3rd Grade - Gateway 2

The materials reviewed for i-Ready Classroom Mathematics Grade 3 meet expectations for developing conceptual understanding of key mathematical concepts, especially where called for in specific content standards or cluster headings.

Within the materials, Program Implementation, Program Overview, i-Ready has three different types of lessons to address the unique approaches of the standards and to support a balance of conceptual understanding, application, and procedural fluency. “Understand Lessons focus on developing conceptual understanding and help students connect new concepts to familiar ones as they learn new skills and strategies. Strategy Lessons focus on helping students persevere in solving problems, discuss solution strategies, and compare multiple representations through the Try- Discuss-Connect routine." The Math in Action Lessons "feature open-ended problems with many points of entry and more than one possible solution." Lessons are designed to support students to explore and develop conceptual understanding of grade-level mathematics. Additional Practice and Interactive Games are also provided so that students can continue to practice the skills taught during lessons if needed.

Students develop conceptual understanding with teacher guidance and support. For example:

• Unit 2, Lessons 5 and 11, students develop conceptual understanding of 3.OA.5 (Apply properties of operations as strategies to multiply and divide). Lesson 5, Multiply with 0, 1, 2, 5, and 10, Session 1, Explore, Connect It, Reflect, Problem 3, “Elisa sees 7 crabs with 10 legs each. What other method besides skip-counting can you use to find the total number of legs?” Session 2, Develop, Connect It, students use the problem, “A company makes a toy robot that has 2 antennas and 5 buttons. How many antennas and buttons are needed for 6 robots?” to solve problems. Problem 3, “If you take the antenna array in the second Model It and turn it, what would the equation be for each way the array is shown?” A picture is shown of a 6 by 2 array, then a 2 by 6 array. Teacher Edition, Facilitate Whole Class Discussion, “Tell students that these problems will show them different ways to think about and represent the number of antennas. Be sure students understand that the two arrays represent the same number grouped in different ways.” Lesson 11, Understand how Multiplication and Division are Connected, Session 1, Explore, Model It, Problem 3, “Ronan collects flag stickers for his scrapbook. He puts 20 stickers on 5 pages in his scrapbook. He puts the same number of stickers on each page. Draw the stickers Ronan puts on the pages. Write a division equation and a multiplication equation for this problem. Division equation: _____ Multiplication equation: ____” A picture is shown of five pages with four dots on each page. A picture is also shown of 20 different flag stickers. Teacher Edition, Model It, “Tell students that they will now think about how to use a related multiplication equation to solve a division problem.” Differentiation, Reteach or Reinforce, Hands On Activity, “If students are unsure about how the model relates division and multiplication, then use this activity to provide a more concrete experience. Tell Students they will act out the sticker problem. Elicit from students how many stickers [20] and pages (index cards) [5] they need. Have students place the 20 stickers on the 5 cards one at a time. Guide students in placing one sticker on the first card, one on the next card, and so on, returning to the first card after placing a sticker on the last card until all stickers are placed.Together count the number of stickers on each card [4]. Write 20\div5=__ and 5\times__=20 on the board. Ask students to explain what each equation means for this situation.”

• Unit 3, Lesson 15, Multiply to Find Area, Session 2, Develop, Hands-On Activity, page 321, students develop conceptual understanding of 3.MD.7a (Find the area of a rectangle with whole-number side lengths by tiling it, and show that the area is the same as would be found by multiplying the side lengths). “Use a ruler and tiles to measure length, width, and area. If students struggle with relating the length and width of a rectangle to the number of square units it can be divided into or covered by, then use the activity below to connect length measured with a ruler and area found by counting square units. Use a ruler and tiles to measure length, width, and area. Have each pair of students choose a rectangular object to measure, such as a picture or a book. Tell each pair to use the ruler to measure the length of the rectangle to the nearest inch. Have them measure the width of the rectangle the same way. Then have students use the tiles to cover the rectangle and count the total number of tiles to find the area. Ask them to verify that the number of tiles along each side of the rectangle corresponds to the length or width as measured with the ruler.” 

• Unit 4, Lessons 22 and 24, students develop conceptual understanding of cluster 3.NF.3 (Explain equivalence of fractions in special cases, and compare fractions by reasoning about their size). Lesson 22, Understand Equivalent Fractions, Session 2, Develop, Model It: Number Lines, Problem 1a, “Complete the number lines by writing the missing fractions.” A picture is shown of 2 number lines from 0 to 1, 1 with tics at each third, and one with tics at each sixth.” Problem 1b, “Use the number lines to write the equivalent fractions. \frac{1}{3}=___ \frac{2}{3}=___.” Teacher Edition, Model It, “As students complete the problems, have them identify that they are being asked to complete the number lines by labeling the tic marks to identify equivalent fractions.” Lesson 24, Understand Comparing Fractions, Session 1, Explore, Model It, Problem 4 shows two circles, one partitioned into thirds, and one partitioned into eighths. Problem 4a, “Which model at the right has more parts?” Problem 4b, “Which model has smaller parts? Problem 4c, “Shade \frac{1}{3} of model A and \frac{1}{8} of model B.” Problem 4d, “Use the fractions \frac{1}{3} and \frac{1}{8} to complete the sentence. ___ is greater than ___.” Teacher Edition, Model It, “Tell students that they will now think about comparing fractions with the same numerator but different denominators. Point out that if the shaded parts in each circle or rectangle are all next to each other, it can be easier to compare the models.”

Students have opportunities to independently demonstrate conceptual understanding. For example:

• Unit 2, Lesson 8, Sessions 1-4, Use Order and Grouping to Multiply, students independently engage with 3.OA.5 (Apply properties of operations as strategies to multiply and divide) as they use multiplication and division strategies to solve problems. Session 1, Additional Practice, Prepare for Using Order and Grouping to Multiply, Problem 3, “Solve the problem. Show your work. At a tamale cart, Sofia buys 4 packages of 2 tamales. Alec buys 2 packages of 4 tamales. How many tamales does each person buy?” Session 2, Develop, Apply It, Problem 9, “Tessa knows that 5\times8=40. What other math fact does this help Tessa know?” Answer choices: 5+8=13, 40-8=32, 8\times5=40, 4\times12=48. Session 3, Additional Practice, Practice Using Grouping to Multiply, Problem 8, “Show two different ways to group 8\times2\times3. Then show the steps to find the product.” Session 4, Additional Practice, Practice Using Order and Grouping to Multiply, Problem 1, “Order and group the number 3, 4, and 2. Then multiply to find the product.”

• Unit 2, Lesson 12 Multiplication and Division Facts, Session 2, Develop, Apply It, Problems 7-8, students independently engage with 3.OA.4 (Determine the unknown whole number in a multiplication or division equation relating three whole numbers) as they use fact families and multiplication/division strategies to help them determine unknowns in multiplication and division problems. Problem 7, “Write the unknown product. Then complete the fact family.  2\times3=____.” Problem 8, “Write two multiplication facts Enrico can use to solve ____\div3=7.” 

• Unit 3, Lesson 14, 15, and 16, students independently engage with 3.MD.7 (Relate area to the operations of multiplication and addition) by utilizing concrete and semi-concrete representations to find area. Lesson 14, Understand Area, Session 1, Model It, Problem 2, “Area is the amount of space a flat shape covers. The area of a rug is the amount of floor space it covers. How do you think you could measure the area of the Zapotec rug at the right?” Session 2, Additional Practice, Practice Finding Area, Problem 6, “Carla’s mom displays his artwork on a board shaped like rectangle B. How can you skip-count to find the area of the board? Write the area.” A picture is shown of a board partitioned into 12 equal parts. Session 3, Refine, Apply It, Problem 4, “Use a ruler and the dot grid below to complete the problems. Part A Draw a rectangle with an area of 8 square units on the grid. Label it with an A. Part B Draw a rectangle with an area greater than 8 square units on the same grid. Label it with a B. Part C How did you know how to draw your rectangle B with an area greater than 8 square units?” Lesson 15, Multiply to Find Area, Session 1, Explore, Connect It, Problem 2, “Look Ahead When you know the length and width of a rectangle, you do not have to count all the unit squares to find the area. You can multiply instead. a. Dakota’s rectangle without the paint spill is an array of squares. What two multiplication equations can you write to describe this array? b. Write an equation to multiply the length and the width of the rectangle. Explain how you can use length and width to find the area of a rectangle. c. Explain how 5\times3 gives you the same area as counting the squares.” Session 2, Develop, Apply It, Problem 11, “Yolanda’s aunt sends her a rectangular piece of mola fabric. The mola fabric has a length of 8 inches and a width of 6 inches. What is the area of the mola fabric? Show your work.” Session 3, Develop, Apply It, Problem 11, “Elisa has a rectangular photo that is 7 inches long and 5 inches wide. How much space will this photo cover in Elisa’s photo album? Show your work.” Session 4, Refine, Apply It, Problem 2, “Ms. Assad is building a rectangular patio that is 4 yards long and 3 yards wide. She has enough bricks to cover an area of 14 square yards. Does Ms. Assad have enough bricks to build the patio? Explain. Show your work.” Lesson 16, Add Areas, Session 1, Additional Practice, Prepare for Adding Areas, students engage independently with semi-concrete representations to develop understanding of adding area. Problem 3, “Solve the problem. Show your work. Tara and Ashur have small Turkish rugs. Tara’s rug is 4 feet long and 3 feet wide. Ashur’s rug is 3 feet long and 2 feet wide. They put the rugs on the floor, as shown. What is the total area of the floor covered by the rugs?” Session 2, Develop, Apply It, Problem 8, “How many 1-meter-square tiles will it take to cover the figure below? Show your work.” Session 3, Develop, Apply It, Problem 7, “Opal draws this model of a picnic table. What is the total area of the picnic table? Show your work.” Session 4, Refine, Apply It, Problem 4, “Mrs. Rivera draws the model below of her new porch and garden. What is the total area of Mrs. Rivera’s new porch and garden?” Answer choices: 22 meters, 22 square meters, 30 meters, 30 square meters.

The materials reviewed for i-Ready Classroom Mathematics Grade 3 meet expectations for giving attention throughout the year to individual standards that set an expectation of procedural skill and fluency.

Within the materials, Program Implementation, Program Overview, i-Ready has three different types of lessons to address the unique approaches of the standards and to support a balance of conceptual understanding, application, and procedural fluency. The materials include problems and questions, interactive practice, and math center activities that help students develop procedural skills and fluency, as well as opportunities to independently demonstrate procedural skills and fluency throughout the grade. Additional Practice and Interactive Games are also provided so that students can continue to practice the skills taught during lessons if needed.

Students develop procedural skills and fluency with teacher guidance and support. There are also interactive tutorials embedded in classroom resources to help develop procedural skills and fluency. For example:

• Unit 1, Lessons 2 and 3, students build procedural skills and fluency of 3.NBT.2 (Fluently add and subtract within 1000 using strategies and algorithms based on place value, properties of operations, and/or the relationship between addition and subtraction.) Lesson 2, Add Three-Digit Numbers, Session 3, Develop, “What is the sum? Use place value to help you add.” Model It, “You can use place value and partial sums to add. Add ones to ones, tens to tens, and hundreds to hundreds.” The problem is written vertically with place values lined up, with each place-value sum written out in expanded form. Model It, “You can record your work in a shorter way. Add ones to ones, tens to tens, and hundreds to hundreds. Record your work by showing regrouping above the problem and writing the sum in one row. A grid can help you keep track of the place value of the digits.” A picture is shown of the steps to add the problem, with a grid to help keep track of the place value. Teacher Edition, Model It, “If no student presented these models, have students analyze key features and then point out the way each model represents: 5 ones + 9 ones = 14 ones, 14 ones = 1 ten + 4 ones Ask What number represents the ones of both addends combined? How does each model show the 14 ones regrouped as 1 ten and 4 ones?” Lesson 3, Subtract Three-Digit Numbers, Session 1, Explore, Connect It, Problem 2 Look Ahead, “You can solve subtraction problems in different ways. Breaking apart numbers is one way to subtract. Suppose you want to find 525-213.” Problem 2a, “Break apart 525 into a sum of hundreds, tens, and ones.” Problem 2b, “Break apart 213 into a sum of hundreds, tens, and ones.” Problem 2c, “Subtract ones from ones, tens from tens, and hundreds from hundreds to find 525-213.” Teacher Edition, Look Ahead, “Point out that just as with addition, breaking apart numbers can help students subtract the numbers more easily. Students should be able to use place-value language to identify and subtract the ones, tens, and hundreds of 525 and 213.”

• Unit 2, Lessons 5 and 12, students build procedural skills of 3.OA.7 (Fluently multiply and divide within 100, using strategies such as the relationship between multiplication and division or properties of operations. By the end of Grade 3, know from memory all products of two one-digit numbers). Lesson 5, Multiply with 0, 1, 2, 5, and 10, Session 1, Explore, Multiplying with 0, 1, 2, 5, and 7, Connect It, Problem 2 Look Ahead, “You can show and solve multiplication problems in different ways, such as using arrays or equal groups. One way to find products when multiplying with 2, 5, or 10 is to use skip-counting. Marco sees 8 black crabs with 10 legs each.” A picture is shown of 8 crabs with 10 legs each. Problem 2a, “Show how you could use skip-counting to find the number of legs Marco sees. 10, 20, ___.” Problem 2b, “Write a multiplication fact to find the number of legs. number of crabs x legs on each crab = total number of legs. ___ \times ___ = ___.” Teacher Edition, Look Ahead, “Point out that skip-counting is a quick method of repeated addition, which is also called multiplication. Students should be able to use skip-counting to count equal groups of items and then model the groups and total as a multiplication equation.” Lesson 12, Multiplication and Division Facts, Session 2, Model It, “You can use a number line to help you understand the problem. Skip-count by fives to find the answer. Start at 0 and jump by fives until you get to 40.” Model It, “You can use fact families and multiplication facts you know. Here are the facts in this family: 5\times__=40, \times5=40, 40\div__=5, 40\div5= Write the multiplication facts for 5: Look for the fact that has the numbers you know from the fact family, 5 and 40, Use that fact to fill in the unknown numbers above.” Session 3, Develop, “Complete the facts. 2\times ___ =10, 24\div6= ___, ___ \times6=48, ___  \div1=8 Picture It You can use a multiplication table to find the numbers in multiplication and division fact families. A multiplication table shows both multiplication and division fact families.” A picture of a multiplication table is shown. Teacher Edition, Facilitate Whole Class Discussion, “Call on students to share selected strategies. Prompt students to think of mistakes as a way to learn. Guide students to Compare and Connect the representations. Ask Where does each model show whether you are finding a factor, a quotient, or a total?” Session 4, Differentiation, Reteach, Hands-On Activity, Use patterns to learn facts for 9, “Have students color the multiples of 9 on their table. Have them describe any patterns they see [e.g. the product digits add to 9; the ones digit decreases by 1 as the tens digit increases by 1]; Illustrate how patterns can be helpful in learning facts using the sum of 9 pattern. Help students reason that the product of 4\times9 must be in the 30s because 4\times10=40 and 4\times9 is 4 less. Ask: What number added to 3 equals 9? [6] Write the competed fact [4\times9=36]; Repeat these steps for other 9 facts such as 8\times9. (The product will be in the 70s and 7+2=9, so the product is 72.)”

• Unit 2, Lesson 7, Interactive Tutorials, contains two 17-minute tutorials to help students develop procedural skills and fluency with multiplying and dividing within 100. The videos focus on breaking apart a number to multiply. (3.OA.7)

The instructional materials provide opportunities for students to independently demonstrate procedural skills and fluency throughout the grade level. Examples include:

• Unit 1, Lessons 2 and 3, students independently demonstrate procedural skill and fluency of 3.NBT.2 (Fluently add and subtract within 1000 using strategies and algorithms based on place value, properties of operations, and/or the relationship between addition and subtraction.). Lesson 2, Add Three-Digit Numbers, Session 2, Develop, Teacher’s Edition, Fluency & Skills Practice, “In this activity students practice adding two- and three-digit numbers using place value.” Fluency & Skills Practice, Problem 2, “640+40, 640+140, 640+150.” Session 4, Refine, Adding Three-Digit Numbers, Problem 2, “Find the sum of 345 and 626. Show your work.” Lesson 3, Subtract Three-Digit Numbers, Session 2, Develop, Teacher’s Edition, Fluency & Skills Practice, “In this activity students practice using place value to subtract three-digit numbers when regrouping is needed.” Fluency & Skills Practice, “Circle all the problems that need regrouping. Then find the differences of only the problems you circled.” Problem 5, “534-221.”  Problem 8, “908-435.”

• Unit 1, Lesson 2 and 3, Learning Games, Hungry Fish, Cupcake, and Pizza help students develop procedural skills and fluency with adding and subtracting within 1000. (3.NBT.2)

• Unit 2, Lesson 5, Interactive Practice, contains one, 15-minute practice session to help students use strategies such as repeated addition and skip counting by twos, fives, and tens to solve multiplication problems involving 0, 1, 2, 5, and 10. (3.OA.7)

• Unit 2, Lessons 5-7 and 12, students independently demonstrate procedural skill and fluency of 3.OA.7 (Fluently multiply and divide within 100, using strategies such as the relationship between multiplication and division or properties of operations. By the end of Grade 3, know from memory all products of two one-digit numbers). Lesson 5, Multiply with 0, 1, 2, 5, and 10, Session 2, Develop, Teacher’s Edition, Fluency & Skills Practice, “In this activity students practice multiplying numbers by 2, 5, and 10.” Problem 2, “2\times5= .” Problem 14, “10\times4=.” Lesson 6, Multiply with 3, 4, and 6, Session 4, Develop, Teacher’s Edition, Fluency & Skills Practice, “In this activity students practice multiplying numbers by 6.” Problem 7, 8 x 6 = .”  Fluency Skills & Practice, Problem 11, “0\times6= .” Lesson 7, Multiply with 7, 8, and 9, Session 4, Develop, Teacher’s Edition, Fluency & Skills Practice, “In this activity students practice multiplying numbers by 9.” Problem 5, “8\times9=.” Problem 9, “4\times9= .” Lesson 12, Multiplication and Division Facts, Session 3, Develop, Teacher’s Edition, Fluency & Skills Practice, “In this activity students practice using a multiplication table to find the missing number in a multiplication or division fact.” Fluency Practice and Skills worksheet, “Write the missing numbers in the boxes to make each multiplication or division problem true.” There are 20 equations to solve. Examples include, “32\div8= , 9\times =27, and \div7=7.” Session 4, Refine, Apply It, Problem 2, “Solve 4\times9= _. Show your work.” 

• Unit 2, Lesson 12, Center Activities, Complete a Fact Family, students work with a partner to develop procedural skills and fluency with multiplying and dividing within 100. (3.OA.7)

The materials reviewed for i-Ready Classroom Mathematics Grade 3 meet expectations for being designed so that teachers and students spend sufficient time working with engaging applications of the mathematics.

Within the materials, Program Implementation, Program Overview, i-Ready has three different types of lessons to address the unique approaches of the standards and to support a balance of conceptual understanding, application, and procedural fluency. “Understand Lessons focus on developing conceptual understanding and help students connect new concepts to familiar ones as they learn new skills and strategies. Strategy Lessons focus on helping students persevere in solving problems, discuss solution strategies, and compare multiple representations through the Try- Discuss-Connect routine.” The Math in Action Lessons “feature open-ended problems with many points of entry and more than one possible solution.” Lessons are designed to support students as they apply grade-level mathematics. Additional Practice and Interactive Games are also provided so that students can continue to practice the skills taught during lessons if needed.

There are multiple routine and non-routine application problems throughout the grade level, including opportunities for students to work with support of the teacher and independently. While single and multi-step application problems are included across various portions of lessons, independent application opportunities are most often found within Additional Practice, Refine, and Math in Action lessons.

Examples of routine applications of the mathematics include:

• Unit 2, Lesson 5, Multiply with 0, 1, 2, 5, and 10, Session 1, Additional Practice, Prepare for Multiplying with 0, 1, 2, 5, and 10, Problem 3, students apply multiplication strategies independently to solve a routine problem. 3.OA.3 (Use multiplication and division within 100 to solve word problems in situations involving equal groups, arrays, and measurement quantities). “Hugo goes to his mom’s house for 5 weeks in the summer.  Each week has 7 days. How many days is Hugo at his moms’s house?”

• Unit 3, Lesson 18, Solve Two-Step Word Problems Using the Four Operations, Session 3, Develop, Apply It, Problem 9, students apply multiplication and subtraction strategies to independently solve a real-world problem. 3.OA.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.) Problem 9, “Yen is earning money to buy a surfboard that costs $289. For the past 6 weeks, Yen has saved $7 each week. How much money, d, does Yen still need to save? Show your work.”

• Unit 6, Lesson 32, Area and Perimeter of Shapes, Sessions 2 and 4, with teacher support, students apply addition and multiplication strategies to solve problems involving perimeter and area. 3.MD.8 (Solve real world and mathematical problems involving perimeters of polygons, including finding the perimeter given the side lengths, finding an unknown side length, and exhibiting rectangles with the same perimeter and different areas or with the same area and different perimeters). Session 2, Develop, p. 715, “Kadeem builds a pen for his sheep. The pen has 6 sides. The perimeter is 23 feet. The lengths of five of the sides are 7 feet, 2 feet, 5 feet, 2 feet, and 2 feet. What is the length of the sixth side?” Teacher Edition, Make Sense of the Problem, “Before students work on Try It, use Notice and Wonder to help them make sense of the problem. Have students respond to the question: What do you notice? Record as many responses as time or interest allows. Ensure student understanding of a pen as an enclosure of small farm animals.” Session 4, Develop, Teacher Edition, p. 728, “Facilitate Whole Class Discussion, To prompt students to use precise academic language, call on volunteers to reword vague or unclear statements. Guide students to Compare and Connect the representations. Prompt students to show the parts of the strategy they think are not correct. Ask them to suggest corrections. Ask How did you find rectangles that you knew would have a perimeter of 12 units? Some students may have realized that the length and width of each rectangle should add up to 6 and looked for the appropriate additional facts. Others may have let the length equal each number, starting with 1 and calculated each corresponding width. Picture It and Model It, If no student presents these models, have students analyze key features and then point out the ways each model represents: the length and width of each rectangle, the fact that the perimeter of each rectangle is 12 units. Ask How are the lengths and widths of the rectangles shown? How do you know each rectangle has a perimeter of 12 units? Listen For, You can count the squares in the drawings to find the dimensions of each rectangle and check that they each have a perimeter of 12 units. The table clearly states the length, width, area and perimeter of each rectangle. You can see from the fourth column that all of the rectangles have the same perimeter. For drawings, prompt students to identify the dimension of each rectangle. What is the length and width of each rectangle shown? What is the perimeter of each rectangle? For the table, prompt students to check whether the table is complete or if additional rectangles can be added. What patterns do you notice in the first two columns? Can a rectangle with a length of 6 or more units have a perimeter of 12 units? Why or why not?”

Examples of non-routine applications of the mathematics include:

• Unit 1, Math in Action, Session 2, Persevere On Your Own, “Ticket Sales”, students independently demonstrate application of addition and rounding through a non-routine problem. 3.NBT.1 (Use place value understanding to round whole numbers to the nearest 10 or 100). “Alex works at a science center. The science center donates money to the Wildlife Protectors for every ticket sold on Saturday morning. They donate $1 for each child ticket and $2 for each adult ticket. Alex looks at ticket records for the past 5 weeks. Estimate how much money the science center will donate to the Wildlife Protectors for Week 6.” A chart is shown with the number of adult and child tickets sold each week for the past five weeks.

• Unit 3, Math In Action, Use the Four Operations,  Session 1, Try Another Approach, Teacher Edition, pp. 444-445, Facilitate Whole Class Discussion, with teacher support, students demonstrate application of the four operations to solve word problems with unknowns. 3.OA.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.) “Review Sweet T’s Tees problem on the previous page. Ask How can you summarize the steps in Sweet T’s solution? Listen For Make a table of shirts and prices. Find the cost of the most expensive shirts plus set-up fee. Add the cost of 2 shirts and the set-up fee. Subtract from available money. Ask What are some different steps you could use to solve the problem? Listen For Find the cost of 2 shirts for one member and then use place value to multiply and find the cost for 10 members. Plan It, Facilitate Whole Class Discussion, Read the questions aloud. Prompt students to recognize that they are being asked to find a different number or combination of shirts and tell how much money is left over. Ask What are some things Sweet T should think about when deciding how many team members to buy for? Listen For The problem says that the number of team members is not exact. Students should be able to identify some issues related to buying too many shirts (wastes money) or too few (not everyone gets shirts). Ask Why might Sweet T want to spend all of the money? Why might he want to have money left over? Listen For If Sweet T spends all of the money, the members get nicer shirts or he can buy extra shirts for more members. Sweet T probably does not get to keep the money himself. If he has some money left over, he could buy snacks or equipment for the members. Solve It, Problem-Solving Checklist, Introduce the Problem-Solving Tips as ideas students may use to explain their thinking. Remind them to also use the Problem- Solving checklist to help organize their work. Have students write their own complete solutions on a copy of Activity Sheet Solution Sheet 1 or a blank sheet of paper. Reflect, As they work, have students share their thinking with a partner and discuss the Reflect questions. Close, As time permits, have students explain their solutions to the class informally or as a brief oral presentation. Use the Oral Presentation Checklist on the Teacher Toolbox. Alternatively, share the solution below and discuss it as a class. Possible Solution, Sweet T needs to buy 2 shirts for 8 to 10 team members. The prices are given. Sweet T has up to $225 to spend but wants to keep some money to spend on other things. Sweet T should buy for 10 team members to be sure he has enough shirts. Sweet T could buy the least expensive shirt with no printing. Then there is no set-up fee. The short-sleeve T-shirts cost 6 each, and the collar shirts cost $7 each. So for each person, the two shirts cost 13. \$13\times10=\$130 and \$225=\$130=\$95. So, Sweet T would have $95 left over.”

• Unit 5, Math in Action, Solve Measurement Problems, Session 2, Soup Snacks, students independently use operations to determine how many 1-liter containers to buy. 3.MD.2 (Measure and estimate liquid volumes and masses of objects using standard units of grams (g), kilograms (kg), and liters (l). Add, subtract, multiply, or divide to solve one-step word problems involving masses or volumes that are given in the same units). “Max plans to make tomato soup. His recipe makes 24 liters of soup. He will freeze the soup in containers. Then he will have plenty of soup snacks ready to go. Max wants to buy some 1-liter containers for the soup. He can buy different packages of 1-liter containers. package of 4 containers, package of 5 containers, package of 6 containers What package should Max buy?”

The materials reviewed for i-Ready Classroom 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.

Within the materials, Program Implementation, Program Overview, i-Ready has three different types of lessons to address the unique approaches of the standards and to support a balance of conceptual understanding, application, and procedural fluency. The materials include problems and questions, interactive practice, and math center activities that help students develop skills. 

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

• Unit 2, Lesson 7, Session 1, Explore, Additional Practice, Problem 3, students develop procedural skill and fluency by fluently multiplying and dividing within 100. 3.OA.7 (Fluently multiply and divide within 100, using strategies such as the relationship between multiplication and division or properties of operations…). “Solve the problem. Show your work. Apone and Galeno are both finding 6\times9. They each break apart the problem in a different way. Show two different ways to break apart 6\times9 and find the product.” 

• Unit 3, Lesson 18, Solve Two-Step Word Problems Using the Four Operations, Sessions 2 and 5, students demonstrate application of the four operations. 3.OA.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.) Session 2, Develop, Apply It, Problem 8, “There are 48 water bottles that are divided equally between 8 jai alai teams. Each team has 2 players. Each player gets an equal number of water bottles. How many water bottles does each player get? Show your work.” Session 5, Refine, Apply It, Problem 7, “Seth is packing a book order. He has already packed 3 boxes with 5 books in each box. There are 210 books left to pack. How many books are in the whole order? Show your work.” Problem 9, “Sara is stocking a shelf with jars of pickles. She has one box with 30 jars and another box with 18 jars. Sara can fit 6 jars in a row on the shelf. Write and solve one equation to find out how many rows she makes using all the jars in both boxes. Explain how you solved the problem.”

• Unit 4, Lesson 20, Session 1, Explore, Model It, Problem 4, students develop conceptual understanding of fractions as numbers 3.NF.1 (Understand a fraction \frac{1}{b} as the quantity formed by 1 part when a whole is partitioned into b equal parts; understand a fraction \frac{a}{b} as the quantity formed by a parts of size \frac{1}{b}.) “Complete the problem below. A unit fraction has a 1 in the numerator. It names 1 part of the whole. Shade \frac{1}{4} of the model below.”

Multiple aspects of rigor are engaged simultaneously throughout the materials in order to develop students’ mathematical understanding of grade-level topics. Examples include:

• Unit 2, Math In Action, Session 2, Persevere On Your Own, Problem Space Creatures, students develop procedural skill and fluency, conceptual understanding, and application as they use multiplication and arrays to solve an open-ended word problem. 3.OA.3 (Use multiplication and division within 100 to solve word problems in situations involving equal groups, arrays, and measurement quantities…). “Brandi doesn’t know how many space creatures to have in the play, but she has some ideas. Space Creature Notes: The creature should march out of the spaceship in equal groups or in equal rows. There should be more than 20. There should not be more than 30. How many space creatures should Brandi use? Solve It Write a plan for Brandi’s space creatures: Decide how many space creatures to use. Tell how many groups or rows of creatures to use. Also tell how many are in each group or row. Describe how the space creature will march out off the spaceship.” 

• Unit 3, Math in Action, Use the Four Operations, Session 2, Persevere on Your Own, students engage with all aspects of rigor as they choose to use multiplication facts and/or addition with appropriate models and strategies to plan and solve multi-step problems with a variety of answers based upon choices. 3.NBT.A (Use place value understanding and properties of operations to perform multi-digit arithmetic.) “Skate Park Sweet T has $80 left after buying items for the team. He wants to buy at least three different items for the skate park he is making. Here are the items Sweet T is looking at, along with the prices. What items should Sweet T buy? Pictured are items with the following labels: “Table: $24, Bench: $15, L-Shaped Box: $15, Box: $18, Pallet: $10, Rail: $22.” Solve It, “Tell which items Sweet T should buy. Give the total cost. Explain why you chose the items you did.”

• Unit 6, Lesson 32, Area and Perimeter of Shapes, Session 3, Additional Practice, Practice Finding Same Area with Different Perimeter, Problem 3, students engage with all aspects of rigor as they apply conceptual understanding of perimeter and area to solve real-world problems involving addition and multiplication. 3.MD.7 (Relate area to the operations of multiplication and addition). “Ria has 16 square-inch tiles. She wants to glue them on cardboard to make two different rectangles, each with the same area but different perimeters. What are the side lengths of two rectangles she can make? Show your work.”

The materials reviewed for i-Ready Classroom 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. 

Within Program Implementation, Standards for Mathematical Practice in Every Lesson, “The Standards for Mathematical Practice (SMPs) are embedded within the instructional design of i-Ready Classroom Mathematics.” Embedded SMPs Within Lessons, “In addition to SMPs 1, 2, 3, 4, 5, and 6, which are integrated into the instructional framework, the Teacher’s Guide includes additional opportunities for students to develop the habits of mind described by the Standards of Mathematical Practice. The table of contents indicates all of the embedded Standards for Mathematical Practice for each lesson (both integrated SMPs and the specific SMPs highlighted within the lesson). The Lesson Overview includes the Standards for Mathematical Practice addressed in each lesson. In the Student Worktext, the Learning Target also highlights the SMPs that are included in the lesson.” SMPS Integrated in Try-Discuss-Connect Instructional Framework, “i-Ready Classroom Mathematics” infuses SMPs 1, 2, 3, 4, 5, and 6 into every lesson through the Try-Discuss-Connect instructional framework (found in the Explore and Develop sessions of Strategy lessons, with a modified framework used in understand lessons).” Try It, “Try It begins with a language routine such as Three Reads, which guides students to make sense of the problem (SMP 1): For the first read, students begin to make sense of the problem (SMP 1) as the teacher reads the problem aloud. For the third read, students read the problem in unison or in pairs. Students reason quantitatively and abstractly (SMP 2) by identifying the important information and quantities, understanding what the quantities mean in context, and discussing relationships among quantities.” Discuss It, “All students reason abstractly and quantitatively (SMP 2) as they find similarities, differences, and connections among the strategies they have discussed and relate them to the problem they are solving.” Connect It, “As students think through the questions and problems, they connect the quantitative, concrete/representational approaches to a more abstract understanding (SMP 2).”

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 modules. Examples include:

• Unit 1, Lesson 2, Add Three-Digit Numbers, Session 2, Develop, Try It, students make sense of the numbers in the problem and persevere in solving it. “Greg and Mora take photos at a gaucho festival. Greg takes 130 photos. Mora takes 280 photos. How many photos do they take in all?” Teacher Edition, Try It, Make Sense of the Problem, “See Connect to Culture to support student engagement. Before students work on Try It, use Three Reads to help them make sense of the problem. After the first read, ask students what the problem is about. After the second read, have students tell what the question is asking them to find. For the third read, ask them to identify important information.”

• Unit 2, Math in Action, Solve Multiplication and Division Problems, Session 1, Robot Prop, Teacher Edition, Purpose, students use a variety of strategies that make sense to solve the problem. “Understand an open-ended, multi-step problem involving multiplication, comparison, and addition of numbers. Choose appropriate models and strategies to plan for and solve the problems.” Student Edition, “Brandi’s play is about space creatures. She wants to make a space robot prop. Brandi has 50 pie plates. She will use the pie plates to make arms and legs for the robot. My Robot Prop Plan Use up to 50 plates. Use the same number of plates for each arm. Use the same number of plates for each leg. Use more plates for each leg than for each arm. How many pie plates should Brandi use for each leg and each arm?” Plan It and Solve It, “Find a solution for the Robot Prop problem. Make a plan for Brandi’s robot.  Tell how many plates to use for each arm and leg. Tell how many plates you need in all. Explain why your plan works.  You may want to use the Problem-Solving Tips to get started.” Reflect, “Use Mathematical Practices As you work through the problem, discuss these questions with a partner. Use Models How could a drawing help you find a solution? Make an Argument How do you know that the numbers you chose work?”

• Unit 5, Lesson 29, Mass, Session 3, Develop, Try It, students analyze and make sense of the problem. “Lateefah’s uncle sends her a maraca and a carved owl from Peru. The maraca has a mass of 70 grams, and the owl has a mass of 40 grams. What is the mass of the maraca and owl together?” Connect It, Problem 4, “Explain how you could estimate to know that your answer makes sense.” Teacher Edition, Facilitate Whole Class Discussion, Problem 4, “Be sure students understand that the problem is asking them how they could estimate using the numbers in the problem. Ask How could you use your equation and estimate to check your answer? Listen for ‘I could use nearby compatible numbers and then add.’ 40 and 70 are close to 50. 50+50=100, which is close to 110 so the answer makes sense.” 

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 modules. Examples include:

• Unit 2, Lesson 7, Multiplying with 7, 8, and, 9, Session 1, Explore, Try It, students consider the units involved in a problem and attend to the meaning of the quantity. “Tameka and Seth are both finding 6\times7. They each break apart the problem in a different way. Show two different ways to break apart 6\times7 and find the product.” Teacher Edition, Try It, Make Sense of the Problem, “Before students work on Try It, use Notice and Wonder to help them make sense of the problem. Students may notice and wonder about information in the problem or about what they see in the array.”

• Unit 4, Math In Action, Use Fractions, Session 1, Study an Example Problem and Solution, 8-Mile Trail, G.O.’s Solution, Deepen Understanding, Understanding the Zero Mark on the Number Line, students understand relationships between the problem scenario and the mathematical representation. “As you discuss the number line in G.O.’s solution, guide students to talk about the meaning of the zero mark both abstractly and in the context of the problem.”

• Unit 5, Lesson 29, Mass, Session 2, Develop, Try It, students consider units involved in a problem and attend to the meaning of quantities. “Read and try to solve the problem below. Diego bought a medium-sized watermelon at the store. Estimate the mass of the watermelon.” Teacher Edition, Make Sense of the Problem, “Before students work on Try It, use Notice and Wonder to help them make sense of the problem. Have students respond to the question, What do you wonder? Record as many responses as time or interest allows making only encouraging comments. Ask What everyday object has a mass of about 1 gram? 1 kilogram?”

The materials reviewed for i-Ready Classroom 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.

Within Program Implementation, Standards for Mathematical Practice in Every Lesson, “The Standards for Mathematical Practice (SMPs) are embedded within the instructional design of i-Ready Classroom Mathematics.” Embedded SMPs Within Lessons, “In addition to SMPs 1, 2, 3, 4, 5, and 6, which are integrated into the instructional framework, the Teacher’s Guide includes additional opportunities for students to develop the habits of mind described by the Standards of Mathematical Practice. The table of contents indicates all of the embedded Standards for Mathematical Practice for each lesson (both integrated SMPs and the specific SMPs highlighted within the lesson). The Lesson Overview includes the Standards for Mathematical Practice addressed in each lesson. In the Student Worktext, the Learning Target also highlights the SMPs that are included in the lesson.” SMPS Integrated in Try-Discuss-Connect Instructional Framework, “i-Ready Classroom Mathematics infuses SMPs 1, 2, 3, 4, 5, and 6 into every lesson through the Try-Discuss-Connect instructional framework (found in the Explore and Develop sessions of Strategy lessons , with a modified framework used in understand lessons).” Discuss It, “Discuss It begins as student pairs explain and justify their strategies and solutions to each other.  Partners listen to and respectfully critique each other’s reasoning (SMP 3). As students/pairs share their different approaches, the teacher facilitates the discussion by prompting students to listen carefully and asking them to repeat or rephrase the explanations to emphasize key ideas (SMP 3).” 

Students construct viable arguments and 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 2, Lesson 13, Understand Patterns, Session 3, Refine, Apply It, Problem 3, students critique the reasoning of another student in the problem, and justify their own reasoning when talking about factors and products. “Parker says an odd factor times an odd factor will always equal an even product. Is he correct? Explain.”

• Unit 3, Lesson 17, Solve One-Step Word Problems Using Multiplication and Division, Session 4, Explore, Discuss It, students justify their own thinking for why they chose a strategy to solve a word problem. “Ask your partner: Why did you choose that strategy? Tell your partner: The strategy I used to find the answer was…”

• Unit 4, Lesson 22, Understanding Equivalent Fractions, Session 1, Explore, Discuss It, students explain their thinking about equivalent fractions on a number line. “How do the number lines in problem 1 show that \frac{1}{2} is equivalent to \frac{2}{4}? I think the wholes need to be the same size to compare fractions because…”

• Unit 4, Lesson 25, Use Symbols to Compare Fractions, Session 2, Develop, Teacher Edition, Discuss It, Support Partner Discussion, students justify their reasoning for choosing models to solve a problem comparing fractions. “Encourage students to use the terms numerator, denominator, greater than, and less than as they discuss their solutions. Support as needed with questions such as: Why did you choose the model you used? What was your first step? How can you label your model to show that your comparison is correct?”

• Unit 6, Lesson 32, Area and Perimeter of Shapes, Session 5, Refine, Apply It, Problem 7, students critique the reasoning of another student in the problem, and justify their own thinking about area and perimeter. “Kateri says that all rectangles with a perimeter of 14 meters have the same area. Is she correct? Explain.”

The materials reviewed for i-Ready Classroom 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.

Within Program Implementation, Standards for Mathematical Practice in Every Lesson, “The Standards for Mathematical Practice (SMPs) are embedded within the instructional design of i-Ready Classroom Mathematics.” Embedded SMPs Within Lessons, “In addition to SMPs 1, 2, 3, 4, 5, and 6, which are integrated into the instructional framework, the Teacher’s Guide includes additional opportunities for students to develop the habits of mind described by the Standards of Mathematical Practice. The table of contents indicates all of the embedded Standards for Mathematical Practice for each lesson (both integrated SMPs and the specific SMPs highlighted within the lesson). The Lesson Overview includes the Standards for Mathematical Practice addressed in each lesson. In the Student Worktext, the Learning Target also highlights the SMPs that are included in the lesson.” SMPs Integrated in Try-Discuss-Connect Instructional Framework, “i-Ready Classroom Mathematics infuses SMPs 1, 2, 3, 4, 5, and 6 into every lesson through the Try-Discuss-Connect instructional framework (found in the Explore and Develop sessions of Strategy lessons , with a modified framework used in understand lessons).” Try It, “Try it continues as students work individually to model important quantities and relationships (SMP 4) and begin to solve the problem (SMP 1). They may model the situation concretely, visually, or using other representations, and they make strategic decisions about which tools or manipulatives may be appropriate (SMP 5).” Connect It, “For each problem students determine which strategies they feel are appropriate, and they model and solve (SMP 4) using pictures, diagrams, or mathematical representations. Students can also choose from a variety of mathematical tools and manipulatives (SMP 5) to support their reasoning.”

MP4 is identified and connected to grade-level content, and there is intentional development of the MP to meet its full intent. Students model with mathematics as they work with support of the teacher and independently throughout the modules. Examples include:

• Unit 2, Lesson 4, Understand the Meaning of Multiplication, Session 1, Explore, Model It, Problem 1, students model situations with appropriate representations and use appropriate strategies. “You can add to find the total number of objects in different groups. When the groups are equal, you can also multiply to find the total. Three pairs of players are flying kites in a game. Draw a picture of 3 equal groups of 2 kites.” 

• Unit 3, Lesson 18, Solve Two-Step Word Problems Using The Four Operations, Session 2, Additional Practice, Practice Solving Two-Step Word Problems Using Two Equations, Problem 6, students model with mathematics as they create representations and equations to solve two-step word problems. “An office building has 4 elevators. There are 8 people in each elevator. Then 10 people get off the elevators. How many people are in the elevators now? Draw a model for the problem. Label the model.”

• Unit 6, Lesson 32, Session 4, Area and Perimeter of Shapes, Develop, Teacher Edition, Discuss It, students create and compare models to solve area and perimeter problems with teacher guidance as they compare models for finding area and perimeter. “One possible order for whole class discussion: concrete models, such as rectangles, made from tiles; drawings on grid paper; lists or tables, without visual models.” Picture It & Model It, “If no student presented these models, have students analyze key features and then point out the ways each model represents: the length and width of each rectangle; the fact that the perimeter of each rectangle is 12 units.”

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

• Unit 2, Lesson 9, Use Place Value to Multiply, Session 1, Explore, Try It, with teacher guidance, students recognize the insight to be gained from using base 10 blocks as they solve multiplication equations. “There are 4 stacks of books on a table. Each stack has 20 books. How many books are there in all?” Connect It, Problem 1 Look Back, “Explain how you found how many books there are in all.” Teacher Edition, Hands-On Activity, Use base-ten blocks to model 4\times20, “If students are unsure how to approach multiplying with a two-digit number, then use this activity to have them model a problem. Materials For each student: base-ten blocks (10 tens rods). Have students model the number 20 with 2 tens rods. Have students model the number 20 three more times to show 4 groups of 20. Instruct students to combine all the tens rods together. Ask: There are 8 tens rods. What is the value of 8 tens? What is the product 4\times20? Repeat the activity for 2\times30, 3\times30 and 2\times50.”

• Unit 3, Lesson 18, Solve Two-Step Word Problems Using The Four Operations, Session 1, Additional Practice, Prepare for Solving Two-Step Word Problems Using The Four Operations, Problem 3, students use different strategies (e.g. models and equations) to solve two-step word problems. “Solve the problem. Show your work. Ade has 356 beads. Then he receives a package with more beads in which: the beads come in 6 different colors (red, orange, yellow, green, blue, purple); there are 9 beads of each color. How many beads does Ade have now?”

• Unit 5, Lesson 29, Mass, Session 1, Explore, Connect It, Problem 1, students recognize both the insight to be gained from different tools/strategies and their limitations as they estimate the mass of an object, then measure the exact mass. “Explain how you could estimate and measure the mass of the eyeglasses.” Teacher Edition, Connect It, Look Back, “Look for understanding that paper clips can be used to estimate and measure the mass of eyeglasses because the mass of a paper clip is known (1 gram).”

The materials reviewed for i-Ready Classroom 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. 

Within Program Implementation, Standards for Mathematical Practice in Every Lesson, “The Standards for Mathematical Practice (SMPs) are embedded within the instructional design of i-Ready Classroom Mathematics.” Embedded SMPs Within Lessons, “In addition to SMPs 1, 2, 3, 4, 5, and 6, which are integrated into the instructional framework, the Teacher’s Guide includes additional opportunities for students to develop the habits of mind described by the Standards of Mathematical Practice. The table of contents indicates all of the embedded Standards for Mathematical Practice for each lesson (both integrated SMPs and the specific SMPs highlighted within the lesson). The Lesson Overview includes the Standards for Mathematical Practice addressed in each lesson. In the Student Worktext, the Learning Target also highlights the SMPs that are included in the lesson.” SMPs Integrated in Try-Discuss-Connect Instructional Framework, “i-Ready Classroom Mathematics infuses SMPs 1, 2, 3, 4, 5, and 6 into every lesson through the Try-Discuss-Connect instructional framework (found in the Explore and Develop sessions of Strategy lessons , with a modified framework used in understand lessons).” Try It, “Multiple students share a word or phrase that describes the context of the problem as the teacher guides them to consider precision (SMP 6) of the mathematical language and communication.” Discuss It, “The teacher guides students to greater precision (SMP 6) in their mathematics, language, and vocabulary.”

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

• Unit 1, Math In Action, Use Rounding and Operations, Session 1, Try Another Approach, Adopt An Animal, students work towards precision as they perform many calculations. “The Wildlife Protectors save endangered animals. Alex helps them raise money. Her goal is to raise at least $750. Alex asks people at companies to buy adoption kits. Here are her notes. My Notes: Two companies will spend up to $200. Two companies will spend about $100. Other companies will spend less than $75. Use the information in the notes. Show what kits and how many of each Alex can sell to reach her goal. Explain your choices.” A graphic with Snowy Owl kits and Tiger kits is shown. Tiger kits are listed as $59, $95, and $199. Snowy Owl kits are listed as $25, $55, and $99. Reflect, “Use Mathematical Practices, As you work through the problem, discuss these questions with a partner. Be Precise, Why do you have to find actual sums to solve this problem? Reason Mathematically, What additional strategies can you use to solve this problem?”

• Unit 3, Lesson 14, Understand Area, Session 1, Model It, Problem 1, with teacher guidance, students attend to precision as they measure areas by counting unit squares. “There are different ways you can measure a rug that is shaped like a rectangle.” Problem 1a, “Use the space at the right to draw a rectangular rug. Use the words length and width to label the rug.” Problem 2b, “How could you measure the length and width of the rug?” Teacher Edition, Model It, “Read the question at the top of the Student Worktext page. Remind students that they already know how to measure the length of a shape. See Connect to Culture to support student engagement. Tell students that they are going to use what they know about measuring the side of a rectangle to think about how they might measure the amount of space inside a rectangle. Then clarify the task and have students complete the problems.”

• Unit 6, Lesson 32, Area and Perimeter of Shapes, Sessions 1 and 3, students attend to precision to find the area and perimeter of shapes. Session 1, Explore, Connect It, Problem 2, “You have already learned about finding area, the amount of space a shape covers. Perimeter is the distance around a shape. The dashed line around the soccer field shows the perimeter.” Problem 2a, “In problem 1 you found the perimeter of the field. What operation did you use?” Problem 2b, “Write an equation you could use to find the perimeter of the field.” Problem 2c, “You can find the perimeter of shapes other than rectangles. Find the perimeter of this shape. Show your work.” A picture of a soccer field and a hexagon with side lengths labeled is provided.” Session 3, Additional Practice, Practice Finding Same Area with Different Perimeter, Problem 3, “Ria has 16 square-inch tiles. She wants to glue all of them on cardboard to make two different rectangles, each with the same area but different perimeters. What are the side lengths of two rectangles she can make? Show your work.”

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

• Unit 3, Lesson 19, Scaled Graphs, Sessions 1 and 4, students use language specific to graphs. Session 1, Explore, “You have had practice modeling and solving word problems. In this lesson, you will use data, or information, from graphs to solve word problems.” Connect It, Problem 2, “The data in a picture graph can also be shown on a bar graph. A bar graph for the data on the previous page is shown. The scale on a bar graph is the difference between any two numbers that are next to each other along the bottom or left side of the graph. These numbers are called scaled numbers.” Problem 2a, “What is the scale of this bar graph?” Problem 2b, “What does the bar for Gil mean? How do you know?” Problem 2c, “How do the key for the picture graph and the scale for the bar graph compare? Explain how each helps you read a graph.” Session 4, Develop, Model It, students work towards using specialized language. “You can use the scale or key and multiplication to help you make a graph. Multiply to find the scale numbers to write on a bar graph or how many symbols to draw on a picture graph. Use a scale of 5.” 

• Unit 4, Lesson 25, Use Symbols to Compare Fractions, Session 1, Explore, Connect it, Problem 2, students attend to the specialized language of mathematics as they compare fractions using symbols. “You can use the symbols <, >, or = to write statements that compare fractions just as you did with whole numbers. < means less than. > means greater than. = means equal to. Remember that the symbol opens to the greater fraction and points to the lesser fraction. greater fraction > lesser fraction and lesser fraction < greater fraction Use words and a symbol to complete the statements below.” Problem 2a, “Compare \frac{1}{2} and \frac{1}{3}.” Problem 2b, “Compare \frac{1}{4} and \frac{1}{2}.” Problem 2c, “Compare \frac{1}{2} and \frac{1}{2}.” Additional Practice, Prepare for Using Symbols to Compare Fractions, Problem 3, “Solve the problem. Show your work. Grace and Ashon each buy same-size sandwiches. Grace eats \frac{6}{8} of her sandwich. Ashon eats \frac{5}{8} of his sandwich. Compare \frac{6}{8} and \frac{5}{8}. Use <, >, or = to write your comparison. Who eats more?”

• Unit 5, Lesson 29, Mass, Session 2, Develop, Teacher Edition, Try It, Make Sense of the Problem, students attend to the specialized language of mathematics. “Before students work on Try It, use Notice and Wonder to help them make sense of the problem. Have students respond to the question, what do you wonder? Record as many responses as time or interest allows making only encouraging comments. Ask, What everyday object has a mass of about 1 gram? 1 kilogram?” Discuss It, “Encourage students to use the term estimate (or estimation) and mass as they talk to each other. Support as needed with questions such as: How did you start? Did you draw a picture or a model to make your estimate? Why or why not? How could writing down what you know about mass help you solve the problem?”

The materials reviewed for i-Ready Classroom 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.

Within Program Implementation, Standards for Mathematical Practice in Every Lesson, “The Standards for Mathematical Practice (SMPs) are embedded within the instructional design of i-Ready Classroom Mathematics.” Embedded SMPs Within Lessons, “In addition to SMPs 1, 2, 3, 4, 5, and 6, which are integrated into the instructional framework, the Teacher’s Guide includes additional opportunities for students to develop the habits of mind described by the Standards of Mathematical Practice. The table of contents indicates all of the embedded Standards for Mathematical Practice for each lesson (both integrated SMPs and the specific SMPs highlighted within the lesson). The Lesson Overview includes the Standards for Mathematical Practice addressed in each lesson. In the Student Worktext, the Learning Target also highlights the SMPs that are included in the lesson.” Structure and Reasoning, “As students make connections between multiple strategies, they make use of structure (SMP7) as they find patterns and use relationships to solve particular problems. Students may also use repeated reasoning (SMP 8) as they construct and explore strategies. SMPs 7 and 8 may be particularly emphasized in selected problems throughout the lesson. As students look for patterns and generalize about strategies, they always consider the reasonableness of their work.”   

MP7 is identified and connected to grade-level content, and there is intentional development of the MP to meet its full intent. Students look for and make use of structure as they work with support of the teacher and independently throughout the modules. Examples include:

• Unit 1, Lesson 2, Add Three-Digit Numbers, Session 2, Teacher Edition, Picture It & Model It, students look at and decompose large numbers into smaller numbers. “If no student presented these models, have students analyze key features and then point out the ways each model represents: 1 hundred + 2 hundreds = 3 hundreds, 3 tens + 8 tens = 11 tens. Ask How does each model represent the hundreds in the addends? in the sum? How does each model represent the tens in the addends? in the sum? Listen For Students should recognize that the number of flats is the same as the digit in the hundreds place, the number of rods is the same as the digit in the tens place, and that the number 110 represents 11 tens. For the base-ten blocks model, prompt students to think about which digits correspond to each place value. Is there any way that this model is more or less helpful than the picture drawn by [student name]? How does the number of hundreds flats relate to the number 130 and 280? How does the number of tens rods relate to the number 130 and 280? For partial sums, prompt students to think of the value represented by each digit of the addends. How many hundreds are in each addend? [1 hundred, 2 hundreds] How do you know? What digit is in the tens place of each addend? [3,8] What is the value of each digit? [30,80] Where do the numbers ‘300’ and ‘110’ come from?” 

• Unit 2, Lesson 10, Understand the Meaning of Division, Session 3, Refine, Apply It, Problem 5, students analyze a problem and look for more than one approach to solve, “Find 45\div9 by describing two ways you can model it using equal groups. Then tell how 9 means something different in each model.”

• Unit 3, Lesson 16, Add Areas, Session 3, Develop, Connect It, Problem 3, students look for patterns or structures to make generalizations and solve problems when breaking up shapes in different ways to find the area. “Amari breaks apart a shape into two smaller shapes. Pablo breaks apart the same shape into two different shapes. Explain how you know that the total area of Amari’s two shapes is the same as the total area of Pablo’s two shapes.”

MP8 is identified and connected to grade-level content, and there is intentional development of the MP to meet its full intent. Students look for and express regularity in repeated reasoning as they work with support of the teacher and independently throughout the modules. Examples include:

• Unit 4, Lesson 23, Find Equivalent Fractions, Session 4, Develop, Connect It, Problem 5, students look for and express regularity in repeated reasoning when writing fractions. “Write a fraction equivalent to 4. Use the fraction below to help you. \frac{number of equal parts described}{number of equal parts in the whole} ____.”

• Unit 4, Lesson 21, Understand Fractions on a Number Line, Session 1, Explore, Model It, Problem 2, students look for repeated reasoning to understand that each equal part on the number line is a fraction of the whole when labeling fractions on a number line. “Fractions are numbers that name equal parts of a whole. You can show fractions on a number line. On the number line below, the whole that is between 0 and 1 is divided into equal parts. Label each part of the area model above the number line with the unit fraction it represents.” A picture is shown of a number line with tics at each fourth.

• Unit 5, Lesson 29,  Mass, Session 2, Develop, with teacher guidance, students look for and express regularity in repeated reasoning when estimating mass. Try It, “Diego bought a medium-sized watermelon at the store. Estimate the mass of the watermelon.” Teacher Edition, Discuss It, “Encourage students to use the terms estimate (or estimation) and mass as they talk to each other. Support as needed with questions such as: How did you start? Did you draw a picture or a model to make your estimate? Why or why not? How could writing down what you know about mass help you solve the problem?” Teacher Edition, Differentiation, Deepen Understanding, Estimating Measurements,  When discussing Picture It, prompt students to consider how using a reference object, such as a book, can help in making estimates. Ask Why can knowing the mass of one of the books help you figure out the mass of the watermelon? Listen For If I know the mass of 1 book, and the watermelon feels about as heavy as 6 of these same books, then I can multiply the mass of the book by 6 to estimate the mass of the watermelon. Generalize How is estimating to find the mass of an object like estimating to find other measurements, such as liquid volume? You make an ‘educated guess’ by using what you know to estimate. For example, I can see how the mass of an object being measured compares with the known mass of another object to estimate the first object to estimate the first object’s mass, or I can see how much of a container is filled by 1 liter of liquid to estimate how much the container can hold when full. These estimates should be close since they are based on something I know and not just a wild guess.”