The materials reviewed for Imagine Learning Illustrative Mathematics Grade 5 meet expectations for developing conceptual understanding of key mathematical concepts, especially where called for in specific content standards or cluster headings.

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

• Unit 1, Finding Volume, Lesson 6, Warm-up, students develop conceptual understanding as they determine equivalence of numerical expressions using place value strategies. Launch, “Display one statement. Give me a signal when you know whether the statement is true and can explain how you know. 1 minute: quiet think time.” Student task statements, “Decide if each statement is true or false. Be prepared to explain your reasoning. $(4\times2)\times5=4\times(2\times5)$. $(2\times5)\times4=2\times20$. $5\times4\times2=10\times40$.” Activity synthesis, “Focus Question: How can you justify your answer without evaluating both sides? (I could see on the first equation that all of the factors are the same so it is true.) Consider asking: Who can restate ___’s reasoning in a different way? Does anyone want to add on to ___’s reasoning? Can we make any generalizations based on the statements?” (5.MD.5)

• Unit 4, Wrapping Up Multiplication and Division with Multi-digit Numbers, Lesson 18, Warm-up, students develop conceptual understanding as they estimate the volume of a milk carton using a photo. Launch, “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 statements, “What is the volume of the milk carton in cubic inches? Record an estimate that is: too low, about right, too high.” Students record their estimates in a table. Activity synthesis, “How can you use what you know about volume to estimate the volume of the milk container? (I can measure to see how many cubic inches it would take to fill the carton. I can measure the length, width, and height and multiply them.) What units do you usually use to measure liquids? (Liters, quarts, cups) We learned in an earlier unit that cubic centimeters or cubic inches are also units for measuring a volume.” (5.MD.5)

• Unit 8, Putting it All Together, Lesson 17, Warm-up, students develop conceptual understanding as they determine if equations involving fraction addition are true or false. Launch, “Display one equation. Give me a signal when you know whether the equation is true and can explain how you know. 1 minute: quiet think time.” Student task statements, “Decide if each statement is true or false. Be prepared to explain your reasoning., $\frac{3}{4}+\frac{3}{8}=\frac{3}{8}+\frac{3}{8}$, $\frac{7}{5}+\frac{2}{3}=\frac{21}{15}+\frac{8}{15}$, $\frac{8}{9}+\frac{5}{12}=\frac{32}{36}+\frac{15}{36}$.” Activity synthesis, “What did the writer of this activity have to pay attention to when they designed this activity? (Some equations are true and some are false. Some terms on both sides are equal. Pay attention to the unlike and like denominators.) Where do we see those things in how the equations change during the True or False? (The first two equations are false, but they use an appropriate common denominator.).” (5.NF.1)

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

• Unit 2, Fractions as Quotients and Fraction Multiplication, Lesson 1, Cool-down, students draw diagrams to represent division and fractions. “1. Draw a diagram to show how much sandwich each person will get., 3 sandwiches are equally shared by 4 people., 2. Explain or show how you know that each person gets the same amount of sandwich.” (5.NF.3)

• Unit 4, Wrapping Up Multiplication and Division with Multi-digit Numbers, Section A Practice Problems, Problem 4, students calculate volume of a prism using an image of a rectangular prism. “What is the volume of this rectangular prism? Explain or show your reasoning.” The image of the prism includes labels for the length, width and height. (5.MD.5)

• Unit 5, Place Value Patterns and Decimal Operations, Lesson 2, Cool-down, students represent decimal numbers using a grid. Students are provided a blank grid of 100 squares labeled to represent 1. Student task statements, “1. Shade the grid to represent 0.149. 2. What is another way you could represent 0.149?” (5.NBT.1)

The materials reviewed for Imagine Learning Illustrative Mathematics Grade 5 meet expectations for giving attention throughout the year to individual standards that set an expectation of procedural skill and fluency.

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

• Unit 4, Wrapping Up Multiplication and Division with Multi-Digit Numbers, Lesson 1, Activity 2, students multiply by 18. Student Task Statements, “Find the value of each expression. Explain or show your reasoning. $18\times9$, $18\times49$, $18\times149$.” (5.NBT.5)

• Unit 4, Wrapping Up Multiplication and Division with Multi-Digit Numbers, Lesson 6, Activity 2, All The Products, students multiply two- and three-digit numbers using the standard algorithm for multiplication. Launch: “You are going to find products with many new composed units. As you work, think carefully about where you place these values.” Student Facing:  “Find the value of each product using the standard algorithm. $647\times9$, $647\times50$, $647\times59$, $264\times38$” (5.NBT.5)

• Unit 8, Putting It All Together, Lesson 3, Warm-up, students multiply two- and three-digit numbers mentally. Student Task Statements, “Find the value of each expression mentally. $230\times10$, $230\times12$, $230\times15$, $232\times15$.” (5.NBT.5)

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

• Unit 4, Wrapping Up Multiplication and Division with Multi-Digit Numbers, Lesson 4, Cool-down, students use the standard algorithm to multiply larger numbers. Student Task Statements, “Use the standard algorithm to find the value of $3,154\times7$.” (5.NBT.5)

• Unit 6, More Decimal and Fraction Operations, Lesson 8, Cool-down, students add fractions with unlike denominators. Student Task Statements, “Find the value of each expression. Explain or show your reasoning. 1. $\frac{5}{6}-\frac{1}{3}$. 2. $\frac{3}{4}+\frac{1}{2}$.” (5.NF.1) 

• Unit 8, Putting It All Together, Lesson 1, Cool-down, students practice using the standard algorithm for products. Student Task Statements, “Find the value of each product. Explain or show your reasoning. 1. $35\times47$. 2. $37\times45$.” (5.NBT.5)

The materials reviewed for Imagine Learning Illustrative Mathematics Grade 5 meet expectations for being designed so that teachers and students spend sufficient time working with engaging applications of the mathematics.

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

Examples of routine applications include:

• Unit 3, Multiplying and Dividing Fractions, Lesson 17, Activity 2, students solve word problems involving multiplication and division of fractions (5.NF.6, 5.NF.7). Student Task Statements, “Solve each problem. Explain or show your reasoning. 1. If 11 grains of rice weigh $\frac{1}{3}$ gram, how much does each grain of rice weigh? 2. Mai’s road is $\frac{9}{10}$ mile long. She ran $\frac{3}{4}$ of the length of her road. How far did she run? 3. If each tennis ball weighs $2\frac{1}{16}$ ounces, how much do 9 tennis balls weigh?”

• Unit 5, Place Value Patterns and Decimal Operations, Lesson 26, Activity 2, students solve  real-world problems by adding and multiplying decimals (5.NBT.7). Student Task Statements, “Price list from the publisher: type of book, price. boxed sets & collections $24.95. comic books 2.60. science books $8.00. chapter books $9.99. history books $14.49. audiobooks, $20.00. activity books $4.50. reference books $12.00. Spanish language books $6.00. biographies $6.05. Plan a book fair: 1. Choose 3–5 types of books you want to order. 2. Decide on the mark-up price for each type of book you chose. 3. Estimate the amount of money your school will raise as a profit with your book sale. 4. Show or explain your reasoning for the estimate. Include the assumptions you made.”

• Unit 6, More Decimal and Fraction Operations, Lesson 9, Activity 3, students solve a real-world problem by finding the difference of fractions (5.NF.1 & 5.NF.2). Student Task Statements, “Jada and Andre compare the growth of their plants. Jada’s plant grew $1\frac{3}{4}$  inches since last week. Andre’s plant grew $\frac{7}{8}$ inches. How much more did Jada’s plant grow? Explain or show your reasoning.”

Examples of non-routine applications include:

• Unit 1, Finding Volume, Lesson 5, Activity 3: What is the Question?, students apply the volume formula to calculate the volume of rectangular prisms (5.MD.5b). Student Task Statements, “This is the base of a rectangular prism that has a height of 5 cubes. These are answers to questions about the prism. Read each answer and determine what question it is answering about the prism.  1. 3 is the answer. What is the question? 2. 5 is the answer. What is the question? 3. $3\times4=12$. The answer is 12. What is the question? 4. $12\times5=60$. The answer is 60 cubes. What is the question? 5. 3 by 4 by 5 is the answer. What is the question?” The Activity Synthesis states, “Ask previously selected students to share their solutions. Connect the informal language to the math terms length, width, height, and area of a base. 'How does the expression $3\times4\times5$ represent the prism described in the second question?’ (The area of the base is 3 4 = 12, and the height is 5, so $3\times4\times5$ represents the product of length, width, and height.)”

• Unit 3, Multiplying and Dividing Fractions, Lesson 8, Activity 2: More Flags, students solve real world problems involving multiplication of fractions and mixed numbers. (5.NF.6) Student Task Statements, “Han has a replica of the flag of Colombia. 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 and the blue and red stripes are each $\frac{1}{4}$ of the width. 2. $\frac{1}{2}\times3\frac{1}{2}=\frac{7}{4}\times\frac{21}{4}=\frac{147}{16}$. The answer is $\frac{147}{16}$ square inches. What is the question?” 

• Unit 7, Shapes on the Coordinate Plane, Lesson 13, Cool-down: Area and Perimeter of a Rectangle, students use a given point to find the area and perimeter of a rectangle (5.G.2). Student Task Statements, “The point represents the length and width of a rectangle. (4, 5). 1. What is the area and perimeter of the rectangle? Explain or show your reasoning. 2. What is a point that represents a different rectangle with the same area? Explain or show your reasoning.” 

The materials reviewed for Imagine Learning Illustrative Mathematics 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. 

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

All three aspects of rigor are present independently throughout Grade 5. Examples where instructional materials attend to conceptual understanding, procedural skill and fluency, or application include:

• Unit 3, Multiplying and Dividing Fractions, Lesson 19, Cool-down, students use conceptual understanding of fractions and division to make the largest and smallest expressions using given numbers to solve a real world problem. Student Task Statements, “Use the numbers 6, 7, 8, and 9 to make the greatest product. Show or explain how you know it is the greatest product. $\frac{?}{?}\times\frac{?}{?}$.” (5.NF.6 and 5.NF.7)

• Unit 4, Wrapping Up Multiplication and Division with Multi-Digit Numbers, Lesson 5, Activity 1, students develop procedural skill and fluency as they use the standard algorithm to multiply up to three-digit numbers and two-digit numbers. Launch, “ Display the algorithms. We are going to learn about a new algorithm today.” Student Task Statements, “Two algorithms for finding the value of  $413\times13$ are shown. 1. How are the two algorithms the same? How are they different? 2. Explain or show where you see each step from the first algorithm in the second algorithm. 3. How do the final steps in the two algorithms compare?” (5.NBT.5) 

• Unit 7, Shapes on the Coordinate Plane, Lesson 12, Activity 2, students apply their understanding using a coordinate plane to find answers to questions. Launch, “‘What do you know about coins? (They're round. I can buy things with them. There are different kinds and they have different values.)’ Record responses for all to see. Display a penny, dime, nickel, and quarter. If no student mentions it, say and record the value of each coin.” Student Task Statements, “The graph shows the number and value of coins some students had with them. 1. Tyler has 1 dime, 3 nickels, and 2 pennies. Which point represents Tyler's coins? Label the point. 2. Lin has 3 quarters, 1 dime, and 1 penny. Which point represents Lin's coins? Label the point. 3. Diego has 1 quarter and 1 dime. Write the coordinates of the point that represents Diego's coins. Explain or show your reasoning. 4. Clare has 5 coins and does not have a quarter. Write the coordinates of the point that represents Clare's coins. 5. Which coins might Clare have? Explain or show your reasoning.” (5.G.2) 

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

• Unit 2, Fractions as Quotients and Fraction Multiplication, Lesson 2, Activity 2, students develop conceptual understanding alongside application as they relate equal shares to division expressions and visual representations of fractions. Launch, “Display image from student workbook. This representation shows how 2 sandwiches can be shared by 5 people equally. How much sandwich does each person get? Be prepared to share your thinking.” Activity, “This set of cards includes diagrams, expressions, and situations. Match each diagram to a situation and an expression. Some situations and expressions will match more than one diagram. Work with your partner to justify your choices. Then, answer the questions in your workbook.” Student Task Statements, “Your teacher will give you a set of cards. Match each representation with a situation and expression. Some situations and expressions will have more than one matching representation. Choose one set of matched cards. 1. Show or explain how the diagram(s) and expression represent the number of sandwiches being shared. 2. Show or explain how the diagram(s) and expression represent the number of people sharing the sandwiches. 3. How much sandwich does each person get in the situation?” (5.NF.3)

• Unit 6, More Decimal and Fraction Operations, Lesson 12, Activity 2, students develop all three aspects of rigor simultaneously, conceptual understanding, procedural skill and fluency, and application, as they solve multi-step problems involving the addition and subtraction of fractions. Student Task Statements, “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? If the recipe requires  $1\frac{1}{3}$ cups of walnuts, how many more cups of walnuts does Jada need to 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? If the trail they are hiking is a total of $2\frac{1}{2}$ miles, how much farther do they have to hike? Explain or show your reasoning. 2. Discuss the problems and solutions with your partner. What is the same about your strategies and solutions? What is different? 3. Revise your work if necessary.” (5.NF.2)

• Unit 8, Putting It All Together, Lesson 4, Activity 1, students develop conceptual understanding alongside procedural skill and fluency as they estimate and find whole-number quotients. Student Task Statements, “1. Circle the most reasonable estimate. Show your reasoning. a. $364\div13$ answer choices include 20, 30, 40. b. $540\div12$ answer choices include 40, 50, 60. c. $1,008\div14$ answer choices include 70, 80, 90.” (5.NBT.6)

The materials reviewed for Imagine Learning Illustrative Mathematics Grade 5 meet expectations for supporting the intentional development of MP1: Make sense of problems and persevere in solving them; and MP2: Reason abstractly and quantitatively, for students, in connection to the grade-level content standards, as expected by the mathematical practice standards. Students have opportunities to engage with the Math Practices across the year, and they are often explicitly identified for teachers in several places including the Instructional Routines (Warm-up Routines and Other Instructional Routines), Activity Narratives, and About this lesson.

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

• Unit 2, Fractions as Quotients and Fraction Multiplication, Lesson 3, Activity 1, students make sense of and persevere in solving a problem about dancers. Activity Narrative, “The purpose of this activity is for students to write and interpret division expressions and equations that represent equal sharing situations. They explain the relationships between the dividend and the numerator and divisor and the denominator. Students may draw diagrams to help them make sense of these relationships (MP1).” Student Task Statements, “1. Three dancers share 2 liters of water. How much water does each dancer get? Write a division equation to represent the situation. 2. Mai said that each dancer gets $\frac{3}{2}$ of a liter of water because 3 divided into 2 equal groups is $\frac{3}{2}$. Do you agree with Mai? Show or explain your reasoning.”

• Unit 3, Multiplying and Dividing Fractions, Lesson 17, Activity 1, students make sense of and persevere to solve a problem using the info gap instructional routine. The Activity Narrative states, “This Info Gap activity gives students an opportunity to determine and request the information needed to solve multi-step problems involving multiplication and division of unit fractions. In both cases, the student with the problem card needs to find out the side lengths of the area being covered and the size of the tiles and from there they can figure out how many tiles are needed. The numbers in the problems are chosen so that students can draw diagrams or perform arithmetic directly with the numbers. In this Info Gap activity, the first problem encourages students to think about multiplying the given fractions. The second problem involves a given area and a missing side length, which may encourage students to represent and solve the problem with a missing factor equation. The Info Gap structure requires students to make sense of problems by determining what information is necessary, and then to ask for information they need to solve it. This may take several rounds of discussion if their first requests do not yield the information they need (MP1). It also allows them to refine the language they use and ask increasingly more precise questions until they get the information they need (MP6).” The Student Task Statement, “Your teacher will give you either a problem card or a data card. Do not show or read your card to your partner. If your teacher gives you the problem card: 1. Silently read your card and think about what information you need to answer the question. 2. Ask your partner for the specific information that you need. 3. Explain to your partner how you are using the information to solve the problem. 4. Solve the problem and explain your reasoning to your partner. If your teacher gives you the data card: 1. Silently read the information on your card. 2. Ask your partner, ‘What specific information do you need?’ and wait for your partner to ask for information. Only give information that is on your card. (Do not figure out anything for your partner!) 3. Before telling your partner the information, ask, ‘Why do you need that information?’ 4. After your partner solves the problem, ask them to explain their reasoning and listen to their explanation. Pause here so your teacher can review your work. Ask your teacher for a new set of cards and repeat the activity, trading roles with your partner.”

• Unit 5, Place Value Patterns and Decimal Operations, Lesson 9, Activity 1, students make sense of and persevere to solve problems using number lines. The Activity Narrative states, “The purpose of this activity is for students to apply what they have learned about comparing decimals to find numbers that lie between two other decimal numbers. Students may draw number line diagrams, if it helps them, or they may use their understanding of place value. In each case, there are many different decimal numbers between the two and this will be brought out in the activity synthesis. The last question in this activity is exploratory. Students may say that there is no number between 1.731 and 1.732 or they may say that it looks like there is and they cannot name it yet. The important observation is that the number line suggests that there are numbers in between but we cannot name any of those numbers yet. This question gives students an opportunity to make sense of a problem and some students may propose an answer, using fractions for example (MP1).” Student Task Statement, “1. Fill in the blank to make each statement true. Be prepared to explain your reasoning. Use the number lines if they are helpful. a. , b. , c. , d. , e. . 2. Kiran says that there is no number between 1.731 and 1.732. Do you agree with Kiran? Use the number line if it is helpful?”

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

• Unit 2, Fractions as Quotients and Fraction Multiplication, Lesson 4, Activity 1, students reason abstractly and quantitatively as they use different representations to solve problems. Student Task Statements, “1. Complete the missing parts of the table. Be prepared to explain your thinking. 2. Discuss both your solutions with your group. What is the same? What is different?” Activity Narrative, “Students go back and forth between equations, situations, and diagrams, interpreting the diagrams and equations and creating situations that these diagrams and equations represent (MP2).”

• Unit 3, Multiplying and Dividing Fractions, Lesson 18, Activity 1, students reason abstractly and quantitatively to solve problems with multiplying and dividing fractions. The Activity Narrative states, “The purpose of this activity is for students to articulate the relationship between multiplication and division explaining how to solve two different problems using multiplication or division. Students have observed that dividing a whole number by a unit fraction gives the same result as multiplying the whole number by the denominator. They have also observed that dividing a unit fraction by a whole number gives the same result as multiplying the fraction by the unit fraction that has the whole number as a denominator. They also know from prior units and courses that the operations of multiplication and division are closely related. This activity brings these two ideas together, making explicit how one situation and one diagram modeling the situation can be interpreted using either multiplication or division (MP2).” The Student Task Statements, “1. Diego’s dad is making hamburgers for the picnic. There are 2 pounds of beef in the package. Each burger uses $\frac{1}{4}$ pound. How many burgers can be made with the beef in the package? 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 are going to equally share $\frac{1}{4}$ pound of potato salad. How many pounds of potato salad will 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.”

• Unit 6, More Decimal and Fraction Operations, Lesson 17, Activity 2, students reason abstractly and quantitatively when they “compare a product to an unknown factor based on the size of the other factor.” Student Task Statements, “Priya ran to her grandmother’s house. Jada ran twice as far as Priya. Han ran $\frac{6}{7}$ as far as Priya. Clare ran $\frac{14}{8}$ as far as Priya. Mai ran $\frac{3}{5}$ times as far as Priya. 1. Which students ran farther than Priya? 2. Which students did not run as far as Priya? 3. List the runners in order from shortest distance run to longest. Explain or show your reasoning. 4. The point P represents how far Priya ran. Write the initial of each student in the blank that shows how far they ran. One of the students will be missing. 5. Label the distance for the missing student on the number line above.” Activity Narrative , “For this part of the activity the expectation is that they will use what they already know about the order of the distances to determine which point corresponds to each student. They might, however, also reason about the quantities. For example, twice Priya’s distance can be found by marking off Priya’s position on the number line a second time (MP2).”

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

Examples of constructing viable arguments include:

• Unit 4, Wrapping Up Multiplication and Division with Multi-digit Numbers, Lesson 4, Activity 1, students construct viable arguments and critique the reasoning of others related to the standard algorithm for multiplication. Activity Narrative, “When students discuss their interpretation of Elena's calculation and improve their explanations they construct viable arguments and critique the reasoning of others (MP3).” Student Task Statements, “Here is how Han calculated $318\times3$ using partial products. (Partial products of 24, 30 and 900 are arranged vertically beneath the multiplication problem for a total of 954.) Here is how Elena calculated $318\times3$ using the standard algorithm. (Elena regrouped the 20 and placed a 2, shown in blue, above the 1 in 318 for a total of 954.) 1. What does the 2 in Elena’s calculation represent? Explain or show your reasoning. 2. What does the 5 in Elena’s solution represent? Explain or show your reasoning.”

• Unit 6, Area and Multiplication, Lesson 14, Cool-down, students construct viable arguments as they use line plots. Student Task Statements, “Here are the weights of a different collection of chicken eggs. What is the combined weight of all the eggs that weigh more than $2\frac{1}{2}$ ounces? Explain or show your reasoning.”

• Unit 7, Shapes on the Coordinate Plane, Lesson 8, Activity 1, students sort triangles and compare their work with others, requiring them to construct viable arguments and critique the reasoning of others. Activity Narrative, “Students think about whether or not such a triangle could exist and present informal arguments to explain their reasoning (MP3). The Activity Synthesis formally introduces the category of right triangles.” Student Task Statements, “1. Find a triangle card that fits in each space on the grid. 2. If you don’t think it is possible to find a triangle that fits certain criteria, explain why not.” The grid has 3 rows: has a 90 degree angle, has an angle that is greater than 90 degrees, all three angles are less than 90 degrees; and 3 columns: all three side lengths are different, exactly two of the sides are the same, all three side lengths are the same.

Examples of critiquing the reasoning of others include:

• Unit 3, Multiplying and Dividing Fractions, Lesson 12, Activity 2, students construct a viable argument and critique the reasoning of others when they use diagrams to show division of a unit fraction. Activity, “5 minutes: independent work time.” Student Task Statements, “2. This is Priya’s work for finding the value of $\frac{1}{3}\div2$. $\frac{1}{3}\div2=\frac{1}{2}$ because I divided $\frac{1}{3}$ into 2 equal parts and $\frac{1}{2}$ of $\frac{1}{3}$ is shaded in. a. What questions do you have for Priya? b. Priya’s equation is incorrect. How can Priya revise her explanation?” The Activity Narrative, “When students decide whether or not they agree with Priya’s work and explain their reasoning, they critique the reasoning of others (MP3).”

• Unit 4, Wrapping Up Multiplication and Division with Multi-digit Numbers, Lesson 14, Activity 1, students analyze errors in partial quotient division problems requiring them to critique the reasoning of others and construct viable arguments. Activity Narrative, “The purpose of this activity is for students to identify and correct common errors in using an algorithm that uses partial quotients. One of the errors involves subtraction and two of them involve multiplication. Students may choose to correct the errors and continue the work that is there or they may choose to find the quotient in a different way that makes sense to them. When students determine where the errors are and explain their reasoning, they critique and construct viable arguments (MP3).” Student Task Statements, “For each problem, describe where you see an error in the calculation. Then find the correct whole number quotient.” Three partial quotient division problems that contain errors are shown.

• Unit 8, Putting It All Together, Lesson 2, Activity 1, students critique the reasoning of others as they use the standard algorithm to multiply multi-digit numbers. Student Task Statements, “2. Below is Kiran’s work finding the value of the product $650\times27$. Is his answer reasonable? Explain your reasoning.” Activity Narrative, “When students determine Kiran's error and make sense of his work, they interpret and critique the work of others (MP3).”

The materials reviewed for Imagine Learning Illustrative Mathematics Grade 5 meet expectations for supporting the intentional development of MP4: Model with mathematics; and MP5: Use appropriate tools strategically, for students, in connection to the grade-level content standards, as expected by the mathematical practice standards. Students have opportunities to engage with the Math Practices across the year and they are often explicitly identified for teachers in several places including the Instructional Routines (Warm-up Routines and Other Instructional Routines), Activity Narratives, and About this Lesson.

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

• Unit 1, Finding Volume, Lesson 12, Activity 2, students model with mathematics when they investigate what happens to garbage and make estimates. Lesson Narrative, “In the second activity, students estimate the number of shipping containers on a fully loaded cargo ship from a picture. When students estimate quantities and make assumptions, they model with mathematics. Students circle back to the question of garbage by computing together how much garbage could fit on a fully loaded cargo ship. When students translate a mathematical answer back into the real-world situation, they model with mathematics (MP4).” Activity 2, Student Task Statements, “1. How many containers are on the cargo ship? [an illustration of a cargo ship with about 1,000 containers] Record an estimate that is: too low about right too high 2. How many containers are on the cargo ship? [an illustration of a cargo ship with substantially more containers, about 5,000] Record an estimate that is: too low, about right, too high 3. What assumptions were you making when you came up with your estimates?”

• Unit 6, More Decimal and Fraction Operations, Lesson 21, Activity 2, students model with mathematics as they analyze line plots using operations of fractions. About this Lesson, “When students define categories, choose and ask questions, collect and analyze data, and tell a story about the situation based on data, they model with mathematics (MP4).” Student Task Statements, “Your teacher will assign a poster with a data set for one of the categories from the previous activity. 1. Create a line plot for the activity. Make sure to label the line plot. 2. Analyze the data and tell the story of your data. Choose at least 3 things. Use the following questions if they are helpful. What is the total number of hours the class spends on this activity? What is the difference between greatest and least time? Is there something surprising? How many data points are there? What does that tell you? What fraction of your classmates spend less than an hour on this activity? More than an hour? Be prepared to share the story with the class.”

• Unit 8, Putting It All Together, Lesson 9, Activity 2, students estimate capacity and model with mathematics. “When students make a list of the different things they do in the house that use water and then estimate how much water is used they model with mathematics (MP4).” Activity Narrative, “The purpose of this activity is for students to find out if the amount of water that falls on the house is sufficient for many of the daily household chores that use water. This will require a lot of estimation and will vary from house to house. How much of the calculations to leave up to the students is an individual teacher choice and this lesson could easily be extended for another day if the students make well reasoned estimates (some values are given in parentheses) for how much water is used for different activities such as: taking baths or showers (150 liters or 80 liters) washing clothes (100 liters) washing dishes (100 liters) washing hands (1 liter) flushing the toilet (10 liters). More estimation comes into play for how often each of these activities happens and this will vary greatly depending on the student. Consider inviting students to check their estimates by looking at one of their monthly water bills. The bill will usually give the number of gallons of water used and there are almost 4 liters in a gallon.” Student Task Statements, “1. What are some of the ways you use water at home? 2. Estimate how much water you use at your home in a month. 3. How much rain would need to fall on your home each month to supply all of your water needs? 4. What challenges might come up if you tried to use the rainwater that falls on the roof of your home? Do you think it makes sense to try to capture the rain that falls on your home?”

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

• Unit 4, Wrapping Up Multiplication and Division with Multi-digit Numbers, Lesson 11, Cool-down, students consider different strategies for dividing large numbers. Lesson Narrative, “In the previous lesson, students found quotients in a way that makes sense to them. In this lesson, students consider notation to record a partial quotients strategy, which they have used with one-digit divisors in a prior course. Students use the notation to record how dividends can be decomposed in different ways to make different partial quotients. Students consider more efficient ways to make partial quotients based on place value understanding and calculations they are able to do mentally.” Student Task Statements, “Find the value of $465\div15$. Explain or show your reasoning.”

• Unit 5, Place Value Patterns and Decimal Operations, Lesson 14, Activity 1, students use appropriate tools strategically to subtract decimals. About this Lesson, “Strategies students may use include using hundredths grids (MP5), using place value and writing equations.” Student task statements, “Find the value of $2.26-1.32$. Explain or show your reasoning.” Activity Narrative, “Students should be encouraged to use whatever strategies make sense to them, including using place value understanding and the relationship between addition and subtraction. ”

• Unit 6, More Decimal and Fraction Operations, Lesson 8, Activity 2, students choose the appropriate tool to show addition and subtraction of fractions. “Students who choose to use the number line or tape diagrams use appropriate tools strategically (MP5).” Activity Narrative, “The purpose of this activity is for students to add and subtract fractions in a way that makes sense to them. Students may use strategies such as drawing tape diagrams or number lines, or they may use computations to find a common denominator. Monitor for and select students with the following strategies to share in the synthesis: use the meaning of fractions to explain why $\frac{2}{3}+\frac{2}{3}=\frac{4}{3}$, use a diagram like a number line to find the value of $\frac{2}{3}-\frac{1}{6}$ and $\frac{2}{3}+\frac{1}{2}$, use equivalent fractions and arithmetic to find the value of $\frac{2}{3}-\frac{1}{6}$ and $\frac{2}{3}+\frac{1}{2}$, ” Student Task Statements, “Find the value of each expression. Explain or show your thinking. 1. $\frac{2}{3}+\frac{2}{3}$, 2. $\frac{2}{3}=\frac{1}{6}$, 3. $\frac{2}{3}+\frac{1}{2}$.”

The materials reviewed for Imagine Learning Illustrative Mathematics Grade 5 meet expectations for supporting the intentional development of MP6: Attend to precision; and attend to the specialized language of mathematics, for students, in connection to the grade-level content standards, as expected by the mathematical practice standards. 

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

• Unit 1, Finding Volume, Lesson 7, Cool-down, students attend to precision when finding the volume of an object. Lesson Narrative, “In previous lessons, students used unit cubes with a side length of 1 unit to determine the volume of right rectangular prisms. In this lesson, the units are now a specific unit of measure. In grade 5, students use words, not exponents, when recording the cubic unit of measure, such as “cubic centimeters (cm),” "cubic feet (ft)" or “cubic inches (in).” The exponents in Grade 5 are limited to powers of 10, which will be addressed in a later unit. In this lesson, students distinguish between different standard unit measures of volume. They examine the distinction between cubic cm, cubic in, and cubic ft. Throughout the lesson, students share their rationale for choosing a unit to measure specific real-world objects and learn the importance of identifying the unit of measure when finding the volume of an object (MP6).” Cool-down Task Statement, “Priya’s family rented a moving truck to move their belongings to their new house. The space inside the back of the moving truck is 15 feet long, 5 feet wide, and 8 feet tall. What is the volume of the back of the moving truck? Explain or show your reasoning. (Remember to include the cubic unit of measure.)”

• Unit 5, Place Value Patterns and Decimal Operations, Lesson 6, Activity 1, students use precision when they plot the same number on different number lines. Student Task Statements, “On each number line: Label all of the tick marks. Locate and label the number 0.001.” Activity Narrative, “The important take-away is that when a decimal does not lie on a tick mark estimation is needed to locate the number and it can be difficult or impossible to locate it precisely (MP6).”

• Unit 6, More Decimal and Fraction Operations, Lesson 15, Activity 1, students use precise language as part of the Info Gap routine to elicit the information they need to solve a problem about the weight of fruit. Activity Narrative, “‘This Info Gap activity gives students an opportunity to solve problems about data represented on line plots. In both sets of cards, there is a partially complete line plot and some missing data. For the first set of cards, the problem card has the missing data and the data card has a partially complete line plot.’ Monitor for students who: request all the information on the data card and create a complete line plot which they may use to answer the question, only request the information they need to answer the question about the heaviest apricot and the most common weight. For the second set of cards, the problem card has the partially complete line plot and the data card has information to determine the missing data. Here students will likely need to communicate with each other as the information about the most common weight is vital to solve the problem, but the student with the problem card may not think to ask about this. The Info Gap allows students to refine the language they use and ask increasingly more precise questions until they get the information they need (MP6).” Student Task Statements, “Your teacher will give you either a problem card or a data card. Do not show or read your card to your partner. Pause here so your teacher can review your work. Ask your teacher for a new set of cards and repeat the activity, trading roles with your partner.”

• Unit 7, Shapes on the Coordinate Plane, Lesson 1, Activity 1, students use the specialized language of mathematics as they draw shapes on a coordinate grid. Activity Narrative, “Students work with a partner to replicate given rectangles. One partner uses precise language and describes the rectangles and the other draws them based on their partner's verbal description.” Student Task Statements, “1. Play three rounds of Draw My Shape using the three sets of cards from your teacher. For each round: Partner A choose a card—without showing your partner—and describe the shape on the card. Partner B draw the shape as described. Partner A reveal the card and partner B reveal the drawing. Compare the shapes and discuss: What’s the same? What’s different? 2. Look at partner B's drawings for each round. When does partner B's drawing look most like the shape on the card? Explain why you think that is so.” Activity Synthesis, “‘Here are some of the words and phrases you used as you worked with your partners. We may add additional words or phrases that are important to include on our display as we continue to share and discuss the activity. You could use the language on the display to explain your thinking.’ As students share responses, update the display by adding (or replacing) language, diagrams, or annotations. Ask previously selected students to explain their thinking. ‘How did the gridlines help you? (They helped us draw the shapes more accurately.) How did the numbers help you? (We could use them to describe where the shape was located.)’ Display the image from the warm-up: ‘This grid, with numbers labeling the gridlines, is called a coordinate grid. We are going to learn more about the coordinate grid in the next few lessons. How would you describe the coordinate grid? (It has vertical lines with numbers on them and horizontal lines with numbers on them. It has squares on it. There are two of each number except 0. The horizontal and vertical lines intersect.)’”

The materials reviewed for Imagine Learning Illustrative Mathematics Grade 5 meet expectations for supporting the intentional development of MP7: Look for and make use of structure; and MP8: Look for and express regularity in repeated reasoning, for students, in connection to grade-level content standards, as expected by the mathematical practice standards. Students have opportunities to engage with the Math Practices throughout the year.

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

• Unit 2, Fractions as Quotients and Fraction Multiplication, Lesson 3, Activity 2, students look for and make use of structure as they write equations to represent division situations. Student Task Statements, “1. Complete the table. Draw a diagram if it is helpful. 2.What patterns do you notice in the table?” Lesson Narrative, “The activity is designed to highlight the relationship between the number of objects being shared and the numerator, on the one hand, and the number of people sharing and the denominator on the other (MP7).” Activity Synthesis, “Display the table from the activity. ‘What are some of the numbers you used for the last two rows?’ Record the answers as additional rows to the table. ‘What are some patterns that you notice in the table?’ Ask previously selected students to share their solution.”

• Unit 3, Multiplying and Dividing Fractions, Lesson 2, Activity 2, students look for and make use of structure as they write and evaluate expressions using products. Student Task Statements, “Priya shaded part of a square. 1. Explain or show how the expression $\frac{1}{5}\times\frac{1}{2}$ represents the area of the shaded piece. 2. Explain or show how the expression $\frac{1}{2}\times\frac{1}{5}$ represents the area of the shaded piece. 3. Write a multiplication expression to represent the area of the shaded piece. Be prepared to explain your reasoning. 4. How much of the whole square is shaded?” Lesson Narrative, “‘The expressions here are products of unit fractions.’ Students start with a diagram and first explain how an expression represents the diagram. Then, they write their own expression representing a different diagram (MP7).”

• Unit 6, More Decimal and Fraction Operations, Lesson 1, Cool-down, students use the structure of the base ten number system to create true equations. Lesson Narrative, “In previous grades, students recognized that in a multi-digit whole number, a digit in one place represents ten times what it represents in the place to its right. In the previous unit, students saw that this same pattern continues to the right of the decimal. In this lesson, students apply what they know about multiplication, division, and place value to express that each digit in a decimal represents ten times as much as it represents in the place to its right and one tenth as much as it represents in the place to its left (MP7).” The Cool-down Task Statement, “Fill in the blank to make each equation true. 1. , 2. , 3. .”

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

• Unit 1, Finding Volume, Lesson 8, Activity 2, students reason that the structure of a solid figure is made up of two non-overlapping right rectangular prisms. Student Task Statement, “Find the volume of each figure. Explain or show your reasoning.” Lesson Narrative, “As students experiment with different ways to group the cubes to efficiently count them, applying what they already know about the volume of rectangular prisms, they are looking for and making use of the structure of geometric objects (MP7). As students begin to generalize the idea that volume is additive, they are looking for and expressing regularity in repeated reasoning (MP8).” Activity Synthesis, “Display Figures c. and d. Ask selected students to share their way of splitting each figure. ‘Who broke the figure up the same way? Who broke it up differently? Can you think of other ways you could break up these figure (I can cut them into several layers—3 horizontal layers for c, and 5 vertical layers for d. Each layer is a rectangular prism.)’”

• Unit 7, Shapes on the Coordinate Plane, Lesson 9, Cool-down, students use repeated reasoning as they generate patterns using two rules. Student Task Statements, “1. List the first 10 numbers for these 2 patterns. Jada’s rule: Start with 0 and keep adding 5. Priya’s rule: Start with 0 and keep adding 10. 2. What number will be in Priya’s pattern when Jada’s pattern has 100? 3. What relationship do you notice between corresponding numbers in the two patterns?” Activity 2, Lesson Narrative, “When students find and explain patterns related rules and relationships, they look for and express regularity in repeated reasoning (MP8).”

• Unit 8, Putting It All Together, Lesson 10, Activity 2, students use repeated reasoning to practice adding fractions with unlike denominators. Activity Narrative, “The purpose of this activity is for students to practice adding fractions with unlike denominators. Monitor for students who notice that the overall strategy in this game is the same as in the previous game except that the numbers that they placed in the numerator in the first game go in the denominator in this game and similarly the numbers that went in the denominator in the first game go in the numerator in this game (MP8).” Student Task Statements, “Use the directions to play Smallest Sum with a partner. 1. Spin the spinner. 2. Each player writes the number that was spun in an empty box for Round 1. Be sure your partner cannot see your paper. 3. Once a number is written down, it cannot be changed. 4. Continue spinning and writing numbers in the empty boxes until all 4 boxes have been filled. 5. Find the sum. 6. The person with the lesser sum wins the round. 7. After all 4 rounds, the player who won the most rounds wins the game. 8. If there is a tie, players add the sums from all 4 rounds and the lesser total sum wins the game.” The spinner contains the numbers 5, 2, 3, 1, 6, 2, 4, 1, and there is space to record 4 fraction addition problems. Activity Synthesis, “What strategies were helpful as you played Smallest Sum? (I tried to make unit fractions with large denominators. I used the opposite strategy to the previous game, trying to put the smallest numbers in the numerator and the largest numbers in the denominator.)”

The materials reviewed for Imagine Learning Illustrative Mathematics Grade 5 meet expectations for providing teacher guidance with useful annotations and suggestions for how to enact the student materials and ancillary materials, with specific attention to engaging students in order to guide their mathematical development.

Materials provide comprehensive guidance that will assist teachers in presenting the student and ancillary materials. Examples include:

• IM Curriculum, Why is the curriculum designed this way?, Design Principles, Coherent Progression, “To support students in making connections to prior understandings and upcoming grade-level work, it is important for teachers to understand the progressions in the materials. Grade level, unit, lesson, and activity narratives describe decisions about the organization of mathematical ideas, connections to prior and upcoming grade-level work, and the purpose of each lesson and activity. When appropriate, the narratives explain whether a decision about the scope and sequence is required by the standards or a choice made by the authors. The basic architecture of the materials supports all learners through a coherent progression of the mathematics based both on the standards and on research-based learning trajectories. Each activity and lesson is part of a mathematical story that spans units and grade levels. This coherence allows students to view mathematics as a connected set of ideas that makes sense. Each unit, lesson, and activity has the same overarching design structure: the learning begins with an invitation to the mathematics, is followed by a deep study of concepts and procedures, and concludes with an opportunity to consolidate understanding of mathematical ideas. The invitation to the mathematics is particularly important because it offers students access to the mathematics. It builds on prior knowledge and encourages students to use their own language to make sense of ideas before formal language is introduced, both of which are consistent with the principles of Universal Design for Learning.”

• IM Curriculum, Scope and sequence information, provides an overview of content and expectations for the units. “The big ideas in grade 5 include: developing fluency with addition and subtraction of fractions, developing understanding of multiplication and division of fractions in limited cases (unit fractions divided by whole numbers and whole numbers divided by unit fractions), extending division to two-digit divisors, developing understanding of operations with decimals to hundredths, developing fluency with whole number and decimal operations, and developing understanding of volume.”

• Unit 4, Wrapping Up Multiplication and Division with Multi-Digit Numbers, Section A, Multi-Digit Multiplication Using the Standard Algorithm, Section Overview, Section Narrative, provides an overview of the content and expectations for the section. For example, “This section introduces the standard algorithm for multiplication, extending students’ earlier work on multiplication. In grade 4, students used diagrams and partial-products algorithms to find the product of a one-digit number and a number up to four digits, and the product of 2 two-digit numbers. They attended to the role of place value along the way. Students revisit these strategies and representations here, but work with factors with more digits than encountered in grade 4. They make connections between the partial products in diagrams and previous algorithms to the numbers in the standard algorithm. They also learn the notation for recording new place-value units that result from finding partial products. When using the standard algorithm to multiply a two-digit number and a three-digit number, students account for the place value of the digits being multiplied, as they had done before. For example, the 3 in 23 represents 3 ones, so $3\times123$ is 369. The 2 in 23, however, represents 2 tens, so the partial product is $2\times10\times123$ or 2,460, instead of $2\times123$ or 246. The partial products 369 and 2,460 can be seen in a diagram as well. Once students have practiced recording products this way, they learn to multiply factors that require composing new units, such as $264\times38$.”

Materials include sufficient and useful annotations and suggestions that are presented within the context of the specific learning objectives. Several components focus specifically on the content of the lesson. Preparation and Lesson Narratives, Warm-up, Activities, and Cool-down narratives all provide useful annotations. IM Curriculum, Why is the curriculum designed this way?, Design Principles, Coherent Progressions, “Each unit starts with an invitation to the mathematics. The first few lessons provide an accessible entry point for all students and offer teachers the opportunity to observe students’ prior understandings. Each lesson starts with a warm-up to activate prior knowledge and set up the work of the day. This is followed by instructional activities in which students are introduced to new concepts, procedures, contexts, or representations, or make connections between them. The lesson ends with a synthesis to consolidate understanding and make the learning goals of the lesson explicit, followed by a cool-down to apply what was learned. 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. In each of the activities, care has been taken to choose contexts and numbers that support the coherent sequence of learning goals in the lesson.” Examples include:

• Unit 1, Finding Volume, Lesson 4, Activity 1, teachers are provided context to help students reason about the volume of prisms. Narrative, “This activity continues to develop the idea of decomposing rectangular prisms into layers. Students explicitly multiply the number of cubes in a base layer by the number of layers. Students can use any layer in the prism as the base layer as long as the height is the number of those base layers.” Launch, “Groups of 2. Display first image from the student workbook. ‘What do you know about the volume of this prism? What would you need to find out to find the exact volume of this prism? You are going to work with prisms that are only partially filled in this activity.’ Give students access to connecting cubes.” Activity, “5 minutes: independent work time. 5 minutes: partner work time As students work, monitor for: students who notice that prisms A and D and prisms B and C are “the same” but they are sitting on different faces so the layers might be counted in different ways. Students who reason about the partially filled prisms by referring to the cubes in one layer they would see if all of the cubes were shown. Students who recognize that there are several different layers they can use to determine the volume of a prism, all of which result in the same volume.”

• Unit 7, Shapes on the Coordinate Plane, Lesson 9, About this Lesson provides notes about the lesson for the teacher. “The purpose of this lesson is for students to generate two different numerical patterns and then compare the terms in the two patterns. In this lesson, the patterns are the multiples of given whole numbers, starting with 0, and one of the numbers is a multiple of the other. This means that one of the patterns is contained inside the other. For example, the list of multiples of 9 is contained inside the list of multiples of 3 since every third multiple of 3 is a multiple of 9. Students express relationships within a pattern and between 2 patterns using multiplication and division. Lesson purpose: The purpose of this lesson is for students to generate patterns, given two rules, and identify relationships between corresponding terms in the different patterns. Teacher reflection question: In what ways did you accept students' everyday way of talking as a starting point for joining the math conversation today?” The section also includes lesson overview, learning goals, learning goals (student facing), materials to gather, materials to copy, and required preparation.

The materials reviewed for Imagine Learning Illustrative Mathematics Grade 5 meet expectations for containing adult-level explanations and examples of the more complex grade-level concepts and concepts beyond the current grade so that teachers can improve their own knowledge of the subject. 

Within the Teacher’s Guide, IM Curriculum, Why is the curriculum designed this way?, there are sections entitled “Further Reading” that consistently link research to pedagogy. There are adult-level explanations including examples of the more complex grade-level concepts and concepts beyond the current grade so that teachers can improve their own knowledge of the subject. Professional articles support teachers with learning opportunities about topics such as ensuring mathematical success for all, early understanding of equality, and repeating patterns. Examples include:

• Why is the curriculum designed this way? Further Reading, Unit 1, “A world without order (of operations). In this blog post, McCallum describes a world with only parentheses to guide the order of operations and discusses why the conventional order of operations is useful.” 

• Why is the curriculum designed this way? Further Reading, Unit 3, “Why is a negative times a negative a positive? supports teachers with context for work beyond the grade. “In this blog post, McCallum discusses how the “rule” for multiplying negative numbers is grounded in the distributive property.”

• Unit 4, Wrapping Up Multiplication and Division with Multi-Digit Numbers, Lesson 21, Food Waste Journal, About this Lesson, “This lesson does provide students with an opportunity to apply precursor skills of mathematical modeling. In previous lessons, students made estimates about the volume of recyclable garbage students at their school produce. In this lesson, students make similar estimates and calculations, but now they estimate the weight of food waste produced. In the first activity students estimate the amount of food waste they produce based on average production by individuals in the United States. In the second activity, students are introduced to a food waste journal and make some initial calculations about their food waste. The last activity is optional and can be used after students use the journal to record their food waste for a week. Students share what they notice and compare the amount of food waste they produce to the national average. Students use this sample data to estimate how much food waste they produce monthly and annually. When students recognize the mathematical features of familiar real world situations and use those features to solve problems, they model with mathematics (MP4).”

• Unit 6, More Decimals and Fraction Operations, Lesson 21, Weekend Investigation, About this Lesson, “This lesson does provide students with an opportunity to apply precursor skills of mathematical modeling. In this lesson, students brainstorm and define categories of how to spend time. Then they collect and represent data on a line plot. They analyze and describe the data to tell a story about the time use. When students define categories, choose and ask questions, collect and analyze data, and tell a story about the situation based on data, they model with mathematics (MP4).”

The materials reviewed for Imagine Learning Illustrative Mathematics Grade 5 meet expectations for including standards correlation information that explains the role of the standards in the context of the overall series. 

Correlation information is present for the mathematics standards addressed throughout the grade level/series and can be found in several places, including the Curriculum Course Guide, within unit resources, and within each lesson. Examples include:

• Grade-level resources, Grade 5 standards breakdown, standards are addressed by lesson. Teachers can search for a standard in the grade and identify the lesson(s) where it appears within materials.

• Course Guide, Lesson Standards, includes all Grade 5 standards and the units and lessons each standard appears in. 

• Unit 3, Resources, Teacher Guide, outlines standards, learning targets and the lesson where they appear. This is present for all units and allows teachers to identify targeted standards for any lesson.

• Unit 6, More Decimal and Fraction Operations, Lesson 5, the Core Standards are identified as 5.MD.A.1 and 5.NBT.A.2. Lessons contain a consistent structure that includes a Warm-up with a Narrative, Launch, Activity, Activity Synthesis. An Activity 1, 2, or 3 that includes Narrative, Launch, Activity, Activity Synthesis, Lesson Synthesis. A Cool-down that includes Responding to Student Thinking and Next Day Supports. This provides an additional place to reference standards, and language of the standard, within each lesson.

Each unit includes an overview identifying the content standards addressed within the unit, as well as a narrative outlining relevant prior and future content connections. Examples include: 

• Unit 5: Place Value Patterns and Decimal Operations, Unit Overview, Unit Learning Goals, full Unit Narrative, include an overview of how the math of this module builds from previous work in math. “In this unit, students expand their knowledge of decimals to read, write, compare, and round decimals to the thousandths. They also extend their understanding of place value and numbers in base ten by performing operations on decimals to the hundredth. In grade 4, students wrote fractions with denominators of 10 and 100 as decimals. They recognized that the notations 0.1 and 110 express the same amount and are both called “one tenth.” They used hundredths grids and number lines to represent and compare tenths and hundredths. Here, students likewise rely on diagrams and their understanding of fractions to make sense of decimals to the thousandths. They see that “one thousandth” refers to the size of one part if a hundredth is partitioned into 10 equal parts, and that its decimal form is 0.001. Diagrams help students visualize the magnitude of each decimal place and compare decimals.”

• Unit 6: More Decimals and Fraction Operations, Unit Overview, full Unit Narrative, include an overview of how the math of this module builds from previous work in math. “In this unit, students deepen their understanding of place-value relationships of numbers in base ten, unit conversion, operations on fractions with unlike denominators, and multiplicative comparison. The work here builds on several important ideas from grade 4. In grade 4, students learned the value of each digit in a whole number is 10 times the value of the same digit in a place to its right. Here, they extend that insight to include decimals to the thousandths. Students recognize that the value of each digit in a place (including decimal places) is $\frac{1}{10}$ the value of the same digit in the place to its left. This idea is highlighted as students perform measurement conversions in metric units. Previously, students learned to convert from a larger unit to a smaller unit. Here, they learn to convert from a smaller unit to a larger unit. They observe how the digits shift when multiplied or divided by a power of 10 and learn to use exponential notation for powers of 10 to represent large numbers.”

The materials reviewed for Imagine Learning Illustrative Mathematics Grade 5 meet expectations for providing explanations of the instructional approaches of the program and identification of the research-based strategies.

Instructional approaches of the program are described within the Curriculum Guide, Why is the curriculum designed this way? Design Principles. “It is our intent to create a problem-based curriculum that fosters the development of mathematics learning communities in classrooms, gives students access to the materials through coherent progression, and provides teachers the opportunity to deepen their knowledge of mathematics, student thinking, and their own teaching practice. Examples from the Design Principles include:

• Curriculum Guide, Why is the curriculum designed this way?, Design Principles, includes information about the 11 principles that informed the design of the materials. Balancing Rigor, “There are three aspects of rigor essential to mathematics: conceptual understanding, procedural fluency, and the ability to apply these concepts and skills to mathematical problems with and without real-world contexts. These aspects are developed together and are therefore interconnected in the materials in ways that support student understanding.  Opportunities to connect new representations and language to prior learning support students in building conceptual understanding. Access to new mathematics and problems prompts students to apply their conceptual understanding and procedural fluency to novel situations. Warm-ups, practice problems, centers, and other built-in routines help students develop procedural fluency, which develops over time.”

• Curriculum Guide, Why is the curriculum designed this way?, Design Principles, Task Complexity, “Mathematical tasks can be complex in different ways, with the source of complexity varying based on students’ prior understandings, backgrounds, and experiences. In the curriculum, careful attention is given to the complexity of contexts, numbers, and required computation, as well as to students’ potential familiarity with given contexts and representations. To help students navigate possible complexities without losing the intended mathematics, teachers can look to warm-ups and activity launches for built-in preparation, and to teacher-facing narratives for further guidance. In addition to tasks that provide access to the mathematics for all students, the materials provide guidance for teachers on how to ensure that during the tasks, all students are provided the opportunity to engage in the mathematical practices. More details are given below about teacher reflection questions, and other fields in the lesson plans help teachers assure that all students not only have access to the mathematics, but the opportunity to truly engage in the mathematics.”

Research-based strategies within the program are cited and described within the Curriculum Guide, within Why is the curriculum designed this way?. There are four sections in this part of the Curriculum Guide including Design principles, Key Structures, Mathematical Representations, and Further Reading. Examples of research-based strategies include: 

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

• Curriculum Guide, Why is the curriculum designed this way?, Further Reading, 3–5, “Fraction Division Parts 1–4. In this four-part blog post, McCallum and Umland discuss fraction division. They consider connections between whole-number division and fraction division and how the two interpretations of division play out with fractions with an emphasis on diagrams, including a justification for the rule to invert and multiply. In Part 4, they discuss the limitations of diagrams for solving fraction division problems. Fraction Division Part 1: How do you know when it is division? Fraction Division Part 2: Two interpretations of division Fraction Division Part 3: Why invert and multiply? Fraction Division Part 4: Our final post on this subject (for now). Untangling fractions, ratios, and quotients. In this blog post, McCallum discusses connections and differences between fractions, quotients, and ratios.”

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

• Curriculum Guide, Why is the curriculum designed this way?, Design Principles, Using the 5 Practices for Orchestrating Productive Discussions, “Promoting productive and meaningful conversations between students and teachers is essential to success in a problem-based classroom. The Instructional Routines section of the teacher course guide describes the framework presented in 5 Practices for Orchestrating Productive Mathematical Discussions (Smith & Stein, 2011) and points teachers to the book for further reading. In all lessons, teachers are supported in the practices of anticipating, monitoring, and selecting student work to share during whole-group discussions. In lessons in which there are opportunities for students to make connections between representations, strategies, concepts, and procedures, the lesson and activity narratives provide support for teachers to also use the practices of sequencing and connecting, and the lesson is tagged so teachers can easily identify these opportunities. Teachers have opportunities in curriculum workshops and PLCs to practice and reflect on their own enactment of the 5 Practices.”

The materials reviewed for Imagine Learning Illustrative Mathematics Grade 5 meet expectations for providing a comprehensive list of supplies needed to support instructional activities. 

The Course Overview, Grade-level resources, provides a Materials List intended for teachers to gather materials for each grade level. Additionally, specific lessons include a Teaching Notes section and a Materials List, which include specific lists of instructional materials for lessons. Examples include:

• Course Overview, Grade Level Resources, Grade 5  Materials List, contains a comprehensive chart of all materials needed for the curriculum. It includes the materials used throughout the curriculum, whether they are reusable or consumable, quantity needed, lessons the materials are used in, and suitable substitutes for the materials. Each lesson listed in the chart and any additional virtual materials noted for a lesson are digitally linked in the materials for quick access. Meter sticks are a reusable material used in lesson 5.4.19, 5.6.3, and 5.6.5. 15 (one per pair) are needed per 30 students. No suitable substitutes are listed. Number cubes are a reusable material used in lessons 5.5.11, 5.5.14, and 5.8.12. 15 are needed for 30 students. No suitable substitutes for the material are listed. Chart paper is a consumable material used in lessons 5.5.1, 5.5.11, 5.5.14, 5.8.14, and 5.8.15. 30 pages are needed per 30 students. Poster Paper is a suitable substitute for the material.

• Unit 1, Finding Volume, Lesson 7, Activity 1: What are the Units?, Teaching Notes, Materials to gather, “Rulers (centimeters), Rulers (inches), Yardsticks.” Launch, “Write the list of objects (moving truck, freezer, etc…) on a display for all students to see. In this activity we are going to consider using different cubic units of measure to find the volume of different sized objects. There is no right or wrong answer in these questions, but be prepared to explain your choice. Give students access to rulers and yardsticks.”

• Unit 5, Place Value Patterns and Decimal Operations, Lesson 12, Activity 2, Teaching Notes, Materials to gather, “Number cubes, Target Numbers Stage 9 Recording Sheet.” Launch, “Give each group 1 number cube. We’re going to play a new stage of the game called Target Numbers. Let’s read through the directions and play one round together. Read through the directions with the class and play a round with the class: Display each roll of the number cube. Think through your choices aloud. Record your move and score for all to see. Now, play the game with your partner.”

The materials reviewed for Imagine Learning Illustrative Mathematics Grade 5 meet expectations for providing strategies and supports for students in special populations to support their regular and active participation in learning grade-level mathematics.

Materials regularly provide strategies, supports, and resources for students in special populations to help them access grade-level mathematics. In the Curriculum Guide, How do the materials support all learners? Access for Students with Disabilities, “These materials empower all students with activities that capitalize on their existing strengths and abilities to ensure that all learners can participate meaningfully in rigorous mathematical content. Lessons support a flexible approach to instruction and provide teachers with options for additional support to address the needs of a diverse group of students, positioning all learners as competent, valued contributors. When planning to support access, teachers should consider the strengths and needs of their particular students. The following areas of cognitive functioning are integral to learning mathematics (Addressing Accessibility Project, Brodesky et al., 2002). Conceptual Processing includes perceptual reasoning, problem solving, and metacognition. Language includes auditory and visual language processing and expression. Visual-Spatial Processing includes processing visual information and understanding relation in space of visual mathematical representations and geometric concepts. Organization includes organizational skills, attention, and focus. Memory includes working memory and short-term memory. Attention includes paying attention to details, maintaining focus, and filtering out extraneous information. Social-Emotional Functioning includes interpersonal skills and the cognitive comfort and safety required in order to take risks and make mistakes. Fine-motor Skills include tasks that require small muscle movement and coordination such as manipulating objects (graphing, cutting with scissors, writing).” 

Examples include:

• Unit 3, Multiplying and Dividing Fractions, Lesson 3, Activity 2, Narrative, Access for Students with Disabilities, “Engagement: Provide Access by Recruiting Interest. Invite students to share a connection between the diagram and something in their own lives that represent the fractional values. Supports accessibility for: Attention, Conceptual Processing.”

• Unit 4, Wrapping Up Multiplication and Division with Multi-Digit Numbers, Lesson 19, Activity 1, Narrative, Access for Students with Disabilities, “Representation: Internalize Comprehension.  Activate or supply background knowledge. Provide students with a visual representation of a kilometer for students who are unfamiliar with the distance. Supports accessibility for: Conceptual Processing, Memory, Attention.”

• Unit 5, Place Value Patterns and Decimal Operations, Lesson 8, Activity 2, Narrative, Access for Students with Disabilities, “Action and Expression: Internalize Executive Functions. Invite students to verbalize their strategy for determining proximity before they begin. Students can speak quietly to themselves, or share with a partner. Supports accessibility for: Organization, Conceptual Processing, Language.”

The materials reviewed for Imagine Learning Illustrative Mathematics Grade 5 meet expectations for providing extensions and/or opportunities for students to engage with grade-level mathematics at higher levels of complexity.

While there are no instances where advanced students do more assignments than classmates, materials do provide multiple opportunities for students to investigate grade-level content at a higher level of complexity. These are found in a section titled, “Exploration Problems” within lessons where appropriate. According to the IM K-5 Teacher Guide, How do you use the materials?, Practice Problems, Exploration Problems, “Each practice problem set also includes exploration questions that provide an opportunity for differentiation for students ready for more of a challenge. There are two types of exploration questions. One type is a hands-on activity that students can do directly related to the material of the unit, either in class if they have free time, or at home. The second type of exploration is more open-ended and challenging. These problems go deeper into grade-level mathematics. They are not routine or procedural, and they are not just “the same thing again but with harder numbers. Exploration questions are intended to be used on an opt-in basis by students if they finish a main class activity early or want to do more mathematics on their own. It is not expected that an entire class engages in exploration problems, and it is not expected that any student works on all of them. Exploration problems may also be good fodder for a Problem of the Week or similar structure.”

Examples include:

• Unit 3, Multiplying and Dividing Fraction, Section B: Fraction Division, Problem 9, Exploration, “It takes Earth 1 year to go around the Sun. 1. During the time it takes Earth to go around the Sun, Mercury goes around the Sun about 4 times. How many years does it take Mercury to make 1 full orbit of the Sun? Write an equation showing your answer. 2. During the time it takes Earth to go around the Sun, Saturn goes $\frac{1}{29}$ of the way around the Sun. How many years does it take Saturn to go around the Sun? Write an equation showing your answer.”

• Unit 4, Wrapping Up Multiplication and Division with Multi-Digit Numbers, Section C: Let’s Put it to Work, Problem 5, Exploration, “The Pentagon has 5 floors and the Empire State Building has 102 floors. Noah says that the Empire State Building is bigger. Do you agree with Noah? Investigate and justify your answer.”

• Unit 5, Place Value Patterns and Decimal Operations, Section D: Divide Decimals, Problem 7, Exploration, “1. The daily recommended allowance of vitamin C for a 5th grader is 0.05 grams. A vitamin C tablet has 1 gram of vitamin C. How many times the daily recommended allowance of vitamin C is one vitamin C tablet? Use the diagram if it is helpful. 2. A large orange has 0.18 grams of vitamin C. How many times the daily recommended allowance of vitamin C is in a large orange? Use the diagram if it is helpful.”

The materials reviewed for Imagine Learning Illustrative Mathematics Grade 5 meet expectations for providing strategies and supports for students who read, write, and/or speak in a language other than English to regularly participate in learning grade-level mathematics.

Guidance is consistently provided for teachers to support students who read, write, and/or speak in a language other than English, providing scaffolds for them to meet or exceed grade-level standards. According to the Curriculum Guide, How do the materials support all learners? Mathematical language development, “Embedded within the curriculum are instructional routines and supports to help teachers address the specialized academic language demands when planning and delivering lessons, including the demands of reading, writing, speaking, listening, conversing, and representing in math (Aguirre & Bunch, 2012). While these instructional routines and supports can and should be used to support all students learning mathematics, they are particularly well-suited to meet the needs of linguistically and culturally diverse students who are learning mathematics while simultaneously acquiring English. Mathematical Language Routines (MLR) are also included in each lesson’s Support for English learners, to provide teachers with additional language strategies to meet the individual needs of their students. Teachers can use the suggested MLRs as appropriate to provide students with access to an activity without reducing the mathematical demand of the task. When selecting from these supports, teachers should take into account the language demands of the specific activity and the language needed to engage the content more broadly, in relation to their students’ current ways of using language to communicate ideas as well as their students’ English language proficiency. Using these supports can help maintain student engagement in mathematical discourse and ensure that struggle remains productive. All of the supports are designed to be used as needed, and use should fade out as students develop understanding and fluency with the English language.” The series provides principles that promote mathematical language use and development: 

• Principle 1. Support sense-making: Scaffold tasks and amplify language so students can make their own meaning. 

• Principle 2. Optimize output: Strengthen opportunities for students to describe their mathematical thinking to others, orally, visually, and in writing. 

• Principle 3. Cultivate conversation: Strengthen opportunities for constructive mathematical conversations. 

• Principle 4. Maximize meta-awareness: Strengthen the meta-connections and distinctions between mathematical ideas, reasoning, and language. 

The series also provides Mathematical Language Routines (MLR) in each lesson. Curriculum Guide, How do the materials support all learners? Mathematical language development, “A mathematical language routine is a structured but adaptable format for amplifying, assessing, and developing students' language. The MLRs were selected because they are effective and practical for simultaneously learning mathematical practices, content, and language. These routines can be adapted and incorporated across lessons in each unit to fit the mathematical work wherever there are productive opportunities to support students in using and improving their English and disciplinary language use. These routines facilitate attention to student language in ways that support in-the-moment teacher, peer, and self-assessment. The feedback enabled by these routines will help students revise and refine not only the way they organize and communicate their own ideas, but also ask questions to clarify their understandings of others’ ideas.” Examples include:

• Unit 1, Finding Volume, Lesson 3, Activity 1, Teaching notes, Access for English Learners, “MLR8 Discussion Supports. During small-group discussion, invite students to take turns sharing their responses. Ask students to restate what they heard using precise mathematical language and their own words. Display the sentence frame: “I heard you say . . . .” Original speakers can agree or clarify for their partner. Advances: Listening, Speaking.”

• Unit 2, Fractions as Quotients and Fraction Multiplication, Lesson 11, Activity 1, Teaching notes, Access for English Learners, “MLR8 Discussion Supports. Display sentence frames to support small-group discussion: “I wonder if . . . ”, “____ and ____ are the same because. . . .”, and “____ and ____ are different because ____.  Advances: Conversing, Representing.”

• Unit 4, Wrapping Up Multiplication and Division With Multi-Digit Numbers, Lesson 9, Synthesis, Teaching notes, Access for English Learners, “MLR1 Stronger and Clearer Each Time. Synthesis: Before the whole-class discussion, give students time to meet with 2–3 partners to share and get feedback on their response to What is the possible range of volumes for each type of birdhouse?. Invite listeners to ask questions, to press for details, and to suggest mathematical language. Give students 2–3 minutes to revise their written explanation based on the feedback they receive. Advances: Writing, Speaking, Listening.”

The materials reviewed for Imagine Learning Illustrative Mathematics Grade 5 meet expectations for providing manipulatives, both virtual and physical, that are accurate representations of the mathematical objects they represent and, when appropriate, are connected to written methods.

The materials consistently include suggestions and/or links, within the lesson notes, for virtual and physical manipulatives that support the understanding of grade-level math concepts. Examples include: 

• Unit 5, Place Value Patterns and Decimal Operations, Lesson 17, Activity 2, students use grid paper to find products of whole numbers and some tenths or hundredths. Launch, “Groups of 2. Make hundredths grids available for students. Activity, ‘Take a few minutes to find the value of the expressions in the first problem.’ Student facing, ‘Find the value of each expression. Explain or show your reasoning. a. $3\times0.5$ b. $5\times0.3$ c. $7\times0.02$’”

• Unit 6, More Decimal and Fraction Operations, Lesson 3, Activity 1, students use meter sticks to help them convert meters to centimeters. Launch, “Give students access to meter sticks. Display image from student workbook. ‘What do you notice? What do you wonder?’ Display additional information about track and field events: The height of a hurdle is 1 meter. The approximate distance between hurdles in 110 meter races is 10 meters. The shortest race in many track competitions is 100 meters. ‘Work with your partner to complete the problems.’”

• Unit 7, Shapes on the Coordinate Plane, Lesson 1, Activity 1, students use coordinate grids to communicate and draw shapes.  Launch, “Groups of 2. ‘We are going to play a drawing game. Decide who will be partner A and who will be partner B.’ Student facing, Play three rounds of Draw My Shape using the three sets of cards from your teacher. For each round: Partner A chooses a card—without showing your partner—and describes the shape on the card. Partner B draws the shape as described. Partner A reveals the card and partner B reveals the drawing. Compare the shapes and discuss: ‘What’s the same? What’s different?’ Activity, Circulate, listen for, and collect the language students use to describe the location of each figure on the coordinate grid. Listen for students who: use the grid to determine the side lengths or area of the rectangle. describe the general location of the rectangle. use the numbers on the axes as reference points when describing the rectangle. Record students’ words and phrases on a visual display and update it throughout the lesson.”