5th Grade - Gateway 2

The instructional materials for Match Fishtank Mathematics Grade 5 meet expectations for developing conceptual understanding of key mathematical concepts, especially where called for in specific standards or cluster headings.

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

• In Unit 2, Lesson 13, Anchor Tasks, students 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 (5.NBT.6). In Anchor Tasks, Problem 1 states, “Find the missing side length of each of the rectangles below. Then find their combined length.” Guiding Questions state, “How can you find the missing side length of each rectangle? How can you find the combined side length? How can you represent the side length of the combined rectangles with an expression? What is another way to write an equivalent expression? (Write $$(60\div3)+(9\div3)=69\div3$$.) The quotients of $$60\div3$$ and $$9\div3$$ are called the partial quotients. What did you do with the partial quotients in order to find the total quotient? How can we record what we did to compute $$69\div3$$ vertically? Recording our product vertically using the partial quotients algorithm, how are our area model and our partial quotients similar? How are they different? Recording our quotient vertically using the standard algorithm, how are the area model and the standard algorithm similar? How are they different?”
• In Unit 4, Lesson 4, students add and subtract fractions with unlike denominators (5.NF.1). Anchor Tasks, Problem 3, students “Solve. Show your work with an area model and a number line. $$\frac{1}{3}+\frac{1}{2}$$ .” Guiding Questions state, “Can we add 1 third plus 1 half? What model can we draw to represent each fraction? How can we make like units in our model? What fractional unit have we made for each whole? How many shaded units are in $$\frac{1}{3}$$? How many shaded units are in $$\frac{1}{2}$$? What is our addition sentence now? What is our sum? Can it be simplified?”
• In Unit 6, Lesson 1, students 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. (5.NBT.7). In Anchor Tasks, Problem 2 state, “Solve. Show or explain your work. a. $$2\times0.04=$$ ______ b. $$3\times0.32=$$ ______ c. $$4\times0.67=$$ ______.” 

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

• In Unit 1, Lesson 7, students build numbers to thousandths by dividing by 10 repeatedly (5.NBT.3). In Target Task, Problem 2, students solve, “Jossie drew a picture to represent 0.024: (visual model provided). She said, ‘The little squares represent tenths and the rectangles represent hundredths, which makes sense because ten little squares make one rectangle, and ten times ten is one hundred.’ a. Explain what is wrong with Jossie's reasoning. b. Name two numbers that Jossie's picture could represent. In each case, what does a little square represent? What does a rectangle represent?” 
• In Unit 3, Lesson 1, students “understand volume as an attribute of solid figures that is measured in cubic units. They find the volume of concrete three-dimensional figures” (5.MD.3). In Problem Set, Problem 2, students “Build 2 different structures with the following volumes using your unit cubes. Then, show your classmate or teacher the figures. a. 4 cubic units b. 7 cubic units c. 8 cubic units.”
• In Unit 3, Lesson 2, students “find the volume of pictorial three-dimensional figures,” (5.MD.4). Centimeter cubes are recommended for students to use if they need more concrete experiences with finding volumes of three-dimensional figures. In Problem Set, Problem 3 states, “Find the total volume of each figure below and explain how you found it. Be sure to include units.” Six pictorial representations are provided. Students fill in a table for the calculated volume and an explanation.
• In Unit 5, Lesson 1, students “model fractions as division using area models and solve word problems involving division of whole numbers with answers in the form of fractions or mixed numbers,” (5.NF.3). In Problem Set, Problem 3, students solve, “Six people are sharing four sandwiches. a. Draw a picture to show how they could equally share the sandwiches. How much of a sandwich does each person get? b. Write an equation using division to show the fraction of a sandwich each person gets. Explain how the equation you wrote represents this situation. c. Write an equation involving multiplication to show how all the parts make up the four sandwiches. Explain how the equation you wrote represents the situation. d. Write an equation involving addition to show how together all the parts make up the four sandwiches. Explain how the equation you wrote represents the situation.”
• In Unit 5, Lesson 18, students solve real world problems involving division of unit fractions by whole numbers using tape diagrams and number lines (5.NF.7a). In Anchor Tasks, Problem 1, students solve, “Nolan gives some Fruit-by-the-Foot to his 3 friends to share equally. a. If he has 3 feet of Fruit-by-the-Foot, how many feet of Fruit-by-the-Foot will each friend receive? b. If he has 1 foot of Fruit-by-the-Foot, how many feet of Fruit-by-the-Foot will each friend receive? c. If he has $$\frac{1}{2}$$ foot of Fruit-by-the-Foot, how many feet of Fruit-by-the-Foot will each friend receive? d. If he has $$\frac{1}{3}$$ foot of Fruit-by-the-Foot, how many feet of Fruit-by-the-Foot will each friend receive?”

The instructional materials for Match Fishtank Mathematics Grade 5 for attending to those standards that set an expectation of procedural skill and fluency.

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

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

The instructional materials develop procedural skill and fluency throughout the grade level. The instructional materials provide opportunities for students to demonstrate procedural skill and fluency independently throughout the grade level, especially where called for by the standards (5.NBT.5). For example:

• In Unit 2, Lesson 4, students “Multiply multiples of powers of ten. Estimate multi-digit products by rounding numbers to their largest place value,” (5.NBT.5). In Anchor Tasks, Problem 2 states, “Solve. $$1. 60\times5=$$ ____   2. $$60\times50=$$ ____ 3. $$60\times500=$$ ____ 4. $$60\times5,000=$$ ____.”
• In Unit 2, Lesson 5, students “multiply two-digit, three-digit, and four-digit numbers by one-digit numbers,” (5.NBT.5). Students are provided with multiple opportunities to practice multiplication using the standard algorithm in the Problem Set, Homework, and Target Tasks. In Problem Set, Problem 1 states, “Solve. Then assess the reasonableness of your answer. a. $$45\times6=$$ b. $$32\times5=$$ c. $$67\times3=$$ d. $$324\times2=$$ e. $$106\times4=$$ f. $$624\times8=$$ g. $$9,856\times3=$$ h. $$4,352\times2=$$ i. $$2,781\times7=$$ .”
• In Unit 2, Lesson 8, students “multiply four-digit numbers by two-digit numbers,” (5.NBT.5). Students are provided with multiple opportunities to practice multiplication using the standard algorithm in the Problem Set, Homework, and Target Tasks. In Anchor Tasks, Problem 1 states, “Solve using the standard algorithm. If you get stuck, use an area model and/or the partial products algorithm to help. Then assess the reasonableness of your answer. a. $$1,634\times74$$  b. $$5,803\times46$$ c. $$65\times2,116$$.”
• In Unit 3, Lesson 4, students solve more complex problems involving volume by applying the formula V = b x h (5.MD.5). Students are provided with multiple opportunities to practice multiplication by finding volume in the Problem Set, Homework, and Target Tasks. In the Homework, Problem 2 states, “A rectangular present has a base area of 40 square inches. What is the volume of the box if it is 6 inches tall?”

The instructional materials for Match Fishtank Mathematics Grade 5 meet expectations for being designed so that teachers and students spend sufficient time working with engaging applications of the mathematics. Engaging applications include single and multi-step problems, routine and non-routine, presented in a context in which the mathematics is applied. 

In Problem Sets and Target Tasks, students engage with real-world problems and have opportunities for application, especially where called for by the standards (5.NF.6, 5.NF.7c). The instructional materials include multiple opportunities for students to engage in routine and non-routine application of mathematical skills and knowledge. Students have opportunities to independently demonstrate the use of mathematics flexibly in a variety of contexts. Examples of routine application include, but are not limited to:

• In Unit 3, Lesson 6, students “solve more complex problems involving volume by applying the formula V = l x w x h,” (5.MD.5). For example, Problem Set, Problem 2 states, “Geoffrey builds rectangular planters. Geoffrey’s first planter is 8 feet long and 2 feet wide. The container is filled with soil to a height of 3 feet in the planter. What is the volume of soil in the planter?”
• In Unit 4, Lesson 12, students solve two- and multi-step word problems involving addition and subtraction of fractions (5.NF.2). For example, the Target Task states, “Each student in a class plays one of three sports: soccer, volleyball, or basketball. $$\frac{3}{5}$$ of the number of students play soccer. $$\frac{1}{4}$$ of the number of students play volleyball. What fraction of the number of students play basketball?”
• In Unit 5, Lesson 6, students solve real-world problems involving multiplication of fractions and whole numbers and create real-world contexts for expressions involving multiplication of fractions and whole numbers (5.OA.2, 5.NF.4,6). For example, Problem Set, Problem 1 states, “The table shows the number of computers donated to a school by each of 4 companies. All the donated computers were shared equally by 5 classrooms. Which expression represents the number of companies each classroom received? a. $$120\times\frac{5}{4}$$ b. $$120\times\frac{1}{4}$$ c. $$120\times\frac{4}{5}$$ d. $$120\times\frac{1}{5}$$”
• In Unit 5, Lesson 7, students “multiply a fraction by a fraction without subdivisions using tape diagrams and number lines” (5.NF.6). For example, Target Task, Problem 2 states, “A newspaper’s cover page is $$\frac{5}{6}$$ text, and photographs fill the rest. If $$\frac{2}{5}$$ of the text is an article about endangered species, what fraction of the cover page is the article about endangered species?”
• In Unit 5, Lesson 19, students divide a whole number by a unit fraction (5.NF.7b, 5.NF.7c). For example, Problem Set, Problem 5 states, “Avery and Megan are cutting paper to make origami stars. They need $$\frac{1}{5}$$ of a sheet of paper in order to make each star. If they have 6 sheets of paper, how many stars can they make? Explain your work and draw a picture to support your reasoning.”
• In Unit 5, Lesson 20, students solve real-world problems involving division with fractions and create real-world contexts for expressions involving division with fractions (5.NF.7c). For example, Target Task, 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 7, Lesson 11, students “solve real-world problems by graphing information given as a description of a situation in the coordinate plane and interpret coordinate values of points in the context of the situation,” (5.OA.3, 5.G.2). For example, Target Task  states, “Jordan has $10 in the bank. Jordan earns $5 each week for doing chores, and he puts the money in the bank. After a certain number of weeks of doing chores, Jordan has $35. A graph is set up so that Jordan can record the total amount of money in the bank each week after putting in $5.” Students are given an incomplete coordinate grid (of quadrant 1) and asked, “Part A: Which ordered pair represents the amount of money Jordan has in the bank before doing any chores? Part B: Which ordered pair represents the amount of money Jordan has after 4 weeks of doing chores? Part C: After how many weeks does Jordan have $35? Show or explain your work.” 

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

• In Unit 2, Lesson 3, students write expressions that represent real-world situations and evaluate them (5.OA.1,2). For example, the Target Task states, “Part A: A jar with 64 fluid ounces of water is used to fill cups. The jar is used to fill 3 cups each with 8 fluid ounces of water and 2 cups each with 9 fluid ounces of water. Write an expression that represents the number of fluid ounces left in the jar after filling all of the cups.Then solve your expression. Part B: A different jar has 42 fluid ounces of water. All of the water in the jar is used to fill cups. Write an expression to show how many cups can be filled if each cup is filled with 7 fluid ounces of water. Use p as the unknown number of cups in your question. Do not solve the equation.” 
• In Unit 3, Lesson 8, students “Understand that volume is additive. Find the volume of composite solid figures when not all dimensions are given and/or they must be decomposed. (5.MD.5c). For example, Problem Set, Problem 3 states, “A rectangular tank with a base area of 24 cm square is filled with water and oil to a depth of 9 cm. The oil and water separate into two layers when the oil rises to the top. If the thickness of the oil layer is 4 cm, what is the volume of the water?”
• In Unit 4, Lesson 12, students apply their understanding of adding and subtracting fractions to solve two- and multi-step, real world, word problems (5.NF.2). For example, Problem Set, Problem 5 states, “The table below shows part of the operating budget of a small dairy farm for last year. The only expense not listed in the table is maintenance.” A table of “Last Year’s Operating Budget” is provided. The problem further states, “This year, the managers of the farm will change the fraction of the budget for housing to 18 but will leave the fraction of the budget for food and medical care the same. Again, the remaining portion of the budget will be for maintenance expenses. What is the difference between the fraction of the budget for maintenance this year and last year?”
• In Unit 5, Lesson 18, students divide unit fractions by whole numbers, (5.NF.7c). For example, Target Task, Problem 2 states, “Larry spends half of his workday teaching piano lessons. If he sees 6 students, each for the same amount of time, what fraction of his workday is spent with each student?”
• In Unit 6, Lesson 17, students solve real-world problems involving multiplication and division of decimals (5.NBT.7). For example, Problem Set, Problem 6 states, “Katie went to a craft store to purchase the supplies she needs to make two types of jewelry. This table shows the cost of the supplies Katie needs. (A table with the cost of beads and charms is provided). This table shows the supplies needed to make each piece of jewelry. (A table with the type of jewelry, with how many beads and charms each piece requires is provided.) Katie purchased the exact amount of supplies to make 1 bracelet and 2 necklaces. Write an expression to determine the cost of supplies to make 1 bracelet. Write an expression to determine the cost of supplies to make 2 necklaces. Katie started with $40. How much did she have left after purchasing the supplies?”
• In Unit 6, Lesson 23, students solve real-world problems involving measurement conversions (5.MD.1). For example, Problem Set, Problem 5 states, “Regina buys 24 inches of trim for a craft project. a. What fraction of a yard does Regina buy? b. If a whole yard of trim costs $6, how much did Regina pay?”

The instructional materials for Match Fishtank Mathematics Grade 5 meet expectations that the three aspects of rigor are not always treated together and are not always treated separately. All three aspects of rigor are present in the instructional materials. Many of the lessons incorporate two aspects of rigor with an emphasis on application. Student practice includes all three aspects of rigor, though there are fewer questions for conceptual understanding. There are instances where all three aspects of rigor are present independently throughout the program materials. 

Examples of Conceptual Understanding include:

• In Unit 1, Lesson 12, students “Use place value understanding to round decimals to the nearest whole.” (5.NBT.4). For example, Lesson 12, Anchor Task, Problem 1 states, “Is 7.6 closer to 7 or closer to 8? Plot 7 and 8 on the two outermost spots on the number line below. Then plot 0.6 to prove your answer.”
• In Unit 2, Lesson 6, students “multiply two-digit numbers by two-digit numbers,” (5.NBT.5).  In Anchor Task, Problem 1 states, “Here are two ways to find the area of a rectangle that is 23 units by 31 units. a. What do the area models have in common? How are they alike? b. How are they different? c. If you were to find the area of a rectangle that is 37 units by 49 units, which of the three ways of decomposing the rectangle would you use? Why?”
• In Unit 4, Lesson 14, students “subtract decimals,” (5.NBT.7). In Anchor Task, Problem 2 states, “Solve. Show your work with an area model. a 0.8 – 0.2 b. 0.94 – 0.6 c.0.7 – 0.13.”
• In Unit 5, Lesson 3, students multiply a fraction by a whole number (5.NF.4a,6) using set models. For example, Anchor Task, Problem 1 states, “a. Kyle has 6 Skittles. $$\frac{1}{3}$$ of them are red. How many of Kyle’s Skittles are red?  B. Samantha has 6 Reese’s Pieces. $$\frac{2}{3}$$ of them are yellow. How many of Samantha’s Reese’s Pieces are yellow?”

Examples of Procedural Skills and Fluency include:

• In Unit 2, Lesson 9, students multiply three- and four-digit numbers by three-digit numbers using the standard algorithm, (5.NBT.5). For example, Problem Set, Problem 5 states, “Solve. Show or explain your work. a. $$8,401\times305$$  b. $$7,481\times350$$ c. $$1,346\times297$$ d. $$1,346\times207$$.”
• In Unit 6, Lesson 6, students multiply a decimal to tenths by a decimal to hundredths (5.NBT.7). For example, Target Task, students are given the following equations to solve independently, “1. $$0.35\times0.4$$   2. $$2.02\times4.2$$ 3. $$2.2\times0.42$$.”

Application:

• In Unit 2, Lesson 20, students apply multi-digit multiplication and division when solving word problems (5.OA.1,2, 5.NBT.5,6). For example, the Target Task states, “Sixteen students in a drama club want to attend a play. The ticket price is $35 for each student, and the transportation and meals for everyone will cost $960. To pay for the trip, the students design sweatshirts to sell for a profit of $18 per sweatshirt. If each student sells the same number of sweatshirts, how many sweatshirts must each student sell so that there will be enough money to pay for the entire cost of the trip?”
• In Unit 5, Lesson 6, students “Solve real-world problems involving multiplication of fractions and whole numbers and create real-world contexts for expressions involving multiplication of fractions and whole numbers” (5.OA.2, 5.NF.4,6). For example, the Target Task states, “In a classroom, $$\frac{1}{4}$$ of the students are wearing blue shirts and 23are wearing white shirts. There are $$\frac{3}{6}$$ students in the class. How many students are wearing a shirt other than a blue or white?”
• In Unit 5, Lesson 20, students “solve real-world problems involving division with fractions and create real-world contexts for expressions involving division with fractions,” (5.NF.7c). In Anchor Task, Problem 1 states, “Jenny buys 2 feet of string. If this is one-third the amount she needs to make a bracelet, how many feet will she need? Draw a diagram to represent the problem. Write an expression to represent the problem. Find how many feet of string Jenny needs.”
• In Unit 6, Lesson 17, students apply the procedure of multiplying and dividing with decimals to solving real-world problems (5.NBT.7). For example, Problem Set, Problem 4 states, “Mr. Hower can buy a computer with a down payment of $510 and 8 monthly payments of $35.75. If he pays cash for the computer, the cost is $699.99. How much money will he save if he pays cash for the computer instead of paying for it in monthly payments?”

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

• In Unit 1, Lesson 4, students, “Explain patterns in the number of zeros of the product when multiplying a whole number by powers of 10” (5.NBT.2). For example, the Target Task, Problem 1 states, “Solve. a. $$450\times1,000=$$ ___ b. $$67 \times 10^4=$$ ____.”  Problem 2 states “Explain the pattern in the number of places the digits shift in each of the products above and relate it to the numbers on the left side of the equations.”
• In Unit 6, Lesson 10, students apply the concept of estimation to the procedure of dividing a decimal by a two-digit multiple of ten (5.NBT.1,2,6,7). For example, Anchor Task, Problem 3 states, students are directed to, “Estimate the following quotients. 1. $$39.1\div17$$   2. $$3.91\div17$$.”
• In Unit 7, Lesson 11, students “Solve real-world problems by graphing information given as a description of a situation in the coordinate plane and interpret coordinate values of points in the context of the situation,” (5.G.2, 5.OA.3).  For example, Problem Set, Problem 3 states, “Three chocolate chip cookies cost $4 at the grocery store. a. Create a table and a graph for how much 3, 6, 9, and 12 cookies cost. b. What does the coordinate (9, 12) represent in the context of this problem? c. How many cookies can you buy with $24? Show or explain your work. d. How much would 15 cookies cost? Show or explain your work.”

The instructional materials reviewed for Match Fishtank Grade 5 meet expectations for identifying the Standards for Mathematical Practice and using them to enrich mathematics content within and throughout the grade-level.

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

• In Unit 2, Lesson 11, Tips for Teachers states, “Throughout Lessons 11-19, students see and make use of structure (MP.7) and attend to precision (MP.6) as they decompose numbers into sums of multiples of base-ten units to multiply or divide them.”
• In Unit 4, Lesson 5, Criteria For Success, students “Solve one-step word problems involving the subtraction of two fractions with unlike denominators whose sum is less than 1 (MP4).”
• In Unit 6, the Unit Summary states, “Reasoning about the placement of the decimal point affords students many opportunities to engage in mathematical practice, such as constructing viable arguments and critiquing the reasoning of others (MP.3) and looking for and expressing regularity in repeated reasoning (MP.8). For example, “students can summarize the results of their reasoning as specific numerical patterns and then as one general overall pattern such as ‘the number of decimal places in the product is the sum of the number of decimal places in each factor’” (NBT Progression, p. 20). “

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

• MP8 is connected to the mathematical content in Unit 1, Lesson 4, Criteria for Success, Anchor Task, Problem 1, as students “Generalize the pattern that multiplying a number by a power of 10 results in the digits in the number shifting one place to the left for each power of 10 (MP.8).” For example, “1. Solve. a. $$4\times10=$$ ____  b. $$4\times100=$$ ____ c. $$4\times1,000=$$ ____ . 2. What do you notice about #1? What do you wonder? 3. Use your conclusions form #2 to solve the following equations. a. $$6\times100,000=$$ ____ b. $$78\times1,000,000=$$ ____ c. $$530\times10,000=$$ ____.” 
• MP7 is connected to the mathematical content in Unit 3, Lesson 9, Anchor Task, Problem 2, as students “classify polygons into a hierarchy based on properties (MP.7).” For example, “Sort the polygons from Anchor Task #1 (provided on Template: Polygons) however you’d like. You can create as many groups as you’d like.”
• MP6 is connected to the mathematical content in Unit 6, Lesson 3, Criteria for Success as students use “estimation when the product is difficult to estimate, such as computing $$8\times0.09$$ (MP.6).” For example, Target Task, Problem 1 states, “Use reasoning to choose the correct value for each expression. a. $$0.51\times2$$  (choices include: 0.102, 1.02, 10.2, 102) b. $$4\times8.93$$ (choices include: 3.572, 35.72, 357.2, 3572.)

The instructional materials reviewed for Match Fishtank Mathematics Grade 5 meet expectations for carefully attending to the full meaning of each practice standard. 

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

• MP1: In Unit 5, Lesson 11, Criteria for Success states, “1. Understand when a problem calls for the use of multiplication and when other operations are called for in a problem involving two fractions (MP.1, MP.4).” For example, Anchor Task, Problem 1 states, “Some of the problems below can be solved by multiplying, 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. Two-fifths of the students in Anya’s fifth-grade class are girls. One-eighth of the girls wear glasses. What fraction of Anya’s class consists of girls who wear glasses? b. A farm is in the shape of a rectangle $$\frac{1}{8}$$ of a mile long and $$\frac{2}{5}$$ of a mile wide. What is the area of the farm? c. There is $$\frac{2}{5}$$ of a pizza left. If Jamie eats another $$\frac{1}{8}$$ of the original whole pizza, what fraction of the original pizza is left over? d. In Sam’s fifth-grade class, $$\frac{1}{8}$$ of the students are boys. Of those boys, $$\frac{2}{5}$$ have red hair. What fraction of the class is red-haired boys? e. Only $$\frac{1}{20}$$ of the guests at the party wore both red and green. If $$\frac{1}{8}$$ of the guests wore red, what fraction of the guests who wore red also wore green? f. Alex was planting a garden. He planted $$\frac{2}{5}$$ of the garden with potatoes and $$\frac{1}{8}$$ of the garden with lettuce. What fraction of the garden is planted with potatoes or lettuce? g. At the start of the trip, the gas tank on the car was $$\frac{2}{5}$$ full. If the trip used $$\frac{1}{8}$$ of the remaining gas, what fraction of a tank of gas is left at the end of the trip? h. On Monday, $$\frac{1}{8}$$ of the students in Mr. Brown’s class were absent from school. The nurse told Mr. Brown that $$\frac{2}{5}$$ of those students who were absent had the flu. What fraction of the absent students had the flu? i. Of the children at Molly’s daycare, $$\frac{1}{8}$$ are boys and $$\frac{2}{5}$$ of the boys are under 1 year old. How many boys at the daycare are under 1 year old? j. The track at school is $$\frac{2}{5}$$ of a mile long. If Jason has run $$\frac{1}{8}$$ of the way around the track, what fraction of a mile has he run?”
• MP2: In Unit 3, Lesson 5, Criteria for Success states, “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).” For example, Anchor Task, Problem 3 states, “Akiko and Philip are finding the volume of the following rectangular prism. Philip says that you have to multiply length by width by height, so you have to multiply $$10\times14\times2$$. Akiko says the computation will be easier if you multiply $$10\times2\times14$$. Is Philip correct? Must the dimensions be multiplied in that order? Show or explain your thinking. Why do you think Akiko thinks that multiplying $$10\times2\times14$$ will be an easier computation? Is it possible to multiply the dimensions in that order? Show or explain your thinking. Use what you’ve concluded from Parts (a) and (b) to explain how you would calculate the volume of a rectangular prism whose length is 4 feet, width is 7 feet, and height is 15 feet.”
• MP4: In Unit 5, Unit Summary states, “Students also solve myriad word problems, seeing the strategies they used to solve word problems with whole numbers still apply but that special attention should be paid to the whole being discussed (5.NF.6, MP.4).”  In Lesson 18, Criteria for Success states, “2. Solve partitive division word problems that involve the division of a unit fraction by a whole number (MP.4).” For example, Anchor Task, Problem 2 states, “Alexis has a lot of studying to do over the holiday break. She wants to complete 14of her homework equally over 2 days. What fraction of her homework will she do on each day?”
• MP5: In Unit 5, Lesson 15, Criteria for Success states, “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).” For example, Anchor Task, Problem 1 states, “Sophia and Zack are multiplying $$2\frac{2}{3}$$ and $$3\frac{1}{4}$$. Sophia decides to multiply using partial products, like we did in yesterday’s lesson. Zack decides to convert the mixed numbers to fractions and multiply. Will their strategies result in the same product? Why or why not?”
• MP6: In Unit 5, Lesson 22, Criteria for Success states, “Evaluate expressions with and without grouping symbols that include fractions using the order of operations (MP.6).” For example, Anchor Task, Problem 2 states, “Write numerical expressions that records the following calculations. Then evaluate them. a. Twice the sum of 35 and $$1\frac{1}{2}$$ b. Half the sum of $$\frac{3}{5}$$ and $$1\frac{1}{2}$$. c. $$\frac{1}{2}$$ subtracted from $$\frac{3}{4}$$ and then divided by 3.”
• MP7: In Unit 3, Lesson 3, Criteria for Success states, “Look for and make use of structure to understand that rectangular prisms can be decomposed into layers in different ways (MP.7).” For example, Target Task, Problem 1 states, “Use the figure to the right to answer the following questions. a. 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.”
• MP8: In Unit 6, Unit Summary states, “Reasoning about the placement of the decimal point affords students many opportunities to engage in mathematical practice, such as constructing viable arguments and critiquing the reasoning of others (MP.3) and looking for and expressing regularity in repeated reasoning (MP.8).” For example, Lesson 13, Anchor Task, Problem 3 states, “Solve. Show or explain your work. a. $$8\div0.01$$ b. $$8.3\div0.01$$ c. $$8.37\div0.01$$” Guiding Questions: “How many hundredths are in 8? How is Part (b) similar to Part (a)? How is it different? How is Part (c) similar to Part (b)? How is it different?Do you notice a pattern when you divide a number by one hundredth?”

The instructional materials reviewed for Match Fishtank Mathematics Grade 5 meet expectations for prompting students to construct viable arguments and analyze the arguments of others concerning key grade-level mathematics. 

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

• In Unit 1, Lesson 1, Problem Set, Problem 4 states, “Jose thinks that $$10\times10\times10$$ is 100 because there are two instances of multiplication, and if you shift the place values over two places, you get 100. Do you agree or disagree? Explain.” Students analyze the argument of others (MP.3). 
• In Unit 2, Lesson 10, Target Task, Problem 2 states, “a. Use estimation to explain why Elmer’s answer is not reasonable. b. What error do you think Elmer made? Why do you think he made that error? c. Find the correct product of $$179\times64$$.” Students analyze the argument of others (MP.3). 
• In Unit 3, Lesson 9, Problem Set, Problem 7 states, “Edwin says that a polygon is always a pentagon and his sister says that he has it backwards. Instead, a pentagon is always a polygon. With whom do you agree? Why?” Students analyze the argument of others (MP.3). 
• In Unit 4, Lesson 1, students “Determine whether two fractions are equivalent using an area model, a number line, or multiplication/division (MP.3).” In the Anchor Task, Problem 1 states, “Ms. Kosowsky makes brownies in two pans of the same size. She cuts the pans in the following ways.” Students are shown two rectangles. One is divided in half, and the other is divided into eighths. They are then given the following prompt, “Ms. Kosowsky gives one brownie from Pan A to Ms. Kohler and keeps four brownies from Pan B for herself. Ms. Kohler thinks it isn’t fair since she go one brownie and Ms. Kosowsky got four. Ms. Kosowsky thinks it’s fair. Who do you agree with, Ms. Kosowsky or Ms. Kohler? Why?” Students critique the reasoning of others (MP.3). 
• In Unit 5, Lesson 9, Anchor Task, Problem 2b states, “Presley and Julia are cutting ft. square poster board to make a sign for the new park. Presley cut her poster so that the length of the top and bottom are $$\frac{1}{2}$$ ft. and the length of the sides are $$\frac{3}{4}$$ ft. Julia cut her poster so that the lengths of the top and bottom are 34 ft. and the length of the sides are $$\frac{1}{2}$$ ft. Draw a diagram of each poster board. Label the values on the diagram. How are their poster boards similar and different? Justify your reasoning.” Students construct a viable argument in order to justify their reasoning of how the poster boards are similar and different. 
• In Unit 6, Lesson 1, Problem Set, Problem 3 states, “Miles incorrectly gave the product of $$7\times2.6$$ as 14.42. What is Miles’ mistake? Find the correct value of $$7\times2.6$$. Show your work or explain your answer.” Students analyze the reasoning of others (MP.3).

The instructional materials reviewed for Match Fishtank Mathematics Grade 5 meet expectations for assisting teachers in engaging students to construct viable arguments and analyze the arguments of others concerning key grade-level mathematics.

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

• In Unit 4, Lesson 1, Criteria for Success, “3. Determine whether two fractions are equivalent using an area model, a number line, or multiplication/division (MP.3).” In Anchor Task, Problem 1, students analyze the arguments of others, “Ms. Kosowsky makes brownies in two pans of the same size. She cuts the pans in the following way: (Pan A and Pan B displayed) 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, but are not limited to, “ 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, Lesson 17, Criteria for Success states, “2. Explain why multiplying a given number by a fraction greater than 1 results in a product greater than the given number, recognizing multiplication by whole numbers greater than 1 as a familiar case (MP.3).” In Anchor Task, Problem 3 states, “Cai, Mark, and Jen were raising money for a school trip. Cai collected $$2\frac{1}{2}$$ times as much as Mark. Mark collected $$\frac{2}{3}$$ as much as Jen. Who collected the most? Who collected the least? Explain.” Guiding Questions include but are not limited to, “What can you draw to represent the relationship between how much money Cai raised and how much money Mark raised? Why is it possible to represent this relationship with a model even though we don’t know the exact quantities? What can you add to your model to represent the relationship between how much money Mark raised and how much money Jen raised? Did Cai or Mark raise more money? How do you know? Did Mark or Jen collect more money? How do you know? Did Cai or Jen raise more money? How do you know? Who raised the most money? Who raised the least? What is the relationship between how much money Cai raised and how much money Jen raised? In other words, how many times more money did Cai raise than Jen?”
• In Unit 7, Lesson 8, Criteria for Success states, “1. Given a set of points, plot them in the coordinate grid and identify and explain what shape they form when connected (MP.3).” In Anchor Task, Problem 2 states, “Alonso wants to create a triangle on the coordinate grid below. Two of the vertices are located at point F (2, 3) and point G (7, 3). a. If Alonso wants to make a right triangle, what could be the ordered pair of the third vertex? b. Alonso decides to use his original two points to form a square instead. Where should he place the other two points, and why?” Guiding Questions include, but are not limited to, “What attributes does a right triangle have? How can you be sure that the shape you construct has these attributes? Is there more than one correct right triangle you could create? Can you create a right triangle where is the longest side? Where would the right angle be? How can you be sure it is a right angle? What attributes does a square have? How can you be sure that the shape you construct has those attributes? Where can you plot the other vertices? Is there more than one correct square you could create?”

The instructional materials reviewed for Match Fishtank Mathematics Grade 5 meet expectations for explicitly attending to the specialized language of mathematics.

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

• At the beginning of each unit, the Unit Prep lists the Vocabulary for the unit. For example, in Unit 2, some of the key vocabulary is, “Equation, expression, grouping symbols (parentheses, brackets, braces), etc.”
• In Unit 3, Lesson 6, Tips for Teachers states, “Lesson 6’s 3-Act task addresses the idea of ‘filling’ volume. As the Geometric Measurement Progression states, ‘solid units are ‘packed,’ such as cubes in a three-dimensional array, whereas liquid ‘fills’ three-dimensional space, taking the shape of the container… The unit structure for liquid measurement may be psychologically one-dimensional for some students’ (GM Progression, p. 26).”
• In Unit 4, Lesson 2, Tips for Teachers states, “‘The term ‘improper’ can be a source of confusion because it implies that this representation is not acceptable, which is false. Instead it is often the preferred representation in algebra. Avoid using this term and instead use ‘fraction’ or ‘fraction greater than one’ (Van de Walle, Teaching Student-Centered Mathematics, 3-5, Vol. 2, p. 217). Further, fractions do not always need to be converted from ‘improper’ fractions to mixed numbers, since the need to do so often depends on the context (e.g., in the case of fractional coefficients in algebra, they are often written as fractions greater than one, which are generally easier to manipulate).”

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

• In Unit 1, Lesson 8, Criteria for Success, students will, “Understand that a digit in one place (including decimal places) represents ten times what it represents in the place to its right, 100 times what it represents two places to its right, etc.”
• In Unit 3, Lesson 1,Criteria for Success, students will, “Define volume as the measurement of how much space an object takes up.”
• In Unit 5, Lesson 18, Tips for Teachers states, “There are two interpretations for division, (a) partitive division, also called equal group with group size unknown division, and (b) measurement division, also called equal group with number of groups unknown.”
• In Unit 6, Lesson 19, Criteria for Success, students will, “Understand the meaning of different prefixes for metric units and how they relate to the base unit (e.g., kilo- means 1,000, so a kilogram is 1,000 times as large as a gram).”