The materials reviewed for Fishtank Plus Math Grade 8 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, Solving One-Variable Equations, Lesson 1, Anchor Problem 3, students write equivalent expressions using properties of operations and verify equivalence using substitution. It states, “Write an expression for the sequence of operations. Then simplify each expression. Add 3 to x, subtract the result from 1, then double what you have. Add 3 to x, double what you have, then subtract 1 from the result.” The following Guiding Questions support discourse and the development of conceptual understanding, “How are the directions for each example similar? How are the directions for each example different? What is the difference between ‘subtract the result from 1’ and ‘subtract 1 from the result’? Where do you need parentheses? Why do you need them? Are the two expressions equivalent? How do you know?” (8.EE.7)

• In Unit 3, Transformations and Angle Relationships, Lesson 4, Anchor Problem 1, students reason about reflections on the coordinate plane and demonstrate a conceptual understanding as they describe the reflections. It states, “Figure ABCDEF is shown in the coordinate plane below. Trace the figure on a piece of patty paper, then use it to investigate the questions that follow.” A figure on a coordinate plane is shown with vertices A(1,4), B(1,1), C(2,1), D(2,3), E(3,3), F(3,4). A point, X, is plotted at (-3,4) and a point Y, is plotted at (3,-4) “a. Use your patty paper to reflect the figure so that point F maps to point X. Draw the reflected image in the coordinate plane. How would you describe the reflection?  b. Use your patty paper to reflect the original figure so that point F maps to point Y. Draw the reflected image in the coordinate plane. How would you describe the reflection? c. What impact does reflecting an image have on its orientation? How is this different from a translation?” (8.G.2)

• In Unit 8, Bivariate Data, Lesson 7, Anchor Problem 2, students reason about bivariate data from a two-way table. It states, “A reporter in a small town polled some residents and asked them if they were in favor of increasing the minimum wage or against it. The two-way table summarizes the data. a. Name two things you notice and two things you wonder about. b. How many people did the reporter survey? c. How many people in the age group of 16–30 years old were against the increase in minimum wage? d. Were more people in favor of the increase or against the increase?” The following, Guiding Questions support discourse and the development of conceptual understanding, “Is this numerical bivariate data or categorical bivariate data? What does the number 52 represent? What does the number 28 represent? Explain to a peer how to read and interpret the data in this table.” (8.SP.4)

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 2, Solving One-Variable Equations, Lesson 3, Target Task Problem 1, students demonstrate conceptual understanding as they justify each step when solving a multi-step equation with variables on one side of the equation. It states, “Solve the equation below. For each step, explain why each line of your work is equivalent to the line before it. $$\frac{1}{2}(-12x+4)+5x=\frac{2}{3}(24)$$.” (8.EE.7a, 8.EE.7b)

• In Unit 4, Functions, Lesson 1, Problem Set, Problem 7, students identify and describe a functional relationship from tables of information. It states, “Molly and Daylon both created tables to represent functions. Did both students create a function? Explain why or why not for each student.” Two tables are provided, one for Molly that shows a function, and one for Daylon that shows a non-function relationship. (8.F.1)

• In Unit 7, Pythagorean Theorem and Volume, Lesson 6, Problem Set, Problem 4, students explain the relationship between sides of a right triangle using the Pythagorean Theorem. It states, “Gina drew a right triangle and labeled the side lengths x, y, and z, as shown below.” The triangle shown includes a hypotenuse labeled y and side lengths x  and z. The problem then states, “Gina wrote the equation $$z^2=x^2+y^2$$ to represent the relationship between the side lengths of the triangle. Did Gina write a correct equation? Explain why or why not.” (8.G.6)

The materials reviewed for Fishtank Plus Math Grade 8 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 1, Exponents and Scientific Notation, Lesson 2, Anchor Problem 2, students evaluate an expression and determine common student errors. The problem states, “Evaluate the following expression: $\frac{2-4(-2-1)^2}{\frac{1}{2}^2}$ Name 3 common errors that might be made when evaluating this expression. Guiding Questions for teachers which support student reflection about procedural execution in the Anchor Problem include, “What do you notice about the structure of the expression? What step will you do first? Is this the only first step you can take, or could you have started in a different way? How many exponents do you see and what is the base for each one?” According to the Unit Summary, “Later in the unit, they learn efficient ways to describe, communicate, and operate with very large and very small numbers. Though there are many procedural elements in this unit, underneath these procedures are strong conceptual understandings.” (8.EE.1)

• In Unit 2, Solving One-Variable Equations, Unit Summary, describes how students build procedural skill and fluency from a conceptual foundation within the unit. It states, “In Unit 2, eighth-grade students hone their skills of solving equations and inequalities. They encounter complex-looking, multi-step equations, and they discover that by using properties of operations and combining like terms, these equations boil down to simple one- and two-step equations. Students also discover that there are many different ways to approach solving a multi-step equation, and they spend time closely looking at their own work and the work of their peers. When solving an equation with variables on both sides of the equal sign, students are challenged with results such as 4=5, and they refine their definition of ‘solution’ to include such examples.” (8.EE.C)

• In Unit 4, Functions, Lesson 2, Anchor Problem 1, students analyze a function table to find a rule for the given inputs and outputs. The problem states, “A function machine takes an input and, based on some rule, produces an output. The tables below show some input-output pairs for different functions. For each table, describe a function rule in words that would produce the given outputs from the corresponding inputs. Then fill in the rest of the table values as inputs and outputs that are consistent with that rule. a. Input values can be any English word. Output values are letters from the English alphabet. b. Input values can be any rational number. Output values can be any rational number.” Two tables are provided. Guiding Questions for teachers which support student reflection about procedural execution in the Anchor Problem include, “How does seeing the input/output visual help you understand functions? What rule is determining the output values in each table? Is it possible for more than one output to be generated from any of the inputs? Why is each table a function? How could you change one of the tables to make it not a function?” (8.F.1)

• In Unit 6, Systems of Linear Equations, Lesson 3, Anchor Problem 3, students inspect a linear equation in order to create a system with a given number of solutions. The problem states, “Consider the equation $y=\frac{2}{5}x+1$. Write a second linear equation to create a system of equations that has: a. Exactly one solution. b. No solutions, c. infinite solutions.” (8.EE.8b)

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 1, Exponents and Scientific Notation, Lesson 7, Problem Set, Problem 4, students use exponent rules to identify an expression that is not equivalent. The problem states, “Which of the following is NOT equivalent to $x^6$? A. $\frac{x^8}{x^2}$ B. $x^4⋅x^2$ C. $x^{12}⋅x^{-6}$ D. $\frac{x^4}{x^2}$” (8.EE.1)

• In Unit 3, Transformations and Angle Relationships, Lesson 20, Problem Set, Problems 1-6, students use the interior angle sum theorem for triangles to calculate missing angles or determine if a set of angles could form a triangle. Problem 2 states, “Triangle EFG is shown below.” Isosceles triangle is shown with angle F shown as 37 degrees. It further states, “Which are possible measurements for ∠E and ∠G? A. m∠E = 43$$\degree$$, m∠G = 90$$\degree$$ B. m∠E = 53$$\degree$$, m∠G = 100$$\degree$$ C. m∠E = 63$$\degree$$, m∠G = 70$$\degree$$ D. m∠E = 83$$\degree$$, m∠G = 60$$\degree$$.” (8.G.5)

• In Unit 7, Pythagorean Theorem and Volume, Lesson 11, Target Task, students solve mathematical problems using the Pythagorean Theorem. The task states, “A rectangular prism is shown below. Use the given information to determine the exact length of $\bar{DF}$. $\bar{AB}=8$ units. $\bar{BF}=6$ units. $\bar{FG}=12$ units.” (8.G.7)

The materials reviewed for Fishtank Plus Math Grade 8 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, Exponents and Scientific Notation, Lesson 1, Anchor Problem 4, students apply the properties of integer exponents to generate equivalent numerical expressions in a routine application problem (8.EE.1). The problem states, “You cut a piece of paper in half to get $$\frac{1}{2}$$ of the paper. You repeat this four more times. What fraction of the paper do you have after these five cuts? What fraction of the paper do you have after 15 cuts? Write your answers as exponentials.”

• In Unit 3, Transformations and Angle Relationships, Lesson 15, Anchor Problem 2, students use scale factor to analyze and create similar figures in a non-routine problem (8.G.4). The problem states, “Are triangles ABC and EFG similar? Explain your answer. Sketch a triangle that is similar to either triangle ABC or triangle EFG. Label the side lengths.” Triangle ABC is shown with side lengths of 3, 5, and 6. Triangle EFG is shown with side lengths of 5, 7, and 8. 

• In Unit 5, Linear Relationships, Lesson 15, Anchor Problem 2, students write a linear equation to model a non-routine real world problem (8.F.4). The problem states, “You have $100 to spend on a barbeque where you want to serve chicken and steak. Chicken costs $1.30 per pound and steak costs $3.50 per pound. You want to know how many pounds of chicken and steak you can afford to buy. a. Write and graph an equation that relates the amount of chicken and the amount of steak you can buy. b. What is the meaning of each intercept in this context? c. What is the meaning of the slope in this context? d. Discuss what your options are for the amounts of chicken and steak you can buy for the barbeque.” 

• In Unit 8, Bivariate Data, Lesson 6, Anchor Problem 1, students interpret the slope and y-intercept of a linear model in a routine real world problem (8.SP.3). The problem states, “At a restaurant, the amount of tip for the waitress or waiter is automatically calculated at 20% of the bill total. The graph below shows the amount a tip would be for 6 different bill totals. 1. Write an equation to represent the amount of tip based on the bill total. 2. What is the rate of change and what does it represent in the context of the problem? 3. What is the initial value and what does it represent in the context of the problem? 4. If you had a bill that came to $75.80, then how much money will you pay for the tip?”

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 2, Solving One-Variable Equations, Lesson 10, Target Task students write an equation representing fixed costs and earnings in a non-routine problem (8.EE.7). The task states, “Create your own business model, similar to Anchor Problem 1. What are you selling? What are your fixed costs? Your variable costs? How much will you sell each item for? How many do you need to sell to break even? Show all of your work clearly and include an equation in your answer.” 

• In Unit 4, Functions, Lesson 12, Problem Set, EngageNY Mathematics, Grade 8 Mathematics > Module 6 > Topic A > Lesson 5 — Exercises 3–5, Problem Set 3–7, Problem Set, Problem 7, students use qualitative descriptions and sketch functions for non-routine problems (8.F.5). The problem states, “Using the axes in Problem 7(b), create a story about the relationship between two quantities. a. Write a story about the relationship between two quantities. Any quantities can be used (e.g., distance and time, money and hours, age and growth). Be creative! Include keywords in your story such as increase and decrease to describe the relationship. b. Label each axis with the quantities of your choice, and sketch a graph of the function that models the relationship described in the story.” 

• In Unit 6, Systems of Linear Equations, Lesson 5, Problem Set, Problem 5, students analyze a given solution strategy for a routine problem involving a system of equations (8.EE.8b). The problem states, “Chad is solving the system of equations below. y = x + 3, y = -2x + 84. He plans to solve the system by graphing because he notices that both equations are written in slope-intercept form, y = mx + b. Do you think Chad’s plan is a good idea? Explain your reasoning.”

• In Unit 7, Pythagorean Theorem and Volume, Lesson 10, Target Task, students use the Pythagorean Theorem to solve a routine real-world problem (8.G.7). The problem states, “Kendrick is interested in purchasing a new television. He has picked out a specific space on a wall on which to mount the television. The wall space measures $$1\frac{3}{4}$$ feet tall and 3 feet wide. Sizes of televisions are given in inches and describe the diagonal length from the top corner of the television to the opposite bottom corner. Can Kendrick fit a 42-inch television on the space that he has picked out? Explain your reasoning.”

The materials reviewed for Fishtank Plus Math Grade 8 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 8. Examples where instructional materials attend to conceptual understanding, procedural skill and fluency, or application include:

• In Unit 2, Solving One-Variable Equations, Lesson 9, Anchor Problem 3, students develop conceptual understanding as they reason about the solutions to equations without solving them. The problem states, “Without solving them, say whether these equations have a positive solution, a negative solution, a zero solution, or no solution. a.3x = 5, b. 5z + 7 = 3, c. 7 − 5w = 3, d. 4a = 9a, e. y = y + 1.” (8.EE.7a: Give examples of linear equations in one variable with one solution, infinitely many solutions, or no solutions. Show which of these possibilities is the case by successively transforming the given equation into simpler forms, until an equivalent equation of the form x = a, a = a, or a = b results [where a and b are different numbers].)

• In Unit 4, Functions, Lesson 7, Target Task, students apply their understanding of functions as they identify and graph linear and nonlinear functions. The task states, “Water flows from a hose at a constant rate of 11 gallons every 4 minutes. The total amount of water that flows from the hose is a function of the number of minutes you are observing the hose. a. Write an equation that describes the amount of water, y, in gallons, that flows from the hose as a function of the number of minutes, x, you observe it. b. Use the equation you wrote in part (a) to determine the amount of water that flows from the hose during an 8-minute period, a 4-minute period, and a 2-minute period. c. The input of the function, x, is time in minutes, and the output of the function, y, is the amount of water that flows out of the hose in gallons. Write the inputs and outputs from part (b) as ordered pairs, and plot them as points on the coordinate plane. d. Is the function linear or nonlinear? Explain your answer.” (8.F.3: Interpret the equation y = mx + b as defining a linear function, whose graph is a straight line; give examples of functions that are not linear.)

• In Unit 7, Pythagorean Theorem and Volume, Lesson 12, Problem Set, Problem 4, students develop procedural skill and fluency as they calculate the distance between points on a coordinate plane using the Pythagorean Theorem. The problem states, “Rank the distances between the following pairs of points in the coordinate plane from smallest to largest. A coordinate plane is provided below if needed.a. (0, 0) and (5, 3), b.(1, 3) and (4, 7), c. (0, 2) and (4, 4), d. (−2, 0) and (3, 2).” (8.G.8: Apply the Pythagorean Theorem to find the distance between two points in a coordinate system.)

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, Exponents and Scientific Notation, Lesson 6, Anchor Problem 4, students develop conceptual understanding alongside procedural skill and fluency as they apply exponent rules to analyze equivalent expressions. The problem states, “How is $$7^27^6$$ different from $$(7^2)^6$$? What is a simplified expression for each one? Use your reasoning to simplify the following:  $$(11^5)^4$$,  $$-(2^3)^6$$,  $$(-1^3)^{12}$$.” (8.EE.1: Know and apply the properties of integer exponents to generate equivalent numerical expressions.)

• In Unit 3, Transformations and Angle Relationships, Lesson 9, Problem Set, Problem 2, students develop conceptual understanding alongside application as they use their knowledge of multiple rigid transformations to describe a sequence of transformations. The problem states, “Stephanie performed a transformation on line segment XY. She recorded the original coordinate points of the line segment and the coordinate points of the line segment after the transformation. Her peer, Jon, looked at the chart she made and asked her if she performed a reflection over the y-axis because the y- coordinates all changed sign. How should Stephanie respond to Jon’s question?” A chart provided shows Point X Original (1,3) and Image (1,-3) and Point Y Original (5,-2) and Image (5,2). (8.G.3: Describe the effect of dilations, translations, rotations, and reflections on two-dimensional figures using coordinates.)

• In Unit 8, Bivariate Data, Lesson 1, Target Task, students develop conceptual understanding alongside application as they analyze and interpret data from scatter plots. The task states, “In the game of Monopoly, as you move around the board, the rental costs of properties with one hotel change. The scatter plot below shows some information about these rental costs compared to the number of spaces you are from the starting place, ‘GO.’ 1. What are the two variables in the scatter plot? 2. What is the cheapest rent for a Monopoly property with one hotel? What is the most expensive?  3. Are there any hotels that have the same rental price but are a different number of spaces from ‘GO’? If so, name the rent cost and number of spaces from ‘GO.’ 4. Do you notice a relationship between the data? Explain.” The scatter plot shows Number of Spaces from “GO” on the x-axis and Rent Cost with Hotel on the y-axis and a positive linear relationship. (8.SP.1: Construct and interpret scatter plots for bivariate data to investigate patterns of association between two quantities.)

The materials reviewed for Fishtank Plus Math Grade 8 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 1, Exponents and Scientific Notations, Lesson 15, students use scientific notation and properties of exponents to solve multi-step problems. Target Task states, “A new movie is being released and it is expected to be a blockbuster. Use the information below to predict how much money the movie will make in ticket prices over the opening weekend. The average movie ticket price in the country is $9. There are approximately $4×10^4$ movie screens in the country. The movie will be playing on 30% of the screens in the country on opening weekend. For each screen, there is an average of 250 seats, and it is expected that each showing will be sold out. There are 8 showings per screen over the opening weekend. Write your answer in scientific notation and standard form.” Criteria for Success states, “1. Outline a solution pathway for a multi-step problem (MP.1) 2. Understand the relationships between numbers in a problem in order to determine how you will use them (MP.1).” 

• In Unit 3, Transformations and Angle Relationships, Lesson 13, students make sense of problems as they reason about dilations and similar figures. Problem Set, Problem 6 states, “Parallelogram P is shown in the coordinate plane below.” A parallelogram is graphed on a coordinate grid. The vertices are labeled A(2,-1), B (5,-1), C(4,-3), D(1,-3). Students respond to the following: “Parallelogram P is dilated from point D using a scale factor of 3. Which of the following are true statements? Select all that apply. A. $\bar{A’B’}$ is 3 units long B. $\bar{C’D’}$ is 9 units long C. m∠A'is equal to the m∠A D. m∠B'is 3 times the m∠B E. $\bar{A’B’}$ is parallel to $\bar{C’D’}$F. The perimeter ABCD of is equal to the perimeter of A’B’C’D’.” 

• In Unit 5, Linear Relationships, Lesson 3, students compare proportional relationships represented by graphs. Anchor Problem 1 states, “The graphs below show the cost y of buying x pounds of fruit. One graph shows the cost of buying x pounds of peaches, and the other shows the cost of buying x pounds of plums.” A graph is shown with the x-axis labeled number of pounds and the y-axis labeled cost. There are two lines graphed; one labeled peaches and one labeled plums; the peaches line is steeper than the plum line. Students respond to the following: “a. Which kind of fruit costs more per pound? Explain. b. Bananas cost less per pound than peaches or plums. Draw a line alongside the other graphs that might represent the cost, y, of buying x pounds of bananas.” Notes after the Guiding Questions provide additional guidance for teachers, “Use this Anchor Problem to engage students in comparing graphs of proportional relationships in the same coordinate plane. Students should reason how the steepness or shallowness of the line helps them interpret the slope and the unit rate for each fruit in order to compare (MP.7). If students struggle with the abstract nature of the graph, have them label a few values to find concrete costs, in particular values corresponding to costs per 1 pound of each fruit (MP.1).” 

• In Unit 7, Pythagorean Theorem and Volume, Lesson 16, students solve real-world problems involving three-dimensional shapes, including cylinders, cones, and spheres. Problem Set, Problem 4 states, “A coin-operated bouncy ball dispenser has a large glass sphere that holds many spherical balls. The large glass sphere has a radius of 9 inches. Each bouncy ball has radius of 1 inch and sits inside the dispenser.If there are 243 bouncy balls in the large glass sphere, what proportion of the large glass sphere’s volume is taken up by bouncy balls? Explain how you know.” One Criteria for Success states, “Map out a solution pathway and use relevant formulas and math concepts to solve complex, real-world problems (MP.1 and MP.4).” 

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, Solving One-Variable Equations, Lesson 4, students solve multi-step equations. Target Task, Problem 1 states, “Given a right triangle, find the measures of all the angles, in degrees, if one angle is a right angle and the measure of the second angle is six less than seven times the measure of the third angle.” Problem 2 states, “School uniform shirts normally cost $15 each but are on sale for 30% off the original price. You also have a coupon for $10 off the cost before you take the percent discount. If you have $150 to spend, which of the following equations will help you to determine how many shirts, x, you can buy? a. 0.7(15x) − 10 = 150, b. 0.7(15x − 10) = 150, c. 15x − 0.7(10) = 150. d. 15(0.7x − 10) = 150.” Criteria for Success states, “Decontextualize a situation to represent it algebraically, and re-contextualize to interpret the solution in context of the problem.”

• In Unit 5, Linear Relationships, Lesson 8, students graph linear equations using slope intercept form. Anchor Problem 2 states, “For each linear equation below, identify the slope and y-intercept and use them to graph the line a. $y=\frac{1}{3}x-2$, b. y = -3x + 4, c. y = -x” . Guiding Questions state, “Which lines have a negative slope and will increase up toward the left? Which lines have a positive slope and will increase up toward the right? Which line(s) will be steep? Shallow? Which lines have a positive y-intercept and will cross the y-axis above 0? Which lines have a negative y-intercept and will cross the y-axis below 0? Do any lines represent a proportional relationship? Explain. Pick a coordinate point on each line and substitute it into the equation to check that it is a solution.” Teacher Notes provide additional guidance, “There is more than one way to use the slope to find additional points on the line. For example, the slope $\frac{1}{3}$ can be used as up 1 and right 3 or as down 1 and left 3. Having a firm understanding of what slope means as the measure of vertical change over the measure of horizontal change will help students use slope in a flexible way in the coordinate plane (MP.2). Highlight different student approaches where possible.” 

• In Unit 6, Systems of Equations, Lesson 7, students make sense of quantities as they write and solve systems of equations. Anchor Problem 1 states, “Farmer Joe has cows and chickens on his farm. One day he counts 76 legs and 24 heads. How many cows and how many chickens are on the farm? Write and solve a system of equations.” Guiding Questions state, “What are the two variables, or unknowns, that you are solving for? What does 76 represent? How does that relate to the variables you identified? How many legs does each cow have? Each chicken? What does 24 represent? How does that relate to the variables you identified? How many heads does each cow have? Each chicken? Is it easier to solve this system using substitution or graphing?” Teacher notes provide additional guidance, “Students must make sense of the quantities involved in this problem in order to write equations. They must think, how are 76 legs represented by cows and chickens? How are 24 heads represented by cows and chickens? How can I represent this mathematically? (MP.2) Ensure students define the variables they use in solving the problem and completely answer the question by providing the correct units. When solving applications where systems of equations are used, it’s common for students to find the values of the variables but not re-contextualize to explain what the solution means for the problem at hand (MP.2).” 

• In Unit 8, Bivariate Data, Lesson 1, students analyze data in a scatter plot. Problem Set, Problem 1 states, “Ciro is studying the temperature of water. He determines the temperature of different bodies of water at different depths and represents each as a point in the graph below. Use the plot to answer the questions that follow. a. Which point represents the greatest depth? b. Which point represents the coldest temperature? c. Which two points represent the same temperature? d. Which two points represent the same depth?” A graph is shown with temperature on the y-axis and depth on the x-axis. There are five points on the graph labeled A through E. Criteria for Success states, “Interpret ordered pairs (x,y) in scatter plots in context of the variables (MP.2).”

The materials reviewed for Fishtank Plus Math Grade 8 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 1, Exponents and Scientific Notation, Unit Assessment, students construct viable arguments and critique the reasoning of others when they simplify expressions with exponents. Problem 13 states, “Jude incorrectly simplified the expression $(\frac{1}{2})^2×\frac{1}{2}×\frac{1}{2}^3$, as shown below. $(\frac{1}{2})^2×\frac{1}{2}×\frac{1}{2}^3=\frac{1}{262,144}$ Describe the mistake that Jude made. In your explanation, provide the correct way to simplify the expression.”

• In Unit 3, Transformations and Angle Relationships, Lesson 12, students construct viable arguments and critique the reasoning of others as they describe and perform dilations. Problem Set, Problem 7 states, “Use Rectangle ABCD in the grid below to answer the questions that follow. [A rectangle is shown on a coordinate grid with vertices at A(2,2), B(2,4), C(6,4), D(6.2).] Create similar rectangle A’B’C’D’ by dilating Rectangle ABCD using a center of dilation at point (0,0) and a scale factor of 2. Create similar rectangle A’’B’’C’’D’’ by dilating Rectangle ABCD  using a center of dilation at point (0,0) and a scale factor of $\frac{1}{2}$. Lonnie thinks that Rectangle A’B’C’D’ and Rectangle A’’B’’C’’D’’ are also similar figures. Do you agree? Explain your reasoning. If you think they are also similar, then provide a center of dilation and scale factor in your response.”

• In Unit 6, System of Linear Equations, Lesson 2, students construct viable arguments and critique the reasoning of others as they solve systems of equations. Problem Set, Problem 6 states, “Use the graph below to answer the following questions. Line m is graphed for you. a. Phillip says that the equation for line m is y = 16x + 1. Is she right? Explain why you agree or disagree. b. Create an equation for line n that meets the following criteria: Line m and line n create a system of equations that has a solution at (0, 1). Line n has a slope between -1 and -2. Line n has a positive y-intercept.”

The materials reviewed for Fishtank Plus Math Grade 8 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, identifying important quantities to make sense of relationships, and representing them mathematically. They model with math as they work with support of the teacher and independently throughout the units. Examples include:

• In Unit 2, Solving One-Variable Equations, Lesson 7, Anchor Problem 1, students model with math as they write and solve multi-step equations to represent situations. “You have a coupon worth $18 off the purchase of a scientific calculator. At the same time, the calculator is offered with a discount of 15%, but no further discounts may be applied. For what tag price on the calculator do you pay the same amount for each discount?” Guiding Questions for the teacher state, “Describe the two different discounts. Which one do you think is the better deal? How does the price of the calculator change the value of each discount? What expression represents each discount? What equation models the situation? What does each part of the equation represent?” Criteria for Success states, “Use expressions and equations to model and solve real-world situations (MP.4).”

• In Unit 5, Linear Relationships, Unit Assessment, Problem 4, students model a linear relationship with an equation. “The graph below shows a relationship between x and y. Which of the following equations best represents this relationship? a.y = 2x b. y = x + 2 c. $y=\frac{1}{2}x+2$ d. $y=2x+\frac{1}{2}$.” A graph with a line passing through points (-4,0), (0,2), and (4,4) is shown. 

• In Unit 7, Pythagorean Theorem and Volume, Lesson 11, Problem Set, Problem 5, students use the Pythagorean Theorem to model a real-world problem. The problem states, “A spider walks on the outside of a box from point A to B to C to D and finally to point E as shown in the picture below a. Draw a net of the box and map out the path of the spider on the net. b. How long is the path of the spider?” Tips for Teachers state, “Lessons 10 and 11 engage students in real-world and mathematical problems that can be modeled and solved using the Pythagorean Theorem. In these two lessons, students tackle problems involving a race and speed, and students will see how to apply the Pythagorean Theorem in three dimensions (MP.4). Depending on time, these two lessons can be combined into one lesson for a longer class period, or they can be kept as two separate lessons.” 

MP5: Use appropriate tools strategically, is identified and connected to grade-level content, and there is intentional development of the MP to meet its full intent. Students have multiple opportunities throughout the materials, with support of the teacher or during independent practice, to identify and use a variety of tools or strategies that support their understanding of grade-level math. Examples include:

• In Unit 3, Transformations and Angle Relationships, Lesson 17, Anchor Problem 2, students determine which tools to use as they reason about congruent angles. The problem states, “In the diagram below, lines a and b are parallel. Line c is a transversal that cuts through the parallel lines. a. Name four pairs of congruent vertical angles. b. ∠2 and ∠8 are congruent. How can you prove this? c. Name three other pairs of corresponding angles.” Problem notes for the teacher state, “Use this Anchor Problem to introduce parallel line diagrams, transversals, and corresponding angles. Allow students to use a strategy of their choice to prove the congruency of ∠2 and ∠8. Have tools such as protractors, tracing paper, graph paper, etc., on hand if students would like to use them (MP.5).” 

• In Unit 4, Functions, Lesson 8, Problem Set, Problem 4, students choose a strategy to determine if functions are linear or nonlinear. The problem states, “Determine whether each equation is linear or nonlinear. y = 6 (4 − 3x), y = $(3x − 4)^2$, $y=\frac{7}{8}x$, $y=\frac{9x}{2}$, $y=\frac{x^2}{y}$. Check one box per column.”  

• In Unit 7, Pythagorean Theorem and Volume, Lesson 1, Problem Set, Problem 4, students use strategies as tools to approximate irrational numbers. The problem states, “Between what two integers do the following square roots fall? a.$$\sqrt{30}$$, b. $\sqrt{75}$, c. $\sqrt{123}$, d. $\sqrt{58}$, e. $\sqrt{0.23}$.”

The materials reviewed for Fishtank Plus Math Grade 8 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, Exponents and Scientific Notation, Lesson 8, students attend to precision as they reason with negative exponents in order to write equivalent, simplified exponential expressions. Anchor Problem 2 states, “Simplify the following expressions to the fewest number of bases possible and no negative exponents. $\frac{15^4}{15{-3}}$, $2^5$, $(ab^2)^{-3}$, $\frac{m^3}{m}$, $$(\frac{1}{3})^{-2}$$, $\frac{1}{(7^{-8})}$.” Problem Notes for the teacher state, “Watch for students applying the rules for exponents with precision, especially with negative exponents, and not over-applying any rules (MP.6)." 

• In Unit 3, Transformations and Angle Relationships, Lesson 6, students attend to precision as they perform rotations between congruent figures. Target Task states, “Triangle QRS was transformed to create triangle Q′R′S′ in the coordinate plane below. a. Describe the transformation that maps QRS to Q′R′S′. b. Rotate figure Q′R′S′ 90° clockwise around the origin and draw the new figure Q″R″S″. Describe a single transformation that maps triangle QRS to Q″R″S″.” 

• In Unit 7, Pythagorean Theorem and Volume, Lesson 8, students attend to precision when they use the Pythagorean Theorem to reason about right triangles. In Problem Set, Problem 4 states, “Is the triangle below a right triangle? If yes, explain how you know. If no, change one of the side lengths to make it a right triangle. Then label the right angle in the triangle.” A triangle is shown with legs of 9 and 11 and hypotenuse of $\sqrt{200}$. Unit 7 Summary states, “Throughout the unit, students must attend to precision in their work, their solutions, and their communication, being careful about specifying appropriate units of measure, using the equals sign appropriately, and representing numbers accurately (MP.6).” 

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 3, Transformations and Angle Relationships, Unit Assessment, students use the specialized language of mathematics as they explain angle congruence and similarity. Problem 11 states, “The figure shows line 𝑅𝑆 parallel to line 𝑈𝑉. The lines are intersected by 2 transversals. a. If the measure of ∠𝑆𝑇𝑉 is 108° and the measure of ∠𝑅𝑆𝑇 is 74°, determine the measure of ∠𝑇𝑉𝑈. b. Explain why triangle 𝑅𝑇𝑆 is similar to triangle 𝑉𝑇𝑈.” The Unit 3 Summary states, “They use precision in their descriptions of transformations and in their justifications for why two figures may be similar or congruent to each other (MP.6).”

• In Unit 4, Functions, Lesson 1, students use the specialized language of mathematics as they identify and describe functions. Target Task states, “In each example below, an arrow is used to show an input mapping to an output. Determine which relationships are functions. For each relationship that is not a function, explain why.”

• In Unit 5, Linear Relationships, Lesson 5, students use the specialized language of mathematics as they graph a linear equation using a table of values. Anchor Problem 2 states, “Emily tells you that she scored 18 points in a basketball game. a. Write down all the possible ways she could have scored 18 points with only two- and three-point baskets. Use the table below to organize your work. Write an equation to represent the situation, with x as the number of two-point baskets and y as the number of three-point baskets Emily scored. Then graph the situation.” Problem Notes for teachers state, “Note that the axes are not drawn in on the graph in the problem. Ask students where they should draw the axes that is most appropriate for the context. Ensure students appropriately label their axes (MP.6).”

The materials reviewed for Fishtank Plus Math Grade 8 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 are given 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, Exponents and Scientific Notation, Lesson 2, students make use of structure as they evaluate numerical and algebraic expressions with exponents. Anchor Problem 1 states, “Two expressions are shown below. Expression A: $(2^2+3^2)$; Expression B: $(2+3)^2$ Are the two expressions equivalent? What is your process to evaluate each one?” Problem Notes for teachers state, “In 6th and 7th grades, students spent a lot of time looking at the structure of expressions to understand what is happening with the numbers and operations. Ask students to describe what is happening in each expression A and B to recall this skill and practice (MP.7).” 

• In Unit 2, Solving-One Variable Equations, Lesson 9, students make use of structure as they generalize patterns in the three types of solutions to equations. In Problem Set, Problem 2 states, “Which values of J and K result in an equation with no solutions? Jx − 25 = Kx + 55. Select all that apply. A. J = 25 and K = −55; B. J = 55 and K = −25; C. J = 25 and K = 25; D. J = -55 and K = −55.” Tips for Teachers state, “In this lesson, students use what they know about equations with no, infinite, or unique solutions to reason about solutions without having to completely solve the equation. For example, if they can simplify each side of an equation to one or two terms, they should be able to determine if a solution is possible and if so, if that solution is unique (MP.7)."

• In Unit 6, System of Linear Equations, Lesson 6, students make use of structure as they reason about the solutions to a system of equations. Target Task states, “Determine by inspection if each system below has a unique solution, no solution, or infinite solutions. If the system has a unique solution, use substitution to find the coordinate point where the two lines intersect. a. y = −3x + 5, 2y = −6x + 10; b. x + y = 8, 2x + y = −6; c. x + y = 3, 2x + 2y = 3.” The Unit 6 Summary states, “Using the structure of the equations in a system, students will determine if systems have one, no, or infinite solutions without solving the system (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 1, Exponents and Scientific Notation, Lesson 4, students use repeated reasoning as they reason about equivalent expressions with exponents. Target Task states, “Jake believes that $\frac{30^5}{30^4}×30=1$. Do you agree with Jake? If yes, explain why he is correct. If not, then give the correct answer and an argument to convince Jake that your answer is correct.” In the Criteria for Success, Criteria 4 states, “Look for and make sense of repeated reasoning with exponentials in order to write equivalent expressions (MP.8).” 

• In Unit 3, Transformations and Angle Relationships, Lesson 9, students use repeated reasoning as they describe multiple rigid transformations on a coordinate plane. Anchor Problem 1 states, “Point A is located at (2,4). Perform the following transformations on point A, and label each new point. Translate point A 2 units to the right and 4 units down. Label it point W. Rotate point A 90° counter-clockwise about the origin. Label it point X. Reflect point A over the x-axis. Label it point Y. Reflect point A over the y-axis. Label it point Z.” Guiding Questions for teachers which support the development of MP.8, include “What are the coordinate points of each new location of point A? How does each transformation change the coordinates of the point? Which type of transformation(s) could move point A to the location (4,−2)? Try out the same transformations on a different point. What patterns do you notice?” Problem Notes for teachers state, “If needed, students can organize their information in a chart or table to support the discovery of patterns or generalizations. One approach to this Anchor Problem could be to have students work in small groups, each group given a different starting point to work with. As a class, students could share the results from their group to compare and look for patterns in the coordinate points for each transformation (MP.8).” 

• In Unit 7, Pythagorean Theorem and Volume, Lesson 6, students use repeated reasoning as they use the Pythagorean Theorem to explore the relationships of side lengths in right triangles. Anchor Problem 1 states, “Several right triangles are shown below (not drawn to scale). In a right triangle, the two side lengths that form the right angle are called legs, and the side length opposite the right angle is called the hypotenuse. Use the triangles to investigate the question: What relationship do you see between the measures of the legs and the measure of the hypotenuse?” Guiding Questions for teachers which support the development of MP.8 include, “Given the definition of legs and hypotenuse, which side length represents the hypotenuse in each triangle? What is your strategy for investigating different relationships? How will you keep track of your work? If you are stuck, ask your teacher for a clue. Does the relationship you find work for all of the right triangles shown?” Problem Notes for teachers state, “The purpose of this Anchor Problem is to allow students time to explore the side lengths of right triangles in search of the relationship that exists between them. After students have enough exploration time, discuss what relationships were discovered and confirmed with repetition (MP.8). Ensure that the discussion ends with the Pythagorean Theorem $a^2+b^2=c^2$, and confirm this relationship holds for all triangles shown above.”

The materials reviewed for Fishtank Plus Math Grade 8 meet expectations for providing teacher guidance with useful annotations and suggestions for how to enact the student materials and ancillary materials, with specific attention to engaging students in order to guide their mathematical development.

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

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

• In Unit 6, Systems of Linear Equations, Unit Summary provides an overview of content and expectations for the unit. Within Unit Prep, View Unit Launch, there is a video detailing the content for teachers. The materials state, “Welcome to the Unit Launch for 8th Grade Math, Unit 6 Systems of Linear Equations. Please watch the video below to get started.” Additionally, the Unit Summary contains Intellectual Prep and Unit Launch with Standards Review, Big ideas, and Content connections. The Standards Review provides teachers an “opportunity to reflect on select standards from the unit. [...] In this section you will examine the language of these standards and reflect on how several problems on the end-of-unit assessment relate to the standard.”  Then, Big Ideas help teachers “understand how these ideas develop throughout the unit by analyzing lessons and problems from the unit, and finally, have the chance to reflect on how you will address your students’ needs around these concepts.” Finally, Content Connections states, “In this final section of the unit launch, you’ll have the chance to zoom out and look at the related content that students study before and after this unit.” This information is included for units 1-5 in Grade 8.

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

• In Unit 1, Exponents and Scientific Notation, Lesson 6, Tips of Teachers provide guidance so teachers are prepared to support students in discovering a general rule for operations with exponents. The materials state, “Similar to Lesson 5, these Anchor Problems can be used in a variety of ways, including having students lead the discovery and seek out a general rule. Once students have experimented with the problems and found a generalization, then provide them with the name of the rule and the general form.”

• In Unit 5, Linear Relationships, Lesson 5, Anchor Problem 1 Notes provide guidance to connect students' prior knowledge to new concepts. The materials state, “Use this Anchor Problem to graph a linear graph outside of the first quadrant, as compared to the proportional graphs students saw in the beginning of this unit. This example is similar to problems that students saw in Lessons 1–4 in that the graph passes through the origin and is linear; however, the rate of the change is negative, which places the graph in the 4th quadrant in the coordinate plane.”

• In Unit 8, Bivariate Data, Lesson 2, Tips for Teachers include guidance for teachers to address common misconceptions as students create scatter plots for given data. The materials state, “A common misconception is to confuse causality with association. For example, students may misunderstand a relationship between two variables to imply that one variable causes another to change, when there is only evidence to show an association between the two variables. As students describe relationships between variables, ensure they use language to imply an association rather than a casual relationship.”

The materials reviewed for Fishtank Plus Math Grade 8 meet expectations for including standards correlation information that explains the role of the standards in the context of the overall series. 

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

• In 8th Grade Math, Standards Map includes a table with each grade-level unit in columns and aligned grade-level standards in the rows. Teachers can easily identify a unit when each grade-level standard will be addressed. 

• In 8th Grade Math, Unit 7, Pythagorean Theorem and Volume, Lesson Map outlines lessons, aligned standards, and the lesson objective for each unit lesson. This is present for all units and allows teachers to identify targeted standards for any lesson.

• In Unit 4, Functions, Lesson 3, the Core Standards are identified as 8.F.A.1, 8.F.A.2, 8.F.B.4. The Foundational Standards are identified as 6.EE.A.2.C, 6.RP.A.2, 7.RP.A.2.B. Lessons contain a consistent structure that includes an Objective, Common Core Standards, Criteria for Success, Tips for Teachers, Anchor Problems, Problem Set, and Target Task. This provides an additional place to reference standards, and language of the standard, within each lesson.

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

• In Unit 1, Exponents and Scientific Notation, Unit Summary includes an overview of how the content in 8th grade connects to mathematics students will learn in high school. The materials state, “In high school, students will need a strong understanding of exponents and exponent properties. They will apply the properties of exponents to exponential equations in order to reveal new understandings of the relationship. They will work with fractional exponents and discover the properties of rational exponents and rational numbers. In general, students’ ability to see the structure in an expression will support them in manipulating quadratic functions, operating with polynomials, and making connections between various relationships.”

• In Unit 3, Transformations and Angle Relationships, Unit Summary includes an overview of how the math of this unit builds from previous work in math. The materials state, “Prior to eighth grade, students developed their understanding of geometric figures and learned how to draw them, calculate measurements, and model real-world situations. In seventh grade, students were introduced to the concept of scaling through scale drawings, and they solved for various measurements using proportional reasoning. Students will draw on these prior skills when they investigate dilations and similar triangles.”

• In Unit 6, Systems of Linear Equations, Unit Summary includes an overview of the Math Practices that are most strongly connected to the content in the unit. The materials state, “Using the structure of the equations in a system, students will determine if systems have one, no, or infinite solutions without solving the system (MP.7). Students also explore the many rich applications that can be modeled with systems of linear equations in two variables (MP.4).”

The materials reviewed for Fishtank Plus Math Grade 8 meet expectations for providing explanations of the instructional approaches of the program and identification of the research-based strategies. This information can be found within Our Approach and Math Teacher Tools. Examples where materials explain the instructional approaches of the program include:

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

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

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

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

• In Math Teacher Tools, Procedural Skill and Fluency, Fluency Expectations by Grade states, “The language of the standards explicitly states where fluency is expected. The list below outlines these standards with the full standard language. In addition to the fluency standards, Model Content Frameworks, Mathematics Grades 3-11 from the Partnership for Assessment of Readiness for College and Careers (PARCC) identify other standards that represent culminating masteries where attaining a level of fluency is important. These standards are also included below where applicable. 8th Grade, 8.EE.7, 8.G.9.”

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

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

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

The materials reviewed for Fishtank Plus Math Grade 8 meet expectations for providing a comprehensive list of supplies needed to support instructional activities. The 8th Grade Course Summary, Course Material Overview, Course Material List 8th Grade Mathematics states, “The list below includes the materials used in the 8th grade Fishtank Math course. The quantities reflect the approximate amount of each material that is needed for one class. For more detailed information about the materials, such as any specifications around sizes or colors, etc., refer to each specific unit.” The materials include information about supplies needed to support the instructional activities. Examples include:

• Scientific calculators are used in Units 1, 2, 3 7, and 8, one per student. 

• Graph paper is used in Units 3, 5, 7, and 8, one ream. 

• Patty paper is used in Units 3 and 5, one box of 1000 sheets. 

• Protractors are used in Unit 3, one per student. In Unit 3, Transformations and Angle Relationships, Lesson 11, students define a dilation as a non-rigid transformation, and understand the impact of scale factor (8.G.A.4 ). Anchor Problem 2 states, “Triangle ABC is dilated to create similar triangle DEF. a.) Indicate the corresponding angles in the diagram. What is the relationship between corresponding angles? b.) Name the corresponding sides in the diagram. What is the relationship between corresponding side lengths?” Notes state, “To prove that the corresponding angle measures are congruent, students may use patty paper to copy and transfer one triangle over the other, or they may use protractors to measure (MP.5).”

• Wikisticks (or raw spaghetti) are used in Unit 8, one per student. In Unit 8, Bivariate Data, Lesson 4, Tips for Teachers states, “Manipulatives that represent a line that can be maneuvered on paper are a valuable tool for this lesson; for example, wikisticks, raw spaghetti, thin pencils, etc.”

The materials reviewed for Fishtank Plus Math Grade 8 meet expectations for having assessment information included in the materials to indicate which standards and mathematical practices are assessed. 

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

• In Unit 2, Solving One-Variable Equations, Expanded Assessment Package, Post-Unit Assessment Analysis denotes content standards addressed for each problem. Problem 1 is aligned to 8.EE.7b and states, “What is the solution to the equation below? $\frac{2}{3}x+5=1$ a. $x= -6$ b. $x=4$  c. $x= -4.5$  d.$.”

• In Unit 4, Functions, Unit Summary, Unit Assessment, Answer Key denotes standards addressed for each question. Question 5 is aligned to 8.F.1 and states, “Which sets of ordered pairs represent a function? Select two correct answers. a. {(1, 0), (2,1), (2, 3)} b. {(1, 3), (1, 4), (1, 5)} c. {(1, 5), (2, 5), (3, 5)} d. {(3, 1), (2, 2), (3, 3)} e. {(3, 2), (5, 1), (4, 0)}.”

• In Unit 5, Linear Relationships, Unit Summary, Post-Unit Assessment, Answer Key denotes Standards for Mathematical Practice addressed for each question. Question 1 is aligned to MP1 and states, “At a local market, the cost of apples is directly proportional to the weight of the apples. Carlos bought 10 pounds of apples for a cost of $15.00. Which graph shows the relationship between the weight of the apples, in pounds, and the cost of the apples, in dollars?” 

• In Unit 7, Pythagorean Theorem and Volume, Unit Summary, Mid-Unit Assessment, Answer Key denotes Standards for Mathematical Practice addressed for each question. Question 2 is aligned to MP3 and states, “Dr. Professor built a math robot to help her with her research. The first thing Dr. Professor asks her robot to do is to find the exact decimal value of $\sqrt{5}$. Can the robot do it? Explain why or why not.”

• In Unit 8, Bivariate Data, Unit Assessment Answer Key includes a constructive response and 2-point rubric with the aligned grade-level standard. Questions 3a & 3b are aligned to 8.SP.1 and state, “In a city, there are several office spaces of different sizes that businesses can rent on a yearly basis. The scatter plot below shows the sizes of some office spaces and their yearly rent in this city. Part A: Does there appear to be a relationship between the size of an office space and the rent? If so, then describe if the relationship is linear or non-linear, positive or negative. If not, then explain why there is no relationship. Part B: Are there any clusters or outliers in the scatter plot? If so, explain what they mean in context of the situation.”

The materials reviewed for Fishtank Plus Math Grade 8 meet expectations for including an assessment system that provides multiple opportunities throughout the grade, course, and/or series to determine students’ learning and sufficient guidance to teachers for interpreting student performance and suggestions for follow-up. 

Each Post-Unit Assessment Analysis provides an answer key, potential rationales for incorrect answers, and a commentary to support analysis of student thinking. According to Math Teacher Tools, Assessment Resource Collection, “commentaries on each problem include clarity around student expectations, things to look for in student work, and examples of related problems elsewhere on the post-unit assessment to look at simultaneously.” Each Mid-Unit Assessment provides an answer key and a 1-, 2-, 3-, or 4-point rubric. Each Pre-Unit Assessment provides an answer key and guide with a potential course of action to support teacher response to data. Each lesson provides a Target Task with a Mastery Response. According to the Math Teacher Tools, Assessment Overview, “Target Tasks offer opportunities for teachers to gather information about what students know and don’t know while they are still engaged in the content of the unit.” Examples from the assessment system include:

• In Unit 1, Exponents and Scientific Notation, Post-Unit Assessment Analysis, Problem 5 states, “The population of California is approximately $4×10^7$ people. The population of South Dakota is approximately 800,000 people. The population of California is about how many times the population of South Dakota? a. 20 b. 50 c. 200 d. 500. Commentary: In this problem, students are given two large numbers, one of which is written in scientific notation, and are asked to compare the values. In order to compare the values, students will need to change one of the numbers into a form that matches the other. Some students may do this efficiently by writing 800,000 in scientific notation, and then using properties of exponents when dividing the two numbers.”

• In Unit 2, Solving One-Variable Equations, Lesson 5, Target Task, students model multi-step equations. The materials state, “Sequel: You want to put up 25 pictures using the same spacing; however, these pictures have been rotated 90 degrees to be in a portrait orientation. How long of a wall, in feet, do you need? How long of a wall would you need for n pictures in portrait orientation using the same spacing?” The Mastery Response states, “2(4.5) + 25(8.5) + 24(5) = ? 9 + 212.5 + 120 = 341.5 inches or 28 ft $5\frac{1}{2}$ in. For n pictures, 9 + 8.5n + 5(n - 1) inches.” 

• In Unit 5, Linear Relationships, Pre-Unit Assessment, Teacher Answer Key & Guide, Problem 1 states, “A photocopier can print 40 pages in 16 seconds. Which equation represents the relationship between 𝑝, the number of pages printed, and 𝑠, the amount of time in seconds? a. $𝑝=\frac{1}{16}$s, b. $𝑝=\frac{2}{5}$s, c. $𝑝=2\frac{1}{2}$s, d. 𝑝 = 24𝑠 Potential Course of Action, If needed, this concept should be addressed early on in the unit, as students study and compare proportional relationships in Lessons 1 – 4, and again later in the unit when students focus on writing linear equations starting in Lesson 10. For example, include a multiple choice problem similar to the one above as a warm-up for Lesson 1 or Lesson 2; or include a problem without multiple choice options as a warm-up for Lesson 10 or Lesson 15. Find problems and other resources in these Fishtank lessons: Grade 7 Unit 1 Lessons 4 – 5. Grade 8 Unit 4 Lesson 4.”

• In Unit 6, System of Linear Equations, Mid-Unit Assessment, Answer Key, Problem 7 states, “Solve the system of equations algebraically. Use the substitution method. Show all your work. 4𝑥 + 2𝑦 = 9; −3𝑥 + 𝑦 = 4.” The Answer Key provides the correct answers and a 3-point rubric for teachers to follow. It states, “3 points. Student response demonstrates an exemplary understanding of the concepts in the task. The student points correctly and completely answers all aspects of the prompt. 2 points. Student response demonstrates a good understanding of the concepts in the task. The student arrived at an acceptable conclusion, showing evidence of understanding of the task, but some aspect of the response is flawed. 1 point. Student response demonstrates a minimal understanding of the concepts in the task. The student arrived at an incomplete or incorrect conclusion, showing little evidence of understanding of the task, with most aspects of the task not completed correctly or containing significant errors or omissions. 0 points. Student response contains insufficient evidence of an understanding of the concepts in the task. Work points may be incorrect, unrelated, illogical, or a correct solution obtained by chance.”

The materials reviewed for Fishtank Plus Math Grade 8 meet expectations for providing assessments that include opportunities for students to demonstrate the full intent of grade-level standards and practices across the series.

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

• In Unit 2, Solving One-Variable Equations, Post-Unit Assessment problems support the full intent of MP5 (Choose tools strategically) as students select strategies for reasoning about linear relationships. Problem 11 states, “The perimeter of a given rectangle is 40 inches. The length of the rectangle is two inches more than three times the width. What are the dimensions of the rectangle?” Problem 15 states, “Denzel has saved $75 in his bank account and saves an additional $12.50 every week. Halle has saved $339 in her bank account, but spends $20.50 each week. After how many weeks will Denzel and Halle have the same amount of money in their bank accounts?” 

• In Unit 4, Functions, Mid-Unit Assessment Problems 4 and 5 and Post-Unit Assessment Problem 12 develop the full intent of 8.F.4 (Construct a function to model a linear relationship between two quantities. Determine the rate of change and initial value of the function from a description of a relationship or from two (x,y) values, including reading these from a table, or from a graph. Interpret the rate of change and initial value of a linear function in terms of the situation it models, and in terms of its graph or a table of values). Mid-Unit Problem 4 states, “A health club charges an up-front, joining fee to its members as well as a monthly fee. The table below shows the total cost of being a member at this health club for different numbers of months. Part A: What is the monthly fee for this health club? a. $40, b. $70, c. $80, d. $110. Part B: What is the annual fee for this health club? a. $30, b. $40, c. $70, d. $110.” Mid-Unit Problem 5 states, “Mr. Johnson is an art teacher and is buying tee-shirts online for an art project at school. The graph below shows the total cost of his order as a function of the number of tee-shirts he buys. a. Mr. Johnson’s online order includes a shipping cost. What is the shipping cost? b. What is the rate of change? What does it mean in context of the situation? c. Write an equation to represent the total cost of Mr. Johnson’s order, y, for x tee-shirts.” Post-Unit Problem 12 Part b states, “Write an equation to represent Function F.” A table is provided with x values as 0, 4, 6, 10 and y values as 6, 18, 24, 36.

• In Unit 5, Linear Relationships, Mid-Unit Assessment Problem 2 and Post-Unit Assessment Problem 6 develop the full intent of 8.EE.5 (Graph proportional relationships, interpreting the unit rate as the slope of the graph. Compare two different proportional relationships represented in different ways). Mid-Unit Problem 2 states, “Dawn is filling three buckets using three different faucets at her house. The water flows from each faucet at a different constant rate, as shown in each representation below. Assuming each bucket is the same size and they all start out empty, which bucket will be filled first? Justify your response with specific rates of change.” Post-Unit Problem 6 states, “Grace and Jackie are both accountants. They each charge their customers an hourly fee for their service, as described below. Grace charges $540 for 18 hours of service. Jackie charges $40 per hour. a. Draw two graphs in the grid below to show the relationship between cost, 𝑦, and hours of service 𝑥, for Grace and for Jackie. Label each line with the accountant’s name. b. Who charges more per hour? Explain your answer using your graph.” 

• In Unit 8, Bivariate Data, Post-Unit Assessment problems support the full intent of MP1 (Make sense of problems and persevere in solving them) as students make sense of bivariate data in real world contexts. Problem 5 states, “The scatter plot below shows the numbers of customers in a restaurant for four hours of the dinner service on two different Saturday nights. The line shown models this relationship, and  𝑥 = 0 represents 7 p.m. What does the value of the 𝑦-intercept represent? a. The average number of customers at 7 p.m. b. The average number of customers at 11 p.m. c. The average change in the number of customers each hour. d. The average change in the number of customers during four hours of the dinner service. Problem 7 states, “Colton is heating a pot of water. He records the temperature of the water in the pot every minute. This equation models Colton’s data, where 𝑥 represents the number of minutes the water has been heated, and 𝑦 represents the temperature of the water in degrees Fahrenheit. 𝑦 = 7.5𝑥 + 40. Part A. What does the coefficient 7.5 in the equation represent in the context of this situation? Part B. What does the value 40 in the equation represent in the context of this situation? Part C. What is the temperature, in degrees Fahrenheit, of Colton’s pot of water after 16 minutes? Show or explain how you got your answer.”