4th Grade - Gateway 2

The materials reviewed for Imagine Learning Illustrative Mathematics Grade 4 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 3, Fraction Operations to Fractions, Lesson 16, Activity 2, students develop conceptual understanding as they represent equations with tenths and hundredths using number lines. Activity, “Take a few quiet minutes to work on the first two problems. Then, share your responses with your partner. 5 minutes: independent work time. 2 minutes: partner discussion. Monitor for the ways students think about the total distance Noah has walked (third problem) given a fraction in tenths and one in hundredths. Now try finding the values of the sums in the last problem. 5 minutes: independent or partner work time.” Student Task Statements, “Noah walks \frac{2}{10} kilometer (km), stops for a drink of water, walks \frac{5}{100} kilometer, and stops for another sip. 1. Which number line diagram represents the distance Noah has walked? Explain how you know. 2. The diagram that you didn’t choose represents Jada’s walk. Write an equation to represent: a. the total distance Jada has walked b. the total distance Noah has walked 3. Find the value of each of the following sums. Show your reasoning. Use number lines if you find them helpful., a. \frac{5}{10}+\frac{1}{10}, b. \frac{50}{100}+\frac{10}{100}, c. \frac{5}{10}+\frac{30}{100}, d. \frac{15}{100}+\frac{4}{10}.” The number line diagrams shown in problem 1 are from 0 to 1 km, and divided into tenths. The first number line shows a jump from 0 to 2 tenths, then 5 more tenths ending on 7 tenths. The second number line shows a jump from 0 to 2 tenths, then a half tenth jump ending on 25 hundredths. The number lines in problem 3 are from 0 to 1 and are broken into tenths but are otherwise blank. Activity synthesis, “Invite students to share how they know which diagram represents Noah’s walk and their equations for the distances Noah and Jada walked. Given the number line diagram for support, students are likely to write \frac{2}{10}+\frac{50}{100}=\frac{25}{100}. Discuss why this is true. How do you know that the sum of \frac{2}{10} and \frac{5}{100} is \frac{25}{100}? Highlight that \frac{2}{10} is equivalent to \frac{20}{100} and another \frac{5}{100} makes \frac{25}{100}. Consider displaying a number line that is partitioned into tenths and hundredths and shows \frac{25}{100} as \frac{20}{100}+\frac{5}{100}.” (4.NF.5)

• Unit 4, From Hundredths to Hundred-thousands, Lesson 9, Warm-up, students develop conceptual understanding as they use commutative and associative properties of addition to compose numbers and determine equivalent sums. Launch, “Display one statement. Give me a signal when you know whether the statement is true and can explain how you know. 1 minute: quiet, think time.” Student Task Statements, “Decide if each statement is true or false. Be prepared to explain your reasoning. 4,000+600+70,000=70,460. 900,000+20,000+3,000=920,000+3,000. 80,000+800+8,000=800,000+80+8.” Activity Synthesis, “Focus question: How can you explain your answer without finding the value of both sides? We can write numbers in different forms. What form is used to represent the numbers in this True or False? (expanded form).” (4.NBT.2)

• Unit 6, Multiplying and Dividing Multi-digit Numbers, Lesson 7, Activity 1, students develop conceptual understanding as they use rectangular diagrams to represent multiplication of three-digit and one-digit numbers. Activity, “Work with your partner on the first problem. Then, try the rest of the activity independently. 2 minutes: group work time on the first problem. 5–7 minutes: independent work time on the rest of the activity. 3 minutes: partner discussion. Monitor for students who: write expressions that show the multi-digit factor decomposed by place value, draw diagrams that partition the multi-digit factor by place value.” Student Task Statement, “1. Clare drew this diagram. a. What multiplication expression can be represented by the diagram? b. Find the value of the expression. Show your reasoning.” A rectangle that is partitioned is shown. Activity synthesis, “Invite several students to share diagrams they drew to represent 6\times252. As each student shares, consider asking the class: Where do you see the parts of the factor that was decomposed? What expressions are represented in the diagram? Record expressions as students share.” (4.NBT.5)

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 2, Fraction Equivalence and Comparison, Lesson 7, Cool-down, students demonstrate conceptual understanding as they generate and explain equivalent fractions. Student Task Statement, “Name two fractions that are equivalent to \frac{5}{3}. Explain or show your reasoning.” (4.NF.1)

• Unit 4, From Hundredths to Hundred-thousands, Lesson 11, Cool-down, students use place value understanding to locate large numbers on a number line. Student Task Statements, “1. Estimate the location of 28,500 on the number line and label it with a point. 2. Which point—A, B, or C—could represent a number that is 10 times as much as 28,500? Explain your reasoning.” For problem 1, an image of a number line is shown with A,B,C on the number line from 0 to 400,000. (4.NBT.1) 

• Unit 6, Multiplying and Dividing Multi-digit Numbers, Lesson 15, Situations Involving Area, Cool Down, Sticky Notes on the Door, students demonstrate conceptual understanding when they find whole number quotients by determining the number of rows of sticky notes needed to cover a door. “Jada’s class is decorating their door with square sticky notes for their teacher. Each sticky note has a drawing or a message from a student. The class used 234 square sticky notes to cover their classroom door completely, leaving no gaps or overlaps between the notes. It takes 9 square notes to cover the width of the door. How many square notes does it take to cover the full height of the door? Show how you know.” (4.NBT.6)

The materials reviewed for Imagine Learning Illustrative Mathematics Grade 4 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 4, From Hundredths to Hundred-thousands, Lesson 21, Activity 2, students use subtraction to calculate characters' ages. Student Task Statements, “Jada recorded the birth year of each of her maternal grandparents for a family history project. As of this year, what is the age of each family member? Show your reasoning. Use the standard algorithm at least once.” The problem includes a table with different characters’ ages listed in two columns with the headings “family member” and “birth year.” (4.NBT.4)

• Unit 7, Angles and Angle Measurement, Lesson 2, Warm-up, students mentally subtract numbers. Student Task Statements, “Find the value of each expression mentally. 90-45, 270-45, 270-135, 360-135.” (4.NBT.4)

• Unit 9, Putting It All Together, Lesson 9, Activity 2, students create problems with a given answer using established constraints. Student Task Statements, “Elena, Noah, and Han each created a problem with an answer of 1,564. Elena used multiplication. Noah used multi-digit numbers and addition only. Han used multiplication and subtraction. Write a problem that each student could have written. Show that the answer to the question is 1,564.” (4.NBT.4, 4.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, From Hundredths to Hundred-thousands, Lesson 18, Centers, Number Puzzles, Standard Algorithm to Add and Subtract, 19, Compose and Decompose to Add and Subtract, 20, Add and Subtract Within 1,000,000. “Students use the digits 0–9 to make each addition or subtraction equation true. Stage 6: Beyond 1,000, Stage Narrative Students use the digits 0–9 to make addition equations true. They work with sums and differences beyond 1,000.” (4.NBT.4)

• Unit 4, From Hundredths to Hundred-thousands, Lesson 20, Cool-down, students use the standard algorithm to solve a subtraction problem. Student Task Statements, “Use the standard algorithm to find the value of the difference. 173,225-114,329.” (4.NBT.4)

• Unit 9, Putting It All Together, Lesson 4, Cool-down, students use the standard algorithm for subtraction. Student Task Statements, “Find the value of each difference. Show your reasoning. 1. 8,050-213. 2. 60,000-1,984.” (4.NBT.4)

The materials reviewed for Imagine Learning Illustrative Mathematics Grade 4 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 3, Extending Operations to Fractions, Lesson 6, Cool-down, students multiply whole numbers by a fraction. (4.NF.4) Student Task Statements, Problem 2, “Han bought 4 cartons of chocolate milk. Each carton contains \frac{5}{8} liter. Did Han buy the same amount of milk as Tyler? Explain or show your reasoning.” 

• Unit 3, Extending Operations to Fractions, Lesson 11, Activity 1, students solve a real-world problem involving subtraction of fractions referring to the same whole (4.NF.3d). Student Task Statements, “Clare, Elena, and Andre are making macramé friendship bracelets. They’d like their bracelets to be 9\frac{4}{8} inches long. For each question, explain or show your reasoning. 2. So far, Elena’s bracelet is 5\frac{1}{8} inches long and Andre’s is 3\frac{5}{8} inches long. How many more inches do they each need to reach 9\frac{4}{8} inches?”

• Unit 5, Multiplicative Comparison and Measurement, Lesson 10, Activity 2, students solve a real-world problem by using metric units of measurement and multiplicative comparison to solve multi-step problems (4.MD.2, 4.OA.2, 4.OA.3). Student Task Statements, “Here are six water bottle sizes and four clues about the amount of water they each hold. One bottle holds 350 mL.  A bottle in size B holds 5 times as much water as the bottle that holds 1 L. The largest bottle holds 20 times the amount of water in the smallest bottle. One bottle holds 1,500 mL, which is 3 times as much water as a bottle in size E. Use the clues to find out the amount of water, in mL, that each bottle size holds. Be prepared to explain or show your reasoning.”

Examples of non-routine applications include:

• Unit 6, Multiplying and Dividing Multi-digit Numbers, Lesson 22, Cool-down, students solve a measurement problem using multiplication (4.MD.2, 4.MD.3, 4.NBT.5). Student Task Statements, “Han has a rectangular piece of paper that is 96 inches by 36 inches. He is using it to create a banner for Awards Day. Last year the banner measured 2,304 square inches. 1. Will the new banner fit in the same area that the old banner was? Show your reasoning. 2. What is the difference in square inches between the area of last year's banner and this year's banner?” 

• Unit 7, Angles and Angle Measurement, Lesson 16, Activity 2, students use their understanding of geometric figures and measurements to draw, describe, and identify two-dimensional figures to solve a real world problem (4.G.1 and 4.G.2). Activity, “5 minutes: independent work time.” Student Task Statements, “1. Create a two-dimensional shape that has at least 3 of the following: ray, line segment, right angle, acute angle, obtuse angle, perpendicular lines, parallel lines.”

• Unit 9, Putting It All Together, Lesson 10, Activity 2, students use estimation to solve a real-world problem (4.OA.3). Student Task Statements, “It’s your turn to create an estimation problem. 1. Think of situations or look around for images that would make interesting estimation problems. Write down 4–5 ideas or possible topics. 2. Choose your favorite idea. Then, write an estimation question that would encourage others to use multiplication of multi-digit numbers to answer. Record an estimate that is ___.”

The materials reviewed for Imagine Learning Illustrative Mathematics Grade 4 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 4. Examples where instructional materials attend to conceptual understanding, procedural skill and fluency, or application include:

• Unit 1, Factors and Multiples, Lesson 2, Cool-down, students apply their understanding of factor pairs to find possible sides of a rectangle. Activity, “What are all of the possible side lengths of a rectangle with an area of 21 square units? What are all of the possible side lengths of a rectangle with an area of 50 square units?” (4.OA.4)

• Unit 2, Fraction Equivalence and Comparison, Lesson 2, Activity 2, students extend their conceptual understanding as they create representations of fractions. Launch, “Give each student a straightedge and access to their fraction strips from a previous lesson. How can you show \frac{3}{4} with fraction strips? (Find the strip showing fourths, highlight 3 parts of fourths.) How can you show \frac{8}{4}? If students say that they don’t have enough strips to show 8 fourths, ask them to combine their strips with another group’s. Invite groups to share their representations of \frac{8}{4}. Students may use different fractional parts (fourths and halves, or fourths and eighths).” Student Task Statements, Problem 2, “Here are four fractions and four blank diagrams. Partition each diagram and shade the parts to represent the fraction. a. \frac{2}{2}, b. \frac{4}{2}, c. \frac{5}{4}, d. \frac{10}{8}.” (4.NF.1)

• Unit 4, From Hundredths to Hundred-thousands, Lesson 20, Activity 1, students develop procedural skill and fluency as they use the standard addition algorithm. Student Task Statements, “1. Use the standard algorithm to find the value of each sum and difference. If you get stuck, try writing the numbers in expanded form. a. 7,106+2,835, b. 8,179+3,599, c. 142,571+10,909, d. 268,322+72,145.” (4.NBT.4) 

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

• Unit 3, Extending Operations to Fractions, Lesson 1, Activity 1, students engage with conceptual understanding, procedural fluency, and application as they extend understanding of multiplication to multiply a fraction by a whole number. Activity: “‘Take a few quiet minutes to think about the first set of problems about crackers. Then, discuss your thinking with your partner.’ 4 minutes: independent work time, 2 minutes: partner discussion, Pause for a whole-class discussion. Invite students to share their responses. If no students mention that there are equal groups, ask them to make some observations about the size of the groups in each image. Discuss the expressions students wrote:, ‘What expression did you write to represent the crackers in Image A? Why?’ ($$6\times4$$, because there are 6 groups of 4 full crackers.), ‘What about the crackers in Image B? Why?’ ($$6\times\frac{1}{4}$$, because there are 6 groups of \frac{1}{4} of a cracker.) Ask students to complete the remaining problems., 5 minutes: independent or partner work time, Monitor for students who reason about the quantities in terms of _____ groups of _____ to help them write expressions.” Student Task Statements: “Here are images of some crackers. a. How are the crackers in image A like those in B? b. How are they different? c. How many crackers are in each image? d. Write an expression to represent the crackers in each image. 2. Here are more images and descriptions of food items. For each, write a multiplication expression to represent the quantity. Then, answer the question. a. Clare has 3 baskets. She put 4 eggs into each basket. How many eggs did she put in baskets? b. Diego has 5 plates. He put \frac{1}{2} of a kiwi fruit on each plate. How many kiwis did he put on plates? c. Priya prepared 7 plates with \frac{1}{8} of a pie on each. How much pie did she put on plates?  d. Noah scooped \frac{1}{3} cup of brown rice 8 times. How many cups of brown rice did he scoop?” (4.NF.4)

• Unit 5, Multiplicative Comparison and Measurement, Lesson 4, Cool-down, students develop conceptual understanding alongside application as they solve for an unknown factor using multiplicative comparisons. Student Task Statements, “Priya read some pages on Monday. Jada read 63 pages, which is 7 times as many pages as Priya read. 1. Write an equation to show the comparison. Use a symbol for the unknown. 2. How many pages did Priya read?” (4.OA.2)

• Unit 9, Putting It All Together, Lesson 9, Cool-down, students use procedural fluency and apply their understanding as they write their own mathematical questions and solve them. Student Task Statements, “The school band wants to raise $1,700 for a music festival. They have raised $175 each week for the past 6 weeks. Write a question that could be asked about this situation and answer it. Show your reasoning.” (4.NBT.4, 4.NBT.5, 4.NBT.6, 4.OA.3)

The materials reviewed for Imagine Learning Illustrative Mathematics Grade 4 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, Extending Operations to Fractions, Lesson 18, Activity 1: Stack Centavos and Pesos, students use a variety of strategies to solve problems involving tenths and hundredths. Student Task Statements, “Diego and Lin each have a small collection of Mexican coins. The table shows the thickness of different coins in centimeters (cm) and how many of each Diego and Lin have. 1. If Diego and Lin each stack their centavo coins, whose stack would be taller? Show your reasoning. 2. If they each stack their peso coins, whose stack would be taller? Show your reasoning. 3. If they each stack all their coins, whose stack would be taller? Show your reasoning. 4. If they combine their coins to make a single stack, would it be more than 2 centimeters tall? Show your reasoning.” Activity Narrative, “Though the mathematics here is not new, the context and given information may be novel to students. Students have a wide variety of approaches available for these problems with no solution approach suggested (MP1).”

• Unit 4, From Hundredths to Hundred-thousands, Lesson 6, Activity 1: Build Numbers, students represent multi-digits in different ways. Student Task Statements, “1. Use two cards to make a two-digit number. Name it and build the number with base-ten blocks. 2. Use a third card to make a three-digit number. Name it and build it with base-ten blocks. 3. Use a fourth card to make a four-digit number. Name it and build it. If you don’t have enough blocks, describe what you would need to build the number. 4. Your teacher will give you one more digit card. Use the last card from your teacher to make a five-digit number. Make the card the first digit. Name it and build it. If you don’t have enough blocks, describe what blocks you would need to build the number.” Activity Narrative, “As students build numbers to the ten-thousands place, they may struggle to name the number. As they make sense of the value of the number, they should realize a need for more base-ten blocks, but should be given space to represent the number in a way that makes sense to them. It is not critical to name the number correctly or accurately describe how to build it. The idea is to create a bit of struggle to motivate another way to make sense of the number (MP1).” 

• Unit 5, Multiplicative Comparison and Measurement, Lesson 13, Activities 1 and 2, students make sense of and persevere in solving problems using the info gap instructional routine. Activity 1 narrative, “The purpose of this activity is to introduce students to the structure of the MLR4 Information Gap routine. This routine facilitates meaningful interactions by positioning some students as holders of information that is needed by other students. Tell students that first, a demonstration will be conducted with the whole class, in which they are playing the role of the person with the problem card. Explain to students that it is the job of the person with the problem card (in this case, the whole class) to think about what information they need to answer the question. For each question that is asked, students are expected to explain what they will do with the information, by responding to the question, ‘Why do you need to know (that piece of information)?’ If the problem card person asks for information that is not on the data card (including the answer!), then the data card person must respond with, ‘I don’t have that information.’ Once the students have enough information to solve the problem, they solve the problem independently. The info gap routine requires students to make sense of problems by determining what information is necessary and then ask for information they need to solve them. This may take several rounds of discussion if their first requests do not yield the information they need (MP1). Data Card, On each school day, Noah spends \frac{1}{4} hour getting to school., Noah’s homework and reading time is 5 times as long as he spends getting to school., Noah spends \frac1}{2} hour on his bedtime routine.”  Student Task Statement, “Problem Card, On a school day, Noah usually spends 40 minutes on his morning routine and 75 minutes on his sports practice. Which takes more time: 1. Noahs’ morning routine or his bedtime routine? 2. Noah’s sports practice or his homework and reading time?”

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, Factors and Multiples, Lesson 5, Activity 1, students reason abstractly and quantitatively about numbers in the context of a problem. Activity Narrative, “The purpose of this activity is to find multiples of two different numbers in context. Students decide possible table sizes for a party based on whether or not a given number of people is a multiple of 6, 8, both, or neither. In situations where a given number is not a multiple of 6 or 8, they reason about what it means in context (MP2).” The Student Task Statement, “Students are preparing for a party. The school has tables where 6 people can sit and tables where 8 people can sit. The students can only choose one type of table and they want to avoid having empty seats. 1. Jada’s class has 18 students. Which tables would you choose for Jada’s class? Explain or show your reasoning. 2. Noah’s class has 30 students. Which tables would you choose for Noah’s class? Explain or show your reasoning. 3. Which tables would you choose for Noah’s and Jada’s classes together? Can you find more than one option? Explain or show your reasoning. 4. If you also want places for Noah’s teacher and Jada’s teacher to sit, which tables would you choose? Explain or show your reasoning.”

• Unit 6, Multiplying and Dividing Multi-digit Numbers, Lesson 5, Activity 2, students consider units and strategies in the context of a real-world problem. The Activity Narrative states, “In this activity, students use strategies and representations that make sense to them to find products beyond 100. As before, the context of stickers lends itself to be represented with an array. The factors are large enough, however, that doing so would be inconvenient, motivating other representations or strategies (MP2). Look for the ways that students extend or generalize previously learned ideas or representations to find multiples of larger two-digit numbers. While many of the student responses are written with expressions, students are not expected to represent their reasoning using equations and expressions at this time. Teachers may choose to represent student reasoning using equations and expressions so students can start connecting representations.” The Student Task Statement, “1. Elena has another sheet of stickers that has 9 rows and 21 stickers in each row. How many stickers does Elena have? Explain or show your reasoning. 2. Noah’s sticker sheet has 3 rows with 48 stickers in each row. Andre’s sticker sheet has 7 rows with 23 stickers in each row. Who has more stickers? Explain or show your reasoning.”

• Unit 9, Putting It All Together, Lesson 8, About this lesson, “In the previous lesson, students solved word problems involving multiplicative comparison. In this lesson, they practice solving a wider variety of problems, with a focus on the relationships among multiple quantities in a situation. Students think about how to represent the relationships with one or multiple equations and using multiple operations. They also interpret their solutions and the solutions of others in context, including interpreting remainders in situations that involve division (MP2).”  Cool-down, “In one week, a train made 8 round trips between its home station and Union Station. At the end of the week, it traveled a few more miles from the home station to a repair center. That week, the train traveled a total of 1,564 miles. 1. Which statement is true for this situation? Explain or show your reasoning. a. The distance traveled for each round trip is 200 miles. The distance to the repair station is 26 miles. b. The distance traveled for each round trip is 195 miles. The distance to the repair station is 4 miles. c. The distance traveled for each round trip is 8 miles. The distance to the repair station is 1,500 miles. d. The distance traveled for each round trip is 193 miles. The distance to the repair station is 8 miles.”

The materials reviewed for Imagine Learning Illustrative Mathematics Grade 4 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 1, Factors and Multiples, Lesson 5, Activity 2, students construct viable arguments as they find multiples. Student Task Statements, “Each package of hot dogs has 10 hot dogs. Each package of hot dog buns has 8 buns. 1. Lin expects to need 50 hot dogs for a class picnic. a. How many packages of hot dogs should Lin get? Explain or show your reasoning. b. Can Lin get exactly 50 hot dog buns? How many packages of hot dog buns should Lin get? Explain or show your reasoning. 2. Diego expects to need 72 hot dogs for a class picnic. a. How many packages of hot dogs should Diego get? Explain or show your reasoning. b. How many packages of hot dog buns should Diego get? Explain or show your reasoning. 3. Is it possible to buy exactly the same number of hot dogs and buns? If you think so, what would that number be? If not, explain your reasoning.” The Activity Narrative, “As multiple answers can be expected, the focus is on explaining why the solutions make sense (MP3).”

• Unit 3, Extending Operations to Fractions, Lesson 3, Activity 2, students construct a viable argument and critique the reasoning of others when they identify patterns using multiplication. Activity, “3 minutes: independent work time on the first set of problems. 2 minutes: group discussion. Select students to explain how they reasoned about the missing numbers in the equations.” Student Task Statements, “2. Your teacher will give you a sheet of paper. Work with your group of 3 and complete these steps on the paper. After each step, pass your paper to your right. Step 1: Write a fraction with a numerator other than 1 and a denominator no greater than 12.  Step 2: Write the fraction you received as a product of a whole number and a unit fraction. Step 3: Draw a diagram to represent the expression you just received. Step 4: Collect your original paper. If you think the work is correct, explain why the expression and the diagram both represent the fraction that you wrote. If not, discuss what revisions are needed.” The Activity Narrative, “As students discuss and justify their decisions they create viable arguments and critique one another’s reasoning (MP3).”

• Unit 5, Multiplicative Comparison and Measurement, Lesson 3, Activity 2, students construct a viable argument and critique the reasoning of others as they represent missing amounts in multiplication situations. Activity Narrative, “Students use the relationship between multiplication and division to write equations to represent multiplicative comparisons. These problems have larger numbers than in previous lessons in order to elicit the need for using more abstract diagrams, which are the focus of upcoming lessons. When students analyze Han's and Tyler's claims they construct viable arguments (MP3).” Student Task Statements, “1. Clare donated 48 books. Clare donated 6 times as many books as Andre. a. Draw a diagram to represent the situation. b. How many books did Andre donate? Explain your reasoning. 2. Han says he can figure out the number of books Andre donated using division. Tyler says we have to use multiplication because it says ‘times as many’. a. Do you agree with Han or Tyler? Explain your reasoning. b. Write an equation to represent Tyler’s thinking. c. Write an equation to represent Han’s thinking. 3. Elena donated 9 times as many books as Diego. Elena donated 81 books. Use multiplication or division to find the number of books Diego donated.”

Examples of critiquing the reasoning of others include:

• Unit 4, From Hundredths to Hundred-thousands, Lesson 2, Activity 2, students critique the reasoning of others as they determine if fractions are equivalent to decimals. The Activity Narrative states, “The last question refers to decimals on a number line and sets the stage for the next lesson where the primary representation is the number line. As students discuss and justify their decisions about the claim in the last question, they critically analyze student reasoning (MP3).” Student Task Statements, “1. Decide whether each statement is true or false. For each statement that is false, replace one of the numbers to make it true. (The numbers on the two sides of the equal sign should not be identical.) Be prepared to share your thinking. \frac{50}{100}=0.50, 0.05=0.5, 0.3=\frac{3}{10}, 0.3=\frac{30}{100}, 0.3=0.30, 1.1=1.10, 3.06=3.60, 2.70=0.27. 2. Jada says that if we locate the numbers 0.05, 0.5, and 0.50 on the number line, we would end up with only two points. Do you agree? Explain or show your reasoning.” Problem 2 is followed by a number line labeled 0 on the left and 1 on the right.

• Unit 6, Multiplying and Dividing Multi-digit Numbers, Lesson 7, Activity 2, students critique the reasoning of others as they multiply four digit numbers by a one digit number. Activity, “‘Take a few quiet minutes to analyze Jada’s errors. Then, share your thinking with your partner.’ 2 minutes: independent work time on the first problem about Jada’s diagram. 2 minutes: partner discussion. Pause for a discussion. ‘What did Jada do correctly? (She multiplied the non-zero digits correctly. She decomposed the 6,489 by place value, which is helpful.) What did Jada miss? (She multiplied only the non-zero digits. She didn’t account for the place value of the digits being multiplied. For example: The partial product of 3 x 6,000 is 18,000, not 18.)’ Display and correct Jada’s diagram as a class. ‘Now complete the remaining problems independently.’ 5 minutes: independent work time on the last two problems.” Student Task Statements, “1. Jada used a diagram to multiply 3\times6,489 and made a few errors. a. Explain the errors Jada made. b. Find the value of 3\times6,489. Show your reasoning.” The Activity Narrative, “When students analyze Jada's work, find her errors and explain their reasoning, they critique the reasoning of others (MP3).”

• Unit 8, Properties of Two-dimensional Shapes, Lesson 5, Activity 1, students critique the reasoning of others as they identify symmetry in a figure. Student Task Statements, “Each shaded triangle is half of a whole figure that has a line of symmetry shown by the dashed line. Clare drew in some segments to show the missing half of each figure. Do you agree that the dashed line is a line of symmetry for each figure Clare completed? Explain your reasoning. If you disagree with Clare's work, show a way to complete the drawing so the dashed line is a line of symmetry.” Activity Narrative, “This activity highlights that having two identical halves on each side of a line doesn’t necessarily make a figure symmetrical. It encourages students to use their understanding of symmetry and the line of symmetry to articulate why this is so as they critique supplied reasoning (MP3).”

The materials reviewed for Imagine Learning Illustrative Mathematics Grade 4 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 3, Extending Operations to Fractions, Lesson 20, Activity 2, students model with mathematics as they use multiplication and fractions to create sticky-note letter designs. About this Lesson, “When students make decisions and choices, analyze real-world situations with mathematical ideas, translate a mathematical answer back into the context of a (real-world) situation, and adhere to constraints, they model with mathematics (MP4).” Student Task Statements, “Design your initial with sticky notes. 1. Plan your design and determine the number of sticky notes that you need. 2. Write at least two equations that show your design will fit on a piece of paper. 3. Take turns sharing your design with your partner. 4. Get the supplies and make your design.”

• Unit 4, From Hundredths to Hundred-thousands, Lesson 17, Activity 2, students model with mathematics as they consider aspects of rounding numbers. Activity Narrative, “In this activity, students continue to consider rounding in the same context as in the first activity. Students think about why rounding the altitudes to the nearest 1,000 may make it appear that two planes are a safe distance apart while the exact altitudes may show otherwise. As they consider different ways and consequences of rounding in this situation, students practice reasoning quantitatively and abstractly (MP2) and engage in aspects of mathematical modeling (MP4).” Student Task Statements, “Use the altitude data table from earlier for the following problems. 1. Look at the column showing exact altitudes. a. Find two or more numbers that are within 1,000 feet of one another. Mark them with a circle or a color. b. Find another set of numbers that are within 1,000 feet of one another. Mark them with a square or a different color. c. Based on what you just did, which planes are too close to one another? 2. Repeat what you just did with the rounded numbers in the last column. If we look there, which planes are too close to one another? 3. Which set of altitude data should air traffic controllers use to keep airplanes safe while in the air? Explain your reasoning. 4. Are there better ways to round these altitudes, or should we not round at all? Explain or show your reasoning.”

• Unit 9, Putting It All Together, Lesson 9, Activity 1, students model with mathematics as they consider mathematical questions that could be asked given certain information and answers. The Activity Narrative states, “In this activity, students analyze a situation and given solutions and think about what questions were asked. They also write their own question that can be answered with the quantities in the situation. As they connect equations to a context, students reason quantitatively and abstractly (MP2) and engage in aspects of modeling (MP4).” Student Task Statements, “George Meegan walked 19,019 miles between 1977 and 1983. He finished at age 31. He wore out 12 pairs of hiking boots. Jean Beliveau walked 46,900 miles between 2000 and 2011 and finished at age 56. Here are the responses Kiran gave to answer some questions about the situation. Write the question that Kiran might be answering. In the last row, write a new question about the situation and show the answer, along with your reasoning.” There is a table showing questions 1-3 along with their response and reasoning, and a blank space for question 4. “1. 1983-1977=6, 12\div6=2, 2 pairs or hiking boots; 2. 56-31=23, 23 years; 3. 2011-2000=11, 11 years, 11\times___=46,900, $$11\times4,000=44,000$$, 11\times200=2,200, 11\times70=770, 44,000+2,200+770=46,970, 4,000+200+70=4,270, 4,270 miles., 4._____.”

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 4, From Hundredths to Hundred-thousands, Lesson 16, Activity 1, students use appropriate tools strategically to round numbers. Activity Narrative, “When they find all of the numbers that round to a given number, students need to think carefully about place value and may choose to use a number line to support their reasoning (MP5).” Student Task Statements, “Noah says that 489,231 can be rounded to 500,000. Priya says that it can be rounded to 490,000. 1. Explain or show why both Noah and Priya are correct. Use a number line if it helps. 2. Describe all the numbers that round to 500,000 when rounded to the nearest hundred-thousand. 3. Describe all the numbers that round to 490,000 when rounded to the nearest ten-thousand. 4. Name two other numbers that can also be rounded to both 500,000 and 490,000.” 

• Unit 8, Properties of Two-Dimensional Shapes, Lesson 4, Activity 2, students choose tools strategically to help them determine symmetry in shapes. Activity Narrative, “In this activity, students practice identifying two-dimensional figures with line symmetry. They sort a set of figures based on the number of lines of symmetry that the figures have. Continue to provide access to patty paper, rulers, and protractors. Students who use these tools to show that a shape has or does not have a line of symmetry use tools strategically (MP5). Consider allowing students to fold the cards, if needed.” Student Task Statements, “Your teacher will give your group a set of cards. 1. Sort the figures on the cards by the number of lines of symmetry they have. 0 lines of symmetry, 1 line of symmetry, 2 lines of symmetry, 3 lines of symmetry 2. Find another group that has the same set of cards. Compare how you sorted the figures. Did you agree with how their figures are sorted? If not, discuss any disagreement.”

• Unit 8, Properties of Two-dimensional Shapes, Lesson 9, Activity 1, students engage with MP5 as they identify line symmetry and solve problems. Student Task Statements, “1. Mai has a piece of paper. She can get two different shapes by folding the paper along a line of symmetry. What is the shape of the paper before it was folded?” Activity Narrative, “The first question offers opportunities to practice choosing tools strategically (MP5). Some students may wish to trace the half-shapes on patty paper, to make cutouts of them, or to use other tools or techniques to reason about the original shape. Provide access to the materials and tools they might need.”

The materials reviewed for Imagine Learning Illustrative Mathematics Grade 4 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 1, Factors and Multiples, Lesson 8, Activity 2, students use precise language to identify shapes that meet certain conditions. The Activity Narrative states, “In this activity, students use their understanding of factor pairs, prime, and composite numbers to analyze their peers’ artwork. They look for rectangles that have the same area and those with a prime number or a composite number for their area. Students practice communicating with precision as they identify rectangles and how they know the rectangles meet these conditions (MP6).” Student Task Statement, “Trade artwork with your partner. Using your partner’s artwork, look for and describe each of the following: 1. Rectangles that have the same area 2. Rectangles with an area that is a prime number 3. Rectangles with an area that is a composite number 4. Which challenge they completed.”

• Unit 2, Fraction Equivalence and Comparison, Lesson 7, Activity 2, students use precise language to discuss fractions. Activity Narrative, “In this activity, students find equivalent fractions for fractions given numerically. They also work to clearly convey their thinking to a partner, which involves choosing and using words, numbers, or other representations with care. In doing so, students practice attending to precision (MP6) as they communicate about mathematics.” Student task statement, “For each fraction, find two equivalent fractions. Partner A 1. \frac{3}{2}, 2. \frac{10}{6}, Partner B 1. \frac{4}{3}, 2. \frac{14}{10}. Next, show or explain to your partner how you know that the fractions you wrote are equivalent to the original. Use any representation that you think is helpful.” Students repeat this independently during the Cool-down. Student Task Statement, “Name two fractions that are equivalent to \frac{5}{3}. Explain or show your reasoning.”

• Unit 4, From Hundredths to Hundred-thousands, Lesson 12, Cool-down, students use precise language to compare numbers and discuss place value. Lesson Narrative, “In this lesson, students use their understanding of place value to compare numbers and articulate how they reason about the size of the numbers. In doing so, they reinforce their understanding of place value and the base-ten number system (MP7). In communicating their thinking, they also practice attending to precision (MP6).” Cool-down Task Statement, “Here are two numbers, each with the same digit missing in different places. 17,__42,1___{_,}742 If the missing digit in both numbers is 1, which number will be greater: the first or the second? 2. Name all the digits from 0 to 9 that will make the second number greater. Explain how you know.”

• Unit 8, Properties of Two-dimensional Shapes, Lesson 1, Activity 1, students use the specialized language of mathematics as they sort shapes into categories. The Activity Narrative, “Students will likely use informal language in describing their categories (for example, “they have all the same sides” or “the sides are slanted the same way”). Encourage students to use more precise mathematical language (MP6) and support them in doing so by revoicing their ideas (for example, “all sides have the same length” for “all sides are the same”).” Student Task Statements, “1. Sort the shapes from your teacher into 3–5 categories. For each category, write a title on a sticky note. 2. Share your categories with another group. Take turns listening to each other’s explanations. Do your categories make sense to them? Do their categories make sense to you? Any suggestions or corrections?” The Activity Synthesis, “‘What were some of the categories you used to sort the shapes?’ As students share responses, update the display by adding (or revising) language, diagrams, or annotations. Point out to students that some categories had to do with the sides of the shapes and others had to do with the angles. Remind students to borrow language from the display in the next activity.”

The materials reviewed for Imagine Learning Illustrative Mathematics Grade 4 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 4, From Hundredths to Hundred-thousands, Lesson 7, Cool-down, students use the structure of the base ten number system to answer questions about a number. Lesson Narrative, “In this lesson, students count to read and write multi-digit numbers up to the ten-thousands place. They also count to develop a sense of the magnitude of 10,000. In the previous lesson, students counted by thousands and created 10 groups of 1,000 to make 10,000. This continues to build awareness of the structure of our number system with the base of ten (MP7). In this lesson, students practice writing numbers up to 100,000, which sets the stage for 100,000 as a new unit in base-ten in the lessons that follow.” Cool-down Task Statement, “Consider the number 57,000. 1. How many thousands are in it? 2. How many ten-thousands are in it? 3. Write the number in words.”

• Unit 5, Multiplicative Comparison and Measurement, Lesson 6, Activity 1, students look for and make use of structure as they represent and solve multiplicative comparison problems involving multiples of 10. Student Task Statements, “Here is a diagram that represents two quantities, A and B. 1. What are some possible values of A and B? 2. Select the equations that could be represented by the diagram. A. 15\times10=150. B. 16\times100=1,600. C. 30\div3=10. D. 5,000\div5=1,000. E. 80\times10=800. F. 12,000\div10=1,200.” Lesson Narrative, “The activity also reinforces what students previously learned about the product of a number and 10—namely, that it ends in zero and each digit in the original number is shifted one place to the left because its value is ten times as much (MP7).” Activity Synthesis, “Display possible values for A and corresponding values for B in a table such as this: ‘What do you notice about the values of each set? How do the values compare? (The values for B are all multiples of 10. Each value for B is ten times the corresponding value for A.)’ Select students to share their responses and reasoning.  For students who used the diagram to reason about the equations, consider asking, ‘How might you label the diagram to show that it represents the equations you selected? (Sample response: For the equation 30\div3=10 I would label A “3” and B “30” I know that 30 is ten times as much as 3, so the diagram represents the equation.)’ For students who used their understanding of numerical patterns to support their reasoning, consider asking, ‘How did you know that the equation could be represented as a comparison involving ten times as many? (Sample response: I know that when we multiply a number by 10, the product will be ten times the value. I also know that division is the inverse of multiplication, so I looked for equations that were multiplying or dividing by 10 or had ten as a quotient.)’”

• Unit 8, Properties of Two-dimensional Shapes, Lesson 8, Warm-up, students look for and make use of structure as they use properties of operations to add, subtract, and multiply fractions including mixed numbers. Student Task Statements, “Decide if each statement is true or false. Be prepared to explain your reasoning.  1\frac{1}{5}+2\frac{2}{5}+3\frac{3}{5}+ 4\frac{4}{5}=12, 10-\frac{1}{2}-\frac{2}{2}-\frac{3}{2}-\frac{4}{2}=5, 1\frac{1}{6}+2\frac{2}{6}+3\frac{3}{6}+4\frac{4}{6}+5\frac{5}{6}=15\frac{3}{6}, \frac{1}{3}+\frac{2}{3}+\frac{3}{3}=3\times\frac{2}{3}.” Lesson Narrative, “The series of equations prompt students to use properties of operations (associative and commutative properties in particular) in their reasoning, which will be helpful when students solve geometric problems involving fractional lengths (MP7).” Activity Synthesis, “What strategies did you find useful for adding or subtracting these numbers with fractions? (Possible strategies: Adding whole numbers separately than fractions. Noticing that 1+2+3+4 is 10 and using that fact to add or subtract fractions. Combine fractions that add up to 1 (such as \frac{1}{5}+\frac{4}{5} and \frac{2}{5}+\frac{3}{5}). In the second equation, add up the fractions and subtract the sum from 10, instead of subtracting each fraction individually.) Consider asking: Who can restate _____’s reasoning in a different way? Did anyone have the same strategy but would explain it differently? Did anyone approach the expression in a different way? Does anyone want to add on to _____’s strategy?”

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 1, Factors and Multiples, Lesson 2, Warm-up, students use repeated reasoning to complete related multiplication problems mentally. Activity Narrative, “The purpose of this Number Talk is to elicit strategies and understandings students have for multiplying single-digit numbers. These understandings help students develop fluency and will be helpful later in this lesson when students find factor pairs of numbers. As students use earlier problems to find the new products, they look for and make use of structure (MP7) and use repeated reasoning (MP8).” Student Task Statement, “Find the value of each expression mentally. 2\times7, 4\times7, 3\times7, 7\times7.” Activity Synthesis, “How did the first three expressions help you find 7\times7? (The 7 breaks apart into 3 and 4, so I could multiply in parts and add them.)’ Consider asking: ‘Who can restate _____’s reasoning in a different way? Did anyone have the same strategy but would explain it differently? Did anyone approach the expression in a different way? Does anyone want to add on to _____’s strategy?’”

• Unit 2, Fraction Equivalence and Comparison, Lesson 3, Activity 1, students reason about the relative sizes of two fractions with the same numerator. Students Facing, “1. This diagram shows a set of fraction strips. Label each rectangle with the fraction it represents. 2. Circle the greater fraction in each of the following pairs. If helpful, use the diagram of fraction strips. a. \frac{3}{4} or \frac{5}{4}. b. \frac{3}{5} or \frac{5}{5}. c. \frac{3}{6} or \frac{6}{6}. d. \frac{3}{8} or \frac{5}{8}. e. \frac{3}{10} or \frac{5}{10}. 3. What pattern do you notice about the circled fractions? How can you explain the pattern?” Lesson Narrative, “When students observe that 5 equal parts are greater than 3 of the same equal part, regardless of the size of those parts, they see regularity in repeated reasoning (MP8).”  Activity Synthesis, “What do you notice about each pair of fractions in the second question? (They all have 3 and 5 for the numerators, and they have the same denominator.) What does it mean when two fractions, say \frac{3}{8} and \frac{5}{8}, have the same denominator? (They are made up of the same fractional part—eighths in this case.) How can we tell which fraction is greater? (Because the fractional parts are the same size, we can compare the numerators. The fraction with the greater numerator is greater.)”

• Unit 8, Properties of Two-dimensional Shapes, Lesson 7, Activity 2, students use repeated reasoning as they find the perimeter of shapes and write a matching expression. Student Task Statements, “1. Find the perimeter of each shape. Write an expression that shows how you find the perimeter. 2. Compare your expressions with your partners’ expressions. Make 1–2 observations.” Lesson Narrative, “In this activity, students find the perimeter of several shapes and write expressions that show their reasoning. Each side of the shape is labeled with its length, prompting students to notice repetition in some of the numbers. The perimeter of all shapes can be found by addition, but students may notice that it is efficient to reason multiplicatively rather than additively (MP8).” Activity Synthesis, “Read Mai’s reasoning aloud. “What do you think Mai means? What is unclear? Are there any mistakes? 1 minute: quiet think time, 2 minute: partner discussion. With your partner, work together to write a revised explanation for Mai. Display and review the following criteria: We can’t find the perimeter of a quadrilateral if one or more side lengths are missing and _____.’ 3–5 minute: partner work time Select 1–2 groups to share their revised explanation with the class. Record responses as students share. ‘How did you know that the perimeter of C can be found by combining twice 4 cm and twice 7\frac{1}{2} cm or (2\times4)+(2\times7\frac{1}{2})? (We saw earlier that when quadrilaterals have two pairs of parallel sides, opposite sides have the same length. If the sides are parallel, they are the same distance apart.) How did you know the unlabeled side in Shape D is also 7\frac{1}{2} cm long? (The figure has a line of symmetry through the midpoint of the top and bottom segment.) What expression can we write for the perimeter of D? (4+5\frac{3}{4}+7\frac{1}{2}+7\frac{1}{2}) or 4+5\frac{3}{4}+(2\times7\frac{1}{2}) or equivalent).’”