The materials reviewed for Open Up Resources K–5 Math 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 the Grade 2 Course Guide, Design Principles, conceptual understanding is a part of the design of the materials. Balancing Rigor states, “There are three aspects of rigor essential to mathematics: conceptual understanding, procedural fluency, and the ability to apply these concepts and skills to mathematical problems with and without real-world contexts. These aspects are developed together and are therefore interconnected in the materials in ways that support student understanding. 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, Student Work Time, students develop conceptual understanding as they learn about the ways a line plot can be used to represent data collected from measuring objects. Student Work Time 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, Section A, 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 minute: partner discussion. Share responses. Display 357 and $300+50+7$. ‘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, Section A, Lesson 5, Warm-up, Student Work Time and Activity Synthesis, 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. In Student Work Time, 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 the Grade 2 Course 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 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, Section B, Lesson 9, Activity 2, Student Work Time, students demonstrate conceptual understanding as they use place value to compare numbers based on different representations. Student Work Time 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. a. Locate and label 420 and 590 on the number line. b. Use <, >, and = to compare 420 and 590. 2. a. 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. b. Use <, >, and = to compare 378 and 387. 3. a. Diego and Jada compared 2 numbers. Use their work to figure out what numbers they compared. then use <, >, and = to compare the numbers. b. 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, Section B, Lesson 8, Activity 2, Student Work Time,  students demonstrate conceptual understanding as they analyze base-ten diagrams and corresponding equations representing sums. Images of base ten blocks and equations are provided.  In Student Work Time, Student Facing states, “a. 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. b. 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, Section B, Lesson 6, Warm-up, Student Work Time, students demonstrate conceptual understanding as they explain why an equation is true based on place value. Student Work Time 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 Open Up Resources K–5 Math Grade 2 meet expectations for giving attention throughout the year to individual standards that set an expectation of procedural skill and fluency. 

According to the Grade 2 Course Guide, 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, Section A, Lesson 1, Activity 1, Launch and Student Work Time, 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.’” In Student Work Time, 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, Section B, Lesson 8, Warm-up, Launch and Student Work Time, 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.” Student Work Time states, “‘What patterns do you see?’ 1–2 minutes: quiet think time. Record responses.” (2.NBT.2)

• Unit 9, Putting It All Together, Section A, Lesson 4, Activity 1, Launch and Student Work Time, 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.” Student Work Time states, “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 the Grade 2 Course 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, Section B, Lesson 10, Activity 1, Launch, 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, Section B, Lesson 9, Activity 1, Student Work Time, students demonstrate procedural skill and fluency as they add and subtract within 100. Student Work Time 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, “a. What number makes this equation true?___. $38-4=?$ Represent your thinking on the number line. b. What number makes this equation true?___. $75-68=?$ Represent your thinking on the number line. c. What number makes this equation true?___. $57-24=?$” (2.MD.6, 2.NBT.5)

• Unit 9, Putting It All Together, Section A, 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 a. $11-5$ b. $12-3$ c. $16-8$ d. $9+3$ e. $8+8$ f. $13-8$.” (2.OA.2)

The materials reviewed for Open Up Resources K–5 Math Grade 2 meet expectations for being designed so that teachers and students spend sufficient time working with engaging applications of mathematics. 

According to the Grade 2 Course Guide, 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 teacher support and independently. According to the Grade 2 Course 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 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, Section B, Lesson 10, Activity 2, Launch and Student Work Time, 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).” Student Work Time 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.” In Student 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, Section B, 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, Section A, 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, Section C, Lesson 16, Activity 1, Student Work Time, 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, Section C, Lesson 13, Activity 2, Student Work Time, students use tape diagrams and equations to represent addition and subtraction story problems within 100. Student Work Time 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, Section A, Lesson 2, Activity 2, Student Work Time, students draw shapes that have a given number of sides or corners, and then compare the shapes. Student Work Time 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, “a. 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) b. Draw a different shape that could be Clare’s shape. c. Andre drew a shape that has 4 corners. Circle shapes that could be Andre’s shape. (6 figures are shown on dot paper) d. 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 Open Up Resources K–5 Math 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, Section C, Lesson 16, Activity 2, Student Work Time, students apply their understanding of addition and subtraction strategies when solving real-world problems. Student Work Time 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, “a. Jada read 10 fewer pages than Noah. Noah read 27 pages. How many pages did Jada read? b. Noah spent 25 minutes reading. Jada spent 30 more minutes reading than Noah. How many minutes did Jada spend reading? c. Jada read 47 pages of the book. Noah read 20 pages of the book. How many fewer pages did Noah read? d. 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, Section B, Lesson 11, Activity 2, Student Work Time, students develop fluency with addition and subtraction within 100. Student Work Time 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 a. Find the value of $59+27$   Find the value of $65-18$. Partner B b. Find the value of $68-39$. Find the value of $22+49$.” (2.NBT.5)

• Unit 5, Numbers to 1000, Section B, Lesson 9, Activity 1, Student Work Time, students deepen their conceptual understanding as they make sense of different methods to compare three-digit numbers. Student Work Time 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. a. What is the same and different about these students’ representations? Discuss with a partner. b. 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. c. 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, Section B, 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. a. 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? b. 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, Section B, Lesson 8, Activity 2, Launch and Student Work Time, students extend conceptual understanding and procedural skills as they use different methods to decompose numbers. Launch states, “Groups of 2, 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.)” In Student Work Time, Student Facing states, “1. Andre found the value of $65-28$. He made a base-ten diagram and wrote equations to show his thinking. 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. $71-53$.” (2.NBT.5, 2.NBT.9)

• Unit 8, Equal Groups, Lesson 1, Activity 2, Launch and Student Work TIme, students use conceptual understanding and apply their understanding of equal groups to find ways to solve routine real-world problems. Launch states, “Groups of 2, Give students access to connecting cubes or counters. ‘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.” In Student Work Time, 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. a. How many marbles would Andre and his sister get? Would there be any marbles left out of the game? Show your thinking. b. What if Andre had 18 marbles? How many would each person get? Would there be any marbles left out? Show your thinking. c. 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 Open Up Resources K–5 Math 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). A chart is provided within this section that highlights several lessons that showcase particular Mathematical Practices. The Mathematical Practices are also identified within specific lessons (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, Section C, Lesson 11, Activity 2, Student Work Time and Activity Narrative, students solve different types of story problems where they compose or decompose tens when adding or subtracting. Student Work Time 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.” In Student Work Time, Student Facing states, “Solve each story problem. Show your thinking. a. Lin had 31 sunflower seeds. She gave Priya 15 seeds. How many seeds does Lin have now? b. 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? c. Elena gathered 50 pumpkin seeds. Andre collected 23 fewer pumpkin seeds than Elena. How many seeds did Andre collect?” Activity 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, Section B, Lesson 14, Warm-up, Student Work Time and Activity Narrative, 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. Activity 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, Section D, 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, Launch and Activity Narrative, students reason about tape diagrams as similar to bar graphs and can be used to represent the same data. Launch states, “Groups of 2, Display the image. ‘What do you notice? What do you wonder?’” Activity 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, Section B, Lesson 11, Activity 1, Launch, Student Work Time, and Activity Narrative, students reason about length measurements and a tape diagram representation. 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.’” Student Work Time 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.” In Student Work Time, Student Facing states, “1. What do you notice? What do you wonder? 2. 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.” Activity 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, Section C, Lesson 12, Activity 1, Student Work Time and Activity Narrative, students write story problems for equations with an unknown value. Student Work Time 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.” In Student Work Time, 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 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 Open Up Resources K–5 Math 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 the Math Practices across the year, and they are often explicitly identified for teachers within the Course Guide (How to Use These Materials). A chart is provided within this section that highlights several lessons that showcase particular Mathematical Practices. The Mathematical Practices are also identified within specific lessons (Lesson Preparation Activity Narratives and Lesson Activities’ Activity Narratives).

According to the Grade 2 Course Guide, Design Principles, Learning Mathematics By Doing Mathematics, “Students learn mathematics by doing mathematics, rather than by watching someone else do mathematics or being told what needs to be done. Doing mathematics can be defined as learning mathematical concepts and procedures while engaging in the mathematical practices - making sense of problems, reasoning abstractly and quantitatively, making arguments and critiquing the reasoning of others, modeling with mathematics, making appropriate use of tools, attending to precision in their use of language, looking for and making use of structure, and expressing regularity in repeated reasoning. By engaging in the mathematical practices with their peers, students have the opportunity to see themselves as mathematical thinkers with worthwhile ideas and perspectives.”

Students 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, Section A, Lesson 4, Activity 1, Launch, Student Work Time and Activity Narrative, students construct viable arguments and critique the reasoning of others as they practice the skill of estimating a reasonable length in centimeters. Launch states, “Groups of 2, 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.” Student Work Time 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.” In Student Work Time, Student Facing states, “a. Record an estimate that is: too low, about right, too high. b. Record an estimate that is: too low, about right, too high. c. Record an estimate for each object on the recording sheet. d. 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. Activity 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, Section A, Lesson 3, Activity 2, Student Work Time and Activity Narrative, students construct arguments when they represent numbers up to 100 on a number line. Student Work Time 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.” In Student Work Time, 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. a. Locate and label 17 on the number line. b. Locate and label 59 on the number line. c. Locate and label 43 on the number line. d. Locate and label 35 on the number line. e. Share your number lines with your partner.” Images of number lines are shown for each problem. Activity 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, Section B, Lesson 8, Activity 1, Launch, Student Work Time, and Activity Synthesis, 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.” In Student Work Time, 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. a. Which of these shows fourths or quarters? Explain and share with your partner. b. Name the shaded piece. Shape A has a ___ shaded. Shape B has a ___ shaded. c. Show 2 different ways to partition the rectangle into quarters or fourths. Shade in a fourth of each rectangle. d. 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, Section B, 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.” Activity 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, Section B, Lesson 12, Activity 1, Student Work Time and Activity Narrative, students critique the reasoning of others as they interpret the order of numbers. In Student Work Time, 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. Student Work Time 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.” Activity 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, Section C, Lesson 16, Activity 1, Launch and Student Work Time, 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.” Student Work Time 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.” In Student Work Time, Student Facing states, “1. a. Discuss how Jada’s equations match Lin’s diagram. b. 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 Open Up Resources K–5 Math 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). A chart is provided within this section that highlights several lessons that showcase particular Mathematical Practices. The Mathematical Practices are also identified within specific lessons (Lesson Preparation Instructional Routines and Lesson Activities’ Instructional Routines).

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, Section C, Lesson 18, Activity 1, Student Work Time, 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. In Student Work Time, Student Facing states, “a. What is your survey question? b. What are your categories? Category 1: ___ Category 2: ___ Category 3: ___ Category 4: ___. c. Record the data. d. 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.” Preparation, 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, Section B, Lesson 15, Activity 1, Student Work Time and Launch, 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. In Student Work Time, Student Facing states, “Solve Kiran’s age riddles. Show your thinking. Use a number line if it helps. a. I’m 7. My sister is 5 years older than I am. How old is she? ___ years old. b. If you add 27 years to my sister’s age, you get my mom’s age. How old is my mom? ___ years old. c. My brother is 24 years younger than my mom. How old is my brother? ___ years old. d. My grandma is 53 years older than my brother. How old is my grandma? ___ years old. e. My uncle is 21 years younger than my grandma. How old is my uncle? ___ years old. f. My uncle is 33 years older than my cousin. How old is my cousin? ___ years old. g. 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, “Groups of 2, ‘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.” Student Work Time 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.” Preparation, 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. Preparation, 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, Section A, Lesson 3, Activity 1, Student Work Time, Launch, Instructional Routine, 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. In Student Work Time, 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, “Groups of 2, 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?’” Student Work Time 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.” Instructional Routine 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, Section B, 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.” 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, Section C, Lesson 12, Activity 1, Student Work Time, Launch, and Instructional Routine, students subtract one-digit and two-digit numbers from a three-digit number using strategies that make sense to them. In Student Work Time, Student Facing states, “Find the value of each expression in any way that makes sense to you. Explain or show your reasoning. a. $354-7$, b. $354-36$, c. $354-48$.” Launch states, “Groups of 2, Give students access to base-ten blocks.” Student Work Time 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.” Instructional Routine 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 Open Up Resources K–5 Math 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 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). A chart is provided within this section that highlights several lessons that showcase particular Mathematical Practices. The Mathematical Practices are also identified within specific lessons (Lesson Preparation Narratives and Lesson Activities’ Narratives).

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, Section B, Lesson 10, Warm-up, Launch, students use specialized language as they describe the features of data representations. Activity 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, Section A, 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?” Activity Narrative  (for Activity 2) 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, Section A, Lesson 4, Activity 2, Student Work Time, students attend to precision when they use prior knowledge of the meaning of the digits in a three-digit number to identify the value to make an equation true. Student Work Time 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.” In Student Work time, Student Facing states, “Find the number that makes each equation true. Use base-ten blocks or diagrams if they help. a. 4 hundreds + 6 tens + 2 ones =  ___ b. 7 ones + 2 hundreds + 6 tens =___ c. 3 tens + 5 hundreds  ___   d. 325 = ___hundreds  ___ + ones  ___ + tens ___  e. $70+300+2=$ f. $836=6+800+$___ g. 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.” Activity 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, Section B, Lesson 6, Activity 1, Student Work Time and Activity Synthesis, students attend to the specialized language as they compose the same shape in different ways. Student Work Time 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.” In Student Work Time, 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.” Activity 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, Section B, Lesson 7, Cool-down, students use specialized language as they work with and describe arrays. Student Facing states, “a. How many rows are in this array? b. How many counters are in each row? c. How many counters are there in all?” Activity Narrative (for Activity 2) 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, Section B, Lesson 7, Warm-up, students use precision as they compare the digits in expressions. Activity 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).” Student Work Time states, “Pick one that doesn’t belong. Be ready to share why it doesn’t belong.” In Student Work Time, 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 Open Up Resources K–5 Math 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). A chart is provided within this section that highlights several lessons that showcase particular Mathematical Practices. The Mathematical Practices are also identified within specific lessons (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, Section A, Lesson 2, Warm-up, Student Work Time and Activity Synthesis, students look for and make use of structure as they reason about quantities within 10. In Student Work Time, 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?’” Activity 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, Section C, Lesson 16, Activity 2, Student Work Time and Activity Synthesis, students look for and make use of structure as they interpret measurement data represented by line plots. In Student Work Time, Student Facing states, “1. The Plant Project. Answer the questions based on your line plot. a. What was the shortest plant height? b. What was the tallest plant height? c. What is the difference between the height of the tallest plant and the shortest plant? Write an equation to show how you know. 2. a. Han looked at this line plot and said that the tallest plant was 29 centimeters. Do you agree with him? Why or why not? b. How many plants were measured in all? c. 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.” Activity 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, Section A, 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, Section C, Lesson 3, Activity 2, students use repeated reasoning to measure the length of rectangles with the rulers created in a previous activity. In Student Work Time, 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?” Student Work Time 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?’” Activity 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, Section D, 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.” Preparation, 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, Section C,  Lesson 9, Warm-up, Student Work Time and Activity Synthesis, students use repeated reasoning to find the value of differences when they may need to decompose a ten. In Student Work Time, 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 $56-6$.)” Activity 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).”

The materials reviewed for Open Up Resources K-5 Math Grade 2 meet expectations for providing teachers guidance with useful annotations and suggestions for how to enact the student materials and ancillary materials, with specific attention to engaging students in order to guide their mathematical development. 

Within the Course Guide, several sections (Design Principles, A Typical Lesson, How to Use the Materials, and Key Structures in This Course) provide comprehensive guidance that will assist teachers in presenting the student and ancillary materials. Examples include but are not limited to:

• Resources, Course Guide, Design Principles, Learning Mathematics by Doing Mathematics, “A problem-based instructional framework supports teachers in structuring lessons so students are the ones doing the problem solving to learn the mathematics. The activities and routines are designed to give teachers opportunities to see what students already know and what they can notice and figure out before having concepts and procedures explained to them. The teacher has many roles in this framework: listener, facilitator, questioner, synthesizer, and more.”

• Resources, Course Guide, A Typical Lesson, “A typical lesson has four phases: 1. a warm-up; 2. one or more instructional activities; 3. the lesson synthesis; 4. a cool-down.” “A warm-up either: helps students get ready for the day’s lesson, or gives students an opportunity to strengthen their number sense or procedural fluency.” An instructional activity can serve one or many purposes: provide experience with new content or an opportunity to apply mathematics; introduce a new concept and associated language or a new representation; identify and resolve common mistakes; etc. The lesson synthesis “assists the teacher with ways to help students incorporate new insights gained during the activities into their big-picture understanding.” Cool-downs serve “as a brief formative assessment to determine whether students understood the lesson.”

• Resources, Course Guide, How to Use the Materials, “The story of each grade is told in eight or nine units. Each unit has a narrative that describes the mathematical work that will unfold in that unit. Each lesson in the unit also has a narrative. Lesson narratives explain: the mathematical content of the lesson and its place in the learning sequence; the meaning of any new terms introduced in the lesson; how the mathematical practices come into play, as appropriate. Activities within lessons also have narratives, which explain: the mathematical purpose of the activity and its place in the learning sequence, what students are doing during the activity, what the teacher needs to look for while students are working on an activity to orchestrate an effective synthesis, connections to the mathematical practices, when appropriate.”

• Resources, Course Guide, Scope and Sequence lists each of the nine units, a Pacing Guide to plan instruction, and Dependency Diagrams. These Dependency Diagrams show the interconnectedness between lessons and units within Grade 2 and across all grades.

• Resources, Course Guide, Course Glossary provides a visual glossary for teachers that includes both definitions and illustrations. Some images use examples and nonexamples, and all have citations referencing what unit and lesson the definition is from.

Materials include sufficient and useful annotations and suggestions that are presented within the context of the specific learning objectives. Several components focus specifically on the content of the lesson. Examples include:

• Unit 4, Addition and Subtraction on a Number Line, Lesson 5, Lesson Narrative, "In previous lessons, students estimated the length of objects using what they know about the size of standard length units and the tools used to measure them. Students have located numbers on number lines using what they know about the structure of a number line and the labeled tick marks. The purpose of this lesson is for students to extend this understanding by estimating numbers on number lines that do not have tick marks to represent each consecutive whole number. Students use their understanding of length and unit intervals on the number line to estimate. Students should be encouraged throughout the lesson to explain why their estimates are reasonable using what they know about number, length, and the structure of the number line."

• Unit 6, Geometry, Time and Money, Section D, Lesson 15, Activity 1, Launch, provides teachers guidance on how to represent coins. "Display the pre-made poster to show front and back images of pennies, nickels, and dimes. 'Each coin has a value in cents. Does anyone know the names or values of these coins?' Share and record responses, Write the name and value of each coin on the poster. 'When we write the total value we use the cent symbol after the number to show that it represents cents.' Demonstrate writing the ¢ symbol next to the amount."

• Unit 9, Putting It All Together, Section A, Lesson 1, Activity 1, “The purpose of this activity is for students to identify the addition facts within 20 that they do not yet know from memory. They write these sums on index cards which can be used to help students build fluency throughout the section. Students should store these cards to use again in an upcoming lesson. The number choices in this activity include some of the facts that students may still be working to recall from memory at this point in the school year. If desired, the inventory of sums that students complete at the beginning of the activity could be replaced with a list of all sums within 20 or a smaller set of sums that best fit the needs of your students.”

The materials reviewed for Open Up Resources K-5 Math Grade 2 meet expectations for containing adult-level explanations and examples of the more complex grade-level concepts and concepts beyond the current grade so that teachers can improve their own knowledge of the subject. 

Unit Overviews and sections within lessons include adult-level explanations and examples of the more complex grade-level concepts. Within the Course Guide, How to Use the Materials states, “Activities within lessons also have narratives, which explain: the mathematical purpose of the activity and its place in the learning sequence, what students are doing during the activity, what the teacher needs to look for while students are working on an activity to orchestrate an effective synthesis, connections to the mathematical practices, when appropriate.” Examples include:

• Unit 4, The Structure of the Number Line, Section A, Lesson 4, Compare Numbers on a Number Line, Lesson Narrative, “In previous lessons, students learned how to locate a number on the number line and represent numbers with labeled tick marks and points. They used multiples of 5 and 10 to help them locate numbers up to 100 on a number line. In this lesson, students recognize that as you move to the right on the number line, numbers increase in value because they are a greater distance from 0. Students also use the relative position of numbers and generalize that a number that is greater than a given number if it is farther to the right on the number line. To demonstrate this understanding, students compare numbers within 100 (a skill from grade 1) and use the number line to explain their comparison (MP7).”

• Unit 5, Numbers to 1,000, Section B, Lesson 12, Warm-up, Instructional Routine, “The purpose of this Number Talk is to elicit strategies and understandings students have for mentally subtracting a multiple of 10 from a number. Building on their understanding of place value, students subtract tens from tens. These understandings help students develop fluency and will be helpful in later lessons when students will need to be able to subtract using strategies based on place value.”

• Unit 8, Equal Groups, Overview, “Later, students transition from working with arrays containing discrete objects to equal-size squares within a rectangle. They build rectangular arrays using inch tiles and partition rectangles into rows and columns of equal-size squares. The work here sets the stage for the concept of area in grade 3.”

Also within the Course Guide, About These Materials, Further Reading states, “The curriculum team at Open Up Resources has curated some articles that contain adult-level explanations and examples of where concepts lead beyond the indicated grade level. These are recommendations that can be used as resources for study to renew and fortify the knowledge of elementary mathematics teachers and other educators.” Examples include:

• Resources, Course Guide, About These Materials, Further Reading, K-2, “Units, a Unifying Idea in Measurement, Fractions, and Base Ten. In this blog post, Zimba illustrates how units ‘make the uncountable countable’ and discusses how the foundation built in K-2 measurement and geometry around structuring space allows for the development of fractional units and beyond to irrational units.”

• Resources, Course Guide, About These Materials, Further Reading, Entire Series, “The Number Line: Unifying the Evolving Definition of Number in K-12 Mathematics. In this article, the authors (Lahme, McLeman, Nakamaye, and Umland) focus their attention on the selection of definitions, notation, and graphical conventions surrounding the development of the real numbers from kindergarten to grade 12, and address the work that students might do in later years.”

The materials reviewed for Open Up Resources K-5 Mathematics Grade 2 meet expectations for including standards correlation information that explains the role of the standards in the context of the overall series. 

 Correlation information can be found within different sections of the Course Guide and within the Standards section of each lesson. Examples include:

• Resources, Course Guide, About These Materials, CCSS Progressions Documents, “The Progressions for the Common Core State Standards describe the progression of a topic across grade levels, note key connections among standards, and discuss challenging mathematical concepts. This table provides a mapping of the particular progressions documents that align with each unit in the K–5 materials for further reading.”

• Resources, Course Guide, How to Use the Materials, Noticing and Assessing Student Progress in the Mathematical Practices, The Standards for Mathematical Practices Chart, “The unit-level Mathematical Practice chart is meant to highlight a handful of lessons in each unit that showcase certain Mathematical Practices. Some units, due to their size or the nature of their content, may have fewer predicted chances for students to engage in a particular Mathematical Practice. A dash in the chart indicates that there may not be enough opportunities to reliably look for this Mathematical Practice in the unit. One primary place Mathematical Practice 4 is tagged is the optional modeling lesson at the end of each unit. Aside from these lessons, optional activities and lessons are not included in this chart.”

• Resources, Course Guide, Scope and Sequence, Dependency Diagrams, All Grades Unit Dependency Diagram identifies connections between the units in grades K-5. Additionally, a “Section Dependency Diagram” identifies specific connections within the grade level.

• Resources, Course Guide, Lesson and Standards, provides two tables: a Standards by Lesson table, and a Lessons by Standard table. Teachers can utilize these tables to identify standard/lesson alignment.

• Unit 1, Adding, Subtracting, and Working With Data, Section B, Lesson 8, Standards, “Building On: 1.OA.C.5, 2.MD.D.10, Addressing: 2.MD.D.10 Draw a picture graph and a bar graph (single-unit scale) to represent data set with up to four categories. Solve simple put-together, take-apart, and compare problems using information presented in a bar graph.”

Explanations of the role of specific grade-level mathematics can be found within different sections of the Resources, Course Guide, Unit Overviews, Section Overviews, and Lesson Narratives. Examples include:

• Resources, Course Guide, Scope and Sequence, each Unit provides Unit Learning Goals, for example, “Students measure and estimate lengths in standard units and solve measurement story problems within 100.” Additionally, each Unit Section provides Section Learning Goals, “measure length in centimeters and meters.”

• Unit 3, Measuring Length, Unit Overview, "Students relate the structure of a line plot to the tools they used to measure lengths. This prepares students for the work in the next unit, where they interpret numbers on the number line as lengths from 0. The number line is an essential representation that will be used in future grades and throughout students’ mathematical experiences."

• Unit 5, Numbers to 1,000, Section A, Lesson 2, Lesson Narrative, "In a previous lesson, students learned that a hundred is composed of 10 tens or 100 ones. In this lesson, students deepen their understanding of a hundred as a unit. They learn that for every 10 tens, they can compose 1 hundred. Students notice that it may be easier to count the hundreds rather than count the tens to find a total value. Students begin to recognize and describe the patterns in the structure of the base-ten system (MP7, MP8). They recognize that 10 tens make 1 hundred, 30 tens make 3 hundreds, 60 tens make 6 hundreds, etc. as they build numbers with tens and exchange them for hundreds. Students identify the multiples of 100 written as numerals and begin to make connections between base-ten blocks and the value of each digit in a three-digit number."

• Unit 6, Geometry, Time, and Money, Section C, Section C Overview, "In this section, students use their understanding of fourths and quarters to tell time. In grade 1, students learned to tell time to the hour and half-hour. Here, they make a connection between the analog clock and circles partitioned into halves or fourths."

The materials reviewed for Open Up Resources K-5 Math Grade 2 meet expectations for providing explanations of the instructional approaches of the program and identification of the research-based strategies. 

The materials explain and provide examples of instructional approaches of the program and include and reference research-based strategies. Both the instructional approaches and the research-based strategies are included in the Course Guide under the Resources tab for each unit. Design Principles describe that, “It is our intent to create a problem-based curriculum that fosters the development of mathematics learning communities in classrooms, gives students access to mathematics through a coherent progression, and provides teachers the opportunity to deepen their knowledge of mathematics, student thinking, and their own teaching practice.” Examples include:

• Resources, Course Guide, Design Principles, “In order to design curriculum and professional learning materials that support student and teacher learning, we need to be explicit about the principles that guide our understanding of mathematics teaching and learning. This document outlines how the components of the curriculum are designed to support teaching and learning aligning with this belief.” Principles that guide mathematics teaching and learning include: All Students are Capable Learners of Mathematics, Learning Mathematics by Doing Mathematics, Coherent Progression, Balancing Rigor, Community Building, Instructional Routines, Using the 5 Practices for Orchestrating Productive Discussions, Task Complexity, Purposeful Representations, Teacher Learning Through Curriculum Materials, and Model with Mathematics K-5.

• Resources, Course Guide, Design Principles, Community Building, “Students learn math by doing math both individually and collectively. Community is central to learning and identity development (Vygotsky, 1978) within this collective learning. To support students in developing a productive disposition about mathematics and to help them engage in the mathematical practices, it is important for teachers to start off the school year establishing norms and building a mathematical community. In a mathematical community, all students have the opportunity to express their mathematical ideas and discuss them with others, which encourages collective learning. ‘In culturally responsive pedagogy, the classroom is a critical container for empowering marginalized students. It serves as a space that reflects the values of trust, partnership, and academic mindset that are at its core’ (Hammond, 2015).”

• Resources, Course Guide, Design Principles, Instructional Routines, “Instructional routines provide opportunities for all students to engage and contribute to mathematical conversations. Instructional routines are invitational, promote discourse, and are predictable in nature. They are ‘enacted in classrooms to structure the relationship between the teacher and the students around content in ways that consistently maintain high expectations of student learning while adapting to the contingencies of particular instructional interactions.’ (Kazemi, Franke, & Lampert, 2009)”

• Resources, Course Guide, Key Structures in This Course, Student Journal Prompts, Paragraph 3, “Writing can be a useful catalyst in learning mathematics because it not only supplies students with an opportunity to describe their feelings, thinking, and ideas clearly, but it also serves as a means of communicating with other people (Baxter, Woodward, Olson & Robyns, 2002; Liedke & Sales, 2001; NCTM, 2000).”

The materials reviewed for Open Up Resources K-5 Math Grade 2 meet expectations for including a comprehensive list of supplies needed to support the instructional activities.

In the Course Guide, Materials, there is a list of materials needed for each unit and each lesson. Lessons that do not have materials are indicated by none; lessons that need materials have a list of all the materials needed. Examples include:

• Resources, Course Guide, Key Structures in This Course, Representations in the Curriculum, provides images and explanations of representations for the grade level. “5-frame and 10-frame (K-2): 5- and 10-frames provide students with a way of seeing the numbers 5 and 10 as units and also combinations that make these units. Because we use a base-ten number system, it is critical for students to have a robust mental representation of the numbers 5 and 10. Students learn that when the frame is full of ten individual counters, we have what we call a ten, and when we cannot fill another full ten, the ‘extra’ counters are ones, supporting a foundational understanding of the base-ten number system. The use of multiple 10-frames supports students in extending the base-ten number system to larger numbers.”

• Resources, Course Guide, Materials, includes a comprehensive list of materials needed for each unit and lesson.. The list includes both materials to gather and hyperlinks to documents to copy. “Unit 5, Lesson 4 - Gather: Base-ten blocks, Number cards 0-10; Copy: Greatest of Them All Stage 1 Recording Sheet, Mystery Number Stage 1 Directions.”

• Unit 6, Geometry, Time and Money, Section D, Lesson 15, Materials Needed, “Activities: Scissors (Activity 2); Centers: Paper (How Are They the Same?, Stage 2), Picture books (Picture Books, Stage 3).”

The materials reviewed for Open Up Resources K-5 Math Grade 2 meet expectations for providing strategies and supports for students in special populations to support their regular and active participation in learning grade-level/series mathematics.

Materials regularly provide strategies, supports, and resources for students in special populations to help them access grade-level mathematics as suggestions are outlined within each lesson. According to the Resources, Course Guide, Universal Design for Learning and Access for Students with Disabilities, “Supplemental instructional strategies that can be used to increase access, reduce barriers and maximize learning are included in each lesson, listed in the activity narratives under ‘Access for Students with Disabilities.’ Each support is aligned to the Universal Design for Learning Guidelines (udlguidelines.cast.org), and based on one of the three principles of UDL, to provide alternative means of engagement, representation, or action and expression. These supports provide teachers with additional ways to adjust the learning environment so that students can access activities, engage in content, and communicate their understanding.” Examples of supports for special populations include: 

• Unit 3, Measuring Length, Section B, Lesson 10, Activity 1, Access for Students with Disabilities, “Action and Expression: Expression and Communication. Give students access to inch tiles to double check their measurement. Reiterate how the measurement is the same regardless of the whole inch they start at on the ruler. Provides accessibility for: Conceptual Processing, Visual-Spatial Processing.”

• Unit 7, Adding and Subtracting Within 1000, Section C, Lesson 13, Activity 2, Access for Students with Disabilities, “Action and Expression: Expression and Communication. Provide students with alternatives to writing on paper. Students can share their learning by creating a video using the base-ten blocks, or writing out their steps and explaining on video. Provides accessibility for: Language, Attention, Social-Emotional Functioning.”

• Unit 8, Equal Groups, Section B, Lesson 12, Activity 2, Access for Students with Disabilities, “Action and Expression: Expression and Communication. Give students access to 1-inch grid paper to get their thinking started, and create an array with the inch tiles. Have students transfer what they made on the grid paper to the open rectangles given. The concrete image transferred to the more abstract image may help some students visually. Provides accessibility for: Visual-Spatial Processing, Organization.”

The materials reviewed for Open Up Resources K-5 Math Grade 2 meet expectations for providing extensions and/or opportunities for students to engage with grade-level/course-level mathematics at higher levels of complexity.

While there are no instances where advanced students do more assignments than classmates, materials do provide multiple opportunities for students to investigate grade-level content at a higher level of complexity. These are found where problems are labeled as “Exploration” at the end of practice problem sets within sections, where appropriate. According to the Resources, Course Guide, How To Use The Materials, Exploration Problems, “Each practice problem set also includes exploration questions that provide an opportunity for differentiation for students ready for more of a challenge. There are two types of exploration questions. One type is a hands-on activity directly related to the material of the unit that students can do either in class if they have free time, or at home. The second type of exploration is more open-ended and challenging. These problems go deeper into grade-level mathematics. They are not routine or procedural, and they are not just “the same thing again but with harder numbers. Exploration questions are intended to be used on an opt-in basis by students if they finish a main class activity early or want to do more mathematics on their own. It is not expected that an entire class engages in exploration problems, and it is not expected that any student works on all of them. Exploration problems may also be good fodder for a Problem of the Week or similar structure.” Examples include:

• Unit 3, Measuring Length, Section B, Practice Problems, Problem 7 (Exploration), “If you and all of your classmates stand side to side with your arms stretched out, about how long of a line do you think you can make? Explain your reasoning including the unit of measure you choose.”

• Unit 5, Numbers to 1,000, Section A, Practice Problems, Problem 10 (Exploration), “a. Can you represent the number 218 without using any hundreds? Explain your reasoning.; b. Can you represent the number 218 without using any tens? Explain your reasoning.; c. Can you represent the number 218 without using any ones? Explain your reasoning.“

• Unit 7, Adding and Subtracting Within 1,000, Section A, Practice Problems, Problem 11 (Exploration), “Tyler says he can find the value of $438-275$ using what he knows about differences of two-digit numbers. “First I find $43-27$ and then I find $8-5$ and that gives me the answer.” Use Tyler’s reasoning to find the value of $438-275$.”

The materials reviewed for Open Up Resources K-5 Math Grade 2 meet expectations for providing strategies and supports for students who read, write, and/or speak in a language other than English to regularly participate in learning grade-level mathematics.

Guidance is consistently provided to teachers to support students who read, write, and/or speak in a language other than English, providing scaffolds for them to meet or exceed grade-level standards. According to the Resources, Course Guide, Mathematical Language Development and Access for English Learners, “In an effort to advance the mathematics and language learning of all students, the materials purposefully engage students in sense-making and using language to negotiate meaning with their peers. To support students who are learning English in their development of language, this curriculum includes instruction devoted to fostering language development alongside mathematics learning, fostering language-rich environments where there is space for all students to participate.” Examples include:

• Unit 2, Adding and Subtracting Within 100, Section C, Lesson 14, Activity 1, “Access for English Learners - Convesing, Representing: MLR8 Discussion Supports. To support both students with an opportunity to produce language, display a question starter ‘How did you do the problem?’ and sentence frames ‘First, I  because _____ because… My method is like yours because …Our methods are different because …’”

• Unit 3, Measuring Length, Lesson 5, Activity 2, Access for English Learners, "Conversing, Reading: MLR2 Collect and Display. Direct attention to words collected and displayed from the previous lesson. Add “meter stick” to the collection. Invite students to borrow language from the display as needed, and update it throughout the lesson."

• Unit 5, Numbers to 1,000, Section A, Lesson 6, Activity 2, "Access for English Learners, Representing, Conversing: MLR7 Compare and Connect, Synthesis: After the Gallery Walk, lead a discussion comparing, contrasting, and connecting the different representations of numbers. To amplify student language and illustrate connections, follow along and point to the relevant parts of the displays as students speak."

The materials reviewed for Open Up Resources K-5 Math Grade 2 meet expectations for providing manipulatives, both virtual and physical, that are accurate representations of the mathematical objects they represent and, when appropriate, are connected to written methods.

Suggestions and/or links to manipulatives are consistently included within materials to support the understanding of grade-level math concepts. Examples include:

• Unit 2, Adding and Subtracting Within 100, Section B, Lesson 5, Activity 1, “The purpose of this activity is for students to subtract in a way that makes sense to them. Students use a method of their choice and share their methods with one another. This can serve as a formative assessment of how students approach finding the value of a difference when a ten must be decomposed when subtracting by place. Although students may use many methods to subtract, including those based on counting or compensation, the synthesis focuses on connecting these methods to those based on place value where a ten is decomposed. Monitor and select students with the following methods to share in the synthesis:  Uses connecting cubes to make 82 and removes 9 blocks. Counts back or counts all to find the difference. Subtracts 2 from 82 to get to a ten, 80, and then subtracts 7 from 80 by counting back (with or without blocks). Uses base-ten blocks to show 82 and decomposes a ten to get 12 ones. Subtracts 9 ones from 12 ones and counts the remaining blocks.”

• Unit 6, Geometry, Time, Money, Lesson 8, Activity 2, “Groups of 2. Give students access to colored pencils. ‘You are going to read some stories with a partner about students sharing pies. Then you will partition and color shapes on your own.’ 5 minutes: partner work time. As students work, encourage them to use precise language when talking with their partners. Consider asking: ‘Is there another way you could say how much of the circle is shaded?’ 10 minutes: independent work time. Monitor for students who accurately shade the circles to share in the synthesis.”

• Unit 8, Equal Groups, Section B, Lesson 7, Activity 1, “The purpose of this activity is for students to create arrays with counters. Students get sets of 6, 7, and 9 counters. Based on their experiences with images that show an even number of objects arranged in 2 equal groups, they may make an array with 3 rows and 2 columns or 2 rows and 3 columns with 6 counters. They may wonder if 7 or 9 counters can be arranged in an array since they are not even numbers.  Encourage students to experiment with other ways of arranging the counters that include more than 2 rows or columns. They may also make an array with 1 row. Arrays with 1 row or 1 column will be studied in future grades, so for the rest of this unit, students should be encouraged to make arrays with more than 1 row and more than 1 column.”