8th Grade - Gateway 1

The materials reviewed for Eureka Math² Grade 8 meet expectations for assessing grade-level content and, if applicable, content from earlier grades.

Assessment System includes lesson-embedded Exit Tickets, Topic Quizzes, and Module Assessments. According to the Implementation Guide, “Exit Tickets are not graded. They are paper based so that you can quickly review and sort them. Typical Topic Quizzes consist of 4-6 items that assess proficiency with the major concepts from the topic. You may find it useful to grade Topic Quizzes. Typical Module Assessments consist of 6-10 items that assess proficiency with the major concepts, skills, and applications taught in the module. Module Assessments represent the most important content taught in the module. These assessments use a variety of question types, such as constructed response, multiple select, multiple choice, single answer, and multi-part. There are two analogous versions of each Module Assessment available digitally. Analogous versions target the same material at the same level of cognitive complexity.” Examples of summative Module Assessments items that assess grade-level standards include:

• Module 1, Module Assessment 2, Item 5, “Evaluate \frac{(4^5)^2\cdot4^{-8}}{4}.” (8.EE.1)

• Module 2, Module Assessment 1, Item 3, “Figure JKLMN is congruent to figure STUVW. Describe a sequence of rigid motions that maps figure JKLMN onto figure STUVW. Drag one response into each box to correctly describe the sequence of rigid motions.” Responses provided, “90\degree clockwise rotation around the origin, 90\degree counterclockwise rotation around the origin, Reflection across the y-axis, Reflection across the x-axis, Translation 6 units left, Translation 6 units down.” (8.G.2)

• Module 4, Module Assessment 1, Item 4, “Enter the repeating decimal 0.\overline{83} as a fraction.” (8.NS.1)

• Module 6, Module Assessment 1, Item 3, “Consider the scatter plot. Part A: Which type of association best describes the data in the scatter plot?” Answers provided, “Strong, negative, linear association; Strong, positive, linear association; Weak, negative, linear association; Weak, positive, linear association. Part B: Drag the points to draw a line that fits the data in the scatter plot.” (8.SP.1, 8.SP.2)

• Module 6, Module Assessment 2, Item 2, “Sara goes for a two-day hike. She hikes for the first day and then camps for the night. The next morning, she hikes at a constant speed. The table shows the number of hours Sara hiked on the second day and the distance that she hiked in the two days (Table provided shows time hiking on the second day in hours as 1.5, 4, and 7 and the total distance hiked in miles as 9.2, 12.2, and 15.8). Part A: Enter an equation for a function that represents the total distance hiked y in miles when Sara has hiked x hours on the second day. Part B: What do the initial value and the rate of change in your function represent?” (8.F.4)

The materials reviewed for Eureka Math² Grade 8 meet expectations for giving all students extensive work with grade-level problems to meet the full intent of grade-level standards.

Each lesson consists of four sections (Fluency, Launch, Learn, and Land) that provide extensive work with grade-level problems and to meet the full intent of grade-level standards. The Fluency section provides opportunities for students to practice previously learned content and activates students’ prior knowledge to prepare for new learning. Launch activities build context for learning goals. Learn activities present new learning through a series of learning segments. During the Land section, teachers facilitate a discussion to address key questions related to the learning goal. Practice pages can be assigned to students for additional practice with problems that range from simple to complex.

Instructional materials engage all students in extensive work with grade-level problems. Examples include:

• Module 1, Topic A, Lesson 4: Adding and Subtracting Numbers Written in Scientific Notation. Fluency, Problem 3, students write numbers in scientific notation, “5200.” Launch, Problem 3, students combine like terms in expressions and then look for similarities and differences to access prior knowledge in preparation for adding and subtracting in scientific notation, “3m+4m+2m. How are the expressions in problems 1 through 6 alike? How are the expressions in problems 1 through 6 different?” Learn, Problem 8, students add and subtract numbers written in scientific notation with the same and different powers of ten, “For problems 8-10, add or subtract. Write the answer in scientific notation, 3\times10^{12}+2x\times10^{12}+4\times10^{12}.” Land, teachers facilitate a discussion of strategies used during the lesson, “How is a strategy we use to find 2\times10^5+3\times10^5 similar to a strategy we use to find 2x^5+3x^5?” Students then compare the differences. Practice, Problem 8, “Write the answer in scientific notation, 6\times10^6+8\times10^6.” Students engage in extensive work with grade-level problems of 8.EE.4 (Perform operations with numbers expressed in scientific notation, including problems where both decimal and scientific notation are used).

• Module 2, Topic C, Lesson 13: Angle Sum of A Triangle. Fluency, Problem 2, students solve for unknown variables given angles at a point to prepare for showing that the sum of the measures of a triangle’s interior angles is 180\degree, “Solve for x.” Students are shown a pair of intersecting lines with a set of vertical angles measuring 52\degree and a set of vertical angles measuring x\degree. Launch, students use a digital platform to analyze interior angles of a triangle to verify that the sum of the angles is always 180 degrees, “What can you say about the triangles’ interior angles?” Learn, students use a concrete mode (sheet of paper) to draw a triangle, rip the angles and explore the sum of the angles, “If we can arrange these angles on a line, what does that mean about the sum of the measures?” Exit Ticket, Problem 2, students find the missing interior angle of a triangle, “What is the measure of \angleA? Explain how you know?” Students are shown a triangle with angles measuring 90\degree and 61\degree. Practice, Problem 3, students find the measure of a given angle for six different triangles, “For problems 1–6, find the measure of the given angle.” Students are shown a triangle with angles measuring 40\degree and 101\degree. Students engage in extensive work with grade-level problems of 8.G.5, “Use informal arguments to establish facts about the angle sum and exterior angle of triangles, about the angles created when parallel lines are cut by a transversal, and the angle-angle criterion for similarity of triangles.” 

• Module 6, Topic A, Lesson 3: Linear Functions and Proportionality. Fluency, Problem 2, students find the slope of a line to prepare for writing equations that represent linear functions, “Find the slope of the line that passes through the given points.” Launch, students watch a video introducing the costs of going to the state fair, there is a $10 admission fee plus $2.50 per ride ticket, “What is the total amount Ava spent if she purchased 4 tickets? Explain.” Learn, Problem 2, students determine whether a situation can be represented by a linear function, “Yu Yan’s 50-gallon bathtub has 8 gallons of water in it. The plug is in the bathtub so water in it or added to it will not drain out. Water begins to flow from the faucet into a bathtub at a constant rate of 7 gallons of water every 2 minutes. a. Describe the inputs and outputs for this situation. Then complete the table to represent the relationship by filing in inputs and their corresponding outputs. b. Can this situation be represented by a function? Explain your thinking. c. Write an equation to represent the situation.d. Can this situation be represented by a linear function? Explain. e. How long does it take to fill Yu Yan’s bathtub when it starts with 8 gallons of water in it? f. What input makes sense in this context?” Land, students discuss how linear functions and proportional relationships are related, “Are all proportional relationships linear functions? Explain.” Exit Ticket, “Maya runs 1.5 miles around the school and then goes to a track to run laps. Each lap on the track is 0.25 miles. a. Complete the table by entering values of x and y to represent the relationship. b. Can this situation be represented by a function? Explain your thinking. c. Write an equation to represent the situation. d. If Maya runs 6 laps at the track, how many total miles does she run during cross country practice? f. How many laps does Maya need to run on the track during cross-country practice to run a total of 5 miles. g. What inputs make sense in this context? Explain.” Practice, Problem 8, “Henry says all proportional relationships are linear functions. Do you agree? Explain.” Students engage in extensive work with grade-level problems of 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). 

Instructional materials provide opportunities for all students to engage with the full intent of grade-level standards. Examples include: 

• Module 2, Topic D, Lesson 21: Applying the Pythagorean Theorem. Fluency, Problem 4, Students find side lengths of right triangles to prepare for applying the Pythagorean theorem to solve real-world and mathematical problems, “A right triangle has leg lengths a and b and hypotenuse length c. Find the unknown side length. b = 5 and c = 7.” Launch, students use calculators to find the square root, “What is the value of \sqrt{3400} rounded to the nearest tenth?” Learn, Problem 3, students apply the Pythagorean Theorem to a mathematical problem, “The area of the right triangle is 26.46 square units. What is the perimeter of the triangle?” A triangle with a height of 6.3 is pictured. Land, teachers facilitate a discussion of the application of the Pythagorean Theorem to real-world situations, “In what real-world and mathematical situations does the Pythagorean theorem apply?” Practice, Problem 4, students apply the Pythagorean Theorem to a real-world problem using three dimensions, “Consider the diagram of a portable soccer goal. The black lines show the frame of the goal. How many feet of framing are needed for the frame of the goal? Round to the nearest tenth of a foot.” A net with a width of 10 feet, height of 6 feet and a depth of 5 feet is pictured. The materials meet the full intent of 8.G.7 (Apply the Pythagorean Theorem to determine unknown side lengths in right triangles in real-world and mathematical problems in two and three dimensions).

• Module 5, Topic A, Lesson 5: Estimating Solutions, Practice, Problem 2, students graph a system of equations and estimate the solution, “Consider the system of equations. 5x+4y=20 and y=\frac{5}{2}x-4 a. Graph the system of equations. Estimate the coordinates of the intersection point of the lines. b. Is your estimate from part (a) the solution to the system? Why?” In Module 5, Topic B, Lesson 7: The Substitution Method, Exit Ticket, students solve a system of equations algebraically, “Solve the system of equations by using the substitution method. Check your solution. 3y-21=2x, y-4x=2.” Module 5, Topic B, Lesson 9: Rewriting Equations to Solve a System of Equations, Launch, Problem 3, students identify the number of solutions to a system by inspection, “Determine by inspection whether each system of equations has only one solution, no solution, or infinitely many solutions.y=-4x+6, y=-4x-7.” Students meet the full intent of 8.EE.8b (Solve systems of two linear equations in two variables algebraically, and estimate solutions by graphing the equations. Solve simple cases by inspection).

• Module 6, Topic B, Lesson 8: Comparing Functions, Fluency, Problem 5, students identify the slope and y-intercept of a line given its graph to prepare for determining the rate of change and initial value of a linear function from its graph, “line q, slope: 0, y-intercept: 3.” Launch, Students match one representation of a function to other representations of the same function with other classmates, “Find your group by matching your function representation to other possible representations of the same function. Once you have identified other members of your group, make sure that everyone agrees that you have representations of the same function.” Learn, Problem 2, students compare two functions represented in different ways, “Studio A and Studio B price memberships for fitness classes differently. Studio A charges $55 per month for unlimited classes. Studio B depends on the number of classes a person attends. The total monthly cost is a linear function of the number of classes that a person attends. The table shows some inputs and the corresponding outputs that the function assigns. (table shows x: number of classes attended and y: total cost in dollars) a. For what number of classes will the total monthly cost for each studio be the same? b. When are classes at Studio A less expensive than classes at Studio B? c. When are classes at Studio B less expensive than classes at Studio A?” Land, students discuss comparing linear functions represented in different ways, “In what ways can we represent a linear function? Which representation makes it easiest to identify the rate of change and the initial value? Why? How can we find the rate of change and the initial value from the other representations?” Practice, Problem 2, “A local park needs to replace a fence. The park commission compares the price plans of two companies. The cost of the fence of both companies is a linear function of the length of the fence. Company A charges $7000 for building materials and $200 per foot for the length of the fence. Company B charges based on the length of the fence. The table represents some inputs and corresponding outputs for the function that represents the amount charged by company B. a. Which company charges a higher rate of dollars per foot of fencing? How do you know? b. For what length of fencing will both companies charge the same amount? How much will they charge? c. Which company is the better choice if the park commission needs 190 feet of fencing?” The materials meet the full intent of 8.F.2 (Compare properties of two functions each represented in a different way [algebraically, graphically, numerically in tables, or by verbal description]).

The materials reviewed for Eureka Math² Grade 8 meet expectations that, when implemented as designed, the majority of the materials address the major work of each grade.

• There are 6 instructional modules, of which 6 modules address major work of the grade or supporting work connected to major work of the grade, approximately 100%.

• There are 127 instructional lessons, of which 105.5 lessons address major work of the grade or supporting work connected to major work of the grade, approximately 83%.

• There are 160 instructional days, of which 135.5 address major work of the grade or supporting work connected to the major work of the grade, approximately 85%. Instructional days include 127 instructional lessons, 27 topic assessments, and 6 module assessments.

A lesson-level analysis is most representative of the instructional materials as the lessons include major work and supporting work connected to major work. As a result, approximately 83% of the instructional materials focus on major work of the grade.

The materials reviewed for Eureka Math² Grade 8 meet expectations that supporting content enhances focus and coherence simultaneously by engaging students in the major work of the grade.

Each lesson contains Achievement Descriptors that provide descriptions and about what the students should be able to do after completing the lesson and lists standards. Materials do not provide information about connections between standards in lessons.

Materials connect learning of supporting and major work to enhance focus on major work. Examples include:

• Module 1, Topic E, Lesson 22: Familiar and Not So Familiar Numbers, Learn, students classify numbers in decimal form by using the descriptions of rational and irrational numbers, “Are all square roots irrational? Invite a variety of opinions and then display the following examples. \sqrt{2}=1.41421356… and \sqrt{25}=5. Give students time to think–pair–share about the following discussion questions: Are these numbers rational or irrational? How do you know? Are all square roots irrational? How do you know? After pairs share their answers with the class, instruct students to add each number, \sqrt{2} and \sqrt{25}, to the Examples row of the table in problem 9.” This connects the supporting work of 8.NS.1 (Know that numbers that are not rational are called irrational. Understand informally that every number has a decimal expansion; for rational numbers show that the decimal expansion repeats eventually, and convert a decimal expansion which repeats eventually into a rational number) to the major work of 8.EE.2 (Use square roots and cube root symbols to represent solutions to equations in the form x^2=p and x^3=p, where p is a positive rational number. Evaluate square roots of small perfect squares and cube roots of small perfect cubes.  Know that \sqrt{2} is irrational).

• Module 4, Topic A, Lesson 6: An Interesting Application of Linear Equations, Part 2, Learn, Problems 7 and 8, students use linear equations to write the fraction form of a decimal with digits that do not all repeat, “For problems 7 and 8, write the fraction form of the decimal. Identify the powers of 10 that both sides of the equation need to be multiplied by to find the fraction form. 7. \overline{2.138} Powers of 10: ____ and ____. 8. \overline{0.74} Powers of 10: ____ and ____.” This connects the supporting work of 8.NS.1 (Know that numbers that are not rational are called irrational. Understand informally that every number has a decimal expansion; for rational numbers show that the decimal expansion repeats eventually,, and convert a decimal expansion which repeats eventually into a rational number) to the major work 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).

• Module 6, Topic E, Lesson 25: Applications of Volume, Learn, Problem 4, students answer a real-world question involving the volumes of a cylinder and a sphere by using an equation to represent a function, “An ice cream scoop makes scoops that are essentially spherical with a 3-inch diameter. This ice cream scoop is used to take scoops out of a full, cylindrical ice cream carton that is 11.5 inches in diameter and 11 inches in height. A function represents the relationship between the approximate volume of ice cream V in cubic inches left in the carton and the number of scoops n that have been removed. a. What is the initial value of this function? What does that number mean in context? Use 3.14 for \pi. Round to the nearest hundredth. b. What is the rate of change of this function? What does that number mean in context? Use 3.14 for \pi. Round to the nearest hundredth. c. Write an equation for a function that describes the approximate volume of ice cream V in cubic inches left in the carton after n scoops have been removed. d. How many full scoops of ice cream can be removed from the carton? Assume that all ice cream in the carton can be scooped.” This connects the supporting work of 8.G.9 (Know the formulas for the volumes of cones, cylinders, and spheres and use them to solve real-world and mathematical problems) to the major work 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).

The materials reviewed for Eureka Math² Grade 8 meet expectations for including problems and activities that serve to connect two or more clusters in a domain or two or more domains in a grade.

Each lesson contains Achievement Descriptors that provide descriptions and about what the students should be able to do after completing the lesson and lists standards. Materials do not provide information about connections between standards in lessons.

Materials provide connections from major work to major work throughout the grade-level when appropriate. Examples include.

• Module 1, Topic D, Lesson 19: Using the Pythagorean Theorem, Learn, Problem 3, students find the length of a rational hypotenuse given the legs of a right triangle when given the lengths of the legs of the triangle. “For problems 1-4, find the length of the hypotenuse c.” Students are given a diagram of a right triangle with side lengths of 0.4 and 0.3 units. and are asked to find the length of a hypotenuse. This connects the major work of 8.EE.A (Work with radicals and integer exponents) to the major work of 8.G.B (Understand and apply the Pythagorean Theorem). 

• Module 6, Topic A, Lesson 3: Linear Functions and Proportionality, Learn, Problem 4, students determine whether a situation can be represented by a linear function, “A pet shelter has 450 cans of pet food that were donated through a school’s canned pet food drive. A local business pledges to donate an additional 15 cans of pet food a week to the shelter. a. Complete the table by entering values of x and y to represent the situation. b. Can this situation be represented by a function? Explain your thinking. c. Write an equation to represent this situation. d. Can this situation be represented by a linear function? Explain. e. What input makes sense in this context? Explain. f. How many weeks does it take to have 705 cans donated?” This connects the major work of 8.F.A (Define, evaluate, and compare functions) to the major work of 8.EE.C (Analyze and solve linear equations and pairs of simultaneous linear equations).

• Module 6, Topic B, Lesson 10: Graphs of Nonlinear Functions, Learn, Problem 5, students analyze equations and graphs to determine if they represent functions. “For problems 5-11, use each equation with its corresponding graph to answer the questions. y=x Does the relationship given by the equation and the graph represent y as a function of x? Explain. b. Is this relationship is a function, is the function linear or nonlinear? Explain.” This connects the major work of 8.F.A (Define, evaluate, and compare functions) to the major work of 8.F.B (Use functions to model relationships between quantities). Materials provide connections from supporting work to supporting work throughout the grade-level when appropriate. Examples include:

• Module 6, Topic E, Lesson 23: Volume of Cones, Learn, Problem 3, students develop and use the formula for the volume of a cone. “Find the approximate volume of the cone shown on card L by using 3.14 for \pi.” Students are shown a cone with a radius and height of three units. This 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).

The materials reviewed for Eureka Math² 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.

Each Module Overview contains Before This Module and After This Module looking forward and back respectively, to reveal coherence across modules and grade levels. The Topic Overview includes information about how learning connects to previous or future content. Some Teacher Notes within lessons enhance mathematical reasoning by providing connections/explanations to prior and future concepts.

Content from future grades is identified and related to grade-level work. Examples include:

• Module 2: Rigid Motions and Congruent Figures, Module Overview, After this Module, Algebra I Module 3, “In Algebra I, students apply rigid motions to describe transformations of functions and to graph functions in the coordinate plane.” Geometry Module 1, “In Geometry, students combine their knowledge of rigid motions and functions to recognize a rigid motion as a function of the plane. They also use the properties of translations, reflections, and rotations to justify triangle congruence theorems.”

• Module 4, Topic A, Lesson 2: Solving Linear Equations, Learn, Teacher Note, “In grade 8, students only encounter examples of the multiplication property of equality in which both sides of an equation are multiplied by a nonzero number. If both sides of an equation such as 6x-3=4x+5 are multiplied by 0, the resulting equation is 0 = 0, which does not have the same solution set as 6x-3=4x+5. In later courses, students will refine their understanding to recognize that although the multiplication property of equality applies when multiplying by any number, multiplying both sides of an equation by 0 changes the solution set of the original equation.”

• Module 6, Topic C: Bivariate Numerical Data, Topic Overview, “The topic begins with constructing scatter plots and looking for patterns in the data. Students find that these patterns are evidence of association, but that association does not imply a cause and effect relationship. They describe the shape of these patterns of association as linear or nonlinear and describe the direction of linear associations as positive or negative. In preparation for work in Algebra I, they also describe the strength of these associations as weak, moderate, or strong.” 

Materials relate grade-level concepts explicitly to prior knowledge from earlier grades. Examples include:

• Module 1, Topic D: Perfect Squares, Perfect Cubes, and the Pythagorean Theorem, Topic Overview, “Topic D introduces the Pythagorean theorem to motivate and support the need for expressing numbers that are not rational and to develop students’ early awareness of the meaning of these numbers. Working with the Pythagorean theorem inherently builds on and extends students' prior exploration in grade 7 of the conditions that determine a unique triangle.”

• Module 3: Dilations and Similar Figures, Module Overview, Before This Module, Grade 7 Module 1, “Students apply their knowledge of proportional reasoning and scale drawings from grade 7 to understand dilations and their properties.” Grade 8 Module 2, “Students build upon their geometric skills of applying rigid motions to the plane. They analyze how the properties of rigid motions are similar to and different from the properties of dilations.”

• Module 5, Topic C, Lesson 14: Back to the Coordinate Plane, Learn, Teacher Note, “If students need support with writing the equation for line , consider having them graph the points (5,1) and (-1,1). Students may also refer to module 4 topic F for more practice.”