3rd Grade - Gateway 2

The materials reviewed for Fishtank Plus Math Grade 3 meet expectations for developing conceptual understanding of key mathematical concepts and provide opportunities for students to independently demonstrate conceptual understanding throughout Grade 3. 

Materials develop conceptual understanding throughout the grade level. According to Course Summary, Learn More About Fishtank Math, Our Approach, “Procedural Fluency AND Conceptual Understanding: We believe that knowing ‘how’ to solve a problem is not enough; students must also know ‘why’ mathematical procedures and concepts exist.” Each lesson begins with Anchor Tasks and Guiding Questions, designed to highlight key learning aligned to the objective and to support the development of conceptual understanding through student discourse and reflection. This is followed by a Problem Set, Homework and Target Task. Examples include: 

• In Unit 2, Multiplication and Division, Part 1, Lesson 8, Anchor Task, Problem 1, students use their ability to skip count and make connections to dividing. “Maureen says the skip-counting sequence ‘10, 20, 30, 40, 50, 60’ to help her solve a problem. a. What multiplication problem might Maureen be trying to solve? How do you know? b. What if Maureen was solving a division problem? What problem might that have been?” Guiding Questions include, “How can I generalize Maureen’s strategy to use skip-counting to solve division problems? How will I know what to count by? When will I know where to stop? Where will I find my quotient?” This problem and the guiding questions help develop conceptual understanding of 3.OA.2 (Interpret whole number quotients of whole numbers).

• In Unit 3, Multiplication and Division, Part 2, Lesson 8, Anchor Tasks, Problem 3 states, “Break 6 into smaller factors in the following expression and multiply the numbers in the order that makes sense to you. 6 × 8.” Guiding Questions include, “Could you break apart the other factor into smaller factors in this expression? What would you break it up into? Will you still get the same product?” This activity supports conceptual understanding of 3.OA.5 (Apply properties of operations as strategies to multiply and divide).

• In Unit 4, Area, Lesson 6, Anchor Task, Problem 2 states, “What are the length, width, and area of the rectangle below? How did you solve?” Guiding Questions include, “What are the side lengths of this rectangle? How do you know? What is the area of this rectangle? How do you know? Why don’t we need to draw the rows and columns to show individual square inches in the rectangle in order to find its area?” This problem shows opportunities for students to engage with teacher support and/or guidance while developing conceptual understanding of 3.MD.6 (Measure areas by counting unit squares [square cm, square m, square in, square ft, and improvised units]).

Materials provide opportunities for students to independently demonstrate conceptual understanding throughout the grade level. Problem Sets and Homework Problems can be completed independently during a lesson. Target Tasks, or end of lesson checks for understanding of key concepts, are designed for independent completion. Many of these problems provide opportunities for students to independently demonstrate conceptual understanding. Examples include:

• In Unit 3, Multiplication and Division, Part 2, Lesson 4, Problem Set, Problem 4 states, “Decompose the following array into arrays whose facts you can use to find the larger product.” This activity supports conceptual understanding of 3.OA.5 (Apply properties of operations as strategies to multiply and divide).

• In Unit 4, Area, Lesson 7, Target Task, Problem 1, students calculate the area of a rectangle in multiple ways. It states, “A rectangle has the measurements shown. Select the three ways to calculate the area of the rectangle in square inches. A. 3 × 3; B. 9 × 9; C. 3 × 9; D. 9 × 3; E. 3 + 3 + 3; F. 9 + 9 + 9.” Through this problem, students show their conceptual understanding of 3.MD.7 (Relate area to the operations of multiplication and addition).  

• In Unit 6, Fractions, Lesson 14, Homework, Problem 5, students identify a fraction on a number line. It states, “Label the point where $$\frac{3}{5}$$ belongs on the number line. Be as exact as possible.” This problem shows concrete representation while developing conceptual understanding of 3.NF.1 (Understand a fraction as a number on the number line; represent fractions on a number line diagram).

The materials reviewed for Fishtank Plus Math Grade 3 meet expectations for developing procedural skills and fluency while providing opportunities for students to independently demonstrate procedural skills and fluency throughout the grade level. 

According to Teacher Tools, Math Teacher Tools, Procedural Skill and Fluency, “In our curriculum, lessons explicitly indicate when fluency or culminating standards are addressed. Anchor Problems and Tasks are designed to address both conceptual foundations of the skills as well as procedural execution. Problem Set sections for relevant standards include problems and resources that engage students in procedural practice and fluency development, as well as independent demonstration of fluency. Skills aligned to fluency standards also appear in other units after they are introduced in order to provide opportunities for continued practice, development, and demonstration.”

Opportunities to develop procedural skill and fluency with teacher support and/or guidance occurs in the Anchor Tasks, at the beginning of each lesson, and the Problem Sets, during a lesson. Additionally, Fluency Activities may be implemented with teacher support. Examples Include:

• In Unit 1, Place Value, Rounding, Addition and Subtraction, Lesson 9, Anchor Task, Problem 2 states,  “Find the sum. Show or explain your work.a. 68 + 75 = __; b. 487 = ◻ - 513; c. 296 + 144 + 35 = __.” In addition to the problem, Guiding Questions include, “How can you solve using the place value chart? How can you solve using the standard algorithm? What is unique about the sum in Part (b)? How did you solve Part (b)? Why did you solve using addition even though it appears to be a subtraction problem? What was unique about the solution? For Part (c), when solving using the standard algorithm, how should we line up our numbers? Why do we line them up this way? Could you solve any of these tasks by converting them to an easier problem? If so, which one(s)?” The problems and Guiding Questions help develop student understanding of 3.NBT.2 (Fluently add and subtract within 1000).

• In Unit 2, Multiplication and Division, Part 1, Lesson 6, Problem Set, Discussion of Problem Set states, “Why do you think it is that when you skip-count by twos you say all the even numbers? Is Rob’s reasoning correct in #11? Why is place value understanding helpful when multiplying by ten?” This activity provides an opportunity for students to develop 3.OA.7 (Fluently multiply and divide within 100).  

• In Unit 6, Fractions, Lesson 8, Anchor Task, Problem 2 states, “Each shape represents the unit fraction. Draw a picture representing a possible whole.” Guiding Questions for the teacher include, “The shape in Part (a) represents $$\frac{1}{3}$$. What might one whole look like? What does the numerator represent? What does the denominator represent? How can you use that to determine what the whole would look like? Is there more than one correct answer? Why can our whole resemble different shapes but still have the same unit fraction? Can we say that the wholes themselves are equal even if they have different shapes? Why?” The problem provides an opportunity for students to develop fluency of 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 $$\frac{1}{b}$$).

The instructional materials provide opportunities for students to independently demonstrate procedural skills and fluency throughout the grade level. Problem Sets and Homework can be completed independently during a lesson. Target Tasks, or end of lesson checks for understanding, are designed for independent completion. Fluency Activities may be completed independently with partners. Examples include:

• In Unit 1, Place Value, Rounding, Addition, and Subtraction, Lesson 12, Fluency Activities state, “Get to 1,000: In this fluency activity, students build and add together multiple-digit numbers in hopes of getting as close to 1,000 as possible. This fluency activity should be completed as a partner game.” Additionally, “Hit the Target: In this fluency activity, students are given a three-digit Target Number. Students then randomly draw 8 digit cards, and decide where to place each digit card in a three-digit plus three-digit addition equation (or three-digit minus three-digit subtraction equation), with the option to discard two cards, in order to get a sum (or difference) as close to the Target Number as possible.This fluency activity should be completed in partners.” These activities provide an opportunity for students to independently demonstrate fluency of 3.NBT.2 (Adding and subtracting within 1,000 using the standard algorithm).

• In Unit 2, Multiplication and Division, Part 1, Lesson 9, Fluency Activities state, “Bump: In this fluency activity, students multiply two single-digit numbers in hopes of bumping other players off the products of those numbers. The objective is to be the player to first use all 8 of one’s game makers.” Additionally, “Flash Cards: In this fluency activity, students practice recalling their multiplication facts with the factors written on one side of a flash card and the product on the other. This fluency activity should be completed as a whole class, in small groups, as a partner activity, or as an individual activity. (Card Set A).” These activities provide an opportunity for students to independently demonstrate fluency of 3.OA.7 (Fluently multiply and divide within 100).

• In Unit 3, Multiplication and Division, Part 2, Lesson 15, Target Task states, “Hector solves 9 × 8 by finding (10×8) − (1 × 8). Gabriella solves 9 × 8 by finding (5 × 8) + (4 × 8). Who is correct, Hector, Gabriella, both of them, or neither of them? Show or explain your reasoning.” The problem allows students to independently demonstrate procedural skill and fluency of 3.OA.5 (Apply properties of strategies to multiply and divide).

The materials reviewed for Fishtank Plus Math Grade 3 meet expectations for being designed so that teachers and students spend sufficient time working with engaging applications of the mathematics.

Materials include multiple routine and non-routine applications of the mathematics throughout the grade level. Anchor Tasks, at the beginning of each lesson, routinely include engaging single and multi-step application problems. Examples include:

• In Unit 2, Multiplication and Division, Part 1, Lesson 9, Anchor Task, Problem 3, students engage in solving routine problems involving multiplication within 100 to solve word problems in situations involving equal groups (3.OA.3). The problem states, “Ms. Glynn decides to put her students into groups of 2 to work on a project together. She has 20 students in her class. How many groups of two did Ms. Glynn make?” Teachers are provided the following guiding questions, “What is happening in this situation? What can you draw to represent this situation? Is this situation about equal groups? How do you know? How does this relate to the other work we’ve done in this unit? What quantities and relationships do we know? What is the question asking you to find out? What equation can we use to represent the problem? Is there more than one equation we can use? How did you find the answer to the question? Did anyone find the answer differently?”

• In Unit 4, Area, Lesson 8, Anchor Task, Problem 4, students engage in solving routine application problems involving area (3.MD.7). The problem states, “Amir is getting hardwood floors in his bedroom, which measures 8 feet by 9 feet. How many square feet of hardwood flooring will Amir need?  The area of Theo’s banner is 28 square feet. If the length of his banner measures 4 feet, how wide is his banner?” Teachers are provided the following guiding questions, “What is happening in this situation? What can you draw to represent this situation? What about the situation in Part (b) makes it difficult to draw accurately? What quantities and relationships do we know? What is the question asking you to find out? What equation can we use to represent each part of the problem? What letter should we use to represent the unknown? How did you find the answer to the question? Did anyone find the answer differently? Is your answer reasonable? How do you know?”

• In Unit 5, Shapes and Their Perimeter, Lesson 10, Anchor Task, Problem 1, students engage in solving non-routine application problems involving perimeters of polygons, including finding 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 (3.MD.8). The problem states, “Claudia’s bedroom is in the shape of a rectangle that is 8 feet long and 9 feet wide. Claudia’s mom needs to order new carpet for her room and is going to put baseboards around the border of her bedroom. 1. How much carpet should Claudia’s mom order? 2. How much baseboard should Claudia’s mom order?” Guiding Questions for teachers include, “What can you draw to help you solve this problem? How will you label your drawing? How will you find the amount of carpet Claudia’s mom needs based on your drawing? Write and solve an equation to represent this. How will you find the amount of baseboard Claudia’s mom needs based on your drawing? Write and solve an equation to represent this. Claudia’s sisters, Gillian and Felicia, have rooms that are the exact same size as Claudia’s. If their mom wanted to carpet and put in baseboards for all three of their rooms, how much of each material will she need? How did you solve?

• In Unit 7, Measurement, Lesson 12, Anchor Task, Problem 4, students engage in solving non-routine application problems, involving measuring and estimating liquid volumes and masses of objects using standard units of grams (g), kilograms (kg), and liters (l) (3.MD.2). The problem states, “Act 4 (the sequel): Raul has orange juice and milk to drink at breakfast. He drinks 237 mL of milk. He also drinks 60 mL less orange juice than milk. How much does Raul drink at breakfast, including milk and juice?” Guiding Questions for teachers include, “What is happening in this situation? What can you draw to represent this situation? What quantities and relationships do we know? What is the question asking you to find out? What equation can we use to represent each part of the problem? What letter should we use to represent the unknown?  How did you find the answer to the question? Did anyone find the answer differently? Is your answer reasonable? How do you know?” 

Materials provide opportunities within Problem Sets and Homework, and Daily Word Problems for students to independently demonstrate multiple routine and non-routine applications throughout the grade level. Target Tasks, or end of lesson checks for understanding, are designed for independent completion. Examples include:

• In Unit 1, Place Value, Rounding, Addition, and Subtraction, Homework, Problem 6, students independently solve non-routine two step word problems involving addition and subtraction (3.NBT.2). The problem states, “Third-grade students took a total of 1,000 pictures for the yearbook during the school year. Ted took 72 pictures. Mary took 48 pictures. a. What is the total number of pictures taken by the rest of the third-grade students during the school year? b. Ella took 8 more pictures than Ted took. How many more pictures did Ella take than Mary?”

• In Unit 3 Multiplication and Division, Lesson 19, Target Task, students independently solve a routine two step word problem with multiple operations (3.OA.8). The problem states, “Solve. Explain why your answer is reasonable. Warren went swimming on Saturday and running on Sunday. Between these two activities, Warren spent 117 minutes exercising over the weekend. On Saturday, Warren swims laps in the pool for 45 minutes. On Sunday, he runs 8 miles. It takes him the same amount of time to run each mile. How long did it take Warren to run each mile?”

• In Unit 5, Shapes and Their Perimeter, Lesson 9, Homework, Problem 6, students independently engage in solving non-routine application 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 (3.MD.8). The problem states, “Imagine all of the rectangles you could build with a perimeter of 32 units. Which one do you think will have the greatest area? Why?” 

• In Unit 7 Measurement, Lesson 9, Target Task, Problem 1, students independently solve a routine problem to measure and estimate liquids volumes and masses (3.MD.2). The problem states, “Mr. Smith, the principal, placed a bag of oranges on the scale as shown. Mr. Smith bought 60 bags of oranges for a school event. How many kilograms of oranges did he buy? A. 2 kilograms B. 62 kilograms C. 80 kilograms D. 120 kilograms.”

The materials reviewed for Fishtank Plus Math Grade 3 meet expectations in that the three aspects of rigor are not always treated together and are not always treated separately. There is a balance of the three aspects of rigor within the grade.

All three aspects of rigor are present independently throughout Grade 3. Examples where instructional materials attend to conceptual understanding, procedural skill and fluency, or application include:

• In Unit 2, Multiplication and Division Part 1, Lesson 5, Anchor Task, Problem 3, students engage in solving non-routine application problems. The problem states, “Write a multiplication equation and a division equation to represent each of the following situations. a. Ross has 15 flowers that he wants to make into flower arrangements. Each flower arrangement will use 5 flowers. How many flower arrangements can he make? b. Heidi has 8 apps that she wants to place into rows of 4. How many apps will there be in each row?” (3.OA.3: Use multiplication and division within 100 to solve word problems in situations involving equal groups, arrays, and measurement quantities, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem.)

• In Unit 2, Multiplication and Division, Part 1, Lesson 8, Anchor Task, Problem 2, students develop procedural skill and fluency as they solve multiplication sentences. The problem states, “Solve. a. 12 ÷ 2 = ____; b. ____ = 35 ÷ 5; c. 90 ÷ 10 = ___; d. ___ × 5 = 45; e. 2 × ___= 16.” The teacher asks, “How can you use skip-counting to solve Parts (a)–(c)? Parts (d) and (e) are unknown factor problems. How can I use skip-counting to solve? The completed equation for Part (e) is 2 × 8 = 16. How can I be sure that is correct if I don’t know how to skip-count by 8s?” (3.NBT.2: Fluently add and subtract within 1000.)

• In Unit 6, Fractions, Lesson 2, Problem Set, Problem 4, students develop conceptual understanding by using fraction strips to solve. The problem states, “Use your fraction strips as tools to help you solve. To make a garage for his toy truck, Zeno bends a rectangular piece of cardboard in half. He then bends each half in half again. a. What fraction of the original cardboard is each part? Draw and label the matching fraction strip below. b. Zeno bends a different piece of cardboard in thirds. He then bends each third in half again. Which of your faction strips best matches this story? Draw and label the matching fraction strip in the space below.” (3.NF.1: Understand a fraction 1/b as the quantity formed by 1 part when a whole is partitioned into b equal parts.)

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

• In Unit 1, Place Value, Rounding, Addition, and Subtraction, Lesson 14, Anchor Task, Problem 3, students develop conceptual understanding alongside application as they solve a real world problem involving addition and subtraction while using rounding strategies. The problem states, “Mrs. Ingall is going to a hockey game. She has $139 in her pocket and wants to take out more cash to be able to pay for everything at the game. She will have to pay $268 for the tickets, $18 for a cab ride to the game, and $55 for food and drinks while she’s there. Approximately how much money should Mrs. Ingall take out of the bank to cover her costs?” (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.)

• In Unit 4, Area, Lesson 10, Target Task, students develop conceptual understanding alongside procedural skill and fluency as they find the area using the distributive property. The task states, “Jason is a carpenter. He designed a door using equally sized squares of wood. He used two different types of wood as shown below. a. Which equation represents the total number of wood squares Jason needed to build the door? a. 4 × (5 + 7) = 4 + 5 + 7,  b. 4 × (5 + 7) = 4× 5 × 7, c. 4 (5 + 7) = (4 × 5) x (4 × 7), d. 4 (5 + 7) = (4 5) + (5 × 7). b. What is the area of Jason’s door? Show your work. c. Jason decides to change the dimensions of his door. He writes the equation (6 × 3) + (6 × 5) to represent the number of wood squares he will use for the new door. Draw Jason’s door in the grid below. Then, explain what each number in Jason’s equation represents.” (3.MD.7c: Use tiling to show in a concrete case that the area of a rectangle with whole-number side lengths a and b + c is the sum of a × b and a × c. Use area models to represent the distributive property in mathematical reasoning.) 

• In Unit 7, Measurement, Lesson 12, Problem Set, Problem 6, students develop all three aspects of rigor simultaneously, conceptual understanding, procedural skill and fluency, and application, as they solve real world problems to find volume using strategies that involve the same unit. The problem states, “Tanner’s beaker has 45 milliliters of water in it at first. After each of his friends poured in 8 milliliters, the beaker contained 93 milliliters. How many friends poured water into Tanner’s beaker?” (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, e.g., by using drawings (such as a beaker with a measurement scale) to represent the problem.)

The materials reviewed for Fishtank Plus Math Grade 3 meet expectations for supporting the intentional development of MP1: Make sense of problems and persevere in solving them; and MP2: Reason abstractly and quantitatively, for students, in connection to the grade-level content standards, as expected by the mathematical practice standards. MPs are explicitly identified for teachers within the Unit Summary and specific lessons (Criteria for Success, Tips for Teachers, or Anchor Task 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 units. Examples include:

• In Unit 3, Multiplication and Division, Part 2, Lesson 19, Homework, Problem 5, students “Assess the reasonableness of an answer (MP.1).” The problem states, “Sharon uses 72 centimeters of ribbon to wrap gifts. She uses 24 centimeters of her total ribbon to wrap a big gift. She uses the remaining ribbon for 6 small gifts. How much ribbon will she use for each small gift if she uses the same amount on each? Solve, then explain why your answer is reasonable.”

• In Unit 4, Area, Lesson 2, Problem Set, Problem 7, students “Find the approximate area of various figures using concrete standard units, including figures with partial units.” The problem states, “The following shapes aren’t rectangles. Use your centimeter cubes to estimate their area to the nearest square centimeter.” Three non-rectangles are provided.

• In Unit 5, Shapes and Their Perimeter, Lesson 8, Target Task, with guidance from the teacher, students, “Determine the dimensions of all possible rectangles with a given area.” The problem states, “Chris is replacing the fence around his rectangular backyard. Chris’s drawing of the backyard is shown below. Chris measured two of the side lengths and labeled them in his drawing. How much fencing does he need to buy?” A backyard drawing is provided which is 10 yards by 15 yards. Guiding Questions include, “Since the question is asking about how much fencing is needed, what measurement is the problem asking for us to find? How do you know? Is it possible to find the perimeter of this rectangle with only two measurements? Why or why not? Based on what we know about rectangles, what are the missing side lengths? Write and solve an equation to find the perimeter of this shape given its side lengths. What are all the possible equations you could write? What is the perimeter of the yard? What unit should we use?”

• In Unit 7, Measurement, Lesson 4, Anchor Task, Problem 2, with guidance from the teacher, students “Solve word problems involving addition and subtraction of time intervals in minutes...” The problem states, “Joe finishes his chores at 5:42 p.m. It took him 28 minutes to complete them. What time did he start doing his chores? Lucia’s math class started at 10:18 a.m. She worked for 33 minutes. What time was it when Lucia’s math class ended? Leslie starts reading at 11:27 a.m. She finishes reading at 11:54 a.m. How many minutes does she read?” Guiding Questions include, “Would you rather use a clock or a number line to solve each part? Why? What makes these problems more difficult than those in Anchor Task #1? Why? How and when can you still count up or back by fives and tens to solve? How are Parts (a)—(c) similar? How are they different?”

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

• In Unit 2, Multiplication and Division, Part 1, Lesson 14, Problem Set, Problem 11, students “Write a word problem that can be solved with a given multiplication or division expression or equation (MP.2).” The problem states, “Mrs. Oro needs to buy 90 corn seeds. The Garden Center sells corn seeds in packs of 10 seeds each. a. Write a division equation showing how many packs of seeds Mrs. Oro should buy. b. Write a multiplication equation showing many packs of seeds Mrs. Oro should buy. c. How many packs of seeds should Mrs. Oro buy?”

• In Unit 5, Shapes and Their Perimeter, Lesson 3, Problem Set, Problem 4, students “Write an equation to represent the perimeter of a shape.” The problem states, “Michael and Jeffrey are trying to find the perimeter of the shape below. Michael says the expression that represents the perimeter is 9 + 9 + 11 + 11, but Jeffrey says the expression is (9 × 2) + (11 × 2). Who is correct? Explain.”

• In Unit 6, Fractions, Lesson 3, Problem Set, Problem 7, students “Determine the unit fraction represented by an abstract description of a situation.” The problem states, “A circle is divided into parts. Each part is of the total area of the circle. Which sentence describes the circle? A. The circle has 1 small part and 3 large parts. B. The circle has 1 small part and 4 large parts. C. The circle has 4 parts that are each the same size. D. The circle has 5 parts that are each the same size.”

• In Unit 7, Measurement, Lesson 8, Anchor Task, Problem 2, with teacher support, students “Estimate the mass of an object when the measurement is not precise.” The problem states, “Estimate the mass of the following objects in grams or kilograms, depending on which is more appropriate. Then measure their actual mass using a scale. a. Pair of scissors; b. Thumbtack; c. Ruler; d.Textbook.” Guiding Questions include, “What do you estimate the mass of the pair of scissors to be? The ruler? Explain how you came up with those estimates. What is the actual mass of each object? Were our estimates reasonable? Why or why not? In Lesson 7 we decided that a thumbtack had a mass of about 1 gram and the textbook had a mass of about 1 kilogram. What is the actual mass of each object? Are they reasonable benchmarks for these measurements? Why or why not?”

The materials reviewed for Fishtank Plus Math Grade 3 meet expectations for supporting the intentional development of MP3: Construct viable arguments and critique the reasoning of others, for students, in connection to the grade-level content standards, as expected by the mathematical practice standards. MP3 is explicitly identified for teachers within the unit summary or specific lessons (criteria for success, tips for teachers, or anchor problem notes) and students engage with the full intent of the MP through a variety of lesson problems and assessment items.

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:

• In Unit 3, Multiplication and Division, Part 2, Lesson 1, Anchor Task, Problem 1, students “Demonstrate and explain the commutativity of multiplication using models.” The problem 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 on each of a different number of shelves. Who do you agree with, Katia or Gerard? Explain.”  Guiding Questions include, “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?”

• In Unit 5, Shapes and Their Perimeter, Lesson 11, Homework, Problem 6, students “Classify quadrilaterals according to their attributes’ like the presence of parallel sides, right angles, and side length; and justify those classifications.”  The problem states, “Byron drew the following shapes. a. Byron says that all of his shapes are rectangles because they all have four sides. Is he correct? Explain your thinking. Be sure to use specific vocabulary in your explanation. b. What name could be used to describe all of Byron’s shapes?”

• In Unit 5, Shapes and Their Perimeter, Lesson 12, Problem Set, students “Classify polygons according to their attributes, like number of sides and angles, and justify this classification.” The problem states, “Use the following shapes to answer the questions below. [11 different polygons provided for students, labeled A, C, I, J, K, N, P, R, S, W, U.] a. Some of the shapes above have attributes in common. Identify one attribute that at least three of the shapes share. Write down the attribute and the shapes that have this property. b. Do the same thing again, but this time, choose a different attribute. You may reuse any of the shapes from #1. c. Find and identify at least two other shapes above that share at least one attribute with the shape below. What attribute do they share? [Shape provided.] d. Compare Polygon S and Polygon N. What is the same? What is different? e. Jenny says, ‘Polygon P and Polygon I are both quadrilaterals!’ Is she correct? Why or why not? Are there other quadrilaterals she didn’t identify? f. ‘I have five equal sides and five equal angles. I have no right angles.’ Write the letter and the name of the polygon described above. Then, draw the same kind of polygon, but with no equal sides.”

The materials reviewed for Fishtank Plus Math 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. The MPs are explicitly identified for teachers within the unit summary or specific lessons (Criteria for Success, Tips for Teachers, or Anchor Tasks).

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

• In Unit 3, Multiplication and Division, Lesson 19, Target Task, students “Solve two-step word problems involving addition, subtraction, multiplication, and division (MP.4).” The task states, “Solve. Explain why your answer is reasonable. Warren went swimming on Saturday and running on Sunday. Between these two activities, Warren spent 117 minutes exercising over the weekend. On Saturday, Warren swims laps in the pool for 45 minutes. On Sunday, he runs 8 miles. It takes him the same amount of time to run each mile. How long did it take Warren to run each mile?”

• In Unit 4, Area, Lesson 8, Target Task, Problem 1, students “Solve word problems that involve finding the area of a rectangle and the missing side length of a rectangle (MP.4). Solve word problems that involve finding the missing side length of a rectangle (MP.4).” The problem states, “Cara wants new carpeting for her bedroom. Her bedroom is an 8 foot by 6 foot rectangle. How much carpeting does she need to buy to cover her entire bedroom floor?” 

• In Unit 7, Measurement, Lesson 9, Anchor Tasks, Problem 4, students “Solve word problems involving masses given in the same unit (MP.4).” The problem states, “Act 4 (the sequel): a. Jessica puts 28 grams of pinto beans and 36 grams of rice into a pot to make rice and beans. What is the total mass of the rice and beans? b. Lindsey bought 845 grams of potatoes at the grocery store. She uses some of them to make mashed potatoes. She now has 392 grams of potatoes left. How many grams of potatoes did Lindsey use to make the mashed potatoes? c. Jerry buys 6 bags of groceries. Each bag has a mass of 4 kilograms. What is the total mass, in kilograms, of Jerry’s grocery bags?” Guiding Questions include, “What is happening in this situation? What can you draw to represent this situation? What quantities and relationships do we know? What is the question asking you to find out? What equation can we use to represent each part of the problem? What letter should we use to represent the unknown? How did you find the answer to the question? Did anyone find the answer differently? Is your answer reasonable? How do you know?”

MP5: Use appropriate tools strategically, 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 identify and use a variety of tools or strategies that support their understanding of grade level math. Examples include:

• In Unit 3, Multiplication and Division, Part 2, Lesson 26, Anchor Task, Problem 1, students “Understand the purpose of a picture graph as a way to represent a data set to be able to see trends and analyze it more easily (MP.5).” The problem states, “Determine the favorite season of your classmates. Record their answers in the following chart. [Chart Provided.] Now, record your classmates’ responses on the bar chart below.” Guiding Questions include, “How many ice cream cones will you draw to represent the number of students whose favorite season is winter? Spring? Summer? Fall? How many students were surveyed overall? How can you see that in the bar graph? Does it match what you recorded in your tally chart? Which season is the most people’s favorite? The least? How can you see that on the bar graph? What if more than 10 students responded with their favorite season as being any of them in particular? How could we modify the bar graph so that all of the responses could be recorded?”

• In Unit 4, Area, Lesson 13, Homework, Problem 5, students “Find the area of a composite figure that must be decomposed into three or more rectangles with some missing side lengths by breaking it into as many rectangles as needed, finding any missing side lengths, finding the area of each rectangle given their respective side lengths, and adding those areas together (MP.5).”  The problem states, “Mr. and Mrs. Jackson are buying a new house. They are deciding between the two floor plans below. [House A and House B dimensions provided.] Which floor plan has the greater area? Show or explain how you found your answer.”

• In Unit 5, Shapes and Their Perimeter, Lesson 1, Anchor Tasks, Problem 1, students “Use string or pipe cleaners to trace around the perimeter of shapes, then compare the lengths of strings/pipe cleaners to determine which shape has greater perimeter (MP.5).” The problem states, “Maureen wants to wrap ribbon around her clock and around her picture frame. Will she need more ribbon for her clock or her picture frame? Both are shown below.” Shapes included are a circular clock and a square picture frame. Guiding Questions include, “How can we compare which object will need more ribbon? What tools could we use to help us? The boundary around a shape is called its perimeter. Which shape has a greater perimeter, the clock or the picture frame? How do you know? How could you find the perimeter of any shape?”

The materials reviewed for Fishtank Plus Math Grade 3 meet expectations for supporting the intentional development of MP6: Attend to precision; and attend to the specialized language of mathematics, for students, in connection to the grade-level content standards, as expected by the mathematical practice standards. MP6 is explicitly identified for teachers within the unit summary or specific lessons (criteria for success, tips for teachers, or anchor problem notes), and students engage with the full intent of the MP through a variety of lesson problems and assessment items.

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

• In Unit 3, Multiplication and Division, Part 2, Lesson 26, Criteria for Success states, “Include all necessary labels in a drawn bar graph, including a title, category labels, and numerical labels (MP.6).” Homework, Problem 5, students include all necessary labels in a drawn bar graph, including a title, category labels, and numerical labels. The problem states, “The picture below shows the number of stamps five friends each have in their stamp collection. Use this data to complete the bar graph below. Remember to label all parts of your graph.”

• In Unit 5, Shapes and Their Perimeter, Lesson 2, Criteria for Success states, “Understand that perimeter is measured in length units (centimeters, inches, etc.) (MP6).” Problem Set, Problem 1 states, “Find the perimeter, in centimeters, of each shape. Show and explain your work. What unit did you use to record the perimeters of the shapes in #1? Why?”

• In Unit 6, Fractions, Lesson 27, Criteria for Success states, “Measure objects to the nearest quarter inch with a ruler whose 0 mark is not at its edge (MP.6). Problem Set, Problem 2, students attend to precision as they use a ruler and create a line plot partitioned into quarter inch segments. The problem states, “Use your ruler to measure the spoons below to the nearest 1/4 inch. Then, add the data to the table and create a line plot to show the length of all of the spoons. Remember to label all parts of your line plot.”

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 units. The “Tips for Teachers” sections provide teachers with an understanding of grade-specific language and how to stress the specialized language during the lesson. Examples include:

• Each Unit Overview provides a link to a Third Grade Vocabulary Glossary. The glossary contains a chart with the columns “Word” and “Definition.” Under the Definition column, is the mathematical definition and an example. For example, for area the definition reads, “The amount of two-dimensional space within a bounded region.” An example is included with a label that reads, “The figure above has an area of 5 square units.” 

• In Unit 1, Place Value, Rounding, Addition, and Subtraction, Lesson 6, Tips For Teachers state, “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 3, Multiplication and Division, Part 2, Lesson 2, Tips For Teachers, “Students are not expected to use the terms ‘zero property’ or ‘identity property,’ but they are included in the objective so that teachers can understand this lesson’s connection to the topic. Teachers may choose to rewrite the objective to be more student-friendly if they use objectives in a student-facing way.”

• In Unit 6, Fractions, Lesson 7, Tips For Teachers state, “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) (NF Progression, p. 7).”

The materials reviewed for Fishtank Plus Math Grade 3 meet expectations for supporting the intentional development of MP7: Look for and make use of structure; and MP8: Look for and express regularity in repeated reasoning, for students, in connection to grade-level content standards, as expected by the mathematical practice standards. MPs are explicitly identified for teachers within the unit summary or specific lessons (criteria for success, tips for teachers, or anchor problem notes).

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

• In Unit 2, Multiplication and Division, Part 1, Lesson 7, Criteria For Success states, “Know that when multiplying numbers, the order in which factors are multiplied doesn’t matter (MP.7).” Anchor Task, Problem 2 states, “Is the following statement true or false? Explain your answer. 4 × 5 = 5 × 4.”

• In Unit 3, Multiplication and Division, Part 2, Lesson 23, Criteria for Success states, “Identify the rule of a growing number pattern. Use the rule of a growing number pattern to extend it to subsequent terms (MP.7). Use the rule of a growing number pattern to find its nth term (MP.7). Identify features of a number pattern that aren’t explicit in the rule itself (such as, in a pattern that starts with 2 and the rule is ‘add 4,’ all of the terms in the pattern are even) (MP.7).” Homework, Problem 4, students use the rule of a growing number pattern to extend it to subsequent terms. The problem states, “Marc-Anthony wrote the number pattern below. It increases by the same amount each time to get the next number. 15, 19, 23, ___, 31 a. What is the missing number in Marc-Anthony’s pattern? b. What is the rule for this pattern? c. What would the next three numbers in his pattern be?”

• In Unit 4, Area, Lesson 4, Criteria For Success, “Understand that length and width are the measurements of two side lengths of a rectangle, understanding that opposite sides have the same length (MP.7).” In Problem Set, Problem 7 states, “How does knowing side lengths A and B help you find side lengths C and D on the rectangle below?”

• In Unit 5, Shapes and Their Perimeter, Lesson 14, Tips For Teachers state, “Problems such as finding all the possible different compositions of a set of shapes involve geometric problem solving and notions of congruence and symmetry (MP.7).” In Problem Set, Problem 2 states, “Use tetrominoes to create at least two rectangles, each with an area of 28 square units. Then color the grid below to show how you created your rectangles. You may use the same tetromino more than once. a. Write a number sentence to show the area of the rectangle above as the sum of the areas of the tetrominoes you used to make the rectangle. b. Write a number sentence to show the area of a rectangle above as the product of its side lengths.” 

MP8: Look for and express regularity in repeated reasoning, 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:

• In Unit 1, Place Value, Rounding, Addition, and Subtraction, Lesson 1, Criteria For Success states, “Look for and express regularity in repeated reasoning in finding unknown numbers on a number grid (MP.8).” Anchor Tasks, Problem 2 states, “Play Number Grid Tic-Tac-Toe. The rules are as follows: Instead of a usual Tic-Tac-Toe board, we’ll use an almost blank number chart (show students an example from the Problem Set). Player A fills in any square on the board with the correct number that should go in that square with their crayon/marker. The different colors will help you keep track of who wrote what since we won’t be using the system of Xs and Os anymore. Player B uses their crayon/marker to fill in any square on the hundreds chart. When one player gets four in a row vertically, horizontally, or diagonally, mark it with a line. Once a square is used for a four-in-a-row, it can’t be played again. Keep track of how many four-in-a-rows you have with a tally chart. Play until the board is filled up.” Guiding Questions include, “What are some strategies that you and your classmates used to fill in grids while playing? What pattern did we find in the counting sequence?”

• In Unit 3, Multiplication and Division, Part 2, Lesson 22, Criteria for Success states, “1. Look for and express regularity in repeated reasoning in the multiplication table to deduce that an odd number times an odd number results in an odd product, an even number times an odd number results in an even product, and an even number times an odd number results in an even product (MP.8). 2. Look for and express regularity in repeated reasoning in the multiplication table to deduce that n×n is a sum of the first n odd numbers (e.g., 16 = 4 × 4 = 1 + 3 + 5 + 7) (MP.8). 3. Look for and express regularity in repeated reasoning to find other patterns in the multiplication table (MP.8).” Homework, Problem 1, students look for and express regularity in repeated reasoning to find other patterns in the multiplication table. The problem states, “Write the products into the chart as fast as you can. a. What numbers occur in the most places in the table? b. What numbers occur an odd number of times in your table? c. Shade the rows and columns with even factors. What do you notice about the factors and products that are left unshaded?”

• In Unit 5, Shapes and Their Perimeter, Lesson 9, Criteria for Success states, “Understand that given a particular perimeter, the most square-like rectangle will have the greatest area, and the most oblong rectangle will have the least area (MP.8).” In Problem Set, Problem 3,  students understand that given a particular perimeter, the most square-like rectangle will have the greatest area, and the most oblong rectangle will have the least area. The problem states, “Find a rectangle with the same perimeter but a smaller area from the rectangle below. Draw the new rectangle on the grid to the right. How did you find a rectangle with the same perimeter but a smaller area in #3? Did you think you’d need a longer and skinnier rectangle or a more square-like one? Why?”

• In Unit 6, Fractions, Lesson 18, Criteria for Success states, “Look for and express regularity in repeated reasoning to generalize that when the numerator can be evenly divided by the denominator in a fraction, the fraction is equivalent to the quotient of that division (MP.8).” Homework, Problem 3 states, “What pattern do you notice in each of the columns of the table above? Following the pattern, how many twelfths are equivalent to 2 wholes? 3 wholes? 4 wholes?”