8th Grade - Gateway 2

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

The structure of the lessons provide several opportunities that address conceptual understanding, and the materials include problems and questions that develop conceptual understanding throughout the grade-level.

• In the Teacher’s Edition, both Modules and Lessons begin with The Three Pillars of Rigor where conceptual understanding for the topic is briefly outlined. For example, Module 1, “In this module, students draw on their knowledge of exponents to develop understanding of the properties of exponents and scientific notation.”
• In Explore & Develop, Explore is “intended to build conceptual understanding through Interactive Presentations that introduce the concept and can be completed by pairs on devices or as a whole class through digital classroom projection.” For example, in Module 4, Lesson 4-1, Explore, “Students will be presented with a rate at which Marcus can download songs from the Internet. Throughout this activity, students will use a table, graph, and ratio to compare the number of minutes and the number of songs downloaded.” (8.EE.5)
• Some Checks address conceptual understanding. For example, Lesson 5-5 Check, "Determine whether the table represents a linear or nonlinear function. Explain." (8.F.3)
• Some Exit Tickets address conceptual understanding. Lesson 9-1 Exit Ticket, “What transformations can be used to show that triangle RST and triangle R’S’T’ are congruent?” (8.G.2)

Examples of the materials providing opportunities for students to independently demonstrate conceptual understanding include: 

• In Lesson 4-3, students develop conceptual understanding about the slope of a line being the same between any two points on the line. Explore - Right Triangles and Slope, “Inquiry question: How does the slope compare between any two pairs of points on a line? You will use Web Sketchpad to explore this problem.” Learn - Similar Triangles and Slope: “You can use the properties of similar triangles to show the ratios of the rise to the run for each triangle are equal.” Practice Question 5, “Multiselect. The graph shows similar slope triangles on a line. Select all of the statements that are true: The slope of the line is negative. The slopes of each triangle are the same because they lie on the same line. Triangle CDE has a greater slope because the triangle is larger. The slope of each triangle is 2/3. The slope of each line is positive.” (8.EE.6)
• In Lesson 5-2, students graph lines from function tables. This is extended in Lesson 5, Examples 1-3, when students identify linear and nonlinear functions from graphs and tables and, in Examples 3-5, functions from equations. (8.F.A)
• Lesson 6-1 Explore - Systems of equations: “Inquiry question: What does it mean when graphs of two linear equations intersect? You will use Web Sketchpad to explore this problem. Situation: Edna leaves a trailhead at dawn to hike toward a lake 12 miles away where her friend, Maria, has been camping. At the same time, Maria leaves the lake to hike toward the trailhead (on the same trail, but in the opposite direction). Edna is walking uphill, so her average speed is 1.5 miles per hour. Maria is walking downhill, so her average speed is 2 miles per hour. Select the Start/Stop Simulation button to see what happens on the hike. Select Reset to return the hikers to their starting points. Record your observations.” Students put the data into a table and a graph. “Talk about it!: What does the point of intersection represent?” (8.EE.8a)

The instructional materials reviewed for Reveal Math Grade 8 meet expectations for attending to those standards that set an expectation of procedural skill and fluency.

The structure of the lessons includes several opportunities to develop these skills. The instructional materials develop procedural skill and fluency throughout the grade-level.

• In the Teacher’s Edition, both Modules and Lessons begin with The Three Pillars of Rigor where procedural skill and fluency for the topic is briefly outlined. For example, Lesson 6-1, “In this lesson, students draw on their knowledge of graphing linear equations to build fluency with solving systems graphically.”
• Some Interactive Presentations (slide format) demonstrate procedures to solve problems. For example, in Lesson 1-3, Explore and Develop - Example 1: Problems include, “Simplify $$(8^6)^3$$” and “Simplify $$(k^7)^5$$.” Students are stepped through using the Power of a Power property. (8.EE.1)
• Some Checks address procedural skills and fluency. For example, Lesson 2-2 Check: “Solve $$y^2$$ = 256” (8.EE.2)

The instructional materials provide opportunities to independently demonstrate procedural skill and fluency throughout the grade-level.

• Lesson 3-1: Students use properties of equality to solve equations with variables on each side algebraically. They have multiple opportunities in the lesson to practice procedures. Example 1: “Solve 3(8x+12) - 15x = 2(3 - 3x). Check your solution. Move through the steps to solve the equation.” The steps include: Write the equation; Distributive Property; Combine like terms; Addition property of Equality; Simplify; Subtraction property of Equality; Simplify; Division property of Equality; Simplify.” (8.EE.7)
• Lesson 6-3: Students solve systems of equations by substitution, rewrite equations to solve by substitution, and use procedures to solve systems with no solutions and infinitely many solutions. (8.EE.8)
• Lesson 10-3, Practice 1-4: “Find the volume of each sphere. Express your answer in terms of $$\pi$$. 2) Given illustration: a sphere with diameter 9 in.” (8.G.9)

The instructional materials for Reveal Math Grade 8 meet expectations for being designed so that teachers and students spend sufficient time working with engaging applications of the mathematics. Engaging applications include single and multi-step problems, routine and non-routine, presented in a context in which the mathematics is applied. 

The instructional materials include multiple opportunities for students to engage in routine and non-routine application of mathematical skills and knowledge of the grade-level.

• In the Teacher’s Edition, Modules and Lessons begin with The Three Pillars of Rigor where application for the topic is briefly outlined. For example, in Lesson 11-3, “In this lesson, students apply their understanding of writing linear equations to real-world problems by interpreting the slope and y-intercept of the line.”
• Each Module includes a Performance Task that addresses application. For example, in Module 6, Performance Task, “Keith and Margo are in the process of remodeling their home. The remodeling process consists of several projects. They will be adding new landscaping, pouring a new concrete patio, adding crown molding, and fixing their kitchen’s pantry. Some of the work they will do themselves, and for some of it, they will hire contractors. Part C. Keith and Margo are going to add crown molding around the perimeters of the ceilings of two rooms. The first room has dimensions and perimeter as shown below. The second room has the same length as the first room, a width that measures twice as much as the first room, and a perimeter, P, of 40 feet. Write a system of equations to represent this situation. Then solve the system of equations using the elimination method. State the dimensions of each room.”(8.EE.8)
• In Lesson 5-3, Reflect and Practice, Apply, Practice Question 2, “The table shows the distance Penelope is from the park as she walks to soccer practice. Assume the relationship between the two quantities is linear. Find and interpret the rate of change and the initial value. Then write the equation of the function in the form y = mx + b.” (8.F.4)
• Some Checks address application. For example, in Lesson 4-5, Check, “Katie wants to attend fitness classes at a local gym. The cost of attending Fitness for Life is represented in the graph shown. Fitness World charges a registration fee of $90 plus $8 per month. Katie wants a membership for 18 months. Which gym charges less for 18 months? How much less?” (8.EE.6)

The instructional materials provide opportunities for students to independently demonstrate the use of mathematics flexibly in a variety of contexts.

• In Lesson 3-2, Practice Questions, “Write and solve an equation for each exercise. Check your solution. 1) Marko has 45 comic books in his collection, and Tamara has 61 comic books. Marko buys 4 new comic books each month. After how many months will Marko and Tamara have the same number of comic books?” (8.EE.7)
• In Lesson 4-3, Practice Question 6, “Lines r and s are parallel, meaning they will never intersect. Draw similar slope triangles on each line and find the slope of each line. What conclusion can you draw about the slopes of parallel lines?” (8.EE.6)
• In Lesson 6-5, “A concession stand sells hot dogs and hamburgers. At the football game, 84 hot dogs and 36 hamburgers were sold for $276. At another football game, 18 hamburgers and 60 hot dogs were sold for $174. What is the cost of each hot dog and each hamburger?” (8.EE.8c)
• In Lesson 7-3, Apply, Maps, “Alma has a motor boat that averages 3 miles per gallon of gasoline, and the tank holds 15 gallons of gasoline. At 9 a.m., Alma left the dock. At 10 a.m., her position was 3 miles west and 4 miles north of the dock. If she continues at this rate, in how many more hours will the tank be out of gasoline?” (8.G.7)

The instructional materials for Reveal Math Grade 8 meet expectations that the three aspects of rigor are not always treated together and are not always treated separately. Many of the lessons incorporate two aspects of rigor, with an emphasis on application, and practice problems for students to address all three aspects of rigor.

All three aspects of rigor are present independently throughout the materials, and examples include:

• In Lesson 9-1, Explore and Develop, Example 1, students develop understanding of transformations. Students determine if each pair of figures are congruent. If so, they describe a sequence of transformations that maps one figure onto the other figure. If not, they explain why they are not congruent. “Think About It!: How do you know that more than one transformation is needed to map ABC onto XYZ?” Students have an interactive eTool to move the triangles around the coordinate plane and record the sequence of their transformations. Finally, students “Talk About It!: Could you have first translated ABC 4 units up and then reflected it across the y-axis? Explain. Why is ABC congruent to XYZ?”. (8.G.2)
• In Lesson 5-2, Reflect and Practice, students develop understanding that a function is a rule that assigns to each input exactly one output. In the first three problems, given a function and the inputs, students complete the output column in a function table. At times, students generate both parts of the table such as Question 4: “A single-engine plane can travel up to 140 miles per hour. The total number of miles m is represented by the function m = 140h, where h is the number of hours traveled. Determine appropriate input values for this situation. Then complete the function table for m = 140.” (8.F.1)
• In Lesson 6-5, Practice Question 6, students use their understanding of simultaneous equations to find solutions to real-life situations. “At a farmer’s market, Amar purchased 4 jars of salsa and 3 cucumbers and spent a total of $12.25. Dylan purchased 1 jar of salsa and 2 cucumbers and spent a total of $4. Dakota purchased 1 jar of salsa and 5 cucumbers. If each jar of salsa costs the same and each cucumber costs the same, how much did Dakota spend?” (8.EE.8)

Examples of the materials integrating at least two aspects of rigor include:

• In Lesson 5-6, Practice Question 3, students use their understanding to sketch a graph that exhibits the qualitative features of a function that has been described verbally in real-world situations: “Ryan’s heart rate was steady before exercising. While exercising, his heart rate increased rapidly and then steadied. During cool down, his heart rate decreased slowly then lowered quickly until becoming steady again. Sketch a qualitative graph to represent the situation. Determine if the graph is linear or nonlinear and where the graph is increasing or decreasing.” (8.F.5) 
• In Lesson 10-2, students develop understanding of the relationship between the volume of a cone and a cylinder by filling each with rice. Using conceptual understanding, students develop the volume formula and practice finding volume of various cones. Finally, they complete an application problem with a picture of a cone and a cylinder, both with 4 inch diameters and heights of 6 inches, “A family-owned movie theater offers popcorn in the sizes shown. Their cost for the popcorn is $0.09 per cubic inch. If each container is filled to the top, what is the difference between the costs of the popcorn in the two containers?” (8.G.9)

The instructional materials reviewed for Reveal Math Grade 8 meet expectations that the Standards for Mathematical Practice are identified and used to enrich mathematics content within and throughout the grade-level.

All 8 MPs are clearly identified throughout the materials, including:

• The materials contain a Correlation to the Mathematical Practices PDF which includes explanations and descriptions of the MPs and examples of MPs located in specific lessons.
• Within the digital module opener and lesson, the Standards tab contains a list of the MPs found in that specific module/lesson. The same list is part of the Teacher Edition PDF. Throughout each lesson, the program indicates each opportunity for students to engage in the practices, with an MP symbol and a description of how to connect the MP to the content within the lesson. 
• In Reflect and Practice, questions intended to engage students in the MPs are specifically noted with an MP symbol. The Teacher Edition states which of the MPs each practice question is intended to align with.
• Performance Task rubrics list which MPs students are intended to engage in during the task.
• Each component of the digital materials (Learn, Explore, Examples, Apply) contains an About this Resource narrative explaining how related MPs should specifically be addressed within the activity. The same information is found in the Teacher Edition PDF in the margin labeled MP Teaching the Mathematical Practices.
• Each lesson includes Launch - Today’s Standards: How can I use these Practices? The Teacher’s Notes recommend that teachers, “Tell students that they will be addressing these content and practice standards in this lesson. You may wish to have a student volunteer read aloud How Can I meet this standard? and How can I use these practices? and connect these to the standards.”

Examples of the MPs being used to enrich the mathematical content include:

• MP1: In Lesson 10-4, Explore and Develop, Apply, Shopping, students make sense of the problem to determine using the formula for the volume of cylinders when solving real-world problems. “A local grocery store sells corn in two different-sized cans. A one-meter wide shelf is being stocked. How many more of the smaller cans will fit on the shelf than the larger cans?”
• MP2: In Lesson 6-2, Explore and Develop, Example 3, students demonstrate reasoning with the prompt, “Talk About It! Describe a method you can use to verify the system has infinitely many solutions.”
• MP8: In Lesson 4-2, Explore and Develop, Explore, Slope of Horizontal and Vertical Lines, students use regularity and repeated reasoning to enhance their understanding of slope. “This Explore spans eight slides. Working in pairs, students will be presented with a series of lines whose slopes approach zero or infinity. Throughout this activity, students will view the patterns of slopes as they approach zero or infinity and make conjectures as to the slopes of all vertical and horizontal lines.”

There are instances where the labeling of MPs is inconsistent, and examples of this include:

• In Lesson 4-1, Example 3 is labeled with MPs 3 and 5 in the digital materials, but in the print materials, the same problem is not labeled with any MPs.

The instructional materials reviewed for Reveal Math Grade 8 meet the expectations for prompting students to construct viable arguments and analyze the arguments of others concerning key grade-level mathematics. 

Examples of the materials prompting students to both construct viable arguments and analyze the arguments of others include:

• Talk About It! in lesson examples are often opportunities for students to create viable arguments. For example, in Lesson 9-4, “Can you assume that the two rectangles are similar just because their corresponding angles are congruent? Explain.”
• In Lesson 1-3, Practice Question 13, “Make an argument for why $$(4^2)^4 = (4^4)^2$$.”  
• In Lesson 8-1, Practice Question 12, “Reason Inductively. Determine whether the following statement is always, sometimes, or never true. Write an argument that can be used to defend your solution: A preimage and its translated image are the same size and the same shape.” 
• Write About It! within lesson examples are often opportunities for students to engage with MP3. In Apply of many lessons, students are prompted to “Write About It! Write an argument that can be used to defend your solution.”
• In Lesson 4-4, Practice Question 10, “Find the Error. The cost of apps varies directly with the number of apps purchased. Aditi bought four apps for a total of $5.16. She found the direct variation equation below for this relationship. Find her mistake and correct it.”
• In Lesson 5-2, Practice Question 10, “Justify Conclusions. Liam’s truck has a 25 gallon tank and uses 0.05 gallon of gas for every mile driven. When creating a table for the function y = -0.05x + 25, Liam argues that he can only use positive rational numbers for the input. Is Liam correct? Justify your answer.”
• In Lesson 8-1, Practice Question 11, “A classmate states that a two-dimensional figure could have each of its vertices translated in different ways and it would still be considered a translation. Explain to your classmate why this is incorrect.”

The instructional materials reviewed for Reveal Math Grade 8 meet the expectations for assisting teachers in engaging students to construct viable arguments and analyze the arguments of others concerning key grade-level mathematics.

There are multiple locations in the materials where teachers are provided with prompts to elicit student thinking.

• In Resources, there is a Correlation to the Mathematical Practices, Grade 8, which defines the Standards for Mathematical Practice. For example, MP3 is defined, there are examples connected to MP3, and states, “Students are required to justify their reasoning and to find the errors in another’s reasoning or work. Look for the Apply problems and the exercises labeled as Make a Conjecture, Find the Error, Use a Counterexample, Make an Argument, or Justify Conclusions. Many Talk About It! question prompts ask students to justify conclusions and/or critique another student’s reasoning. In the Teacher Edition, look for the Teaching the Mathematical Practices tips labeled as this mathematical practice.”
• There are Questions for Mathematical Discourse in Develop and Explore of each lesson. For example, in Lesson 11-3, Extra Example 2, the teacher notes suggest, “Use these questions for mathematical discourse, “Why are choices A and C incorrect?; How different would your conjecture be for a 65 year old?”.
• Talk About It! is designed to elicit student justification. For example, in Lesson 6-3: “As students discuss the Talk About It! question (In Step 2 you substituted -1 for x into the equation y = 3x + 8. You can also substitute -1 for x into the other equation 8x + 4y = 12. Why is either method correct? Which do you prefer? Explain.), encourage them to create a plausible argument to defend why either method is correct, and why they prefer their chosen method.”
• The materials also prompt teachers to have students share their responses to Write about it!. Teacher guidance throughout the materials states, “As students respond to the Write About It! prompt, have them make sure their argument uses correct mathematical reasoning. If you choose to have them share their responses with others, encourage the listeners to ask clarifying questions to verify that the reasoning is correct.” The Write About It! prompts typically read, “Write an argument that can be used to defend your solution.”
• The Teacher Edition includes Teaching the Mathematical Practices tips which involve developing arguments. For example, in Lesson 8-3, Learn, “While discussing the Talk About It! question on Slide 2, encourage students to create a plausible argument and draw a counterexample to illustrate why these coordinate notations are only valid for rotations about the origin.”
• Teacher’s Notes often give prompts and suggestions for facilitating arguments. For example, in Lesson 5-1, Practice Question 8, the teacher notes state, “In Exercise 8, students will explain why the relationship is a function. Encourage students to identify the information that identifies the relationship as a function.”

The instructional materials reviewed for Reveal Math Grade 8 meet the expectations for explicitly attending to the specialized language of mathematics.

The materials use precise and accurate mathematical terminology and definitions, and the materials support students in using them. Teacher’s guides, student books, and supplemental materials explicitly attend to the specialized language of mathematics.

• In Resources, there is a Correlation to the Mathematical Practices, Grade 8, which defines the Standards for Mathematical Practice. For example, MP6 is defined, there are examples where MP6 can be found, and states, “Students are routinely required to communicate precisely to partners, the teacher, or the entire class by using precise definitions and mathematical vocabulary. Look for the exercises labeled as Be Precise. Many Talk About It! prompts ask students to clearly and precisely explain their reasoning. In the Teacher Edition, look for the Teaching the Mathematical Practices tips labeled as this mathematical practice.”
• In each Module introduction, What Vocabulary Will You Learn? prompts teachers to lead students through a specific routine to learn the vocabulary of the unit.
• Many Lessons have a “Language Objective.” For example, in Lesson 7-1, “Students will state and apply the definition of angle relationships formed by parallel lines cut by a transversal: exterior angles, interior angles, alternate exterior angles.”
• In each lesson, Math Background briefly describes key concepts/vocabulary or directs teachers to an online component to learn background. Definitions are not included, but are accessible in the glossary. Glossary definitions are precise and accurate, and there are definitions for math content and math models. In addition, the glossary references the lesson where the vocabulary is introduced.
• The lesson Launch includes a vocabulary section that introduces new vocabulary for the lesson. During Develop and Explore, the new vocabulary is always bolded and defined. For example, in Lesson 4-5, Learn: “The y-intercept of a line is the y-coordinate of the point where the line crosses the y-axis.”
• In Lesson 5-1, What Vocabulary Will You Learn?, Teacher questions include: “What does the prefix co- mean? What is the everyday meaning of the word constant? What part of speech is the word like in like terms? How does it help you understand what like terms might be?”
• When students see vocabulary in successive lessons, What Vocabulary Will You Use? assists teachers in facilitating discussions that help students apply the vocabulary they have previously learned.
• In Lesson 3-3, Teaching Notes for Interactive Slideshows state, “When discussing the Talk About It! question on Slide 2 (How can you make sure that you solved the equation correctly?), encourage students to use clear and precise mathematical language, such as substitute  or replace, when describing how they can check to verify they solved the equation correctly.”
• In Lesson 1-2, Example 3, the teacher notes prompt, “Encourage students to use academic vocabulary, such as Product of Powers Property to explain how to simplify the expression.”
• Each Module includes a Vocabulary Test. “This summative assessment asset is designed for students to demonstrate their knowledge, understanding, and proficiency of the vocabulary covered in this module.”