3rd Grade - Gateway 2

The instructional materials for Match Fishtank Mathematics Grade 3 meet expectations for developing conceptual understanding of key mathematical concepts, especially where called for in specific standards or cluster headings.

The materials include problems and questions that develop conceptual understanding throughout the grade-level, for example: 

• In Unit 2, students are provided multiple opportunities to develop the concept of multiplication through arrays, equal groups, skip counting, and tape diagrams (3.OA.1). For example, in Lesson 10, Anchor Tasks, Problem 2, Guiding Questions state, “How can the skip-counting sequence help you solve Part (a)? Why can I skip-count by threes even though the size of each group is 8? How do you know when you’ve reached your solution when you skip-count on your paper? What about on your fingers? How can the skip-counting sequence help you solve Part (b)? Part (c)? How do you know when you’ve reached your solution when you skip-count on your paper? What about on your fingers? If you don’t know the skip-counting sequence, what could you draw to solve?”
• In Unit 3, Lesson 17, students multiply a one-digit whole number by a multiple of ten (3.NBT.3) using base ten blocks, number lines, and properties of operations. For example, Anchor Tasks, Problem 3, Guiding Questions state, “How can you represent this problem with base ten blocks? How can you represent this problem on a number line?”
• In Unit 4, Lessons 2-4, students use square-inch and square-centimeter tiles to explore the concept of area (3.MD.5). For example, Lesson 2, Anchor Tasks, Problem 3, students make connections through the Guiding Questions, “Which square unit has greater area, a square centimeter or a square inch? How is that related to the amount of space taken up by each shape constructed from each of those units?”
• In Unit 6, Lesson 26, students measure lengths to the nearest quarter inch (3.MD.4) using square inch tiles and creating a ruler. For example, in Anchor Tasks, Problem 1, students make connections between concepts through the Guiding Questions, “How is a ruler similar to a number line? How is it different?”  
• In Unit 7, Lesson 7, students use various objects (thumbtacks, dictionary, snap cubes, paper clips, etc.) to develop benchmarks for 1 kilogram and 1 gram (3.MD.2). For example, Anchor Tasks, Problem 3, Guiding Questions state, “About how many paperclips would it take to balance with a dictionary? How do you know?”

The materials provide opportunities for students to independently demonstrate conceptual understanding throughout the grade, for example: 

• In Unit 2, Lesson 4, students identify, interpret, and create situations involving an unknown number of groups and find the number of groups in situations (3.OA.1,2). In Problem Set, Problem 5, students solve, “Leah’s favorite marbles come in bags of 10. If she wants to purchase 60 marbles, how many bags will she need to buy? a. Write a multiplication equation to represent the problem. b. Create a visual model of the marbles in the bags. c. How many bags of marbles are there?”
• In Unit 3, Lesson 5, students use arrays to explore the associative property (3.OA.5). In Anchor Tasks, Problem 3, students “a. Decompose the array into smaller arrays whose product you know from memory to be able to find the total product represented by the array. b. Find a second way to decompose the array.”
• In Unit 4, Lesson 2, students “find the area of a figure using square-inch and square-centimeter tiles, which can be used as concrete standard units” (3.MD.5). In Target Task, Problem 2, students, “Construct a rectangle whose area is 16 square inches. Then show your teacher.”
• In Unit 6, Lesson 1, students develop the concept of fractional parts (3.NF.1) by using pattern blocks to construct shapes (3.G.2) with fractional parts. In Anchor Task, Problem 3, students “Create pattern block shapes where one piece represents each of the following fractions. a. 1 half  b. 1 fourth c. 1 third d. 1 sixth.” Guiding questions state, “When I construct the whole, what do I need to make sure is true of each of my parts in order for them to represent 1 half, or 1 fourth or any other fraction? In order for each part to represent 1 half, how many equal parts must your whole be? What does that tell us about the number of pattern blocks we’ll need to construct a shape whose fractional unit is halves?” 
• Unit 7, Lesson 3, students relate clocks to number lines (3.MD.1). In the Target Task, students “Plot points on the following number line that correspond with each time below. Label each point with its corresponding letter a - d. a. 5:45 p.m. b. (picture of a clock provided). c. 6:12 p.m. d. (picture of a clock provided).”

The instructional materials for Match Fishtank Mathematics Grade 3 meet expectations for attending to those standards that set an expectation of procedural skill and fluency.

The structure of the lessons includes several opportunities to develop these skills, for example:

• In the Unit Summary, procedural skills for the unit are identified.
• Throughout the materials, Anchor Tasks provide students with a variety of problem types to practice procedural skills.
• Problem Sets provide students with a variety of resources or problem types to practice procedural skills.
• There is a Guide to Procedural Skills and Fluency under teachers tools and mathematics guides. 

The instructional materials develop procedural skill and fluency throughout the grade level. The instructional materials provide opportunities for students to demonstrate procedural skill and fluency independently throughout the grade level, especially where called for by the standards (3.OA.7 and 3.NBT.2). For example:

• In Unit 1, Lessons 8 and 9, students have multiple opportunities to develop fluency in adding two numbers (3.NBT.2) with multiple compositions within 100. Problem Set, Problem 1 states, “Solve. Show or explain your work.  a. $$46 + 5 =$$ _____ b. $$46 + 25 =$$ _____ c. $$46 + 125 =$$ _____   d. _____ $$= 59 + 30$$ e. $$509 + 83 =$$ _____   f. _____ $$= 597 + 30$$ g. $$29 + 63 = $$ _____   h. _____ $$= 345 + 294$$ i. $$480 + 476 =$$ _____.”
• In Unit 1, Lesson 11, students subtract two numbers with up to one decomposition within 1,000 (3.NBT.2). In Anchor Task, Problem 2, students, “Find the difference. Show or explain your work. a. $$82 - 56 =$$ _____   b. $$287 +$$ _____ $$= 308$$ c. _____ $$= 748 - 93$$.” Also, in Homework, Problem 1h, students independently solve, “$$803 + 542$$.”
• In Unit 1, Lesson 12, students subtract two numbers with up to one decomposition within 1,000 (3.NBT.2). In Target Task, Problem 1, students, “Solve. Show or explain your work. a. $$346 - 187 =$$ _____   b._____ $$= 1,000 - 592$$ c. $$239 = 305 -$$ _____.” Also, in Problem Set, Problem 1, students independently solve, “a. _____ $$= 340 - 60$$ b. $$340 - 260 =$$ _____     c. $$540 - 260 =$$ _____ d. $$513 -$$ ____ $$= 148$$ e. $$387 +$$ _____ $$= 641$$   f. $$934 - 488 =$$ _____ g. _____ $$= 700 - 52$$ h. $$700 - 452 =$$ _____  i. $$452 = 1,000 -$$ ____.”
• In Unit 2, Lessons 6, 8, and 10 address developing multiplication fluency (3.OA.7). Students multiply using arrays, skip counting, and independent practice. For example, in Lesson 10, Problem Set, Problem 3c, students “Solve. a. $$2 \times 3 =$$ ___   b. ___ $$= 3 \times 3$$ c. $$5 \times 3 =$$ ___ d. ___ $$= 10 \times 3$$”
• In Unit 2, Lesson 11, students develop fluency with multiplication and division facts using units of 4 (3.OA.7). In the Homework, Problem 9a, students independently solve the following equation, “$$8 \div 4 =$$ ___.”
• In Unit 2, Lesson 13, students determine the unknown whole number in a multiplication or division equation, including equations with a letter standing for the unknown quantity (3.OA.4,7,8). The Target Task, Problem Set and Homework sections provide students many opportunities to independently practice finding the unknown. For example, Target Task states, “1. $$z = 5 \times 9$$,  $$z=$$ ___,   2. $$20v = 5$$, $$v=$$ ___,   3. $$3 \times w = 24$$, $$w =$$___,   4. $$7=y \div 4$$, $$y=$$___.”
• In Unit 3, Lesson 6, students have multiple opportunities to skip-count to build fluency with multiplication facts using units of 6 (3.OA.7). Problem Set, Problem 1 states, “Skip-count by six to fill in the blanks. Match each number in the count-by with its multiplication fact. Then, use the multiplication expression to write the related division fact directly to the right.” 
• In Unit 3, Lesson 17, students “multiply one-digit whole numbers by multiples of 10,” (3.NBT.3) as Guided Practice in the Anchor Tasks. Anchor task, Problem 1 states, “There are 6 tables in Mrs. Potter's classroom. There are 4 students sitting at each table. Each student has a dime to use as a place marker on a board game. a. What is the value of the dimes at each table? b. What is the value of the dimes in Mrs. Potter’s classroom?” In the Homework and Problem Set, students have numerous opportunities to independently practice multiplying by a multiple of 10. Homework, Problem 2e, “_____ $$= 60 \times 3$$.”

The instructional materials for Match Fishtank Mathematics 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 problems, routine and non-routine, presented in a context in which the mathematics is applied. 

In Problem Sets and Target Tasks, students engage with real-world problems and have opportunities for application, especially where called for by the standards (3.OA.3 and 3.OA.8). The instructional materials include multiple opportunities for students to engage in routine and non-routine application of mathematical skills and knowledge. Students have opportunities to independently demonstrate the use of mathematics flexibly in a variety of contexts. Examples of routine application include, but are not limited to:

• In Unit 1, Lesson 7, Target Task, students use place value understanding to round whole numbers to the nearest 10 or 100. For example, “Mrs. Needham knows there are 283 students in Grades 3, 4, and 5 at Match Community Day. She says that there are about 280 students. a. Did Mrs. Needham round to the nearest ten or hundred? b. Mrs. Needham rounded the number of Grade 3, 4, and 5 students to be able to set up seats in the gym for an assembly with those grades. Do you think Mrs. Needham made a good decision about how she rounded the number of students? Why or why not?” (3.NBT.1)
• In Unit 2, Lesson 9, students apply their understanding of multiplication to solve one-step word problems (3.OA.3). For example, Target Task, Problem 1 states, “Ms. McCarty has 8 stickers. She puts 2 stickers on each homework paper and has no more left. How many homework papers does she have?”
• In Unit 3, Lesson 3, students understand the role of parentheses and apply to solving problems (3.OA.5, 3.OA.8). For example, Target Task, Problem 2, students solve, “Marcos solves $$24 \div 6 + 2 =$$ ____. He says it equals 6. Iris says it equals 3. Show how the position of parentheses in the equation can make both answers true.” 
• In Unit 5, Lesson 5, students solve word problems involving finding perimeter given the side lengths (3.MD.8). For example, Problem Set, Problem 7 states, “Kevin’s yard is a square. He wants to build a fence around it. Each side of the square is 12 feet long. His house is one side of the square, so he doesn’t need a fence on that side. Decide how many feet of fencing Kevin needs. Explain your thinking.”
• In Unit 6, Lesson 9, students identify a shaded fractional part in different ways, depending on the designation of the whole (3.NF.1). For example, Problem Set, Problem 5 states, “Mrs. Ingall has two apples. She cuts each one into 4 pieces. Mrs. Ingall thinks that one piece is $$\frac {1}{4}$$. Mr. Silver says one piece is $$\frac {1}{8}$$. What’s the reason for their disagreement?”
• In Unit 7, Lesson 6, students solve real world word problems involving all cases of elapsed time in minutes (3.MD.1). For example, Problem Set, Problem 4 states, “Tessa walks her dog for 47 minutes. Jeremiah walks his dog for 30 minutes. How many more minutes does Tessa walk her dog than Jeremiah?” 
• In Unit 7, Lesson 12, students apply their understanding of volume in real world contexts (3.MD.2). For example, Target Task, Problem 2 states, “Elijah uses 275 mL of milk for a recipe. He has 367 mL of milk left. How much milk did Elijah have before using some for his recipe?” 

Examples of non-routine application include, but are not limited to:

• In Unit 1, Lesson 14, students solve one- and two-step problems involving addition and subtraction, using rounding to assess the reasonableness of the solution. For example, Problem Set, Problem 6 states, “Third-grade students took a total of 1,000 pictures during the school year. Ted took 72 pictures. Mary took 48 pictures. Part A: What is the total number of pictures taken by the rest of the third-grade students during the school year? Part B: Ella took 8 more pictures than Ted took. How many more pictures did Ella take than Mary?” (3.OA.8).
• In Unit 3, Lesson 16, students solve one- and two-step word problems involving units up to 9 (3.OA.8). For example, Problem Set, Problem 7 states, “Solve. Show or explain your work. a. Roland ate three pancakes at IHOP. His sister Janice ate two more than that. The total price for the pancakes was $24. How much did each pancake cost? b. Janice’s dad decided to order 3 pancakes of his own and 2 glasses of juice. The bill, including the cost of Roland’s and Janice’s pancakes, now comes to $29. How much does each glass of juice cost?”
• In Unit 4, Lesson 8, students solve word problems involving area (3.MD.7b). For example, Problem Set, Problem 4 states, “A rectangular garden has a total area of 48 square meters. What are the possible length and width of the garden? Come up with two possibilities.”
• In Unit 5, Lesson 10, students solve a variety of word problems involving area and perimeter (3.MD.8). For example, Problem Set, Problem 5 states, “A path is built around a pool in the shape of a rectangle. The width of the pool is 7 yards. The area of the pool is 70 square yards. Find the length, in yards, of the pool. Find the perimeter, in yards, of the pool.” 
• In Unit 6, Lesson 13, students place any fraction on a number line with endpoints greater than 0, (3.NF.2). For example, Problem Set, Problem 2 states, “A student was asked to place 6/2 on a number line from 2 to 6. Here is their work.” A picture of student work is shown, and the problem states, “Evaluate their work by thinking about: Is it correct? Why? If it is incorrect, what mistake did the student make?”

The instructional materials for Match Fishtank Mathematics Grade 3 meet expectations that the three aspects of rigor are not always treated together and are not always treated separately. All three aspects of rigor are present in the instructional materials. Many of the lessons incorporate two aspects of rigor with an emphasis on application. Student practice includes all three aspects of rigor, though there are fewer questions for conceptual understanding. There are instances where all three aspects of rigor are present independently throughout the program materials. 

Examples of Conceptual Understanding include:

• In Unit 2, Lesson 3, students “identify and create situations involving unknown group size and find group size in situations,” (3.OA.2). In Anchor Task, Problem 1 states, “Split 18 counters equally into two groups. a. How many counters do you have in each group? b. Write a multiplication equation to represent this situation.”
• In Unit 3, Lesson 1, students use arrays, number lines and tape diagrams to explore commutativity to find known facts of 6, 7, 8, and 9 (3.OA.5).  In Anchor Task, Problem 1 states, “Katia and Gerard are stocking shelves at the grocery store. Katia stocks 3 shelves with 6 boxes of cereal on each shelf. Gerard stocks 6 shelves with 3 boxes of cereal on each shelf. Katia says they put the same number of cereal boxes on each shelf. Gerard says they didn’t, since they stocked a different number of boxes of each of a different number of shelves. Who do you agree with, Katia or Gerard? Explain.” The Guiding Questions state, “Do you agree with Katia or Gerard? Why?  What could you have drawn to support your argument? Write an equation that represents your argument. Can you generalize this to more numbers? If you know the value of 8 fours, do you know the value of 4 eights? Why?” The Teacher’s Notes state, “Students will likely draw an array for this problem, since that is the representation encouraged by the problem context. But, they could also use a number line or tape diagram to support their answer.”
• In Unit 4, Lesson 9, students “compose and decompose rectangles, seeing and making use of the idea that the sum of the areas of the decomposed rectangles is equal to the area of the composed rectangle,” (3.MD.7c). In Anchor Task, Problem 1 states, “a. Cut out the large rectangle. Then cut it into two smaller rectangles along the darkened border. b. Find the area of each rectangle. c. Push the rectangles back together. Find the area of the new rectangle you made. What do you notice? What do you wonder?” Guiding Questions include but are not limited to, “What are the length and width of the combined rectangle? How are the areas of the separate rectangles related to the area of the combined rectangle?”
• In Unit 6, Lesson 3, students “partition a whole into equal parts, identifying and counting unit fractions using pictorial area models and tape diagrams, identifying the unit fraction numerically,” (3.G.2, 3.NF.1). In Anchor Task, Problem 2 states, “Make a model in which the shaded part represents the corresponding unit fraction. a. $$\frac {1}{3}$$ b. $$\frac {1}{2}$$ c. $$\frac {1}{6}$$” Students deepen their conceptual understanding with Guiding Questions, “How many pieces should we partition the whole into? How did you decide that based on the fraction? How many pieces should you shade? How did you decide that based on the fraction itself? Can we partition any of these shapes in different ways? Why is it possible to have equal parts of the same whole be different shapes? Can they be different sizes or just different shapes? Count each fractional unit for each model. (Count 1 third, 2 thirds, 3 thirds for Part (a) and similarly for Parts (b) and (c).)”

Examples of Procedural Skills and Fluency include:

• In Unit 1, Lesson 8, students “add two numbers with up to one composition within 1,000,” (3.NBT.2). Anchor Task, Problem 1 states, “Mr. Silver and Mrs. Ingall want to know the largest number they can represent with their base ten blocks. Mr. Silver has 4 tens, 2 hundreds, and 6 ones. Mrs. Ingall has 2 tens, 3 hundreds, and 8 ones. What’s the largest number they could represent with their base ten blocks? Show or explain how you know.”
• In Unit 2, Lesson 6, students “build fluency with multiplication facts using units of 2, 5, and 10,” (3.OA.7).  Anchor Task, Problem 2 states, “Solve. a. $$9 × 2 =$$ _____ b. _____ $$= 5 × 2$$ c. $$2 × 5 =$$ _____ d. _____ $$= 9 × 5$$ e. $$4 × 10 =$$ _____ f. _____ $$= 10 × 10$$.”
• In Unit 3, Lesson 12, students skip-count to build fluency with multiplication facts using units of 8 and 9 (3.OA.4,7). In Problem Set, Problem 3 states, “Skip count by nine to fill in the blanks. Match each number in the count-by with its multiplication fact. Then, use the multiplication expression to write the related division fact directly to the right.” 

Examples of Application include:

• In Unit 1, Lesson 10, students “solve word problems involving addition, using rounding to assess the reasonableness of the solution,” (3.OA.8, 3.NBT.1,2). For example, the Target Task states, “Jesse practices the trumpet for a total of 165 minutes during the first week of school. He practices for 245 minutes during the second week. a. Estimate the total amount of time Jesse practices. b. How much time did Jesse actually spend practicing?”
• In Unit 2, Lesson 14, students “solve one-step word problems involving multiplication and division and write problem contexts to match expressions and equations” (3.OA.1-3). In Anchor Task, Problem 2 states, “Write a word problem that can be solved using the following expressions.  a.$$4 \times 6$$  b.$$15 \div 5$$” 
• In Unit 3, Lesson 20, students “solve two-step word problems involving all four operations and assess the reasonableness of solutions,” (3.OA.8, 3.NBT.3). For example, Target Task states, “Solve. Explain why your answer is reasonable. There were 80 adults and 20 children at a school play. The school collected $8 for each adult’s ticket and $3 for each child’s ticket. The school donated $125 of the money from the tickets to a local theater program and used the remaining money to buy supplies for next year’s school play. How much money does the school have to buy supplies for next year’s play?”
• In Unit 5, Lesson 7, students “solve more complex word problems involving perimeter, such as finding a missing side length given perimeter and other side lengths,”  (3.MD.8). For example, Target Task, Problem 1 states, “Maya’s rectangular rug has a perimeter of 16 feet. The length of the rug is 5 feet. What is the width of the rug? a. 3 feet b. 9 feet c. 11 feet d. 13 feet”

Examples of multiple aspects of rigor engaged simultaneously to develop students’ mathematical understanding of a single topic/unit of study include:

• In Unit 3, Lesson 8, students use and apply the associative property to develop fluency with multiplication (3.OA.5,7,9). Problem Set, Problem 4 states, “What expression is another way to show $$7 \times 6$$? a. $$7 + (3 + 2)$$  b. $$7 \times (3 + 2)$$ c. $$7 + (3 \times 2)$$ d. $$7 \times (3 \times 2)$$.” 
• In Unit 7, Lesson 9, students apply their understanding of mass to real-world problems (3.MD.2). In Target Task, Problem 2 states, “Carla buys apples and peaches at the store. The mass of the apples is 724 grams. The mass of the peaches is 471 grams. How much greater is the mass, in grams, of the apples than the mass of the peaches?”

The instructional materials reviewed for Match Fishtank Grade 3 meet expectations for identifying the Standards for Mathematical Practice and using them to enrich mathematics content within and throughout the grade-level.

All Standards for Mathematical Practice are clearly identified throughout the materials in numerous places, that include but are not limited to: Unit Summaries, Criteria for Success, and Tips for Teachers. Examples include: 

• In Unit 3, the Unit Summary states, “students deepen their understanding of multiplication and division, including their properties. ‘Mathematically proficient students at the elementary grades use structures such as…the properties of operations…to solve problems’ (MP.7) (Standards for Mathematical Practice: Commentary and Elaborations for K–5, p. 9). Students use the properties of operations to convert computations to an easier problem (a Level 3 strategy), as well as construct and critique the reasoning of others regarding the properties of operations (MP.3). Lastly, students model with mathematics with these new operations, solving one- and two-step equations using them (MP.4).”
• In Unit 2, Lesson 16, Criteria For Success states, “1. Make sense of a 3-Act Task and persevere in solving it (MP.1). 2. Solve two-step word problems involving addition, subtraction, multiplication, and division (MP.4). 3. Assess the reasonableness of a solution (MP.1).”
• In Unit 6, Lesson 7, Tips for Teachers states, “Too often, when students are asked questions about what fraction is shaded, they are shown regions that are portioned into pieces of the same size and shape. The result is that students think that equal shares need to be the same shape, which is not the case. On the other hand, sometimes visuals do not show all of the partitions (Van de Walle, Teaching Student-Centered Mathematics, Grades 3–5, Vol. 2, p. 211). Thus, this lesson tries to address both of these potential misconceptions and deepen students’ conceptual understanding of fractions. Having students explain what it meant by ‘equal parts’ also provides opportunities for students to attend to precision. (MP.6).”

Examples of the MPs being used to enrich the mathematical content include:

• MP4 is connected to mathematical content in Unit 2, Lesson 9, Target Task, Problem 2, as students “solve one-step word problems involving multiplication or division with units of 2, 5, or 10, using a tape diagram to represent the problem if necessary (MP.4).” For example, “Jonathan is making lemonade for his lemonade stand. The recipe says you need 2 cups of sugar in each pitcher of lemonade. Jonathan wants to make 10 pitchers of lemonade. How many cups of sugar will he need?”
• MP3, MP5, and MP6 are connected to the mathematical content in Unit 4, Lesson 2, Anchor Tasks, Problem 1, as students “Find the area of various figures by covering a space with concrete standard units without gaps or overlaps (MP.5, MP.6). Explain how two figures with the same number of square units but with different sized square units differ in size (MP.3).” For example, “Which of the following rectangles has the greatest area?” Show or explain your thinking.” Three different size rectangles are provided. Guiding Questions state, “Who decomposed and recomposed their rectangles to determine which one had the greatest area? Who tried to use some sort of area unit to find the area? What did you use for your area unit? I think Rectangle C has the greatest area because it is the longest. Do you agree or disagree? Why?”
• MP1 and MP4 are connected to the mathematical content in Unit 7, Lesson 5, Anchor Tasks, Problem 2, as students “solve word problems involving addition and subtraction of time intervals in minutes, e.g., by representing the problem on a number line diagram, where the times cross the hour mark, including the following three cases: a. Where the end time is unknown. b. Where the start time is unknown. c. Where the duration is unknown (MP.1 and MP.4).” For example, “a. Ms. Banta leaves school at 4:52 p.m. She gets home at 5:13 p.m. How long was Ms. Banta’s commute home? b. Katherine wakes up from a nap at 2:26 p.m. Her watch tells her that she slept for 43 minutes. What time did Katherine fall asleep? c. Fernando leaves home at 7:48 a.m. It takes him 19 minutes to walk to school. What time will Fernando arrive at school?”

The instructional materials reviewed for Match Fishtank Mathematics Grade 3 meet expectations for carefully attending to the full meaning of each practice standard. 

The materials attend to the full meaning of each of the 8 Mathematical Practices (MPs). The MPs are discussed in both the Unit and Lesson Summaries as they relate to the overall content. The MPs are also explained, when applicable, within specific parts of each lesson, including but not limited to the Criteria for Success and Tips for Teachers. Each practice is addressed multiple times throughout the year. Over the course of the year, students have ample opportunity to engage with the full meaning of every MP. Examples include, but are not limited to:

• MP1: In Unit 5, Lesson 8, Criteria for Success states, “Determine the dimensions of all possible rectangles with a given area (e.g., a 1-by-12 rectangle, 2-by-6 rectangle, and 3-by-4 rectangle all have an area of 12 square units) (MP.1).” For example, Anchor Task, Problem 1 states, “a. Create as many rectangles as you can with an area of 12 square units. b. Find the perimeter of each rectangle that you found. c. Record the information from Parts (a) and (b) in the table below. d. What do you notice about the rectangles you created? What do you wonder?”
• MP2: In Unit 6, Lesson 3, Criteria For Success states, “Determine the unit fraction represented by an abstract description of a situation (MP.2).” For example, Problem Set, Problem 7 states, “A circle is divided into parts. Each part is 14 of the total area of the circle. Which sentence describes the circle? The circle has 1 small part and 3 large parts. The circle has 1 small part and 4 large parts. The circle has 4 parts that are each the same size. The circle has 5 parts that are each the same size.”
• MP4: In Unit 3, Lesson 28, Criteria for Success states, “Solve one- and two-step word problems using information presented in scaled picture and bar graphs, including “how many more” and “how many less” problems (MP.4).” For example, Anchor Task, Problem 1 states, “The picture graph below shows how many trees of each kind were in the arboretum. a. How many more evergreen trees than maple trees are there? (Fir trees and spruce trees are evergreen trees. b. In total, there were 111 trees in the arboretum. All of the trees were either fir, spruce, maple or oak trees. How many oak trees are there in the arboretum? 
• MP5: In Unit 4, Lesson 1, Criteria for Success states, “Find the area of various figures by covering a space with concrete non-standard units without gaps or overlaps (MP.5, MP.6).”  For Example, Anchor Task, Problem 2 states, “Area is the measure of how much flat space an object takes up. Compare the areas of the medium triangle and the parallelogram. Guiding Questions: Which shape has greater area? How do you know? What makes comparing the areas of the medium triangle and the parallelogram more difficult than comparing the area of the large triangle and the area of the square from Anchor Task #1?”
• MP6: In Unit 6, Lesson 7, Tips for Teachers states, “Having students explain what it meant by ‘equal parts’ also provides opportunities for students to attend to precision (MP.6).” For example, Problem Set, Problem 3 states, “Draw a shape below such that the shaded part of it represents 3/8 of the whole shape. Try to make it as complex as possible. Then explain how you know that the shaded part represents 3/8 .”
• MP7: In Unit 2, Lesson 2, Criteria For Success states, “Relate arrays to equal groups, relating rows to the number of groups and the number of objects in each row to the size of groups (MP.7).” For example, Anchor Task, Problem 2 states, “Can you rearrange the following groups so that they are in an array? Why or why not?” 
• MP8: In Unit 3, Lesson 2, Criteria for Success states, “Look for and express regularity and repeated reasoning when multiplying by 1 to generalize the pattern of multiplying any number by 1 results in a product that is just that number (i.e., $$1\times n=n$$) (MP.8).” For example, Anchor Task, Problem 1 states, “a. Rudy is waiting patiently for his new hen to lay some eggs. On Monday, he went to the henhouse to check their nests. He has 12 hens in the house, and each hen had laid 0 eggs. How many eggs did Rudy have in total?” b. On Tuesday when he went outside, he discovered that each of his 12 hens had laid 1 egg. How many eggs does he have now?” Guiding Questions include but are not limited to, “What if Rudy had 6 hens that each laid 1 egg? How many eggs would he have in total? Can you write an equation to represent this situation? What if Rudy had 20 hens that each laid 1 egg? How many eggs would he have in total? Can you write an equation to represent this situation?”

The instructional materials reviewed for Match Fishtank Mathematics Grade 3 meet expectations for prompting students to construct viable arguments and analyze the arguments of others concerning key grade-level mathematics. 

The student materials consistently prompt students to construct viable arguments and analyze the arguments of others, for example:

• In Unit 1, Lesson 7, Anchor Tasks, students “construct and critique reasoning for rounding two- and three-digit numbers to the nearest ten or hundred.” Problem 1 states, “A pair of pants cost $52. Mrs. Ingall says that the pants are about $50. Mr. Silver says they are about $100. Who is correct, Mr. Silver, Mrs. Ingall, both of them, or neither of them? Explain your answer.” 
• In Unit 3, Lesson 1, Target Task, students “demonstrate and explain the commutativity of multiplication using models (MP.3).”  Problem 2 states, “Karen says, ‘If I know $$8 \times 3 = 24$$, then I know the answer to $$3 \times 8!$$’ Explain why this is true.” 
• In Unit 4, Lesson 1, Anchor Task, students construct a viable argument to justify which shape takes up more space. Problem 1 states, “Does the large triangle or square take up more space? Justify your answer.” 
• In Unit 5, Lesson 12, Anchor Task states, “Sort these shapes (cut out from Template: Polygons) into groups. You may sort them any way you want and into as many groups as you want.” Guiding Questions state, “What is true about all of these shapes?” and “Which shapes are quadrilaterals? For those shapes that are not quadrilaterals, are they still polygons? Why or why not?” Students “Classify polygons according to their attributes, like number of sides and angles, and justify this classification (MP.3).” 
• In Unit 6, Lesson 21, Anchor Tasks, students compare fractions in order to choose which snack they would take a piece from, and then construct a viable argument to defend their choice. Problem 1 states, “Kiana bought two Fruit-by-the-Foot snacks to share with friends. She splits one of them into 3 equal-sized pieces and the other into 8 equal- sized pieces. If Kiana were sharing a piece of Fruit-by-the-Foot with you, which snack would you take a piece from, the 3-piece snack or the 8-piece snack? Explain why.” 

The instructional materials reviewed for Match Fishtank Mathematics Grade 3 meet expectations for assisting teachers in engaging students to construct viable arguments and analyze the arguments of others concerning key grade-level mathematics.

The teacher materials assist teachers in engaging students in constructing viable arguments and analyzing the arguments of others through the Criteria for Success, Guiding Questions, and Tips for Teachers, for example:

• In Unit 1, Lesson 7, Tips for Teachers states, “This whole lesson provides an excellent opportunity for students to construct viable arguments and critique the reasoning of others regarding when to round and to what level of precision, based on the context of the problem (MP.3). ‘How close an estimate must be to the actual computation is a matter of context,’ and thus ‘the goal of computational estimation is to be able to flexibly and quickly produce an approximate result that will work for the situation and give a sense of reasonableness’ (Van de Walle, p. 195). Thus, these tasks offer an opportunity for a rich discussion where one’s decision about the degree of precision of an estimate should be supported by reasoning.”
• In Unit 3, Lesson 2, Criteria for Success states, “4. Explain why any number divided by 0 results in an impossible product by rewriting the division sentence as a multiplication one to see that no such value exists (e.g., $$6\div0 = a \rightarrow a\times0 = 6$$, and no such a exists since any number multiplied by 0 would have a product of 0) (MP.3).” In Anchor Tasks, Problem 3, Guiding Questions include, but are not limited to, “Imagine the equation $$9\div0 = e$$. What do you think the solution is? How can the relationship between multiplication and division help you to think about this problem? (Rewrite $$9\div0 = e$$ as $$e\times0=9$$, which will help students to see that no such number exists.)”
• In Unit 5, Lesson 9, Tips for Teachers states, “With strong and distinct concepts of both perimeter and area established, students can work on problems to differentiate their measures. For example, they can find and sketch rectangles with the same perimeter and different areas or with the same area and different perimeters and justify their claims” (MP.3).”
• In Unit 6, Lesson 2, Criteria for Success states, “5. Determine whether a model represents equal sharing/fractions and explain why or why not (MP.3).” In Anchor Task, Problem 3 states, “Noah believes the shape below represents 3 eighths. Explain why Noah is incorrect in his reasoning. Draw a correct model to represent 3 eighths. Explain why your model is correct.” Guiding Questions include, but are not limited to: “What must be true about the pieces in order for them to represent fractional parts of the whole? Is that true of Noah’s model? Let’s construct our own model to represent 3 eighths. How can we use our fraction strips from Anchor Task #2 to help us? How many equal pieces must our model have? How do you know? How many pieces are counted? How can we indicate the counted pieces in our model? Do our shaded pieces have to be right next to each other? Why or why not?”

The instructional materials reviewed for Match Fishtank Mathematics Grade 3 meet expectations for explicitly attending to the specialized language of mathematics.

Examples of the materials providing explicit instruction in how to communicate mathematical thinking using words, diagrams, and symbols include:

• In Unit 1, Lesson 5, Tips for Teachers states, “You’ll want to avoid using terms like ‘round up’ and ‘round down’, since these terms can be confusing for students. ‘Rounding up’ a number results in a change in the value of the place to which you’re rounding, where ‘rounding down’ does not. Often students will change the value mistakenly as a result.” 
• In Unit 2, Lesson 7, Tips for Teachers states, “Students need not use formal terms for the properties of operations, including the terms ‘commutative’ or commutative property.’ However, exposure to the term is helpful so that students can develop and use a common language and thus introduced in this lesson.” 
• In Unit 3, Lesson 17, Tips for Teachers states, “Make sure to be precise in language use when discussing how basic multiplication facts are related to multiplication by multiples of ten (e.g., how $$2 \times 4$$ is related to $$2 \times 40$$). Avoid saying ‘add a zero,’ and instead discuss how the units shift. This serves two purposes: (1) it doesn’t conflate two operations, multiplication with addition, and (2) it is aligned to the work they’ll do in later grades of seeing how digits shift places when multiplying or dividing numbers by powers of ten, including decimals, where in fact ‘adding a zero’ after a decimal point won’t change its value.”
• In Unit 5, Lesson 11, Tips for Teachers states, “Students have learned much of the vocabulary used in today’s lesson in prior grade levels, but not all. New vocabulary includes parallel, right angle, parallelogram, and quadrilateral. The term quadrilateral in particular provides ‘the raw material for thinking about the relationships between classes. For example, students can form larger, superordinate, categories, such as the class of all shapes with four sides, or quadrilaterals, and recognize that it includes other categories, such as squares, rectangles, rhombuses, parallelograms, and trapezoids. They also recognize that there are quadrilaterals that are not in any of those subcategories’ (Geometry Progressions, p. 13).”
• In Unit 7, Lesson 7, Tips For Teachers states, “As noted in the Progressions, ‘the Standards do not differentiate between weight and mass. Technically, mass is the amount of matter in an object. Weight is the force exerted on the body by gravity. On the earth’s surface, the distinction is not important (on the moon, an object would have the same mass, would weigh less due to the lower gravity) (GM Progression, p. 2).’ Thus, the discussion is excluded from the lesson and the use of ‘mass’ and ‘weight’ interchangeably is avoided. But, it could be discussed if a student raises the issue.”

Examples of the materials using precise and accurate terminology and definitions when describing mathematics, and supporting students in using them, include:

• At the beginning of each unit, the Unit Prep provides vocabulary for the unit. As found in Unit 1, “Digit, estimate, interval, place, reasonable, value, etc.”
• In Unit 1, Lesson 3, Criteria for Success, students will, “Understand that a number line is a straight line with equally spaced intervals used to plot numbers.”
• In Unit 2, Lesson 1, Criteria for Success, students will, “Use the multiplication symbol to represent equal groups.” 
• In Unit 5, Lesson 1, Criteria for Success, students will, “Understand perimeter to be the boundary of a shape.”
• In Unit 6, Lesson 1, Criteria for Success, students will, “Understand that a fraction is an equal share of a whole.”
• In Unit 7, Lesson 7, Anchor Task Problem 1, Guiding Question states, “The mass of an object is its heaviness. Which object has more mass, the thumbtack or the textbook? How do you know?”