5th Grade - Gateway 2

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

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

• Unit 1, Finding Volume, Lesson 1, Activity 1, 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, Lesson 5, Warm-up, 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, Lesson 10, Activity 1, students develop conceptual understanding as they practice adding fractions with unlike denominators and reason about how the size of the numerators and denominators impact the value of a fraction. Student 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 IM K-5 Math Teacher Guide, materials were designed to include opportunities for students to independently demonstrate conceptual understanding, when appropriate. Design Principles, Coherent Progress states, “Each activity starts with a launch that gives all students access to the task. This is followed by independent work time that allows them to grapple with problems individually before working in small groups. The activity ends with a synthesis to ensure students have an opportunity to consolidate their learning by making connections between their work and the mathematical goals.” A Typical IM Lesson states, “The cool-down task is to be given to students at the end of the lesson. Students are meant to work on the cool-down for about 5 minutes independently and turn it in. The cool-down serves as a brief formative assessment to determine whether students understood the lesson. Students’ responses to the cool-down can be used to make adjustments to further instruction.” Examples include:

• Unit 1, Finding Volume, 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, Lesson 12, Activity 2, 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, Lesson 20, Activity 2, students demonstrate conceptual 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 Kendall Hunt's Illustrative Mathematics Grade 5 meet expectations for giving attention throughout the year to individual standards that set an expectation of procedural skill and fluency. 

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

• Unit 4, Wrapping Up Multiplication and Division with Multi–Digit Numbers, Lesson 4, Warm-Up, students develop procedural skill and fluency as they notice the patterns in calculations within the number talk, leading towards the standard algorithm. Student 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, Lesson 9, Warm-up, 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, Lesson 2, Activity 1, 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.’” Student Facing states, “1. Find the value of the product. 650\times27. 2. Below is Kiran’s work finding the value of the product 650\times27. Is his answer reasonable? Explain your reasoning. 3. What parts of the work do you agree with? Be prepared to explain your reasoning. 4. What parts of the work do you disagree with? Be prepared to explain your reasoning. 5. Look at your solution to problem 1. Is there anything you want to revise? Be prepared to explain.” (5.NBT.5) 

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

• Unit 4, Wrapping Up Multiplication and Division with Multi-Digit Numbers, 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, 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. 1. 0.06\times10=___. 2. 60=___$$\times0.6$$. 3. ___$$= 6\div100$$.” (5.NBT.A)

• Unit 8, Putting It All Together, Lesson 1, Activity 2, students demonstrate procedural skill and fluency as they practice using the standard algorithm to find products. Student Facing states, “1. 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.’ 5-7 minutes: work time.” (5.NBT.5)

The materials reviewed for Kendall Hunt's Illustrative Mathematics Grade 5 meet expectations for being designed so that teachers and students spend sufficient time working with engaging applications of the mathematics. According to IM Curriculum, Design Principles, Balancing Rigor, “Access to new mathematics and problems prompts students to apply their conceptual understanding and procedural fluency to novel situations.” Multiple routine and non-routine applications of the mathematics are included throughout the grade level and these single- and multi-step application problems are included within Activities or Cool-downs. 

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

Examples of routine applications of the math include:

• Unit 3, Multiplying and Dividing Fractions, Lesson 18, Activity 1, students work with real-world problems involving multiplication and division of fractions. 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, 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 a. Han cut 5 feet of ribbon into pieces that are \frac{1}{4} foot long. How many pieces are there? b. 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, Lesson 12, Cool-Down, students solve real-world problems that involve adding and subtracting fractions with unlike denominators. Student facing states, “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. 2. 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, Lesson 5, Activity 3, students interpret equations in order to match information given about rectangular prisms. Activity 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. 1. 3 is the answer. What is the question? 2. 5 is the answer. What is the question? 3. 3\times4=12. The answer is 12. What is the question? 4. 12\times5=60. The answer is 60 cubes. What is the question? 5. 3 by 4 by 5 is the answer. What is the question?” (5.MD.5b)

• Unit 5, Place Value Patterns and Decimal Operations, Lesson 26, Activity 2, students solve a  real-world problem including operations with numbers in decimal form. 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: 1. Choose 3–5 types of books you want to order. 2. Decide on the mark-up price for each type of book you chose. 3. Estimate the amount of money your school will raise as a profit with your book sale. Record an estimate that is: too low, about right, too high. 4. Show or explain your reasoning for the estimate. Include the assumptions you made.” (5.NBT.7)

• Unit 7, Shapes on the Coordinate Plane, Lesson 13, Activity 1, students plot points that represent the length and width of a rectangle with a given perimeter. Activity states, “2 minutes: independent think time. 5 minutes: partner work time.” Student Facing states, “1. 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. 2. Plot the length and width of each rectangle on the coordinate grid. 3. If Jada drew a square, how long and wide was it? 4. If Jada’s rectangle was 2.5 cm long, how wide was it? Plot this point on the coordinate grid. 5. 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 Kendall Hunt's Illustrative Mathematics Grade 5 meet expectations in that the three aspects of rigor are not always treated together and are not always treated separately. There is a balance of the three aspects of rigor within the grade.

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, 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, “1. Which of these expressions does not represent the volume of the rectangular prism in cubic units? Explain or show your reasoning. 2. 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, Lesson 5, Activity 2, students develop procedural skill and fluency as they use the standard algorithm to multiply three-digit numbers by two-digit numbers. 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, Lesson 12, Activity 1, students apply their understanding of the coordinate plane as they interpret data about a series of coin flips. 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. 1. How many heads did Jada get? How many tails did Jada get? Explain or show how you know. 2. How many heads did Han get? How many tails did Han get? Explain or show how you know. 3. 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. 4. 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. 5. Do any of the points you plotted lie on the horizontal axis? What would a point on the horizontal axis mean in this situation? 6. 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, 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, “1. Write a multiplication expression to represent the area of the shaded region. 2. 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, Lesson 12, Activity 2, students develop all three aspects of rigor, conceptual understanding, procedural skill and fluency, and application, as they solve multi-step problems involving the addition and subtraction of fractions. Student 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 1\frac{1}2} miles and took a rest. Then they hiked another \frac{4}{10} mile before stopping for lunch. How many miles have they hiked so far? If the trail they are hiking is a total of 2\frac{1}{2} miles, how much farther do they have to hike? Explain or show your reasoning. 2. Discuss the problems and solutions with your partner. What is the same about your strategies and solutions? What is different? 3. Revise your work if necessary.” (5.NF.2)

• Unit 7, Shapes on the Coordinate Plane, Lesson 6, Activity 1, students use conceptual understanding and application to construct quadrilaterals and explain their attributes. Student Facing states, “1. Build a square with your toothpicks. How do you know it is a square? 2. Use the same four toothpicks to build this shape. (a parallelogram is shown) What stayed the same? What changed? 3. Build a rectangle with six toothpicks. How do you know it is a rectangle? 4. 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 Kendall Hunt's Illustrative Mathematics Grade 5 meet expectations for supporting the intentional development of MP1: Make sense of problems and persevere in solving them; and MP2: Reason abstractly and quantitatively, for students, in connection to the grade-level content standards, as expected by the mathematical practice standards. Students have opportunities to engage with the Math Practices across the year and they are often explicitly identified for teachers within the Course Guide (How to Use These Materials) and within specific lessons (Instructional Routines, Lesson Preparation Narratives, and Lesson Activities’ Narratives).

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

• Unit 1, Finding Volume, Lesson 5, Warm-up, students reason about the attributes of a rectangular prism. 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.” 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, 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. 1. Write a division expression to represent the situation. 2. How many groups of 8 will there be? Explain or show your thinking.”

• Unit 8, Putting It All Together, Lesson 8, Activity 2, students make sense of problems as they reason about multiplication and division. 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.” 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, Lesson 7, Activity 2, students reason abstractly and quantitatively as they match expressions and diagrams. Student Facing states, “Han, Lin, Kiran, and Jada together ran a 3 mile relay race. (They each ran the same distance.) 1. ”Find the expressions and diagrams that match this situation. Be prepared to explain your reasoning. 2. How far did each person run?” 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, Lesson 14, Activity 2, students reason about information presented in a line plot. Student Facing states, “1. Here are the weights of some eggs, in ounces. Use them 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} 2. Jada said that \frac{1}{4} of the eggs weigh 1\frac{7}{8} ounces. Do you agree? Explain or show your reasoning. 3. How much heavier is the heaviest egg than the lightest egg? Explain or show your reasoning.” 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, Lesson 10, Cool-down, students think abstractly as they determine rules for given patterns. 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, “1. 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. 2. 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 Kendall Hunt's Illustrative Mathematics Grade 5 meet expectations for supporting the intentional development of MP3: Construct viable arguments and critique the reasoning of others, for students, in connection to the grade-level content standards, as expected by the mathematical practice standards. 

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

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

• Unit 2, Fractions as Quotients and Fraction Multiplication, 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, Lesson 1, Activity 1, students construct a viable argument as they reason about appropriate estimates for multi-digit multiplication calculations. 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).” Activity 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.” Student Facing states, “Which estimate for the product 18\times149 is most reasonable? Explain or show your reasoning. A. 2,000  B. 4,000  C. 3,000  D. 1,500 2. Are any of the estimates unreasonable? Explain or show your reasoning. 3. 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, Lesson 5, Activity 1, students construct an argument and critique the reasoning of others as they defend a strategy to solve a division problem. 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).” Activity 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.” Student Facing states, “1. Find the value of the quotient. 6773\div13$ 2. 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. 3. What parts of the work do you agree with? Be prepared to explain your reasoning. 4. What parts of the work do you disagree with? Be prepared to explain your reasoning. 5. 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, Lesson 7, Activity 1, students construct arguments and critique the reasoning of others as they reason about the volume measurements of different items. 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).” Activity 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.” 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, Lesson 3, Activity 1, students critique the reasoning of others when working with fraction division. 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).” Activity 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.”  Student Facing states, “Three dancers share 2 liters of water. How much water does each dancer get? Write a division equation to represent the situation. 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, Lesson 15, Activity 3, 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. Student Facing states, “1. Find the value of 622.35-71.4 Explain or show your reasoning. 2. Elena and Andre found the value of 622.35-71.4. Who do you agree with? Explain or show your reasoning.” 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 Kendall Hunt's Illustrative Mathematics 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) and within specific lessons (Instructional Routines, Lesson Preparation Narratives, and Lesson Activities’ Narratives).

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

• Unit 2, Fractions as Quotients and Fraction Multiplication, Lesson 17, Activity 1, students model with mathematics as they use the context of art to multiply fractions. 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).” Student Facing states, “1. 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. 2. What is the area of one of your rectangles? Show your reasoning. 3. 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, 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, “1. 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, students model with math by selecting the appropriate unit of measure as well as what the estimate means within the real-world situation. Narrative 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).” Activity 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.”  Student Facing states, “1. What measurements would you take of the wagon to accurately estimate its volume? 2. What units would you use to measure the wagon? Explain your reasoning. 3. Record an estimate for the volume of the wagon that is: too low, about right, too high. 4. 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, Lesson 9, Activity 1, students choose an appropriate strategy to find the area of a shaded region. Narrative 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.)” 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, Lesson 14, Activity 1, students use appropriate tools strategically to subtract decimals. Student Facing states, “Find the value of 2.26-1.32. Explain or show your reasoning.” Lesson 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, 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. 1. \frac{5}{6}-\frac{1}{3}. 2. \frac{3}{4}+\frac{1}{2}.” Lesson Narrative states, “Students 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 Kendall Hunt's Illustrative Mathematics Grade 5 meet expectations for supporting the intentional development of MP6: Attend to precision; and attend to the specialized language of mathematics, for students, in connection to the grade-level content standards, as expected by the mathematical practice standards. 

Students have many opportunities to attend to precision and 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, Lesson 5, Activity 2, students use precise language as they complete a table showing multiplication expressions for the volume of prisms. 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.” Lesson Narrative 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, Lesson 2, Activity 2, students attend to precision of language when connecting mathematical representations for a real world problem. 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).” Activity 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.” 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. 1. Show or explain how the diagram(s) and expression represent the number of sandwiches being shared. 2. Show or explain how the diagram(s) and expression represent the number of people sharing the sandwiches. 3. How much sandwich does each person get in the situation?”

• Unit 4, Wrapping Up Multiplication and Division with Multi-Digit Numbers, Lesson 6, Cool-down, students attend to precision when they apply the standard algorithm for multiplication. Student Facing states, “Use the standard algorithm to find the product 251\times34.” Lesson Narrative 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, Lesson 7, Warm-up, students attend to precision when working with weight measurements. 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).” 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, 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.” Narrative 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, Lesson 5, Cool-down, students use accurate mathematical language to classify quadrilaterals as trapezoids. Student Facing states, “1. When is a quadrilateral also a trapezoid?  2. Which of the following shapes are trapezoids? Show or explain your reasoning.” Student Response sample states, “1. A quadrilateral is a trapezoid if it has at least one pair of opposite sides that are parallel. 2. 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 Kendall Hunt's Illustrative Mathematics Grade 5 meet expectations for supporting the intentional development of MP7: Look for and make use of structure; and MP8: Look for and express regularity in repeated reasoning, for students, in connection to grade-level content standards, as expected by the mathematical practice standards. Students have opportunities to engage with the Math Practices across the year and they are often explicitly identified for teachers within the Course Guide (How to Use These Materials) and within specific lessons (Instructional Routines, Lesson Preparation Narratives, and Lesson Activities’ Narratives).

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

• Unit 1, Finding Volume, Lesson 3, Cool-down, students look for and make use of structure while they consider the layered structure of a prism to find its volume. Lesson 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).” Student Facing states, “Jada’s prism has 4 layers and each layer has 9 cubes. 1. Circle the prism that is Jada's. 2. 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, Lesson 2, Warm-up, students look for and make use of structure as they use a hundreds grid to estimate a shaded region. 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.” 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, students look for and make use of structure as they plot points on the coordinate grid. Student Facing states, “Partner A. 1. Estimate the location of each point. A(5,1) B(5,2) C(5,3) D(5,4). 2. Plot and label the points on the coordinate grid. 3. What do the points have in common? 4. Plot the point with coordinates (5,0) on the coordinate grid. Partner B. 1. Estimate the location of each point. A(4,3) B(5,3) C(6,3) D(7,3). 2. Plot and label the points on the coordinate grid. 3. What do the points have in common? 4. Plot the point with coordinates (0,3) on the coordinate grid.” Lesson 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, students use repeated reasoning as they use partial quotients to divide up to four-digit dividends by a two-digit divisor. Student Facing states, “Estimate the value of each quotient. Then, use an algorithm using partial quotients to find the value. 1. A reasonable estimate for 612\div34 is: ___. 2. A reasonable estimate for 529\div23 is: ___. 3. A reasonable estimate for 1,044\div29 is: ___.” 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, Lesson 2, Warm-up, students use repeated reasoning as they analyze a diagram and make connections to exponents. 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 10^310^3. 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, 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. Jada’s rule: Start with 0 and keep adding 5. Priya’s rule: Start with 0 and keep adding 10. 2. What number will be in Priya’s pattern when Jada’s pattern has 100? 3. What relationship do you notice between corresponding numbers in the two patterns?” Activity 2 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.