Kindergarten - Gateway 2

The materials reviewed for Eureka Math² Kindergarten meet expectations for developing conceptual understanding of key mathematical concepts, especially where called for in specific content standards or cluster headings. 

Materials provide opportunities for students to develop conceptual understanding throughout the grade level. These opportunities are most often found within the Launch and Learn portions of lessons. Examples include:

• Module 1, Topic B, Lesson 9: Conserve number regardless of the arrangement of objects, Learn, students develop conceptual understanding as they count objects and learn that the arrangement of the objects does not impact the number of objects counted. “Count and use numeral cards to describe a set before and after it is rearranged. Display 3 Unifix Cubes in a linear configuration and cards 1–5. ‘How many cubes are there?’ (3) ‘Which number card tells how many cubes there are?’ (The 3 card) Move the 3 card below the cubes. Move the cubes slowly, deliberately letting students see you create a scattered configuration. ‘How many are there now? How do you know?’ Some students will know right away that there are still 3 cubes. Others will need to count to make sure. Validate both strategies by calling on students to share how they know there are still 3 cubes. (I know there are still 3 because I counted them again. I know there are still 3 cubes because I saw you move them. If there were 3 before, then that means there are still 3 now.) Which number card tells how many there are?” (K.CC.4)

• Module 2, Topic B, Lesson 6: Distinguish between flat and solid shapes, Learn, students develop conceptual understanding about flat and three-dimensional shapes. “Partner students and give each pair a bag of geometric solids and 2D shapes. ‘Look inside your bag. Take out a square and a shape that looks like a die. Put them on your table. Pretend you are an ant. Put your eyes near the top of your table to look at these two shapes. What is different about them?’ (The square lies on the table. This one is taller. It stands up.) ‘In your bag, you have shapes that are flat like this. (Hold up a flat shape.) We call these flat shapes. You also have shapes that are tall like this. (Hold up a solid shape.) We call these solid shapes.’ Distribute a work mat and a set of Hide Zero cards to each pair. Briefly orient students to the sorting materials and procedure: Partners sort their shapes into flats and solids on the work mat. Each partner counts one group. They select the Hide Zero card that tells how many are in the group.” (K.G.4)

• Module 5, Topic A, Lesson 2: Relate number sentences and number bonds through story problems, Learn, Relate Representations, students develop conceptual understanding as they reason about different representations for the same addition and subtraction situations. “Display two student work samples, one that uses a number bond and one that uses a number sentence. Invite the students who own the work to tell their story about the pigeons. ‘Sam, tell us about your math story.’ (Here are the birds that were there at the beginning. There are 2. These are the birds that came flying. There are 3. 2 and 3 is 5. I wrote that in the number bond.) ‘Jacob, tell us about your math story.’ (There were 7 birds on the playhouse. Then 2 came walking up. Now there are 9 birds.) Focus attention on the sample that uses a number bond. ‘Where are the parts and the total in Sam’s math drawing?’ As students explain, point to the parts and total in the picture and in the number bond. Connect the numbers in the bond to their referents in the picture. Focus attention on the sample that uses a number sentence. ‘Turn and talk: Does Jacob have two parts and a total in his math drawing? What about in his number sentence?’ Listen as students discuss. ‘I heard Lorena and Kailey say they see parts in the drawing. Where do you see parts?’ (The birds on top were already there. The birds on the bottom came over to play. Those are the parts.) ‘Where are the parts in the number sentence?’ (7 and 2. The 7 birds are on top there. Points.) And there are 2 birds are on the bottom. Points.) ‘Where do we see the total in Jacob’s work?’ (The total is all of the birds, all of the circles. 9. There are 9 birds in the drawing and 9 in the number sentence.) As students share their thinking, connect the parts and the total in the picture to the numbers in the number sentence.” (K.OA.1)

Materials provide opportunities for students to independently demonstrate conceptual understanding throughout the grade level. The Problem Set, within Learn, consistently includes these opportunities. When appropriate, teachers may use other portions of lessons for independent demonstration of conceptual understanding. Examples include:

• Module 1, Topic B, Lesson 8: Count sets in linear, array, and scattered configurations, Learn, Problem Set, students demonstrate conceptual understanding as they independently count to answer “how many?” questions about as many as 20 things arranged in a line, a rectangular array, or a circle. “Transition to this segment by practicing your routine for passing out the student book and finding the correct page. Prompt students to look at the first group of snails. Demonstrate by using cubes to mark and count the snails. Invite students to do the same. Move the cubes from the snails to the number path while counting again. Pause and have students do the same. ‘How many snails are there?’ (3) ‘How do we know there are 3?’ (We used our cubes to count and there are 3 cubes. There are 3 cubes on our number path.) Prompt students to circle the numeral 3. If time permits, they may color in the squares on the number path. Invite students to complete the next problem independently. Use systematic modeling (see Teacher Note) for the second Problem Set page, but this time demonstrate crossing out the butterflies. Allow students to complete both problems on the third practice page independently if they are ready. Encourage use of their preferred counting strategy.” (K.CC.5)

• Module 2, Topic A, Lesson 2: Classify shapes as triangles or nontriangles, Learn, Problem Set, students independently demonstrate conceptual understanding as they reason about 3-sided closed shapes. Students are first shown Item 3, a picture of three triangles of different sizes, positions, and angles and a shape that resembles a triangle with a portion of one side missing. “Display the four figures. Point to the first figure and ask students to decide whether it is a triangle. If it is a triangle, color it. If it is not a triangle, cross it out. Allow students to complete the second page of the Problem Set independently, following these instructions.” Item 4 shows a variety of shapes including three triangles in different sizes and positions and six other shapes. (K.G.2) 

• Module 6, Topic B, Lesson 7: Decompose numbers 10–20 with 10 as a part, Launch, students demonstrate conceptual understanding as they decompose a teen number into 10 ones and another group of ones. “Distribute a set of Hide Zero cards to each student. ‘Put your cards in order from 1 to 10 and so that you can see the side that has numbers. Stand up when you’ve done this.’ Have students sit down when everyone’s cards are organized. ‘I’ll show you dots. Make the number that matches my dots.’ Show the cards with 10 dots and 9 dots. Move them together to form a total of 19 dots. Have students place their cards directly in front of them or on a work mat. Scan them to check for accuracy and quickly provide feedback. Have students return the cards to prepare for the next problem. Repeat with other teen numbers. ‘This time I’ll show you some cubes. Make the number that matches my cubes.’ Display the 13 Unifix Cubes in a scattered configuration. Expect students to respond with uncertainty and then invite them to share their reactions. ‘What’s the matter?’ (It’s so messy! I can’t tell how many there are. I wish I could do touch and count or line them up. I know there’s 3 yellow for sure. Maybe there are 10 blue?) ‘Why didn’t we have this problem when I showed dot cards?’ (The dot cards are in 5-groups so it’s easy to see how many. The dot cards are organized.) Transition to the next segment by framing the work. ‘Today, we will think about what makes some things easier to count than other things.’” (K.NBT.1)

The materials reviewed for Eureka Math² Kindergarten meet expectations for giving attention throughout the year to individual standards that set an expectation of procedural skill and fluency. 

The materials develop procedural skill and fluency throughout the grade level, within various portions of lessons, including Fluency, Launch, and Learn. There are also opportunities for students to independently demonstrate procedural skill and fluency. Examples include:

• Module 1, Topic C, Lesson 10: Count out a group of objects to match a numeral, Land, Debrief, students develop procedural skill and fluency as they work with the teacher to count a number of objects to match the numeral they are shown. “Bring students together in a place where they can all see Puppet, the 3 card, and a pile of beans. ‘Puppet got an order for 3 apples. Watch Puppet count out the apples.’ Have Puppet count beyond 3, stopping when all the beans are counted. ‘Uh oh. What is wrong?’ (There are too many! Puppet didn’t stop at 3.) Have students think–pair–share about how to help Puppet. ‘Inside your head, think about how Puppet could remember when to stop counting. Turn to your partner, and tell what Puppet could do.’ Invite one or two students to share their responses. ‘How can Puppet remember when to stop counting?’ (Puppet can look at the number on the card. If it says 3, stop at 3. Puppet can hold the number the customer asked for in their head and stop counting there.)”  (K.CC.3)

• Module 4, Topic C, Lesson 13: Choose a math tool to solve put together with total unknown story problems, Learn, Share, Compare, and Connect, students develop procedural skill and fluency as they use math tools to solve an addition story problem and then write a number sentence to represent the problem. “‘Libby, tell us what tools you used to show the story.’ (I used cubes on a 10-frame mat. I put 5 red cubes on the top. Then I put 4 blue cubes on the bottom.) ‘What part of the story do the red cubes show?’ (The red cubes are the boy’s pennies and the blue cubes are the girl’s pennies.) ‘Okay. You also drew a number bond. Can you tell us about it?’ (I put 5 in one part for the red cubes. Then I put 4 in the other part for the blue cubes. I put 9 in the total.) ‘How did you get 9?’ (I got the 9 because I counted the red and blue cubes.) ‘What does 9 tell us about from the video story?’ (It tells about the children’s pennies when they put them together.) ‘Libby says the children have 9 pennies altogether. Turn and talk to your partner: Do they have enough money for the pencil? How do you know?’ Modeling with a Drawing (Jason’s Way) Invite a student who drew a picture to share.’ Jason used another kind of tool, a picture. Tell us about it.’ (I drew 5 circles and circled them, and then I drew 4 circles and circled them.) ‘What part of the story do the 5 circles tell us about? The 4 circles?’ (The 5 circles show the boy’s pennies. The 4 circles show the girl’s pennies.) ‘What did you do next?’ (I counted all the circles to find the total. Ask the class to say the number sentence. Write 5 and 4 make 9.) ‘Let’s write the number sentence the way mathematicians do. (Write 5+4=9 below.) Does this number sentence match Libby’s work? Does it match Jason’s work? How do you know?’ (It matches Libby’s work. Her parts are 5 and 4 and her total is 9. Jason’s matches! He has 5 and 4 make 9.) ‘We all used different tools to show the story in different ways, but the same number sentence tells us about the story. The total is always 9 pennies. Are 9 pennies enough to buy the pencil?’ (Yes! They need 9 pennies.)” (K.OA.5)

• Module 5, Topic A, Lesson 7: Find the total in an addition sentence, Learn, Find a Different Way, students develop procedural skill and fluency with addition by using a variety of strategies to find the solution to a problem. “Write 3+6= ___ and tell students they will find the total in a different way than they did before. Remind students of the available tools, such as cubes, 10-frames, number paths, whiteboards, fingers, or drawing. Invite students to share their thinking with a partner. Then ask them to use their chosen way to find the total. ‘What is the total of 3+6?’ (9) Ask a few students to share their strategy for finding the total. ‘Turn and talk to your partner: Which way was easier for you to solve, the way you solved 4+3 or the way you solved 3+6? Why?’” (K.OA.5)

Materials provide opportunities for students to independently demonstrate procedural skill and fluency throughout the grade level. The Problem Set, within Learn, consistently includes these opportunities. When appropriate, teachers may use other portions of lessons for independent demonstration of procedural skill and fluency. Examples include:

• Module 4, Topic B, Lesson 7: Find partners to 5, Fluency, Shake Those Disks, students demonstrate procedural skill and fluency with decomposing 5 as they record a total and parts in a number bond in more than one way. “Form student pairs. Distribute the Shake Those Disks removable in a personal whiteboard, a marker, and cup of 5 counters to each pair and have them play according to the following rules. Consider doing a practice round with students. Partner A: Shake and spill the cup of counters. Partner A: Place the counters on the number path and count. Partner B: Write the total in the number bond. Partner B: Count the number of red and yellow counters, and then write the numbers in each part. Switch roles after each turn.” (K.OA.3)

• Module 5, Topic C, Lesson 15: Find the difference in a subtraction sentence, Problem Set, students demonstrate procedural skills and fluency as they look at a number sentence and use strategies and tools to subtract. “Invite students to self-select tools to complete the Problem Set. Space is provided for drawing, but students may or may not choose to draw. Before releasing the class to work independently, ask students to notice what is different about the last two number sentences on the back page. Fill in the number sentence. 3-1=__ , 5-4=__ , 6-3=__, 4-1=__, 5-0=__.” Six additional problems are included. (K.OA.5)

• Module 6, Topic A, Lesson 1: Describe teen numbers as 10 ones and ___ ones, Learn, Problem Set, Problem 1, students demonstrate procedural skill and fluency with teen numbers as they count pictured objects, circle groups of 10, and record how many tens and ones there are. “Discuss strategies for finding and circling a group of 10 before letting students work independently on the Problem Set. In some configurations, students may be able to see a group of 10 as two sets of 5. In other configurations, students might mark and count to find a group of 10.” Students see seven groups of various objects and directions, “Circle a group of ten. Fill in the blanks.” Problem 1 includes 13 crabs (2 rows of 5 crabs and 1 row of 3 crabs) and students complete, “10 ones and ___ ones.” (K.NBT.1)

The materials reviewed for Eureka Math² Kindergarten meet expectations for being designed so that teachers and students spend sufficient time working with engaging applications of the mathematics. 

There are multiple routine and non-routine application problems throughout the grade level, including opportunities for students to work with support of the teacher and independently. While single and multi-step application problems are included across various portions of lessons, independent application opportunities are most often found within Problem Sets or the Lesson Debrief, Learn and Land sections respectively.

Examples of routine applications of the math include:

• Module 1, Topic D, Lesson 17: Model story problems, Launch, students solve routine addition and subtraction problems with teacher support. “Make a bus by placing 5 chairs in a row. Look at the bus! ‘How many people can ride on our bus?’ (5 ) ‘Everyone will get to ride the bus today. Count with me as I tap 3 people to get on the bus.’ Tap 3 students to get on the bus. Reassure students that everyone will ride the bus. (1, 2, 3) ‘How many people got on the bus?’ (3) Tap 2 more students to get on the bus. How many people are on the bus now? (5) Tap 1 student to get off the bus. ‘How many people got off the bus?’ (1) ‘How many people are on the bus now?’ (4) Continue as long as it takes to give every student the opportunity to ride the bus and answer the how many questions. Vary the number of students that get on and get off the bus. Use the questions below to engage the class in thinking about adding to or taking away. ‘How many people got on the bus? How many people got off the bus? How many people are on the bus now?’ Transition to the next segment by framing the work. ‘Today, let’s think about some more story problems!’” (K.OA.1)

• Module 2, Topic C, Lesson 14: Compose Flat Shapes, Fluency, Numeral Writing, students solve routine number problems and build proficiency with numeral formation from 0-10. “Make sure students have a personal whiteboard with a Scoreboard removable inside. Display the Baseball Bears digital interactive. ‘The blue and red teddy bears are having a home run competition. They get 1 point for every home run they hit. Our job is to keep score. Write the numbers on your scoreboard to keep track of the points. The bears don’t have any points yet. Write the number of points for each bear on the scoreboard.’ Give students time to work. When most students are ready, signal for students to show their whiteboards. Provide immediate and specific feedback. If students need to revise, briefly return to validate their corrections. Show the score: 0 to 0. Use the digital interactive to have the bears take turns at bat. If a bear hits a home run, prompt students to change their scoreboards. If a bear does not hit a home run, it gets an out. Continue the process to 10 points, demonstrating numeral formation while saying the number rhyme as needed.” (K.CC.3)

• Module 5, Topic B, Lesson 10: Represent and solve take from with result unknown story problems, Learn, Represent and Solve, students solve routine story problems independently and write matching number sentences. “Distribute student books and help students turn to the cookies problem. ‘Listen to the next cookie story: Edwin has 7 cookies. He eats 2 cookies. How many cookies does Edwin have now? You can use any tools that you want. When you have solved the problem, write a number sentence to match it. Use the space on the page to show your thinking.’ Observe as students work. Take a picture or make note of the strategies and tools they use. Select one or two students who used different representations to share their work.” (K.OA.2)

Examples of non-routine applications of the math include:

• Module 2, Topic A, Lesson 5: Communicate the position of flat shapes using position words, Land, Debrief, students solve non-routine problems where they identify different shapes. “Display the picture of birthday candles and toothpicks. ‘Do you see any shapes in the picture?’ (Yes.) ‘Whisper the names of the shapes you see to your partner. What words can we use to describe where the shapes are?’ (Above, below, beside, in front of, behind, around) ‘I’m thinking of the shape that is beside the hexagon. What shape am I thinking about?’ (The triangle that’s made out of candles) Call on a few students to think about a shape in the picture and use position words to describe it to the class. Shapes may include: the rectangular window on the candle box, the square candle box, without the hanging tab, the circles next to the words, or the hexagon around the box.” (K.G.1)

• Module 5, Topic A, Lesson 2: Relate number sentences and number bonds through story problems, Launch, students solve non-routine word problems independently and create their own problems with partners. “‘Cover your eyes and make a movie in your mind as I tell a story. There are some pigeons on our playground. Then some more pigeons land on our playground. Open your eyes. Turn and talk about how you saw the pigeons in your mind.’ Give students a moment to talk about what they saw in their mind. Then use the following prompts to help them with the mathematical parts of the story. ‘Tell your neighbor how many pigeons you saw at first. Tell your neighbor how many pigeons landed next.’ Distribute paper and crayons. ‘Make a math drawing of how you saw the pigeons. Use a number bond or a number sentence to tell about the picture.’ As students work, support them as needed. Identify two student work samples for use in Learn: one that uses a number bond and one that uses a number sentence. ‘Look at your picture. Do you know the total number of pigeons? How do you know?’ (I know the total. I counted all of them. The total is right here in my number bond.) Transition to the next segment by framing the work. ‘Today, we will use our pictures to see how our number bonds and number sentences are the same and different.’” (K.OA.1 and K.OA.2)

• Module 6, Topic D, Lesson 24: Organize, count, and represent a collection of objects, Launch, students solve non-routine problems as they classify objects into given categories, count the numbers of objects in each category, and sort the categories by count. “Students discuss ways to find the total of a collection. Display the picture of boxed colored pencils.’How many pencils are in the box? Show thumbs-up when you know.’ (10 pencils) Model counting to confirm that there are 10 pencils. ‘Each box can hold 10 pencils, like a 10-frame.’ Display the picture of the 3 pencil boxes. ‘How can we find the total?’ (We can count each pencil. We can count by tens. 10, 20, 30 We can add. It’s 10+10=10.) Invite students to think-pair-share about the following question. ‘Suppose there are 6 boxes. How could we find the total then?’ (We could keep adding tens. It’s 3 more tens. There would be 6 tens because each box is 1 ten. We can Say Ten count, like 1 ten, 2 ten, 3 ten. I know 3 and 3 is 6. So 3 tens and 3 tens is 6 tens or 60.)  Transition to the next segment by framing the work. ‘Today, we will count a collection that has already been put into groups, like the pencils. You can use math tools to help you count and record.’” (K.MD.3)

The materials reviewed for Eureka Math² 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:

• Module 2, Topic C, Lesson 14: Compose Flat Shapes, Fluency, Happy Counting Within Ten, students develop procedural skill and fluency as they count forward and backward from a given number. “‘When I give this signal, count up. (Demonstrate.) When I give this signal, count down. (Demonstrate.) Let’s count by ones. The first number you say is 3. Ready?’ Signal up or down accordingly for each count. Continue counting by ones within 10. Change directions occasionally, emphasizing crossing over 5 and where students hesitate or count inaccurately.” (K.CC.2) 

• Module 5, Topic A, Lesson 2: Relate number sentences and number bonds through story problems, Learn, Problem Set, students demonstrate conceptual understanding when creating a number sentence to represent a picture. Students see five pictures in the problem set. Each picture shows objects that are in two locations, allowing the student to create a number sentence to represent each situation. “The Problem Set directions follow the work of the previous segment to help students transition to independent work. Circulate and assist as needed. Use the following questions and prompts to assess and advance student thinking: ‘Tell me a story about this picture. What part of the story does this number show? (Point to a number in the number sentence or bond.) Where are the parts in your number sentence? Where is the total?’” (K.OA.1, K.OA.2)

• Module 5, Topic B, Lesson 9: Represent take from with result unknown story problems by using drawings and numbers, Learn, Represent a Subtraction Situation, students solve routine application problems with addition and subtraction. “‘Cover your eyes and make a movie in your mind as I tell my orange story. I went to the store and bought 9 oranges. (Pause.) I was really hungry when I got home. I ate 4 oranges. (Pause.) How many oranges are left? Open your eyes. Draw a picture of what you saw in your mind.’ Distribute paper and crayons. Give students a few minutes to draw. Invite a few students to share about their drawings. Select samples that show obviously different drawing styles, such as the following: Crossing out one by one, or all at once. Circling, labeling, or drawing a line to show the two parts. Erasing the part that was taken away. For each sample, have students indicate which part was taken away and which part is left. Students will continue to use their work in the next segment.” (K.OA.2)

Multiple aspects of rigor are engaged simultaneously throughout the materials in order to develop students’ mathematical understanding of grade-level topics. Examples include:

• Module 2, Topic B, Lesson 7: Name solid shapes and discuss their attributes, Learn, Shape Hunt, students solve a non-routine application problem and develop conceptual understanding as they identify real-world objects around the classroom that look like solid shapes. “Place a set of geometric solids where students can see them. ‘Let’s go on a solid shape hunt in our room. When I say go, look around our room and try to find something that looks like one of the solid shapes we talked about today. Make sure your objects can fit inside this box.’ Give students about a minute to look for objects in the room that are like solid shapes. If the activity is difficult, display the picture of classroom objects to stimulate students’ thinking If some students don’t find an object in the time allotted, simply give them an object from the classroom. ‘Bring your object and put it in the box.’ Show an object from the collection and ask students to compare it with a solid shape. As needed, use a sentence frame such as the following: It looks like a ___ because ___. ‘What solid shape does this pad of sticky notes look like?’ (It looks like a cube because the top is a square. The sides are not square. I think it’s a rectangular prism.) Continue with other objects in the box.” (K.G.4)

• Module 4, Topic B, Lesson 5: Sort to decompose a number in more than one way, Learn, Sort and Record, students develop conceptual understanding alongside procedural skill and fluency as they decompose a number in more than one way and represent the decompositions with number bonds. “Arrange the paper plates to resemble a number bond. Put all the bears on one plate. Draw an empty number bond on chart paper, leaving room for other number bonds. Select a student to be the recorder. ‘Look at the bears all together. (Point to the total.) What do we call the place in the number bond that shows how many are in the whole group?’ (The total) ‘How many bears do we have in total?’ (5) ‘Do we have to draw the bears, or can we write a number to tell about the bears in the total?’ (We can write numbers. It’s faster.) Invite students to turn and talk about ways to sort the bears. ‘I heard a few partners say that we could sort by color.’ Sort the bears by color, moving each color group to its own plate. ‘What do we call these places in the number bond?’ (Point to the parts.) (Parts) Yes. We sorted, or broke, the total into two parts. Invite the recorder to write the numbers in each part of the number bond. Then label the number bond color to remember the attribute used for the sort. Return the bears to the total plate. Draw another number bond. Repeat the process by using another attribute that produces different partners to 5. This time let’s sort by tigers and bears. Guide students to a final sort that produces 0 and 5 as parts. Record the sort with numerals in a number bond for the next segment.” (K.OA.3)

• Module 4, Topic B, Lesson 6: Organize, count, and represent a collection of objects, Fluency, Whisper-Shout Counting, students develop conceptual understanding alongside application as they understand that the last number name tells the number of objects counted. “‘Let me hear you whisper 1, 2, 3. Make it just loud enough so I can hear you.’ (Emphasize with a finger to your lips.) (1, 2, 3, in a whisper voice) ‘Great! Now let me hear you shout 1, 2, 3. Make it an “indoor shout” so we don’t disturb the other classes.’ (Emphasize with cupped hands around your mouth and a corresponding facial expression.) (1, 2, 3 ,shouting) Display a stick of 3 Unifix Cubes. Using a dry-erase marker, make a dot on the last cube. ‘I’ll touch, and you’ll count. (Point to the last cube.) We will whisper, but when you get to the last one, shout the number!’ [1 (whisper), 2 (whisper), 3 (shout) ‘How many cubes?’ (3) Repeat the process a few times with 3 cubes and then 5 cubes.” (K.CC.4)

The materials reviewed for Eureka Math² 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 MP1 and MP2 across the year and they are identified for teachers within margin notes called, “Promoting the Standards for Mathematical Practice.” According to the Kindergarten Implementation Guide, “‘Promoting the Standards for Mathematical Practice’ highlights places in the lesson where students are engaging in or building experience with the Standards for Mathematical Practice (MPs). Although most lessons offer opportunities for students to engage with more than one Standard for Mathematical Practice, this guidance identifies a focus MP within each lesson. The notes also provide lesson-specific information, ideas, and questions that you can use to deepen students’ engagement with the focus MP.”

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 modules. Examples include:

• Module 1, Topic B, Lesson 6: Organize, count, and represent a collection of objects, Learn, Share, Compare and Connect, students build experience with MP1 as described in the margin notes for teachers, Promoting the Standards for Mathematical Practice, “When students count a collection of objects, they make sense of problems and persevere in solving them (MP1). They plan how to count a collection, carry out the plan, and adjust the plan as needed.” Teacher directions state, “Gather the class to view and discuss the selected work samples. Invite each selected student pair to share their counting process. Name the counting strategies each pair used. Use the examples below to guide your class discussion. Touch and Count (Oscar and Audrey’s Way), Invite a pair who used a touch and count strategy to demonstrate, using their collection or a photo of their work. When the pair are finished counting, help the class discuss their strategy. ‘How many blocks are in their collection? How do you know?’ (8 The last number was 8.) ‘What did Oscar and Audrey do to be sure they counted all the blocks?’ (They touched all the blocks. They didn’t miss any numbers.) ‘We have a lot of ways to make sure we count correctly. Let’s call those ways strategies. Oscar and Audrey used the touch and count strategy to make sure they said one number for each block. Oscar and Audrey, how did lining up your blocks help you count?’ (We started at the bottom and went up.) ‘So the line helped you know where to start counting and stop counting?’ (Yes.) Move and Count (Alaina and Campbell’s Way), Invite a pair who used a move and count strategy to demonstrate. Stop the pair after they count about 5 items and ask the following question, ‘I see that some of your alligators are in a line. (Point to the line.) Some of your alligators are in a pile. Why?’ (These are the ones we counted. We didn’t count the ones in the pile yet.) ‘Ah. You are moving the alligators as you count them.’ Allow the pair to finish counting, uninterrupted. ‘How did the move and count strategy help Alaina and Campbell find out how many alligators are in their collection?’ (They only counted each alligator once. They knew what they already counted.)”

• Module 3, Topic A, Lesson 3: Compare lengths of complex objects by using longer than, shorter than, and about the same as, Learn, Sort by Length, students build experience with MP1 as described in the margin notes for teachers, Promoting the Standards for Mathematical Practice, “When students try to cut a piece of yarn that is the same length as the bracelet, they make sense of problems and persevere in solving them (MP1). Since students cannot compare the yarn directly with the bracelet, encourage them to persevere in finding a strategy to develop their best guess. Using the question in this section to analyze the chart showing all of the students’ pieces of yarn encourages students to evaluate why this task is challenging.” Teacher directions state, “‘Look at my bracelet. Cut a piece of yarn that is the same length as this bracelet.’  Playfully explain to students that they can’t touch the bracelet or hold it next to their piece of yarn. Pass around balls of yarn so students can cut a piece that is about the same length as the bracelet. ‘Now you will hold your yarn next to the bracelet and see if it’s longer than, shorter than, or about the same length as the bracelet.’ Have one student at a time test their yarn. Students may lay the bracelet straight or wrap their yarn around the bracelet. Ensure that they align the endpoints. Then invite them to place their yarn on the chart and say the comparison statement starting with, ‘My yarn is ….’ Once all students place their yarn on the chart, generate discussion with questions, such as: Which group has many pieces of yarn? What does that tell us? Which group has few pieces of yarn? What does that tell us? Why do you think there aren’t many pieces of yarn that are about the same length as the bracelet?” 

• Module 5, Topic C, Lesson 15: Identify the action in a problem to represent and solve it, Launch, students build experience with MP1 as described in the margin notes for teachers, Promoting the Standards for Mathematical Practice, “As students work to understand the story at hand before moving on to solve the problem, they make sense of problems and persevere in solving them.” (The teacher tells a math story, adjusting the context to centers in the classroom so students can act it out.) Teacher directions state, “‘Listen to my story. 5 students are reading in the library. Some of those students go to the computer center. Let’s talk about what we know. What can you tell me?’ (We know there are 5 students at the library. Some left to go to the computer center.) Repeat the second line of the story. Then invite students to think–pair–share about the following question. ‘Are there more students or fewer students in the library than before? How do you know?’ (There are fewer because some left. Students are going away. There aren’t as many in the library now.) Transition to the next segment by framing the work. ‘Today, we will practice paying close attention to what happens in math stories. That will help us show problems and solve them.’”

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 modules. Examples include:

• Module 3, Topic C, Lesson 16: Count and compare sets with unlike units, Land, Debrief, students build experience with MP2 as described in the margin notes for teachers, Promoting the Standards for Mathematical Practice, “When students use numbers to compare the two sets of objects that they can’t see, they reason abstractly and quantitatively (MP2). Here students see that it’s possible to compare the number of objects in two different sets when they can’t actually see the objects, and in time students will see that this method is often necessary or more convenient. For example, if the sets of objects are very large or can’t be manipulated, counting and comparing the numerals is often the most efficient course.” Teacher directions state, “Place Puppet in a visible location along with the sorting bag used to demonstrate at the beginning of Learn. Take the items out of the bag and place each group in a line along the number path. ‘Puppet wanted to use the number path to compare groups. Which group has more?’ (The cubes go all the way to 7, so they have more. There are more cubes than crayons.) ‘Which group has fewer?’ (There are fewer crayons.) If students do not mention the number of cubes and crayons in their responses, ask the class to tell how many are in each group. Move each group of objects into a pile and label with Hide Zero cards. Keep the number path in sight. ‘Which group has more?’ (It’s still the cubes.) ‘Which group has fewer?’ (There are fewer crayons than cubes. It didn’t change.) Place a piece of paper over the groups so only the numbers are showing. ‘Which group has more?’ (Still the cubes! There are still 7 cubes.) ‘Can you tell which group has more things and which has fewer things just by looking at the number? How?’ (Yes. It’s the same as before. It is 7 and 4.) ‘If you can see the number, do you have to see the groups?’ (I’m not sure. No. 7 comes after 4, so the 7 group has more. No. I remember that there are 7 cubes.) ‘How can we compare groups of different things?’ (We can match them and see which group has some left. You can just look at the numbers. The number path can help you see which number is more. You can count and see which number comes after.)”

• Module 4, Topic C, Lesson 15: Choose a math tool to solve take apart with both addends unknown situations, Learn, Zoo Story, students build experience with MP2 as described in the margin notes for teachers, Promoting the Standards for Mathematical Practice, “As students work to represent the take apart with both addends unknown story, they reason abstractly and quantitatively (MP2). While it can be difficult to imagine the story without being told how many meerkats go in each truck, students decontextualize to decompose the total as a number and recontextualize to understand how their work shows the story. The questions in this section are designed to promote MP2.”  Teacher directions state, “Display the picture of 8 meerkats. Remind students that they can use tools to show and explain their thinking about stories. Tools may include objects, drawings, or something in their minds (such as a known fact) that helps them show and explain their thinking. ‘Listen to my story: There are 8 meerkats moving to a new zoo. Two trucks drive them to their new home. How could the zookeeper put the meerkats in the trucks?’ Invite students to turn and talk about which tool they will use to show this story. Have them self-select math tools and model the story. Use the following questions and prompts to assess and advance student thinking: Where are the parts in your work? The total? Is there another way the meerkats could go on the trucks? How did the tool you chose help you? Circulate and observe student strategies. Select two or three students to share in the next segment. Look for work samples that help advance the lesson’s objective by using the count all and count on strategies to find a total.”

• Module 6, Topic B, Lesson 8: Represent teen number compositions and decompositions as addition sentences, Learn, Share, Compare and Connect, students build experience with MP2 as described in the margin notes for teachers, Promoting the Standards for Mathematical Practice, “As students work to solve the bracelets problem, they reason abstractly and quantitatively (MP2). Students decontextualize by using representations such as cubes, numbers, number bonds, and number sentences to represent the bracelets. They recontextualize by explaining which parts of the story the referents represent. In this instance, students are also asked to reason abstractly by making connections between the different representations, recognizing that the parts and total can be represented in different, but equivalent, ways.” In an earlier lesson segment students have solved an add to with result unknown story problem involving bracelets, self-selecting tools and strategies. In this segment specific students are asked to share, and a sample dialogue is provided. Teacher directions state, “Refer to the Talking Tool for other ideas to support student-to-student discussion. ‘Samuel, how did you use your Hide Zero cards to solve?’ (I took the 10 to show Ko’s 10 bracelets. Then I took the 7 to show Isaac’s 7 bracelets. I put the cards together to make 17.) ‘Zaden, tell us how you used the number bond.’ (I put 10 cubes in one part and 7 cubes in another part. Then I counted the cubes and there were 17.) ‘Tasha, how did you use a number sentence to solve?’ (I wrote 10 plus 7 because I knew I needed to add their bracelets to get the answer. 10 plus 7 is 17, so I knew they had 17 bracelets.) Display the selected work samples so all students can see them. If none of the students wrote a number sentence, use the work samples to write an addition sentence as a class. Ask the class to find the referents in each drawing or number sentence. They should identify where they see Ko’s bracelets, Isaac’s bracelets, and the total number of bracelets.”

The materials reviewed for Eureka Math² 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 explicitly identified for teachers within materials in margin notes called, “Promoting the Standards for Mathematical Practice.” According to the Kindergarten Implementation Guide, “‘Promoting the Standards for Mathematical Practice’ highlights places in the lesson where students are engaging in or building experience with the Standards for Mathematical Practice (MPs). Although most lessons offer opportunities for students to engage with more than one Standard for Mathematical Practice, this guidance identifies a focus MP within each lesson. The notes also provide lesson-specific information, ideas, and questions that you can use to deepen students’ engagement with the focus MP.”

Students construct viable arguments and 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:

• Module 3, Topic D, Lesson 20: Compare two numbers in story situations, Learn, Julie’s Pennies, students build experience with MP3 as described in the margin notes for teachers, Promoting the Standards for Mathematical Practice, “Students construct viable arguments and critique the reasoning of others (MP3) when they discuss these story problems. Each problem allows for multiple solution paths, giving students multiple opportunities to explain how they arrived at their answer. If students disagree about problems, promote MP3 by asking: What don’t you understand about Braxton’s thinking? What questions can you ask about Tao’s thinking?” Teacher directions state, “Display the picture with the three items and their prices. ‘Julie has 6 pennies. What can Julie buy? How do you know?’ Invite students to select tools and give them time to work. As you circulate, notice which strategies students use. Select three students who use different number comparison strategies to share their work with the class. Gather the class for discussion. Invite each selected student to share their thinking. Name the comparison strategy each student used. ‘Braxton, tell us how you used cubes.’ (I used brown cubes to show Julie’s 6 pennies. Then I took 2 red cubes to show the pennies for the star and matched them to the brown cubes and there were enough. There were 4 brown cubes left, which means that Julie can buy the star and the eraser. She can’t buy the bubbles though because she doesn’t have enough.) ‘Tao, tell us how you used your fingers to find what Julie can and can’t buy.’ (Both 2 and 4 are before 6, so Julie has enough money to buy the star or the eraser. She could buy both if she wanted. I know because I used my fingers to count 2 and 4 together and got 6. She can’t buy the bubbles though. 7 comes after 6.) ‘Mayson, show us how you used your tool.’ (I used the number path. I lined up 6 cubes for Julie’s pennies. I saw that the 2 and 4 are on the path before the 6. So Julie had enough to buy those things. She doesn’t have enough to buy the bubbles because 7 is more.)”

• Module 4, Topic A, Lesson 1: Compose flat shapes and count the parts, Learn, Gallery Walk, students build experience with MP3 as described in the margin notes for teachers, Promoting the Standards for Mathematical Practice, “As students point out similarities and differences in the hexagons, parallelograms, and triangles they view during the gallery walk, they construct viable arguments and critique the reasoning of others (MP3).” Teacher directions state, “Gather students away from their puzzles and remind them of the protocol for a gallery walk. Remind students to look but not to touch, as they would in a museum or gallery. They can hold their hands behind their backs as a reminder. Prompt students to be very quiet as they look. Encourage them to think about the shape puzzles. ‘While we take a gallery walk, pay attention to the hexagons. What do you notice about them?’ Observe students as they walk around. Once they have completed their walk around the room, gather the class. ‘What did you notice about our hexagons?’ (They didn’t look the same. We didn’t all use the same shapes to make them.) ‘Xavier noticed that the hexagons were made of different shapes, or parts. How do we know that they are all hexagons?’ (We all have the same puzzles. The gray shape is a hexagon in all our puzzles. We just put other shapes on top. They all have 6 sides and 6 corners.) Prepare for Land by gathering two or three student samples that show the triangle composed of different parts. If time allows, offer more practice by inviting students to use pattern block puzzles. As students work, ask them to tell how many shapes, or parts, they used.”

• Module 5, Topic B, Lesson 13: Tell subtraction story problems starting from number sentence models, Land, Debrief, students build experience with MP3 as described in the margin notes for teachers, Promoting the Standards for Mathematical Practice, “As students explain why one equation matches the baseballs and the other doesn’t, they construct viable arguments and critique the reasoning of others (MP3). The argument presented by the teacher, that the baseballs match 4-2=2, gives students an opportunity to find the flaw in someone else’s reasoning. Encouraging students to use precise language, such as part and total, helps them to see how their specific mathematical knowledge can be used to explain the teacher’s mistake.” Teacher directions state, “Show the baseball card with 6-2=4 and 4-2=2 as shown. Establish 6 as the total by chorally counting all the baseballs. Then ask the following question. ‘Which number sentence matches the baseball picture?’ (6-2=4) Hold up the card 4-2=2. ‘Why doesn’t this match?’ (There are 6 baseballs, but there’s no 6 in the number sentence.) Playfully challenge students’ assumptions about matching number sentences. ‘But wait, I see 4 baseballs here. (Point to the picture.) And I see 2 baseballs here. (Point to the picture.) This is just like in the number sentence. Are you sure this is not the right number sentence?’ (Yes, but there’s not 2 left like when you do 4 minus 2. There’s 4 left. 4 is not the total. The 4 comes first in the number sentence. That means there were 4 in the beginning and 2 got taken away. That’s not what you see in the picture. 6 were there at first. There’s still no 6 in that number sentence.) Hold up the card 6-2=4. ‘Use the words part and total to explain why this number sentence matches.’ (2 is the part that was taken away. 4 is the part that is left. We counted all the baseballs and the total was 6.) ‘How do you know when a subtraction sentence matches a story or a picture?’ (You have to match the numbers to the things in the picture. When you see the right total minus the crossed off part, you know it matches.)”

• Module 6, Topic C, Lesson 18: Count within and across decades when counting by ones, part 1, Land, Debrief, students build experience with MP3 as described in the margin notes for teachers, Promoting the Standards for Mathematical Practice, “As students consider Puppet’s work and determine whether the numbers are in the right order, they construct viable arguments and critique the reasoning of others (MP3). The questions in Land are designed to promote MP3. As needed, use the following questions to further engage students in critiquing Puppet’s work: What is confusing about Puppet’s work? What questions could you ask Puppet about this work?” In the activity students are shown an incorrect sequence of numbers (47, 48, 49, 40, 41) and critique Puppet’s reasoning. Teacher directions state, “‘Puppet put numbers in order today, just like you did. How could we check Puppet’s work?’ (We can count. We can look at the chart. We can check on the number path.) Invite students to whisper count from 45 to 50 the regular way. Ask students to stand if they see a number in the wrong place. Invite students to correct the sequence. Count from 47 to check the new sequence. ‘How can you figure out where to put a number on the number path?’ (I can count to see if the numbers are in the right place. You can add 1 more if you need to find the next number from one you know. You know the number before is 1 less, so you could take away.)”

The materials reviewed for Eureka Math² 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 MP4 and MP5 across the year and they are explicitly identified for teachers within materials in margin notes called, “Promoting the Standards for Mathematical Practice.” According to the Kindergarten Implementation Guide, “‘Promoting the Standards for Mathematical Practice’ highlights places in the lesson where students are engaging in or building experience with the Standards for Mathematical Practice (MPs). Although most lessons offer opportunities for students to engage with more than one Standard for Mathematical Practice, this guidance identifies a focus MP within each lesson. The notes also provide lesson-specific information, ideas, and questions that you can use to deepen students’ engagement with the focus MP.”

MP4 is identified and connected to grade-level content, and there is intentional development of the MP to meet its full intent. Students model with mathematics as they work with support of the teacher and independently throughout the modules. Examples include:

• Module 1, Topic A, Lesson 4: Classify objects into three categories and count, Learn, Problem Set, students build experience with MP4 as described in the Teacher Note, Promoting the Standards of Mathematical Practice, “When students draw their sort and compare their drawing to the physical objects, they are modeling with mathematics (MP4).” Teacher directions state, “Distribute sorting bags to individual students or pairs. Make number paths available for students who want to use them for counting. Invite students to sort the objects in their bags into three groups. Allow time for them to consider the differences between the objects in a bag and develop their own sorting rule. Support students by providing a way to sort only if necessary. When circulating, ask partners to answer questions about their groups. The following dialogue shows sample questions and sentence stems. ‘How did you sort into groups?’ (We sorted by …) ‘How did you decide where to put this? (Holding object.) Why didn’t it fit into the other group(s)?’ (It fits in this group because …It doesn’t fit in that group because …) ‘How many are in that group?’ If time permits, invite students to draw their groups on the Problem Set page after they have sorted and counted.”

• Module 3, Topic B, Lesson 10: Use a balance scale to compare an object to different units, Learn, Balance and Record, students build experience with MP4 as described in the Teacher Note, Promoting the Standards of Mathematical Practice, “Students model with mathematics (MP4) when they make drawings that show their scale with the sides balanced. Creating this representation moves students toward understanding the more abstract concept of the weight of an object.” Teacher directions state, “Students weigh a single object and compare its weight by using different units. Group students and give each group a scale. Help students turn to the Comparing Weights Recording Sheet in their student books. Invite groups to choose one classroom object that stays in the scale for every comparison. ‘Put your object on one side of the balance scale. Leave it there. Your job is to make the sides of the scale balance. When you get a bag like this, use the things inside to balance the sides of the scale. (Hold up a bag of objects.) Use your book to record, or show, your scale once the sides are balanced.’ It may be helpful to clarify that their completed recording should resemble the class chart. Distribute a bag of units, such as cubes, blocks, pennies, or beans, to each group. Circulate as students weigh the object they chose several times by using other objects as units. Encourage them to use numbers in their recordings.”

• Module 4, Topic B, Lesson 5: Sort to decompose a number in more than one way, Launch, students build experience with MP4 as described in the Teacher Note, Promoting the Standards of Mathematical Practice, “As students use number bonds to understand part–total relationships, they model with mathematics (MP4). Moving from pictures to circles to numerals takes students from more concrete models to more abstract models. This is an important progression in their ability to model with mathematics.” Teacher directions state, “Students consider different ways to represent a situation by using a number bond. Display the fish and dot number bonds. ‘At the beginning of the year, we talked about things that are exactly the same. Are these number bonds exactly the same?’ (No.) ‘Think inside your head: What is the same about these number bonds? What is different?’ … (One has fish and one has circles. The fish are green and yellow. The circles are just blue. The circles are in a line, but the fish are swimming all around.) ‘What is the same about these number bonds?’ (They both have the total on the top. There are 5 in the total and 3 and 2 in the parts. Display the picture of three number bonds.) ‘The fish and the circle number bonds have the same total and parts. Look at this number bond. (Point to the number bond with numerals.) Is it the same as the other number bonds?’ Invite students’ observations about how the new number bond compares to the other two.”

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

• Module 1, Topic D, Lesson 16: Decompose a set shown in a picture, Learn, Dog Picture, students build experience with MP5 as described in the Teacher Note, Promoting the Standards of Mathematical Practice, “Students gain familiarity with using appropriate tools strategically (MP5) when they use their fingers to sort and count the dogs in the picture and again when they use drawings to model the crayons story problem.” Teacher directions state, “Continue to display the dog picture from Launch. ‘How many dogs are in this picture? Show me with your fingers.’ (Shows 4 fingers) ‘Let’s sort the 4 dogs into two groups. How many dogs are wearing collars? Show me with your fingers.’ (Shows 3 fingers) ‘Keep those fingers up. (Wave open hand.) Show me on your other hand: How many dogs are not wearing a collar?’ (Shows 1 finger) Raise 3 fingers on one hand and 1 finger on the other. ‘Show me how many dogs are wearing a collar.’ (Waves a hand showing 3 fingers) ‘Show me how many dogs are not wearing a collar.’ (Waves a hand showing 1 finger) ‘Put them together.’ (Moves the hands together) ‘How many dogs are your fingers showing now?’ (All 4 dogs) ‘Finish my number sentence: 4 is …’ (Move the hands showing 3 and 1 apart.) ‘What is another way we could sort the dogs in this picture?’ (We can sort by color. I see some dogs with spots and some without spots.) Use a student idea to sort the dogs a different way and show both groups on fingers as shown above. Support students to say a number sentence to match their sort (e.g., 4 is ___ and ___) while separating the two hands to model the decomposition.”

• Module 3, Topic C, Lesson 14: Use number to compare sets with like units, Learn, Number Path Comparison, students build experience with MP5 as described in the Teacher Note, Promoting the Standards of Mathematical Practice, “Students use appropriate tools strategically (MP5) when they find a way to use a number path to help them compare two cube sticks. Giving students room to explore how to use this tool allows them to strategize and find a way to use the number path that makes sense to them.” Teacher directions state, “Students compare the number of cubes in two sticks by using the number path. Ask each pair to pick two sticks and put the rest of the number stairs away. ‘It’s time to experiment. How can the number path help you compare the number of cubes in each stick?’ Give students a few uninterrupted minutes to experiment. If a pair is close to losing interest or getting frustrated, invite them to listen and watch a pair that is having success by using the number path. If a pair comes up with a way to use the number path very quickly, ask them to think of another way. Select a few work samples that use the number path in different ways. Gather the class around the samples and facilitate a discussion by asking the following questions for each sample: ‘Which stick has more cubes? Which has fewer cubes? How does the number path help you compare the sticks? How does the number path help you know how many cubes are in each stick?’”

• Module 6, Topic C, Lesson 15: Count by tens by using math tools, Learn, Scavenger Hunt, students build experience with MP5 as described in the Teacher Note, Promoting the Standards for Mathematical Practice, “Students choose appropriate tools strategically (MP5) when they search for a tool that makes it easy to see 10. Specifically instructing students to look for a tool with this property will help them choose tools more strategically when problem solving in the future. The discussion in this segment is designed to highlight MP5.” In this activity students look for an appropriate tool on a scavenger hunt and teacher directions state, “Place assorted math tools that clearly show tens around the room. Depending on the number of tools you place, consider pairing students so that everyone, or each student pair, has something to find. Invite students to participate in a scavenger hunt. Designate which areas of the classroom are part of the hunt and which are not. ‘Let’s look for tools that help us count by tens.’ Consider pretending to search as if by looking through binoculars. Invite students to do the same. ‘You’re looking for a certain kind of math tool: One that makes it easy to see 10. Show me 10 on your hands. (Shows 10 fingers) The tool you find could show exactly 10, or it could show lots of tens. Find one kind of tool and bring it back to the meeting area.’ Be prepared for unexpected finds and accept them as long as students have solid justification. Have students demonstrate to a partner how to use their tool to count by tens. Circulate and select a few students or student pairs to share. ‘Shu’aib, how does the two-hands mat make it easy to see 10?’ (There are two hands and you have 5 fingers on each hand. 5 and 5 makes 10.) ‘How could we use this tool to count by tens?’ (You need a few of them. Then you can count like this: 10, 20, 30, 40. Sets down one mat at a time.) ‘I heard counting to 40. What does 40 tell us about?’ (40 fingers, 40 dots) Invite other students to share their tools. Close this segment by introducing a nonexample. Hold up a dot card in which 10 is not as easily recognizable. ‘Is it easy to see 10 on this tool?’ (No.)”

The materials reviewed for Eureka Math² 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 opportunities to engage with MP6 across the year and it is explicitly identified for teachers within materials in margin notes called, “Promoting the Standards for Mathematical Practice.” According to the Kindergarten Implementation Guide, “‘Promoting the Standards for Mathematical Practice’ highlights places in the lesson where students are engaging in or building experience with the Standards for Mathematical Practice (MPs). Although most lessons offer opportunities for students to engage with more than one Standard for Mathematical Practice, this guidance identifies a focus MP within each lesson. The notes also provide lesson-specific information, ideas, and questions that you can use to deepen students’ engagement with the focus MP.”

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

• Module 2, Topic C, Lesson 12: Construct solid shapes by using a square base, Learn, Count Faces, Edges, and Corners, students build experience with MP6 as described in the Teacher Note, Promoting the Standards for Mathematical Practice, “Students attend to precision (MP6) when they count the different parts of solid shapes. Students display precision when they are careful to count each part once without counting the same part multiple times. Ask the following questions to promote MP6: What do you have to be extra careful about when counting the parts of solid shapes? What counting strategies can help you count the faces, edges, and corners?” Teacher directions state, “‘If you made a cube, stand up with your shape. If you did not make a cube, follow along with your red solid shape.’ (Show a straw and clay cube.) ‘Imagine that the faces of the cube are filled in. Let’s count the faces together.’ (1, 2, 3, 4, 5, 6) ‘How many faces are on a cube?’ (6 faces)  ‘What shape are the faces of a cube?’ (Square) ‘Let’s count the edges. The edges are the straws. Start with the bottom, then the sides, and finally the top.’ Point to each edge. (1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12)  ‘How many edges are on a cube?’ (12 edges)  ‘The corners are the bits of clay that hold the edges together. Let’s count the corners. Start with the bottom.’ Point to each corner. (1, 2, 3, 4, 5, 6, 7, 8) ‘How many corners are on a cube?’ (8 corners) Repeat the process of counting faces, edges, and corners for the pyramid and rectangular prism. As time allows, ask the following questions to help students relate flat shapes with solid shapes.” 

• Module 4, Topic B, Lesson 9: Compose shapes in more than one way, Land, Debrief, students build experience with MP6 as described in the Teacher Note, Promoting the Standards for Mathematical Practice, “Students who explain why the same shape can be made with different parts attend to precision (MP6). In particular, students who notice that one part (the red trapezoid) can be replaced by several smaller parts (3 triangles) are being precise in explaining how they look for structure (MP7). The questions in Land are designed to get students to explain this relationship.” Teacher directions state, “Display the picture of three triangles. ‘Look at some of the different ways we made the same triangle. We changed the parts, but the whole shape didn’t change.’ Invite students to think–pair–share about the following question. ‘Why can different parts be used to make the same whole shape?’ (There are a lot of different parts that fit inside the triangle. The blue diamond and the green triangle can make a trapezoid. 3 green triangles can also make a trapezoid. The trapezoid is bigger than the other shapes, so you don’t need as many. You can use smaller shapes to make bigger shapes.) Display the picture of the fish. Ask students to turn and talk about the parts they see. Display the number bonds. Invite students to find and point to a number bond that matches the way they see the parts. ‘Look at the different ways to make 3. We came up with different parts, but the total didn’t change. Invite students to think–pair–share about the following question. ‘Why can different parts be used to make the same total?’ (You can sort in different ways. There are lots of ways to make a number.)”

• Module 6, Topic C, Lesson 14: Count by tens, Learn, Count by Ones and Tens, students build experience with MP6 as described in the Teacher Note, Promoting the Standards for Mathematical Practice, “As students take inventory of the bracelet strings and the beads, they attend to precision (MP6). Students are precise as they think about which part of the bracelet they’re counting and whether to count by ones or tens when counting that part. Naming what’s being counted as a unit (e.g., 1 string, 2 strings, … and 10 beads, 20 beads, …) draws attention to the precision needed. This precision becomes increasingly important in later grades when students are introduced to different place value and measurement units.” Teacher directions state, “Gather students with their completed bracelets. Ask the class to help inventory the bracelet materials they used so that the materials can be reordered for next year’s class. ‘What do we need to count?’ (We need to count the bracelet strings. We also need to count all the beads.) ‘Let’s count the bracelet strings first. Stand up with your bracelet. Let’s go around the room and count the bracelet strings. After we count your string, sit down.’ Begin the choral count by pointing to a student. Continue around the room until the bracelet strings are counted. (1 string, 2 strings, 3 strings, …) ‘Now let’s count the beads. Can we count the beads the same way we counted the bracelet strings?’ (I think maybe yes. There are more beads. No, because there is only 1 string but there are 10 beads.) ‘Counting by ones would be very slow. Do we know another way to count that could help?’ (Counting by tens—we know how to do that.) ‘Stand up with your bracelet. Let’s count again, but this time we will count by tens. After we count your beads, sit down.’ (Start the choral count with a different student. Expect to support students with the count across 100, and across 200 if needed. Continue around the room until all the beads are counted. 10 beads, 20 beads, 30 beads, …) ‘Turn and talk: Why did we count the bracelet strings and the beads differently?’”

Students attend to the specialized language of mathematics, in connection to grade-level content, as they work with support of the teacher and independently throughout the units. Examples include:

• Module 1, Topic A, Lesson 1: Compare objects based on their attributes, Learn, Exactly the Same or the Same But…, students build experience with MP6 as described in the Teacher Note, Promoting the Standards for Mathematical Practice, “Students attend to precision (MP6) when they use the phrases exactly the same and the same but … to describe two objects. They precisely communicate common attributes among the objects rather than simply describing them as the same or different.” Teacher directions state, “Display the picture of the two ducks. ‘Look at the two ducks. Are they the same? How do you know they are the same?’ (Yes. They are both yellow. They are the same size.) ‘Everything about them is the same. We can say they are exactly the same.’ Show the picture of the apples. ‘Look at the two apples. Are they exactly the same?’ (No.) ‘What is the same about both apples? Turn and talk to your neighbor about the things that are the same about both apples.’ (They are both red. They are both round.) ‘What’s not the same? What’s different about them? Turn and talk to your neighbor about how the apples are different.’ (One is big and one is small.) ‘They are the same because they are both red, round apples, but they are different because one is big and one is small. We can say they have the same name and are the same color and shape, but they are different sizes.’ Repeat with each picture: the glasses of juice, the dice, and the dogs. To support students, summarize their thoughts by using a repetitive sentence structure. Everything about them is the same. They are exactly the same. They are the same because ___ but different because ___. They are the same ___ but different ___.”

• Module 2, Topic A, Lesson 1: Find and describe attributes of flat shapes, Learn, Shape Sort, students build experience with MP6 as described in the Teacher Note, Promoting the Standards of Mathematical Practice, “By correctly using mathematical attributes, such as straight side, corners, and closed to sort and describe shapes, students attend to precision (MP6). Eventually students will understand that these attributes are mathematically relevant because they can be used to categorize shapes, in contrast with nonmathematical attributes such as color. For a shape to be a triangle, it must have 3 sides and 3 corners, but the shape’s color does not affect whether a shape is a triangle.” Teacher directions state, “Have students stand where they have room to move. From the set of flat shapes, show the trapezoid. ‘If you see straight sides, show me with your body. If you don’t see straight sides, make an X with your arms like this.’ (Cross arms to form an X.) Repeat for corners and curves. Show a 3-column chart for sorting shapes. Describe each column. ‘This shape has straight sides and corners but no curves. (Run your finger along the sides and corners.) Where should I put it on the chart?’ (Put it in the place for straight sides. (Points to straight sides.)) Have Puppet hold up the black shape from the set of flat shapes. ‘Puppet says its shape has straight sides and a corner. Watch where Puppet puts the shape.’ Place it incorrectly in the straight sides column. ‘Do you think Puppet put its shape in the right place?’ Support students in sharing and discussing their ideas by providing a sentence frame, such as: ‘I think (we should move the shape or the shape should stay) because ___.’ Recap the discussion as follows and have Puppet move the shape to the both column. ‘We should move this shape to the column that says both because it has straight sides and a curve.’ Distribute one foam shape or shape card to each student. ‘It’s your turn to place a picture in our chart. Look at it. Feel or trace it with your finger. Does it have straight sides? Curves?’ Invite a student to show and tell about their picture. Let the student sort their picture into the chart. ‘If you agree with where the picture is, show thumbs-up. If you disagree, put your hand on your head.’ Give students a moment to decide whether they agree or disagree with the picture’s placement. Facilitate discussion and encourage students to use the words straight sides, curves, and corners as much as possible. Continue until all students have placed their picture in the chart. To keep students engaged, have groups of students come up together.”

• Module 3, Topic A, Lesson 1: Align endpoints to compare lengths by using taller than and shorter than, Launch, students build experience with MP6 as described in the Teacher Note, Promoting the Standards for Mathematical Practice, “As students move from describing objects by using general words like big to using terminology like taller and shorter, they attend to precision (MP6).” Teacher directions state, “Students reason about pictures to notice the measurable attribute of length. Display the picture of tall and short objects. Use the guess my rule routine to focus students on a particular attribute. Circle the dog and the giraffe. (Point.) ‘If you think you know why I circled these pictures, put your hands on your head. What’s my rule?’ (They are animals and the rest of the pictures aren’t. They are both animals! So my rule is things that are animals.)... Erase. Circle the building, tree, and giraffe. (Point.) ‘If you know why I circled these pictures, put your hands on your head. What’s my rule?’ (Your rule is to circle the big pictures.) Encourage students to clarify what they mean by big in this case. (The building goes up and the tree is up but not as much and the giraffe is up. The things you circled are tall.) ‘You are talking about height. The building, tree, and giraffe are tall.’ (Raise one hand high and the other low.) ‘The dog, flower, and mailbox are short.’ (Bring hands closer together.) ‘In these pictures, it’s easy to see that the flower is shorter than the tree. Sometimes it’s not so easy to tell.’ Transition to the next segment by framing the work. ‘Today, we will learn how to tell if something is taller or shorter than something else.’”

The materials reviewed for Eureka Math² 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 the grade-level content standards, as expected by the mathematical practice standards. 

Students have opportunities to engage with MP7 and MP8 across the year and they are explicitly identified for teachers within materials in margin notes called, “Promoting the Standards for Mathematical Practice.” According to the Kindergarten Implementation Guide, “‘Promoting the Standards for Mathematical Practice’ highlights places in the lesson where students are engaging in or building experience with the Standards for Mathematical Practice (MPs). Although most lessons offer opportunities for students to engage with more than one Standard for Mathematical Practice, this guidance identifies a focus MP within each lesson. The notes also provide lesson-specific information, ideas, and questions that you can use to deepen students’ engagement with the focus MP. Often, the suggested questions for a particular MP repeat. This intentional repetition supports students in understanding the MPs in different contexts.”

MP7 is identified and connected to grade-level content, and there is intentional development of the MP to meet its full intent. Students look for and make use of structure as they work with support of the teacher and independently throughout the modules. Examples include:

• Module 1, Topic E, Lesson 20: Count objects in 5-group and array configurations and match to a numeral, Learn, Relate Counting the Math Way to 5-Groups, students build experience with MP7 as described in the Teacher Note, Promoting the Standards for Mathematical Practice, “As students look for five to help them count, they look for and make use of structure (MP7). For example, they see five on their hands, in 5-groups, and in other arrangements around the classroom. While it may be easier to let students count however they like, encourage them to look for and make use of five. Planting this seed now will help ensure that students have a proper foundation in place when they are expected to count on in grade 1.” Teacher directions state, “Display 6 cubes on a mat, with 5 on top and 1 on bottom. ’Take your cubes off your fingers and make them look like this.’ Show Hide Zero 6 card. ‘With just fingers, show me 6 the math way. Look at your fingers. Look at your cubes. Fingers, cubes, fingers, cubes!’ Repeat playfully a few times, and then take the Hide Zero 6 card out of view. ‘Where’s the five on your hand? Hold it up high.’ (Raises left hands, showing all 5 fingers) ‘Where’s the five on your mat? Circle it with your finger.’ (Circles the top row) (Show Hide Zero 6 card.) ‘Show me 6 the math way. Where do you see 1 on your hands? Hold it up high.’ (Raises right hands, showing the thumb) ‘Where do you see 1 on your mat? Circle it with your finger.’ (Circles the single cube on the bottom row) ’Look at your cubes. Put your hand up when you know how many are on your mat.’ Allow time to count, and then signal for a choral answer. (Point to the top row.) ‘We showed 6 as 5 and 1 more. (Point to the bottom row.) We call that a 5-group. A what?’ (5-group)”

• Module 4, Topic C, Lesson 16: Compose and decompose numbers and shapes, Land, Debrief, students build experience with MP7 as described in the Teacher Note, Promoting the Standards for Mathematical Practice, “Students look for and make use of structure (MP7) when they connect the different ways parts and wholes/totals were seen in the different stations. Seeing the relationship between the parts/whole in a shape puzzle and the parts/total in a number bond can help students gain a deeper understanding of the part-total relationship going forward. The questions in Land are designed to promote MP7.” Teacher directions state, “‘Where did you see parts and totals, or wholes, today?’ (We saw parts in the pictures. We wrote a number bond with the parts and total. There were tall bears and short bears that were parts. If you put the tall and short bears together, you get the total. The little shapes were part of the whole shape puzzle.) Invite students to think–pair–share about the following questions. ‘What have you learned about parts and total for numbers?’ (I learned that the parts go together to make the total. I learned how to write a number bond to show parts and total. I know the partners to 5. ‘What have you learned about parts and whole for shapes?’ (Sometimes little shapes are hiding inside big shapes. The little shapes are the parts that make up the whole puzzle. ‘What math tools can you use to show your thinking about parts and totals, or wholes?’ (Shapes; Number bonds; My brain)”

• Module 5, Topic D, Lesson 22: Identify and extend linear patterns, Launch, students build experience with MP7 as described in the Teacher Note, Promoting the Standards for Mathematical Practice, “This lesson supports the Standard for Mathematical Practice (MP7), look for and make use of structure. This lesson focuses on discerning the structure of a pattern, which is foundational to recognizing number patterns.” In this activity students isolate attributes to describe and extend a pattern. Teacher directions state, “Display the line of bears. Invite students to share what they notice. Use student language to focus attention on size. ‘When I point to a bear, say whether it’s big or small. Ready?’ (Big, small, big, small, …) Incorporate movement such as swaying or head bobbing to attach a musical quality to the pattern. Have students continue to state the pattern and continue movement to communicate the idea that patterns can be extended. ‘We knew which words to say even after the line of bears stopped because the bears make a pattern. There’s something that we keep saying, something that repeats. What is it? (Big, small) ‘We keep repeating big, small. The part of that pattern that repeats is called the pattern unit. Let’s find the pattern unit.’ Ask students to state the pattern again. Have them pause as you circle the pattern unit each time they say it. Ask students to extend the pattern by having them tell what comes next. Erase the circles to reset. ‘There’s a different pattern in the same bears. Look again. Show thumbs-up when you find it.’ If necessary, encourage students to think about color. Identify the pattern unit and verbally extend the pattern as before.”

MP8 is identified and connected to grade-level content, and there is intentional development of the MP to meet its full intent. Students look for and express regularity in repeated reasoning as they work with support of the teacher and independently throughout the modules. Examples include:

• Module 1, Topic G, Lesson 31: Model the pattern of 1 less in the backward count sequence, Land, Debrief, students build experience with MP8 as described in the Teacher Note, Promoting the Standards for Mathematical Practice, “Students look for and express regularity in repeated reasoning (MP8) when they notice that when a group has 1 less item, they always get the counting number that comes just before. Expressing this verbally helps students think about the pattern as a rule they can trust and use going forward.” Teacher directions state, “‘What happened to the group of apples as Farmer Brown plucked them from the tree?’ (The group got smaller.) ‘Who can talk about it using the words 1 less?’ (When Farmer Brown plucked an apple, there was 1 less apple on the tree. We took off 1 cube each time, so that was 1 less.) ‘What happened to the number stairs as we counted backward from 10 to 0?’ (The stairs got smaller and smaller.) ‘Who can talk about it using the words 1 less?’ (We took 1 cube off each time, so there was 1 less cube. 1 less is the next number when counting back.) ‘What happens to the numbers as we count back from 10 to 0?’ (They get smaller.) ‘Who can talk about it using the words 1 less?’ (Each number is 1 less.)”

• Module 5, Topic B, Lesson 8: Understand taking away as a type of subtraction, Launch, students build understanding of MP8 as described in the Teacher Note, Promoting the Standards for Mathematical Practice, “Students look for and express regularity in repeated reasoning (MP8) when they recognize that take away stories such as this one, where the amount taken away is greater than zero, result in the number of objects getting smaller.” In this activity students are shown two pictures in which the number of apples on a tree are different. “‘One thing is different in these pictures. Can you find it?’ (One tree has more apples than the other.) ‘1 apple is missing from the tree in this picture.’ Point to the picture on the right. ‘What do you think happened to the apple?’ (Maybe the boy put it in his backpack. The squirrels ate it.) ‘How many apples are on the tree in this picture?’ Point to the picture on the left. (6) ‘Are there more apples or fewer apples in this picture?’ Point to the picture on the right. (There are less—fewer apples.) Display the picture of the boy holding 1 apple. ‘There are fewer apples because the boy took 1 of the apples. 6 take away 1 is 5. When something gets taken away and we figure out how many are left, we call it subtraction.’”

• Module 6, Topic A, Lesson 3: Write Numerals 11-20, Learn, Inventory Demonstration, students build experience with MP8 as described in the Teacher Note, Promoting the Standards for Mathematical Practice, “As students use Hide Zero cards to help them write the numbers 11-19, they look for and express regularity in repeated reasoning (MP8). After counting and representing several collections of objects, they come to understand that the 1 in the numbers 11-19 represents a group of 10. This lays the foundation for place value understanding that students will come to rely on in grade 1 and beyond. Here the group of 10 is always thought of as being made up of 10 distinct objects. In grade 1, students will learn to unitize this as a ten.” Teacher directions state, “Introduce the concept of taking inventory or making a complete list of items in a particular place. Invite the class to consider why an inventory of classroom materials might be helpful. Hold up a set of books or other materials to count. Invite students to count with you as you place them into a group of 10 ones and 3 ones. ‘How many books?’ (13) ‘How many are in this group? (Point to the stack of 10 books.)’ (10) Place the 10 Hide Zero card next to the stack of 10 books. Repeat for the stack of 3 books. ‘How do we say the total number of books the Say Ten way?’ (Ten 3) Move the 10 and 3 cards together to show 13. 10 and 3 make ten 3, or 13. What happened to the 0 of the 10?’ (The 0 is hiding under the 3. It is covered by the 3.) ‘Turn and ask your partner: Does the 1 in 13 tell us about 1 book or 10 books? How do you know?’ (1 tells about 10 books. You can see the 10 books. The 1 comes from the 10. We just covered the 0 with the 3 card.) ‘Why do you think these cards are called Hide Zero cards? (Hold up the Hide Zero cards.)’ (I think they are called Hide Zero cards because you put the 3 over the 0 in 10. We’ve been making big numbers by putting the smaller number on top of the 0. It hides the 0.) Demonstrate how to write 13 by using the Classroom Inventory page. Use pictures or words to show what was counted.”