8th Grade - Gateway 2

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.”