3rd Grade - Gateway 2

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

• Unit 1, Introducing Multiplication, Lesson 12, Warm-up, students develop conceptual understanding as they use grouping strategies to describe the images they see. An image is provided that shows groups of dots. Launch, “Groups of 2. How many do you see? How do you see them? Flash image. 30 seconds: quiet think time, Student Task Statements, How many do you see? How do you see them?” Activity Synthesis, “What pattern was helpful in finding the total number of dots? Consider asking: Who can restate the way _____ saw the dots in different words? Did anyone see the dots the same way but would explain it differently? Does anyone want to add an observation to the way ______ saw the dots?” (3.OA.1)

• Unit 4, Relating Multiplication to Division, Lesson 2, Activity 1, students develop conceptual understanding as they discuss how many in each group division questions with objects and drawings. Activity, “‘Solve these problems and show your thinking using objects, a drawing, or a diagram. 5–7 minutes: independent work time. As student work, consider asking: How can you represent what you are thinking? ‘Where can you see the boxes in your work? Where can you see how many apples are in each box in your work? Monitor for students who solve the first problem in the same way. Arrange them into groups of 2 to create a poster together. Now you are going to create a poster to show your thinking on the first problem. You are going to work with a partner who solved the problem in the same way you did. Give each group tools for creating a visual display. 5–7 minutes: partner work time.” Student Task Statements, “Solve each problem. Show your thinking using objects, a drawing, or a diagram. 1. If 20 apples are packed into 4 boxes with each box having the same number of apples, how many apples are in each box?, 2. If 36 apples are packed into 6 boxes with each box having the same number of apples, how many apples are in each box?, 3. If 45 apples are packed into 9 boxes with each box having the same number of apples, how many apples are in each box?” (3.OA.2)

• Unit 8, Putting It All Together, Lesson 2, Warm-up, students develop conceptual understanding as they compare fractions on a number line. An image of different four number lines with fractions is provided. Launch, “Groups of 2. Display the image. Pick one that doesn’t belong. Be ready to share why it doesn’t belong. 1 minute: quiet think time.” Activity synthesis, “Let’s find at least one reason why each one doesn’t belong.” (3.NF.2)

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

• Unit 2, Area and Multiplication, Lesson 9, Cool-down, students independently demonstrate conceptual understanding as they measure a rectangle using rulers and then calculate its area. “Use your ruler to find the area of the rectangle in square inches.” The Cool-down includes one rectangle image. (3.MD.7)

• Unit 4, Relating Multiplication to Division, Section A, Practice Problems, students interpret arrays as multiplication expressions and equations. Problem 1 shows an array with 4 rows and 5 columns of blue circles, “a. Write a multiplication expression that represents the array., b. Write a multiplication equation that represents the array.” (3.OA.1)

• Unit 5, Fractions as Numbers, Lesson 3, Activity 1, students write and read fractions that relate to their images. Student Task Statements, “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)

The materials reviewed for Imagine Learning Illustrative Mathematics Grade 3 meet expectations for giving attention throughout the year to individual standards that set an expectation of procedural skill and fluency.

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

• Unit 3, Wrapping Up Addition and Subtraction Within 1,000, Lesson 16, Warm-up, students use strategies and understanding for finding the products of 4 and 6. Student Task Statements, “Find the value of each expression mentally. 5\times7, 4\times7, 6\times7, 4\times8.” (3.OA.7)

• Unit 4, Relating Multiplication to Division, Lesson 3, Warm-up, students add two three-digit numbers. Student Task Statements, “Find the value of each expression mentally. 120+120, 121+119, 125+115, 129+111.” (3.NBT.2)

• Unit 7, Two-dimensional Shapes and Perimeter, Lesson 7, Warm-up, students use understanding as they add multi-digit numbers. Student Task Statements, “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) 

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

• Unit 3, Wrapping Up Addition and Subtraction Within 1,000, Lesson 3, Cool-down, students add two three-digit numbers. Student Task Statements, “Find the value of 258+217. Explain or show your reasoning.” (3.NBT.2)

• Unit 6, Measuring Length, Time, Liquid Volume, and Weight, Lessons 14, Centers 15 and 16. In the Center, Compare (1–5), students work in pairs to find the product of two factors before their partner. The center is described in the teacher resource pack. “Both partners flip over a card, and the partner whose card has the greater value takes both cards. The game is over when each partner runs out of cards to flip over. The partner with the most cards wins. Stage 3: Multiply within 100.” (3.OA.7)

• Unit 8, Putting It All Together, Lesson 8, Activity 1, students quiz their partner and rate how well they know basic multiplication facts. Student Task Statements, “Quiz your partner on their multiplication facts and sort your partner’s facts into one of these columns: 1. know it right away, 2. can find it quickly, 3. don’t know it yet. Multiplication expressions I’m going to practice: 1., 2., 3., 4., 5.” (3.OA.7)

The materials reviewed for Imagine Learning Illustrative Mathematics Grade 3 meet expectations for being designed so that teachers and students spend sufficient time working with engaging applications of the mathematics.

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

Examples of routine applications include:

• Unit 1, Introducing Multiplication, Lesson 19, Cool-down, students solve real world problems involving multiplication (3.OA.3). Student Task Statements, “Clare has 3 rows of baseball cards. Each row has 10 cards. How many cards does she have? 1. Write an equation with a symbol for the unknown number to represent the situation. 2. Find the number that makes the equation true. Explain or show your reasoning.” 

• Unit 4, Relating Multiplication to Division, Lesson 1, Cool-down, students solve real world problems involving division (3.OA.3). “Lin has 30 apples to share with her friends. She is putting them in bags, with 6 apples in each bag. How many bags does she need? Explain or show your reasoning.”

• Unit 7, Two-dimensional Shapes and Perimeter, Lesson 9, Cool-down, students solve a word problem involving perimeter (3.MD.8). Student Task Statements, “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.”

Examples of non-routine applications include:

• Unit 3, Wrapping Up Addition and Subtraction Within 1,000, Lesson 17, Cool-down, students solve a word problem involving two steps (3.OA.8). Student Task Statement, “In the bin there are 124 beads. Ninety-six more beads are dumped in the bin. Then 53 beads are used to make a bracelet. Tyler says there are 273 beads in the bin now. Explain why Tyler’s statement doesn’t make sense.” 

• Unit 6, Measuring Length, Time, Liquid Volume, and Weight, Lesson 15, Activity 1, students solve real-world problems involving time, weight, and volume (3.MD.1, 3.MD.2, 3.OA.3). Launch, “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.” Student Task Statements, “You spent a day at the fair. Solve four problems about your day and create a poster to show your reasoning and solutions. 1. 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? 2. 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? 3. What was next? (Choose one.) You visited a life-size sculpture of a cow made of butter. The butter cow weighs 273 kilograms, which is 277 kilograms less than 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?”

• Unit 8, Putting It All Together, Lesson 4, Activity 2, students relate area and perimeter to operations of multiplication and addition while solving real world problems (3.MD.7,3.MD.8). Launch states, “Groups of 2 “Work independently to write two questions that could be answered using your tiny house design. One question should be about area and the other about perimeter.” 3–5 minutes: independent work time.” Activity states, “Share your questions with your partner and answer them together. Revise your questions if needed.” 5 minutes: partner work time “Find a new partner and answer each other’s questions. Be sure to share your tiny house design with your new partner.” Student Task Statements state, “1. Write two questions about your tiny house design: a. one question that involves area b. one question that involves perimeter. 2. Work with a partner to answer your own questions about your tiny house design. Make any revisions to your questions if needed. 3. Find a new partner. Answer their questions about their tiny house design.”

The materials reviewed for Imagine Learning Illustrative Mathematics 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. 

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

All three aspects of rigor are present independently throughout Grade 3. 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, Lesson 4, Activity 2, students develop procedural skill and fluency as they add within 1,000 using strategies and algorithms. Student Task Statements, “Try using an algorithm to find the value of each sum. Show your thinking. Organize it so it can be followed by others. 1. 475+231, 2. 136+389, 3. 670+257.” (3.NBT.2) 

• Unit 5, Fractions as Numbers, Lesson 3, Activity 2, students extend their conceptual understanding as they match fractions and diagrams. Launch, “‘We’re going to play a game in which you match fractions and diagrams. Read the directions to the game with your partner and discuss any questions you have about the game.’ Answer any questions about the game. Give each group one set of cards created from Fraction Match Part 1.” Activity, “‘Play one round of Fraction Match with your partner.’ 5–7 minutes: partner work time. Give each group one set of Fraction Match Part 2. ‘Before you play another round, work with your partner to create 4 new pairs of cards to add to the set. Partition and shade a diagram to match each fraction.’ 3–5 minutes: partner work time. ‘Now, play another round of fraction match with your partner using all the cards.’” Student Task Statements, “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)

• Unit 7, Two-dimensional Shapes and Perimeter, Lesson 9, Activity 1, students apply their understanding to find unknown side lengths given the perimeter of a shape. Launch, “In an earlier lesson, we found the perimeter of shapes when not all the side lengths were labeled. Now, let’s find some missing side lengths when we know the perimeter.” Student Task Statements, Problem 2, “This rectangle has a perimeter of 56 feet. What are the lengths of the unlabeled sides? Explain or show your work. A rectangle shows a width of 8 feet.” (3.MD.8)

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

• Unit 3, Wrapping Up Addition and Subtraction Within 1,000, Lesson 17, Activity 2, students develop conceptual understanding alongside application as they use estimation strategies to see if two-step word problems make sense. Launch, “Now you are going to solve for the exact answer to some problems. As you do so, think about how estimating could help you decide if an answer makes sense.” Activity, “‘Work with your partner and decide who will solve each problem. Then, work independently to solve your problem.’’ 3–5 minutes: independent work time. “‘Now, trade work with your partner and decide whether their answer for the problem they solved makes sense. Record your thoughts on your partner’s paper for them to refer back to if they want to adjust their answer.’ 3–5 minutes: independent work time. ‘Take turns sharing your thoughts on your partner’s work. Give your partner a chance to share how they solved their problem.’” Student Task Statements, “1. Solve one of the problems. Explain or show your reasoning. a. Jada has 326 beads. She gives her friend 32 beads. Then, Jada uses 84 beads to make a bracelet for her cousin. How many beads does Jada have now? b. Noah starts an art project on Monday and uses 624 beads. On Tuesday he uses 132 more beads. Finally, on Wednesday he finishes the project by using 48 more beads. How many beads did Noah use on his art project? 2. Trade work with a partner. Decide whether your partner’s answer for their problem makes sense. On their paper, explain your reasoning.” (3.OA.8)

• Unit 4, Relating Multiplication to Division, Lesson 19, Activity 1, students develop conceptual understanding alongside procedural skill and fluency as they represent quotients greater than 10. Student Task Statements, Problem 2. “Find the value of each expression. Use base-ten blocks if you find them helpful. a. 63\div3. b. 84\div7. c. 100\div5.” (3.OA.5 and 3.OA.7)

• Unit 7, Two-dimensional Shapes and Perimeter, Lesson 6, Activity 2, students use procedural fluency as they apply their understanding to find the perimeter of shapes. Launch, “Earlier, we used paper clips to measure the distance around shapes. What are some other units we could use to measure distances or lengths? (The side length of a square on grid paper, the distance between points on dot paper, centimeter, inch, foot). Let’s find the length of the perimeter of some shapes on dot paper and some shapes whose side lengths are shown with tick marks.”  Student Task Statements, “Find the perimeter of each shape. Explain or show your reasoning.” (3.MD.8)

The materials reviewed for Imagine Learning Illustrative Mathematics 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 in several places including the Instructional Routines (Warm-up Routines and Other Instructional Routines), Activity Narratives, and About this lesson.

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

• Unit 1, Introducing Multiplication, Lesson 8, Activity 2, students use data from a scaled bar graph to answer questions. Student Task Statements, “The bar graph shows how many of each type of tree Clare saw on the way home. Use the graph to answer the questions. Show your thinking using expressions or equations.1. How many more pine trees did Clare see than fir trees? 2. How many more pine trees did Clare see than oak or maple trees? 3. How many fewer oak trees did Clare see than pine trees? 4. How many fewer maple or oak trees did Clare see than fir trees?” Activity Narrative, “The Three Reads routine has students read a problem three times for different purposes to support them to make sense of the problem and persevere in solving it (MP1).” 

• Unit 4, Relating Multiplication to Division, Lesson 21, Activity 1, students consider relationships between different quantities as they solve two-step problems using the four operations. Activity, “‘A list of numbers is shown in the activity. Work with your partner to choose 4 numbers that would make sense together in this situation. If you find one combination of numbers that works, you can look for other combinations.’ 8–10 minutes: partner work time. Groups of 4. ‘Share with another group of students how your number choices make sense.’ 2–3 minutes: small-group discussion.” Student Task Statements, “A farmer picked some apples. Some of the apples are packed into boxes and some are not. From the list, choose 4 numbers that would make sense together in this situation. Write your choices in the table. Be ready to explain how your numbers make sense together.” The Narrative states, “Students who do not choose a matching set of numbers quickly make sense of and persevere in solving the problem as they consider the relationship between the different quantities and the restrictions that puts on which numbers can describe the situation (MP1).”

• Unit 6, Measuring Length, Time, Liquid Volume, and Weight, Lesson 13, Activity 1, students discuss and make sense of the weight of a pumpkin using the info gap structure. The Activity Narrative, “This activity uses MLR4 Information Gap. The Info Gap structure requires students to make sense of problems by determining what information is necessary, and then to ask for information they need to solve it. This may take several rounds of discussion if their first requests do not yield the information they need (MP1). It also allows them to refine the language they use and ask increasingly more precise questions until they get the information they need (MP6). This Info Gap activity provides students an opportunity to solve multiplication and division problems involving weight.” The teacher instructs the student, “‘I will give you either a problem card or a data card. Silently read your card. Do not read or show your card to your partner.’ Distribute cards. 1–2 minutes: quiet think time, Remind students that after the person with the problem card asks for a piece of information the person with the data card should respond with ‘Why do you need to know (restate the information requested)?’ The first problem card says, ‘A fair is holding a pumpkin weigh-off. The farmer who grew the winning pumpkin says during some days in August, his pumpkin gained a lot of weight each day. How much did the weight of the pumpkin increase during this time?’ The first data card says, ‘The pumpkin’s weight increased 13 kilograms each day during these days in August. There were 5 days in August when the pumpkin gained this much weight each day.’” 

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

• Unit 3, Wrapping Up Addition and Subtraction within 1,000, Lesson 19, Activity 2, students reason abstractly and quantitatively when they create diagrams and equations based on a situation. The Activity Narrative, “Previously, students matched diagrams and equations to situations with an unknown quantity. Here, they generate such equations, using a letter for the unknown quantity, solve problems, and explain how they know their answers make sense. Students should be encouraged to use any solving strategy they feel comfortable with. If not yet addressed, mention that any letter can be used for the unknown quantity in their equation. While this activity is focused on independent practice, encourage students to discuss the problem with a partner if needed. Though the task asks students to write an equation first, students may complete the task in any order that makes sense to them. Students reason abstractly and quantitatively when they write an equation that represents the situation (MP2).” The Student Task Statement, “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? 1. Write an equation to represent the situation. Use a letter for the unknown quantity. 2. Solve the problem. Explain or show your reasoning. 3. Explain how you know your answer makes sense.”

• Unit 6, Measuring Length, Time, Liquid Volume, and Weight, Lesson 5, About this lesson, “In a previous lesson, students analyzed line plots that included measurements in halves and fourths of an inch. In this lesson, students collect measurement data, represent them on a line plot, and analyze line plots that represent different data sets (MP2).” Cool-down, students use data and represent it on a line. “The list shows lengths of leaves in inches. Use the measurements to complete the line plot. 2\frac{3}{4}, 3, 2, 3\frac{1}{4}, 4\frac{1}{2}, 3\frac{1}{4}, 2\frac{3}{4}, 2\frac{1}{2} .”

• Unit 7, Two-dimensional Shapes and Perimeter, Lesson 9, Activity 2, students reason abstractly and quantitatively when they solve problems about perimeter. Activity Narrative, “The purpose of this activity is for students to solve problems in situations that involve perimeter (MP2). Students may draw diagrams with length labels or simply reason arithmetically. They also explain how each problem does or does not involve perimeter. The activity synthesis provides an opportunity to begin discussing the difference between area and perimeter, which will be fully explored in upcoming lessons.” The Student Task Statements, “Solve each problem. Explain or show your reasoning. 1. A rectangular park is 70 feet on the shorter side and 120 feet on the longer side. How many feet of fencing is needed to enclose the boundary of the park? 2. Priya drew a picture and is framing it with a ribbon. Her picture is square and one side is 9 inches long. How many inches of ribbon will she need? 3. A rectangular flower bed has a fence that measures 32 feet around. One side of the flower bed measures 12 feet. What are the lengths of the other sides? 4. Kiran took his dog for a walk. Their route is shown. How many blocks did they walk?”

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

Examples of constructing viable arguments include:

• Unit 1, Introducing Multiplication, Lesson 16, Activity 1, students construct a viable argument and critique the reasoning of others when they compare a drawing of four groups of five to a four by five array. The Activity Narrative says, “The purpose of this activity is for students to describe an array as an arrangement of objects into rows with an equal number of objects in each row and into columns with an equal number in each column. This will be helpful in the next activity when students arrange objects into arrays and describe arrays in terms of multiplication. When students decide whether or not they agree with Noah about seeing equal groups in the array and explain their reasoning, they construct a viable argument and critique the reasoning of others (MP3).” Student Task Statements include 2 illustrations, the first of 4 groups of 5 shown in individual circles, the second of a 4 by 5 array. “1. How does arranging the dots into an array affect how you see the number? 2. Noah says he sees equal groups in the drawing with 4 circles and 5 dots in each circle, but says there are no equal groups in the array. Do you agree with Noah? Explain your reasoning.”

• Unit 2, Area and Multiplication, Lesson 1, Activity 1, students construct viable arguments as they compare shapes. Launch, “‘Display or sketch the two triangles in the first problem. Which triangle do you think is larger? Be prepared to explain your reasoning.’ 1 minute: quiet think time. ‘Share and record responses. How could you decide for sure which shape is larger?’ (I could think about putting one shape on top of the other. I could measure which is longer. I could cut one out to see if it fits inside the other.)” Student Task Statements, “1. Here are two triangles. Which triangle is larger? 2. In each pair of shapes, which shape is larger? Be prepared to explain your reasoning.” The Activity Narrative, “If students disagree about which shape is larger, encourage them to share their reasoning so that the class can consider multiple ideas and come to a resolution together (MP3).”

• Unit 4, Relating Multiplication to Division, Lesson 5, Activity 1, students construct viable arguments as they write division expressions. Student Task Statements, “1. Your teacher will give you a set of cards that show situations. Sort the cards into 2 categories of your choosing. Be prepared to explain the meaning of your categories. A. Mole crickets have special legs for digging. Ten special legs belong to 5 mole crickets. How many special legs does each mole cricket have? B. A beetle has a pair of antennae for sensing heat, touch, smell, and more. If there are 8 antennae, how many beetles are there? C. Fourteen antennae belong to a group of bees. If each bee has 2 antennae, how many bees are there? D. There are 12 wings. If each dragonfly has 4 wings, how many dragonflies are there? E. Thirty legs belong to 5 ants. If all the ants have the same number of legs, how many legs does each ant have? F. There are 50 spots on 5 butterflies. If each butterfly has the same number of spots, how many spots does each butterfly have? 2. Write a division expression to represent each situation. Be ready to explain your reasoning.” The Activity Narrative, “As students discuss and justify their decisions, they share a mathematical claim and the thinking behind it (MP3).”

Examples of critiquing the reasoning of others include:

• Unit 3, Wrapping Up Addition and Subtraction within 1,000, Lesson 8, Cool-down, students compare a base ten diagram showing 386 - 267 to the same problem completed using the standard vertical algorithm. The work is completed which requires them to analyze the reasoning of others, and their explanation requires them to create their own argument. Cool-down, “Explain how the diagram matches the algorithm.” The Cool-down is similar to Activity 1, Activity Narrative, “The purpose of this activity is for students to use their knowledge of base-ten diagrams and place value to make sense of a subtraction algorithm. Students notice that in both the base-ten drawing and the algorithm, the subtraction happens by place. We can find the difference of two numbers by subtracting ones from ones, tens from tens, and hundreds from hundreds, and adding these partial differences to find the overall difference. Students also recall that sometimes a place value unit needs to be decomposed before subtracting. For example, a ten may first need to be decomposed into 10 ones. This decomposition can be seen in both the base-ten drawing and in the algorithm. In the synthesis, students interpret the work and reasoning of others (MP3).”

• Unit 5, Fractions as Numbers, Lesson 16, Activity 1, students critique the reasoning of others as they compare fractions with the same numerator. Activity, “‘Talk to your partner about who you agree with. Use diagrams or number lines to show your thinking.’ 3–5 minutes: partner discussion. As students work, consider asking: ‘How does your representation show which fraction is greater? How do you know that eighths are smaller than sixths?’ Monitor for students who use diagrams and those who use number lines. Pause for a discussion. Select two students, one who uses each representation, to share. Display their work side-by-side for all to see.” 5–7 minutes: independent or partner work time. Student Task Statements, ‘1. Priya says that \frac{5}{6} is greater than \frac{5}{8}. Tyler says that \frac{5}{8} is greater than \frac{5}{6}. Who do you agree with? Show your thinking using diagrams or number lines.’”

• Unit 7, Two-dimensional Shapes and Perimeter, Lesson 10, Activity 1, students critique the reasoning of others and construct viable arguments. The Activity Narrative states, “The purpose of this activity is for students to differentiate methods for finding perimeter from those for finding area. While addition and multiplication are both involved in various ways, students need to understand the problem situation and think about whether the operations performed will provide the desired information. As in earlier problems, students can find perimeter in various ways. The emphasis should be on how understanding the problem situation and the information given should inform the solution method. When students analyze claims about how to use addition and multiplication to find the perimeter of a rectangle they construct viable arguments (MP3).” Student Task Statements, “Andre wants to know how much rope is needed to enclose the new rectangular school garden. The length of the garden is 30 feet. The width of the garden is 8 feet. Clare says she can use multiplication to find the length of rope Andre needs. Diego says he can use addition to find the length of rope Andre needs. Who do you agree with? Explain or show your reasoning.”

The materials reviewed for Imagine Learning Illustrative Mathematics Grade 3 meet expectations for supporting the intentional development of MP4: Model with mathematics; and MP5: Use appropriate tools strategically, for students, in connection to the grade-level content standards, as expected by the mathematical practice standards. Students have opportunities to engage with the Math Practices across the year and they are often explicitly identified for teachers in several places including the Instructional Routines (Warm-up Routines and Other Instructional Routines), Activity Narratives, and About this Lesson.

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

• Unit 1, Introducing Multiplication, Lesson 21, Activity 1, students model with mathematics as they design a seating arrangement. Activity Narrative, “Students make their own decisions about other aspects of the scenario before planning their seating arrangement and also choose how to represent their seating arrangement (MP4).” Student Task Statements, “Your club is planning a game night. Guests can play one of four different games that require a different number of players: Game A - 2 players. Game B - 4 players. Game C - 5 players. Game D - 10 players. The game room has 16 identical square tables, where one person can sit on each side. 1. Make a seating plan that shows the table arrangement so that each guest can play one of the games. 2. Make a poster that includes: a. a seating chart,b. an explanation about how you decided on your seating plan, c. how many people can play games in the room with your seating plan.” 

• Unit 5, Fractions as Numbers, Lesson 18, Activity 2, students model with mathematics as they use geometric designs to demonstrate their understanding of fractions. About this Lesson, “To mark a given length, students apply their experience with partitioning a segment into equal parts. To mark a fractional length, they decide which endpoint of each side to use as a starting point, whether to always mark the points in the same direction (clockwise or counterclockwise), how many iterations are practical, and so on (MP4).” Student Task Statements, “1. Here is another square. On each side, mark a point to show \frac{1}{4} of its length. Connect each point to the point on the two sides next to it. What shape did you create? 2. Look at the new shape you created. On each side, mark a point to show \frac{1}{4} of its length. Connect the points again. What shape did you create? 3. Repeat the steps you just did at least two more times. Make some observations about the design you just created.”

• Unit 8, Putting It All Together, Lesson 5, Activity 1, students model with mathematics as they use operations to calculate the costs of finishing a tiny house. About this Lesson, “Students engage in aspects of mathematical modeling as they make decisions about quantities, relate measurements and costs, and interpret their results in context (MP4).” Student Task Statements, “Choose a room from your tiny house to finish. Use the cost sheet to calculate the cost of finishing the room in your tiny house. Your budget is $1,000.” A table is provided with the costs and various items.

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

• Unit 3, Wrapping Up Addition and Subtraction within 1,000, Lesson 3, Activity 1, students select the appropriate tool to add numbers. The Activity Narrative states, “The purpose of this activity is for students to add within 1,000 using any strategy that makes sense to them. The expressions in this activity give students a chance to use different strategies, such as adding hundreds to hundreds, tens to tens, and ones to ones, reasoning with numbers close to a hundred, or using a variety of representations. Students who use base-ten blocks or draw number line diagrams choose appropriate tools strategically (MP5).” Student Task Statements, “Find the value of each sum in any way that makes sense to you. Explain or show your reasoning. 1. 325+102, 2. 301+52, 3.276+118, 4. 298+305.” The Cool-down provides another opportunity for students to show their thinking using the appropriate tools, “Find the value of 258+217. Explain or show your reasoning.”

• Unit 4, Relating Multiplication to Division, Lesson 13, Activity 1, students use appropriate tools strategically when they multiply within 100. Activity Narrative, “This is the first time students have worked with problems with numbers in this range, so they should be encouraged to use the tools provided to them during the lesson if they choose (MP5).” Student Task Statements, “Solve each problem. Show your thinking using objects, a drawing, or a diagram. 1. A seller at a farmers market has 7 dozen eggs when they close for the day. How many eggs does the seller have? 2. At the farmers market there’s a space for performers to play music with some chairs for people to sit and listen. There are 5 rows of chairs and each row has 15 chairs. How many chairs are there? 3. A booth at a farmers market has a table top that has lengths of 4 feet and 16 feet. What is the area of the table top? Students should also be encouraged to use strategies and representations from the previous section.”

• Unit 6, Measuring Length, Time, Liquid Volume, and Weight, Lesson 4, Activity 1, students engage with MP5 as they analyze line plots. Activity Narrative, “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).” Students are given the heights of the seedlings in inches and a line plot. Student Task Statements, “1. Write 3 statements about the measurements represented in the line plot. 2. What questions could be answered more easily with the line plot than the list? Write at least 2 questions.”

The materials reviewed for Imagine Learning Illustrative Mathematics 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 many opportunities to attend to precision and to attend to the specialized language of mathematics in connection to grade-level work. This occurs with the support of the teacher as well as independent work throughout the materials. Examples include:  

• Unit 1, Introducing Multiplication, Lesson 14, Activity 1, students use precise language as they explain how numbers and symbols match their equations. Launch, “Display: 4\times5=? ‘What might this equation mean? Different symbols can be used to represent the unknown number in an equation. Some that are common are question marks, blank spaces, and boxes. For example, in the equation 80=8\times10, if we didn’t know the product we could write ?=8\times10. Display these equations as you explain. If we didn’t know one of the factors, what is an equation you could write using a symbol for the unknown number?’ Distribute one set of pre-cut cards to each group of students.” Student Task Statements, “Your teacher will give you a set of cards. Match each equation with a situation or diagram.” Activity Narrative, “Students explain their matches to their peers and revise their language for precision and clarity when they describe how the numbers and symbols in the equations match the representations (MP3, MP6).” In the Synthesis, students explain the meaning of the factors and products, and what a symbol in an equation represents.

• Unit 5, Fractions as Numbers, Lesson 9, Activity 1, students use precision as they use a number line to locate fractions and 1. Student Task Statements, “2. Use any of the number lines to explain how you located 1.” Activity Narrative, “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).”

• Unit 7, Two-dimensional Shapes and Perimeter, Lesson 4, Cool-down, students use precise language to explain how they know which shapes are rhombuses. Lesson Narrative, “In previous lessons, students learned how to compare and describe shapes using geometric attributes. In this lesson, students analyze examples and non-examples of rectangles, rhombuses, and squares in order to identify their defining attributes. As they discern and describe features that define these quadrilaterals, students practice looking for structure (MP7) and communicating with precision (MP6).” Cool-down Task Statement, “‘Select all of the quadrilaterals that are rhombuses. Explain your reasoning.’ The images include a. [a kite shape], b. [a square rotated 45 degrees as if it is balancing on a vertex], c. [a parallelogram with a longer width than height], d. [a rhombus that is slightly askew], e. [a quadrilateral with 4 different-length sides and 1 right angle]. In the same lesson, students also use precise language during the warm-up when they describe which shape doesn’t belong. Activity Narrative, “This warm-up prompts students to compare four shapes. It gives students a reason to use language precisely (MP6). It gives the teacher an opportunity to hear how students use terminology and talk about the characteristics of the items in comparison to one another. During the synthesis, emphasize that three of the shapes are quadrilaterals, even though they look very different.”

• Unit 8, Putting it All Together, Lesson 3, Activity 1, students attend to precise language as they reason about statements related to fractions. Activity Narrative, “The purpose of this activity for students is to think about and discuss statements that address their understanding of important ideas about fractions. Students will consider ideas about how fractions are defined, comparing fractions, and how fractions relate to whole numbers. It is not necessary for each group to discuss all of the statements, but if there are any you’d like to make sure each group discusses, let them know at the start of the activity. Students construct viable arguments to explain their choices (MP3) and in order to do so they need to use key fraction language, such as whole and equal-size pieces, precisely (MP6).” Student Task Statements, “Discuss each statement in 3 rounds with your group. Round 1: Go around the group and state whether you agree, disagree, or are unsure about the statement and justify your choice. You will be free to change your response in the next round. Round 2: Go around the group and state whether you agree, disagree, or are unsure about the statement you or someone else made in the first round. You will be free to change your response in the next round. Round 3: State and circle the word to show whether you agree, disagree, or are unsure about the statement now that discussion has ended. Repeat the rounds for as many statements as you can.” The statements include, “a, A fraction is a number less than 1, b. A fraction can be located on a number line, c. The numerator tells us the size of the part, d. The denominator tells us the number of parts, e. Whole numbers are fractions, f. Fractions are whole numbers, g. One half is always greater than one third, h. Fractions can be used to describe a length.”

The materials reviewed for Imagine Learning Illustrative Mathematics 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 throughout the year.

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

• Unit 4, Relating Multiplication to Division, Lesson 2, Activity 3, students use the structure of multiplication problems to compare multiplication situations. Activity Narrative, “The purpose of this activity is for students to consider what is the same and what is different about the ’how many groups?’ and ‘how many in each group?’ problems they solved in a previous lesson and in this lesson. The discussion should highlight that in ‘how many groups?’ problems we know the size of each group and in ‘how many in each group?’ problems we know how many groups there are. In order to describe how the problems are the same and how they are different, students attend to the structure of the problems, that is what is given in each situation and what is unknown (MP7).” Student Task Statements, “If 24 apples are put into boxes with 8 apples in each box, how many boxes are there? If 20 apples are packed into 4 boxes with each box having the same number of apples, how many apples are in each box? Discuss with your partner: How are these problems alike? How are they different? What is alike and what is different about how these problems are represented and solved?” Activity Synthesis, “‘What did you and your partner notice was alike? What did you and your partner notice was different? Share and record responses’. As students share, encourage them to use the posters to show examples of what they notice.”

• Unit 5, Fractions as Numbers, Lesson 8, Warm-up, students use the structure of known division facts to find unknown answers when mentally solving division problems. Activity Narrative, “This Number Talk encourages students to rely on their knowledge of multiplication, place value, and properties of operations to mentally solve division problems. The reasoning elicited here helps to develop students' fluency with multiplication and division within 100. To find the quotients of larger numbers, students need to look for and make use of structure in quotients that are smaller or more familiar, or to rely on the relationship between multiplication and division (MP7).” Student Task Statements, “Find the value of each expression mentally. 12\div4, 24\div4, 60\div4, 72\div4.” Activity Synthesis, “How did the earlier expressions help you find the value of the later expressions? Consider asking: Did anyone have the same strategy but would explain it differently? Did anyone approach the problem in a different way?”

• Unit 7, Two-dimensional Shapes and Perimeter, Lesson 2, Warm-up, students look for and make use of structure as they use strategies to multiply a one digit number by a multiple of ten. Student Task Statements, “Decide if each statement is true or false. Be prepared to explain your reasoning. 3\times60=9\times10, 180=3\times60, 6\times40=24\times10, 24\times10=240.” 

• Lesson Narrative, “When students use place value or properties of operations as strategies to divide, they look for and make use of structure (MP7).” Activity Synthesis, “How can you explain your answer without finding the value of both sides? Consider asking: Who can restate _____’s reasoning in a different way? Does anyone want to add on to _____’s reasoning?”

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

• Unit 1, Introducing Multiplication, Lesson 11, Warm-up, students notice patterns as they count by 5’s and 2’s. Lesson Narrative, “When students notice patterns in the count, such as in the count by 5 that the ones place alternates between 0 and 5, they look for and express regularity in repeated reasoning (MP8).” Launch states, “‘Count by 5, starting at 0.’ Record as students count. See Student Responses for recording structure. ‘Stop counting and recording at 50.’” Activity states, “‘What patterns do you see?’ 1-2 minutes: quiet think time.’Record responses. Repeat activity. Count by 2, start at 0 and stop at 20.’”  Activity Synthesis, “How could some of the patterns help you with counting by these numbers? (I know that the next count by 5 should end in 5. I know that the next count by 2 should have a 2 in the ones place.) 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?”

• Unit 4, Relating Multiplication to Division, Lesson 9, Cool-down, students use repeated reasoning as they fill in missing numbers on a multiplication chart. Lesson Narrative, “Students may have worked with the multiplication table in an optional lesson in a previous unit. In this lesson, they observe patterns and structures in the multiplication table that highlight properties of multiplication and are helpful for multiplying numbers. Although there is an opportunity to highlight multiple properties, the focus of this lesson is the commutative property (though students are not expected to name the property). Students notice that multiplying two numbers in any order gives the same product and make use of this observation to find unknown products (MP8).” Cool-down Task Statement, “‘What number should replace the question mark? Explain or show your reasoning.’ 10 by 10 multiplication chart is included that shows 1 product in the first row, 2 in the second, 3 in the third, and so on. The question mark is placed on a blank square representing 4 times 8.’”

• Unit 6, Measuring Length, Time, Liquid Volume, and Weight, Lesson 10, Warm-up students use repeated reasoning as they choral count by 15-minute increments. Activity Narrative, “The purpose of this Choral Count is to invite students to practice counting time 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, “‘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.’” Activity Synthesis, “‘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?’”