K-2nd Grade - Gateway 2

The materials reviewed for Illustrative Mathematics® v.360 Kindergarten through Grade 2 meet expectations for supporting the intentional development of students’ conceptual understanding of key mathematical concepts, especially where called for in specific content standards or clusters.

Multiple conceptual understanding problems are embedded throughout the grade level within Warm-ups, Activities, or Cool-downs. Students have opportunities to engage with these problems both independently and with teacher support. According to the Illustrative Mathematics® v.360 Course Guide, the Key Structures in This Course, Principles of IM Curriculum Design section emphasizes the role of purposeful representations in developing conceptual understanding. The Purposeful Representations section states, “Across lessons and units, students are systematically introduced to representations and encouraged to use those that make sense to them. As their learning progresses, students make connections between different representations and the concepts and procedures they show.” 

An example in Kindergarten includes:

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

An example in Grade 1 includes:

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

An example in Grade 2 includes:

• Unit 8, Equal Groups, Lesson 5, Warm-up, students develop conceptual understanding by using grouping strategies to describe and determine if the groups of dots have an even or odd number of members. Student Task Statements show images of 12 dots, 13 dots, and 14 dots, “How many do you see? How do you see them?” Activity Synthesis states, “Which images show a group with an even number of dots? (Image 1 and Image 3) How can you tell using the equations we recorded?” (2.OA.C)

According to Illustrative Mathematics® v.360 Course Guide, the materials are designed to provide students with opportunities to independently demonstrate conceptual understanding, when appropriate. 7. Key Structures in This Course, Principles of IM Curriculum Design, Coherent Progression states, “Each activity starts with a Launch that gives all students access to the task. Independent work time follows, allowing them to grapple with problems individually before working in small groups. In the Activity Synthesis at the end, students consolidate their learning by making connections between their work and the mathematical goals.” 

An example in Kindergarten includes:

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

An example in Grade 1 includes:

• Unit 5, Adding Within 100, Lesson 12, Activity 2, students demonstrate conceptual understanding as they use what they know about the base-ten structure of numbers to create different expressions. Students are provided counting cubes, and the Student Task Statements state, “37, 22, 18, 56, 41. Write an addition expression with 2 numbers to make each statement true. Use only the numbers above. 1. This sum has the smallest possible value. Expression: ____, 2. This sum has the largest possible value. Expression: ____, 3. You do not need to make a new ten to find the value of this sum. Expression: ____, 4. If you make a new ten to find the value of this sum, you will still have some more ones. Expression: ____, 5. If you make a new ten to find the value of this sum, you will have no more ones. Expression: ____. If you have time: Choose 2 numbers from above and write an addition expression where the value is closest to 95. How do you know the value is closest to 95?” Activity Synthesis states, “Are there other numbers you could use? How do you know?” (1.NBT.C)

An example in Grade 2 includes:

• Unit 5, Numbers to 1000, Lesson 9, Activity 2, students demonstrate conceptual understanding as they use place value to compare numbers based on different representations. Activity states, “‘In the last activity, we saw a way to use the number line to explain that 371 is greater than 317. In this activity, you will compare three-digit numbers and explain your thinking using the number line.’ 6 minutes: independent work time. ‘Compare your answers with a partner and use the number line to explain your reasoning.’ 4 minutes: partner discussion.” Student Task Statements states, “1. Locate and label 420 and 590. Use <, >, and = to compare 420 and 590. 2. Estimate the location of 378 and 387. Mark each number with a point. Label the point with the number it represents. Use <, >, and = to compare 378 and 387. 3. Diego and Jada compared 2 numbers. Use their work to figure out what numbers they compared. Then use <, >, and = to compare the numbers. 4. Which representation helped you the most? Explain your reasoning.” Number lines are included for the numbers 1, 2, and 3, along with base ten representations for the number 3. (2.MD.6, 2.NBT.1, 2.NBT.4)

The materials reviewed for Illustrative Mathematics® v.360 Kindergarten through Grade 2 meet expectations for providing intentional opportunities for students to develop procedural skills and fluencies, especially where called for in specific content standards or clusters.

According to the Course Guide, 7. Key Structures of This Course, Principles of IM Curriculum Design, Conceptual Understanding and Procedural Fluency, “Warm-up routines, practice problems, centers, and other built-in activities help students develop procedural fluency, which develops over time.” 

An example in Kindergarten includes: 

• Unit 6, Numbers 0-20, Lesson 3, Activity 3, students develop fluency with addition and subtraction within 5 as they find the number that makes 5 when added to a given number. Launch states, “Groups of 2. Give each student a set of cards, a recording sheet, and access to two-color counters, 5-frames, and 10-frames. ‘We’re going to learn a center called Find the Pair. Put your cards in a pile in the middle of the table. You and your partner will both draw 5 cards. Keep your cards hidden from your partner.’ Demonstrate drawing 5 cards. Invite a student to act as the partner and draw 5 cards. ‘I am going to look at my cards. I need to choose 1 card and figure out which number I need to make 5 with the card.’ Display a card with the number 4. ‘My card says 4. What card do I need to go with it to make 5? (1) I need a 1 card. I’m going to ask my partner if they have a 1 card. If my partner has a 1 card, they will give it to me. I will put the 4 card and 1 card down as a match and write an expression. If I have a 4 card and a 1 card, what expression should I write?’ ($$4+1$$ or 1+4).” (K.OA.5)

An example in Grade 1 includes: 

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

An example in Grade 2 includes: 

• Unit 9, Putting It All Together, Lesson 4, Activity 1, students develop fluency in working with data as they add and subtract to answer questions about the data in the table. Launch, “Groups of 3–4. Give each student an unsharpened pencil and a centimeter ruler. ‘Without measuring it, estimate the length of a brand new pencil.’ 30 seconds: quiet think time. Share responses. ‘Measure the pencil to the nearest centimeter.’ (18 cm) 1 minute: group work time. Share responses.” Activity, “Display the table. ‘The table shows the length of pencils from 4 different student groups.’ ‘Find the length of your own pencil and share it with your group. Record your group’s measurements in the table.’ 4 minutes: group work time. ‘Use the table to find the total length of each group’s pencils.’” Student Task Statement, “1. Measure the length of your pencil. ___ cm. 2. Write the lengths of your group’s pencils in the last row of the table. 3. Find the total length of each group’s pencils.” (2.MD.1, 2.NBT.5, 2.OA.2)

According to the Course Guide, 7. Key Structures of This Course, Principles of IM Curriculum Design, Coherent Progression, materials were designed to include opportunities for students to independently demonstrate procedural skill and fluency, when appropriate. “Each activity starts with Launch that gives all students access to the task. Independent work time follows, allowing them to grapple with problems individually before working in small groups. In the Activity Synthesis at the end, students consolidate their learning by making connections between their work and the mathematical goals.” 

An example in Kindergarten includes:

• Unit 7, Solid Shapes All Around Us, Lesson 6, Activity 3, students demonstrate procedural skill and fluency as they use addition and subtraction within 5. Launch states, “Groups of 2. Give each group of students 1 cup, 5 two-color counters, and 2 copies of the blackline master. ‘We are going to learn a new way to do the Shake and Spill center. It is called Shake and Spill, Cover. Let’s play a round together.’ ‘I am going to put 3 counters in the cup and shake them up. You will close your eyes so I can spill the counters and cover all the yellow ones with the cup. Then you will open your eyes and figure out how many counters are under the cup.’ Put 3 counters in a cup and shake them up. ‘Close your eyes.’ Spill the counters and cover 1 yellow counter. Leave 2 red counters on the table. ‘Open your eyes. Look at the counters on the table. How many counters are under the cup? How do you know?’ (1 because there are 2 on the table and 2 and 1 more makes 3.) 30 seconds: partner discussion. Share responses. Pick up the cup showing the 1 counter that was covered. ‘Now we fill in the recording sheet. We had 3 counters total. Then we fill in the expression that matches the parts we broke 3 into. There were 2 counters outside the cup and 1 counter in the cup.’ Demonstrate completing the recording sheet. ‘Take turns with your partner spilling counters and covering the yellow ones. On each turn you can decide to use 3, 4, or 5 counters. Make sure you and your partner agree on how many total counters to use before you shake, spill, and cover.’”(K.OA.5)

An example in Grade 1 includes:

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

An example in Grade 2 includes:

• Unit 2, Adding and Subtracting within 100, Lesson 10, Activity 1, students demonstrate fluency as they practice adding and subtracting within 100. Launch states, “Groups of 2. Give each student a copy of the recording sheet. Give each group 3 number cubes and access to base-ten blocks. We are going to learn a new way to play Target Numbers. You and your partner will start with 100 and see who can get closer to 0 in four rounds of subtracting.’ Instead of using cards to decide whether to take away tens or ones, you will use number cubes to create a two-digit number and then subtract that number.’ ‘First, represent 100 with base-ten blocks.’ As needed, invite students to count by 10 to 100 using the base-ten blocks or invite students to share how they might represent 100 with the blocks. ‘When it’s your turn, roll all 3 number cubes. Pick 1 number to represent the tens and one number to represent the ones. Then show the subtraction with your blocks and write an equation on your recording sheet.’ ‘Take turns rolling and subtracting for 4 rounds. The player who ends closer to 0, without going below 0, wins the game.’ As needed, demonstrate a round with a student volunteer.” (2.NBT.5)

The materials reviewed for Illustrative Mathematics® v.360 Kindergarten through Grade 2 meet expectations for supporting the intentional development of students’ ability to utilize mathematical concepts and skills in engaging applications, especially where called for in specific content standards or clusters. 

Multiple routine and non-routine applications of mathematics are included throughout the grade level, with single- and multi-step application problems embedded within Activities or Cool-downs. Students have opportunities to engage with these applications both with teacher support and independently. According to the Course Guide, materials are designed to provide students with opportunities to independently demonstrate their understanding of grade-level mathematics when appropriate. 7. Key Structures in This Course, Principles of IM Curriculum Design, Coherent Progression states, “Each activity starts with a Launch that gives all students access to the task. Independent work time follows, allowing them to grapple with problems individually before working in small groups. In the Activity Synthesis at the end, students consolidate their learning by making connections between their work and the mathematical goals.” 

An example of a routine application of the math in Kindergarten includes:

• Unit 4, Understanding Addition and Subtraction, Lesson 11, Activity 1, students draw a picture to represent and solve a story problem. Student Task Statement, “7 kids play soccer in the park. 3 kids leave to go play on the swings. How many now?” There is an image of four kids playing soccer. Activity states, “Monitor for students who draw pictures with details to represent the story. Monitor for students who use symbols, such as circles.” (K.OA.2)

An example of a non-routine application of the math in Kindergarten includes:

• Unit 7, Story Problems about Shapes, Lesson 5, Activity 2, students connect the action in the story to the meaning of the addition and subtraction signs. Activity states, “Reread the first story problem. ‘Show your thinking using objects, drawings, numbers, or words.’ 2 minutes: independent work time. 2 minutes: partner discussion. Display 8=____+____‘Andre began writing this equation but didn’t finish it. Finish her equation to show what happened in the story problem.’ 2 minutes: independent work time. Repeat the steps with the second story problem. Display 9-3=___ for students to complete the equation.” Student Task Statement,”1. Andre has 4 pattern blocks. He adds 4 more. How many pattern blocks? ___ equation: 8=___+___ 2. Elena makes a train with 9 pattern blocks. She takes away 3 pattern blocks. How many pattern blocks make up the train? ___ equation 9-3=___.” (K.OA.1, K.OA.2)

An example of a routine application of the math in Grade 1 includes:

• Unit 2, Addition and Subtraction Story Problems, Lesson 2, Activity 2, students solve and write equations for Result Unknown word problems. Student Task Statement, “1. 9 books are on a cart. The librarian takes 2 of the books. How many books are still on the cart? Show your thinking using drawings, numbers, or words. Equation: ___ 2. 7 children work on an art project. 2 children join them. How many children work on the art project now? Show your thinking using drawings, numbers, or words. Equation:___ ” (1.OA.1)

An example of a non-routine application of the math in Grade 1 includes:

• Unit 3, Adding and Subtracting Within 20, Lesson 26, Activity 2, students generate, articulate, and solve their own addition and subtraction problems. Activity states, “Arrange groups together so each larger group has students who have solved each of the four problems. ‘Share your poster with your group. Explain how the equations you wrote match the story. As each group shares, discuss how the problems are the same and different. Make a list of equations you used for each problem.’” Student Task Statement, “Story Problem 1, Han has some pencils. He gets 9 more. Now he has 15 pencils. How many pencils did Han have to start? Story Problem 2, Han has 15 pencils. He gives some to friends. Now he has 9 pencils. How many pencils did Han give to his friends?” (1.OA.1, 1.OA.4, 1.OA.6)

An example of a routine application of the math in Grade 2 includes:

• Unit 4, Addition and Subtraction on the Number Line, Lesson 13, Cool-down, students solve word problems involving addition or subtraction. Student Task Statement, “Clare made a train that was 15 cubes long. Then she added some more cubes. Now her train is 28 cubes long. How many cubes did she add to her train? Show your thinking. Use a number line or diagram if it helps.” A number line with 0 to 50 labeled is included. (2.MD.5, 2.OA.1)

An example of a non-routine application of the math in Grade 2 includes:

• Unit 2, Adding and Subtracting within 100, Lesson 13, Activity 2, students use tape diagrams and equations to represent addition and subtraction story problems within 100. Activity states, “‘Now you get a chance to draw diagrams and write equations that represent story problems. Read the story carefully. Then solve each problem and show your thinking.’ 8 minutes: independent work time. 5 minutes: partner discussion. For the second problem, monitor for students who: Use an addition equation to represent Andre’s seeds. Subtract to find the number of seeds Andre won using a base-ten diagram or equations.” Student Task Statement, “1. Lin plays a game with seeds. She starts with some seeds. Then she wins 36 more. Now she has 64 seeds. How many seeds did Lin have at first? a. Write an equation. Use a ? for the unknown value. b. Solve the equation. Show your thinking using drawings, numbers, or words. Andre started a game with 32 seeds. Then he wins more. Now he has 57 seeds. How many seeds did Andre win? a. Label the diagram to represent the story. b. Write an equation. Use a ? for the unknown value. c. Solve the equation. Show your thinking using drawings, numbers, or words. Diego gathers 22 seeds from yellow flowers. He gathers 48 seeds from blue flowers. How many seeds does he gather in all? a. Label the diagram to represent the story. b. Write an equation. Use a ? for the unknown value. c. Solve the equation. Show your thinking using drawings, numbers, or words. Noah and Kiran gather 92 pumpkin seeds. Noah gathers 53. How many seeds did Kiran gather? a. Draw a diagram to represent the story. b. Write an equation. Use a ? for the unknown value. c. Solve the equation. Show your thinking using drawings, numbers, or words.” Tape diagrams are included for problems 1, 2 and 3. (2.NBT.5, 2.OA.1)

The materials reviewed for Illustrative Mathematics® v.360 Kindergarten through Grade 2 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 as reflected by the standards.

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

An example in Kindergarten includes:

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

An example in Grade 1 includes:

• Unit 1, Adding, Subtracting, and Working with Data, Lesson 8, Activity 2, students develop conceptual understanding alongside procedural fluency as they sort shapes into categories and explain their strategies for sorting. Launch states, “Give students access to colored pencils or crayons and copies of the 3-column sorting mat.” Student Task Statement, “1. Show how you sorted the shape cards. Be sure that someone else who looks at your paper can see how many shapes are in each category. 2. Complete the sentences: a. The first category has ___ shapes. b. The second category has ___ shapes. c. The third category has ___ shapes. d. There are ___ shapes all together.” (1.MD.4)

An example  in Grade 2 includes:

• Unit 8, Equal Groups, Lesson 1, Activity 2, students engage in all three aspects of rigor, conceptual understanding, procedural fluency, and application as they find ways to solve routine real-world problems. Launch states, “Andre has a collection of 17 marbles. He wants to play a game with his sister. To play, they both need to start with the same number of marbles, and they want to use as many as they can. Use the counters, diagrams, symbols, or other representations to show how they could start the game.’ 2 minutes: independent work time. Monitor for different ways students group the counters or objects or draw the groupings in the diagrams they create.” Student Task Statement, “Andre has 17 marbles. He wants to play a game with his sister. They each need to start with the same number of marbles. They want to use as many as they can. 1. How many marbles do Andre and his sister each get? Do they use all the marbles? Show your thinking using drawings, numbers, or words. 2. What if Andre had 18 marbles? How many would each player get? Would they use all of the marbles? Show your thinking using drawings, numbers, or words. 3. What if Andre had 20 marbles? How many would each player get? Would there be any marbles left over?” (2.OA.C)

The materials reviewed for Illustrative Mathematics® v.360 Kindergarten through Grade 2 meet expectations for supporting the intentional development of MP1: Make sense of problems and persevere in solving them, for students, in connection to the grade-level content standards, as expected by the mathematical practice standards. 

Students have opportunities to engage with MP1 across the year, and it is often explicitly identified for teachers within the Course Guide (Standards for Mathematical Practice) and within specific lessons (Preparation Narratives and Lesson Activities’ Narratives). According to the Course Guide, Standards for Mathematical Practice, “The Standards for Mathematical Practice (MP) describe the types of thinking and behaviors in which students engage as they do mathematics. Throughout the curriculum, the Teacher Guide identifies lessons and activities in which to observe the different MPs. The unit-level Mathematical Practice chart is meant to highlight a handful of lessons in each unit that showcase certain MPs. Some units, due to their size or the nature of their content, have fewer predicted chances for students to engage in a particular MP, indicated in the chart by a dash.” 

An example in Kindergarten includes:

• Unit 2, Numbers 1-10, Lesson 4, Activity 1, students identify a group of objects that has more. Student Task Statement, “There are 4 people. There are 6 spoons. Can each person get 1 spoon?” Launch states, “Groups of 2. Give each group of students access to connecting cubes and two-color counters. ‘We have been learning about different tools that we use at home and in our classroom. What kind of tools do you use when you eat at home?’ (spoons, forks, chopsticks, plates, bowls, napkins, cups, straws). 30 seconds: quiet think time. 1 minute: partner discussion. Share and record responses. ‘We use many different tools when we eat.’ Display and read the story. More context can be added to the story so there is more to act out. ‘What is the story about?’ (a meal, setting a table, spoons for a meal). 30 seconds: quiet think time. Share responses. Read the story again. ‘How can you act out this story?’ (We can pretend we are sitting at the table and pretend to hand out spoons. We can use the cubes to show the people and the counters to show the spoons. We can draw a picture). 30 seconds: quiet think time. 1 minute: partner discussion. Share responses.” Activity Narrative, “‘Act out the story with your partner.’ 3 minutes: partner work time. ‘Are there more people or spoons? How do you know?” (There are more spoons than people. Each person gets 1 spoon and then there are some more spoons.)’ 2 minutes: partner work time. Monitor for students who matched 1 spoon to each person to see if there were enough spoons and which there was more of.” Activity Narrative, “The context of family mealtimes that is introduced in this activity will be revisited throughout the unit. Acting it out gives students an opportunity to make sense of a context (MP1).”

An example in Grade 1 includes:

• Unit 5, Adding within 100, Lesson 8, Activity 3, students add within 100, using place value and properties of operations in their reasoning. Narrative, “When students create representations and expressions for the context, they develop ways to model the mathematics of a situation and strategies for making sense of and persevering to solve problems (MP1, MP4).” Student Task Statement, “1. Priya watches a football game. The home team scores 35 points. Then they score 6 more points. How many points do they score all together? Show your thinking using drawings, numbers, or words. 2. At the football game, 9 fans cheer for the visiting team. There were 45 fans who cheer for the home team. How many fans are at the game all together? Show your thinking using drawings, numbers, or words.”

An example in Grade 2 includes:

• Unit 6, Geometry, Time, and Money, Lesson 18, Cool-down, students make sense of problems that require them to add or subtract money. Student Task Statement, “Mai has these coins to buy school supplies: 3 nickels, 1 dime, and 2 quarters are shown. If Mai buys a pencil for 27¢, how much money will she have left? Show your thinking using drawings, numbers, words, or an equation. If it helps, you can use a diagram.”

The materials reviewed for Illustrative Mathematics® v.360 Kindergarten through Grade 2 meet expectations for supporting the intentional development of 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 MP2 across the year, and it is often explicitly identified for teachers within the Course Guide (Standards for Mathematical Practice) and within specific lessons (Preparation Narratives and Lesson Activities’ Narratives). According to the Course Guide, Standards for Mathematical Practice, “The Standards for Mathematical Practice (MP) describe the types of thinking and behaviors in which students engage as they do mathematics. Throughout the curriculum, the Teacher Guide identifies lessons and activities in which to observe the different MPs.The unit-level Mathematical Practice chart is meant to highlight a handful of lessons in each unit that showcase certain MPs. Some units, due to their size or the nature of their content, have fewer predicted chances for students to engage in a particular MP, indicated in the chart by a dash.” 

An example in Kindergarten includes:

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

An example in Grade 1 includes:

• Unit 2, Addition and Subtraction Story Problems, Lesson 6, Activity 2, students consider two different equations that represent the same story problem. Student Task Statement: “Tyler and Clare want to know how many pets they have together. Tyler has 2 turtles. Clare has 4 dogs. Tyler writes 2+4. Clare writes 4+2. Who do you agree with? Show your thinking using objects, drawings, numbers, or words.” Launch states, “Groups of 2. Give students access to 10-frames and connecting cubes or two-color counters. ‘We just solved a problem about pet fish. What else do you know about pets?’ 30 seconds: quiet think time. 1 minute: partner discussion. If needed ask, ‘What other kinds of pets are there?’” Activity Narrative, “Students contextualize the problem and see that each number represents a specific object’s quantity, no matter which order it is presented, and connect these quantities to written symbols (MP2).”

An example in Grade 2 includes:

• Unit 1, Adding, Subtracting, and Working with Data, Lesson 14, Warm-up, students reason that tape diagrams as similar to bar graphs and can be used to represent the same data. Launch, “Display the image. ‘What do you notice? What do you wonder?’” Activity Narrative, “When students make connections between the different ways the representations represent the same categories and quantities, they reason abstractly and quantitatively and look for and make use of structure (MP2, MP7).”

The materials reviewed for Illustrative Mathematics® v.360 Kindergarten through Grade 2 meet expectations for supporting the intentional development of MP3: Construct viable arguments and critique the reasoning of others, for students, in connection to the grade-level content standards, as expected by the mathematical practice standards. 

Students have opportunities to engage with MP3 across the year, and it is often explicitly identified for teachers within the Course Guide (Standards for Mathematical Practice) and within specific lessons (Preparation Narratives and Lesson Activities’ Narratives). According to the Course Guide, Standards for Mathematical Practice, “The Standards for Mathematical Practice (MP) describe the types of thinking and behaviors in which students engage as they do mathematics. Throughout the curriculum, the Teacher Guide identifies lessons and activities in which to observe the different MPs.  The unit-level Mathematical Practice chart is meant to highlight a handful of lessons in each unit that showcase certain MPs. Some units, due to their size or the nature of their content, have fewer predicted chances for students to engage in a particular MP, indicated in the chart by a dash.” 

An example in Kindergarten includes: 

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

An example in Grade 1 includes: 

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

An example in Grade 2 includes: 

• Unit 7, Adding and Subtracting within 1000, Lesson 16, Activity 1, students construct viable arguments and critique the reasoning of others as they interpret and connect different representations for subtraction methods. Launch. “Give students access to base-ten blocks. Display Lin’s diagram. ‘Take a minute to make sense of Lin’s subtraction.’ 1–2 minutes: quiet think time. ‘Discuss Lin’s work with your partner.’ 1–2 minutes: partner discussion. Share and record responses. Highlight that a ten was decomposed and discuss students’ ideas about the numbers in the subtraction.” Activity, “‘Jada and Lin both found the value of 582-145. Work with your partner to compare Lin and Jada's work. Then complete Jada's work to find the value of 582-145.’ 3–5 minutes: partner work time. ‘Jada found the value of 402-298 with a different method. Work with your partner to make sense of Jada's thinking. Discuss if you agree or disagree with Jada’s reason for why she chose this method.’” Student Task Statement: “1. Discuss how Jada’s equations match Lin’s diagram. Finish Jada’s work to find the value of 582-145 2. Jada is thinking about how to find the value of 402-298 a. Jada knows a way to count on to find the difference. She shows her thinking, using a number line. Explain Jada’s thinking. b. Jada says you can’t decompose to find the value of 402-298, because there aren’t any tens. Do you agree? Show your thinking, using objects, drawings, numbers, or words.” Lesson Narrative, “In this lesson, students attend to the relationships between numbers in expressions to flexibly subtract. Although the focus of this section has been on interpreting and using methods based on place value, the number choices in this lesson also are intended to encourage the strategies students used in prior sections. Throughout this lesson, students explain their thinking and listen to and critique the reasoning of others (MP3).”

The materials reviewed for Illustrative Mathematics® v.360 Kindergarten through Grade 2 meet expectations for supporting the intentional development of MP4: Model with mathematics, for students, in connection to the grade-level content standards, as expected by the mathematical practice standards.

Students have opportunities to engage with MP4 across the year, and it is often explicitly identified for teachers within the Course Guide (Standards for Mathematical Practice) and within specific lessons (Preparation Narratives and Lesson Activities’ Narratives). According to the Course Guide, Standards for Mathematical Practice, “The Standards for Mathematical Practice (MP) describe the types of thinking and behaviors in which students engage as they do mathematics. Throughout the curriculum, the Teacher Guide identifies lessons and activities in which to observe the different MPs. The unit-level Mathematical Practice chart is meant to highlight lessons in each unit that showcase certain MPs. Some units, due to their size or the nature of their content, have fewer predicted chances for students to engage in a particular MP, indicated in the chart by a dash.” 

An example from Kindergarten includes: 

• Unit 7, Solid Shapes All Around Us, Lesson 3, Section A Checkpoint, students ask and answer mathematical questions about shapes composed of pattern blocks. The teacher observes to capture evidence of student thinking on the checkpoint checklist. Observation, Look Fors, “Count all to find the total. Use objects, drawings, or equations to represent a story problem.” Preparation, Lesson Narrative, “In previous lessons, students have answered ‘how many?’ questions and comparison questions about shapes composed of pattern blocks. In this lesson, students create a shape out of pattern blocks and brainstorm questions that they could ask about other students’ shapes. Students create and solve story problems about shapes made out of pattern blocks (MP4).”

 An example from Grade 1 includes: 

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

An example from Grade 2 includes: 

• Unit 9, Putting It All Together, Lesson 10, Cool-down, Student Task Statement, students analyze a tape diagram and number line diagram, and determine a question that could be answered based on the representations. “Tyler put 26 apples into his basket. Clare put 35 apples into her basket. Ask and answer a math question about this situation. Preparation, Lesson Narrative, “In previous lessons, students sorted and solved a variety of story problems. In this lesson, students use given information to ask math questions and figure out what question was asked when presented with student work. Students interpret the context of a story and analyze tape diagrams to determine what question is being asked (MP2, MP4). Students then pose a math question and use a representation of their choice to answer it.”

The materials reviewed for Illustrative Mathematics® v.360 Kindergarten through Grade 2 meet expectations for supporting the intentional development of MP5: Choose tools strategically, for students, in connection to the grade-level content standards, as expected by the mathematical practice standards. 

Students have opportunities to engage with MP5 across the year, and it is often explicitly identified for teachers within the Course Guide (Standards for Mathematical Practice) and within specific lessons (Preparation Narratives and Lesson Activities’ Narratives). According to the Course Guide, Standards for Mathematical Practice, “The Standards for Mathematical Practice (MP) describe the types of thinking and behaviors in which students engage as they do mathematics. Throughout the curriculum, the Teacher Guide identifies lessons and activities in which to observe the different MPs. The unit-level Mathematical Practice chart is meant to highlight a handful of lessons in each unit that showcase certain MPs. Some units, due to their size or the nature of their content, have fewer predicted chances for students to engage in a particular MP, indicated in the chart by a dash.” 

An example from Kindergarten includes:

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

An example from Grade 1 includes:

• Unit 6, Length Measurements within 120, Lesson 3, Activity 1, students compare two lengths indirectly. Launch, “Give students access to connecting cubes in towers of 10 and singles, string, unsharpened pencils, and scissors. ‘Clare and Mai walk to school every day. You can see their paths on the map. Who has the shorter walk? Choose a tool to use. Be ready to explain your thinking so that others will understand.’ Make sure students know that the two arrowed lines on the image represent the two paths that Clare and Mai walk to school.” Activity, “Monitor for students who use a single tool to compare the two paths and mark the tool to show one length, such as bending string or breaking off cubes to compare the other length.” Student Task Statement: “Clare and Mai walk to school. Whose walk is shorter? Be ready to explain your thinking.” Activity Narrative: “The purpose of this activity is for students to compare two lengths indirectly. Since a third object is not given, students choose a third object strategically and share different ways to use a third object to compare (MP5).”

An example in Grade 2 includes:

• Unit 2, Adding and Subtracting within 100, Lesson 3, Activity 1, students interpret and solve a story problem by adding or subtracting within 100. Students solve an Add To, Start Unknown problem, one of the more difficult problem types from grade 1. Student Task Statement: “Some students sit on a bus. 34 more students get on the bus. Now there are 55 students. How many students were on the bus at first?” An image of two students at a zoo is shown. Launch, “Give students access to connecting cubes and base-ten blocks. ‘Have you ever been on a field trip? Where did you go? Did everyone on your field trip stay together the whole time or did you split into smaller groups?’” Activity states, “Monitor for students who: use base-ten blocks or base-ten diagrams to show adding tens to tens or ones to ones, use base-ten blocks or base-ten diagrams to show subtracting from tens or ones from ones.” Activity Narrative, “Students who choose to use connecting cubes or base-ten blocks or who draw a diagram to represent the situation are using tools strategically (MP5). During the Synthesis, invite all students to explain why these methods work using their understanding of place value.”

The materials reviewed for Illustrative Mathematics® v.360 Kindergarten through Grade 2 meet expectations for supporting the intentional development of MP6: Attend to precision, 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 often explicitly identified for teachers within the Course Guide (Standards for Mathematical Practice) and within specific lessons (Preparation Narratives and Lesson Activities’ Narratives). According to the Course Guide, Standards for Mathematical Practice, “The Standards for Mathematical Practice (MP) describe the types of thinking and behaviors in which students engage as they do mathematics. Throughout the curriculum, the Teacher Guide identifies lessons and activities in which to observe the different MPs. The unit-level Mathematical Practice chart is meant to highlight a handful of lessons in each unit that showcase certain MPs. Some units, due to their size or the nature of their content, have fewer predicted chances for students to engage in a particular MP, indicated in the chart by a dash.” 

An example in Kindergarten includes:

• Unit 5, Composing and Decomposing Numbers to 10, Lesson 5, Section B Checkpoint, students use grade-appropriate math terms to restate and represent story problems. Observation, Look Fors, “Explain how objects or drawings represent a story problem. Retell a story problem in their own words. Use labels, colors, numbers, or other methods to represent the two groups in a story problem. Use objects or drawings to represent a story problem.” Preparation, Lesson Narrative, “Students may have used some of these representations in the previous unit. While it is not important for students to use any particular method, they should be able to communicate how their representation shows the story (MP6).”

An example in Grade 1 includes:

• Unit 4, Numbers to 99, Lesson 10, Cool-down, students attend to precision as they write numbers using their knowledge of base-ten representations. Student Task Statement: “Write the number that matches each representation. 1. 30+9, 2. an image of 65 base ten blocks, 3. 7 ones + 9 tens.” Activity 1 Narrative: “Students must attend to the units in each representation and the meaning of the digits in a two-digit number. Students shouldn’t always write the number they see on the left in the tens place and the number they see on the right in the ones place (MP6).”

An example in Grade 2 includes:

• Unit 8, Equal Groups, Lesson 7, Cool-down, students use specialized language as they work with and describe arrays. Student Task Statement, “1. How many rows are in this array? 2. How many counters are in each row? 3. How many counters are there in all?” Activity 2 Narrative: “The purpose of this activity is for students to describe the number of rows in an array, the number of objects in each row, and the total number of objects. They use this vocabulary to describe arrays and to create arrays given a number of counters and a number of rows (MP6). They may use trial and error to build these arrays.”

The materials reviewed for Illustrative Mathematics® v.360 Kindergarten through Grade 2 meet expectations for supporting the intentional development of MP7: Look for and make use of structure, for students, in connection to the grade-level content standards, as expected by the mathematical practice standards. 

Students have opportunities to engage with MP7 across the year, and it is often explicitly identified for teachers within the Course Guide (Standards for Mathematical Practice) and within specific lessons (Preparation Narratives and Lesson Activities’ Narratives). According to the Course Guide, Standards for Mathematical Practice, “The Standards for Mathematical Practice (MP) describe the types of thinking and behaviors in which students engage as they do mathematics. Throughout the curriculum, the Teacher Guide identifies lessons and activities in which to observe the different MPs. The unit-level Mathematical Practice chart is meant to highlight a handful of lessons in each unit that showcase certain MPs. Some units, due to their size or the nature of their content, have fewer predicted chances for students to engage in a particular MP, indicated in the chart by a dash.” 

An example in Kindergarten includes:

• Unit 4, Understanding Addition and Subtraction, Lesson 15, Unit 4, Section C Checkpoint, students look for and make use of structure while they connect expressions to drawings. Observation, Look Fors, ”Explain how an expression connects to a drawing or story problem. Fill in an expression to represent a drawing.” Activity 1 Narrative: “The purpose of this activity is for students to match drawings to expressions. Students use the structure of the dots to decide whether they represent an addition or subtraction expression and then identify that expression (MP2, MP7).” 

An example in Grade 1 includes:

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

An example in Grade 2 includes:

• Unit 3, Measuring Length, Lesson 16, Activity 2, students look for and make use of structure as they interpret measurement data represented by line plots. Student Task Statements, “The Plant Project. Answer the questions using your line plot. 1. What is the shortest plant height? 2. What is the tallest plant height? 3. What is the height difference between the tallest and shortest plants? Write an equation.” Activity Synthesis, “Invite 1–2 students to share how they found the difference between the height of the tallest and shortest plants on their line plot. ‘How does the line plot help you see differences in the measurements that are collected?’ (Each tick mark is the same length apart. You can count the distance between each. You can see if there’s a big or small difference between the measurements by how they are spread out.)” Activity Narrative: “The purpose of this activity is to interpret measurement data represented by line plots. Students use the line plots they created in the previous activity and another line plot about plant heights to answer questions. In the Activity Synthesis, students share how they found the difference between two lengths using the line plot and discuss how the structure of the line plot helps to show differences (MP7).”

The materials reviewed for Illustrative Mathematics® v.360 Kindergarten through Grade 2 meet expectations for supporting the intentional development of 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 MP8 across the year and it is often explicitly identified for teachers within the Course Guide (Standards for Mathematical Practice) and within specific lessons (Preparation Narratives and Lesson Activities’ Narratives). According to the Course Guide, Standards for Mathematical Practice, “The Standards for Mathematical Practice (MP) describe the types of thinking and behaviors in which students engage as they do mathematics. Throughout the curriculum, the Teacher Guide identifies lessons and activities in which to observe the different MPs. The unit-level Mathematical Practice chart is meant to highlight a handful of lessons in each unit that showcase certain MPs. Some units, due to their size or the nature of their content, have fewer predicted chances for students to engage in a particular MP, indicated in the chart by a dash.” 

An example in Kindergarten includes:

• Unit 6, Numbers 0–20, Lesson 4, Warm-up, students count collections of objects and understand that the number of objects in a collection stays the same, regardless of how they are arranged. Student Task Statement, “What do you notice? What do you wonder?” Launch states, “Groups of 2. Display the image. ‘What do you notice? What do you wonder?’ 1 minute: quiet think time.” Activity states, “‘Discuss your thinking with your partner.’ 1 minute: partner discussion. Share and record responses.” Activity Synthesis states, “Which arrangements do you think would be easier to count? Why? (The lined up dots would be easier to count. I could count one line and then the other line.).” Preparation, Lesson Narrative states, “Students count the same collection of objects in different arrangements to build this conservation of number, which develops through experience over time. While developing conservation of number, students may need to recount the objects each time they are rearranged. With repeated practice, some students may know that the number of objects is the same without recounting (MP8).”

An example in Grade 1 includes:

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

An example in Grade 2 includes:

• Unit 9, Putting It All Together, Lesson 9, Warm-up, students use repeated reasoning to find the value of differences when they may need to decompose a ten. Student Task Statement, “Find the value of each expression mentally. 10-6, 14-6, 54-6, 54 - 26.” Activity Synthesis states, “How can you use the result of 14-6 to find the value of 54-6? (54 has 4 more tens than 14 so add 4 tens or 40 to the result of 14-6.) How can you use the result of 54-6 to find the value of 54-26? (26 has 2 more tens than 6 so that means 2 tens need to be taken away from the answer to 54-6.)” Activity narrative states, “When students consider how they can use known differences, like 10-6 or 14-6, to find the values of the other expressions, they look for and make use of structure and express regularity in repeated reasoning (MP7, MP8).”