3rd Grade - Gateway 2

The materials reviewed for ORIGO Stepping Stones 2.0 Grade 3 meet expectations for giving attention throughout the year to individual standards that set an expectation of procedural skill and fluency. Materials attend to the Grade 3 expected fluencies, single-digit products and quotients (products from memory by end of Grade 3) and add/subtract within 1000.

The instructional materials develop procedural skills and fluencies throughout the grade-level. Opportunities to formally practice procedural skills are found throughout practice problem sets that follow the units. Practice problem sets also include opportunities to use and practice emerging fluencies in the context of solving problems. Ongoing practice is also found in Assessment Interviews, Games, and Maintaining Concepts and Skills.

The materials attend to the Grade 3 expected fluencies: 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. For example, In Module 3, Lesson 1 students build fluency through completing Maintaining Concepts and Skills. In Maintaining Concepts and Skills, students identify addition, subtraction and multiplication facts primarily of fives and tens. In addition, the instructional materials embed opportunities for students to independently practice procedural skills and fluency. Examples include:

• Maintaining Concepts and Skills lessons incorporate practice of previously learned skills from the prior grade level. For example, Maintaining Concepts and Skills in Module 1, Lesson 2, Number: Identifying three-digit numbers on a number line, provides practice for adding and subtracting within 20 (2.NBT.2).

• Each module contains a summative assessment called Interviews. According to the program, “There are certain concepts and skills , such as the ability to route count fluently, that are best assessed by interviewing students.”  For example, in Module 4’s Interview 1 has students demonstrate fluency of 2’s multiplication facts and Interview 2 has students demonstrate fluency of 4’s multiplication facts.

• “Fundamentals Games” contain a variety of computer/online games that students can play to develop grade level fluency skills. For example Double Bucket, students demonstrate fluency of 2’s multiplication facts and on Interview 2 students demonstrate fluency of 4’s multiplication facts (3.OA.7).

• Some lessons provide opportunities for students to practice the procedural fluency of the concept being taught in the “Step Up” section of the student journal.

• Activities provide practice for skills learned earlier in the grade such as, Module 9, Lesson 6, Subtraction: Exploring subtraction involving zero, where students practice multiplication facts (3.OA.7).

The materials reviewed for ORIGO Stepping Stones 2.0 Grade 3 meet expectations for being designed so that teachers and students spend sufficient time working with engaging applications of the mathematics.

Engaging applications include single and multi-step word problems presented in contexts in which mathematics is applied. There are routine problems, and students also have opportunities to engage with non-routine application problems. Thinking Tasks found at the end of Modules 3, 6, 9, and 12, provide students with problem-solving opportunities that are complex and non-routine with multiple entry points.

Examples of routine application problems include:

• Module 1, Lesson 7, Multiplication: Introducing the symbol, Student Journal, Step Up, page 25, Problem 2d, addresses standard 3.OA.3, “Henry cut 5 lengths of rope. Each piece was 4 meters long. What was the total length of rope?"

• Module 4, Lesson 6, Division: Introducing the twos and fours facts, Student Journal, Step Up, page 135, Problem 3b, addresses standard 3.OA.3, “32 chicken nuggets are shared equally among 8 friends. How many nuggets are in each share?”

• Maintaining Concepts and Skills includes some application problems and addresses standard 3.OA.8, for example Module 3, Lesson 8, Number: Working with place value, Student Journal, Maintaining concepts and skills, Ongoing Practice, page 105, Problem 2a, “Samuel’s mom bought 3 tickets for the roller coaster. Tickets are $4 each. What was the total cost?”

• Module 6, Lesson 4, Multiplication: Solving word problems, Student Journal, Maintaining concepts and skills, Words at Work, page 206, addresses standard 3.OA.8, “Michelle’s grandmother gave her $40 to spend at the county fair. Michelle had 6 rides on the Mega Drop and 4 rides on the Rollercoaster. Rides on the Mega Drop cost $5 each and rides on the Rollercoaster cost $8 each. She also bought lunch for $12. At the end of the day, she has $2 left. How much of her own money did she take to the fair?”

• Module 1, Lesson 11, Multiplication: Introducing the five facts, Problem Solving Activity 3, addresses standard 3.OA.3, “Stella has been collecting baseball cards. Every week she doubles the number of baseball cards she has. Stella has 120 cards. How many cards did she have three weeks ago? How many cards will Stella have next week?”

• Module 8, Lesson 12, Mass/capacity: Solving word problems, Problem Solving Activity 4 has eight story problems and addresses standard 3.OA.3, “A farmer planted fruit trees in rows of 9. He planted 81 trees in total. How many rows did he plant?”

Examples of non-routine application problems with connections to real-world contexts include:

• Module 2, Lesson 7, Time: Relating past and to the hour, Investigation 2, students brainstorm a list of real-life situations where it would be necessary to read and write time to the nearest minute. In the extension, students brainstorm situations where it would be necessary to write time to the nearest second (3.MD.1).

• Module 3, Lesson 12, Number: Rounding three- and four-digit numbers, Thinking Task, Question 2 asks, “A class of Grade 3 students is raising money for a field trip. They decide to run a car wash as a fundraiser. Customers can decide between three different types of car washes (A chart with car wash prices is provided). At the end of the carwash, the Premium Wash option raised $90. The Quick Wash option raised the same amount. How many cars were washed with the Quick Wash option?”

• Module 6, Lesson 12, Angles: Estimating and Calculating, Thinking Task, Question 1 states, “This year, the PTA raised $300 to plant a school garden. The PTA president announces that this year they raised $124 more dollars than last year. How much money did they raise last year.”

• Module 9, Lesson 12: Common fractions: Solving comparison word problems, Thinking Task, Question 1 provides a diagram of where students and families will sit during a choir performance in the gym. Problems include, “How many students will fit in each full row of the risers? What is the greatest number of students who can stand on all the risers to perform all at once?”

• Module 12, Perimeter/area: Solving word problems,  Thinking Task, “Mrs Chopra’s Grade 3 class has been asked to hang their drawings on a folding display board, the board has four rectangular panels, each panel is 3 feet x 6 feet, drawings can be posted on the front and the back of each panel, all the drawings were made on rectangular paper in three different sizes.” A table of drawing sizes is provided with the following information: 32 small 1 x 1, 16 medium 1 x 2, 8 large 2 x 2. Problem 1: “What is the area of each panel?” Students must use the information from the table to answer the question. Question 2 states, “Write an expression that shows how to find the total area for the front and the back panels of the display board."

The materials reviewed for ORIGO Stepping Stones 2.0 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:  Mathematical Practice Overview, Module Mathematical Practice documents and within specific lessons, alongside the learning targets or embedded within lesson notes.

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:

• Module 1, Lesson 8, Multiplication: Using the turnaround idea with arrays, Step 2 Starting the Lesson and Step 3 Teaching the Lesson, students make sense of and persevere in using multiple representations of arrays to identify multiplication turnaround facts. “Organize students into pairs and distribute the resources. Ask, How can we show 24 cubes in equal groups? What can we write? (MP1) Encourage students to make observations on the different representations, the different structure of the groups/arrays, and the equations (being multiplication and/or addition).” Step 3, “Project the Step In discussion from Student Journal 1.8 and work through the questions with the whole class. Read the Step Up and Step Ahead instructions with the students. Explain that in Question 3 they must color the items to match the word problem, then write the two turnaround facts that give the same product (MP1).”

• Module 2, Lesson 5, Addition: Solving word problems, Step 3 Teaching the Lesson and Student Journal, page 57, students make sense of word problems involving addition of two and three-digit numbers and persevere in determining solution methods. “Project the word problem (slide 2) and read it out loud to the group. Discuss the points below (MP1): Noah and Daniela are collecting pennies for a local charity. Noah has collected 124 pennies and Daniela has collected 132. How many pennies have they collected?” The teacher asks “What is the problem asking you to do? What do you need to find out? How will you calculate the total number of pennies?  Will you choose a mental or written method to calculate the total? What numbers help you to decide?” Students are then guided to solve problems in Student Journal, page 57, independently before sharing answers during Step 4 Reflecting on the work. Question 2a, “Solve each problem. Show your thinking. 199 hotdogs were sold in the first half and 175 hotdogs in the second half. How many hotdogs were sold?”

• Module 4, Lesson 12, Common fractions: Relating models, Step 3 Teaching the Lesson, students analyze word problems, consider the different fraction models they could use to represent each problem, and persevere to find a solution. “Tell the students that they are now going to solve word problems that involve fractions and that it might be helpful to use a model to help them solve the problem. Project slide 1 as shown. Then discuss the points below (MP1): Each load of washing uses 14 of a cup of washing powder. Victor does 3 loads of washing. How many cups of washing powder does he use? Organize students into pairs and have them work together to solve the problem. Direct them to pages 156 and 157 of the Student Journal to show their thinking. If some students struggle in the process, ask questions such as What could you do differently? What other model could you try? What is a similar problem using whole numbers that you can solve? (MP1)”

• Module 7, More Math, Problem Solving Activity 4, Word Problems, students make sense and persevere in solving one and two-step word problems using various operations. “Project slide 1 and read the word problem with the students. Ask, What information do we need to solve this problem? What operations will we use? What will we do first? What will we do next? How could you show your thinking? Allow time for students to find a solution. Then invite students to share their solution (72) and explain their thinking. Slide 1: One box holds 9 muffins. Two boxes hold a total of 18 muffins. Three boxes hold a total of 27 muffins. How many muffins will be in 8 boxes?”

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:

• Module 3, Lesson 7, Multiplication: Solving word problems, Step 3 Teaching the Lesson and Student Journal, Step Up, page 100, students reason abstractly and quantitatively as they decontextualize word problems and represent them symbolically. Step 3, “Discuss the students’ answers to Student Journal 3.7. Invite students to write their equations for Question 1 on the board (MP2).” Student Journal, Question 1, “This recipe makes one large bowl of fruit gelatin. Write the answers. Fruit Gelatin: 1 packet of gelatin, 2 sliced peaches, 10 strawberries, 1 can of pineapple, 4 bananas.” Question 1a-d, “How many sliced peaches are needed to make four bowls of fruit gelatin? Hailey bought 20 strawberries. How many bowls of fruit gelatin could she make? How many cans of pineapple are needed to make four bowls of fruit gelatin? Andre has 16 bananas. How many bowls of fruit gelatin could he make?”

• Module 6, Lesson 9, Data: Working with many-to-one picture graphs, Step 3 Teaching the Lesson, students reason abstractly as they interpret a picture graph and connect information to the context of the problem. “Project the picture graph as shown. Explain that this picture graph shows the number of cans that were recycled by three different grade levels. If necessary, review the definition of recycle.” Students answer teacher questions. Then, “Have the students identify the number of cans that have been recycled by each grade. They can then write equations to figure out the differences between the numbers of cans that were recycled. Ask questions such as, How many more cans did Grade 3 recycle than Grade 1? What is the difference between the number of cans recycled by Grade 1 and Grade 2? How many cans were recycled in total? How do you know? (MP2)”

• Module 7, Lesson 1, Multiplication: Introducing the sixes facts, Step 3 Teaching the Lesson, students reason abstractly and quantivity about multiplication as they write equations to match arrays. “Project slide 1 and ask, What do you know about this array? What multiplication fact could we write to match? Interpret each array with the students. Encourage them to describe how a known multiplication fact (8 x 5 = 40) can be used to figure out an unknown multiplication fact (6 x 8 = 48). Say 5 rows of 8 is 40, 6 rows of 8 is 8 more, 6 rows of 8 is 48. Remind the students that the array shows another fact when it is rotated by a quarter turn. Ask a volunteer to write the matching equations 6 x 8 = 48 and 8 x 6 = 48  on the board. (MP2)”

• Module 10, Lesson 4, Area: Identifying dimensions of rectangles, Student Journal, Step Ahead, page 377, students reason abstractly and quantivity as they relate length to area in rectangles. “Use the grid to help solve this problem. A garden in a park is shaped like a rectangle. It has an area of 60 square units. The longer side of the garden is 15 units long. how long is the shorter side?” Step 4 Reflecting on the work, “Ask students to explain why all of the answers were the same. Have students discuss the possible dimensions of other parks that have an area of 60 square units where both dimensions are unknown. Finally, as time allows, discuss what the square units might be if the area is a park.(MP2)”

The materials reviewed for ORIGO Stepping Stones 2.0 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 as it is explicitly identified for teachers in several places: Mathematical Practice Overview, Module Mathematical Practice documents and within specific lessons, and alongside the learning targets or embedded within lesson notes.

Teacher guidance, questions, and sentence stems for MP3 are found in the Steps portion of lessons. In some lessons, teachers are given questions that prompt mathematical discussions and engage students to construct viable arguments. In some lessons, teachers are provided questions and sentence stems to help students critique  the reasoning of others and justify their thinking. Convince a friend, found in the Student Journal at the end of each module and Thinking Tasks in modules 3, 6, 9, and 12, provide additional opportunities for students to engage in MP3. 

Students engage with MP3 in connection to grade level content, as they work with support of the teacher and independently throughout the units. Examples include:

• Module 1, Lesson 10, Multiplication: Introducing the tens facts, Step 3 Teaching the lesson, students justify their own thinking and critique the reasoning of others as they multiply tens. “Project the two groups of marbles (slide 5) and ask, How are the two collections of marbles the same? How are they different? (MP3) If needed, encourage students to think about the number of groups, the number in each group, and the product. Project the next two groups of marbles (slide 6) and repeat the discussion and questions with the two collections side-by-side. When students have established that the totals are the same in the two collections (four groups of ten is the same amount as ten groups of four), project slide 7 showing the summaries. Say, For the first collection you were able to count by tens to get the product. Does that work for the second collection of marbles? Is there another number you can skip count by to get the product? (Fours.) Emphasize that counting by ones, fours, or tens yields the same amount for either collection. If needed, ask volunteers to verify that it is true (MP3). Ask, Is there a way to figure out how many there are in total without having to skip count by tens? For example, is there a way to just know how many there will be in four groups of ten? Allow time for students to share their strategies on how they can remember without skip counting by tens (MP3).”

• Module 4, Lesson 8, Common fractions: Reviewing unit fractions, Step 3 Teaching the lesson, students justify examples and non-examples of unit fractions while critiquing the reasoning of others. “Discuss examples and non-examples created. If necessary, fold and mark one of the shapes so that it shows eight sections that are not equal in area (or length). Encourage a class debate about whether the sections are eighths or not. Make sure students justify their thinking and support their arguments with other examples. Prompt respectful critique by providing sentence stems such as (MP3): I agree/disagree with ___ because … That makes sense. I also think …  I understand why you think that, but …”

• Module 8, Student Journal, page 319, Convince a friend, students construct viable arguments and critique the reasoning of others as they determine equivalent fractions in real-word and mathematical problems. “Felipe cuts modeling clay into equal pieces for his art students. As he hands one piece to each of his 14 students he says, “Each person has one quarter of a block. Use it wisely, as there is not enough for everyone to have a second piece. Samantha figures out there must have been five blocks of clay to begin with. Do you agree or disagree with Samantha? Explain why. Share your thinking with another student to get feedback. Discuss how constructive feedback can help your learning.” 

• Module 10, Lesson 6, Area: Solving word problems, Students Journal, page 373, Step Ahead and Step 4 Reflecting on the work, students construct a viable argument and critique the reasoning of others when they solve word problems using units of measure and analyze solutions and answers of classmates. Step Ahead, “Write an area word problem to match this equation. Then write the area. A = 4 cm x 7 cm.” Step 4, “Invite students to share the word problems they wrote for Step Ahead. Remind other students to listen carefully and critique the problems (MP3). Prompt discussion with sentence stems such as: I can identify the numbers from the equation in … problem… I don’t think that problem matches because… We could change the problem by… The area is not correct because...” 

• Module 12, More Math, Thinking Tasks, Question 3, students construct viable arguments as they use geometric measurements to solve real world problems. “Students use black trim to put a border around the outside of the front of the display board. They have 40 feet of trim left over. Will they have enough trim for the back of the display board? If so, how much trim will be left? If not, how much more will they need? Explain and show your thinking.”

The materials reviewed for ORIGO Stepping Stones 2.0 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 throughout the year. The MPs are often explicitly identified for teachers in several places: Mathematical practice overview, Module Mathematical practice documents, Mathematical modeling tasks, Thinking tasks, and within specific lessons, alongside the learning targets or embedded within lesson notes.

MP4 is identified and connected to grade level content, and there is intentional development of the MP to meet its full intent. Students have many opportunities to solve real-world problems, identify important quantities to make sense of relationships, and represent them mathematically. Students model with mathematics as they work with support of the teacher and independently throughout the modules. Examples include:

• Module 1, Student Journal, page 43, Mathematical modeling task, students model with math as they solve a word problem, explain their thinking, and then explain how others could use that thinking to solve similar problems. “To encourage recycling, Teresa promises 5 cents for every can or bottle the children in the neighborhood give to her. John lives down the street. He gives Teresa some bottles and collects a little over two dollars. Question a “Figure out the number of cans or bottles that could have been given to Teresa. Show your thinking.” Question b “Explain how the children in the neighborhood can apply the same thinking to figure out the amount they could collect for any number of cans or bottles.”

• Module 2, Lesson 3, Addition: Two- and three-digit numbers (with composing), Step 3 Teaching the lesson, students model addition strategies and explain their thinking about the strategy they chose. “Distribute a copy of the support page to each pair of students. Then project slide 4 as shown and discuss the points below: $38, $64 What is your estimate of the total cost of these two items? How did you form your estimate? How can you calculate the sum of the two items? What tool (base-10 blocks, number line, or written method) would you use? What steps would you follow? Have the students work in their pairs to calculate the sum. Although number lines and base-10 blocks are provided, stress that other strategies are also encouraged. It is important that students are given the freedom to choose a strategy/model that best reflects their thinking (MP5). Invite some students to show and explain their strategy (MP4).”

• Module 3, Lesson 9, Number: Comparing and ordering three-digit numbers, Step 3 Teaching the lesson, students model with math as they determine the order of three-digit numbers by placing them on a number line and by using the appropriate symbols to compare them. “Have two students mark the locations of the numbers on the number line (MP4). Highlight that when comparing two numbers on a number line, the number to the right will be greater than the number to the left. Then ask, What symbols can we use to show that a number is greater than or less than another number? Allow a student to write the symbols > and < on the board with the words greater than and less than labeled appropriately (MP4). Ask, ``Do any of you have a way to remember which symbol to use?”

• Module 7, Student Journal, page 280, Mathematical modeling task, students model a real- world problem involving calculations with perimeter. “Isaac and Naomi are building a sandbox with 7 sides like the one below. There is one post at each corner and there are 3 wooden planks between the posts on each side. Each wooden plank needs 2 bolts to fix it into place. Write a list to show how many wooden planks and how many bolts Isaac and Naomi need to buy.”

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

• Module 4, Lesson 3, Division: Introducing the tens facts, Step 3 Teaching the lesson, students use the appropriate strategy as a tool in order to solve problems in multiple ways, explain how methods are connected, and why solutions make sense. “Project the diagram shown above (slide 1) and say, Each set of three numbers belongs to a fact family. Why do you think there is a circle around one of the numbers in each set? (It represents the product.) Why does it represent the product? What do the other numbers represent? (Either the number of groups or the number in each group.) Organize students into pairs and distribute the resources. Have them work together to calculate and write the missing number in each group of three on the support page. Remind them that for each example, they can use strategies including acting it out with cubes or counters, drawing pictures, or using words to help find the solution (MP5).”

• Module 5, Lesson 9, Subtraction: Counting back to subtract two- and three-digit numbers (with decomposing), Step 3 Teaching the lesson, students choose strategies as tools in order to model subtraction of two and three-digit numbers. “Project the next equation (slide 3). Organize students into pairs to discuss how they would most likely solve the problem mentally or find a reasonable estimate. Encourage students to choose a strategy and/or tool to record or step through their thinking (MP5). Invite students to explain and demonstrate their preferred method, using Flare Place Value or simply drawing on the board for any discussion with base-10 blocks.”

• Module 10, Student Journal, page 356, Mathematical modeling task, students choose from tools or strategies they have learned to solve a multi-step real-world problem. “Paul is buying materials for a new building project. He notices 35 long wood screws are sold in each packet. He needs 300 wood screws in total and wonders if 8 packets are enough. How many packets of wood screws should Paul buy? Explain your thinking.” 

• Module 12, More math, Problem solving activity 1, students choose an appropriate strategy as a tool to solve a real-world problem. “Oliver is planning a birthday party and can order tables that can seat 4 or 6. 76 people are coming to the party and he wants all the tables to be filled. How many tables should Oliver order?” Directions to the teacher state, “Discuss points such as, Is there a way to organize the information to help you answer the question? Could you draw a picture to help? Could you use a table? What operation (addition, subtraction, multiplication, division) could you use? Can all the groups be equal in size? Is there more than one possible solution? Organize students in pairs to solve the problem. Observe whether they use a systematic approach to finding a solution. Afterwards, invite pairs of students to share the different solutions.”

The materials reviewed for ORIGO Stepping Stones 2.0 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. MP6 is explicitly identified for teachers in several places: Mathematical practice overview, Module Mathematical practice documents, Mathematical modeling tasks, Thinking tasks, and within specific lessons, alongside the learning targets or embedded within whole class lesson notes. 

Students have many opportunities to attend to precision in connection to grade level content as they work with support of the teacher and independently throughout the modules. Examples include:

• Module 1, Lesson 6, Number: Locating four-digit numbers on a number line, Step 3 Teaching the lesson, students attend to precision as they label 4-digit numbers on a numberline and check their work. “Project the Step In discussion from Student Journal 1.6 and work through the questions with the whole class. Read the Step Up and Step Ahead instructions with the students. Make sure they know what to do, then have them work independently to complete the tasks. Remind students to use the number lines with precision and to take the time to check their answers before moving on (MP6).”

• Module 7, Lesson 4, Multiplication: Working with all facts, Student Journal, page 252, Step Up, students attend to precision as they find multiplication facts that are close to a given number. Question 1, ”Write four multiplication facts to match each of these. a. Facts with a product close to 39. b. Facts with a product close to 52. c. Facts with a product close to 46. d. Facts with a product close to 11.”

• Module 9, Thinking Tasks, Preparing for a School Assembly, Question 1, students attend to precision as they multiply to solve a real world problem. “Every year in the gym, the school hosts two choir assemblies where students perform for their families. Half the families come on the first night and half the families come on the second night. Each grade helps prepare for the choir performance. Grade 3 has two tasks: Set up the risers where students stand to sing. 6 students can fit on one riser. Set up the chairs for the families in the audience. 48 chairs fit in each section. Students will stand on the risers to sing. How many students will fit in each full row of risers? What is the greatest number of students who can stand on all the risers to perform all at once? Show your thinking.”  

Students have frequent opportunities to attend to the specialized language of math in connection to grade level content as they work with support of the teacher and independently throughout the modules. Examples include:

• Module 3, Module overview, Vocabulary development, students can attend to the specialized language of math as teachers are provided a list of vocabulary terms. “The bolded vocabulary below will be introduced and developed in this module. These words are also defined in the student glossary at the end of each Student Journal. A support page accompanies each module where students create their own definition for each of the newly introduced vocabulary terms. The unbolded vocabulary terms below were introduced and defined in previous lessons and grades. Addition, array, closest, commutative property of addition, commutative property of multiplication, compare, distance, double, equation, fact, fraction, greater than, greater, greatest, hundreds, least, less than, longer, meter (m), multiplication, multiply, nearest, number line, one-fourth, one-half, one-third, order, place value, polyhedron, position, product, pyramid, rectangle, round (a number), rows of, shorter, split, tens.” Students are provided with a Building Vocabulary support page. The page includes: Vocabulary term (the bolded terms), Write it in your own words, and Show what it means.

• Module 8, Lesson 8, Common fractions: Exploring equivalent fractions, Student Journal, page 304,  Words at Work, students use the specialized language of mathematics to write equivalent fractions and explain how they are equivalent. “Write two equivalent fractions. Write how you know the two fractions are equivalent.”

• Module 10, Lesson 2, Area: Calculating the area of rectangles (metric units), Step 2 Starting the lesson, students use the specialized language of mathematics as they reason about and explain area problems. “Review what the students know about the area by asking, What does area mean? Encourage students to share their understanding (MP6). Establish that area relates to the amount of space that an object covers.” Student Journal, Step Ahead, age 361, “a. Measure the area of this rectangle using these blocks and write the number of each. _ orange pattern blocks _ base-10 ones blocks. b. What do you notice? c. Why do you think this happened?”

The materials reviewed for ORIGO Stepping Stones 2.0 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 and they are often explicitly identified for teachers in several places: Mathematical practice overview, Module Mathematical practice documents, Mathematical modeling tasks, Thinking tasks, and within specific lessons, alongside the learning targets or embedded within whole class lesson notes.

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 modules to look for, describe, and make use of patterns within problem-solving as they work with support of the teacher and independently. Examples include:

• Module 1, Lesson 3, Number: Representing four-digit numbers, Step 2 Starting the lesson, students look at and use base-10 blocks and expanders to apply their understanding of two- and three-digit numbers to four-digit numbers. “Open Flare Place Value showing the base-10 tens block and a base-10 ones block. Refer to the tens block and ask, How many ones would we need to show the same number? (Ten.) Why do we need exactly ten ones? Bring out that the total amount is the same (MP7). Say, One ten is the same amount as ten ones. Repeat the discussion for a tens block and a hundreds block. Show a thousands block and a ones block and ask, How many ones would you estimate are in the big block? (Note: Researchers have noted that many students, even many in Grade 6, will estimate that there are around six hundred ones represented in a thousands block. Many students see the thousands block as composed of six hundreds blocks, one for each face.) Demonstrate with blocks how a thousands block represents 10 hundreds blocks stacked on top of each other. Have students count by hundreds for each layer that is stacked. Say, Ten hundreds is the same amount as one thousand. Encourage students to think about a thousands cube as having ten layers of the hundreds blocks (MP7). Ask, How many tens blocks are in a thousands block? Allow time for students to figure out the amount then say, One hundred tens is the same amount as one thousand. Organize students into groups and distribute the resources. Have them work in groups to build a thousands block with tens blocks to demonstrate the relationship. (MP7)”

• Module 4, Lesson 6, Division: Introducing the twos and fours facts, Step 3 Teaching the lesson, students use the structure of multiplication to think about the relationship between multiplication and division. “Project slide 1 as shown. Clarify that 12 dots have been arranged into equal rows, and that some of the dots have been covered. Then discuss the points below: 12 dots, 2 equal rows, What do you know about this picture? What do you see? What do you need to figure out? (The number of dots in each row.)  How could we calculate the number of dots in each row? What equation could we write to help figure out the number in each row? Encourage students to use thinking such as “2 rows of ___ is 12,” “Double ___ is 12,” “12 divided equally into 2 rows is ___,” or “Half of 12 is ___.” (MP7) Write the multiplication equation with a missing factor, 2 × ___ = 12 and the related division equation 12 ÷ 2 = ___ on the board. Ask, Which equation is easier to solve? Invite individuals to share and justify their thinking (MP3). Complete the equations on the board. Repeat the activity for diagrams showing 16 dots, 2 equal rows (slide 2), and 10 dots, 5 in each row. (slide 3)”

• Module 7, Lesson 1, Multiplication: Introducing the sixes facts, Step 4 Reflecting on the work, students make use of structure as they connect strategies for sixes and nines multiplication facts. “Project slide 7. Ask students to describe the thinking they would use to figure out the products. Where possible, encourage them to suggest more than one strategy. Highlight the strategies that use the distributive property of multiplication. Ask, How is this strategy the same as (different from) the strategy we used for the nines multiplication facts? How is it different? Help students recognize the similarities between the structure and models used for the two strategies. (MP7)” Slide 7, “6\times7 = __  6\times8 = __ 6\times4 = __ .”

• Module 9, Lesson 8, Common fractions: Comparing unit fractions (length model), Student Journal, page 340, Step Up, Question 1, students make use of structure as they recognize patterns in numbers. “a. Color one part in each row of this fraction chart. b. Circle the fraction that is greater in each pair. \frac{1}{2} or \frac{1}{4}, \frac{1}{8} or \frac{1}{2}, \frac{1}{4} or \frac{1}{8}.” Step 4 Reflecting on the work, “Discuss the students’ answers to Student Journal 9.8. Refer to Step Up and ask, What is the same about all the fractions? (The numerator is always one.) Write the term unit fraction on the board and explain that unit fractions always have a numerator of one. Ask, What happens to the size of unit fractions when the denominators increase? Discuss the points below: As the denominator increases, the size of the unit fraction decreases. The denominator helps determine the size of the unit fraction and the numerator helps count how many unit fractions there are. For example, \frac{3}{4}means there are 3 (count) unit fractions of \frac{1}{4}(size). There is a pattern to the relationship between the number of parts and the size of the parts needed to fill the whole. (MP7) 2 one-halves make one whole, 3 one-thirds make one whole, 4 one-fourths make one whole.”

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:

• Module 2, Lesson 1, Addition: Investigating patterns, Step 3 Teaching the lesson, students extend knowledge of number patterns to find unknowns in equations. “Project the next pan balance picture (slide 5) and ask, What will happen to the pan balance? How can we keep it level (even)? Encourage students to explain that one of the parts will need to be changed. Project the related equation 26 = 15 + ___ (slides 6 and 7) and have a volunteer write the unknown part (MP8). Repeat the discussion to change the total to 27 and then 28. (slides 8 to 10)”

• Module 7, More math, Investigation 3, Using written methods to add, students use repeated reasoning as they look for shortcuts to solve problems. “What are all the different written methods that could be used to calculate 537 + 374? How are the same methods (different)? What are the advantages (disadvantages) of the methods? Are there times when you would use one method over the other? Why?” 

• Module 8, Lesson 5, Common fractions: Counting beyond one whole, Student Journal, page 294, Step Up, Question 1, students use repeated reasoning as they represent fractions visually. “Each strip is one whole. Color parts to show each fraction. a. \frac{1}{3} b. \frac{2}{3} c. \frac{3}{3} d. \frac{4}{3} e. \frac{5}{3} f. \frac{6}{3} g. \frac{7}{3}.” Step 4 Reflecting on the work, “Some students may relate the length models to equal groups of two, four, and eight, and use multiplication to calculate the numbers in six wholes. For example, "Six wholes is six groups of two halves, or 12 halves." Others may use repeated addition or skip counting to figure it out. (MP8)”

• Module 12, Lesson 1, Division: Two-digit numbers, Student Journal, page 435, Step Up, Question 2a, students extend their fact families knowledge to writing equations. “Draw blocks in the large part to show the number being shared. Then draw blocks in the small parts to show the number in each share. Complete the equation. 48\div 2.”  Step 4 Reflecting on the work states, “Refer to Question 2a. Remind students of the earlier lessons about fact families. Say, We know that 48\div 2 = 24. What other related equations can we write? Invite volunteers to record the related equations (48\div 2 = 24, 24\times 2 = 48, 2\times 24 = 48) on the board. (MP8)”