The materials reviewed for Kendall Hunt's 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 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 2, Numbers 1-10, Lesson 9, Warm-up, students develop conceptual understanding as they identify groups that have more, less, or the same number as a given group of images. Activity states, “‘How can we figure out how many students like apples better?’ 30 seconds: quiet think time. 30 seconds: partner discussion. Share responses. Demonstrate or invite students to demonstrate counting. ‘How many students like apples better? How can we figure out how many students like bananas better?’ 30 seconds: quiet think time. 30 seconds: partner discussion. Share responses. Demonstrate or invite students to demonstrate counting. ‘How many students like bananas better?’” (K.CC.B)

• Unit 3, Flat Shapes Around Us, Lesson 1, Activity 2, students develop conceptual understanding as they use informal language to describe shapes and share what they know about different shapes. An image of a Backgammon game is shown. Launch states, “‘What games do you play with your family?’ 30 seconds: quiet think time. 30 seconds: partner discussion. Share responses. ‘Backgammon is a popular game in many different countries, such as Iraq, Lebanon, Egypt, and Syria. Lots of people play backgammon in our country, too. Have you ever played this game or a game like this? Tell your partner about a shape you see in the backgammon game. Take turns describing the shapes you see in the picture with your partner.’ 30 seconds: quiet think time.” (K.G.4)

• Unit 8, Putting It All Together, Lesson 2, Warm-up, students develop conceptual understanding of 10 as they subitize or use grouping strategies to describe the images they see. Dot images are provided and Student Facing states, “How many do you see? How do you see them?” Activity Synthesis states, “‘How is the 10-frame helpful when figuring out how many dots there are?”’ (I know that there are 10 dots on the 10-frame and 10 and 5 is 15. I start counting at 10 and count the rest of the dots.)” (K.NBT.1)

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 1, Math In Our World, Lesson 12, Activity 1, students demonstrate conceptual understanding as they count collections of objects and say one number for each object. Activity states, “Give each student a bag of objects. Give students access to 5-frames and a counting mat. ‘Figure out how many objects are in your collection. Use the tools if they are helpful.’ 2 minutes: independent work time. ‘Switch collections with a partner. Figure out how many objects are in your new collection.’ 2 minutes: independent work time. Monitor for students who say one number for each object.” (K.CC.4a)

• Unit 4, Understanding Addition and Subtraction, Lesson 16, Cool-down, students demonstrate conceptual understanding as they find the value of and represent an expression. Student Facing states, “Find the value of the expression $1+4$. Show your thinking using objects, drawings, numbers, or words.” (K.OA.1)

• Unit 7, Solid Shapes All Around Us, Lesson 14, Activity 2, students demonstrate conceptual understanding as they build and describe figures composed of solid shapes. Launch states, “Groups of 2. Give students access to solid shapes and geoblocks. ‘Choose who will build first. The first partner will use the solid shapes to build something. Watch as your partner builds.’ 2 minutes: independent work time. ‘Use the solid shapes to build the same thing as your partner.’ 1 minute: independent work time. Repeat the steps above, with students switching roles.” Activity Synthesis states, “Invite students to share how they changed their building using positional words and names of shapes.” (K.G.1, K.G.6)

• Unit 8, Putting It All Together, Lesson 21, Activity 2, students demonstrate conceptual understanding as they compose and decompose numbers 11–19. Launch states, “Give students access to connecting cubes or two-color counters, 10-frames, and bead tools. Display the student book. ‘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.’” Student Facing states, “___. $19=$___. $10+$___. ___. $11=$___. $15=10+$___.” (K.NBT.1)

The materials reviewed for Kendall Hunt's Illustrative Mathematics Kindergarten 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 2, Numbers 1-10, Lesson 6, Warm-Up, students develop procedural skill and fluency as they recognize quantities represented with fingers. Launch states, “Groups of 2. ‘How many do you see? How do you see them?’ Display 4 fingers.” Activity states, “‘Discuss your thinking with your partner.’ 30 seconds: partner discussion. Share responses. Repeat with 8 fingers and 10 fingers.” An image of two hands is shown with one hand showing four fingers and one hand balled in a fist. (K.CC)

• 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. Launch states, “Groups of 2. Give each student a set of cards, a recording sheet, and access to two-color counters, 5-frames, and 10-frames. ‘We’re going to learn a center called Find the Pair. Put your cards in a pile in the middle of the table. You and your partner will both draw 5 cards. Keep your cards hidden from your partner.’ Demonstrate drawing 5 cards. Invite a student to act as the partner and draw 5 cards. ‘I am going to look at my cards. I need to choose 1 card and figure out which number I need to make 5 with the card.’ 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?’ ( or $1+4$).” (K.OA.5)

• Unit 8, Putting It All Together, Lesson 1, Warm-Up, students develop procedural skills and fluency as they practice counting and finding patterns in the count. Launch states, “‘Count by 1, starting at 57.’ Record as students count. Stop counting and recording at 77.” (K.CC.2, K.CC.4c)

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 4, Understanding Addition and Subtraction, Lesson 17, Activity 1, students demonstrate fluency as they find the value of addition expressions with +0 and +1. Launch states, “‘Mai has a tower with 3 cubes. Mai wants to add 0 cubes to the tower. What should Mai do?’ (Nothing. When you add 0, you don’t add anything.) 30 seconds: quiet think time. Share responses. 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 7, Solid Shapes All Around Us, Lesson 6, Activity 3, students demonstrate procedural skill and fluency as they use addition and subtraction within 5. Launch states, “Groups of 2. Give each group of students a cup, 5 two-color counters, and 2 copies of the blackline master. ‘We are going to learn a new way to do the Shake and Spill center. It is called Shake and Spill, Cover. Let’s play a round together.’ ‘I am going to put 3 counters in the cup and shake them up. Before I spill the counters, you will close your eyes so I can cover all the yellow counters with the cup. Then you will open your eyes and figure out how many counters are under the cup.’ Put 3 counters in a cup and shake them up. ‘Close your eyes.’ Spill the counters and cover 1 yellow counter. Leave 2 red counters on the table. ‘Open your eyes. Look at the counters on the table. How many counters are under the cup? How do you know?’ (One because there are 2 on the table and 2 and 1 more makes 3.) 30 seconds: partner discussion. Share responses. Pick up the cup showing the 1 counter that was covered. ‘Now we fill in the recording sheet. We had 3 counters total. Then we fill in the expression that matches the parts we broke 3 into. There were 2 counters outside the cup and 1 counter in the cup.’ Demonstrate completing the recording sheet. ‘Take turns with your partner spilling and covering the yellow counters. On each turn you can decide to use 3, 4, or 5 counters. Make sure you and your partner agree on how many total counters you are using before you shake, spill, and cover.’”(K.OA.5)

• 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)

The materials reviewed for Kendall Hunt's Illustrative Mathematics Kindergarten 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 3, Flat Shapes All Around Us, Lesson 4, Activity 2, students consider and describe attributes of shapes and sort shapes into categories. Launch states, “Give each group a set of shape cards. ‘In the last activity we sorted our objects into groups. We put the objects together based on something that was the same about them.’ Display a couple of shape cards. ‘You and your partner will sort the shape cards into two groups. You can decide how to sort the shapes. Put each shape in one of your groups. Talk to your partner about why each shape fits in the group.’” Activity states, “Monitor for groups that sort the shapes in different ways. ‘Write a number to show how many shapes are in your groups.’ 2 minutes: independent work time. ‘Which group has more shapes? How do you know?’” (K.G.4, K.MD.3)

• Unit 4, Understanding Addition and Subtraction, Lesson 11, Activity 1, students draw a picture to represent and solve a story problem. Student Facing states, “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?” There is an image of four kids playing soccer. Activity states, “Monitor for students who draw pictures with details to represent the story. Monitor for students who use symbols such as circles.” (K.OA.2)

• Unit 8, Putting It All Together, Lesson 18, Cool-down, students solve a real-world problem by composing and decomposing within 10. Student Facing states, “There are 10 birds on the wire. Some of the birds are red. The rest of the birds are blue. How many of the birds are red? Then how many of the birds are blue? Show your thinking using objects, drawings, words, or numbers. Find more than 1 solution to the problem.” (K.OA.2, K.OA.3)

Examples of non-routine applications of the math include:

• Unit 2, Numbers 1- 10, Lesson 5, Activity 1, students make groups that have more, fewer, or the same number of objects as another group. Launch states, ”Give each group a mat and access to collections of between 2–9 objects and connecting cubes. ‘Choose a group of objects and place them in the box at the top of the mat. Use cubes to make a new group of objects for each box below. Make a group that has fewer objects, a group that has the same number of objects, and a group that has more objects. Discuss with your partner how you know each group has more, fewer, or the same number of objects.’” (K.CC.6)

• Unit 7, Story Problems about Shapes, Lesson 5, Activity 2, students connect the action in the story to the meaning of the addition and subtraction signs. Activity states, “Reread the first story problem. ‘Show your thinking using objects, drawings, numbers, or words.’ 2 minutes: independent work time. 2 minutes: partner discussion. ‘Lin began writing this equation but didn’t finish it. Finish her equation to show what happened in the story problem.’ 2 minutes: independent work time. Repeat the steps with the second story problem. Display $9-3=$___ for students to complete the equation.” Student Facing states,”1. 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? ___ equation: $8=$______ 2. 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. How many pattern blocks are in Elena's train now? ___ equation $9-3=$___.” (K.OA.1, K.OA.2)

• Unit 8, Putting It All Together, Lesson 3, Activity 1, students use their knowledge of the count sequence to solve Add To, Result Unknown and Take From, Result Unknown story problems where one is added or taken away. Launch states, “Give students access to connecting cubes and 10-Frames. ‘Today you are going to solve two story problems about people on a bus.’” Student Facing states, “1. There were 7 people on the bus. Then 1 more person got on the bus. How many people are on the bus now? Show your thinking using objects, drawings, numbers, or words. 2. There were 10 people on the bus. Then 1 person got off the bus. How many people are on the bus now? Show your thinking using objects, drawings, numbers, or words.” (K.CC.2, K.CC.4, K.OA.2)

The materials reviewed for Kendall Hunt's 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.

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 4, Understanding Addition and Subtraction, Lesson 4, Activity 1, students extend their conceptual understanding as they use objects to represent addition. Launch states, “Give each student at least 10 counters and access to 5-frames. ‘Count out 3 counters. You can use a 5-frame if it is helpful.’ 30 seconds: independent work time. ‘Now add 4 more counters.’ 30 seconds: independent work time. ‘How many counters are there altogether?’ Write ‘There are 7 counters altogether.’ ‘This sentence now says There are 7 counters altogether. We are going to continue to work on problems where we add more counters to the group we started with. Let’s add and find the total number of counters.’” (K.CC.5, K.OA.1)

• Unit 5, Composing and Decomposing Numbers to 10, Lesson 1, Activity 1, students develop procedural skill and fluency as they decompose 6 into two parts using connecting cubes. Activity states, “‘You have 6 cubes. Put some of the cubes in your hand and some on your desk.’ 30 seconds: independent work time. ‘Tell your partner how many cubes are in your hand. Show them the cubes. Tell your partner how many cubes are on your desk. Show them the cubes. Tell your partner how many cubes you have altogether.’” (K.OA.3, K.OA.5)

• Unit 8, Putting It All Together, Lesson 11, Activity 1, students apply their understanding of addition and subtraction as they represent and solve a story problem about their school community. Launch states, “Give each student a piece of chart paper and access to connecting cubes or two-color counters and crayons. ‘Tell your partner the story problem that you came up with yesterday. Today you are going to make a poster to show your story problem. Solve the story problem. Show your thinking using drawings, numbers, or words.’” Activity states, “10 minutes: independent work time. ‘If you have time, you may want to show different ways to solve the problem using pictures, numbers, words, or symbols.’ 10 minutes: independent work time.” (K.OA.1, K.OA.2)

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 2, Number 1-10, Lesson 2, Activity 2, students develop conceptual understanding alongside procedural skill and fluency as they recognize that the arrangement of a group of objects does not change the number of objects. Launch states, “‘We are going to learn a new center called Shake and Spill. Let's play a round together. Choose who will go first and start with all of the counters in the cup. Shake the cup and spill the counters on the table.’ 30 seconds: partner work time. ‘Take turns figuring out how many counters there are. When you know how many counters there are, tell your partner and see if you both agree.’ 1 minute: partner work time. ‘Put the counters back into the cup, shake them and spill them again. Take turns figuring out how many counters there are and share with your partner.’ 1 minute: partner work time. ‘Now you can take turns playing with your partner. Take some of the counters out of the cup and put them away so that you are using a different number of counters this time. Remember to spill the counters, figure out how many there are, spill the counters again, and figure out how many there are.’” (K.CC.4, K.CC.5)

• Unit 4, Understanding Addition and Subtraction, Lesson 14, Activity 1, students develop conceptual understanding alongside application as they explain how a subtraction expression represents a story problem. Launch states, “Give students access to connecting cubes or two-color counters. Read and display the task statement. ‘Tell your partner what happened in the story.’ 30 seconds: quiet think time. 1 minute: partner discussion. Monitor for students who accurately retell the story. Choose at least one student to share with the class. Write the expression 10 - 6. ‘How does this expression show what happens in the story problem?’” Student Facing states, “There were 10 people riding bikes in the park. Then 6 of the people stopped riding to have lunch. How many people are riding bikes now?” (K.OA.1, K.OA.2)

• Unit 6, Numbers 0-20, Lesson 2, Activity 2, students develop conceptual understanding alongside procedural skill and fluency as they keep track of objects counted in order to accurately count groups up to 20. Launch states, “Give each student a collection of objects and access to 10-frames and a counting mat. Display a 10-frame mat and a counting mat. ‘How can you use the counting mat to help you figure out how many objects are in your collection?’ (I can put all of the objects on one side and say a number as I move each object to the other side.). 30 seconds: quiet think time. 30 seconds: partner discussion, Share responses. ‘How can you use the 10-frame to help you figure out how many objects there are in your collection?’ (I can put one object in each box and line up the rest of the objects. Then I can count them.) 30 seconds: quiet think time. 30 seconds: partner discussion. Share responses. ‘Choose either the 10-frame or the counting mat to help you figure out how many objects are in your collection.’” Activity states, “4 minutes: independent work time. ‘Find a partner who used a different tool to help them count their collection. If you used a 10-frame to help you count your collection, find a partner who used the counting mat. Show your new partner how you counted your collection.’ 4 minutes: partner work time.” (K.CC.4a)

The materials reviewed for Kendall Hunt's 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 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 1, Math In Our World, Lesson 11, Activity 1, students think of different ways to represent a story. Student Facing states, “4 little speckled frogs sat on a speckled log, eating the most delicious bugs. Yum! Yum! 1 jumped into the pool, where it was nice and cool. Now there are 3 green speckled frogs. Glub! Glub!” An image of a green frog is shown. Narrative states, “Acting out gives students opportunities to make sense of a context (MP1). Monitor for suggestions of acting out the story with concrete objects such as cubes, fingers, or students, as well as representing the story with pictures.”

• Unit 2, Numbers 1-10, Lesson 4, Activity 1, students identify a group of objects that has more. Student Facing states, “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?” Launch states, “Groups of 2. 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). 30 seconds: quiet think time. 1 minute: partner discussion. Share and record responses. ‘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). 30 seconds: quiet think time. Share responses. 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.). 30 seconds: quiet think time. 1 minute: partner discussion. Share responses.” Activity states, “‘Act out the story with your partner.’ 3 minutes: partner work time. ‘Are there more people or spoons? How do you know?’ (There are more spoons than people. Each person gets one spoon and then there are some more spoons.). 2 minutes: partner work time. Monitor for students who matched one spoon to each person to see if there were enough spoons and which there was more of.” Narrative states, “The context of family mealtimes that is introduced in this activity will be revisited throughout the unit. Acting it out gives students an opportunity to make sense of a context (MP1).”

• Unit 7, Solid Shapes All Around Us, Lesson 5, Warm-up, students reason about a problem context involving quantities within ten. Student Facing states, “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.” Narrative states, “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 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 1, Math in Our World, Lesson 5, Warm-up, students reason about tools and consider different representations. Launch states, “Groups of 2. Display the image.” Student Facing states, “What do you notice? What do you wonder?” Narrative states, “The purpose of this warm-up is for students to consider how different tools can be used to represent the same thing. When students describe how each object represents a house and make connections between the objects, they show their ability to reason abstractly and quantitatively (MP2).”

• Unit 3, Flat Shapes All Around Us, Lesson 10, Activity 2, students use pattern blocks to fill in simple puzzles. Student Facing shows an image of a pattern puzzle, with different pattern blocks shown to make the figure. Launch states, “Groups of 2. Give each group of students pattern blocks. ‘In the last activity we put together pattern blocks to make quilts. We can also use pattern blocks to make things that we see in real life. Close your eyes and think about something that you see at home or in your community. Use the pattern blocks to make what you see.’ 2 minutes: independent work time. ‘Tell your partner about what you made and why.’ 2 minutes: partner discussion. Share responses. ‘Each puzzle looks like something we see in real life. Use the pattern blocks to fill in each puzzle. Write a number to show how many of each pattern block you used. Ask your partner a question about each puzzle using the word “fewer”.’ Activity states, “6 minutes: partner work time.” Narrative states, “When students make connections between the pattern blocks and the shape outlines in the puzzle, they show their ability to reason abstractly and quantitatively (MP2).”

• Unit 6, Numbers 0-20, Lesson 11, Activity 2, students show that numbers 11-19 consist of 10 ones and and some more ones as they color images to match expressions. Student Facing states, “1. Color the squares to show $10+2$. $10+2=$____. An image of 12 squares is shown. 2. Color the triangles to show $10+8$. $10+8=$____. An image of 18 triangles is shown. 3. Color the hexagons to show $10+4$. $10+4=$____. An image of 14 hexagons is shown. 4. Color the circles to show $10+9$. $10+9+$____. Two images are shown: a rectangular shape with 10 circles and 9 circles.” Launch states, “Groups of 2. Give each student access to at least two different colored crayons. ‘Color the shapes to show each expression. Then complete the equation to show how many shapes there are altogether.’” Activity states, “2 minutes: independent work time. 3 minutes: partner work time. Monitor for students who count on from 10.” Narrative states, “Because students are coloring in the shapes to show $10+$____, students may count on from 10 to determine the total number of shapes. It is important that students connect their equations to the corresponding representations (MP2).”

The materials reviewed for Kendall Hunt's 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 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 1, Math in Our World, Lesson 13, Warm-up, students construct viable arguments as they count collections of objects, focused on keeping track of which objects have been counted. Launch states, “‘We need to figure out how many of us are here. How can we make sure that we count each person one time?’ 30 seconds: quiet think time. Share responses. Monitor for students who suggest a way to organize the students, such as having all of the students line up.” Activity states, “Count the students using two of the methods suggested by students. ‘How many of us are here today?’” Narrative states, “As students share answers to questions such as ‘How can we figure out how many of us are here?’ and ‘Did I count the students correctly?’ they are beginning to construct viable arguments and attend to precision (MP3, MP6).”

• Unit 3, Flat Shapes Around Us, Lesson 1, Warm-up, students construct arguments as they compare four different images and analyze the characteristics or attributes of the images. Launch states, “Display the image. ’What is the same and what is different about the teddy bears?’ 30 seconds: quiet think time. 30 seconds: partner discussion. Share responses.” Activity states, “Which one doesn’t belong? Display the image. 30 seconds: quiet think time. ‘Tell your partner which teddy bear doesn’t belong and why.’ 30 seconds: partner discussion. Student Facing states, “Which one doesn’t belong?” Images of teddy bears are provided. Narrative states, “In this warm-up, students only work with three images of teddy bears. By the end of the section, students will compare four images of shapes. Emphasize to students that there is no right answer to the question and that it is important to explain their choice. Listen to how students create an argument and use or revise their language to make their argument clear to others (MP3).”

• Unit 7, Solid Shapes All Around Us, Lesson 16, Activity 2, students construct arguments as they use solid shapes to make a model of the classroom. Launch states, “When a classmate comes to your model, tell them all about your model and what the shapes represent.” Activity states, “Invite half of the class to stand by their models while the other half walks around. 5 minutes: gallery walk. Switch groups. 5 minutes: gallery walk. ‘Were there any things that you saw in your classmates’ models that gave you an idea for things you want to add to or change about your model?’ 1 minute: quiet think time. 2 minutes: partner discussion. Share responses. ‘Work on your model.’ 4 minutes: independent work time. Monitor for changes students make to their models including changing the shapes that they use, changing the relative position of the shapes, or putting in more shapes to represent additional features.” Activity Synthesis states, “Invite selected students to share changes that they made to their models and why they made them.” Narrative states, “Describing their model to their peers and seeing other models helps students develop ideas for how to add to or change their model (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 3, Flats Shapes All Around Us, Lesson 5, Activity 2, students construct arguments and begin to critique the reasoning of others as they distinguish triangles from other shapes. Launch states, “Groups of 4, Give each group a set of cards. ‘Work with your group to sort the shapes into 2 groups. Put the shapes that are triangles on the left side of your page. Put the shapes that are not triangles on the right side of your page. When you place a shape, tell your group why you think the shape belongs in that group.’” Activity states, “4 minutes: small-group work time. Monitor for students who discuss attributes of triangles when sorting. ‘Write a number to show how many shapes are in each group.’ 1 minute: independent work time. ‘Walk around to see how the other groups organized their shapes. Did they organize them the same way that your group did?’ 6 minutes: work time.” Student Facing states, “Let’s put the shapes into 2 groups.Triangle, Not a Triangle” Activity Synthesis states, “Display cards O, K, and G next to each other. ‘Noah says that the shape in the middle is not a triangle because it is pointing down and triangles have to point up. Do you agree with Noah? Why or why not?’”

• Unit 5, Composing and Decomposing Numbers to 10, Lesson 1, students construct arguments and begin to critique the reasoning of others as they decompose numbers into two groups. Launch states, “Groups of 2. Give students access to connecting cubes. ‘Diego and Lin also put some cubes in their hands and some on their desks. Diego has 3 in his hand and 1 on his desk. He says he has 4 cubes altogether. Lin has 2 in her hand and 2 on her desk. She also says she has 4 cubes total. Can they both have 4 cubes altogether?’” Activity states, “3 minutes: partner discussion. Monitor for students who count the groups to determine that both students have 4 even though they are broken into different parts.” Activity Synthesis states, “Do Diego and Lin both have 4 cubes? Invite previously identified students to share. ‘What parts did Diego break 4 into? (3 and 1)’ Write $3+1$. ‘What parts did Lin break 4 into?’ (2 and 2) Write $2+2$. ‘Diego and Lin showed us that we can break numbers apart in different ways.’”

• Unit 8, Putting It All Together, Lesson 7, Cool-down, students critique the work of others as they use numbers to create a number book using objects in their environment. Student Facing states, “Choose 1 object in our classroom. Create a number book page about the object. Include a number, a drawing, and letters, a word, or words.” Narrative states, “Students share their work with a partner, receive feedback, and then improve their work (MP3).”

The materials reviewed for Kendall Hunt's Illustrative Mathematics Kindergarten 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 3, Flat Shapes All Around Us, Lesson 14, Activity 2, students put shapes together to form larger shapes, Launch states, “Give each student a sheet of construction or white paper. Give students access to cut out shapes, glue, crayons, colored pencils, and markers. ‘We noticed that artists use shapes in different ways to create art. Some artists make patterns and designs. Some put shapes together to form people or animals. Now you are going to make your own piece of artwork using shapes. You can use any of these materials. Think about how you can draw or put shapes together to make larger shapes.’” Activity states, “8 minutes: independent work time.” Narrative states, “When students recognize mathematical features of objects in the real world, they model with mathematics (MP4).”

• Unit 7, Solid Shapes All Around Us, Lesson 3, Cool-down, Section A Checkpoint, students ask and answer mathematical questions about shapes composed of pattern blocks. The teacher observes to capture evidence of student thinking on the checkpoint checklist. Student Response states, “Count all to determine the total. Use objects, drawings, or equations to represent a story problem.” Lesson Narrative states, “In this lesson, students create a shape out of pattern blocks and brainstorm questions that they could ask about other students’ shapes. Students create and solve story problems about shapes made out of pattern blocks (MP2, MP4).”

• Unit 8, Putting It All Together, Lesson 8, Activity 2, students recognize different ways math is all around them in their community. Student Facing states, “Find something that you can count. Find 2 objects that you can compare the weight of. Find something that you know how many there are without counting. Find something that there are 5 of. Find 2 groups of objects that make 10 objects altogether. Find a group of objects that you could use to fill in a 10-frame. Find something that you could make using solid shapes. Find 2 groups of objects that you can compare the number of. Find something that has a number on it. Find 2 objects that you can compare the length of.” Launch states, “Give students access to 10-frames, geoblocks, and solid shapes. ‘Work with your partner to find an object or objects that goes with each prompt.’” Activity states, “‘Find something that you can count.’ 30 seconds: quiet think time. 2 minutes: partner work time. ‘Now count what you found.’ 1 minute: partner work time. Repeat the steps with the rest of the prompts.” Narrative states, “When students identify objects in the classroom that fit different constraints they are taking an important step toward modeling 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 1, Math In Our World, Lesson 13, Activity 1, students count collections of objects and the focus is saying one number for each object. Launch states, “Today you’re going to count another collection of objects. As you’re working, think about how to make sure you count each object.” Activity states, “Give each student a bag of objects. Give students access to 5-frames and a counting mat. ‘Figure out how many objects are in your collection.’ 2 minutes: independent work time. ‘Switch collections with a partner. Figure out how many objects are in your new collection.’ 2 minutes: independent work time. Monitor for students who have a method of keeping track of which objects have been counted, such as moving and counting or lining up the objects and counting them in order.” Narrative states, “Students use appropriate tools strategically as they choose which tools help them count their collections (MP5).”

• Unit 2, Numbers 1–10, Lesson 21, Activity 1, students compare numbers in a way that makes sense to them, Student Facing states, “Circle the number that is more. 1. 5, 8, 2. 9, 4,” Launch states, “Give students access to connecting cubes or counters. ‘Work with your partner to figure out which number is more. Circle the number that is more.’” Activity states, “5 minutes: partner work time. Monitor for students who create representations of the numbers using cubes or a drawing and use these representations to compare. Monitor for students who counted to figure out which number is more.” Narrative states, “Students can use physical objects or make drawings to represent each number (MP5).”

• Unit 4, Understanding Addition and Subtraction, Lesson 8, Cool-down, Section B checkpoint, students represent and solve story problems using a strategy that makes sense to them. Teachers observe and capture evidence of student thinking on the checkpoint checklist. Student Response states, “Accurately retell a story problem in their own words. Understand the action in a story problem and act it out or demonstrate it with objects or drawings. Use objects or drawings to represent a story problem.” Narrative states, “Students may use objects, math tools, or drawings to represent and solve the story problem (MP5).”

The materials reviewed for Kendall Hunt's 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 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, Math in Our World, Lesson 4, Warm-up, students use specialized language to describe shapes. Narrative states, “The purpose of this activity is to elicit ideas students have about geoblocks. This allows teachers to see what language students use to describe shapes (MP6). There is no need to introduce formal geometric language at this point since this will happen in a later unit.” Launch states, “Groups of 2. Give each student a few geoblocks and display a collection of geoblocks or the image in the student book. ‘What do you notice?’ 30 seconds: quiet think time.” Activity states, “‘Tell your partner what you noticed.’ 1 minute: partner discussion. Share and record responses. ‘What do you wonder?’ 1 minute: quiet think time. ‘Tell your partner what you wondered.’ 30 seconds: quiet think time. 1 minute: partner discussion. Share and record responses.” Activity Synthesis states, “These are called geoblocks. What is one thing that you think you could do or make with the geoblocks?”

• Unit 1, Math In Our World, Lesson 8, Activity 2, students attend to precision as they recognize, name, and match groups with the same number of images. Launch states, “Groups of 2. Display the image from the student book. ‘When I point to each group, show your partner with your fingers and tell your partner how many things there are.’ Point to the ducks. 30 seconds: partner work time. Repeat the steps with the cats and dogs. ‘Which groups have the same number of things? How do you know?’ (There are 3 ducks and 3 dogs. They are both 3.) 30 seconds: quiet think time. Share responses. Display or write “3”. ‘There are 3 ducks and 3 dogs. They both have the same number of things.’” Activity states, “Give each group of students a set of cards. ‘Work with your partner to match the cards that have the same number of things. Explain to your partner how you know.’ 4 minutes: partner work time.”  Narrative states, “When students say that two cards match because they have the same number of objects, they attend to precision in their language (MP6).”

• Unit 2, Numbers 1- 10, Lesson 6, Cool Down, students attend to precision as they compare groups of objects and describe their comparisons using “more,” “fewer,” and “the same number.” Student Facing states, “Compare the number of objects in groups. Use ‘more,’ ‘fewer,’ and ‘the same number’ to describe comparisons. Make groups with more, fewer, or the same number of objects than a given group.” Narrative states, “In making comparisons, students have a reason to use language precisely (MP6).”

• Unit 3, Flat Shapes All Around Us, Lesson 9, Warm-up, students attend to precision as they compare attributes of shapes to determine which one does not belong. Launch states, “Groups of 2. Display the image. ‘Pick one that doesn’t belong. Be ready to share why it doesn’t belong.’ 1 minute: quiet think time.” Narrative states, “This warm-up prompts students to carefully analyze and compare the attributes of 4 shapes. In making comparisons, students have a reason to use language precisely (MP6). The activity also enables the teacher to hear the words students know and how they talk about attributes of shapes.” Activity Synthesis states, “Display the image of the square. ‘Noah said that this shape doesn’t belong because it is not a rectangle. What do you think?’ 30 seconds: quiet think time. Share responses. ‘A square is a special kind of rectangle.’”

• Unit 5, Composing and Decomposing Numbers to 10, Lesson 5, Cool-down, students use grade appropriate math terms to restate and represent story problems. Student Facing states, “Accurately retell a story problem in your own words. Use objects or drawings to represent a story problem. Explain how objects or drawings represent a story problem. Use labels, colors, numbers, or other methods to represent the two groups in a story problem.” Narrative states, “Students are encouraged to use clear and precise language to explain how their representation shows the story problem (MP6).”

• Unit 7, Solid Shapes Around Us, Lesson 13, Activity 1, students use specific mathematical language to describe the solid shapes. Launch states, “Groups of 2. Give students access to solid shapes. ‘Choose 2 solid shapes.’ 30 seconds: independent work time.” Activity states, “‘We are going to go for a walk. Your job is to look for objects that look like your solid shapes. Tell your partner about the shapes you find.’ 10 minutes: shape walk. Monitor for students who use positional words to describe the location of shapes. ‘Tell your partner about your favorite object. Where did you see it?’ 30 seconds: quiet think time. 1 minute: partner discussion. Share responses.” Narrative states, “The purpose of this activity is for students to identify and describe solid shapes in their environment (MP4, MP6).” Activity Synthesis states, “Invite students who used positional words to describe the location of shapes to share. ‘____ saw a round light bulb below the lamp shade. It looked like a sphere. _____ saw a book on the bookshelf. It looked like a box.’ Display image: ‘Which shape does this clock look like?’ (Students say “cylinder” or hold up a cylinder.) Display image: ‘Which shape does this party hat look like? (Students say “cone” or pick up a cone.) In the next activity, we are going to use clay to make shapes that show the objects we saw.’”

The materials reviewed for Kendall Hunt's 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 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, Numbers 1–10, Lesson 8, Activity 2, students look for and make use of structure while they compare groups of images. Launch states, “Groups of 2. Display the student page. ‘How are these pictures different from the ones we worked with in the first activity?’ (There are different pictures. The pictures aren’t matched up.)” Activity states, “‘Are there enough cartons of milk for each student? How do you know?’ 30 seconds: quiet think time. 30 seconds: partner discussion. ‘Are there more students or cartons of milk? How do you know?’ 30 seconds: independent work time. 30 seconds: partner discussion. ‘There are more cartons of milk than students. How many students are there? How many cartons of milk are there?’ 1 minute: independent work time. ‘8 cartons of milk is more than 7 students.’ Repeat the steps with each group of images. Switch between asking students ‘Are there more _____ or _____?’ and ‘Are there fewer _____ or _____?’ Monitor for students who draw lines to match each image. Narrative states, “Matching the images helps students relate the comparisons to the situation they just worked with where the images were already matched (MP7).”

• Unit 4, Understanding Addition and Subtraction, Lesson 15, Cool-down: Unit 4, Section C Checkpoint, students look for and make use of structure while they connect expressions to drawings. Student Facing states, ”Lesson observations. Explain how an expression connects to a drawing or story problem. Fill in an expression to represent a drawing.” Activity 1 Narrative states, “The purpose of this activity is for students to match drawings to expressions. Students use the structure of the dots to decide whether they represent an addition or subtraction expression and then identify that expression (MP2, MP7).” The cool-down also assesses students’ ability to connect expressions to drawings. Teacher Instructions state, “For this Checkpoint Assessment, a full checklist for observation of students can be found in the Assessments for this unit. The content assessed is listed below for reference. Relate addition and subtraction expressions to story problems. Explain how an expression connects to a drawing or story problem. Fill in an expression to represent a drawing. Find the value of addition and subtraction expressions within 5. Use fingers, objects, or drawings to find the value of an expression. Count all to determine the total when 0 or 1 are added. Use knowledge of the count sequence to determine the total when 1 is added.”

• Unit 5, Composing and Decomposing Numbers to 10, Lesson 13, Warm-up, students look for and make use of structure while 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 states, “Display image. ‘Discuss your thinking with your partner.’ 1 minute: partner discussion. Record responses. Repeat for each image.” Activity Synthesis states, “Display the hands showing 8 fingers. ‘How many fingers are up? (8) How many fingers need to go up so there are 10 fingers?’ (2). Repeat the steps with the rest of the images.” Narrative states, “When students think about quantities in relation to 5 and 10, they look for and make use of structure (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 1, Math in Our World, Lesson 11, Cool-down, students use repeated reasoning using one-to-one correspondence as they make groups of objects. Student Facing states, “Lesson observations.” Sample Student Responses include, “Say the count sequence to 10. Say one number for each object. Answer how many without counting again. Recognize and name groups of 1, 2, or 3 objects or images without counting. Recognize and name groups of 4 objects or images without counting. Show quantities on fingers.Identify groups with the same number of objects (for groups of up to 4 objects).” Lesson Narrative states, “As students notice that when you get enough of an object for each student to have one, the number of students and the number of objects are the same, they look for and express regularity in repeated reasoning (MP8).”

• Unit 5, Composing and Decomposing Numbers to 10, Lesson 12, Activity 2, students use repeated reasoning to find how many counters are needed to fill a 10-frame. Launch states, “Groups of 2. Give students access to two-color counters. ‘Figure out how many counters are needed to fill each 10-frame. Write a number to show how many counters are needed to fill it. Circle the equation that shows the number of counters in the 10-frame and the number of counters needed to fill the 10-frame.’” Student Facing states, “1. (A ten frame with 7 counters is shown), $10=8+2$, $10=5+5$ 2. (A ten frame with 4 counters is shown) $10=8+2$, $10=1+9$, $10=4+6$ 3. (A ten frame with 9 counters is shown) $10=9+1$, $10=5+5$, $10=7+3$ 4. (A ten frame with 3 counters is shown) $10=5+5$, $10=3+7$, $10=2+8$ 5. (A ten frame with 5 counters is shown) $10=9+1$, $10=6+4$, $10=5+5$ 6. (A ten frame with 2 counters is shown) $10=1+9$, $10=2+8$, $10=4+6$.” Narrative states, ”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 4, Warm-up, students count collections of objects and understand that the number of objects in a collection stays the same, regardless of how they are arranged, Student Facing states, “What do you notice? What do you wonder?” 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, “Which arrangements do you think would be easiest to count? Why? (The lined up dots would be easy to count. I could count one line and then the other line.).” Lesson Narrative states, “Students will count the same collection of objects in different arrangements to build this conservation of number, which develops through experience over time. While developing conservation of number, students may need to recount the objects each time they are rearranged. With repeated practice, some students may know that the number of objects is the same without recounting (MP8).”

The materials reviewed for Kendall Hunt's 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. 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 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. 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 2, Numbers 1–10, Lesson 2, Activity 1, teachers are provided context to support students in understanding that the arrangement of objects does not change the total. Narrative states, “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 states, “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 states, “5 minutes: partner work time. Monitor for students who notice that the number of objects is the same after they are rearranged.”

• Unit 8, Putting It All Together, Lesson 18, Lesson Synthesis provides guidance around strategies for composing and decomposing within 10, “Display the chart with solutions to the story problem. ‘Tyler and Priya recorded the different ways that the pigeons could be in the fountain and on the bench. What do you notice? What patterns do you see?’ (There are lots of ways to make 10. The numbers in one column are counting up and the numbers in the other column are counting down. I see that there are 7 and 3 and 3 and 7.)”

The materials reviewed for Kendall Hunt's 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, 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 1, When is a number line not a number line?, “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.” 

• Unit 4, Understanding Addition and Subtraction, Lesson 10, Preparation, Lesson Narrative states, “In a previous lesson, students solved Add To and Take From, Result Unknown story problems and explained how both objects and drawings represented the story. In this lesson, students solve story problems and compare how different drawings represent the story. Students interpret both drawings that correctly and incorrectly represent the story problem, as well as unorganized and organized drawings. While students are not expected to produce a drawing to represent and solve a story problem in this lesson, students make sense of various drawings, which will help them be prepared to create drawings in a future lesson. The purpose of the lesson synthesis is for students to discuss how it can be easier to see what happens in the story problem in an organized drawing.”

• IM K-5 Math Teacher Guide, About These Materials, Kindergarten, 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 7, Solid Shapes All Around Us, Lesson 1, Preparation, Lesson Narrative states, “In previous units, students put together pattern blocks to form larger shapes and filled in puzzles. They counted groups of up to 20 objects and images and wrote numbers to record their count. Students use only 1 kind of pattern block to fill in puzzles and eventually create given shapes without outlines provided, which requires students to think informally about the attributes of shapes. Students need to change the orientation of the pattern blocks and align the sides of the pattern blocks. Students may be able to visualize how to turn or flip the shape to fill a particular space or may need to use trial and error. In both activities, students count and write a number to record how many pattern blocks they used.”

The materials reviewed for Kendall Hunt's 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 K, 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) addressed within.

• Grade K, Course Guide, Lesson Standards, includes all Kindergarten 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 1, Resources, Teacher Guide, outlines standards, learning targets and the lesson where they appear. This is present for all units and allows teachers to identify targeted standards for any lesson.

• Unit 3, Flat Shapes All Around Us, Lesson 5, the Core Standards are identified as K.G.B.4 and K.MD.A.2. 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; and a Lesson Synthesis.

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 K, Course Guide, Scope and Sequence, Unit 4, Understanding Addition and Subtraction, Unit Learning Goals, includes an overview of how the math of this module builds from previous work in math, “In this unit, students develop their understanding of addition and subtraction as they represent and solve story problems. Previously, students built their counting skills and represented quantities in a group with their fingers, objects, drawings, and numbers. Here, they relate counting to the result of two actions: putting objects together or taking objects away. Students enact addition by counting the total number of objects in two groups, and subtraction by counting what remains after some objects are taken away. (The word ‘total’ is used here instead of ‘sum’ to reduce potential confusion with the word ‘some’ or part of a whole.) Students then make sense of stories without questions and later solve story problems of two types—Add To, Result Unknown and Take From, Result Unknown. Students represent the problems in different ways, by acting them out, drawing, using numbers, or using objects. Connecting cubes should be accessible in all lessons for students who wish to use them, including for cool-downs. All story problems should be read aloud by the teacher, multiple times if needed. Students are also introduced to expressions, a symbolic way to represent addition and subtraction. Initially, the teacher records the process of adding and subtracting using words such as ‘5 and 3’ or ‘4 take away 1.’ Later, students see that ‘5 and 3’ and ‘4 take away 1’ can be expressed by $5+3$ and $4+1$ , respectively. They learn that these expressions are read ‘5 plus 3’ and ‘4 minus 1.’ (Students are not expected to read expressions out loud or to use precise language at this point.) Later in the section, students connect expressions to pictures and story problems. They find the value of addition and subtraction expressions within 10. In a future unit, students will compose and decompose numbers up to 10 and solve other types of addition and subtraction problems.”

• Grade K, Course Guide, Scope and Sequence, Unit 7, Solid Shapes All Around Us, Unit Learning Goals, includes an overview of how the math of this module builds from previous work in math, “In this unit, students explore solid shapes while reinforcing their knowledge of counting, number writing and comparison, and flat shapes. They compose figures with pattern blocks and continue to count up to 20 objects, write and compare numbers, and solve story problems. In an earlier unit, students investigated two-dimensional shapes. They named shapes (circle, triangle, rectangle, and square) and described the ways the shapes are different. Students used pattern blocks to build larger shapes and used positional words (above, below, next to, beside) along the way. Here, students distinguish between flat and solid shapes before focusing on solid shapes. They consider the weight and capacity of solid objects and identify solid shapes around them. Geoblocks, connecting cubes, and everyday objects are used throughout the unit. Standard geoblock sets do not include cylinders, spheres, and cones. When these shapes are required, ‘solid shapes’ are indicated as required materials. If solid shapes are not available, students can work with everyday items that represent each shape. Students use their own language to describe attributes of solid shapes as they identify, sort, compare, and build them, while also learning the names for cubes, cones, spheres, and cylinders.” 

The materials reviewed for Kendall Hunt's Illustrative Mathematics Kindergarten 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, In the Beginning: Unit 1 in Kindergarten, Alex Clayton, Exploring our math tools, “Unit 1, Section A is titled Exploring Our Math Tools, but really it is all about exploring our new math community as well! During this section, we are introduced to many of the tools that will be used throughout the year: connecting cubes, two-color counters, pattern blocks, 5-frames, and geoblocks. Students get to explore these materials freely. They discover how the materials work and try out their own ideas, before they are ever asked to use them for a specific mathematical purpose.Equally important, students get to practice sharing their ideas (‘What do you want to make or do with the connecting cubes?’) and listening to the ideas of others. These are some of the first steps in building a mathematical community where everyone has valuable mathematical ideas. We learn from each other.”

• 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 Kindergarten 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 or on the “Lesson” tab in a section called “Required Materials.” Examples include:

• Course Guide, Required Materials for Kindergarten, Materials Needed for Unit 1, Lesson 5, teachers need, “Connecting cubes, Materials from previous centers, Pattern blocks, Connecting Cubes Stage 2 Cards (groups of 2), Pattern Blocks Stage 2 Mat (groups of 2).” 

• Unit 3, Flat Shapes All Around Us, Lesson 3, Activity 3, Required Materials, “Counters, Materials from previous centers, Which One Stage 1 Gameboard.” Launch states, “We are going to learn a center called Which One. Let’s play one round together. Pick a shape on the board to be your mystery shape. I’ve chosen a shape that is on the board. Your job is to ask me questions that will help you figure out which shape I chose. You can only ask questions that I can answer with a ‘yes‘ or a ‘no.’ For example, you cannot ask, How many sides does your shape have? But you can ask, Does your shape have more than 3 sides? After each question, ask students to share which shapes they can rule out based on the question and place a counter on those shapes. When students feel ready to guess your shape, invite students to guess the shape, asking them to explain why they think it’s your shape. Take turns choosing a mystery shape and asking questions with your partner.” Activity states, “Now you can choose another center. You can also continue playing Which One. Display the center choices in the student book. Invite students to work at the center of their choice.”

• Course Guide, Required Materials for Kindergarten, Materials Needed for Unit 5, Lesson 10, teachers need, “Glue, Materials from previous centers, Scissors, Two-color counters, Numbers on Fingers and 10-frames (groups of 1), 5-Frames to Cut Out (groups of 1).” 

• Unit 4, Understanding Addition and Subtraction, Lesson 9, Activity 1, Required Materials, “Connecting cubes or two-color counters, makers.” Launch states, “Groups of 2. Give students access to two-color counters, connecting cubes, and markers. Read and display the task statement. ‘Tell your partner what happened in the story.’ 30 seconds: quiet think time. 1 minute: partner discussion. Monitor for students who accurately retell the story. Choose at least one student to share with the class. Reread the task statement. ‘Show your thinking using drawings, numbers, words, or objects.’”

The materials reviewed for Kendall Hunt's 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 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 2, Numbers 1–10, Lesson 18, Activity 2, Narrative, Access for Students with Disabilities, “Engagement: Develop Effort and Persistence. Students might benefit from counting the first tower that was built to determine how many cubes they need to create a tower that is 1 fewer or 1 more. Invite students to count in sequence the number of cubes and remind them to stop at the number that is 1 less or 1 more. Supports accessibility for: Memory, Attention, Organization.”

• Unit 3, Flat Shapes All Around Us, Lesson 5, Activity 1, Narrative, Access for Students with Disabilities, “Representation: Access for Perception. Synthesis: Students might need extra support determining that the oval and the pizza slice shapes are not circles or triangles. Hold up a triangle and circle next to the shapes to visually show that the oval and pizza slice shapes do not match the triangle and circle. Supports accessibility for: Visual-Spatial Processing.”

• 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 8, Putting It All Together, Lesson 5, Activity 2, Narrative, Access for Students with Disabilities, “Engagement: Internalize Self-Regulation. Provide students an opportunity to self-assess and reflect on the number clue and if that number clue matches the number they will stand by. For example, students can choral count together to check that the number 9 is 1 less than 10. Supports accessibility for: Memory, Conceptual Processing.”

The materials reviewed for Kendall Hunt's 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 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 2, Numbers 1–10, Section C: Connect Quantities and Numbers, Problem 6, Exploration, “Han says he sees 5. Lin says she sees 4. Tyler says he sees 3. Explain or show how Han, Lin, and Tyler can all be correct.”

• Unit 4, Understanding Addition and Subtraction, Section A: Count to Add and Subtract, Problem 7, Exploration, “Pick a number from the list to put in the blank space. 2, 7, 6, 3. Then try the problem you made. Count out 8 counters. Take away ___ counters. How many counters are left? After you try the problem you made, try it again with a different number in the blank space. Do you think your answer will be the same or different? Explain.”

• Unit 6, Numbers 0–20, Section B: 10 Ones and Some More, Problem 8, Exploration, “1. Arrange 18 dots in a way that helps you see there are 18. 2. Arrange 18 dots in a way that makes it hard to see how many there are. 3. Explain why you chose your arrangements. Try again with other numbers up to 19.”

• Unit 7, Solid Shapes All Around Us, Section B: Describe, Compare, and Create Solid Shapes, Problem 4, Exploration, “1. Can you find an object in the classroom that fits the description? I am not flat. I am heavy. You can see some rectangles on me. Can you find more than one object? 2. Can you find an object in the classroom that fits the description? I am flat. I have lots of colors and different shapes. I have some rectangles. Can you find more than one object?”

The materials reviewed for Kendall Hunt's 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 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 1, Math in Our World, Lesson 15, Activity 1, Teaching notes, Access for English Learners, “MLR8 Discussion Supports. Provide multiple opportunities for verbal output. Invite students to chorally repeat each count in unison. Advances: Listening, Speaking.”

• Unit 3, Flat Shapes All Around Us, Lesson 14, Activity 1, Teaching Notes, Access for English Learners, “MLR7 Compare and Connect. Synthesis: To amplify student language and illustrate connections, follow along and point to the relevant parts of the images as students compare how they are alike and different. Advances: Representing, Conversing.”

• Unit 4, Understanding Addition and Subtraction, Lesson 13, Activity 1, Teaching notes, Access for English Learners, “MLR2 Collect and Display. Circulate, listen for and collect the language students use as they create story problems. On a visible display, record words and phrases such as: ‘more,’ ‘joined,’ ‘went away,’ ‘take away,’ and ‘less.’ Review the language on the display, then ask, “Which of these words tell you the story is about addition?” and “Which of these words tell you the story is about subtraction?” Advances: Representing, Listening.”

The materials reviewed for Kendall Hunt's Illustrative Mathematics Kindergarten 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, Math in Our World, Lesson 5, Activity 1, identifies two-color counters and 5-frames as strategies for students to engage in the math of the lesson. Launch states, “Give each student a 5-frame. ‘As you explore the two-color counters, you will also explore a new tool called a 5-frame.’ Display the 5-frame. ‘Why do you think we call this a 5-frame?’ (Because it has five spaces or squares in it.) Share responses. Give each group of students a container of two-color counters. ‘Let’s explore two-color counters and 5-frames.’”

• Unit 2, Numbers 1–10, Lesson 14, Activity 1, references the use of cards and counters to count out objects and match the quantity to a number. Launch states, “Groups of 2. Give each group of students a set of number cards and counters. ‘What is your favorite pizza topping?’ Display the student book and a number card. ‘If my partner showed me this card, how many pizza toppings should I add to my pizza?’ 30 seconds: quiet think time. 1 minute: partner discussion. Share responses.”

• Unit 3, Flat Shapes All Around Us, Lesson 11, Activity 1, identifies pattern blocks for use in identifying shapes that are the same, regardless of orientation. Launch states, “Groups of 2. Give students pattern blocks. Display the student book. ‘What do you notice? What do you wonder?’ (There are lots of different pattern blocks. I wonder why they are all missing a piece.) 30 seconds: quiet think time. 30 seconds: partner discussion. Share responses. ‘Figure out which pattern block is missing from each puzzle. Tell your partner how you know.’”

• Unit 4, Understanding Addition and Subtraction, Lesson 6, Activity 3, describes the use of connecting cubes and a number mat to support understanding of subtraction problems. Launch states, “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.”