The materials reviewed for Open Up Resources K–5 Math Grade 4 meet expectations for developing conceptual understanding of key mathematical concepts, especially where called for in specific content standards or cluster headings.

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

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

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

According to the Grade 4 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 4, From Hundredths to Hundred-thousands, Section B, Lesson 11, Cool-down, students demonstrate conceptual understanding of place value to locate large numbers on a number line. Student Facing states, “a. Estimate the location of 28,500 on the number line and label it with a point. b. Which point - A, B, or C - could represent a number that is 10 times as much as 28,500? Explain your reasoning.” For the problem, an image of a number line is shown with A,B,C on the number line from 0 to 400,000. (4.NBT.1) 

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

• Unit 6, Multiplying and Dividing Multi-digit Numbers, Section B, Lesson 7, Activity 1, Student Work Time, students demonstrate conceptual understanding as they use rectangular diagrams to represent multiplication of three-digit and one-digit numbers. Student Facing states, “1. Clare drew this diagram. a. What multiplication expression can be represented by the diagram? b. Find the value of the expression. Show your reasoning.” A rectangle that is partitioned is shown. (4.NBT.5)

The materials reviewed for Open Up Resources K–5 Math Grade 4 meet expectations for giving attention throughout the year to individual standards that set an expectation of procedural skill and fluency. 

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

• Unit 6, Multiplying and Dividing Multi-digit Numbers, Section C, Lesson 19, Activity 1, students develop procedural skill and fluency as they use partial quotients and interpret remainders. Student Work Time states, “3 minutes: independent work time on the first 2 problems. Pause after problem 2 to discuss students’ responses. Display the different ways that students decompose 389 to divide it by 7. ‘Most other calculations we’ve seen so far end with a 0, but this one ends with a 4. What does the 4 tell us?’ (We cannot make a group of 7 with 4 leftover. 389 is not a multiple of 7, and there are leftovers.) ‘When we divide and end up with leftovers, we call them remainders, because they represent what is remaining after we divide into equal groups.’ Display: $389=7\times55+4$ ‘How does this equation show that $389\div7$ has a remainder?’ (It shows that 389 is not a multiple of 7. It also shows that 7 and 55 make a factor pair for 385, and 389 is 4 more than that.) 3 minutes: independent work time on the last 2 problems. As students work on the last two problems monitor for students who: start with the largest multiple of 3 and 10 within 702 that they can think of to decompose the dividend (690, 600), use the fewest steps to find the quotient.” Student Facing states, “Jada used partial quotients to find out how many groups of 7 are in 389. Analyze Jada’s steps in the algorithm. (A vertical representation of partial quotients is shown.)  a. Look at the three numbers above 389. What do they represent? b. Look at the three subtractions below 389. What do they represent? c. What is another way you can decompose 389 to divide by 7? d. Is 389 a multiple of 7? Explain your reasoning. e. Use an algorithm that uses partial quotients to find out how many groups of 3 are in 702. f. Is 702 a multiple of 3? Explain your reasoning.” (4.NBT.6)

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

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

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

• Unit 9, Putting It All Together, Section B, Lesson 4, Activity 1, Student Work Time, students demonstrate procedural skill and fluency as they subtract multi-digit numbers. Student Work Time states, “6–8 minutes: independent work time. 3-4 minutes: partner discussion” Student Facing states, “1. Find the value of each difference. a. $700-16$. b. $7,000-16$. c. $70,000-16$. d. $700,000-16$.” (4.NBT.4)

The materials reviewed for Open Up Resources K–5 Math Grade 4 meet expectations for being designed so that teachers and students spend sufficient time working with engaging applications of mathematics. 

According to the Grade 4 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 4 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, Extending Operations to Fractions, Section A, Lesson 6, Cool-down, students apply their understanding about multiplication of a fraction by a whole number to solve real-world problems. Student Facing states, “a. Tyler bought 5 cartons of milk. Each carton contains $\frac{3}{4}$ liter. How many liters of milk did Tyler buy? Explain or show your reasoning. b. Han bought 3 cartons of chocolate milk. Each carton contains $\frac{5}{8}$ liter. Did Han buy the same amount of milk as Tyler? Explain or show your reasoning.” (4.NF.4c)

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

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

Examples of non-routine applications of the math include:

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

• Unit 5, Multiplicative Comparison and Measurement, Section B, Lesson 10, Activity 2, Student Work Time, students solve multi-step problems by using metric units of measurement and multiplicative comparison. In Student Work Time, Student Facing states, “Here are six water bottle sizes and four clues about the amount of water they each hold. One bottle holds 350 mL. A bottle in size B holds 5 times as much water as the bottle that holds 1 L. The largest bottle holds 20 times the amount of water in the smallest bottle. One bottle holds 1,500 mL, which is 3 times as much water as a bottle in size E. Use the clues to find out the amount of water, in mL, that each bottle size holds. Be prepared to explain or show your reasoning.” (4.MD.2, 4.OA.2, 4.OA.3)

• Unit 7, Angles and Angle Measurement, Section C, Lesson 16, Activity 2, Student Work Time, students use their understanding of geometric figures and measurements to draw, describe, and identify two-dimensional figures. Student Work Time states, “5 minutes: independent work time. 8–10 minutes: partner work time. Monitor for diagrams that reflect a variety of geometric features. Monitor for students who consider both geometric features and measurement in the description.” Student Facing states, “a. Create a two-dimensional shape that has at least 3 of the following: 1. ray, 2. line segment, 3. right angle, 4. acute angle, 5. obtuse angle, 6. perpendicular lines, 7. parallel lines. b. Without showing your partner, describe the figure so that your partner is able to draw it as best as possible. c. Switch roles, and draw your partner’s shape based on their description.” (4.G.1, 4.G.2)

The materials reviewed for Open Up Resources K–5 Math Grade 4 meet expectations in that the three aspects of rigor are not always treated together and are not always treated separately. There is a balance of the three aspects of rigor within the grade.

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

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

• Unit 3, Extending Operations to Fractions, Section B, Lesson 11, Activity 1, Student Work Time, students solve routine real-world problems that involve subtracting mixed numbers where it is necessary to decompose one or both numbers. In Student Work Time, Student Facing states, “Clare, Elena, and Andre are making macramé friendship bracelets. They’d like their bracelets to be $9\frac{4}{8}$ inches long. For each question, explain or show your reasoning. a. Clare started her bracelet first and has only $\frac{7}{8}$ inch left until she finishes it. How long is her bracelet so far? b. So far, Elena’s bracelet is $5\frac{1}{8}$ inches long and Andre’s is $3\frac{5}{8}$ inches long. How many more inches do they each need to reach $9\frac{4}{8}$inches? c. How much longer is Elena’s bracelet than Andre’s at the moment?” (4.NF.3d)

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

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

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

• Unit 3, Extending Operations to Fractions, Section A, Lesson 1, Activity 1, Student Work Time, students engage in all three aspects of rigor, conceptual understanding, procedural fluency, and application as they interpret situations involving equal groups. Student Work Time states, “‘Take a few quiet minutes to think about the first set of problems about crackers. Then, discuss your thinking with your partner.’ 4 minutes: independent work time. 2 minutes: partner discussion. Pause for a whole-class discussion. Invite students to share their responses. If no students mention that there are equal groups, ask them to make some observations about the size of the groups in each image. Discuss the expressions students wrote: ‘What expression did you write to represent the crackers in Image A? Why? ($6\times4$, because there are 6 groups of 4 full crackers.) What about the crackers in Image B? Why? ($6\times\frac{1}{4}$, because there are 6 groups of $\frac{1}{4}$ of a cracker.)’ Ask students to complete the remaining problems. 5 minutes: independent or partner work time. Monitor for students who reason about the quantities in terms of ‘___ groups of ___’ to help them write expressions.” Student Facing states, “Here are images of some crackers. a. How are the crackers in image A like those in B? b. How are they different? c. How many crackers are in each image? d. Write an expression to represent the crackers in each image. 2. Here are more images and descriptions of food items. For each, write a multiplication expression to represent the quantity. Then, answer the question. a. Clare has 3 baskets. She put 4 eggs into each basket. How many eggs did she put in baskets? b. Diego has 5 plates. He put $\frac{1}{2}$ of a kiwi fruit on each plate. How many kiwis did he put on plates? c. Priya prepared 7 plates with $\frac{1}{8}$ of a pie on each. How much pie did she put on plates? d. Noah scooped $\frac{1}{3}$ cup of brown rice 8 times. How many cups of brown rice did he scoop?” (4.NF.4)

• Unit 7, Angles and Angle Measurement, Section B, Lesson 10, Activity 1, Launch and Student Work Time, students develop conceptual understanding alongside procedural skill and fluency as they use a protractor to measure angles and understand perpendicular lines. Launch states, “Groups of 2, Give each student a protractor and access to rulers or straightedges.” Student Work Time states, “5 minutes: independent work time, 1–2 minutes: partner discussion Monitor for students who: align the rays of the angle to tick marks on the protractor and count from one ray to the other, don’t align either ray of an angle to $0\degree$ or $180\degree$ on the protractor and instead find the difference of the numbers where the two rays land on the protractor, always align one ray of an angle with the $0\degree$ or $180\degree$ line on the protractor and always read from the scale that starts with $0\degree$” Student Facing, “Problem 1, Use a protractor to find the value of each angle measurement in degrees.”(4.G.1, 4.MD.6)

The materials reviewed for Open Up Resources K–5 Math Grade 4 meet expectations for supporting the intentional development of MP1: Make sense of problems and persevere in solving them; and MP2: Reason abstractly and quantitatively, for students, in connection to the grade-level content standards, as expected by the mathematical practice standards. 

Students have opportunities to engage with the Math Practices across the year, and they are often explicitly identified for teachers within the Course Guide (How to Use These Materials). 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 2, Fraction Equivalence and Comparison, Section C, Lesson 14, Activity 1, Student Work Time, students reason about fractions given descriptive clues. Preparation, Lesson Narrative states, “In the first activity, students compare sets of fractions with like and unlike denominators. They do so by using benchmarks, writing equivalent fractions, or reasoning about the numerators and denominators. In the second activity, students interpret and solve problems involving fractional measurements in context. Both activities present a new setup, structure, or context, requiring students to make sense of the given information and the problems, and to persevere in solving them (MP1).” In Student Work Time, Student Facing states, “Six friends are each given a list of 5 fractions. They each chose one fraction quietly and wrote clues about their choice. Use their clues to identify the fractions they chose.”

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

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

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

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

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

• Unit 9, Putting It All Together, Section C, Lesson 8, Cool-down, students solve multi-step problems involving all operations. Preparation, Lesson Narrative states, “In the previous lesson, students solved word problems involving multiplicative comparison. In this lesson, they practice solving a wider variety of problems, with a focus on the relationships among multiple quantities in a situation. Students think about how to represent the relationships with one or multiple equations and using multiple operations. They also interpret their solutions and the solutions of others in context, including interpreting remainders in situations that involve division (MP2).” Student Facing states, “In one week, a train made 8 round trips between its home station and Union Station. At the end of the week, it traveled a few more miles from the home station to a repair center. That week, the train traveled a total of 1,564 miles. a. Which statement is true for this situation? Explain or show your reasoning. A. The distance traveled for each round trip is 200 miles. The distance to the repair station is 26 miles. B. The distance traveled for each round trip is 195 miles. The distance to the repair station is 4 miles. C. The distance traveled for each round trip is 8 miles. The distance to the repair station is 1,500 miles. D. The distance traveled for each round trip is 193 miles. The distance to the repair station is 8 miles. b. Explain why one of the choices could not be true.”

The materials reviewed for Open Up Resources K–5 Math Grade 4 meet expectations for supporting the intentional development of MP3: Construct viable arguments and critique the reasoning of others, for students, in connection to the grade-level content standards, as expected by the mathematical practice standards. 

Students have opportunities to engage with 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 4 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, Fraction Equivalence and Comparison, Section B, Lesson 9, Cool-down, students construct viable arguments as they determine if two fractions are equivalent. Student Facing states, “Problem b, Diego wrote $\frac{11}{5}$ and $\frac{55}{10}$ as equivalent fractions. Are those fractions equivalent? Explain or show how you know. Use a number line, if it helps.”

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

• Unit 9, Putting It All Together, Section C, Lesson 8, Activity 1, Student Work Time and Activity Synthesis, students construct an argument and critique the reasoning of others as they interpret a problem involving equal groups. In Student Work Time, Student Facing states, “Here are three situations. Which ones are true? Which ones are not true? Show how you know. Situation A: A high-rise building has 53 stories. The first floor is 17 feet tall, but all other stories are each 11 feet tall. The building is 589 feet tall. Situation B: A window washer has 600 seconds to wash 17 windows of a building. It takes 54 seconds to wash each window. The washer will finish washing all the windows and have 11 seconds to spare. Situation C: Eleven students set a goal to raise at least $600 for charity. Each student raised $17 each day. After 3 days of fundraising, the group will still be short by $54.” Activity Synthesis states, “Select students to share their responses and reasoning. Record the expressions or equations they wrote to represent the situations. Highlight different ways of representing the same situation. ‘For which situations did you need to find the actual values in order to tell if they were true or not true? Why is that?’ (I tried to estimate on Situation A and I knew it would be close, so I did the multiplication and added to find out if the total height was really 610 feet. I needed to write equations to make sense of Situation C. I just did the multiplication and subtraction to check and see if it could really be 54.) ‘For which stories was it possible to tell by estimation and mental math?’ (I could do some estimation for all of them, but for Situation B I could tell that even if he washed 10 windows the window washer would almost be out of time and couldn’t do 17.)”

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

• Unit 3, Extending Operations to Fractions, Section A, Lesson 3, Activity 2, Student Work Time and Activity Narrative, students construct a viable argument and critique the reasoning of others when they identify patterns using multiplication. Student Work Time states, “3 minutes: independent work time on the first set of problems. 2 minutes: group discussion. Select students to explain how they reasoned about the missing numbers in the equations.” In Student Work Time, Student Facing states, “2. Your teacher will give you a sheet of paper. Work with your group of 3 and complete these steps on the paper. After each step, pass your paper to your right. Step 1: Write a fraction with a numerator other than 1 and a denominator no greater than 12. Step 2: Write the fraction you received as a product of a whole number and a unit fraction. Step 3: Draw a diagram to represent the expression you just received. Step 4: Collect your original paper. If you think the work is correct, explain why the expression and the diagram both represent the fraction that you wrote. If not, discuss what revisions are needed.” Activity Narrative states, “As students discuss and justify their decisions they create viable arguments and critique one another’s reasoning (MP3).”

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

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

The materials reviewed for Open Up Resources K–5 Math Grade 4 meet expectations for supporting the intentional development of MP4: Model with mathematics; and MP5: Choose tools strategically, for students, in connection to the grade-level content standards, as expected by the mathematical practice standards. 

Students have opportunities to engage with the Math Practices across the year, and they are often explicitly identified for teachers within the Course Guide (How to Use These Materials). 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 1, Factors and Multiples, Section B, Lesson 8, Activity 1, Student Work Time, students model with mathematics as they use art and concepts of area. Preparation, Lesson Narrative states, “When students isolate and describe the mathematical elements in art and adhere to mathematical constraints to create art, they model with mathematics (MP4).” In Student Work Time, Student Facing states, “Create an outline for art in the Mondrian style, starting with an 18-by-24 grid. Your artwork should: be partitioned into at least 12 rectangles, include two different rectangles that have the same area, include at least one rectangle whose area is a prime number. Try at least one of these challenges. Make a design where: all but two of the rectangles have a prime number for its area, no two rectangles share a side entirely.”

• Unit 4, From Hundredths to Hundred-Thousands, Section D, Lesson 23, Activity 1, Student Work Time, students model mathematical concepts and apply them to real life situations. Preparation, Lesson Narrative states, “Students make decisions and choices, adhere to mathematical constraints, use mathematical ideas to analyze real-world situations, and interpret a mathematical answer and whether it makes sense in the context of a situation. In doing so, they model with mathematics (MP4).” Student Work Time states, “2 minutes: independent work time. 8 minutes: partner work time. Monitor for students who: think of multiple quantities that the same number might represent, use estimation in their reasoning.” In Student Work Time, Student Facing states, “Here is some information about insects: Termites, Size of a colony: 100–1,000,000. A queen lives for 30–50 years. There are 3,000–3,500 species of termites. The length of a termite is 4 to 15 millimeters. In some species, the mature queen may produce around 40,000 eggs a day. Odorous House Ants Size of colony: up to 100,000. A queen lives for 300–1,800 days. The length of an ant is 1.5–3.2 millimeters. Foraging ants travel up to 700 feet from their nests. There are 12,000–22,000 possible species. Honey Bees Size of a hive: 10,000–60,000. There are around 500 drones in a hive. A queen can lay about 1,5002,000 eggs each day. A hive produces 7–40 liters of honey in a season. The length of a bee is 10–20 millimeters. a. Here are some numbers that could represent facts about termites, house ants, and honey bees. What might each number represent?  Number, What it Might Represent, 2.4, 8, 4878, 1,794, 6,905, 20,799, 530,097  b. Add another number to the list. What about the insects might this number represent? c. Discuss your answers with your partner. Be prepared to show or explain your reasoning.”  

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

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

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

• Unit 4, From Hundredths to Hundred-thousands, Section C, Lesson 16, Activity 1, Student Work Time and Instructional Routine, students use appropriate strategies as tools when rounding numbers. In Student Work Time, Student Facing states, “Noah says that 489,231 can be rounded to 500,000. Priya says that it can be rounded to 490,000. a. Explain or show why both Noah and Priya are correct. Use a number line if it helps. b. Describe all the numbers that round to 500,000 when rounded to the nearest hundred-thousand. c. Describe all the numbers that round to 490,000 when rounded to the nearest ten-thousand. d. Name two other numbers that can also be rounded to both 500,000 and 490,000.”Instructional Routine states, “When they find all of the numbers that round to a given number, students need to think carefully about place value and may choose to use a number line to support their reasoning (MP5).”

• Unit 8, Properties of Two-dimensional Shapes, Section B, Lesson 9, Activity 1,Student Work Time and Activity Narrative, students use tools strategically as they identify line symmetry and solve problems. In Student Work Time, Student Facing states, “a. Mai has a piece of paper. She can get two different shapes by folding the paper along a line of symmetry. What is the shape of the paper before it was folded?” Activity Narrative states, “The first question offers opportunities to practice choosing tools strategically (MP5). Some students may wish to trace the half-shapes on patty paper, to make cutouts of them, or to use other tools or techniques to reason about the original shape. Provide access to the materials and tools they might need.”

The materials reviewed for Open Up Resources K–5 Math Grade 4 meet expectations for supporting the intentional development of MP6: Attend to precision; and attend to the specialized language of mathematics, for students, in connection to the grade-level content standards, as expected by the mathematical practice standards. 

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

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

• Unit 2, Fraction Equivalence and Comparison, Section B, Lesson 9, Warm-up, Activity Narrative, Launch, and Student Work Time, students use accuracy and precision when they describe strategies in finding the value of multiplication problems. Activity Narrative states, “The strategies of doubling and halving elicited here will be helpful later in the lesson when students generate equivalent fractions. In describing strategies, students need to be precise in their word choice and use of language (MP6).” Launch states, “Display one expression. ‘Give me a signal when you have an answer and can explain how you got it.’ 1 minute: quiet think time” In Student Work Time, Student Facing states, “Find the value of each expression mentally. $10\times6$, $10\times12$, $10\times24$, $5\times24$.” 

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

• Unit 5, Multiplicative Comparison and Measurement, Section B, Lesson 13, Cool-down, students compare the ingredients needed to make cookies, using precision when comparing units of measure including pounds and ounces (MP6). Student Facing states, “Priya needs oats and raisins to make cookies. She needs 3 pounds of oats. That amount is 4 times as much as the amount of raisins that she needs. How many ounces of raisins does she need? Explain or show your reasoning.”

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

The materials reviewed for Open Up Resources K–5 Math Grade 4 meet expectations for supporting the intentional development of MP7: Look for and make use of structure; and MP8: Look for and express regularity in repeated reasoning, for students, in connection to grade-level content standards, as expected by the mathematical practice standards. 

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

• Unit 4, From Hundredths to Hundred-thousands, Section B, Lesson 9, Activity 1, Student Work Time and Activity Synthesis, students look for and make use of structure while they compare numbers to determine value. Activity Narrative states, “When students sort the cards, they look for how the numbers are the same and different, including their overall value or the digits that make up the numbers (MP7).” Student Work Time states, “Give each group a set of cards from the blackline master. 5 minutes: partner and group work time on the first two problems. As students work, listen for place-value language such as: value of the digit, ten times, thousands, ten-thousands, and hundred-thousands. Record any place-value language students use to describe how they sorted the numbers and display for all to see. ‘Now work independently to write the numbers in the next problem in expanded form. Then, talk with your partner about the value of the digits.’ 3 minutes: independent work time. 5 minutes: partner work time. Monitor for students who: accurately write the numbers in expanded form, describe the relationship between the value of the digits in multiplicative terms (“ten times”).” In Student Work Time, Student Facing states, “Your teacher will give you and your partner a set of cards with multi-digit numbers on them. a. Sort the cards in a way that makes sense to you. Be prepared to explain your reasoning. b. Join with another group and explain how you sorted your cards. 2. Write each number in expanded form. a. 4,620 b. 46,200 c. 462,000. 3. Write the value of the 4 in each number. 4. Compare the value of the 4 in two of the numbers. Write two statements to describe what you notice about the values.56. How is the value of the 2 in 46,200 related to the value of the 2 in 462,000?” Activity Synthesis states, “Invite students to share their expressions in expanded form and what they noticed about the value of the 4. ‘What do you notice about the value of the 6 in each number? The value of the 2?’ (The value of the 6 is different in each number. It is first 600, then 6,000, then 60,000.) Students may talk about the number of zeros in each number. Shift their focus to the place value of the 6— hundreds, thousands, ten-thousands. ‘How is the value of the 2 in 46,200 related to the value of the 2 in 462,000?’ (The value of the 2 in 462,000 is 2,000 and the same digit in 46,200 has a value of 200. 2,000 is ten times the value 200.) ‘What multiplication equation could we write to represent the relationship between the 2 in 46,200 and 462,000?’ $(2,000=200\times10)$ ‘We can also write this equation using division: $2,000\div200=10$.’”

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

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

• Unit 3, Extending Operations to Fractions, Section B, Lesson 7, Activity 2, Student Work Time and Activity Synthesis, students use repeated reasoning as they decompose fractions to find sums. Activity Narrative states, “In the previous activity, students saw that a fraction can be decomposed into a sum of fractions with the same denominator and that it can be done in more than one way. In this activity, they record such decompositions as equations. The last question prompts students to consider whether any fraction can be written as a sum of smaller fractions with the same denominator. Students see that only non-unit fractions (with a numerator greater than 1) can be decomposed that way. Students observe regularity in repeated reasoning as they decompose the numerator, 9, into different parts while the denominator in all cases is 5 (MP8).” Student Work Time states, “‘Take a few quiet minutes to complete the activity. Then, share your responses with your partner.’ 5–6 minutes: independent work time. 3–4 minutes: partner discussion. Monitor for different explanations students offer for the last question.” In Student Work Time, Student Facing states, a1. Use different combinations of fifths to make a sum of $\frac{9}{5}$. 1. $\frac{9}{5}$ = ___ + ___ + ___ + ___ + ___  2.  $\frac{9}{5}$ = ___ + ___ + ___ + ___  3. $\frac{9}{5}$ = ___ + ___ + ___  4. $\frac{9}{5}$= ___ + ___   b. Write different ways to use thirds to make a sum of . How many can you think of? Write an equation for each combination. c. Is it possible to write any fraction with a denominator of 5 as a sum of other fifths? Explain or show your reasoning.” Activity Synthesis states, “Invite students to share their equations. Display or record them for all to see. Next, discuss students' responses to the last question. Select students with different explanations to share their reasoning. If not mentioned by students, highlight that fractions with a numerator of 1 (unit fractions) cannot be further decomposed into smaller fractions with the same denominator because it is already the smallest fractional part. Other fractions with a numerator other than 1 (non-unit fractions) can be decomposed into fractions with the same denominator.”

• Unit 8, Properties of Two-dimensional Shapes, Section B, Lesson 7, Cool-down, students use repeated reasoning as they find the perimeter of shapes and write matching expressions. Student Facing states, “Here is a rectangle with two lines of symmetry. Find its perimeter. Write an expression to show how you find it.” Activity 1 Lesson Narrative states, “In this activity, students find the perimeter of several shapes and write expressions that show their reasoning. Each side of the shape is labeled with its length, prompting students to notice repetition in some of the numbers. The perimeter of all shapes can be found by addition, but students may notice that it is efficient to reason multiplicatively rather than additively (MP8).” Students have the opportunity to demonstrate this same reasoning within the Cool-down.

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

The materials reviewed for Open Up Resources K-5 Math Grade 4 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 4 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 2, Fraction Equivalence and Comparison, Section B, Lesson 8, Warm Up, “The purpose of an Estimation Exploration is to practice estimating a reasonable answer based on experience and known information. Students can identify fractions represented by the shaded portions in tape diagrams in which unit or non-unit fractions are marked. To estimate the shaded parts in an unmarked tape, students may rely on the size of benchmark fractions— $\frac{1}{2}$, $\frac{1}{3}$, or $\frac{1}{4}$—and partition those parts mentally until it approximates the size of the shaded ports. They may also estimate how many copies of the shaded part could fit in the entire diagram.”

• Unit 3, Extending Operations to Fractions, Section B, Lesson 7, Activity 1, provides teachers guidance in building students' understanding of fractions. "Previously, students considered non-unit fractions in terms of equal groups of unit fractions or as a product of a unit fraction and a whole number. This activity prompts students to think about non-unit fractions as being sums of other fractions. The given context—about measuring fractional amounts using measuring cups of certain sizes—allows students to continue thinking in terms of equal groups, but also invites them to consider a fractional quantity as a sum of two or more fractions with the same denominator."

• Unit 5, Multiplicative Comparison and Measurement, Overview, Throughout this Unit, assists teachers in presenting materials. “The Number Talks in this unit allow students to use what they previously learned about numbers and operations to support their current learning. Students use the relationship between multiplication and division to solve missing factor equations, which is helpful for multiplicative comparison work. They mentally find the value of expressions involving multiplication by 100 and 1,000, which is helpful when converting metric units of measurement. Some Number Talks offer ongoing practice toward end-of-year fluency goals, such as in multi-digit addition. Students also practice multiplying fractions by whole numbers, which supports the problem solving work in the unit.”

The materials reviewed for Open Up Resources K-5 Math Grade 4 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, Fraction Equivalence and Comparison, Overview, “In this unit, students extend their prior understanding of equivalent fractions and comparison of fractions. In grade 3, students partitioned shapes into parts with equal area and expressed the area of each part as a unit fraction. They learned that any unit fraction $\frac{1}{b}$ results from a whole partitioned into b equal parts. They used unit fractions to build non-unit fractions, including fractions greater than 1, and represent them on fraction strips and tape diagrams. The denominators of these fractions were limited to 2, 3, 4, 6, and 8. Students also worked with fractions on a number line, establishing the idea of fractions as numbers and equivalent fractions as the same point on the number line. Here, students follow a similar progression of representations. They use fraction strips, tape diagrams, and number lines to make sense of the size of fractions, generate equivalent fractions, and compare and order fractions with denominators 2, 3, 4, 5, 6, 8, 10, 12, and 100. Students generalize that a fraction $\frac{s}{b}$ is equivalent to fraction $\frac{(n\times a)}{(n\times b)}$ because each unit fraction is being broken into n times as many equal parts, making the size of the part n times as small $\frac{1}{(n\times b)}$ and the number of parts in the whole n times as many $(n\times a)$. For example, we can see $\frac{3}{5}$ is equivalent to $\frac{6}{10}$  because when each fifth is partitioned into 2 parts, there are 2 x 3 or 6 shaded parts, twice as many as before, and the size of each part is half as small $\frac{1}{(2\times5)}$ or $\frac{1}{10}$.”

• Unit 4, From Hundredths to Hundred-Thousands, Section A, Lesson 1, Preparation, Lesson Narrative, “In this lesson, students rely on their knowledge of fractions to express tenths and hundredths as decimals. They begin to see connections between fraction notation, the names of fractions in words, and decimal notation. They also start to notice the structure of the decimal notation and how it relates to place value. Students use increasingly precise language to read decimals through this section (MP6). Students will develop this new understanding over several lessons so they are not expected to name the value of each place of a decimal at this time.”

• Unit 6, Multiplying and Dividing Multi-Digit Numbers, Section B, Lesson 9, Preparation, Lesson Narrative, “Students engage in quantitative and abstract reasoning (MP2) as they relate the partial products in a diagram and in an algorithm. Because this lesson offers an initial exposure to the new notation, students are not required to use an algorithm that uses partial products to multiply. They can rely on other methods they have learned so far.”

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 4 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, From Hundredths to Hundred-Thousands, Section B, Lesson 6, Standards, “Addressing: 4.NBT.A.1 Recognize that in a multi-digit whole number, a digit in the one place represents ten times what it represents in the place to its right. For example, recognize that $700\div70=10$ by applying concepts of place value and division. Building Towards: 4.NBT.A.1.”

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 apply understanding of multiplication and area to work with factors and multiples.” Additionally, each Unit Section provides Section Learning Goals, “determine if a number is prime or composite.”

• Unit 2, Fraction Equivalence and Comparison, Overview, “In this unit, students extend their prior understanding of equivalent fractions and comparison of fractions. In grade 3, students partitioned shapes into parts with equal area and expressed the area of each part as a unit fraction. They learned that any unit fraction $\frac{1}{b}$ results from a whole partitioned into b equal parts.”

• Unit 5, Multiplicative Comparison and Measurement, Section A, Overview, "In this section, students expand on these concepts to convert measurements within the same system (metric or customary) from larger units to smaller units. These conversions require an understanding of the multiplicative relationship between units. Students begin by exploring lengths in metric units. To develop a sense of the multiplicative relationship between centimeters and meters, students build a length of 1 meter from centimeter grid paper. They recognize that 1 meter is 100 times as long as 1 centimeter and use this reasoning to convert meters to centimeters. Later, they make sense of 1 kilometer by relating it to multiples of shorter measurements, such as the length of a basketball court or a soccer field. Later, students learn the relationships between grams and kilograms, milliliters and liters, ounces and pounds, and hours, minutes, and seconds. As they solve problems and use multiplication to perform conversion, they develop a sense of the relative size of the units." (4.MD.1)

• Unit 7, Angles and Angle Measurement, Section B, Lesson 9, "Students then make sense of one-degree angles in terms of a fraction of a turn and are introduced to the protractor as a tool of measurement. They make sense of the numbers on the tool and how angles are shown. They learn to read the measurement of angles whose vertices have been pre-aligned to the center point of a protractor. Students will continue to add new vocabulary to their personal word walls. In the next lesson, students will further develop their ability to use a protractor by measuring a variety of angles with less support." (4.MD.6)

The materials reviewed for Open Up Resources K-5 Math Grade 4 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 4 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. “Fraction Strips (3-4): Fraction strips are rectangular pieces of paper or cardboard used to represent different parts of the same whole. They help students concretely visualize and explore fraction relationships. As students partition the same whole into different-size parts, they develop a sense for the relative size of fractions and for equivalence. Experience with fraction strips facilitates students’ understanding of fractions on the number line.” 

• 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 4, Lesson 10 - Gather: Number cards 0-10; Copy: Greatest of Them All Stage 3 Recording Sheet, Mystery Number Stage 4 Gameboard.”

• Unit 2, Fraction Equivalence and Comparison, Section B, Lesson 8, Materials Needed, “Activities: Tape (painter’s or masking) (Activity 1); Centers: Dry erase markers (Get Your Numbers in Order, Stage 4), Sheet protectors (Get Your Numbers in Order, Stage 4).”

The materials reviewed for Open Up Resources K-5 Math Grade 4 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 2, Fraction Equivalence and Comparison, Section C, Lesson 12, Activity 1, Access for Students with Disabilities, “Engagement: Sustaining Effort and Persistence, Chunk this task into more manageable parts. Invite students to look at column A first, then column B, then row 1. Provide access to pre-made fraction strips for thirds and fifths to help them get started. Check in with students to provide feedback and encouragement after each chunk, particularly in terms of looking for and making use of structure. Provides accessibility for: Conceptual Processing, Organization, Social-Emotional Functioning.”

• Unit 5, Multiplicative Comparison and Measurement, Section C, Lesson 17, Access for Students with Disabilities, “Representation: Language and Symbols. Invite students to represent each problem as an equation to help them identify strategies for solving and to give them practice interpreting mathematical language. Provides accessibility for: Conceptual Processing, Language.”

• Unit 7, Angles and Angle Measurement, Section A, Lesson 5, Access for Students with Disabilities, “Engagement: Recruiting Interest Synthesis: Optimize meaning and value. Ask, “How might thinking about angles be useful in our lives?” Consider making a connection to sports. For example, it might be easier to score in soccer if the ball is in front of the goal rather than off to the side, because of the angles involved. Show pictures if applicable and possible. (Consider drawing or labeling a picture in which the soccer ball is the vertex and the posts are points along the rays.) Provides accessibility for: Conceptual Processing, Attention, Social- Emotional Functioning.”

The materials reviewed for Open Up Resources K-5 Math Grade 4 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, Extending Operations to Fractions, Section A, Practice Problems, Problem 12 (Exploration), “Diego walked the same number of miles to school each day. He says that he walked  48/5 miles in total, but does not say how many days that distance includes. What are some possible number of days Diego counted and the distance he walked each of those days?”

• Unit 4, From Hundredths to Hundred-Thousands, Section B, Practice Problems, Problem 7 (Exploration), “For each question, use only the digits 1, 0, 5, 9, and 3. You may not use a digit more than once and you do not need to use all the digits. a. Can you make three numbers greater than 3,000 but less than 3,500?; b. Can you make three numbers greater than 9,000 but less than 10,000?; c. Which numbers can you make that are greater than 39,500 but less than 40,000?”

• Unit 8, Properties of Two-Dimensional Shapes, Section A, Problem 11 (Exploration), “Draw each shape and all the lines of symmetry you can find in it. a. rectangle; b. rhombus; c. square.”

The materials reviewed for Open Up Resources K-5 Math Grade 4 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 2, Fraction Equivalence and Comparison, Section B, Lesson 8, Activity 1, “Access for English Learners - Listening, Speaking: MLR8 Discussion Supports. During partner work, invite students to take turns sharing their responses. Ask students to restate what they heard using precise mathematical language and their own words. Display the sentence frame: ‘I heard you say …’ Original speakers can agree or clarify for their partner.”

• Unit 3, Extending Operations to Fractions, Section A, Lesson 4, Activity1, “Access for English Learners - Reading, Representing: Reading: MLR6 Three Reads. ‘We are going to read this 3 times.’ After the 1st Read: ‘Tell your partner what this situation is about.’ After the 2nd Read: ‘List the quantities. What can be counted or measured?’ (number of jars, number of friends, number of cups of jam). After the 3rd Read: ‘What strategies can we use to solve this problem?’”

• Unit 5, Multiplicative Comparison and Measurement, Lesson 3, Activity 1, "Access for English Learners, Representing - Conversing: MLR7 Compare and Connect; Synthesis: After all strategies have been presented, lead a discussion comparing, contrasting, and connecting the different approaches. Ask, ‘What did the strategies have in common?’, ‘How were they different?’, and ‘Why did the different approaches lead to the same outcome?’”

The materials reviewed for Open Up Resources K-5 Math Grade 4 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 2, Fraction Equivalence and Comparison, Section A, Lesson 1, Activity 1, “The purpose of this activity is for students to use fraction strips to represent halves, fourths, and eighths. The denominators in this activity are familiar from grade 3. The goal is to remind students of the relationships between fractional parts in which one denominator is a multiple of another. Students should notice that each time the unit fractions on a strip are folded in half, there are twice as many equal-size parts on the strip and that each part is half as large. Groups of 2. Give each group 4 paper strips and a straightedge. Hold up one strip for all to see. ‘Each strip represents 1.’ Label that strip with ‘1’ and tell students to do the same on one of their strips. ‘Take a new strip. How would you fold it to show halves?’ 30 seconds: partner think time. ‘Think about how to show fourths on the next strip and eighths on the last strip.’”

• Unit 4, From Hundredths to Hundred-Thousands, Section A, Lesson 2, Activity 1, “In this activity, students reinforce their understanding of equivalent fractions and decimals by sorting a set of cards by their value. The cards show fractions, decimals, and diagrams. A sorting task gives students opportunities to analyze different representations closely and make connections (MP2, MP7). ‘Work with your group to sort the set of cards by their value. One diagram has no matching cards. Write the fraction and decimal it represents.’ 6–7 minutes: group work on the first two problems. Monitor for the ways students sort the cards and the features of the representations to which they attend. ‘Work on the last problem independently.’ 2–3 minutes: independent work on the last problem. Your teacher will give you a set of cards. Each large square on the cards represents 1. Sort the cards into groups so that the representations in each group have the same value. Record your sorting decisions. Be prepared to explain your reasoning. One of the diagrams has no matching fraction or decimal. What fraction and decimal does it represent? Are 0.20 and 0.2 equivalent? Use fractions and a diagram to explain your reasoning.”

• Unit 6, Multiplying and Dividing Multi-Digit Numbers, Section B, Lesson 6, Activity 1, “This activity prompts students to make sense of base-ten diagrams for representing multiplication. The representation supports students in grouping tens and ones and encourages them to use place value understanding and to apply the distributive property (MP7). This activity is an opportunity for students to build conceptual understanding of partial products in a more concrete way. In the next activity, students will notice that working with these drawings can be cumbersome and transition to using rectangular diagrams, which are more abstract.”