The materials reviewed for Open Up Resources K–5 Math 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 the Grade 5 Course Guide, Design Principles, conceptual understanding is a part of the design of the materials. Balancing Rigor states, “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.” Additionally, Purposeful Representations states, “Across lessons and units, students are systematically introduced to representations and encouraged to use representations that make sense to them. As their learning progresses, students are given opportunities to make connections between different representations and the concepts and procedures they represent.” Examples include:

• Unit 1, Finding Volume, Section A, Lesson 1, Activity 1, Student Work Time, students develop conceptual understanding of volume as they recognize that objects with the same volume take up the same amount of space. Students are given different pictures of pattern block formations. Student Facing states, “1. Which is bigger? Explain or show your reasoning. 2. Which is bigger? Explain or show your reasoning. 3. What does it mean for an object to be ‘bigger’?” (5.MD.3)

• Unit 5, Place Value Patterns and Decimal Operations, Section A, Lesson 5, Warm-up, Student Work Time, students develop conceptual understanding as they use place value understanding to compare decimals to the thousandths place. Student Facing states, “Decide if each statement is true or false. Be prepared to explain your reasoning. $7.06=7.006$, $7.06=7.060$, $7.06=7.600$.” (5.NBT.3)

• Unit 8, Putting It All Together, Section C, Lesson 10, Activity 1, Student Work Time and Activity Synthesis, students develop conceptual understanding as they practice adding fractions with unlike denominators and reason about how the size of the numerators and denominators impact the value of a fraction. In Student Work Time, Student Facing states, “Use the directions to play Greatest 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 greater 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 highest total sum wins the game. Total sum of all 4 rounds: ___.” Activity Synthesis states, “‘What strategies were helpful as you played Greatest Sum?’ (I tried to make fractions that have a larger numerator than denominator so they would be greater than one. I tried to make sure the ones and twos were in the denominator and put bigger numbers in the numerator.) ‘How did you add your fractions?’ (My denominators were 1, 2, 3, and 4 so I used 12 as a common denominator for all of them.)” (5.NF.1)

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

• Unit 1, Finding Volume, Section A, Lesson 2, Cool-down, students demonstrate conceptual understanding of volume when they use their understanding of volume as the amount of unit cubes that fill a space. Students see a picture of a rectangular prism, and Student Facing states, “Find the volume of the rectangular prism. Explain or show your reasoning.” (5.MD.4)

• Unit 4, Wrapping Up Multiplication and Division with Multi-Digit Numbers, Section B, Lesson 12, Activity 2, Student Work Time, students demonstrate conceptual understanding as they deepen their understanding of an algorithm that uses partial quotients. Students are provided three division problems, and Student Facing states, “Use Elena’s strategy to complete the following problems: $492\div12$, $630\div15$, $364\div14$.” (5.NBT.6)

• Unit 6, More Decimal and Fraction Operations, Section C, Lesson 20, Activity 2, Student Work Time, students demonstrateconceptual understanding as they compare a product of fractions to one of the factors. Student Facing states, “Andre says: When you multiply any fraction by a number less than 1, the product will be less than the fraction. When you multiply any fraction by a number greater than 1, the product will be greater than the fraction. Each partner choose one of the statements and describe why it is true. You may want to include details such as notes, diagrams, and drawings to help others understand your thinking.” (5.NF.5)

The materials reviewed for Open Up Resources K–5 Math Grade 5 meet expectations for giving attention throughout the year to individual standards that set an expectation of procedural skill and fluency. 

According to the Grade 5 Course Guide, 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, Section A, Lesson 4, Warm-up, Student Work Time and Activity Synthesis, students develop procedural skill and fluency as they notice the patterns in calculations within the number talk, leading towards the standard algorithm. In Student Work Time, Student Facing states, “Find the value of each product mentally. $3\times3$, $3\times20$, $3\times600$, $3\times623$.” Activity Synthesis states, “‘How is the last product related to the first three?’ (It is the sum of the first three.) ‘Did the first three calculations help you find the last product?’ (Yes, I was able to add them together to find.)” (5.NBT.5)

• Unit 6, More Decimal and Fraction Operations, Section B, Lesson 9, Warm-up, Student Work Time, students develop procedural skill and fluency with adding and subtracting fractions with different denominators. Student Facing states, “Decide if each statement is true or false. Be prepared to explain your reasoning. $\frac{1}{4}+\frac{2}{4}=\frac{3}{4}$, $\frac{1}{2}+\frac{1}{4}=\frac{2}{4}$, $\frac{3}{4}+\frac{1}{2}=\frac{2}{4}$.” (5.NF.1)

• Unit 8, Putting It All Together, Section A, Lesson 2, Activity 1, Launch and Student Work Time, students develop procedural skill and fluency as they find mistakes when they multiply large numbers. Launch states, “Display or write for all to see. $650\times27$. Display each number in a different corner of the room: 14,000, 18,000, 13,000, 19,000. ‘When I say go, stand in the corner with the number that you think is the most reasonable estimate for $650\times27$. Be prepared to explain your reasoning.’ 1 minute: quiet think time. Ask a representative from each corner to explain their reasoning. ‘Does anyone want to switch corners?’ Ask a student who switched corners to explain their reasoning. ‘Now you are going to find this product and analyze some work.’” In Student Work Time, Student Facing states, “a. Find the value of the product. $650\times27$. b. Below is Kiran’s work finding the value of the product $650\times27$. Is his answer reasonable? Explain your reasoning. c. What parts of the work do you agree with? Be prepared to explain your reasoning. d. What parts of the work do you disagree with? Be prepared to explain your reasoning. e. Look at your solution to problem a. Is there anything you want to revise? Be prepared to explain.” (5.NBT.5) 

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

• Unit 4, Wrapping Up Multiplication and Division with Multi-Digit Numbers, Section A, Lesson 5, Cool-down, students demonstrate fluency when multiplying a multi-digit number. Student Facing states, “Use the standard algorithm to find the value of $203\times23$.” (5.NBT.5)

• Unit 6, More Decimal and Fraction Operations, Section A, Lesson 1, Cool-down, students demonstrate procedural skill and fluency as they use place value patterns when multiplying and dividing whole numbers and numbers in decimal form. Student Facing states, “Fill in the blank to make each equation true. a. $0.06\times10=$___. b. $60=$___$\times0.6$. c. ___$= 6\div100$.” (5.NBT.A)

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

The materials reviewed for Open Up Resources K–5 Math Grade 5 meet expectations for being designed so that teachers and students spend sufficient time working with engaging applications of mathematics. 

According to the Grade 5 Course Guide, Design Principles, Balancing Rigor, “Access to new mathematics and problems prompts students to apply their conceptual understanding and procedural fluency to novel situations.” Multiple routine and non-routine applications of the mathematics are included throughout the grade level, and these single- and multi-step application problems are included within Activities or Cool-downs. 

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

Examples of routine applications of the math include:

• Unit 3, Multiplying and Dividing Fractions, Section C, Lesson 18, Activity 1, Student Work Time, students work with real-world problems involving multiplication and division of fractions. In Student Work Time, Student facing states, “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.” (5.NF.4, 5.NF.6, 5.NF.7)

• Unit 3, Multiplying and Dividing Fractions, Section B, Lesson 15, Cool-down, students solve problems involving division of whole numbers and unit fractions. Student facing states, “Match each expression to a situation. Answer each question. $5\div\frac{1}{4}$, $\frac{1}{4}\div5$ 1. Han cut 5 feet of ribbon into pieces that are $\frac{1}{4}$ foot long. How many pieces are there? 2. Han cut a $\frac{1}{4}$ foot long piece of ribbon into 5 equal pieces. How long is each piece?” (5.NF.7c)

• Unit 6, More Decimal and Fraction Operations, Section B, Lesson 12, Cool-down, students solve real-world problems that involve adding and subtracting fractions with unlike denominators. Student facing states, “a. Priya hiked $1\frac{2}{3}$ miles. Diego hiked $\frac{1}{2}$ mile. How much farther did Priya hike than Diego? Explain or show your reasoning. b. On Monday, Andre hiked $\frac{3}{4}$ mile in the morning and $1\frac{1}{3}$ miles in the afternoon. How far did Andre hike on Monday? Explain or show your reasoning.” (5.NF.1, 5.NF.2)

Examples of non-routine applications of the math include:

• Unit 1, Finding Volume, Section B, Lesson 5, Activity 3, Student Work Time, students interpret equations in order to match information given about rectangular prisms. Student Work Time states, “2 minutes: quiet think time. 4 minutes: partner work time. Monitor for students who: use informal language, such as layers, use the terms length, width, height, and base in their questions.” Student facing states, “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. a. 3 is the answer. What is the question? b. 5 is the answer. What is the question? c.  $3\times4=12$. The answer is 12. What is the question? d. $12\times5=60$. The answer is 60 cubes. What is the question? e. 3 by 4 by 5 is the answer. What is the question?” (5.MD.5b)

• Unit 5, Place Value Patterns and Decimal Operations, Section D, Lesson 26, Activity 2, Student Work Time, students solve a  real-world problem including operations with numbers in decimal form. In Student Work Time, Student Facing states, “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: a. Choose 3–5 types of books you want to order. b. Decide on the mark-up price for each type of book you chose. c. Estimate the amount of money your school will raise as a profit with your book sale. Record an estimate that is: too low, about right, too high. d. Show or explain your reasoning for the estimate. Include the assumptions you made.” (5.NBT.7)

• Unit 7, Shapes on the Coordinate Plane, Section C, Lesson 13, Activity 1, Student Work Time, students plot points that represent the length and width of a rectangle with a given perimeter. Student Work Time states, “2 minutes: independent think time. 5 minutes: partner work time.” Student Facing states, “a. Jada drew a rectangle with a perimeter of 12 centimeters. What could the length and width of Jada’s rectangle be? Use the table to record your answer. b. Plot the length and width of each rectangle on the coordinate grid. c. If Jada drew a square, how long and wide was it? d. If Jada’s rectangle was 2.5 cm long, how wide was it? Plot this point on the coordinate grid. e. If Jada’s rectangle was 3.25 cm long, how wide was it? Plot this point on the coordinate grid.” (5.G.2, 5.NBT.7, 5.OA.3)

The materials reviewed for Open Up Resources K–5 Math 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.

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

• Unit 1, Finding Volume, Section B, Lesson 6, Cool-down, students demonstrate conceptual understanding as they use their understanding of volume to identify and explain the correct expression. Student Facing states, “a. Which of these expressions does not represent the volume of the rectangular prism in cubic units? b. Explain or show your reasoning. c. Choose one of the expressions from above and explain why it represents the volume of the prism in cubic units.” (5.MD.5b)

• Unit 4, Wrapping Up Multiplication and Division with Multi-Digit Numbers, Section A, Lesson 5, Activity 2, Student Work Time, students develop procedural skill and fluency as they use the standard algorithm to multiply three-digit numbers by two-digit numbers. In Student Work Time, Student Facing states, “Use the standard algorithm to find the value of each expression. 1. $202\times12$, 2. $122\times33$, 3. $321\times24$. 4. Diego found the value of $301\times24$  Here is his work. Why doesn’t Diego’s answer make sense?” The answer shown is 1,806. (5.NBT.5) 

• Unit 7, Shapes on the Coordinate Plane, Section C, Lesson 12, Activity 1, Student Work Time, students apply their understanding of the coordinate plane as they interpret data about a series of coin flips. In Student Work Time, Student facing states, “Han and Jada flipped a penny several times and counted how many times it came up heads and how many times it came up tails. Their results are plotted on the graph. a. How many heads did Jada get? How many tails did Jada get? Explain or show how you know. b. How many heads did Han get? How many tails did Han get? Explain or show how you know. c. Flip the coin 10 times and record how many heads and tails you get. Plot the point on the coordinate grid that represents your coin flips. d. Show your partner the point you plotted on the coordinate grid. Look at your partner's coordinate grid. How many heads did your partner flip? How many tails did your partner flip? Explain or show your reasoning. e. Do any of the points you plotted lie on the horizontal axis? What would a point on the horizontal axis mean in this situation? f. If time allows, toss the coin 10 more times and record your results and your partner’s results on the coordinate grid.” (5.G.2) 

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

• Unit 2, Fractions as Quotients and Fraction Multiplication, Section C, Lesson 10, Cool-down, students use conceptual understanding and procedural fluency as they compute the area of rectangles when there is one non-unit fractional side length and one whole number side length. Student Facing states, “a. Write a multiplication expression to represent the area of the shaded region. b. Find the area of the shaded region.” An image shows a rectangle with a length of 5 and a width of $\frac{3}{4}$. (5.NF.4)

• Unit 6, More Decimal and Fraction Operations, Section B, Lesson 12, Activity 2, Student Work Time, students develop all three aspects of rigor, conceptual understanding, procedural skill and fluency, and application, as they solve multi-step problems involving the addition and subtraction of fractions. In Student Work Time, Student Facing states, “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 $1\frac{2}{3}$ 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 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. a. Discuss the problems and solutions with your partner. What is the same about your strategies and solutions? What is different? b. Revise your work if necessary.” (5.NF.2)

• Unit 7, Shapes on the Coordinate Plane, Section B, Lesson 6, Activity 1, Student Work Time, students use conceptual understanding and application to construct quadrilaterals and explain their attributes. Student Facing states, “a. Build a square with your toothpicks. How do you know it is a square? b. Use the same four toothpicks to build this shape. (a parallelogram is shown) What stayed the same? What changed? c. Build a rectangle with six toothpicks. How do you know it is a rectangle? d. Use the same six toothpicks to build this shape. (a parallelogram is shown) What stayed the same? What changed?” (5.G.3)

The materials reviewed for Open Up Resources K–5 Math 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 within the Course Guide (How to Use These Materials). A chart is provided within this section that highlights several lessons that showcase particular Mathematical Practices. The Mathematical Practices are also identified within specific lessons (Lesson Preparation Narratives and Lesson Activities’ Narratives).

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

• Unit 1, Finding Volume, Section B, Lesson 5, Warm-up, Activity Narrative and Student Work Time, students reason about the attributes of a rectangular prism. Activity Narrative states, “The purpose of this Warm-up is for students to notice that each face of a prism can be the base, which will be useful when students use a base of a prism to find the prism’s volume in a later activity. While students may notice and wonder many things about these images, the relationship between the images of the prism and the images of the rectangles are the important discussion points.” In Student Work Time, Student Facing states, “What do you notice? What do you wonder?” Three rectangular prisms and three rectangles are shown.

• Unit 4, Wrapping up Multiplication and Division with Multi-Digit Numbers, Section B, Lesson 10, Cool- Down, students make sense of multi-digit division problems. Preparation, Lesson Narrative states, “In this lesson, students explore a context to make sense of division with multi-digit numbers (MP1). This builds on work students did in grade 4 where they divided with up to 4-digit dividends and single-digit divisors. Students used place value understanding, the relationship between multiplication and division and partial quotients to divide. The work in this lesson gives teachers an opportunity to see how students apply their prior understanding, including multiplying multi-digit numbers in the last section. In future lessons, students work toward using more efficient methods to divide multi-digit numbers, including partial quotients.” Student Facing states, “A different group of 4,632 dancers make groups of 8. a. Write a division expression to represent the situation. b. How many groups of 8 will there be? Explain or show your thinking.”

• Unit 8, Putting It All Together, Section B, Lesson 8, Activity 2, Student Work Time and Activity Narrative, students make sense of problems as they reason about multiplication and division. In Student Work Time, Student Facing states, “The Radio Flyer wagon is 27 feet long 13 feet wide and 2 feet deep. The wagon is being used to deliver 4,000 boxes that each have the side lengths 2 feet by 2 feet by 2 feet. How many trips will the wagon have to make? Explain or show your reasoning.” Activity Narrative states, “The purpose of this activity is for students to solve another problem about the Radio Flyer using multiplication and division. Instead of filling the wagon with sand, they consider filling the wagon with boxes and determine how many boxes will fill the wagon. Unlike with the sand, the boxes do not fill the wagon completely and the number of boxes that do fit is not a divisor of the total number of boxes. Accounting for these considerations will be the focus of the synthesis. When students account for these constraints of the situation, they persevere in solving the problem (MP1).”

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

• Unit 2, Fractions as Quotients and Fraction Multiplication, Section B, Lesson 7, Activity 2, Student Work Time and Activity Narrative, students reason abstractly and quantitatively as they match expressions and diagrams. In Student Work Time, Student Facing states, “Han, Lin, Kiran, and Jada together ran a 3 mile relay race. (They each ran the same distance.) a. ”Find the expressions and diagrams that match this situation. Be prepared to explain your reasoning. b. How far did each person run?” Activity Narrative states, “Students reason abstractly and quantitatively (MP2) when they relate the story to the diagrams and expressions. All of the diagrams and expressions involve the same set of numbers so students need to carefully analyze the numbers in the story, the diagrams, and the expressions in order to choose the correct matches.”

• Unit 6, More Decimal and Fraction Operations, Section B, Lesson 14, Activity 2, Student Work Time and Activity Narrative, students reason about information presented in a line plot. In Student Work Time, Student Facing states, “a. Use the egg weights listed to make a line plot. $1\frac{7}{8}$, $2\frac{1}{2}$, $2\frac{3}{8}$, $1\frac{3}{4}$, $2\frac{1}{4}$, $2\frac{4}{8}$, $2\frac{1}{8}$, $1\frac{7}{8}$, $2\frac{1}{4}$, $1\frac{6}{8}$, $2\frac{1}{8}$, $1\frac{7}{8}$ b. Jada said that $\frac{1}{4}$ of the eggs weigh $1\frac{7}{8}$ ounces. Do you agree? Explain or show your reasoning. c. How much heavier is the heaviest egg than the lightest egg? Explain or show your reasoning.” Activity Narrative states, “The purpose of this activity is for students to use measurement data to make a line plot and then solve problems about the data presented in the line plot (MP2).”

• Unit 7, Shapes on the Coordinate Plane, Section C, Lesson 10, Cool-down, students think abstractly as they determine rules for given patterns. In Preparation, Lesson Narrative states, “In this lesson students continue to generate two patterns and observe relationships between their corresponding terms. Most of the relationships are more complex in this lesson, involving either multiplication by a fractional amount or both multiplication and addition or subtraction. Students begin to express the relationships between patterns using equations (MP2).” Student Facing states, “a. Jada and Priya are creating rules for patterns. Follow each rule to complete the patterns. Jada’s rule: start with 0 and add 3. Priya’s rule: start with 0 and add 4. b. Kiran says that when Jada’s number is 45, Priya’s corresponding number will be 90. Do you agree? Why or why not?”

The materials reviewed for Open Up Resources K–5 Math Grade 5 meet expectations for supporting the intentional development of MP3: Construct viable arguments and critique the reasoning of others, for students, in connection to the grade-level content standards, as expected by the mathematical practice standards. 

Students have opportunities to engage with the Math Practices across the year, and they are often explicitly identified for teachers within the Course Guide (How to Use These Materials). A chart is provided within this section that highlights several lessons that showcase particular Mathematical Practices. The Mathematical Practices are also identified within specific lessons (Lesson Preparation Activity Narratives and Lesson Activities’ Activity Narratives).

According to the Grade 5 Course Guide, Design Principles, Learning Mathematics By Doing Mathematics, “Students learn mathematics by doing mathematics, rather than by watching someone else do mathematics or being told what needs to be done. Doing mathematics can be defined as learning mathematical concepts and procedures while engaging in the mathematical practices - making sense of problems, reasoning abstractly and quantitatively, making arguments and critiquing the reasoning of others, modeling with mathematics, making appropriate use of tools, attending to precision in their use of language, looking for and making use of structure, and expressing regularity in repeated reasoning. By engaging in the mathematical practices with their peers, students have the opportunity to see themselves as mathematical thinkers with worthwhile ideas and perspectives.”

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

• Unit 2, Fractions as Quotients and Fraction Multiplication, Section A, Lesson 5, Cool-down, students construct viable arguments as they write division expressions and equations that represent real world situations. Student Facing states, “Explain why $8\div5=\frac{8}{5}$.” Preparation, Lesson Narrative states, “In this lesson, students generalize their understanding that a fraction can be interpreted as division of the numerator by the denominator. They interpret situations where a certain amount of pounds of blueberries is shared with a certain number of people when the pounds of blueberries each person gets is equal to 1, greater than 1, and less than 1. Then, they construct arguments about why an equation would make sense for any numerator and for any denominator.”

• Unit 4, Wrapping Up Multiplication and Division with Multi-Digit Numbers, Section A, Lesson 1, Activity 1, students construct a viable argument as they reason about appropriate estimates for multi-digit multiplication calculations. Activity Narrative states, “The purpose of this activity is for students to make a reasonable estimate for a given product. In addition to estimating the product, students also decide whether the estimate is too large or too small. In the activity synthesis, students consider how far their estimate is from the actual product. In the next activity, students will evaluate the expressions using a strategy of their choice. Students choose between several different possible estimates and justify their choice before they calculate the product (MP3).” Student Work Time states, “5–7 minutes: independent work time. 2–3 minutes: partner discussion. Monitor for students who: relate the given expression to each proposed answer by rounding or changing one or both factors, estimate by rounding the factors, use benchmark numbers, use place value reasoning or the properties of operations to explain why their estimate is reasonable.” In Student Work Time, Student Facing states, “a. Which estimate for the product $18\times149$ is most reasonable? Explain or show your reasoning. 2,000, 4,000, 3,000, 1,500 b. Are any of the estimates unreasonable? Explain or show your reasoning. c. Do you think the actual product will be more or less than your estimate? Explain or show your reasoning.”

• Unit 8, Putting It All Together, Section A,  Lesson 5, Activity 1, Activity Narrative and Student Work Time, students construct an argument and critique the reasoning of others as they defend a strategy to solve a division problem. Activity Narrative states, “The purpose of this activity is for students to revisit the partial quotients method to find whole number quotients. Students compare their strategy with Elena's strategy and reason about the similarities and differences using their understanding of place value. They may use estimation to identify that Elena's answer is not reasonable while they may also use parts of their own calculation to identify Elena's error (MP3).” Student Work Time states, “‘Work with your partner to complete the second, third, and fourth problems.’ 5–7 minutes: partner work time. ‘Now you will have a chance to revisit your work from the first problem.’ 1–2 minutes: independent work time. Monitor for students who: revised their original solution, used different partial quotients.” In Student Work Time, Student Facing states, “a. Find the value of the quotient. $6773÷13$ b. Here is how Elena found the quotient. Is her answer reasonable? (Students see work done by Elena using partial quotient strategy.) Explain or show your reasoning. c. What parts of the work do you agree with? Be prepared to explain your reasoning. d. What parts of the work do you disagree with? Be prepared to explain your reasoning. e. Look at your solution to problem 1. Is there anything you want to revise? Be prepared to explain.”

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

• Unit 1, Finding Volume, Section B, Lesson 7, Activity 1, Activity Narrative and Student Work Time, students construct arguments and critique the reasoning of others as they reason about the volume measurements of different items. Activity Narrative states, “The purpose of this activity is for students to consider how the size of an object impacts the unit we use to measure the volume of that object. Since this is the students’ first experience with these cubic units of measure, it may be helpful for them to see the actual length of a centimeter, inch and foot. Have rulers or cubes available to provide extra support to visualize the size of the cubic units of measure. Because there are no mathematically correct or incorrect answers, this activity provides a rich opportunity for students to discuss and defend different points of view (MP3).” Student Work Time states, “2 minutes: independent work time. 5 minutes: partner discussion. As students work, monitor for students who discuss how big or small the object is when choosing the size of the unit of measure. Ask these students to share during the synthesis. If students finish early, ask them to find other objects they would measure the volume of using the different cubic units of measure. If the objects are in the room, they could estimate and check their estimates.” In Student Work Time, Student Facing states, “For each object, choose the cubic unit you would use to measure the volume: cubic centimeter, cubic inch, or cubic foot.” A table is included with the following objects: the volume of a moving truck, the volume of a freezer, the volume of a juice box, the volume of a classroom, the volume of a dumpster, and the volume of a lunch box.

• Unit 2, Fractions as Quotients and Fraction Multiplication, Section A, Lesson 3, Activity 1, Activity Narrative and Student Work Time, students critique the reasoning of others when working with fraction division. Activity Narrative states, “The last problem provides an opportunity for students to think critically about a proposed solution to a problem (MP3). Different ways to think about the proposed solution include: estimation: with 3 friends sharing 2 liters, each friend gets less than 1 liter thinking about the meaning of the numerator (how many liters are being shared) and denominator (how many people are sharing the water).” Student Work Time states, “5 minutes: independent work time. 5 minutes: partner discussion. As students work, monitor for students who: draw a diagram to determine the amount of water each dancer drinks if 3 dancers share 2 liters of water, revise their solution for how much water each dancer gets after explaining why Mai’s answer doesn’t make sense.”  In Student Work Time, Student Facing states, “a. Three dancers share 2 liters of water. How much water does each dancer get? Write a division equation to represent the situation. b. 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 5, Place Value Patterns and Decimal Operations, Section B, Lesson 15, Activity 3, Student Work TIme and Activity Narrative, students construct a viable argument and critique the reasoning of others as they analyze a common error when using the standard algorithm to subtract decimals. In Student Work Time, Student Facing states, “a. Find the value of $622.35-71.4$ Explain or show your reasoning. b. Elena and Andre found the value of $622.35-71.4$. Who do you agree with? Explain or show your reasoning.” Activity Narrative states, “When students identify and correct Elena's error they construct viable arguments and critique the reasoning of others (MP3).”

The materials reviewed for Open Up Resources K–5 Math Grade 5 meet expectations for supporting the intentional development of MP4: Model with mathematics; and MP5: Choose tools strategically, for students, in connection to the grade-level content standards, as expected by the mathematical practice standards. Students have opportunities to engage with the Math Practices across the year and they are often explicitly identified for teachers within the Course Guide (How to Use These Materials). A chart is provided within this section that highlights several lessons that showcase particular Mathematical Practices.  The Mathematical Practices are also identified within specific lessons (Lesson Preparation Instructional Routines and Lesson Activities’ Instructional Routines).

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

• Unit 2, Fractions as Quotients and Fraction Multiplication, Section C, Lesson 17, Activity 1, Instructional Routine and Student Work Time, students model with mathematics as they use the context of art to multiply fractions. Preparation, Lesson Narrative states, “When students make decisions and choices, analyze contextual objects with mathematical ideas, and translate a mathematical answer back into the context of a situation, they model with mathematics (MP4).” In Student Work Time, Student Facing states, “a. Use the colored paper and scissors to cut identical rectangles. Make sure the measurement of one side of the rectangle is a whole number and the other is a fraction greater than one. b. What is the area of one of your rectangles? Show your reasoning. c. Use the rectangles from your group to make a group mosaic by arranging some of the different colored rectangles on a blank piece of paper.” 

• Unit 4, Wrapping Up Multiplication and Division with Multi-Digit Numbers, Section C, Lesson 20, Cool-down, students model with math when they make estimates and solve complex problems. Lesson Narrative states, “Throughout the lesson, students make estimates and simplifying assumptions in order to answer complex mathematical questions (MP4).” Student facing states, “A different shipping container is 40 feet long, 9 feet wide, and 8 feet tall. a. What is the volume of this container? Explain or show your thinking. b. A school makes 24 cubic feet of recyclable plastic each day. How many days does it take the school to fill this container? Explain or show your thinking.”

• Unit 8, Putting it All Together, Lesson 7, Activity 1, Instructional Routine and Student Work Time, students model with math by selecting the appropriate unit of measure as well as what the estimate means within the real-world situation. Instructional Routine states, “Choosing an appropriate unit of measure for an estimation and understanding how that choice affects both the calculations and the meaning of the estimate are important aspects of applying mathematics to solve real world problems (MP4).” Student Work Time states, “3 - 5 minutes: quiet work time. 5 minutes: partner discussion time. Monitor for students who: notice that the wagon has a rectangular prism shape, roughly, and recognize that we need to know the side lengths of the wagon in order to make a reasonable estimate about its volume, use references, such as the size and number of people in the wagon, to help estimate the wagon’s volume, choose different units of length and volume for their estimates.” In Student Work Time, Student Facing states, “a. What measurements would you take of the wagon to accurately estimate its volume? b. What units would you use to measure the wagon? Explain your reasoning. c. Record an estimate for the volume of the wagon that is: too low, about right, too high. d. What can you use in the picture to refine your estimate?”

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

• Unit 2, Fractions as Quotients and Fraction Multiplication, Section C, Lesson 9, Activity 1, Instructional Routine, Launch, and Student Work Time,  students choose an appropriate strategy to find the area of a shaded region. Instructional Routine states, “The purpose of this activity is for students to find the area of rectangles with one fractional side length and one whole number side length. Students begin by considering a rectangle with whole number side lengths and then look at a series of rectangles with unit fraction side length. All of the rectangles have the same whole number width to help students see how the area changes when the fractional width changes. Students should use a strategy that makes sense to them. These strategies might include counting the individual shaded parts in the diagram or thinking about moving them to fill in unit squares. Some students may use multiplication or division. These ideas will be brought out in future lessons. During discussion, connect the different strategies students use to calculate the areas. As they choose a strategy, they have an opportunity to use appropriate tools, whether it be expressions that represent the shaded area or physical manipulations of the diagrams, strategically (MP5).”  Launch states, “Groups of 2. Display the images of the shaded rectangles. ‘What is the same about all of the rectangles? What is different?’ (They are all shaded. They have different amounts shaded. They have different widths.) ‘We are going to figure out how much of each rectangle is shaded. We call this finding the area of the shaded region. What are some strategies we could use to find the area of each of the shaded regions?’ (Move the pieces around to make full squares, count the number of blue pieces and multiply the number of pieces by their size.)” In Student Work Time, Student facing states, “Find the area of the shaded region. Explain or show your reasoning.” The following representations would be shown: 6 by 1, 6 by $\frac{1}{2}$, 6 by $\frac{1}{3}$, and 6 by $\frac{1}{4}$.

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

• Unit 6, More Decimal and Fraction Operations, Section B, Lesson 8, Cool-down, students use an appropriate strategy as a tool to find solutions to problems involving addition and subtraction of fractions and then explain their strategy. Student facing states, “Find the value of each expression. Explain or show your reasoning. a. $\frac{5}{6}-\frac{1}{3}$. b. $\frac{3}{4}+\frac{1}{2}$.” Lesson Narrative states, “Students describe how the situations are different and find the sums and differences in a way that makes sense to them. The denominators of the fractions used in this lesson are familiar from grade 3, inviting students to use a variety of different familiar representations.”

The materials reviewed for Open Up Resources K–5 Math 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 opportunities to engage with the Math Practices across the year, and they are often explicitly identified for teachers within the Course Guide (How to Use These Materials). A chart is provided within this section that highlights several lessons that showcase particular Mathematical Practices. The Mathematical Practices are also identified within specific lessons (Lesson Preparation Narratives and Lesson Activities’ Narratives).

Students have many opportunities to attend to precision and the specialized language of math, in connection to grade-level content, as they work with support of the teacher and independently throughout the units. Examples include:

• Unit 1, Finding Volume, Section A, Lesson 5, Activity 2,Student Work Time and Activity Narratives, students use precise language as they complete a table showing multiplication expressions for the volume of prisms. In Student Work Time, Student Facing states, “Here is a base of a rectangular prism. 1. Fill out the table for the volumes of rectangular prisms with this base and different heights.” Activity Narratives states, “Students may still use informal language, such as layers, to describe the prisms and find their volume. During the lesson synthesis, connect their informal language to the more formal math language of length, width, height, and area of the base.”

• Unit 2, Fractions as Quotients and Fraction Multiplication, Section A, Lesson 2, Activity 2, Activity Narrative and Student Work Time, students attend to precision of language when connecting mathematical representations for a real-world problem. Activity Narrative states, “This sorting task gives students opportunities to analyze and connect representations, situations, and expressions (MP2, MP7). As students work, encourage them to refine their descriptions of how the diagrams represent the situations and expressions using more precise language and mathematical terms (MP6).” Student Work Time states, “‘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.’ 5–8 minutes: partner work time. Monitor for students who: notice that the number of large rectangles in the picture and the dividend in the expressions represent the number of sandwiches, notice that the number of pieces in each whole and the divisor in the expressions represent the number of people sharing the sandwiches.” In Student Work Time, Student Facing states, “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. a. Show or explain how the diagram(s) and expression represent the number of sandwiches being shared. b. Show or explain how the diagram(s) and expression represent the number of people sharing the sandwiches. c. How much sandwich does each person get in the situation?”

• Unit 4, Wrapping Up Multiplication and Division with Multi-Digit Numbers, Section A, Lesson 6, Cool-down, students attend to precision when they apply the standard algorithm for multiplication. Student Facing states, “Use the standard algorithm to find the product $251\times34$.” Activity Narrative (from Activity 2) states, “Because these calculations have new units composed in almost every place value, students will need to locate and use the composed units carefully. It gives students a reason to attend to the features of their calculation and to use language precisely (MP6).”

• Unit 5, Place Value Patterns and Decimal Operations, Section A, Lesson 7, Warm-up, Activity Narrative, Student Work Time, and Activity Synthesis, students attend to precision when working with weight measurements. Activity Narrative states, “The weights on the scale total 12.32 ounces, but the scale reads 12.3 ounces. There are different possible explanations for this discrepancy. For example, the scale might be inaccurate. Or the scale might only give readings in tenths of an ounce. In the discussion, students consider the idea that the value shown on the scale is not always exact. It may just show the closest value that it is capable of reading, which is the nearest tenth of an ounce in this case (MP6).” In Student Work Time, Student Facing states, “What do you notice? What do you wonder?” Activity Synthesis states, “What do you notice about the weights on the scale and the reading of the scale? (They aren’t the same. The weights are 12.32 ounces and the scale says 12.3 ounces.) Why do you think the scale and the weights don’t agree? (The scale could be wrong.) What if the scale only shows tenths of an ounce, and it can’t show hundredths of an ounce? (The value is still not accurate but it’s the best the scale can do.) In today’s lesson we will look at scales that show different numbers of decimals and see how that influences what they show.”

• Unit 6, More Decimal and Fraction Operations, Section A, Lesson 5, Cool-down, students attend to precision when they compare two measurements. Student Facing states, “Jada ran 15.25 kilometers. Han ran 8,500 meters. Who ran farther? How much farther? Explain or show your reasoning.” Activity Narrative (from Activity 1) states, “This gives students an opportunity to think about which units are most helpful for communicating a distance (MP6).”

• Unit 7, Shapes on the Coordinate Plane, Section B, Lesson 5, Cool-down, students use accurate mathematical language to classify quadrilaterals as trapezoids. Student Facing states, “a. When is a quadrilateral also a trapezoid? b. Which of the following shapes are trapezoids? Show or explain your reasoning.” Student Response sample states, “a. A quadrilateral is a trapezoid if it has at least one pair of opposite sides that are parallel. b. All of the shapes except D are trapezoids because they have at least one pair of opposite sides that are parallel.”

The materials reviewed for Open Up Resources K–5 Math 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 across the year, and they are often explicitly identified for teachers within the Course Guide (How to Use These Materials). A chart is provided within this section that highlights several lessons that showcase particular Mathematical Practices. The Mathematical Practices are also identified within specific lessons (Lesson Preparation Narratives and Lesson Activities’ Narratives).

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

• Unit 1, Finding Volume, Section A, Lesson 3, Cool-down, Student Work Time, students look for and make use of structure while they consider the layered structure of a prism to find its volume. Preparation, Activity Narrative states, “In previous lessons, students built objects, including rectangular prisms, with unit cubes and counted the number of cubes. In this lesson, students continue to count the number of unit cubes needed to build a rectangular prism, but now they are presented with images of prisms instead of the objects themselves. To encourage students to develop a systematic way to count the cubes, they are shown prisms made from larger numbers of cubes. As students use horizontal or vertical layers to measure the volume, they make use of the layered structure of prisms (MP7).” In Student Work Time, Student Facing states, “Jada’s prism has 4 layers and each layer has 9 cubes. a. Circle the prism that is Jada's. b. Find the volume of Jada’s prism. Explain or show your reasoning.” Students see four prisms with two layers of 12, four layers of 9, three layers of 9, and three layers of 8.

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

• Unit 7, Shapes on the Coordinate Plane, Lesson 3, Activity 1, Student Work Time and Activity Synthesis, students look for and make use of structure as they plot points on the coordinate grid. In Student Work Time, Student Facing states, “1. Partner A. a. Estimate the location of each point. A (5,1) B (5,2) C (5,3) D (5,4). b. Plot and label the points on the coordinate grid. c. What do the points have in common? d. Plot the point with coordinates (5,0) on the coordinate grid. 2. Partner B. a. Estimate the location of each point. A (4,3) B (5,3) C( 6,3) D (7,3). b. Plot and label the points on the coordinate grid. c. What do the points have in common? d. Plot the point with coordinates (0,3) on the coordinate grid.” Activity Narrative states, “The purpose of this activity is for students to plot several points with the same vertical or horizontal coordinate and observe that they lie on a horizontal or vertical line respectively (MP7). Students also plot points on the axes for the first time. Before plotting the points on a grid with grid lines, students first estimate the location of the points. This encourages them to think about the coordinates as distances (from the vertical axis for the first coordinate and from the horizontal axis for the second coordinate).” Activity Synthesis states, “Ask previously identified students to share their thinking. ‘What can we say about a set of points when they share the same first coordinate?’ (They will be on the same vertical line.) Display image from student solution showing points with first coordinate 5. ‘How did you know where to put the point with coordinates (5,0)?’ (I put it on the horizontal axis. I went over 5 but did not go up at all.) ‘What happens when a set of points share the same second coordinate?’ (They will be on the same horizontal line.) Display image from student solution showing points with second coordinate 3. ‘What does the zero in (0,3) tell us?’ (It means the point will be on line zero of the horizontal axis, which is the vertical axis.) (0,0) is an important point because it's where we start when we plot a point on the coordinate grid. Find (0,0) on the grid you have been working with.”

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

• Unit 4, Wrapping Up Multiplication and Division with Multi-Digit Numbers, Lesson 13, Activity 2, Student Work Time and Activity Synthesis, students use repeated reasoning as they use partial quotients to divide up to four-digit dividends by a two-digit divisor. In Student Work Time, Student Facing states, “Estimate the value of each quotient. Then, use an algorithm using partial quotients to find the value. a. A reasonable estimate for $612\div34$ is: ___. b. A reasonable estimate for $529\div23$ is: ___. c. A reasonable estimate for $1,044\div29$ is: ___.” Activity Narrative states, “Before finding the quotient, students estimate the value of the quotient which both helps students decide which partial quotients to use and helps them evaluate the reasonableness of their solution (MP8).” Activity Synthesis states, “Ask 2 - 3 students to share their work for the same problem that shows different partial quotients. ‘How can you make sure that the whole number quotient you got at the end is reasonable?’ (It should be close to my estimate. I can multiply the quotient and divisor and that should give me the dividend.) If students pair and share with other partners, ask, ‘How did explaining your work to others help you today?’ or ‘What did someone say today that helped you in your understanding of division?’ (I learned that it’s ok to take more steps because I was comfortable with the multiples I used.)”

• Unit 6, More Decimal and Fraction Operations, Section A, Lesson 2, Warm-up, Launch and Activity Synthesis, students use repeated reasoning as they analyze a diagram and make connections to exponents. Activity Narrative states, “When students analyze the diagram and determine how many segments there are of each length, they are observing and making use of the repeated structure of ten segments joining at the different vertices (MP7, MP8).” Launch states, “Groups of 2. ‘How many do you see? How do you see them?’ Display the image. 1 minute: quiet think time.” Student Facing states, “How many do you see? How do you see them?” Activity Synthesis states, “Invite students to share their estimates for how many of the smallest line segments there are in the diagram. ‘How can you find out exactly how many there are?’ (I can count the number of long segments and then the number of medium size segments on one long segment and then the number of tiny segments on one medium size one. Then I multiply those numbers.) Invite students to count and then display the expression: $10\times10\times10$. ‘How does the expression relate to the diagram?’ (It’s the total number of tiny segments.) ‘Another way to write $10\times10\times10$ is $10^3$. This is called a power of ten. The number 3 tells us how many factors of 10 there are, or how many times we multiply 10 to get the number.’”

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

The materials reviewed for Open Up Resources K-5 Math Grade 5 meet expectations for providing teachers 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. 

Within the Course Guide, several sections (Design Principles, A Typical Lesson, How to Use the Materials, and Key Structures in This Course) provide comprehensive guidance that will assist teachers in presenting the student and ancillary materials. Examples include but are not limited to:

• Resources, Course Guide, Design Principles, Learning Mathematics by Doing Mathematics, “A problem-based instructional framework supports teachers in structuring lessons so students are the ones doing the problem solving to learn the mathematics. The activities and routines are designed to give teachers opportunities to see what students already know and what they can notice and figure out before having concepts and procedures explained to them. The teacher has many roles in this framework: listener, facilitator, questioner, synthesizer, and more.”

• Resources, Course Guide, A Typical Lesson, “A typical lesson has four phases: 1. a warm-up; 2. one or more instructional activities; 3. the lesson synthesis; 4. a cool-down.” “A warm-up either: helps students get ready for the day’s lesson, or gives students an opportunity to strengthen their number sense or procedural fluency.” An instructional activity can serve one or many purposes: provide experience with new content or an opportunity to apply mathematics; introduce a new concept and associated language or a new representation; identify and resolve common mistakes; etc. The lesson synthesis “assists the teacher with ways to help students incorporate new insights gained during the activities into their big-picture understanding.” Cool-downs serve “as a brief formative assessment to determine whether students understood the lesson.”

• Resources, Course Guide, How to Use the Materials, “The story of each grade is told in eight or nine units. Each unit has a narrative that describes the mathematical work that will unfold in that unit. Each lesson in the unit also has a narrative. Lesson narratives explain: the mathematical content of the lesson and its place in the learning sequence; the meaning of any new terms introduced in the lesson; how the mathematical practices come into play, as appropriate. Activities within lessons also have narratives, which explain: the mathematical purpose of the activity and its place in the learning sequence, what students are doing during the activity, what the teacher needs to look for while students are working on an activity to orchestrate an effective synthesis, connections to the mathematical practices, when appropriate.”

• Resources, Course Guide, Scope and Sequence lists each of the nine units, a Pacing Guide to plan instruction, and Dependency Diagrams. These Dependency Diagrams show the interconnectedness between lessons and units within Grade 5 and across all grades.

• Resources, Course Guide, Course Glossary provides a visual glossary for teachers that includes both definitions and illustrations. Some images use examples and nonexamples, and all have citations referencing what unit and lesson the definition is from.

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. Examples include:

• Unit 1, Finding Volume, Section A, Lesson 1, Preparation, Math Community guides teachers in developing a Math Community. “Prepare a space, such as a piece of poster paper, titled “Mathematical Community”​ ​and a T-chart with the headers “Doing Math”​ ​and “Norms.” Partition each of the columns into two sections: students and teacher. The two sections encourage the students and teacher to be mindful that both respective parties are responsible for the way math is done in the classroom.”

• Unit 3, Multiplying and Dividing Fractions, Section C, Lesson 19, Preparation, Lesson Narrative, explains how students will apply knowledge and work together to solve problems. “Students work together with expressions involving a unit fraction divided by a whole number and a whole number divided by a unit fraction. In both activities, students write multiplication and division expressions, given specific digits to choose from. In Activity 1, students are applying what they learned to strategically write expressions that represent the greatest product or quotient. In Activity 2, they are trying to write expressions that represent the smallest product or quotient.”

• Unit 6, More Decimal and Fraction Operations, Overview, Throughout this Unit, “The Number Talk routine is used throughout the unit to support students’ developing fluency and to see multiplicative structures present in the base-ten system, adding and subtracting of fractions, and multiplication of fractions. Students use benchmark fractions and equivalent fractions to reason about the value of the expressions.” It then gives a sampling of the Number Talk Warm-ups. It also explains how students will use this routine to reason values of expressions.

The materials reviewed for Open Up Resources K-5 Math 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. 

Unit Overviews and sections within lessons include adult-level explanations and examples of the more complex grade-level concepts. Within the Course Guide, How to Use the Materials states, “Activities within lessons also have narratives, which explain: the mathematical purpose of the activity and its place in the learning sequence, what students are doing during the activity, what the teacher needs to look for while students are working on an activity to orchestrate an effective synthesis, connections to the mathematical practices, when appropriate.” Examples include:

• Unit 2, Fractions as Quotients and Fraction Multiplication, Section B, Lesson 6, Relate Division and Multiplication, Lesson Narrative, “In previous lessons, students interpreted a fraction as division of the numerator by the denominator, and equivalently, as a whole number divided into equal sized pieces. In this lesson, students relate division of two whole numbers to multiplying a whole number by a unit fraction. In the first activity, students are given an opportunity to solve a division problem using any strategy and, in the synthesis, they examine how the solution can be interpreted in terms of multiplication or division.  In the second activity, students continue to explore the relationship between a fraction, a division expression, and a multiplication expression. In grade 4, students multiplied a unit fraction by a whole number and in this lesson they begin to explore how to interpret a whole number multiplied by a unit fraction.”

• Unit 5, Place Value Patterns and Decimal Operations, Section D, Lesson 22, Preparation, Lesson Narrative, “In prior lessons, students represented decimals to the thousandths with diagrams, words, numbers, and expressions. They also added, subtracted and multiplied decimals using place value understanding, properties of operations, and relationships between operations. In this lesson, students begin to work with decimals and division. They divide whole numbers by one tenth and one hundredth and notice and explain patterns they observe. Students apply their understanding of division as ‘how many groups’ to hundredths grids where the entire grid represents one whole. This allows them to visualize how many tenths or hundredths are in one or several wholes while also preparing students to find quotients of more complex decimals in future lessons.” 

• Unit 7, Shapes on the Coordinate Plane, Overview, “In this unit, students learn about the coordinate grid, deepen their knowledge of two-dimensional shapes, and use the coordinate grid to study relationships of pairs of numbers in various situations. Here, students learn about grids that are numbered in two directions. They see that the structure of a coordinate grid allows us to precisely communicate the location of points and shapes. Students also continue to study two-dimensional shapes and their attributes. In grade 3, they classified triangles and quadrilaterals by the presence of right angles and sides of equal length. In grade 4, they learned about angles and parallel and perpendicular lines, which allowed them to further distinguish shapes. In this unit, students use these insights to make sense of the hierarchy of shapes.”

Also within the Course Guide, About These Materials, Further Reading states, “The curriculum team at Open Up Resources has curated some articles that contain adult-level explanations and examples of where concepts lead beyond the indicated grade level. These are recommendations that can be used as resources for study to renew and fortify the knowledge of elementary mathematics teachers and other educators.” Examples include:

• Resources, Course Guide, About These Materials, 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.”

• Resources, Course Guide, About These Materials, 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.”

The materials reviewed for Open Up Resources K-5 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 can be found within different sections of the Course Guide and within the Standards section of each lesson. Examples include:

• Resources, Course Guide, About These Materials, CCSS Progressions Documents, “The Progressions for the Common Core State Standards describe the progression of a topic across grade levels, note key connections among standards, and discuss challenging mathematical concepts. This table provides a mapping of the particular progressions documents that align with each unit in the K–5 materials for further reading.”

• Resources, Course Guide, How to Use the Materials, Noticing and Assessing Student Progress in the Mathematical Practices, The Standards for Mathematical Practices Chart, “The unit-level Mathematical Practice chart is meant to highlight a handful of lessons in each unit that showcase certain Mathematical Practices. Some units, due to their size or the nature of their content, may have fewer predicted chances for students to engage in a particular Mathematical Practice. A dash in the chart indicates that there may not be enough opportunities to reliably look for this Mathematical Practice in the unit. One primary place Mathematical Practice 4 is tagged is the optional modeling lesson at the end of each unit. Aside from these lessons, optional activities and lessons are not included in this chart.”

• Resources, Course Guide, Scope and Sequence, Dependency Diagrams, All Grades Unit Dependency Diagram identifies connections between the units in grades K-5. Additionally, a “Section Dependency Diagram” identifies specific connections within the grade level.

• Resources, Course Guide, Lesson and Standards, provides two tables: a Standards by Lesson table, and a Lessons by Standard table. Teachers can utilize these tables to identify standard/lesson alignment.

• Unit 4, Wrapping Up Multiplication and Division with Multi-Digit Numbers, Section A, Lesson 4, Warm Up, “Addressing 5.NBT.B.5. The purpose of this Number Talk is to highlight the calculations that students will make when they use the standard algorithm. The first three calculations are partial products. The fourth calculation is the sum of the first three and this is the number that is recorded when performing the standard multiplication algorithm to which students will be introduced in this lesson.”

Explanations of the role of specific grade-level mathematics can be found within different sections of the Resources, Course Guide, Unit Overviews, Section Overviews, and Lesson Narratives. Examples include:

• Resources, Course Guide, Scope and Sequence, each Unit provides Unit Learning Goals, for example, “Students find the volume of right rectangular prisms and solid figures composed of two right rectangular prisms.” Additionally, each Unit Section provides Section Learning Goals, “Describe volume as the space taken up by a solid object.”

• Unit 3, Multiplying and Dividing Fractions, Overview, “In this unit, students find the product of two fractions, divide a whole number by a unit fraction, and divide a unit fraction by a whole number. Previously, students made sense of multiplication of a whole number and a fraction in terms of the side lengths and area of a rectangle. Here, they make sense of multiplication of two fractions the same way. Students interpret area diagrams with two unit fractions for their side lengths, then a unit fraction and a non-unit fraction, and then two non-unit fractions.”

• Unit 5, Place Value Patterns and Decimal Operations, Section B, Section Overview, "In this section, students add and subtract decimals to the hundredths. They begin by adding and subtracting in ways that make sense to them, which prompts them to relate the operations to those on whole numbers. It also allows the teacher to take note of the strategies and algorithms they choose, including the standard algorithm and those that use expanded form. Adding and subtracting decimals using the standard algorithm brings up a new question in terms of how the digits should be aligned. To highlight this consideration, students analyze a common error. Before using the standard algorithm, students use place-value reasoning to decide whether sums and differences are reasonable and to ensure that the digits in the numbers are aligned correctly. As they take care to align tenths with tenths and hundredths with hundredths, students practice attending to precision (MP6)."

• Unit 7, Shapes on the Coordinate Plane, Lesson 6, Lesson Narrative, "The purpose of this lesson is for students to first relate squares and rhombuses and then relate rectangles and parallelograms. They see that if a shape is a square then it is also a rhombus and if a shape is a rectangle then it is also a parallelogram. But there are rhombuses that are not squares and there are parallelograms that are not rectangles. Students record these observations on the anchor chart from previous lessons. This gives students a chance to organize the quadrilaterals in a hierarchy and highlight the relationships they see between the properties of the shapes they worked with in this lesson. Students should have access to straight edges, protractors, and patty paper throughout this lesson. When students define shapes and make explicit connections between shapes and categories, they reason abstractly and quantitatively (MP2)."

The materials reviewed for Open Up Resources K-5 Math Grade 5 meet expectations for providing explanations of the instructional approaches of the program and identification of the research-based strategies. 

The materials explain and provide examples of instructional approaches of the program and include and reference research-based strategies. Both the instructional approaches and the research-based strategies are included in the Course Guide under the Resources tab for each unit. Design Principles describe that, “It is our intent to create a problem-based curriculum that fosters the development of mathematics learning communities in classrooms, gives students access to mathematics through a coherent progression, and provides teachers the opportunity to deepen their knowledge of mathematics, student thinking, and their own teaching practice.” Examples include:

• Resources, Course Guide, Design Principles, “In order to design curriculum and professional learning materials that support student and teacher learning, we need to be explicit about the principles that guide our understanding of mathematics teaching and learning. This document outlines how the components of the curriculum are designed to support teaching and learning aligning with this belief.” Principles that guide mathematics teaching and learning include: All Students are Capable Learners of Mathematics, Learning Mathematics by Doing Mathematics, Coherent Progression, Balancing Rigor, Community Building, Instructional Routines, Using the 5 Practices for Orchestrating Productive Discussions, Task Complexity, Purposeful Representations, Teacher Learning Through Curriculum Materials, and Model with Mathematics K-5.

• Resources, Course Guide, Design Principles, Community Building, “Students learn math by doing math both individually and collectively. Community is central to learning and identity development (Vygotsky, 1978) within this collective learning. To support students in developing a productive disposition about mathematics and to help them engage in the mathematical practices, it is important for teachers to start off the school year establishing norms and building a mathematical community. In a mathematical community, all students have the opportunity to express their mathematical ideas and discuss them with others, which encourages collective learning. ‘In culturally responsive pedagogy, the classroom is a critical container for empowering marginalized students. It serves as a space that reflects the values of trust, partnership, and academic mindset that are at its core’ (Hammond, 2015).”

• Resources, Course Guide, 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)”

• Resources, Course Guide, Key Structures in This Course, Student Journal Prompts, Paragraph 3, “Writing can be a useful catalyst in learning mathematics because it not only supplies students with an opportunity to describe their feelings, thinking, and ideas clearly, but it also serves as a means of communicating with other people (Baxter, Woodward, Olson & Robyns, 2002; Liedke & Sales, 2001; NCTM, 2000).”

The materials reviewed for Open Up Resources K-5 Math Grade 5 meet expectations for including a comprehensive list of supplies needed to support the instructional activities.

In the Course Guide, Materials, there is a list of materials needed for each unit and each lesson. Lessons that do not have materials are indicated by none; lessons that need materials have a list of all the materials needed. Examples include:

• Resources, Course Guide, Key Structures in This Course, Representations in the Curriculum, provides images and explanations of representations for the grade level. “Base-ten Blocks (2-5): Base-ten blocks are used after students have had the physical experience of composing and decomposing towers of 10 cubes. The blocks offer students a way to physically represent concepts of place value and operations of whole numbers and decimals. Because the blocks cannot be broken apart, as the connecting cube towers can, students must focus on the unit. As students regroup, or trade, the blocks, they are able to develop a visual representation of the algorithms. The size of relationships among the place value blocks and the continuous nature of the larger blocks allow students to investigate number concepts more deeply. The blocks are used to represent whole numbers and, in grades 4 and 5, decimals, by defining different size blocks as the whole.” 

• Resources, Course Guide, Materials, includes a comprehensive list of materials needed for each unit and lesson. The list includes both materials to gather and hyperlinks to documents to copy. “Unit 2, Lesson 11 - Gather: Number cards 0-10; Copy: How Close? Stage 6 Recording Sheet, How Close? Stage 7 Recording Sheet.”

• Unit 1, Finding Volume, Section A, Lesson 3, Materials Needed, “Activities: Connecting cubes (Activity 1); Centers: Connecting cubes (Can You Build It?, Stage 3), Folders (Can You Build It?, Stage 3), Paper clips (Five in a Row: Multiplication, Stage 3), Two-color counters (Five in a Row: Multiplication, Stage 3).”

The materials reviewed for Open Up Resources K-5 Math 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/series mathematics.

Materials regularly provide strategies, supports, and resources for students in special populations to help them access grade-level mathematics as suggestions are outlined within each lesson. According to the Resources, Course Guide, Universal Design for Learning and Access for Students with Disabilities, “Supplemental instructional strategies that can be used to increase access, reduce barriers and maximize learning are included in each lesson, listed in the activity narratives under ‘Access for Students with Disabilities.’ Each support is aligned to the Universal Design for Learning Guidelines (udlguidelines.cast.org), and based on one of the three principles of UDL, to provide alternative means of engagement, representation, or action and expression. These supports provide teachers with additional ways to adjust the learning environment so that students can access activities, engage in content, and communicate their understanding.” Examples of supports for special populations include: 

• Unit 3, Multiplying and Dividing Fractions, Section B, Lesson 13, Activity 2, Access for Students with Disabilities, “Representation: Perception. Provide access to strips of paper for students to cut and fold. Ask students to identify correspondences between the number of pieces/folds and the fraction they represent. Provides accessibility for: Conceptual Processing, Memory.”

• Unit 5, Place Value Patterns and Decimal Operations, Section D, Lesson 22, Activity 1, Access for Students with Disabilities, “Representation: Comprehension. Begin by asking, “Does this problem/situation remind anyone of something we have seen/read/done before?” Provides accessibility for: Memory, Attention.”

• Unit 7, Shapes on the Coordinate Plane, Section B, Lesson 5, Activity 2, Access for Students with Disabilities, “Engagement: Recruiting Interest. Synthesis: Optimize meaning and value. Invite students to share the meaning of a trapezoid and the similarities and differences in the two definitions of a trapezoid with a classmate who missed the lesson. Provides accessibility for: Conceptual Processing, Language.”

The materials reviewed for Open Up Resources K-5 Math Grade 5 meet expectations for providing extensions and/or opportunities for students to engage with grade-level/course-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 where problems are labeled as “Exploration” at the end of practice problem sets within sections, where appropriate. According to the Resources, Course Guide, How To Use The Materials, 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 directly related to the material of the unit that students can do 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 Fractions, Section B, Practice Problems, Problem 9 (Exploration), “It takes Earth 1 year to go around the Sun. a. 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. b. During the time it takes Earth to go around the Sun, Saturn goes 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 5, Place Value Patterns and Decimal Operations, Section B, Practice Problems, Problem 8 (Exploration), “Lin is trying to use the digits 1, 3, 4, 2, 5, and 6 to make 2 two-digit decimals whose sum is equal to 1. a. Explain why Lin can not make 1 by adding together 2 two-digit decimal numbers made with these digits. b. What is the closest Lin can get to 1? Explain how you know.”

• Unit 7, Shapes on the Coordinate Plane, Section C, Practice Problems, Problem 7 (Exploration), “Andre starts from 2 and counts by 6s. Clare starts at 1,000 and counts back by 7s. a. List the first 6 numbers Andre and Clare say. b. Do Andre and Clare ever say the same number in the same spot on their lists? Explain or show your reasoning.”

The materials reviewed for Open Up Resources K-5 Math 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 to 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 Resources, Course Guide, Mathematical Language Development and Access for English Learners, “In an effort to advance the mathematics and language learning of all students, the materials purposefully engage students in sense-making and using language to negotiate meaning with their peers. To support students who are learning English in their development of language, this curriculum includes instruction devoted to fostering language development alongside mathematics learning, fostering language-rich environments where there is space for all students to participate.” Examples include:

• Unit 3, Multiplying and Dividing Fractions, Section A, Lesson 4, Activity 1, “Access for English Learners - Speaking, Conversing, Representing: MLR8 Discussion Supports. Synthesis: At the appropriate time, give groups 2–3 minutes to plan what they will say when they present to the class. ‘Practice what you will say when you share your drawing with the class. Talk about what is important to say, and decide who will share each part.’”

• Unit 4, Wrapping Up Multiplication and Division with Multi-Digit Numbers, Section A, Lesson 9, Activity 2, "Access for English Learners - Writing, Speaking, Listening: MLR1 Stronger and Clearer. 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."

• Unit 6, More Decimal And Fraction Operations, Section A, Lesson 2, Activity 1, “Access for English Learners - Conversing, Reading: MLR2 Collect and Display. Circulate, listen for and collect the language students use as they use exponential notation to represent large numbers. On a visible display, record words and phrases such as: million, thousands, billion, powers of 10, exponential notation, represent, times, multiply by ten, number of zeros. Invite students to borrow language from the display as needed, and update it throughout the lesson.”

The materials reviewed for Open Up Resources K-5 Math 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.

Suggestions and/or links to manipulatives are consistently included within materials to support the understanding of grade-level math concepts. Examples include:

• Unit 3, Multiplying and Dividing Fractions, Section A, Lesson 9, students use paper, rulers, markers, crayons or colored pencils to use the principles of flag design from the North American Vexillological Association to design their own flag. “The purpose of this activity is for students to make their own flags and analyze them. Students will use their experience with multiplying fractions to answer area questions related to their flag. Some students may include non-rectangular designs. Encourage them to relate the area of their shape to a rectangle and estimate.” Launch “Give each student white paper. ‘Use the design principles we discussed in the last activity to make your own flag. As you make the design, think about the meaning of each symbol and color you use.’ Student Work Time “15 minutes: independent work time. 5 minutes: partner discussion. a.) Design your flag; b.) Imagine you are making your flag with fabric. About how much of each color fabric will you need in square inches? c.) Switch flags with a partner. Describe the meaning of each symbol and color you used ; d.) How do you see each of the design principles in your partner’s flag?”

• Unit 4, Wrapping Up Multiplication and Division with Multi-Digit Numbers, Section C, Lesson 19, Activity 1, “Groups of 2. Lay a meter stick on the ground. ‘Tyler walked from his classroom to the cafeteria. He said, ‘I think that’s about a kilometer.’ Do you agree with Tyler?’ (No, a kilometer is a long distance, it’s 1,000 meters, and it is not that far from a classroom to the cafeteria.). 1 minute: quiet think time. 1–2 minutes: partner discussion. Give students access to meter sticks.”

• Unit 6, More Decimal and Fraction Operations, Section A, Lesson 7, Activity 1, “Groups of 2. Give each group of students one set of pre-cut cards. Display a yardstick. ‘What do you notice? What do you wonder?’ (It shows feet and inches. It shows 36 inches. I wonder if it’s the same length as a meter stick.). ‘In this activity, you will sort some cards into categories of your choosing. When you sort the measurements, you should work with your partner to come up with categories.’ 4 minutes: partner work time. Select groups to share their categories and how they sorted their cards. Choose as many different types of categories as time allows, but ensure that one set of categories identifies the way the quantity is written (whole number, mixed number, fraction). ‘Now work with your partner to match the cards with equal measurements. Then, list the groups of matching measurements in increasing order.’ 3 minutes: partner work time.”