2021

Eureka Math²

Publisher
Great Minds
Subject
Math
Grades
K-8
Report Release
03/29/2023
Review Tool Version
v1.5
Format
Core: Comprehensive

EdReports reviews determine if a program meets, partially meets, or does not meet expectations for alignment to college and career-ready standards. This rating reflects the overall series average.

Alignment (Gateway 1 & 2)
Meets Expectations

Materials must meet expectations for standards alignment in order to be reviewed for usability. This rating reflects the overall series average.

Usability (Gateway 3)
Meets Expectations
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About This Report

Report for 8th Grade

Alignment Summary

The materials reviewed for Eureka Math² Grade 8 meet expectations for Alignment to the CCSSM. In Gateway 1, the materials meet expectations for focus and coherence. In Gateway 2, the materials meet expectations for rigor and meet expectations for practice-content connections.

8th Grade
Alignment (Gateway 1 & 2)
Meets Expectations
Gateway 3

Usability

24/27
0
17
24
27
Usability (Gateway 3)
Meets Expectations
Overview of Gateway 1

Focus & Coherence

The materials reviewed for Eureka Math² Grade 8 meet expectations for focus and coherence. For focus, the materials assess grade-level content and provide all students extensive work with grade-level problems to meet the full intent of grade-level standards. For coherence, the materials are coherent and consistent with the CCSSM.

Criterion 1.1: Focus

06/06

Materials assess grade-level content and give all students extensive work with grade-level problems to meet the full intent of grade-level standards.

The materials reviewed for Eureka Math² Grade 8 meet expectations for focus as they assess grade-level content and provide all students extensive work with grade-level problems to meet the full intent of grade-level standards.

Indicator 1A
02/02

Materials assess the grade-level content and, if applicable, content from earlier grades.

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 (45)2484\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°90\degree clockwise rotation around the origin, 90°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 0.830.\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)

Indicator 1B
04/04

Materials give all students extensive work with grade-level problems to meet the full intent of grade-level standards.

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+2m3m+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, 3×1012+2x×1012+4×10123\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 2×105+3×1052\times10^5+3\times10^5 similar to a strategy we use to find 2x5+3x52x^5+3x^5?” Students then compare the differences. Practice, Problem 8, “Write the answer in scientific notation, 6×106+8×1066\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°180\degree, “Solve for x.” Students are shown a pair of intersecting lines with a set of vertical angles measuring 52°52\degree and a set of vertical angles measuring x°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 ? Explain how you know?” Students are shown a triangle with angles measuring 90°90\degree and 61°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°40\degree and 101°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+by=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 3400\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=205x+4y=20 and y=52x4y=\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. 3y21=2x3y-21=2x, y4x=2y-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+6y=-4x+6, y=4x7y=-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]).

Criterion 1.2: Coherence

08/08

Each grade’s materials are coherent and consistent with the Standards.

The materials reviewed for Eureka Math² Grade 8 meet expectations for coherence. The materials: address the major clusters of the grade, have supporting content connected to major work, make connections between clusters and domains, and have content from prior and future grades connected to grade-level work.

Indicator 1C
02/02

When implemented as designed, the majority of the materials address the major clusters of each grade.

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.

Indicator 1D
02/02

Supporting content enhances focus and coherence simultaneously by engaging students in the 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. 2=1.41421356\sqrt{2}=1.41421356… and 25=5\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, 2\sqrt{2} and 25\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 x2=px^2=p and x3=px^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 2\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. 2.138\overline{2.138} Powers of 10: ____ and ____. 8. 0.74\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 π\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 π\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).

Indicator 1E
02/02

Materials include problems and activities that serve to connect two or more clusters in a domain or two or more domains in a grade.

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=xy=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 π\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).

Indicator 1F
02/02

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.

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 6x3=4x+56x-3=4x+5 are multiplied by 0, the resulting equation is 0 = 0, which does not have the same solution set as 6x3=4x+56x-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.”

Indicator 1G
Read

In order to foster coherence between grades, materials can be completed within a regular school year with little to no modification.

The materials reviewed for Eureka Math2 Grade 8 foster coherence between grades and can be completed within a regular school year with little to no modification.

Recommended pacing information is found in the Implementation Guide on page 21. The instructional materials include pacing for 129 lessons. 

  • Instructional Days: There are six instructional modules with 129 lessons. The Implementation Guide states, “Plan to teach one lesson per day of instruction. Each lesson is designed for an instructional period that lasts 45 minutes. Grade levels and courses have fewer lessons than the typical number of instructional days in a school year. This provides some flexibility in the schedule for assessment and responsive teaching, and it allows for unexpected circumstances.”

  • Modules: There are six learning modules organized by related lessons into modules.

  • Assessments: There are six summative module assessments and formative assessments for each topic. The Implementation Guide states, “In addition to the lessons referenced in the table above, Eureka Math2 provides assessments that can be given throughout the year at times you choose. You can also flexibly use class periods to address instructional needs such as reteaching and preteaching.”

  • Optional Lessons: Two lessons are designated as optional: Module 1, Topic B, Lesson 10 and Module 5, Topic B, Lesson 8. The Implementation Guide states, “Some lessons in each grade level or course are optional. Optional lessons are clearly designated in the instructional sequence, and they are included in the total number of lessons per grade level or course. Assessments do not include new learning from optional lessons. Lessons may be optional for the following reasons: The lesson is primarily for enrichment. The lesson offers more practice with skills, concepts, or applications. The lesson bridges gaps between standards.”

Overview of Gateway 2

Rigor & the Mathematical Practices

The materials reviewed for Eureka Math² Grade 8 meet expectations for rigor and balance and practice-content connections. The materials help students develop procedural skills, fluency, and application. The materials also make meaningful connections between the Standards for Mathematical Content and the Standards for Mathematical Practice (MPs).

Criterion 2.1: Rigor and Balance

08/08

Materials reflect the balances in the Standards and help students meet the Standards’ rigorous expectations, by giving appropriate attention to: developing students’ conceptual understanding; procedural skill and fluency; and engaging applications.

The materials reviewed for Eureka Math² Grade 8 meet expectations for rigor. The materials develop conceptual understanding of key mathematical concepts, give attention throughout the year to procedural skill and fluency, and spend sufficient time working with engaging applications of mathematics. There is a balance of the three aspects of rigor within the grade.

Indicator 2A
02/02

Materials develop conceptual understanding of key mathematical concepts, especially where called for in specific content standards or cluster headings.

The materials reviewed for Eureka Math2 Grade 8 meet expectations for developing conceptual understanding of key mathematical concepts, especially where called for in specific content standards or cluster headings.

The Learn portion of the lesson presents new learning through instructional segments to develop conceptual understanding of key mathematical concepts. Students independently demonstrate conceptual understanding in Exit Tickets, formative assessments that close the learning, and Practice, additional practice problems aligned to the lesson’s learning objectives.

Materials develop conceptual understanding throughout the grade level. Examples include:

  • Module 1, Topic D, Lesson 16: Perfect Squares and Cubes, Learn, students apply prior knowledge of area and volume to explore perfect squares and perfect cubes. “Have students check answers with a partner. Then use the following prompts to facilitate a discussion about problems 1−4. If you know the side length of a square, how do you find its area? If you know the edge length of a cube, how do you find its volume? Problem 2 gives the area of a square. Describe how you can determine the square’s side length from its area. Problem 4 gives the volume of a cube. Describe how you can determine the cube’s edge length from its volume. When we multiply two equal factors, we say we are squaring the number. When we multiply three equal factors, we say we are cubing the number.” This activity supports the conceptual understanding of 8.EE.2 (Use square root and cube root symbols to represent solutions to equations of the form x2=px^2=p, where p is a positive rational number. Evaluate square roots of small perfect squares and cube roots of small perfect cubes. Know that 2\sqrt{2} is irrational).

  • Module 4, Topic C, Lesson 13: The Graph of a Linear Equation in Two Variables, Learn, Problem 5, students define and use the x-intercept and y-intercept points to graph a linear equation in standard form. “Use the table to find two ordered pairs that satisfy the equation 4x3y=124x-3y=12. Then use the ordered pairs to graph the equation in the coordinate plane. What x- or y-value did you use to find an ordered pair that satisfies the equation? What ordered pair resulted from that value? Why did you use that value? Why would we want to use 0 as a value for x or y to find an ordered pair that satisfies the equation 4x3y=124x-3y=12? What is special about the locations of these points? These points are called intercept points because the line meets the axes at these points. An x-intercept point of a graph has the coordinates (a,0) and is a point where the graph intersects the x-axis. A y-intercept point of a graph has the coordinates (0,b) and is a point where the graph intersects the y-axis.” This activity supports the conceptual understanding of 8.EE.6 (Use similar triangles to explain why the slope m is the same between any two distinct points on a non-vertical line in the coordinate plane; derive the equation y=mxy=mx for a line through the origin and the equation y=mx+by=mx+b for a line intercepting the vertical axis at b).

  • Module 6, Topic A, Lesson 2: Definition of a Function, Learn, Problem 2, students make sense of functions in terms of inputs and outputs and formalize the definition. Teachers state, “This table includes the distance traveled by a falling fish after a specified number of seconds since its release from a height of 256 feet. The function describing this situation assigns exactly one distance to each moment in time the fish is falling. 2. A fish is released from a height of 256 feet. Complete each statement by using the rule for the function that relates the distance traveled in feet to the time in seconds. When students have completed the table, use th following prompts to facilitate a discussion: The values of x, which correspond to the values of the independent variable, are referred to as the input values, or sometimes simply, the input. What do you think we call the value of the dependent variable or the number that is assigned to a given input? We can say that a function assigns to each input exactly one output. This means that each input corresponds to one and only one output. This understanding helps us build toward a more formal definition of a function.” This activity supports the conceptual understanding of 8.F.3 (Interpret the equation y=mx+by=mx+b as defining a linear function, whose graph is a straight line; give examples of functions that are not linear).

Materials provide opportunities for students to independently demonstrate conceptual understanding throughout the grade level. Examples include:

  • Module 2, Topic D, Lesson 17: Proving the Pythagorean Theorem, Exit Ticket, with a partner, students engage in a digital lesson to complete a puzzle that is a visual representation of the Pythagorean Theorem. “1. How did your partner use rigid motions in their proof? 2. How did your partner use the sum of the interior angle measures of a triangle in their proof?” Students independently demonstrate conceptual understanding of 8.G.6 (Explain a proof of the Pythagorean Theorem and its converse).

  • Module 3, Topic B, Lesson 4: Using Lined Paper to Explore Dilations, Exit Ticket, students draw and label a dilated image. “The diagram shows a dilation with center O and scale factor 117\frac{11}{7}. a. Place a point R on PQ\overline {PQ}. Draw and label the image of R under a dilation with center O and scale factor 117\frac{11}{7}. b. How do you know the image of R is in the correct location?” Students independently demonstrate conceptual understanding of 8.G.3 (Describe the effect of dilations, translations, rotations, and reflections on two-dimensional figures using coordinates).

  • Module 5, Topic A, Lesson 3: Identifying Solutions, Practice, Problem 1, students determine if a system of equations has a solution and explain. “Consider the system of equations. y=12x+4y=\frac{1}{2}x+4 and y=12x+1y=\frac{1}{2}x+1 a. Without graphing, determine whether the system of equations has a solution. Explain how you know. b. Graph the system of equations to confirm your answer from part (a). If the system has a solution, estimate the coordinates of the point of intersection of the two lines. Then check whether your estimate is the solution to the system.” Students independently demonstrate conceptual understanding of 8.EE.8 (Analyze and solve pairs of simultaneous linear equations).

Indicator 2B
02/02

Materials give attention throughout the year to individual standards that set an expectation for procedural skill and fluency.

The materials reviewed for Eureka Math2 Grade 8 meet expectations for giving attention throughout the year to individual standards that set an expectation of procedural skill and fluency.

The Learn portion of the lesson presents new learning through instructional segments to develop procedural skill of key mathematical concepts. Students independently demonstrate procedural skill in Exit Tickets, formative assessments that close the learning, and Practice, additional practice problems aligned to the lesson’s learning objectives.

Materials develop procedural skills and fluency throughout the grade level. Examples include:

  • Module 1, Topic B, Lesson 8: Making Sense of Integer Exponents, Learn, Problem 4, students define negative exponents. “For problems 4–9, use the definition of negative exponents to write an equivalent expression. Assume that x is nonzero. (5)9(-5)^{-9}” Students develop procedural skill of 8.EE.1 (Know and apply the properties of integer exponents to generate equivalent numerical expressions).

  • Module 1, Topic D, Lesson 16: Perfect Squares and Perfect Cubes, Exit Ticket, Problems 1 - 2, “For problems 1 - 3, state whether the number is a perfect square, a perfect cube, both, or neither. Explain. 1. 196; 2. 150.” Students develop procedural skill and fluency of 8.EE.2 (Use square root and cube root symbols to represent solutions to equations in the form of x2=px^2=p and x3=px^3=p, where p is a positive rational number. Evaluate square roots of small perfect squares and cube roots of small perfect cubes…). 

  • Module 4, Topic A, Lesson 2: Solving Linear Equations, Learn, Problems 3-5, students solve linear equations with variables on both sides. “Direct students’ attention to the equations in problems 3–5 and ask the following questions. How are the equations in problems 3–5 similar to the equation in problem 2? How are some of these equations different from the equation in problem 2? For problems 3–5, solve the equation. Check your solution. 3. 273a=8a+527-3a=8a+5, 4. 9(c2)=3c289(c-2)=3c-28, 5. 84m=2(m1)+108-4m=2(m-1)+10. Confirm answers as a class. If there are any discrepancies, consider having pairs share their solution steps. Emphasize that checking solutions is an important step to ensure accuracy. If time allows, consider asking students to identify instances in their work where they applied the distributive property and, in more subtle ways, the commutative and associative properties.” Students develop procedural skill of 8.EE.7 (Solve linear equations in one variable).

Materials provide opportunities for students to independently demonstrate procedural skills and fluency throughout the grade level. Examples include:

  • Module 1, Topic E, Lesson 21: Approximating Values of Roots and π2\pi^2, Practice, Problems 1-4, students approximate square and cube roots, “For problems 1–4, determine the two consecutive whole numbers each value is between. 1. ___ ___ 2. ___ ___ 3. ___ ____ 4. ______.” Students independently demonstrate procedural skill 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).

  • Module 5, Topic B, Lesson 6: Solving System of Linear Equations without Graphing, Exit Ticket, students solve systems of linear equations by using the substitution method to write the systems as linear equations in one variable. “Solve the system of equations by using the substitution method. Check your solution. 3y12=2x,y+8=2x{3y-12=2x,y+8=2x}” Students independently demonstrate procedural skill 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 5, Topic B, Lesson 10: Choosing A Solution Method, Practice, Problem 2, students determine the number of solutions in a system of equations and solve, “For problems 1–5, determine the number of solutions to the system of equations. If the system has only one solution, solve the system of equations by graphing or by using the substitution method, and check your answer. If you want to solve by graphing, use the graphs provided at the end of problem 5. y=23xy=-\frac{2}{3}x and 4x+2y=64x+2y=6.” Students independently demonstrate procedural skill of 8.EE.8 (Analyze and solve pairs of simultaneous linear equations).

Each lesson begins with Fluency problems that provide practice of previously learned material. The Implementation Guide states, “Fluency provides distributed practice with previously learned material. It is designed to prepare students for new learning by activating prior knowledge and bridging small learning gaps. Fluency activities are included with each lesson, but they are not accounted for in the overall lesson time. Use them as bell ringers, or, in a class period longer than 45 minutes, consider using the facilitation suggestions in the Resources to teach the activities as part of the lesson.” For example, Module 2, Topic C, Lesson 12: Lines Cut By a Transversal, Fluency, Problem 4, students solve for unknown variables given linear pairs. “Directions: Solve for x.” Students are shown a straight line with angle measures 63°63\degree and x°x\degree. Students practice fluency 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).

Indicator 2C
02/02

Materials are designed so that teachers and students spend sufficient time working with engaging applications of the mathematics.

The materials reviewed for Eureka Math2 Grade 8 meet expectations for being designed so that teachers and students spend sufficient time working with engaging applications of the mathematics.

The Learn portion of the lesson presents new learning through instructional segments to develop application of mathematical concepts. Students independently demonstrate routine application of the mathematics in Exit Tickets, formative assessments that close the learning, and Practice, additional practice problems aligned to the lesson’s learning objectives.

Materials include multiple routine and non-routine applications of the mathematics throughout the grade level. Examples include:

  • Module 1, Topic B, Lesson 7: Making Sense of the Exponent 0, Learn, Problems 2-3, students test their predictions about the value of an expression with an exponent of 0. “2. Could 100=1010^0=10? Use the product 10010310^0\cdot10^3 to show whether it upholds the property xmxn=xm+n3x^m\cdot x^n=x^{m+n}3. Could 100=110^0=1? Use the product 10010310^0\cdot10^3 to show whether it upholds the property xmxn=xm+nx^m\cdot x^n=x^{m+n}.” In this non-routine problem, students apply the mathematics of 8.EE.1 (Know and apply the properties of integer exponents to generate equivalent numerical expressions).

  • Module 2, Topic D, Lesson 22: On the Right Path, Learn, students use a grid map of an amusement park and a table with wait and ride times to find the distance between two points. “Suppose you have 1 hour and 30 minutes left until closing time to explore the rides at this section of the amusement park. Starting at the entrance, use the map to determine a path you can take to ride any 4 rides and make it back to the entrance by closing time. Guidelines: You cannot repeat a ride. You run between rides at 5 miles per hour. a. Sketch your path on the map. b. How long does it take you to make it back to the entrance? c. Do you have extra time before the park closes? If so, how much?” In this non-routine problem, students apply the mathematics of 8.G.7, “Apply the Pythagorean Theorem to determine unknown side lengths in the right triangles in real-world and mathematical problems in two and three dimensions.”

  • Module 4, Topic B, Lesson 10: Using Linear Equations to Solve Real-World Problems, Learn Problem 2, students work together to solve routine problems that involve linear equations. “A new pizza place is selling discount cards to gain customers. One card offers the cardholder 12 pizzas at a discounted price plus 6 pizzas for $1 each. Another card offers the cardholder 6 visits where they get 3 pizzas per visit: 2 pizzas for a discounted price and 1 pizza for $1. At what discounted price would the deals on the cards be the same?” In this routine problem, students apply the mathematics of 8.EE.7 (Solve linear equations in one variable).

Materials provide opportunities for students to independently demonstrate multiple routine and non-routine applications of the mathematics throughout the grade level. Examples include:

  • Module 3, Topic B, Lesson 5: Figures and Dilations, Practice, Problem 5, students draw images of figures under dilations with various scale factors. “Kabir’s Cookie Shop is creating flyers with a cookie-shaped logo. Kabir applies a dilation with center C and a scale factor of 3 to enlarge the size of the logo. He finds the image of five points. What is a problem Kabir could have with this dilation?” In this non-routine problem, students independently apply the mathematics of 8.G.3 (Describe the effect of dilations, translations, rotations, and reflections on two-dimensinal figures using coordinates).

  • Module 5, Topic C, Lesson 14: Back to the Coordinate Plane, Practice, Problem 6, students write and linear equations when given information about two lines to identify points. “Line q has y-intercept -10 and slope 49\frac{4}{9}. Line l has x-intercept -18 and y-intercept -4. What ordered pairs, if any, satisfy the equations of lines q and l?” In this routine problem, students independently apply the mathematics of 8.EE.8 (Analyze and solve pairs of simultaneous linear equations).

  • Module 6, Topic B, Lesson 9: Increasing and Decreasing Functions, Practice, Problem 5, students describe qualitative features of a function by analyzing a graph. “Create your own graph story. a. Write a story about the relationship between two quantities. Any quantities can be used, such as distance and time, money and hours, or age and growth. Include keywords in your story, such as increase and decrease, to describe the relationship. b. Label each axis with the quantity of your choice. Then sketch a graph of the function that represents the relationship described in your story.” In this non-routine problem, students independently apply the mathematics of 8.F.5 (Describe quantitatively the functional relationship between two quantities by analyzing a graph. Sketch a graph that exhibits the qualitative features of a function that has been described verbally).

Indicator 2D
02/02

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.

The materials reviewed for Eureka Math2 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 the grade. Examples where the materials attend to conceptual understanding, procedural skill and fluency, or application include:

  • Module 2, Topic D, Lesson 22: On The Right Path, Practice, Problem 8, students use the Pythagorean Theorem to find a missing side length. “Find the unknown side length.” Pictured is a right triangle with a leg of 9 and hypotenuse of 15. Students attend to the procedural skill 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 4, Topic B, Lesson 10: Using Linear Equations to Solve Real-World Problems, Exit Ticket, students use linear equations to solve a real-world problem. “At a trampoline park, the individual rate is $6.00 per person plus an additional $2.00 per person for socks. The group rate is a $25.00 rental fee plus $8.00 per person, and socks are included. How many people must be in a group for the group rate to cost the same amount as the individual rate?” Students attend to the application of 8.EE.7 (Solve linear equations in one variable).

  • Module 6, Topic D, Lesson 18: Bivariate Categorical Data, Land, students construct and interpret a two-way table summarizing a bivariate categorical data set. “What do bivariate categorical data represent? How can we organize and display bivariate categorical data? How are scatter plots and two-way tables similar? How are they different?” Students develop conceptual understanding of 8.SP.4 (Use measures of center and measures of variability for numerical data from random samples to draw informal comparative references about two populations).

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

  • Module 1, Topic E, Lesson 24: Revisiting Equations with Squares and Cubes, Practice, Problem 16, students solve squared and cubed equations involving squares and cubes of rational numbers with real number solutions. “Fill in the boxes with any digits 1–9 to create an equation that has the described solutions. Each digit can be used only once. a. Two rational solutions b. One irrational solution.” Students are shown boxes with x to the ___ power + ____ = ____. Students engage in conceptual understanding and procedural skill of 8.NS.1 (Know that numbers that are not rational are called irrational). 

  • Module 3, Topic D, Lesson 15: Applications of Similar Figures, Exit Ticket, students use properties of similar figures to solve real-world problems. “Lily stands near a streetlamp. The light from the streetlamp causes her to cast a shadow. Lily is 5.5 feet tall. The shadow she casts is 3 feet long. Lily is 10 feet away from the streetlight. What is the height of the streetlamp? Round your answer to the nearest foot.” Students engage in procedural skill and application 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 B, Lesson 6: Linear Functions and Rate of Change, Exit Ticket, students calculate rate of change for a linear function and interpret the rate of change in context. “The tables show values of two functions. The functions represent the number of downloads for two different songs for a given number of days after their release. a. Is the function representing downloads for song B a linear function? Explain. b. If the function representing downloads for song B is a linear function, what is the rate of change? What does the rate of change mean in context? c. Is the function representing downloads for song C a linear function? Explain d. If the function representing downloads for song C is a linear function, what is the rate of change? What does the rate of change mean in context?” Students engage in procedural skill, conceptual understanding, and application 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 graph).

Criterion 2.2: Math Practices

10/10

Materials meaningfully connect the Standards for Mathematical Content and Standards for Mathematical Practice (MPs).

The materials reviewed for Eureka Math² Grade 8 meet expectations for practice-content connections. The materials meaningfully connect the Standards for Mathematical Content and the Standards for Mathematical Practice (MPs).

Indicator 2E
02/02

Materials support 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.

The materials reviewed for Eureka Math2 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. 

Each lesson provides opportunities for students to engage with multiple Standards for Mathematical Practice (MP) with a focus on one MP within each lesson. Margin Notes, Promoting the Standards for Mathematical Practice, highlights student engagement with the MP focused in the lesson. These notes provide specific lesson information, ideas, and questions to deepen student engagement. 

Materials provide intentional development of MP1 to meet its full intent in connection to grade-level content. Students make sense of problems and persevere in solving them as they work with the support of the teacher and independently throughout the units. Examples include:

  • Module 1, Topic C, Lesson 13: Applications with Numbers in Scientific Notation, Learn, students make sense of problems as they examine a new context to determine missing information. “What do you notice? What do you wonder? What information do you need to answer the question, ‘How many breaths would it take to reach the goal?’ How did you figure out the answers to these questions? What is another operation we could use to figure out the answers? Do we have enough information to figure out how many breaths it takes to reach the goal? How do you know?” Teacher margin note states, “Ask the following questions to promote MP1: What information or facts do you need to solve this problem? What are some things you could try to solve the problem? What is your plan to solve the problem?”

  • Module 4, Topic A, Lesson 4, Launch, Problem 1, students use a variety of strategies to solve consecutive integer and geometric relationship problems by defining variables and writing equations. “The sum of three consecutive integers is 372. What are the integers?” Teacher margin note states, “Ask the following questions to promote MP1: What is your plan for defining the variable? What information or facts do you need to write the expressions that represent the situation? What do you need to write the equation? Does the solution make sense? Why?”

  • Module 5, Topic A, Lesson 5, Launch, Problem 1, students monitor and evaluate their progress as they use lines in the coordinate plane to estimate the solution for a system of linear equations. “Consider the graph of the system of linear equations {x3y=9,y=3x+2x-3y=9,y=-3x+2}. a. Estimate the coordinates of the intersection point of the lines. b. Determine whether your estimate from part (a) is the solution to the system.” Teacher margin note states, “Ask the following questions to promote MP1: Does your estimated solution make sense? why? What is your plan to determine whether you have found the actual solution to the system?”

Materials provide intentional development of MP2 to meet its full intent in connection to grade-level content. Students reason abstractly and quantitatively as they work with the support of the teacher and independently throughout the units. Examples include:

  • Module 1, Topic D, Lesson 19: Finding the Length of the Hypotenuse, Learn, Problem 2, students apply the Pythagorean theorem to represent the situation symbolically. “For problems 1–4, find the length of the hypotenuse c.” Shown is a right triangle with leg lengths of 5 and 12. Teacher margin note states, “Ask the following questions to promote MP2: Do both solutions of −10 and 10 make sense in the context of side lengths? What real-world situations are modeled by right triangles?”

  • Module 3, Topic D, Lesson 15: Application of Similar Figures, Learn, Problems 3 and 4, “When students decontextualize problems by drawing a diagram to represent real-world scenarios and use facts about similar triangles to solve these problems, they are reasoning quantitatively and abstractly (MP2). For problems 3 - 5, use the space provided to take notes and draw a diagram from the video. Problem 3, What is the height of the building to the nearest foot? Problem 4, What is the length of the flagpole to the nearest tenth of a foot?”

  • Module 5, Topic C, Lesson 12: Solving Historical Problems with Systems of Equations, Learn: Balancing Birds, students consider units and attend to quantities as they solve systems of equations. “This scale has a 1-jin weight attached. A jin is an ancient Chinese unit of weight. Can you balance the scale using the swallows and sparrows?” Teacher margin note states, “Ask the following questions to promote MP2: What is the problem asking you to do? Does the solution you found make sense mathematically?”

Indicator 2F
02/02

Materials support 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.

The materials reviewed for Eureka Math2 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. 

Each lesson provides opportunities for students to engage with multiple Standards for Mathematical Practice (MP) with a focus on one MP within each lesson. Margin Notes, Promoting the Standards for Mathematical Practice, highlights student engagement with the MP focused in the lesson. These notes provide specific lesson information, ideas, and questions to deepen student engagement. 

Materials provide support for the intentional development of MP3 by providing opportunities for students to construct viable arguments in connection to grade-level content. Examples include:

  • Module 1, Topic D, Lesson 17: Solving Equations with Squares and Cubes, Learn, Problem 14, students construct viable arguments as they solve equations of the form x2=px^2=p, where p is a perfect square, “w2=4w^2=-4.” Students engage in the “Take a Stand Routine” by standing in the corner of the room near the sign that shows what they believe to be the correct solution. The solution signs posted are 2 and -2, -2, 2, and No Solution. Teachers are prompted to ask, “Is your choice of sign a guess, or do you know for sure? How do you know for sure? What questions can you ask your peers who are standing by other signs to make sure you understand the reason for their choice?”

  • Module 3, Topic C, Lesson 13: Similar Triangles, Learn, Problem 4, students justify their thinking as they use known angle relationships to help determine whether two triangles are similar by the angle–angle criterion. “Consider KLM\triangle{KLM} and STU\triangle{STU} STU\triangle{STU} shown in the diagram. a. Complete the table with the angle measures. Then identify the angle relationship that allowed you to find the measure. b. Is KLM\triangle{KLM} similar to STU\triangle{STU} by the angle–angle criterion? Explain.”

  • Module 6, Topic B, Lesson 10: Graphs of Nonlinear Functions, Learn, Problem 5, students create their own conjectures as they analyze equations and graphs to determine which are functions. “For problems 5–11, use each equation and its corresponding graph to answer the questions. y=xy=x a. Does the relationship given by the equation and the graph represent y as a function of x? Explain. b. If this relationship is a function, is the function linear or nonlinear? Explain.”

Materials provide support for the intentional development of MP3 by providing opportunities for students to critique the reasoning of others in connection to grade-level content. Examples include:

  • Module 1, Topic A, Lesson 3: Time to Be More Precise-Scientific Notation, Learn, Problem 12, students critique the reasoning of others as they write numbers presented in scientific notation in standard form. “Logan writes the number 6.7×1036.7\times10^3 in standard form. He writes 67,000 because 10310^3 represents thousands. Do you agree with Logan? Explain.”

  • Module 4, Topic D, Lesson 18: Slopes of Falling Lines, Learn, Problem 6 students conduct error analysis of others as they graph falling lines given the slope and a point on each line. “A line with slope 43-\frac{4}{3} passes through the point (−6,1). Four students use the slope and the given point to locate a second point and graph the line. Their work is shown. Three of the students made an error. a. Who graphed the line correctly? b. Describe the error made by each of the other three students.” Teachers are prompted to ask, “What parts of this student’s work do you question? Why? How would you change this student’s work to make it more accurate?”

  • Module 6, Topic C, Lesson 13: Informally Fitting a Line to Data, Exit Ticket, Problem 1, students critique the reasoning of others as they fit a line to data displayed in a scatter plot and use it to make predictions. “The scatter plot shows the mean precipitation per year and the mean temperature in July for selected midwestern cities. Eve draws a line to fit the data. a. Use the line to predict the mean temperature in July for a city that has a mean precipitation per year of 36 inches. b. Do you think Eve’s line fits the data well? Explain.”

Indicator 2G
02/02

Materials support 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.

The materials reviewed for Eureka Math2 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. 

Each lesson provides opportunities for students to engage with multiple Standards for Mathematical Practice (MP) with a focus on one MP within each lesson. Margin Notes, Promoting the Standards for Mathematical Practice, highlights student engagement with the MP focused in the lesson. These notes provide specific lesson information, ideas, and questions to deepen student engagement. 

Materials provide intentional development of MP4 to meet its full intent in connection to grade-level content. Students model with mathematics to solve real-world problems, identify important quantities to make sense of relationships, and represent them mathematically as they work with the support of the teacher and independently throughout the units. Examples include:

  • Module 3, Topic B, Lesson 6: The Shadowy Hand, Practice, Problem 1, students use dilations to determine the distance of a light source. “An activity is set up as shown in the diagram. The figure is not drawn to scale. A light source is placed 30 feet from a wall and turned on. On the wall is an outline of a hand. The height of the hand from the wrist to the tip of the longest finger is 15 inches. When your classmate’s hand is held parallel to the wall, the height of their hand from the wrist to the tip of their longest finger is 6 inches. a. Label the diagram with the given measurements. b. How far should your classmate’s hand be from the light source so their hand’s shadow matches the hand outline on the wall? c. Explain your answer by using dilations.” 

  • Module 4, Topic B, Lesson 11: Planning a Trip, Learn, students write linear equations to represent the time it takes to fly or drive to a destination, and adjust their equations based on changes to the scenario they are given. “Students write and solve an equation to find the distance for which the times are the same. Using speeds of 65 and 575 miles per hour for the car and plane, respectively, and an airport waiting time of 2 hours and 30 minutes, students define a variable to represent the distance for the trip. They then write and solve an equation.” Teacher prompts include, “What math can you write to represent the total time it takes to travel to your destination by plane? By car? What assumptions could you make to help write an expression that represents the total time it takes to travel to your destination by plane? By car?”

  • Module 6, Assessment 2, Item 4, students describe a situation that a function graph represents. “Consider the function represented by the given graph. Describe a situation that the function could describe. Define each variable.” The graph provided begins decreasing, then is neither increasing or decreasing, then decreases again.

Materials provide intentional development of MP5 to meet its full intent in connection to grade-level content. Students use appropriate tools strategically as they work with the support of the teacher and independently. Examples include:

  • Module 1, Topic C, Lesson 12: Operations with Numbers in Scientific Notation, Learn, Problem 2, students calculate cubes of numbers written in scientific notation using a calculator. “For problems 2-6, use the properties of exponents to evaluate the expression. Write the answer in scientific notation. Check the answer with a calculator. (3.72×105)2(3.72\times10^5)^2” Teachers are prompted to ask, “Is it appropriate to use a calculator to help evaluate this expression? Why? Is it appropriate to use a calculator to multiply 2 and 4.5? Why? Your calculator says the answer is ____. Does that seem about right?”

  • Module 2, Topic A, Lesson 1: Motions of the Plane, Launch, Problem 1, students create a pattern of repeated shapes. “Study the pattern. a. Use any of the given tools and only Figure A to create the pattern. What tools did you use? What strategy did you use?” Students are shown a four petal flower shape with petals labeled A, B, C, and D. Teachers are instructed, “Present the available tools: a dry-erase marker, a piece of paper, a protractor, a ruler, a straightedge, scissors, and a transparency. Allow students time to choose their tools, discuss their strategy, and try to create the pattern shown in problem 1.” Teachers are prompted to ask, “Why did you choose the tools you choose? Did they work well? Which tool would be the most helpful to recreate the pattern? Why?” 

  • Module 6, Topic B, Lesson 8: Comparing Functions, Learn, Problem 1, students compare two different functions represented in two different ways. “Abdul and Maya each drive from city A to city B. The two cities are 147 miles apart. Abdul and Maya take the same route and drive at constant speeds. Abdul begins driving at 1:40 p.m. and arrives at city B at 4:15 p.m. Maya’s trip can be described by the equation y=64xy=64x, where y is the distance traveled in miles and x is the time spent traveling in hours. Who gets to city B faster?” Teachers are prompted to ask, “Can you use a graph, table, or equation to help you compare the functions? Which representation is the most helpful to compare the functions? Why?”

Indicator 2H
02/02

Materials attend to 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.

The materials reviewed for Eureka Math2 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. 

Each lesson provides opportunities for students to engage with multiple Standards for Mathematical Practice (MP) with a focus on one MP within each lesson. Margin Notes, Promoting the Standards for Mathematical Practice, highlights student engagement with the MP focused in the lesson. These notes provide specific lesson information, ideas, and questions to deepen student engagement. Margin Notes, Language Support, provide suggestions for student-to-student discourse, support of new and familiar content-specific terminology or academic language, or support of multiple-meaning words.

Materials provide intentional development of MP6 to meet its full intent in connection to grade-level content Examples include:

  • Module 2, Topic A, Lesson 4: Translations and Reflections on the Coordinate Plane, Launch, students attend to precision as they describe a translation so it can be replicated by others. “Begin the lesson by having students view the animation and describe the translation as precisely as possible. Expect qualitative responses for now.” Teachers are prompted to ask, “When describing a rigid motion, what steps do you take to ensure the description is precise? What details are important to think about in this work?”

  • Module 5, Topic B, Lesson 6: Solving Systems of Linear Equations without Graphing, Learn, Problem 6, students attend to precision as they solve systems of linear equations using substitution. “For problems 6–9, determine the number of solutions to the system of equations. If the system has only one solution, solve the system of equations by using the substitution method and check your solution. 3x=3y6,3x=2y{3x=3y-6,3x=2y}.” Teachers are prompted to ask, “What details are important to think about when writing a system of equations as an equation in one variable? Where is it easy to make mistakes when solving for the values of x and y?“

  • Module 6, Topic E, Lesson 21: Volumes of Prisms and Pyramids, Learn, Problem 3, students attend to precision as they use a given dimension to find volume. “For problems 3–5, find the volume of the solid.” A pyramid is shown with a height of 4.5 cm and a base area of 22cm222cm^2.. Teachers are prompted to ask, “In problem 3, is it exactly correct to say the volume is 33? What can we add or change to be more precise? What does the variable B represent in the volume formula? What details are important to think about when substituting values in the volume formula?” 

The instructional materials attend to the specialized language of mathematics. Examples include:

  • Module 1, Topic B, Lesson 9: Writing Equivalent Expressions, Learn, Language Support, “Students are familiar with the term simplify when working with fractions, but it may be new for them to consider simplifying expressions. Tell students they have been simplifying exponential expressions throughout this topic.”

  • Module 3, Topic D, Lesson 14: Using Similar Figures to Find Unknown Side Lengths, Learn, Teacher Note, “The similarity of the three triangles given in problem 5 is an example of the transitive property of similarity, which states that if A BA~B  and B CB~C, then A CA~C. Consider referring to this property by name if students are ready.”

  • Module 5, Topic A, Lesson 4: More Than One Solution, Learn, Language Support, “Support students’ understanding of the word coincide so they are comfortable using the term. Consider drawing a line with points A and B on the line. Draw two more points on the line and label them C and D. Then tell students that AB\overline {AB} and CD\overline {CD} coincide because they are the same line.”

Indicator 2I
02/02

Materials support 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 the grade-level content standards, as expected by the mathematical practice standards.

The materials reviewed for Eureka Math2 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. 

Each lesson provides opportunities for students to engage with multiple Standards for Mathematical Practice (MP) with a focus on one MP within each lesson. Margin Notes, Promoting the Standards for Mathematical Practice, highlights student engagement with the MP focused in the lesson. These notes provide specific lesson information, ideas, and questions to deepen student engagement. 

Materials provide intentional development of MP7 to meet its full intent in connection to grade-level content. Students look for and make use of structure as they work with the support of the teacher and independently throughout the units. Examples include:

  • Module 1, Topic A, Lesson 2: Comparing Large Numbers, Learn, Problem 3, students look for patterns or structures to make generalizations and solve problems as they determine how many times as much one number is as another. “9 million is how many times as much as 3,000,000?” Teachers are prompted to ask, “Can you break 9×1093×103\frac{9\times10^9}{3\times10^3} into easier problems? How are 9billion3million\frac{9 billion}{3 million} and 9×1093×103\frac{9\times10^9}{3\times10^3} related? How could that help you find the quotient? How are 9million3million\frac{9 million}{3 million} and 9×1063×106\frac{9\times10^6}{3\times10^6} related? How could that help you find the quotient?”

  • Module 4, Topic B, Lesson 8: Another Possible Number of Solutions, Learn, Problem 3, students create, describe, and explain a general process as they engage in a card sort activity to determine if the linear equations have one solution, infinitely many solutions, or no solution. “m6=6m9m-6=6m-9, 10r2=10r+110r-2=10r+1, and 4x+5=54x-4x+5=5-4x Complete the table.” The table contains columns for Number of Solutions, Equations, and What I Notice About the Structure. Teachers are prompted to ask, “How can you use what 4x+5-4x+5 and 54x5-4x have in common to determine the number of solutions to the equation 4x+5=54x-4x+5=5-4x? How can you use what 10r210r-2 and 10r+110r+1 have in common to determine the number of solutions to the equation 10r2=10r+110r-2=10r+1?”

  • Module 5, Topic B, Lesson 8: Using Tape Diagrams to Solve Systems of Equations, Learn, Problem 4, students look for and explain the structure within mathematical representations to analyze a tape diagram used to solve a system of linear equations where neither variable term is isolated. “Draw a tape diagram and use it to solve the system of equations. x+3y=10,3x+8y=28{x+3y=10,3x+8y=28} Pedro drew the following tape diagrams to solve the system of equations. First tape diagram: Second tape diagram: Third tape diagram: a. Explain Pedro’s work. b. Use Pedro’s third tape diagram to solve the system of equations.” Teachers are prompted to ask, “How did Pedro use what the two tapes have in common to redraw the bottom tape? How can what you know about and help you find the value of ?“

Materials provide intentional development of MP8 to meet its full intent in connection to grade-level content. Students look for and express regularity in repeated reasoning to make generalizations and build a deeper understanding of grade-level math concepts.. Examples include:

  • Module 2, Topic B, Lesson 7: Working Backwards, Learn, Problem 1, students explain a general process as they recognize and describe how any single rigid motion can be undone. “For problems 1–4, complete the table by describing the rigid motion that maps the original figure onto its image and the rigid motion that maps the image back onto its original figure.” Students are shown two wiggly lines on either side of the y-axis. The three columns of the table are Diagram, Maps Figure onto Image, and Maps Image onto Figure. Teachers are prompted to ask, “What is the same about the rigid motion that maps a figure onto its image and the rigid motion that maps the image back onto its original figure? What is different about these rigid motions?What patterns do you notice when you map an image back onto its original figure? Will those patterns always be true?”

  • Module 3, Topic A, Lesson 8: Dilations on the Coordinate Plane, Learn, Problem 2, students describe and explain a method as they identify the relationship between the coordinates of a point and the coordinates of its image under a dilation centered at the origin. “2. Consider points O and B in the coordinate plane. a. Choose a Scale Factor card. Then plot and label the image of point B under a dilation with center O and scale factor r from the card. b. Check one another’s images. Then complete the table for the group’s four images. c. Look for a pattern within each row of the table. What do you notice about the relationship between the scale factor and the coordinates of points B and B′?” Teachers are prompted to ask, “What is the same about how each group member located their point B′? What arithmetic patterns do you notice when you compare the coordinates of the point and its image? Will the pattern always work?”

  • Module 6, Topic A, Lesson 1: Motion and Speed, Learn, students explain a process as they explore a nonlinear motion situation by using average speed and describe the function by using patterns.“We found a rule for the function that describes car A. Can we use a function with the same rule to describe car B?” Teachers are prompted to ask, “When you calculate the average speed over different time intervals for a proportional relationship, does anything repeat? How can that help you determine average speed over any interval more efficiently? What patterns did you notice when you calculated average speed over -second intervals for car B?”

Overview of Gateway 3

Usability

The materials reviewed for Eureka Math² Grade 8 meet expectations for Usability. The materials meet expectations for Criterion 1, Teacher Supports; partially meet expectations for Criterion 2, Assessment; and meet expectations for Criterion 3, Student Supports.

Criterion 3.1: Teacher Supports

09/09

The program includes opportunities for teachers to effectively plan and utilize materials with integrity and to further develop their own understanding of the content.

The materials reviewed for Eureka Math² Grade 8 meet expectations for Teacher Supports. The materials: provide teacher guidance with useful annotations and suggestions for enacting the student and ancillary materials; contain adult-level explanations and examples of the more complex grade-level concepts and concepts beyond the current grade so that teachers can improve their own knowledge of the subject; include standards correlation information that explains the role of the standards in the context of the overall series; provide explanations of the instructional approaches of the program and identification of the research-based strategies; and provide a comprehensive list of supplies needed to support instructional activities. 

Indicator 3A
02/02

Materials provide teacher guidance with useful annotations and suggestions for how to enact the student materials and ancillary materials, with specific attention to engaging students in order to guide their mathematical development.

The materials reviewed for Eureka Math2 Grade 8 meet expectations for providing teacher guidance with useful annotations and suggestions for how to enact the student materials and ancillary materials, with specific attention to engaging students in order to guide their mathematical development.

Materials provide comprehensive guidance that will assist teachers in presenting the student and ancillary materials. These are found in a variety of sections within the Implementation Guide, including the Overview, Why, Achievement Descriptors Overview, and Lesson Structure. Examples include:

  • Grade 6-9 Implementation Guide, Inside Teach, Module-Level Components, Overview, “Your Teach book begins with the Overview, a topic-by-topic summary that shows the development of learning throughout the module. It also provides connections to work done before and after the module, helping you understand the module’s place in the overall development of learning in and across the grade levels.”

  • Grade 6-9 Implementation Guide, Inside Teach, Module-Level Components, Why, “The Why section gives insight into the decisions made during the writing of the module. This insight helps you understand the underlying structure of the module, flow of the content, and coherence of the different parts of the curriculum.”

  • Grade 6-9 Implementation Guide, Inside Teach, Module-Level Components, Achievement Descriptors, “The Achievement Descriptors: Overview section is a helpful guide that describes what Achievement Descriptors (ADs) are and briefly explains how to use them. It identifies specific ADs for the module, with more guidance provided in the Achievement Descriptors: Proficiency Indicators resource at the end of each Teach book.”

  • Grade 6-9 Implementation Guide, Inside Teach, Module-Level Components, Lesson Structure, “Each lesson is structured in four sections: Fluency, Launch, Learn, and Land. Lessons are designed for one 45-minute instructional period. Fluency provides distributed practice with previously learned material. Launch creates an accessible entry point to the day’s learning through activities that build context and often create productive struggle that leads to a need for the learning that follows. Learn presents new learning related to the lesson objective, usually through a series of instructional segments. This lesson component takes most of the instructional time. Land helps you facilitate a brief discussion to close the lesson and provides students with an opportunity to complete the Exit Ticket.”

Materials include sufficient and useful annotations and suggestions that are presented within the context of specific lessons. This guidance can be found for teachers within boxes called Differentiation, UDL, and Teacher Notes. The Implementation Guide states, “There are six types of instructional guidance that appear in the margins. These notes provide information about facilitation, differentiation, and coherence. Teacher Notes communicate information that helps with implementing the lesson. Teacher Notes may enhance mathematical understanding, explain pedagogical choices, five background information, or help identify common misconceptions. Universal Design for Learning (UDL) suggestions offer strategies and scaffolds that address learner variance. These suggestions promote flexibility with engagement, representation, and action and expression, the three UDL principles described by CAST. These strategies and scaffolds are additional suggestions to complement the curriculum’s overall alignment with the UDL Guidelines.” Examples include: 

  • Module 1, Topic A, Lesson 2: Comparing Large Numbers, Learn, Teacher Note, “Writing the factors of 10 helps make the division simpler by allowing students to pair factors of 10 in the numerator and denominator to create quotients of 1.”

  • Module 3, Topic C, Lesson 11: Similar Figures, Launch, Differentiation: Support, “Some students may have difficulty recognizing the difference between a rotation and a reflection. Encourage students to trace a figure and label its vertices on a transparency and to use the transparency to review how the orientation of a figure changes under a reflection but remains the same under a rotation.” 

  • Module 5, Topic B, Lesson 7: The Substitution Method, UDL: Action and Expression, “Consider providing access to the Equation Assistant interactive during this lesson. This interactive is meant to support students with efficiently and accurately rewriting equations in preparation for solving systems of equations with substitution.”

Indicator 3B
02/02

Materials contain adult-level explanations and examples of the more complex grade-level/course-level concepts and concepts beyond the current course so that teachers can improve their own knowledge of the subject.

The materials reviewed for Eureka Math2 Grade 8 meet expectations for containing adult-level explanations and examples of the more complex grade-level concepts and concepts beyond the current grade so that teachers can improve their own knowledge of the subject.

Materials consistently contain adult-level explanations, examples of the more complex grade/ course-level concepts, and concepts beyond the course within Topic Overviews and/or Module Overviews. According to page 7 of the Grade 6-9 Implementation Guide, “Your Teach book begins with the Overview, a topic-by-topic summary that shows the development of learning throughout the module. It also provides connections to work done before and after the module, helping you understand the module’s place in the overall development of learning in and across the grade levels.” Page 9 outlines the purpose of the Topic Overview, “Each topic begins with a Topic Overview that is a summary of the development of learning in that topic. It typically includes information about how learning connects to previous or upcoming content.” Examples include:

  • Module 1: Scientific Notation, Exponents, and Irrational Numbers, Module Overview, Why, “Why are operations with numbers written in scientific notation in topics A and C? Topic A sparks students’ interest through engaging contexts that utilize their exponent and place value understanding. Having students write the many factors of 10 foreshadows and drives the need for the properties and definitions of exponents. In topic C, students become fluent in applying the properties and definitions of exponents by operating with numbers written in scientific notation. Why are exponential expressions written as 105+210^{5+2}  and 3543^{5\cdot4}? In lesson 5, the expression 10510210^5\cdot10^2  is intentionally shown as the equivalent expression 105+210^{5+2}  to emphasize the use of the product of powers with like bases property. Writing 105+210^{5+2} has more instructional value than writing 10710^7. When students are ready, ask them to express the sum of the exponents as an integer.”

  • Module 4: Linear Equations in One and Two Variables, Module Overview, Why, “Why do we say the value of the ratio? In rade 6, students learn that a ratio is an ordered pair of numbers that are not both zero. For a ratio A : B, the value of the ratio is the quotient AB\frac{A}{B} as long as B is not zero. Therefore, when students determine the slope of a line, they find the value of the height-to-base ratio of a slope triangle for the line.”

  • Module 6: Functions and Bivariate Statistics, Module Overview, Why, “The goal of introducing functions to students is to carefully connect prior knowledge so that students build new understanding as an extension of previous learning. In this module, students explore the connection of functions to average speed and linear relationships. Functions represent real-world situations, whether they be numeric or nonnumeric. Choosing to present functions in real-life contexts grounds students’ work with functions as a tool to understand the world around them.”

Indicator 3C
02/02

Materials include standards correlation information that explains the role of the standards in the context of the overall series.

The materials reviewed for Eureka Math2 Grade 8 meet expectations for including standards correlation information that explains the role of the standards in the context of the overall series. 

The Achievement Descriptors, found in the Overview section, identify, describe, and explain how to use the standards. The lesson overview includes content standards addressed in the lesson. Additionally, a Proficiency Indicators resource at the end of each Teach book, helps assess student proficiency. Correlation information and explanations are present for the mathematics standards addressed throughout the grade level in the context of the series. Examples include:

  • Module 1: Scientific Notation, Exponents, and Irrational Numbers, Achievement Descriptors and Standards, “8.Mod1.AD1 Determine whether numbers are rational or irrational. (8.NS.A.1, 8.EE.A.2)”

  • Module 2, Topic D, Lesson 19: Using the Pythagorean Theorem and Its Converse, Achievement Descriptors and Standards, “8.Mod2.AD7 Explain a proof of the Pythagorean theorem and its converse geometrically. (8.G.B.6)”

  • Module 5: Percents and Application of Percent, “Students graph systems of linear equations in two variables, estimate the coordinates of the intersection point on the graph, and verify that the ordered pair is a solution to the system. They also analyze systems of linear equations to determine the number of solutions. Students find that estimating solutions from a graph is difficult for solutions composed of one or more fractional values. So they use the substitution method to write a system of linear equations in two variables as one linear equation in one variable. Now equipped with various solution methods, students are challenged to write and solve systems resulting from numerical, geometrical, historical, and real-world contexts.”

Indicator 3D
Read

Materials provide strategies for informing all stakeholders, including students, parents, or caregivers about the program and suggestions for how they can help support student progress and achievement.

The materials reviewed for Eureka Math2 Grade 8 partially provide strategies for informing all stakeholders, including students, parents, or caregivers about the program and suggestions for how they can help support student progress and achievement. 

The student book, Learn, contains a Recap (Level 6-Algebra I) that “outlines key learning from the lesson and provides examples with supporting notes. The Recap summarizes the main learning in the lesson. Definitions of any terms introduced in the lesson are included. Each Recap also shows problems like those completed in class and examples of the thinking that helps students solve the problems. For middle and high school students, Recaps are the activities designed to be completed at home with families. Whether your student is missing class or could use additional support at home, Recaps can help students preview or review lesson concepts.” The Implementation Guide states, “You may use the Recaps as a guide to support practice outside of class. Recaps are also useful for anyone supporting the student’s learning, including family members, tutors, and special educators.”

Families can support students with a resource that includes additional grade-level problems aligned to lessons. Practice (Level 1–Algebra I) states, “Practice problems for each lesson include mixed practice of related skills. This helps students solidify their conceptual understanding and procedural skills, transfer knowledge to new applications, and build fluency. Each Practice is structured as two pages. The front page includes problems that represent learning from class that day. The second page includes Remember problems. These problems help students recall previously learned concepts and skills. While Practice problems related to the day’s lesson help solidify new learning, Remember problems keep students sharp with familiar concepts. In level 6–Algebra I, Practice is included in the Learn book.”

Indicator 3E
02/02

Materials provide explanations of the instructional approaches of the program and identification of the research-based strategies.

The materials reviewed for Eureka Math2 Grade 8 meet expectations for providing explanations of the instructional approaches of the program and identification of the research-based strategies. 

Materials explain the instructional approaches of the program. According to the Grades 6-9 Implementation Guide, “Eureka Math2 features a set of instructional routines that optimize equity by increasing access, engagement, confidence, and students’ sense of belonging. The following is true about Eureka Math2 instructional routines: Each routine presents a set of teachable steps so students can develop as much ownership over the routine as the teacher. The routines are flexible and may be used in additional math lessons or in other subject areas. Each routine aligns to the Stanford Language Design Principles (see Works Cited): support sense-making, optimize output, cultivate conversation, maximize linguistic and cognitive meta-awareness.” Examples of instructional routines include:

  • Instructional Routine: Always Sometimes Never, students make justifications and support their claims with examples and nonexamples. Implementation Guide states, “Present a mathematical statement to students. This statement may hold true in some, all, or no contexts, but the goal of the discussion is to invite students to explore mathematical conditions that affect the truth of the statement. Give students an appropriate amount of silent think time to evaluate whether the statement is always, sometimes, or never true. Have students discuss their thinking with a partner. Circulate and listen as they talk. Identify a few students to share their thinking. Then facilitate a class discussion. Invite students to share their thinking with the whole group. Encourage them to provide examples and nonexamples to support their claim. Encourage use of the Talking Tool. Conclude by bringing the class to consensus that the statement is [always/sometimes/never] true [because …].”

  • Instructional Routine: Critique a Flawed Response, students communicate with one another to critique others’ work, correct errors, and clarify meanings. Implementation Guide states, “Present a prompt that has a partial or broken argument, incomplete or incorrect explanation, common calculation error, or flawed strategy. The work presented may either be authentic student work or fabricated work. Give students an appropriate amount of time to identify the error or ambiguity. Invite students to share their thinking with the class. Then provide an appropriate amount of time for students to solve the problem based on their own understanding. Circulate and identify a few students to share their thinking. Purposefully choose work that allows for rich discussion about the prompt given. Then facilitate a class discussion by inviting students to share their solutions with the whole group. Encourage use of the Talking Tool. Lead the class to a consensus about how best to correct the flawed response.”

  • Instructional Routine: Stronger, Clearer Each Time, students revise and refine their written responses. Implementation Guide states, “Present a problem, a claim, or a solution path and prompt students to write an explanation or justification for their solution path, response to the claim, or argument for or against the solution path. Give students an appropriate amount of time to work independently. Then pair students and have them exchange their written explanations. Provide time for students to read silently. Invite pairs to ask clarifying questions and to critique one another’s response. Circulate and listen as students discuss. Ask targeted questions to advance their thinking. Direct students to give specific verbal feedback about what is or is not convincing about their partner’s argument. Finally, invite students to revise their work based on their partner’s feedback. Encourage them to use evidence to improve the justification for their argument.”

Materials include and reference research-based strategies. The Grades 6-9 Implementation Guide states, “In Eureka Math2 we’ve put into practice the latest research on supporting multilanguage learners, leveraging Universal Design for Learning principles, and promoting social-emotional learning. The instructional design, instructional routines, and lesson-specific strategies support teachers as they address learner variance and support students with understanding, speaking, and writing English in mathematical contexts. A robust knowledge base underpins the structure and content framework of Eureka Math2. A listing of the key research appears in the Works Cited for each module.”

Indicator 3F
01/01

Materials provide a comprehensive list of supplies needed to support instructional activities.

The materials reviewed for Eureka Math2 Grade 8 meet expectations for providing a comprehensive list of supplies needed to support instructional activities. 

Each module and individual lesson contains a materials list for the teacher and student. The lesson preparation identifies materials teachers need to create or assemble in advance. Examples include:

  • Module 2, Topic A, Lesson 3: Reflections, Materials, “Teacher: Translation or Reflection Cards (1 set), Transparency film. Students: Translation or Reflection Cards (1 set per student pair), Transparency film, Sticky notes (2 per student pair), Straightedge. Lesson Preparation: Copy and cut out the Translation or Reflection Cards (in the teacher edition). Prepare enough sets for 1 per student pair and 1 for display during discussion.”

  • Module 5: Systems of Linear Equations, Module Overview, Materials, “The following materials are needed to implement this module. The suggested quantities are based on a class of 24 students and one teacher. (1) Chart paper, tablet, (6) Construction paper, 12″ x 18″ sheets, (25) Dry-erase markers, (24) Grid paper, sheets, (24) Learn books, (1) Marker, (25) Pencils, (25) Personal whiteboards, (25) Personal whiteboard erasers, (1) Projection device, (96) Sticky notes, (25) Straightedges, (12) Student computers or devices, (1) Tape, transparent roll, (1) Teach book, (1) Teacher computer or device.”

  • Module 6, Topic B, Lesson 8: Comparing Functions, Materials, “Teacher: None. Students: Function Representations Match card. Lesson Preparation: Copy and cut out the Function Representations Match cards (in the teacher edition). Prepare enough cards so each student has one card.”

Indicator 3G
Read

This is not an assessed indicator in Mathematics.

Indicator 3H
Read

This is not an assessed indicator in Mathematics.

Criterion 3.2: Assessment

07/10

The program includes a system of assessments identifying how materials provide tools, guidance, and support for teachers to collect, interpret, and act on data about student progress towards the standards.

The materials reviewed for Eureka Math² Grade 8 partially meet expectations for Assessment. The materials identify the content standards assessed in formal assessments, but do not identify the mathematical practices for some of the formal assessments. The materials provide multiple opportunities to determine students' learning and sufficient guidance to teachers for interpreting student performance, but do not provide specific suggestions for following-up with students. The materials include opportunities for students to demonstrate the full intent of grade-level standards and mathematical practices across the series.

Indicator 3I
01/02

Assessment information is included in the materials to indicate which standards are assessed.

The materials reviewed for Eureka Math2 Grade 8 partially meet expectations for having assessment information included in the materials to indicate which standards are assessed. The materials identify the standards assessed for all of the formal assessments, but the materials do not identify the practices assessed for some of the formal assessments.

According to the Grade 6-9 Implementation Guide, Core Assessment Components, Exit Tickets (p. 47), “Exit Tickets are short, paper-based assessments that close lessons. These assessments use at least one problem, question, or writing prompt to assess whether a student has learned the basic skills and concepts needed for success in upcoming lessons. Items reflect the minimum that students must demonstrate to meet the lesson objective. You may look for evidence of the Standard for Mathematical Practice (MP) identified as the focus MP for the lesson in student work on the Exit Ticket.” Topic Quizzes (p. 48), “Typical Topic Quizzes consist of 4–6 items that assess proficiency with the major concepts from the topic. Many items allow students to show evidence of one or more of the Standards for Mathematical Practice (MPs). You may use the Standards and Achievement Descriptors at a Glance charts to find which MPs you may be more likely to see from your students on a given assessment item in relation to the content that is assessed. For example, you may be likely to see evidence of MP2 and MP8 on the Level 7 Module 1 Topic A Quiz as those are the MPs explicitly identified in the lessons of that topic.” Module Assessments (p.48), “Typical Module Assessments consist of 6–10 items that assess proficiency with the major concepts, skills, and applications taught in the module. Many items allow students to show evidence of one or more of the Standards for Mathematical Practice (MPs). You may use the Standards and Achievement Descriptors at a Glance charts to find which MPs you may be more likely to see from your students on a given assessment item in relation to the content that is assessed. Module Assessments represent the most important content, but they may not assess all the strategies and standards taught in the module.” 

Additionally, within the Grade 6-9 Implementation Guide (pp. 52), Achievement Descriptors, Standards and Achievement Descriptors at a Glance, “Every module in grade 6 through Algebra 1 has a Standards and Achievement Descriptors at a Glance chart. These charts identify the location and show the frequency of the content standards, Standards for Mathematical Practice, and Achievement Descriptors (ADs) in each module.” Within the Proficiency Indicators section (p. 52), “Each AD has its own set of proficiency indicators. Proficiency indicators are more detailed than ADs and help you analyze and evaluate what you see or hear in the classroom as well as what you see in students’ written work. Each AD has up to three indicators that align with a category of proficiency: Partially Proficient, Proficient, or Highly Proficient. Proficiency Indicators use language that offers insights about which MPs may be observed as students engage with assessment items. For example, Proficiency Indicators that begin with justify, explain, or analyze likely invite students to show evidence of MP3: Construct viable arguments and critique the reasoning of others. Proficiency Indicators that begin with create or represent likely invite students to show evidence of MP2: Reason abstractly and quantitatively.” 

The Standards and Achievement Descriptors at a Glance chart is provided within each grade level’s Implementation Resources, within the Maps section. “How to use the Standards and Achievement Descriptors at a Glance; Identity Where Content is Taught before Teaching” states, “The Standards and Achievement Descriptors at a Glance charts identify the location and show the frequency of the content standards, Standards for Mathematical Practice, and Achievement Descriptors (ADs) in each module.” While these documents align the MPs to specific lessons and corresponding Exit Tickets, the MPs are not identified within Topic Quizzes or Module Assessments. Examples include but are not limited to:

  • Module 1: Scientific Notation, Exponents, and Irrational Numbers, Topic C, Quiz 1, Item 4, “A technology company designs a phone that weighs about 0.068125 pounds. What is the weight of the phone rounded to the nearest hundredth pound? Enter your answer in scientific notation. (8.EE.A.3)”

  • Module 3: Dilations and Similar Figures, Module Assessment 1, Item 6, “A volleyball player hits the ball when it is h feet above the ground. The volleyball net is 8 feet tall. The ball lands 15 feet away from the base of the net, as shown. What is the height of the ball when the player hits it?  (8.G.A.5)” The ball is shown landing 15 feet from the net with a hypotenuse of 20.825 feet.

  • Module 5: Systems of Linear Equations, Module Assessment 1, Item 6, “A bike rental company rents out scooters for $8.50 per day and bikes for $15.20 per day. On Tuesday, they rent out a  combined total of 8 scooters and bikes. They make $101.50 for the rentals. How many scooters and bikes does the bike rental company rent on Tuesday? (8.EE.C.8.c)”

  • Module 6: Functions and Bivariate Statistics, 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 total distance that she hiked in two days. Part A  Enter an equation for a function that represents the total distance hiked y in miles when Sara has hikedx hours on the second day. Part B What do the initial value and the rate of change in your function represent? (8.F.B.4)”

Indicator 3J
02/04

Assessment system provides multiple opportunities throughout the grade, course, and/or series to determine students' learning and sufficient guidance to teachers for interpreting student performance and suggestions for follow-up.

The materials reviewed for Eureka Math2 Grade 8 partially meet expectations for including an assessment system that provides multiple opportunities throughout the grade, course, and/or series to determine students’ learning and sufficient guidance to teachers for interpreting student performance and suggestions for follow-up.

The assessment system provides multiple opportunities to determine students' learning, and sufficient guidance for teachers to interpret student performance is reinforced by the Proficiency Indicators. However, suggestions to teachers for following up with students are general and minimal, for example, “Look back at those lessons to select guidance and practice problems that best meet your students’ needs.” While teachers can refer back to specific lessons, it is incumbent on the teacher to determine which guidance and practice problems meet the needs of their individual students. Examples include:

  • Grade 6-A1 Implementation Guide, Resources, Achievement Descriptors: Proficiency Indicators (p. 15), “Achievement Descriptors (ADs) are standards-aligned descriptions that detail what students should know and be able to do based on instruction they receive. The number of ADs addressed in each lesson varies depending on the content. This resource includes proficiency indicators for each AD. Proficiency indicators are descriptions of work that is partially proficient, proficient, or highly proficient. Proficiency indicators help you assess your students’ level of proficiency.”

  • Grade 6-A1 Implementation Guide, Assessment, Standards and Achievement Descriptors at a Glance (pp. 52), “Every module in grade 6 through Algebra 1 has a Standards and Achievement Descriptors at a Glance chart. These charts identify the location and show the frequency of the content standards, Standards for Mathematical Practice, and Achievement Descriptors (ADs) in each module. Use these charts to quickly determine where and when standards and ADs are taught within and across modules to help you target observations. You may also use these charts in conjunction with assessment data to identify targeted ways to help meet the needs of specific learners. Use assessment data to determine which ADs and Proficiency Indicators to revisit with students. Use the examples provided with the Proficiency Indicator(s) as the basis for responsive teaching or use the modules’ Standards and Achievement Descriptors at a Glance chart to identify lessons that contain guidance and practice problems to support student follow up.”

  • Grade 6-A1 Implementation Guide, Assessment, Respond to Student Assessment Performance (p. 60), “After administering an assessment, use the assessment reports in the Analyze space of the Great Minds Digital Platform to view student performance by Achievement Descriptor (AD). Analyze the student-performance data and select one or both of the following methods to address learning needs.” Proficiency Indicators: “Proficiency indicators increase in cognitive complexity from partially proficient (PP) to proficient (P) to highly proficient (HP). If a student has difficulty with content of the P indicator of a given AD, follow-up with the student by revisiting the content at the PP indicator of the same AD as shown in the AD proficiency indicator charts. Select the Student Performance report in the Analyze space of the Great Minds Digital Platform. Filter by proficiency indicator and any individual or group of assessments. When the report indicates proficiency of an AD has not been met, refer to the module’s Achievement Descriptors: Proficiency Indicator resource and use the lower-complexity task to build toward full understanding. Use the examples provided with the Proficiency Indicator(s) as the basis for responsive teaching. Example: For students who do not meet the Proficient indicator (4.Mod1.AD1.P), consider focusing on the Partially Proficient indicator (4.Mod1.AD1.PP). In this case, strengthen student foundational understanding of creating one comparison statement to build towards proficient understanding with two comparison statements.”

  • Grades 6-A1 Implementation Guide, Assessment, The Standards and Achievement Descriptors at a Glance Charts (p. 61), “Select the Student Performance report in the Analyze space of the Great Minds Digital Platform. Filter by proficiency indicator and any individual or group of assessments. When the report indicates proficiency of an AD has not been met, refer to the Standards and Achievement Descriptors at a Glance charts to identify lessons that teach the concepts of that AD. Navigate to those lessons to find guidance and practice problems to follow up with students. Example: If students struggle with 4.Mod1.AD1, use the Standards and Achievement Descriptors at a Glance chart to find that lessons 1, 2, 3, 4, and 6 address the AD. Look back at those lessons to select guidance and practice problems that best meet your students’ needs.”

The assessment system provides guidance to teachers for interpreting student performance within Scoring Guides for Module Assessments and Topic Quizzes. Examples include:

  • Module 2, Assessment 2, Item 1, “Consider the diagram. Which point is the image of point I under a dilation with center O and scale factor 3? A. F, B. G, C. H, D. J” Sample Solution states, “A. Incorrect. The student may have selected the point that is the image of point G under a dilation centered at O with scale factor 3. B. Incorrect. The student may have selected the point that is the same distance from O as I is from O. C. Incorrect. The student may have selected the point that is the image of point I under a dilation centered at O with scale factor 13\frac{1}{3}. D. Correct.”

  • Module 4, Topic D, Quiz 1, Item 3, “A line passes through the points (5, 4) and (2, -3). What is the slope of the line? A. 71\frac{7}{1} B. 17\frac{1}{7} C. 37\frac{3}{7} D. 73\frac{7}{3}”  Sample Solution states, “A. Incorrect. The student may have added the y-values and the x-values instead of subtracting them and calculated the slope by using the formula m=x2+x1y2+y1m=\frac{x_2+x_1}{y_2+y_1}. B. Incorrect. The student may have added the y-values and the x-values instead of subtracting them. C. Incorrect. The student may have calculated the slope by using the formula m=x2x1y2m=\frac{x_2-x_1}{y_2}.  D. Correct.”  

  • Module 6, Topic B, Lesson 10: Graphs of Nonlinear Functions, Exit Ticket, “A car travels at a constant speed for a while and then comes to a stop at a red light. The light stays red for a few minutes. When the light changes to green, the car speeds up and then maintains a slightly faster constant speed. a. Sketch a graph of a function to match the description b. Identify whether the function is linear or nonlinear. How do you know?” Sample solution shows a graph of the function and states, “b. The function is nonlinear because the graph is not a line, so it cannot be the graph of a linear function.”

Indicator 3K
04/04

Assessments include opportunities for students to demonstrate the full intent of grade-level/course-level standards and practices across the series.

The materials reviewed for Eureka Math2 Grade 8 meet expectations for providing assessments that include opportunities for students to demonstrate the full intent of grade-level/ course-level standards and practices across the series.

Assessments identify standards and include opportunities for students to demonstrate the full intent of grade-level/course-level standards. Examples include: 

  • Module 2, Module Assessment 1, Item 5, students use a digital applet to prove the Pythagorean theorem and explain their reasoning. “Part A: Rearrange the triangles to help prove the Pythagorean theorem. Drag the four triangles onto the blank square on the right. Part B: Explain how your rearrangement helps prove the Pythagorean theorem. (8.G.B.6)”

  • Module 4, Topic B, Quiz 1, Item 3, students solve linear equations. “Consider the equation 3(1.5x0.75)1=ax+b3(1.5x-0.75)-1=ax+b. The equation has infinitely many solutions. What are the values of a and b? The value of a is ____. The value of b is _____. (8.EE.C.7.a)”

  • Module 6, Topic C, Quiz 1, Item 1, students model real world data using a scatter plot. “Mrs. Banks coaches a softball team. She records the number of home runs batted in for each player. Create a scatter plot of the number of home runs and the number of runs batted in. (8.SP.A.1)”

Assessments do not identify mathematical practices in either teacher or student editions. Although assessment items do not clearly label the MPs, students are provided opportunities to engage with the mathematical practices to demonstrate full intent. Examples include: 

  • Module 1, Module Assessment 2, Item 7, “Yu Yan knows that the value of π\pi rounded to the nearest tenth is 3.1. She says the value of π1.76\sqrt{\pi}\approx1.76 because 3.11.76\sqrt{3.1}\approx1.76. Yu Yan enters π\sqrt{\pi} into a computer program to check her answer. The program displays this number line. Part A: Based on the number line, what is the value of π\sqrt{\pi} rounded to the nearest hundredth? Part B: Why is Yu Yan's value of π\sqrt{\pi} different from the computer's value of π\sqrt{\pi}? Justify your answer.” This item addresses  MP3, construct viable arguments and critique the reasoning of others.

  • Module 3, Module Assessment 2, Item 6, “A table tennis player hits the ball when it is h inches above the table. The table tennis net is 6 inches tall. The ball lands 8 inches away from the base of the net, as shown. What is the height of the ball when the player hits it?” This item addresses MP2, reason abstractly and quantitatively.

  • Module 5, Topic C, Quiz 1, Item 1, “Eve's aunt is 7 years older than twice Eve's age. The sum of Eve's age and her aunt's age is 52. How old is Eve? How old is Eve's aunt?” This item addresses MP1, make sense of problems and persevere in solving them.

Indicator 3L
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Assessments offer accommodations that allow students to demonstrate their knowledge and skills without changing the content of the assessment.

The materials reviewed for Eureka Math2 Grade 8 do not provide assessments which offer accommodations that allow students to demonstrate their knowledge and skills without changing the content of the assessment.

Materials provide three analogous versions of each Topic Quiz and two analogous versions of each Module Assessment. According to the Implementation Guide, “Analogous versions target the same material at the same level of cognitive complexity. However, typical items on analogous versions are not clones of the original version. Use the analogous versions to give retakes, with reteaching or additional practice between takes, until students score proficient or above.”

Criterion 3.3: Student Supports

08/08

The program includes materials designed for each student’s regular and active participation in grade-level/grade-band/series content.

The materials reviewed for Eureka Math² Grade 8 meet expectations for Student Supports. The materials provide: strategies and supports for students in special populations and for students who read, write, and/or speak in a language other than English to support their regular and active participation in learning grade-level mathematics; multiple extensions and/or opportunities for students to engage with grade-level mathematics at higher levels of complexity; and manipulatives, both virtual and physical, that are accurate representations of the mathematical objects they represent and, when appropriate, are connected to written methods.

Indicator 3M
02/02

Materials provide strategies and supports for students in special populations to support their regular and active participation in learning grade-level/series mathematics.

The materials reviewed for Eureka Math2 Grade 8 meet expectations for providing strategies and supports for students in special populations to support their regular and active participation in learning grade-level/series mathematics.

Materials provide strategies, supports, and resources for students in special populations to support their regular and active participation in grade-level mathematics. According to the Implementation Guide, “There are six types of instructional guidance that appear in the margins. These notes provide information about facilitation, differentiation, and coherence.” Additionally, “Universal Design for Learning (UDL) is a framework based on current research from cognitive neuroscience that recognizes learner variance as the norm rather than the exception. The guiding principles of the UDL framework are based on the three primary networks of the brain. Although the concept of UDL has roots in special education, UDL is for all students. When instruction is designed to meet the needs of the widest range of learners, all students benefit. Eureka Math2 lessons are designed with these principles in mind.” Examples include:

  • Module 2, Topic B, Lesson 9: Ordering Sequences of Rigid Motions, Learn, Language Support, “Consider using the Talking Tool throughout this lesson to support student discussion by using the following examples. Ask students to use prompts from the Share Your Thinking section when discussing whether order matters in their given sequence of rigid motions. As students circulate for the gallery walk, consider directing students to use the Say It Again section to rephrase how other students describe whether the order matters in each sequence of rigid motions.”

  • Module 3, Topic A, Lesson 2: Enlargements, Learn, UDL: Representation, “Consider gesturing to the points when asking about the distance between O and P′ and the distance between O and P. This will connect the auditory cue to a visual cue for students.”

  • Module 4, Topic F, Lesson 26: Linear Equations from Word Problems, Learn, Differentiation: Support, “If the y-intercept’s meaning in this context is difficult for students to interpret, consider having them write the y-intercept point (0,140). Then have them interpret the x-value and y-value in this context.”

Indicator 3N
02/02

Materials provide extensions and/or opportunities for students to engage with grade-level/course-level mathematics at higher levels of complexity.

The materials reviewed for Eureka Math2 Grade 8 meet expectations for providing extensions and/or opportunities for students to engage with grade-level/course-level mathematics at higher levels of complexity.

Materials do not require advanced students to do more assignments than their classmates. Instead, students have opportunities to think differently about learning with alternative questioning, or extension activities. Specific recommendations are routinely highlighted as Teacher Notes within parts of each lesson, as noted in the following examples: 

  • Module 2, Topic B, Lesson 7: Working Backward, Differentiation: Challenge, “For students who finish early, ask if they can find another rigid motion that maps a figure onto its image for problems 3 and 4.”

  • Module 6, Topic B, Lesson 8: Comparing Functions, Learn, Differentiation: Challenge, “Challenge students who finish early to create other representations for problems 1–4. Have them compare the written description, equation, graph, and table.”

  • Module 5, Topic A, Lesson 4:  More Than One Solution, Learn, Differentiation: Challenge, “If pairs finish early, challenge them to write three of their own systems: one with only one solution, one with no solution, and one with infinitely many solutions.”

Indicator 3O
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Materials provide varied approaches to learning tasks over time and variety in how students are expected to demonstrate their learning with opportunities for students to monitor their learning.

The materials reviewed for Eureka Math2 Grade 8 provide varied approaches to learning tasks over time and variety in how students are expected to demonstrate their learning with opportunities for students to monitor their learning. 

Students engage with problem-solving in a variety of ways within a consistent lesson structure: Fluency, Launch, Learn, Land. According to the Implementation Guide, Lesson Structure, “Each lesson is structured in four sections: Fluency, Launch, Learn, and Land. Lessons are designed for one 45-minute instructional period. Fluency provides distributed practice with previously learned material. It is designed to prepare students for new learning by activating prior knowledge and bridging small learning gaps. Launch creates an accessible entry point to the day’s learning through activities that build context and often create productive struggle that leads to a need for the learning that follows. Every Launch ends with a transition statement that sets the goal for the day’s learning. Learn presents new learning related to the lesson objective, usually through a series of instructional segments. This lesson component takes most of the instructional time. Suggested facilitation styles vary and may include direct instruction, guided instruction, group work, partner activities, interactive video, and digital elements. Land helps you facilitate a brief discussion to close the lesson and provides students with an opportunity to complete the Exit Ticket. In the Debrief portion of Land, suggested questions, including key questions related to the objective, help students synthesize the day’s learning. The Exit Ticket provides a window into what students understand so that you can make instructional decisions.”

Examples of varied approaches across the consistent lesson structure include:

  • Module 4, Topic B, Lesson 8: Another Possible Number of Solutions, Learn, students work in pairs on a card sort activity. “Students solve and compare the structure of linear equations with only one solution, infinitely many solutions, and no solution. Have students work in pairs on the card sort activity. Distribute one set of Linear Equations cards to each pair. Have students review the cards, identify relationships among them, and then sort them into three groups. Encourage students to use the space in their book for work. Tell students they should be prepared to describe their groups to the class.”

  • Module 4, Topic D, Lesson 17: Slopes of Rising Lines, Debrief, students complete a graphic organizer and participate in a class discussion. “Have students complete the Slope of a Rising Line section of their Slope of a Line graphic organizer started in lesson 16 as shown. Then facilitate a discussion by asking the following questions. Can we find the slope of a rising line that does not pass through the origin? If so, how? What information determines a line?”

  • Module 6, Topic C, Lesson 16: Using the Investigative Process, Learn, UDL: Engagement, “Consider offering feedback that emphasizes how finding and understanding difficulties can help students learn to be successful in the future. For example, if students sometimes use centimeters to measure a variable but at other times use inches, encourage them to use this error as an opportunity to improve their efforts to attend to precision when collecting their data.”

Indicator 3P
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Materials provide opportunities for teachers to use a variety of grouping strategies.

The materials reviewed for Eureka Math2 Grade 8 provide opportunities for teachers to use a variety of grouping strategies. 

The materials provide opportunities for teachers to use a variety of grouping strategies. Teacher suggestions include guidance for a variety of groupings, including whole group, small group, pairs, or individual. Examples include:

  • Module 1, Topic E, Lesson 23: Ordering Irrational Numbers, Launch, “Arrange students in pairs and hand out a set of Battle Cards to each pair. Instruct pairs to shuffle the cards and split them equally into two piles laid facedown, with one pile in front of each student. Have students flip over the top card in their pile. The student who turns over the card with the greater value collects both cards and sets them aside in their own winning pile. The game ends when students have gone through all their cards once. Students should keep the cards in their winning pile to use in the next activity. Circulate to listen for strategies that pairs of students use to compare their Battle Cards. When most pairs have finished playing, discuss the comparison strategies as a class.”

  • Module 4, Topic B, Lesson 9: Writing Linear Equations, Learn, Problem 1, “Organize students into groups of three. Use the Numbered Heads routine and assign each student a number, 1 through 3. Direct students to problem 1 in row 1 of the table in their book. Give groups a few minutes to write equations with the given number of solutions. Remind students that any one of them could be the spokesperson for the group, so they should be prepared to answer.” Teachers use the students’ assigned numbers to call on students as a part of a whole class discussion, “Continue to use the Numbered Heads routine to facilitate discussion about the structure of the written equations and their respective solutions. Encourage students to voice the complete equation instead of just the values for a and b.”

  • Module 6, Topic C, Lesson 14: Determining an Equation of a Line Fit to Data, Launch, Problem 1, “Have students think–pair–share about the following question. What do you notice about the association between fuel efficiency and the independent variable?” After a whole class discussion, students complete Problem 1 independently. “Have students complete problem 1 independently. Then have them compare their predictions with a partner.”

Indicator 3Q
02/02

Materials provide strategies and supports for students who read, write, and/or speak in a language other than English to regularly participate in learning grade-level mathematics.

The materials reviewed for Eureka Math2 Grade 8 meet expectations for providing strategies and supports for students who read, write, and/or speak in a language other than English to regularly participate in learning grade-level mathematics.

Support for active participation in grade-level mathematics is consistently included within a Language Support Box embedded within parts of lessons. The Implementation Guide explains supports for language learners, “Multilingual learners, or learners who speak a language other than English at home, require specific learning supports for gaining proficiency with the English needed to access the mathematics. Research suggests that best practices for these learners include opportunities and supports for student discourse and for using precise terminology. In addition to precise domain-specific terminology, high-impact academic terminology that supports learners across learning domains is explicitly introduced and used repeatedly in various contexts to build familiarity and fluency across the grade levels. Eureka Math2 is designed to promote student discourse through classroom discussions, partner or group talk, and rich questions in every lesson.” 

Examples include:

  • Module 2, Topic D, Lesson 22: On the Right Path, Learn, Language Support, “In addition to playing the video, consider supporting the context further by drawing on student experience. For example, show picture examples of each ride to facilitate a discussion about which ride might take up the greatest amount of total time based on the wait and ride times. Discuss and clarify possible assumptions needed to solve the problem, such as wait times remaining constant over the 1.5 hours or deciding whether to account for the time taken to get on and off a ride.”

  • Module 3, Topic B, Lesson 6: The Shadowy Hand, Learn, Language Support, “As groups discuss solution strategies, some students may benefit from additional support in the form of sample questions to drive group discourse. Consider providing the following examples for students to ask one another during group work time: What are we trying to find? What information do we need? What can we measure? What do we need to calculate?”

  • Module 4, Topic D, Lesson 15: Comparing Proportional Relationships, Learn, Language Support, “Students may benefit from support with the term steeper line. Consider displaying the graphs of two linear equations and discussing which line is steeper.”

Indicator 3R
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Materials provide a balance of images or information about people, representing various demographic and physical characteristics.

The materials reviewed for Eureka Math2 Grade 8 provide a balance of images or information about people, representing various demographic and physical characteristics.

Images are included in the student materials as clip art. These images represent different races and portray people from many ethnicities in a positive, respectful manner, with no demographic bias for who achieves success based on the problem contexts and grade-level mathematics. There are also a variety of people captured in video clips that accompany the Launch portion of lessons. Examples include: 

  • Module 1, Topic B, Lesson 5: Products of Exponential Expressions with Whole-Number Exponents, Exit Ticket, Problem 5, “Abdul finds an equivalent expression for 43464^3\cdot4^6. He writes 4346=(444)(444444)4^3\cdot4^6=(4\cdot4\cdot4)\cdot(4\cdot4\cdot4\cdot4\cdot4\cdot4). Then he counts the 4’s to determine the exponent to use in his answer. Explain a quicker way for Abdul to get the same result.” 

  • Module 3, Topic A, Lesson 3: Reductions and More Enlargements, Learn, Problem 5, “In the diagram, Eve identified the scale factor as −2 for the dilation centered at point O. Jonas identified the scale factor as 2 for the dilation. Do you agree with Eve or Jonas? Why?” 

  • Module 4, Topic F, Lesson 27: Get to Work, Learn, Problem 2, “Mr. Adams, Mrs. Banks, Mrs. Kondo, and Mr. Jacobs each have one of the following jobs: carpenter, electrician, painter, or plumber. Each person charges a different hourly rate and a different one-time fee. Use the clues to determine each person’s job, hourly rate, and one-time fee. Clue 1: Neither Mr. Adams nor Mr. Jacobs is an electrician. Clue 2: Mr. Adams charges an hourly rate of $20. He charges a total of $75 for 3 hours of work, which includes the one-time fee. Clue 3: Mrs. Kondo charges an hourly rate of $24. Clue 4: The electrician charges an hourly rate of $25 and no one-time fee. Clue 5: The carpenter charges an hourly rate of $21 and a one-time fee of $25. Clue 6: The plumber charges a total of $68 for 2 hours of work and $116 for 4 hours of work. The total charges include the one-time fee.”

Indicator 3S
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Materials provide guidance to encourage teachers to draw upon student home language to facilitate learning.

The materials reviewed for Eureka Math2 Grade 8 provide guidance to encourage teachers to draw upon student home language to facilitate learning. 

The Grades 6-9 Implementation Guide states, “Multilingual learners, or learners who speak a language other than English at home, require specific learning supports for gaining proficiency with the English needed to access the mathematics. Research suggests that best practices for these learners include opportunities and supports for student discourse and for using precise terminology. In addition to precise domain-specific terminology, high-impact academic terminology that supports learners across learning domains is explicitly introduced and used repeatedly in various contexts to build familiarity and fluency across the grade levels. Eureka Math2 is designed to promote student discourse through classroom discussions, partner or group talk, and rich questions in every lesson. Learn more about supporting multilingual learners in Eureka Math2 here.” This statement provides a link to Multilingual Learner English Support, “How to Support Multilingual Learners in Engaging in Math Conversation in the Classroom,” which provides teachers with literature on research-based supports for Multilingual Learners.

Additionally, for grades 6-9, Eureka Math2 provides Lesson Recaps, “You may use the Recaps as a guide to support practice outside of class. Recaps are also useful for anyone supporting the student’s learning, including family members, tutors, and special educators.” Lesson Recaps include:

  • Summaries of the main learning of the lesson.

  • Definitions of any terms introduced in the lesson.

  • Problems like those completed in class and examples of the thinking that helps students solve the problems.

Indicator 3T
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Materials provide guidance to encourage teachers to draw upon student cultural and social backgrounds to facilitate learning.

The materials reviewed for Eureka Math2 Grade 8 partially provide guidance to encourage teachers to draw upon student cultural and social backgrounds to facilitate learning. 

While Spanish materials are accessible within lessons, there are few specific examples of drawing upon student cultural and social backgrounds. Examples include: 

  • Module 1, Topic A, Lesson 3: Time to Be More Precise, Launch, “Archimedes had to develop a new way to write very large numbers. He called this new notation a myriad, represented by the capital letter M. He used the symbol M for the number 1⁢0,0⁢0⁢0”.

  • Module 5, Overview, Math Past, “One of the most influential Chinese mathematical texts of all time is the Jiuzhang Suanshu, translated as Nine Chapters on the Mathematical Art. The text is composed of 246 problems that provide methods to offer solutions and guidance in the contexts of land surveying, the exchange of goods, engineering, and taxation. The original authors are unknown, but the text is estimated to have been written during the late Qin or early Han dynasties.”

Indicator 3U
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Materials provide supports for different reading levels to ensure accessibility for students.

The materials reviewed for Eureka Math2 Grade 8 partially provide supports for different reading levels to ensure accessibility for students.

The Grades 6-9 Implementation Guide states, “A student’s relationship with reading should not affect their relationship with math. All students should see themselves as mathematicians and have opportunities to independently engage with math text. Readability and accessibility tools empower students to embrace the mathematics in every problem. Lessons are designed to remove reading barriers for students while maintaining content rigor. Some ways that Eureka Math2 clears these barriers are by including wordless context videos, providing picture support for specific words, and limiting the use of new, non-content-related vocabulary, multisyllabic words, and unfamiliar phonetic patterns.” For example:

  • Module 2, Topic B, Lesson 10: Congruent Figures, Language Support, “As the class defines the term congruent, consider providing students with examples and nonexamples as a scaffold for the definition.” Examples of congruent and non congruent shapes to display are provided.

  • Module 4, Topic F, Lesson 26: Linear Equations from Word Problems, Learn, UDL: Representation, “Presenting the water park problem in a video format supports students in understanding the problem context by removing barriers associated with written and spoken language.”

  • Module 5, Topic B, Lesson 7: The Substitution Method, Language Support, “Students’ experience with the term isolate has been in the context of rewriting equations to get either x or yby itself on one side of an equation. As they begin to apply the multiplication property of equality, students will encounter cases where they must first isolate an expression before substitution. Support students use of isolate in this new context by displaying the solution to problem 6 and annotating the step where an expression is isolated as “isolated expression.”

Indicator 3V
02/02

Manipulatives, both virtual and physical, are accurate representations of the mathematical objects they represent and, when appropriate, are connected to written methods.

The materials reviewed for Eureka Math2 Grade 8 meet expectations for providing manipulatives, both virtual and physical, that are accurate representations of the mathematical objects they represent and, when appropriate, are connected to written methods.

Manipulatives provide accurate representations of mathematical objects. Examples Include:

  • Module 3, Topic A, Lesson 2: Enlargements, Learn, “Ensure each student has a transparency and a dry-erase marker. Then direct students to problem 2. Model the use of the transparency as a tool to apply a dilation as students recreate your actions with their own transparency.” 

  • Module 5, Topic B, Lesson 7: The Substitution Method, Learn, UDL: Action & Expression, “Consider providing access to the Equation Assistant interactive during this lesson. This interactive is meant to support students with efficiently and accurately rewriting equations in preparation for solving systems of equations with substitution.” 

  • Module 6, Topic C, Lesson 13: Informally Fitting a Line to Data, Learn, “Students informally fit a line to data in a scatter plot. Direct students to work in pairs to complete problems 1(b)–(d). Allow for student choice in their approach to fitting a line to the data by providing access to a variety of tools, such as a ruler, a transparency, and uncooked spaghetti. 1. The scatter plot shows the sizes and prices of 16 houses for sale in a midwestern city. a. How much do you think the price of a 3000-square-foot house might be? Explain your reasoning. b. Draw a line in the scatter plot that you think fits the pattern of the data. c. Use your line to predict the price of a 3000-square-foot house. d. Use your line to predict the price of a 1500-square-foot house.”

Criterion 3.4: Intentional Design

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The program includes a visual design that is engaging and references or integrates digital technology, when applicable, with guidance for teachers.

The materials reviewed for Eureka Math² Grade 8 integrate technology such as interactive tools, virtual manipulatives/objects, and/or dynamic mathematics software in ways that engage students in the grade-level standards; include or reference digital technology that provides opportunities for teachers and/or students to collaborate with each other; have a visual design that supports students in engaging thoughtfully with the subject that is neither distracting nor chaotic; and provide teacher guidance for the use of embedded technology to support and enhance student learning. 

Indicator 3W
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Materials integrate technology such as interactive tools, virtual manipulatives/objects, and/or dynamic mathematics software in ways that engage students in the grade-level/series standards, when applicable.

The materials reviewed for Eureka Math2 Grade 8 integrate technology such as interactive tools, virtual manipulatives/objects, and/or dynamic mathematics software in ways that engage students in the grade-level/series standards, when applicable. 

According to the Grades 6-9 Implementation Guide, “Each Eureka Math² lesson provides projectable slides that have media and content required to facilitate the lesson…” Examples include: 

  • Fluency activities

  • Digital experiences such as videos, teacher-led interactives, and demonstrations

  • Images and text from Teach or Learn cued for display by prompts such as display, show, present, or draw students’ attention to 

  • Pages from Learn including Classwork, removables, Problem Sets, and Exit Tickets. 

Additionally, Inside the Digital Platform, “Lessons that include digital interactives are authored so that while you demonstrate the digital interactive, students engage with the demonstrations as a class. Eureka Math² digital interactives help students see and experience mathematical concepts interactively. You can send slides to student devices in classroom settings where it feels appropriate to do so. Use Teacher View to present, send slides to students, monitor student progress, and create student discussions. If you send interactive slides to students from this view, you can choose to view all students’ screens at once or view each student’s activity individually.”

Indicator 3X
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Materials include or reference digital technology that provides opportunities for teachers and/or students to collaborate with each other, when applicable.

The materials reviewed for Eureka Math2 Grade 8 include or reference digital technology that provides opportunities for teachers and/or students to collaborate with each other, when applicable. 

According to the Implementation Guide, “To encourage student discussion and collaboration, provide one device per student pair. This is indicated in the Materials section.” Examples include:

  • Module 1, Topic D, Lesson 18: The Pythagorean Theorem, “In this digital lesson, students follow the plot of the book What’s Your Angle, Pythagoras? to explore the Pythagorean theorem. Students consider what combinations of side lengths create a right triangle with Pythagoras’s knotted rope.”

  • Module 3, Topic D, Lesson 17: Similar Triangles on a Line, “In this digital lesson, students are introduced to right triangles with horizontal and vertical legs and with hypotenuses that lie on the same line by first generating an example of this type of triangle. They then compare their right triangle to those of their classmates and realize that the triangles are similar.”

  • Module 4, Topic C, Lesson 12: Solution to Linear Equations in Two Variables, “In this digital lesson, students graph and examine solutions to linear equations in two variables. Students begin by considering how to score exactly 32 points in a basketball challenge. Then they examine these different ways to score 32 points to determine what makes an ordered pair a solution to an equation.”

  • Digital Lesson Teacher View: The Implementation Guide states, “Use Teacher View to present, monitor student progress, and create student discussions. From this view, you can choose to view all students’ screens at once or view each student’s activity individually. Toggle among Guidance, Monitoring, and Gallery modes to present, monitor student progress, and create student discussion points. Pacing gates restrict students from moving to the next slide so that you can facilitate discussion about a concept or discovery. You can track the pace of the class and pause students as needed.”

Indicator 3Y
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The visual design (whether in print or digital) supports students in engaging thoughtfully with the subject, and is neither distracting nor chaotic.

The materials reviewed for Eureka Math2 Grade 8 have a visual design (whether in print or digital) that supports students in engaging thoughtfully with the subject, and is neither distracting nor chaotic. Examples include:

  • Learn Book: The Implementation Guide states, “Lesson pages are completed by students during the lesson. The pages are organized in the order they are used in the lesson, starting with Launch, and are labeled with the segment titles in the lesson. Exit Tickets are completed during the Land segment of the lesson. The Exit Ticket is a brief, formative assessment of key learning in the lesson. The Recap outlines key learning from the lesson and provides examples with supporting notes. Practice pages provide a bank of problems organized from simple to complex.”

  • Module 3, Topic B, Lesson 6: The Shadowy Hand, Learn, Differentiation Support, “Consider displaying the following diagram for students who have difficulty finding a model for the situation. Ask the following questions to promote the connection between the activity and dilations: What does this diagram remind you of? What are some relationships among the elements in the diagram? How is this diagram useful?”

  • Module 5, Topic A, Topic Opener, “In mathematics, the word intersection has a special meaning: It is the shared part two objects have in common. Picture the middle part of a Venn Diagram, and you’ll get the idea. So what is the intersection of two roads? Usually, it’s just that: It’s the place where they meet. What about parallel roads? They have no intersection at all because they never cross. And what is the intersection of a road with itself? Well, it’s the entire road. The next topic isn’t about intersecting roads. Instead, it’s about the mathematical version: pairs of lines in the coordinate plane. Because every line goes by so many names, finding the intersection point of two lines is sometimes a tricky matter. They may intersect just once. Or they may never intersect. Or the two lines may turn out to be the same line in disguise.”

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Materials provide teacher guidance for the use of embedded technology to support and enhance student learning, when applicable.

The materials reviewed for Eureka Math2 Grade 8 provide teacher guidance for the use of embedded technology to support and enhance student learning, when applicable. Examples include:

  • Teacher View: The Implementation Guide states, “Use Teacher View to present, monitor student progress, and create student discussions. From this view, you can choose to view all students’ screens at once or view each student’s activity individually. Student View Digital lessons contain interactives that students access on their devices. Students use the interactives to engage directly with mathematical concepts and receive immediate feedback.”

  • Digital Lessons: The Implementation Guide states,“Every module contains digital lessons that are accessed on the digital platform. They are part of the module’s sequence of lessons and have objectives that advance key learning. Digital lessons provide you with immediate access to every student’s response, and they create easy ways to use student work to facilitate discussion. Within the lesson overview, the Lesson at a Glance and icons in the lesson agenda identify digital lessons.”

  • Module 2, Topic B, Lesson 8: Sequencing the Rigid Motions, Teach Book, “In this digital lesson, students examine sequences of rigid motions by creating and describing rigid motions that map a figure onto its image.”