2nd Grade - Gateway 2

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

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

• Unit 5, Numbers to 1000, Lesson 8, Activity 2, students develop conceptual understanding as they use their place value understanding to locate numbers on a number line. Students are given number lines where the tick marks are not labeled with a starting number, 10 length units marked with tick marks, and an ending number. Students determine the size of the units based on the range of the number line. “‘Locate and label each number on the number line. Label the tick marks with the numbers they represent if it helps.’ 1. 700 Number line shown has a range of 0 to 1000; 2. 472 Number line shown a range of 470 to 480.” (2.MD.6)

• Unit 7, Adding and Subtracting within 1,000, Lesson 8, Warm-up, students develop conceptual understanding as they use grouping strategies to describe amounts represented with base-ten diagrams. An image is provided that shows base ten blocks in hundreds, tens, and ones. Student Task Statements, “How many do you see? How do you see them?” (2.NBT.7)

• Unit 8, Equal Groups, Lesson 2, Warm-up, students develop conceptual understanding as they compare four images to determine which group of two does not belong with the other pairs. An image of different colored socks are shown. “Pick one that doesn’t belong. Be ready to share why it doesn’t belong. Discuss your thinking with your partner.” (2.OA.C)

Materials provide opportunities for students to independently demonstrate conceptual understanding throughout the grade level. Design Principles, Coherent Progress, “Each activity starts with a launch that gives all students access to the task. This is followed by independent work time that allows them to grapple with problems individually before working in small groups. The activity ends with a synthesis to ensure students have an opportunity to consolidate their learning by making connections between their work and the mathematical goals.” The Cool-down part of the lesson includes independent work.  Curriculum Guide, How Do You Use the Materials, A Typical Lesson, Four Phases of a Lesson, Cool-down, “The cool-down task is to be given to students at the end of a lesson. Students are meant to work on the cool-down for about 5 minutes independently and turn it in.” Independent work could include practice problems, problem sets, and time to work alone within groups. Examples include:

• Unit 5, Numbers to 1,000, Lesson 3, Cool-down, students use base-ten representations to demonstrate their understanding of the digits in three-digit numbers. “How many of each? 1. There are ___ hundreds. 2. There are ___ tens. 3. There are ___ ones. 4. Draw a base-ten diagram to represent the same total value with the fewest number of blocks.” An image shows base-ten blocks with 2 hundreds, 11 tens, and 12 ones. (2.NBT.1) 

• Unit 7, Adding and Subtracting Within 1,000, Lesson 13, Decompose Tens and Hundreds, Cool-Down, students demonstrate conceptual understanding as they add and subtract within 1,000 using concrete models or drawings and strategies based on place value. “Find the value of 519 - 236. Show your thinking. Sample student response, 283. Sample responses: Students draw a base-ten diagram that shows 519 as 5 hundreds, 1 ten, and 9 ones. Students show decomposing a hundred to make 10 tens. Students cross out 2 hundreds, 3 tens, and 6 ones. Labels or equations clearly show the difference as 283.” (2.NBT.7)

• Unit 8, Equal Groups, Lesson 12, Cool-down, students demonstrate conceptual understanding as they use addition to find the total number of objects arranged in rectangular arrays and write equations to express the total as a sum of equal addends. “Write an equation that represents the number of squares in the rectangle?” An image of a rectangle is shown to demonstrate they have partitioned into rows and columns in a previous problem. (2.OA.4)

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

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

• Unit 1, Adding, Subtracting, and Working with Data, Lesson 3, Warm-up, students use mental strategies for adding and subtracting. Student Task Statements, “Find the value of each expression mentally. 7+3, 10-7, 10-2, 10-4.” (2.OA.2) 

• Unit 2, Add and Subtract Within 100, Lesson 5, Warm-up, students use mental strategies to subtract. Launch, “Find the value of each expression mentally. 17-7; 17-8; 26-6; 26-8.” (2.NBT.5, 2.OA.2)

• Unit 9, Putting It All Together, Lesson 8, Warm-up, students have an opportunity to strengthen number sense and procedural fluency. Launch, “Find the value of each expression mentally: 9+5; 20+30; 29+35; 229+435.” (2.NBT.5)

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

• Unit 4, Addition and Subtraction on the Number Line, Lesson 11, Activity 2, students develop fluency with addition and subtraction within 100. Student Task Statements, “Partner A 1. Find the value of 59+27.  2. Find the value of 65-18. Partner B 1. Find the value of 68-39.  2. Find the value of 22+49.” (2.NBT.5)

• Unit 5, Number to 1,000, Lesson 7, Activity 2, Narrative, “Before playing, students remove the cards that show 0 and set them aside. Students use digit cards to make addition and subtraction equations true. They work with sums and differences within 100 with composing and decomposing. Each digit card may only be used one time on a page.” (2.NBT.5)

• Unit 9, Putting It All Together, Lesson 1, Cool-down, students add and subtract to find the value of each expression. Student Task Statements, “Find the value of each expression 1. 11-5. 2. 12-3. 3. 16-8. 4. 9+3. 5. 8+8. 6. 13-8.” (2.OA.2)

The materials reviewed for Imagine Learning Illustrative Mathematics Grade 2 meet expectations for being designed so that teachers and students spend sufficient time working with engaging applications of the mathematics.

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

Examples of routine applications include:

• Unit 2, Adding and Subtracting Within 100, Lesson 11, Activity 1, students use addition and subtraction to represent and solve real-world problems. (2.OA.1) Activity, “6 minutes: independent work time. As students work, consider asking: What do you need to find to answer the question? How do you know? How did you show Diego’s seeds? How did you show Jada’s seeds? How will you find the difference?” Student Task Statements, Problem 1, “Diego gathered 42 orange seeds. Jada gathered 16 apple seeds. How many more seeds did Diego gather than Jada? Show your thinking.”

• Unit 4, Addition and Subtraction on a Number Line, Lesson 13, Activity 2, students represent lengths on a number line diagram. (2.MD.6) Student Task Statements, Problem 1, “Clare started with 24 cubes and added on some more. Clare made a train with 42 cubes. How many cubes did Clare add on?” Images of a number line from 0-80 with intervals of 5, and a blank tape diagram are included.

• Unit 6, Geometry, Time, and Money, Lesson 11, Activity 1, students select the clock that represents a  given time (2.MD.7). Student Task Statements, “1. Circle the clock that shows 4 o’clock. Why doesn’t the other clock show 4 o’clock? 2. Circle the clock that shows half past 7. Why doesn’t the other clock show half past 7? You are going to look at some analog clocks and think about how they work to show different times. As needed, review how to read time presented in a digital format.”

Examples of non-routine applications include:

• Unit 1, Adding, Subtracting, and Working with Data, Lesson 18, Activity 2, students create their own relevant mathematical questions and use their understanding of addition and subtraction to answer questions about their own and their peers' survey data. Activity Narrative, “Switch graphs. Use the sentence stems or create your own questions about the other group’s graphs. 1–2 minutes: independent work time 4 minutes: partner work time. Take turns to ask each other the questions you came up with and use your own graph to answer.  When possible, write down an equation to show your reasoning. 5 minutes: group discussion. Monitor for students who write equations to show the categories they combine or compare when answering their peers' questions.” Student Task Statements, “1. Trade graphs. Create questions about the other group’s graph. Sentence stems: How many students picked ___ and ___ all together? How many more students picked ___ than ___? 2. Take turns asking and answering questions.” (2.MD.10, 2.OA.1, 2.OA.2)

• Unit 7, Adding and Subtracting Within 1000, Lesson 16, Activity 1, students add and subtract within 1000 using place value strategies and explain why the strategies work using  real-world problems (2.NBT.7, 2.NBT.9). Launch, “Take a minute to make sense of Lin’s subtraction. 1–2 minutes: quiet think time.” Student Task Statements, “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.”

• Unit 9, Putting It All Together, Lesson 5, Activity 1, students use place value to represent three digit numbers (2.NBT.1). Student Task Statements, “1. Start with 2 hundreds. Grab a handful of tens and of ones. a. What number do your base-ten blocks represent? _____ b. Represent the same number in another way. Show your thinking using diagrams, symbols, or other representations. 2. Combine your blocks with your partner’s blocks. a. What number do your base-ten blocks represent? _____ b. Represent the same number in another way. Show your thinking using diagrams, symbols, or other representations. 3. Represent your group’s number in the following ways: a. without hundreds b. without tens c. without hundreds or tens.”

The materials reviewed for Imagine Learning Illustrative Mathematics Grade 2 meet expectations in that the three aspects of rigor are not always treated together and are not always treated separately. There is a balance of the three aspects of rigor within the grade. 

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

All three aspects of rigor are present independently throughout Grade 2. Examples where instructional materials attend to conceptual understanding, procedural skill and fluency, or application include:

• Unit 5, Numbers to 1000, Lesson 9, Activity 2, students deepen their conceptual understanding as they compare three-digit numbers using different representations and represent numbers on a line diagram. Activity, “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.” Student Task Statements, “1. Locate and label 420 and 590 on the number line. 2. Estimate the location of 378 and 387 on the number line. Mark each number with a point. Label the point with the number it represents. 3. Diego and Jada compared 2 numbers. Use their work to figure out what numbers they compared. Then use <, >, and = to compare the numbers. 4. Which representation was most helpful to compare the numbers? Why?” (2.MD.6, 2.NBT.4)  

• Unit 8, Equal Groups, Lesson 5, Activity 2, students apply their understanding as they test conjectures about the effect of adding 1 and 2 to groups of objects to determine if the groups of objects are even or odd. Launch, “Give students recording sheets and access to counters. Draw: An image of 5 dots are shown. ‘If we add 1 more circle to this group, will it change if the group has an even or odd number?’ (Yes. It’s odd, so if you add 1 circle you’d make another pair and it’d be even.) 30 seconds: quiet think time. Share responses. ‘Does adding 1 always change whether a number of objects is even or odd?’ (Yes. If you add 1 to an odd number, you’d always make a new pair, and the sum would be even. If it’s even, and you add 1, you’d have a leftover, so the sum would be odd. No. I think it works with some numbers, but maybe not all numbers.) 30 seconds: quiet think time. 1 minute: partner discussion. Share and record responses.” Activity, “‘Let’s test our ideas. Complete the first two columns of the table. You can test other numbers if you have time.’ 4 minutes: independent work time. 2 minutes: partner discussion. ‘If we add 2 more to a group, will it change if the group has an even or odd number?” (No. For even, it’d be like counting by 2, the next number is even too. When we counted on 2 to odd, we made a list of odd numbers. Yes. I think if you add to a number, it’s going to change some numbers.)’ 30 seconds: quiet think time. 1 minute: partner discussion. Share and record responses. ‘Let’s test our thinking. Complete the table for the “add 2 counters” column. You can test other numbers if you have time.’ 4 minutes: independent work time. 2 minutes: partner discussion.” Student Task Statements, “1. In the first column of your recording sheet, decide whether each student has an even or odd number of counters. Show your reasoning and circle your choice. 2. Complete the gray column. Does adding 1 change whether the number of counters is even or odd? Explain. 2. Complete the last column. Does adding 2 change whether the number of counters is even or odd? Explain.” (2.OA.2, 2.OA.3)

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

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

• Unit 1, Adding and Subtracting and Working With Data, Lesson 16, Activity 2, students apply their conceptual understanding of addition and subtraction to solve problems within 100. Activity, “‘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 Task Statements, “1. Jada read 10 fewer pages than Noah. Noah read 27 pages. How many pages did Jada read? 2.Noah spent 25 minutes reading. Jada spent 30 more minutes reading than Noah. How many minutes did Jada spend reading? 3.Jada read 47 pages of the book. Noah read 20 pages of the book. How many fewer pages did Noah read? 4.Noah stacked 14 more books than Jada. Jada stacked 28 books. How many books did Noah stack?” (2.OA.1) 

• Unit 4, Addition and Subtraction on the Number Line, Lesson 12, Equations with Unknowns, Activity 1: Number Line Riddles, students develop conceptual understanding and procedural fluency as they solve addition and subtraction problems within 100 with the unknown in all positions. Student Task Statements, “Solve riddles to find the mystery number. For each riddle: Write an equation that represents the riddle and write a ? for the unknown. Write the mystery number. Represent the equation on the number line. 1. I started at 15 and jumped 17 to the right. Where did I end? Equation: ____. Mystery number: ____. 2. I started at a number and jumped 20 to the left. I ended at 33. Where did I start? Equation: ____.  Mystery number: ____. 3. I started on 42 and ended at 80. How far did I jump? Equation:____. Mystery number:____. 4. I started at 76 and jumped 27 to the left. Where did I end? Equation: ____. Mystery number: ____.  5. I started at a number and jumped 19 to the right. I ended at 67. Where did I start? Equation: ____. Mystery number: ____. 6. I started at 92 and ended at 33. How far did I jump? Equation: ____. Mystery number: ____.” (2.MD.6, 2.NBT.5, 2.OA.1)

• Unit 8, Equal Groups, Lesson 1, Cool-down, students use conceptual understanding as they apply their understanding to find ways to share numbers while determining if groups of numbers are even or odd. Student Task Statements, “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.” (2.OA.3)

The materials reviewed for Imagine Learning Illustrative Mathematics Grade 2 meet expectations for supporting the intentional development of MP1: Make sense of problems and persevere in solving them; and MP2: Reason abstractly and quantitatively, for students, in connection to the grade-level content standards, as expected by the mathematical practice standards. Students have opportunities to engage with the Math Practices across the year, and they are often explicitly identified for teachers in several places including the Instructional Routines (Warm-up Routines and Other Instructional Routines), Activity Narratives, and About this lesson.

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

• Unit 3, Measuring Length, Lesson 6, Activity 1, students “interpret and solve Compare problems involving length where the language suggests an incorrect operation.” Activity “‘What are different ways we could represent this problem?’ (tape diagram, equations, base ten blocks).” Activity Narrative, “At the end of the launch, students open their books and work to find the diagram that matches the story problem. This further helps them to visualize the quantities in the problem before they work to find a solution (MP1).” Student Task Statements, “1. Lin's pet lizard is 62 cm long. It is 19 cm shorter than Jada's. How long is Jada's pet lizard? b. Whose pet is longer? ___. b. Circle the diagram that matches the story. c. Solve. Show your thinking.”

• Unit 4, Addition and Subtraction on the Number Line, Lesson 5, Activity 1, students make sense of number line diagrams. Activity, “‘Look at each number line and record an estimate of the number that the point represents. 5 minutes: independent work time. ‘Compare each estimate with your partner and explain why you believe your answer is reasonable.’ 7 minutes: partner work time. Monitor for students who add tick marks or labels, including multiples of 10 or 5, to help identify the number. This activity continues on the next card.” Student Task Statements, “1. What number could this be? ____ (Number line. Scale 30 to 60 by 5's. Evenly spaced tick marks. Point plotted between 50 and 55.) 2. What number could this be? ____ (Number line. Scale 0 to 30, by 10's. Point plotted between 20 and 30.)” Activity Narrative, “For each successive number line, the given tick marks are farther apart so students need to rely more on their understanding of properties of the number line and the accuracy with which they can locate the given numbers depends on how much extra work they do thinking about other numbers which they can locate accurately (MP1).”

• Unit 6, Geometry, Time, and Money, Lesson 3, Activity 2, students “recognize and draw shapes that have a specific number of sides and corners, and specific side lengths”. Activity, “Now you will get a chance to pick your own attributes, draw your own shapes, and guess which attributes your partner picked.” Activity Narrative, “Students may persevere in problem solving if they look for or choose particular attributes that do not go together (MP1).” Student Task Statements, “Choose your own attributes. Circle an attribute from each row. Draw and name a shape with the attributes you chose. If you cannot draw the shape, explain why.”

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

• Unit 1, Adding, Subtracting, and Working With Data, Lesson 15, Activity 2, students connect quantities to structures in the story problem. Activity, “8 minutes: partner work time. Monitor for students who explain how each diagram and equation matches the quantities in the context of the story problem. ‘Compare your matches with the matches of another group. If you have different matches, work together to explain which cards belong or why a card could belong to different groups.’ 4 minutes: small-group work time.” Student Task Statements, “Lin and Diego want to compare other things they collected and did at the beach. Student on the beach. Read a card with a story problem. Find cards that match the story problem. Explain why the cards match.” Activity Narrative, “When students analyze and connect the quantities and structures in the story problems, diagrams, and equations, they think abstractly and quantitatively (MP2) and make use of structure (MP7).” 

• Unit 7, Add and Subtract within 1,000, Lesson 15, Activity 1, students “use their knowledge of base-ten diagrams and place value to make sense of a written method.” Activity, “Elena is finding the value of 726-558. Use base-ten blocks or a base-ten diagram to show Elena’s steps. Then finish Elena’s work. If you have time, work together to show a different way Elena could use numbers or equations to show her steps.” Activity Narrative, students “describe how numbers, words, and equations can be used to represent the steps they have used with other representations (MP2).”

• Unit 9, Putting It All Together, Lesson 3, Activity 1, students “measure lengths to the nearest centimeter and to find the total distance each student moves on the map”. Student Task Statements, “4. Find the total length of each student’s trip. Represent the total with an equation”. Activity Narrative, “When students measure and represent the trips with equations and find the total distance on the map, they reason abstractly and quantitatively (MP2).”

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

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

Examples of constructing viable arguments include:

• Unit 3, Measuring Length, Lesson 9, Warm-up, students construct viable arguments as they practice the skill of estimating a reasonable length based on their experience and known information. Launch, “Groups of 2. Display the image. ‘About how long do you think the fish in the picture is in inches? What is an estimate that’s too high? Too low? About right?’ 1 minute: quiet think time.” Student Task Statements, “‘How long is this Cobia fish in inches? Record an estimate that is: too low, about right; too high.’ Image of Cobia fish included.” Activity Narrative, “This gives students an opportunity to share a mathematical claim including the assumptions they made when interpreting the image with limited information (MP3, MP4).”

• Unit 4, Addition and Subtraction on the Number Line, Lesson 5, Activity 2, Narrative, “Students use what they know about multiples of 10, the relative position of numbers on the number line, and comparing length to locate and label a set of numbers on the number line.” Lesson Narrative, students “construct viable arguments for how they placed the numbers and to critique the reasoning of others (MP3)”. Activity Narrative, “‘You will be working with your group to arrange the number cards on the number line. Take turns picking a card and placing it near its spot on the number line. Explain how you decided where to place your card.If you think you need to rearrange other cards, explain why. When you agree that you have placed all the numbers in the right spots, mark each of the numbers on your cards with a point on the number line. Label each point with the number it represents.’ 10 minutes: small-group work time.”

• Unit 6, Geometry, Time, and Money, Lesson 13, Activity 1, students construct viable arguments and critique the reasoning of others as they make sense of a visual representation of the hours in 1 day. Activity, “‘Cut out the two parts of the day and glue them together. Circle and label when you eat breakfast, lunch, and dinner on the diagram. Then shade in all the times you might be sleeping.’ 5 minutes: independent work time. ‘Share responses. This activity continues on the next card.’” Student Task Statements, “Use the materials your teacher gives you to create your own representation for the hours in a day. Circle and label when you eat breakfast, lunch, and dinner on the diagram. Shade in when you might be sleeping.” Activity Narrative, “Students have opportunities to develop logical arguments for why an event may happen during a.m. or p.m. hours and critique the arguments of others (MP3).”

Examples of critiquing the reasoning of others include:

• Unit 2, Add and Subtract within 100, Lesson 6, Activity 1, the narrative states, students “interpret and compare representations that show decomposing a ten to subtract by place.” Narrative, “Students compare and make connections between the representations and a set of equations that also shows how to find the value of the difference (MP3).” Activity, “‘Tyler found the value by using equations. Diego says Tyler’s equations match his diagram. Elena says the equations match her diagram. Who do you agree with?’ 2 minutes: independent work time.”

• Unit 5, Numbers to 1,000, Lesson 9, Activity 1, students critique the reasoning of others as they make sense of different methods they can use to compare three-digit numbers.” Activity, “‘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.’” Student Task Statements, “‘Diego: I see 3 hundreds for each number. 317 only has 1 ten, but 371 has 7 tens. 371>317.’ Image of base ten blocks: 3 hundreds, 7 tens, and 1 one and image of base ten blocks: 3 hundreds, 1 ten, 7 ones.” Activity Narrative, “They analyze the thinking of others and make connections across representations (MP2, MP3).” 

• Unit 9, Putting It All Together, Lesson 13, Activity 2, students critique the reasoning of others as they make revisions to their own work after seeing their peers’ work. The narrative states that students ask “mathematical questions or leave feedback using precise math language (MP3).” Activity Narrative, “Use your sticky notes to leave comments or questions about the stories and solutions, including things that helped you understand the problem and solutions and any other representations you might add to the poster.” 7 minutes: partner work time “Make revisions to your own poster based on what you saw and discussed.” 3 minutes: independent work time.”

The materials reviewed for Imagine Learning Illustrative Mathematics Grade 2 meet expectations for supporting the intentional development of MP4: Model with mathematics; and MP5: Use appropriate tools strategically, for students, in connection to the grade-level content standards, as expected by the mathematical practice standards. Students have opportunities to engage with the Math Practices across the year and they are often explicitly identified for teachers in several places including the Instructional Routines (Warm-up Routines and Other Instructional Routines), Activity Narratives, and About this Lesson.

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

• Unit 1, Adding, Subtracting and Working With Data, Lesson 8, Activity 2, students model with mathematics as they interpret data represented in a picture graph. Student Task Statements, “Circle the 4 questions that can be answered using the graph. a. How many kids chose pizza? b. How many chose tacos or grilled cheese? c. Why did so many kids choose spaghetti? d. How many more kids chose pizza than tacos? e. What is the total number of kids who chose spaghetti or pizza?” Activity Narrative, “This type of reasoning helps students make sense of the mathematical elements in a context and interpret and use the data presented in the picture graph to answer questions (MP4).”

• Unit 8, Equal Groups, Lesson 8, Warm-up, students organize objects to find the total quicker. Activity Narrative, “Making an estimate or a range of reasonable answers with incomplete information is a part of modeling with mathematics (MP4).” Student Task Statements, students are shown many red and yellow counters in a random order. Launch, “How many counters do you see? What is an estimate that’s too high? Too low? About right?” Synthesis, “We saw different arrangements of the same number of counters. Which one makes it easier to tell how many there are altogether? Explain. (In the last way, it is easy to see it is 10 and another 10. It looks like 2 10-frames.) Refer to the counters in an array. Organizing the circles into arrays can help us see ways to find the total more quickly. How could I use skip counting to find the total number of counters? (We could count by 5 or 2.)”

• Unit 9, Putting It All Together, Lesson 10, Activity 2, students analyze answers and determine  what the question could have been. Student Task Statements, “Clare picked 51 apples. Lin picked 18 apples and Andre picked 19 apples.” Lesson Narrative, “Determining the relationships between quantities and using them to ask questions and solve problems is an aspect of modeling with mathematics (MP4).” Synthesis, “How did you know the student was trying to find a total amount? (The tape diagram and student work shows addition.) Why do you think the student added 51 and 19 rather than 51 and 18? (They make 70 together. That way, there is no need to make a ten when you add the third number.)” Synthesis, “How did you know the question might be about comparing? Why not a question about taking away? (The operation is subtraction but Lin’s apples and Andre’s apples aren’t taken away from Clare’s apples. The diagram helps see that it is a comparison.) What strategies can you use to calculate ? (Use the number line. Make a drawing. Subtract 20 and add 1, subtract 20 more and add 2.)” 

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

• Unit 3, Measuring Length, Lesson 5, Activity 1, students choose appropriate tools strategically as they experience the need for a longer length unit and measuring tool. Activity Narrative, “They can choose to measure with centimeter cubes, 10-centimeter tools, their self-made rulers, or centimeter rulers (MP5).” Student Task Statements, “1. Measure to find the length of each reptile. Don’t forget the unit. a. What is the length of a gila monster? b. What is the length of a baby alligator? c. What is the length of a baby cobra? d. What is the length of a komodo dragon?” 

• Unit 4, Addition and Subtraction of the Number Line, Lesson 1, Activity 2, “Students choose their own length unit to make equally spaced tick marks and label them 0–20. In order to make an accurate number line, students will need to make strategic use of materials in order to measure the units on their number line. This could be a paper clip or a staple or the equally spaced lines on a lined sheet of paper (MP5).” Student Task Statements, “1. Make a number line that goes from 0 to 20. 2. Locate 13 on your number line. Mark it with a point. 3. Locate 3 on your number line. Mark it with a point. 4. Compare your number line with your partner’s.” Activity Narrative, “Now you’re going to create your own number line. You can use any of the tools provided to create a number line that represents the numbers from 0 to 20.”

• Unit 5, Numbers to 1,000, Lesson 12, Activity 2, students use tools strategically as they consider insights gained from number lines. Activity Narrative, “Students reflect on how the number line can help us organize numbers (MP5).” Student Task Statements, “Estimate the location of 839, 765, 788, 815, and 719 on the number line. Mark each number with a point. Label the point with the number it represents. Order the numbers from least to greatest. _____, _____, _____, _____, _____.”

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

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

• Unit 3, Measuring Length, Lesson 14, Activity 2, students use precision when analyzing a given line plot and its features (MP6). Launch, “‘What do the numbers on our line plot represent? What does the way the numbers are arranged remind you of? (The numbers represent lengths in inches. It reminds me of a ruler. It has tick marks and each tick mark is the same length apart.) The line on a line plot represents the unit you use to measure. It shows numbers in order and the same length apart, just like on a ruler. What length unit do the numbers on our line plot represent? How could we label this? (The lengths of our hand spans in inches, measurement in inches)’ 1 minute: quiet think time. ‘Share responses and record a label. The length of the line between two numbers does not have to match the unit you used, so it's important to label the line on the line plot with the unit.’”

• Unit 4, Addition and Subtraction on the Number Line, Lesson 11, Warm-up, Activity Narrative, “In reasoning together about the number line representation, and connecting the strategy of making a ten to jumping to the nearest ten, students need to be precise in their word choice and use of language (MP6).” Launch, “‘Display one expression. Give me a signal when you have an answer and can explain how you got it.’ 1 minute: quiet think time.” Synthesis, “For 52−7 some students decomposed the 7 to make it easier to get to a ten. How does this number line representation connect to that strategy? Draw a number line showing 52 represented with a point, a jump of 2, and then a jump of 5.”  

• Unit 6, Geometry, Time, and Money, Lesson 1, Activity 3, students use precise language as they recognize and describe the attributes of triangles. Activity Narrative, “As students work, encourage them to refine their descriptions of their shape using more precise language (MP6).” Activity, “‘Pick one shape card. Think about how you would describe your shape to a partner without naming it.’ 1 minute: quiet think time. ‘You’re going to find a partner and describe your shape without showing them your card. Your partner will guess the name of your shape using triangle, quadrilateral, pentagon, or hexagon. After you both name the shapes, find one way your shapes are alike and one way they are different.’ Give a signal for students to find a partner. 2 minutes: partner discussion.” Synthesis, “What clues did your partner give you that made it easy to guess their shape? (number of sides, number of corners) Were there any clues that did not help you guess the name of the shape? (color, size).”

• Unit 8, Equal Groups, Lesson 7, Activity 2, Activity Narrative, “They use this vocabulary to describe arrays and create arrays given a number of counters and a number of rows (MP6).” Student Task Statements, Image shows an array with 3 rows of 2. “a. How many rows are in this array? b. How many counters are in each row? c. How many counters are there in all?”

The materials reviewed for Imagine Learning Illustrative Mathematics Grade 2 meet expectations for supporting the intentional development of MP7: Look for and make use of structure; and MP8: Look for and express regularity in repeated reasoning, for students, in connection to grade-level content standards, as expected by the mathematical practice standards. Students have opportunities to engage with the Math Practices throughout the year.

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

• Unit 1, Adding, Subtracting, and Working with Data, Lesson 3, Cool-down, students look for and make use of structure as they find the missing numbers in given equations. Lesson Narrative, “Throughout the lesson, students have opportunities to use what they know about the structure of whole numbers and the relationship of addition and subtraction to find the unknown numbers and explain their methods (MP3, MP7).” Student Task Statements, “Find the number that makes each equation true. 1. ___ + 17 = 20, 2. 20 - 9 = ___ , 3. ___ + 5 = 20, 4. 20 - ___ = 8.”

• Unit 2, Adding and Subtracting Within 100, Lesson 8, Warm-up, students look for and make use of structure as they use the relationship between addition and subtraction to find the value of expressions. Activity Narrative, “When they describe ways to use the value of the sums to find the value of the differences, they look for and make use of the structure of expression and the relationship between addition and subtraction (MP7).” Student Task Statements, “Find the value of each expression mentally. 18+10+10; 18+20+10; 38−20; 48−30.” Launch, “‘Display one expression. Give me a signal when you have an answer and can explain how you got it.’ 1 minute: quiet think time.” Activity, “Record answers and strategy. Keep expressions and work displayed. Repeat with each expression. This activity continues on the following cards.” Activity Synthesis, “How are the addition expressions related to the subtraction expressions? (The second expression is the opposite of the last expression. They are in the same fact family. The first expression helped me solve the third expression because I know 18+20=38, so 38−20 must be 18.)” 

• Unit 4, Addition and Subtraction on the Number Line, Lesson 4, Activity 1, students look for and make use of structure as they notice that the number farthest to right (on a number line) has a greater value. Activity Narrative, “Students recognize that given any two numbers, the number farther to the right represents a greater value than the number to the left (MP7).” Launch, “Groups of 2. Give each group 3 number cubes and 2 counters. Assign Partner A and B.” Activity, “‘You will use the number line you created and work with a partner. Decide with your partner whose number line you will use.’ As needed, demonstrate the task with a student. ‘I am Partner A. I am going to roll the 3 number cubes and find the sum. Then, I take a counter and place it on the number line to represent the sum. Now it’s my partner's turn. They do the same thing and put their counter on the same number line to represent the sum of their numbers.’” This activity continues on the next card. “Then, we decide which number is greater and explain how we know. Last, we use the < , >, or = symbols to record our comparison.” Activity Synthesis, “Invite 2–3 previously identified groups to share comparisons and their explanations. ‘What do you notice about the numbers that are farther to the right? (They were greater. They represent a greater length from zero.)’” 

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

• Unit 3, Measuring Length, Lesson 3, Warm-up, students look for and express regularity in repeated reasoning as they use one expression to help them find the value of another expression. Activity Narrative, “When they share how each expression helps them find the value of the next, they look for and express regularity in repeated reasoning (MP8).” Student Task Statements, “Find the value of each expression mentally. 63−3; 63−20; 63−23; 63−24” Launch, “Display one expression. ‘Give me a signal when you have an answer and can explain how you got it.’ 1 minute: quiet think time.” Activity, “Record answers and strategy. Keep problems and work displayed. Repeat with each expression. This activity continues on the following cards.”  Activity Synthesis, “Which expressions were easier to find mentally? Why? How did the third expression help you think about the fourth one?” 

• Unit 5, Numbers to 1,000, Lesson 8, Activity 2, students look for and express regularity in repeated reasoning as they notice the tick marks on a number line are equally distanced apart. Activity Narrative, “Students may begin to notice that when the two ticks at the right and left of the number line are 100 apart, the individual tick marks go up by 10 and when the two tick marks at the right and left of the number line are 10 apart, the individual tick marks go up by 1 (MP8).” Launch, “Groups of 2” Activity, “‘Now you are going to locate and label three-digit numbers on the number line. Take a few minutes to try them on your own and be ready to explain to your partner.’ 5 minutes: independent work time. ‘Now compare with a partner and share your thinking.’ If students are not finished, they can work together. 5 minutes: partner discussion. Monitor for different ways students determine the unit represented on the number line for representing 940 such as: counting by ones, tens, and hundreds to see which one gets them to the ending number using the starting and ending numbers to determine what the unit must be.” Activity Synthesis, “Display the number line showing 900–1,000. Invite a student to demonstrate their strategy of trial and error to determine how to label the tick marks. Invite another student to demonstrate their strategy of reasoning about the starting and ending numbers. ‘(I know the difference between 900 and 1,000 is 100 and there are 10 length units. Each one must be 10 because there are 10 tens in a hundred.) What is the same and different about how they decided the unit on this number line? (They both counted to see how many tick marks were there. ____ tried counting by 1 and then 10, but ____ counted by 10 right away.)’” 

• Unit 6, Geometry, Time, and Money, Lesson 12, Activity 2, students notice and use patterns as they practice telling and writing time. Activity Narrative, “When students look for shortcuts to tell the time (for example, counting on from 30 rather than 0 or counting back from 60), they are looking for and expressing regularity in repeated reasoning (MP8).” Student Task Statements, “‘Write the time shown on each clock.’ Students are then given 6 pictures of analog clocks.” Activity Synthesis, “How did ____ know you could start counting at 30? Why does this work? (They know it is 2:30 when the minute hand points to 6, so they can just start there. It works because you will still count the same numbers. If you start at 30 you still say 35, 40. It is just faster.)”