6th Grade - Gateway 2

The materials reviewed for Fishtank Plus Math Grade 6 meet expectations for developing conceptual understanding of key mathematical concepts, especially where called for in specific content standards or cluster headings.

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 Problems 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. Examples include: 

• In Unit 2, Unit Rates and Percents, Lesson 8, Anchor Problem 1, students develop conceptual understanding of part to whole ratios. The problem states, "Robb’s Fruit Farm consists of 100 acres on which three different types of apples grow. On 25 acres, the farm grows Empire apples. Macintosh apples grow on 30 acres of the farm. The remainder of the farm grows Fuji apples. Shade in the grid below to represent the portion of the farm each apple type occupies. Use a different color or pattern for each type of apple. What percent of the fruit farm is taken up by each type of apple? Fill in the table below." The table includes columns: Type of Apple, Color in Grid, Part:Whole Ratio, Rate per 100 Acres, % of Fruit Farm. (6.RP.3c)

• In Unit 5, Numerical and Algebraic Expressions, Lesson 6, Anchor Problem 2, students develop conceptual understanding as they match algebraic expressions with their verbal statements. The problem states, “Match each algebraic expression with a verbal statement that describes it. Algebraic Expressions: 1. $$(n+2)^2$$, 2. $$n^2+2^2$$, 3. $$n^2+2$$, 4. $$n+2^2$$. Verbal Statements: a. the sum of n and 2 squared, b. the square of the sum of n and 2, c. the square of n increased by 2, d. the sum of the squares of n and 2.” Teachers use the following Guiding Questions to support the development of conceptual understanding through student discourse, “Compare and contrast the algebraic expressions. How are they similar? How are they different? Mark up each statement on the right. How can that help you match the expressions? Can you think of other verbal statements to describe the algebraic expressions?” (6.EE.2a)

• In Unit 8, Statistics, Lesson 4, Anchor Problem 1, students develop conceptual understanding by describing the center, spread, and overall shape of a data set. The problem states, “Three histograms are shown below. a. Describe the shape of each distribution and explain what it means about the data set. Which graph is skewed left? Skewed right? Symmetrical? b. If these histograms represented the wages that people at a company earned, which company would you want to work at? Why? (Assume the same scale in each graph.)” The following Guiding Questions support discourse and the development of conceptual understanding, “How are the first two graphs similar? How are they different from the third graph? Which graph would you describe as symmetrical? Why? What features make it symmetrical? A skewed distribution has values that are not typical of the rest of the data. These skewed data points can be on the low or the high end. Which graph would you say is skewed left (is skewed toward the smaller values or has a ‘tail’ to the left)? Which graph would you say is skewed right (is skewed toward the larger values or has a ‘tail’ to the right)?” Teacher notes state, “In explaining their choice for part (b), students should use structural features of the distributions in constructing their arguments. They should not only explain why they chose one specific histogram, but also why they did not choose the other two.” (6.SP.2)

Materials provide opportunities for students to independently demonstrate conceptual understanding throughout the grade level. Problem Sets, or student practice 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. Both problem types, when appropriate, provide opportunities for students to independently demonstrate conceptual understanding. Examples include:

• In Unit 4, Rational Numbers, Lesson 4, Target Task, students develop conceptual understanding of integers and their opposites as they analyze four written statements. It states, “Determine if each statement below is sometimes, always, or never true. Explain your reasoning for each statement. a. The opposite of a number is 0. b. The opposite of a negative number is positive. c. The opposite of the opposite of a positive number is negative. d. If two numbers are on opposite sides of 0, then they are opposites.” (6.NS.6)

• In Unit 6, Equations and Inequalities, Lesson 11, Problem Set, Problem 6, students solve a one-step inequality and explain what the solution means as it relates to the problem. It states, “Verena has at most 90 minutes to read 60 pages for homework. How many pages does Verena need to read per minute to stay within her time limit? Write and solve an inequality. Explain what the solution means in the context of the problem.” (6.EE.6, 6.EE.8)

• In Unit 7, Geometry, Lesson 11, Problem Set, Problem 5, students develop conceptual understanding as they reason about the relationship between the sides and volume. It states, “A rectangular prism is shown below. If you use the formula V= B x h to find the volume of the prism, does it matter which face of the prism you use as the base? Explain your reasoning.” (6.G.2)

The materials reviewed for Fishtank Plus Math Grade 6 meet expectations for giving attention throughout the year to individual standards that set an expectation of procedural skill and fluency.

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 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.” Examples Include:

• In Unit 3, Multi-digit and Fraction Computation, Unit Summary, describes how students build fluency within the unit and across the year. It states, “Throughout this unit, students will develop, practice, and demonstrate fluency with decimal operations; however, practice and demonstration opportunities should continue throughout the year with the goal of fluency by the end of the year. Several opportunities are already built into future units, such as the unit on Expressions and the unit on Equations, but additional opportunities need to be planned for and included.“ (6.NS.A, 6.NS.B)

• In Unit 3, Multi-Digit and Fraction Computation, Lesson 7, Anchor Problem 3, students evaluate each solution using the standard algorithm for addition and subtraction of multi-digit decimals. The problem states, “The two problems below have been solved incorrectly. Explain the error in each problem and then correctly solve each one.  244.038 + 18.65, 95.75 − 37.424. 244.038 + 18.65 = 245.903, 95.75 - 37.42 = 58.334.” (6.NS.3)

• In Unit 5, Numerical and Algebraic Expressions, Lesson 4, Anchor Problem 3, students evaluate expressions by replacing the variable in expressions, then simplifying. The problem states, “Evaluate the algebraic expressions for the given values of the variables. a. x^3+18y^2 where x is \frac{1}{2} and y is \frac{1}{3} b. 2(x^2-1)+4y, where x is 6 and y is 3. Guiding Questions for teachers which support student reflection about procedural execution in the Anchor Problem include, “Describe the structure of each expression. What do you see happening? What is the numerical expression you get once you substitute in the values for the variables? How will you evaluate your numerical expression? What will you do first and why? What role do parentheses play in these examples?” (6.EE.2)

• In Unit 7, Geometry, Lesson 7, Anchor Problem 3, students calculate area and perimeter with an irregular polygon. The problem states, “Find the area and perimeter of the polygon in the coordinate plane below.” An irregular polygon is provided on a coordinate plane. Guiding Questions for teachers which support student reflection about procedural execution in the Anchor Problem include, “How can you use composition or decomposition to find the area? What is an efficient way to find the perimeter?” (6.G.3)

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

• In Unit 4, Rational Numbers, Lesson 5, Problem Set, Problem 5, students plot rational numbers on a number line, building fluency with these numbers. The problem states, “Use the number line below to plot the location of each point. Label each point with its letter. Point A: 2\frac{1}{2} Point B: -\frac{3}{4} Point C: -\frac{7}{2} Point D: -4\frac{1}{4} Point E: \frac{17}{4}.” (6.NS.6)

• In Unit 5, Numerical and Algebraic Expressions, Lesson 8, Problem Set, Problem 1, students generate equivalent expressions, building fluency within the grade. The problem states, “Write an algebraic expression equivalent to each of the ones below. A. x + x + x + 3 + 3 + 3 B. y + y − 6 − 6 C. 2m + 2m + 2m + 2m + 8 + 8 + 8 + 8 D. n + n + 2n − 4 − 4 − 8.” (6.EE.3, 6.EE.4)

• In Unit 8, Statistics, Lesson 7, Target Task, students calculate and analyze measures of center by analyzing data in a histogram. The task states, “The histogram below represents the number of minutes some students studied for a science quiz.” The histogram shows “number of minutes” from 0 to 30 and “number of students” from 0 to 18. Students respond to the following: “a. How many students are represented in the histogram? b. Determine the median range of minutes that students spent studying for the quiz. c. Determine the mode range of minutes that students spent studying for the quiz. d. Explain why you cannot determine the mean number of minutes students studied for the quiz.” (6.SP.2)

The materials reviewed for Fishtank Plus Math Grade 6 meet expectations for being designed so that teachers and students spend sufficient time working with engaging applications of the mathematics.

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

• In Unit 1, Understanding and Representing Ratios, Lesson 3, Anchor Problem 1, students solve a routine real-world problem by defining and identifying equivalent ratios (6.RP.1). The problem states, “On Saturday morning, you decide to make pancakes for your family. To make a batch of pancakes, your recipe calls for 1 cup of milk and 2 cups of flour. a. Draw a diagram to represent the ratio of milk to flour in one batch of pancakes. b. Your sister invites some friends to your house for pancakes. You decide that you need to make 2 batches of pancakes. Draw a diagram to represent the flour and milk needed for 2 batches. Write a ratio statement. c. Your neighbors hear that you’re making pancakes and they come over as well. Now you need to make 3 batches of pancakes to feed everyone. Draw a diagram and write a ratio statement to represent the flour and milk needed for 3 batches. d. In general, if you were to make c batches of pancakes, then how many cups of milk and flour would you need? What ratio statement would describe this?”

• In Unit 2, Unit Rates and Percent, Lesson 1, Anchor Problem 2, students use ratio reasoning to solve a non-routine real-world problem (6.RP.3). The problem states, “On your way home from school, you stop at the corner store and pick up some eggs and apple juice. However, you don’t need as many eggs or juice cans as come in the package. The cashier lets you buy only part of the package and pay only for what you buy. About how much is your total bill?” Students are provided two images. Image one is five sodas with a price tag that says $3.29 for six-pack. Image two shows seven eggs with a price tag that says $2.20 a dozen. 

• In Unit 3, Multi-digit and Fraction Computation, Lesson 1, Anchor Problem 2, students write and solve routine multiplication and division problems for given scenarios (6.NS.1). The problem states, “For each problem, write a division and a multiplication problem to represent the situation. Then solve the problem and explain what it means. a. You make 10 cups of pudding and pour it equally into 6 containers. How many cups of pudding are in each container? b. You mix yellow and blue paint to make 126 ounces of green paint. You pour the green paint into some smaller buckets, with 18 ounces in each one. How many buckets did you fill? c. A recipe for pancakes calls for 3 cups of flour for one batch. If you have 11 cups of flour, how many batches of pancakes can you make? d. A florist is arranging 13 dozen flowers into vases. Each vase can hold 15 flowers. How many vases can be completely filled? How many more flowers are needed to fill the last vase?”

• In Unit 8, Statistics, Lesson 6, Anchor Problem 3, students create a data set to fit a given set of constraints in a non-routine problem (6.SP.2, 6.SP.5C). The problem states, “Create a data set of at least 10 numbers such that: All of the numbers in the data set are whole numbers. The median is not a whole number. The median is not part of the data set.” 

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

• In Unit 3, Multi-Digit and Fraction Computation, Lesson 5, Target Task, students solve and write non-routine problems involving division with fractions (6.NS.1). The task states, “There are other ways to think about division of fractions. Try these two questions. They both use division, but why? And how do you know what to divide by what? 1. Analise’s hair grew $$1\frac{3}{4}$$ inches in $$3\frac{1}{2}$$ months. On average, how many inches did Analise’s hair grow per month? 2. Write your own story problem that involves division of fractions. Then solve your problem.”

• In Unit 4, Rational Numbers, Lesson 3, Problem Set, Problem 3, students create and write situations for positive and negative numbers in a non-routine problem (6.NS.5). The problem states, “Create and write a situation for each of the integers in the table below.” Numbers are: 35, -8, -500.”  

• In Unit 6, Equations and Inequalities, Lesson 7, Target Task, students solve routine multi-part equations leading to the form  x + p  = q and px = q (6.EE.6, 6.EE.7). The task states, “A town's total allocation for firefighters’ wages and benefits in a new budget is $600,000. If wages are calculated at $40,000 per firefighter and benefits at $20,000 per firefighter, write an equation whose solution is the number of firefighters the town can employ if they spend their whole budget. Solve the equation.” 

• In Unit 7, Geometry, Lesson 17, Problem Set, Problem 4, students find surface area in a routine real-world problem (6.G.2, 6.G.4). The problem states, “Lulu was asked by her math teacher to find the surface area of a storage box in their classroom so they can paint it. The storage box is in the shape of a rectangular prism and measures 5 feet long by 3.25 feet wide by 2.5 feet tall. If the bottom of the storage box does not need to be painted, how many square inches should Lulu tell her teacher need to be painted?”

The materials reviewed for Fishtank Plus Math Grade 6 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 6. Examples where instructional materials attend to conceptual understanding, procedural skill and fluency, or application include:

• In Unit 2, Unit Rates and Percents, Lesson 14, Target Task, students use rates and unit rates to solve application problems. The task states, “For lunch, you order a pizza and drink some milk. The pizza is cut into 8 slices, and you eat 3 of the slices. The pizza box says that the serving size is 2 slices, and there are 570 calories per serving. The milk is in a quart container, and you drink 1 cup of it. The milk carton says that there are 580 calories in the entire quart. How many calories did you eat for lunch? If you are trying to stick to a limit of 2,000 calories for the day, then what percent of your daily calories did you just eat for lunch?” (6.RP.2: Understand the concept of a unit rate a/b associated with a ratio a:b with b ≠ 0 and 6.RP.3: Use ratio and rate reasoning to solve real-world and mathematical problems.)

• In Unit 5, Numerical and Algebraic Expressions, Lesson 1, Target Task, students develop procedural skill and fluency as they evaluate equations with exponents. The task states, “Decide whether each equation is true or false, and explain how you know. a. $$2^4=2⋅4$$, b. $$3+3+3+3+3=3^5$$, c. $$5^3=5⋅5⋅5$$, d. $$2^3=3^2$$, e. $$16^1=8^2$$, f. $$(1+3)^2=1^2+3^2$$ g. $$2⋅2⋅2⋅3⋅3⋅3=6^3$$.” (6.EE.1: Write and evaluate numerical expressions involving whole-number exponents.)

• In Unit 8, Statistics, Lesson 11, Problem Set, Problem 2, students develop conceptual understanding as they use box plots to represent data. The problem states, “Several students, from middle and high-school, attended a 6th grade basketball game. The list below shows the ages of twenty students who attended the game. 11, 11, 12, 12, 12, 12, 12, 12, 12, 13, 13, 13, 13, 14, 14, 15, 15, 16, 17, 18. Create a box plot to represent the data.” (6.SP.4: Display numerical data in plots on a number line, including dot plots, histograms, and box plots, and 6.SP.5: Summarize numerical data sets in relation to their context.)

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

• In Unit 1, Understanding and Representing Ratios, Lesson 6, Problem Set, Problem 5, students develop conceptual understanding alongside application as they analyze a double number line and solve ratio problems. The problem states, “Use the double number line below to answer the questions that follow. a. Write a complete sentence that describes the relationship between water and lemonade powder that is used to make lemonade. b. How much lemonade powder is needed if 10 cups of water is used? c. How much water is needed if 12 scoops of lemonade powder is used?” One line of the double number line is labeled water (cups) 0, 2, 4, 6 and the other line is labeled lemonade powder (scoops) 0, 1.5, 3, 4.5. (6.RP.3: Use ratio and rate reasoning to solve real-world and mathematical problems.)

• In Unit 3, Multi-Digit and Fraction Computation, Lesson 3, Target Task, students develop conceptual understanding alongside application as they use a model to divide a whole number by a fraction in a real-world problem. The task states, “A jar has 5 tablespoons of honey in it. One serving of honey is $$\frac{3}{4}$$ of a tablespoon. How many servings of honey are in the jar? Draw a diagram to solve the problem and explain how your diagram shows the solution.” (6.NS.1: Interpret and compute quotients of fractions, and solve word problems involving division of fractions by fractions, e.g., by using visual fraction models and equations to represent the problem.)

• In Unit 7, Geometry, Lesson 1, Anchor Problem 1, students develop conceptual understanding alongside procedural skill and fluency as they experiment with GeoGebra and generalize a strategy to find the area of a parallelogram. The problem states, “A parallelogram is shown below. a. What strategies could you use to find the area of the parallelogram? b. Follow Steps 1–4 of this GeoGebra applet Area of Parallelogram to explore the area of parallelograms. Try out different parallelograms by moving the red and blue dots. c. In general, how can you find the area of any parallelogram?” (6.G.1: Find the area of right triangles, other triangles, special quadrilaterals, and polygons by composing into rectangles or decomposing into triangles and other shapes; apply these techniques in the context of solving real-world and mathematical problems.)

The materials reviewed for Fishtank Plus Math Grade 6 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 or specific lessons (criteria for success, tips for teachers, or anchor problem 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, Unit Rates and Percents, Lesson 9, students solve percent problems using benchmark fractions. Anchor Problem 3 states, “Answer the questions below using mental math. Use your double number lines from Anchor Problem 1 if you need to. 5 is 50% of what number? 12 is 10% of what number? 15 is 25% of what number? 30 is 75% of what number? 400 is 20% of what number? A coat is on sale for 10% off. You save $18. What was the original price of the coat?” Tips for Teachers state, “The problems in this lesson all lend themselves toward being answered using mental strategies and do not necessarily require the use of a tool. Knowing how to work with these benchmark percentages will support students in reflecting on the reasonableness of their answers (MP1).” 

• In Unit 5, Numerical and Algebraic Expressions, Lesson 10, students use the distributive property to rewrite expressions. Anchor Problem 2 states, “An area diagram is shown below. Write two equivalent expressions to represent the shaded area in the diagram.” Teachers Guiding Questions include, “Describe what you see in the diagram in your own words. What are the dimensions of the outer rectangle? What are the dimensions of the two smaller rectangles? How can you find the area of the shaded rectangle by subtracting from the larger outer rectangle? Check your expressions by substituting in a value for x.” Teacher Notes provide additional guidance, “If students struggle with the introduction of a variable, they can revisit Anchor Problem 1 and make connections between the value of 50 and the value of x, or they can simplify the problem by replacing the variable with a value first. (MP.1)” 

• In Unit 7, Geometry, Lesson 12, students apply volume concepts for rectangular prisms to solve a real world problem. Problem Set, Problem 5 states, “Lorvo completely fills a bucket in the shape of a rectangular prism with water. His little sister, Loana, bumps into the bucket and some water splashes out. The picture below shows the bucket after the water splashed out.” A rectangular prism is shown with dimensions 5 in, 6 in, and 8 in. The prism is filled nearly to the top. Students respond to the following: “a. If 15 in^3 of water splashed out, what is the volume of water remaining in the bucket? b. What is the height of the water, after the water splashes out, as shown in the picture above?” 

• In Unit 8, Statistics, Lesson 5, students determine a fair share payment by analyzing data from a table. Anchor Problem 1 states, “A company hires five people for the same job for one week. The amount that each person is paid for the week is shown in the table below. Person D states that the payments are not fair since each person is doing the same job and brings the same set of skills to the job. Everyone agrees that they should all get paid the same amount. How much should each person get paid so that everyone gets the same amount? Assume that the company will spend the same amount as it currently is.” The table shows Person A $360, Person B $340, Person C $300, Person D $200, and Person E $400. Teacher Notes provide additional guidance, including, “Look for ways that students are trying out different approaches to make sense of the problem and working collaboratively as a group. For example, are any students simplifying the problem to consider how they would solve it if only 2 or 3 workers were involved? How are students using the play money to act out the situation? How are students checking the reasonableness of their group’s answers? Identify and showcase different examples to the whole class (MP.1)” 

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 1, Understanding and Representing Ratios, Lesson 5, students use reasoning to identify equivalent ratios on a double number line. Anchor Problem 3 states, “To make green-colored water, Brian mixes drops of green food dye and cups of water in a ratio of 4:3. a. Draw a double number line to represent the ratio of drops of green food dye to cups of water. b. Use your double number line to find 2 equivalent ratios. c. Brian’s friend, Evan, uses a ratio of 20 drops of green food dye to 15 cups of water. Will Evan’s water be the same color green as Brian’s? Explain your reasoning” Notes for the teacher provide additional guidance, including, “Students engage in MP.2 in this problem, representing the context on a double number line, and then interpreting the information in the double number line to determine if Brian and Evan have made the same color green. For students who struggle with this problem, check to see where their understanding may have broken down, either in decontextualizing or re-contextualizing.” 

• In Unit 2, Unit Rates and Percents, Lesson 11, students reason about percent in a real-world problem. Anchor Problem 3 states, “Nicky bought a hardcover book for $24. She went to an event where the author signed her book. The book is now worth 175% of its original value. How much is the signed copy of the book worth?” Tips for Teachers notes state, “In Lessons 11 – 13, students engage in MP.2 as they make sense of how the part, percent, and whole relate to one another, and how they can use ratio reasoning to find any one of the values when given the other two.” 

• In Unit 6, Equations and Inequalities, Lesson 3, students write equations for real-world situations. Target Task states, “Hodan is planting flowers around her apartment building. The total distance around her building is 120 feet, and she wants to plant a flower every 4\frac{1}{2} feet. Let x represent the number of flowers Hodan plants around her apartment building. Write an equation she can use to determine how many flowers she’ll need.” Tips for Teachers states, “In this lesson, students encounter real-world problems and either identify or write equations to represent the situations. This requires students to abstract the situations and use symbols in place of verbal descriptions (MP.2).” 

• In Unit 8, Statistics, Lesson 8, students reason about measures of central tendency. Anchor Problem 1 states, “Bobbie is a sixth grader who competes in the 100-meter hurdles. In eight track meets during the season, she recorded the following times (to the nearest one hundredth of a second). 18.11, 31.23, 17.99, 18.25, 17.50, 35.55, 17.44, 17.85. a. What is the mean of Bobbie’s times for these track meets? What does the mean tell you in terms of the context? b. What is the median of Bobbie’s times? What does the median tell you in terms of the context? c. Explain why the mean is higher than the median. d. Which measure of center do you think best represents Bobbie’s 100-meter hurdle time? Explain your reasoning. e. The two times that were over 30 seconds were times when Bobbie fell on a hurdle. She decides to take those times out to get a sense of her typical time without falling. What is the new mean? The new median? How do those compare to the original mean and median?” Teacher Notes provide additional guidance, including, “Students engage in MP.2 in this problem by contextualizing the measures of center they find in parts (a) and (b). They also must take into account all of the values in the data set to determine why one measure of center may be more representative as a typical value than the other.”

The materials reviewed for Fishtank Plus Math Grade 6 meet expectations for supporting the intentional development of MP3: Construct viable arguments and critique the reasoning of others, for students, in connection to the grade-level content standards, as expected by the mathematical practice standards. MP3 is explicitly identified for teachers within the unit summary or specific lessons (criteria for success, tips for teachers, or anchor problem notes) and students engage with the full intent of the MP through a variety of lesson problems and assessment items.

Students construct viable arguments and critique the reasoning of others in connection to grade-level content as they work with support of the teacher and independently throughout the units. Examples include:

• In Unit 3, Multi-Digit and Fraction Computation, Lesson 6, students construct viable arguments and critique the reasoning of others as they use division of fractions to solve problems. Problem Set, Problem 6 states, “Mai, Clare, and Tyler are hiking from a parking lot to the summit of a mountain. They pass a sign that gives distances. Parking lot: $$\frac{3}{4}$$ mile. Summit: $$1\frac{1}{2}$$ miles. Mai says: ‘We are one third of the way there.’ Clare says: ‘We have to go twice as far as we have already gone.’ Tyler says: ‘The total hike is three times as long as what we have already gone.’ Do you agree with any of them? Explain your reasoning.”

• In Unit 7, Geometry, Mid-Unit Assessment, students critique the reasoning of others and construct a viable argument as they reason about the area of a parallelogram. Problem 3 states, “Mateo and Sanjay are comparing their work to find the area of the parallelogram below. Mateo says you have to split the parallelogram into a rectangle and two triangles, find the area of each piece, and add them together. Sanjay says Mateo’s method will work, but he can find the area in fewer steps. a. What could be Sanjay’s method for finding the area of the parallelogram? b. Explain why Sanjay’s method works.”

• In Unit 8, Statistics, Lesson 1, students construct viable arguments and critique the reasoning of others as they identify and describe statistical questions. Anchor Problem 2 states, “Which of the following are statistical questions? (A statistical question is one that can be answered by collecting data and where there will be variability in that data.) a. How many days are in March? b. How old is your dog? c. On average, how old are the dogs that live on this street? d. What proportion of the students at your school like watermelon? e. Do you like watermelon? f. How many bricks are in this wall? g. What was the temperature at noon today at City Hall?” Guiding Questions provide support to the teacher for development of MP3, “How are questions B and C similar? How are they different? Which question is a statistical question? For which of the questions would you expect the same response? What is another example of a statistical question? What is another example of a non-statistical question? How could you change the non-statistical questions to be statistical ones?” Problem Notes state, “This is a good opportunity to have students work in pairs in order for each student to have a chance to explain why a question is/is not a statistical question and to listen to other students explain their reasoning as well (MP.3).”

The materials reviewed for Fishtank Plus Math Grade 6 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 problem notes).

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 1, Understanding and Representing Ratios, Lesson 17, Anchor Problem 1, students model with math as they solve complex ratio problems. The problem states, “At a concert, the ratio of the number of boys to the number of girls is 2:7. If there are 250 more girls at the concert than boys, how many boys are at the concert?” Tips for Teachers suggest, “Students encounter more challenging real-world problems in this lesson providing them with the opportunity to apply their ratio reasoning and strategies to model and solve the problems (MP.4). Tape diagrams are an effective tool to use to solve these problems, however, students may also use other strategies of their choice.” 

• In Unit 2, Unit Rates and Percent, Lesson 14, Problem Set, Problem 1, students model a real- world situation with a percent. The problem states, “A jet plane can carry up to 200,000 liters of fuel. It used 130,000 liters of fuel during a flight. What percentage of the fuel capacity did it use on this flight?” 

• In Unit 8, Statistics, Lesson 2, Target Task, students model with math as they represent data in dot plots. The task states, “A city planner for a neighborhood of Boston is trying to determine if a new park should be built in the neighborhood. She goes around the neighborhood, knocking on doors, asking, ‘How many children under the age of 7 live in this household?’ The data she collected from 28 households is shown in the frequency table below. a. Create a dot plot to represent the data. Describe your dot plot in words. b. What number would you use to represent the center of the data? Explain why you chose that number. c. Based on the data, what would you recommend to the city planner about the new park? Does this neighborhood seem like a good place to put the new park? Explain why or why not using the data.” Unit 8 Summary states, “Students learn various ways to represent the data, including frequency tables, histograms, dot plots, box plots, and circle graphs, and they analyze each representation to determine what information and conclusions they can glean from each one (MP.4).” 

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 1, Understanding and Representing Ratios, Unit Assessment, Problem 8, after learning about diagrams, double number lines, ratio tables and tape diagrams, students solve ratio problems using any strategy. The problem states, “The community garden in your neighborhood has three different types of vegetable plants: peppers, tomatoes, and cucumbers. The ratio of the number of peppers to tomatoes to cucumbers is 1:5:3. If there are 48 cucumber plants in the garden, how many total vegetable plants are there in the garden? Use any method to solve. Show your work.”  

• In Unit 4, Rational Numbers, Lesson 12, Anchor Problem 1, students reason about reflections and the signs of ordered pairs. The problem states, “Two pairs of coordinate points are shown in the table below. For each pair of points: a. Locate and label the points in the coordinate plane. b. Record your observations for each row in the table. (3,4) and (-3,4); (3,4) and (3,-4): How are the coordinate points similar? How are the coordinate points different? What do you notice about their locations? How far is each point from each axis? Which axis could you fold along to make the points match up?” Guiding Questions for the teacher support the development of MP5, including, “What is the absolute value of each x−coordinate? Of each y−coordinate? What does it mean if two points are symmetrical about an axis? How far is each point from the axis? Would the points (−2,5) and (2,5) line up vertically or horizontally? Which axis would you fold along to match (−2,5) with (2,5)?” Problem Notes for the teacher state, “Introduce the idea of a reflection by visualizing a fold along the axis. For additional support, patty paper (transparency paper) can be used (MP.5).” 

• In Unit 7, Geometry, Lesson 3, Problem Set, Problem 2, students use strategies as tools to find the area of acute triangles. The problem states, “In each triangle below, one side has been identified as the base, b. Draw a line segment in each triangle that represents the height and label it h.” Four different triangles are provided. Students use tools such as rulers, index cards, or other strategies.

The materials reviewed for Fishtank Plus Math Grade 6 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 1, Understanding and Representing Ratios, Mid-Unit Assessment, students attend to precision when they write ratios describing a real-world situation. Problem 1 states, “Jaden combined blueberries and raspberries in a bowl. For every 3 tablespoons of blueberries he added to the bowl, he also put in 2 tablespoons of raspberries. What is the ratio of blueberries to raspberries in the bowl? a. 2:3 b. 3:2 c. 3:5 d. 5:2.”  

• In Unit 3, Multi-Digit and Fraction Computation, Lesson 13, students attend to precision when computing with decimals. Problem Set, Problem 3 states, “40 ounces of granola costs $6.52. a. How much does granola cost per ounce? (Give your answer to the nearest cent.) b. How much will 4.5 ounces cost? (Give your answer to the nearest cent.) Criteria for Success 2 states, “Apply the standard algorithms for adding, subtracting, multiplying, and dividing with decimals to solve problems accurately and efficiently (MP.6).” 

• In Unit 7, Geometry, Lesson 9, students attend to precision as they use appropriate symbols and units of measurement with real-world problems involving distance, area, and perimeter of polygons on and off the coordinate plane. Anchor Problem 2 states, ”You are responsible for a small plot of land that measures 9 ft. x 9 ft. in the community garden. You want to include 48 square feet of gardening space, and you want your garden to have a rectangle shape and a triangle shape. Draw a possible plan for your garden on the grid below. Make sure you do not go outside of the 9 ft. x 9 ft. space." 

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. Examples include:

• In Unit 2, Unit Rates and Percent, Lesson 3, students use the specialized language of mathematics as they find unit rates and use them to explain real world problems. Anchor Problem 1 states, “The grocery store sells beans in bulk. The grocer's sign above the beans says, ‘5 pounds for $4.’ At this store, you can buy any number of pounds of beans at this same rate, and all prices include tax. Alberto said, ‘The ratio of the number of dollars to the number of pounds is 4:5. That's $0.80 per pound.’ Beth said, ‘The sign says the ratio of the number of pounds to the number of dollars is 5:4. That's 1.25 pounds per dollar.’ Are Alberto and Beth both correct? Explain.”

• In Unit 4, Rational Numbers, Lesson 11, students use the specialized language of mathematics and knowledge of the structure of a coordinate plane to plot and describe coordinate points. Target Task states, “Use the coordinate plane provided below to answer questions below. Label the second quadrant on the coordinate plane, and then answer the following questions. a. Write the coordinates of one point that lies in the second quadrant of the coordinate plane. b. What must be true about the coordinates of any point that lies in the second quadrant? c. Plot and label the following points: A: (-4, 6) B: (6,-4) C: (0,-2) D: (2,0) E: (-5,-3).” Tips for Teachers state, “A common misconception students may have prior to extending beyond the first quadrant is that an x-coordinate indicates how far to move right, and a y-coordinate indicates how far to move up. Discuss how the addition of negative values in an ordered pair create the need for a more precise way to describe movement in the coordinate plane (MP.6).” 

• In Unit 8, Statistics, Unit Assessment, students use the specialized language of mathematics as they investigate and describe measures of center and variability. In Problem 4, Part C states,  “Which statements about the measures of center are true? Select two correct answers. a. There is a mode of this data set and it is 22 minutes. b. In any data set, the mean is always a value in the data set. c. In any data set, the median is always a value in the data set. d. In any data set, the mode is always a value in the data set. e. In any data set, the median and mode are always the same value.” 

The materials reviewed for Fishtank Plus Math Grade 6 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 1, Understanding and Representing Ratios, Lesson 13, students make use of structure as they analyze information in tables in order to compare ratios. In Problem Set, Problem 3 states, “The tables below show three different recipes for making orange juice using orange concentrate and water. Order the recipes from strongest orange flavor to weakest orange flavor. Justify your answer.” Recipe 1 Table shows 3, 6, 9, 10 Cups of orange con. and 5, 10, 15, 16$$\frac{2}{3}$$ Cups of water. Recipe 2 table shows 2, 4, 6, 8 Cups of orange con. and 3, 6, 9, 12 Cups of water. Recipe 3 table shows 5, 10, 15, 20 Cups of orange con. and 8, 16, 24, 32 Cups of water. 

• In Unit 5, Numerical and Algebraic Expressions, Lesson 2, students make use of structure as they use the order of operations to evaluate numerical expressions. Target Task states, “Evaluate the expressions. a. 24 ÷ (2 + 1) + (11 −8) × 2. b. 3^2× 4−10+(12−3)^2. c.  (3×4)^2−10×(12−3^2).” Unit 5 Summary states, “Throughout this unit, students pay close attention to the structure of expressions, understanding the role of parentheses, the order of operations, and the way expressions are described verbally (MP.7).” 

• In Unit 8, Statistics, Lesson 11, students make use of structure as they describe and create representations for statistical data sets. Anchor Problem 1 states, “A dot plot is shown below. Use it to answer the questions that follow. a. What statistical question could this data be answering? b. How many people were surveyed? c. Find the five-number summary of the data set, and draw a vertical line through each of those values on the dot plot. d. Draw a box plot for the data on the number line below. How is it similar to the dot plot? How is it different?” Problem Notes for the teacher state, “Discuss the structures of the two graphical representations with students. How is the same information shown in different ways in the two plots? What information can you see clearly in each representation? What are some advantages/ disadvantages of using either? (MP.7)” 

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 2, Unit Rates and Percents, Lesson 9, students use repeated reasoning and look for patterns when solving percent problems. Anchor Problem 2 states, “Answer the questions below using mental math. Use your double number lines from Anchor Problem 1 if you need to. What is \frac{1}{2} of 20? What is 50% of 20? What is 10% of 60? What is 20% of 60? What is \frac{1}{4} of 48? What is 25% of 48? What is 75% of 48? What is 10% of 125? A textbook that costs $36 is on sale for 25% off. How much money will you save?”  Problem Notes for teachers state, “If students struggle with doing this mentally, start by asking similar questions but with using fractions instead of percentages. You can also use repetition in your questions to support students in seeing the patterns and structure; for example, asking what is 10% of five different numbers in a row (MP.8).” 

• In Unit 3, Multi-Digit and Fraction Computation, Lesson 4, students use repeated reasoning with visual models in order to develop a general rule for division with fractions. Anchor Problem 1 states, “The number 3 is divided by unit fractions \frac{1}{2}, \frac{1}{3}, \frac{1}{4}, and \frac{1}{5}. For each division problem, draw a visual model to represent the problem and to find the solution. Then complete the rest of the chart (headings: Division Problem, Visual Model, Quotient, Multiplication Problem) and answer the questions that follow. What pattern do you notice? What generalization can you make? Explain your reasoning.” Guiding Questions which support the development of MP.8 include, “State each division problem as a question asking “How many _s in ?” Explain how your visual model represents each division problem. What impact does changing the denominator have on each visual model? On the quotient? Where do you see the quotient in your model? What multiplication pattern do you notice? Why does it make sense? Complete this sentence: Dividing by a unit fraction is the same as _________.”  

• Unit 4, Rational Numbers, Unit Assessment, students use repeated reasoning and make generalizations about integers on a number line throughout the unit. Problem 3 provides the opportunity for students to apply those strategies on the unit assessment. The problem states, “Point A and point B are placed on a number line. Point A is located at -20 and point B is 5 less than point A. Which statement about point B is true? a. It is located at -25 and is to the right of point A on the number line. b. It is located at -15 and is to the right of point A on the number line. c. It is located at -25 and is to the left of point A on the number line. d. It is located at -15 and is to the left of point A on the number line.”