3rd-5th Grade - Gateway 2

The materials reviewed for Illustrative Mathematics® v.360 Grade 3 through Grade 5 meet expectations for supporting the intentional development of students’ conceptual understanding of key mathematical concepts, especially where called for in specific content standards or clusters.

Multiple conceptual understanding problems are embedded throughout the grade level within Warm-ups, Activities, or Cool-downs. Students have opportunities to engage with these problems both independently and with teacher support. According to the Illustrative Mathematics® v.360 Course Guide, the Key Structures in This Course, Principles of IM Curriculum Design section emphasizes the role of purposeful representations in developing conceptual understanding. The Purposeful Representations section states, “Across lessons and units, students are systematically introduced to representations and encouraged to use those that make sense to them. As their learning progresses, students make connections between different representations and the concepts and procedures they show.” 

An example in Grade 3 includes:

• Unit 2, Area and Multiplication, Lesson 7, Warm-Up, students develop conceptual understanding of measurement units, larger square units can be useful in situations involving larger areas. Students see a picture of a girl on a playground holding a large square and Student Task Statements states, “What do you notice? What do you wonder?” Activity Synthesis states, “If needed, ‘What could you measure with this square?’ (You could measure the area of big areas, like the playground.) ‘Why might you want this square instead of square centimeters or square inches?’ (It takes fewer squares of this size to measure an area that is a lot larger like a playground or a room.)” (3.MD.6)

An example in Grade 4 includes:

• 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 Task Statements states, “1. Priya’s mom makes 85 gulab jamuns for the class to share. Priya gives 5 gulab jamuns to each student in the class. How many students are in Priya’s class? Explain or show your reasoning. 2. Han’s uncle sends 108 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)

An example in Grade 5 includes:

• Unit 8, Putting It All Together, Lesson 10, Activity 1, students develop conceptual understanding as they practice adding fractions with unlike denominators and reason about how the size of the numerators and denominators impact the value of a fraction. Student Task Statements states, “Use the directions to play Greatest Sum with a partner. 1. Spin the spinner. 2. Write the number in an empty box for Round 1. Be sure your partner can’t see your paper. Once a number is written down, it can’t be changed. 3. Take turns. Spin and write numbers in the empty boxes until all 4 boxes have been filled. 4. Find the sum. 5. The partner with the greater sum wins the round. 6. The partner who won the most rounds wins the game. If there is a tie, players add the sums from all 4 rounds. The greater total sum wins the game.” Activity Synthesis states, “‘What strategies were helpful as you played Greatest Sum?’ (I tried to make fractions that have a larger numerator than denominator so they would be greater than one. I tried to make sure the ones and twos were in the denominator and put bigger numbers in the numerator.) ‘How did you add your fractions?’ (I wrote equivalent fractions with a common denominator)” (5.NF.1)

According to Illustrative Mathematics® v.360 Course Guide, the materials are designed to provide students with opportunities to independently demonstrate conceptual understanding, when appropriate. 7. Key Structures in This Course, Principles of IM Curriculum Design, Coherent Progression states, “Each activity starts with a Launch that gives all students access to the task. Independent work time follows, allowing them to grapple with problems individually before working in small groups. In the Activity Synthesis at the end, students consolidate their learning by making connections between their work and the mathematical goals.” 

An example in Grade 3 includes:

• Unit 5, Fractions as Numbers, Lesson 3, Activity 2, students demonstrate conceptual understanding of fractions as they match fractions to shaded diagrams. Student Task Statements states, “Your teacher will give you a set of cards. 1. Match each fraction to a diagram. Be ready to explain why some fractions do not have a matching diagram. 2. Compare your matches with another group. 3. Use Cards R, T, V, and X to create a shaded diagram for each fraction card that did not have a match. Be ready to share your new cards and explain why they match.” (3.NF.1)

An example in Grade 4 includes:

• 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 Task Statements 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, and C on the number line from 0 to 400,000. (4.NBT.1)

An example in Grade 5 includes:

• Unit 6, More Decimal and Fraction Operations, Lesson 20, Activity 2, students demonstrate conceptual understanding as they compare a product of fractions to one of the factors. Student Task Statements states, “Andre says: When you multiply any fraction by a number less than 1, the product will be less than the fraction. When you multiply any fraction by a number greater than 1, the product will be greater than the fraction. Each partner chooses a different statement and describes why it is true. Show your thinking, using diagrams, symbols, or other representations.” (5.NF.5)

The materials reviewed for Illustrative Mathematics® v.360 Grade 3 through Grade 5 meet expectations for providing intentional opportunities for students to develop procedural skills and fluencies, especially where called for in specific content standards or clusters.

According to the Course Guide, 7. Key Structures of This Course, Principles of IM Curriculum Design, Conceptual Understanding and Procedural Fluency, “Warm-up routines, practice problems, centers, and other built-in activities help students develop procedural fluency, which develops over time.” 

An example in Grade 3 includes: 

• Unit 7, Two-dimensional Shapes and Perimeter, Lesson 7, Warm-up, students develop procedural skill and fluency as they use strategies they have learned to add multi-digit numbers. Student Task Statement, “Decide whether each statement is true or false. Be prepared to explain your reasoning. 123+75+123+75=100+100+70+70+5+5+3+3, 123+75+123+75=(2\times123)+(2\times75), 123+75+123+75=208+208, 123+75+123+75=246+150.” (3.NBT.2)

An example in Grade 4 includes: 

• 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 strategies. Keep expressions and work displayed. Repeat with each expression.” Student Task Statement, “Find the value of each expression mentally. 5,000-403, 5,300-473, 25,300-493, 26,000-1,493.” (4.NBT.4)

An example in Grade 5 includes: 

• Unit 4, Wrapping Up Multiplication and Division with Multi–Digit Numbers, Lesson 4, Warm-up, students develop procedural skill and fluency as they notice the patterns in calculations within the number talk, leading towards the standard algorithm. Student Task Statement, “Find the value of each product mentally. 3\times3, 3\times20, 3\times600, 3\times623.” Activity Synthesis states, “‘How is the last product related to the first three?’ (It is the sum of the first three.) ‘Did the first three calculations help you find the last product?’ (Yes, I was able to add them together to find 3\times623.)” (5.NBT.5)

According to the Course Guide, 7. Key Structures of This Course, Principles of IM Curriculum Design, Coherent Progression, materials were designed to include opportunities for students to independently demonstrate procedural skill and fluency, when appropriate. “Each activity starts with Launch that gives all students access to the task. Independent work time follows, allowing them to grapple with problems individually before working in small groups. In the Activity Synthesis at the end, students consolidate their learning by making connections between their work and the mathematical goals.” 

An example in Grade 3 includes:

• Unit 4, Relating Multiplication to Division, Lesson 9, Activity 2, students demonstrate fluency as they identify patterns in multiplication. Activity states, “‘In the right column, write at least two multiplication facts you can figure out because you know the given multiplication fact in the left column.’ 3–5 minutes: independent work time. ‘Now, share the facts that you found with your partner. Record any facts that your partner found that you didn’t find. Be sure to explain your reasoning.’” Student Task Statement, “1. In each row, write at least 2 multiplication facts you can figure out because you know the given multiplication fact in the left column. Be prepared to share your reasoning. If I know… 2\times4, then I also know 4\times2, 4\times4, 2\times8.” (3.OA.7) 2. If time permits, complete the rest of the multiplication table from the previous activity. Use the multiplication facts you know to find those you don’t know.”

An example in Grade 4 includes:

• 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 Task Statement, “Use the standard algorithm to find the value of the difference. 173,225-114,329.” (4.NBT.4)

An example in Grade 5 includes:

• Unit 8, Putting It All Together, Lesson 1, Activity 2, students demonstrate procedural skill and fluency as they practice using the standard algorithm to find products. Student Task Statement, “1. Use the digits 7, 3, 2, and 5 to make the greatest product. Use each digit only once” Launch states, “Groups of 2. Display: 7, 3, 2, 5. ‘Using only these digits, what multiplication expressions could we write?’ 1 minute: quiet think time. Record answers for all to see. ‘Which of these expressions do you think would make the greatest product? Be prepared to explain your reasoning.’ (I think the three-digit by one-digit expression would make the greatest product because you can put the 7 in the hundreds place.) ‘Use the digits 7, 3, 2, and 5 to make the greatest product.’” (5.NBT.5)

The materials reviewed for Illustrative Mathematics® v.360 Grade 3 through Grade 5 meet expectations for supporting the intentional development of students’ ability to utilize mathematical concepts and skills in engaging applications, especially where called for in specific content standards or clusters. 

Multiple routine and non-routine applications of mathematics are included throughout the grade level, with single- and multi-step application problems embedded within Activities or Cool-downs. Students have opportunities to engage with these applications both with teacher support and independently. According to the Course Guide, materials are designed to provide students with opportunities to independently demonstrate their understanding of grade-level mathematics when appropriate. 7. Key Structures in This Course, Principles of IM Curriculum Design, Coherent Progression states, “Each activity starts with a Launch that gives all students access to the task. Independent work time follows, allowing them to grapple with problems individually before working in small groups. In the Activity Synthesis at the end, students consolidate their learning by making connections between their work and the mathematical goals.” 

An example of a routine application of the math in Grade 3 includes:

• Unit 3, Wrapping Up Addition and Subtraction within 1000, Lesson 19, Activity 2, students solve a multi-step real-world problem and then write an equation to represent the problem. Activity states, “‘Take some independent time to work on this problem. You can choose to solve the problem first or write the equation first.’ 5–7 minutes: independent work time, Monitor for different ways students: Write an equation. Represent the problem, such as by using a tape diagram. Decide their answer makes sense, such as by thinking about the situation or by rounding.” Student Task Statement, “Kiran is setting up a game of mancala (mahn-KAH-lah). He has a jar of 122 stones. From the jar, he takes 3 stones to put in each of the 6 pits on his side of the board. How many stones are in the jar now? 1. Write an equation to represent the situation. Use a letter for the unknown quantity. 2. Solve the problem. Explain or show your reasoning. 3. Explain how you know your answer makes sense.” (3.OA.8)

An example of a non-routine application of the math in Grade 3 includes:

• Unit 6, Measuring Length, Time, Liquid Volume, and Weight, Lesson 15, Activity 1, students solve a real-world problem by using concepts of time, weight, and volume. Launch states, “‘We’re going to solve some problems about a day at the fair. What are some things you could do during a day at the fair?’ (Go on rides, walk around, play games, look at some of the animals) 30 seconds: quiet think time. Share responses. Give each group tools for creating a visual display.” Student Task Statement, “You spent a day at the fair. Solve 4 problems about your day and create a poster to show your reasoning and solutions. 1. You arrived at the fair! Entry to the fair is $9 a person. You went there with 6 other people. How much did it cost your group to enter the fair? 2. How did you start your day? (Choose 1.) You arrived at the giant pumpkin weigh-off at 11:12 a.m. and left at 12:25 p.m. How long were you there? You spent 48 minutes at the carnival and left at 12:10 p.m. What time did you get to the carnival? 3. What was next? (Choose 1.) You visited a barn with 7 sheep. Together, the sheep are given 91 liters of water a day. Each sheep is given the same amount. How much water does each sheep get each day? You visited a life-size sculpture of a cow made of butter. The butter cow weighs 273 kilograms, which is 277 kilograms less than the actual cow. How much does the actual cow weigh? 4. Before you went home, you played some games. At the balloon pop game, there were 72 balloons arranged in 9 equal rows. How many balloons were in each row?” (3.MD.1, 3.MD.2, 3.OA.3)

An example of a routine application of the math in Grade 4 includes:

• 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 Task Statement, “1. 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. 2. 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)

An example of a non-routine application of the math in Grade 4 includes:

• 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. ‘Work in your group to answer each question. Consider the representations you created or saw earlier to help you think about the problems.’” Student Task Statement, “Tyler’s class plays the same locker game again. Your goal this time is to find out which lockers are touched as each of the 20 students takes their turns. 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 touched by only 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)

An example of a routine application of the math in Grade 5 includes:

• Unit 3, Multiplying and Dividing Fractions, Lesson 18, Activity 1, students work with real-world problems involving multiplication and division of fractions. Student Task Statement, “1. Diego and some classmates paint one wall of a long hallway. They have 2 gallons of paint to share equally in paint trays. Each paint tray holds 1⁄4 gallon of paint. How many paint trays can they fill with the 2 gallons of paint? a. Draw a diagram to represent the situation. b. Write a division equation to represent the situation. c. Write a multiplication equation to represent the situation. 2. Diego and Clare decide to equally share 1⁄4 gallon of a special paint that glows in the dark. How many gallons of paint does each person get? a. Draw a diagram to represent the situation. b. Write a division equation to represent the situation. c. Write a multiplication equation to represent the situation.” (5.NF.4, 5.NF.6, 5.NF.7)

An example of a non-routine application of the math in Grade 5 includes:

• Unit 7, Shapes on the Coordinate Plane, Lesson 13, Activity 1, students plot points that represent the length and width of a rectangle with a given perimeter. Student Task Statement, “1. Jada draws a rectangle with a perimeter of 12 centimeters. What could be the length and width of the rectangle if all the side lengths are whole numbers? Use the table to record 5 possible answers. 2. Represent the length and width of each rectangle as a point on the coordinate grid. 3. If Jada draws a square, how long and wide will it be? 4. If Jada’s rectangle is 2.5 cm long, how wide is it? Represent this rectangle as a point on the coordinate grid. 5. If Jada’s rectangle is 3.25 cm long, how wide is it? Represent this rectangle as a point on the coordinate grid.” (5.G.2, 5.NBT.7, 5.OA.3)

The materials reviewed for Illustrative Mathematics® v.360 Grade 3 through Grade 5 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 as reflected by the standards.

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. 

An example in Grade 3 includes:

• Unit 4, Relating Multiplication to Division, Lesson 19, Activity 2, students engage in all three aspects of rigor, conceptual understanding, procedural fluency, and application as they represent division within 100. Launch states, “Give base-ten blocks to each group. Ask students to keep their books closed. ‘Use base-ten blocks to find the value of 60\div5.’ 1–2 minutes: independent work time.” Student Task Statement, “Jada and Han used base-ten blocks to represent 60\div5. 1. Make sense of Jada’s and Han’s work. a. What did they do differently? b. Where do you see the value of 60\div5 in each person’s work? 2. How would you use base-ten blocks to represent these expressions and find their value? Be prepared to explain your reasoning. a. 64\div4: Would you make 4 groups or groups of 4? b. 72\div6: Would you make 6 groups or groups of 6? c. 75\div15: Would you make 15 groups or groups of 15?” (3.OA.2, 3.OA.7)

An example in Grade 4 includes:

• 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 Task Statement, “Here are images of some crackers. a. How are the crackers in image A and image B alike? 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 question, 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 scoops \frac{1}{3} cup of brown rice 8 times. How many cups of brown rice does he scoop?” (4.NF.4)

An example in Grade 5 includes:

• Unit 6, More Decimal and Fraction Operations, Lesson 12, Activity 2, students develop all three aspects of rigor, conceptual understanding, procedural skill and fluency, and application, as they solve multi-step problems involving the addition and subtraction of fractions. Student Task Statement, “1. Choose a problem to solve. Problem A: Jada is baking protein bars for a hike. She adds \frac{1}{2} cup of walnuts and then decides to add another \frac{1}{3} cup. How many cups of walnuts has she added altogether? The recipe requires 1\frac{1}{3} cups of walnuts. How many more cups of walnuts does Jada add? Explain or show your reasoning. Problem B: Kiran and Jada hiked 1\frac{1}2} miles and took a rest. Then they hiked another \frac{4}{10} mile before stopping for lunch. How many miles have they hiked so far? The trail is a total of 2\frac{1}{2} miles. hHow much farther do they hike? Explain or show your reasoning. 2. Discuss the problems and solutions with your partner. How are your strategies alike? How are they different? 3. Revise your work if necessary.” (5.NF.2)

The materials reviewed for Illustrative Mathematics® v.360 Grade 3 through Grade 5 meet expectations for supporting the intentional development of MP1: Make sense of problems and persevere in solving them, for students, in connection to the grade-level content standards, as expected by the mathematical practice standards. 

Students have opportunities to engage with MP1 across the year, and it is often explicitly identified for teachers within the Course Guide (Standards for Mathematical Practice) and within specific lessons (Preparation Narratives and Lesson Activities’ Narratives). According to the Course Guide, Standards for Mathematical Practice, “The Standards for Mathematical Practice (MP) describe the types of thinking and behaviors in which students engage as they do mathematics. Throughout the curriculum, the Teacher Guide identifies lessons and activities in which to observe the different MPs. The unit-level Mathematical Practice chart is meant to highlight a handful of lessons in each unit that showcase certain MPs. Some units, due to their size or the nature of their content, have fewer predicted chances for students to engage in a particular MP, indicated in the chart by a dash.” 

An example in Grade 3 includes:

• Unit 2, Area and Multiplication, Lesson 10, Activity 1, students solve a real-world problem involving area. Student Task Statement, “Noah is painting a wall in a community garden. The wall is shaped like a rectangle. A diagram of the wall is shown here. Paint is sold in 3 different sizes: A small container will cover 3 square meters. A medium container will cover 10 square meters. A large container will cover 40 square meters. What should Noah buy? Explain your reasoning.” Activity Narrative, “The purpose of this activity is for students to solve a real-world problem involving area. The activity includes a rectangle in which the side lengths are labeled. When students solve problems with multiple solutions and have to choose and justify a solution, they make sense of problems and persevere in solving them (MP1).”

An example in Grade 4 includes:

• Unit 2, Fraction Equivalence and Comparison, Lesson 14, Activity 1, students reason about fractions given descriptive clues. Preparation, Lesson Narrative, “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 Task Statement, “Six friends were each given a list of 5 

An example in Grade 5 includes:

• Unit 8, Putting It All Together, Lesson 6, Activity 2, students make sense of problems as they calculate the volume of different objects. Student Task Statement, “1. The Great Pyramid of Giza was built in Egypt more than 4,000 years ago. Today, it is 137 meters tall. The base of the pyramid is a square. Each side of the base is 230 meters long. If the pyramid was shaped like a rectangular prism, what would the volume of the prism be? 2. The Empire State Building is in New York City. The base is 129 meters by 57 meters. The building is 381 meters tall. Estimate the volume of the Empire State Building. 3. Which do you think is larger, the Great Pyramid or the Empire State Building? Explain or show your reasoning.” Activity Narrative, “The purpose of this activity is for students to solve problems about the volume of different structures. While students can find products of the given numbers, those products do not represent the actual volumes of the structures because neither the Great Pyramid of Giza nor the Empire State Building is a rectangular prism. The pyramid steadily decreases in size as it gets taller, while the Empire State Building also decreases in size at higher levels but not in the same regular way as the pyramid. With not enough information to make a definitive conclusion, students can see that both structures are enormous and that their volumes are roughly comparable, close enough that more studying would be needed for a definitive conclusion (MP1).”

The materials reviewed for Illustrative Mathematics® v.360 Grade 3 through Grade 5 meet expectations for supporting the intentional development of 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 MP2 across the year, and it is often explicitly identified for teachers within the Course Guide (Standards for Mathematical Practice) and within specific lessons (Preparation Narratives and Lesson Activities’ Narratives). According to the Course Guide, Standards for Mathematical Practice, “The Standards for Mathematical Practice (MP) describe the types of thinking and behaviors in which students engage as they do mathematics. Throughout the curriculum, the Teacher Guide identifies lessons and activities in which to observe the different MPs.The unit-level Mathematical Practice chart is meant to highlight a handful of lessons in each unit that showcase certain MPs. Some units, due to their size or the nature of their content, have fewer predicted chances for students to engage in a particular MP, indicated in the chart by a dash.” 

An example in Grade 3 includes:

• Unit 5, Fractions as Numbers, Lesson 4, Activity 2, students use diagrams to represent the fractional amount in a given situation. Activity Narrative, “The purpose of this activity is for students to use diagrams to represent situations that involve non-unit fractions. The Activity Synthesis focuses on how students partition and shade the diagrams and how the end of the shaded portion could represent the location of an object. When students interpret the different situations in terms of the diagrams, they reason abstractly and quantitatively (MP2).” Activity, “‘In the activity, each strip represents the length of a street where Pilolo is played. Work independently to represent each situation on a diagram.’ 3–5 minutes: independent work time. ‘With a partner, choose one of the situations and make a poster to show how you represented the situation with a fraction strip. You may want to include details, such as notes, drawings, labels, and so on, to help others understand your thinking.’ Give students materials for creating a display.” Student Task Statement: “Here are 4 situations about playing Pilolo (PIH-loh-loh) and 4 diagrams. Each diagram represents the length of a street where the game is played. Represent each situation on a diagram. Be prepared to explain your reasoning. 1. A student walks \frac{4}{8} the length of the street and hides a rock. 2. A student walks \frac{2}}{3} the length of the street and hides a penny. 3. A student walks \frac{3}{4} the length of the street and hides a stick. 4. A student walks \frac{5}{6} the length of the street and hides a penny. 5. This diagram represents the location of a hidden stick. About what fraction of the length of the street did the student walk to hide it? Be prepared to explain your reasoning.”

An example in Grade 4 includes:

• 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 Task Statement: “While on an outing, a group of friends had a stone-stacking contest to see who could build the tallest stone tower. Andre’s tower is 3 times as tall as Diego’s, but Diego didn’t build the shortest tower. Tyler built the tallest tower at 4 feet and 2 inches. 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? 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.” Activity Narrative: “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).”

An example in Grade 5 includes:

• Unit 2, Fractions as Quotients and Fraction Multiplication, Lesson 7, Activity 2, students reason abstractly and quantitatively as they match expressions and diagrams. Student Task Statement: “Your teacher will give you a set of cards that show expressions and diagrams. 1. Find the cards that match this situation. Be ready to explain your reasoning. Han, Lin, Kiran, and Jada run a 3-mile relay race as a team. They each run the same distance. 2. How far does each person run?” Activity Narrative, “Students reason abstractly and quantitatively (MP2) when they relate the story to the diagrams and expressions. All of the diagrams and expressions involve the same set of numbers so students need to carefully analyze the numbers in the story, the diagrams, and the expressions in order to choose the correct matches.”

The materials reviewed for Illustrative Mathematics® v.360 Grade 3 through Grade 5 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 (Standards for Mathematical Practice) and within specific lessons (Preparation Narratives and Lesson Activities’ Narratives). According to the Course Guide, Standards for Mathematical Practice, “The Standards for Mathematical Practice (MP) describe the types of thinking and behaviors in which students engage as they do mathematics. Throughout the curriculum, the Teacher Guide identifies lessons and activities in which to observe the different MPs. The unit-level Mathematical Practice chart is meant to highlight a handful of lessons in each unit that showcase certain MPs. Some units, due to their size or the nature of their content, have fewer predicted chances for students to engage in a particular MP, indicated in the chart by a dash.” 

An example in Grade 3 includes: 

• Unit 3, Wrapping Up Addition and Subtraction within 1,000, Lesson 9, Cool-down, students construct viable arguments and critique the reasoning of others as they solve subtraction problems using the algorithm. Preparation, Lesson Narrative, “Previously, students learned to record subtraction, using an algorithm in which the numbers are written in expanded form. They made connections between the structure and the steps of the algorithm and those of the base-ten diagram that represents the same subtraction. In this lesson, students take a closer look at the algorithm and use it to find differences. Students also examine a common error in subtracting numbers when decomposition of a place-value unit is required. When students discuss shown work, they construct viable arguments and critique the reasoning of others (MP3).” Student Task Statement, (Students see the thinking of a student on the problem with regrouping.) “Andre found the value of 739 - 255. His work is shown. Explain how he subtracted and the value he found for 739 - 255.”

An example in Grade 4 includes: 

• Unit 6, Multiplying and Dividing Multi-digit Numbers, Lesson 9, Activity 1, students construct viable arguments and critique the reasoning of others as they use partial products. Activity, “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 Task Statement, “1. Noah draws a diagram and writes expressions to multiply 2 numbers. How does each expression represent Noah’s diagram? 2. Noah learns another way to record the multiplication.” An image of Noah’s work is shown along with his calculations. Make sense of each step of the calculations and record your thoughts.” Activity Narrative: “The purpose of this activity is to analyze an algorithm that uses partial products. Students are not required to use a specific notation, but analyzing each algorithm deepens their understanding of the structure of place value in multiplication. When students interpret and make sense of Noah's work, they construct viable arguments and critique the reasoning of others (MP3).”

An example in Grade 5 includes: 

• Unit 5, Place Value Patterns and Decimal Operations, Lesson 15, Activity 3, students construct viable arguments and critique the reasoning of others as they analyze a common error when using the standard algorithm to subtract decimals. Student Task Statement, “1. Find the value of 622.35-71.4. Explain or show your reasoning. 2. Elena and Andre try to find the value of 622.35-71.4. Who do you agree with? Explain or show your reasoning.” Activity Narrative, “When students share their explanation of Han's calculations with a partner and revise their work after receiving feedback, they critique the reasoning of others and improve their arguments (MP3).”

The materials reviewed for Illustrative Mathematics® v.360 Grade 3 through Grade 5 meet expectations for supporting the intentional development of MP4: Model with mathematics, for students, in connection to the grade-level content standards, as expected by the mathematical practice standards.

Students have opportunities to engage with MP4 across the year, and it is often explicitly identified for teachers within the Course Guide (Standards for Mathematical Practice) and within specific lessons (Preparation Narratives and Lesson Activities’ Narratives). According to the Course Guide, Standards for Mathematical Practice, “The Standards for Mathematical Practice (MP) describe the types of thinking and behaviors in which students engage as they do mathematics. Throughout the curriculum, the Teacher Guide identifies lessons and activities in which to observe the different MPs. The unit-level Mathematical Practice chart is meant to highlight lessons in each unit that showcase certain MPs. Some units, due to their size or the nature of their content, have fewer predicted chances for students to engage in a particular MP, indicated in the chart by a dash.” 

An example from Grade 3 includes: 

• Unit 6, Measuring Length, Time, Liquid Volume, and Weight, Lesson 14, Cool-down, students reason about situations involving elapsed time. Student Task Statement, “1. A show at the carnival starts at 2:45 p.m. and lasts 47 minutes. What time does the show end? Explain or show your reasoning. 2. Another show that is 24 minutes long ends at 5:10 p.m. Kiran says that the show starts before 4:40 p.m. Do you agree? Explain or show your reasoning.” Preparation, Lesson Narrative, “In earlier lessons, students encountered and solved problems about time and liquid volume. In this lesson, students model with mathematics (MP4) as they determine quantities, questions, and solutions that make sense in given situations and that adhere to mathematical and real-world constraints when solving problems.”

An example from Grade 4 includes: 

• Unit 1, Factors and Multiples, Lesson 6, Activity 1, students model with mathematics as they apply their knowledge of factors, multiples, prime numbers, and composite numbers to solve problems about a game involving opening and closing of lockers. Activity Narrative: “The purpose of this activity is for students to visualize and make sense of the context of problems they will solve in the next activity. They will also consider representations that can be used to model the quantities and actions in the situation (MP4) and try creating them.” Student Task Statement: “The picture shows lockers in a school hallway. The 20 students in Tyler’s fourth-grade class play a game in a hallway that has 20 lockers in a row. The lockers are numbered from 1 to 20. The 1st student starts with the 1st locker, and while going down the hallway, opens all the lockers. The 2nd student starts with the 2nd locker, and while going down the hallway, shuts every other locker. The 3rd student stops at every 3rd locker and opens the locker if it is closed or shuts the locker if it is open. The 4th student stops at every 4th locker and opens the locker if it is closed or shuts the locker if it is open. This process continues through the 20th student, so that all 20 students in the class touch the lockers. Create a representation to show what you understand about this problem. Consider: How does your representation show lockers? How does your representation keep track of students who touch lockers? How does your representation show which lockers are open or closed?”

An example from Grade 5 includes: 

• Unit 3, Multiplying and Dividing Fractions, Lesson 9, Activity 2, students model with mathematics as they apply fraction multiplication to problems. Activity Narrative, “The goal of this activity is to examine calculations with measurements of a flag and try to figure out what question the calculations answer. The answers include units and this can serve as a guide to students. Since the first calculation has an answer in inches, the question it answers must ask for a length. Since the second calculation has an answer in square inches, the question it answers must ask for an area. This is an important step in solving the problems as students can then look at the diagram and the measurements and decide what the question could be. One important part of the modeling cycle (MP4) is interpreting information. That information may be presented in words or graphs or with mathematical symbols. In this case, students interpret equations in light of given numerical relationships and diagrams.” Student Task Statement, “Han has a replica of the flag of Columbia. It is 3\frac{1}{2} inches wide and 5\frac{1}{4} inches long. The yellow stripe is \frac{1}{2} of the width of the flag. The blue and red stripes are each \frac{1}{4} of the width. 1. \frac{1}{4} \times 3\frac{1}{2}=\frac{7}{8}.  The answer is \frac{7}{8} inch. What is the question? 2. \frac{1}{2}\times 3\frac{1}{2}=\frac{7}{4} and \frac{7}{4}\times \frac{21}{4}=\frac{147}{16}.The answer is \frac{147}{16} square inches. What is the question?”

The materials reviewed for Illustrative Mathematics® v.360 Grade 3 through Grade 5 meet expectations for supporting the intentional development of 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 MP5 across the year, and it is often explicitly identified for teachers within the Course Guide (Standards for Mathematical Practice) and within specific lessons (Preparation Narratives and Lesson Activities’ Narratives). According to the Course Guide, Standards for Mathematical Practice, “The Standards for Mathematical Practice (MP) describe the types of thinking and behaviors in which students engage as they do mathematics. Throughout the curriculum, the Teacher Guide identifies lessons and activities in which to observe the different MPs. The unit-level Mathematical Practice chart is meant to highlight a handful of lessons in each unit that showcase certain MPs. Some units, due to their size or the nature of their content, have fewer predicted chances for students to engage in a particular MP, indicated in the chart by a dash.” 

An example from Grade 3 includes:

• Unit 4, Relating Multiplication to Division, Lesson 13, Cool-down, students use appropriate tools strategically when they multiply within 100. Student Task Statement: “There were 6 buckets of sunflowers at the farmers market. Each bucket had 11 sunflowers. How many sunflowers were in the buckets? Show your thinking using objects, a drawing, or a diagram.” Activity 1 Narrative, “This is the first time students have solved multiplication problems with numbers in this range, so they should be encouraged to use the tools provided to them during the lesson if they choose (MP5). Students should also be encouraged to use strategies and representations from the previous section.” This outlines the goal of working with tools throughout this lesson.

An example from Grade 4 includes:

• Unit 8, Properties of Two-dimensional Shapes, Lesson 9, Activity 1, students use tools strategically as they identify line symmetry and solve problems. Student Task Statement: “1. Mai has a piece of paper. She can get each of these 2 shapes by folding the paper once along a line of symmetry. What is the shape of the unfolded paper?” Activity Narrative, “The first question offers opportunities to practice choosing tools strategically (MP5). Some students may wish to trace the half-figures on patty paper, to make cutouts of them, or to use other tools or techniques to reason about the original figure. Provide access to the materials and tools they might need.”

An example from Grade 5 includes:

• Unit 6, More Decimal and Fraction Operations, Lesson 8, Cool-down, students use an appropriate strategy as a tool to find solutions to problems involving addition and subtraction of fractions and then explain their strategy. Student Task Statement, “Find the value of each expression. Explain or show your reasoning. 1. \frac{5}{6}-\frac{1}{3}. 2. \frac{3}{4}+\frac{1}{2}.” Activity 2 Narrative, “The approaches are sequenced from more concrete to more abstract to help students connect a variety of different, but familiar, representations as they make sense of adding and subtracting fractions with unlike denominators. Students, who choose to draw number lines or tape diagrams, use appropriate tools strategically (MP5). Aim to elicit both key mathematical ideas and a variety of student voices, especially students who haven’t shared recently.”

The materials reviewed for Illustrative Mathematics® v.360 Grade 3 through Grade 5 meet expectations for supporting the intentional development of MP6: Attend to precision, for students, in connection to the grade-level content standards, as expected by the mathematical practice standards. 

Students have opportunities to engage with MP6 across the year, and it is often explicitly identified for teachers within the Course Guide (Standards for Mathematical Practice) and within specific lessons (Preparation Narratives and Lesson Activities’ Narratives). According to the Course Guide, Standards for Mathematical Practice, “The Standards for Mathematical Practice (MP) describe the types of thinking and behaviors in which students engage as they do mathematics. Throughout the curriculum, the Teacher Guide identifies lessons and activities in which to observe the different MPs. The unit-level Mathematical Practice chart is meant to highlight a handful of lessons in each unit that showcase certain MPs. Some units, due to their size or the nature of their content, have fewer predicted chances for students to engage in a particular MP, indicated in the chart by a dash.” 

An example in Grade 3 includes:

• Unit 3, Wrapping Up Addition and Subtraction Within 1,000, Lesson 8, Cool-down, students use accuracy and precision when they use base-ten diagrams to make sense of a written subtraction algorithm. Student Task Statement: “Explain how the diagram matches the algorithm.” Activity 2 Narrative: “As students work, encourage them to refine their descriptions of what is happening in both the diagrams and the algorithms, using more precise language and mathematical terms (MP6).”

An example in Grade 4 includes:

• Unit 7, Angle and Angle Measurement, Lesson 1, Cool-down, students attend to precise mathematical language as they describe a drawing to a partner. Activity Narrative: “The purpose of this activity is to motivate a need for more precise geometric language. Students work with a partner to replicate given geometric images—one partner describes the images and the other draws them, solely based on the verbal descriptions from their partner. Students do this over several rounds, switching roles after two rounds. As students attempt to produce more accurate drawings, they try to fine-tune their descriptions. They notice that more specific language or terminology is needed to better describe the features in the images (MP6).” Student Task Statement: “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 response: 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.”

An example in Grade 5 includes:

• Unit 4, Wrapping Up Multiplication and Division with Multi-Digit Numbers, Lesson 6, Cool-down, students attend to precision when they apply the standard algorithm for multiplication. Student Task Statement, “Use the standard algorithm to find the product 251\times34.” Activity Narrative: “The goal of this activity is to multiply numbers with no restrictions on the number of new units composed. Students first multiply a three-digit number by a one-digit number and a three-digit number by a two-digit number, with no ones. They then can put these two results together to find the product of a three-digit number and a two-digit number, with many compositions. They then solve another three-digit-number-by-two-digit-number example, with no scaffolding. Because these calculations have new units composed in almost every place value, students will need to locate and use the composed units carefully. It gives students a reason to attend to the features of their calculation and to use language precisely (MP6).”

The materials reviewed for Illustrative Mathematics® v.360 Grade 3 through Grade 5 meet expectations for supporting the intentional development of MP7: Look for and make use of structure, for students, in connection to the grade-level content standards, as expected by the mathematical practice standards. 

Students have opportunities to engage with MP7 across the year, and it is often explicitly identified for teachers within the Course Guide (Standards for Mathematical Practice) and within specific lessons (Preparation Narratives and Lesson Activities’ Narratives). According to the Course Guide, Standards for Mathematical Practice, “The Standards for Mathematical Practice (MP) describe the types of thinking and behaviors in which students engage as they do mathematics. Throughout the curriculum, the Teacher Guide identifies lessons and activities in which to observe the different MPs. The unit-level Mathematical Practice chart is meant to highlight a handful of lessons in each unit that showcase certain MPs. Some units, due to their size or the nature of their content, have fewer predicted chances for students to engage in a particular MP, indicated in the chart by a dash.” 

An example in Grade 3 includes:

• Unit 8, Putting It All Together, Lesson 1, Cool-down, students use structure to determine if the three representations all show the same fractional value. Preparation, Lesson Narrative: “In previous lessons, students learned how to represent fractions with area diagrams, fraction strips, and number lines. In this lesson, students revisit each representation in an estimation context. Students have an opportunity to think about how to partition each representation to decide what fraction is shown (MP7). Additionally, if time allows and it seems of benefit to students’ understanding, there is an option after each activity to find the exact value of the fraction in the Task Statement.” Student Task Statement: “Could the shaded part of the shape, the point on the number line, and the shaded part of the diagram all represent the same fraction? Explain your reasoning.” Students see a diamond, a number line, and a diagram that do not all represent the same fraction.

An example in Grade 4 includes:

• 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. Preparation, Lesson Narrative, “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 Task Statement: “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.”

An example in Grade 5 includes:

• Unit 5, Place Value Patterns and Decimal Operations, Lesson 2, Warm-up, students look for and make use of structure as they use a hundredths grid to estimate a shaded region. Activity Narrative, “When students reflect about how the hundredths grid could help refine their estimate, they observe the value and power of its structure (MP7).” Launch states, “Groups of 2. Display the image. ‘What is an estimate that’s too high? Too low? About right?’ 1 minute: quiet think time.” Student Task Statement: “How much of the square is shaded?” Activity Synthesis, “‘Why is estimating the shaded region more difficult without the grid lines of a hundredths grid?’ (The grid lines show me the tenths and hundredths. Without grid lines, I can only guess or estimate.)”

The materials reviewed for Illustrative Mathematics® v.360 Grade 3 through Grade 5 meet expectations for supporting the intentional development of MP8: Look for and express regularity in repeated reasoning, for students, in connection to the grade-level content standards, as expected by the mathematical practice standards. 

Students have opportunities to engage with MP8 across the year and it is often explicitly identified for teachers within the Course Guide (Standards for Mathematical Practice) and within specific lessons (Preparation Narratives and Lesson Activities’ Narratives). According to the Course Guide, Standards for Mathematical Practice, “The Standards for Mathematical Practice (MP) describe the types of thinking and behaviors in which students engage as they do mathematics. Throughout the curriculum, the Teacher Guide identifies lessons and activities in which to observe the different MPs. The unit-level Mathematical Practice chart is meant to highlight a handful of lessons in each unit that showcase certain MPs. Some units, due to their size or the nature of their content, have fewer predicted chances for students to engage in a particular MP, indicated in the chart by a dash.” 

An example in Grade 3 includes:

• Unit 5, Fractions as Numbers, Lesson 2, Cool-down, students use repeated reasoning as they partition shapes into equal parts. Student Task Statement, “1. Label each part with the correct fraction. 2. Partition and shade the rectangle to show \frac{1}{4}.” Activity 1 Narrative states, “When students make halves, fourths, and eighths, they observe regularity in repeated reasoning as each piece is subdivided into two equal pieces. They observe the same relationship between thirds and sixths (MP8). ” The Cool-down provides an opportunity to demonstrate this reasoning.

An example in Grade 4 includes:

• Unit 3, Extending Operations to Fractions, Lesson 7, Activity 2, students use repeated reasoning as they decompose fractions to find sums. Activity 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 Task Statement, “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 find? 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.”

An example in Grade 5 includes:

• Unit 4, Wrapping Up Multiplication and Division with Multi-Digit Numbers, Lesson 12, Activity 2, students use repeated reasoning as they use partial quotients to divide up to four-digit dividends by a two-digit divisor. Student Task Statement: “Estimate the value of each quotient. Then use a partial-quotients algorithm to find the value. 1. A reasonable estimate for 612\div34 is: 2. A reasonable estimate for 536\div23 is: 3. A reasonable estimate for 1,044\div29 is: ” Activity Narrative states, “Before finding the quotient, students estimate the value of the quotient, which helps students both decide which partial quotients to use and evaluate the reasonableness of their solution (MP8). ” Activity Synthesis states, “Ask 2–3 students to share their work showing different partial quotients for the same problem. ‘How can you make sure that the whole-number quotient you got at the end is reasonable?’ (It should be close to my estimate. I can multiply the quotient and divisor and that should give me the dividend.) If students share with other partners, ask: How did explaining your work to others help you today?’ or ‘What did someone say today that helped you in your understanding of division?’ (I learned that it’s okay to take more steps because I was comfortable with the multiples I used.)”