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

Materials develop conceptual understanding throughout the grade level. According to IM Curriculum, Design Principles, Purposeful Representations, “Across lessons and units, students are systematically introduced to representations and encouraged to use representations that make sense to them. As their learning progresses, students are given opportunities to make connections between different representations and the concepts and procedures they represent.” Each lesson begins with a Warm-up, designed to highlight key learning aligned to the objective and to support the development of conceptual understanding through student discourse and reflection. Examples include: 

• Unit 1, Math in Our World, Lesson 6, Look for Small Groups, Activity 2, Introduce Picture Books, Explore, students develop conceptual understanding as they recognize and name quantities in picture books. “Give each group of students access to at least one picture book. Look for groups of things in your book. Use your fingers to show your partner and tell your partner how many things there are in the groups you find.” (K.CC.4)

• Unit 5, Composing and Decomposing Numbers to 10, Lesson 3, Warm-up, students develop conceptual understanding as they demonstrate that quantities can be broken apart in different ways. An image of connecting cubes shows 6 connecting cubes broken apart in different ways. “What do you notice? What do you wonder?” Students may notice, “There are connecting cubes. The towers look like they are broken into 2 pieces. There are 6 connecting cubes in each image.” Students may wonder, “ Why are all of the connecting cube towers broken? Why aren’t there the same number of cubes in each smaller tower? What is the same about each set of cubes?” (K.OA.3)

• Unit 7, Solid Shapes All Around Us, Lesson 5, Activity 2, students develop conceptual understanding by solving add to, result unknown and take from, result unknown story problems. Students fill in an equation, which encourages them to connect the action in the story to the meaning of the + and - signs. “Andre put together 4 pattern blocks to make a shape. Then Andre put 4 more pattern blocks on the shape. How many pattern blocks are in Andre’s shape?” (K.OA.1)

Materials provide opportunities for students to independently demonstrate conceptual understanding throughout the grade level. Design Principles, Coherent Progress, “Each activity starts with a launch that gives all students access to the task. This is followed by independent work time that allows them to grapple with problems individually before working in small groups. The activity ends with a synthesis to ensure students have an opportunity to consolidate their learning by making connections between their work and the mathematical goals.” Independent work includes practice problems, problem sets, and time to work alone within groups. Examples include:

• Unit 2, Numbers 1–10, Lesson 7, Activity 1, students independently demonstrate conceptual understanding as they count groups of up to 10 images and notice that the order counted does not change the number of images. “At each station, there is a card with dots or fingers on it. Take turns figuring out how many things are on the card. Show your group how you figured out how many things are on the card. When I give the signal, move to the next station. Display the card with 8 dots in an array. How would you figure out how many dots there are? Invite two students to demonstrate counting the dots, with one student counting across the rows and one student counting down each column. There are 8 dots. Even if we count the dots in a different order, there are still 8 dots.” (K.CC.4, K.CC.5)

• Unit 3, Flat Shapes All Around Us, Lesson 6, Centers: Shake and Spill, students independently demonstrate conceptual understanding as they count objects and connect counting to cardinality. Task Statement, “Students put some counters in a cup. They shake, spill, and count the counters. They may choose to use the 5-frame to organize the counters. Both partners count the counters. Then, they choose a different number of counters and repeat.” (K.CC.4)

• Unit 8, Putting It All Together, Lesson 21, Activity 2, students demonstrate the composition and decomposition of numbers 11–19. “Give students access to connecting cubes or two-color counters, 10-frames, and bead tools. Kiran wrote equations to show the total number of students and how many students sat at the table and how many sat on the rug, but he didn’t finish the equations. Finish filling in each equation. You can use connecting cubes or two-color counters if they are helpful. . . . . . .” (K.NBT.1)

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

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

• Unit 5, Composing and Decomposing Numbers to 10, Lesson 9, Warm-up, students extend the verbal count sequence to 70 and count on from a given number as pointed by the teacher. Launch, “Let’s count to 70. Count to 70 1–2 times as a class.” Launch, “Now, start at the number 7 and count to 30. Count on from 7 to 30. Repeat 3–4 times starting with other numbers within 10.” Activity Synthesis, “When I say a number, tell your partner what number comes next when we count. What numbers come after 11 when we count? 30 seconds: partner discussion Repeat 3–4 times with numbers 1–20.” (K.CC.1, K.CC.2)

• Unit 6, Numbers 0-20, Lesson 3, Activity 3, students develop fluency with addition and subtraction within 5 as they find the number that makes 5 when added to a given number. Activity Launch, “Display a card with the number 4.My card says 4. What card do I need to go with it to make 5? (1) I need a 1 card. I’m going to ask my partner if they have a 1 card. If my partner has a 1 card, they will give it to me. I will put the 4 card and 1 card down as a match and write an expression. If I have a 4 card and a 1 card, what expression should I write? $4+1$ or $1+4$.” (K.OA.5)

• Unit 8, Putting It All Together, Lesson 15, Warm-up, students analyze and compare expressions and equations to strengthen their number sense and procedural fluency. Activity, “Which one doesn’t belong? a. $3+1$, b. $3=2+1$, c. $3+0$, d. $4-1$” (K.OA.2, K.OA.5)

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

• Unit 4, Understanding Addition and Subtraction, Lesson 17, Activity 1, students find the value of addition expressions with +0 and +1. Launch, “Give each group of students a copy of the blackline master and a connecting cube. Give students access to connecting cubes and two-color counters. ‘Take turns with your partner. Roll the cube to figure out if you need to add 0 or 1. Fill in the expression. Find the value of the expression and write the number on the line. You can use objects or drawings if they are helpful.’” (K.OA.5).

• Unit 6, Numbers 0-20, Lesson 7, Centers, Narrative, “Before playing, students remove the cards that show numbers greater than 5 and set them aside. Partner A asks their partner for a number that would make 5 when added to the number on one of their cards. If Partner B has the card, they give it to Partner A and Partner A gets a match. If not, Partner A chooses a new card. When students make the target number 5, they put down those two cards and write an expression to represent the combination. Students continue playing until one player runs out of cards. The player with the most pairs wins.” (K.OA.5) 

• Unit 8, Putting It All Together, Lesson 15, Cool-down, students practice adding and subtracting within 5. Student Task Statements, “Find the value of each expression. $2+3$, $4-1$, $5-3$.” (K.OA.5)

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

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

Examples of routine applications include:

• Unit 2, Numbers 1-10, Lesson 13, Activity 1, students count and “match groups of images to numbers” (K.CC.5). Student Task Statements provide students with images of counters and written numbers. Students match the numbers to the amount of counters they represent. “Draw a line from each number to the group of dots that it matches.” 5 minutes: independent work time.

• Unit 4, Understanding Addition and Subtraction, Lesson 9, Activity 1, students represent and solve an Add To, Result Unknown story problem (K.OA.2). Activity, “2 minutes: quiet work time.” Student Task Statements, “There are 4 markers at school. Elena brought 3 more markers to school. How many markers are at school now?”

• Unit 8, Putting It All Together, Lesson 18, Activity 1, students solve a Put Together, Total Unknown story problem and a related Put Together/Take Apart, Both Addends Unknown story problem (K.OA.2, K.OA.3). Student Task Statements, Problem 1, “There are 6 pigeons in the fountain. There are 4 pigeons on the bench. How many pigeons are there?” Launch, “Tell your partner what happened in the story.”

Examples of non-routine applications include:

• Unit 1, Math in Our World, Lesson 9, Activity 2, students understand the relationship between numbers and quantities. (K.CC.4) “You are going to make a page for a picture book like the ones we looked at earlier. There are two dots at the top of the page, so on this page you should draw things that there are two of in our classroom. 3 minutes: independent work time.”

• Unit 5, Composing and Decomposing Number to 10, Section C Practice Problems, Problem 9, students solve a real-world problem (K.CC.3, K.OA.4). About this Lesson, “Teachers may decide to assign the practice problems for in-class practice, homework, or as a means to assess certain learning on a particular concept.” Student Task Statements, “Diego is playing What’s Behind My Back? He has a tower of 10 cubes. He accidentally snaps the tower into 3 pieces. He shows this tower.” The image is a tower of 3 cubes. “How many cubes could be in Diego’s other two towers?”

• Unit 6, Numbers 0-20, Lesson 3, Activity 2, students understand that counting the same collection should yield the same result each time as they discuss real-world problems (K.CC.4). Student Task Statements, “Clare, Andre, and Noah all counted these cubes. Clare says there are 15 cubes. Andre says there are 16 cubes. Noah says there are 17 cubes. Can they all be right?”

The materials reviewed for Imagine Learning Illustrative Mathematics Kindergarten meet expectations in that the three aspects of rigor are not always treated together and are not always treated separately. There is a balance of the three aspects of rigor within the grade. 

In the K-5 Curriculum Guide, Why is the curriculum designed this way?, Design Principles, Balancing Rigor, “opportunities to connect new representations and language to prior learning support students in building conceptual understanding. Access to new mathematics and problems prompts students to apply their conceptual understanding and procedural fluency to novel situations. Warm-ups, practice problems, centers, and other built-in routines help students develop procedural fluency, which develops over time.”

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

• Unit 2, Numbers 1-10, Lesson 4, Activity 1, students apply their understanding as they identify the group of objects that has more. Launch, “Give each group of students access to connecting cubes and two-color counters. We have been learning about different tools that we use at home and in our classroom. What kind of tools do you use when you eat at home? (Spoons, forks, chopsticks, plates, bowls, napkins, cups, straws).  We use many different tools when we eat. Display and read the story. What is the story about? (A family eating dinner, Priya’s family, spoons for dinner) Read the story again. How can you act out this story? (We can pretend we are sitting at the table and pretend to hand out spoons. We can use the cubes to show the people and the counters to show the spoons. We can draw a picture.)” Student Task Statements, “Priya and her family are sitting down at the table for dinner. There are 4 people sitting at the table. There are 6 spoons. Are there enough spoons for each person to get one?” (K.CC.6)

• Unit 6, Numbers 0-20, Lesson 2, Activity 1, students extend their conceptual understanding as they count their collection in a way that makes sense to them and keep track of which objects have been counted. Students are given a collection of objects to count. Activity, “Give each student a collection of objects and access to 10-frames and a counting mat. How many objects are in your collection?” Activity, “Monitor for students who use the 10-frame or the counting mat to organize and count their objects. Tell your partner how many objects are in your collection. Synthesis of Counting Collection, “What do you notice about how they counted? (They said one number for each object. They used the counting mat/10-frame to organize their objects. They counted all of the objects one time.)” (K.CC.4)

• Unit 8, Putting It All Together, Lesson 12, Cool-down: Unit 8, Section C Checkpoint, students demonstrate their fluency as they use strategies to find sums and differences. Student response states, “Students count all to find the sum. Students use their knowledge of the count sequence to find certain sums. Students know certain sums. Students represent all, then cross off or remove to find the difference. Students use their knowledge of the count sequence to find certain differences. Students know certain differences.” (K.OA.5)

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

• Unit 4, Understanding Addition and Subtraction, Lesson 12, Activity 2, students develop conceptual understanding alongside application as they solve story problems within 10. Launch, “Give students access to connecting cubes or counters. Reread the story problem from the first activity. What is the same about the story problems? What is different about them? (They are both about ducks in the pond. They are different because in the first one, more ducks came to swim in the pond, and in the second one, some of the ducks left the pond.)” Student Task Statements, “There were 9 ducks swimming in the pond. Then 4 of the ducks waddled onto the grass. How many ducks are swimming in the pond now?” (K.OA.2)

• Unit 5, Composing and Decomposing Numbers, Lesson 11, Activity 2, students extend their conceptual understanding and procedural fluency as they represent equations on fingers. Launch, “Give each student at least 2 different colored crayons. Color the fingers to show each equation.” Activity, “As you continue working, tell your partner about the total and the 2 parts you colored in each set of fingers.” Student Task Statements, ". $10=9+1$. $10=5+5$. $10=3+7$. $10=8+2$. $10=1+9$.” (K.OA.1)

• Unit 8, Putting it All Together, Lesson 19, Activity 2, students develop conceptual understanding alongside procedural skill and fluency as they find a number that makes 10 when added to the given number. Launch, “Give students access to connecting cubes or two-color counters, bead tools, and 10-frames. ‘Fill in each equation so that they show a way to make 10.’” Student Task Statements, “Fill in the equation to show ways to make 10.” Equations included are 10 = 9 + ___; 10 = 3 + ___ ; 10 = 5 + ___; 10 = 4 + ___; 10 = 8 + ___; 10 = 7 + ___. Activity Synthesis, “How did you choose which tool to use to help you figure out which number you needed to make 10? Were there any problems that you didn’t need to use a tool to figure out?” (K.OA.4)

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

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

• Unit 1, Math In Our World, Lesson 8, Warm-up, students consider different ways of acting out a story. Student Task Statements, “3 little ducks went out one day, over the hill and far away. Mother duck said, “Quack, quack, quack. Then 3 little ducks came back. What is the story about?” Activity Narrative, “Acting out gives students opportunities to make sense of a context (MP1).”

• Unit 3, Flat Shapes All Around Us, Lesson 12, Activity 1, students use pattern blocks to complete puzzles that do not show each individual pattern block. Activity, 4 minutes: independent work time. “Monitor for students who fill in the puzzle with different pattern blocks.” Activity Launch, “Groups of 2. Give each student pattern blocks. ‘We are going to learn a new way to do the Pattern Blocks center.’ Display the book. ‘What do you notice? What do you wonder? (They are puzzles. Some of the lines in the middle are not there.)’ 30 seconds: quiet think time. 30 seconds: partner discussion. Share responses. ‘Use the pattern blocks to fill in the puzzle. Write a number to show how many of each pattern block you used.’” Activity Narrative, “In either case, as they fill in the puzzle, each choice they make will influence which shapes they can use and whether or not they can fill in the entire puzzle so students will need to persevere and likely go back and make changes (MP1).”

• Unit 5, Composing and Decomposing Numbers to 10, Lesson 5, Warm-up, students make sense of problems before solving them. Activity, “‘Discuss your thinking with your partner.’ 1 minute: partner discussion. Share and record responses.” Student Task Statements, “What do you notice? What do you wonder? Elena used 9 pattern blocks to make a train. Then she took 3 of the pattern blocks off of the train and put them back in the bucket.” Activity Narrative, “This warm-up prompts students to make sense of a problem before solving it by familiarizing themselves with a context and the mathematics that might be involved (MP1).”

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

• Unit 2, Numbers 1-10, Lesson 16, Activity 2, students make connections between counting, recognizing, and writing numbers of 1-10 objects. Activity, “Give each group of students 4 bags filled with 1-10 objects and sticky notes. ‘Work with your group to figure out how many objects are in each bag. Write a number on the sticky note to show how many objects are in each bag.’ 5 minutes: small-group work time. Each group trades their bags with another group. ‘Now you have another group’s bags. Look at their sticky notes and see if you can figure out how many objects are in the bag. Then check in the bag to see how many objects are really in the bag.’” 5 minutes: small-group work time. Student Task Statements, “Work with your group to figure out how many objects are in each bag.” Activity Narrative, “In the activity synthesis, students determine how many objects are in a bag based on the number label, which encourages them to connect numbers to quantities (MP2).”

• Unit 6, Numbers 0-20, Lesson 10, Activity 1, students relate the ten-frame images to equations. About this lesson, “In this lesson, students interpret equations and fill in the missing numbers to complete equations for numbers 11–19 (MP2).” Activity launch, “Groups of 2. “Use the dots to find the numbers that make each equation true.” Activity, “2 minutes: independent work time. 3 minutes: partner work time” Student problems: 10 + 8 = ___; 10 + 3 ___; 10 + 4 = ___ ; ___ + ___ = 16 (each problem has an image of a ten-frame with black dots and some more dots to represent the problem). Activity Narrative, “When students relate the parts of the 10-frame representations and the equations they reason abstractly and quantitatively (MP2).”

• Unit 8, Putting It All Together, Lesson 13, Activity 1, students “sort dominoes into groups by total as they identify different compositions and decompositions of numbers to 5.” Activity, “‘Work together to sort the dominoes into groups based on the total number of dots. As you work together, tell your partner the parts that you see and how many total dots you see.’ 3 minutes: partner work time ‘Choose one of the groups that you sorted the dominoes into. Write an expression to show each domino.’ Activity Narrative, Students represent each domino with an expression (MP2).” Student Task Statements, “Sort the dominoes based on the total number of dots. Choose 1 group. Write an expression for each domino.”

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

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

Examples of constructing viable arguments include:

• Unit 3, Flat Shapes All Around Us, Lesson 12, Activity 2, students construct viable arguments as they put together pattern blocks to form larger shapes in more than one way. Launch, “Groups of 2. Give each group of students pattern blocks. Work with your partner to find many different ways to make hexagons with pattern blocks. Tell your partner about the shapes that you use each time using ‘more’, ‘fewer’, or ‘the same number’. Student Task Statements, “Work with your partner to find many different ways to make hexagons with pattern blocks.” Activity Narrative, “In the synthesis, students discuss whether the same pattern blocks in different orientations should be considered as different ways to make a hexagon (MP3).”

• Unit 5, Composing and Decomposing Numbers to 10, Lesson 8, Activity 2, students construct viable arguments and critique the reasoning of others as they solve a Put Together/Take Apart, Both Addends Unknown story problem about dates stuffed with cheese or almonds in more than one way. Activity, “2 minutes: quiet work time. 2 minutes: partner discussion. Write an expression to show how many of the dates were stuffed with cheese and how many were stuffed with almonds. 1 minute: independent work time. ‘As you walk around, look to see if you can find other ways to solve the story problem.’ 5 minutes: gallery walk.” Student Task Statements, “Andre and his older brother have 8 dates. They stuff some of the dates with cheese. They stuff the rest of the dates with almonds. How many of the dates did they stuff with cheese? Then how many of the dates did they stuff with almonds?” Activity Narrative, “Some students may determine that both solutions are the same because they both showed 5 and 3 while other students may determine that the solutions are different because there are different numbers of dates stuffed with cheese and dates stuffed with almonds (MP3).”

• Unit 7, Solid Shapes All Around Us, Lesson 3, Warm-up, students produce questions about shapes composed of pattern blocks. As students discuss and justify their questions and answers, they “share a mathematical claim and the thinking behind it (MP3).” Student Task Statements, “Mai used pattern blocks to make this shape. What do you notice? What do you wonder?” Activity Narrative, “‘Discuss your thinking with your partner.’ 1 minute: partner discussion ‘Share and record responses.’”

Examples of critiquing the reasoning of others include:

• Unit 2, Numbers 1-10, Lesson 17, Activity 2, students critique the reasoning of others as they put cube towers and numbers in order in a way that makes sense to them. Launch, “Groups of 2. Give groups access to number cards and the cube towers created in the previous activity.” Student Task Statements, “Put your cube towers and numbers in order in a way that makes sense to you.” Activity Narrative, “These different strategies are discussed in the synthesis, giving students a chance to articulate different ways they made sense of ordering the towers and numbers (MP3).”

• Unit 7, Solid Shapes All Around Us, Lesson 9, Activity 1, students critique the reasoning of others as they think about and compare the capacities of containers. Activity, “Display 2 cups and give each student a sticky note. ‘Which of these cups do you think would hold more lemonade? Put your sticky note by the cup that you think would hold more lemonade.’ 3 minutes: independent work time. ‘People had different answers about which cup would hold more lemonade. What can we do to figure out which cup can hold more lemonade?’ 1 minute: quiet think time. 1 minute: partner discussion. ‘Share and record responses.’ Demonstrate filling one of the cups with water and then slowly pour that water into the other cup. ‘I filled up the red cup and poured the same water into the blue cup, but the blue cup overflowed. Which cup do you think can hold more lemonade?’ 30 seconds: quiet think time. 1 minute: partner discussion. ‘Share responses. The red cup can hold more lemonade than the blue cup.’” Launch states, “Diego’s class needs a lot of lemonade for a lemonade sale they are going to have at school. Which container do you think they should use to hold the lemonade? Why do you think that?” Activity Narrative, “As students make predictions and then discuss and justify their comparisons, they share a mathematical claim and the thinking behind it (MP3).”

• Unit 8, Putting It All Together, Lesson 7, Activity 1, students critique the reasoning of others as they “create a number book about their school community.” Narrative, “Students share their work with a partner, receive feedback, and then improve their work (MP3).” Launch, “‘Look through your recording sheet to decide what you would like to put on the first page of your number book about our school.’ 1 minute: quiet think time. Activity Narrative, ‘Think of one or two things that your partner could add or change to make their book even better.’ 30 seconds: quiet think time. 2 minutes: partner discussion. ‘Share responses. Think about your partner’s suggestions as you continue working on your number book.’ 7 minutes: independent work time.”

The materials reviewed for Imagine Learning Illustrative Mathematics Kindergarten meet expectations for supporting the intentional development of MP4: Model with mathematics; and MP5: Use appropriate tools strategically, for students, in connection to the grade-level content standards, as expected by the mathematical practice standards. Students have opportunities to engage with the Math Practices across the year and they are often explicitly identified for teachers in several places including the Instructional Routines (Warm-up Routines and Other Instructional Routines), Activity Narratives, and About this Lesson.

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

• Unit 5, Composing and Decomposing Numbers to 10, Lesson 7, Activity 1, students model with mathematics as they learn there is more than one way to solve a Put Together/Take Apart, Both Addends Unknown story problem.” Student Task Statements, “Jada made 6 paletas with her brother. They made two flavors, lime and coconut. How many of the paletas were lime? Then how many of the paletas were coconut?” Activity Narrative, “When students attend to the mathematical features of a situation, adhere to mathematical constraints, make choices, and translate a mathematical answer back into the context, they model with mathematics (MP4).”

• Unit 7, Solid Shapes All Around Us, Lesson 3, Activity 1, students model with mathematics as they develop math questions about shapes created from pattern blocks. Launch, “Walk around and look at the shapes that everyone created. Think of at least one question that you can ask about each shape that you see.” Activity Narrative, “When students ask mathematical questions and recognize the mathematical features of the shapes and the pattern blocks they are made of, they model with mathematics (MP4).”

• Unit 8, Putting It All Together, Lesson 6, Activity 2, students “identify important objects or features in their school community and connect them to numbers”. Activity Narrative, “When students identify objects around them that they can count they make a first step toward quantifying their world (MP4)”. Launch, “We’re going to take a walk around the school. As we’re walking, look for things that you would like to include in your number book. Use your recording sheet so that you remember your ideas. If I wanted to write about how many tables are in our class, what could I put on my recording sheet so I remember?”

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

• Unit 1, Math in Our World, Lesson 17, Activity 1, students “count their collection in a way that makes sense to them and to answer how many questions without recounting the collection.” Students use appropriate tools strategically as they choose which tools help them count their collections (MP5).” Activity Narrative, “‘Figure out how many cubes are in your collection. Show how you counted your collection. Show your thinking using objects, drawings, numbers, or words.’ 2 minutes: independent work time, ‘How many cubes are in your collection? Tell your partner how many cubes are in your collection without counting them again.’ 2 minutes: partner discussion.”

• Unit 2, Numbers 1-10, Lesson 20, Activity 1, students “represent a number from 1–10 in different ways. Students use appropriate tools strategically as they choose which objects to use and how to organize them to represent their number (MP5)”. Launch, “‘Give each group a half-sheet of chart paper and access to crayons or colored pencils and connecting cubes or counters. Work with your partner to choose a number from 1–10 to show in many different ways.’ 1 minute: partner discussion, ‘Show your number in as many different ways as you can.’ Synthesis,’ What are some different ways you and your partner showed your number? (We counted out 5 cubes. We wrote the number 5. We drew groups of 5 images. We showed that it is 1 more than 4 and 1 less than 6.)’” 

• Unit 4, Understanding Addition and Subtraction, Lesson 11, Activity 1, students have access to tools that they can choose from as they draw a picture to represent and solve a story problem. Student Task Statements, “There were 7 kids playing soccer in the park. 3 of the kids left to go play on the swings. How many kids are playing soccer in the park now?(includes image of students playing soccer.)” Activity Narrative, “Students should have access to connecting cubes and two-color counters to help them represent the story (MP5).”

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

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

• Unit 2, Numbers 1-10, Lesson 20, Activity 2, Activity Narrative, “When students share how they compare their numbers, they use their own mathematical vocabulary and listen to others' thinking (MP6).” Launch, “We are going to go on a gallery walk. We will look at two charts showing different numbers that our classmates made. First, talk to your group about what number each chart shows. Then, compare the numbers using the words 'more', 'less', and 'the same number'.”

• Unit 3, Flat Shapes All Around Us, Lesson 8, Activity 2, students use precision to “describe and draw shapes”. Activity Narrative, “Students learn that they need to be precise in describing the shape in order for their partner to draw the shape accurately and have opportunities to use the language they have learned to describe shape attributes (MP6).” Launch, “Today we’re going to play a new game called ‘Draw the Mystery Shape’. One partner will choose a shape and describe it. The other partner will draw the shape.”

• Unit 4, Understanding Addition and Subtraction, Lesson 7, Activity 2, Activity Narrative, “While in the first activity, the story was provided, in this activity students create the action in the story, which is an opportunity to hear what language students associate with addition and subtraction (MP6).” Student Task Statements, “There were 7 kids playing tag on the field.” Activity Launch, “Groups of 2. ‘We have heard and acted out some stories about students playing at school. Where else in your community do you see people playing outside? Describe it to your partner.’ 30 seconds: quiet think time. 1 minute: partner discussion. ‘Share and record responses. Display images from the student book. Some of the places where we play and walk around outside are parks and playgrounds. These stories all take place in different parks and playgrounds. How are these pictures the same as parks and playgrounds that you have been to? How are they different?’ Give each student a bag of 10 two-color counters. ‘We’re going to use our counters to show what is happening in our stories, but this time, the stories aren’t finished yet.’” Synthesis, “Tell your partner how your objects or drawings show what happened in the story.”

• Unit 7, Solid Shapes All Around Us, Lesson 14, Activity 2, students use the specialized language of mathematics to recreate a building using blocks. Activity Narrative, “Students practice using names of solid shapes and positional words as they try to recreate a building (MP6).” Activity, “‘This time I am going to show you something that I built, and you and your partner will work together to try to make the same thing. But you will only get to look at what I built for one minute, so look closely and try to remember where the shapes go.’ Display the building, built with 6–8 solid shapes. Allow students to look at the building closely. Then cover the building. ‘Work with your partner to build the same thing that I built.’ 3 minutes: partner work time.” Synthesis, “‘Now that you have seen what I built again, tell your partner where to put the shapes to revise what you built.’ 2 minutes: partner work time. Invite students to share how they changed their building using positional words and names of shapes.”

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

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

• Unit 4, Understanding Addition and Subtraction, Lesson 17, Warm-up, students make use of structure as they elicit the idea that adding 1 results in the next number in the count sequence. Activity Narrative, “While students may notice and wonder many things about these towers, 1 more being added to each tower and how that affects the total number of cubes are the important discussion points.” Student Task Statements, “What do you notice? What do you wonder? (3 images: 2 stacks of linking cubes, one row is labeled 2 and one row is labeled 3; 2 stacks of linking cubes, one row is labeled 7 and one row is labeled 8; 2 stacks of linking cubes, one row is labeled 4 and one row is not labeled.)” Activity, “‘Discuss your thinking with your partner.’ 1 minute: partner discussion. ‘Share and record responses.’” Activity Synthesis, “What changed from this tower (point to the tower with 2 cubes) to this tower (point to the tower with 3 cubes)? (There are still 2 blue cubes, but 1 more yellow cube was put on top.)” This Activity Synthesis continues on the next card. “What expression can we write to show what changed from this tower (point to the tower with 2 cubes) to this tower (point to the tower with 3 cubes, $2+1$.)” This activity synthesis continues on the next card. “How many cubes do you think are in the last tower? How do you know? (5. I counted them. I know that there is 1 more than the last tower, and 5 is one more than 4.)”

• Unit 7, Solid Shapes All Around Us, Lesson 7, Activity 2, students look for and make use of structure when identifying and sorting flat and solid shapes. Lesson Narrative, “A sorting task gives students opportunities to analyze the structure of the shapes and identify common properties and characteristics (MP7)”. Activity Narrative, “‘Work with your partner to sort the shapes into two groups. Write a number to show how many shapes are in each group. 3 minutes: partner work time. Pair up with another group. Show them how you sorted your shapes. Did you sort all of the shapes in the same way?’ 3 minutes: small-group work time ‘What could you call each group of shapes to show why you put those shapes together?’” Activity Synthesis, “Invite previously selected students to share the way they sorted the shapes into flat shapes and solid shapes. ‘This group has flat shapes. This group has solid shapes.’ Display a square. ‘Should this shape go with the flat shapes or the solid shapes? Why? (It is a flat shape. If we put it on our desk, it doesn’t stick up.)’ Display a cube. ‘Should this shape go with the flat shapes or the solid shapes? Why? (It is a solid shape. It sticks up and takes up space.)’”

• Unit 8, Putting it All Together, Lesson 16, Activity 2, students look for and make use of structure as they find the missing value in addition and subtraction equations. Lesson Narrative, “Students may just know some of the answers or they may use counting forward or backward (MP7) or they may draw a picture”. Student Task Statements, “Fill in the missing part of each equation. 3 - ___ = 2, 2 + ___ = 2, 5 - ___ = 2, 1 + ___ = 2.” 

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

• Unit 2, Number 1-10, Lesson 11, Numbers 1–10, Activity 1, students notice patterns as they “draw groups of images that have more, fewer, or the same number of images as a group drawn by their partner.” Activity Narrative, “Students see that when creating a group that is more than another group, you first have to make the same amount and then add more (MP8).” Launch, “You are going to draw a group of things. Then show your group to your partner and say one of the sentences. Draw a group that has more things than my group. Draw a group that has fewer things than my group. Draw a group that has the same number of things as my group. Your partner will draw a group next to yours, tell you how many things are in the group, and say a sentence using ‘more’, ‘fewer’, or ‘the same number’. Switch roles and repeat.” Activity Synthesis, “‘I need to draw a group of things that has more than this group.’ Draw 2 circles. Does this group have more things? How can you tell? (No, I know 2 is less than 4.)’ Draw 2 more circles. ‘Does this group have more things? How can you tell? (No, they both have 4 circles so they are the same.)’ Draw 1 more circle. ‘Does this group have more things? How can you tell? (Yes, because they both have 4, but your group has 1 more.) What if I drew another circle? (You would still have more. You could keep drawing as many circles as you want and you will always have more than 4.)’”

• Unit 5, Composing and Decomposing Numbers to 10, Lesson 12, Cool-down, students look for and express regularity in repeated reasoning as they compose and decompose 10 in multiple ways. Student Task Statements, “Recognize that a full 10-frame contains 10 counters and that 2 hands have 10 fingers. Relate equations to compositions and decompositions of 10. Given a number, use the structure of 10-frames or fingers to determine how many more are needed to make 10.” Activity Narrative, “With repeated experience composing 10 in many ways, students may begin to know the combinations to make 10 (MP8).”

• Unit 6, Numbers 0-20, Lesson 3, Activity 1, students look for and express regularity in repeated reasoning as they, Activity Narrative, “​​Students notice that the number of objects stays the same when a collection is counted multiple times (MP8).” Launch, “Groups of 2. Give each student a collection of objects and access to 10-frames and a counting mat. ‘How many objects are in your collection?’” Activity Narrative, “3 minutes: independent work time. ‘Tell your partner how many objects are in your collection.’ 30 seconds: partner discussion. ‘Switch collections with your partner. Do you agree with your partner about how many objects are in the collection?’ 3 minutes: independent work time.” Activity Synthesis, “Show your partner how you counted and tell your partner how many objects are in your new collection. Did you and your partner count the collection the same way? Did you agree about how many objects are in the collection? We might count the objects differently than our partner, but we should get the same number if we count every object one time.”

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

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

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

• IM Curriculum, Scope and sequence information, provides an overview of content and expectations for the units. “The big ideas in kindergarten include: representing and comparing whole numbers, initially with sets of objects; understanding and applying addition and subtraction; and describing shapes and space. More time in kindergarten is devoted to numbers than to other topics.”

• Unit 3, Flat Shapes All Around Us, Section B, Making Shapes, Section Overview, Section Narrative, provides an overview of the content and expectations for the section. For example, “In this section, students develop spatial reasoning by manipulating shapes and solving geometric puzzles while using geometric language from earlier work. Students use pattern blocks to compose geometric figures, explore shapes in different orientations, find shapes that match exactly, and complete puzzles that require reorienting shapes. Throughout the section, students use their own language to describe how the shapes they are working with are alike and different, including descriptions of the side lengths of shapes in their comparison.”

Materials include sufficient and useful annotations and suggestions that are presented within the context of the specific learning objectives. Several components focus specifically on the content of the lesson. Preparation and Lesson Narratives, Warm-up, Activities, and Cool-down Narratives all provide useful annotations. IM Curriculum, Why is the curriculum designed this way?, Design Principles, Coherent Progressions, “Each unit starts with an invitation to the mathematics. The first few lessons provide an accessible entry point for all students and offer teachers the opportunity to observe students’ prior understandings. Each lesson starts with a warm-up to activate prior knowledge and set up the work of the day. This is followed by instructional activities in which students are introduced to new concepts, procedures, contexts, or representations, or make connections between them. The lesson ends with a synthesis to consolidate understanding and make the learning goals of the lesson explicit, followed by a cool-down to apply what was learned. Each activity starts with a launch that gives all students access to the task. This is followed by independent work time that allows them to grapple with problems individually before working in small groups. The activity ends with a synthesis to ensure students have an opportunity to consolidate their learning by making connections between their work and the mathematical goals. In each of the activities, care has been taken to choose contexts and numbers that support the coherent sequence of learning goals in the lesson.” Examples include:

• Unit 2, Numbers 1–10, Lesson 2, Activity 1, teachers are provided context as they help students recognize the arrangement of groups does not change the number in each group. Narrative, “Students grab a handful of connecting cubes and count to see how many they have. They then rearrange the connecting cubes using a 5-frame and discover that although the connecting cubes are arranged differently, the number of connecting cubes stays the same. This understanding develops over time with repeated experience working with quantities in many different arrangements. Students may continue to recount the objects in this and future lessons until they understand and are confident that the number of objects remains the same when they are rearranged.” Launch, “Groups of 2. Give each group of students connecting cubes. ‘We are going to play a game with our connecting cubes and 5-frame. One person will grab a handful of connecting cubes and figure out and tell their partner how many there are. Then the other partner will organize the connecting cubes using the 5-frame, and figure out and tell their partner how many there are. Take turns playing with your partner.’” Activity, “5 minutes: partner work time. Monitor for students who notice that the number of objects is the same after they are rearranged.”

• Unit 5, Composing and Decomposing Numbers to 10, Lesson 4, Find All the Ways, Warm-up, Lesson Plan, Teaching Notes provide information to the teacher for teaching specific parts of the lesson. For example, “Pacing: 10 minutes for warm-up activity and synthesis; About the warm-up: Warm-ups help students get ready for the day's lesson, or give students an opportunity to strengthen their number sense or procedural fluency. Activity Narrative: The purpose of this warm-up is to count on from a given number. As students count, point to the numbers posted so that students can follow along.”

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

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

• Why is the curriculum designed this way? Further Reading, Unit 1, When is a number line not a number line?, supports teachers with context for work beyond the grade. “In this blog post, McCallum shares why the number line is introduced in grade 2 in IM K–5 Math, emphasizing the importance of foundational counting skills.”

• Why is the curriculum designed this way? Further Reading, Unit 7, What is a Measurable Attribute?, “In this blog post, Umland wonders about what counts as a measurable attribute and discusses how this interesting and important mathematical idea begins to develop in kindergarten.”

• Unit 1, Math in Our World, Lesson 6, Look for Small Groups, About this Lesson, “This skill (subitizing) is essential to students’ number work. Students communicate how many there are by showing quantities on their fingers and saying number words (MP6). Although some students may count to determine how many, the focus of this lesson is on recognizing groups of objects without counting. Students learn two new routines that will be used throughout the year to develop counting concepts. These routines will continue to be developed throughout the section and will be used across the year Throughout the section, observe students for the look-fors on the Unit 1, Sections A-D Checkpoint. In the Lesson Synthesis, students practice saying the verbal count sequence to 10 in About this Lesson for counting objects in an upcoming section. Add variety to the counting by adding movement. For example, students can count as they clap, stomp their feet, or jump.”

• Unit 5, Composing and Decomposing Numbers to 10, Lesson 15, About this Lesson, “This lesson does provide students with an opportunity to apply precursor skills of mathematical modeling. In previous lessons, students represented and solved Put Together/Take Apart, Both Addends Unknown story problems. This lesson builds on students’ experience in the Math Stories center. In this lesson, students use familiar contexts to generate and solve Put Together/Take Apart, Both Addends Unknown story problems. In the second activity, students are encouraged to find all possible solutions and use reasoning based on patterns explored in previous lessons (MP8). When students attend to the mathematical features of a situation, adhere to mathematical constraints, make choices, and translate a mathematical answer back into the context they model with mathematics (MP4).”

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

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

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

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

• 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 4, Understanding Addition and Subtraction, Lesson 18, the Core Standards are identified as K.CC.1, K.CC.2, K.OA.1, K.OA.2. Lessons contain a consistent structure that includes a Warm-up with a Narrative, Launch, Activity, Activity Synthesis. An Activity 1, 2, or 3 that includes Narrative, Launch, Activity, Activity Synthesis, Lesson Synthesis. A Cool-down that includes Responding to Student Thinking and Next Day Supports. This provides an additional place to reference standards, and language of the standard, within each lesson.

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

• Unit 2, Numbers 1-10, Unit Overview, Unit Learning Goals, full Unit Narrative, includes an overview of how the math of this module builds from previous work in math. “In this unit, students continue to develop counting concepts and skills, including comparing, while learning to write numbers. Previously, students answered “how many” and “are there enough” questions and counted groups of up to 10 objects. They also learned the structures and routines for activities and centers. Here, students rely on familiar activity structures to build their counting skills and concepts. First, they count and compare the number of objects, and then do the same with groups of images. The images are given in different arrangements—in lines, arrays, number cube patterns, on 5-frames—to help students connect different representations to the same number.”

• Unit 6, Numbers 0-20, Unit Overview, Unit Learning Goals, full Unit Narrative, includes an overview of how the math of this module builds from previous work in math. “In this unit, students count and represent collections of objects and images within 20. They apply previously developed counting concepts—such as one-to-one correspondence, keeping track of what has been counted, and conservation of numbers—to larger numbers. Previously, students have counted, composed, and decomposed numbers up to 10, using tools such as counters, connecting cubes, 5-frames, 10-frames, drawings, and their fingers. They wrote expressions to record compositions and decompositions. Here, students use the 10-frame to organize groups of 11-19 objects and images. This tool encourages students to see teen numbers as 10 ones and some more ones, emphasizing the $10 + n$ structure of the numbers 11–19. They use this structure as they represent teen numbers with their fingers, objects, drawings, expressions, and equations. Students see equations with the addend written first, such as $10 + 6 = 16$.”

The materials reviewed for Imagine Learning Illustrative Mathematics Kindergarten meet expectations for providing explanations of the instructional approaches of the program and identification of the research-based strategies.

Instructional approaches of the program are described within the Curriculum Guide, Why is the curriculum designed this way? Design Principles. “It is our intent to create a problem-based curriculum that fosters the development of mathematics learning communities in classrooms, gives students access to the materials through coherent progression, and provides teachers the opportunity to deepen their knowledge of mathematics, student thinking, and their own teaching practice. Examples from the Design principles include:

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

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

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

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

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

• Curriculum Guide, Why is the curriculum designed this way?, Key Structures in these materials, 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.”

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

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

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

• Course Overview, Grade Level Resources, Grade K Materials List, contains a comprehensive chart of all materials needed for the curriculum. It includes the materials used throughout the curriculum, whether they are reusable or consumable, quantity needed, lessons the materials are used in, and suitable substitutes for the materials. Each lesson listed in the chart and any additional virtual materials noted for a lesson are digitally linked in the materials for quick access. Geoblocks are a reusable material used in lessons K.1.4, K.1.5, K.1.6, K.1.7, K.1.8, K.1.9, K.1.10, K.1.11, K.1.12, K.1.13, K.1.14, K.1.15, K.1.16, K.1.17, K.2.14, K.3.10, K.3.11, K.3.12, K.3.13, K.3.14, K.7.1, K.7.2, K.7.3, K.7.4, K.7.5,… 15 sets with at least 10 shapes in each set are needed per 30 students. Block set or cardboard boxes are suitable substitutes. Brown paper bags are a consumable material used in lessons K.2.3, K.2.6, K.2.12, and K.2.16. 45 brown bags are needed per 30 students. No suitable substitutes for the material are listed. Play dough or modeling clay is a reusable material used in lessons K.3.7, K.7.7, K.7.8, K.7.9, K.7.10, K.7.11, K.7.12, and  K.7.13. 15 are needed per 30 students. No suitable substitutes are listed.

• Course Overview, Grade Level Resources, Grade K Picture Books, contains a “list of suggested picture books to read throughout the curriculum.” Unit 2, The Little Red Hen (Makes a Pizza) by Philomen Sturges is suggested. Unit 3, Stitchin’ and Pullin’: A Gee’s Bend Quilt by Patricia McKissack is suggested. Unit 7, The Seesaw by Judith Koppens is suggested.

• Unit 8, Putting It All Together, Lesson 14, Activity 1, Teaching Notes, Materials to gather, “Colored pencils, crayons, or markers.” Launch, “Display the student book. This code tells us which color to use. If the group of dots or expression shows 5, you are going to color that section brown. This section says $2+0$. What color should I color this section? How do you know? (You should color it green. $2+0$ is 2.) Figure out the total number of dots in each image. Find the value of each expression. Check the key to determine which color to use in this section. If the expression is $2+1$, you would color that section red, because in the key it says that ‘3’ should be colored red.”

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

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

Examples include: 

• Unit 4, Understanding Addition and Subtraction, Lesson 2, Activity 2, Narrative, Access for Students with Disabilities, “Action and Expression: Develop Expression and Communication. Some students may benefit from using 5-frames to help count the number of green and red apples. Give students access to 5-frames and counters to represent the apples in each problem. Invite students to use the 5-frames to figure out how many apples there are altogether. Supports accessibility for: Organization, Conceptual Processing.”

• Unit 5, Composing and Decomposing Numbers to 10, Lesson 4, Activity 3, Narrative, Access for Students with Disabilities, “Engagement: Provide Access by Recruiting Interest. Use visible timers or audible alerts to help learners anticipate and prepare to transition between activities. Supports accessibility for: Social-Emotional Functioning, Organization.”

• Unit 7, Solid Shapes All Around Us, Lesson 9, Activity 2, Narrative, Access for Students with Disabilities, “Engagement: Develop Effort and Persistence. Invite students to generate a list of shared expectations for group work. Ask students to share explicit examples of what those expectations would look like in this activity. Supports accessibility for: Social-Emotional Functioning.”

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

While there are no instances where advanced students do more assignments than classmates, materials do provide multiple opportunities for students to investigate grade-level content at a higher level of complexity. These are found in a section titled, “Exploration Problems” within lessons where appropriate. According to the IM K-5 Teacher Guide, How do you use the materials?, Practice Problems, Exploration Problems, “Each practice problem set also includes exploration questions that provide an opportunity for differentiation for students ready for more of a challenge. There are two types of exploration questions. One type is a hands-on activity that students can do directly related to the material of the unit, either in class if they have free time, or at home. The second type of exploration is more open-ended and challenging. These problems go deeper into grade-level mathematics. They are not routine or procedural, and they are not just “the same thing again but with harder numbers. Exploration questions are intended to be used on an opt-in basis by students if they finish a main class activity early or want to do more mathematics on their own. It is not expected that an entire class engages in exploration problems, and it is not expected that any student works on all of them. Exploration problems may also be good fodder for a Problem of the Week or similar structure.”

Examples include:

• Unit 2, Numbers 1–10, Section B: Count and Compare Groups of Images, Problem 7, Exploration, “‘Are there fewer students than chairs? Explain how you know.​​​​​’ An image is provided of a classroom with students and chairs.”

• Unit 4, Understanding Addition and Subtraction, Section B: Represent and Solve Story Problems, Problem 8, Exploration, “There are 6 dolphins swimming around the boat. Complete the story in two different ways. Solve your problems or share with a partner and solve your partner's problems.”

• Unit 6, Numbers 0–20, Section A: Count Groups of 11-20 Objects, Problem 1, Exploration, “​​​How many shapes do you see on the soccer ball?”

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

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

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

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

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

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

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

• Unit 2, Numbers 1-10, Lesson 9, Activity 1, Teaching Notes, Access for English Learners, “MLR8 Discussion Supports. Make sure students can explain how they know which card has more. Invite groups to rehearse what they will say when they share with the whole class. Advances: Speaking, Conversing.”

• Unit 6, Numbers 0-20, Lesson 13, Activity 2, Teaching Notes, Access for English Learners, “MLR8 Discussion Supports. Pair gestures with verbal directions to clarify the meaning of any unfamiliar terms. Students may benefit from discussing possible strategies they can use to determine order before they begin. Advances: Listening, Representing.”

• Unit 8, Putting It All Together, Lesson 5, Activity 1, Teaching Notes, Access for English Learners, “MLR8 Discussion Supports. Invite each group to chorally read numbers 1–20 in order once the group agrees on the order. Listen for and clarify questions. Advances: Speaking, Conversing.”

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

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

• Unit 4, Understanding Addition and Subtraction, Lesson 6, Activity 3, students use connecting cubes, and a number mat to support understanding of subtraction. Launch, “Give each group of students 10 connecting cubes and a number mat. ‘We are going to learn a center called Subtraction Towers.’ Display a connecting cube tower with 7 cubes. ‘How many cubes are in the tower? If I have to subtract, or take away, 3 cubes from my tower, what should I do?’ (Break off 3 cubes, take off 1 cube at a time as you count.) ‘One partner uses up 5-10 cubes to build a tower. Then the other partner rolls to figure out how many cubes to take away, or subtract, from the tower. Then work together to figure out how many cubes are left in the tower. Take turns building the tower.’”

• Unit 5, Composing and Decomposing Numbers to 10, Lesson 7, Activity 1, students use connecting cubes or two color counters to support solving word problems. Launch, “Give students access to connecting cubes or two-color counters. ‘Many families and cultures make special desserts. Are there desserts that you make with your family?’ 30 seconds: quiet think time 1 minute: partner discussion Share responses. Display the image. ‘Paletas are a type of ice pop popular in Mexico. They are usually made with fruit. Read and display the task statement. Tell your partner what happened in the story. What are we trying to figure out? (How many of the paletas had lime and how many had coconut.) Show your thinking using drawings, numbers, words, or objects.’”

• Unit 8, Putting It All Together, Lesson 13, Activity 2, students use domino cards and sorting cards to compare groups. Launch, “Invite each student to make a pile with half of the domino cards. ‘We are going to play a comparing game with our dominoes. You and your partner will both flip over one card. One partner will compare the number of dots using 'fewer' or 'the same number' and explain how they know. The other partner will compare the number of dots using 'more' or 'the same number. Let’s play one round together.’ Display 2 domino cards. Choose who will go first. Compare the number of dots using 'fewer' or 'the same number' and explain how you know. ‘The domino with 2 dots and 1 dot has fewer dots than the domino with 2 dots and 2 dots. 3 is less than 4. If you did not go first, compare the number of dots using 'more' or 'the same number' and explain how you know.’”