8th Grade - Gateway 2

The materials reviewed for Reveal Math 2025 Grade 8 meet expectations for developing conceptual understanding of key mathematical concepts, especially where called for in specific content standards or cluster headings.

Materials provide opportunities for students to develop conceptual understanding throughout the grade level. Unit Overviews outline the three parts of rigor–conceptual understanding, procedural skill & fluency, and application. The Be Curious activities, which occur during the Launch, focus on sense making with different routines, such as the Notice and Wonder^TM. During the Explore & Develop (Activity-Based and Guided Exploration), instruction links the sense-making activity to conceptual understanding, ensuring students understand the “why” behind operations and other important mathematical skills. Additionally, the eToolkit provides eTools to help students develop a conceptual understanding of math concepts.” Examples include:

• Unit 2, Unit Overview: Congruence and Similarity, Lesson 2-2: Explore Reflections, Rigor, Conceptual Understanding, “Students verify experimentally the properties of reflections. Students describe the effects of reflections on two-dimensional figures using coordinates.” (8.G.3)

• Unit 4: Understand and Analyze Functions, Lesson 4-1: Describe Qualitative Relationships, Session 1, Practice, Items 1 - 3, students build conceptual understanding of the relationship between two quantities as they interpret and describe a graph. The materials state, “1. How can you describe the relationship between the volume of water in the reservoir and time? 2. Integral A represents a period of rainy weather. How can you interpret interval B? 3. How can you explain the different slopes that are visible during intervals A and C?” Students are given a graph labeled Time on the x-axis and Volume of Water on the y-axis. The graph has 3 line segments labeled A, B, and C. (8.F.5)

• Unit 6: Angles, Triangles, and the Pythagorean Theorem, Lesson 6-3: Understand and Use Square Roots, Session 1, Guided Exploration, Let’s Explore More, students build conceptual understanding as they explore squares, square roots, and different ways to represent solutions. The materials state, “a. What is the relationship between squares and square roots? b. What other squares less than 125 are there? Express their factors as square roots. c. What are the different ways you can represent the solution to the equation x^2= 121?” (8.EE.2)

• Unit 8: Systems of Linear Equations, Lesson 8-4: Use Substitution to Solve Systems of Equations, Session 1, Launch, Be Curious: Notice & Wonder, students build conceptual understanding of solving systems. The materials ask, “What do you notice? What do you wonder? What do you think the different shapes represent? How might the figures in the columns and rows be related to the numbers at the end of each column and row? What strategies might you use to find the value of each figure?” Students are given a 4x4 chart, each box has either a triangle, circle, or star in it. At the end of each row and column is a number. (8.EE.8)

• Unit 9: Irrational Numbers, Exponents, and Scientific Notation, Lesson 9-4: Explore Patterns of Exponents, Session 1, Guided Exploration, Let’s Explore More, students explore the properties of integer exponents through the metric system. The materials ask, “How can you compare units in the metric system?” then “a. What does it mean when a power has an exponent of zero? b. What is the value of 10^{-5} written as a fraction and as a decimal? c. What is the difference between 10^3 and 10^{-3}? d. how 8^{-1} do 8^{-2} and relate?” (8.EE.1)

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

• Unit 2: Congruence and Similarity, Lesson 2-3: Explore Rotations, Session 1, Guided Exploration, students explore the properties of rotations by moving furniture in a room. The materials state, “a. Where would the love seat be placed if it were rotated 360° clockwise about the center of rotation? b. A classmate concludes that if the love seat was rotated 270° counterclockwise it will be in the same location as a 90° clockwise rotation. Is your classmate correct? Explain your thinking.” (8.G.1)

• Unit 4: Understand and Analyze Functions, Lesson 4-1, Describe Qualitative Relationships, Session 1, Launch, Be Curious: Notice & Wonder, students share their thinking about information shown on the treadmill screen while building towards describing qualitative relationships. The materials state, “What question can you ask? What information does the screen show? What does the height of each section in the graph represent? Why does the height of the graph change? What happens when the graph is going up or down?” Pause & Reflect states, “Students think about how the quantities shown on the treadmill are related.” (8.F.5)

• Unit 5: Patterns of Association, Lesson 5-3: Analyze Linear Associations in Bivariate Data, Session 1, Activity-Based Exploration, students construct a line of fit for bivariate data and determine the closeness of the association. The materials state, “Digital: Students explore how shifting points on a scatterplot affects the line of fit. Before students begin the first activity, have them explore the Websketch^TM tools they will be using. They ensure that they can move a point on the coordinate plane.” then the materials state, “Hands-On: Students individually construct a straight line that best describes the data for graphs A-D on the Scatter Plots Teaching Resource. Then, students then compare their lines, debating who has the better line and why.” (8.SP.2)

• Unit 7: Volume, Lesson 7-2: Solve Problems Involving Cylinders, Session 1, Activity-Based Exploration, students demonstrate conceptual understanding as they find the volume of a cylinder is like finding the volume of a prism. The materials state, “Digital: Students explore how a cylinder’s height and base affect its volume. Before students begin the first activity, have them explore the Websketch^TM tools they will be using. They ensure that they understand how to use the calculation widget with the given values for the area of the base and for the height.” The problem states, Practice, Item 1, “How would you explain to a classmate how to find the volume of a cylinder?” (8.G.9)

• Unit 8: Systems of Linear Equations, Lesson 8-3: Determine the Number of Solutions, Session 1, Practice, Item 7, students demonstrate understanding of identifying the number of solutions to a system of equations. The materials state, “7. Which of these describes a system of equations that has no solutions? A. They have the same y-intercept. B. They are perpendicular. C. They have the same slope and different y-intercepts. D. They have the same x-intercept and different y-intercepts.” (8.EE.8)

The materials reviewed for Reveal Math 2025 Grade 8 meet expectations for giving attention throughout the year to individual standards that set an expectation of procedural skill and fluency.

Materials provide opportunities for students to develop procedural skill and fluency throughout the grade level. Reveal Math provides students with multiple opportunities to revisit concepts and develop these areas of fluency within each unit. Implementation Guide (page 58) “Number Routines provide students with daily opportunities to develop number sense, deepening their understanding of number relationships. In addition, every unit reviews a computational strategy previously learned to revisit concepts and strategies adding to students’ flexibility when choosing methods.” Examples include:

• Unit 3: Linear Relationships and Equations, Lesson 3-6: Solve Linear Equations, Session 2, Guided Exploration, students explore the concept of the Distributive Property through a problem involving costs for an end-of-year party. The problem states, “The students in eighth grade are planning an end-of-year party for 45 classmates. They have a budget of $1500 for the party. The party room costs $170.25. Each guest will get a $6 gift. How much can the students spend per guest on food? Step 1, Make sense of the quantities. The budget is $1500. The party room is $170.25. Each of the 45 guests receive a $6 party favor. You need to determine how much the class can spend per guest on food. 1500=170.25+45(x+6) Step 2, Solve the equation. Simplify grouping symbols first.” Students develop procedural fluency in 8.EE.7 (Solve linear equations in one variable).

• Unit 8: Systems of Linear Equations, Lesson 8-2: Estimate Solutions to Systems of Equations by Graphing, Session 2, Lesson Quiz, Items 1-3, students write and graph systems of equations to find which florists the wedding coordinator should hire. The problem states, “A wedding coordinator is hiring a florist. To decorate for a wedding, Petals Florist charges $22 per hour plus $55 for supplies. Red Roses Florist charges $11 per hour plus $120 for supplies. 1. Let x represent the number of hours the florist works and y represent the total cost. Write a system of equations that models the situation. 2. Graph the systems of equations on the coordinate plane. 3. What is an approximate solution to the system? Explain the meaning of the solution.” Students develop procedural fluency with 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.)

• Unit 9: Irrational Numbers, Exponents, and Scientific Notation, Lesson 9-4: Explore Patterns of Exponents, Session 2, Guided Exploration, students explore and compare values of negative and zero exponents. The Lesson states, “Rules of Exponents: A student is using a calculator to explore negative and zero exponents. How do the values on the calculators compare?” A worked example of the negative exponent rule and zero exponent rule is provided. “Let’s Explore More a. What is the difference between the expressions (-3)^2 and 3^{-2}? b. Without calculating, which value is greater: 7^{-3} or 8^{-3}?” Students develop procedural fluency of 8.EE.1 (Know and apply the properties of integer exponents to generate equivalent numerical expressions).

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

• Unit 5: Patterns of Association, Fluency Practice, the materials state, “Fluency practice helps students develop procedural fluency, that is, the “ability to apply procedures accurately, efficiently, and flexibly.” Because there is no expectation of speed, students should not be timed when completing the practice activity.” Build Fluency section states, “Objective Students build fluency with relating unit rate with slope. As students work to develop fluency with unit rates, have them reflect on and share with classmates the strategies they find the most useful.” Fluency Talk states, “How would you describe the relationship between the slope of a linear graph and the unit rate for the relationship it shows?”

• Unit 6: Angles, Triangles, and the Pythagorean Theorem, Lesson 6-3: Understand and Use Square Roots, Differentiate, Reinforce Understanding, Independent Work, Exercise 5-8, students practice solving problems involving square roots. “For exercises 5-8, solve each equation and check the solutions. 5. x^2=144 6. t^2=\frac{25}{9} 7. m^2=-25 8. g^2=\frac{100}{225}” This activity provides an opportunity for students to develop procedural skill and fluency of 8.EE.1 (Know and apply the properties of integer exponents to generate equivalent numerical expressions.)

• Unit 8: Systems of Linear Equations, Lesson 8-4: Use Substitution to Solve Systems of Equations, Number Routines, If I Know This…, students build proficiency with operations as they use the solution to an equation to solve equations with the same digits with different base ten values. The materials state, “This routine is similar to the Number String Matrix routine. Students are given a single fact and four or five equations that are related in some way. Students explain how they used the given fact to determine the solutions to the equations.” This activity provides an opportunity for students to develop procedural skill and fluency of 8.EE.8 (Analyze and solve pairs of simultaneous linear equations.)

The materials reviewed for Reveal Math 2025 Grade 8 meet expectations for being designed so that teachers and students spend sufficient time working with engaging applications of the mathematics.

There are multiple routine and non-routine application problems throughout the grade level, including opportunities for students to work with teacher support and independently. The materials state, “While single and multi-step application problems are included across various portions of lessons, independent application opportunities are most often found within the Apply section. Many Apply problems provide multiple options, helping to build student agency through choice.” Materials provide opportunities for students to engage with routine application problems throughout the grade level. Examples include:

• Unit 4: Understand and Analyze Functions, Lesson 4-4: Explore Types of Functions, Session 2, Summarize & Apply, students interpret the equation y = mx + b as a linear function and apply it to real-world scenarios. The materials state, “Apply: Types of Functions Fatima wants to invest $200. She is considering two options. For option A, the interest rate is a simple interest rate of 4%. For option B, the interest rate is 3.25% compounded annually. Question: Which option should Fatima choose for her investment if she plans to invest for less than 20 years? Which option is best if Fatima wants to invest for more than 20 years?” (8.F.3)

• Unit 6: Angles, Triangles, and the Pythagorean Theorem, Lesson 6-4: Understand the Pythagorean Theorem, Session 2, Guided Exploration, Let’s Explore More, students consider how to modify the Pythagorean Theorem equation to find the length of a leg. The materials state, “a. How can you use the Pythagorean Theorem to determine the length of a missing leg, given the lengths of one leg and the hypotenuse? b. Ancient Egyptians discovered that a triangle with side lengths 3, 4, and 5 is a right triangle. How could you use this fact to solve the bridge cable length problem above? c. Using the same 3, 4, 5 relationship, can you name another set of three whole numbers that make a right triangle? Use the Pythagorean Theorem to check your answer.” (8.G.7)

• Unit 9: Irrational Numbers, Exponents, and Scientific Notation, Mathematical Modeling, Wind Energy is Trending, students develop a proposal to build a wind turbine for a city. The materials state, “You are the owner of a wind turbine company that services single households and cities. An average household in the United States uses 893 kilowatt-hours (kWh) of electricity per month. An average wind turbine generates 8.43 x 10^5 kWh per month. The following cities are considering being powered by wind energy. Which city should your company submit a proposal to?” A chart with 4 cities, their electrical usage (kWh) and population is provided. (8.EE.1)

Within the Implementation Guide, Focus, Coherence, Rigor, Application, “Students encounter real- world problems throughout each lesson. The On My Own exercises include rich, application-based question types, including Error Analysis and Extend Thinking. Lesson differentiation provides opportunities for application through the STEM Adventures. The unit performance task and the Mathematical Modeling Project, both found in the Student Edition offer additional opportunities for students to apply their knowledge of math concepts to solve non-routine application problems.” Examples of non-routine application problems include

• Unit 3: Linear Relationships and Equations, Lesson 3-3: Use Similar Triangles to Determine Slope, Differentiate, STEM Adventures, students compare two different proportional relationships represented in different ways. The materials state, “In this STEM Adventure, biosystems engineers are exploring erosion control solutions for different project sites. Use your linear relationship and equation knowledge to test and interpret which erosion control solutions are most efficient.” (8.EE.5)

• Unit 5: Patterns of Association, Describe Patterns in Two-Way Tables, Session 2, Lesson Quiz, Exercises 7-11, STEM Connection, students understand that patterns of association can also be seen in bivariate categorical data by displaying frequencies and relative frequencies in a two-way table. The materials state, “An experiment investigates possible side effects of a headache medicine. Study participants are given either a dose of the medicine or a placebo (an inactive treatment). The table shows the results of the study. 7. Complete the two-way table by filling in the totals for each category. 8. Which side effect is most common among participants taking the medicine? Taking the placebo? Among all study participants? 9. What patterns of association do you notice between the two categories? 10. How could this information help the company that makes the headache medicine? 11. Error Analysis One of the researchers says that nausea and fever are the main side effects from the medicine. How do you respond to that researcher?” (8.SP.4)

• Unit 7: Volume, Mathematical Modeling, There's Volume in Green Buildings, Project Two, students explore the volume of a cylinder through determining how many boards a tree can produce. The materials state, “Green buildings conserve natural resources, which enhances our quality of life. Wood products are a good choice for most new construction green building projects. Wood is renewable, sustainable and recyclable. A company building a green housing development needs to estimate how many trees will be needed to build a home. 1 square foot of a home needs approximately 6.3 boards. It is estimated the mature pine trees used have a height of 70 feet and a circumference of 6 feet. Number of boards = Volume of cylindrical tree trunk \times3. Design a proposal recommending the number of mature pine trees needed for the company to build a 2,000 square foot home.” (8.G.9)

The materials reviewed for Reveal Math 2025 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 instructional materials attend to conceptual understanding, procedural skill and fluency, or application include:

• Unit 4: Understand and Analyze Functions, Lesson 4-2: Explore Functions, Session 1, Launch, Be Curious: Notice & Wonder, students build conceptual understanding as they learn that a function is a rule that assigns to each input exactly one output. The materials state, “Students discuss various aspects of a vending machine. What do you notice? What do you wonder? What do the letters and numbers on the keypad mean? How is the keypad related to the snacks in the vending machine? How can you use the keypad to get a particular snack? Why is it important that each combination of letters and numbers gives only one type of snack?” (8.F.1)

• Unit 6: Angles, Triangles, and the Pythagorean Theorem, Lesson 6-6: Determine Distance on the Coordinate Plane, Practice, students develop procedural fluency while using the Pythagorean Theorem to determine the length of the hypotenuse of a right triangle constructed on the coordinate plane. The materials state, “For exercises 8-14, use the graph to answer the questions. 8. What is the distance between points A and B? 9. What is the distance between points D and F? 10. Which points are the closest to each other? What is the distance between those points?” (8.G.8)

• Unit 7: Volume, Lesson 7-4: Solve Problems Involving Spheres, Mathematical Modeling, There’s Volume in Green Buildings, students apply their knowledge of volume by developing a proposal for an alternative energy source for the town and determine the amount of space needed for the energy source. The materials state, “Project One One way to conserve water for sustainable living is to use a rainwater harvesting system that stores rainwater in a cylindrical tank. A home development company in Seattle in hiring you to design the tank. Recommended Tank Size =0.05\timesA\timesD\timesR A = Area of Roof D = 0.8 = Drainage Coefficient R = Annual Rainfall” Students are provided a chart with years and annual rainfall in inches. (8.G.9)

Multiple aspects of rigor are engaged simultaneously to develop students’ mathematical understanding of a single unit of study throughout the materials. Implementation Guide, Lesson Walk-Through, Rigor, states “Every lesson describes the main rigor focus of each lesson based on the goals and expectations of the standards.” The Apply section states, “The Apply offers students a non-routine problem to solve. Many Apply problems provide multiple options, helping to build student agency though choice. The Teacher Edition offers scaffolded prompts that the teacher can ask students who may need support getting started.” Practice & Reflect states, “Practice & Reflect provides students with practice that address all elements of rigor.” Many lessons include more than one aspect of rigor. Examples include:

• Unit 2: Congruence and Similarity, Lesson 2-6: Understand Similarity, Session 1, Guided Exploration, Reimagining the Patio, students build conceptual understanding and develop procedural fluency through transformations to show that one two-dimensional figure is similar to another, then describe a sequence of transformations that exhibits the similarity of the figures. The material states, “A client has requested a larger table for his patio. The landscape architect also suggests a new location for the table. How can the architect describe the position and size of the larger table compared to the smaller one?” Let’s Explore More, “a. Are all the rectangles similar? Why or why not? b. How can the side measures help you determine if the tables are similar? c. What does it mean if the scale factor of the dilation in a sequence of transformations is one?” (8.G.4)

• Unit 3: Linear Relationships and Equations, Lesson 3-5: Describe Solutions to Linear Equations, Session 1, Exit Ticket, students build conceptual understanding, develop procedural fluency, and use application by representing nonproportional linear relationships with an equation and applying the concept of the equation y=mx+b to real-world situations. The materials state, “For items 1 and 2, use the graph representing the total cost to ship a package to complete the exercise. 1. Does the graph represent a proportional relationship? Explain. 2. Identify the slope and y-intercept. Explain what the slope represents.” Students are given a graph with an x-axis of Weight (lbs) and a y-axis of Cost ($). There is a ray drawn on the graph. (8.EE.6)

• Unit 6: Angles, Triangles, and the Pythagorean Theorem, Lesson 6-5: Understand the Converse of the Pythagorean Theorem, Summarize & Apply, students build conceptual understanding, develop procedural fluency, and apply the converse of the Pythagorean Theorem to determine if a triangle is a right triangle. The materials state, “The Ancient Egyptians used a rope loop with 12 equally-spaced knots to check for right angles. Question: How do you think they used this rope loop to make square corners when they surveyed land or built structures?” (8.G.7)

The materials reviewed for Reveal Math 2025 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. 

Within the Implementation Guide, Math Practices states, “To think like mathematicians, students must build thinking habits that help them develop a problem-solving frame of mind. Reveal Math helps students build proficiency with these important thinking habits and problem-solving skills through the Math is... Prompts found in every lesson. These prompts model the kinds of questions students can ask themselves to become proficient problem solvers and doers of math.” The Standards for Mathematical Practice are identified for teachers in the Lesson Overviews, and within the lesson margins labeled in orange as “Math Practices and Processes” or “MPP”. Although most lessons offer opportunities for students to engage with more than one Standard for Mathematical Practice, this guidance identifies a focus MP within each lesson. The notes also provide lesson-specific information, ideas, and questions teachers can use to deepen students’ engagement with the focus MP. 

MP1 is identified and connected to grade-level content, and there is intentional development of the MP to meet its full intent. Students make sense of problems and persevere in solving them as they work with support of the teacher and independently throughout the modules. Examples include:

• Unit 1: Math is…, Unit Overview, Math Practices and Processes, Promoting students’ sense-making, thinking, and reasoning states, “In this unit, Lessons 1-2 through 1-5 focus students’ learning on the mathematical habits of mind that are integral to proficiency in mathematics. Each lesson focuses on two specific habits of mind: Lesson 1-2 - Math Is Exploring and Thinking: Students refine their thinking habits related to problem solving. They discuss how they make sense of a problem, analyze givens, and determine an appropriate solution strategy for the problem. They consider how they monitor their progress towards a solution. Students also consider the meaning of quantities and the relationship among the quantities and values in a problem.” 

• Unit 2-6: Congruence and Similarity, Lesson 2-6: Understand Similarity, Session 1, Activity- Based Exploration, students analyze and make sense of sequences of transformations to show similarity. The materials state, “Encourage students to be flexible in their approach to problems. If a strategy is not leading to a reasonable solution, it is important to regroup and try a new approach. Have students complete the Concluding Questions in their Activity Exploration Journal. How can you tell which types of transformations you should use to show that two triangles are similar? If you change the order of the transformations in a sequence, does that change the image? Explain.” 

• Unit 4: Understand and Analyze Functions, Lesson 4-6: Compare Functions, Session 1, Guided Exploration, Orthodontic Braces, students analyze and make sense of features of linear functions represented in two different ways. The materials state, “Have students think about factors that make up payment plans. What is the monthly payment for the clear plastic braces? for the metal braces? Students work with a partner to determine the monthly cost of each type of braces. Consider the properties of functions that can be used to compare the pricing for the two types of braces. How can you organize the information to make sense of the problem? Help students see that information presented differently can be organized in the same way to help compare features of functions. Display the following two statements: The cost of metal braces is more expensive than the cost of the clear plastic braces. 1. Students work independently to respond to the statement, providing arguments to support their answers. 2. Students discuss their responses and reasoning with a classmate. Based on feedback from their partner, students revise and strengthen their responses. 3. Ask student-pairs to present their responses to the class. Other students should listen and ask clarifying questions about their justifications.” 

MP2 is identified and connected to grade-level content, and there is intentional development of the MP to meet its full intent. Students reason abstractly and quantitatively as they work with support of the teacher and independently throughout the modules. Examples include:

• Unit 6: Angles, Triangles, and the Pythagorean Theorem, Lesson 6-7: Apply the Pythagorean Theorem, Session 1, Guided Exploration, Handicap Ramps, students understand the relationships between problem scenarios and mathematical representations of the Pythagorean Theorem and a real-world problem. The materials state, “Have students work with a partner to determine the shapes they see in the sections of the ramp. Students should recognize that the first section is a right triangle, the second and last sections are rectangles, and the third section is a trapezoid. Have students describe the quantities and their importance to solving the problem to a partner. Encourage students to identify the height and the length of each portion of the ramp.” Let’s Explore More states, “1. Set Up: Give students time to respond to the Let’s Explore More question b and to think about what they will say to their partner to explain and justify their strategies and responses 2. What is similar, What is Different: Place students into pairs and ask them to share their responses, looking for similarities and differences. 3. Mathematical Focus: Students work in groups to discuss the relationship between the horizontal and vertical distances covered by the ramp and the total length of the ramp.” 

• Unit 8: Systems of Linear Equations, Unit Overview, Math Practices, Reason Abstractly and Quantitatively states, “Writing and solving systems of equations is a foundational skill for higher-level mathematics. Helping students to reason abstractly and quantitatively in order to represent and solve mathematical and real-world situations with systems of linear equations will provide them with the skill set they need to be successful in high-school mathematics and beyond. Encourage students to identify the important quantities in a problem and how the quantities are connected. Focus students’ attention to the relationship between quantities in a problem and how they can use the relationships to determine an appropriate solution method. For example, if the coefficients on the same variable in two equations are opposites, then elimination is a good choice.” 

• Unit 9: Irrational Numbers, Exponents, and Scientific Notation, Lesson 9-5: Use Product and Quotient of Powers Properties, Session 1, Activity-Based Exploration, Properties of Powers, students discuss what the numbers in a Product of Powers and Quotient of Powers Properties represent. The materials state, “Hands On: Students work in groups to find multiple solutions to x^{0}\cdot x^{0}=x^{10}. Be sure to tell students to use positive and negative integers for the exponents of x. Students compare their solutions with other groups and look for patterns. Then students repeat this process for the equation. How can you connect these problems to math that you already know? Students already know how to write numbers as powers. Encourage students to make connections between the expanded forms of powers and the properties used to find the products of numbers with the same exponential bases. Have students complete the Concluding Questions in their Activity Exploration Journal. How can exponential expressions be simplified when multiplying expressions with the same bases? How can exponential expressions be simplified when dividing expressions with the same bases?”

The materials reviewed for Reveal Math 2025 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.

Within the Implementation Guide, Math Practices states, “To think like mathematicians, students must build thinking habits that help them develop a problem-solving frame of mind. Reveal Math helps students build proficiency with these important thinking habits and problem-solving skills through the Math is... Prompts found in every lesson. These prompts model the kinds of questions students can ask themselves to become proficient problem solvers and doers of math.”

Students construct viable arguments and critique the reasoning of others in connection to grade-level content as they work with support of the teacher and independently throughout the units. Examples include:

• Unit 2: Congruence and Similarity, Lesson 2-5: Explore Dilations, Sessions 1, Activity-Based Exploration, Can You Shrink That?, students construct mathematical arguments as they explore dilations of two-dimensional figures. The materials state, “What changes and what remains the same when you enlarge or shrink a figure? Encourage students to think about strategies they can use to determine whether their answers are reasonable. Ask students to explain how they can use these strategies to reinforce or revise their initial ideas. Have students complete the Concluding Questions in their activity Exploration Journal. How are the triangles in this activity alike? How are they different? How does enlarging or reducing a figure compare to the previous transformations we have studied?” 

• Unit 3: Linear Relationships and Equations, Lesson 3-3: Use Similar Triangles to Determine Slope, Session 1, Guided Exploration, Triangles on a line, students construct viable arguments while exploring the hypotenuse of similar triangles having the same slope. The materials state, “Triangles ABC and CDE have the measurements shown. How do the slopes of line segments AC and CE compare?” Let’s Explore More states, “a. Why do you think similar triangles are sometimes referenced as slope triangles? b. Would the slope be the same if a third triangle is drawn? Why or why not? There are 2 similar triangles shown with the dimensions of 3,4,5 and 6,8,10.”

• Unit 4: Understand and Analyze Functions, Lesson 4-3: Represent Functions, Session 1, Activity-Based Exploration, Representing Functions, students critique the reasoning of others as they explore different ways to represent linear functions and determine initial value and rate of change in each representation. The materials state, “Have students respond to the Introductory Question in their Activity Exploration Journal. How can you represent a function in different ways? As student-pairs explore the activities, check that all pairs understand the task. If students need guidance or support, ask: What tools can you use to display the relationship between the number of cups stacked and the height of the cups? What has to be true for a function to have a constant rate of change? How is the initial value represented on a graph, table, and equation? What patterns do you notice in the table and graph? Have students discuss the patterns they notice in the table and graph. Ask volunteers to share their thoughts with the class. Encourage students to think about the relationships between the patterns they see in the table and graph. Have students complete the Concluding Questions in their Activity Exploration Journal. What are the different ways to represent a function? What characteristics can you use to describe a representation of a linear function?”

• Unit 6: Angles, Triangles, and the Pythagorean Theorem, Lesson 6-3: Understand and Use Square Roots, Session 2, Guided Exploration, Table Size, students justify their choices and estimates while finding the length of a side of a square with an area that is not a perfect square. The materials state, “A table has an area of 2 square meters. How long is each side of the table?” Let’s Explore More, “a. How can you use your knowledge of square numbers to estimate square roots of numbers that are not perfect squares? b. What values would you try that are in the hundredths? c. What do you notice each time you refine your estimates to find identical factors that equal 2?”

The materials reviewed for Reveal Math 2025 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. 

Within the Implementation Guide, Math Practices states, “To think like mathematicians, students must build thinking habits that help them develop a problem-solving frame of mind. Reveal Math helps students build proficiency with these important thinking habits and problem-solving skills through the Math is... Prompts found in every lesson. These prompts model the kinds of questions students can ask themselves to become proficient problem solvers and doers of math.”

MP4 is identified and connected to grade-level content, and there is intentional development of the MP to meet its full intent. Students model with mathematics as they work with teacher support and independently throughout the modules. Examples include:

• Unit 4: Understand and Analyze Functions Lesson 4-5: Model Linear Relationship with Functions, Session 2, Activity-Based Exploration, students construct a function to model a real-world situation and then apply the function to answer questions. The materials state, “Facilitate a whole-class discussion of the activity. Using the evidence of student thinking that you gathered, sequence students’ findings to highlight different approaches and thinking strategies used to respond to the Concluding Questions. How can you use the rate of change and initial value to find values for a linear function that are not shown in a table? How can you write an equation for a linear function represented by a table? What are some other representations you could use?”

• Unit 5: Patterns of Association, Lesson 5-6: Interpret Two-Way Relative Frequency Tables, Session 1, Activity-Based Exploration, Interpret Two-Way Tables, students create and describe what they do with the two-way relative frequency table model and how it relates to the problem situation. The material states, “What questions can you ask to better define the problem? MPP Have students focus on asking questions about how each value in the table relates to the total and the other values in the row or column. Have students complete the Concluding Questions in their Activity Exploration Journal. What information does a two-way table’s relative frequencies provide? How can you tell if there is an association between the two categorical variables represented in a two-way table”

• Unit 6: Angles, Triangles, and the Pythagorean Theorem, Lesson 6-6: Determine Distance on the Coordinate Plane, Session 1, Guided Exploration, Mountain Biking, students apply the Pythagorean Theorem and check to see whether an answer makes sense. The materials state, “Does your answer/solution make sense in the context of the situation? MPP Students should identify the total bike ride includes three parts. Discuss why the three sides of the triangle represent the length of the bike ride and why the distance from A to B is represented by the longest side of the right triangle.”

• Implementation Guide, Unit Walk-Through, Mathematical Modeling states, “As part of the STEM focus, each unit ends with a Mathematical Modeling project that offers students the opportunity to apply the math concepts they have learned. Each Unit contains two options from which students can choose, promoting engagement and student agency. These STEM-focused projects also encourage students to make decisions about how to approach the project, what mathematics to use, and how to present their project findings.”

MP5 is identified and connected to grade-level content; however, there are only 4 opportunities for students to develop MP5, as identified in the materials. One of the opportunities is in Unit 1 and focuses on a prior grade-level standard. Two examples are found in Unit 2 and one example in Unit 5. Students choose tools strategically as they work with teacher support and independently throughout the modules. Examples include:

• Unit 2: Congruence and Similarity, Unit Overview states, “Use Appropriate Tools Strategically The predominant tool that students use throughout this unit is the coordinate plane. Students use this tool to explore and make generalizations about transformations and to show that two figures are congruent or similar. Students also use paper and pencil, concrete models such as patty paper, and rulers or straightedges. Proficiency in choosing tools is a skill that applies to all areas of mathematics. As students use various tools throughout the unit, engage them in conversation about what insight they expect to gain from the tool and any limitations the tool might have.”

• Unit 2: Congruence and Similarity, Lesson 2-7: Use Angle-Angle Similarity, Session 1, Activity-Based Exploration, For Four Triangles, students consider how protractors can be helpful when completing the activity. The materials state, “As students-pairs explore the activities, check that all the pairs understand the task. If students need guidance or support ask: What do you know about the three interior angles of a triangle? How do two angles determine the shape of a triangle? How did a protractor help you complete the activity? MPP: Make sure students understand that a protractor can help compare the corresponding side lengths and angle measures. Have students complete the Concluding Question in their Activity Exploration Journal. If two angles of one triangle are congruent to two angles of another triangle, what do you know about the two triangles? Why?”

• Unit 5: Patterns of Association, Lesson 5-1: Construct Scatter Plots, Session 1, Guided Exploration, Lemonade Sales, students gain insights and recognize imitations from using different tools. The materials state, “The manager of a lemonade stand kept track of the daily high temperatures and daily sales of lemonade at a local park. How can you determine whether there is a relationship between temperature and lemonade sales? You can use a scatter plot to determine relationships between two variables. A scatter plot is a graph on the coordinate plane in which values from two variables are plotted.” Below the question is a two way table followed by a scatterplot graph of Temperature versus Lemonade Sales. Let’s Explore More states, “a. What would the managers learn from making this scatter plot and how could they use that information to plan? b. Based on the scatter plot, what is a reasonable estimate of lemonade sales on a day when the temperature is predicted to be 76°F?”

The materials reviewed for Reveal Math 2025 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. 

Within the Implementation Guide, Math Practices states, “To think like mathematicians, students must build thinking habits that help them develop a problem-solving frame of mind. Reveal Math helps students build proficiency with these important thinking habits and problem-solving skills through the Math is... Prompts found in every lesson. These prompts model the kinds of questions students can ask themselves to become proficient problem solvers and doers of math.”

Students attend to precision in mathematics, in connection to grade-level content, as they work with teacher support and independently throughout the units. Examples include:

• Unit 2: Congruence and Similarity, Lesson 2-1: Explore Translations, Session 1, Activity-Based Exploration, Can You Translate that?, students attend to precision as they explore the characteristics of transformations and translations. They observe what changes and what remains the same when a figure slides. The materials state, “Have students read and respond to the Introductory Question in their Activity Exploration Journal. What changes and what remains the same when a figure is slid? What characteristics can you use to describe the figure before it slides? Which of those characteristics remain the same after the figure slides? Which direction will the figure be facing after a slide is performed? What level of precision does the solution require?”

• Unit 7: Volume, Lesson 7-3: Solve Problems Involving Cones, Session 2, Activity-Based Exploration, Coney Island, students attend to precision as they explore how the volume of a cylinder compares to the volume of a cone. The materials state, “What must be the same about a cylinder and a cone for the volume of the cone to be one-third the volume of the cylinder? How can you find the volume of a cone?” “Which symbols are appropriate to solve this problem?” 

• Unit 8: Systems of Linear Equations, Lesson 8-2: Estimate Solutions to Systems of Equations by Graphing, Session 2, Guided Exploration, students attend to precision as they explore systems of equations. The materials state, “Rafael wants to determine the solution to a system of equations, but the intersection point was not on a grid line. What is an estimated solution to the system of equations y=2.5x-1 and y=0.5x+2?” Teacher guidance states, “Does the problem call for an estimate or an exact solution? As students think about if the solution requires an estimate or an exact solution, have them consider if there are better ways to find an estimate and what to do with that estimate.”

Students attend to the specialized language of mathematics, in connection to grade-level content, as they work with teacher support and independently throughout the units. Examples include:

• Unit 5: Patterns of Association, Lesson 5-4: Use Linear Models to Solve Problems, Session 1, Guided Exploration, Advertising Costs and Sales, students attend to the specialized language of mathematics as they write an equation that represents a line of fit, use it to make a prediction, and explain their thinking. The materials state, “The scatter plot shows the relationship between the amount of money a company spends on advertising and the amount of money they make in sales over several months. Jada drew a line she thinks represents a linear fit to the data. How much should the company expect to make in sales if they spend $50,000 in advertising?” Let’s Explore More states, “a. What is the meaning of the slope in this situation? b. What is the meaning of the y-intercept in this situation? c. The company is considering a $3 million ad during the Super Bowl. What would the company project for sales and do you think the linear relationship will still be valid? Explain your thinking.”

• Unit 8: Systems of Linear Equations, Lesson 8-2: Estimate Solutions to Systems of Equations by Graphing, Lesson Overview, Focus, Language Objectives states, “Students will distinguish between, and use, coordinating and subordinating conjunctions. To optimize output, students will participate in MLR: Co-Craft Questions and Problems and MLR: Information Gap.” Session 1, Launch, Be Curious: Notice & Wonder, Students attend to the specialized language of mathematics as they share their thinking, “What trends or patterns do you see in the graph? What do you think the graph could be representing? What do you think might occur if the lines on the graph continue farther to the right?” 

• Unit 9: Irrational Numbers, Exponents, and Scientific Notation, Lesson 9-3: Compare and Order Rational and Irrational Numbers, Lesson Overview, Focus, Language Objectives states, “Students will organize steps in sequential order orally and in writing. To optimize output, students will participate in MLR: Stronger and Clearer Each Time, MLR: Discussion Supports and MLR Co-Craft Problems” Session 1, Launch, Be Curious: Notice & Wonder, students attend to precision as they discuss what is the same and different about the two circles, “How are they the same? How are they different? What do the different marks and labels suggest about the circles? How are the two numbers related to the circles? How are the measurements related to one another? Are these circles more alike than different?”

The materials reviewed for Reveal Math 2025 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.

Within the Implementation Guide, Math Practices states, “To think like mathematicians, students must build thinking habits that help them develop a problem-solving frame of mind. Reveal Math helps students build proficiency with these important thinking habits and problem-solving skills through the Math is... Prompts found in every lesson. These prompts model the kinds of questions students can ask themselves to become proficient problem solvers and doers of math.”

MP7 is identified and connected to grade-level content, and there is intentional development of the MP to meet its full intent. Students look for and make use of structure as they work with teacher support and independently throughout the modules. Examples include:

• Unit 2: Congruence of Similarity, Lesson 2-4: Understand Congruence, Session 2, Guided Exploration, Transforming Figures, students look for patterns while transforming figures on the coordinate plane to determine whether two figures are congruent. The materials state, “Is quadrilateral EFGH congruent to quadrilateral IJKL? Two figures are congruent if there is a sequence of rigid motion transformations that can be used to generate one figure from the other.” Let's Explore More states, “a. How can you determine whether quadrilateral A is congruent to quadrilateral B? b. What sequence of transformations can be used to transform quadrilateral A onto quadrilateral B?”

• Unit 3: Linear Relationships and Equations, Lesson 3-5: Describe Nonproportional Linear Relationships, Summarize & Apply: Describe Nonproportional Linear Relationships, Apply: Erosion Control, students look for and explain structure as they write a linear equation in the form y=mx+b. The materials state, “A lakeshore is eroding each year and there are two mitigation options available. A development team is deciding which method of erosion control they wish to use to cover 5,000 square feet of lakeside over the next 20 years. Mulch must be replaced every two years. The erosion blankets must be replaced every five years. Question: Which method should the team use if they have a budget of $5,000 per year? What limitations will this have?” Students are given a chart of cost per square feet for the mulch and the erosion blankets. 

• Unit 4: Understand and Analyze Functions, Lesson 4-4: Explore Types of Functions, Practice, Item 12, students look for and explain structure as they identify if a function is linear or nonlinear based on multiple representations. STEM Connection states: “The Mars 2020 Perseverance Rover is traveling across the surface of Mars collecting samples and information to learn about ways to support life on the planet. The rover moves at a constant speed of 120 meters per hour for an hour. It stops to collect soil samples and take pictures for 20 minutes The rover then starts moving to ist next location over a rougher terrain at a constraint speed of 60 meters per hour for 40 minutes.” Error Analysis states, “A classmate says that if Perseverance had continued moving at a constant speed of 120 meters per hour after it stopped to collect samples, then the relationship would represent a linear function. How would you respond to the student?” 

MP8 is identified and connected to grade-level content, and there is intentional development of the MP to meet its full intent. Students look for and express regularity in repeated reasoning as they work with teacher support and independently throughout the modules. Examples include:

• Unit 3: Linear Relationships and Equations, Lesson 3-4: Describe Proportional Linear Relationships, Session 1, Activity-Based Exploration, students look for and express regularity in repeated reasoning as they explore the relationship between slope, the equation of a line, and ordered pairs on a line. The materials state, “When you calculate the slope of a line, does it matter which two points on the line you pick?” Teacher Guidance states, “As student-pairs explore the activities, check that all pairs understand the task. If students need guidance or support, ask: How can you use two points to determine the slope of the line? How can you use the slope to find any ordered pair on the graph? Math is… Looking for Patterns How is looking for patterns helpful in solving this problem? Have students discuss the patterns they notice. Connect the patterns they notice to their understanding of proportional relationships. Encourage students to consider how they may use these patterns to create an equation that represents a proportional relationship. Have students complete the Concluding Question in their Activity Exploration Journal. How can you create an equation that represents a proportional relationship?” 

• Unit 7: Volume, Lesson 7-1: Understand and Use Cube Roots, Session 1, Guided Exploration, Puzzle Cube, students evaluate the reasonableness of their answers and thinking as they find the side length of a cube given its volume. The materials state, “How can you find the side length of any cube if you know the volume?” Let’s Explore More states, “a. How are cube roots and square roots similar? How are they different? b. What are the perfect cubes from 1 to 1,000?” MPP, “Have students discuss the calculations involved when finding the volume of a cube. Make sure students recognize that raising a value to the third power involves repeated multiplication. The base of the power is multiplied by itself three times.”

• Unit 9: Irrational Numbers, Exponents, and Scientific Notation, Lesson 9-1: Represent Rational Numbers in Decimal Form, Session 1, Activity-Based Exploration, Fractions as Decimals, students evaluate the reasonableness of their answers and thinking as they explore the relationship between rational numbers and repeating decimals. Students convert a series of fractions to decimals and record the patterns that they find. The materials state, “Have students review the steps used to convert fractions to decimals. They should note any calculations that are repeated. Make sure students understand that if the remainder equals the dividend when dividing a fraction by a decimal, the pattern will repeat. Have students complete the Concluding Question in their Activity Exploration Journal. How can you express a rational number as a decimal?” Hands-On, “Students work with a partner to write unit fractions ranging from \frac{1}{2} to \frac{1}{25} and then convert each fraction to a decimal. Students record patterns and draw conclusions about unit fractions written as decimals.”