4th Grade - Gateway 2

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

• Unit 2, Fraction Equivalence and Comparison, Lesson 7, Warm-up, students develop conceptual understanding as they use previous knowledge of equivalence and strategies for comparing fractions. Student Facing states, “Decide if each statement is true or false. Be prepared to explain your reasoning. \frac{4}{8}=\frac{7}{8}, \frac{3}{4}=\frac{6}{8}, \frac{2}{6}=\frac{2}{8}, \frac{6}{3}=\frac{4}{2}.” (4.NF.1)

• Unit 4, From Hundredths to Hundred-Thousands, Lesson 7, Warm-Up, students develop conceptual understanding of place value with larger numbers and notice patterns in the count. Launch states, “‘Count by 1,000, starting at 3,400.’ Record as students count. Stop counting and recording at 23,400.” Activity Synthesis states, “What parts of the numbers stay the same each time we count? (The digits in the hundreds, tens, and one place remain the same each time.) When will these digits change? (The digit in the hundreds, tens, and ones place will never change because we are counting by 1,000 each time.)” (4.NBT.2)

• Unit 6, Multiplying and Dividing Multi-Digit Numbers, Lesson 13, Activity 2, students develop conceptual understanding of dividing multi-digit numbers in the context of real-life situations. Student Facing states, “1. Priya’s mom made 85 gulab jamuns for the class to share. Priya gave 5 to each student in the class. How many students are in Priya’s class? Explain or show your reasoning. 2. Han’s uncle sent in 110 chocolate-covered breadsticks for a snack. The students in Han's class are seated at 6 tables. Han plans to give the same number of breadsticks to each table. How many breadsticks does each table get? Explain or show your reasoning.” (4.NBT.6)

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

• Unit 4, From Hundredths to Hundred-thousands, Lesson 11, Cool-down, students demonstrate conceptual understanding of place value to locate large numbers on a number line. Student Facing states, “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 5, Multiplicative Comparison and Measurement, Lesson 8, Cool-down, students demonstrate conceptual understanding while comparing and converting metric measurements. Student Facing states, “1. Kiran lives 7 kilometers from school. How many meters from school does he live? Explain or show your reasoning. 2. A classmate of Kiran’s lives 800 meters from school. Does he live closer or farther away from school than Kiran? Explain your reasoning.” Responding to Student Thinking states, “Students may say that Kiran’s classmate lives farther from school (or that 800 meters is greater than 7 kilometers) if they mistake 7 kilometers to be 700 meters instead of 7,000 meters, or if they confuse the relationship between kilometers and meters with that between meters and centimeters.” (4.MD.1, 4.MD.2)

• Unit 6, Multiplying and Dividing Multi-digit Numbers, Lesson 7, Activity 1, students demonstrate conceptual understanding as they use rectangular diagrams to represent multiplication of three-digit and one-digit numbers. Student Facing states, “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. (4.NBT.5)

The materials reviewed for Kendall Hunt's Illustrative Mathematics Grade 4 meet expectations for giving attention throughout the year to individual standards that set an expectation of procedural skill and fluency. 

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

• Unit 4, From Hundredths to Hundred-thousands, Lesson 10, Warm-up, students develop procedural skill and fluency as they use strategies and understanding of adding and subtracting multi-digit numbers. Student Facing states, “Find the value of each expression mentally. 650+75, 5,650+75, 50,650+75, 500,650+75.” (4.NBT.4) 

• Unit 6, Multiplying and Dividing Multi-digit Numbers, Lesson 19, Activity 1, students develop procedural skill and fluency as they use partial quotients and interpret remainders. Activity states, “3 minutes: independent work time on the first 2 problems. Pause after problem 2 to discuss students’ responses. Display the different ways that students decompose 389 to divide it by 7. ‘Most other calculations we’ve seen so far end with a 0, but this one ends with a 4. What does the 4 tell us?’ (We cannot make a group of 7 with 4 leftover. 389 is not a multiple of 7, and there are leftovers.) ‘When we divide and end up with leftovers we call them remainders, because they represent what is remaining after we divide into equal groups.’ Display: 389=7x55+4 ‘How does this equation show that 389\div7 has a remainder?’ (It shows that 389 is not a multiple of 7. It also shows that 7 and 55 make a factor pair for 385, and 389 is 4 more than that.) 3 minutes: independent work time on the last 2 problems. As students work on the last two problems monitor for students who: start with the largest multiple of 3 and 10 within 702 that they can think of to decompose the dividend (690, 600), use the fewest steps to find the quotient. Student Facing states, “Jada used partial quotients to find out how many groups of 7 are in 389. Analyze Jada’s steps in the algorithm. (A vertical representation of partial quotients is shown.) 1. a. Look at the three numbers above 389. What do they represent? b. Look at the three subtractions below 389. What do they represent? c. What is another way you can decompose 389 to divide by 7? 2. Is 389 a multiple of 7? Explain your reasoning. 3. Use an algorithm that uses partial quotients to find out how many groups of 3 are in 702. 4. Is 702 a multiple of 3? Explain your reasoning.”

• Unit 9, Putting It All Together, Lesson 9, Warm-Up, students develop procedural skill and fluency with subtraction. Activity states, “1 minute: quiet think time. Record answers and strategy. Keep expressions and work displayed. Repeat with each expression.” Student Facing states, “Find the value of each expression mentally. 5,000-403, 5,300-473, 25,300-493, 26,000-1,493.” (4.NBT.4)

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

• Unit 4, From Hundredths to Hundred-thousands, Lesson 20, Cool-Down, students demonstrate procedural skill and fluency as they use the standard algorithm for subtraction. Student Facing states, “Use the standard algorithm to find the value of the difference. 173,225-114,329.” (4.NBT.4)

• Unit 7, Angles and Angle Measurement, Lesson 15, Activity 1, students demonstrate procedural skill and fluency as they find angle measurements. Activity states, “5 minutes: independent work time. 2 minutes: partner discussion. Monitor for students who: use symbols or letters to represent unknown angles, write equations to help them reason about the angle measurements.” Student Facing states, “Find the measurement of each shaded angle. Show how you know. (A. A right angle is shown, with 62 degrees and the unknown shaded part.)” (4.MD.7)

• Unit 9, Putting It All Together, Lesson 4, Activity 1, students demonstrate procedural skill and fluency as they subtract multi-digit numbers. Activity states, “6–8 minutes: independent work time.” Student Facing states, “1. Find the value of each difference. a. 700-16. b. 7,000-16. c. 70,000-16. d. 700,000-16.” (4.NBT.4)

The materials reviewed for Kendall Hunt's Illustrative Mathematics Grade 4 meet expectations for being designed so that teachers and students spend sufficient time working with engaging applications of the mathematics. According to IM Curriculum, Design Principles, Balancing Rigor, “Access to new mathematics and problems prompts students to apply their conceptual understanding and procedural fluency to novel situations.” Multiple routine and non-routine applications of the mathematics are included throughout the grade level and these single- and multi-step application problems are included within Activities or Cool-downs. 

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

Examples of routine applications of the math include:

• Unit 3, Extending Operations to Fractions, Lesson 6, Cool-down, students apply their understanding about multiplication of a fraction by a whole number to solve real-world problems. Student Facing states, “Tyler bought 5 cartons of milk. Each carton contains \frac{3}{4} liter. How many liters of milk did Tyler buy? Explain or show your reasoning. Han bought 3 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.” (4.NF.4c)

• Unit 6, Multiplying and Dividing Multi-digit Numbers, Lesson 23, Activity 1, students solve real-world problems as they reason about distance and use multiple operations to find a solution. Student Facing states, “Mai’s cousin is in middle school. She travels from her homeroom to math, then English, history, and science. When she finishes her science class, she takes the same path back to her homeroom. Mai’s cousin makes the same trip 5 times each week. The distances between the classes are shown. 1. How far does Mai’s cousin travel each round trip—from her homeroom to the four classes and back? Write one or more expressions or equations to show your reasoning. 2. Each week, Mai’s cousin makes 3 round trips from her homeroom to her music class. The total distance traveled on those 3 round trips is 2,364 feet. How far away is the music room from her homeroom? Show your reasoning. 3. Mai thinks her cousin travels 2 miles each week just going between classes. Do you agree? Explain or show your reasoning.” A diagram showing distances between locations, in feet, is shown. (4.NBT.4, 4.NBT.5, 4.NBT.6)

• Unit 9, Putting It All Together, Lesson 8, Activity 2, students solve a real-world problem as they interpret a situation involving equal groups and make sense of a remainder. Student Facing states, “A school is taking everyone on a field trip. It needs buses to transport 375 people. Bus Company A has small buses with 27 seats in each. Bus Company B has large buses with 48 seats in each. 1. What is the smallest number of buses that will be needed if the school goes with: Bus Company A? Show your reasoning. Bus Company B? Show your reasoning. 2. Which bus company should the school choose? Explain your reasoning. 3. Bus Company C has large buses that can take up to 72 passengers. Diego says, ‘If the school chooses Bus Company C, it will need only 6 buses, but the buses will have more empty seats.’ Do you agree? Explain your reasoning.” (4.OA.3)

Examples of non-routine applications of the math include:

• Unit 1, Factors and Multiples, Lesson 6, Activity 2, students examine factors of numbers from 1 to 20 and use them to solve problems. Launch states, “‘Let’s solve some problems about a game you read about earlier, where students take turns opening and closing lockers. Silently read and think about each question.’ 1 minute: quiet think time.” Student Facing states, “The 20 students in Tyler’s fourth-grade class are playing a game in a hallway with 20 lockers in a row. Your goal is to find out which lockers will be touched as all 20 students take their turn touching lockers. 1. Which locker numbers does the 3rd student touch? 2. Which locker numbers does the 5th student touch? 3. How many students touch locker 17? Explain or show how you know. 4. Which lockers are only touched by 2 students? Explain or show how you know. 5. Which lockers are touched by only 3 students? Explain or show how you know. 6. Which lockers are touched the most? Explain or show how you know. If you have time: Which lockers are still open at the end of the game? Explain or show how you know.” (4.OA.4)

• Unit 5, Multiplicative Comparison and Measurement, Lesson 10, Activity 2, students solve multi-step problems by using metric units of measurement and multiplicative comparison. Student Facing states, “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.” (4.MD.2, 4.OA.2, 4.OA.3)

• 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. Activity states, “5 minutes: independent work time. 8–10 minutes: partner work time. Monitor for diagrams that reflect a variety of geometric features. Monitor for students who consider both geometric features and measurement in the description.” Student Facing states, “1. Create a two-dimensional shape that has at least 3 of the following: a. ray, b.line segment, c. right angle, d. acute angle, e. obtuse angle, f. perpendicular lines, g. parallel lines. 2. Without showing your partner, describe the figure so that your partner is able to draw it as best as possible. 3. Switch roles, and draw your partner’s shape based on their description.” (4.G.1, 4.G.2)

The materials reviewed for Kendall Hunt's 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.

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

• Unit 2, Fraction Equivalence and Comparison, Lesson 2, Cool-down, students demonstrate conceptual understanding as they create visual representation of non-unit fractions. Student Facing states, “Use a blank diagram to create a representation for each fraction. Both blank diagrams represent the same quantity. 1. \frac{5}{8} 2. \frac{9}{8}” Bar models broken into two wholes are provided for each problem. (4.NF.A)

• Unit 3, Extending Operations to Fractions, Lesson 11, Activity 1, students solve routine real-world problems that involve subtracting mixed numbers where it is necessary to decompose one or both numbers. Student Facing states, “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. 1. Clare started her bracelet first and has only \frac{7}{8} inch left until she finishes it. How long is her bracelet so far? 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? 3. How much longer is Elena’s bracelet than Andre’s at the moment?” (4.NF.3d)

• Unit 4, From Hundredths to Hundred-thousands, Lesson 19, Activity 1, students develop procedural skill and fluency as they use the addition algorithm. Student Facing states, “1. Find the value of each sum. a. 8,299+1, b. 8,299+11, c. 8,299+111, d. 8,299+1111. 2. Use the expanded form of both 8,299 and 1,111 to check the value you found for the last sum.” (4.NBT.4)

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

• Unit 1, Factors and Multiples, Lesson 1, Cool-down, students use procedural fluency with multiples and apply their understanding of area. Student Facing states, “If a rectangle is 6 tiles wide, what could be its area? Name three possibilities. Explain or show your reasoning.” (4.OA.4)

• Unit 3, Extending Operations to Fractions, Lesson 1, Activity 1, students engage in all three aspects of rigor, conceptual understanding, procedural fluency, and application as they interpret situations involving equal groups. Activity states, “‘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 Facing states, “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 7, Angles and Angle Measurement, Lesson 10, Activity 2, students develop conceptual understanding alongside procedural skill and fluency as they use a protractor to measure angles and understand perpendicular lines. Launch states, “Give each student 2 pieces of paper and colored pencils. Provide access to straightedges or rulers, in case requested. Read the opening prompts and the first question. ‘What do you think Lin did with her paper? Mark a point on a piece of paper and try folding it as Lin might have done.’ 2–3 minutes: quiet think time on the first problem. Pause for a discussion. Invite a couple of students to share how they think Lin met the challenge.” Student Facing states, “Tyler gave Lin a challenge: ‘Without using a protractor, draw four 90\degree angles. All angles have their vertex at point P.’ Lin folded the paper twice, making sure each fold goes through point P. Then, she traced the creases. 1. Your teacher will give you a sheet of paper. Draw a point on it. Then, show how Lin might have met the challenge. 2. When Lin folded the paper, the creases formed a pair of perpendicular lines. What do you think ‘perpendicular lines’ mean? 3. Use Lin’s method to create a new pair of perpendicular lines through the same point. Trace the creases with a different color. Be prepared to explain how you know the lines you created are perpendicular. 4. Which shapes have sides that are perpendicular to one another? Mark the perpendicular sides. Be prepared to explain how you know the sides are perpendicular.” A is a regular pentagon, B is a rectangle, C is a right triangle, and D is a parallelogram. (4.G.1, 4.MD.6)

The materials reviewed for Kendall Hunt's 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 within the Course Guide (How to Use These Materials) and within specific lessons (Instructional Routines, Lesson Preparation Narratives, and Lesson Activities’ Narratives).

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

• Unit 2, Fraction Equivalence and Comparison, Lesson 14, Activity 1, students reason about fractions given descriptive clues. Preparation, Lesson Narrative states, “In the first activity, students compare sets of fractions with like and unlike denominators. They do so by using benchmarks, writing equivalent fractions, or reasoning about the numerators and denominators. In the second activity, students interpret and solve problems involving fractional measurements in context. Both activities present a new setup, structure, or context, requiring students to make sense of the given information and the problems, and to persevere in solving them (MP1).” Student Facing states, “Six friends are each given a list of 5 fractions. They each chose one fraction quietly and wrote clues about their choice. Use their clues to identify the fractions they chose.”

• Unit 6, Multiplying and Dividing Multi-Digit Numbers, Lesson 12, Cool-down, students persevere to solve and make sense of a real-world problem involving multi-digit multiplication. Preparation, Lesson Narrative states, “This lesson gives students the opportunity to apply the multiplication strategies they have learned to solve various contextual problems involving measurement. The problems vary in format and complexity—some involve a single computation and others require multiple steps to solve. The work here prompts students to make sense of problems and persevere in solving them (MP1) and to reason quantitatively and abstractly (MP2).” Student Facing states, “In a leap year, the month of February has 29 days. How many hours are in that month? Show your reasoning.”

• Unit 7, Angles and Angle Measurement, Lesson 15, Activity 2, students use their knowledge of angles to make sense of a problem and persevere in solving it. Narrative states, “In this Info Gap activity, students solve abstract multi-step problems involving an arrangement of angles with several unknown measurements. By now students have the knowledge and skills to find each unknown value, but the complexity of the diagram and the Info Gap structure demand that students carefully make sense of the visual information and look for entry points for solving the problems. They need to determine what information is necessary, ask for it, and persevere if their initial requests do not yield the information they need (MP1).” Launch states, “Groups of 2. MLR4 Information Gap Display the task statement, which shows a diagram of the Info Gap structure.1–2 minutes: quiet think time. Read the steps of the routine aloud. ‘I will give you either a problem card or a data card. Silently read your card. Do not read or show your card to your partner.’ Distribute the cards. ‘The diagram is not drawn accurately, so using a protractor to measure is not recommended.’ 1–2 minutes: quiet think time. Remind students that after the person with the problem card asks for a piece of information, the person with the data card should respond with ‘Why do you need to know (restate the information requested)?’”  Activity states, “5 minutes: partner work time. After students solve the first problem, distribute the next set of cards. Students switch roles and repeat the process with Problem Card 2 and Data Card 2.”

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

• Unit 4, From Hundredths to Hundred-thousands, Lesson 1, Activity 2, students reason abstractly and quantitatively about decimals and their representations. Narrative states, “In this activity, students practice representing and writing decimals given another representation (fraction notation or a diagram). The idea that two decimals can be equivalent, just like two fractions can be equivalent, is made explicit here. When students make connections between quantities in word form, decimal form, and fraction form, they reason abstractly and quantitatively (MP2).” Student Facing states, “Each large square represents 1. 1. Write a fraction and a decimal that represent the shaded parts of each diagram. Then, write each amount in words. 2. Shade each diagram to represent each given fraction or decimal. a. Fraction: ___ Decimal: 0.78 b. Fraction: \frac{8}{10} Decimal: ___ c. Fraction: \frac{55}{100} Decimal: ___ d. Fraction: \frac{107}{100} Decimal: ___ e. Fraction: ___ Decimal: 1.6  3. Han and Elena disagree about what number the shaded portion represents. Han says that it represents 0.60 and Elena says it represents 0.6. Explain why both Han and Elena are correct.”

• Unit 5, Multiplicative Comparison and Measurement, Lesson 15, Activity 2, students reason abstractly and quantitatively when they convert feet and inches and solve a logic puzzle. Student Facing states, “While on an outing, a group of friends had a stone-stacking contest to see who could build the tallest stone tower. Andre’s stone tower is 3 times as tall as Diego’s, but Diego didn’t build the shortest tower. The tallest tower is 4 feet and 2 inches tall and belongs to Tyler. One person built a tower that is 39 inches tall. Tyler’s tower is 5 times as tall as the shortest tower. 1. How tall is each person’s stone tower? Be prepared to explain or show your reasoning. 2. Elena came along and built a tower that is 5 times as tall as Diego’s tower. Is Elena’s tower more than 6 feet? Show your reasoning.” Narrative states, “In this activity, students apply their knowledge of multiplicative comparison and ability to convert feet and inches to solve a logic puzzle. They use several given clues to determine the heights of four objects. As they use the clues to reason about the heights of the towers and who built them, students reason abstractly and quantitatively (MP2).”

• Unit 9, Putting It All Together, Lesson 8, Cool-down, students solve multi-step problems involving all operations. Preparation, Lesson Narrative states, “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).” Student Facing states, “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. Explain why one of the choices could not be true.”

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

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

• Unit 2, Fraction Equivalence and Comparison, Lesson 10, Cool-down, students construct viable arguments as they determine if two fractions are equivalent. Student Facing states, “Problem 2, Diego wrote \frac{11}{5} and \frac{55}{10} as equivalent fractions. Are those fractions equivalent? Explain or show how you know. Use a number line, if it helps.”

• Unit 7, Angles and Angle Measurement, Lesson 3, Activity 2, students construct a viable argument as they create and reason about perpendicular and parallel lines. Narrative states, “In this activity, students are prompted to draw intersecting and parallel lines, and to explain how they know a pair of parallel lines would never intersect. Students are not expected to formally justify that two lines are parallel. They are expected to make a case that goes beyond appearance (such as ‘it looks like they would never cross’) and notice that the parallel lines maintain the same distance apart (MP3). Students are also introduced to the convention of naming lines with letters to support precision when describing and comparing lines. They are not expected to formally name lines or line segments with letters.” Launch states, “Groups of 2. Give each student access to a ruler or a straightedge. Display a field of dots. Select a student to draw a line in the field. ‘Sometimes we label lines to help communicate about different parts of a figure.’ Demonstrate labeling the line with a letter. ‘We can call this “line a” because we labeled it with an “a”.’”  Student Facing states, “Here is another field of dots. Each dot represents a point. 1. Draw a line through at least 2 points. Label it line h. 2. Draw another line that goes through at least 2 points and intersects your first line. Label it line g. 3. Can you draw a new line that you think would never intersect: a. line h?  b, line g? If so, draw the line. Be prepared to explain or show how you know the lines would never cross. If not, explain or show why it can’t be done. 4. Here is a trapezoid. Do you think its top and bottom sides are parallel? What about its left and right sides? Explain or show how you know. If you have time: Can you draw a new line that you think would never intersect either line h or line g? If so, draw the line and be prepared to explain or show how you know the lines would never cross. If not, explain why it can’t be done.”

• Unit 9, Putting It All Together, Lesson 8, Activity 1, students construct an argument and critique the reasoning of others as they interpret a problem involving equal groups. Student Facing states, “A school is taking everyone on a field trip. It needs buses to transport 375 people. Bus Company A has small buses with 27 seats in each. Bus Company B has large buses with 48 seats in each. 1. What is the smallest number of buses that will be needed if the school goes with: Bus Company A? Show your reasoning. Bus Company B? Show your reasoning. 2. Which bus company should the school choose? Explain your reasoning. 3. Bus Company C has large buses that can take up to 72 passengers. Diego says, “If the school chooses Bus Company C, it will need only 6 buses, but the buses will have more empty seats.” Do you agree? Explain your reasoning.” Activity Synthesis states, “Select students to share their responses and reasoning. Record the different representations students used to solve the problems. Consider asking: ‘How did you decide how many buses you would need from each company? Do all the buses carry the same amount of passengers? How can you see that in your representations or equations? How did you decide what operations to use to answer each question?”

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

• Unit 3, 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 states, “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 Facing states, “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.” Narrative states, “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 7, Activity 2, students convert measurement from meters to centimeters and critique student work to identify and describe errors. Narrative states, “In this activity, students analyze student work converting meters to centimeters to develop the understanding that a meter is ‘100 times as long’ as a centimeter. They correct errors in reasoning centering around place value (MP3).” Activity states, “‘Take 5 quiet minutes to spot and correct Priya’s errors and find the missing measurement. Then, share your thinking with your partner.’ 5 minutes: independent work time. 3–4 minutes: partner discussion. Monitor for students who place zeros for the measurement in centimeters and those who explicitly reason in terms of 100 times the value in meters.” Student Facing states, “Priya took some measurements in meters and recorded them in the table, but she made some errors when converting them to centimeters. She also left out one measurement.”  Students are given a table with the headings “Measurement in Meters, Measurement in Centimeters” and then the following: ”a. height of door 2 and 200 b. height of hallway 3 and 30  c. width of hallway 5 and 500  d. length of gym.18 and 180 e. length of hallway 27 and 2,700 f. length of playground 50 and ____ 1. Find and correct Priya’s conversion errors. Be prepared to explain how you know. 2. Fill in the length of the playground in centimeters. Write an equation to represent your thinking.”

• Unit 6, Multiplying and Dividing Multi-digit Numbers, Lesson 9, Activity 1, students critique the reasoning of others as they analyze an algorithm that uses partial products. Activity states, “4 minutes: independent work time on the first problem about Noah’s diagram. 4 minutes: partner discussion. 5 minutes: group work time on the rest of the activity. Monitor for students who include the place value of each digit in 124 in explaining what is happening in the algorithm.” Student Facing states, “1. Noah drew a diagram and wrote expressions to show his thinking as he multiplied two numbers. How does each expression represent Noah’s diagram? Be prepared to share your thinking with a partner. 2. Later, Noah learned another way to record the multiplication, as shown here. Make sense of each step of the calculations and record your thoughts. Be prepared to explain Noah’s steps to a partner.” An image of Noah’s work is shown along with his calculations. Narrative states, “When students interpret and make sense of Noah's work, they construct viable arguments and critique the reasoning of others (MP3).”

The materials reviewed for Kendall Hunt's Illustrative Mathematics Grade 4 meet expectations for supporting the intentional development of MP4: Model with mathematics; and MP5: Choose tools strategically, for students, in connection to the grade-level content standards, as expected by the mathematical practice standards. Students have opportunities to engage with the Math Practices across the year and they are often explicitly identified for teachers within the Course Guide (How to Use These Materials) and within specific lessons (Instructional Routines, Lesson Preparation Narratives, and Lesson Activities’ Narratives).

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

• Unit 1, Factors and Multiples, Lesson 8, Activity 1, students model with mathematics as they use art and concepts of area. Lesson Narrative states, “When students isolate and describe the mathematical elements in art and adhere to mathematical constraints to create art, they model with mathematics (MP4).” Student Facing states, “Create an outline for art in the Mondrian style, starting with an 18-by-24 grid. Your artwork should: be partitioned into at least 12 rectangles, include two different rectangles that have the same area, include at least one rectangle whose area is a prime number. Try at least one of these challenges. Make a design where: all but two of the rectangles have a prime number for its area, no two rectangles share a side entirely.”

• Unit 4, From Hundredths to Hundred-Thousands, Lesson 23, Activity 1, students model mathematical concepts and apply them to real life situations. Lesson narrative states, “Students make decisions and choices, adhere to mathematical constraints, use mathematical ideas to analyze real-world situations, and interpret a mathematical answer and whether it makes sense in the context of a situation. In doing so, they model with mathematics (MP4).”  Activity states, “2 minutes: independent work time. 8 minutes: partner work time. Monitor for students who: think of multiple quantities that the same number might represent, use estimation in their reasoning.” Student Facing states, “Here is some information about insects: Termites, Size of a colony: 100–1,000,000. A queen lives for 30–50 years. There are 3,000–3,500 species of termites. The length of a termite is 4 to 15 millimeters. In some species, the mature queen may produce around 40,000 eggs a day. Odorous House Ants Size of colony: up to 100,000. A queen lives for 300–1,800 days. The length of an ant is 1.5–3.2 millimeters. Foraging ants travel up to 700 feet from their nests. There are 12,000–22,000 possible species. Honey Bees Size of a hive: 10,000–60,000. There are around 500 drones in a hive. A queen can lay about 1,500–2,000 eggs each day. A hive produces 7–40 liters of honey in a season. The length of a bee is 10–20 millimeters. 1. Here are some numbers that could represent facts about termites, house ants, and honey bees. What might each number represent?  Number, What it Might Represent, 2.4, 8, 4878, 1,794, 6,905, 20,799, 530,097  2. Add another number to the list. What about the insects might this number represent? 3. Discuss your answers with your partner. Be prepared to show or explain your reasoning.”  

• Unit 6, Multiplying and Dividing Multi-digit Numbers, Lesson 25, Activity 3, students model with mathematics as they create their own flower pattern and multi-step problem. Lesson Narrative states, “When students ask and answer questions that arise from a given situation, use mathematical features of an object to solve a problem, make choices, analyze real-world situations with mathematical ideas, interpret a mathematical answer in context, and decide if an answer makes sense in the situation, they model with mathematics (MP4).” Student Facing states, “1. Write a multi-step problem about making paper flowers. 2. Exchange the problem with your partner and solve each other’s problems.” Student Response states, “Sample response: It takes 1 sheet of tissue paper to make a big flower and \frac{1}{2} sheet to make a small flower. How much tissue paper is needed to make a garland that has 7 small and 7 large garlands? (7 sheets for big flowers and 4 sheets for small flowers, where half a sheet will not be used. 11 sheets are needed, with \frac{1}{2} a sheet left over).”

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

• Unit 2, Fraction Equivalence and Comparison, Lesson 15, Cool-down, students use appropriate tools strategically when they compare two fractions using a strategy of their choice. Student Facing states, “In each pair of fractions, which fraction is greater? Explain or show your reasoning. 1. \frac{3}{10} or \frac{2}{6}. 2. \frac{99}{100} or \frac{9}{10}.” 

• Unit 4, From Hundredths to Hundred-thousands, Lesson 16, Activity 1, students use appropriate strategies as tools when rounding numbers. Student Facing states, “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.” Lesson Narrative states, “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).”

• Unit 8, Properties of Two-dimensional Shapes, Lesson 9, Activity 1, students use tools strategically as they identify line symmetry and solve problems. Student Facing states, “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?” Lesson Narrative states, “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 Kendall Hunt's 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 the specialized language of math, in connection to grade-level content, as they work with support of the teacher and independently throughout the units. Examples include:

• Unit 1, Factors and Multiples, Lesson 7, Activity 1, students attend to the specialized language of math as they accurately describe factors and multiples. Student Facing states, “1. Complete a statement using the word “factor” and a statement using the word “multiple” for each number. 2. As you compare statements with your partner, discuss one thing you notice and one thing you wonder.” Lesson Narrative states, “The purpose of this activity is for students to find factors and multiples of a given number and make statements that use the terms “factors” and “multiples.” This work prompts students to use language precisely (MP6).”

• Unit 2, Fraction Equivalence and Comparison, Lesson 2, Warm-up, students use precise mathematical language as they describe how four shapes are partitioned and shaded. Narrative states, “This warm-up prompts students to carefully analyze and compare the features of four partitioned shapes. It allows the teacher to hear the terminologies students use to talk about fractions and fractional parts. In making comparisons, students have a reason to use language precisely (MP6).”  Launch states, “Groups of 2. Display the image. ‘Pick one that doesn’t belong. Be ready to share why it doesn’t belong.’ 1 minute: quiet think time.” Student response sample states,  A is the only one not partitioned into 3 parts. B is the only one that does not have straight edges. C is the only one not partitioned into equal parts. D is the only one whose parts are not all clear or unshaded.”  Activity Synthesis states,  “‘What does the shaded part in D represent?’ ($$\frac{1}{3}$$ or one-third of the shape). Shade one part of B and C. ‘Is each shaded part one-third of the shape as well? (Yes for B, no for C.)  Why is the shaded part not one-third of the square in C?’ (The parts aren’t equal in size.)  Shade one part of A. ‘Is it a third of the square?’ (No, it is \frac{1}{4} or one-fourth.)”

• Unit 2, Fraction Equivalence and Comparison, Lesson 9, Warm-up, students use accuracy and precision when they describe strategies in finding the value of multiplication problems. The Narrative states, “strategies of doubling and halving elicited here will be helpful later in the lesson when students generate equivalent fractions. In describing strategies, students need to be precise in their word choice and use of language (MP6).” Launch states, “Display one expression. ‘Give me a signal when you have an answer and can explain how you got it.’” Student Facing states, “Find the value of each expression mentally. 10\times6, 10\times12, 10\times24, 5\times24.” 

• Unit 3, Extending Operations to Fractions, Lesson 13, Activity 1, students use accuracy and precision when they measure pencils to the nearest \frac{1}{4} and \frac{1}{8} inch. Student Facing states, “Your teacher will give your group a set of colored pencils. 1. Work with your group to measure each colored pencil to the nearest \frac{1}{4} inch. Check each other’s measurements. Record each measurement in the table. 3. Work with your group to measure each colored pencil to the nearest \frac{1}{8} inch. Check one another’s measurements. Record each measurement in the table.” The Lesson Narrative states, “Students attend to precision when they measure the pencils to the appropriate fractional unit (MP6).” The Activity Synthesis states, “Allow students to record their two sets of data on two different class line plots. (If dot stickers are available, consider using them—one sticker for each data point.) ‘How did your data and line plots change when you measured colored pencils to the nearest \frac{1}{8} inch?’ (Sample responses: We got different numbers. The marks or points on the line plots are distributed differently. The points for some of the same pencils show up as different lengths in the second line plot.) ‘What is challenging about measuring to the nearest \frac{1}{8} inch?’ (The tick marks are smaller and harder to see on the ruler.) ‘Why do you think we measure to the nearest \frac{1}{8} inch?’ (We measure to be more accurate.) “Let’s look at some other length data with measurements in halves, fourths and eighths of an inch.”

• Unit 5, Multiplicative Comparison and Measurement, Lesson 13, Cool-Down, students compare the ingredients needed to make cookies, using precision when comparing units of measure including pounds and ounces (MP6). Student Facing states, “Priya needs oats and raisins to make cookies. She needs 3 pounds of oats. That amount is 4 times as much as the amount of raisins that she needs. How many ounces of raisins does she need? Explain or show your reasoning.” Student Section Summary states, “In this section, we learned about various units for measuring length, distance, weight, capacity, and time. We saw how different units that measure the same property are related. Here are the relationships that we saw: One meter (m) is 100 times as long as 1 centimeter (cm). One kilometer (km) is 1,000 times as long as 1 meter (m). One kilogram (kg) is 1,000 times as heavy as 1 gram (g). One liter (L) is 1,000 times as much as 1 milliliter (mL). One pound (lb) is 16 times as heavy as 1 ounce (oz). One hour is 60 times as long as 1 minute. One minute is 60 times as long as 1 second. When given a measurement in one unit, we can find the value in another unit by reasoning and writing equations. Throughout the section, we used these relationships to convert measurements from one unit to another, to compare and order measurements, and to solve problems in different situations.”

• Unit 7, Angle and Angle Measurement, Lesson 1, Cool-down, students attend to precise mathematical language as they describe a drawing to a partner. Student Facing states, “Here is a drawing on a card: Write a description of the drawing that could be used by a classmate to make a copy.” Student Response sample states, “Draw two diagonal lines: one from the top left corner to the bottom right, and another from the bottom left corner to the top right. Draw a line that goes up and down through the point where the two diagonal lines cross. From the top of that line, draw a line to the bottom right corner. The bottom segment of the up-and-down line is thicker than the rest of the lines. The lines make a lot of triangles of different sizes.”

The materials reviewed for Kendall Hunt's 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 across the year and they are often explicitly identified for teachers within the Course Guide (How to Use These Materials) and within specific lessons (Instructional Routines, Lesson Preparation Narratives, and Lesson Activities’ Narratives).

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

• Unit 1, Factors and Multiples, Lesson 4, Warm-up, students use structure and knowledge of math facts to find larger products. Narrative states, “The purpose of this Number Talk is to elicit strategies and understandings students have for multiplying within 100 with one factor larger than 10. These understandings help students develop fluency and will be helpful when students find factor pairs of numbers later in the lesson. In this activity, students have an opportunity to look for and make use of structure (MP7) as they use a combination of products of smaller factors to find products of larger factors.” Launch states, “Display one expression. ‘Give me a signal when you have an answer and can explain how you got it.’  1 minute: quiet think time.” Student Facing states, “Find the value of each expression mentally. 10\times6, 3\times6, 13\times6, \12\times4.” Activity Synthesis states, “How can knowing the value of the first two expressions help you find the value of the third expression? (I can multiply in parts and add the smaller parts together to find a larger product.)”

• Unit 4, From Hundredths to Hundred-thousands, Lesson 9, Activity 1, students look for and make use of structure while they compare numbers to determine value. Narrative states, “When students sort the cards, they look for how the numbers are the same and different, including their overall value or the digits that make up the numbers (MP7).” Activity states, “Give each group a set of cards from the blackline master. 5 minutes: partner and group work time on the first two problems. As students work, listen for place-value language such as: value of the digit, ten times, thousands, ten-thousands, and hundred-thousands. Record any place-value language students use to describe how they sorted the numbers and display for all to see. ‘Now work independently to write the numbers in the next problem in expanded form. Then, talk with your partner about the value of the digits.’ 3 minutes: independent work time. 5 minutes: partner work time. Monitor for students who: accurately write the numbers in expanded form, describe the relationship between the value of the digits in multiplicative terms (“ten times”).” Student Facing states, “Your teacher will give you and your partner a set of cards with multi-digit numbers on them. 1. Sort the cards in a way that makes sense to you. Be prepared to explain your reasoning. 2. Join with another group and explain how you sorted your cards. 3. Write each number in expanded form. a. 4,620 b. 46,200 c. 462,000. 4. Write the value of the 4 in each number. 5. Compare the value of the 4 in two of the numbers. Write two statements to describe what you notice about the values. 6. How is the value of the 2 in 46,200 related to the value of the 2 in 462,000?” Activity Synthesis states, “Invite students to share their expressions in expanded form and what they noticed about the value of the 4. ‘What do you notice about the value of the 6 in each number? The value of the 2?’ (The value of the 6 is different in each number. It is first 600, then 6,000, then 60,000.) Students may talk about the number of zeros in each number. Shift their focus to the place value of the 6— hundreds, thousands, ten-thousands. ‘How is the value of the 2 in 46,200 related to the value of the 2 in 462,000?’ (The value of the 2 in 462,000 is 2,000 and the same digit in 46,200 has a value of 200. 2,000 is ten times the value 200.) ‘What multiplication equation could we write to represent the relationship between the 2 in 46,200 and 462,000?’ ($$2,000=200\times10$$) ‘We can also write this equation using division: 2,000\div200=10.’”

• Unit 9, Putting It All Together, Lesson 1, Cool-down, students look for and make use of structure as they reason about sums of fractions. Lesson Narrative states, “In this lesson, students practice multiplying a fraction and a whole number and adding and subtracting fractions, including mixed numbers. They rely on their understanding of equivalence and the properties of operations to decompose fractions, whole numbers, and mixed numbers to enable comparison, addition, subtraction, and multiplication (MP7).” Student Facing states, “Here are some fractions: \frac{15}{10}, \frac{13}{10}, \frac{53}{100}, \frac{9}{10}. 1. Select two fractions that have a sum greater than 2. Explain or show your reasoning. 2. Use all four fractions to write an expression that has a value greater than 1 but less than 2.”

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

• Unit 3, Extending Operations to Fractions, Lesson 7, Activity 2, students use repeated reasoning as they decompose fractions to find sums. Narrative states, “In the previous activity, students saw that a fraction can be decomposed into a sum of fractions with the same denominator and that it can be done in more than one way. In this activity, they record such decompositions as equations. The last question prompts students to consider whether any fraction can be written as a sum of smaller fractions with the same denominator. Students see that only non-unit fractions (with a numerator greater than 1) can be decomposed that way. Students observe regularity in repeated reasoning as they decompose the numerator, 9, into different parts while the denominator in all cases is 5 (MP8).” Activity states, “‘Take a few quiet minutes to complete the activity. Then, share your responses with your partner.’ 5–6 minutes: independent work time. 3–4 minutes: partner discussion. Monitor for different explanations students offer for the last question.” Student Facing states, “1. Use different combinations of fifths to make a sum of \frac{9}{5}. a. \frac{9}{5} = ___ + ___ + ___ + ___ + ___  b.  \frac{9}{5} = ___ + ___ + ___ + ___  c. \frac{9}{5} = ___ + ___ + ___  d. \frac{9}{5}=___ + ___   2. Write different ways to use thirds to make a sum of \frac{4}{3}}. How many can you think of? Write an equation for each combination. 3. Is it possible to write any fraction with a denominator of 5 as a sum of other fifths? Explain or show your reasoning.” Activity Synthesis states, “Invite students to share their equations. Display or record them for all to see. Next, discuss students' responses to the last question. Select students with different explanations to share their reasoning. If not mentioned by students, highlight that fractions with a numerator of 1 (unit fractions) cannot be further decomposed into smaller fractions with the same denominator because it is already the smallest fractional part. Other fractions with a numerator other than 1 (non-unit fractions) can be decomposed into fractions with the same denominator.”

• Unit 8, Properties of Two-dimensional Shapes, Lesson 7, Cool-down, students use repeated reasoning as they find the perimeter of shapes and write matching expressions. Student Facing states, “Here is a rectangle with two lines of symmetry. Find its perimeter. Write an expression to show how you find it.” Activity 1 Lesson Narrative states, “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).” Students have the opportunity to demonstrate this same reasoning within the Cool-down.

• Unit 9, Putting It All Together, Lesson 4, Warm-up, students work through a number talk, using repeated reasoning to solve increasingly challenging addition problems. Narrative states, “This Number Talk encourages students to think about the base-ten structure of whole numbers and properties of operations to mentally solve subtraction problems. The reasoning elicited here will be helpful later in the lesson when students find differences of multi-digit numbers.” Activity states, “1 minute: quiet think time. Record answers and strategy. Keep expressions and work displayed. Repeat with each expression.” Student Facing states, “Find the value of each difference mentally. 87-24, 387-124, 6,387-129, 6,387-4,329.” Activity Synthesis states, “How is each expression related to the one before it? How might the first expression help us find the value of the last expression?”