Kindergarten - Gateway 2

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

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

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

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

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

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

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

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

• Unit 8, Putting It All Together, Lesson 21, Activity 2, students demonstrate the composition and decomposition of numbers 11–19. “Give students access to connecting cubes or two-color counters, 10-frames, and bead tools. Kiran wrote equations to show the total number of students and how many students sat at the table and how many sat on the rug, but he didn’t finish the equations. Finish filling in each equation. You can use connecting cubes or two-color counters if they are helpful. 17=10+___. 19=___+9. 10+___=14. ___+2=12. 11=___+1. 15=10+___.” (K.NBT.1)

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

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

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

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

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

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

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

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

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

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

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

Examples of routine applications include:

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

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

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

Examples of non-routine applications include:

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

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

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

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

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

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

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

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

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

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

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

• Unit 5, Composing and Decomposing Numbers, Lesson 11, Activity 2, students extend their conceptual understanding and procedural fluency as they represent equations on fingers. Launch, “Give each student at least 2 different colored crayons. Color the fingers to show each equation.” Activity, “As you continue working, tell your partner about the total and the 2 parts you colored in each set of fingers.” Student Task Statements, "$$10=6+4$$. 10=9+1. 10=5+5. 10=3+7. 10=8+2. 10=1+9.” (K.OA.1)

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

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

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

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

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

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

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

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

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

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

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

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

Examples of constructing viable arguments include:

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

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

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

Examples of critiquing the reasoning of others include:

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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