## Imagine Learning Illustrative Mathematics K-5 Math

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

Title ISBN Edition Publisher Year
Imagine Learning Illustrative Mathematics Student Workbook - Grade 1 Unit 1 978-1-63870-088-3 Imaging Learning 2021
Imagine Learning Illustrative Mathematics Student Workbook - Grade 1 Unit 2 978-1-63870-089-0 Imaging Learning 2021
Imagine Learning Illustrative Mathematics Student Workbook - Grade 1 Unit 3 978-1-63870-090-6 Imaging Learning 2021
Imagine Learning Illustrative Mathematics Student Workbook - Grade 1 Unit 4 978-1-63870-091-3 Imaging Learning 2021
Imagine Learning Illustrative Mathematics Student Workbook - Grade 1 Unit 5 978-1-63870-092-0 Imaging Learning 2021
Imagine Learning Illustrative Mathematics Student Workbook - Grade 1 Unit 6 978-1-63870-093-7 Imaging Learning 2021
Imagine Learning Illustrative Mathematics Student Workbook - Grade 1 Unit 7 978-1-63870-094-4 Imaging Learning 2021
Imagine Learning Illustrative Mathematics Student Workbook - Grade 1 Unit 8 978-1-63870-095-1 Imaging Learning 2021
Imagine Learning Illustrative Mathematics Student Workbook - Grade 1 Unit 1 978-1-64885-802-4 Imaging Learning 2021
Imagine Learning Illustrative Mathematics Student Workbook - Grade 1 Unit 2 978-1-64885-803-1 Imaging Learning 2021
Imagine Learning Illustrative Mathematics Student Workbook - Grade 1 Unit 3 978-1-64885-804-8 Imaging Learning 2021
Imagine Learning Illustrative Mathematics Student Workbook - Grade 1 Unit 4 978-1-64885-805-5 Imaging Learning 2021
Imagine Learning Illustrative Mathematics Student Workbook - Grade 1 Unit 5 978-1-64885-806-2 Imaging Learning 2021
Imagine Learning Illustrative Mathematics Student Workbook - Grade 1 Unit 6 978-1-64885-807-9 Imaging Learning 2021
Imagine Learning Illustrative Mathematics Student Workbook - Grade 1 Unit 7 978-1-64885-808-6 Imaging Learning 2021
Imagine Learning Illustrative Mathematics Student Workbook - Grade 1 Unit 8 978-1-64885-809-3 Imaging Learning 2021
Imagine Learning Illustrative Mathematics Teacher's Guide - Grade 1 Unit 1 978-1-64885-852-9 Imaging Learning 2021
Imagine Learning Illustrative Mathematics Teacher's Guide - Grade 1 Unit 2 978-1-64885-853-6 Imaging Learning 2021
Imagine Learning Illustrative Mathematics Teacher's Guide - Grade 1 Unit 3 978-1-64885-854-3 Imaging Learning 2021
Imagine Learning Illustrative Mathematics Teacher's Guide - Grade 1 Unit 4 978-1-64885-855-0 Imaging Learning 2021
Imagine Learning Illustrative Mathematics Teacher's Guide - Grade 1 Unit 5 978-1-64885-856-7 Imaging Learning 2021
Imagine Learning Illustrative Mathematics Teacher's Guide - Grade 1 Unit 6 978-1-64885-857-4 Imaging Learning 2021
Imagine Learning Illustrative Mathematics Teacher's Guide - Grade 1 Unit 7 978-1-64885-858-1 Imaging Learning 2021
Imagine Learning Illustrative Mathematics Teacher's Guide - Grade 1 Unit 8 978-1-64885-859-8 Imaging Learning 2021
Imagine Learning Illustrative Mathematics Teacher's Guide - Grade 1 Teacher's Course Guide 978-1-64885-895-6 Imaging Learning 2021
Imagine Learning Illustrative Mathematics Student Workbook - Grade 2 Unit 1 978-1-63870-096-8 Imaging Learning 2021
Imagine Learning Illustrative Mathematics Student Workbook - Grade 2 Unit 2 978-1-63870-097-5 Imaging Learning 2021
Imagine Learning Illustrative Mathematics Student Workbook - Grade 2 Unit 3 978-1-63870-098-2 Imaging Learning 2021
Imagine Learning Illustrative Mathematics Student Workbook - Grade 2 Unit 4 978-1-63870-099-9 Imaging Learning 2021
Imagine Learning Illustrative Mathematics Student Workbook - Grade 2 Unit 5 978-1-63870-100-2 Imaging Learning 2021
Imagine Learning Illustrative Mathematics Student Workbook - Grade 2 Unit 6 978-1-63870-101-9 Imaging Learning 2021
Imagine Learning Illustrative Mathematics Student Workbook - Grade 2 Unit 7 978-1-63870-102-6 Imaging Learning 2021
Imagine Learning Illustrative Mathematics Student Workbook - Grade 2 Unit 8 978-1-63870-103-3 Imaging Learning 2021
Imagine Learning Illustrative Mathematics Student Workbook - Grade 2 Unit 9 978-1-63870-104-0 Imaging Learning 2021
Imagine Learning Illustrative Mathematics Student Workbook - Grade 2 Unit 1 978-1-64885-810-9 Imaging Learning 2021
Imagine Learning Illustrative Mathematics Student Workbook - Grade 2 Unit 2 978-1-64885-811-6 Imaging Learning 2021
Imagine Learning Illustrative Mathematics Student Workbook - Grade 2 Unit 3 978-1-64885-812-3 Imaging Learning 2021
Imagine Learning Illustrative Mathematics Student Workbook - Grade 2 Unit 4 978-1-64885-813-0 Imaging Learning 2021
Imagine Learning Illustrative Mathematics Student Workbook - Grade 2 Unit 5 978-1-64885-814-7 Imaging Learning 2021
Imagine Learning Illustrative Mathematics Student Workbook - Grade 2 Unit 6 978-1-64885-815-4 Imaging Learning 2021
Imagine Learning Illustrative Mathematics Student Workbook - Grade 2 Unit 7 978-1-64885-816-1 Imaging Learning 2021
Imagine Learning Illustrative Mathematics Student Workbook - Grade 2 Unit 8 978-1-64885-817-8 Imaging Learning 2021
Imagine Learning Illustrative Mathematics Student Workbook - Grade 2 Unit 9 978-1-64885-818-5 Imaging Learning 2021
Imagine Learning Illustrative Mathematics Teacher's Guide - Grade 2 Unit 1 978-1-64885-860-4 Imaging Learning 2021
Imagine Learning Illustrative Mathematics Teacher's Guide - Grade 2 Unit 2 978-1-64885-861-1 Imaging Learning 2021
Imagine Learning Illustrative Mathematics Teacher's Guide - Grade 2 Unit 3 978-1-64885-862-8 Imaging Learning 2021
Imagine Learning Illustrative Mathematics Teacher's Guide - Grade 2 Unit 4 978-1-64885-863-5 Imaging Learning 2021
Imagine Learning Illustrative Mathematics Teacher's Guide - Grade 2 Unit 5 978-1-64885-864-2 Imaging Learning 2021
Imagine Learning Illustrative Mathematics Teacher's Guide - Grade 2 Unit 6 978-1-64885-865-9 Imaging Learning 2021
Imagine Learning Illustrative Mathematics Teacher's Guide - Grade 2 Unit 7 978-1-64885-866-6 Imaging Learning 2021
Imagine Learning Illustrative Mathematics Teacher's Guide - Grade 2 Unit 8 978-1-64885-867-3 Imaging Learning 2021
Imagine Learning Illustrative Mathematics Teacher's Guide - Grade 2 Unit 9 978-1-64885-868-0 Imaging Learning 2021
Imagine Learning Illustrative Mathematics Teacher's Guide - Grade 2 Teacher's Course Guide 978-1-64885-896-3 Imaging Learning 2021
Imagine Learning Illustrative Mathematics Student Workbook - Grade K Unit 1 978-1-63870-080-7 Imagine learning 2021
Imagine Learning Illustrative Mathematics Student Workbook - Grade K Unit 4 978-1-63870-083-8 Imagine learning 2021
Imagine Learning Illustrative Mathematics Student Workbook - Grade K Unit 7 978-1-63870-086-9 Imagine learning 2021
Imagine Learning Illustrative Mathematics Student Workbook - Grade K Unit 7 978-1-64885-800-0 Imagine learning 2021
Imagine Learning Illustrative Mathematics Teacher's Guide - Grade K Unit 2 978-1-64885-845-1 Imagine learning 2021
Imagine Learning Illustrative Mathematics Student Workbook - Grade 5 Unit 1 978-1-63870-122-4 Imaging Learning 2021
Imagine Learning Illustrative Mathematics Student Workbook - Grade 5 Unit 2 978-1-63870-123-1 Imaging Learning 2021
Imagine Learning Illustrative Mathematics Student Workbook - Grade 5 Unit 3 978-1-63870-124-8 Imaging Learning 2021
Imagine Learning Illustrative Mathematics Student Workbook - Grade 5 Unit 4 978-1-63870-125-5 Imaging Learning 2021
Imagine Learning Illustrative Mathematics Student Workbook - Grade 5 Unit 5 978-1-63870-126-2 Imaging Learning 2021
Imagine Learning Illustrative Mathematics Student Workbook - Grade 5 Unit 6 978-1-63870-127-9 Imaging Learning 2021
Imagine Learning Illustrative Mathematics Student Workbook - Grade 5 Unit 7 978-1-63870-128-6 Imaging Learning 2021
Imagine Learning Illustrative Mathematics Student Workbook - Grade 5 Unit 8 978-1-63870-129-3 Imaging Learning 2021
Imagine Learning Illustrative Mathematics Student Workbook - Grade 5 Unit 1 978-1-64885-836-9 Imaging Learning 2021
Imagine Learning Illustrative Mathematics Student Workbook - Grade 5 Unit 2 978-1-64885-837-6 Imaging Learning 2021
Imagine Learning Illustrative Mathematics Student Workbook - Grade 5 Unit 3 978-1-64885-838-3 Imaging Learning 2021
Imagine Learning Illustrative Mathematics Student Workbook - Grade 5 Unit 4 978-1-64885-839-0 Imaging Learning 2021
Imagine Learning Illustrative Mathematics Student Workbook - Grade 5 Unit 5 978-1-64885-840-6 Imaging Learning 2021
Imagine Learning Illustrative Mathematics Student Workbook - Grade 5 Unit 6 978-1-64885-841-3 Imaging Learning 2021
Imagine Learning Illustrative Mathematics Student Workbook - Grade 5 Unit 7 978-1-64885-842-0 Imaging Learning 2021
Imagine Learning Illustrative Mathematics Student Workbook - Grade 5 Unit 8 978-1-64885-843-7 Imaging Learning 2021
Imagine Learning Illustrative Mathematics Teacher's Guide - Grade 5 Unit 1 978-1-64885-886-4 Imaging Learning 2021
Imagine Learning Illustrative Mathematics Teacher's Guide - Grade 5 Unit 2 978-1-64885-887-1 Imaging Learning 2021
Imagine Learning Illustrative Mathematics Teacher's Guide - Grade 5 Unit 3 978-1-64885-888-8 Imaging Learning 2021
Imagine Learning Illustrative Mathematics Teacher's Guide - Grade 5 Unit 4 978-1-64885-889-5 Imaging Learning 2021
Imagine Learning Illustrative Mathematics Teacher's Guide - Grade 5 Unit 5 978-1-64885-890-1 Imaging Learning 2021
Imagine Learning Illustrative Mathematics Teacher's Guide - Grade 5 Unit 6 978-1-64885-891-8 Imaging Learning 2021
Imagine Learning Illustrative Mathematics Teacher's Guide - Grade 5 Unit 7 978-1-64885-892-5 Imaging Learning 2021
Imagine Learning Illustrative Mathematics Teacher's Guide - Grade 5 Unit 8 978-1-64885-893-2 Imaging Learning 2021
Imagine Learning Illustrative Mathematics Teacher's Guide - Grade 5 Teacher's Course Guide 978-1-64885-899-4 Imaging Learning 2021
Imagine Learning Illustrative Mathematics Student Workbook - Grade 3 Unit 1 978-1-63870-105-7 Imaging Learning 2021
Imagine Learning Illustrative Mathematics Student Workbook - Grade 3 Unit 2 978-1-63870-106-4 Imaging Learning 2021
Imagine Learning Illustrative Mathematics Student Workbook - Grade 3 Unit 3 978-1-63870-107-1 Imaging Learning 2021
Imagine Learning Illustrative Mathematics Student Workbook - Grade 3 Unit 4 978-1-63870-108-8 Imaging Learning 2021
Imagine Learning Illustrative Mathematics Student Workbook - Grade 3 Unit 5 978-1-63870-109-5 Imaging Learning 2021
Imagine Learning Illustrative Mathematics Student Workbook - Grade 3 Unit 6 978-1-63870-110-1 Imaging Learning 2021
Imagine Learning Illustrative Mathematics Student Workbook - Grade 3 Unit 7 978-1-63870-111-8 Imaging Learning 2021
Imagine Learning Illustrative Mathematics Student Workbook - Grade 3 Unit 8 978-1-63870-112-5 Imaging Learning 2021
Imagine Learning Illustrative Mathematics Student Workbook - Grade 3 Unit 1 978-1-64885-819-2 Imaging Learning 2021
Imagine Learning Illustrative Mathematics Student Workbook - Grade 3 Unit 2 978-1-64885-820-8 Imaging Learning 2021
Imagine Learning Illustrative Mathematics Student Workbook - Grade 3 Unit 3 978-1-64885-821-5 Imaging Learning 2021
Imagine Learning Illustrative Mathematics Student Workbook - Grade 3 Unit 4 978-1-64885-822-2 Imaging Learning 2021
Imagine Learning Illustrative Mathematics Student Workbook - Grade 3 Unit 5 978-1-64885-823-9 Imaging Learning 2021
Imagine Learning Illustrative Mathematics Student Workbook - Grade 3 Unit 6 978-1-64885-824-6 Imaging Learning 2021
Imagine Learning Illustrative Mathematics Student Workbook - Grade 3 Unit 7 978-1-64885-825-3 Imaging Learning 2021
Imagine Learning Illustrative Mathematics Student Workbook - Grade 3 Unit 8 978-1-64885-826-0 Imaging Learning 2021
Imagine Learning Illustrative Mathematics Teacher's Guide - Grade 3 Unit 1 978-1-64885-869-7 Imaging Learning 2021
Imagine Learning Illustrative Mathematics Teacher's Guide - Grade 3 Unit 2 978-1-64885-870-3 Imaging Learning 2021
Imagine Learning Illustrative Mathematics Teacher's Guide - Grade 3 Unit 3 978-1-64885-871-0 Imaging Learning 2021
Imagine Learning Illustrative Mathematics Teacher's Guide - Grade 3 Unit 4 978-1-64885-872-7 Imaging Learning 2021
Imagine Learning Illustrative Mathematics Teacher's Guide - Grade 3 Unit 5 978-1-64885-873-4 Imaging Learning 2021
Imagine Learning Illustrative Mathematics Teacher's Guide - Grade 3 Unit 6 978-1-64885-874-1 Imaging Learning 2021
Imagine Learning Illustrative Mathematics Teacher's Guide - Grade 3 Unit 7 978-1-64885-875-8 Imaging Learning 2021
Imagine Learning Illustrative Mathematics Teacher's Guide - Grade 3 Unit 8 978-1-64885-876-5 Imaging Learning 2021
Imagine Learning Illustrative Mathematics Teacher's Guide - Grade 3 Teacher's Course Guide 978-1-64885-897-0 Imaging Learning 2021
Imagine Learning Illustrative Mathematics Student Workbook - Grade 4 Unit 1 978-1-63870-113-2 Imaging Learning 2021
Imagine Learning Illustrative Mathematics Student Workbook - Grade 4 Unit 2 978-1-63870-114-9 Imaging Learning 2021
Imagine Learning Illustrative Mathematics Student Workbook - Grade 4 Unit 3 978-1-63870-115-6 Imaging Learning 2021
Imagine Learning Illustrative Mathematics Student Workbook - Grade 4 Unit 4 978-1-63870-116-3 Imaging Learning 2021
Imagine Learning Illustrative Mathematics Student Workbook - Grade 4 Unit 8 978-1-63870-120-0 Imaging Learning 2021
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### Overall Summary

The materials reviewed for Imagine Learning Illustrative Mathematics Grade 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 meet expectations for practice-content connections.

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

### Focus & Coherence

The materials reviewed for Imagine Learning Illustrative Mathematics Grade 2 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 2 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.

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

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

The materials reviewed for Imagine Learning Illustrative Mathematics Grade 2 meet expectations for assessing grade-level content and if applicable, content from earlier grades. The materials for Grade 2 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, Adding and Subtracting within 100, End-of-Unit Assessment, Problem 2, “Jada has 40 stickers. She gets 13 more stickers. How many stickers does she have? Jada gave 15 stickers to Noah. How many stickers does Jada have now? Show your thinking using drawings, numbers, words.” (2.NBT.5, 2.OA.1)

• Unit 6, Geometry, Time, and Money, End-of-Unit Assessment, Problem 4, “1. Split the circle into 4 equal parts. 2. Explain why 4 fourths of the circle is the whole circle.” (2.G.3)

• Unit 7, Adding and Subtracting within 1,000, End-of-Unit Assessment, Problem 6, students “Find the value of each difference. Show your thinking. Use base-ten blocks if it helps. a. 528-315, b. 471-124, c. 600-594.” (2.NBT.7)

• Unit 8: Equal Groups, End-of-Unit Assessment, Problem 4, “For each number, decide whether the number is even or odd. Write each even number as the sum of 2 equal addends. 1. 6, 2. 11, 3. 14.” (2.OA.3)

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

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 2 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 2 standards by including in every lesson a Warm Up, one to three instructional activities, and Lesson Synthesis. Within Grade 2, students engage with all CCSS standards.

Examples of extensive work include:

• Unit 3, Measuring Length, Lessons 4 and 7, engage students in extensive work with 2.MD.3 (Estimate lengths using units of inches, feet, centimeters, and meters). In Lesson 4, Measure and Estimate in Centimeters, Cool-down, students estimate the length of a picture of a pencil in centimeters. “1. Estimate: I think the length of the pencil is about ___cm.” In Lesson 4, Measure and Estimate in Centimeters, Activity 1: Estimate Length in Centimeters, students are provided with different objects and are asked to estimate their lengths in centimeters. “Now look at the objects I gave each group and think about how long they are. Record your estimates on the recording sheet on your own.” In Lesson 7, Center Day 1, Activity 1: Introduce Estimate and Measure, students estimate and measure the length of objects in centimeters, inches, and feet. “Choose an object to measure and hold it up for students to see. What tool would you use to measure this object? Before measuring the object, you and your partner will both estimate the length of the object using the unit you chose. Your estimates do not have to be the same. Record your estimate.”

• In Unit 4, Addition and Subtraction on the Number Line, Lesson 2, Features of a Number Line, and Unit 5 Numbers to 1000, Lesson 1, How Do We Compose a Hundred? students engage with extensive work with 2.NBT.2 (Count within 1000; skip-count by 5s, 10s, and 100s)., as students choral count from 0-100 by 5s. Teacher Guidance states, “‘Count by 5, starting at 0.’ Record as students count. Stop counting and recording at 100.” In Lesson 3, Unlabeled Tick Marks, Cool Down Problem 1, students are provided a number line that begins at 15, includes large tick marks in intervals of 5 and small tick marks in intervals of 1. The numbers 15, 20 and 45 are included on the number line, and there are blanks below the other multiples of 5. “‘Complete each number line by filling in the missing labels with the number the tick mark represents.’ Part b, ‘Locate and label 37 on the number line.’” Problem 2 is similar, but includes blanks only below multiples of ten. Students are asked to locate and label 35 on the number line. In Unit 5, Lesson 1, How Do We Compose a Hundred?, students choral count from 0 to 300 by 10. The Teacher Guidance instructs the teacher to say, “‘Count by 10, starting at 0.’ Record as students count. ‘Record 10 numbers in each row. Then start a new row directly below. Stop counting and recording at 300.’” In Unit 6, Geometry, Time, and Money, Lesson 12, Count by 5 to Tell Time, students count by 5s on a clock. The Warm Up includes an image of an analog clock with the labels 5, 10, 15, etc. shown outside the clock adjacent to 1, 2, 3, etc. In Unit 7, Add and Subtract within 1,000, Lesson 1: Compare, Count on, and Count Back, students count by 100s. In Activity 2, students complete number lines. “‘1. Fill in the missing numbers. Does this number line show counting on by 10 or counting on by 100?’ The problem includes an image of a number line with labels at 502, 702, and 902, and blanks at 602, 802. Problems 2 and 3 are similar, but include different images on the number lines. On the Cool Down, students count by 100. Problem 2, ‘Complete the list of numbers to show counting on by 100. 552, ___, ___, 852, 952 Explain how you know your list shows counting on by 100 and not counting on by 10.’”

• Unit 6, Geometry, Time, and Money, Lesson 11, Tell Time with Halves and Quarters, and Lesson 12, Count by 5 to Tell Time, engages students in extensive work with 2.MD.7 (Tell and write time from analog and digital clocks to the nearest five minutes, using a.m. and p.m.) In Lesson 11, Tell Time with Halves and Quarters, Activity 2: Card Sort: Halves and Quarters, students use their understanding of halves and quarters of circles to match clock faces, partitioned circles, and the phrases “quarter past,” “half past,” and “quarter till.”  Student Task Statements, “Problem 1. Find matching sets of cards. Each set should have 3 cards. Be prepared to explain why they match. Problem 2. Write the time shown on each clock using the words half past, quarter past, or quarter till.” In Lesson 12, Count by 5 to Tell Time, Activity 1: Count by 5 on the Clock, students count by fives and tell time on an analog clock. The Teacher Guide, “Display the image of the clock that shows 4:30 with the minutes labeled on the outside. ‘Tell your partner 2 ways to read this time.’ (4:30 or half past 4) 30 seconds: partner discussion ‘How could you prove that the time is 4:30?’ (Each tick mark shows one minute. You could count by 1 to 30. You can count the minutes by 5. Start at the 12 and count by 5 for each number until you get to 6. 5, 10, 15, 20, 25, 30. So it is 4:30.)’ Display the image of the clock that shows 4:15. ‘What time does this clock show?’ (4:15 or quarter past 4). ‘When telling time, we can count by 5 to determine how many minutes have passed since the hour.’ Use a clock to demonstrate starting at 4:00 and moving the minute hand to the 1, 2, then 3, as you say, “4:00, 4:05, 4:10, 4:15. Give each group a set of cards. ‘You are going to continue counting by 5 to tell time. Take turns telling the time on your cards. Work together to put the cards in order based on the times they show.’” Student Task Statements, “1. Discuss 2 ways to read the time on this clock. 2. A clock is shown with the time 4:15. What time does this clock show? 3. Read the time on each clock card with your partner. Put the clocks in order based on the times they show.”

Examples of full intent include:

• Unit 3, Measuring Length, Lesson 4, Measure and Estimate in Centimeters, Lesson 5, Measure in Meters, Lesson 8, What is an Inch, and Lesson 9, From Feet to Inches meet the full intent of 2.MD.3 (Estimate lengths using units of inches, feet, centimeters, and meters.) Lesson 4, Measure and Estimate in Centimeters, Activity 1, students make estimates of objects lengths in centimeters, beginning with a notebook. The Teacher Guidance instructs the teacher to show the notebook next to a 10 cm tool, “‘Let’s look at another image of the object.’ Display the image or hold a folder next to a 10-centimeter tool.” After launching the activity, the teacher instructs students to estimate the length of different objects, “‘Now look at the objects I gave each group and think about how long they are. Record your estimates on the recording sheet on your own. When you and your partner finish, compare your estimates and explain why you think they are ‘about right.’” In Activity 2, students measure the objects and compare their estimates to the measurements. In Lesson 5, Measure in Meters, students estimate the length of an object in meters during the Cool Down, “Noah held a gecko at the zoo. The length of the gecko fit in his hands. He measured it and said it was about 13 meters long. Do you think his  measurement is correct? Why or why not?” In Lesson 8, What is an Inch?, Activity 2, students estimate the length of the sides of different shapes that are pictured in the materials, “1. Here is a rectangle. How long is the long side of the rectangle in inches? Estimate: ___ Measure the long side of the rectangle. Actual length: ___.” Problem 2 follows the same structure but includes a square, while Problem 3 is about a triangle. Finally, in Lesson 9, From Feet to Inches, Activity 2, students estimate lengths in feet, “Estimate the length of objects around the room. Say if you will measure in inches or feet.” The Lesson 9 Cool Down also provides an opportunity to estimate length in feet, “Tyler told Han that a great white shark is about 16 inches long, but Han disagrees. Han believes it would be about 16 feet long. Who do you agree with? Explain.”

• Unit 5, Numbers to 1000, Lesson 4, Write Three-digit Numbers, and Lesson 6, Represent Numbers in Different Ways, engages students with the full intent of 2.NBT.3 (Read and write numbers to 1000 using base-ten numerals, number names, and expanded form.) In Lesson 4, Write Three-digit Numbers, Activity 1: Place Value Riddles, students write the number of hundreds, tens, and ones, and represent the value as a three-digit number. The Teacher Guide, “Give students access to base-ten blocks. ‘I have 4 hundreds, 3 ones, and 2 tens. Which of these shows the total value written as a three-digit number? Explain how you know.’ Display 432, 234, 423. ‘You are going to solve number riddles using base-ten blocks.’” Student Task Statements, “Solve each riddle and write the three-digit number. Use the table to help you organize the digits. 1. I have 2 ones, 7 tens, and 6 hundreds. 2. I have 3 ones, 5 tens, and 2 hundreds. 3. I have 7 hundreds, 5 ones, and 3 tens. 4. I have 5 hundreds, no tens, and 9 ones. 5. I have 4 ones, 6 tens, and 3 hundreds. 6. I have 8 tens, 1 hundred, and no ones.” Lesson 5, Expanded Form of Numbers, Cool-down: Three-digit Numbers in Expanded Form, “1. Represent the number 375 as the sum of hundreds, tens, and ones. Expanded form: 2. Represent 200+40+7 as a three-digit number. Three-digit number:” Lesson 6, Represent Numbers in Different Ways, Activity 1: Numbers as Words, students use words to represent three-digit numbers. The Teacher Guide, “Display the anchor chart that shows the different forms of 253. Complete the chart together. “This number has ____ hundreds, ____ tens, and ____ ones.’ (2, 5, 3) ‘The expanded form of this number is ___. The three-digit number is ___. These other forms can help us think about writing a number using number names. What is this number?’ (two hundred fifty-three) Write the number name as the students say two hundred fifty-three. ‘Fifty-three has a hyphen because numbers with tens and ones representing 21 through 99 use a hyphen to show the 2 parts of a two-digit number.’” Activity 2: Represent the Numbers, Student Task Statements, “Represent the number on your poster. Be sure to represent the number using: a three-digit number, a base-ten diagram, expanded form, words.” Cool-down: Words and Other Ways, “1. Represent 147 with words. 2. Represent 147 in one other way.”

• Unit 8, Equal Groups, Lesson 9 and Lesson 10, students engage in the full intent of 2.OA.4 (Use addition to find the total number of objects arranged in rectangular arrays with up to 5 rows and up to 5 columns; write an equation to express the total as a sum of equal addends). In Lesson 9, A Sum of Equal Addends, Activity 1: Sums of Rows and Sums of Columns, students recognize that an expression with equal addends can represent the sum of the number of objects in each row or the sum of the number of objects in each column. “‘Mai and Diego represented the number of objects in the same array with different expressions. Diego wrote 2+2+2+2+2+2. Mai wrote 6+6. Who do you agree with? Work with a partner to decide who you agree with and be prepared to explain your reasoning.’” In Lesson 10, Write Expressions and Equations to Represent Arrays, Activity 1: Build Arrays and Write Equations, students write equations that represent the number of objects in the rows or columns of an array. “‘First, you will arrange counters to make an array. Then you will write an equation that has equal addends. There are 2 equations that match each array. To find the total number of counters, you can use any method that makes sense to you.’”

#### Criterion 1.2: Coherence

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

The materials reviewed for Imagine Learning Illustrative Mathematics Grade 2 meet expectations for coherence. The materials: address the major clusters of the grade, have supporting content connected to major work, make connections between clusters and domains, and have content from prior and future grades connected to grade-level work.

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

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

The materials reviewed for Imagine Learning Illustrative Mathematics Grade 2 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 7 out of 9, approximately 78%.

• The number of lessons devoted to major work of the grade (including assessments and supporting work connected to major work) is 122 lessons out of 155 lessons, approximately 79%. The total number of lessons include 114 lessons plus 8 assessments for a total of 122 lessons.

• The number of days devoted to major work of the grade (including assessments and supporting work connected to major work) is 131 days out of 163 days, approximately 80%.

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 79% of the instructional materials focus on major work of the grade.

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

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 2 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, Measuring Length, Lesson 15, Create Line Plots, Activity 1: Measure and Plot Pencil Lengths connects the supporting work of 2.MD.9 (Generate measurement data by measuring lengths of several objects to the nearest whole unit, or by making repeated measurements of the same object. Show the measurements by making a line plot, where the horizontal scale is marked off in whole- number units) to the major work of 2.MD.1 (Measure the length of an object by selecting and using appropriate tools such as rulers, yardsticks, meter sticks, and measuring tapes.) Student Task Statements, the teacher provides the groups with 10-12 pencils of various lengths, “1. Measure the pencils in centimeters. Work with a partner and check each other’s measurements. Record each measurement in the table. 2. Create a line plot to represent the lengths of all the pencils in your group.”

• Unit 6, Geometry, Time, and Money, Lesson 18, Money Problems, Activity 1: Shop for School Supplies, connects the supporting work of 2.MD.8 (Solve word problems involving dollar bills, quarters, dimes, nickels, and pennies, using \$ and ¢ symbols appropriately) to the major work of 2.OA.1 (Use addition and subtraction within 100 to solve one- and two-step word problems involving situations of adding to, taking from, putting together, taking apart, and comparing with unknowns in all positions, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem). Students engage in the task, “Lin has these coins: 2 quarters, 3 dimes, 1 nickel. a. How much money does Lin have for supplies? b. If Lin buys an eraser, how much money will she have left? Show your thinking.”

• Unit 8, Equal Groups, Lesson 5, Patterns with Even and Odd Numbers, Cool-down: Odd One Out, connects the supporting work of 2.OA.3 (Determine whether a group of objects (up to 20) has an odd or even number of members, e.g., by pairing objects or counting them by 2s; write an equation to express an even number as a sum of two equal addends.) to the major work of 2.OA.2 (Fluently add and subtract within 20 using mental strategies.) Student Task Statements, “1. Elena has 8 counters. Does she have an even or odd number of counters? Explain or show your reasoning. 2. Without adding, explain which one of these expressions represents an odd number. A. 4+4, B. 8+1, C. 8+2.”

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

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

The materials reviewed for Imagine Learning Illustrative Mathematics Grade 2 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, Measuring Lengths, Lesson 11, Saree Silk Stories, Necklaces and Bracelets connects the major work of 2.MD.B (Relate addition and subtraction to length) to the major work of 2.NBT.B (Use Place Value Understanding and Properties of Operations to Add and Subtract). In Cool-down: More Saree Ribbon, students solve subtraction problems within 100 with the unknown in all positions. Student Task Statements, “Priya had a piece of ribbon that was 74 inches long. She cut off 17 in. How long is Priya’s ribbon now? Show your thinking. Use a diagram if it helps. Don’t forget the unit in your answer.”

• Unit 4, Addition and Subtraction on the Number Line, Lesson 2, Features of a Number Line connects the major work of 2.MD.B (Relate addition and subtraction to length) to the major work of 2.NBT.A (Understand place value). In Activity 2: Analyze Number Lines, students analyze number lines to determine whether they represent numbers within 10 as lengths from 0. Student Task Statements, “1. How should Jada revise her number line? 2. How should Andre revise his number line? 3. Elena’s number line. How should Elena revise her number line? 4. Fill in the numbers to create your own number line.”

• Unit 8, Equal Groups, Lesson 12, Partition Rectangles Into Squares connects the supporting work of 2.OA.C (Work with equal groups of objects to gain foundations for multiplication) to the supporting work of 2.G.A (Reason with shapes and their attributes). In Activity 1, How Many Squares?, students partition rectangles into rows and columns. Teaching Notes, “Use 8 tiles to make a rectangle. Your tiles should be in 2 rows now, draw lines in the rectangle to show the squares. It should have the same number of equal-size squares as the rectangle you made out of tiles. You may use a ruler if it helps you.”

• Unit 9, Putting It All Together, Lesson 3, Measure on a Map, Activity 2, connects the major work of 2.MD.A (Measure and estimate lengths in standard units) with the major work of 2.MD.B (Relate addition and subtraction to length). Students use maps to answer measurement questions. Student Task Statements, “Use your map and the stories from the previous activity to answer the questions. Represent each story with an equation with a symbol for the unknown length. 1. How much longer is the total length of Diego’s trip than the total length of Lin’s trip? 2. How much longer is the total length of Diego’s trip than the total length of Noah’s trip? 3. How much shorter is the total length of Noah’s trip than the total length of Lin’s trip?”

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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 2 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 3, Measuring Length, Section B, Customary Measurement, Section narrative, connects work with 2.MD.1 (Measure the length of an object by selecting and using appropriate tools such as rulers, yardsticks, meter sticks, and measuring tapes.), 2.MD.2 (Measure the length of an object twice, using length units of different lengths for the two measurements; describe how the two measurements relate to the size of the unit chosen.), 2.MD.3 (Estimate lengths using units of inches, feet, centimeters, and meters.), 2.MD.4 (Measure to determine how much longer one object is than another, expressing the length difference in terms of a standard length unit.), 2.MD.5 (Use addition and subtraction within 100 to solve word problems involving lengths that are given in the same units, e.g., by using drawings (such as drawings of rulers) and equations with a symbol for the unknown number to represent the problem.), 2.NBT.5 (Fluently add and subtract within 100 using strategies based on place value, properties of operations, and/or the relationship between addition and subtraction.), 2.OA.2 (Fluently add and subtract within 20 using mental strategies) to work done in grade 3. “As in the previous section, students make choices about the tool to use based on the length of the object being measured (MP5) and measure the length of the same object in both feet and inches. They begin to generalize that when they use a longer length unit, fewer of those units are needed to span the full length of the object. This understanding is a foundation for their work with fractions in grade 3 and beyond.”

• Unit 6, Geometry, Time, and Money, Lesson 9: You Ate the Whole Thing, About this lesson, connects 2.G.3 (Partition circles and rectangles into two, three, or four equal shares, describe the shares using the words halves, thirds, half of, a third of, etc., and describe the whole as two halves, three thirds, four fourths. Recognize that equal shares of identical wholes need not have the same shape) and 2.NBT.2 (Count within 1000; skip-count by 5s, 10s, and 100s) to the work in grade 3. “In this lesson, students continue to practice partitioning circles and describe halves, thirds, and quarters of circles using the language a half of, a third of, and a quarter of to describe a piece of the shape. They also use this language to describe the whole shape as a number of equal pieces. Students recognize that a whole shape can be described as 2 halves, 3 thirds, or 4 fourths. This understanding is the foundation for students' work with a whole and fraction equivalency in grade 3.”

• Unit 8, Equal Groups, Lesson 9, Activity 3: Add It All Up connects 2.OA.3 (Determine whether a group of objects (up to 20) has an odd or even number of members, e.g., by pairing objects or counting them by 2s; write an equation to express an even number as a sum of two equal addends) and 2.OA.4 (Use addition to find the total number of objects arranged in rectangular arrays with up to 5 rows and up to 5 columns; write an equation to express the total as a sum of equal addends) to work in grade 3. Activity 3, “The purpose of this activity is for students to determine the total number of objects in an array and match expressions to arrays by paying attention to the number of objects in each row and the number of objects in each column. For example, students recognize that 3 rows with 4 in each row would be 4+4+4. The arrays in this task provide students opportunities to compare different ways an array could be decomposed to find the total number of objects” when students compare the different ways they find the total number of objects in the array to expressions that use equal addends to represent the sums of rows or sums of columns. (3.OA.1)

Examples of connections to prior knowledge include:

• Unit 1, Adding, Subtracting, and Working with Data, Lesson 1, Add and Subtract Within 10, Warm-up: Notice and Wonder, A Picture of Shapes connects 2.G.1 (Recognize and draw shapes having specified attributes, such as a given number of angles or a given number of equal faces…) to the work in grade 1, “Students develop fluency with addition and subtraction within 10 in grade 1. This lesson provides an opportunity for formative assessment of students' fluency within 10, including recognizing sums with a value of 10 (1.OA.6).”

• Unit 4, Addition and Subtraction on the Number Line, Lesson 4, Compare Numbers on a Number Line, About this lesson, connects 2.MD.6 (Represent whole numbers as lengths from 0 on a number line diagram with equally spaced points corresponding to the numbers 0, 1, 2, ..., and represent whole-number sums and differences within 100 on a number line diagram.), 2.NBT.5 (Fluently add and subtract within 100 using strategies based on place value, properties of operations, and/or the relationship between addition and subtraction.) to work done in grade 1. “In this lesson, students recognize that as you move to the right on the number line, numbers increase in value because they are a greater distance from 0. Students also use the relative position of numbers and generalize that a number that is greater than a given number if it is farther to the right on the number line. To demonstrate this understanding, students compare numbers within 100 (a skill from grade 1) and use the number line to explain their comparison (MP7).”

• Unit 5, Numbers to 1,000, Lesson 1, How Do We Compose a Hundred?, About this lesson, connects 2.NBT.1 (Understand that the three digits of a three-digit number represent amounts of hundreds, tens, and ones; e.g., 706 equals 7 hundreds, 0 tens, and 6 ones) and 2.NBT.2 (Count within 1000; skip-count by 5s, 10s, and 100s) to the work in grade 1. “In grade 1, students were introduced to a ten as a unit made of 10 ones. They used that understanding to represent two-digit numbers and add within 100. Students used connecting cubes to make and break apart two-digit numbers. In previous units in grade 2, students used the words compose and decompose as they made and broke apart tens when they added and subtracted within 100.”

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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 2 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 2, there are 163 days of instruction including:

• 145 lesson days

• 18 days of assessments

There are 9 units with each unit containing 10 to 28 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 2 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 2 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 2 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 5, Numbers to 1000, Lesson 8, Activity 2, students develop conceptual understanding as they use their place value understanding to locate numbers on a number line. Students are given number lines where the tick marks are not labeled with a starting number, 10 length units marked with tick marks, and an ending number. Students determine the size of the units based on the range of the number line. “‘Locate and label each number on the number line. Label the tick marks with the numbers they represent if it helps.’ 1. 700 Number line shown has a range of 0 to 1000; 2. 472 Number line shown a range of 470 to 480.” (2.MD.6)

• Unit 7, Adding and Subtracting within 1,000, Lesson 8, Warm-up, students develop conceptual understanding as they use grouping strategies to describe amounts represented with base-ten diagrams. An image is provided that shows base ten blocks in hundreds, tens, and ones. Student Task Statements, “How many do you see? How do you see them?” (2.NBT.7)

• Unit 8, Equal Groups, Lesson 2, Warm-up, students develop conceptual understanding as they compare four images to determine which group of two does not belong with the other pairs. An image of different colored socks are shown. “Pick one that doesn’t belong. Be ready to share why it doesn’t belong. Discuss your thinking with your partner.” (2.OA.C)

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 5, Numbers to 1,000, Lesson 3, Cool-down, students use base-ten representations to demonstrate their understanding of the digits in three-digit numbers. “How many of each? 1. There are ___ hundreds. 2. There are ___ tens. 3. There are ___ ones. 4. Draw a base-ten diagram to represent the same total value with the fewest number of blocks.” An image shows base-ten blocks with 2 hundreds, 11 tens, and 12 ones. (2.NBT.1)

• Unit 7, Adding and Subtracting Within 1,000, Lesson 13, Decompose Tens and Hundreds, Cool-Down, students demonstrate conceptual understanding as they add and subtract within 1,000 using concrete models or drawings and strategies based on place value. “Find the value of 519 - 236. Show your thinking. Sample student response, 283. Sample responses: Students draw a base-ten diagram that shows 519 as 5 hundreds, 1 ten, and 9 ones. Students show decomposing a hundred to make 10 tens. Students cross out 2 hundreds, 3 tens, and 6 ones. Labels or equations clearly show the difference as 283.” (2.NBT.7)

• Unit 8, Equal Groups, Lesson 12, Cool-down, students demonstrate conceptual understanding as they use addition to find the total number of objects arranged in rectangular arrays and write equations to express the total as a sum of equal addends. “Write an equation that represents the number of squares in the rectangle?” An image of a rectangle is shown to demonstrate they have partitioned into rows and columns in a previous problem. (2.OA.4)

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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 2 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 1, Adding, Subtracting, and Working with Data, Lesson 3, Warm-up, students use mental strategies for adding and subtracting. Student Task Statements, “Find the value of each expression mentally. 7+3, 10-7, 10-2, 10-4.” (2.OA.2)

• Unit 2, Add and Subtract Within 100, Lesson 5, Warm-up, students use mental strategies to subtract. Launch, “Find the value of each expression mentally. 17-7; 17-8; 26-6; 26-8.” (2.NBT.5, 2.OA.2)

• Unit 9, Putting It All Together, Lesson 8, Warm-up, students have an opportunity to strengthen number sense and procedural fluency. Launch, “Find the value of each expression mentally: 9+5; 20+30; 29+35; 229+435.” (2.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, Addition and Subtraction on the Number Line, Lesson 11, Activity 2, students develop fluency with addition and subtraction within 100. Student Task Statements, “Partner A 1. Find the value of 59+27.  2. Find the value of 65-18. Partner B 1. Find the value of 68-39.  2. Find the value of 22+49.” (2.NBT.5)

• Unit 5, Number to 1,000, Lesson 7, Activity 2, Narrative, “Before playing, students remove the cards that show 0 and set them aside. Students use digit cards to make addition and subtraction equations true. They work with sums and differences within 100 with composing and decomposing. Each digit card may only be used one time on a page.” (2.NBT.5)

• Unit 9, Putting It All Together, Lesson 1, Cool-down, students add and subtract to find the value of each expression. Student Task Statements, “Find the value of each expression 1. 11-5. 2. 12-3. 3. 16-8. 4. 9+3. 5. 8+8. 6. 13-8.” (2.OA.2)

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

The materials reviewed for Imagine Learning Illustrative Mathematics Grade 2 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 2, Adding and Subtracting Within 100, Lesson 11, Activity 1, students use addition and subtraction to represent and solve real-world problems. (2.OA.1) Activity, “6 minutes: independent work time. As students work, consider asking: What do you need to find to answer the question? How do you know? How did you show Diego’s seeds? How did you show Jada’s seeds? How will you find the difference?” Student Task Statements, Problem 1, “Diego gathered 42 orange seeds. Jada gathered 16 apple seeds. How many more seeds did Diego gather than Jada? Show your thinking.”

• Unit 4, Addition and Subtraction on a Number Line, Lesson 13, Activity 2, students represent lengths on a number line diagram. (2.MD.6) Student Task Statements, Problem 1, “Clare started with 24 cubes and added on some more. Clare made a train with 42 cubes. How many cubes did Clare add on?” Images of a number line from 0-80 with intervals of 5, and a blank tape diagram are included.

• Unit 6, Geometry, Time, and Money, Lesson 11, Activity 1, students select the clock that represents a  given time (2.MD.7). Student Task Statements, “1. Circle the clock that shows 4 o’clock. Why doesn’t the other clock show 4 o’clock? 2. Circle the clock that shows half past 7. Why doesn’t the other clock show half past 7? You are going to look at some analog clocks and think about how they work to show different times. As needed, review how to read time presented in a digital format.”

Examples of non-routine applications include:

• Unit 7, Adding and Subtracting Within 1000, Lesson 16, Activity 1, students add and subtract within 1000 using place value strategies and explain why the strategies work using  real-world problems (2.NBT.7, 2.NBT.9). Launch, “Take a minute to make sense of Lin’s subtraction. 1–2 minutes: quiet think time.” Student Task Statements, “Jada says you can’t decompose to find the value of 402−298 because there aren’t any tens. Do you agree with Jada? Use base-ten blocks, diagrams, or other representations to show your thinking.”

• Unit 9, Putting It All Together, Lesson 5, Activity 1, students use place value to represent three digit numbers (2.NBT.1). Student Task Statements, “1. Start with 2 hundreds. Grab a handful of tens and of ones. a. What number do your base-ten blocks represent? _____ b. Represent the same number in another way. Show your thinking using diagrams, symbols, or other representations. 2. Combine your blocks with your partner’s blocks. a. What number do your base-ten blocks represent? _____ b. Represent the same number in another way. Show your thinking using diagrams, symbols, or other representations. 3. Represent your group’s number in the following ways: a. without hundreds b. without tens c. without hundreds or tens.”

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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 2 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 2. Examples where instructional materials attend to conceptual understanding, procedural skill and fluency, or application include:

• Unit 5, Numbers to 1000, Lesson 9, Activity 2, students deepen their conceptual understanding as they compare three-digit numbers using different representations and represent numbers on a line diagram. Activity, “In the last activity, we saw that Jada found it helpful to use the number line to explain that 371 is greater than 317. In this activity, you will compare three-digit numbers and explain your thinking using the number line. 6 minutes: independent work time. Compare your answers with a partner and use the number line to explain your reasoning.” Student Task Statements, “1. Locate and label 420 and 590 on the number line. 2. Estimate the location of 378 and 387 on the number line. Mark each number with a point. Label the point with the number it represents. 3. Diego and Jada compared 2 numbers. Use their work to figure out what numbers they compared. Then use <, >, and = to compare the numbers. 4. Which representation was most helpful to compare the numbers? Why?” (2.MD.6, 2.NBT.4)

• Unit 9, Putting It All Together, Lesson 4, Activity 1, students develop fluency as they answer questions in a data table and add and subtract. Launch, “Groups of 3–4. Give each student an unsharpened pencil and a centimeter ruler. Without measuring it, estimate the length of a brand new pencil. 30 seconds: quiet think time. Share responses. Measure the pencil to the nearest centimeter. (18 cm) 1 minute: group work time. Share responses.” Activity, “Display the table. The table shows the length of pencils from 4 different student groups. ‘Find the length of your own pencil and share it with your group. Record your group’s measurements in the table.’ 4 minutes: group work time. ‘Use the table to find the total length of each group’s pencils.’” Student Task Statements, “1. Measure the length of your pencil. ___ cm. 2. Write the lengths of your group’s pencils in the table. 3. Find the total length of each group’s pencils.” (2.MD.1, 2.NBT.5, 2.OA.2)

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

• Unit 4, Addition and Subtraction on the Number Line, Lesson 12, Equations with Unknowns, Activity 1: Number Line Riddles, students develop conceptual understanding and procedural fluency as they solve addition and subtraction problems within 100 with the unknown in all positions. Student Task Statements, “Solve riddles to find the mystery number. For each riddle: Write an equation that represents the riddle and write a ? for the unknown. Write the mystery number. Represent the equation on the number line. 1. I started at 15 and jumped 17 to the right. Where did I end? Equation: ____. Mystery number: ____. 2. I started at a number and jumped 20 to the left. I ended at 33. Where did I start? Equation: ____.  Mystery number: ____. 3. I started on 42 and ended at 80. How far did I jump? Equation:____. Mystery number:____. 4. I started at 76 and jumped 27 to the left. Where did I end? Equation: ____. Mystery number: ____.  5. I started at a number and jumped 19 to the right. I ended at 67. Where did I start? Equation: ____. Mystery number: ____. 6. I started at 92 and ended at 33. How far did I jump? Equation: ____. Mystery number: ____.” (2.MD.6, 2.NBT.5, 2.OA.1)

• Unit 8, Equal Groups, Lesson 1, Cool-down, students use conceptual understanding as they apply their understanding to find ways to share numbers while determining if groups of numbers are even or odd. Student Task Statements, “Noah and Lin want to share 11 connecting cubes equally. How many will each student get? Will there be any leftovers? Show your thinking using diagrams, symbols, or other representations. You may use cubes if it helps.” (2.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 2 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 2 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, Measuring Length, Lesson 6, Activity 1, students “interpret and solve Compare problems involving length where the language suggests an incorrect operation.” Activity “‘What are different ways we could represent this problem?’ (tape diagram, equations, base ten blocks).” Activity Narrative, “At the end of the launch, students open their books and work to find the diagram that matches the story problem. This further helps them to visualize the quantities in the problem before they work to find a solution (MP1).” Student Task Statements, “1. Lin's pet lizard is 62 cm long. It is 19 cm shorter than Jada's. How long is Jada's pet lizard? b. Whose pet is longer? ___. b. Circle the diagram that matches the story. c. Solve. Show your thinking.”

• Unit 4, Addition and Subtraction on the Number Line, Lesson 5, Activity 1, students make sense of number line diagrams. Activity, “‘Look at each number line and record an estimate of the number that the point represents. 5 minutes: independent work time. ‘Compare each estimate with your partner and explain why you believe your answer is reasonable.’ 7 minutes: partner work time. Monitor for students who add tick marks or labels, including multiples of 10 or 5, to help identify the number. This activity continues on the next card.” Student Task Statements, “1. What number could this be? ____ (Number line. Scale 30 to 60 by 5's. Evenly spaced tick marks. Point plotted between 50 and 55.) 2. What number could this be? ____ (Number line. Scale 0 to 30, by 10's. Point plotted between 20 and 30.)” Activity Narrative, “For each successive number line, the given tick marks are farther apart so students need to rely more on their understanding of properties of the number line and the accuracy with which they can locate the given numbers depends on how much extra work they do thinking about other numbers which they can locate accurately (MP1).”

• Unit 6, Geometry, Time, and Money, Lesson 3, Activity 2, students “recognize and draw shapes that have a specific number of sides and corners, and specific side lengths”. Activity, “Now you will get a chance to pick your own attributes, draw your own shapes, and guess which attributes your partner picked.” Activity Narrative, “Students may persevere in problem solving if they look for or choose particular attributes that do not go together (MP1).” Student Task Statements, “Choose your own attributes. Circle an attribute from each row. Draw and name a shape with the attributes you chose. If you cannot draw the shape, explain why.”

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 1, Adding, Subtracting, and Working With Data, Lesson 15, Activity 2, students connect quantities to structures in the story problem. Activity, “8 minutes: partner work time. Monitor for students who explain how each diagram and equation matches the quantities in the context of the story problem. ‘Compare your matches with the matches of another group. If you have different matches, work together to explain which cards belong or why a card could belong to different groups.’ 4 minutes: small-group work time.” Student Task Statements, “Lin and Diego want to compare other things they collected and did at the beach. Student on the beach. Read a card with a story problem. Find cards that match the story problem. Explain why the cards match.” Activity Narrative, “When students analyze and connect the quantities and structures in the story problems, diagrams, and equations, they think abstractly and quantitatively (MP2) and make use of structure (MP7).”

• Unit 7, Add and Subtract within 1,000, Lesson 15, Activity 1, students “use their knowledge of base-ten diagrams and place value to make sense of a written method.” Activity, “Elena is finding the value of 726-558. Use base-ten blocks or a base-ten diagram to show Elena’s steps. Then finish Elena’s work. If you have time, work together to show a different way Elena could use numbers or equations to show her steps.” Activity Narrative, students “describe how numbers, words, and equations can be used to represent the steps they have used with other representations (MP2).”

• Unit 9, Putting It All Together, Lesson 3, Activity 1, students “measure lengths to the nearest centimeter and to find the total distance each student moves on the map”. Student Task Statements, “4. Find the total length of each student’s trip. Represent the total with an equation”. Activity Narrative, “When students measure and represent the trips with equations and find the total distance on the map, they reason abstractly and quantitatively (MP2).”

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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 2 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 3, Measuring Length, Lesson 9, Warm-up, students construct viable arguments as they practice the skill of estimating a reasonable length based on their experience and known information. Launch, “Groups of 2. Display the image. ‘About how long do you think the fish in the picture is in inches? What is an estimate that’s too high? Too low? About right?’ 1 minute: quiet think time.” Student Task Statements, “‘How long is this Cobia fish in inches? Record an estimate that is: too low, about right; too high.’ Image of Cobia fish included.” Activity Narrative, “This gives students an opportunity to share a mathematical claim including the assumptions they made when interpreting the image with limited information (MP3, MP4).”

• Unit 4, Addition and Subtraction on the Number Line, Lesson 5, Activity 2, Narrative, “Students use what they know about multiples of 10, the relative position of numbers on the number line, and comparing length to locate and label a set of numbers on the number line.” Lesson Narrative, students “construct viable arguments for how they placed the numbers and to critique the reasoning of others (MP3)”. Activity Narrative, “‘You will be working with your group to arrange the number cards on the number line. Take turns picking a card and placing it near its spot on the number line. Explain how you decided where to place your card.If you think you need to rearrange other cards, explain why. When you agree that you have placed all the numbers in the right spots, mark each of the numbers on your cards with a point on the number line. Label each point with the number it represents.’ 10 minutes: small-group work time.”

• Unit 6, Geometry, Time, and Money, Lesson 13, Activity 1, students construct viable arguments and critique the reasoning of others as they make sense of a visual representation of the hours in 1 day. Activity, “‘Cut out the two parts of the day and glue them together. Circle and label when you eat breakfast, lunch, and dinner on the diagram. Then shade in all the times you might be sleeping.’ 5 minutes: independent work time. ‘Share responses. This activity continues on the next card.’” Student Task Statements, “Use the materials your teacher gives you to create your own representation for the hours in a day. Circle and label when you eat breakfast, lunch, and dinner on the diagram. Shade in when you might be sleeping.” Activity Narrative, “Students have opportunities to develop logical arguments for why an event may happen during a.m. or p.m. hours and critique the arguments of others (MP3).”

Examples of critiquing the reasoning of others include:

• Unit 2, Add and Subtract within 100, Lesson 6, Activity 1, the narrative states, students “interpret and compare representations that show decomposing a ten to subtract by place.” Narrative, “Students compare and make connections between the representations and a set of equations that also shows how to find the value of the difference (MP3).” Activity, “‘Tyler found the value by using equations. Diego says Tyler’s equations match his diagram. Elena says the equations match her diagram. Who do you agree with?’ 2 minutes: independent work time.”

• Unit 5, Numbers to 1,000, Lesson 9, Activity 1, students critique the reasoning of others as they make sense of different methods they can use to compare three-digit numbers.” Activity, “‘Diego, Jada, and Clare were asked to compare 371 and 317. They each represented their thinking differently. Take some time to look over their methods.’ 2 minutes: independent work time. ‘Discuss with your partner how their methods are the same and different.’ 4 minutes: partner discussion. ‘Now try Jada’s way.’” Student Task Statements, “‘Diego: I see 3 hundreds for each number. 317 only has 1 ten, but 371 has 7 tens. 371>317.’ Image of base ten blocks: 3 hundreds, 7 tens, and 1 one and image of base ten blocks: 3 hundreds, 1 ten, 7 ones.” Activity Narrative, “They analyze the thinking of others and make connections across representations (MP2, MP3).”

• Unit 9, Putting It All Together, Lesson 13, Activity 2, students critique the reasoning of others as they make revisions to their own work after seeing their peers’ work. The narrative states that students ask “mathematical questions or leave feedback using precise math language (MP3).” Activity Narrative, “Use your sticky notes to leave comments or questions about the stories and solutions, including things that helped you understand the problem and solutions and any other representations you might add to the poster.” 7 minutes: partner work time “Make revisions to your own poster based on what you saw and discussed.” 3 minutes: independent work time.”

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

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

The materials reviewed for Imagine Learning Illustrative Mathematics Grade 2 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 1, Adding, Subtracting and Working With Data, Lesson 8, Activity 2, students model with mathematics as they interpret data represented in a picture graph. Student Task Statements, “Circle the 4 questions that can be answered using the graph. a. How many kids chose pizza? b. How many chose tacos or grilled cheese? c. Why did so many kids choose spaghetti? d. How many more kids chose pizza than tacos? e. What is the total number of kids who chose spaghetti or pizza?” Activity Narrative, “This type of reasoning helps students make sense of the mathematical elements in a context and interpret and use the data presented in the picture graph to answer questions (MP4).”

• Unit 8, Equal Groups, Lesson 8, Warm-up, students organize objects to find the total quicker. Activity Narrative, “Making an estimate or a range of reasonable answers with incomplete information is a part of modeling with mathematics (MP4).” Student Task Statements, students are shown many red and yellow counters in a random order. Launch, “How many counters do you see? What is an estimate that’s too high? Too low? About right?” Synthesis, “We saw different arrangements of the same number of counters. Which one makes it easier to tell how many there are altogether? Explain. (In the last way, it is easy to see it is 10 and another 10. It looks like 2 10-frames.) Refer to the counters in an array. Organizing the circles into arrays can help us see ways to find the total more quickly. How could I use skip counting to find the total number of counters? (We could count by 5 or 2.)”

• Unit 9, Putting It All Together, Lesson 10, Activity 2, students analyze answers and determine  what the question could have been. Student Task Statements, “Clare picked 51 apples. Lin picked 18 apples and Andre picked 19 apples.” Lesson Narrative, “Determining the relationships between quantities and using them to ask questions and solve problems is an aspect of modeling with mathematics (MP4).” Synthesis, “How did you know the student was trying to find a total amount? (The tape diagram and student work shows addition.) Why do you think the student added 51 and 19 rather than 51 and 18? (They make 70 together. That way, there is no need to make a ten when you add the third number.)” Synthesis, “How did you know the question might be about comparing? Why not a question about taking away? (The operation is subtraction but Lin’s apples and Andre’s apples aren’t taken away from Clare’s apples. The diagram helps see that it is a comparison.) What strategies can you use to calculate ? (Use the number line. Make a drawing. Subtract 20 and add 1, subtract 20 more and add 2.)”

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 3, Measuring Length, Lesson 5, Activity 1, students choose appropriate tools strategically as they experience the need for a longer length unit and measuring tool. Activity Narrative, “They can choose to measure with centimeter cubes, 10-centimeter tools, their self-made rulers, or centimeter rulers (MP5).” Student Task Statements, “1. Measure to find the length of each reptile. Don’t forget the unit. a. What is the length of a gila monster? b. What is the length of a baby alligator? c. What is the length of a baby cobra? d. What is the length of a komodo dragon?”

• Unit 4, Addition and Subtraction of the Number Line, Lesson 1, Activity 2, “Students choose their own length unit to make equally spaced tick marks and label them 0–20. In order to make an accurate number line, students will need to make strategic use of materials in order to measure the units on their number line. This could be a paper clip or a staple or the equally spaced lines on a lined sheet of paper (MP5).” Student Task Statements, “1. Make a number line that goes from 0 to 20. 2. Locate 13 on your number line. Mark it with a point. 3. Locate 3 on your number line. Mark it with a point. 4. Compare your number line with your partner’s.” Activity Narrative, “Now you’re going to create your own number line. You can use any of the tools provided to create a number line that represents the numbers from 0 to 20.”

• Unit 5, Numbers to 1,000, Lesson 12, Activity 2, students use tools strategically as they consider insights gained from number lines. Activity Narrative, “Students reflect on how the number line can help us organize numbers (MP5).” Student Task Statements, “Estimate the location of 839, 765, 788, 815, and 719 on the number line. Mark each number with a point. Label the point with the number it represents. Order the numbers from least to greatest. _____, _____, _____, _____, _____.”

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

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

The materials reviewed for Imagine Learning Illustrative Mathematics Grade 2 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 3, Measuring Length, Lesson 14, Activity 2, students use precision when analyzing a given line plot and its features (MP6). Launch, “‘What do the numbers on our line plot represent? What does the way the numbers are arranged remind you of? (The numbers represent lengths in inches. It reminds me of a ruler. It has tick marks and each tick mark is the same length apart.) The line on a line plot represents the unit you use to measure. It shows numbers in order and the same length apart, just like on a ruler. What length unit do the numbers on our line plot represent? How could we label this? (The lengths of our hand spans in inches, measurement in inches)’ 1 minute: quiet think time. ‘Share responses and record a label. The length of the line between two numbers does not have to match the unit you used, so it's important to label the line on the line plot with the unit.’”

• Unit 4, Addition and Subtraction on the Number Line, Lesson 11, Warm-up, Activity Narrative, “In reasoning together about the number line representation, and connecting the strategy of making a ten to jumping to the nearest ten, students need to be precise in their word choice and use of language (MP6).” Launch, “‘Display one expression. Give me a signal when you have an answer and can explain how you got it.’ 1 minute: quiet think time.” Synthesis, “For 52−7 some students decomposed the 7 to make it easier to get to a ten. How does this number line representation connect to that strategy? Draw a number line showing 52 represented with a point, a jump of 2, and then a jump of 5.”

• Unit 8, Equal Groups, Lesson 7, Activity 2, Activity Narrative, “They use this vocabulary to describe arrays and create arrays given a number of counters and a number of rows (MP6).” Student Task Statements, Image shows an array with 3 rows of 2. “a. How many rows are in this array? b. How many counters are in each row? c. How many counters are there in all?”

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

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

The materials reviewed for Imagine Learning Illustrative Mathematics Grade 2 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 1, Adding, Subtracting, and Working with Data, Lesson 3, Cool-down, students look for and make use of structure as they find the missing numbers in given equations. Lesson Narrative, “Throughout the lesson, students have opportunities to use what they know about the structure of whole numbers and the relationship of addition and subtraction to find the unknown numbers and explain their methods (MP3, MP7).” Student Task Statements, “Find the number that makes each equation true. 1. ___ + 17 = 20, 2. 20 - 9 = ___ , 3. ___ + 5 = 20, 4. 20 - ___ = 8.”

• Unit 2, Adding and Subtracting Within 100, Lesson 8, Warm-up, students look for and make use of structure as they use the relationship between addition and subtraction to find the value of expressions. Activity Narrative, “When they describe ways to use the value of the sums to find the value of the differences, they look for and make use of the structure of expression and the relationship between addition and subtraction (MP7).” Student Task Statements, “Find the value of each expression mentally. 18+10+10; 18+20+10; 38−20; 48−30.” Launch, “‘Display one expression. Give me a signal when you have an answer and can explain how you got it.’ 1 minute: quiet think time.” Activity, “Record answers and strategy. Keep expressions and work displayed. Repeat with each expression. This activity continues on the following cards.” Activity Synthesis, “How are the addition expressions related to the subtraction expressions? (The second expression is the opposite of the last expression. They are in the same fact family. The first expression helped me solve the third expression because I know 18+20=38, so 38−20 must be 18.)”

• Unit 4, Addition and Subtraction on the Number Line, Lesson 4, Activity 1, students look for and make use of structure as they notice that the number farthest to right (on a number line) has a greater value. Activity Narrative, “Students recognize that given any two numbers, the number farther to the right represents a greater value than the number to the left (MP7).” Launch, “Groups of 2. Give each group 3 number cubes and 2 counters. Assign Partner A and B.” Activity, “‘You will use the number line you created and work with a partner. Decide with your partner whose number line you will use.’ As needed, demonstrate the task with a student. ‘I am Partner A. I am going to roll the 3 number cubes and find the sum. Then, I take a counter and place it on the number line to represent the sum. Now it’s my partner's turn. They do the same thing and put their counter on the same number line to represent the sum of their numbers.’” This activity continues on the next card. “Then, we decide which number is greater and explain how we know. Last, we use the < , >, or = symbols to record our comparison.” Activity Synthesis, “Invite 2–3 previously identified groups to share comparisons and their explanations. ‘What do you notice about the numbers that are farther to the right? (They were greater. They represent a greater length from zero.)’”

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 3, Measuring Length, Lesson 3, Warm-up, students look for and express regularity in repeated reasoning as they use one expression to help them find the value of another expression. Activity Narrative, “When they share how each expression helps them find the value of the next, they look for and express regularity in repeated reasoning (MP8).” Student Task Statements, “Find the value of each expression mentally. 63−3; 63−20; 63−23; 63−24” Launch, “Display one expression. ‘Give me a signal when you have an answer and can explain how you got it.’ 1 minute: quiet think time.” Activity, “Record answers and strategy. Keep problems and work displayed. Repeat with each expression. This activity continues on the following cards.”  Activity Synthesis, “Which expressions were easier to find mentally? Why? How did the third expression help you think about the fourth one?”

• Unit 5, Numbers to 1,000, Lesson 8, Activity 2, students look for and express regularity in repeated reasoning as they notice the tick marks on a number line are equally distanced apart. Activity Narrative, “Students may begin to notice that when the two ticks at the right and left of the number line are 100 apart, the individual tick marks go up by 10 and when the two tick marks at the right and left of the number line are 10 apart, the individual tick marks go up by 1 (MP8).” Launch, “Groups of 2” Activity, “‘Now you are going to locate and label three-digit numbers on the number line. Take a few minutes to try them on your own and be ready to explain to your partner.’ 5 minutes: independent work time. ‘Now compare with a partner and share your thinking.’ If students are not finished, they can work together. 5 minutes: partner discussion. Monitor for different ways students determine the unit represented on the number line for representing 940 such as: counting by ones, tens, and hundreds to see which one gets them to the ending number using the starting and ending numbers to determine what the unit must be.” Activity Synthesis, “Display the number line showing 900–1,000. Invite a student to demonstrate their strategy of trial and error to determine how to label the tick marks. Invite another student to demonstrate their strategy of reasoning about the starting and ending numbers. ‘(I know the difference between 900 and 1,000 is 100 and there are 10 length units. Each one must be 10 because there are 10 tens in a hundred.) What is the same and different about how they decided the unit on this number line? (They both counted to see how many tick marks were there. ____ tried counting by 1 and then 10, but ____ counted by 10 right away.)’”

• Unit 6, Geometry, Time, and Money, Lesson 12, Activity 2, students notice and use patterns as they practice telling and writing time. Activity Narrative, “When students look for shortcuts to tell the time (for example, counting on from 30 rather than 0 or counting back from 60), they are looking for and expressing regularity in repeated reasoning (MP8).” Student Task Statements, “‘Write the time shown on each clock.’ Students are then given 6 pictures of analog clocks.” Activity Synthesis, “How did ____ know you could start counting at 30? Why does this work? (They know it is 2:30 when the minute hand points to 6, so they can just start there. It works because you will still count the same numbers. If you start at 30 you still say 35, 40. It is just faster.)”

### Usability

The materials reviewed for Imagine Learning Illustrative Mathematics Grade 2 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 2 meet expectations for Teacher Supports. The materials: provide teacher guidance with useful annotations and suggestions for enacting the student and ancillary materials; contain adult-level explanations and examples of the more complex grade-level concepts and concepts beyond the current grade so that teachers can improve their own knowledge of the subject; include standards correlation information that explains the role of the standards in the context of the overall series; provide explanations of the instructional approaches of the program and identification of the research-based strategies; and provide a comprehensive list of supplies needed to support instructional activities.

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

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

The materials reviewed for Imagine Learning Illustrative Mathematics Grade 2 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 2 include: extending understanding of the base-ten number system, building fluency with addition and subtraction, using standard units of measure, and describing and analyzing shapes.”

• Unit 3, Measuring Length, Section A, Metric Measurement, Section Overview, Section Narrative, provides an overview of the content and expectations for the section. “This section introduces two metric units: centimeter and meter. Students use base-ten blocks, which have lengths of 1 centimeter and 10 centimeters, to measure objects in the classroom and to create their own centimeter ruler. Students iterate the 1-centimeter unit Just as they had done with non-standard units in grade 1. Students relate the side length of a centimeter cube to the distance between tick marks on their ruler. They see that each tick mark notes the distance in centimeters from the 0 mark, and that the length units accumulate as they move along the ruler and away from 0. Students then compare the ruler they created to a standard centimeter ruler. They learn the importance of placing the end of an object at 0 and discuss how the numbers on the ruler represent lengths from 0. Students also learn about a longer unit in the metric system, meter, and use it to estimate lengths. They have opportunities to choose measurement tools and to do so strategically (MP5), by considering the lengths of objects being measured. Students also measure the length of longer objects in both centimeters and meters, which prompts them to relate the size of the unit to the measurement. To close the section, students apply their knowledge of measurement to compare the lengths of objects and solve Compare story problems involving lengths within 100, measured in metric units.”

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, Numbers to 1,000, Lesson 5, provides teachers guidance on how to represent numbers up to 1,000. 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.” Activity, “‘Share and record answers and strategy.’ Repeat with each statement.” Activity Synthesis, “What is different about the last equation? (It’s not decomposed into hundreds, tens, and ones. 22 shows some tens and some ones and 10 shows another ten).”

• Unit 7, Adding and Subtracting within 1,000, Lesson 12, Lesson Synthesis provides teachers guidance on ways to decompose 10 to subtract within 1,000. “Today we saw that we can subtract by place with larger numbers, and sometimes a ten is decomposed. How did you know when a ten would be decomposed when you subtracted three-digit numbers? (I could tell when I looked at the ones place and saw I didn't have enough ones to subtract ones from ones.) How was this the same as when you subtracted two-digit numbers? How was it different? (It was just like when we subtracted two-digit numbers. It's different because one of the numbers has hundreds.)”

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

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

The materials reviewed for Imagine Learning Illustrative Mathematics Grade 2 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 4, The Nuances of Understanding a Fraction as a Number supports teachers with context for work beyond the grade. “In this blog post, Gray discusses the role the number line plays in student understanding of fractions as numbers.”

• Why is the curriculum designed this way? Further Reading, Unit 8, What is Multiplication?, “In this blog post, McCallum discusses multiplication beyond repeated addition—as equal groups. The foundation of this understanding is laid in this unit of grade 2.”

• Unit 1, Adding, Subtracting, and Working with Data, Lesson 16, Solve All Kinds of Compare Problems, “The number choices in the Compare problems in this lesson encourage students to use methods based on place value to find the unknown value. Students may look for ways to compose a ten or subtract multiples of ten when finding unknown values within 100. Students will subtract numbers other than multiples of ten within 100 in future lessons. Encourage students to use a tape diagram to make sense of the problem if it is helpful.”

• Unit 5, Numbers to 1,000, Lesson 14, Hundreds of Objects, 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 previous understandings and experiences with representations of numbers between 100 and 999. Students use their understanding of the base-ten structure of numbers to count and represent quantities of real-world objects (MP7). When students investigate the advantages and disadvantages of different methods of counting a large number of objects and then choose a method to use they critique the reasoning of others and model with mathematics (MP3, MP4).”

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

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

The materials reviewed for Imagine Learning Illustrative Mathematics Grade 2 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 2 standards breakdown, standards are addressed by lesson. Teachers can search a standard in the grade and identify the lesson(s) where it appears within materials.

• Course Guide, Lesson Standards, includes all Grade 2 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 8, Equal Groups, Lesson 10, the Core Standards are identified as 2.NBT.A.2, 2.OA.B.2, 2.OA.C.3, and 2.OA.C.4. 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 5, Numbers to 1,000, 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 this unit, students extend their knowledge of the units in the base-ten system to include hundreds. In grade 1, students learned that a ten is a unit made up of 10 ones, and two-digit numbers are formed using units of tens and ones. Here, they learn that a hundred is a unit made up of 10 tens, and three-digit numbers are formed using units of hundreds, tens, and ones. To make sense of numbers in different ways and to build flexibility in reasoning with them, students work with a variety of representations: base-ten blocks, base-ten diagrams or drawings, number lines, expressions, and equations.”

• Unit 8, Equal Groups, Unit Learning Goals, full Unit Narrative, include an overview of how the math of this module builds from previous work in math. “In this unit, students develop an understanding of equal groups, building on their experiences with skip-counting and with finding the sums of equal addends. The work here serves as the foundation for multiplication and division in grade 3 and beyond. Students begin by analyzing even and odd numbers of objects. They learn that any even number can be split into 2 equal groups or into groups of 2, with no objects left over. Students use visual patterns to identify whether numbers of objects are even or odd. Next, students learn about rectangular arrays. They describe arrays using mathematical terms (rows and columns). Students see the total number of objects as a sum of the objects in each row and as a sum of the objects in each column, which they express by writing equations with equal addends. They also recognize that there are many ways of seeing the equal groups in an array.”

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

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

The materials reviewed for Imagine Learning Illustrative Mathematics Grade 2 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 3, Measuring Length, Addition 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 2 Unit 3. In this unit, students measure and estimate lengths in standard units, and solve measurement problems within 100. Section A: Metric Measurement, Section B: Customary Measurement, Section C: Line Plots.” The guide also includes a Spanish language version.

• Course Overview, Unit 6, Geometry, Time, and Money, Additional Resources, Home School Connection, Family Support Material, Try It At Home!, “Near the end of the unit, ask your student to do the following tasks: Find different shapes around the house (bonus points for finding non-traditional shapes!). Tell time on an analog clock. Pull out some coins and determine the value of the coin combination. Questions that may be helpful as they work: How did you know it was (shape name)? How did you determine the time? What kind of coin is this? How much is it worth? How did you figure out the total value of the coin combination?”

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

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 2 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 4, “To learn more about the essential nature of the number line (which is introduced in this unit) in mathematics beyond grade 2, see: The Nuances of Understanding a Fraction as a Number. In this blog post, Gray discusses the role the number line plays in student understanding of fractions as numbers. Why is 3–5=3+(-5)? In this blog post, McCallum discusses the use of the number line in introducing negative numbers.” Unit 8, “What is Multiplication? In this blog post, McCallum discusses multiplication beyond repeated addition—as equal groups. The foundation of this understanding is laid in this unit of grade 2.”

• 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.”

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

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

The materials reviewed for Imagine Learning Illustrative Mathematics Grade 2 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 2 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. Pattern blocks are reusable materials used in lessons 2.6.6, 2.6.10, and 2.6.21. Included in the pattern blocks list is a note that 180 triangles and 120 each of other shapes are needed per 30 students. Cut out shapes from paper or cardstock and Virtual Pattern Blocks are suitable substitutes. Inch tiles are a reusable material used in lessons 2.3.8, 2.3.9, and 2.4.1. 360 inch tiles are needed for 30 students. Cut out inch squares from grid paper and Virtual Grid Paper are suitable substitutes for the material. Sticky notes are a consumable material used in lessons 2.2.18, 2.3.14, 2.4.1, and 2.5.14. Sticky notes are needed per 30 students. No suitable substitute is listed.

• Unit 4, Addition and Subtraction on the Number Line, Lesson 4, Activity 1: Compare the Numbers, Teaching Notes, Materials to gather, “Give each group 3 number cubes and 2 counters.” Activity, “You will use the number line you created and work with a partner. Decide with your partner whose number line you will use. As needed, demonstrate the task with a student. I am Partner A. I am going to roll the 3 number cubes and find the sum. Then, I take a counter and place it on the number line to represent the sum. Now it’s my partner's turn. They do the same thing and put their counter on the same number line to represent the sum of their numbers. Then, we decide which number is greater and explain how we know. Last, we use the  <, =, or > symbols to record our comparison.”

• Unit 6, Geometry, Time, and Money, Lesson 7, Activity 1, Teaching Notes, Materials to gather, “Construction paper, Rulers, Scissors.” Launch, “Give each student 3 paper rectangles and access to scissors and rulers. In an earlier lesson, we thought about how shapes could be composed using equal-size smaller shapes. Today, we are going to decompose shapes into equal pieces and name the pieces. Each of you has 3 rectangles. First, cut out each rectangle. Next, fold each rectangle in different ways. You can use a ruler to draw lines first, if it is helpful. You each have 2 pieces. How can you check to see if they are equal? (If you lay them on top of each other, they are the same size.)” Activity, “Have extra paper on hand if students want to try again when making thirds.”

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

This is not an assessed indicator in Mathematics.

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

This is not an assessed indicator in Mathematics.

#### Criterion 3.2: Assessment

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

The materials reviewed for Imagine Learning Illustrative Mathematics Grade 2 partially meet expectations for Assessment. The materials identify the standards and the mathematical practices assessed in formal assessments. The materials provide multiple opportunities to determine students' learning and sufficient guidance to teachers for interpreting student performance but do not provide suggestions for follow-up. The materials include opportunities for students to demonstrate the full intent of grade-level standards and mathematical practices across the series.

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

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

The materials reviewed for Imagine Learning Illustrative Mathematics Grade 2 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 3, Adding and Subtracting within 100, End-of-Unit Assessment answer key, denotes standards addressed for each problem. Problem 3, 2.NBT.5, “Find the number that makes each equation true. Show your thinking using drawings, numbers, or words. a. 23+19=___. b. 75-36= ___.”

• Unit 4, Addition and Subtraction on the Number Line, End-of-Unit Assessment answer key, denotes standards addressed for each problem. Problem 3, 2.MD.6 and 2.NBT.5, “a. Locate and label 43 and 38 on the number line. (number line image with intervals of 5 denoted up to 50.); b. Explain how to use the number line to find the value of 43-38.” Image of a number line with intervals of 5 denoted up to 50 with 43 and 38 plotted.

• Unit 9, Putting it All Together,  End-of-Course Assessment answer key, denotes standards addressed for each problem. Problem 9, 2.NBT.7, “Find the value of each expression. Show your thinking. a. 347+583. b. 612-174.”

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 2, MP1 is found in Unit 3, Lessons 6, 11, and 12.

• IM K-5 Curriculum Guide How do you assess progress?, Standards For Mathematical Practice, Standards for Mathematical Practices Chart, Grade 2, MP7 is found in Unit 9, Lessons 2, 3, 5, 6, and 9.

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

• IM K-5 Curriculum Guide, How do you assess progress?, Standards For Mathematical Practice, Standards for Mathematical Practice Student Facing Learning Targets, “MP6 I Can Attend to Precision. I can use units or labels appropriately. I can communicate my reasoning using mathematical vocabulary and symbols. I can explain carefully so that others understand my thinking. I can decide if an answer makes sense for a problem.”

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

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

The materials reviewed for Imagine Learning Illustrative Mathematics Grade 2 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 4, Addition and Subtraction on the Number Line, End-of-Unit Assessment, Problem 1, “Select 3 true statements about the numbers on the number line. A. P + 18 = Q. B. Q + 18 = P. C. P = Q - 18. D. Q = P - 18 E. Q - P = 18.” End-of-Unit Assessment Answer Key, “Students interpret equations relating numbers given on a number line. Students who select B or fail to select A, or who select D or fail to select C, need further practice with interpreting the operations of addition and subtraction on the number line. Students may fail to select E if they do not accurately calculate the difference between P and Q but the other responses should provide support in this direction.” The answer key aligns this question to 2.MD.6.

• Unit 6, Geometry, Time, and Money, End-of-Unit Assessment, Problem 3, “Select 2 drawings that have one third of the square shaded.” End-of-Unit Assessment Answer Key, “Students identify squares partitioned into thirds with one third shaded. The distractors are a square that is divided into 3 unequal parts, with one part shaded, and a square that is divided into 4 equal parts with one part shaded. Students who select either of the distractors need further work partitioning shapes into equal parts in different ways.” The answer key aligns this question to 2.G.1.

• Unit 8, Equal Groups, End-of-Unit Assessment, Problem 6, “Can Jada and Diego share all of the pattern blocks so that they each have the same set of pattern block shapes? Explain or show your reasoning.” End-of-Unit Assessment Answer Key, “Students decide if a collection of pattern blocks can be split into two identical groups. It is important for them to analyze each individual shape and make sure that there are an even number in each case. Some students may answer the final question incorrectly, giving Jada and Diego the same number of pattern blocks but different blocks.” The answer key aligns this question to 2.OA.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 3, Measuring Length, Lesson 4, Cool-down, Student Task Statements, “Diego collected sticks for an art project and measured them. His data is shown in this line plot. Answer the questions based on Diego’s line plot. 1. How many sticks collected were 22 cm? 2. How many sticks did Diego collect? 3. How long was the longest stick? 4. How many sticks were 21 cm?” Responding to Student Thinking, “Students write 22 as the longest stick.” Next Day Supports, “Before the warm-up, have students share the things that are helpful when representing and interpreting data in a line plot.” This problem aligns to 2.MD.9.

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

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 2 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 1, Adding, Subtracting, and Working with Data, End-of-Unit Assessment problems support the full intent of 2.OA.2, fluently add and subtract within 20 using mental strategies. Problem 3, “Find the number that makes each equation true. a. 7+___$$=18$$. b. 20-___$$=12$$. c. 9+7=___. d. 9+7=___. e. 19-14=___.”

• Unit 3, Measuring Length, End-of-Unit Assessment problems support the full intent of MP6, attend to precision, as students measure each of the rectangles and determine how much longer one rectangle is than the other. Problem 4, “How many centimeters longer is rectangle A than rectangle B? Explain or show your reasoning. (Images included of 2 rectangles that are different lengths).”

• Unit 6, Geometry, Time, and Money, End-of-Unit Assessment, develops the full intent of 2.G.1, recognize and draw shapes having specified attributes, such as a given number of angles or a given number of equal faces. Students identify triangles, quadrilaterals, pentagons, hexagons, and cubes. Problem 1, “Draw a quadrilateral with one square corner and two equal sides.” and Problem 2, “Choose the name of the shape. (image of a pentagon) A. Hexagon, B. Triangle, C. Quadrilateral, D. Pentagon”

• Unit 9, Putting It All Together, End-of-Course Assessment supports the full intent of MP2, reason abstractly and quantitatively, as students compare numbers within 1,000. Problem 2, “Fill in each blank with <, =, or > to make the statements true. a. 675 ___ 576 b. 98 ___ 205 c. 500+40+3___$$543$$ d. 675___400+70+1.”

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

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 2 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 you assess progress? Summative assessment opportunity, "In K-2, the assessment may be read aloud to students, as needed."

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

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

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

The materials reviewed for Imagine Learning Illustrative Mathematics Grade 3 meet expectations for providing strategies and supports for students in special populations to support their regular and active participation in learning grade-level mathematics.

Materials regularly provide strategies, supports, and resources for students in special populations to help them access grade-level mathematics. 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 4, Relating Multiplication to Division, Lesson 3, Activity 3, Narrative, Access for Students with Disabilities, “Engagement: Provide Access by Recruiting Interest. Leverage choice around perceived challenge. Invite students to select at least 3 of the 6 problems to complete. Supports accessibility for: Organization, Attention, Social-emotional skills.

• Unit 5, Fractions as Numbers, Lesson 15, Activity 2, Narrative, Access for Students with Disabilities, “Representation: Access for Perception. To support understanding, begin by demonstrating how to play one round of “Spin to Win.” Supports accessibility for: Memory, Social-Emotional Functioning.

• Unit 8, Putting It All Together, Lesson 3, Activity 1, Narrative, Access for Students with Disabilities, “Engagement: Develop Effort and Persistence: Chunk this task into more manageable parts. Check in with students to provide feedback and encouragement after each round. Supports accessibility for: Organization, Focus.”

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

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 2 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, Adding and Subtraction Within 100, Section B: Decompose to Subtract, Problem 7, Exploration, “Here is Jada’s method for finding the value of 73-58. 73-60=13. 13+2. 1. Explain why Jada’s method works. 2. Use Jada’s method to find the value of 85-49.”

• Unit 4, Addition and Subtraction on the Number Line, Section A: The Structure of the Number Line, Problem 11, Exploration, “1. Here is a picture of a thermometer. How is the thermometer the same as a number line? How is it different? 2. Here is a picture of a rain gauge. How is the rain gauge the same as a number line? How is it different?”

• Unit 5, Numbers to 1000, Lesson 12, Section A: The Value of Three Digits, Problem 10, Exploration, “1. Can you represent the number 218 without using any hundreds? Explain your reasoning. 2. Can you represent the number 218 without using any tens? Explain your reasoning. 3. Can you represent the number 218 without using any ones? Explain your reasoning.”

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

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

The materials reviewed for Imagine Learning Illustrative Mathematics 2 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 4, Addition and Subtraction on the Number Line, Lesson 2, Cool-down, students explain how to revise a given number line. Student Task Statements, “Mai made a number line to show the numbers 0–10. How should Mai revise her number line?”

• Unit 6, Geometry, Time, and Money, Lesson 4, Warm-up, Launch, students describe and analyze shapes. “Display the image. What do you notice? What do you wonder?”

• Center, How Are They the Same? (1–5), Stage 2: Grade 2 Shapes, students find shapes that have shared attributes. Narrative, “Students lay six shape cards face up. One student picks two cards that have an attribute in common. All students draw a shape that has a shared attribute with the two shapes. Students get a point if they draw a shape that no other student drew. It is possible that students will draw a shape with a different shared attribute than what the original student chose. This can be an interesting discussion for students to have.”

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

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

The materials reviewed for Imagine Learning Illustrative Mathematics Grade 2 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 4, Addition and Subtraction on the Number Line, Lesson 5, Activity 2, Launch “Groups of 3. Give each group chart paper, markers, and a set of number cards.” Activity: “You will be working with your group to arrange the number cards on the number line. Take turns picking a card and placing it near its spot on the number line. Explain how you decided where to place your card. If you think you need to rearrange other cards, explain why. When you agree that you have placed all the numbers in the right spots, mark each of the numbers on your cards with a point on the number line. Label each point with the number it represents.” 10 minutes: small-group work time. “Consider asking: Why did you place your card there? Where would you draw a point to represent this number? Which cards did you choose to place first? Why?”

• Unit 5, Numbers to 1,000, Lesson 4, Activity 1, Launch, “Groups of 2. Give students access to base-ten blocks. ‘I have 4 hundreds, 3 ones, and 2 tens. Which of these shows the total value written as a three-digit number? Explain how you know.’ Display 432, 234, 423.” 30 seconds: quiet think time. 1 minute: partner discussion. ‘Share responses.’” Activity, “Solve each riddle and write the three-digit number. Use the table to help you organize the digits. (Table Columns: Riddle; Hundreds; Tens; Ones; three-digit number; Rows: 1-6 listed) ‘You are going to solve number riddles using base-ten blocks.’ As needed, demonstrate the task with a student. ‘Take turns reading the clues, while your partner uses blocks to make the number.’” “Make sure you agree before adding each number to the table.” 10 minutes: partner work time. If students finish early, ask them to write their own riddles and trade them with other groups to solve. Monitor for students who recognize they need a zero when writing the three-digit number in places where there were no tens or no ones.”

• Unit 9, Putting It All Together, Lesson 3, Activity 2, Launch “Groups of 2. Activity: ‘Now answer the questions about the length of the path the students traveled on the map. Be ready to share your thinking with your partner.’” 4 minute: independent work time. 2 minutes: partner discussion. “As students find the difference of Diego and Noah’s trips, monitor for students who: use a known addition fact, explain or show counting up or back to make ten, explain or show methods that create equivalent, but easier or known differences.”

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

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

The materials reviewed for Imagine Learning Illustrative Mathematics Grade 2 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 4, Addition and Subtraction on the Number Line, Lesson 3, Activity 1, Teaching Notes, Access for English Learners, “MLR2 Collect and Display. Collect the language students use as they work with the number lines and discuss the number patterns. Display words, phrases, and representations such as: number line, distance from zero, in order, interval, spaces, tick mark, point, and pattern. During the synthesis, invite students to suggest ways to update the display: What are some other words or phrases we should include? Invite students to borrow language from the display as needed. Advances: Conversing, Reading.

• Unit 5, Numbers to 1,000, Lesson 6, Activity 2, Teaching Notes, Access for English Learners, “MLR7 Compare and Connect. Synthesis: After the Gallery Walk, lead a discussion comparing, contrasting, and connecting the different representations of numbers. To amplify student language and illustrate connections, follow along and point to the relevant parts of the displays as students speak. Advances: Representing, Conversing.

• Unit 6, Geometry, Time, and Money, Lesson 2, Activity 2, Teaching Notes, English Learners, “MLR8 Discussion Supports. Invite students to begin partner interactions by repeating the question, Which attributes match the shape I drew? This gives both students an opportunity to produce language. Advances: Conversing.”

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

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 2 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.

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

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

The materials reviewed for Imagine Learning Illustrative Mathematics Grade 2 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.

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

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 2 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, Adding and Subtracting within 100, Lesson 12, Warm-up, students are introduced to the game of Mancala. Activity Synthesis, “The picture shows a type of a game called Mancala. It is one of the world’s oldest games. Mancala was created in Africa. The game has over 800 different names and can be played in many different ways. Most games are played with a board that has different pits or holes in it. Each player uses a certain amount of seeds that they place on their side of the board. Players might use real seeds or they may use shells, rocks, or beads. Each player takes turns placing their seeds on the board. In most games, you try to ‘capture’ more seeds than the other player. In Ghana and the Caribbean, one popular mancala game is called Oware. The board has 12 pits, 6 for each player, and the game uses 32 seeds. In Sudan, one popular mancala game is called Bao. The board for Bao has 28 pits, 14 pits for each player, and the game uses 64 seeds. The largest mancala game is called En Gehé and is played in Tanzania. The board can have up to 50 pits and the players use 400 seeds! Mancala is played all over the world. This board shows a game played in India called Pallanguzhi. The board has 14 pits and uses 70 seeds. What math questions could we ask about this image? (How many seeds are there in all? How many seeds are in the holes? How many more seeds are on the top than on the bottom?)”

• Unit 3, Measuring Length, Lesson 11, Activity 2, students create saree silk ribbon necklaces by cutting ribbon. Lesson Narrative, “The purpose of this activity is for students to solve Take From problems within 100 with the unknown in all positions. Students label tape diagrams and use them to make sense of the story problems before solving them (MP2). In Difference Unknown and Change Unknown problems, students may not be able to anticipate whether the unknown length will be longer or shorter than the length of the part they know. It is okay if they do not accurately label the smaller part of the tape diagram with the smaller length as long as they are accurately making sense of the problem.” Launch, “Groups of 2. Give each group access to base-ten blocks. The kids in Priya’s class are all making saree silk ribbon necklaces, so they are cutting ribbons to share with each other.”

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

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

The materials reviewed for Imagine Learning Illustrative Mathematics Grade 2 provide some supports for different reading levels to ensure accessibility for students.

• Unit 2, Add and Subtract Within 100, Lesson 3, Activity 1, MLR6: Three Reads, Launch, “Display only the story problem, without revealing the question.  ‘We are going to read this problem 3 times.’ 1st Read: ‘Some students were on the bus to go to the zoo. Then 34 more students got on. Now there are 55 students on the bus. What is this story about?’ 1 minute: partner discussion Listen for and clarify any questions about the context.” 2nd Read: “‘Some students were on the bus to go to the zoo. Then 34 more students got on. Now there are 55 students on the bus. What are all the things we can count in this story?’ (number of students who started the story on the bus, number of students who got on next, the total number of students on the bus, the number of buses) 30 seconds: quiet think time 2 minutes: partner discussion  Share and record all quantities. Reveal the question.” 3rd Read: “‘Read the entire problem, including the question aloud. ‘What are different ways we can solve this problem? (We could subtract the number of students who got on the bus second from the total. We could add to the number of students who got on the bus until we get to the total.)’ 30 seconds: quiet think time 1–2 minutes: partner discussion.”

• Unit 4, Addition and Subtraction on the Number Line, Lesson 5, Activity 1, Access for Students with Disabilities, “Engagement: Provide Access by Recruiting Interest. Give students a context to relate the number line to. For example, a frog jumping on lily pads, or a rabbit hopping. The counters can represent the animal hopping along the number line.”

• Unit 7, Adding and Subtracting within 1,000, Lesson 10, Activity 1, Access for Students with Disabilities, “Action and Expression: Internalize Executive Functions. Check for understanding by inviting students to rephrase directions in their own words. Allow students to check off each task as it is completed.”

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

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

The materials reviewed for Imagine Learning Illustrative Mathematics Grade 2 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 2, Add and Subtract within 100, Lesson 1, Activity 2, students use connecting cubes to compare problems. Launch, “Give students access to towers of ten and loose connecting cubes. Display the image of the cubes. ‘What do you notice? What do you wonder?’ (Lin has more cubes. They have 40 cubes all together. Lin has ten more cubes.) Monitor for students who notice the groups of ten cubes and use this structure to find the total number of cubes or the difference.”

• Unit 8, Equal Groups, Lesson 7, Activity 1, students use counters to create arrays. Launch, “Groups of 2. Give each group 3 sets of counters with 6, 7, and 9. Display A from the warm-up or arrange counters to show: (Image of 4 rows: first and third row have 4 counters, second and fourth row have 2 counters.) ‘The red counters are arranged in rows, but it is not an array. How could we rearrange the counters to make an array like image B?’ (We could move the bottom two counters to the middle row. We could move one from the top row to the next row. We could move 1 from the third row to the bottom row.)” Activity, “‘Arrange each of your sets of counters into an array. Your arrays should have the same number of counters in each row with no extra counters.’ Be prepared to explain how you made an array out of each set. ‘If you have time, try to figure out a different way to make an array out of each set of counters.’”

• Unit 9, Putting It All Together, Lesson 8, Activity 1, students play a game called Heads Up to add and subtract within 100. Launch, “Give students number cards. Activity, ‘We are going to play a game called Heads Up.’ Demonstrate with 2 students. ‘Players A and B pick a card and put it on their foreheads without looking at it. I am Player C. My job is to find the value of the sum and tell my group. Players A and B use the other player’s number and the value of the sum to determine what number is on their head. Finally, each player writes the equation that represents what they did. Demonstrate writing an equation for each of the players. After each round switch roles and play again.’”

#### 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 2 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.

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

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

The materials reviewed for Imagine Learning Illustrative Mathematics Grade 2 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 4, Addition and Subtraction on the Number Line, Section B: Practice Problems, Problem 2, students type an equation in a box and then draw in another box to explain their answer, “Here is a number line. a. Write an equation that the number line represents. Type your answer in the box. b. Explain how your equation matches the number line. Draw in the box. Select T to type.”

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

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

The materials reviewed for Imagine Learning Illustrative Mathematics Grade 2 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.”

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

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

The materials reviewed for Imagine Learning Illustrative Mathematics Grade 2 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.

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

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

The materials reviewed for Imagine Learning Illustrative Mathematics Grade 2 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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