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)

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)

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 1, Adding, Subtracting, and Working with Data, Lesson 18, Activity 2, students create their own relevant mathematical questions and use their understanding of addition and subtraction to answer questions about their own and their peers' survey data. Activity Narrative, “Switch graphs. Use the sentence stems or create your own questions about the other group’s graphs. 1–2 minutes: independent work time 4 minutes: partner work time. Take turns to ask each other the questions you came up with and use your own graph to answer.  When possible, write down an equation to show your reasoning. 5 minutes: group discussion. Monitor for students who write equations to show the categories they combine or compare when answering their peers' questions.” Student Task Statements, “1. Trade graphs. Create questions about the other group’s graph. Sentence stems: How many students picked ___ and ___ all together? How many more students picked ___ than ___? 2. Take turns asking and answering questions.” (2.MD.10, 2.OA.1, 2.OA.2)

• 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.”

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 8, Equal Groups, Lesson 5, Activity 2, students apply their understanding as they test conjectures about the effect of adding 1 and 2 to groups of objects to determine if the groups of objects are even or odd. Launch, “Give students recording sheets and access to counters. Draw: An image of 5 dots are shown. ‘If we add 1 more circle to this group, will it change if the group has an even or odd number?’ (Yes. It’s odd, so if you add 1 circle you’d make another pair and it’d be even.) 30 seconds: quiet think time. Share responses. ‘Does adding 1 always change whether a number of objects is even or odd?’ (Yes. If you add 1 to an odd number, you’d always make a new pair, and the sum would be even. If it’s even, and you add 1, you’d have a leftover, so the sum would be odd. No. I think it works with some numbers, but maybe not all numbers.) 30 seconds: quiet think time. 1 minute: partner discussion. Share and record responses.” Activity, “‘Let’s test our ideas. Complete the first two columns of the table. You can test other numbers if you have time.’ 4 minutes: independent work time. 2 minutes: partner discussion. ‘If we add 2 more to a group, will it change if the group has an even or odd number?” (No. For even, it’d be like counting by 2, the next number is even too. When we counted on 2 to odd, we made a list of odd numbers. Yes. I think if you add to a number, it’s going to change some numbers.)’ 30 seconds: quiet think time. 1 minute: partner discussion. Share and record responses. ‘Let’s test our thinking. Complete the table for the “add 2 counters” column. You can test other numbers if you have time.’ 4 minutes: independent work time. 2 minutes: partner discussion.” Student Task Statements, “1. In the first column of your recording sheet, decide whether each student has an even or odd number of counters. Show your reasoning and circle your choice. 2. Complete the gray column. Does adding 1 change whether the number of counters is even or odd? Explain. 2. Complete the last column. Does adding 2 change whether the number of counters is even or odd? Explain.” (2.OA.2, 2.OA.3)

• 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 1, Adding and Subtracting and Working With Data, Lesson 16, Activity 2, students apply their conceptual understanding of addition and subtraction to solve problems within 100. Activity, “‘Today, you’re going to solve problems with your partner. Show your thinking using drawings, numbers, words, or an equation. Remember to ask yourselves questions as you make sense of the problem and create representations.’ 12 minutes: partner work time.” Student Task Statements, “1. Jada read 10 fewer pages than Noah. Noah read 27 pages. How many pages did Jada read? 2.Noah spent 25 minutes reading. Jada spent 30 more minutes reading than Noah. How many minutes did Jada spend reading? 3.Jada read 47 pages of the book. Noah read 20 pages of the book. How many fewer pages did Noah read? 4.Noah stacked 14 more books than Jada. Jada stacked 28 books. How many books did Noah stack?” (2.OA.1) 

• 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)

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).”

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.”

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. _____, _____, _____, _____, _____.”

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 6, Geometry, Time, and Money, Lesson 1, Activity 3, students use precise language as they recognize and describe the attributes of triangles. Activity Narrative, “As students work, encourage them to refine their descriptions of their shape using more precise language (MP6).” Activity, “‘Pick one shape card. Think about how you would describe your shape to a partner without naming it.’ 1 minute: quiet think time. ‘You’re going to find a partner and describe your shape without showing them your card. Your partner will guess the name of your shape using triangle, quadrilateral, pentagon, or hexagon. After you both name the shapes, find one way your shapes are alike and one way they are different.’ Give a signal for students to find a partner. 2 minutes: partner discussion.” Synthesis, “What clues did your partner give you that made it easy to guess their shape? (number of sides, number of corners) Were there any clues that did not help you guess the name of the shape? (color, size).”

• 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?”

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.)”

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.)”

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).”

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.”

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.”

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.”

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.”

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.”

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.

Guidance is consistently provided for teachers to support students who read, write, and/or speak in a language other than English, providing scaffolds for them to meet or exceed grade-level standards. According to the Curriculum Guide, How do the materials support all learners? Mathematical language development, “Embedded within the curriculum are instructional routines and supports to help teachers address the specialized academic language demands when planning and delivering lessons, including the demands of reading, writing, speaking, listening, conversing, and representing in math (Aguirre & Bunch, 2012). While these instructional routines and supports can and should be used to support all students learning mathematics, they are particularly well-suited to meet the needs of linguistically and culturally diverse students who are learning mathematics while simultaneously acquiring English. Mathematical Language Routines (MLR) are also included in each lesson’s Support for English learners, to provide teachers with additional language strategies to meet the individual needs of their students. Teachers can use the suggested MLRs as appropriate to provide students with access to an activity without reducing the mathematical demand of the task. When selecting from these supports, teachers should take into account the language demands of the specific activity and the language needed to engage the content more broadly, in relation to their students’ current ways of using language to communicate ideas as well as their students’ English language proficiency. Using these supports can help maintain student engagement in mathematical discourse and ensure that struggle remains productive. All of the supports are designed to be used as needed, and use should fade out as students develop understanding and fluency with the English language.” The series provides principles that promote mathematical language use and development: 

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

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

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

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

The series also provides Mathematical Language Routines (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.”

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.’”