The materials reviewed for Fishtank Plus Math Grade 5 meet expectations for developing conceptual understanding of key mathematical concepts and provide opportunities for students to independently demonstrate conceptual understanding throughout Grade 5. 

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 1, Place Value with Decimals, Lesson 2, Problem Set, Problems 4 and 5. Problem 4 states, “Solve. a. 10 × 10 = ____; b. 100 × 1,000 = ____; c. 10 × 100,000 = ____; d. 1,000 × 1,000 = ____; e. 10,000 × 100,000 = ____.” Problem 5 states, “What do you notice about the factors and products in #4?” In addition, the “Discussion of Problem Set,” provides teachers with questions to ask students. For example, “Look at #4e. How did you solve this even though it’s beyond what we’ve done already?” This activity supports conceptual understanding of 5.NBT.2 (Explain patterns in the number of zeros of the product when multiplying a number by powers of 10, and explain patterns in the placement of the decimal point when a decimal is multiplied or divided by a power of 10. Use whole-number exponents to denote powers of 10).

• In Unit 2, Multiplication and Division of Whole Numbers, Lesson 12, Anchor Task, Problem 2 states, “Estimate the following quotients. a. 4,212 ÷ 52 b. 1,232 ÷ 28 c. 5,427 ÷ 81.” Guiding Questions include, “What should we round our divisor to? What is a compatible number that we can use in place of the dividend? What is our estimated quotient? Of all three estimates, which one do you think is closest to the actual quotient? Which one do you think is an overestimate? Which one do you think is an underestimate? Why? Why do we round the divisor first, then replace the dividend with a compatible number? What if we rounded the dividend first, then replaced the divisor with a compatible number? Let’s try that with one (or a few) of these computations, then compare it to the actual quotient. For which problems above would rounding both factors to their largest place value give an estimate that we can compute? How does this demonstrate that using compatible numbers is a more reliable strategy to estimate quotients?” This problem allows students to develop, with teacher support, conceptual understanding of 5.NBT.6 (Find whole-number quotients of whole numbers with up to four-digit dividends and two-digit divisors, using strategies based on place value, the properties of operations, and/or the relationship between multiplication and division. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models).

• In Unit 4, Addition and Subtraction of Fractions/Decimals, Lesson 4, Anchor Tasks, Problem 1 states,  “Solve. a.1 orange + 3 oranges = ____; b. 1 child + 3 adults = ____. 2. What do you notice about #1 above? What do you wonder?” This problem reinforces the importance of comparing like items, as students will need to know/remember to use a common denominator when adding or subtracting fractions. Guiding Questions include, “What do you notice about #1? What do you wonder? Can we use some other ‘unit’ for (b) that would make it possible to add them?” This problem and the accompanying questions guide students to develop conceptual understanding of 5.NF.1 (Add and subtract fractions with unlike denominators).

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, Shapes and Volume, Lesson 4, Problem Set, Problem 8 states, “A student filled a right rectangular prism-shaped box with one inch cubes to find the volume, in cubic inches. The student’s work is shown.” The following student work is provided, “Student’s Work - I pack my box full of cubes. Each cube has a volume of 1 cubic inch. I counted 63 cubes in the top layer. Since there are 9 layers of cubes below the top layer, I solved 63 x 9 = 567. So there are 567 cubes. The volume of my box is 567 cubic inches.” Students are then asked the following questions, “a. Explain why the student’s reasoning is incorrect. Provide the correct volume, in cubic inches, of the box. b. A second box also has a base of 63 square inches, but it has a volume of 756 cubic inches. What is the height in inches, of the second box? Explain or show how you determined the height.” This problem builds conceptual understanding of 5.MD.5 (Relate volume to the operations of multiplication and addition and solve real world and mathematical problems involving volume).

• In Unit 4, Addition and Subtraction of Fractions/Decimals, Lesson 15, Target Task states, “Lina brought $10 to the fair. She spent $2.59 for cotton candy. She spent $3.49 for a toy. How much money did Lina have left?” This activity provides an opportunity for students to demonstrate conceptual understanding of 5.NBT.7 (Add, subtract, multiply, and divide decimals to hundredths, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method and explain the reasoning used).

• In Unit 5, Multiplication and Division of Fractions, Lesson 10, Problem Set, Problem 1 states, “Solve each problem in two different ways as modeled in the example. You may draw a model to help you. a. $\frac{6}{7}×\frac{5}{8}$; b. $\frac{4}{5}×\frac{5}{8}$; c. $\frac{2}{3}×\frac{6}{7}$; d. $\frac{4}{9}×\frac{3}{10}$”. These problems allow students to independently demonstrate conceptual understanding of 5.NF.4 (Apply and extend previous knowledge of multiplication to multiply a fraction by a fraction).

The materials reviewed for Fishtank Plus Math Grade 5 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, the Problem Sets, during a lesson, and Fluency Activities. Examples Include:

• In Unit 2, Multiplication and Division of Whole Numbers, Lesson 7, Fluency Activities, Number Talks, Multi-Digit Whole-Number Multiplication. The teacher guides students in a Number Talk, focusing on multiplying multi-digit whole numbers. The materials state, “Number talks are an opportunity to solve computational problems using mental strategies. They are typically a sequence of related computations that allow students to apply strategies from earlier computations to later ones. Number Talks can be completed as a whole class or in a small group with the teacher.” A Number Talks example states, “Multiplying multi-digit numbers using mental strategies” includes “Doubling and Halving.” One set of numbers to engage in a Number Talk includes, “1 × 72; 2 × 36; 4 × 18; 8 × 9.” This activity helps students develop 5.NBT.5 (Fluently multiply multi-digit whole numbers using the standard algorithm).

• In Unit 6, Multiplication and Division of Decimals, Lesson 4, Anchor Task, Problem 1 states, “Solve. Show or explain your work. a. 60 × 3 = ; 60 × 0.3 = ; 60 × 0.03 = __. b. What do you notice about #1? What do you wonder? c. Use what you noticed in #1 to solve 600 × 0.3 and 600 × 0.03.” Guiding Questions prompt teachers to ask, “How are the factors in these problems similar? How are they different? How are the products in these problems similar? How are they different? Why does 6 tens times 3 tenths result in a product whose unit is ones? Why does 6 tens times 3 hundredths result in a product whose unit is tenths? Why are 60 × 0.3 and 600 × 0.03 equivalent? What about 60 × 3 and 600 × 0.3?” These problems and guiding questions help students develop 5.NBT.7 (Add, subtract, multiply, and divide decimals to hundredths).

• In Unit 5, Multiplication and Division of Fractions, Lesson 10, Anchor Task, Problem 2, students multiply a fraction by a fraction. The problem states, “Solve. Show or explain your work. a. $$\frac{3}{5}×\frac{1}{2}$$ b. $$\frac{1}{12}×\frac{8}{9}$$ c. $$\frac{5}{9}×\frac{3}{5}$$ d. $$\frac{15}{4}×\frac{8}{5}$$.” The Guiding Questions prompt teachers to ask, “How can we use what we learned in Anchor Task #1 to compute $$\frac{3}{5}×\frac{1}{2}$$? When I got to the point of computing $$\frac{1×8}{12×9}$$ in Part (b), I thought multiplying 12 by 9 was kind of a pain. Is there some way I can simplify this fraction before computing that product? (Follow a similar process to compute parts (c) and (d), seeing if it’s possible to simplify before computing the product.) Is it easier to simplify before or after performing the computation? Why?” This problem provides an opportunity for students to develop fluency of 5.NF.4 (Apply and extend previous understandings of multiplication to multiply a fraction or whole number by a fraction).

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 or with a partner. Examples include:

• In Unit 4, Addition and Subtraction of Fractions/Decimals, Lesson 1, Target Task, students use equivalent fractions as a strategy to add and subtract fractions. The task states, “Find three fractions that are equivalent to each of the following fractions. Use pictures and an equation to explain why the fractions are equivalent. 1. $$\frac{5}{8}$$ 2. $$\frac{7}{4}$$”. These problems provide an opportunity for students to independently demonstrate procedural skill and fluency of 5.NF.1 (Use equivalent fractions as a strategy to add and subtract fractions).

• In Unit 5, Multiplication and Division of Fractions, Lesson 10, Target Task, students “Solve each problem in two different ways. Show or explain your work. 1. $$\frac{2}{3}×\frac{5}{6}$$ 2. $$\frac{4}{9}×\frac{3}{8}$$.” These problems provide an opportunity for students to independently demonstrate procedural skill and fluency of 5.NF.4 (Apply and extend previous understandings of multiplication to multiply a fraction or whole number by a fraction).

• In Unit 6, Multiplication and Division of Decimals, Lesson 6, Target Task, Problem 3 states, “Solve. Show or explain your work. a. 0.35 × 0.4;  b. 2.02 × 4.2.” These problems provide an opportunity for students to independently demonstrate procedural skill and fluency of 5.NBT.7 (Add, subtract, multiply, and divide decimals to hundredths, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method and explain the reasoning used).

The materials reviewed for Fishtank Plus Math Grade 5 meet expectations for being designed so that teachers and students spend sufficient time working with engaging applications of the mathematics.

Materials provide opportunities, within Problem Set and Homework problems for students to independently demonstrate multiple routine and non-routine applications throughout the grade level. Problem Set, or student practice problems, and Homework can be completed independently during a lesson. Target Task, or end of lesson checks for understanding, are designed for independent completion. Examples include:

• In Unit 2, Multiplication and Division of Whole Numbers, Lesson 9, Anchor Task, Problem 2, students solve routine application problems involving multiplying multi-digit numbers (5.NBT.5). The problem states, “On Friday, April 13, 2018, there were 749 outbound flights each carrying approximately 101 passengers. If there are usually 58,112 daily outbound passengers from Logan, how many more passengers flew out of Logan that Friday than usual?” Guiding Questions for teachers include, “Can you draw something to represent this problem? Is our answer reasonable? Why or why not? Why do you think this was a record-breaking day? What usually happens around that time of year?”

• In Unit 5, Multiplication and Division of Fractions, Lesson 6, Anchor Task, Problem 1, students engage in non-routine application problems as they solve real world problems involving multiplication of fractions and mixed numbers (5.NF.6), and apply and extend previous understanding of multiplication to multiply a fraction or a whole number by a fraction (5.NF.4). The problem states, “Some of the problems below can be solved by multiplying $\frac{3}{5}×15$, while others need a different operation. Select the ones that can be solved by multiplying these two numbers. For the remaining, tell what operation is appropriate. In all cases, solve the problem (if possible) and include appropriate units in the answer. a. There are 15 people at a party. $\frac{3}{5}$ of them are boys. How many people at the party are boys? b. Wesley ran 15 miles on Monday and $\frac{3}{5}$ mile on Tuesday. How many miles did Wesley run? c. If each person at a party eats $\frac{3}{5}$ of a pound of roast beef and there are 15 people at the party, how many pounds of roast beef are needed? d. Nathaniel has 15 cups of soup split into 5 equal-sized portions. He’ll serve three of the portions for dinner tonight. What is the total amount of soup, in cups, that Nathaniel will serve tonight? e. 15 students in the fifth grade want to play soccer. $\frac{3}{5}$ of the students in fifth grade want to play basketball. How many students want to play either soccer or basketball? f. Helena is carpeting a long corridor. It is $\frac{3}{5}$ yards wide and 15 yards long. How much carpeting, in square yards, does Helena need? g. Lisbeth has $\frac{3}{5}$ of a pound of chocolate that she wants to share evenly with 15 people. How much chocolate will everyone get? h. Tiffany has $15. She spends $\frac{3}{5}$ of her money on a teddy bear. How much money does she have left?” Guiding questions for teachers include, “How can you connect what you know about multiplication with whole numbers to multiplication with fractions? What is the whole that is being referred to in the problem? Why does looking for keywords in a problem not always work as a strategy? (You could use Part (e) as an example here - ‘of'’ does not imply that these quantities should be multiplied together.)”

• In Unit 6, Multiplication and Division of Decimals, Lesson 2, Anchor Task, Problem 1, students engage in non-routine problems that involve adding, subtracting, multiplying, and dividing decimals to hundredths, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction (5.NBT.7). The problem states, “Mr. Wynn, the art teacher, is creating a mural on the side of the school building. The mural will be 2 meters tall and 6 meters long. After finishing the mural, he decided he wants to add a small section to the side of it, and the new section is also 2 meters tall but just 0.4 meters long. How many square meters is Mr. Wynn’s mural? What was the total length of the mural Mr. Wynn created?” Guiding questions for teachers include, “What model can you draw to represent the problem? How can you record that work with equations? Does finding the area of each separate piece of the mural and adding those areas together give the same result as finding the overall length and width of the space Mr. Wynn needed to cover and finding the overall area? Why?”

• Unit 7, Patterns and the Coordinate Plane, Lesson 11, Anchor Task, students engage in routine problems that involve graphing points in the first quadrant of the coordinate plane, and interpret coordinate values of points in the context of the situation (5.G.2). The problem states, “Jessica has $15 saved. She earns $6 per hour for babysitting. a. Construct a graph to show how much money Jessica will have after each hour of babysitting. b. What ordered pair corresponds with how much money Jessica has before she does any babysitting? c. What does the ordered pair (2, 27) represent in the context of this problem? d. Jessica babysits for 4 hours. How much money does she now have? e. Jessica wants to buy a video game that costs $45. How many hours does Jessica need to babysit in order to be able to buy it?”  Guiding Questions for teachers to ask include, “How can we represent this situation on a coordinate grid? How much money does Jessica have before she does any babysitting? Where do you see that on the graph? What is the coordinate pair? What does the coordinate pair (2, 27) mean in this context? How much money does Jessica have after babysitting for 4 hours? Where do you see that represented in the graph? How long will it take Jessica to earn $45 to buy the video game? Where do you see that represented in the graph?”

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

• In Unit 1, Place Value with Decimals, Lesson 3, Additional Practice, Word Problem Practice, students independently solve routine problems when comparing zeros in place values by using additive to compare with difference unknown (5.NBT.2). “Santiago owns a small bread company. Saturday his company baked 3,860 loaves of bread. On Sunday his company baked 4,820 loaves of bread. How many more loaves did they bake on Sunday than on Saturday?”

• In Unit 4, Addition and Subtraction of Fractions/Decimals, Lesson 15, Problem Set, Problem 4, students independently solve routine addition problems with decimal numbers to the hundredths place (5.NBT.7). The problem states, “Van Cortlandt Park’s walking trail is 1.02 km longer than Marine Park’s. Central Park’s walking trail is 0.242 km longer than Van Cortlandt’s. Marine Park’s walking trail is 1.28 km. If a tourist walks all 3 trails in a day, how many kilometers would he or she have walked?”

• In Unit 5 Multiplication and Division of Fractions, Lesson 20, Target Task, students independently solve non-routine word problems that involve the division of whole numbers and fractions (5.NF.7). Problem 2 states, “There are 7 math folders on a classroom shelf. This is $\frac{1}{3}$ of the total number of math folders in the classroom. What is the total number of math folders in the classroom?”

• In Unit 6, Multiplication and Division of Decimals, Lesson 24, Homework, Problem 6, students independently solve routine problems involving converting among different-sized standard measurement units within a given measurement system (5.MD.1). The problem states, “Sara and her dad visit Yo-Yo Yogurt again. This time, the scale says that Sara has 14 ounces of vanilla yogurt in her cup. Her father’s yogurt weighs half as much. How many pounds of frozen yogurt did they buy altogether on this visit? Express your answer as a mixed number.”

The materials reviewed for Fishtank Plus Math Grade 5 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 5. Examples where instructional materials attend to conceptual understanding, procedural skill and fluency, or application include:

• In Unit 1, Place Value with Decimals, Lesson 13, Anchor Task, Problem 1, students develop conceptual understanding of rounding decimals as they use number lines. The problem states, “a. The number 8.263 lies between 8 and 9 on the number line. Label all the other tick marks between 8 and 9. Is 8.263 closer to 8 or 9 on the number line? b. Which tenth is 8.263 nearest to on the number line? Determine what values the two outermost spots on the number line below should be to help you determine which tenth 8.263 is closest to. Then plot 8.263 to prove your answer. c. Which hundredth is 8.263 nearest to on the number line? Determine what values the two outermost spots on the number line below should be to help you determine which hundredth 8.263 is closest to. Then plot 8.263 to prove your answer. ” (5.NBT.4: Use place value understanding to round decimals to any place.)

• In Unit 2, Multiplication and Division of Whole Numbers, Lesson 1, Fluency Tasks, Number Bowling, students develop procedural skill and fluency by playing a game with cards to write and evaluate numerical expressions. The problem states, “In this fluency activity, students use digits chosen at random to create expressions equivalent to as many digits, 1-10, as possible to knock down those bowling pins.” (5.OA.1: Use parentheses, brackets, or braces in numerical expressions, and evaluate expressions with these symbols.)

• In Unit 7, Patterns and the Coordinate Plane, Lesson 10, Target Task, students engage in solving application problems as they interpret coordinate values of points in the context of the situation. The problem states, “The line graph below tracks the water level of Plainsview Creek, measured each Sunday for 8 weeks. Use the information in the graph to answer the questions that follow. a. About how many feet deep was the creek in Week 1? b. According to the graph, which week had the greatest change in water depth? c. It rained hard throughout the sixth week. During what other weeks might it have rained? Explain why you think so.“ (5.G.2: Represent real world and mathematical problems by graphing points in the first quadrant of the coordinate plane, and interpret coordinate values of points in the context of the situation.)

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 2, Multiplication and Division of Whole Numbers, Lesson 20, Anchor Task, Problem 4, develop conceptual understanding and procedural skill and fluency as they solve real world problems using multiplication and division. The problem states, “On Saturday, the owner of a department store gave away a $15 gift card to every 25th customer. A total of 8,879 customers came to the store on Saturday. What is the total value of the gift cards the owner gave away? How many additional customers would need to have come in for another gift card to be given away?” Guiding Questions include, “Can you draw something to represent this problem? What does the remainder mean in the context of this problem? How would you interpret it? How did you determine how many more customers would need to have come in for another gift card to be given away? Is your answer reasonable? Why or why not?” (5.NBT.5: Fluently multiply multi-digit whole numbers using the standard algorithm and 5.NBT.6: Find whole-number quotients of whole numbers with up to four-digit dividends and two-digit divisors, using strategies based on place value, the properties of operations, and/or the relationship between multiplication and division. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models.)

• In Unit 5, Multiplication and Division of Fractions, Lesson 6, Problem Set, Problem 4, students develop conceptual understanding and application as they solve real world problems involving multiplication of fractions. The problem states, “Mrs. Diaz makes 5 dozen cookies for her class. One-ninth of her 27 students are absent the day she brings the cookies. If she shares the cookies equally among the students who are present, how many cookies will each student get?” (5.NF.6: Solve real world problems involving multiplication of fractions and mixed numbers, e.g., by using visual fraction models or equations to represent the problem.)

• In Unit 7, Patterns and the Coordinate Plane, Lesson 8, Target Task, students develop procedural skill and fluency alongside application as they solve problems involving ordered pairs. The task states, “Jillian plotted points Q and R on a coordinate grid, as shown below. Jillian wants to plot point S so that when points Q, R, and S are connected they form the vertices of a right triangle. Write an ordered pair that represents where Jillian should plot point S.” (5.G.2: Represent real world and mathematical problems by graphing points in the first quadrant of the coordinate plane, and interpret coordinate values of points in the context of the situation.)

The materials reviewed for Fishtank Plus Math Grade 5 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 2, Multiplication and Division of Whole Numbers, Lesson 19, Problem Set, Problem 3, students “Assess the reasonableness of an answer by rounding and/or using the relationship between multiplication and division to check answers (MP.1).”  The problem states, “ a. At a school carnival there is an egg toss. There are 314 students in the school. Twelve eggs are in one carton. How many cartons are needed so that each student gets an egg to try the egg toss? b. The principal wants to give every student two tries at the egg toss. How will this decision affect the number of cartons he needs to buy?”

• In Unit 4, Addition and Subtraction of Fractions/Decimals, Lesson 10, Anchor Task, Problem 2, students “Assess the reasonableness and/or correctness of an answer based on number sense (MP.1).” The problem states, “Joe is baking cookies. He needs a total of 2 cups of sugar for the recipe. Joe bought a $$4\frac{1}{2}$$ cup bag of sugar and has used $$2\frac{3}{4}$$ cups already. Without solving the problem, does Joe have enough sugar? Explain your thinking.” Guiding Questions include, “How can we use what we did in Anchor Task #1 to help us? Does Joe have enough sugar? How did you figure that out without actually solving the problem?”

• In Unit 5, Multiplication and Division of Fractions, Lesson 20, Problem Set, Problem 2, students “Understand when a problem calls for the use of division and when other operations are called for in a problem involving a unit fraction and a whole number. (MP.1, MP.4)” The problem states, “Larry had $$\frac{1}{2}$$ of a submarine sandwich left over from a field trip. He decided to give it to his 6 friends to share after school. How much of the original sandwich did each person get? Show with numbers, words, and a picture or diagram.”

• In Unit 6, Multiplication and Division of Decimals, Lesson 18, Anchor Task, Problem 4, students “Assess the reasonableness of a solution by rounding to estimate or by checking a solution using multiplication (MP.1).” The problem states, “Act 4 (the sequel): Bella has 6.3 kilograms of berries. She packs 0.35 kilogram of berries into each container. She then sells each container for $2.99. How much money will Bella earn if she sells all the containers? a. Write an expression to determine how much money Bella will earn if she sells all the containers. b. Find the amount of money Bella will earn if she sells all the containers.” Guiding Questions include, “What quantities and relationships do we know? What is the question asking you to find out? How did you find the answer to the question? Did anyone find the answer differently? Is your answer reasonable? How do you know?”

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 of Whole Numbers, Lesson 2, Target Task, Problem 3, students “Interpret expressions without evaluating them (MP.2).” The problem states, “Which of the following expressions represents a number that is 3 times larger than the sum of 8105 and 186?; a. (8105 + 186) ÷ 3;  b. 3 × (8105 + 186); c. 8105 + 186 ÷ 3; d. 3 × 8105 + 186.”   

• In Unit 3, Shapes and Volume, Lesson 5, Target Task, Problem 1, students “Reason abstractly and quantitatively to see that the dimensions can be multiplied together in any order and the volume will remain the same (MP.2).” The problem states, “The right rectangular prism below has a length of 4 units, a width of 3 units, and a height of 6 units. Select the three equations that can be used to find the volume of the prism. a. 12 × 6 b. 4 × (3 × 6) c. 7 × 6 d. (4 + 3) x 6 e. 4 × 3 × 6.”

• In Unit 5, Multiplication and Division of Fractions, Lesson 11, Anchor Task, Problem 3, students, “Write a story context to match a given expression involving the multiplication of two fractions (MP.2). Write an expression to match a story context involving the multiplication of two fractions (MP.2).” The problem states, “One section of a beach has a total of 180 people. Of these 180 people, $$\frac{4}{9}$$ are wearing a hat and $$\frac{2}{5}$$ of the people wearing hats are also wearing sunglasses. How many people in that section of beach are wearing both a hat and sunglasses? a. Write an expression to determine how many people in that section of the beach are wearing both a hat and sunglasses. b. Find the number of people in that section of the beach that are wearing both a hat and sunglasses. c. Find the number of people in that section of the beach that are wearing a hat but not wearing sunglasses.” Guiding Questions include, “What can you draw to represent the problem? What expression can you write to represent the situation? Is there more than one correct expression? How can you use your expression to find the number of people in that section of the beach that are wearing both a hat and sunglasses? How can you find the number of people in that section of the beach that are wearing a hat but no sunglasses?” 

• In Unit 7, Patterns and the Coordinate Plane, Lesson 11, Anchor Tasks, students “Contextualize and decontextualize a coordinate pair based on the situation.” The task states, “Jessica has $15 saved. She earns $6 per hour for babysitting. a. Construct a graph to show how much money Jessica will have after each hour of babysitting. b. What ordered pair corresponds with how much money Jessica has before she does any babysitting? c. What does the ordered pair (2, 27) represent in the context of this problem? d. Jessica babysits for 4 hours. How much money does she now have? e. Jessica wants to buy a video game that costs $45. How many hours does Jessica need to babysit in order to be able to buy it?” Guiding Questions include, “How can we represent this situation on a coordinate grid? How much money does Jessica have before she does any babysitting? Where do you see that on the graph? What is the coordinate pair? What does the coordinate pair (2, 27) mean in this context? How much money does Jessica have after babysitting for 4 hours? Where do you see that represented in the graph? How long will it take Jessica to earn $45 to buy the video game? Where do you see that represented in the graph?”

The materials reviewed for Fishtank Plus Math Grade 5 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 2, Multiplication and Division of Whole Numbers, Lesson 4, Problem Set, Problem 2, students “Multiply multiples of powers of ten with multiples of powers of ten.” The problem states, “Ripley told his mom that multiplying whole numbers by multiples of 10 was easy because you just count zeros in the factors and put them in the product. He used these two examples to explain his strategy. 7,000 × 600 = 4,200,000; 800 × 700 = 560,000. Ripley’s mom said his strategy will not always work. Why not? Give an example.”

• In Unit 4, Addition and Subtraction of Fractions/Decimals, Lesson 1, Anchor Tasks, Problem 1, students, “Determine whether two fractions are equivalent using an area model, a number line, or multiplication/division.” The problem states, “Ms. Kosowsky makes brownies in two pans of the same size. She cuts the pans in the following way: [Pan A is cut in halves; Pan B is cut in eighths.] Ms. Kosowsky gives one brownie from Pan A to Ms. Kohler and keeps four brownies from Pan B for herself. Ms. Kohler thinks this isn’t fair since she got one brownie and Ms. Kosowsky got four. Ms. Kosowsky thinks it’s fair. Who do you agree with, Ms. Kosowsky or Ms. Kohler? Why?” Guiding Questions include, “How much of each brownie pan did each teacher get? Do you agree with Ms. Kosowsky or Ms. Kohler? Equivalent fractions are fractions that represent the same portion of the whole and the wholes are equal-sized. Are these two fractions equivalent? How can we represent that with an equation? What do you notice about the numerators and denominators of the equivalent fractions? How can you use the area models to explain why this happens? How can you represent this using multiplication or division?”

• In Unit 5, Multiplication and Division of Fractions, Lesson 17, Homework, Problem 5, students “Compare the size of a product to the size of one factor on the basis of the size of the other factor, without performing the indicated multiplication.” The problem states, “Lisa claims that when multiplying any number between 0 and 10 by 100, the product is greater than 100. What is a possible number that can be multiplied by 100 to show that Lisa’s claim is not correct?”

The materials reviewed for Fishtank Plus Math Grade 5 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. 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 2, Multi-Digit Multiplication and Division of Whole Numbers, Lesson 20, Anchor Task, Problem 4, students “Solve multi-step word problems involving all four operations, including those that require the interpretation of the remainder (MP.1, MP.4).” The problem states, “On Saturday, the owner of a department store gave away a $15 gift card to every 25th customer. A total of 8,879 customers came to the store on Saturday. What is the total value of the gift cards the owner gave away? How many additional customers would need to have come in for another gift card to be given away?” Guiding Questions include, “Can you draw something to represent this problem? What does the remainder mean in the context of this problem? How would you interpret it? How did you determine how many more customers would need to have come in for another gift card to be given away? Is your answer reasonable? Why or why not?”

• In Unit 6, Multiplication and Division of Decimals, Lesson 24, Problem Set, Problem 10, students “Solve multi-step word problems involving measurement conversions (MP.4).” The problem states, “Tanya bought 12 water bottles. Of those bottles, 5 hold 300 milliliters each and 7 hold 1.5 liters each. How much water, in liters, does Tanya buy?”

• In Unit 7, Patterns and the Coordinate Plane, Lesson 10, Anchor Tasks, Problem 1, students “Answer simple word problems regarding data represented in a coordinate graph (MP.4).” The problem states, “a. The following ordered pairs show the weight of a typical male greyhound, a breed of dog, during the first 28 months of his life. Graph the corresponding points, then connect the points in the order they are given to form a line graph. b. What do you notice? What do you wonder?” Guiding Questions include, “How did you decide what scale to use for the x-axis? The y-axis? How can you label the x-axis? The y-axis? What title can you give it to convey what is represented on the coordinate grid? What do you notice about the points you plotted? What do you wonder? What can we learn about the typical male greyhound’s weight by looking at the graph, especially compared with looking at the same information in the table?”

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 2, Multiplication and Division of Whole Numbers, Lesson 7, Anchor Tasks, Problem 1, students multiply two-digit by three-digit products using the method of their choice (i.e., standard algorithm, partial products, or area model). The problem states, “Find the products using any method. Then assess the reasonableness of your answer. a. 814 x 39, b. 715 x 53, c. 78 x 266.”

• In Unit 4, Addition and Subtraction of Fractions/Decimals, Lesson 1, Homework, Problem 6, students “Generate equivalent fractions using an area model, a number line, or multiplication/ division (MP.5).” The problem states, “Sammie took a really long bike ride through the mountains. She planned on taking breaks at equal points along the ride, as shown below. Sammie is $$\frac{3}{4}$$ of the way along her bike ride. a. Explain how you can use the number line to show $$\frac{3}{4}$$. b. Write a fraction that is equivalent to $$\frac{3}{4}$$.”

• In Unit 5, Multiplication and Division of Fractions, Lesson 15, Homework, Problem 2, students “Decide which generalized method for computing products of mixed numbers will be most efficient for a particular problem and use it to compute the product (MP.5).” The problem states, “Circle the solution strategy that is more efficient in each of the problems above. Then, explain any similarities you notice between problems for which one strategy was more efficient.”

The materials reviewed for Fishtank Plus Math Grade 5 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 2, Multiplication and Division of Whole Numbers, Lesson 1, Criteria For Success states, “Understand that the order of operations is a convention that says operations should be performed from left to right in the following order (MP.6): a. Grouping symbol; b. Multiplication and division; C. Addition and subtraction.” Target Task, Problem 1 states, “What is the value of this expression? 100 − [5 × (3 + 4)].”

• In Unit 3, Shapes and Volume, Lesson 1, Criteria for Success, teachers are provided the following criteria for the lesson, “Use correct units and notation when recording the volume of a three-dimensional figure, including cubic units, cubic centimeters, and cm³ (MP.6).” Anchor Tasks, Problem 3, students find the volume of figures, making sure to accurately label the units of measure. The problem states, “Construct a figure that is 1 centimeter tall, 2 centimeters wide, and 2 centimeters long. What is its volume? Construct another figure with the same volume. Does the following figure have the same volume? Why or why not?” Guiding Questions include, “What is the volume of the figure you constructed? What might other figures with the same volume look like? (Present an example like the ones shown below.) Does this figure have a volume of 4 cubic centimeters? Why or why not? (Present a non-example with gaps.) Does this figure have a volume of 4 cubic centimeters? Why or why not? Does the figure shown in Part (c) have a volume of 4 cubic centimeters? Why or why not? Use the definition of volume to support your argument.”

• In Unit 4, Addition and Subtraction of Fractions/Decimals, Lesson 13, Criteria For Success, teachers are provided the following criteria for the lesson, “Add like units with decimals using an area model or the standard algorithm, aligning place values correctly (which results in the decimal points being aligned) (MP.5, MP.6).” Target Task states, “Solve. a. 2.40 + 1.8 = __ b. 36.25 + 8.67 = __.”

• In Unit 7, Patterns and The Coordinate Plane, Lesson 4, Criteria for Success, “Plot points whose x- and/or y-coordinate is/are not a multiple of the scale of the corresponding axis (i.e., the point will not be located on intersecting gridlines) (MP.6).” Anchor Tasks, Problem 1, students, with support, label points on a coordinate plane whose locations are not intersecting points on the grid. The problem states, “Geraldo is plotting points on the following coordinate plane. Geraldo needs to plot the following four points: (0.8,12) (1.6,18) (1,30) (2,3). Is the graph large enough to fit his points? How do you know? What would be a point that he couldn't fit on the coordinate plane above? How do you know? How would the graph have to change in order for that point to fit on it?” Guiding Questions include, “Can Geraldo fit all of those points on the coordinate grid? How do you know? What point couldn’t he fit on the coordinate grid? How do you know? Let’s say you had no more space to extend the coordinate grid. How would you have to change the coordinate grid in order to be able to fit the point (18,6)? What about (45,0.6)? What about (80,4)?”

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 Fifth Grade Vocabulary Glossary. The glossary contains a chart with the columns “Word” and “Definition.” 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 the word equation the definition reads, “A math statement that has an equal sign” and includes an example that reads, “10 × 5 = 50 is an equation.” 

• In Unit 2, Multiplication/Division of Whole Numbers, Lesson 1, Anchor Task, Problem 1, Guiding Questions include, “An expression is a mathematical phrase that contains symbols for operations (like plus, minus, etc.), numbers, and/or letters that represent unknowns, as opposed to an equation, which is two expressions that are equal to one another. How did Felix evaluate, or solve, the expression? How can we record what he did as separate equations?”

• In Unit 3, Shapes and Volume, Lesson 1, Tips for Teachers states, “It is unclear whether students need to be familiar with the exponential notation for volume, namely units unit3, cm3, in3, etc. There are no released items from standardized tests or other reliable sources (e.g., Illustrative Mathematics) that use this notation. However, since students have seen exponents in the context of powers of ten in Unit 1, it is reasonable to assume that they can make sense of it here. If you choose to introduce the notation, you can eventually relate it to the idea of units being repeatedly multiplied together and therefore the notation representing that idea (e.g., a prism with measurements 2cm, 3cm, and 4cm has a volume of 2cm x 3cm x 4cm, or (2 × 3 × 4) (cm × cm × cm), or 24$cm^3$.”

• In Unit 7, Coordinate Plane, Lesson 1, Tips for Teachers state, “An ordered pair is a pair of two things written in a certain order. A coordinate pair is a pair of two coordinates written in a certain order, x then y. This distinction is not important for Grade 5 students and thus the terms are used interchangeably throughout the unit.”

The materials reviewed for Fishtank Plus Math Grade 5 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 of Whole Numbers, Lesson 2, Criteria for Success states, “Write numerical expressions based on verbal/written descriptions of calculations (e.g., write 2 × (8 + 7) to express the calculation ‘add 8 and 7, then multiply by 2’) (MP.7). Write descriptions of calculations based on numerical expressions (e.g., write ‘add 8 and 7, then multiply by 2’ to describe the expression 2 × (8 + 7)) (MP.7).” In Anchor Tasks, Problem 2 states, “For each problem below, write an expression that records the calculations described below, but do not evaluate. a. 3 times the sum of 26 and 4; b. The quotient of 15 and 3 subtracted from 60.” Guiding Questions include, “How can you represent Part (a) with a picture? How can that picture help you write a numerical expression? Is there more than one correct way to write an expression for Part (a)? How many can we come up with? Why can we write expressions with the addends of the addition expression in either order? Or with the factors of the multiplication problem in either order? Why are parentheses necessary around the 26 and 4 in all of the expressions we wrote for Part (a)? How can you represent Part (b) with a picture? How can that picture help you write a numerical expression? Can we write expressions with the values of the subtraction expression in either order? Or with the values of the division problem in either order? Why not? Do we need parentheses anywhere in the expression in Part (b)? Why or why not?” 

• In Unit 3, Shapes and Volume, Lesson 5, Criteria for Success states, “Look for and make use of structure to find the volume of concrete rectangular prisms by finding the number of cubes in a layer by multiplying its length times width, then multiplying by the number of layers (MP.7).” In Target Tasks, Problem 1 states, “Use the figure below to answer the following questions. 1. How many layers are in the figure to the right? b. How many cubes are in each layer? c. What is the volume of the figure? d. Explain how you could find the volume of the figure using a different number of layers.”

• In Unit 4, Addition and Subtraction of Fractions/Decimals, Lesson 11, Criteria For Success states, “Use the properties of addition and subtraction to make a multi-term computation easier (e.g., to add $\frac{1}{3}+\frac{3}{5}+\frac{2}{3}$, first add $\frac{1}{3}$ and $\frac{2}{3}$ to make 1, making the computation easier) (MP.7).” In Problem Set, Problem 3 states, “Erin jogged $2\frac{1}{4}$ miles on Monday. Wednesday, she jogged $3\frac{1}{3}$ miles, and on Friday, she jogged $2\frac{2}{3}$ miles. How far did Erin jog altogether?”

• In Unit 6, Multiplication and Division of Decimals, Lesson 19, Criteria For Success states, “Write descriptions of calculations involving decimals based on numerical expressions (e.g., write ‘add 8 and 7, then divide 3 tenths by that sum’ to describe the expression 0.3 ÷ (8 + 7)) (MP.7).” In Target Task, Problems 1 and 2 state, “1. Write an equivalent expression in numerical form. Then evaluate it. Twice as much as the difference between 7 tenths and 5 hundredths; 2. Write an equivalent expression in word form. Then evaluate it. 9 ÷ (0.16 + 0.2).”

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 with Decimals, Lesson 1, Criteria for Success states, “Look for and express regularity in repeated calculations of multiplying 10 by itself some number of times, and explain patterns in the number of zeros in the product (MP.8).” Problem Set, Problem 2, students use repeated reasoning and multiples of 10 to solve problems looking for and applying knowledge of patterns. The problem states, “Solve.

  a. 10 × 10 × 10 × 10 = ___

  b. 10 × 10 × 10 = ___

  c. 10 × 10 × 10 × 10 × 10 × 10 = ___

  d. 10 × 10 × 10 × 10 × 10 = ___.”

• In Unit 5, Multiplication and Division of Fractions, Lesson 10, Criteria For Success states, “Deduce the generalized method for multiplying a fraction times a fraction (MP.8).” Target Task states, “Solve each problem in two different ways. Show or explain your work. a. $\frac{2}{3}×\frac{5}{6}$ 2. $\frac{4}{9}×\frac{3}{8}$.”

• In Unit 6, Multiplication and Division of Decimals, Lesson 7, Criteria for Success states, “Deduce the general pattern about the placement of the decimal point when multiplying decimals, namely that the number of decimal places in the product is the sum of decimal places in each factor (MP.8).” Homework, Problem 7 states, “Gerard solves 2.34 × 9.2 by computing 234 × 92, then moving the decimal point three places to the left. Why does Gerard’s method make sense?”

• In Unit 7, Patterns and the Coordinate Plane, Lesson 12, Criteria for Success states, “Look for and express repeated reasoning to identify relationships between corresponding terms (MP.8).” Anchor Tasks, Problem 1 states, “Melissa and Joe are reading their books during independent reading. Because Joe’s book has more pictures in it, he can read more pages in the same amount of time. When Melissa is done with 2 pages of reading, Joe is done with 4 pages. Complete the table below to see how many pages Joe and Melissa will each read. Create a coordinate graph using the information in the table above. What do you notice about the number of pages Melissa and Joe will read? What do you wonder?” Guiding Questions include, “What did you notice about the table and graph? What do you wonder? When Joe has read 16 pages, how many will Michelle have read? When Michelle has read 10 pages, how many will Joe have read? What can you say in general about how many pages Joe reads compared to how many pages Melissa reads?”

The materials reviewed for Fishtank Plus Math Grade 5 meet expectations for providing teacher guidance with useful annotations and suggestions for how to enact the student materials and ancillary materials, with specific attention to engaging students in order to guide their mathematical development.

Materials provide comprehensive guidance that will assist teachers in presenting the student and ancillary materials. Examples include:

• In Teacher Tools, Math Teacher Tools, Preparing to Teach Fishtank Math, Preparing to Teach a Math Unit recommends seven steps for teachers to prepare to teach each unit as well as the questions teachers should ask themselves while preparing. For example step 1 states, “Read and annotate the Unit Summary-- Ask yourself: What content and strategies will students learn? What knowledge from previous grade levels will students bring to this unit? How does this unit connect to future units and/or grade levels?”

• In Unit 4, Addition and Subtraction of Fractions/Decimals, Unit Summary provides an overview of content and expectations for the unit. Within Unit Prep, Intellectual Prep, there is Unit-Specific Intellectual Prep detailing the content for teachers. The materials state, “When referred to fractions and decimals throughout Units 4-6, use unit language as opposed to ‘out of’ or ‘point’ language (e.g.,$$\frac{3}{4}$$ should be described as ‘3 fourths’ rather than ‘3 out of 4’ and 0.8 should be described as ‘8 tenths’ rather than ‘zero point eight’). To understand why this is important for fractions (which can be extrapolated to decimals), read the following blog post: Say What You Mean and Mean What You Say by William McCallum on Illustrative Mathematics. Read the following table that includes models used in this unit.” Additionally, the Unit Summary contains Essential Understandings. It states, “Quantities cannot be added or subtracted if they do not have like units. Just like one cannot add 4 pencils and 3 bananas to have 7 of anything of meaning (unless one changes the unit of both to ‘objects’), the same applies for the units of fractions (their denominators) and the units of decimals (their place values). This explains why one must find a common denominator to be able to add fractions with unlike denominators and why one must align corresponding places correctly (which in turn aligns the decimal points) when adding and subtracting decimals. ‘It is not necessary to find a least common denominator to calculate sums of fractions, and in fact the effort of finding a least common denominator is a distraction from understanding algorithms for adding fractions’ (NF Progressions, p. 11). Similar to computational estimates with whole numbers, some computational estimates with fractions can be better than others, depending on what numbers are chosen to use in place of the actual values. Computational estimates can be too high or too low, depending on how the numbers were originally estimated, what computation is being performed, and where in the number sentence it’s located. ‘Because of the uniformity of the structure of the base-ten system, students use the same place value understanding for adding and subtracting decimals that they used for adding and subtracting whole numbers’ (NBT Progression, p. 19).”  

Materials include sufficient and useful annotations and suggestions that are presented within the context of the specific learning objectives. Teacher Tools, Math Tools, Preparing to Teach Fishtank Math, Components of a Math Lesson, states, “Each math lesson on Fishtank consists of seven key components: Objective, Standards, Criteria for Success, Tips for Teachers, Anchor Tasks/Problems, Problem Set, and Target Task. Several components focus specifically on the content of the lesson, such as the Standards, Anchor Tasks/Problems, and Target Task, while other components, like the Tips for Teachers, serve to ensure teachers have the support and knowledge they need to successfully implement the content.” Examples include:

• In Unit 1, Place Value With Decimals, Lesson 8, Tips for Teachers include guidance to address common misconceptions with decimal patterns. The materials state, “A common misconception when multiplying decimals by 10 is that one just ‘adds a zero’ to the end of the number, which with decimals does not change the value of the number. If you notice this misconception come up, address it directly with the whole class, using various models and arguments to dispute it. The Target Task should give the teacher helpful feedback on whether this misconception persists by the end of the lesson.”

• In Unit 4, Addition and Subtraction of Fractions/Decimals, Lesson 2, Anchor Tasks Problem 1 Notes provide teachers guidance about how to set students up to solve the problems. The materials state, “The contexts for the problems were intentionally chosen to encourage students to use one type of model—an area model for part (a) and a number line for part (b). Students, of course, can use them interchangeably. Since students relied more heavily on tape diagrams and number lines for fraction addition and subtraction in Grade 4, they may gravitate towards those, in which case you can decide whether to introduce the area model here or wait for Lesson 4 when they become more useful to find a common unit. An area model and number line for Parts (a) and (b), respectively, are shown below. If students struggle to see why the numerators are added and the denominators stay the same, relate it to the idea of the denominators being the units of the fractions. Just like 3 bananas and 4 bananas together are 7 bananas, 3 eighths and 4 eighths together are 7 eighths.”

• In Unit 6, Multiplication and Division of Decimals, Lesson 9, Tips for Teachers provide context when students divide decimals by a single-digit whole number. The materials state, “Because students will encounter cases of decimal division for which they have not seen the equivalent fraction division, students cannot use fraction division to reason about where to place the decimal point in all cases of decimal division that students will encounter. Thus, it is not included in this lesson since students have not yet performed the corresponding fraction division related to today’s cases. This lesson is analogous to Lessons 1 and 2 but with decimal division. It has been consolidated into one lesson, assuming that students are more easily able to use reasoning about the placement of the decimal point. However, if students would benefit from this lesson being broken up into two days, including if they may benefit from it based on students’ reasoning skills when placing the decimal point in a decimal quotient or based on their procedural comfort with multi-digit division, it can be split into two lessons.”

The materials reviewed for Fishtank Plus Math Grade 5 meet expectations for including standards correlation information that explains the role of the standards in the context of the overall series.  

Correlation information is present for the mathematics standards addressed throughout the grade level/series and can be found in several places, including the course summary standards map, unit summary lesson map, and within each lesson. Examples include:

• In 5th Grade Math, Standards Map includes a table with each grade-level unit in columns and aligned grade level standards in the rows. Teachers can easily identify a unit when each grade level standard will be addressed. 

• In 5th Grade Math, Unit 2, Multiplication and Division of Whole Numbers, Lesson Map outlines lessons, aligned standards and the objective for each lesson. This is present for all units and allows teachers to identify targeted standards for any lesson.

• In Unit 3, Shapes and Volume, Lesson 1, the Core Standard is identified as 5.MD.C.3 and 5.MD.C.4. The Foundational Standard is identified as 3.MD.C.5. Lessons contain a consistent structure that includes an Objective, Common Core Standards, Criteria for Success, Tips for Teachers, Anchor Tasks, Problem Set & Homework, Target Task, and Additional Practice. This provides an additional place to reference standards, and language of the standard, within each lesson.

Each Unit Summary includes an overview of content standards addressed within the unit as well as a narrative outlining relevant prior and future content connections for teachers. Examples include:

• In Unit 1, Place Value with Decimals, Unit Summary includes an overview of how the math of this unit builds from previous work in math. The materials state, “In Grade 4, students developed the understanding that a digit in any place represents ten times as much as it represents in the place to its right (4.NBT.1). With this deepened understanding of the place value system, students read and wrote multi-digit whole numbers in various forms, compared them, and rounded them (4.NBT.2—3).”

• In Unit 3, Shapes and Volume, Unit Summary includes an overview of the Math Practices that are connected to the content in the unit. The materials state, “Throughout Topic A, students have an opportunity to use appropriate tools strategically (MP.5) and make use of structure of three-dimensional figures (MP.7) to draw conclusions about how to find the volume of a figure.”

• In Unit 7, Patterns and the Coordinate Plane, Unit Summary includes an overview of how the content in 5th grade connects to mathematics students will learn in middle grades. The materials state, “This work is an important part of “the progression that leads toward middle-school algebra” (6—7.RP, 6—8.EE, 8.F) (K–8 Publishers’ Criteria for the Common Core State Standards for Mathematics, p. 7). This then deeply informs students’ work in all high school courses. Thus, Grade 5 ends with additional cluster content, but that designation should not diminish its importance this year and for years to come.”

The materials reviewed for Fishtank Plus Math Grade 5 meet expectations for providing explanations of the instructional approaches of the program and identification of the research-based strategies. This information can be found within Our Approach and Math Teacher Tools. Examples where materials explain the instructional approaches of the program include:

• In Fishtank Mathematics, Our Approach, Guiding Principles include the mission of the program as well as a description of the core beliefs. The materials state, “Content-Rich Tasks, Practice and Feedback, Productive Struggle, Procedural Fluency Combined with Conceptual Understanding, and Communicating Mathematical Understanding.” Productive Struggle states, “We believe that students develop essential strategies for tackling complex problems, and build non-cognitive skills such as perseverance and resilience, through productive struggle. Productive struggle happens when students are asked to use multiple familiar concepts and procedures in unfamiliar applications, and the process for solving problems is not immediately apparent. Productive struggle can occur, and should occur, in multiple settings: whole class, peer-to-peer, and individual practice. Through instruction and high-quality tasks, students can develop a toolbox of strategies, such as annotating and drawing diagrams, to understand and attack complex problems. Through discussion, evaluation, and revision of problem-solving strategies and processes, students build interest, comfort, and confidence in mathematics.”

• In Math Teacher Tools, Preparing To Teach Fishtank Math, Understanding the Components of a Fishtank Math Lesson helps to outline the purpose for each lesson component. The materials state, “Each Fishtank math lesson consists of seven key components, such as the Objective, Standards, Criteria for Success, Tips for Teachers, Anchor Tasks/Problems, Problem Set, the Target Task, among others. Some of these connect directly to the content of the lesson, while others, such as Tips for Teachers, serve to ensure teachers have the support and knowledge they need to successfully implement the content.” 

• In Math Teacher Tools, Academic Discourse, Overview outlines the role discourse plays within Fishtank Math. The materials state, “Academic discourse is a key component of our mathematics curriculum. Academic discourse refers to any discussion or dialogue about an academic subject matter. During effective academic discourse, students are engaging in high-quality, productive, and authentic conversations with each other (not just the teacher) in order to build or clarify understanding of a topic.” Additional documents are provided titled, “Preparing for Academic Discourse, Tiers of Academic Discourse, and Strategies to Support Academic Discourse.” These guides further explain what a teacher can do to help students learn and communicate mathematical understanding through academic discourse.

While there are many research-based strategies cited and described within the Math Teacher Tools, they are not consistently referenced for teachers within specific lessons. Examples where materials include and describe research-based strategies:

• In Math Teacher Tools, Procedural Skill and Fluency, Fluency Expectations by Grade states, “The language of the standards explicitly states where fluency is expected. The list below outlines these standards with the full standard language. In addition to the fluency standards, Model Content Frameworks, Mathematics Grades 3-11 from the Partnership for Assessment of Readiness for College and Careers (PARCC) identify other standards that represent culminating masteries where attaining a level of fluency is important. These standards are also included below where applicable. 5th Grade, 5.NBT.5, 5.OA.1, 5.NBT.3, 5.NBT.4, 5.NBT.5, 5.NBT.6, 5.NBT.7, 5.NF.1, 5.NF.4, 5.NF.7, and 5.MD.1, among others.”

• In Math Teacher Tools, Academic Discourse, Tiers of Academic Discourse, Overview states, “These components are inspired by the book Classroom Discussions in Math: A Teacher’s Guide for Using Talk Moves to Support the Common Core and More. (Chapin, Suzanne H., Catherine O’Connor, and Nancy Canavan Anderson. Classroom Discussions in Math: A Teacher’s Guide for Using Talk Moves to Support the Common Core and More, 3rd edition. Math Solutions, 2013.)”

• In Math Teacher Tools, Supporting English Learners, Scaffolds for English Learners, Overview states, “Scaffold categories and scaffolds adapted from ‘Essential Actions: A Handbook for Implementing WIDA’s Framework for English Language Development Standards,’ by Margo Gottlieb. © 2013 Board of Regents of the University of Wisconsin System, on behalf of the WIDA Consortium, p. 50. https://wida.wisc.edu/sites/default/files/resource/Essential-Actions-Handbook.pdf”

• Math Teacher Tools, Assessments, Overview, Works Cited lists, “Wiliam, Dylan. 2011. Embedded formative assessment.” and “Principles to Action: Ensuring Mathematical Success for All. (2013). National Council of Teachers of Mathematics, p. 98.”

The materials reviewed for Fishtank Plus Math Grade 5 meet expectations for providing a comprehensive list of supplies needed to support instructional activities. The 5th Grade Course Summary, Course Material Overview, Course Material List 5th Grade Mathematics states, “The list below includes the materials used in the 5th grade Fishtank Math course. The quantities reflect the approximate amount of each material that is needed for one class. For more detailed information about the materials, such as any specifications around sizes or colors, etc., refer to each specific unit.” The materials include information about supplies needed to support the instructional activities. Examples include:

• Tape or staplers are used in Units 1 and 3, one per group. In Unit 1, Place Value and Decimals, Lesson 1, students work with place value to millions. The materials state, “For this task, the teacher will need paper hundreds flats (20 count) cut to show 1 one, 1 ten, 1 hundred. Students will need a lot of copies of the paper hundreds flats (20 count), cut along the lines to make hundreds, and tape or staples to group the hundreds together.”

• Base-ten blocks are used in Unit 1, maximum of 80 ones, 70 tens, 80 hundreds, and 8 thousands per individual, pair, or group of students.

• Centimeter cubes are used in Unit 3, two hundred per pair or group of students. 

• Cardstock is used in Unit 3, three sheets per student. 

• A pair of scissors is used in Unit 5, one per student. In Unit 5, Multiplication and Division of Fractions, Lesson 1, students model fractions as division using area models. The materials state, “Students may need pieces of paper and scissors for this task (optional: see note below).”

• Markers or crayons are used in Units 3 and 7, three different colors per student. 

• Graph paper is used Unit 7, two sheets per student.

The materials reviewed for Fishtank Plus Math Grade 5 meet expectations for having assessment information included in the materials to indicate which standards and mathematical practices are assessed. 

Mid- and Post-Unit Assessments within the program consistently and accurately reference grade-level content standards and Standards for Mathematical Practice in Answer Keys or Assessment Analysis. Mid- and Post-Unit Assessment examples include:

• In Unit 1, Place Value with Decimals, Expanded Assessment Package, Post-Unit Assessment Analysis denotes content standards addressed for each problem. Problem 10 is aligned to 5.NBT.3b and states, “Select the two correct comparisons. A. 0.057 < 0.008, B. 0.057 < 0.57, C. 0.57 = 0.570, D. 0.57 > 1.001, E. 0.057 < 0.049.”

• In Unit 2, Multiplication and Division of Whole Numbers, Unit Summary, Mid-Unit Assessment, Answer Key denotes Standards for Mathematical Practice addressed for each question. Question 5 is aligned to MP4 and states, “The ground floor of a hotel measures 74 feet long and 126 feet wide. There is 15 times as much square footage in the whole hotel as on the ground oor. What is the total square footage of the hotel? Show or explain your work.”

• In Unit 4, Addition and Subtraction of Fractions/Decimals, Unit Summary, Unit Assessment, Answer Key denotes standards addressed for each question. Problem 6 is aligned to 5.NF.1 and states, “Solve. $7\frac{5}{6}$ + $3\frac{4}{9}$. Select the two correct answers. A. $10\frac{15}{54}$, B. $10\frac{9}{15}$, C. $11\frac{15}{54}$, D. $10\frac{5}{18}$, E. $11\frac{5}{18}$, F. $10\frac{3}{5}$.” 

• In Unit 6, Multiplication and Division of Decimals, Post-Unit Assessment, Answer Key denotes Standards for Mathematical Practice addressed for each question. Question 1 is aligned to MP2 and states, “At a café, the cost of a turkey sandwich is 1 less than twice the cost of a side salad. A side salad costs 3.50. Which of the following expressions can be used to find the cost, in dollars, of a turkey sandwich at the café? A. , B. $3.50\times2+1$, C. $(3.50−1)\times2$, D. $(3.50+1)\times2$.”

• In Unit 7, Patterns and the Coordinate Plane, Unit Assessment Answer Key includes a constructed response and 2-point rubric with the aligned grade-level standard. Problem 5d is aligned to 5.G.2 and states, “The data table below shows the height of a typical meerkat at different times during their first 20 months of life. Decide whether the meerkat grew more from month 0 to month 10 or from month 10 to month 20. Explain your answer.”

The materials reviewed for Fishtank Plus Math Grade 5 meet expectations for including an assessment system that provides multiple opportunities throughout the grade, course, and/or series to determine students’ learning and sufficient guidance to teachers for interpreting student performance and suggestions for follow-up. 

Each lesson provides a Target Task with a Mastery Response. According to the Math Teacher Tools, Assessment Overview, “Target Tasks offer opportunities for teachers to gather information about what students know and don’t know while they are still engaged in the content of the unit.” Each Pre-Unit Assessment provides an answer key and guide with a potential course of action to support teacher response to data. Each Mid-Unit Assessment provides an answer key and a 2-, 3-, or 4-point rubric. Each Post-Unit Assessment Analysis provides an answer key, potential rationales for incorrect answers, and a commentary to support analysis of student thinking. According to Math Teacher Tools, Assessment Resource Collection,“commentaries on each problem include clarity around student expectations, things to look for in student work, and examples of related problems elsewhere on the post-unit assessment to look at simultaneously.” Examples from the assessment system include:

• In Unit 2, Multiplication and Division of Whole Numbers, Pre-Unit Assessment, Problem 2 states, “Translate the following into numbers and symbols. Do not solve. a. Six more than one; b. Five times as much as four; c. The quotient of eight and two; d. Three subtracted from seven.” The Answer Key includes the following teacher guidance, “Translating verbal statements to numeric expressions (K.OA.1, 1.OA.1, 3.OA.1, 3.OA.2, 4.OA.1). Students started to learn about addition and subtraction starting in Kindergarten, developing an understanding of how addition and subtraction situations translate to numerical expressions, and vice versa. This understanding grew to encompass additive comparison, as expressed in Part (a) above as ‘six more than one’, in 1st grade. Simultaneously, their vocabulary to describe addition and subtraction situations grew, including use of terms like ‘sum’, ‘difference’, and others. Then, in 3rd grade, students develop an understanding of how multiplication and division situations translate to numerical expressions, and vice versa. In 4th grade, students expand their understanding of multiplication to include cases of multiplicative comparison, as expressed in Part (b) above as ‘five times as much as four’, in 4th grade. As students deepened their understanding of these operations, their vocabulary to describe them grew as well, such as terms like ‘product’ and ‘quotient’, among others. Students will rely on this understanding to write simple expressions to record calculations with numbers, such as expressing the calculation ‘add 8 and 7, then multiply by two’ as ‘2×(8+7)’, among other possibilities. Notice how this skill is dependent on an understanding of the role of parentheses and the order of operations to record a numerical expression that is actually equivalent to the verbal description. Potential Course of Action If needed, this concept should be reviewed early in the unit, since students will write expressions that record calculations with numbers in Lesson 2.  For example, include a task similar to the one above as a warm-up for Lesson 2. Find problems and other resources in these Fishtank lessons: Grade 3, Unit 2, Lessons 1—5 (Note: these just encompass multiplication and division; adapt these resources for analogous tasks with addition and subtraction).”

• In Unit 6, Multiplication and Division of Decimals, Lesson 24, Target Task, Problem 2 states, “Solve. Show or explain your work. Yolanda took a bus to visit her grandmother. She brought a CD to listen to on the bus. The CD is 78 minutes long. The bus ride was $2\frac{1}{2}$ hours long. How many minutes longer was the bus ride than the CD?” A Mastery Response is provided, which states, “$$2\frac{1}{2}$$ x 1hr = $$2\frac{1}{2}$$ × 60 minutes =120 + 30 minutes = 150 minutes. 150 - 78 = 72. The bus ride was 72 minutes longer than the CD.” 

• In Unit 3, Shapes and Volume, Post Assessment, Problem 9 states, “An art piece is made using two rectangular prisms. The side lengths of the art piece are shown. The length of one of the sides, m, is missing. What is the length of side m? What is the volume of the art piece? Show or explain your work.” The Post-Unit Assessment Answer Key provides the answer, “m=14 feet; the volume is 1,088 cubic feet. See 4-point rubric on the last page.” The 4-Point Rubric states, “4 points - Student response demonstrates an exemplary understanding of the concepts in the task. The student correctly and completely answers all aspects of the prompt. 3 points - Student response demonstrates a good understanding of the concepts in the task. The student arrived at an acceptable conclusion, showing evidence of understanding of the task, but some aspect of the response is flawed. 2 points - Student response demonstrates a fair understanding of the concepts in the task. The student arrived at a partially acceptable conclusion, showing mixed evidence of understanding of the task, with some aspects of the task completed correctly, while others not. 1 point - Student response demonstrates a minimal understanding of the concepts in the task. The student arrived at an incomplete or incorrect conclusion, showing little evidence of understanding of the task, with most aspects of the task not completed correctly or containing significant errors or omissions. 0 points - Student response contains insufficient evidence of an understanding of the concepts in the task. Work may be incorrect, unrelated, illogical, or a correct solution obtained by chance.” 

• In Unit 4, Addition and Subtraction of Fractions and Decimals, Mid-Unit Assessment, Question 4 states, “Elbert is subtracting $1\frac{5}{8}-\frac{2}{3}$. He thinks the answer is $1\frac{3}{8}$ because 5-2 is 3 and 8 is the larger denominator. Explain the error in reasoning Elbert made. Find the correct answer. Be sure to show or explain your work.” The Mid-Unit Assessment Answer Key provides scoring guidance, “Answers may vary, e.g., ‘Elbert’s mistake was that he just subtracted the numerators and used the bigger denominator for the difference. Instead, he could have computed $1\frac{5}{8}-\frac{2}{3}=1\frac{15}{24}-\frac{16}{24}=1-\frac{1}{24}=\frac{23}{24}$.’” The 2-Point Scoring Rubric states, “2 points: Students' response demonstrates an exemplary understanding of the concepts in the task. The student correctly and completely answers all aspects of the prompt. 1 point: Student response demonstrates a fair understanding of the concepts in the task. The student arrived at a partially acceptable conclusion, showing mixed evidence of understanding of the task, with some aspects of the task completed correctly, while others not. 0 points: Student response contains insufficient evidence of an understanding of the concepts in the task. Work may be incorrect, unrelated, illogical, or a correct solution obtained by chance.”

The materials reviewed for Fishtank Plus Math Grade 5 meet expectations for providing assessments that include opportunities for students to demonstrate the full intent of grade-level standards and practices across the series.

The Expanded Assessment Package includes the Pre-Unit, Mid-Unit, and Post-Unit Assessments. While content standards are consistently identified for teachers within Answer Keys for each assessment, practice standards are not identified for teachers or students. Pre-Unit items may be aligned to standards from previous grades. Mid-Unit and Post-Unit Assessments regularly demonstrate the full intent of grade-level content and practice standards through a variety of item types, including multiple choice, short answer, and constructed response. Examples include:

• In Unit 1, Place Value with Decimals, Post-Unit Assessment, Problem 6, supports the full development of MP3 (Construct viable arguments and critique the reasoning of others). The materials state, “Terry made an error while finding the product 0.4 × 100. He writes, When I multiply by ten I just add a zero to the end of my number, so since multiplying by 100 is the same as multiplying by ten twice, I add two zeros to the end of my number. So, 0.4 × 100 = 0.400. Identify Terry’s mistake. Explain what he should do to get the correct answer and include the correct answer in your response.”

• In Unit 4, Addition and Subtraction of Fractions/Decimals, Post-Unit Assessment, Problems 2, 5, and 7 develop the full intent of 5.NBT.7 (Add, subtract, multiply and divide decimals to hundredths, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction). Problem 2 states, “The decimal grids below are shaded to model and expression. What is the value of the expression modeled by the decimal grids? A. 3.29; B. 3.32; C. 4.10; D. 4.13.” Problem 5 states, “Fred is going to the movies. A movie ticket costs $15, and Fred wants to buy a bucket of popcorn for $3.50 and a candy bar for $0.89. Fred has $20. Does Fred have enough money for a movie ticket, a bucket of popcorn, and a candy bar? If so, how much change will he receive? If not, how much more money does he need? Show or explain your work.” Problem 7 states, “Solve. 14.4 - 5.63.” Additionally, in Unit 6, Multiplication and Division of Decimals, Mid-assessment, Problem 2 states, “Multiply or divide. Show or explain your work. a. 8 × 4.35 b. 3.89 ÷ 5.”

• In Unit 5, Multiplication and Division of Fractions, post assessment, Problem 12, supports the full development of MP2 (Reason abstractly and quantitatively, as students interpret multiplication as scaling). The materials state, “The students in Raul’s class were growing grass seedlings in different conditions for a science project. He noticed that Pablo’s seedlings were $1\frac{1}{2}$ times as tall as his own seedlings. He also saw that Celina’s seedlings were $\frac{3}{4}$ as tall as his own. Which of the seedlings below must belong to which student? Explain your reasoning.” 

• In Unit 7, Patterns and the Coordinate Plane, Mid-Unit Assessment, Problem 2 and Post-Unit Assessment, Problems 4 and 5, meet the full intent of 5.OA.3 (Represent real world and mathematical problems by graphing points in the first quadrant of the coordinate plane, and interpret coordinate values of points in the context of the situation). Mid-unit Assessment Problem 2 states, “Sonia and Ahmal plotted the point F on the coordinate grid below. a. Sonia wants to plot a point G so that F and G form a horizontal line. Write an ordered pair that represents where Sonia could plot point G. b. Ahmal wants to plot a point H so that F and H form a vertical line. Write an ordered pair that represents where Ahmal could plot point H.” Post-Unit Assessment, Problem 4 states, “Select the three statements that correctly describe the coordinate system. A. The x- and y-axes intersect at 10. B. The x- and y-axes intersect at the origin. C. The x- and y-axes are parallel number lines. D. The x- and y-axes are perpendicular number lines. E. The x- and y-coordinates are used to locate points on a coordinate plane.” Problem 5 states, “The data table below shows the height of a typical meerkat at different times during their first 20 months of life. A. Graph the data on the grid below. B. How many inches did the meerkat grow between month 4 and month 12? C. How many months did it take for the meerkat to grow from 7 inches to 12 inches?”