike? How are they different? What strategy did you use to compare? Is there a different strategy that you could use to compare?”

• Course Overview, Unit 5, Multiplicative Comparison and Measurement, Additional Resources, Home School Connection, Family Support Material, “Section B: Measurement Conversion In this section, students expand their knowledge of units of measurement from earlier grades. Previously, they learned that there are 100 centimeters in 1 meter. Here, they relate centimeters and meters in terms of multiplication—1 meter is 100 times as long as 1 centimeter—and use this reasoning to convert any number of meters to centimeters. Students also relate other units of measurement in terms of multiplication: meters and kilometers, grams and kilograms, milliliters and liters, ounces and pounds, and seconds, minutes, and hours. They then solve problems that involve converting a larger unit to a smaller unit.”

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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 Imagine Learning Illustrative Mathematics Grade 4 meet expectations for providing explanations of the instructional approaches of the program and identification of the research-based strategies.

Instructional approaches of the program are described within the Curriculum Guide, Why is the curriculum designed this way? Design Principles. “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 materials through coherent progression, and provides teachers the opportunity to deepen their knowledge of mathematics, student thinking, and their own teaching practice. Examples from the Design principles include:

• Curriculum Guide, Why is the curriculum designed this way?, Design Principles, includes information about the 11 principles that informed the design of the materials. Balancing Rigor, “There are three aspects of rigor essential to mathematics: conceptual understanding, procedural fluency, and the ability to apply these concepts and skills to mathematical problems with and without real-world contexts. These aspects are developed together and are therefore interconnected in the materials in ways that support student understanding.  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. Warm-ups, practice problems, centers, and other built-in routines help students develop procedural fluency, which develops over time.”

• Curriculum Guide, Why is the curriculum designed this way?, Design Principles, Task Complexity, “Mathematical tasks can be complex in different ways, with the source of complexity varying based on students’ prior understandings, backgrounds, and experiences. In the curriculum, careful attention is given to the complexity of contexts, numbers, and required computation, as well as to students’ potential familiarity with given contexts and representations. To help students navigate possible complexities without losing the intended mathematics, teachers can look to warm-ups and activity launches for built-in preparation, and to teacher-facing narratives for further guidance. In addition to tasks that provide access to the mathematics for all students, the materials provide guidance for teachers on how to ensure that during the tasks, all students are provided the opportunity to engage in the mathematical practices. More details are given below about teacher reflection questions, and other fields in the lesson plans help teachers assure that all students not only have access to the mathematics, but the opportunity to truly engage in the mathematics.”

Research-based strategies within the program are cited and described within the Curriculum Guide, within Why is the curriculum designed this way?. There are four sections in this part of the Curriculum Guide including Design Principles, Key Structures, Mathematical Representations, and Further Reading. Examples of research-based strategies include:

• Curriculum Guide, Why is the curriculum designed this way?, Further Reading, Entire Series, The Number Line: Unifying the Evolving Definition of Number in K–12 Mathematics. “In this article, the authors (Lahme, McLeman, Nakamaye, and Umland) focus their attention on the selection of definitions, notation, and graphical conventions surrounding the development of the real numbers from kindergarten to grade 12, and address the work that students might do in later years.”

• Curriculum Guide, Why is the curriculum designed this way?, Further Reading, Unit 2, “Fractions: Units and Equivalence. In this blog post, McCallum discusses equivalent fractions as the same numbers in different units.” Unit 7, ”Making Peace with the Basics of Trigonometry. In this blog post, Phillips highlights how student exploration in trigonometry allows them to see that trigonometric ratios come from measuring real triangles, fostering conceptual understanding. This blog is included in this unit as an example of how concepts of angle come into play in mathematics beyond elementary school.”

• Curriculum Guide, Why is the curriculum designed this way?, Design Principles, Instructional Routines, “Instructional routines provide opportunities for all students to engage and contribute to mathematical conversations. Instructional routines are invitational, promote discourse, and are predictable in nature. They are “enacted in classrooms to structure the relationship between the teacher and the students around content in ways that consistently maintain high expectations of student learning while adapting to the contingencies of particular instructional interactions.” (Kazemi, Franke, & Lampert, 2009)

• Curriculum Guide, Why is the curriculum designed this way?, 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.”

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

The materials reviewed for Imagine Learning Illustrative Mathematics Grade 4 meet expectations for providing a comprehensive list of supplies needed to support instructional activities.

The Course Overview, Grade-level resources, provides a Materials List intended for teachers to gather materials for each grade level. Additionally, specific lessons include a Teaching Notes section and a Materials List, which include specific lists of instructional materials for lessons. Examples include:

• Course Overview, Grade Level Resources, Grade 4 Materials List, contains a comprehensive chart of all materials needed for the curriculum.  It includes the materials used throughout the curriculum, whether they are reusable or consumable, quantity needed, lessons the materials are used in, and suitable substitutes for the materials. Each lesson listed in the chart and any additional virtual materials noted for a lesson are digitally linked in the materials for quick access. Base 10 Blocks are a reusable material used in lesson 4.4.6, 4.4.8, 4.6.16, 4.6.17, and 4.6.18. 60 hexagons and trapezoids, 15 thousands, 300 hundreds, 600 tens, 600 ones are needed per 30 students. Paper cut-outs or Virtual Base-ten blocks are suitable substitutes.  Paper clips are a reusable material used in lessons 4.2.17 and 4.5.9. 150 paper clips are needed for 30 students. No suitable substitutes for the material are listed. Grid paper is a consumable material used in lessons 4.1.1, 4.1.2, 4.1.3, 4.4.18, 4.4.19, 4.4.20, 4.4.21, 4.4.22, and (4.6.3). 15 are needed per 30 students. Paper or Virtual Grid Paper are suitable substitutes for the material.

• Unit 2, Fraction Equivalence and Comparison, Lesson 17, Activity 1: Paper Clip Tossing Game, Teaching Notes, Materials to gather, “Markers, Paper, Paper clips, Tape (painter's or masking).” Launch, “Give each group strips of paper, markers, paper clips. Work with your group to play a version of the paper-clip tossing game. To play the game, you will toss some paper clips and use fractions to label where they land. First, we’ll make the game board. Fold your paper strip in half and then in half again. Carefully tape it down to your workspace (desk or floor can work) and label the benchmark fractions.”

• Unit 7, Angles and Angle Measurement, Lesson 12, Activity 3, Teaching Notes, Materials to gather, “Pattern blocks, Protractors.” Launch, “Give students access to protractors and pattern blocks.”

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

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

The materials reviewed for Imagine Learning Illustrative Mathematics Grade 4 meet expectations for having assessment information included in the materials to indicate which standards are assessed. The materials consistently identify the content standards assessed for formal assessments, and the materials provide guidance, including the identification of specific lessons, as to how the mathematical practices can be assessed across the series.

End-of-Unit Assessments and End-of-Course Assessments consistently and accurately identify grade-level content standards within each End-of-Unit Assessment answer key. Examples from formal assessments include:

• Unit 4, From Hundredths to Hundred-thousands, End-of-Unit Assessment answer key, denotes standards addressed for each problem. Problem 3, 4.NBT.1, “The distance between New York City and Boston is 225 miles. The distance between New York City and Salt Lake City is 10 times as far. How many miles is it between New York City and Salt Lake City? Explain or show your reasoning.”

• Unit 6, Multiplying and Dividing Multi-digit Numbers, End-of-Unit Assessment answer key, denotes standards addressed for each problem. Problem 5, 4.NBT.6, “Find the value of each quotient. Explain or show your reasoning. a. 714\div6. b. 3,626\div7.”

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

Guidance is provided within materials for assessing progress of the Mathematical Practices. According to IM K-5 Curriculum Guide, How do you assess progress?, Standards For Mathematical Practice, “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 Curriculum Guide How do you assess progress? Standards For Mathematical Practice, Standards for Mathematical Practices Chart, Grade 4, MP1 is found in Unit 3, Lessons 11, 14, 15, and 18.

• IM K-5 Curriculum Guide How do you assess progress? Standards For Mathematical Practice, Standards for Mathematical Practices Chart, Grade 4, MP8 is found in Unit 2, Lessons 3, 5, 8, and 11.

• IM K-5 Curriculum Guide, How do you assess progress? Standards For Mathematical Practice, Standards for Mathematical Practice Student Facing Learning Targets, “MP2 I Can Reason Abstractly and Quantitatively. I can think about and show numbers in many ways. I can identify the things that can be counted in a problem. I can think about what the numbers in a problem mean and how to use them to solve the problem. I can make connections between real-world situations and objects, diagrams, numbers, expressions, or equations.”

• IM K-5 Curriculum Guide, How do you assess progress? Standards For Mathematical Practice, Standards for Mathematical Practice Student Facing Learning Targets, “MP4 I Can Model with Mathematics. I can wonder about what mathematics is involved in a situation. I can come up with mathematical questions that can be asked about a situation. I can identify what questions can be answered based on data I have. I can identify information I need to know and don’t need to know to answer a question. I can collect data or explain how it could be collected. I can model a situation using a representation such as a drawing, equation, line plot, picture graph, bar graph, or a building made of blocks. I can think about the real-world implications of my model.”

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

Each End-of-Unit Assessment and End-of-Unit Course Assessment provides an answer key and standard alignment. According to the Curriculum Guide, How do you assess progress?, “All summative assessment problems include a complete solution and standard alignment. Multiple choice and multiple response problems often include a reason for each potential error a student might make.” Examples from the assessment system include:

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

• Unit 5, Multiplicative Comparison and Measurement, End-of-Unit Assessment, Problem 5, “Complete the table showing the number of ounces for the measurements given in pounds.” A table chart shows pounds as the title with the numbers 1, 5, 10, 20, 50, then ounces as the other title with no numbers. End-of-Unit Assessment Answer Key, “Students complete a table converting pounds to ounces. Because they need to multiply by 16, the numbers being converted have been kept friendly so that students can use place value understanding to find the values efficiently. Students who forget the number of ounces in a pound can still show arithmetic fluency and an understanding of how conversions work with an incorrect conversion factor.” The answer key aligns this question to 4.MD.1, 4.NBT.5.

• Unit 9, Putting It All Together, End-of-Course Assessment, Problem 11, “The line plot shows the wingspans of some butterflies in inches. a. How much greater is the longest wingspan than the shortest wingspan? Explain or show your reasoning. b. How much greater is the longest wingspan than the most common wingspan? Explain or show your reasoning.” End-of-Unit Assessment Answer Key, “Students subtract mixed numbers which they read from a line plot. The line plot is a convenient way of presenting the information and also a situation where mixed numbers occur naturally. Students can reason about the differences abstractly or they may use the horizontal axis which can play the role of a number line.” The answer key aligns this question to 4.MD.4, 4.NF.3.

While assessments provide guidance to teachers for interpreting student performance, suggestions for follow-up with students are minimal or absent. Cool Downs, at the end of each lesson, include some suggestions for teachers. According to the Curriculum Guide, 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 6, Multiplying and Dividing Multi-digit Numbers, Lesson 6, Cool-down, Student Task Statements, “Find the value of 6\times83. Use a diagram if it is helpful.” Responding to Student Thinking, “Students use a diagram, but find a value other than 498 as the product.” Next Day Supports, “Launch warm-up or activities by highlighting important representations from previous lessons.” This problem aligns to 4.NBT.5.

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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 Imagine Learning Illustrative Mathematics Grade 4 meet expectations for providing assessments that include opportunities for students to demonstrate the full intent of grade-level standards and practices across the series.

Formative assessment opportunities include instructional tasks, practice problems, and checklists in each section of each unit. 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 of summative assessment items include:

• Unit 2, Fraction Equivalence and Comparison, End-of-Unit Assessment problems support the full intent of MP1, make sense of problems and persevere in solving them, as students compare fractions to benchmarks \frac{1}{2} and 1. Problem 2, “Select all fractions that are greater than \frac{1}{2} but less than 1. A. \frac{4}{5} B. \frac{1}{3} C. \frac{5}{4} D. \frac{4}{7} E. \frac{5}{10}.”

• Unit 4, From Hundredths to Hundred-Thousands, End-of-Unit Assessment develops the full intent of 4.NBT.4, fluently add and subtract multi-digit whole numbers using the standard algorithm. Problem 5, “Find the sum or difference. (Both problems are written vertically.) a. 324,567+34,762; b. 827,419-80,125.”

• Unit 6, Multiplying and Dividing Multi-digit Numbers, End-of-Unit Assessment develops the full intent of 4.OA.3, solve multistep word problems posed with whole numbers and having whole-number answers using the four operations, including problems in which remainders must be interpreted, represent these problems using equations with a letter standing for the unknown quantity, and assess the reasonableness of answers using mental computation and estimation strategies including rounding. Problem 7, “Jada’s family is getting soil for a garden. The garden will be 160 square feet. Each bag of soil covers 6 square feet. How many bags of soil does Jada’s family need? Explain or show your reasoning.”

• Unit 9, Putting It All Together, End-of-Course Assessment problems supports the full intent of MP4, model with mathematics, as students find a whole number multiple of a fraction and a difference of a whole number and a fraction. Problem 13, “a. Mai’s house is \frac{5}{8} mile from school. She walked to school all 5 days of the week. How many miles did Mai walk altogether from home to school? Explain or show your reasoning. b. Mai wants to walk 6 miles total for the week. How much farther does she need to walk? Explain or show your reasoning.”

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

The materials reviewed for Imagine Learning Illustrative Mathematics Grade 4 offer accommodations that allow students to demonstrate their knowledge and skills without changing the content of the assessment. The general accommodations are provided within the Teacher Guide in the section, “Universal Design for Learning and Access for Students with Disabilities.” These accommodations are provided at the program level and not specific to each assessment throughout the materials.

Examples of accommodations to be applied throughout the assessments include:

• Curriculum Guide, How do the materials support all learners?, Access for students with disabilities, UDL Strategies to Enhance Access, “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.”

• Curriculum Guide, How do you assess progress? End-of-Unit Assessments, “Teachers may choose to grade these assessments in a standardized fashion, but may also choose to grade more formatively by asking students to show and explain their work on all problems. Teachers may also decide to make changes to the provided assessments to better suit their needs. If making changes, teachers are encouraged to keep the format of problem types provided, and to include problems of different types and different levels of difficulty.”

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

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

The materials reviewed for Imagine Learning Illustrative Mathematics Grade 4 meet expectations for providing strategies and supports for students in special populations to support their regular and active participation in learning grade-level mathematics.

Materials regularly provide strategies, supports, and resources for students in special populations to help them access grade-level mathematics. In the Curriculum Guide, How do the materials support all learners? 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 include:

• Unit 7, Angles and Angle Measurement, Lesson 9, Activity 2, Narrative, Access for Students with Disabilities, “Action and Expression: Internalize Executive Functions. Invite students to estimate the size of the angle before finding each precise measurement. Offer the sentence frame: “This angle will be greater than _____ and less than _____. It will be closer to _____.” Supports accessibility for: Conceptual Processing, Visual-Spatial Processing, Attention”

• Unit 8, Properties of Two-dimensional Shapes, Lesson 2, Activity 1, Narrative, Access for Students with Disabilities, “Action and Expression: Internalize Executive Functions. Invite students to discuss the steps they might take to complete the task. For example, students may decide to look at one triangle at a time and decide which attributes it has. Alternatively, they may decide to look at each triangle through the lens of one attribute at a time. Supports accessibility for: Organization.

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

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

The materials reviewed for Imagine Learning Illustrative Mathematics Grade 4 meet expectations for providing extensions and/or opportunities for students to engage with grade-level mathematics at higher levels of complexity.

While there are no instances where advanced students do more assignments than classmates, materials do provide multiple opportunities for students to investigate grade-level content at a higher level of complexity. These are found in a section titled, “Exploration Problems” within lessons where appropriate. According to the IM K-5 Teacher Guide, How do you use the materials?, Practice Problems, 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 that students can do directly related to the material of the unit, 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, Fraction Equivalence and Comparison, Section C: Fraction Comparison, Problem 7, Exploration, “Jada lists these fractions that are all equivalent to \frac{1}{2}: \frac{2}{4}, \frac{3}{6}, \frac{4}{8}, \frac{5}{10}. She notices that each time the numerator increases by 1 and the denominator increases by 2. Will the pattern Jada notices continue? Explain your reasoning.”

• Unit 6, Multiplying and Dividing Multi-digit Numbers, Section A: Features of Patterns, Problem 11, Exploration, “Tyler draws this picture and writes the equation 1+3+5=9. 1. How do you think the equation relates to the picture? 2. Tyler keeps drawing circles to make larger squares. How many new circles does he need to draw to make a 4-by-4 square, and then a 5-by-5 square? 3. What pattern do you notice in the number of circles Tyler adds each time? 4. Why do you think the number of circles is increasing that way?”

• Unit 7, Angles and Angle Measurement, Section A: Points, Lines, Segments, Rays, and Angles, Problem 8, Exploration, “Here is a riddle. Can you solve it? “I am a capital letter made of more than 1 segment with no curved parts. I have no perpendicular segments or parallel segments. What letter could I be?’”

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

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

Students engage with problem-solving in a variety of ways: Warm-up, Instructional Activities, Cool-down, and Centers, which is a key component of the program. According to the Curriculum Guide, Why is the curriculum designed this way? Design principles, Coherent Progression, “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.”

Examples of varied approaches include:

• Unit 2, Fraction Equivalence and Comparison, Lesson 16, Cool-down, students compare fractions to solve problems. Student Task Statements, “Jada, Kiran, and Lin tried to run as far as possible before they had to stop and rest. Jada ran \frac{3}{4} mile. Kiran ran \frac{7}{12} mile. Lin ran \frac{4}{6} mile. Who ran the farthest before stopping? Explain or show your reasoning.”

• Unit 3, Extending Operations to Fractions, Lesson 4, Warm-up, Launch, students “examine a diagram representing equal groups of non-unit fractions. Display the image.” Activity, “What do you notice? What do you wonder?”

• Unit 5, Multiplicative Comparison and Measurement, Lesson 17, Activity 1, students “solve geometric problems using their understanding of length measurements, unit conversion, multiplicative comparison, and addition or subtraction of fractions.” Launch, “Let’s solve some problems about the side lengths and perimeter of a room.” Student Task Statements, “A rectangular room has a perimeter of 39 feet and a length of 10\frac{1}{2} feet. What is the width of the room in feet? Explain or show your reasoning. An ant walked along two walls of the room, always in a straight line. It started in one corner and ended up in a corner opposite of where it started. How many inches did it travel? Explain or show your reasoning.”

• Center, Compare (1–5), Stage 5: Fractions, students flip cards over and compare the value of the cards. Narrative, “Students use cards with fractions. They may use either deck of fraction cards or combine them together to play.”

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

The materials reviewed for Imagine Learning Illustrative Mathematics Grade 4 provide opportunities for teachers to use a variety of grouping strategies. Suggested grouping strategies are consistently present within activity launch and include guidance for whole group, small group, pairs, or individual. Examples include:

• Unit 1, Factors and Multiples, Lesson 8, Warm-up, Launch, “Groups of 2. Display the images. ‘What do you notice? What do you wonder?’ 1 minute: quiet think time.” Activity: “Discuss your thinking with your partner. 1 minute: partner discussion. ‘Share and record responses.’”

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

• Unit 5, Multiplicative Comparison and Measurement, Lesson 2, Activity 1, Launch, “Groups of 2. Give students access to connecting cubes. Display the image of Mai's cubes and Kiran's cubes. ‘How do these cubes represent 3 times as many?” (Mai has 6 cubes and Kiran has 2. Mai has 3 groups of 2 cubes. Mai has 6 cubes and Kiran has 2. Three times as many as 2 is 6, or 3 times 2 is 6.)’ Give students access to cubes. This activity launch continues on the next card.” Activity, “‘Jada has 4 times as many cubes as Kiran. Draw a diagram to represent the situation. Diego has 5 times as many cubes as Kiran. Draw a diagram to represent the situation. Lin has 6 times as many cubes as Kiran. How many cubes does Lin have? Explain or show your reasoning.’ (Image of Mai’s cubes and image of Kiran’s cubes included.) MLR7 Compare and Connect. ‘Create a visual display that shows your thinking about the cubes in each problem and include details to help others understand your thinking.’ 6–8 minutes: independent or group work. 3 minutes: gallery walk. ‘How does each representation show times as many?’ 30 seconds, quiet think time. 1 minute: partner discussion. Monitor for students who create diagrams that are similar to connecting cube images and discrete tape diagrams to share in the synthesis.”

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

The materials reviewed for Imagine Learning Illustrative Mathematics Grade 4 meet expectations for providing strategies and supports for students who read, write, and/or speak in a language other than English to regularly participate in learning grade-level mathematics.

• 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 (MLR) in each lesson. Curriculum Guide, How do the materials support all learners? Mathematical language development, “A mathematical language routine is a structured but adaptable format for amplifying, assessing, and developing students' language. The MLRs were selected because they are effective and practical for simultaneously learning mathematical practices, content, and language. These routines can be adapted and incorporated across lessons in each unit to fit the mathematical work wherever there are productive opportunities to support students in using and improving their English and disciplinary language use. These routines facilitate attention to student language in ways that support in-the-moment teacher, peer, and self-assessment. The feedback enabled by these routines will help students revise and refine not only the way they organize and communicate their own ideas, but also ask questions to clarify their understanding of others’ ideas.” Examples include:

• Unit 2, Fraction Equivalence and Comparison, Lesson 13, Synthesis, Teaching Notes, Access for English Learners, “MLR7 Compare and Connect. Synthesis: After each strategy has been presented, lead a whole-class discussion comparing, contrasting, and connecting the different approaches. Ask, “Did anyone solve the problem the same way, but would explain it differently?” and “Why did the different approaches lead to the same outcome?” Advances: Representing, Conversing.”

• Unit 6, Multiplying and Dividing Multi-digit Numbers, Lesson 4, Activity 1, Teaching Notes, Access for English Learners, “MLR8 Discussion Supports. Use multimodal examples to show the patterns of both columns. Use verbal descriptions along with gestures, drawings, or concrete objects to show the connection between the multiples of 9 and 10. Advances: Listening, Representing.”

• Unit 7, Angles and Angle Measurement, Lesson 10, Activity 1, Teaching Notes, Access for English Learners, “MLR8 Discussion Supports. Synthesis: For each strategy that is shared using the protractor, invite students to turn to a partner and restate what they heard using precise mathematical language. Advances: Listening, Speaking.

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

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

The characters in the student materials represent different races and portray people from many ethnicities in a positive, respectful manner, with no demographic bias for who achieves success in the context of problems. Characters in the program are illustrations of children or adults with representation of different races and populations of students. Names include multi-cultural references such as Kiran, Mai, Elena, Diego, and Han. Problem settings vary from rural to urban and international locations.

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

The materials reviewed for Imagine Learning Illustrative Mathematics Grade 4 partially provide guidance to encourage teachers to draw upon student home language to facilitate learning.

The student materials are available in Spanish. Directions for teachers are in English with prompts for students available in Spanish. The student materials including Warm ups, Activities, Cool-downs, Centers, and Assessments are in Spanish for students.

The IM K-5 Curriculum Guide includes a section titled, “Mathematical Language Development” which outlines the program’s approach towards language development in conjunction with the problem-based approach to learning mathematics. This includes the regular use of Mathematical Language Routines, “A mathematical language routine is a structured but adaptable format for amplifying, assessing, and developing students' language. The mathematical language routines were selected because they are effective and practical for simultaneously learning mathematical practices, content, and language. These routines can be adapted and incorporated across lessons in each unit to fit the mathematical work wherever there are productive opportunities to support students in using and improving their English and disciplinary language use. These routines facilitate attention to student language in ways that support in-the-moment teacher, peer, and self-assessment. The feedback enabled by these routines will help students revise and refine not only the way they organize and communicate their own ideas, but also ask questions to clarify their understanding of others’ ideas.” While Mathematical Language Routines (MLRs) are regularly embedded within lessons and support mathematical language development, they do not include specific suggestions for drawing on a student’s home language.

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

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

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

• Unit 2, Fraction Equivalence and Comparison, Lesson 14, Activity 2, students solve fraction comparison problems. In the activity, students are investigating measurements in different units such as Chinese “li” and kilometer. Student Task wtatements, “In China and some East Asian countries, the unit “li” is used for measuring distance. Here are the walking distances between the home of a student in China and the places he visits regularly. School: \frac{7}{5} li, library: \frac{23}{10} li, market: \frac{7}{4} li, badminton club: \frac{23}{12} li. 2. A student in America walks \frac{4}{5}  kilometer (km) to school. These number lines show how 1 kilometer compares to 1 li. Which student walks a longer distance to school? Use the number lines to show your reasoning.”

• Unit 9, Putting It All Together, Lesson 1, Activity 1, students multiply fractions by whole numbers and compare fractions. In this activity, students look at an image of head wraps. Launch, “Look at the picture of the two women with head wraps. What do you notice? What do you wonder? Collect observations and questions from 1–2 students. In many African cultures, women wrap their hair with colorful fabric when they dress for the day. Have you seen a similar practice such as this one? What is your routine for dressing for the day? Allow 1–2 students to share. We will be thinking about the length of head wraps in this activity.” Student task statements, “Jada and Lin saw a picture of head wraps made of African wax print fabric and would like to make their own.”

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

The materials reviewed for Imagine Learning Illustrative Mathematics Grade 4 provide some supports for different reading levels to ensure accessibility for students.

• Unit 4, From Hundredths to Hundred-thousands, Lesson 4, Activity 2, Launch, “MLR6 Three Reads. Display only the opening paragraph, the eight running times, and the table, without revealing the questions. ‘We are going to read this problem 3 times.’” 1st Read: “‘The table shows eight of the top runners in the Women’s 400-Meter event. Their best running times, listed here, put the runners in the world’s top 25 in this event. What is this story about?’ 1 minute: partner discussion Listen for and clarify any questions about the context.” 2nd Read: “‘Read the opening paragraph a second time. Name the quantities. What can we count or measure in this situation?’ (times in seconds, years) 30 seconds: quiet think time Share and record all quantities. Reveal the questions.” 3rd Read: “‘Read the entire problem aloud, including the questions. How might we go about matching the times to the right runners?’ (Arrange the times in order, from shortest to longest.) Explain that in track and field, runners compete to run different distances: 100 meters, 200, 400, 800, and more. ‘The United States, Jamaica, and the Bahamas have produced some of the fastest track runners in the world.’”

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

• Unit 7, Angles and Angle Measurement, Lesson 1, Activity, Narrative, “As students attempt to produce more accurate drawings, they try to fine-tune their descriptions. They notice that more specific language or terminology is needed to better describe the features in the images (MP6).”

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

The materials reviewed for Imagine Learning Illustrative Mathematics Grade 4 meet expectations for providing manipulatives, both virtual and physical, that are accurate representations of the mathematical objects they represent and, when appropriate, are connected to written methods.

The materials consistently include suggestions and/or links, within the lesson notes, for virtual and physical manipulatives that support the understanding of grade-level math concepts. Examples include:

• Unit 4, Extending Operations to Fractions, Lesson 2, Activity 1, students use their understanding of equivalent fractions and decimals by sorting a set of cards by their value. Launch, “Groups of 2. Give each group a set of cards from the blackline master. Activity, ‘Work with your partner to match each expression to a diagram that represents the same equal-group situation and the same amount. Be prepared to explain how you know the two representations belong together.’”

• Unit 5, Multiplicative Comparison and Measurement, Lesson 7, Activity 1, students use scissors, tape, and centimeter grid paper to help them visualize a meter. Launch, “Give each group 2–3 sheets of centimeter grid paper, 2–3 pairs of scissors, and some tape. ‘Each grid on the paper is 1 centimeter long. Work with your group to cut the centimeter grid paper into strips and then join them to make a strip of paper that is exactly 1 meter long.’”

• Unit 7, Angles and Angle Measurement, Lesson 8, Activity 2, students construct a protractor- like tool that has benchmark angles. Launch, “Groups of 2 to 4. Give each student one paper half-circle and access to rulers or straightedges. ‘Your sheet of paper is in the shape of half a circle. It shows a ray on the bottom right and two angles ($$120\degree$$ and 180\degree) measured from the ray. We see the 120\degree label. Where is the 120\degree angle? Where are the two rays that make this angle?’ 1 minute: quiet think time for the first problem 1 minute: group discussion ‘Where do you think the second ray of a 90\degree angle would be?’ (Between 0 and 120, but closer to 120.)”

#### 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 Imagine Learning Illustrative Mathematics Grade 4 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 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 provide some teacher guidance for the use of embedded technology to support and enhance student learning.

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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 Imagine Learning Illustrative Mathematics Grade 4 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. For example:

• Lessons can be shared with students or provide “Live Learn” with slides and lessons presented to students digitally. In the Curriculum Guide, Feature Highlights, Recent Updates, LearnZillion Platform Updates, Enhanced Features and Functionality, “Live Learn is a new teacher-initiated feature in LearnZillion and allows for synchronous instruction and moderation virtually within the platform. You can transition from asynchronous work time to a live session with one click and connect to students in real-time whether they are learning in the classroom, at home, or anywhere in between. ​​Live Learn provides these benefits for you and your and students: Connects students and teachers in real-time​ and enables immediate feedback, offers a way to moderate synchronous instruction virtually, supports learning in the classroom or at home​, ease of use- transition from asynchronous work time to live instruction with one click​.”

Every lesson includes a “Live Lesson” that allows students to work collaboratively without a teacher’s support. For example:

• Unit 6, Multiplying and Dividing Multi-digit Numbers, Lesson 25, Digital Student Task Statements, Activity 2, Problem 1, students solve problems by analyzing and continuing the pattern in a box, “Priya and Jada are making paper flower garlands for their friend’s quinceañera. Each garland uses 12 flowers. Priya wants 2 big flowers, followed by 2 small flowers. Jada wants 1 big flower, followed by 2 small flowers. Use their patterns to draw the garlands. Draw in the box. Select T to type.”

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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 Imagine Learning Illustrative Mathematics Grade 4 include or reference digital technology that provides opportunities for teachers and/or students to collaborate with each other, when applicable.

“LearnZillion’s platform is constantly improving with new features and instantly available to teachers and students. We have developed some big improvements for back to schools 2020-2021!” Examples include:

• Curriculum Guide, Feature Highlights, Recent Updates, LearnZillion Platform Updates, Enhanced Features and Functionality, “New Reporting Capabilities for Teachers: NOW LIVE. New reports on student progress and performance. New data dashboard that organizes and displays performance metrics at the school, class, and student level. ​The Data Dashboard makes student performance data easy to see, understand, and manage for a more effective instructional experience.” ​

• Curriculum Guide, Feature Highlights, Recent Updates, LearnZillion Platform Updates, Enhanced Features and Functionality, “New Tools to Streamline Teacher Feedback: NOW LIVE Google Classroom grade pass back to optimize assignment grading and evaluation Updates to the My Assignments dashboard page (for students too!) New options for teachers to provide student feedback by item or by assignment.”

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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 Imagine Learning Illustrative Mathematics Grade 4 have a visual design (whether in print or digital) that supports students in engaging thoughtfully with the subject, and is neither distracting nor chaotic.

There is a consistent design within units and lessons that supports student understanding of the mathematics. According to the IM K-5 Curriculum Guide, Why is the curriculum designed this way?, 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.

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

The materials reviewed for Imagine Learning Illustrative Mathematics Grade 4 provide some teacher guidance for the use of embedded technology to support and enhance student learning, when applicable.

Imagine Learning Illustrative Mathematics provides videos for teachers to show how to use embedded technology. Examples include:

• Curriculum Guide, How do I navigate and use the LearnZillion platform? “We've compiled a few videos and lessons to help you learn more about navigating and using the materials. To get started, check out this video to learn more about how to navigate a LearnZillion Illustrative Math unit.

• Curriculum Guide, How do I navigate and use the LearnZillion platform? “Ready for more? Check out these resources which highlight features of the LearnZillion platform.” Videos include, “How do I navigate and use the features of a LearnZillion lesson? How do I personalize Illustrative Mathematics lessons on the LearnZillion platform?” A description of a video includes, “This page provides how-to's for copying lessons and making customizations for in-person and distance learning.”

• Curriculum Guide, How do I navigate and use the LearnZillion platform? Warming Up to Digital Items, “Looking for a way to prepare your students for digital activities and assessments? Check out this assessment, which is designed to expose students and teachers to the different question types you may encounter in a digital assessment. You can assign it to your students to give them practice with assessments and to also explore the data and information you receive back.”

## Report Overview

### Summary of Alignment & Usability for Imagine Learning Illustrative Mathematics K-5 Math | Math

#### Math K-2

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

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

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### Overall Summary

###### Alignment
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###### Usability
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