8th Grade - Gateway 2

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 x^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 \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 4x-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 4x-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=mx for a line through the origin and the equation y=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+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 \frac{11}{7}. a. Place a point R on \overline {PQ}. Draw and label the image of R under a dilation with center O and scale factor \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=\frac{1}{2}x+4 and y=\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).

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}” 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 x^2=p and x^3=p, where p is a positive rational number. Evaluate square roots of small perfect squares and cube roots of small perfect cubes…). 

• 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. 27-3a=8a+5, 4. 9(c-2)=3c-28, 5. 8-4m=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 \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. ___ <\sqrt{17}<___ 2. ___<\sqrt{88}< ___ 3. ___<\sqrt{8}< ____ 4. ___<\sqrt[3]{30}<___.” 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. {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=-\frac{2}{3}x and 4x+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\degree and 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).

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 10^0=10? Use the product 10^0\cdot10^3 to show whether it upholds the property x^m\cdot x^n=x^{m+n}3. Could 10^0=1? Use the product 10^0\cdot10^3 to show whether it upholds the property x^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 \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).

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).

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 {x-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?”

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 x^2=p, where p is a perfect square, “w^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 \triangle{KLM} and \triangle{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 \triangle{KLM} similar to \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=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\times10^3 in standard form. He writes 67,000 because 10^3 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 -\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.”

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\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=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?”

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=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 22cm^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~B  and B~C, then A~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 \overline {AB} and \overline {CD} coincide because they are the same line.”

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 \frac{9\times10^9}{3\times10^3} into easier problems? How are \frac{9 billion}{3 million} and \frac{9\times10^9}{3\times10^3} related? How could that help you find the quotient? How are \frac{9 million}{3 million} and \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. “m-6=6m-9, 10r-2=10r+1, and -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 and 5-4x have in common to determine the number of solutions to the equation -4x+5=5-4x? How can you use what 10r-2 and 10r+1 have in common to determine the number of solutions to the equation 10r-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} 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?”