The materials reviewed for Imagine Learning Illustrative Mathematics Grade 1 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 2, Addition and Subtraction Story Problems, Lesson 7, Warm-up, students develop conceptual understanding as they use grouping strategies to represent numbers. An image of red and yellow counters are shown. “How many do you see? How do you see them? If needed, “What equation represents this image?” (1.OA.6)

• Unit 5, Adding Within 100, Lesson 6, Activity 1, students develop conceptual understanding as they determine the unknown addend in equations with sums that are multiples of 10. Students are given access to connection cubes in towers of 10. “Display the first image in the student workbook. (Image is a picture of 4 ten frames filled in with 10 red chips and 1 ten frame with 5 chips) What number makes this equation true? How do you know? (5. I see 4 tens 5 ones and 5 more would fill up the 10-frame.)” (1.NBT.4)

• Unit 6, Length Measurements Within 120 Units, Lesson 12, Warm-up, students develop conceptual understanding by discussing “the idea that addition and subtraction are related operations. Use these equations to explain how addition can help with subtraction.” (If you know an addition fact like $6+8=14$, then you also know two subtraction facts, $14-8=6$ and $14-6=8$.) (1.OA.6)

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 3, Adding and Subtracting Within 20, Lesson 2, Cool-down, students demonstrate how counting relates to addition. “How does knowing $7+2=9$ help you with ? Show your thinking using drawings, numbers, or words.” (1.OA.3) 

• Unit 4, Numbers to 99, Lesson 15, Section C Practice Problems, Problem 2, students demonstrate conceptual understanding as they compare two-digit numbers based on the meaning of the tens and ones. “Decide if each statement is true or false. Show your thinking using drawings, numbers, or words. a. b. c. $81>77$.” (1.NBT.3)

• Unit 5, Adding Within 100, Lesson 14, Activity 1, students demonstrate conceptual understanding as they apply their understanding of place value and properties of operations to solve two-digit addition problems. A table is shown with students' names and the number of cans they collected. Students work independently for six minutes and then check in with their partner to answer the following questions. “The table shows the number of cans four students collected for their class’s food drive. Partner A: Write an equation to represent your thinking. How many cans did Lin and Priya collect altogether? How many cans did Han and Tyler collect altogether? How many cans did all four students collect altogether? Partner B: Write an equation to represent your thinking. How many cans did Tyler and Priya collect altogether? How many cans did Lin and Han collect altogether? How many cans did all four students collect altogether?”  (1.NBT.4)

The materials reviewed for Imagine Learning Illustrative Mathematics Grade 1 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 5, Activity 1, students subtract within 10 using different strategies. Launch, “Give each group a set of number cards, two recording sheets, and access to two-color counters and 10-frames. ‘We are going to learn a new way to play Check it Off. This time, instead of adding to find each number on the recording sheet, you will subtract.’ Demonstrate choosing two number cards. ‘Now I find the difference between the two numbers. The difference is the result when one number is subtracted from another. What is the difference between these two numbers? How do you know?’ Demonstrate checking off the number on the recording sheet. ‘After you check off the number, write a subtraction expression to show the difference.’ Demonstrate writing the subtraction expression. ‘Continue taking turns with your partner. The person who checks off the most numbers wins.’” (1.OA.5 and 1.OA.6).

• Unit 5, Adding Within 100, Lesson 6, Warm-up, Student Task Statements, students use different strategies to add within 20. “Find the value of each expression mentally. $8+2$, $8+5$, $9+8$, $7+6$.” (1.OA.6).

• Unit 8, Putting it All Together, Lesson 1, Warm-up, students have an opportunity to strengthen their number sense and procedural fluency. Activity, “How many do you see? How do you see them?” An image of a  10-frame is shown with 5 red dots and 4 yellow dots. Teaching Notes, Student Response, “9: I see 1 empty space and 1 less than 10 is 9.” (1.OA.6)

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 2, Addition and Subtraction Story Problems, Lesson 5, Activity 2, Narrative, “Partner A asks their partner for a number that would make 10 when added to the number on one of their cards. If Partner B has the card, they give it to Partner A. If not, Partner A chooses a new card. When students make the target number 10, they put down those two cards and write an equation to represent the combination. Students continue playing until one player runs out of cards. The player with the most pairs wins.” (1.OA.6)

• Unit 3, Adding and Subtracting Within 20, Lesson 4, Activity 2, students justify that they have found all the ways to make 10. Teaching Notes, “You will have some time to work on this problem on your own, and then share your thinking with a partner. 5 minutes: independent work time” Student Task Statements, “1. Show all the ways to make 10.” An image of a ten frame with 5 red dots and 5 yellow dots is displayed.  “2. How do you know you have found all the ways? Be ready to explain your thinking in a way that others will understand.”  (1.OA.3, 1.OA.6)

• Unit 8, Putting It All Together, Lesson 2, Cool-down, students use related facts for addition to solve and addition equation. Student Task Statements, “Mai is still working on . Write an addition equation she can use to help figure out the difference. Addition equation: ____.” (1.OA.6)

The materials reviewed for Imagine Learning Illustrative Mathematics Grade 1 meet expectations for being designed so that teachers and students spend sufficient time working with engaging applications of the mathematics.

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

Examples of routine applications include:

• Unit 3, Adding and Subtracting Within 20, Lesson 15, Activity 2, students solve real-world story problems with 3 addends (1.OA.2, 1.OA.3, 1.OA.5, 1.OA.6). Activity, “10 minutes: independent work time.” Student Task Statements, Problem 1, “Noah collected 3 bird picture cards. Clare collected 4 cards. Jada collected 7 cards. How many cards did they collect altogether? Show your thinking using drawings, numbers, or words. Equation:___.”

• Unit 4, Numbers to 99, Lesson 4, Activity 1, students solve addition and subtraction problems (1.NBT.4, 1.NBT.6). Activity, “Read the task statement. 7 minutes: independent work time.” Problem 2, Student Task Statements, “Tyler is counting a collection of cubes. In Bag C there are 7 towers of 10. He takes 40 cubes out of the bag. How many cubes does he have left in the bag? Show your thinking using drawings, numbers, or words.”

• Unit 8, Putting It All Together, Lesson 6, Activity 1, students add and subtract within 20 to solve Compare, Difference Unknown story problems, (1.OA.1, 1.OA.6). Student Task Statements, Problem 2, “The cotton candy booth sold 17 bags of blue cotton candy. They sold 7 bags of pink cotton candy. How many more bags of blue candy did they sell than pink candy?”

Examples of non-routine applications include:

• Unit 2, Addition and Subtraction Story Problems, Lesson 17, Activity 1, students compare and contrast two different story problems (1.OA.1) Student Task Statements, “1. Compare these stories about playing 4 corners. There are 6 students playing 4 corners. Some more students come to play. Now there are 9 students playing 4 corners. How many students came to play? 9 students are playing 4 corners. 7 students are waiting in a corner. The other students are still deciding which corner to pick. How many students are still deciding which corner to pick? How are these problems alike? How are they different?” Activity: “Talk to your partner about how the two problems are the same and different.” 4 minutes: partner discussion. Share responses.”

• Unit 5, Adding Within 100, Lesson 7, Activity 1, students use place value reasoning and properties of operations to determine whether they would compose a ten when adding a two-digit and a one-digit number in a real-world problem (1.NBT.4). Student Task Statements, “Jada likes to look for ways to make a new ten when she adds. Would she be able to a make a new ten when she adds to find the value of these sums? If Jada could make a new ten, circle ‘Yes.’ If Jada could not make a new ten, circle ‘No.’ Does the expression make a new ten? $45+5$ Yes, No, Explain how you know. Find the value. Write equations to show how you found the value of the sum.”

• Unit 8, Putting it All Together, Lesson 5, Activity 1, Problem 1, students use addition and subtraction to solve real-world problems (1.OA.1, 1.OA.6). Activity, “8 minutes: independent work time.” Student Task Statements, “Solve each problem. Show your thinking using drawings, numbers, or words. There are 7 first graders and some second graders at the planetarium. There are 18 students at the planetarium. How many second graders are at the planetarium?”

The materials reviewed for Imagine Learning Illustrative Mathematics Grade 1 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 1. Examples where instructional materials attend to conceptual understanding, procedural skill and fluency, or application include:

• Unit 3, Adding and Subtracting Within 20, Lesson 4, Activity 1, students apply properties of operations as strategies to add and subtract. Launch, “Give each group a cup, 10 two-color counters, and two recording sheets.” Activity, “Today you will play Shake and Spill with 10 counters. When you write the equation to represent your counters, make sure it shows how many red counters and how many yellow counters you got.” Activity Synthesis, “Display six red counters and four yellow counters. Here are the counters from a round of Shake and Spill. There are six red counters and four yellow counters. What equations can I write to represent the counters? Why do both equations represent the counters? (You can start with the red or start with the yellow and there are still 10 total counters.” (1.OA.3)

• Unit 4, Numbers to 99, Lesson 4, Cool-down, students develop procedural fluency as they subtract and add multiples of 10 in the range of 10-90. Student Task Statements, “Find the value of the expressions. 1. $50+20$, 2. $70-50$, 3. $60-30$.”(1.NBT.6)

• Unit 5, Adding Within 100, Section C Practice Problems, Lesson 12, students extend their conceptual understanding as they use place value understanding and properties of operations to add and subtract. Problem 4, “Choose a two-digit number to add to 46 so that you make a new 10. Add the numbers. Write equations to show your thinking.” (1.NBT.4)

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

• Unit 1, Adding, Subtracting, and Working with Data, Lesson 8, Activity 1, students develop conceptual understanding alongside application as they sort shapes into categories and interpret the data. Launch, “Give each group a set of shape cards and access to copies of the three-column table. Look at all of your shape cards. Take a minute to look over the cards by yourself first and think about how you would sort them.” Activity, “Work with your partner to sort the cards into three categories in any way that you want. You do not need to use all of the cards. Share with another group. Explain how you sorted the shapes. Tell how many shapes are in each category and how many shapes there are altogether. (There are 5 white shapes. There are 12 shapes altogether.)” (1.MD.4)

• Unit 2, Addition and Subtraction Story Problems, Lesson 1, Activity 1, students develop conceptual understanding alongside application as they add and subtract within 20 to solve word problems. Launch, “Give students access to 10-frames and connecting cubes or two-color counters. Display the image from the warm-up. ‘This is a picture of a library. Talk to your partner about what you know and what you wonder about libraries.’ 3 minutes: partner discussion. Share and record what students know and wonder about libraries. ‘We are going to solve a lot of story problems about libraries.’ Display and read the numberless story. 30 seconds: quiet think time. 1 minute: partner discussion. Share responses. If not already mentioned, ask, ‘Are there more or fewer kids at the library after some go home?’” Activity, “Ask students to open their books. Read the problem with numbers. 2 minutes: independent work time. ‘Share your thinking with your partner.’ 2 minutes: partner discussion. Monitor students who solve or represent the problem in the following ways: objects, drawings, count back, an expression $(9-2)$.” Student Task Statements, “1. Some kids were at the library. Then some of the kids went home. What do you notice? What do you wonder? 2. There were 9 kids at the library. Then 2 of the kids went home. How many kids are at the library now? Show your thinking using drawings, numbers, or words.” An image of a male student in a library is shown. (1.OA.1) 

• Unit 5, Adding Within 100, Lesson 2, Activity 1, students use conceptual understanding alongside procedural skill and fluency as they add 2 two-digit numbers within 100 composing a ten in a way that makes sense to them. Launch, “Groups of 2. Give students access to connecting cubes in towers of 10 and singles.” Activity, “Read the task statement. 5 minutes: independent work time. 2 minutes: partner discussion. Monitor for students who use the methods described in the Activity Narrative.” Student Task Statements, “Find the value of $23+45$. Show your thinking using drawings, numbers, or words. An image of two female students working together is shown.” (1.NBT.4)

The materials reviewed for Imagine Learning Illustrative Mathematics Grade 1 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 2, Addition and Subtraction Story Problems, Lesson 1, Activity 2, students solve addition and subtraction story problems in ways that make sense to them. Synthesis, “How are these problems the same? How are they different?” (They have the same numbers, but different answers. Problem 1 is addition, problem 4 is subtraction.)” Activity Narrative, “The purpose of this activity is for students to solve Add To and Take From, Result Unknown problems in a way that makes sense to them (MP1).” Student Task Statements, Problem 1, “5 books were on a shelf. Clare put 2 more books on the shelf. How many books are on the shelf now? Show your thinking using drawings, numbers, or words.”

• Unit 3, Adding and Subtracting Within 20, Lesson 11, Activity 1, students represent and solve problems in ways that make sense to them. Activity, “‘Read the task statement.’ 3 minutes: independent work time. ‘Share your thinking with your partner.’ 2 minutes: partner work time. Monitor for students who represent their thinking using 10-frames to show 14 and then add 3 more.” Student Task Statements, “Kiran collects rocks. So far he has 14 rocks. He goes on a hike and collects 3 more rocks. How many rocks does Kiran have? Show your thinking using drawings, numbers, or words. Equation _____.” Activity Narrative, “Students represent and solve the problem in a way that makes sense to them (MP1).”

• Unit 6, Length Measurements Within 120 Units, Lesson 17, Activity 2, students "solve addition and subtraction word problems by acting out the stories.” Launch, “Take turns reading a problem you came up with in the previous activity. Your partner group will act out the story with connecting cubes, then solve the problems. Then switch roles.” Activity Narrative, “Acting out gives students opportunities to make sense of a context (MP1).” Student Task Statements, “Group A: Read your problems to your partner group. Group B: Act out and solve the problems. Show your thinking using drawings, numbers, or words. Write an equation to represent each story problem. What do you notice about the story problems and the equations you wrote? Switch roles.”

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 2, Activity 1, students “relate dot images to addition expressions”. In the Student Task Statements, students are given 5 pairs of dots and a table with 5 expressions: “Match each pair of dots to an expression. Then, find the total.” Activity Narrative, “When students match dot images and expressions and write expressions to match dot images, they reason abstractly and quantitatively (MP2).”

• Unit 6, Length Measurements Within 120, Lesson 11, Activity 1, students find sums and differences using measurement. Activity, “‘Solve the problems using your measurements.’ 3 minutes: independent work time. 2 minutes: partner discussion. Monitor for a student who represents the third problem with: two towers of cubes, one to represent the length of their shoe and one to represent the length of the teacher’s shoe. a drawing that directly compares shoe lengths. an addition equation. a subtraction equation.” Student Task Statements, “Solve these problems about the length of your group’s shoes. Show your thinking using drawings, numbers, words, or equations. What is the length of your shoe and your partner’s shoe together? Whose shoe is longer, yours or your partner’s? How much longer? Whose shoe is shorter, your teacher’s shoe or your shoe? How much shorter?” Activity Narrative, “When students find sums and distances using their measurements they reason abstractly and quantitatively (MP2).”

• Unit 7, Geometry and Time, Lesson 16, Activity 3, students relate time (quantitative) to a schedule (abstract). Activity, “‘Fill in the blanks for your ideal Sunday schedule. Then share with your partner.’ 4 minutes: independent work time. 2 minutes: partner discussion. Monitor for a student who has an activity at 12:30.” Student Task Statements, “Fill in the blanks to show your ideal Sunday schedule. (Words listed: Time, Activity, Clock; image of a blank digital clock; a blank analog clock with numbers listed for hours and tick marks for minutes)” Activity Narrative, “The task gives an opportunity for students to relate time and telling time to their Sunday schedule (MP2).”

The materials reviewed for Imagine Learning Illustrative Mathematics Grade 1 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 2, Addition and Subtraction Story Problems, Lesson 11, Activity 2, students construct viable arguments as they solve Compare, Difference Unknown story problems in the context of connecting cube towers and answer the question "how do you know?” Launch, “Groups of 2. Give each group four towers of ten connecting cubes. Display one red tower of eight connecting cubes, one yellow tower of three connecting cubes, and the handful of yellow connecting cubes. ‘I have two towers and I need to make them the same number of cubes. But I only have these yellow cubes. How can I make them the same?’ 1 minute: quiet think time. 1 minute: partner discussion. ‘Share and record responses.’” Problem 4, Student Task Statements, “Lin is making 2 cube towers. The yellow tower has 7 cubes. The red tower has 3 cubes. She only has red cubes. How can she make the towers have the same number of cubes? Show your thinking using drawings, numbers, or words. If you have time: Write your own problem about 2 cube towers. Trade problems with a partner and solve.” Activity Narrative, “When students answer the question, How do you know? they are beginning to explain their reasoning and construct viable arguments (MP3).” 

• Unit 6, Length Measurements Within 120 Units, Lesson 7, Cool-down, students construct viable arguments as they measure lengths of objects using different length units. Student Task Statements, “Priya says that the length of the shoe is 5 paper clips. Is her measurement accurate? Why or why not?”

• Unit 8, Putting It All Together, Lesson 8, Activity 2, students “interpret representations of numbers up to 100. As students look through each others' work, they discuss how the representations are the same and different and can defend different points of view (MP3).” Synthesis, “How are the representations of your number the same as your classmates? How are they different? If you’d like, you can add to or revise the representations on your page.”

Examples of critiquing the reasoning of others include:

• Unit 1, Adding, Subtracting, and Working with Data, Lesson 9, Activity 3, students critique the reasoning of others as they interpret representations of class data. Narrative, “Students discuss how the representations they see are the same or different (MP3). ‘With your partner, find a group that represented the data in a different way from how you represented it. One person from each group switch papers with someone from the other group. With your partner, talk about what you notice is the same about each representation and what you notice is different.’ 3 minutes: partner discussion. ‘Share your thinking with the other group. What do you agree about?’ (We agree that each representation shows the same number of votes in each category and the same total number of votes.) 3 minutes: small group discussion.”

• Unit 4, Numbers to 99, Lesson 7, Activity 2, students critique the reasoning of others as they think about the value of tens and ones and consider a representation where the tens are not presented to the left of the ones. Launch, “Groups of 2. Give students access to connecting cubes in towers of 10 and singles.” Student Task Statements, “Clare says that this shows 68 (sixty-eight). Diego says that this shows 86 (eighty-six). Base ten diagram. 6 ones. 8 tens. Who do you agree with? How do you know they are correct? I agree with ____because.” Activity Narrative, “When students decide who they agree with and explain their reasoning, they critique the reasoning of others (MP3).” 

• Unit 7, Geometry and Time, Lesson 11, Activity 2, students construct viable arguments and critique the reasoning of others as they generalize that partitioning the same-size shape into fourths creates smaller pieces than partitioning it into halves. Activity, “‘Read the task statement.’ 5 minutes: partner work time. Monitor for a student who shows and can explain that a half is bigger than a fourth. ‘This activity continues on the next card.’” Student Task Statements, “Priya and Han are sharing roti. Priya says, ‘I want half of the roti because halves are bigger than fourths.’ Han says, ‘I want a fourth of the roti because fourths are bigger than halves because 4 is bigger than 2.’ ‘Who do you agree with?’” Activity Narrative, “When students decide whether they agree with Priya's or Han's statement and justify their choice with diagrams and words, they construct viable arguments and critique the reasoning of others (MP3).”

The materials reviewed for Imagine Learning Illustrative Mathematics Grade 1 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 2, Addition and Subtraction Story Problems, Lesson 18, Cool-down, students choose equations that match the given story problem. (MP4). Student Task Statements, “Lin has 5 bingo chips on her board. She also has some chips on the table. All together she has 9 bingo chips. How many bingo chips does Lin have on the table? Circle 2 equations that match the story problem. 9 - 5 = ___  , 5 + = 9, 5 - 9 = ___  , 5 + 9 = ___  ." The Activity 2 Narrative states, “When students interpret different equations in terms of a story problem, they model with mathematics (MP4).”

• Unit 5, Adding Within 100, Lesson 8, Activity 3, students model with mathematics when they experience different contexts in which someone adds a two-digit and a one-digit number. Student Task Statements, “1. Priya watched a football game. The home team scored 35 points in the first half. In the second half they scored 6 more points. How many points did they score all together? Show your thinking using drawings, numbers, or words. 2. At the football game, 9 fans cheered for the visiting team. There were 45 fans who cheered for the home team. How many fans were at the game all together? Show your thinking using drawings, numbers, or words.” Activity Narrative, “When students create representations and expressions for the context, they develop ways to model the mathematics of a situation and strategies for making sense of and persevering to solve problems (MP1, MP4).”

• Unit 6, Length Measurements Within 120 Units, Lesson 2, Activity 2, Lesson Narrative, students “compare the length of a side of their desk to the length of one of the legs of their desk indirectly using a string. This lesson helps students use a familiar object in their classroom and encourages them to mathematize their environment (MP4).” Student Task Statements, students are presented with a drawing of a desk. “Compare the length of the side of your desk and the length of one of the legs of your desk using the string. Use a drawing or words to explain how you know which is longer.”

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, Adding and Subtracting Within 20, Lesson 20, Cool-down, students choose a strategy that helps them solve an addition story problem. Student Task Statements, “Jada visited the primate exhibit. She saw 8 monkeys, 4 gorillas, and 7 orangutans. How many primates did she see? Show your thinking using drawings, numbers, or words.” Activity Narrative, “They think strategically and may either choose to use a double 10-frame or decompose and compose the numbers in a way that helps them see the sum as 10 and some ones (MP1, MP5).”

• Unit 4, Numbers to 99, Lesson 6, Activity 1, students use appropriate tools as they organize, count, and represent a collection of 52 objects. Student Task Statements, “You and your partner will get a bag of objects. Figure out how many are in the bag. Work with your partner to count the collection. Each partner will show on paper how many there are and show how you counted them.” Activity Narrative, “Other students may apply what they learned in previous lessons and create groups of ten using double-ten frames or other tools (MP5).”

• Unit 5, Adding Within 100, Lesson 9, Activity 1, students “find the sum of 2 two-digit numbers in a way that makes sense to them.” Narrative, “Students may represent these methods in different ways, including using connecting cubes in towers of 10 and singles. Monitor for students who use connecting cubes or base-ten drawings to show making a new unit of ten as part of their method (MP5).” Student Task Statements, “Find the value of $17+36$. Show your thinking using drawings, numbers, or words.” Synthesis, “Invite previously identified students to share in the order in the Activity Narrative. As each student shares, record their thinking with drawings and numbers. After each student shares, ask: “How did _____ find the value of $17+36$? Does anyone have any questions for _____?”

The materials reviewed for Imagine Learning Illustrative Mathematics Grade 1 meet expectations for supporting the intentional development of MP6: Attend to precision; and attend to the specialized language of mathematics, for students, in connection to the grade-level content standards, as expected by the mathematical practice standards. 

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

• Unit 1, Adding, Subtracting, and Working with Data, Lesson 7, Activity 1, students use the specialized language of mathematics to “sort math tools, name the groups they used to sort, and tell the number of objects in each group.” Activity Narrative, “When students share how they sorted with their partner, they use their own mathematical vocabulary and listen to and understand their partner's thinking (MP3, MP6).” Activity Narrative “Explain to another group how you sorted your tools. Make sure to tell them the groups you used and how many objects are in each group.” Synthesis, “‘Are there any words or phrases that are important to include on our display?’ Use this discussion to update the display, by adding (or removing) language, diagrams, or annotations. Remind students to borrow language from the display as needed in the next activity. The label that tells how objects in a group are alike is called a category. ‘One category I saw today was shapes with straight sides. Another category I saw was shapes that have four sides. We will continue to sort into categories’.”

• Unit 5, Adding Within 100, Lesson 2, Warm-up, Activity Narrative, “When students describe repeated patterns they see using the language of place value, they look for and make use of the base-ten structure of numbers and connect it to the counting sequence (MP6, MP7, MP8).” Launch, “‘Count backwards by 1, starting at 70.’ Record as students count. ‘Stop counting and recording at 20.’” Synthesis, “Who can restate the pattern in different words?”

• Unit 6, Length Measurements within 120 units, Lesson 1, Activity 1, students use precision when they compare the length of two objects. Activity Narrative, “Students discuss why it is important to line objects up at their endpoints when comparing their length and they make comparisons using precise language (MP3, MP6).” Student Task Statements, “Share your thinking with your partner. Choose 2 objects and compare their lengths. Choose 2 different objects and compare their lengths. Write down your answers. 1. Choose an object that you could write with and find the tower of 3 connecting cubes. Which is longer? Draw the 2 objects to show which is longer. 2. Choose a different object and find the tower of 8 connecting cubes. Which is shorter? Draw the 2 objects to show which is shorter. 3. Find an object from the collection that is shorter than your foot. Fill in the blank. The ___ is shorter than my foot. 4. Find an object from the collection that is longer than your pointer finger. Fill in the blank. The ___ is longer than my pointer finger.” Synthesis, “What would happen if the objects were not lined up? (One might look longer even though it wasn’t.) What statement can we make to compare the length of _____ and _____? Use the phrases 'longer than' and 'shorter than.' The ___ is longer than the ___. The ____ is shorter than the ____.”

• Unit 7, Geometry and Time, Lesson 10, Activity 1, Activity Narrative, “All responses should be shared and compared in the synthesis to help build students' understanding of the new vocabulary and the concept of fractional pieces of a whole (MP6).” Student Task Statements, “Split the square into halves.” Image included in the materials of a square. “Color in one of the halves. How much of the square is colored in?” Activity Synthesis, “Invite previously identified students to share for each problem. Sequence the students in the order described in the narrative. ‘One piece of a shape split into two pieces that are the same size is called a half. One piece of a shape split into four pieces that are the same size is called a fourth.’ Formal definitions and visuals of the terms are on the following cards. For each shape, invite students to describe how much is colored in. (A half of the square is colored in. A fourth of the circle is colored in.)”

The materials reviewed for Imagine Learning Illustrative Mathematics Grade 1 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, Warm-up, students look for and make use of structure when they determine the number of dots in an arrangement without counting each dot. Activity Narrative, “When students use the dot images to relate addition to counting on, they look for and make use of the structure of whole numbers (MP7).” Student Task Statements, “How many do you see? How do you see them?” Students are given 8 images with a group of dots between 1 and 5. Activity Synthesis, “‘How did you know how many dots there are in all?’ Consider asking: ‘Who can restate the way ____ saw the dots in different words? Did anyone see the dots the same way but would explain it differently? Does anyone want to add an observation to the way ____ saw the dots?’”

• Unit 2, Addition and Subtraction Story Problems, Lesson 14, Activity 1, students look for and make use of structure as they explore the relationship between addition and subtraction through a Compare, Difference Unknown story problems. Activity Narrative, “This also helps them relate addition and subtraction and see that often either operation can be used to solve a problem (MP7).” Launch, “Groups of 2. Give students access to connecting cubes or two-color counters. Display the image in the student book. ‘Tell a story about this picture.’ 1 minute: quiet think time. 2 minutes: partner discussion. ‘Share responses.’” Activity, “‘Read the task statement.’ 5 minutes: partner work time. Encourage students to use the representation to make sense of both equations. Monitor for a group who uses the representation to explain the addition equation and one who explains the subtraction equation.” Activity Synthesis, “‘What are we trying to find out in this story problem?” (How many fewer scissors there are than glue sticks. The difference between the number of scissors and glue sticks.)’ Invite previously identified groups to share. ‘What is the same? What is different? (3, 5, and 8 are in each equation. The numbers represent the same things in both. The 5 is boxed in both. One uses addition and the other uses subtraction. The boxed number is in a different place.)’” 

• Unit 4, Numbers to 99, Lesson 20, Warm-up, students look for and make use of structure as they use groups of 10 to estimate large numbers of objects. Activity Narrative, “When students notice that they can make a more accurate estimate when the single cubes are grouped into 10s they make use of base-ten structure (MP7).” Student Task Statements, “What is an estimate that’s too high? Too low? About right? (Base ten diagram.)” Launch, “Groups of 2. Display the image. ‘What is an estimate that’s too high? Too low? About right?’ 1 minute: quiet think time.” Activity, “‘Discuss your thinking with your partner.’ 1 minute: partner discussion. ‘Record responses.’” This activity continues on the next card. “‘Let’s look at another image of the same collection.’ Display image. ‘Based on the second image, do you want to revise, or change, your estimates?’” Activity Synthesis, “Did anyone change their original ‘about right’ estimate? Why did you change it?” (I changed it because I see there are at least 50 cubes in the 5 towers.) Let’s look at our revised estimates. Why were our estimates more accurate the second time? (Some of the cubes are organized.) There are 76 cubes.” 

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, Adding and Subtracting Within 20, Lesson 9, Activity 1, students look for and express regularity in repeated reasoning as they notice the $10+n$ pattern in teen numbers. Activity Narrative, “When students notice the relationship between teen numbers and the $10+n$ pattern, they look for and make use of structure (MP7).” Launch, “Groups of 2. Give each group a set of cards, a double 10-frame, and access to at least 20 connecting cubes or two-color counters. ‘We’re going to use our double 10-frames to build teen numbers today. Let's do one together.’ Choose a card. ‘What number is on my card? Let's build that number on the double 10-frame.’ Demonstrate building the teen number. ‘Now we write an equation to show how we built the number.’ Write an equation such as $10+4=14$.” Activity, “‘Now you will build more teen numbers with your partner. Make sure you both agree on how to build the number and what equation to write.’ 10 minutes: partner work time. Monitor for students who: build a new ten each time, count the 10 each time, change the ones only.” Activity Synthesis, “When you were building these numbers, what part of the equation was the same? What part was different?” (There was always 10 in each equation. I was adding each time. The total changed and was always a teen number. The number I was adding to 10 changed.)” 

• Unit 6, Length Measurements Within 120 Units, Lesson 9, Warm-up, students look for and express regularity in repeated reasoning as they notice patterns in the base-ten number system when counting beyond 99. Activity Narrative, “When students notice the patterns in the digits after counting beyond 99 and explain the patterns based on what they know about the structure of the base-ten system, they look for and express regularity in repeated reasoning (MP7, MP8).” Launch, “‘Count by 1, starting at 90.’ Record as students count. ‘Stop counting and recording at 120.’” Activity, “‘What patterns do you see?’ 1-2 minutes: quiet think time. ‘Record responses.’” Activity Synthesis, “What do you notice about the numbers we counted? (Some only have two digits and some have three. After 100, I see the numbers 1–20 again.)” 

• Unit 7, Geometry and Time, Lesson 9, Cool-down, students look for and express regularity in repeated reasoning as they notice size and shape of the pieces when it is folded are all the same. Student Task Statement, “1. Split the square into halves. 2. Split the circle into fourths. Image of a square and a circle are shown.”

The materials reviewed for Imagine Learning Illustrative Mathematics Grade 1 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 1 include: developing understanding of addition, subtraction, and strategies for addition and subtraction within 20; developing understanding of whole-number relationships and place value, including grouping in tens and ones; developing understanding of linear measurement and measuring lengths as iterating length units; and reasoning about attributes of, and composing and decomposing geometric shapes.”

• Unit 4, Numbers to 99, Section C, Compare Numbers to 99, Section Overview, Section Narrative, provides an overview of the content and expectations for the section. “In this section, students use their understanding of the base-ten structure to compare and order numbers to 99. They notice that if a two-digit number has more tens it will be greater than another number with fewer tens, no matter how many ones there are. They then generalize this insight to compare numbers based on the digits. The < and > symbols are introduced here. Before using the symbols to write true comparison statements, students gain familiarity by reading and interpreting statements with these symbols. They have opportunities to work with the symbols throughout the section. The lesson activities intentionally use mathematical language to support students in recalling how to read or write the symbols. For example, initially students are encouraged to notice that the side of the symbol with the greater amount of space between the top and the bottom segments faces the greater number. Avoid using non-mathematical or imaginative language that may distract from the focus of the unit and delay fluency with reading and writing the symbols.”

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 1, Adding, Subtracting, and Working with Data, Lesson 4, Activity 1, teachers are provided context as they help students subtract. Narrative, “In this stage, students subtract one or two from a number within 10. Some students may count back and some may count all then count back or remove 1 or 2 then count the remaining objects. Provide access to 10-frames and counters and encourage students to use them only if needed.” Launch, “Groups of 2. Give each group a set of number cards, a game board, two-color counters, and access to 10-frames. ‘We are going to learn a new way to play, Five in a Row. Last time we played, we added one or two to the number on our card. This time, you will take turns flipping over a card and choosing whether to subtract one or two from the number.’ Then put a counter on the number on the game board. ‘The first person to get five counters in a row wins. Remember, your counters can be in a row across, up and down, or diagonally.’” Activity, “10 minutes: partner work time. As students work, consider asking: ‘How did you subtract? How did you decide whether to subtract 1 or 2?’ Monitor for students who: represent the number, remove 1 or 2, count all that are left, represent the number, remove 1 or 2, know how many are left without recounting, count back 1 or 2, use the counting sequence to find the difference.”

• Unit 7, Geometry and Time, Lesson 3, Lesson Synthesis provides teachers guidance on ways to sort and describe two-dimensional shapes. “Display Card A. ‘Today we looked at flat shapes and described them in different ways in order to sort shapes. How might you describe this shape? (There are three sides that are the same. There are three corners. It is a triangle.) Display Card Q. How might you describe this shape?” (There are four sides. Three of the sides are the same length and the bottom side is long.)’ Continue with shape cards U and C as time allows.”

The materials reviewed for Imagine Learning Illustrative Mathematics Grade 1 meet expectations for containing adult-level explanations and examples of the more complex grade-level concepts and concepts beyond the current grade so that teachers can improve their own knowledge of the subject. 

Within the Teacher’s Guide, IM Curriculum, Why is the curriculum designed this way?, there are sections entitled “Further Reading” that consistently link research to pedagogy. There are adult-level explanations including examples of the more complex grade-level concepts and concepts beyond the current grade so that teachers can improve their own knowledge of the subject. Professional articles support teachers with learning opportunities about topics such as ensuring mathematical success for all, early understanding of equality, and repeating patterns. Examples include:

• Why is the curriculum designed this way? Further Reading, Unit 2, Representing Subtraction of Signed Numbers: Can You Spot the Difference?, supports teachers with context for work beyond the grade. “In this blog post, Anderson and Drawdy discuss how counting on to find the difference plays a foundational role in understanding subtraction with negative numbers on the number line in middle school.”

• Why is the curriculum designed this way? Further Reading, Unit 4, Rethinking Instruction for Lasting Understanding: An Example. “In this blog post, Nowak uses the progression of inequalities as an example of how to build reliable mathematical understanding.”

• Unit 2, Addition and Subtraction Story Problems, Lesson 4, Result or Change Unknown, About this Lesson, “Since this lesson includes all three of the problem types introduced to the students at this point, students need to pay close attention to each problem to determine the action in the story and the question that is being asked. This lesson provides an opportunity to assess student progress on making sense of different types of story problems, the methods they use to solve, and the equations they write to match the problems.”

• Unit 5, Adding Within 100, Lesson 14, Food Drive, About this Lesson, “This lesson does provide students with an opportunity to apply precursor skills of mathematical modeling. In previous lessons, students found the value of sums within 100 using methods based on place value and the properties of operations, including adding tens and tens and ones and ones, and adding on by place. In this lesson, students apply these methods to make sense of and solve real-world problems within 100. Students may use base-ten representations or equations to represent their thinking. In the warm-up, they are introduced to a food drive context. In the first activity, they solve problems which involve combining quantities of collected cans in various ways. In the second activity, students make choices about which numbers to combine based on their values and the constraints of the problem. Students may use trial and error to reach the target value. This gives them an opportunity to persevere in problem solving (MP1). When students make and articulate mathematical choices and adhere to mathematical constraints, they model with mathematics (MP4).”

The materials reviewed for Imagine Learning Illustrative Mathematics Grade 1 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 1 standards breakdown, standards are addressed by lesson. Teachers can search for a standard in the grade and identify the lesson(s) where it appears within materials.

• Course Guide, Lesson Standards, includes all Grade 1 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 3, Adding and Subtracting Within 20, Lesson 11, the Core Standards are identified as MP8, MP1, 1.OA.D.8, 1.OA.D.7, 1.OA.C.6, and 1.OA.A.1. 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 4, Numbers to 99, Unit Overview, Unit Learning Goals, full Unit Narrative, includes an overview of how the math of this module builds from previous work in math. “This unit develops students’ understanding of the structure of numbers in base ten, allowing them to see that the two digits of a two-digit number represent how many tens and ones there are. Previously, students counted forward by one and ten within 100 in the Choral Counting routine. They learned that 10 ones make a unit called a ten and that a teen number is a ten and some ones. Here, as they count and group quantities, students generalize the structure of two-digit numbers in terms of the number of tens and ones. This understanding enables students to transition from counting by one to counting by ten and then counting on. For example, to count to 73, they may count 7 tens and count on—71, 72, 73.”

• Unit 7, Geometry and Time, 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 focus on geometry and time. They expand their knowledge of two- and three- dimensional shapes, partition shapes into halves and fourths, and tell time to the hour and half of an hour. Center activities and warm-ups continue to enable students to solidify their work with adding and subtracting within 20 and adding within 100. In kindergarten, students learned about flat and solid shapes. They named, described, built, and compared shapes. They learned the names of some flat shapes (triangle, circle, square, and rectangle) and some solid shapes (cube, sphere, cylinder, and cone). Here, students extend those experiences as they work with shape cards, pattern blocks, geoblocks, and solid shapes. They develop increasingly precise vocabulary as they use defining attributes (“squares have four equal length sides”) rather than non-defining attributes (“the square is blue”) to describe why a specific shape belongs to a given category. Students should, however, focus on manipulating, comparing, and composing shapes and using their own language, rather than learning the formal definitions of shapes.”

The materials reviewed for Imagine Learning Illustrative Mathematics Grade 1 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?, 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?, Further Reading, Unit 2, The Power of Small Ideas, “In this blog post, McCallum discusses, among other ideas, the use of a letter to represent a number. The foundation of this idea is introduced in this unit when students first represent an unknown with an empty box.” Representing Subtraction of Signed Numbers: Can You Spot the Difference?, “In this blog post, Anderson and Drawdy discuss how counting on to find the difference plays a foundational role in understanding subtraction with negative numbers on the number line in middle school.” Unit 3, “Russell, S.J., Schifter D., & Bastable, V. (2011). Connecting Arithmetic to Algebra: Strategies for Building Algebraic Thinking in the Elementary Grades. Heinemann. This book explains how generalizing the basic operations, rather than focusing on isolated computations, strengthens students’ fluency and understanding which helps prepare them for the transition from arithmetic to algebra. Chapter 1, Generalizing in Arithmetic, is available as a free sample from the publisher.” Unit 4, Rethinking Instruction for Lasting Understanding: An Example. In this blog post, Nowak uses the progression of inequalities as an example of how to build reliable mathematical understanding.

• 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 1 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 1 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.10 frames are a reusable material used in lessons 1.2.1, 1.2.2, 1.2.3, 1.2.4, 1.2.6, 1.2.9, (1.2.10), 1.2.11, 1.2.12, 1.2.13, 1.2.15, (1.2.16), 1.2.18, 1.2.19, 1.2.20…. 30 10 frames per 30 students. Drawn frames on a piece of paper are suitable substitutes. Centimeter cubes are a reusable material used in lessons 1.2.16, (1.2.21), (1.3.7), (1.3.14), 1.3.17, (1.3.21), 1.3.22, 1.3.23, (1.3.27), 1.4.7, 1.4.9, 1.4.10, (1.4.12), (1.4.13), 1.4.16, (1.4.22), 1.5.6, (1.5.8), and (1.5.13). 960 centimeter cubes are needed for 30 students. No suitable substitutes for the material are listed. Number cards 0-10 are a reusable material used in lessons 2.2.18, 2.3.14, 2.4.1, and 2.5.14. 15 are needed per 30 students. A spinner with numbers 0-10 cut from paper with a paperclip is a suitable substitute. 

• Course Overview, Grade Level Resources, Grade 1 Picture Books, contains a “list of suggested picture books to read throughout the curriculum.” Unit 3, The Sky Painter by Margarita Engle is used. Unit 6, Sadako and the Thousand Paper Cranes by Eleanor Coer is used.

• Unit 5, Adding Within 100, Lesson 2, Activity 3, Teaching Notes, Materials to gather, “Paper clips, Two-color counters, Five in a Row Addition and Subtraction Stage 5 Gameboard.” Launch, “Give each group two paper clips, a gameboard, and two-color counters. We are going to learn a new way to play Five in a Row. Display the gameboard. The first player chooses one number from each row to add together. They place a paperclip on each number. Demonstrate putting a paperclip on a one-digit number and on a two-digit number. Then that player finds the sum of the numbers and puts a counter on the sum on the gameboard. Demonstrate finding the sum of the two numbers and placing the counter on the gameboard.  The next player only moves one of the paper clips to a new number. Then they find the sum of their two numbers and cover it with a counter on the gameboard. Continue taking turns moving one paper clip and covering numbers on the gameboard until someone gets five counters in a row. They are the winner.”

The materials reviewed for Imagine Learning Illustrative Mathematics Grade 1 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 2, Addition and Subtraction Story Problems, Lesson 11, Activity 2, Narrative, Access for Students with Disabilities, “Engagement: Provide Access by Recruiting Interest. Provide choice and autonomy. In addition to connecting cubes, provide access to red, yellow, and blue crayons or colored pencils they can use to represent and solve the story problems. Supports accessibility for: Visual-Spatial Processing, Conceptual Processing.”

• Unit 6, Length Measurements Within 120 Units, Lesson 15, Activity 2, Narrative, Access for Students with Disabilities, “Action and Expression: Internalize Executive Functions. Check for understanding by inviting students to rephrase directions in their own words. Keep a display of directions visible throughout the activity. Supports accessibility for: Memory, Organization.”

• Unit 8, Putting It All Together, Lesson 4, Activity 1, Narrative, Access for Students with Disabilities, “Representation: Access for Perception. Provide appropriate reading accommodations and supports to ensure student access to word problems and other text-based content. Supports accessibility for: Language, Visual-Spatial Processing, Attention.”

The materials reviewed for Imagine Learning Illustrative Mathematics Grade 1 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 4, Numbers to 99, Section A: Units of Ten, Problem 8, Exploration, “You can use towers of 10 cubes to help you with these questions. 1. Noah has 70 cubes in towers of 10. He gave some towers of 10 to Clare. Then he gave some towers of 10 to Andre. Now Noah has no cubes left. What is one way Noah could have done this? Show your thinking using drawings, numbers, or words. Write equations to represent the problem. 2. What is another way Noah could have done this? Show your thinking using drawings, numbers, or words. Write equations to represent the problem. 3. Diego has 10 cubes in a tower. Elena gave him some more towers of 10. Then Mai gave him some more towers of 10. Now Diego has 60 cubes in towers of 10. What is one way this could have happened? Show your thinking using drawings, numbers, or words. Write equations to represent the problem. 4. What is another way this could have happened? Show your thinking using drawings, numbers, or words. Write equations to represent the problem.”

• Unit 6, Length Measurements Within 120 Units, Section B: Measure by Iterating up to 120 Length Units, Problem 6, Exploration, “Priya and Noah want to measure their classroom in steps. Priya takes 28 steps to cross the room and Noah takes 26 steps. 1. How could Priya and Noah get different measurements? 2. Measure the length of your classroom in steps.”

• Unit 7, Geometry and Time, Section A: Flat and Solid Shapes, Problem 12, Exploration, “1. What is the smallest number of pattern blocks you can use to fill in the puzzle? 2. What is the largest number of pattern blocks you can use to fill in the puzzle? 3. Can you fill in the puzzle using exactly 12 pattern blocks?”

The materials reviewed for Imagine Learning Illustrative Mathematics Grade 1 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 2, Addition and Subtraction Story Problems, Lesson 3, Synthesis, Teaching Notes, Access for English Learners, “MLR7 Compare and Connect. Synthesis: After all methods have been presented, lead a discussion comparing, contrasting, and connecting the different approaches. Ask, How were the different methods the same? How were they different?” Advances: Representing, Conversing.”

• Unit 3, Adding and Subtracting Within 20, Lesson 2, Activity 1, Teaching Notes, Access for English Learners, “MLR6 Three Reads. Keep books or devices closed. To launch this activity, display only the problem stem, without revealing the question. “We are going to read this story problem three times. After the 1st Read: Tell your partner what happened in the story. After the 2nd Read: What are all the things we can count in this story? Reveal the question. After the 3rd Read: What are different ways we can solve this problem? Advances: Reading, Representing.”

• Unit 7, Geometry and Time, Lesson 2, Activity 1, Teaching Notes, Access for English Learners, “MLR8 Discussion Supports. Display sentence frames to support small-group discussion: I made a… and The shapes I used were… Advances: Speaking, Conversing.”

The materials reviewed for Imagine Learning Illustrative Mathematics Grade 1 meet expectations for providing manipulatives, both virtual and physical, that are accurate representations of the mathematical objects they represent and, when appropriate, are connected to written methods.

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

• Unit 4, Numbers to 99, Lesson 20, Activity 1, students use connecting cubes in towers of 10 and singles to make given numbers with different combinations of tens and ones. Launch, “Groups of 2. Give each group access to connecting cubes in towers of 10 and singles. Activity, ‘Today’s challenge is to find as many ways as you can to make 94 using tens and ones. You can use cubes if they will help you. Each way you make 94 should have a different number of tens.’” 

• Unit 6, Length Measurements Within 120 Units, Lesson 4, Activity 2, students use Target Numbers, Five in a Row and Get Your Numbers in Order to add and subtraction of two digit numbers. Launch, “‘Now you will choose from centers we have already learned.’ Display the center choices in the student book.” Target Numbers, “On your turn: Start at 55. Roll the number cube. Add that number to your starting number and write an equation to represent the sum. Take turns until you’ve played 6 rounds. Each round, the sum from the previous equations is the starting number in the new equation.The partner to get a sum closest to 95 without going over wins.”

• Unit 7, Geometry and Time, Lesson 7, Activity 1, students use pattern blocks to compose two-dimensional shapes into larger shapes in different ways. Launch, “Give students pattern blocks and the flat shape puzzles. Activity, ‘Use the pattern blocks to fill the outline in different ways. Each time, record how you filled in the shape with pictures, numbers, or words.’”