The materials reviewed for Open Up Resources K–5 Math Grade 3 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 3 Course Guide, Design Principles, conceptual understanding is a part of the design of the materials. Balancing Rigor states, “There are three aspects of rigor essential to mathematics: conceptual understanding, procedural fluency, and the ability to apply these concepts and skills to mathematical problems with and without real-world contexts. These aspects are developed together and are therefore interconnected in the materials in ways that support student understanding. Opportunities to connect new representations and language to prior learning support students in building conceptual understanding. Access to new mathematics and problems prompts students to apply their conceptual understanding and procedural fluency to novel situations.” Additionally, Purposeful Representations states, “Across lessons and units, students are systematically introduced to representations and encouraged to use representations that make sense to them. As their learning progresses, students are given opportunities to make connections between different representations and the concepts and procedures they represent.” Examples include:

• Unit 1, Introducing Multiplication, Section B, Lesson 9, Activity 2, Student Work Time, students develop conceptual understanding as they represent situations involving equal groups in a way that makes sense to them. Student Facing states, “Represent each situation. a. There are 4 people wearing shoes. Each person is wearing 2 shoes. b. There are 2 boxes of markers. Each box has 10 markers. c. There are 3 basketball teams. Each team has 5 players.” (3.OA.1)

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

• Unit 8, Putting It All Together, Section A, Lesson 2, Warm-up, Student Work Time, students develop conceptual understanding as they compare fractions on a number line. An image of four different number lines with fractions is provided and Student Facing states, “Which one doesn’t belong?” (3.NF.2)

According to the Grade 3 Course Guide, materials were designed to include opportunities for students to independently demonstrate conceptual understanding, when appropriate. Design Principles, Coherent Progress states, “Each activity starts with a launch that gives all students access to the task. This is followed by independent work time that allows them to grapple with problems individually before working in small groups. The activity ends with a synthesis to ensure students have an opportunity to consolidate their learning by making connections between their work and the mathematical goals.” A Typical Lesson states, “The Cool-down task is to be given to students at the end of the lesson. Students are meant to work on the Cool-down for about 5 minutes independently and turn it in. The Cool-down serves as a brief formative assessment to determine whether students understood the lesson. Students’ responses to the Cool-down can be used to make adjustments to further instruction.” Examples include:

• Unit 1, Introducing Multiplication, Section B, Lesson 11, Cool-down, students demonstrate conceptual understanding as they write expressions for equal groups. Student Facing states, “There were 6 envelopes. Each envelope had 2 notes in it. Write a multiplication expression to represent the situation. Explain or show your reasoning. Create a drawing or diagram if it’s helpful.” (3.OA.1)

• Unit 2, Area and Multiplication, Section B, Lesson 8, Cool-down, students demonstrate conceptual understanding as they reason about the area of a rectangle. Students are provided a drawing of a rectangle with tick marks rather than a completed grid. Student Facing states, “The tick marks on the sides of the rectangle are 1 foot apart. What is the area of the rectangle? Explain or show your reasoning.” (3.MD.7b)

• Unit 5, Fractions as Numbers, Section A, Lesson 3, Activity 2, Student Work Time, students demonstrate conceptual understanding of fractions as they match fractions to shaded diagrams. Student Facing states, “Your teacher will give you a set of cards for playing Fraction Match. Two cards are a match if one is a diagram and the other a number, but they have the same value. 1. To play Fraction Match: Arrange the cards face down in an array. Take turns choosing 2 cards. If the cards match, keep them and go again. If not, return them to where they were, face down. You can’t keep more than 2 matches on each turn. After all the matches have been found, the player with the most cards wins. 2. Use the cards your teacher gives you to create 4 new pairs of cards to add to the set. 3. Play another round of Fraction Match using all the cards.” (3.NF.1)

The materials reviewed for Open Up Resources K–5 Math Grade 3 meet expectations for giving attention throughout the year to individual standards that set an expectation of procedural skill and fluency. 

According to the Grade 3 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 1, Introducing Multiplication, Section A, Lesson 4, Activity 2, Student Work Time, students develop procedural skill and fluency with data as they create a scaled picture graph. Student Work Time states, “‘Represent the data that you collected in your own scaled picture graph where each picture represents 2 students.’ 10 minutes: independent work time, Circulate as students work: Encourage them to include a title, category labels, and key. Pay attention to how students are grouping by 2. Support students with questions they may have (especially around representing odd number amounts).” Student Facing states, “Represent our survey data in a scaled picture graph where each picture represents 2 students.” (3.MD.B)

• Unit 3, Wrapping Up Addition and Subtraction Within 1,000, Section C, Lesson 16, Warm-up, Student Work Time, students develop fluency as they use strategies for finding the products of 4 and 6 as they relate to products of 5. Student Facing states, “Find the value of each expression mentally. $5\times7$, $4\times7$, $6\times7$, $4\times8$.” (3.OA.7) 

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

According to the Grade 3 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 3, Wrapping Up Addition and Subtraction Within 1,000, Section B, Lesson 10, Cool-down, students demonstrate procedural skill and fluency as they use an algorithm to subtract within 1,000. Student Facing states, “Use an algorithm of your choice to find the value of $419-267$.” (3.NBT.2) 

• Unit 4, Relating Multiplication to Division, Section B, Lesson 9, Activity 2, Student Work Time, students demonstrate fluency as they identify patterns in multiplication. Student Work Time states, “‘In the right column, work independently to write down at least two multiplication facts you can figure out because you know the given multiplication fact in the left column.’ 3–5 minutes: independent work time. ‘Now, share the facts that you found with your partner. Record any facts that your partner found that you didn’t find. Be sure to explain your reasoning.’” Student Facing states, “a. In each row, write down at least two multiplication facts you can figure out because you know the given multiplication fact in the left column. Be prepared to share your reasoning. If I know…  $2\times4$, then I also know $4\times2$, $4\times4$, $2\times8$.” (3.OA.7) 

• Unit 8, Putting It All Together, Section D, Lesson 15, Activity 1, Student Work Time, students demonstrate procedural skill and fluency as they reason about subtraction and write a subtraction expression. Student Work Time states, “‘How would you find the value of each expression, without writing? For each expression, think of at least two ways. Then, share your thinking with your group.’ Reiterate to students that they are to consider how someone might reason about each difference, rather than only finding the value. 4 minutes: independent work time. 4 minutes: small-group discussion.” Student Facing states, “Here are three subtraction expressions. $600-400$, $600-399$, $500-399$. 1. Think of at least two different ways to find the value of each difference mentally. 2. Write a fourth subtraction expression whose value can be found using one of the strategies you thought of.” (3.NBT.2)

The materials reviewed for Open Up Resources K–5 Math Grade 3 meet expectations for being designed so that teachers and students spend sufficient time working with engaging applications of mathematics. 

According to the Grade 3 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 3 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 1, Introducing Multiplication, Section B, Lesson 12, Activity 1, Launch and Student Work Time, students solve a real-world problem involving multiplication. Launch states, “Groups of 2. MLR5 Co-Craft Questions. Display only the problem stem, ‘Tyler has 3 boxes.’ without revealing the question. ‘Write a list of mathematical questions that could be asked about this situation.’ (What’s in the boxes? How many things are in the boxes? How many things does he have altogether?) 2 minutes: independent work time. 2–3 minutes: partner discussion. Invite several students to share one question with the class. Record responses. ‘What do these questions have in common? How are they different?’ Reveal the task (students open books), and invite additional connections.” In Student Work Time, Student Facing states, “Tyler has 3 boxes. He has 5 baseballs in each box. How many baseballs does he have altogether? Show your thinking using diagrams, symbols, or other representations.” (3.OA.3)

• Unit 3, Wrapping Up Addition and Subtraction within 1000, Section D, Lesson 19, Activity 2, Student Work Time, students solve a multi-step real-world problem and then write an equation to represent the problem. Student Work Time states, “‘Take some independent time to work on this problem. You can choose to solve the problem first or write the equation first.’ 5–7 minutes: independent work time, Monitor for different ways students: write an equation, represent the problem, such as by using a tape diagram, decide their answer makes sense, such as thinking about the situation or by rounding.” Student Facing states, “Kiran is setting up a game of mancala. He has a jar of 104 stones. From the jar, he takes 3 stones for each of the 6 pits on his side of the board. How many stones are in the jar now? a. Write an equation to represent the situation. Use a letter for the unknown quantity. b. Solve the problem. Explain or show your reasoning. c. Explain how you know your answer makes sense.” (3.OA.8)

• Unit 7, Two-dimensional Shapes and Perimeter, Section C, Lesson 10, Cool-down, students solve a real-world problem involving the perimeter of a rectangle. Student facing states, “Lin is building a fence around her rectangular garden. A diagram is shown. The area of the garden is 36 square feet. How many feet of fencing material will she need to enclose the whole garden?” (3.MD.8)

Examples of non-routine applications of the math include:

• Unit 4, Relating Multiplication to Division, Section C, Lesson 17, Activity 2, Student Work Time, students develop understanding of multiplication and its relation to division to solve real-world problems. Student Work Time states, “‘Work independently to solve these problems and write an equation with a letter for the unknown quantity to represent each situation. You can choose to solve the problem first or write the equation first.’ 5–7 minutes: independent work time’ Share your solutions and your equations with your partner. Also, tell your partner if you think their solutions and equations make sense or why not.’5–7 minutes: partner discussion.” Student facing states, “For each problem: 1. Write an equation to represent the situation. Use a letter for the unknown quantity. 2. Solve the problem. Explain or show your reasoning. a. Kiran is making paper rings each day to decorate for a party. From Monday to Thursday he was able to complete 156 rings. Friday, Kiran and 2 friends worked on making more rings. Each of them made 9 more rings. How many rings did they make over the week? b. Mai has 168 muffins. She put 104 of the muffins in a basket. She packed the rest of the muffins into 8 boxes with the same number of muffins. How many muffins were in each box? c. There are 184 cups on a table. Three tables with 8 people at each table come up to get drinks and each use a cup. How many cups are on the table now?” (3.OA.8)

• Unit 5, Fractions as Numbers, Section B, Lesson 6, Activity 1, Launch and Student Work Time, students solve a non-routine problem as they partition a number line that extends beyond one. Launch states, “Groups of 2, ‘Today we are going to partition number lines to locate unit fractions. Take a minute to look at how Clare, Andre, and Diego have partitioned their number lines into fourths.’ 1-2 minutes: quiet think time.” in Student Work Time, Student Facing states, “Three students are partitioning a number line into fourths. Their work is shown. Whose partitioning makes the most sense to you? Explain your reasoning.” Clare’s number line partitioned into halves;  Andre’s number line partitioned into fourths; and Diego’s number line partitioned into fifths are shown. (3.NF.2)

• Unit 6, Measuring Length, Time, Liquid Volume, and Weight, Section C, Lesson 15, Activity 1, Launch and Student Work Time, students solve a real-world problem by using concepts of time, weight, and volume. Launch states, “Groups of 2, ‘We’re going to solve some problems about a day at the fair. What are some things you could do during a day at the fair?’ (go on rides, walk around, eat fair food, look at some of the animals) 30 seconds: quiet think time. Share responses. Give each group tools for creating a visual display.” In Student Work Time, Student Facing states, “You spent a day at the fair. Solve four problems about your day and create a poster to show your reasoning and solutions. a. You arrived at the fair! Entry to the fair is $9 a person. You went there with 6 other people. How much did it cost your group to enter the fair? b. How did you start your day? (Choose one.) You arrived at the giant pumpkin weigh-off at 11:12 a.m. and left at 12:25 p.m. How long were you there? You spent 48 minutes at the carnival and left at 12:10 p.m. What time did you get to the carnival? c. What was next? (Choose one.) You visited a barn with 7 sheep. The sheep drink 91 liters of water a day, each sheep drinking about the same amount. How much does each sheep drink a day? You visited a life-size sculpture of a cow made of butter. The butter cow weighs 273 kilograms, which is 277 kilograms less that the actual cow. How much does the actual cow weigh? 4. Before you went home . . . You stopped for some grilled corn on the cob. On the grill, there were 54 ears of corn arranged in 9 equal rows. How many ears of corn were in each row?” (3.MD.1, 3.MD.2, 3.OA.3)

The materials reviewed for Open Up Resources K–5 Math Grade 3 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 3, Wrapping Up Addition and Subtraction Within 1,000, Section A, Lesson 4, Activity 2, Launch and Student Work Time, students develop procedural skill and fluency as they use the addition algorithm. Launch states, “Groups of 2, Give students access to base-ten blocks. ‘Now you are going to have a chance to try the algorithms that Lin and Han used in the last activity. Take a minute to think about which algorithm you want to use for each problem.’” In Student Work Time, Student Facing states, “Try using an algorithm to find the value of each sum. Show your thinking. Organize it so it can be followed by others. a. $475+231$, b. $136+389$, c. $670+257$.” (3.NBT.2) 

• Unit 5, Fractions as Numbers, Section A, Lesson 3, Activity 1, Launch and Student Work Time, students extend their conceptual understanding to read and write fractions that represent images. Launch states, “Groups of 2, Display the table. ‘Let's look at the first table. The first three images are the squares we saw earlier. Let's name them again. (One-fourth, three-fourths, four-fourths) Let’s complete the second row of the table together. This is the square we just worked with in the Warm-up and the number that represents the total amount shaded is already in the table. How many of the parts are shaded? (Three) What is the size of each part? Write ‘three-fourths’ to record how we read this fraction.’” In Student Work Time, Student Facing states, “Each shape in each row of the table represents 1. Use the shaded parts to complete the missing information in the table. Be prepared to explain your reasoning.” A table is provided with the headings: number of shaded parts, size of each part, word name for the shaded parts, and number name for the shaded parts. (3.NF.1)

• Unit 7, Two-dimensional Shapes and Perimeter, Section B, Lesson 9, Cool-down, students apply their understanding of perimeter to solve a real-world problem. Student Facing states, “A rectangular swimming pool has a perimeter of 94 feet. If it is 32 feet on one side, what are the lengths of the other three sides? Explain or show your reasoning.” (3.MD.8)

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, Introducing Multiplication, Section C, Lesson 19, Cool-down, students use all three aspects of rigor, conceptual understanding, procedural skill and fluency, and application, as they use an equation to represent an array. Student Facing states, “Clare has 3 rows of baseball cards. Each row has 10 cards. How many cards does she have? a. Write an equation with a symbol for the unknown number to represent the situation. b. Find the number that makes the equation true. Explain or show your reasoning.” (3.OA.3)

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

• Unit 7, Two-dimensional Shapes and Perimeter, Section B, Lesson 6, Activity 1, Launch and Student Work Time, students use procedural fluency and apply their understanding of perimeter of shapes to solve a non-routine real-world problem. Launch states, “Groups of 4, Give each group a copy of the blackline master and 25–50 paper clips. ‘Make a prediction: Which shape do you think will take the most paper clips to build?’ 30 seconds: quiet think time. Poll the class on whether they think shape A, B, C, or D would take the most paper clips to build.” Student Work Time states, “Work with your group to find out which shape takes the most paper clips to build. You may need to take turns with the paper clips.” In Student Work Time, Student Facing states, “Your teacher will give you four shapes on paper and some paper clips. Work with your group to find out which shape takes the most paper clips to build. Explain or show how you know. Record your findings here. Draw sketches if they are helpful.” (3.MD.8)

The materials reviewed for Open Up Resources K–5 Math Grade 3 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, Area and Multiplication, Section B, Lesson 10, Activity 1, Student Work Time and Activity Narrative, students solve a real-world problem involving area. In Student Work Time, Student Facing states, “Noah is painting a wall in a community garden. The wall is shaped like a rectangle. A diagram of the wall is shown here. Paint is sold in 3 different sizes: A small container will cover 3 square meters. A medium container will cover 10 square meters. A large container will cover 40 square meters. What should Noah buy? Explain your reasoning.” Activity Narrative states, “The activity includes a rectangle where the side lengths are labeled. When students solve problems with multiple solutions and have to choose and justify a solution, they make sense of problems and persevere in solving them (MP1).” 

• Unit 6, Measuring Length, Time, Liquid, Volume, and Weight, Section D, Lesson 13, Cool-down, students make sense of problems involving weight and justify their reasoning. Preparation, Lesson Narrative states, “In this lesson, students solve problems involving weight in two Information Gap activities. They interpret descriptions of situations involving all four operations and in which one or more quantities are missing. Students determine the information that they need to answer the questions and then reason about the solutions.” Student facing states, “The winning pig weighed 48 kilograms when his owner decided to raise him to show at the fair. At the fair weigh-off, the pig weighed 124 kilograms. How much weight did the pig gain? Explain or show your reasoning.”

• Unit 7, Two-dimensional Shapes and Perimeter, Section B, Lesson 6, Warm-up, Launch, Student Work Time, and Activity Narrative, students make sense of perimeter concepts. Students are given an image of a shape and paper clips. Launch states, “Groups of 2, Display the image, ‘What do you notice? What do you wonder?’ 1 minute: quiet think time.” In Student Work Time, Student Facing states, “What do you notice? What do you wonder?” Activity Narrative states, “The purpose of this Warm-up is for students to visualize the idea of perimeter and elicit observations about distances around a shape. It also familiarizes students with the context and materials they will be working with in the next activity, where they will use paper clips to form the boundary of shapes and compare or quantify their lengths.”

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 1, Introducing Multiplication, Section A, Lesson 2, Activity 2, Student Work Time and Activity Narrative, students reason abstractly and quantitatively as they interpret data from a bar graph. Student Work Time states, “‘Now you’re going to use your bar graph to decide if statements are true or false.’ 1–2 minutes: independent work time.” In Student Work Time, Student Facing states, “1. Decide if each statement is true or false about how our class gets home. Explain your reasoning to your partner. a. More students walk than go home any other way. b. More students ride home on a bus than in a car. c. Fewer students walk home than ride their bikes. d. More students walk or ride their bikes than ride in a van. 2. Fill in the blanks as directed by your teacher, then answer each question. a. ‘How many more students ___  than ___?’ b. ‘How many more students ___  or ___ than ___?’” Activity Narrative states, “When students use expression, equations, or describe adding or subtracting to find how many more or less, they show they can decontextualize and recontextualize the data to make sense of and solve the problems (MP2). You will generate the questions students answer in this task from the class graph.”

• Unit 5, Fractions as Numbers, Section A, Lesson 4, Activity 2, students use diagrams to represent the fractional amount in a given situation. Activity Narrative states, “The purpose of this activity is for students to use diagrams to represent situations that involve non-unit fractions. The synthesis focuses on how students partition and shade the diagrams and how the end of the shaded portion could represent the location of an object. When students interpret the different situations in terms of the diagrams they reason abstractly and quantitatively (MP2).” Student Work Time states, “‘In the activity, each strip represents the length of a street where Pilolo is played. Work independently to represent each situation on a diagram.’ 3–5 minutes: independent work time. ‘With a partner, choose one of the situations and make a poster to show how you represented the situation with a fraction strip. You may want to include details such as notes, drawings, labels, and so on, to help others understand your thinking.’ Give students materials for creating a visual display. 5–7 minutes: partner work time.” In Student Work Time, Student Facing states, “Here are four situations about playing Pilolo and four diagrams. Each diagram represents the length of a street where the game is played. Represent each situation on a diagram. Be prepared to explain your reasoning. a. A student walks $\frac{4}{8}$ the length of the street and hides a rock .b. A student walks the length of the street and hides a penny. c. A student walks $\frac{3}{4}$ the length of the street and hides a stick. d. A student walks $\frac{5}{6}$ the length of the street and hides a penny. e. This diagram represents the location of a hidden stick. About what fraction of the length of the street did the student walk to hide it? Be prepared to explain how you know.”

• Unit 6, Measuring Length, Time, Liquid, Volume, and Weight, Section A, Lesson 2, Warm-up, Activity Narrative, Launch, and Student Work Time, students practice estimation strategies with measurements. Activity Narrative states, “The purpose of this Estimation Exploration is to practice the skill of estimating a reasonable answer based on experience and known information. The Warm-up also draws students' attention to a length between a full inch and one-half of an inch, preparing students to work with such lengths later.” Launch states, “Groups of 2, Display the image. ‘What is an estimate that’s too high? Too low? About right?’ 1 minute: quiet, think time.” In Student Work Time, Student Facing states, “What is the length of the paper clip?” A paper clip is shown next to a ruler, and the length is between the 1 and 2 inch mark on the ruler.

The materials reviewed for Open Up Resources K–5 Math Grade 3 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 K 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, Area and Multiplication, Section B, Lesson 10, Cool-down, students construct viable arguments as they find the area of a rectangle. Student Facing, “Kiran bought two pieces of fabric. The black fabric is 9 yards by 2 yards. The purple fabric is 4 yards by 5 yards. Which piece of fabric has the larger area? Explain or show your reasoning.”

• Unit 2, Area and Multiplication, Section C, Lesson 12, Activity 2, Activity Narrative, Launch, and Student Work Time, students find the area of a “figure” by decomposing the figure into rectangles and then critique the reasoning of others. Activity Narrative states, “Some students may partition diagonally to split the figure into what looks like 2 symmetrical parts, or cut the figure up into more than 2 parts. These are both acceptable ways of finding the area. Ask students who partition diagonally to find the area in the way they partitioned, but then encourage them to find a second way that has partitions on one of the grid lines. As students look through each others' work, they discuss how the representations are the same and different and can defend different points of view (MP3).”  Launch states, “Groups of 2. Display the image of the gridded figure. ‘What do you notice? What do you wonder?’ (Students may notice: It looks like 2 rectangles. It looks like a big rectangle with a chunk missing. There are squares. Students may wonder: What is this shape called? Could we find the area of the shape? How would we find the area?) 1 minute: quiet think time. Share responses. ‘This isn’t a shape that we have a name for like a square or triangle. Because of this, we’ll call it a “figure” as we work with it in this activity. This word will be helpful in describing other shapes that we don’t have a name for. Talk with your partner about different ways you could find the area of this figure.’ 1 minute: partner discussion.” In Student Work Time, Student Facing states, “What do you notice? What do you wonder? Find the area of this figure. Explain or show your reasoning. Organize it so it can be followed by others.”

• Unit 5, Fractions as Numbers, Section C, Lesson 11, Activity 1, Student Work Time and Activity Narrative, students construct a viable argument and critique the reasoning of others when they reason about fraction equivalence. In Student Work Time, Student Facing states, “1. The diagram represents 1.​​​​​​ a. What fraction does the shaded part of the diagram represent? b. Jada says it represents $\frac{4}{8}$. Tyler is not so sure. Do you agree with Jada? If so, explain or show how you would convince Tyler that Jada is correct. If not, explain or show your reasoning.” Activity Narrative states, “In the first problem, students construct a viable argument in order to convince Tyler that $\frac{4}{8}$ of the rectangle is shaded (MP3).” 

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

• Unit 1, Introducing Multiplication, Section A, Lesson 6, Activity 1, Student Work Time and Activity Narrative, students construct a viable argument and critique the reasoning of others as they create a scaled bar graph. In Student Work Time, Student Facing states, “Here is a collection of pattern blocks. Mai, Noah, and Priya want to make a bar graph to represent the number of triangles, squares, trapezoids, and hexagons in the collection. Mai says the scale of the bar graph should be 2. Noah says the scale of the bar graph should be 5. Priya says the scale of the bar graph should be 10. a. Who do you agree with? Explain your reasoning. b. Use the scale that you chose to create a scaled bar graph to represent the collection of pattern blocks.” Activity Narrative states, “They consider three students’ ideas, choose a scale of 2, 5, or 10, and create a scaled bar graph to represent the categorical data. Students must justify why they agree that a particular scale would be best. During the activity and whole-class discussion, students share their thinking and have opportunities to listen to and critique the reasoning of their peers (MP3).” 

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

• Unit 4, Relating Multiplication to Division, Section C, Lesson 15, Activity 2, Activity Narrative and Student Work Time, students construct a viable argument and critique the reasoning of others when they participate in a gallery walk and agree or disagree with other students’ work. Activity Narrative states, “The purpose of this activity is for students to see how other students solved one of the problems that involves a factor of a teen number. While students look at each other’s work, they will leave sticky notes describing why they think the answer does or does not make sense (MP3). The synthesis will look specifically at examples of how students used the area diagram to represent the problem.” Student Work Time states, “‘As you visit the posters with your partner, discuss what is the same and what is different about the thinking shown on each poster. Also, leave a sticky note describing why you think the solution does or does not make sense.’ 8 - 10 minutes: gallery walk. Monitor for different uses of the area diagram to highlight, specifically, a fully gridded area diagram with no labels and no decomposition, a gridded area diagram that was gridded, but also decomposed into parts or labeled along the sides or in the parts of the rectangle, a partitioned rectangle that was drawn with no grid, but labeled with side lengths or the area of the parts of the rectangle.”

The materials reviewed for Open Up Resources K–5 Math Grade 3 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 4, Relating Multiplication to Division, Lesson 22, Section D, Activity 1, Student Work Time, students use multiplication and division to determine the arrangement of strawberry plants in a garden. Preparation, Lesson Narrative states, “Students model with mathematics (MP4) as they consider constraints, make assumptions and decisions about quantities, think about how to represent the relationships among quantities, and check their solutions in terms of the situation.” Student Work Time states, “2 minutes: independent work time. 10 minutes: partner work time. Monitor for students who: write multiplication or division expressions or equations, are able to represent the same situation with both multiplication and division.” In Student Work Time, Student Facing states, “For each situation, draw a diagram and write an equation or expression. 1. A strawberry patch has 7 rows with 8 strawberry plants in each row. a. How many strawberry plants are in the patch? b. To grow strawberries in the best way, the rows should be 4 feet apart. Each plant in the row should be 2 feet apart. How long and wide is the strawberry patch? c. You can harvest 12 strawberries per plant. How many strawberries will grow in each row? 2. With your partner, take turns explaining where you see the numbers in the expression or equation you wrote in your diagram.”

• Unit 6, Measuring Length, Time, Liquid Volume, and Weight, Section D, Lesson 16, Warm-up, Launch, Student Work Time, and Lesson Narrative, students make decisions on how to make a good carnival game. Students see a picture of a carnival game using coins and marbles. Launch states, “Groups of 2, Display the image. ‘What do you notice? What do you wonder?’ 1 minute: quiet think time” Student Work Time states, “‘Discuss your thinking with your partner.’ 1 minute: partner discussion, Share and record responses.”  Preparation, Lesson Narrative states, “When students make choices about quantities and rules, analyze constraints in situations, and adjust their work to meet constraints, they model with mathematics (MP4).”

• Unit 7, Two-Dimensional Shapes and Perimeter, Section D, Lesson 13, Activity 1, Student Work Time and Instructional Routine, students apply what they have learned about perimeter and area to design a small park. Instructional Routine states, “The purpose of this Student Work Time is to provide students an opportunity to apply what they’ve learned about perimeter and area to design a small park. Since diagonal lines that connect the dots are not one length unit, students should use vertical and horizontal lines to design the park. When students make and describe their own choices for how they represent real-world objects, they model real-world problems with mathematics (MP4).” Student Work Time states, “‘Work independently to design your small park.’ 5 - 7 minutes: independent work time. ‘You can work with a partner or small group for the last few minutes or continue working on your own. Even if you choose to work alone, be available if your partner wants to think through something together.’ 3 - 5 minutes: partner, small group, or independent work time.”  In Student Work Time, Student facing states, “Your teacher will give you some dot paper for drawing. a. The distance from 1 dot to another horizontally or vertically represents 1 yard. Connect dots on the grid horizontally or vertically to design a small park that has 5 of these features: 1. basketball court  2. soccer goal  3. swings  4. a slide  5. an open area  6. picnic table  7. water play area  8. skate park  9. a feature of your choice  b. Describe the area and the perimeter of 3 features in the park.”

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 3, Wrapping Up Addition and Subtraction Within 1,000, Section B, Lesson 7, Activity 1, students use appropriate tools strategically to subtract within 1,000. In Student Work Time, Student Facing states, “Find the value of each difference in any way that makes sense to you. Explain or show your reasoning. a. $428-213$. b. $505-398$. c. $394-127$.”  Instructional Routine states, “Students may also use a variety of representations, which will be the focus of the Student Work Time synthesis. Students who choose to use base-ten blocks or number lines to represent their thinking use tools strategically (MP5).”

• Unit 4, Relating Multiplication to Division, Section C,Lesson 13, Cool-down, students use appropriate tools strategically when they multiply within 100. Student Facing states, “There are 6 bags of oranges and each bag has 11 oranges. How many oranges are in the bags? Show your thinking using objects, a drawing, or a diagram.” Students should also be encouraged to use strategies and representations from the previous section.” This outlines the goal of working with tools throughout this lesson.

• Unit 6, Measuring Length, Time, Liquid Volume, and Weight, Section A, Lesson 4, Activity 1, Student Work Time and Instructional Routine, students analyze a line plot as a tool for representing data. Students are given the heights of seedlings in inches on a line plot. In Student Work Time, Student Facing states, “a. Write 3 statements about the measurements represented in the line plot. b. What questions could be answered more easily with the line plot than the list? Write at least 2 questions.” Instructional Routine states, “When students recognize how organizing data helps to read the information and to answer questions, they learn that line plots are a powerful tool to present data (MP5).”

The materials reviewed for Open Up Resources K–5 Math Grade 3 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 2, Area and Multiplication, Section A, Lesson 4, Activity 1, Student Work Time and Activity Synthesis, students use specific language and precision to describe the rectangle they create. Activity Narrative states, “The purpose of this activity is for students to create and describe rectangles of a certain area. Students work in groups of 2. One partner creates a rectangle and describes it, and the other partner creates a matching rectangle based on the description. Then students compare how their rectangles are the same and different. Students should describe their rectangle to their partner without revealing the total number of squares they used, so that the focus is on understanding the rectangular structure. In the synthesis, students share language that helped them understand the rectangle their partner built. When students revise their language to be more precise in the descriptions of their rectangle, they attend to precision (MP6).” Student Work Time states, “‘The goal of this activity is to get both partners in a group to draw the same rectangle without looking at each other’s drawing. ‘If you are partner A, draw a rectangle and describe it to your partner. You can’t tell them how many squares you used to draw your rectangle. If you are partner B, draw the rectangle that you think your partner is describing and then compare the drawings. After you finish describing and drawing the first rectangle, switch roles and repeat.’ 10–12 minutes: partner work.”  In Student Work Time, Student Facing states, “a. Can you and your partner draw the same rectangle without looking at each other's drawing?  Partner A: Draw a rectangle on one of the grids provided. Describe it to your partner without telling them the total number of squares. Partner B: Draw the rectangle your partner describes to you. b. Place your two rectangles next to each other. Discuss: What is the same? What is different? c. Switch roles and repeat.”  Activity synthesis states, “‘What language did your partner use that was most helpful for you to draw the same rectangle they drew?’ (The number of squares in each row or column and the number of rows or columns.)”

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

• Unit 5, Fractions as Numbers, Section B, Lesson 9, Activity 1, Student Work Time and Activity Synthesis, students attend to precision when using a number line to locate fractions. Student Work Time states, “‘Take a few minutes to locate 1 on these number lines. Then use any of the number lines to explain how you located 1.’ 5–7 minutes: independent work time.” In Student Work Time, Student Facing states, “2. Use any of the number lines to explain how you located 1.” Activity Narrative states, “In the second problem, they reinforce their knowledge that the denominator of a fraction tells us the number of equal parts in a whole and the size of a unit fraction, and that the numerator gives the number of those parts (MP6).” Activity Synthesis states, “‘Share your written reasoning for one of the number lines with your partner. Take turns being the speaker and the listener. If you are the speaker, share your ideas and writing so far. If you are the listener, ask questions and give feedback to help your partner improve their work.’ 3 - 5 minutes: structured partner discussion. Repeat with 2 - 3 different partners. ‘Revise your initial draft based on the feedback you got from your partners.’ 2 - 3 minutes: independent work time. Invite students to share their revised explanations of how they located 1 on the number lines.”

• Unit 6, Measuring Length, Time, Liquid Volume, and Weight, Section A, Lesson 1, Activity 1, Activity Synthesis, students use the specialized language of mathematics as they discuss how to describe lengths using fractions of an inch. Activity Narrative states, “In the synthesis, discuss the need for fractions of an inch to describe lengths more precisely (MP6).” Activity Synthesis states, “Display the inch ruler and an object that wasn’t exactly a whole number of inches. ‘What is the length of this object? (Between 3 and 4 inches. More than 3 but less than 4. Three-and-a-half inches.) If needed, Could we say that the length of this object is (a whole number of) inches. (No, It's between 3 inches and 4 inches.) We need a way to make our measurements more precise. We'll think about this more in the next activity.’”

• Unit 7, Two-Dimensional Shapes and Perimeter, Section A, Lesson 3, Cool-down, students attend to the specialized language of math as they describe shapes. Student Facing states, “a. Which quadrilateral is being described? Hint 1: It has 4 sides. Hint 2: All of its sides are the same length. Hint 3: It has no right angles. b. Which hints do you need to guess the quadrilateral? Explain your reasoning.” Students see four different quadrilaterals with different features. Activity Narrative (for Activity 2) states, “As students decide which questions to ask they think about important attributes such as side lengths and angles and have an opportunity to use language precisely (MP6, MP7).”

• Unit 8, Putting It All Together, Section A, Lesson 7, Warm-up, Activity Narrative, Launch, and Activity Synthesis, students attend to precision as they use a bar graph and see the importance of precise labels and titles. Activity Narrative states, “The purpose of this Warm-up is to elicit the idea that bar graphs need a title and a scale in order to be able to communicate information clearly (MP6), which will be useful when students draw a scaled bar graph in a later activity. During the synthesis, focus the discussion on the missing scale.”  Launch states, “Groups of 2. Display the graph. ‘What do you notice?  What do you wonder?’ 1 minute: quiet think time.” Activity Synthesis states, “Could each unit or each space between two lines on the graph represent 1 student? Why or why not? (No, because that would mean half of a student likes broccoli, cauliflower, and peas.) If each unit on the graph represents 2 students, how many students have broccoli as their favorite vegetable? (13) What if it represents 4 students? (26) How should you decide on a scale for your graph? (Think about how many people you surveyed and use a scale that will fit them on your graph. Use a scale that will make the bar graph easy to read.)”

The materials reviewed for Open Up Resources K–5 Math Grade 3 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, Introducing Multiplication, Section C, Lesson 16, Warm-up, students look for and make use of structure while they notice the arrangement of a dozen eggs in relation to arrays. Activity Narrative states, “The purpose of this Warm-up is to elicit ideas students have about objects arranged in an array, which will be useful when students arrange equal groups into arrays in a later activity. While students may notice and wonder many things about this image, ideas around arrangement and equal groups are the important discussion points. When students notice the arrangement of the eggs they look for and make use of structure (MP7).” Launch states, “Groups of 2. Display the image. ‘What do you notice? What do you wonder?’” Student Facing states, “What do you notice? What do you wonder?” An egg carton with a dozen eggs is shown. Activity Synthesis states, “‘How does having the eggs in a carton help you see equal groups?’ (I can see how they could be split into equal groups. I can see 6 eggs in each row. I can see 6 groups of 2.) The eggs are arranged in an array. An array is an arrangement of objects in rows and columns. Each column must contain the same number of objects as the other columns, and each row must have the same number of objects as the other rows.”

• Unit 3, Wrapping Up Addition and Subtraction Within 1,000, Section A, Lesson 1, Activity 1, students look for and make use of structure as they represent numbers using base-ten blocks, base-ten diagrams, expanded form, numerals, and word form. Student Work Time states, “‘This set of cards includes numbers in different forms. Find the cards that match. Work with your partner to explain your matches.’ 8 minutes: partner work time” In Student Work Time, Student Facing states, “Your teacher will give you a set of cards that show numbers in different forms. Group the cards that represent the same number. Record your matches here. Be ready to explain your reasoning.” Lesson Narrative states, “As they make matches, students use their understanding of base-ten structure represented in many different ways (MP7).”

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

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, Wrapping Up Addition and Subtraction Within 1,000, Section C, Lesson 14, Activity 2, Student Work Time, students use repeated reasoning as they analyze the position of numbers relative to their immediate multiples of 10 and 100. Student Work Time states, “‘Work with your partner to complete these problems.’ 5 - 7 minutes: partner work time. Monitor for students who use the following strategies to highlight: Reason about the midpoint between a multiple of 10 or a multiple of 100 (5 or 50) to determine which multiple is closer, such as, ‘568 is closer to 570 because 565 would be the middle point between 560 and 570’; Use place value patterns to determine which multiple is closer, such as, ‘Since the 1 in 712 is less than 5, it tells me that the number is closest to 700’. Pause for a brief discussion before students complete the last problem. Select previously identified students to share the strategies they used to find the nearest multiple of 100 and the nearest multiple of 10. ‘Now take a few minutes to complete the last problem.’ 2 - 3 minutes: independent work time.” In Student Work Time, Student Facing states, “1a. Is 349 closer to 300 or 400? 1b. Is 349 closer to 340 or 350? 2a. Is 712 closer to 700 or 800? 2b. Is 712 closer to 710 or 720? 2a. Is 568 closer to 500 or 600? 2b. Is 568 closer to 560 or 570? 3a. Is 712 closer to 700 or 800? 3b. Is 712 closer to 710 or 720? 4. Without locating a given number on a number line, how did you decide: a. the nearest multiple of 100? b. the nearest multiple of 10?” Activity Narrative states, “When students notice and describe patterns in the relationship between the numbers and the nearest multiples of 10 or 100, they look for and express regularity in repeated reasoning (MP8).”

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

• Unit 6, Measuring Length, Time, Liquid Volume, and Weight, Section C, Lesson 10, Warm-up, Launch, Student Work Time, and Activity Synthesis, students use repeated reasoning as they choral count, using 15 minutes as the increment of time. Activity Narrative states, “The purpose of this Choral Count is to invite students to practice counting times by 15 minutes and notice patterns in the count. This will be helpful later in this section when students will solve problems involving addition and subtraction of time intervals. Students have an opportunity to notice regularity through repeated reasoning (MP8) as they count by 15 minutes over a span of 3 hours.” Launch states, “‘Count by 15 minutes, starting at 12:00.’ Record as students count. Record times in the count in a single column. Stop counting and recording at 3:00.” Student Work Time states, “‘What patterns do you see?’ 1–2 minutes: quiet think time. Record responses.” Activity Synthesis states, “‘How much time passed between 1:15 and 1:45? (30 minutes) 1:15 and 2:30?’ (75 minutes) Consider asking: ‘Who can restate the pattern in different words? Does anyone want to add an observation on why that pattern is happening here? Do you agree or disagree? Why?’”

The materials reviewed for Open Up Resources K-5 Math Grade 3 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 eight units, a Pacing Guide to plan instruction, and Dependency Diagrams. These Dependency Diagrams show the interconnectedness between lessons and units within Grade 3 and across all grades.

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

Materials include sufficient and useful annotations and suggestions that are presented within the context of the specific learning objectives. Several components focus specifically on the content of the lesson. Examples include:

• Unit 1, Introduction to Multiplication, Overview, “Students learn that multiplication can mean finding the total number of objects in a groups of b objects each, and can be represented by $a\times b$. They then relate the idea of equal groups and the expression $a\times b$ to the rows and columns of an array. In working with arrays, students begin to notice the commutative property of multiplication. In all cases, students make sense of the meaning of multiplication expressions before finding their value, and before writing equations that relate two factors and a product.”

• Unit 5, Fractions as Numbers, Section B, Lesson 7, Activity 1, “The purpose of this activity is for students to practice identifying fractional intervals along a number line. This is Stage 2 of the center activity, Number Line Scoot. This activity encourages students to count by the number of intervals (the numerator). Students have to land exactly on the last tick mark, which represents 4, to encourage them to move along different number lines. While this activity does not focus on equivalence, it gives students exposure to this idea before they work more formally with it in the next section. In the synthesis, students relate counting on a number line marked off in whole numbers to their number lines marked off in fractional-sized intervals. It may be helpful to play a few rounds with the whole class to be sure students are clear on the rules of the game. Keep the number line game boards for center use.”

• Unit 6, Measuring Length, Time, Liquid Volume, and Weight, Section C, Lesson 9, Activity 2, Activity Synthesis, provides teachers guidance on student telling and writing time. "Invite students to share the times they drew the clocks. Emphasize how they distinguish between the hour and minute hands for someone else to be clear on the time they are showing. Consider asking: 'Were there any times that confused you at first or were harder to show?' (For 3:18 I had to draw the hands really close together.) 'Does anyone have suggestions for how to handle some of the times that might be hard to show? When you were drawing a time for your partner, what did you have to keep in mind?' "

The materials reviewed for Open Up Resources K-5 Math Grade 3 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 1, Introduction to Multiplication, Overview, Throughout this Unit, “Number Talks are likewise designed to help students build fluency with equal groups and multiplication expressions. The sequence of expressions encourages students to relate multiplication to skip-counting. For example, in the sequence $1\times10$, $2\times10$, $3\times10$, $4\times10$, students can discover that the products increase in the same way as in skip-counting by 10. Some Number Talks elicit students’ understanding of addition and subtraction within 100 in preparation for the work in an upcoming unit.”

• Unit 4, Relating Multiplication to Division, Section C, Lesson 12, Multiply Multiples of Ten, Lesson Narrative, “The work of this lesson connects to previous work because students have used strategies based on properties of operations to multiply within 100. Now, students extend this work and consider place value to multiply one-digit numbers by multiples of 10. Students complete a problem in context in which they explore how 180 can be grouped into multiples of ten in different ways. Students analyze two strategies for multiplying a single-digit number by a multiple of ten, then complete similar problems using the strategy of their choice. Throughout the lesson the associative property is used as a strategy to think of problems like $3\times60$  as 18 tens or $18\times10$.”

• Unit 7, Two-Dimensional Shapes and Perimeter, Section A, Lesson 1, Preparation, Lesson Narrative, “In previous grades, students sorted shapes into categories based on the attributes of the shape. In this lesson, students revisit this work and learn the terms angle in a shape and right angle in a shape to describe the corners of shapes. This will be helpful in later lessons as students further sort triangles and rectangles by additional attributes.”

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 3 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 3, Wrapping Up Addition and Subtraction Within 1,000, Section D, Lesson 17, Standards, “Addressing: 3.OA.D.8 Solve two-step word problems using the four operations. Represent these problems using equations with a letter standing for the unknown quantity. Assess the reasonableness of answers using mental computation and estimation strategies including rounding. Building Towards: 3.OA.D.9.”

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 represent and solve multiplication problems through the context of picture and bar graphs that represent categorical data.” Additionally, each Unit Section provides Section Learning Goals, “Interpret scaled picture and bar graphs.”

• Unit 2, Area and Multiplication, Section B, Lesson 10, Lesson Narrative, “In previous lessons, students found the area of rectangles with tiles, grids, partial grids, or linear measurements marked along the sides of the rectangle.  Students also used rulers to find the area of rectangles. The problems in this lesson are about a community garden. Consider launching the lesson with a read-aloud of City Green by DyAnne DiSalvo-Ryan to get students thinking about different aspects of a community garden. Students might draw squares within rectangles, draw tick marks on side lengths, count groups, or multiply to find area in the lesson. Any reasoning that makes sense to them is acceptable. 

• Unit 5, Fractions as Numbers, Unit Overview, “Students develop an understanding of fractions as numbers and of fraction equivalence by representing fractions on a diagram and number lines, generating equivalent fractions, and comparing fractions.” 

• Unit 6, Measuring Length, Time, Liquid Volume, and Weight, Section C Overview, “In this section, students learn to tell and write time to the nearest minute and to show given time on an analog clock. They also solve elapsed time problems with an unknown start time, unknown duration, or unknown end time.”

The materials reviewed for Open Up Resources K-5 Math Grade 3 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 3 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 2, Lesson 13 - Gather: Paper Clips, Two-color counters; Copy: Five in a Row Addition and Subtraction Stage 8 Gameboard, Five in a Row Multiplication and Division Stage 2 Gameboard.”

• Unit 7, Two-Dimensional Shapes and Perimeter, Section D, Lesson 14, Materials Needed, “Activities: Colored pencils, crayons, or markers (Activity 1); Centers: Folders (Can You Draw It?, Stage 4), Number cards 0-10 (How Close?, Stage 5).”

The materials reviewed for Open Up Resources K-5 Math Grade 3 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 4, Relating Multiplication to Division, Section B, Lesson 8, Activity 1, Access for Students with Disabilities, “Representation: Comprehension. To support working memory, provide students with sticky notes or mini whiteboards. Provides accessibility for: Memory, Organization.”

• Unit 5, Fractions as Numbers, Section C, Lesson 13, Access for Students with Disabilities, “Engagement: Recruiting Interest. Leverage choice around perceived challenge. Invite students to select at least 3 of the 5 problems in each question to complete. Provides accessibility for: Organization, Attention, Social-Emotional Functioning.” 

• Unit 7, Two-Dimensional Shapes and Perimeters, Section A, Lesson 2, Access for Students with Disabilities, “Representation: Perception. Synthesis: Use gestures during explanation of triangle sorting to emphasize side lengths of triangles. Provides accessibility for: Visual-Spatial Processing.”

The materials reviewed for Open Up Resources K-5 Math Grade 3 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 2, Area and Multiplication, Section A, Problem 11 (Exploration), “How many different rectangles can you make with 36 square tiles? Describe or draw the rectangles. How are the rectangles the same? How are they different?”

• Unit 4, Relating Multiplication to Division, Section B, Practice Problems, Problem 7 (Exploration), “Noah finds $9\times8$ by calculating $(10\times8)-(1\times8)$. a. Make a drawing showing why Noah’s calculation works.; b. Use Noah’s method to calculate $9\times8$.”

• Unit 7, Two-Dimensional Shapes and Perimeter, Section B, Practice Problems, Problem 6, (Exploration), “a. Draw some different shapes that you can find the perimeter of. Then find their perimeters.; b. Can you draw a rectangle whose perimeter is odd? Explain or show your reasoning.; c. Can you draw a pentagon or hexagon (or a figure with even more sides) whose perimeter is odd?”

The materials reviewed for Open Up Resources K-5 Math Grade 3 meet expectations for providing strategies and supports for students who read, write, and/or speak in a language other than English to regularly participate in learning grade-level mathematics.

Guidance is consistently provided to teachers to support students who read, write, and/or speak in a language other than English, providing scaffolds for them to meet or exceed grade-level standards. According to the Resources, Course Guide, Mathematical Language Development and Access for English Learners, “In an effort to advance the mathematics and language learning of all students, the materials purposefully engage students in sense-making and using language to negotiate meaning with their peers. To support students who are learning English in their development of language, this curriculum includes instruction devoted to fostering language development alongside mathematics learning, fostering language-rich environments where there is space for all students to participate.” Examples include:

• Unit 3, Wrapping Up Addition and Subtraction Within 1,000, Section B, Lesson 12, Activity 1, “Access for English Learners - Conversing, Representing: MLR8 Discussion Supports. Display sentence frames to support partner discussion: ‘Can you say more about …? and Why did you …?’”

• Unit 4, Relating Multiplication to Division, Section C, Lesson 13, Activity 2, "Access for English Learners - Representing, Conversing: MLR7 Compare and Connect. Synthesis: After the Gallery Walk, lead a discussion comparing, contrasting, and connecting the different representations. How did the number of chairs show up in each method? Why did the different approaches lead to the same outcome? To amplify student language, and illustrate connections, follow along and point to the relevant parts of the displays as students speak."

• Unit 7, Two-Dimensional Shapes and Perimeter, Lesson 1, Activity 1, “Access for English Learners - Conversing, Reading: MLR2 Collect and Display. Collect the language students use to sort the cards into categories. Display words and phrases such as: “equal sides,” “equal lengths,” “corners,” “diagonal,” “straight,” “curved,” “slanted,” and “shaded.” During the synthesis, invite students to suggest ways to update the display: ‘What are some other words or phrases we should include?” etc. Invite students to borrow language from the display as needed.’"

The materials reviewed for Open Up Resources K-5 Math Grade 3 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, Area and Multiplication, Section B, Lesson 9, Activity 2, “The purpose of this activity is for students to create a rectangle with a given area. Students use what they know about area and the structure of rectangles to decide on the side lengths of the rectangle. Students use tape (painter’s or masking) to create the rectangles. They should have enough tape to create square feet within the rectangle, but should be encouraged to mark the 1 foot intervals to help them visualize the square feet inside the rectangle, if needed. In the synthesis, each group shares strategies for creating a rectangle and how they know the area is the given number of square feet. When students think about the structure of a rectangle and use it to create a rectangle with a given area they are looking for and making use of structure (MP7).”

• Unit 4, Relating Multiplication to Division, Section B, Lesson 10, Activity 2, “Groups of 2. ‘Take a minute to read the directions of the activity. Then, talk to your partner about what you are asked to do.’ 1 minute: quiet think time. 1 minute: partner discussion. Answer any clarifying questions from students. Give students access to colored pencils, crayons, or markers. ‘Mark or shade each diagram to represent how each student found the area.’ 3–5 minutes: independent work time. ‘Share with your partner how you used the rectangles to show each expression.’ 3–5 minutes: partner discussion.”

• Unit 8, Putting It All Together, Section C, Lesson 10, Activity 1, “The purpose of this activity is for students to relate multiplication and division using a variety of representations. Students are given a card with a base ten diagram, tape diagram, area diagram, multiplication equation with a missing factor, or division equation. Students need to find the other student who has the card that matches their card. Each pair of cards includes a division equation. After students find the student with the matching card, they work together to create another diagram and a division situation that their cards could represent (MP2).”