2nd Grade - Gateway 2

The materials reviewed for Kendall Hunt's Illustrative Mathematics Grade 2 meet expectations for developing conceptual understanding of key mathematical concepts, especially where called for in specific content standards or cluster headings.

Materials develop conceptual understanding throughout the grade level. According to IM K-5 Math Teacher Guide, Design Principles, conceptual understanding is a part of the design of the materials. Balancing Rigor states, “Opportunities to connect new representations and language to prior learning support students in building conceptual understanding. Access to new mathematics and problems prompts students to apply their conceptual understanding and procedural fluency to novel situations.” Additionally, Purposeful Representations states, “Across lessons and units, students are systematically introduced to representations and encouraged to use representations that make sense to them. As their learning progresses, students are given opportunities to make connections between different representations and the concepts and procedures they represent.” Examples include:

• Unit 3, Measuring Length, Lesson 14, Activity 1, students develop conceptual understanding as they learn about the ways a line plot can be used to represent data collected from measuring objects. Activity states, “‘We are going to continue measuring in inches. Each of you will measure your hand span.’ Display the image of the traced hand. ‘You are going to measure your hand span, which is the length from your pinky to your thumb. First, you’ll trace your hand and then measure it to the nearest inch. After measuring your own hand span, check your partner’s measurements.’ 6 minutes: partner work time. ‘Now, we are going to make a representation to show everyone’s hand span measurements.’ Give each student a sticky note that is the same size. ‘Now we need to represent the data we have collected. Draw a big x on your sticky note.’ As needed, demonstrate drawing an x on a sticky note. Display the blank line plot. ‘If we want this display to show others the lengths of all our measurements, where do you think the length of your hand span should go?’ 30 seconds: quiet think time. 1 minute: partner discussion. Invite students to come up to add their sticky notes to the chart above the corresponding measurement. Consider asking students to explain how they place their sticky notes.” Student Facing states, “Trace your hand. (Spread your fingers wide.) Draw a line from your thumb to your pinky. This line represents your hand span. Measure the length of your hand span in inches. My hand span is ___ inches.” An image of a hand is shown. (2.MD.1)

• Unit 5, Numbers to 1,000, Lesson 5, Activity 1, students develop conceptual understanding as they write three-digit numbers as the sum of the value of each digit. Launch states, “Groups of 2. Display Andre, Tyler, and Mai’s situation and the image of their blocks. ‘What would the expression look like?’ 1 minute: independent work time. 1 minutes: partner discussion. Share responses. Display 357 and 300+5+7. ‘We can represent the value of the blocks by writing a three-digit number. A number can also be represented as a sum of the value of each of its digits. This is called expanded form. Like a three-digit number, expanded form shows the sum starting with the place that has the greatest value on the left to the place with the least value on the right.’ As needed, discuss reasons why any expressions generated in the launch would or would not be examples of expanded form.” (2.NBT.1, 2.NBT.3)

• 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 Facing shows 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 even groups of dots? (image 1 and image 3) How can you tell using the equations we recorded?” (2.OA.C)

According to IM K-5 Math Teacher Guide, materials were designed to include opportunities for students to independently demonstrate conceptual understanding, when appropriate. Design Principles, Coherent Progress states, “Each activity starts with a launch that gives all students access to the task. This is followed by independent work time that allows them to grapple with problems individually before working in small groups. The activity ends with a synthesis to ensure students have an opportunity to consolidate their learning by making connections between their work and the mathematical goals.” A Typical IM Lesson states, “The cool-down task is to be given to students at the end of the lesson. Students are meant to work on the cool-down for about 5 minutes independently and turn it in. The cool-down serves as a brief formative assessment to determine whether students understood the lesson. Students’ responses to the cool-down can be used to make adjustments to further instruction.” Examples include:

• Unit 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 that Jada found it helpful 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 Facing states, “1. Locate and label 420 and 590 on the number line. Use <, >, and = to compare 420 and 590. 2. Estimate the location of 378 and 387 on the number line. 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 was most helpful to compare the numbers? Why?” Number lines are included for numbers 1 and 2 while base ten representations are shown for number 3. (2.MD.6, 2.NBT.1, 2.NBT.4)

• Unit 7, Adding and Subtracting within 1000, Lesson 8, Activity 2, students demonstrate conceptual understanding as they analyze base-ten diagrams and corresponding equations representing sums. Images of base ten blocks and equations are provided and Student Facing states, “1. Priya and Lin were asked to find the value of 358+67. What do you notice about their work? What is the same and different about their representations? Be prepared to explain your thinking. 2. Find the value of 546+86. Show your thinking. Use base-ten blocks if it helps.” (2.NBT.7)

• Unit 9, Putting It All Together, Lesson 6, Warm-up, students demonstrate conceptual understanding as they explain why an equation is true based on place value. Activity states, “Share and record answers and strategies. Repeat with each statement.” Student Facing states, “Decide if each statement true or false. Be prepared to explain your reasoning. 5 hundreds + 2 tens + 7 ones = 527, 4 hundreds + 12 tens + 7 ones = 527, 5 hundreds + 7 ones + 2 tens = 527.” (2.NBT.1)

The materials reviewed for Kendall Hunt's Illustrative Mathematics Grade 2 meet expectations for giving attention throughout the year to individual standards that set an expectation of procedural skill and fluency. 

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

• Unit 1, Adding, Subtracting and Working with Data, Lesson 1, Activity 1, students develop procedural skill and fluency as they demonstrate methods for adding and subtracting within 10. Launch states, “Groups of 2, Give each group a set of number cards. Give students access to connecting cubes or counters. ‘We are going to play Check It Off: Add or Subtract within 10. The goal is to be the first person to write an expression for each number. I’m going to pick two cards. (Show 10 and 7.) I have to decide whether I want to add or subtract. I don’t want a value greater than 10, so I’m going to subtract.’ Write 10-7. What is the value of the difference?’ 30 seconds: quiet think time. Share responses. ‘I record the expression I made on my recording sheet next to the value of the difference and check off the number. Now it’s my partner’s turn. Take turns picking cards, making an addition or subtraction expression, finding the value of the sum or difference, and showing your partner how you know. If you run out of cards before someone checks off all the numbers, shuffle them and start again.’” Student Facing states, “1. Pick 2 cards and find the value of the sum or difference. 2. Check off the number you found and write the expression. 3. The person who checks off the most numbers wins.” A table numbered from 0-10 with the headings Found It! and Expressions are shown. (2.OA.2)

• Unit 5, Numbers to 1,000, Lesson 8, Warm-up, students develop procedural fluency as they practice counting by 10 and 100 and notice patterns. Launch states, “‘Count by 10, starting at 0.’ Record in a column as students count. Stop counting and recording at 100. ‘Count by 100, starting at 0.’ Record the count in a new column next to the first. Stop counting and recording at 1,000.” Activity states, “‘What patterns do you see?’ 1–2 minutes: quiet think time. Record responses.” (2.NBT.2)

• 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 Facing states, “1. Measure the length of your pencil. ___ cm. 2. Write the lengths of your group’s pencils in the table. 3. Find the total length of each group’s pencils.” (2.MD.1, 2.NBT.5, 2.OA.2)

According to IM K-5 Math Teacher Guide, materials were designed to include opportunities for students to independently demonstrate procedural skill and fluency, when appropriate. Design Principles, Coherent Progress states, “Each activity starts with a launch that gives all students access to the task. This is followed by independent work time that allows them to grapple with problems individually before working in small groups. The activity ends with a synthesis to ensure students have an opportunity to consolidate their learning by making connections between their work and the mathematical goals.” A Typical IM Lesson states, “The cool-down task is to be given to students at the end of the lesson. Students are meant to work on the cool-down for about 5 minutes independently and turn it in. The cool-down serves as a brief formative assessment to determine whether students understood the lesson. Students’ responses to the cool-down can be used to make adjustments to further instruction.” Examples include:

• Unit 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 race to see who can reach a number less than 10 first.’ ‘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 until the first person reaches a number less than 10.’ As needed, demonstrate a round with a student volunteer.” (2.NBT.5) 

• Unit 4, Addition and Subtraction on the Number Line, Lesson 9, Activity 1, students demonstrate procedural skill and fluency as they add and subtract within 100. Activity states, “You are going to find the number that makes each equation true in a way that makes sense to you.’ ‘Then, use the number line to show your thinking.’ 6 minutes: independent work time. ‘Compare your methods, solutions, and number line representations with a partner.’ 4 minutes: partner discussion.” Student Facing states, “1. What number makes this equation true?___. 38-4=? Represent your thinking on the number line. 2. What number makes this equation true?___. 75-68=? Represent your thinking on the number line. 3. What number makes this equation true?___. 57-24=?” (2.MD.6, 2.NBT.5)

• Unit 9, Putting It All Together, Lesson 1, Cool-down, students demonstrate procedural skill and fluency as they solve addition and subtraction equations. Student Facing states, “Find the value of each expression 1. 11-5 2. 12-3 3. 16-8 4. 9+3 5. 8+8 6. 13-8.” (2.OA.2)

The materials reviewed for Kendall Hunt's Illustrative Mathematics Grade 2 meet expectations for being designed so that teachers and students spend sufficient time working with engaging applications of the mathematics. According to IM Curriculum, Design Principles, Balancing Rigor, “Access to new mathematics and problems prompts students to apply their conceptual understanding and procedural fluency to novel situations.” Multiple routine and non-routine applications of the mathematics are included throughout the grade level and these single- and multi-step application problems are included within Activities or Cool-downs. 

Students have the opportunity to engage with applications of math both with support from the teacher and independently. According to IM K-5 Math Teacher Guide, materials were designed to include opportunities for students to independently demonstrate application of grade-level mathematics, when appropriate. Design Principles, Coherent Progress states, “Each activity starts with a launch that gives all students access to the task. This is followed by independent work time that allows them to grapple with problems individually before working in small groups. The activity ends with a synthesis to ensure students have an opportunity to consolidate their learning by making connections between their work and the mathematical goals.” A Typical IM Lesson states, “The cool-down task is to be given to students at the end of the lesson. Students are meant to work on the cool-down for about 5 minutes independently and turn it in. The cool-down serves as a brief formative assessment to determine whether students understood the lesson. Students’ responses to the cool-down can be used to make adjustments to further instruction.”

Examples of routine applications of the math include:

• Unit 1, Adding, Subtracting, and Working with Data, Lesson 10, Activity 2, students solve problems as they represent data in a bar graph. Launch states, “Groups of 2. Display ‘Fruits We Love’ bar graph with the categories covered. ‘What do you think the labels are on the bottom? Why?’ 2 minutes: partner discussion. Share and record responses. ‘What are some features of this graph that help us understand the data?’ Share responses. Highlight the important features (title, labels, numbers/scale).” Activity states, “Give each student a copy of the graph template. Prompt students to trade data tables with their partner or another student. ‘When you make your own bar graph, use the grid to draw a bar graph that represents the data in your data table. Write the number of the data table on your page. After you have made your bar graph, compare with a partner.’ 10 minutes: independent work time.” Student Facing states, “A group of students were asked, ‘What fruit do you love to eat?’ Their responses are shown in this bar graph. Represent the data shown in your table in a bar graph. Table # ___.” A bar graph labeled “Fruits We Love,” showing student responses, is shown. An image of fruits with prices is also included next to the problem. (2.MD.10)

• Unit 4, Addition and Subtraction on the Number Line, Lesson 13, Cool-down, students solve word problems involving addition or subtraction. Student Facing states, “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)

• Unit 8, Equal Groups, Lesson 2, Cool-down, students pair all of the objects in a group in order to demonstrate their understanding of equal groups. Student Facing states, “Nine students need to pair up to play a game. Will everyone have one partner? Show your thinking using a diagram, symbols, or other representations.” (2.OA.3)

Examples of non-routine applications of the math include:

• Unit 2, Adding and Subtracting within 100, Lesson 16, Activity 1, students use addition and subtraction strategies. Student Facing states, “You sell 3 kinds of items in a store. At the beginning of each day you have: a total of 100 items, less than 10 of one of the items, more than 10 for the other 2. 1. Choose 3 items to sell at your market. Write the names of the items in the first row. 2. Fill in the second row to show how much of each item you begin the day with. 3. Share your store set-up with your partner pair. Discuss: the amount you have for each item, how you know that you have a total of 100 items at your store." (2.NBT.5, 2.NBT.6, 2.OA.1)

• 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. 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 Facing states, “1. Lin played a game with seeds. She started the game with some seeds. Then she won 36 seeds. Now she has 64 seeds. How many seeds did Lin have at first? a. Write an equation using a question mark for the unknown value. b. Solve. Show your thinking using drawings, numbers, or words. 2. Andre started a game with 32 seeds. Then he won more seeds. Now he has 57 seeds. How many seeds did Andre win? a. Label the diagram to represent the story. b. Write an equation using a question mark for the unknown value. c. Solve. Show your thinking using drawings, numbers, or words. 3. Diego gathered 22 seeds from yellow flowers and 48 seeds from blue flowers. How many seeds did he gather in all? a. Label the diagram to represent the story. b. Write an equation using a question mark for the unknown value. c. Solve. Show your thinking using drawings, numbers, or words.” Tape diagrams are included for each part of the problem. (2.NBT.5, 2.OA.1)

• Unit 6, Geometry, Time, and Money, Lesson 2, Activity 2, students draw shapes that have a given number of sides or corners, and then compare the shapes. Activity states, “‘Clare, Andre, and Han drew shapes. Using the clues, see if you can figure out which shapes might belong to each student. Then draw a different shape based on the clues.’ 7 minutes: independent work time. Monitor for examples of Han’s shape that have different numbers of sides, number of corners, side lengths, and angles to share in the synthesis.” Student Facing states, “1. Clare drew a shape that has fewer than 5 sides. Circle shapes that could be Clare’s shape. (5 figures are shown on dot paper) 2. Draw a different shape that could be Clare’s shape. 3. Andre drew a shape that has 4 corners. Circle shapes that could be Andre’s shape. (6 figures are shown on dot paper) 4. Draw a different shape that could be Andre’s shape. 5. Han drew a shape that has more corners than Andre’s shape. Draw two shapes that could be Han’s shape.” (2.G.1)

The materials reviewed for Kendall Hunt's Illustrative Mathematics 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.

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

• Unit 1, Adding, Subtracting, and Working With Data, Lesson 16, Activity 2, students apply their understanding of addition and subtraction strategies when solving real-world problems. Activity states, “‘Today, you’re going to solve problems with your partner. Show your thinking using drawings, numbers, words, or an equation. Remember to ask yourselves questions as you make sense of the problem and create representations.’ 12 minutes: partner work time.” Student Facing states, “1. Jada read 10 fewer pages than Noah. Noah read 27 pages. How many pages did Jada read? 2. Noah spent 25 minutes reading. Jada spent 30 more minutes reading than Noah. How many minutes did Jada spend reading? 3. Jada read 47 pages of the book. Noah read 20 pages of the book. How many fewer pages did Noah read? 4. Noah stacked 14 more books than Jada. Jada stacked 28 books. How many books did Noah stack?” (2.OA.1)

• Unit 4, Addition and Subtraction on the Number Line, Lesson 11, Activity 2, students develop fluency with addition and subtraction within 100. Activity states, “‘Find the value of the sum and difference. You may continue to try Diego or Tyler's method or use any other way that makes sense to you. Use the number line if it helps to show your thinking.’ 5 minutes: independent work time. 3 minutes: partner discussion.” Student Facing states, “Partner A 1. Find the value of 59+27  2. Find the value of 65-18. Partner B 1. Find the value of 68-39.  2. Find the value of 22+49.” (2.NBT.5)

• Unit 5, Numbers to 1000, Lesson 9, Activity 1, students deepen their conceptual understanding as they make sense of different methods to compare three-digit numbers. Activity states, “‘Diego, Jada, and Clare were asked to compare 371 and 317. They each represented their thinking differently. Take some time to look over their methods.’ 2 minutes: independent work time. ‘Discuss with your partner how their methods are the same and different.’ 4 minutes: partner discussion. ‘Now try Jada’s way.’ 6 minutes: partner work time. Student Facing states, “Each student compared 371 and 317, but represented their thinking in different ways. 1. What is the same and different about these students’ representations? Discuss with a partner. 2. Try Jada’s way. Estimate the location of 483 and 443 on the number line. Mark each number with a point. Label the point with the number it represents. 3. Use >, =, or < to compare 483 and 443.” Student work includes a mixture of representation with base ten diagrams, descriptions, number sentences, and a number line. (2.MD.6, 2.NBT.1, 2.NBT.4) 

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

• Unit 1, Adding, Subtracting, and Working with Data, Lesson 8, Cool-down, students use all three aspects of rigor, conceptual understanding, procedural fluency, and application as they read and interpret a picture graph. Student Facing states, “A group of students were asked, “What is your favorite pet?” Their responses are shown in this picture graph. 1. Circle the 2 questions that can be answered by the picture graph. A. How many students chose a cat as their favorite pet? B. How many more students like rabbits than dogs? C. Who owns a lizard? D. How many more students chose cats than dogs? E. Why don’t more students like dogs? 2. Pick a question that can't be answered by the data on the graph. Explain why it can’t be answered.” (2.MD.10) 

• Unit 2, Adding and Subtracting within 100, Lesson 8, Activity 2, students extend conceptual understanding and procedural skills as they use different methods to decompose numbers. Launch states, “Give students access to base-ten blocks. ‘Andre found the value of 65-28. Take a minute to look at his work.’ 1 minute: quiet think time. ‘Do you think it’s more like Clare or Lin’s method? Discuss with your partner.’ (It’s more like Lin’s because he drew all the tens first. It’s more like Clare’s because he took away tens first, he just drew them out.)” Student Facing states, “Andre found the value of 65-28. He made a base-ten diagram and wrote equations to show his thinking. 1. Do you think Andre’s method is more like Clare’s or Lin’s method? Explain. 2. Find the value of each difference. Show your thinking. a. 34-18. b. 82-37. c. 7-53.” (2.NBT.5, 2.NBT.9)

• Unit 8, Equal Groups, Lesson 1, Activity 2, students use conceptual understanding and apply their understanding of equal groups to 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 in the diagrams they create.” Student Facing states, “Andre has a collection of 17 marbles. He wants to play a game with his sister. They both need to start with the same number of marbles and they want to use as many as they can. 1. How many marbles would Andre and his sister get? Would there be any marbles left out of the game? Show your thinking. 2. What if Andre had 18 marbles? How many would each person get? Would there be any marbles left out? Show your thinking. 3. What if Andre had 20 marbles? How many would each person get? Would there be any marbles left out?” (2.OA.C)

The materials reviewed for Kendall Hunt's Illustrative Mathematics Grade 2 meet expectations for supporting the intentional development of MP1: Make sense of problems and persevere in solving them; and MP2: Reason abstractly and quantitatively, for students, in connection to the grade-level content standards, as expected by the mathematical practice standards. Students have opportunities to engage with the Math Practices across the year and they are often explicitly identified for teachers within the Course Guide (How to Use These Materials) and within specific lessons (Instructional Routines, Lesson Preparation Narratives, and Lesson Activities’ Narratives).

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

• Unit 2, Adding and Subtracting Within 100, Lesson 11, Activity 2, students solve different types of story problems where they compose or decompose tens when adding or subtracting. Activity states, “‘Work with your partner to make sense of each story problem and solve it. Show your thinking using drawings, numbers, or words.’ 8 minutes: partner work time. Monitor for different ways students use labels and diagrams to make sense of the last problem.” Student Facing states, “Solve each story problem. Show your thinking. 1. Lin had 31 sunflower seeds. She gave Priya 15 seeds. How many seeds does Lin have now? 2. Noah used yellow and blue corn seeds to make a design. He used 37 seeds altogether. He used 28 yellow seeds. How many blue seeds did he use? 3. Elena gathered 50 pumpkin seeds. Andre collected 23 fewer pumpkin seeds than Elena. How many seeds did Andre collect?” Narrative states, “Monitor for a variety of different ways students use drawings, diagrams, or equations to make sense of or solve the problems for sharing in the lesson synthesis. Look and listen for examples of ways students make sense of what they need to find, such as a tape diagram or base-ten blocks, before they use methods to calculate unknown values (MP1).”

• Unit 5, Numbers to 1,000, Lesson 14, Warm-up, students make sense of problems during a notice and wonder routine. Student Facing states, “What do you notice? What do you wonder?” A jar partially filled with beans is shown. Narrative states, “This warm-up prompts students to make sense of a problem before solving it, by familiarizing themselves with a context and the mathematics that might be involved. In the next activity, students will see three different ways the amount of beans in a cup are counted.”

• Unit 6, Geometry, Time, and Money, Lesson 18, Cool-down, students make sense of problems that require them to add or subtract money. Student Facing, “Mai has these coins to buy school supplies: 3 nickels, 1 dime, and 2 quarters are shown. a. How much money does Mai have for supplies? b. 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.”

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

• Unit 1, Adding, Subtracting, and Working with Data, Lesson 14, Warm-up, students reason about tape diagrams as similar to bar graphs and can be used to represent the same data. Launch states, “Display the image. ‘What do you notice? What do you wonder?’” Narrative states, “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).”

• Unit 3, Measuring Length, Lesson 11, Activity 1, students reason about length measurements and a tape diagram representation. Student Facing states, “What do you notice? What do you wonder? Priya had a ribbon that was 44 inches long. She cut off 18 inches. How long is Priya’s ribbon now? Andre drew this diagram to help him think about the problem. 1. What does the “?” represent in the story? 2. Why do you think there is a dotted line between the parts? 3. Find the unknown value. Show your thinking. 4. Priya’s ribbon is ____ long.” Launch states, “Groups of 2. Give students access to base-ten blocks. Display the image. ‘What do you notice? What do you wonder?’ 1 minute: quiet think time. 1 minute: partner discussion. Share responses. ‘These girls from India are wearing saree dresses. Sarees are usually worn by women and girls and are made by wrapping 5–7 meters of fabric in a special way. Many sarees are made from brightly colored silk, which is a soft fabric. Sometimes when sarees get too small or are worn out, they are cut into strips to make saree ribbon.’” Activity states, “‘Priya and her friends are planning to make saree silk ribbon necklaces. They want to make sure they get their measurements correct. Read the problem. Then look at Andre’s diagram and discuss the first two questions with a partner.’ 1 minute: independent work time. 3–4 minutes partner discussion. ‘Work independently to find the unknown value and compare your answer with your partner. Don’t forget to include the units.’ 4–5 minutes: independent work time. 2 minutes: partner discussion.” Narrative states, “Students use the diagram to make sense of the context and help guide their calculations as they solve the problem (MP2).”

• Unit 9, Putting It All Together, Lesson 12, Activity 1, students write story problems for equations with an unknown value. Student Facing states, “Your teacher will assign you A or B. For each of your equations, write a story problem that fits the equation. A Equations, 23+___$$=37$$, ___$$+9=45$$. B Equations, 73-___$$=28$$, ___$$-15=18$$.” Activity states, “Split the class into two groups, A and B. The students in group A will work with the equations labeled A and the students in group B will work with the equations labeled B. ‘You will write stories for the 2 equations in A or the 2 equations in B. Consider using the same context for both of your stories. It might make it easier for others to make sense of your stories if they are about the same thing.’ 5 minutes: independent work time. ‘Share your stories with your partner.’ 5 minutes: group work time.” Narrative states, “When students contextualize the equations and make connections between the stories their peers share and the equations, they reason abstractly and quantitatively (MP2).”

The materials reviewed for Kendall Hunt's Illustrative Mathematics 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 (How to Use These Materials) and within specific lessons (Instructional Routines, Lesson Preparation Narratives, and Lesson Activities’ Narratives). According to the Course Guide, Instructional Routines, Other Instructional Routines, 5 Practices, “Lessons that include this routine are designed to allow students to solve problems in ways that make sense to them. During the activity, students engage in a problem in meaningful ways and teachers monitor to uncover and nurture conceptual understandings. During the activity synthesis, students collectively reveal multiple approaches to a problem and make connections between these approaches (MP3).”

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

• Unit 3, Measuring Length, Lesson 4, Activity 1, students construct viable arguments and critique the reasoning of others as they practice the skill of estimating a reasonable length in centimeters. Student Facing states, “1. Record an estimate that is: too low, about right, too high. 2. Record an estimate that is: too low, about right, too high. 3. Record an estimate for each object on the recording sheet. 4. Tell your partner why you think your estimates are “about right.” A recording sheet with columns labeled “object, estimate, measurement, choose an object” is shown. Launch states, “Give objects to each group. Display the image or a real notebook. ‘Andre wanted to measure the length of his notebook, but he didn’t have any tools to measure it. He made a guess that he thought would be close. Look at the notebook and think about how long you think it is in centimeters. What is an estimate that’s too high? Too low? About right?’ 1 minute: quiet think time. 1 minute: partner discussion. Record responses. ‘Let’s look at another image of the object.’ Display the image or hold a folder next to a 10-centimeter tool. ‘Based on the second image, do you want to revise, or change, your estimates?’ 1 minute: quiet think time. 1 minute: partner discussion. Record responses. ‘How did your estimation change?’ 30 seconds: quiet think time. Share responses.” Activity states, “As needed, display the names of the objects that students will estimate. ‘Now look at the objects I gave each group and think about how long they are. Record your estimates on the recording sheet on your own. When you and your partner finish, compare your estimates and explain why you think they are “about right”.’ 5 minutes: independent work time. 2 minutes: partner discussion.” Narrative states, “When students compare and explain their estimates in pairs and in the full class discussion they make, interpret, and defend mathematical claims (MP3).”

• Unit 4, Addition and Subtraction on the Number Line, Lesson 3, Activity 2, students construct arguments when they represent numbers up to 100 on a number line. Student Facing states, “Complete each number line by filling in the labels with the number the tick mark represents. Then, locate each number, mark it with a point, and label it with the number it represents. 1. Locate and label 17 on the number line. 2. Locate and label 59 on the number line. 3. Locate and label 43 on the number line. 4. Locate and label 35 on the number line. 5. Share your number lines with your partner.” Images of number lines are shown for each problem. Activity states, “‘On your own, complete each number line by filling in the missing labels with the number the tick mark represents. Then, locate each number, mark it with a point, and label the point with the number it represents. When you finish, think of how you can explain to your partner how you know your labels and points are at the right spots on the number lines.’ 5 minutes: independent work time. ‘Share your work with a partner. Make sure you agree on your answers.’ 5 minutes: partner discussion. Monitor for students who: explain their labeled tick marks based on counting by 5 or 10, explain their labeled tick marks based on the equal lengths between each labeled tick mark, use labeled tick marks to explain how they locate numbers.” Narrative states, “When students explain to one another how they located different numbers on the number lines they construct viable arguments and may critique each other's reasoning (MP3).”

• Unit 6, Geometry,Time and Money, Lesson 8, Activity 1, students construct an argument and critique the reasoning of others as they explore different ways to partition rectangles into halves and fourths. Launch states, “Groups of 2. ‘Lin wanted to partition this square into quarters. She started by splitting the square into halves.’ Display the square partitioned into halves. ‘After she drew the first line, she tried 3 different ways to make fourths.’ Display the 3 squares split into 4 pieces. ‘Which of these shows fourths or quarters? Explain.’ (B is the only one that shows four equal pieces, so they are fourths. The other 2 show 4 parts, but they are not equal.) 30 seconds: quiet think time. 1 minute: partner discussion. Share and record responses.” Student Facing states, “Lin wanted to partition this square into quarters. She started by splitting the square into halves. After she drew the first line, she tried 3 different ways to make fourths. 1. Which of these shows fourths or quarters? Explain and share with your partner. 2. Name the shaded piece. Shape A has a ___shaded. Shape B has a ___shaded. 3. Show 2 different ways to partition the rectangle into quarters or fourths. Shade in a fourth of each rectangle. 4. Show 2 different ways to partition the square into halves. Shade in a half of each square.” Activity Synthesis states, “Invite previously identified students to share their rectangles partitioned to make fourths. Display students’ work. ‘Each of these students believe they have split the rectangle into fourths or quarters. Who do you agree with? Explain.’ Students explain why the equal pieces of the same whole could look very different even though they have the same size, so long as the original shape was split into the same number of equal pieces (MP3).”

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

• Unit 3, Measuring Length, Lesson 9, Cool-down, students critique the reasoning of others and construct an argument as they reason about a new customary measurement unit, the foot. Student Facing states, ”Tyler told Han that a great white shark is about 16 inches long, but Han disagrees. Han believes it would be about 16 feet long. Who do you agree with? Explain.” Narrative states, “Students  are given an illustration of a boy and a fish and are asked to give an estimate for the length of the fish in inches. This gives students an opportunity to share a mathematical claim including the assumptions they made when interpreting the image with limited information (MP3).”

• Unit 5, Numbers to 1,000, Lesson 12, Activity 1, students critique the reasoning of others as they interpret the order of numbers. Student Facing states, “Kiran and Andre put a list of numbers in order from least to greatest. Kiran, 207, 217, 272, 269, 290. Andre, 207, 217, 269, 272, 290. Andre disagreed with Kiran, so he used a number line to justify his answer. Who do you agree with? Why? Be prepared to explain your thinking. Use what you know about place value or the number line to justify your reasoning.” An image of a number line is shown. Activity states, “‘Kiran and Andre put some numbers in order from least to greatest. Andre disagreed with Kiran, so he used a number line to justify his answer. Whom do you agree with? Think about this on your own and be prepared to explain your thinking.’ 3 minutes: independent work time. ‘Discuss with a partner using what you know about place value or the number line to justify your reasoning.’ 5 minutes: partner work time. Monitor for students who: use precise place value language to describe the correct placement of 269 and 272 in the list use the number line to explain that a list of numbers from least to greatest should match the placement of the numbers on the number line from left to right.” Narrative states, “The purpose of this activity is for students to analyze a mistake in ordering numbers (MP3). When placing numbers in order from least to greatest, students can compare using their understanding of place value.”

• Unit 7, Adding and Subtracting within 1000, Lesson 16, Activity 1, students construct arguments and critique other’s reasoning as they interpret and connect different representations for subtraction methods. Launch states, “Groups of 2. 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 student ideas about the numbers being subtracted.” Activity states, “‘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.’” MLR8 Discussion Supports, “Display sentence frames to support partner discussion: ‘I agree because . . . I disagree because . . .’ 7–8 minutes: partner work time. Monitor for students who share why they agree with some (or all) of what Jada says and those that disagree and use a diagram to show decomposing to subtract by place.” Student Facing states, “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 says she knows a way to count on to find the difference. She showed 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 with Jada? Use base-ten blocks, diagrams, or other representations to show your thinking.” Preparation, Lesson Narrative states, “Throughout this lesson, students explain their thinking and listen to and critique the reasoning of others (MP3).”

The materials reviewed for Kendall Hunt's Illustrative Mathematics Grade 2 meet expectations for supporting the intentional development of MP4: Model with mathematics; and MP5: Choose tools strategically, for students, in connection to the grade-level content standards, as expected by the mathematical practice standards. Students have opportunities to engage with the Math Practices across the year and they are often explicitly identified for teachers within the Course Guide (How to Use These Materials) and within specific lessons (Instructional Routines, Lesson Preparation Narratives, and Lesson Activities’ Narratives).

MP4 is identified and connected to grade-level content, and there is intentional development of the MP to meet its full intent. Students have many opportunities to solve real-world problems, model situations with appropriate representations, and describe what they do with the model and how it relates to the problem. Students model with mathematics as they work with support of the teacher and independently throughout the units. Examples include:

• Unit 1, Adding, Subtracting, and Working with Data, Lesson 18, Activity 1, students use what they have learned about data, bar graphs, and tape diagrams to create a survey and to organize, collect, and represent data. Students use their understanding of adding and subtracting to ask and answer questions related to the data. Student Facing states, “What is your survey question? What are your categories? Category 1: ___ Category 2: ___ Category 3: ___ Category 4: ___. Record the data. Organize and represent the data in a picture graph or bar graph.” Student Response states, “What is your favorite pet? cat, dog, fish, hamster. Students may record their collected data in a table using numbers or tallies. Students create a bar graph with a title, categories, and scale. Or they create a picture graph with stars to represent each vote or dots.” Lesson Narrative states, “This lesson supports the development of mathematical modeling skills by providing students opportunities to make choices about their approach for collecting data, determine appropriate equations to represent the situation, and choose ways to best represent their analysis (MP4).”

• Unit 4, Addition and Subtraction on the Number Line, Lesson 15, Activity 1, students use what they know about representing addition and subtraction problems on the number line to solve problems about the differences in family members’ ages. Student Facing states, “Solve Kiran’s age riddles. Show your thinking. Use a number line if it helps. 1. I’m 7. My sister is 5 years older than I am. How old is she? ___ years old. 2. If you add 27 years to my sister’s age, you get my mom’s age. How old is my mom? ___ years old. 3. My brother is 24 years younger than my mom. How old is my brother? ___ years old. 4. My grandma is 53 years older than my brother. How old is my grandma? ___ years old. 5. My uncle is 21 years younger than my grandma. How old is my uncle? ___ years old. 6. My uncle is 33 years older than my cousin. How old is my cousin? ___ years old. 7. There is a 50 year difference between my grandpa’s age and my cousin’s age. How old is my grandpa? ___ years old.” Launch states, “‘Kiran wrote some riddles based on the ages of people in his family. Let’s solve them.’ Give each student a copy of the blackline master.” Activity states, “‘Work with your partner to read each riddle carefully. You may use a number line if it is helpful. As you work, think about whether you are using addition or subtraction.’ 10 minutes: partner work time. Monitor for students who use a number line or write an expression or equation to show their thinking. Monitor for students who locate and label each family member's age and name on the number line.” Lesson Narrative states, “In this lesson, when students decide what quantities are important in a real-world situation, use these quantities to develop their own story problems, and choose math that matches a simplified situation, they build the precursor skills they need to model with mathematics (MP4).”

• Unit 9, Putting It All Together, Lesson 10, Cool-down, Student Facing states,”Tyler put 26 apples into his basket. Clare put 35 apples into her basket. Ask and answer a math question about this situation. Lesson Narrative states, “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 use a representation of their choice to answer a math question which they pose.”

MP5 is identified and connected to grade-level content, and there is intentional development of the MP to meet its full intent. Students have multiple opportunities to identify and use a variety of tools or strategies, working with the support of the teacher and independently, throughout the units to support their understanding of grade-level math. Examples include:

• Unit 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 Facing states, “Some students were waiting on the bus to go to the zoo. Then 34 more students got on. Now there are 55 students on the bus. How many students were on the bus at first?” An image of two students at a zoo is shown. Launch states, “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, “5 minutes: independent work time. 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.” Narrative states, “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).”

• Unit 5, Numbers to 1000, Lesson 12, Cool-down, students order numbers from least to greatest and greatest to least. Student Facing states, “1. Estimate the location and label 748, 704, 762, 789, and 712 on the number line. 2. Order the numbers from least to greatest.” Narrative states, “students may order the numbers using any method that makes sense to them. Students reflect on how the number line can help us organize numbers (MP5). Monitor for the way students explain their reasoning based on place value and the relative position of numbers on the number line.”

• Unit 7, Adding and Subtracting within 1,000, Lesson 12, Activity 1, students subtract one-digit and two-digit numbers from a three-digit number using strategies that make sense to them. Student Facing states, “Find the value of each expression in any way that makes sense to you. Explain or show your reasoning. 1. 354-7, 2. 354-36, 3. 354-48.” Launch states, “Give students access to base-ten blocks.” Activity states, “‘Find the value of each expression in any way that makes sense to you. Explain or show your reasoning.’ 3–4 minutes: independent work time. 3–4 minutes: partner discussion. Monitor for an expression that generates a variety of student methods or representations to share in the synthesis, such as: using base-ten blocks. drawing a number line. writing their reasoning in words. writing equations.” Narrative states, “When students use base-ten blocks, number lines, or equations to find the value of each difference they use appropriate tools strategically (MP5).”

The materials reviewed for Kendall Hunt's Illustrative Mathematics Grade 2 meet expectations for supporting the intentional development of MP6: Attend to precision; and attend to the specialized language of mathematics, for students, in connection to the grade-level content standards, as expected by the mathematical practice standards. 

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

• Unit 1, Adding, Subtracting, and Working with Data, Lesson 10, Warm-up, students use specialized language as they describe the features of data representations. Narrative states, “Students use and revise their language to clearly describe the features of each data representation and explain how they are the same and how they are different (MP6).” Launch states, “Groups of 2. Display image. ‘Pick one that doesn’t belong. Be ready to share why it doesn’t belong.’ 1 minute: quiet think time.”

• Unit 3, Measuring Length, Lesson 1, Cool-down, students attend to precision as they practice measuring, iterating same-size length units, and identifying the need for standard units of measurement. Student Facing states, ”1. How long is the rectangle? Use centimeter cubes to measure. 2. Clare got 6 when she measured the same rectangle. Why might her measurement be different?” Narrative states, “The purpose of this activity is for students to understand why it is important to be precise about the length of the unit used to measure (MP6).”

• Unit 5, Numbers  to 1000, Lesson 4, Activity 2, students attend to precision when they use what they know about the meaning of the digits in a three-digit number to identify the value to make an equation true. Activity states, “‘Find the number that makes each equation true.’ 6 minutes: partner work time. Monitor for students who agree with Elena because: 37 would mean 3 tens and 7 ones, if there are 3 hundreds, you need 3 digits.” Student Facing states, “Find the number that makes each equation true. Use base-ten blocks or diagrams if they help. 1. 4 hundreds + 6 tens + 2 ones =  ___ 2. 7 ones + 2 hundreds + 6 tens =___ 3. 3 tens + 5 hundreds  ___   4. 325 = ___hundreds  ___ + ones  ___ + tens ___  5. 70+300+2= 6. 836=6+800+___ 7. Clare and Elena worked to find the number that makes the equation true: 7 ones + 3 hundreds ___. They wrote different answers. Clare wrote 7 ones +  3 hundreds  = 37. Elena wrote 7 ones + 3 hundreds = 307. Who do you agree with? Explain.” Narrative states, “Throughout the activity, encourage students to explain how they know they have made true equations using precise language about the meaning of each digit in a 3-digit number (MP3, MP6).”

• Unit 6, Geometry, Time and Money, Lesson 6, Activity 1, students attend to the specialized language as they compose the same shape in different ways. Activity states, “Mai used pattern blocks to make this design. ‘Work with a partner to make the same design without using any yellow hexagons. Try to use as many different shape combinations as you can to make each hexagon. For each hexagon, draw the lines inside the shape to show how you composed it. Pick one of your hexagons. Use words and numbers to explain how you composed it.’ 10 minutes: partner work time. Monitor for students who: compose a hexagon using equal-size shapes: 2 trapezoids, 6 triangles, or 3 blue rhombuses, compose hexagons using different shapes.” Student Facing states, ”Mai used pattern blocks to make this design. Work with a partner to make the same design without using any yellow hexagons.” Narrative states, “Throughout the activity, listen for the ways students notice and describe how they can compose a shape from or decompose shapes into smaller shapes (MP6).” Activity Synthesis states, “Invite previously identified students to display their hexagons. Begin with the examples of hexagons composed of the same shape. Then select students to share other examples of hexagons composed of different shapes. If possible, display student hexagons as they share. Keep the hexagons displayed into two groups like the following: You found a lot of different ways to compose a butterfly design without using hexagons. What do you notice about these two groups of hexagons? (In the first group, they are made using the same shape. 6 triangles, 2 trapezoids, or 3 rhombuses. Each hexagon in the second group is made using more than 1 shape.)”

• Unit 8, Equal Groups, Lesson 7, Cool-down, students use specialized language as they work with and describe arrays. Student Facing states, “1. How many rows are in this array? 2. How many counters are in each row? 3. How many counters are there in all?” Narrative states, “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 create arrays given a number of counters and a number of rows (MP6). They may use trial and error to build these arrays.”

• Unit 9, Putting It All Together, Lesson 7, Warm-up, students use precision as they compare the digits in expressions. Narrative states, “This warm-up prompts students to carefully analyze and compare expressions. In making comparisons, students have a reason to use language precisely. Listen for the language students use to describe and compare the expressions with a focus on descriptions of the digits, the operations, place value, and whether or not units may be composed or decomposed when using methods based on place value (MP6).” Activity states, “Pick one that doesn’t belong. Be ready to share why it doesn’t belong.” Student Facing states, “Which one doesn’t belong? 74-23, 24+37, 4 tens + 2 ones + 3 tens + 7 ones, 60+19.”

The materials reviewed for Kendall Hunt's Illustrative Mathematics Grade 2 meet expectations for supporting the intentional development of MP7: Look for and make use of structure; and MP8: Look for and express regularity in repeated reasoning, for students, in connection to grade-level content standards, as expected by the mathematical practice standards. Students have opportunities to engage with the Math Practices across the year and they are often explicitly identified for teachers within the Course Guide (How to Use These Materials) and within specific lessons (Instructional Routines, Lesson Preparation Narratives, and Lesson Activities’ Narratives).

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

• Unit 1, Adding, Subtracting, and Working with Data, Lesson 2, Warm-up, students look for and make use of structure as they reason about quantities within 10. Student Facing states, “What do you know about 10?” Launch states, “Display the number. ‘What do you know about 10?’ 1 minute: quiet think time.” Activity Synthesis states, “If needed, ‘How could we represent the number 10?’” Narrative states, “When students share about numbers that are close to 10 when counting, relate 10 ones to the unit ten, and sums and differences with the value of 10, they show what they know about the structure of whole numbers, place value, and the properties of operations (MP7).”

• 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 Facing states, “The Plant Project. Answer the questions based on your line plot. 1. What was the shortest plant height? 2. What was the tallest plant height? 3. What is the difference between the height of the tallest plant and the shortest plant? Write an equation to show how you know. 4. Han looked at this line plot and said that the tallest plant was 29 centimeters. Do you agree with him? Why or why not? 5. How many plants were measured in all? 6. Write a statement based on Han’s line plot.” Activity Synthesis states, “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.” Narrative states, “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).”

• Unit 8, Equal Groups, Lesson 1, Cool-down, students look for and make use of structure while they arrange a number of objects into two equal groups and reason about numbers that form two equal groups without any objects left over. Student Facing states, “Noah and Lin want to share 11 connecting cubes equally. How many will each student get? Will there be any leftovers? Show your thinking using diagrams, symbols, or other representations. You may use cubes if it helps.” Activity 1 Narrative states, “When students notice that some collections of objects can be shared equally while others can not, they observe an important mathematical structure (MP7) which they will name in a future lesson.” They demonstrate this same understanding within the lesson Cool-down.

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

• Unit 3, Measuring Length, Lesson 3, Activity 2, students use repeated reasoning to measure the length of rectangles with the rulers created in a previous activity. Student Facing states, “1. Use your ruler to measure the length of each rectangle. Don’t forget to label your measurements. Images of different lengths are shown for A -F. 2. How many centimeters longer is rectangle A than rectangle B? 3. How many centimeters longer is rectangle F than rectangle D? 4. Which two rectangles are the longest? How long would the rectangle be if you joined them together?” Activity states, “‘Measure the length of each rectangle with your ruler. You can use the centimeter cubes and 10-centimeter blocks to check your measurement if it helps you. When you finish, check your measurements with your partner and work together to answer the questions.’ 3 minutes: independent work time. 5–7 minutes: partner work time. Monitor for students who find the difference between the longest and shortest length by: directly measuring the length from the end of the shortest rectangle to the end of the longest rectangle, measuring both rectangles and finding the difference.” Activity Synthesis states, “Share measurements for each rectangle. Discuss any differences in measurement. ‘How was the number 0 helpful when you measured each rectangle?’ (It showed us where to put the tool. If you start with 0 then the length is the closest number to the end of the rectangle.) Invite previously identified students to share how they found the difference between the shortest and longest rectangles. ‘How can we use our ruler to prove that the longest rectangle is 10 cm longer than the shortest rectangle?’” Narrative states, “Students notice that each labeled tick mark on the ruler represents a length in centimeters from zero (MP8).”

• Unit 6, Geometry, Time, and Money, Lesson 16, Cool-down, students use repeated reasoning to identify quarters and find the total value of a set of coins including quarters. Student Facing states, “Tyler had 6 pennies, 2 dimes, 2 quarters, and 2 nickels in his pocket. How many cents does Tyler have? Show your thinking using drawings, numbers, words, or an equation.” Lesson Narrative states, “Throughout the lesson, students make connections between quarters and combinations of other coins and notice that if they look for ways to use coins with a larger value first, they can be more certain they are using the fewest amount of coins (MP8).” After repeated reasoning about the value of coins in this lesson and other lessons, the Cool-down provides an opportunity for students to demonstrate their understanding.

• 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 Facing states, “Find the value of each expression mentally. 10-6, 14-6, 56-6, 56-26.” 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 56-26? (26 has 2 more tens than 6 so that means 2 tens need to be taken away from the answer to 56-6.)” Narrative states, “When students consider how they can use known differences, like 10-6 or 14-6, to find the value of the other expressions, they look for and make use of structure and express regularity in repeated reasoning (MP7, MP8).”