The materials reviewed for Kendall Hunt's 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 K-5 Math Teacher Guide, Design Principles, conceptual understanding is a part of the design of the materials. Balancing Rigor states, “Opportunities to connect new representations and language to prior learning support students in building conceptual understanding. Access to new mathematics and problems prompts students to apply their conceptual understanding and procedural fluency to novel situations.” Additionally, Purposeful Representations states, “Across lessons and units, students are systematically introduced to representations and encouraged to use representations that make sense to them. As their learning progresses, students are given opportunities to make connections between different representations and the concepts and procedures they represent.” Examples include:

• Unit 1, Adding, Subtracting, and Working with Data, Lesson 2, Activity 2, students develop conceptual understanding as they write addition expressions for sums within ten. Launch states, “Groups of 2. Give each group a set of cards, two recording sheets, and access to 10-frames and two-color counters. ‘We are going to learn a game called Check It Off. Let’s play a round together. First we take out all of the cards greater than five. We will not use those cards in this game. Now I am going to pick two number cards and find the sum of the numbers. The sum is the total when adding two or more numbers.’ Choose two cards. ‘What is the sum of the numbers? How do you know?’ 30 seconds: quiet think time. 1 minute: partner discussion. Share responses. ‘Now I check off the sum. What addition expression represents the sum of the numbers?’ 30 seconds: quiet think time. Share responses. ‘I record the expression on my recording sheet next to the sum. Now it’s my partner’s turn.’” (1.OA.5, 1.OA.6)

• Unit 3, Adding and Subtracting Within 20, Lesson 13, Activity 1, students develop conceptual understanding as they solve Take From, Change Unknown story problems using a method of their choice. Launch states, “Groups of 2. Give students access to double 10-frames and connecting cubes or two-color counters. Display and read the numberless and questionless story problem. ‘What do you notice? What do you wonder?’ 30 seconds: quiet think time. 1 minute: partner discussion. Record responses. If needed, ‘What question could we ask?’ Student Facing states, “1. There are students standing in the classroom. Some of the students sit down on the rug. There are still some students standing. 2. There are 15 students standing in the classroom. Some of the students sit down on the rug. There are still 5 students standing. How many students sat down on the rug? Show your thinking using drawings, numbers, or words. Equation: ___.” (1.OA.1, 1.OA.5, 1.OA.6)

• Unit 8, Putting It All Together, Lesson 9, Warm-up, students develop conceptual understanding as they use true and false statements to compare two-digit numbers. Student Facing states, “Decide if each statement is true or false. Be prepared to explain your reasoning. $65>35$, $65=75-10$, $65>35+30$.” (1.NBT.3)

According to IM K-5 Math Teacher Guide, materials were designed to include opportunities for students to independently demonstrate conceptual understanding, when appropriate. Design Principles, Coherent Progress states, “Each activity starts with a launch that gives all students access to the task. This is followed by independent work time that allows them to grapple with problems individually before working in small groups. The activity ends with a synthesis to ensure students have an opportunity to consolidate their learning by making connections between their work and the mathematical goals.” A Typical IM Lesson states, “The cool-down task is to be given to students at the end of the lesson. Students are meant to work on the cool-down for about 5 minutes independently and turn it in. The cool-down serves as a brief formative assessment to determine whether students understood the lesson. Students’ responses to the cool-down can be used to make adjustments to further instruction.” Examples include:

• Unit 3, Adding and Subtracting Within 20, Lesson 2, Cool-down, students show a conceptual understanding of addition facts. Student Facing states, “How does knowing $7+2=9$ help you with $2+7=$___? Show your thinking using drawings, numbers, or words.” (1.OA.3) 

• Unit 5, Adding Within 100, Lesson 12, Activity 2, students demonstrate conceptual understanding as they use what they know about the base-ten structure of numbers to create different expressions. Students are provided counting cubes and Student Facing states, “37, 22, 18, 56, 41. Choose 2 numbers from above and write an addition expression to make each statement true. This sum has the smallest possible value. Expression: ____ This sum has the largest possible value. Expression: ____ You do not need to make a new ten to find the value of this sum. Expression: ____ If you make a new ten to find the value of this sum, you will still have some more ones. Expression: ____ If you make a new ten to find the value of this sum, you will have no more ones. Expression: ____ Be ready to explain your thinking in a way that others will understand. If you have time: Choose 2 numbers from above and write an addition expression where the value is closest to 95. How do you know the value is closest to 95?” Activity Synthesis states, “Are there other numbers you could use? How do you know?” (1.NBT.C)

• Unit 8, Putting It All Together, Lesson 8, Activity 1, students demonstrate conceptual understanding as they represent numbers within 100 using drawings, words, numbers, expressions, and equations. Launch states, “Give each student a piece of blank paper and access to connecting cubes in towers of 10 and singles. ‘We are going to create a class book. First you will plan out your page. Pick your favorite number between 20 and 100. You will represent your number in as many different ways as you can. You need to include at least three expressions. Let’s make a page together.’ Display the number 84. ‘What are some ways that I can represent this number?” (I can draw 8 tens and 4 ones, 7 tens and 14 ones, $80+4$, $10+10+10+10+10+10+10+10+4.70+14$) 30 seconds: quiet think time. 1 minute: partner discussion. Record responses. If needed, ask: ‘How can we represent 84 using only 6 tens? What other addition expressions could we write?’” (1.NBT.B)

The materials reviewed for Kendall Hunt's Illustrative Mathematics Grade 1 meet expectations for giving attention throughout the year to individual standards that set an expectation of procedural skill and fluency. 

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

• Unit 1, Adding, Subtracting, and Working with Data, Lesson 2, Activity 1, students develop procedural fluency as they match expressions to dot images and find the total. Activity states, “In this activity, draw a line to connect each dot image to its matching expression. Then, find the total. On the second page, complete the missing expressions or the missing dot images.” Student Facing states, “Match each pair of dots to an expression. Then, find the total. Draw the missing dots to match the expression. Then, find the total. Write the missing expression to match the dots. Then, find the total.” Expressions include: $3+2$, $4+2$, $5+3$, $6+4$, $4+3$. (1.OA.6)

• Unit 3, Adding and Subtracting Within 20, Lesson 5, Warm-up, students develop procedural fluency as they select numbers that make an equation true. Student Facing states, “Find the number that makes each equation true. $6+$___, $10-6=$___, =10$$, $10-2=$___.” (1.OA.6, 1.OA.8) 

• Unit 6, Length Measurements Within 120 Units, Lesson 2, Warm-up, students develop procedural fluency as they add numbers within 100. Student Facing states, “Find the value of each expression mentally. $35+20$, $35+25$, $30+45$, $37+45$.” (1.NBT.4) 

According to IM K-5 Math Teacher Guide, materials were designed to include opportunities for students to independently demonstrate procedural skill and fluency, when appropriate. Design Principles, Coherent Progress states, “Each activity starts with a launch that gives all students access to the task. This is followed by independent work time that allows them to grapple with problems individually before working in small groups. The activity ends with a synthesis to ensure students have an opportunity to consolidate their learning by making connections between their work and the mathematical goals.” A Typical IM Lesson states, “The cool-down task is to be given to students at the end of the lesson. Students are meant to work on the cool-down for about 5 minutes independently and turn it in. The cool-down serves as a brief formative assessment to determine whether students understood the lesson. Students’ responses to the cool-down can be used to make adjustments to further instruction.” Examples include:

• Unit 3, Adding and Subtracting Within 20, Lesson 2, Activity 3, students demonstrate fluency by finding the value of sums within 10. Student Facing states, “Find the value of each sum. 1. $7+2=$___ 2. $3+5=$___ 3. ___ 4. $3+6=$___ 5. $5+2=$___ 6. ___ 7. $2+6=$___ 8. ___.” Activity states, “Read the task statement. 5 minutes: independent work time. 3 minutes: partner discussion.” (1.OA.5, 1.OA.6, 1.OA.8)

• Unit 4, Numbers to 99, Lesson 4, Activity 2, students demonstrate procedural skill and  fluency as they practice adding and subtracting multiples of 10 from multiples of 10. Student Facing states, “5. $20+60=$___ 6. $70-20=$___ 7. $90-70=$___ 8. $40+40=$___.” Activity states, “5 minutes: independent work time. 5 minutes: partner work time.” (1.NBT.2c, 1.NBT.4, 1.NBT.6) 

• Unit 8, Putting It All Together, Lesson 2, Cool-down, students demonstrate procedural fluency by using the subtraction and addition relationship to add or subtract within 10. Student Facing states, “Mai is still working on $9-6=$___. Write an addition equation she can use to help figure out the difference.  Addition equation: ___.” (1.OA.6)

The materials reviewed for Kendall Hunt's Illustrative Mathematics Grade 1 meet expectations for being designed so that teachers and students spend sufficient time working with engaging applications of the mathematics. According to IM Curriculum, Design Principles, Balancing Rigor, “Access to new mathematics and problems prompts students to apply their conceptual understanding and procedural fluency to novel situations.” Multiple routine and non-routine applications of the mathematics are included throughout the grade level and these single- and multi-step application problems are included within Activities or Cool-downs. 

Students have the opportunity to engage with applications of math both with support from the teacher and independently. According to IM K-5 Math Teacher Guide, materials were designed to include opportunities for students to independently demonstrate application of grade-level mathematics, when appropriate. Design Principles, Coherent Progress states, “Each activity starts with a launch that gives all students access to the task. This is followed by independent work time that allows them to grapple with problems individually before working in small groups. The activity ends with a synthesis to ensure students have an opportunity to consolidate their learning by making connections between their work and the mathematical goals.” A Typical IM Lesson states, “The cool-down task is to be given to students at the end of the lesson. Students are meant to work on the cool-down for about 5 minutes independently and turn it in. The cool-down serves as a brief formative assessment to determine whether students understood the lesson. Students’ responses to the cool-down can be used to make adjustments to further instruction.”

Examples of routine applications of the math include:

• Unit 2, Addition and Subtraction Story Problems, Lesson 2, Activity 2, students solve and write equations for Result Unknown word problems. Activity states, “‘Now you will solve the problems and write equations to match. You can solve the problems in any way that makes sense to you.’ Read problems aloud. 5 minutes: partner work time. ‘Find another group and discuss each problem. Share the equation you wrote and how it matches the story.’ 5 minutes: small-group discussion.” Student Facing states, “1. There was a stack of 6 books on the table. Someone put 4 more books in the stack. How many books are in the stack now? Show your thinking using drawings, numbers, or words. Equation: ___ 2. 9 books were on a cart. The librarian took 2 of the books and put them on the shelf. How many books are still on the cart? Show your thinking using drawings, numbers, or words. Equation: ___ 3. 2 kids were working on an art project. 7 kids join them. How many kids are working on the art project now? Show your thinking using drawings, numbers, or words. Equation:___ 4. The librarian had 8 bookmarks. He gave 5 bookmarks to kids at the library. How many bookmarks does he have now? Show your thinking using drawings, numbers, or words. Equation: ___.” (1.OA.1)

• Unit 3, Adding and Subtracting within 20, Lesson 20, Cool-Down, students solve real-world word problems with three addends. Student Facing states, ”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. Equation: ___.” (1.OA.2, 1.OA.6)

• Unit 4, Numbers to 99, Lesson 4, Activity 1, students solve story problems involving adding and subtracting multiples of 10. Launch states, “Groups of 2. Give students access to connecting cubes in towers of 10 and singles and double 10-frames.” Activity states, “Read the task statement. 7 minutes: independent work time. 3 minutes: partner discussion. Monitor for students who show: towers of 10, base-ten drawings, __ tens and __ tens, expressions or equations.” Student Facing states, “1. Jada is counting collections of cubes. In Bag A there are 30 cubes. In Bag B there are 2 towers of 10. How many cubes are in the two bags all together? Show your thinking using drawings, numbers, or words. 2. 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.” (1.NBT.2c, 1.NBT.4, 1.NBT.6)

Examples of non-routine applications of the math include:

• Unit 3, Adding and Subtracting Within 20, Lesson 28, Activity 1, students generate, articulate, and solve their own addition and subtraction problems. Launch states, “Display and read the questionless story problem. ‘What is this story missing? What kind of questions could you ask? (How many pencils did they have altogether? How many more pencils does Noah have than Elena?)’ 30 seconds: quiet think time. 1 minute: partner discussion. Share and record responses. ‘We have been solving different kinds of story problems. Today, you and your partner will write and solve addition and subtraction story problems using objects we have in our classroom.’” Activity states, “‘Partner A will pick a number less than 20. Partner B will use objects in the room to write a story problem and ask a question for which the number Partner A picked is the answer. Together, solve the story problem and write an equation. Switch roles for problem 2.’ 10 minutes: partner work time.” Student Facing states, “Noah had 8 pencils. Elena had 5 pencils. Han had 4 pencils.1. Addition story problem: Solve the story problem. Show your thinking using drawings, numbers, or words. Equation: ___ 2. Subtraction story problem: Solve the story problem, Show your thinking using drawings, numbers, or words. Equation: ___.” (1.OA.1, 1.OA.2, 1.OA.3, 1.OA.6)

• Unit 5, Adding Within 100, Lesson 8, Activity 3, students solve story problems involving addition with two-digit and one-digit numbers. Activity states, “Read the task statement. 3 minutes: independent work time. 3 minutes: partner discussion.” Student Facing states, “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.” (1.NBT.4)

• Unit 6, Length Measurements Within 120 Units, Lesson 7, Cool-down, students solve a problem by reasoning about measurements with different units. Student Facing states, “Priya says that the length of the shoe is 5 paper clips. Is her measurement accurate? Why or why not?” An image of a high-top sneaker is shown with 5 paper clips. (1.MD.2)

The materials reviewed for Kendall Hunt's 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.

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

• Unit 2, Addition and Subtraction Story Problems, Lesson 14, Activity 1, students analyze and solve addition and subtraction application problems. Launch states, “​​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.” Student Facing states, “There are 8 glue sticks and 3 scissors at the art station. How many fewer scissors are there than glue sticks? Mai created a picture. (An Image of eight  red dots and three yellow dots is provided.) She is not sure which equation she should use to find the difference. $8-3=5$, $3+5=8$, Help her decide. Show your thinking using drawings, numbers, or words.” (1.OA.1, 1.OA.5, 1.OA.7)

• Unit 3, Adding and Subtracting Within 20, Lesson 4, Activity 2, students develop conceptual understanding as they justify that they have found all the ways to make 10. An image of 10 counters in a ten frame is displayed. Students have access to counters, and a 10-frame. Student Facing states, “1. Show all the ways to make 10.  2. How do you know that you have found all the ways? Be ready to explain your thinking in a way that others will understand.” Activity Synthesis states, “‘How do you know that you found all of the ways?’ (I started by filling my 10-frame with red counters and then flipped over the first red counter to make it yellow. That was $1+9$. I kept flipping over a one red counter at a time to make it yellow and kept writing expressions.)” (1.OA.3, 1.OA.6)

• Unit 7, Geometry and Time, Lesson 15, Activity 2, students develop procedural fluency as they write times after reading one or both hands on a clock. Launch states, ”Give students their Half Past Clock Cards. ‘Write the times on the new clock cards that show half past.’ 2 minutes: independent work time.” Activity states, “What time is shown on each clock? Work on the questions by yourself and then compare your work with your partner’s.” Student Facing states, “1. For each clock, write the time. a. Clock shows 2:00. b. Clock shows 4:30. c. Clock shows 6:30. d. Clock shows 12:00. e. Clock shows 8:00.” (1.MD.3)

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

• Unit 1, Adding, Subtracting, and Working with Data, Lesson 8, Activity 2, students develop conceptual understanding alongside procedural fluency as they sort shapes into categories and explain their strategies for sorting. Launch states, “Give students access to colored pencils or crayons and copies of the three-column table.” Student Facing states, “1. Show how you sorted the shape cards. Be sure that someone else who looks at your paper can see how many shapes are in each category. 2. Complete the sentences: a. The first category has ___ shapes. b. The second category has ___ shapes. c. The third category has ___ shapes. d. There are ___ shapes all together.” (1.MD.4)

• Unit 3, Adding and Subtracting Within 20, Lesson 6, Activity 1, students develop conceptual understanding alongside application as they use addition to solve routine problems. Student Facing states, “Han is playing Shake and Spill. He has some counters in his cup. Then he puts 3 more counters in his cup. Now he has 10 counters in his cup. How many counters did he start with? Show your thinking using drawings, numbers, or words. Equation: ___.” (1.OA.1, 1.OA.5, 1.OA.6)

• Unit 6, Length Measurements Within 120 Units, Lesson 1, Activity 2, students develop all three aspects of rigor, conceptual understanding, procedural skill and fluency, and application as they order objects by length. Launch states, “Give each group 10–12 objects.” Student Facing states, “1. Pick 3 objects. With your partner, put the objects in order from shortest to longest. Trace or draw your objects. 2. Pick 3 new objects. With your partner, put them in order from longest to shortest. Write the names of the objects in order from longest to shortest.” Activity Synthesis states, “Invite previously identified students to demonstrate how they ordered three objects from shortest to longest. Display the three objects with the endpoints lined up so all students can see. ‘What statements can you make to compare the length of their objects?’ (The ___ is longer than the ___ and ___.)” (1.MD.1)

The materials reviewed for Kendall Hunt's 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 within the Course Guide (How to Use These Materials) and within specific lessons (Instructional Routines, Lesson Preparation Narratives, and Lesson Activities’ Narratives).

MP1 is identified and connected to grade-level content, and there is intentional development of the MP to meet its full intent. Students make sense of problems and persevere in solving them as they work with support of the teacher and independently throughout the units. Examples include:

• Unit 2, Addition and Subtraction Story Problems, Lesson 4, Cool-down, students make sense of story problems and write equations. Student Facing, “Mai has 3 books. She gets some more books from the library. Now she has 7. How many more books did she get? Show your thinking using drawings, numbers, or words. Equation:___.” Preparation, Lesson Narrative states, “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 3, Adding and Subtracting Within 20, Lesson 15, Activity 1, students solve a story problem with three addends in which two of the addends make 10. Student Facing states, “7 blue birds fly in the sky. 8 brown birds sit in a tree. 3 baby birds sit in a nest. How many birds are there altogether? Show your thinking using objects, drawings, numbers, or words.” Launch states, “Groups of 2. Give students access to double 10-frames and connecting cubes or two-color counters. ‘What kind of birds do you see where you live? Where do you see the birds?’ (I see pigeons on wires. I see a big bird in the park. I see red birds at the bird feeder. I hear loud birds in the morning.). 30 seconds: quiet think time. 1 minute: partner discussion. Share and record responses. Write the authentic language students use to describe the birds they see and where they see them. ‘Louis Fuertes was a bird artist. When he was a child, he loved to paint the birds he saw.’ Consider reading the book The Sky Painter by Margarita Engle. ‘We are going to solve some problems about birds.’'' Activity states, “3 minutes: independent work time. 2 minutes: partner discussion. As students work, consider asking: ‘How are you finding the total number of birds? How did you decide the order to add the numbers? Is there another way you can add the numbers?’ Monitor for students who use the methods described in the narrative.” An image of a blue bird is shown. Narrative states, “Students are given access to double 10-frames and connecting cubes or two-color counters. Students read the prompt carefully to identify quantities before they start to work on the problem. They have an opportunity to think strategically about which numbers of birds to combine first since 3 and 7 make 10. They also may choose to use appropriate tools such as counters and a double 10-frame strategically to help them solve the problem (MP1, MP5).”

• Unit 6, Length Measurements Within 120 Units, Lesson 17, Activity 2, students make sense of addition and subtraction word problems. Launch states, “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.” Student Facing states, “Group A: Read your problems to your partner group. Group B: 1. Act out and solve the problems. Show your thinking using drawings, numbers, or words. 2. Write an equation to represent each story problem. 3. What do you notice about the story problems and the equations you wrote? Switch roles.” Narrative states, “The purpose of this activity is for students to solve addition and subtraction word problems by acting out the stories. Acting out gives students opportunities to make sense of a context (MP1).”

MP2 is identified and connected to grade-level content, and there is intentional development of the MP to meet its full intent. Students reason abstractly and quantitatively as they work with support of the teacher and independently throughout the units. Examples include:

• Unit 2, Addition and Subtraction Story Problems, Lesson 6, Activity 2, students consider two different equations that represent the same story problem. Student Facing states, “Tyler and Clare want to know how many pets they have together. Tyler has 2 turtles. Clare has 4 dogs. Tyler wrote the equation  $4+2=6$. Clare wrote the equation $2+4=6$. Do both equations match the story? Why or why not? Show your thinking using drawings, numbers, or words.” Launch states, “Groups of 2. Give students access to 10-frames and connecting cubes or two-color counters. ‘We just solved a problem about pet fish. What else do you know about pets?’ 30 seconds: quiet think time. 1 minute: partner discussion. If needed ask, ‘What kinds of pets are there?’ Activity states, “Read the task statement. 3 minutes: independent work time. 2 minutes: partner discussion. Monitor for a student who uses objects or drawings to show that each equation matches the story.” Narrative states, “Students contextualize the problem and see that each number represents a specific object’s quantity, no matter which order it is presented, and connect these quantities to written symbols (MP2).”

• Unit 5, Adding Within 100, Lesson 14, Activity 1, students solve two-digit addition word problems. Launch states, “Give students access to connecting cubes in towers of 10 and singles. The table shows the number of cans four students collected for their class’ food drive. ‘What do you notice? What do you wonder?’ (They collected a lot of cans. Tyler collected the most. Han collected the least. I wonder how many they collected all together.)” Activity states, “Read the task statement. 6 minutes: independent work time. ‘Check in with your partner. Be prepared to show or explain your thinking.’ 5 minutes: partner discussion.” Student Facing states, “Partner A: Write an equation to represent your thinking.1. How many cans did Lin and Priya collect altogether? 2.How many cans did Han and Tyler collect altogether? 3. How many cans did all four students collect altogether? Partner B: Write an equation to represent your thinking. 1. How many cans did Tyler and Priya collect altogether? 2. How many cans did Lin and Han collect altogether? 3. How many cans did all four students collect altogether?” Narrative states, “The purpose of this activity is for students to apply their understanding of place value and properties of operations to solve two-digit addition real world problems (MP2). Students may use any method and representation that helps them make sense of the problems in context.”

• Unit 8, Putting It All Together, Lesson 5, Activity 1, students solve addition and subtraction story problems. Student Facing states, “Solve each problem. Show your thinking using drawings, numbers, or words. 1. 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? 2. When the show started, 18 stars lit up in the sky. 13 stars were bright. Some of the stars were dim. How many stars were dim? 3. Together, Diego and Tyler saw 15 shooting stars during the show. Diego saw 6 shooting stars. Tyler saw the rest. How many shooting stars did Tyler see? 4. In the gift shop, Elena bought 12 star stickers. She also bought some planet stickers. Elena bought 20 stickers. How many planet stickers did she buy?” Launch states, “Groups of 2. Give each group access to connecting cubes in towers of 10 and singles. Display the image in the student book. ‘What do you notice in this picture? What do you wonder? (There are bright colors. This looks like stars in the sky. Why is there red in the sky? Where is this?). This is a picture of something called the Helix Nebula. It is one of many interesting things that can be seen in our sky. People who are interested in learning more about stars, planets, or anything else that is found in the sky, can visit a planetarium to learn all about these things. We are going to solve some problems about a field trip to the planetarium.’” Activity states, “8 minutes: independent work time. 4 minutes: partner discussion. Monitor for students who solve the problem about bright and dim stars with addition and for students who solve the same problem with subtraction.” Narrative states, “The purpose of this activity is for students to make sense of and solve Put Together/Take Apart, Addend Unknown story problems (MP2). In the synthesis, students discuss different methods used to solve these problems, including using addition and subtraction.”

The materials reviewed for Kendall Hunt's 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 engage with MP3 across the year and it is often explicitly identified for teachers within the Course Guide (How to Use These Materials) and within specific lessons (Instructional Routines, Lesson Preparation Narratives, and Lesson Activities’ Narratives). According to the Course Guide, Instructional Routines, Other Instructional Routines, 5 Practices, “Lessons that include this routine are designed to allow students to solve problems in ways that make sense to them. During the activity, students engage in a problem in meaningful ways and teachers monitor to uncover and nurture conceptual understandings. During the activity synthesis, students collectively reveal multiple approaches to a problem and make connections between these approaches (MP3).”

Students construct viable arguments, in connection to grade-level content, as they work with support of the teacher and independently throughout the units. Examples include:

• Unit 3, Adding and Subtracting Within 20, Lesson 6, Activity 2, students construct arguments as they solve addition and subtraction stories. Student Facing states, “1. Noah is playing Shake and Spill with 10 counters. 4 of the counters fall out of the cup. How many counters are still in the cup? Show your thinking using drawings, numbers, or words. Equation: ____.” Narrative states, “During the synthesis, students focus on sharing equations and comparing the start and change unknown problems, as well as how the commutative property can help them solve story problems with an unknown start. As students discuss and justify their decisions, they share a mathematical claim and the thinking behind it (MP3).”

• Unit 5, Adding Within 100, Lesson 1, Activity 1, students construct viable arguments as they apply their place value understanding to add an amount of tens or ones to a two-digit number. Launch states, “Groups of 2. Give each group a set of number cards and a paperclip. Give students access to connecting cubes in towers of 10 and singles. ‘Remove the 0, 6, 7, 8, 9 and 10 from the number cards. We are going to play a game where you must figure out the number your partner added. Let’s play a round together. All of you are partner A and I am partner B.’ Invite a student to spin. ‘You spun (43). I will draw a number card and decide whether to add that many ones or that many tens. I will say the sum aloud. The sum is (93). What number did I add? Talk with your partner. Be ready to explain how you know.’ (You added 50. In order to get from 43 to 93 you add 5 tens. 53, 63, 73, 83, 93.) 1 minute: partner discussion. Share responses.” Activity states, “‘Now you will play with your partner. For each round, decide whether you will add tens or ones and see if your partner can guess what you added.’ 15 minutes: partner work time. As students work, consider asking: ‘How did you choose to add tens or ones? How did you determine the number your partner added?’” Student Facing states, “Partner B: Pick a number card without showing your partner. Choose whether to add that many ones or tens to your starting number. Make sure you don't go over 100. Tell your partner the sum. Partner A: Tell your partner what number you think they added and explain your thinking. Switch roles and repeat.” Narrative states, “Students explain how they add and how they determined the unknown addend with an emphasis on place value vocabulary (MP3).”

• Unit 8, Putting It All Together, Lesson 8, Cool-down, students construct arguments as they interpret representations of numbers up to 100. Student Facing states, “Represent numbers to show the base-ten structure. Represent the same number with different amounts of tens and ones.” Narrative states, “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).”

Students critique the reasoning of others, in connection to grade-level content, as they work with support of the teacher and independently throughout the units. Examples include:

• Unit 1, Adding, Subtracting, and Working with Data, Lesson 7, Activity 1, students begin to critique the reasoning of others as they sort math tools and understand how a partner sorted the tools. Launch states, “Give each group a bag of math tools and access to the blackline masters.” Activity states, “‘Sort your math tools. Use the tables if they are helpful.’ 4 minutes: partner work time. ‘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.’ 3 minutes: small group discussion. MLR2 Collect and Display. Circulate, listen for, and collect the language students use to describe how they sorted. Listen for categories, the number of shapes in each category, and math tool names. Record students’ words and phrases on a poster titled ‘Words to describe how we sorted’ and update throughout the lesson.” Narrative states, “Students identify attributes of the objects and sort them into two or more groups. Students may choose to use one of the blackline masters to organize as they sort. 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).”

• Unit 3, Adding and Subtracting Within 20, Lesson 19, Activity 1, students create an argument and critique the reasoning of others as they analyze methods for adding within 20 and use those methods flexibly to find sums. Student Facing states, “Lin, Han, and Kiran are finding the value of $8+7$. (An image of a double ten frame, eight red counters, and seven yellow counters are shown.) Lin thinks about  $8+2+5$. Han thinks about $7+7+1$. Kiran thinks about $8+8-1$. Explain how each student’s method works. Show your thinking using drawings, numbers, or words.” Launch states, “Give students access to double 10-frames and connecting cubes or two-color counters.” Activity states, “Read the task statement. ‘Use double 10-frames and counters to determine how each method works. Show your thinking in a way that others will understand.’ 10 minutes: partner work time. 3 minutes: partner discussion. Monitor for students who can explain each method using 10-frames.” Narrative states, “Students must justify and explain the work of the given characters. Students share their thinking and have opportunities to listen to and critique the reasoning of their peers (MP3).”

• Unit 4, Numbers to 99, Lesson 6, Activity 2, students construct an argument and critique reasoning when they analyze a collection of connecting cubes arranged in towers of 10. Launch states, “Groups of 2. Give students access to connecting cubes in towers of 10 and singles. ‘Noah counted a collection of connecting cubes. He says there are 50 cubes. Do you agree or disagree? Explain how you know. You will have a chance to think about it on your own and talk to your partner about Noah’s thinking before you write your response.’” Activity states, “1 minute: quiet think time. ‘Share your thinking with your partner.’ 2 minutes: partner discussion. ‘Explain why you agree or disagree with Noah. Write the word “agree” or “disagree” in the first blank. Then write why you agree or disagree.’ 3 minutes: independent work time.” Student Facing states, “Noah organized his collection of connecting cubes. He counts and says there are 50 cubes. Do you agree or disagree? Explain how you know. I ___ with Noah because.” Narrative states, “When students explain that they disagree with Noah because a ten must include 10 ones, they show their understanding of a ten and the foundations of the base-ten system (MP3).”

The materials reviewed for Kendall Hunt's Illustrative Mathematics Grade 1 meet expectations for supporting the intentional development of MP4: Model with mathematics; and MP5: Choose tools strategically, for students, in connection to the grade-level content standards, as expected by the mathematical practice standards. Students have opportunities to engage with the Math Practices across the year and they are often explicitly identified for teachers within the Course Guide (How to Use These Materials) and within specific lessons (Instructional Routines, Lesson Preparation Narratives, and Lesson Activities’ Narratives).

MP4 is identified and connected to grade-level content, and there is intentional development of the MP to meet its full intent. Students have many opportunities to solve real-world problems, model situations with appropriate representations, and describe what they do with the model and how it relates to the problem. Students model with mathematics as they work with support of the teacher and independently throughout the units. Examples include:

• Unit 2, Addition and Subtraction Story Problems, Lesson 1, Cool-down, Section A Checkpoint, students represent and solve Add To and Take From, Result Unknown story problems using a strategy that makes sense to them. They also write an expression to represent the action in a story problem. Teachers observe in order to capture evidence of student thinking using the checkpoint checklist. Student Response states, “Retell the story. Represent a story problem with objects or drawings. Explain how a representation matches the story.” Narrative states, “When students connect expressions back to the story problem and explain the connection, they model with mathematics (MP4).”

• Unit 4, Numbers to 99, Lesson 23, Activity 1, students apply their place value understanding to estimate quantities of objects and accurately count familiar objects. Student Facing states, “Experiment 1: How many objects are in 2 handfuls? Record an estimate that is: too low, about right, too high. Now find the exact number ___. Experiment 2: How many objects are in 2 handfuls? Record an estimate that is: too low, about right, too high. Now find the exact number ___. Experiment 3: How many objects are in 2 handfuls? Record an estimate that is: too low, about right, too high. Now find the exact number ___.”  Launch states, “Display for all to see approximately 15–25 beans or other small objects. ‘How many objects do you think are in this pile?’ 1 minute: partner discussion. Share responses. ‘How could we find out exactly?’ (Count them.).” Activity states, “‘How many objects are in 2 handfuls? Let's do an experiment.’ Give each group a bag of objects. ‘Take turns and grab a handful. Estimate how many objects you both grabbed altogether. Then find out how many you have exactly. You will do this experiment three times.’ 5 minutes: partner work time. Monitor for students who: count by ones group the objects into groups of 10 and then count the tens and ones.” Lesson Narrative states, “When students recognize the mathematical features of familiar real world objects and solve problems, they model with mathematics (MP4).”

• Unit 8, Putting It All Together, Lesson 6, Activity 2, students use given information to ask and answer different questions. Student Facing states, “Write and answer 2 questions using the information. Use the picture for the first one if it is helpful. 1. Diego went on 7 rides. Priya went on 11 rides. 2. Jada went on 3 rides. Kiran went on 6 rides. Noah went on 9 rides.” Narrative states, “When students recognize the mathematical features of things in the real world and ask questions that arise from a presented situation, they model with mathematics (MP4).”

MP5 is identified and connected to grade-level content, and there is intentional development of the MP to meet its full intent. Students have multiple opportunities to identify and use a variety of tools or strategies, working with the support of the teacher and independently, throughout the units to support their understanding of grade-level math. Examples include:

• Unit 4, Numbers to 99, Lesson 1, Activity 2, students organize and count a collection of 40 objects. Launch states, “Groups of 2. Give each group a bag of objects. Give students access to double 10-frames, cups, paper plates, or other tools to help organize a count. ‘Now you will work with your partner to count more collections. Each partner will show on paper how many there are and show how you counted.’” Activity states, “5 minutes: partner work time. ‘Switch bags with another group. Work with your partner to count the collection. Each partner will show on paper how many there are and show how you counted them.’ 5 minutes: partner work time. As students work, consider asking: ‘How can you use what we learned in the last activity to help you organize your count? Tell me about what you have written here. How many does it show? Does your representation match how you counted?’ Monitor for students who organize objects into groups of ten using cups, paper plates, or other tools, groups using double 10-frames.” Narrative states, “Students choose how to count their collection and determine how to represent their count. They may count by one, using double 10-frames or other tools to keep track of tens (MP5).”

• Unit 5, Adding Within 100, Lesson 6, Cool-down, Section B Checkpoint, students add one-digit and two-digit numbers and deepen their understanding of place value. The teacher observes and collects evidence of student thinking with the checkpoint checklist. Student Response states, “Add within 100 by counting on. Make a ten to add within 100. Add within 100 by combining ones and ones. Explain their addition method orally in a way others will understand. Represent their addition method on paper in a way others will understand.” Lesson Purpose shows the focus on student choice of strategy as it states, “The purpose of this lesson is for students to add one-digit and two-digit numbers, with composing a ten, using place value understanding and the properties of operations. Students also make sense of equations that represent addition methods.”

• Unit 6, Length Measurements within 120, Lesson 8, Activity 1, students measure a length that is over 100 length units long and count the number of units using grouping methods. Launch states, “Groups of 3–4. Give each group 120 base-ten cubes, string, and scissors. ‘Today we are going to measure the height of one of your group members. Choose whose height you will measure and cut a piece of string that is the same length as their height.’ 2 minutes: small-group work.” Activity states, “‘Measure the length of the string using small cubes. Represent the measurement using drawings, numbers, or words.’ 15 minutes: partner work time. Monitor for groups who: have measurements between 100–110 cubes, created groups of ten to organize the cubes.” Student Facing states, “Represent your measurement using drawings, numbers, or words.” Lesson Narrative states, “The purpose of this lesson is for students to count a quantity between 100 and 110. In the first activity, students measure how tall they are using base-ten cubes and represent their work in a way that makes sense to them (MP5).”

The materials reviewed for Kendall Hunt's 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 the specialized language of math, in connection to grade-level content, as they work with support of the teacher and independently throughout the units. Examples include:

• Unit 1, Adding, Subtracting and Working with Data, Lesson 7, Warm-up, students describe attributes of mathematical objects. Narrative states, “The activity provides an opportunity for students to describe mathematical objects in different ways, including non-mathematical characteristics such as color as well as mathematical characteristics such as the number of corners and the category or properties of the shapes. (MP6). If possible, display the objects themselves rather than the image or provide students with a set of the objects. Some students may not know the names of the shapes. Prompt them to use the language that makes sense to them.” Launch states, “Groups of 2. Display the image. ‘What do you notice? What do you wonder?’ 1 minute: quiet think time.” Activity Synthesis states, “How are the shapes alike? How are they different?”

• Unit 4, Numbers to 99, Lesson 10, Cool-down, students attend to precision as they write numbers using their knowledge of base-ten representations. Student Facing states, “Write the number that matches each representation. 1. $30+9$,  2. an image of 63 base ten blocks, 3. 7 ones + 9 tens.” Narrative states, “Students must attend to the units in each representation and the meaning of the digits in a two-digit number, rather than always writing the number they see on the left in a representation in the tens place and the number they see on the right in a representation in the ones place (MP6).”

• Unit 4, Numbers to 99, Lesson 14, Activity 1, students use precise mathematical language as they determine which number is greater and represent their number in any way they choose. Student Facing states, “Each partner spins a spinner. Each partner shows the number any way they choose. Compare with your partner. Which number is more?” Launch states, “Give each group two paper clips and access to connecting cubes in towers of 10 and singles. Display 35 and 52. ‘Which number is more? Show your thinking using math tools. Be ready to explain your thinking to your partner.’ 2 minutes: independent work time. 2 minutes: partner discussion. ‘Which is more and how do you know?’ (53 is more because it has more tens than 35.).” Activity states, “Read the task statement. ‘Each partner can choose to use Spinner A or B for each turn.’ 10 minutes: partner work time.” Narrative states, “Listen for the way students use place value understanding to compare the numbers and the language they use to explain how they know one number is more than the other (MP3, MP6). In the synthesis, students are introduced to the terms greater than and less than.” Activity Synthesis states, “‘Are there any other words or phrases that are important to include on our display?’ As students share responses, update the display by adding (or replacing) language, diagrams, or annotations. Remind students to borrow language from the display as needed. Display 93 and 26. ‘Which is more? How do you know? (93 is more because 9 tens is more than 2 tens.) We can say, ‘93 is greater than 26.’ We can also say, ‘26 is less than 93.’” Display 62 and 64. ‘Which number is more? How do you know? (64 is more. They both have 6 tens but 64 has 4 ones and that is more than the 2 ones in 62.) We can say that 64 is greater than 62. We can also say 62 is less than 64.’”

• Unit 5, Adding Within 100, Lesson 3, Cool-down, students use precision as they explain how to add expressions. Student Facing states, “Find the value of $14+53$. Show your thinking using drawings, numbers, or words. Write equations to show how you found the value.” Lesson Narrative states, “In this lesson, students add two-digit numbers using methods of their choice and write equations to match their thinking. Students interpret and compare different methods for finding the value of the same sums. Students also practice explaining their own methods and listening to the methods of their peers. Students have opportunities to revise how they explain their own and others' methods and consider how representations of their own thinking (for example, drawings or equations) can help them explain or interpret their work (MP3, MP6).”

• Unit 6, Length Measurements Within 120 Units, Lesson 3, Activity 3, students attend to precision when they compare the length of two objects using a third object. Student Facing states, “Will the teacher’s desk fit through the door? Show your thinking using drawings, numbers, or words. Will a student desk fit through the door? Show your thinking using drawings, numbers, or words. Which is longer, the bookshelf or the rug? Show your thinking using drawings, numbers, or words. Which is longer, the file cabinet or the bookshelf? Show your thinking using drawings, numbers, or words. Which is shorter, the bookshelf or the teacher’s desk? Show your thinking using drawings, numbers, or words. Will the teacher’s desk fit next to the bookshelf? Show your thinking using drawings, numbers, or words.” Launch states, “Groups of 2. Give students access to measuring materials. ‘Have you ever seen someone move a large piece of furniture, like a couch, from one room to another? Is it easy to move big pieces of furniture? Why or why not?’ 30 seconds: quiet think time. Share responses. ‘I have been thinking about getting a new desk. If I do, I will have to move this desk out of the room. I am not sure if this desk will fit through the door. How can we check to see if it will fit?’ (We could measure with a string.). 30 seconds: quiet think time. 1 minute: partner discussion. Share responses. ‘You are all going to check to see if my desk will fit through the door. You are also going to compare the length of some other objects in the room.” Activity states, “15 minutes: partner work time. Monitor for a group that measures the width of the teacher's desk and one that measures the length.” Narrative states, “When students decide if the teacher's desk will fit through the door or compare other large pieces of furniture, they will need to be precise about which lengths they are measuring as objects like the teacher's desk, a rug, and a bookcase, have a length, width, and in some cases a height (MP6).”

• Unit 7, Geometry and Time, Lesson 3, Warm-up, students use specialized language as they compare attributes of shapes. Narrative states, “This warm-up prompts students to compare four shapes. It gives students a reason to use language precisely (MP6). It gives the teacher an opportunity to hear how students use terminology and talk about characteristics of two- and three-dimensional shapes.” Launch states, “Groups of 2. Display the image ‘Pick one that doesn’t belong. Be ready to share why it doesn’t belong.’” Activity Synthesis states, “Let’s find at least one reason why each one doesn’t belong. What solid shapes do the images for A, B, and C show? (cube, cone, and cylinder) Does D show a solid shape? Why or why not? (Maybe it is supposed to be a sphere. It looks like it is just a circle.) A circle is not one of our solid shapes. We call it a flat shape.”

The materials reviewed for Kendall Hunt's 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 across the year and they are often explicitly identified for teachers within the Course Guide (How to Use These Materials) and within specific lessons (Instructional Routines, Lesson Preparation Narratives, and Lesson Activities’ Narratives).

MP7 is identified and connected to grade-level content, and there is intentional development of the MP to meet its full intent. Students have many opportunities to look for, describe, and make use of patterns within problem-solving as they work with support of the teacher and independently throughout the units. Examples include:

• Unit 2, Addition and Subtraction Story Problems, Lesson 18, Activity 1, students look for and make use of structure as they interpret equations with a symbol for the unknown and connect them to story problems. Activity states, “‘You have two sets of cards. One set of cards has the story problems we used in the last lesson. The other set of cards has equations with unknown values. Work with your partner to match the story problems to the equations. One story has more than one equation. Be sure you can explain how you know they match.’ 10 minutes: partner work time.” Activity Synthesis states, “‘Which equation matches Card C? How do you know?’ (__. 9 represents how many students were sliding. 6 represents how many students leave so that is $9-6$. The box represents how many are left, which is the answer to the problem.) Repeat for problems F and H. Display equation cards 6 and 8. ‘What do you notice about these equations? (They both have a total of 9 and one part is 4. The other part is the unknown. They both match problem G.). How does each of these equations match the story problem?’ (There are 9 students jumping Double Dutch and 4 students jumping on their own. I need to find the difference, so I can subtract $9-4$ to find the answer or I can say that $9=4+$___. 9 equals 4 plus some more students.).” Narrative states, “The purpose of this activity is for students to match story problems to equations with a symbol for the unknown (MP2). Each equation is written to match the way the numbers are presented in the story problem. Problem G has more than one equation, which prompts students to discuss the relationship between addition and subtraction (MP7). During the synthesis, students discuss how an equation with a symbol for the unknown matches a Take From, Result Unknown story problem.”

• Unit 5, Adding Within 100, Lesson 1, Warm-up, students look for and make use of structure as they subitize or use grouping strategies to describe the images they see. Student Facing states, “How many do you see? How do you see them?” Activity Synthesis states, “How did we describe the second image using tens and ones? How many tens do you see? How many ones? (Some people said they saw it as 3 tens and 5 ones.) How could we describe the last image using tens and ones? (3 tens and 9 ones) How could we write equations to go with the last image? ( or $30+9$).” Lesson Narrative states, “When students look for ways to see and describe numbers as groups of tens and ones and connect this to two-digit numbers, they look for and make use of the base-ten structure (MP7).”

• Unit 7, Geometry and Time, Lesson 14, Cool-down, students look for and make use of structure as they learn about the position of the hands on an analog clock at half past the hour, Student Facing states, “Circle the clock that shows 2:30.” Lesson Narrative states, “Students connect their understanding of half of a circle to the minute hand moving halfway around the face of a clock (MP7).”

MP8 is identified and connected to grade-level content, and there is intentional development of the MP to meet its full intent. Students have multiple opportunities to use repeated reasoning in order to make generalizations and build a deeper understanding of grade-level math concepts as they work with support of the teacher and independently throughout the units. Examples include:

• Unit 3, Adding and Subtracting Within 20, Lesson 17, Cool-down, students use repeated reasoning to add within 20. Students see that they can decompose one addend in order to make a ten. Student Facing states, “8 birds are sitting in a tree. 6 birds are sitting on the grass. How many birds are there all together? Show your thinking using drawings, numbers, or words. Equation: ___.” Lesson Narrative states, “When students identify and use equivalent expressions, they look for and make use of structure (MP7) and here they repeatedly make a 10 to find the value of expressions (MP8).”

• Unit 4, Numbers to 99, Lesson 7, Activity 1, students use repeated reasoning to extend their understanding of teen numbers as a ten and some ones to an understanding of all two-digit numbers as some tens and some ones. Student Facing states, “Partner 1 draws 2 number cards and uses them to make a two-digit number. Each partner says the number. Partner 2 builds the number using cubes. Partner 1 checks to see if they agree. Each partner makes a drawing of the number and records how many tens and ones. Switch roles and repeat.” Activity states, “10 minutes: partner work time. As students work, consider asking: ‘How do you say this two-digit number? What is your plan for building the number? How many tens does this number have? How many ones does this number have?’” Activity Synthesis states, “Display the number 24 and a base-ten drawing of 4 tens and 2 ones. ‘Tyler made a drawing of 24. Do you agree with how he showed 24? Why or why not? (No, because he drew 4 tens and 2 ones instead of 2 tens and 4 ones. He made the number 42 instead of 24.). Tyler’s drawing shows 42, not 24. They both have the digits 2 and 4, but they are in different places, which makes them different numbers.’” Narrative states, “Students choose two number cards and create a two-digit number. As they build the two-digit numbers with towers of 10 and singles, students see that each two-digit number is composed of a number of tens and a number of ones (MP8).”

• Unit 6, Length Measurements Within 120 Units, Lesson 4, Warm-up, students use repeated reasoning to make ten to find sums within 50. Student Facing states, “Find the value of each expression mentally. $9+6$, $29+6$, $39+7$, $39+9$.” Narrative states, “When students notice how they can make a ten when finding the value of each expression or when they use one sum to find the value of the next sum, they look for and make use of structure and express regularity in repeated reasoning (MP7, MP8).” Activity Synthesis states, “Did anyone have the same method but would explain it differently? Did anyone approach the problem in a different way?”

The materials reviewed for Kendall Hunt's 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. This is located within IM Curriculum, How to Use These Materials, and the Course Guide, Scope and Sequence. Examples include:

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

• Course Guide, Scope and Sequence, provides an overview of content and expectations for the units, “The big ideas in grade 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. In these materials, particularly in units that focus on addition and subtraction, teachers will find terms that refer to problem types, such as Add To, Take From, Put Together or Take Apart, Compare, Result Unknown, and so on. These problem types are based on common addition and subtraction situations, as outlined in Table 1 of the Mathematics Glossary section of the Common Core State Standards.”

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

• Unit 3, Adding and Subtracting Within 20, Lesson 17, Activity 1, teachers are provided guidance to support students in transitioning from using 10 frames to writing mathematical equations. Narrative states, “The purpose of this activity is for students to find sums when one addend is nine. Students represent sums on the 10-frame to encourage them to use the structure of a ten. During the launch, the teacher demonstrates playing a round of the game. It is important to let students discover patterns as they play the game. For example, when finding the sum of $9+5$, some students may represent each addend on a separate 10-frame and count to find the sum. Other students may use the associative property and move one counter from the five, and add it to the nine to make a ten. (A picture of a 10-frame is provided.) Students may generalize that when they take one from an addend to make 10, the sum has one less one than that addend. When students build this understanding, they may no longer need to show their thinking on the 10-frame and can just write an equation. By repeatedly making the ten by taking one from an addend, students may see and use the structure of ten to add on (MP7, MP8).”

• Unit 4, Numbers to 99, Lesson 7, Warm-up, provides teachers guidance about how two-digit numbers are composed of tens and ones. Launch states, “Groups of 2. Display the image. ‘What do you notice? What do you wonder?’ 1 minute: quiet think time.” Activity states, “‘Discuss your thinking with your partner.’ 1 minute: partner discussion. Share and record responses.” Activity Synthesis states, “‘The numbers in Set B are called two-digit numbers.’ Display 89. ‘This is one number, the number eighty-nine. This number has two digits, an 8 and a 9. In the number 89, the 8 tells us how many tens are in the number and the 9 tells us how many ones are in the number. Today you will work on making two-digit numbers.’”

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

• IM K-5 Math Teacher Guide, About These Materials, Unit 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.” 

• IM K-5 Math Teacher Guide, About These Materials, Unit 3, Connection to a book by Russell, S.J., Schifter D., & Bastable, V. (2011), supports teachers with context for work beyond the grade, “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, Numbers to 99, Lesson 1, Preparation, Lesson Narrative states, “In the previous unit, students learned that a ten is a unit made up of 10 ones. Students learned that teen numbers are made up of 1 ten and some more ones, using 10-frames, drawings, and expressions . In kindergarten, students learned the counting sequence by ones and tens up to 100. The purpose of this lesson is for teachers to formatively assess how students count objects up to 60 through two counting activities. In the first activity, students count objects and represent how many in a way that makes sense to them, then compare the ways they counted. In the second activity, students count bags of different quantities that are multiples of 10, and begin to make sense of grouping objects into tens. Suggested objects include pennies, paper clips, buttons, connecting cubes, inch tiles, counters, or any other objects around the classroom. Students should also be given access to cups, paper plates and double 10-frames to help them organize their collections if they would like.”

• Unit 6, Length Measurements Within 120 Units, Lesson 5, Preparation, Lesson Narrative states, “In previous lessons, students ordered a set of three objects by length. Students also compared lengths of objects indirectly by using a third object. The purpose of this lesson is for students to describe lengths of objects in terms of connecting cubes. Students measure by using connecting cube towers because the units are lined up without gaps or overlaps, a concept they will explore in future lessons. In the first activity, students use connecting cube towers to measure the length of different animals. Students build towers that are exactly the same length as the animals and make a comparison statement (‘The grasshopper is the same length as a tower of 7 cubes’). In the second activity, students use connecting cube towers to measure the length of more animals and describe the length as ‘___ cubes long.’ Even though the side-length of the cube is the unit, it’s appropriate for students to describe length in terms of ‘x cubes long.’ This transition in language helps students understand that the length of objects can be described as a number of length units (MP6). In this lesson, the length unit is the length of a single connecting cube.”

The materials reviewed for Kendall Hunt's 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 1, Course Guide, Lesson Standards includes a table with each grade-level lesson (in columns) and aligned grade-level standards (in rows). Teachers can search any lesson for the grade and identify the standard(s) that are addressed within.

• Grade 1, Course Guide, Lesson Standards, includes all Grade 1 standards and the units and lessons each standard appears in. Teachers can search a standard for the grade and identify the lesson(s) where it appears within materials.

• Unit 3, 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 6, Length Measurements Within 120 Units, Lesson 3, the Core Standard is identified as 1.MD.A.1. Lessons contain a consistent structure: a Warm-up that includes Narrative, Launch, Activity, Activity Synthesis; Activity 1, 2, or 3 that includes Narrative, Launch, Activity; an Activity Synthesis; a Lesson Synthesis; and a Cool-down that includes Responding to Student Thinking and Next Day Supports. This provides an additional place to reference standards, and language of the standard, within each lesson.

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

• Grade 1, Course Guide, Scope and Sequence, Unit 2: Addition and Subtraction Story Problems, Unit Learning Goals, includes an overview of how the math of this module builds from previous work in math, “In kindergarten, students solved a limited number of types of story problems within 10 (Add To/Take From, Result Unknown, and Put Together/Take Apart, Total Unknown, and Both Addends Unknown). They represented their thinking using objects, fingers, mental images, and drawings. Students saw equations and may have used them to represent their thinking, but were not required to do so. Here, students encounter most of the problem types introduced in grade 1: Add to/Take From, Change Unknown, Put Together/Take Apart, Unknowns in All Positions, and Compare, Difference Unknown. The numbers are kept within 10 so students can focus on interpreting each problem and the relationship between counting and addition and subtraction. This also allows students to continue developing fluency with addition and subtraction within 10.”

• Grade 1, Course Guide, Scope and Sequence, Unit 7: Geometry and Time, Unit Learning Goals, 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).”

The materials reviewed for Kendall Hunt's Illustrative Mathematics Grade 1 meet expectations for providing explanations of the instructional approaches of the program and identification of the research-based strategies. 

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

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

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

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

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

• IM Certified, Blog, The Story of Grade 1, Brianne Durst, Deep Dive into New Learning in Unit 4 (Numbers to 99) and Unit 5 (Adding within 100), “In Unit 4, students use what they have learned about teen numbers and the unit of ten to generalize the structure of two-digit numbers, relating the two digits to the number of tens and ones. They interpret and use multiple representations of numbers up to 99, such as connecting cubes, base-ten diagrams, words, and expressions. Connecting cubes in towers of 10 and singles are used throughout grade 1, rather than base-ten blocks, so that units of ten can be physically composed and decomposed with the cubes. Although students work physically with connecting cubes, they interpret base-ten diagrams, recognizing the diagram as a simplified image of the connecting cubes. This helps students make sense of a more efficient way of drawing diagrams to match their connecting cubes. As students develop their understanding of place value and work with each of these representations, they are able to compare any two-digit numbers by comparing the number of tens, and, if needed, the number of ones.”

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

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

The materials reviewed for Kendall Hunt's Illustrative Mathematics Grade 1 meet expectations for providing a comprehensive list of supplies needed to support instructional activities. 

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

• Unit 1, Adding, Subtracting, and Working with Data, Lesson 4, Activity 1, Required Materials, “10-frames, Materials from a previous activity, Number cards 0–10, Two-color counters.” Launch states, “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.” 

• Course Guide, Required Materials for Grade 1, Materials Needed for Unit 3, Lesson 4, teachers need, “10-frames, Crayons, Cups, Materials from previous centers, Two-color counters, Shake and Spill Stage 3 Recording Sheet Grade 1 (groups of 1).” 

• Unit 4, Numbers to 99, Lesson 7, Activity 1, Required Materials, “Connecting cubes in towers of 10 and singles, Number cards 0-10, Materials to copy (Make It, Two-Digit Numbers Recording Sheet Number, Drawing, Words).” Launch states, “Groups of 2. Give each group a set of number cards, connecting cubes in towers of 10 and singles, and recording sheets. Ask students to take out the cards with 10 on them. ‘We are going to play a game called Make It. You will work with your partner to make a two-digit number and represent the number in different ways.’ Display two number cards and the recording sheet. ‘First, one partner picks two number cards and makes a two-digit number. I picked a [3] and a [5]. What two-digit numbers can I make?’ (35 or 53). Demonstrate writing one of the numbers on the recording sheet. ‘Now both partners say the number. Then, the partner who made the number watches the other partner build the number with connecting cubes. Make sure you both agree on how to build the number. Then both partners complete the recording sheet with a drawing and the number of tens and ones.”

• Course Guide, Required Materials for Grade 1, Materials Needed for Unit 7, Lesson 1, teachers need, “Bags (brown paper), Geoblocks, Materials from a previous activity, Solid shapes.”

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

Examples of supports for special populations include: 

• Unit 1, Adding, Subtracting, and Working with Data, Lesson 9, Activity 1, Narrative, Access for Students with Disabilities, “Representation: Access for Perception. Students with color blindness will benefit from verbal emphasis, gestures, or labeled displays to distinguish between colors of connecting cubes. Supports accessibility for: Visual-Spatial Processing.”

• 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 5, Adding Within 100, Lesson 4, Activity 1, Narrative, Access for Students with Disabilities, “Representation: Develop Language and Symbols. Make connections between the representations visible. For example, ask students to identify correspondences between the visual representation and the expression $37+26$. Supports accessibility for: Visual-Spatial Processing, Conceptual Processing.”

• Unit 7, Geometry and Time, Lesson 9, Activity 3, Narrative, Access for Students with Disabilities, “Action and Expression: Develop Expression and Communication. Give students access to a straight edge or ruler. Supports accessibility for: Fine Motor Skills, Visual-Spatial Processing.”

The materials reviewed for Kendall Hunt's 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 To Use The Materials, Exploration Problems, “Each practice problem set also includes exploration questions that provide an opportunity for differentiation for students ready for more of a challenge. There are two types of exploration questions. One type is a hands-on activity directly related to the material of the unit that students can do either in class if they have free time, or at home. The second type of exploration is more open-ended and challenging. These problems go deeper into grade-level mathematics. They are not routine or procedural, and they are not just the same thing again but with harder numbers. Exploration questions are intended to be used on an opt-in basis by students if they finish a main class activity early or want to do more mathematics on their own. It is not expected that an entire class engages in exploration problems, and it is not expected that any student works on all of them. Exploration problems may also be good fodder for a Problem of the Week or similar structure.” Examples include:

• Unit 1, Adding, Subtracting, and Working with Data, Section C: What Does the Data Tell Us?, Problem 7, Exploration, “Gather data at home or school and make a display of the data. Ask a math question about the data. Trade displays and questions with a partner and answer your partner’s question.”

• Unit 2, Addition and Subtraction Story Problems, Section A: Add To and Take From Story Problems, Problem 8, Exploration, “Choose one of the equations. 1. $5+$___. 2. $8-3=$___. 3. $3+$___. 4. $5+3=$___. Write a story problem that the equation matches. Trade with a partner and decide which equation matches your partner’s story.”

• Unit 5, Adding Within 100, Section B: Make a Ten: Add One- and Two-digit Numbers, Problem 7, Exploration, “Priya is playing the game Target Numbers. Priya starts at 25 and picks these six cards: 1, 2, 3, 5, 6, 8. She chooses whether to add that many tens or ones for each card. What is the highest score she can get without going over 95? Use equations to show your thinking.”

• Unit 7, Geometry and Time, Section C: Tell Time in Hours and Half Hours, Problem 4, Exploration, “Show the time during the day when you might do each of these things. 1. wake up in the morning, 2. go to school, 3. have a snack, 4. go for recess, 5. have lunch.”

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

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

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

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

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

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

• Unit 2, Addition and Subtraction Story Problems, Lesson 9, Activity 1, Teaching notes, Access for English Learners, “MLR8 Discussion Supports. Some students may benefit from the opportunity to act out the scenario. Listen for and clarify any questions about the context of each problem. Advances: Speaking, Representing.”

• Unit 5, Adding Within 100, Lesson 1, Activity 1, Teaching notes, Access for English Learners, “MLR2 Collect and Display. Circulate, listen for and collect the language students use as they talk with their partners. On a visible display, record words and phrases such as: tens, ones, sum, equation, starting number, secret number. Invite students to borrow language from the display as needed, and update it throughout the lesson. Advances: Conversing, Speaking, Listening.”

• Unit 6, Length Measurements within 120 Units, Lesson 12, Activity 1, Launch, “MLR6 Three Reads, Display only the problem stem, without revealing the question. ‘We are going to read this problem three times.’ 1st Read: ‘Priya and Han are comparing the lengths of their friendship bracelets. Han’s bracelet is 14 cubes long. The length of Priya’s bracelet is 4 cubes fewer than Han’s bracelet. What is this story about?’ 1 minute: partner discussion. Listen for and clarify any questions about the context. 2nd Read: ‘Priya and Han are comparing the lengths of their friendship bracelets. Han’s bracelet is 14 cubes long. The length of Priya’s bracelet is 4 cubes fewer than Han’s bracelet. What can be counted or measured?’ (The length of Priya's bracelet. The length of Han's bracelet. The difference in length between the two bracelets.) 30 seconds: quiet think time. 1 minute: partner discussion. Share and record all quantities. 3rd Read: Read the entire problem, including the question, aloud. ‘What are different ways we can solve this problem?’ (use connecting cubes to represent the bracelets, draw a picture, think about the numbers), 30 seconds: quiet think time. 1–2 minutes: partner discussion.”

The materials reviewed for Kendall Hunt's Illustrative Mathematics Grade 1 meet expectations for providing manipulatives, physical but not virtual, that are accurate representations of the mathematical objects they represent and, when appropriate, are connected to written methods.

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

• Unit 1, Adding, Subtracting, and Working with Data, Lesson 8, Activity 1, describes the use of shape cards and a three-column table to help students sort shapes into categories. Launch states, “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 states, “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.”

• Unit 3, Adding and Subtracting Within 20, Lesson 21, Activity 1, identifies number cards, 10-frames, connecting cubes or two-color counters as strategies for students to add numbers. Launch states, “Give each group a set of number cards, two recording sheets, and access to double 10-frames and connecting cubes or two-color counters. ‘We are going to learn a game called How Close? Add to 20. Let's play the first round together. First we take out any card that has the number 10. We will not use those cards for the game.’ Display 5 cards. ‘I can choose two or three of these cards to add to get as close to 20 as I can. What cards should I choose? I write an equation with the numbers I chose and the sum of the numbers.’ Demonstrate writing the equation on the recording sheet. ‘The person who gets a sum closer to 20 gets a point for the round. Then you each get more cards so you always have five cards to choose from. Play again. The first person to get 10 points wins.’”

• Unit 5, Adding Within 100, Lesson 10, Activity 2, identifies connecting cubes in towers of 10 and singles to support understanding of the associative and commutative properties when adding 2 two-digit numbers. Launch states, “Groups of 2. Give students access to connecting cubes in towers of 10 and singles. Read the first problem. 4 minutes: partner work time. ‘What is the difference between how you solved  $28+56$ and $27+44$.’ (For $28+56$, I added the tens first, then the ones. For $27+44$ I added the ones first, then the tens.) 1 minute: partner discussion. Share responses.”

• Unit 6, Length Measurements Within 120 Units, Lesson 4, Activity 2, references number cubes and game recording handouts to support understanding during centers. Launch states, “‘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.’”