8th Grade - Gateway 1

The materials reviewed for Fishtank Plus Math Grade 8 meet expectations for assessing grade-level content and, if applicable, content from earlier grades. The curriculum is divided into eight units and each unit contains a Pre-Unit Assessment, Mid-Unit Assessment, and Post-Unit Assessment. Pre-Unit assessments may be used “before the start of a unit, either as part of class or for homework.” Mid-Unit assessments are “designed to assess students on content covered in approximately the first half of the unit” and may also be used as homework. Post-Unit assessments “are designed to assess students’ full range of understanding of content covered throughout the whole unit.” Examples of Post-Unit Assessments include: 

• In Unit 1, Post-Unit Assessment, Exponents and Scientific Notation, Problem 3 states, “What is the value of n in the equation shown below? 2² × 2ⁿ = (2⁴)³ a. 5, b. 6, c. 10, d. 12.” (8.EE.1)

• In Unit 2, Post-Unit Assessment, Solving One-Variable Equations, Problem 5 states, “What is the value of 𝑚 that satisfies the equation below? 3(𝑚 + 4) − 2(2𝑚 + 3) = −4.” (8.EE.7b) 

• In Unit 4, Post-Unit Assessment, Functions, Problem 6 states, “Mia paddled her canoe from the shore of a lake to an island. She stopped on the island to eat lunch, and then paddled her canoe back to the shore. The graph below shows Mia’s distance, in kilometers, from the shore over time.” A graph is shown with time on the x-axis and distance on the y-axis. From 0 to 40 minutes the graph increases in a straight line to 3 kilometers; the graph then stays at 3 kilometers for 30 minutes; the graph shows a straight line decrease for 30 minutes to 0 km. The problem continues, “Based on the graph, which of the following statements is true? a. Mia paddled a total distance of 3 kilometers. b. Mia paddled for a total of 100 minutes. c. Mia paddled faster on the way back to the shore than on the way to the island. d. Mia paddled faster on the way to the island than on the way back to the shore.” (8.F.5)

• In Unit 7, Post-Unit Assessment, Pythagorean Theorem and Volume, Problem 12 states, “Aaron drew a map showing the locations of two cities, Oden and Lundy, on a grid. The map and its scale are shown. Aaron drew a straight line from Oden to Lundy. Which of the following is closest to the distance between Oden and Lundy along the straight line? a. 8 miles b. 10 miles c. 12 miles d. 14 miles” Aaron’s map shows Oden and Lundy connected with the hypotenuse of a triangle and the scale of 1 unit = 1 mile.” (8.G.8)

The instructional materials reviewed for Fishtank Plus Math Grade 8 meet expectations for the materials giving all students extensive work with grade-level problems to meet the full intent of grade-level standards.

The instructional materials provide extensive work in Grade 8 by including Anchor Problems, Problem Sets, and Target Tasks for all students in each lesson. Within Grade 8, students engage with all CCSS standards. Examples of problems include:

• In Unit 1, Exponents and Scientific Notation, Lessons 10, 11, and 12 engage students in extensive work with 8.EE.3 (Use numbers expressed in the form of a single digit times an integer power of 10 to estimate very large or very small quantities, and to express how many times as much one is than the other). In Lesson 10, Target Task, students reason about powers of 10. Problem 1 states, “A large construction crane weighs one million pounds. A large truck weighs 10,000 pounds. How many times greater is the weight of the crane than the weight of the truck?” Problem 2 states, “Complete the chart below. The first row has been completed for you as an example.” Students complete a table including columns for the Power of 10, Decimal Notation, Place Value, and an example of something measured in the quantity.

• In Unit 4, Functions, Lesson 5 engages students in extensive work with 8.F.1 (Understand that a function is a rule that assigns to each input exactly one output. The graph of a function is the set of ordered pairs consisting of an input and the corresponding output). In Problem Set, Problem 5, students read inputs and outputs in graphs of functions and decide if graphs are functions. It states, “Point A is plotted on the coordinate plane below. A. Point B (not shown) has integer coordinates. The graph of the line through points A and B is not a function. Draw a possible location for point B on the coordinate plane above. B. Point C (not shown) has integer coordinates. The graph of the line through points A and C is a function. Draw a possible location for point C on the coordinate plane above.” An image of the coordinate plane has point A plotted at (-5,2).

• In Unit 8, Bivariate Data, Lesson 4 engages students with extensive work of 8.SP.2 (Know that straight lines are widely used to model relationships between two quantitative variables. For scatter plots that suggest a linear association, informally fit a straight line, and informally assess the model fit by judging the closeness of the data points to the line). In the Target Task, students analyze bivariate data with a line of best fit. It states, “The plot below is a scatter plot of mean temperatures in July and mean inches of rain per year for a sample of Midwestern cities. A line is drawn to fit the data. 1. Choose a point in the scatter plot and explain what it represents. 2. Use the line provided to predict the mean number of inches of rain per year for a city that has a mean temperature of 70°F in July. 3. Do you think the line provided is a good one for this scatter plot? Explain why or why not. 4. A fellow classmate uses the strategy of drawing a line so that half of the points are above the line and half of the points are below the line. Explain or show how this could result in a line that was not representative of the data.” The image of the scatter plot on a coordinate plane is provided.

The instructional materials provide opportunities for all students to engage with the full intent of Grade 8 standards through a consistent lesson structure, including anchor problems, problem sets, and target tasks. Anchor Problems include a connection to prior knowledge, multiple entry points to new learning, and guided instruction support. Problem Set Problems engage all students in practice that connects to the objective of each lesson. Target Task Problems can be used as formative assessment. Examples of meeting the full intent include:

• In Unit 2, Solving One Variable Equations, Lessons 3 and 4 engage students with the full intent of 8.EE.7b (Solve linear equations with rational number coefficients, including equations whose solutions require expanding expressions using the distributive property and collecting like terms). In Lesson 3, Problem Set, Problem 3, students solve linear equations with rational coefficients. It states, “Solve the following equations. Justify each step you take in solving the equation. A. 11 = -3(2y - 2) - 7. B. $$\frac{1}{3}$$(9 - 3x) = -4.” In Lesson 4, Anchor Problem 4 states, “Solve the equations. a. $$\frac{2(x-4)}{9}=\frac{3-7}{6}$$. b. x - 0.8x(3) + 14.12 = 0.75(8).”  

• In Unit 5, Linear Relationships, Lessons 2 and 4 engage students with 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). In Lesson 2, Anchor Problem 1, students graph proportional relationships and interpret slope as the unit rate. It states, “Water flows at a constant rate out of a faucet. Suppose the volume of water that comes out in three minutes is 10.5 gallons. a. Write a linear equation to represent the volume of water, v, that comes out of the faucet in t minutes. b. Find the volume of water out of the faucet after 0 minutes, 1 minute, and 4 minutes. c. Graph the equation in the coordinate plane. d. What is the slope of the graph? What does it mean as a unit rate?” In Lesson 4, Anchor Problem 2, students compare a proportional relationship from a graph to one represented in a table. It states, “Kristina and her sister, Tracee, are painting rooms in their house. The graph below represents the rate at which Kristina paints, and the table below shows how many square feet Tracee painted for given amounts of time. Both sisters paint at a constant pace. Who paints at the faster rate? Justify your answer.” A linear graph passing through the origin and the points (1,4), (2,8), (3,12), (4,16) is shown as well as a table with the values of area painted (square feet) and time (minutes) 18.75, 5; 30,8; 45,12; 75, 20.

• In Unit 7, Pythagorean Theorem and Volume, Lessons 2 and 3 engage students with the full intent of 8.NS.2 (Use rational approximations of irrational numbers to compare the size of irrational numbers, locate them approximately on a number line diagram, and estimate the value of expressions). In Lesson 2, Problem Set, Illustrative Mathematics Estimating Square Roots Task, students estimate square roots. It states, “Without using the square root button on your calculator, estimate $$\sqrt{800}$$ as accurately as possible to 2 decimal places.” In Lesson 3, Anchor Problem 2, students use a number line to approximately locate and compare values of irrational numbers. “Estimate the value of each irrational expression below. Then plot a point to approximate each location on the number line. a. 3$$\pi$$ b. $$\sqrt{50}+1$$ c. $$\frac{\sqrt{140}}{2}$$ d. $$2\sqrt{10}$$.”

The materials reviewed for Fishtank Plus Math Grade 8 meet expectations that, when implemented as designed, the majority of the materials address the major clusters of the grade. The instructional materials devote at least 65 percent of instructional time to the major clusters of the grade: 

• The approximate number of units devoted to major work of the grade (including assessments and supporting work connected to the major work) is 6.5 out of 8, approximately 81%.

• The number of lessons devoted to major work of the grade (including supporting work connected to the major work) is 102 out of 120, approximately 85%. The total number of lessons include: 112 lessons plus 8 assessments for a total of 120 lessons.

• The number of days devoted to major work (including assessments, flex days, and supporting work connected to the major work) is 120 out of 143, approximately 84%. There are a total of 23 flex days and 18 of those days are included within units focused on major work. By adding 18 flex days focused on major work to the 102 lessons devoted to major work, there is a total of 120 days devoted to major work.

• The number of days devoted to major work (excluding flex days, while including assessments and supporting work connected to the major work) is 107 out of 120, approximately 89%. While it is recommended that flex days be used to support major work of the grade within the program, there is no specific guidance for the use of these days.

A lesson-level analysis is most representative of the instructional materials as the lessons include major work, supporting work connected to major work, and the assessments embedded within each unit. As a result, approximately 85% of the instructional materials focus on major work of the grade.

The instructional materials reviewed for Fishtank Plus Math Grade 8 meet expectations that supporting work enhances focus and coherence simultaneously by engaging students in the major work of the grade. Materials are designed so supporting standards/clusters are connected to the major standards/clusters of the grade. These connections are sometimes listed for teachers as “Foundational Standards'' on the lesson page. Examples of connections include:

• In Unit 7, Pythagorean Theorem and Volume, Lesson 13, Problem Set connects the supporting work of 8.NS.2 (use rational approximations of irrational numbers to compare the size of irrational numbers) with the major work of 8.EE.2 (use square root and cube root symbols to represent solutions to equations of the form x² = p and x³ = p). In Problem 7, students evaluate rational and irrational numbers. It states, “Order the following from least to greatest: \sqrt[3]{530}, \sqrt[2]{48}, \pi, \sqrt{121}, \sqrt[3]{27}, \frac{19}{2}.”

• In Unit 7, Pythagorean Theorem and Volume, Lesson 15, Target Task connects the supporting work of 8.G.9 (know the formulas for the volume of cones, cylinders, and spheres and use them to solve real-world and mathematical problems) to the major work of 8.EE.7 (solve linear equations in one variable). In the Target Task, students solve linear equations in one variable as they find the volume of spheres. It states, “A standard soccer ball measures 22 cm in diameter. What is the volume of a standard soccer ball? Give your answer to the nearest whole centimeter.”

• In Unit 8, Bivariate Data, Lesson 3, Anchor Problem 1 connects the supporting work of 8.SP.1 (construct and interpret scatter plots for bivariate measurement data to investigate patterns of association between quantities) to the major work of 8.F.5 (describe qualitatively the functional relationship between two quantities by analyzing a graph). In Anchor Problem 1, students identify and describe associations in scatter plots including linear/nonlinear associations, positive/negative associations, clusters, and outliers. It states, “For the following five scatter plots, answer the following questions: a. Does there appear to be a relationship between x and y? b. If there is a relationship, does it appear to be linear? c. If the relationship appears to be linear, is it a positive or negative linear relationship? d. If applicable, circle the correct words in this sentence: There is a (positive/negative) association between x and y because as x increases, then y tends to (increase/decrease).” 

• In Unit 8, Bivariate Data, Lesson 6, Problem Set, Problem 4 connects the supporting work of 8.SP.3 (use the equation of a linear model to solve problems in the context of bivariate measurement data) to the major work of 8.F.4 (construct a function to model a linear relationship between two quantities). In Problem 4, students interpret the slope and y-intercept from a linear model. It states, “A company uses the equation, y = 15.75x + 5.95, to determine the cost, y, of purchasing, x, calculators, including the flat fee for shipping. a. If you were to create a graph representing the cost and quantity of calculators purchased, what would your graph look like? b. How much would it cost a customer to purchase 12 calculators at this company? c. Is your answer to part B an estimate or an exact value? Explain.”

The instructional materials for Fishtank Plus Math Grade 8 meet expectations that materials include problems and activities that serve to connect two or more clusters in a domain or two or more domains in a grade. Materials are coherent and consistent with the Standards. Examples of connections include:

• In Unit 3, Transformations and Angle Relationships, Lesson 19, Anchor Problem 1 connects the major work of 8.G.A (understand congruence and similarity using physical models, transparencies, or geometry software) to the major work or 8.EE.C (analyze and solve linear equations and pairs of simultaneous linear equations). In Anchor Problem 1, students “solve for missing angle measures in parallel line diagrams using equations. Lines L₁ and L₂ are parallel and cut by transversal p.” A diagram is given with lines L₁ and L₂  horizontal and line p intersecting those lines. The angle to the right of the transversal and above L₁ is labeled 4x + 11; the angle to the right of the transversal and above L₂ is labeled 8x - 25. It states, “What is the measure of the angles shown by the algebraic expressions?”

• In Unit 4, Functions, Lesson 6, Problem Set, Problem 3 connects the major work of 8.F.A (define, evaluate, and compare functions) to the major work or 8.F.B (use functions to model relationships between quantities). In Problem Set, Problem 3, students “Identify properties of functions represented in graphs.” It states, “The graph below shows George’s earnings, working a minimum wage job in Maryland in 2015.” A graph is shown with the title George’s Earnings, x-axis labeled hours worked from 0 to 8, y-axis labeled Dollars earned from 0 to 60. A line is drawn from (0,0) to (7,56) and there are 6 points plotted on the line at the x values from 1-6. Students complete the following: “A. How much does George earn for 4 hours of work? B. How long does George work to earn $40? C. What is the rate of change for this function? What does it mean in context? D. What is the initial value of this function? What does it mean in context? E. Write an equation to represent this function. F. How much would George earn after an 8-hour day? G. How many hours would George need to work to make $100?”

• In Unit 5, Linear Relationships, Lesson 7, Anchor Problem 2 connects the major work of 8.EE.B (understand the connections between proportional relationships, lines, and linear equations) to the major work of 8.F.B (use functions to model relationships between quantities). In Anchor Problem 2, students find the slope of a line that passes through two coordinate points and use proportional reasoning to determine if a given point is on the same line. It states, “Find the slope of the line between the points (-1, 3) and (5, 11). Is the point (-4, -1) on the same line as the other two points? Use slope to justify your answer.”

• In Unit 7, Pythagorean Theorem and Volume, Lesson 14, Target Task connects the supporting work of 8.G.C (solve real-world and mathematical problems involving volume of cylinders, cones, and spheres) to the supporting work of 8.NS.A (know that there are numbers that are not rational, and approximate them by rational numbers). In the Target Task, students find the volume of cylinders and cones and use their knowledge of irrational numbers to approximate solutions. It states, “In pottery class, Asher and Brandi make three-dimensional shapes out of clay. a. Asher makes a cylinder with a radius of 3 inches and a height of $$6\frac{1}{2}$$ inches. How many cubic inches of clay did Asher use? b. Brandi makes a cone and uses approximately 64$$in^3$$ of clay. The height of Brandi’s cone is 4 inches. What is the radius of the circular base of Brandi’s cone?”

The materials reviewed for Fishtank Plus Math Grade 8 meet expectations that content from future grades is identified and related to grade-level work, and materials relate grade-level concepts explicitly to prior knowledge from earlier grades.  

Content from future grades is identified within materials and related to grade-level work. These references are consistently included within the Unit Summary. Examples include:

• In Unit 3, Transformations and Angle Relationships, Unit Summary connects 8.G.A (Understand congruence and similarity using physical models, transparencies, or geometry software) to the work of high school. Unit Summary states, “In high school geometry, students spend significant time studying congruence and similarity in-depth. They build off of the informal proofs and reasoning developed in eighth grade to hone their definitions of transformations, prove geometric theorems, and derive trigonometric ratios.” (HSG.CO.A, HSG.CO.B, HSG.CO.C, HSG.SRT.A, HSG.SRT.B)

• In Unit 5, Linear Relationships, Unit Summary connects 8.EE.B (Understand the connections between proportional relationships, lines, and linear equations), 8.F.A (Define, evaluate and compare functions), and 8.F.B (Use functions to model relationships between quantities) to the work of high school. Unit Summary states, “In high school, students will continue to build on their understanding of linear relationships and extend this understanding to graphing solutions to linear inequalities as half-planes in the coordinate plane.” (HSA.CED.A, HSA.REI.D)

• In Unit 6, System of Linear Equations, Unit Summary connects 8.EE.8 (Analyze and solve linear equations and pairs of simultaneous linear equations) to the work in high school. Unit Summary states, “In high school, students will continue their work with systems, working with linear, absolute value, quadratic, and exponential functions. They will also graph linear inequalities and consider what the solution of a system of linear inequalities looks like in the coordinate plane.” (HSA-REI.C, HSA-REI.D)

• In Unit 8, Bivariate Data, Unit Summary, connects 8.SP.A (Investigate patterns of association in bivariate data) to the work of high school. Unit Summary states, “In high school, students’ understanding of statistics is formalized. They analyze bivariate data using functions, design and carry out experiments, and make predictions about outcomes based on probabilities. Students use their knowledge of association between variables as a basis for correlation. They develop nonlinear models for data and formally analyze how closely the model fits the data.” (HSS.ID.B, HSS.ID.C)

Materials relate grade-level concepts from Grade 8 explicitly to prior knowledge from earlier grades. These references can be found within materials in the Unit Summary or within Lesson Tips for Teachers. Examples include:

• In Unit 1, Exponents and Scientific Notation, Unit Summary connects 8.EE.A (work with radical and integer exponents) to the work from sixth grade (6.EE.A), fifth grade (5.NBT.A) and fourth grade (4.NBT.A). It states, “In sixth grade, students wrote and evaluated expressions with exponents using the order of operations. They identified the parts of an expression, distinguishing a term from a factor from a coefficient. In fourth and fifth grades, students investigated patterns in powers of ten and how those patterns related to place value.” In this unit, students “Learn to simplify complex-looking exponential expressions, and they learn efficient ways to describe, communicate, and operate with very large and very small numbers.”

• In Unit 2, Solving One Variable Equations, Lesson 6, Tips for Teachers connects 8.EE.7 (Solve linear equations in one variable) to the work from seventh grade (7.EE.B). It states, “Solving equations with variables on both sides of the equal sign is new for 8th graders. Though the approach to solving these equations (simplifying, using inverse operations) is the same as what students are used to, there are now more ways that an equation can be solved and additional places where students may have misconceptions (such as combining terms across the equal sign). Continue to emphasize the importance of maintaining balance in the equation through each ‘move’ students make.” In this lesson, students “Solve equations with variables on both sides of the equal sign.” 

• In Unit 4, Functions, Lesson 6, Tips for Teachers connects 8.F.4 (Construct a function to model a linear relationship between two quantities) to the work from seventh grade (7.RP.2). It states, “In seventh grade, students found the constant of proportionality in proportional graphs. This lesson prepares students to compare functions across multiple representations in upcoming lessons in this unit, and to determine slope of linear functions in Unit 5.” In this lesson, students “Identify properties of functions represented in graphs.”

• In Unit 7, Pythagorean Theorem and Volume, Lesson 4, Tips for Teachers connects 8.NS.1 (Know that numbers that are not rational are called irrational) to the work from seventh grade (7.NS.2d). It states, “Students have prior experience from seventh grade with writing decimal expansions for rational numbers using long division.” In this lesson, students “Represent rational numbers as decimal expansions.”