5th Grade - Gateway 2

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 24cm^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?”