8th Grade - Gateway 2

The instructional materials reviewed for enVision Mathematics Common Core Grade 8 meet expectations that the materials develop conceptual understanding of key mathematical concepts, especially where called for in specific standards or cluster headings.

Materials include problems and questions that develop conceptual understanding throughout the grade level. According to the Teacher Resource Program Overview, “The Solve & Discuss It in Step 1 of the lesson helps students connect what they know to new ideas embedded in the problem. When students make these connections, conceptual understanding takes seed. In Step 2 of the instructional model, teachers use the Visual Learning Bridge, either in print or online, to make important lesson concepts explicit by connecting them to students’ thinking and solutions from Step 1.” Examples include:

• Lesson 1-2, Understand Irrational Numbers, Visual Learning, Example 1, students develop conceptual understanding of classifying rational and irrational numbers, “The Venn diagram shows the relationships among rational numbers. How would you classify the number 0.24758326… ?” Teachers then ask, “Is the given decimal a terminating decimal? Explain. Suppose the given decimal was written as 0.24758326. Would this decimal be a rational number? Explain.” (8.NS.1)
• Lesson 2-9, Analyze Linear Equations: y = mx + b, Visual Learning, Example 1, students develop conceptual understanding of recognizing the mathematical relationship between the equation of a line and the graph of that line, “The Middle School Student Council is organizing a dance that has $500 to pay for a DJ. DJ Dave will charge $200 for a set-up fee and the first hour, or $425 for a set-up fee and four hours. How can the Student Council determine whether they can afford to have DJ Dave play for 5 hours?” The teacher asks, “Why does 125 represent the y-intercept? Why does 75 represent the slope?” (8.EE.6)
• Lesson 3-2, Connect Representations of Functions, Visual Learning, Example 2, students deepen their understanding of functions as they explore graphs of linear and nonlinear functions, “How can you determine whether the relationship between side lengths and area is a function? How can you determine whether a relation in the table is a function? What do you notice about the shape of this graph? Is it a function?” (8.F.1)
• Lesson 4-4, Interpret Two-Way Frequency Tables, Visual Learning, Example 3, students develop conceptual understanding of representing, finding, and comparing sub-categories of data, “Two hundred people responded to a survey. Of those who had green eyes, 7 had blonde hair, 9 had brown hair, and 2 had red hair. Of those who had brown eyes, 76 had blonde hair, 89 had brown hair, and 17 had red hair. Construct a two-way table to display these data. Then identify the least common combination of eye and hair color. Explain.” (8.SP.4)
• Lesson 6-5, Understand Congruent Figures, Solve and Discuss It!, students develop conceptual understanding of congruent geometric figures produced through reflections, rotations, and translations, “Simone plays a video game in which she moves shapes into empty spaces. After several rounds, her next move must fit the blue piece into the dashed space. How can Simone move the blue piece to fit in the space?” (8.G.2, 8.G.3)

Materials provide opportunities for students to independently demonstrate conceptual understanding throughout the grade. Practice and Problem Solving exercises found in the student materials provide opportunities for students to demonstrate conceptual understanding. Try It! provides problems that can be used as formative assessment of conceptual understanding following Example problems. Do You Understand?/Do You Know How? problems have students answer the Essential Question and determine students’ understanding of the concept. Examples include:

• Lesson 1-1, Rational Numbers as Decimals, Practice & Problem Solving, Item 10, students independently demonstrate conceptual understanding of writing repeating decimals as fractions, “Thomas asked 15 students whether summer break should be longer. He used his calculator to divide the number of students who said yes by the total number of students. His calculator showed the result as 0.9333…. A. Write this number as a fraction. B. How many students said that summer break should be longer?” (8.NS.1)
• Lesson 2-2, Solve Equations with Variables on Both Sides, Do You Know How?, Item 4, students independently demonstrate conceptual understanding of solving equations with like terms on both sides of the equation, “Maria and Liam work in a banquet hall. Maria earns a 20% commission on her food sales. Liam earns a weekly salary of $625 puls a 10% commission on his food sales. What amount of food sales will result in Maria and LIam earning the same amount for the week?” (8.EE.7b)
• Lesson 3-1, Understand Relations and Functions, Do You Understand?, Item 3, students independently demonstrate their understanding of whether functions are relations, “Is a relation always a function? Is a function always a relation? Explain.” (8.F.1)
• Lesson 4-1, Construct and Interpret Scatter Plots, Visual Learning, Example 3, Try It!, students independently demonstrate understanding of scatter plots, “Avery also tracks the number of minutes a player plays and the number of points the player scored. Describe the association between the two data sets. Tell what the association suggests.” A graph of Basketball Scoring with minutes played and points scored is shown. (8.SP.1)
• Lesson 7-1, Understand the Pythagorean Theorem, Practice & Problem Solving, Item 9, students independently demonstrate conceptual understanding of the Pythagorean Theorem, “What is the length of the hypotenuse of the triangle when x = 15? Round your answer to the nearest tenth of a unit.” A picture of a triangle with legs 4x + 4 and 3x is shown. (8.G.6 and 8.G.7)

The instructional materials for enVision Mathematics Common Core Grade 8 meet expectations that they attend to those standards that set an expectation of procedural skill and fluency.

The instructional materials develop procedural skill and fluency throughout the grade level. According to the Teacher Resource Program Overview, “Students develop skill fluency when the procedures make sense to them. Students develop these skills in conjunction with understanding through careful learning progressions.” Try It!, And Do You Know How? Provide opportunities for students to build procedural fluency from conceptual understanding. Examples Include:

• Lesson 1-2, Understand Irrational Numbers, Do You Know How?, Item 4, students identify rational and irrational numbers, “Is the number 65.4349224... rational or irrational? Explain.” (8.NS.1)
• Lesson 2-2, Solve Equations with Variables on Both Sides, Visual Learning, Example 3, Try It!, students solve equations with variables on both sides, “Solve the equation. 96 - 4.5y - 3.2y = 5.6y + 42.80.” (8.EE.7)
• Lesson 3-1, Understand Relations and Functions, Do You Know How?, Item 5, students identify whether a relation is a function, “Is the relation shown below a function? Explain.” Students are shown a table with inputs 3, 4, 1, 5, 2 and corresponding outputs of 4, 6, 2, 8, 5. (8.F.1)
• Lesson 4-4, Interpret Two-Way Frequency Tables, Do You Know How?, Item 4, students explain how to display and interpret relationships between paired categorical data, “A basketball coach closely watches the shots of 60 players during basketball tryouts. Complete the two-way frequency table to show her observations.” A partially completed table of grade levels and basketball shots is provided. (8.SP.4)
• Lesson 6-4, Compose Transformations, Visual Learning, Example 1, Try It!, students describe a sequence of transformations involving floor plans using a coordinate plane, “Ava decided to move the cabinet to the opposite wall. What sequence of transformations moves the cabinet to its new position?” (8.G.3)

The instructional materials provide opportunities to independently demonstrate procedural skill and fluency throughout the grade level. Practice and Problem Solving exercises found in the student materials provide opportunities for students to independently demonstrate procedural skill and fluency. Additionally, at the end of each Topic is a Fluency Practice page which engages students in fluency activities. Examples include:

• Topic 1 Review, Fluency Practice, students solve one-step equations, including those involving square roots and cube roots, “Crisscrossed: Solve each equation. Write your answers in the cross-number puzzle below. Each digit, negative sign, and decimal point of your answer goes in its own box. A. -377 = x - 1,000; C. x$$^3$$= -8; D. x + 7 = -209; F. x + 19 = -9.” (8.EE.7)
• Lesson 1-3, Compare and Order Real Numbers, Practice & Problem Solving, Item 10, students compare real numbers in various forms, “Does $$\frac{1}{6}$$, -3, $$\sqrt7$$, -6/5, or 4.5 come first when the numbers are listed from least to greatest? Explain.” (8.NS.2)
• Lesson 2-3, Solve Multistep Equations, Practice & Problem Solving, Item 12, students solve multi-step equations using the distributive property, “What is the solution of the equation 3(x + 2) = 2(x + 5)?” (8.EE.7).
• Lesson 3-4, Construct Functions to Model Linear Relationships, Practice & Problem Solving, Item 7, students calculate slope in order to write a linear equation, “A line passes through the points (4, 19) and (9, 24). Write a linear function in the form y = mx + b for this line.” (8.F.4)
• Lesson 4-3, Use Linear Models to Make Predictions, Practice & Problem Solving, Item 6, students calculate rates of change and initial values in order to make predictions in linear relationships, “If x represents the number of years since 2000 and y represents the gas price, predict what the difference between the gas price in 2013 and 2001 is? Round to the nearest hundredth.” (8.SP.3)
• Lesson 7-3, Apply the Pythagorean Theorem to Solve Problems, Practice & Problems Solving, Item 9, students solve problems using a$$^2$$ + b$$^2$$ = c$$^2$$, “A stainless steel patio heater is shaped like a square pyramid. The length of one side of the base is 19.8 inches. The slant height is 92.8 inches. What is the height of the heater? Round to the nearest tenth of an inch.” (8.G.7)

The instructional materials reviewed for enVision Mathematics Common Core Grade 8 meet expectations that the materials are 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 mathematics is applied. 

The instructional materials include multiple opportunities for students to independently engage in routine and non-routine application of mathematical skills and knowledge of the grade level. According to the Teacher Resource Program Overview, “In each topic, students encounter a 3-Act Mathematical Modeling lesson, a rich, real-world situation for which students look to apply not just math content, but math practices to solve the problem presented.” Additionally, each Topic provides a STEM project that presents a situation that addresses real social, economic, and environmental issues. For example:

• Topic 1, STEM Project, Going, Going, Gone, students use real numbers such as rational and irrational numbers to show the depletion rate of a natural resource, "Natural resources depletion is an important issue facing the world. Suppose a natural resource is being depleted at the rate of 1.333% per year. If there were 300 million tons of this resource in 2005, and there are no new discoveries, how much will be left in year 2045?” (8.NS.1, 8.NS.2, 8.EE.1,and 8.EE.2)
• Topic 3, STEM Project, Modeling Population Growth, students use population data to develop linear equations that model growth in urban areas. "Develop a linear model that presents the data on urbanization in India and the U.S. How does the model compare with those of population growth developed in the last topic?” (8.F.4)
• Topic 1, 3-Act Mathematical Modeling: Hard-Working Organs, Question 2, students compare large numbers, “How many times does your heartbeat in a decade? How does that number compare to the number of breaths you take in a decade?” (8.EE.1 and 8.EE.3)
• Topic 5, 3-Act Mathematical Modeling: Up and Downs, Questions 2, students develop a mathematical model to represent and propose a solution to a problem situation involving a system of equations. Students are presented with the main question, “Which route is faster?” Teachers ask, “Does your answer match the answer in the video? If not, what are some reasons that would explain the difference? Would you change your model now that you know the answer? Explain.” (8.EE.8)
• Topic 6, STEM Project, Forest Health, students use similar triangles and ratios to gather and interpret data, "Students use ratios and similar triangles to measure the health of various forest elements. Using what they know about similar triangles will allow students to measure tree heights. Students can use equivalent ratios to help generalize data to larger sections of forests.” (8.G.3 and 8.G.4)
• Topic 8, 3-Act Mathematical Modeling: Measure Up, students determine whether the liquid in one container will fit into a container with a different shape. Question 15, "Suppose you have a graduated cylinder half the height of the one in the video. How wide does the cylinder need to be to hold the liquid in the flask?" (8.G.9)

The instructional materials provide opportunities for students to independently demonstrate the use of mathematics flexibly in a variety of contexts. Pick a Project is found in each Topic and students select from a group of projects that provide open-ended rich tasks that enhance mathematical thinking and provide choice. Additionally, Practice and Problem Solving exercises found in the student materials provide opportunities for students to independently demonstrate mathematical flexibility in a variety of contexts. For example:

• Lesson 1-3, Compare and Order Real Numbers, Practice & Practice Solving, Item 11, students compare irrational numbers, “A museum director wants to hang the painting on a wall. To the nearest foot, how tall does the wall need to be?” A painting is shown with a height of $$\sqrt90$$ ft. (8.NS.2)
• Topic 3, Pick a Project 3D, students design a video game element describing how perimeter and area dimensions change in regards to linear and non-linear relationships. “Make mock-ups of your designs and describe how the game will use the boxes. Calculate the interior and exterior perimeters and areas of each text box. Describe how perimeter and area relate to linear and nonlinear relationships.” (8.F.4)
• Lesson 3-3, Compare Linear and Nonlinear Functions, Practice & Problem Solving, Item 15, students apply their understanding of functions to solve real-world problems using equations and a table, “The students in the After-School Club ate 12 grapes per minute. After 9 minutes, there were 32 grapes remaining. The table shows the number of carrots remaining after different amounts of time. Which snack did the students eat at a faster rate? Explain.” (8.F.2 and 8.F.3)
• Lesson 4-3, Use Linear Models to Make Predictions, Practice & Problem Solving, Item 10, students make predictions using slope of a trend line, “The graph shows the temperature, y, in a freezer x minutes after it was turned on. Five minutes after being turned on, the temperature was actually three degrees from what the trend line shows. What values could the actual temperature be after the freezer was on for five minutes? What is the equation that best represents the data in the graph? Use the equation for the trend line, how cold will the temperature be after 30 minutes? Do you think that will happen? Explain.” (8.SP.3)
• Lesson 5-3, Solve Systems by Substitution, Practice & Problem Solving, Item 15, students use substitution to solve real-world problems, “The members of the city cultural center have decided to put on a play once a night for a week. Their auditorium holds 500 people. By selling tickets, the members would like to raise $2,050 every night to cover all expenses. Let d represent the number of adult tickets sold at $6.50. Let s represent the number of student tickets sold at $3.50 each. a. If all 500 seats are filled at the performance, how many of each type of ticket must have been sold for the member to raise exactly $2,050? b. At one performance there were 3 times as many student tickets sold as adult tickets. If there were 480 tickets sold at that performance, how much below the goal of $2,050 did ticket sales fall?” (8.EE.8)
• Topic 7, Pick a Project 7A, students find the distances in a coordinate plane they mapped of their community route, “On a coordinate grid, map out a metric bike route through your community, increase at least 5 stops. Use at least three diagonal line segments to represent different parts of your route. Calculate the distance between the stops.” (8.G.8)

The instructional materials reviewed for enVision Mathematics Common Core Grade 8 meet expectations that the three aspects of rigor are not always treated together and are not always treated separately. 

All three aspects of rigor are present independently throughout the program materials. Examples where instructional materials attend to conceptual understanding, procedural skill and fluency, and application include:

• Topic 5 Review, Fluency Practice, students solve multi-step equations using the distributive property, “Pathfinder: Shade a path from START to FINISH. Follow the solutions to the equations from least to greatest. You can only move up, down, right, or left. 2(d + 1) = -38; 7x - 2(x - 11) = -103; 2w - 5(w + 4) = -14; -4(h - 2) = 8; 4(3 - 5k) = 92.” (8.EE.7)
• Lesson 6-1, Analyze Transitions, Solve & Discuss It!, students develop conceptual understanding of translating two dimensional figures, “Ashanti draws a trapezoid on the coordinate plane and labels it Figure 1. Then she draws Figure 2. How can she determine whether the figures have the same side lengths and the same angle measure?” (8.G.1 and 8.G.3)
• Lesson 7-4, Find Distance in the Coordinate Plane, Practice & Problem Solving, Item 11, students use application of the Pythagorean Theorem to calculate distance on the coordinate plane, “Suppose a park is located 3.6 miles east of your home. The library is 4.8 miles north of the park. What is the shortest distance between your home and the library?” (8.G.8)

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

• Lesson 1-6, Use Properties of Integer Exponents, Do You Know How?, Item 7, students use properties of integer exponents, “A billboard has the given dimensions, 7$$^2$$ ft and 4$$^2$$ ft. Using exponents, write two equivalent expressions for the area of a rectangle.” This question develops conceptual understanding and procedural skill of 8.EE.1, know and apply the properties of integer exponents to generate equivalent numerical expressions.
• Lesson 2-1, Combine Like Terms to Solve Equations, Do You Know How?, Item 4, students solve equations in real-world context, “Henry is following the recipe card (shown) to make a cake. He has 95 cups of flour. How many cakes can Henry make?” This question develops procedural skill and application of 8.EE.7b, solve linear equations with rational number coefficients.
• Lesson 3-5, Intervals of Increase and Decrease, Practice & Problem Solving, Item 10, students describe the behavior of a function in different intervals, “The graph shows the speed of a car over time. What might the constant intervals in the function represent?” This question develops conceptual understanding and application of 8.F.5, describe qualitatively the functional relationship between two quantities by analyzing a graph.
• Lesson 4-1, Construct and Interpret Scatter Plots, Practice & Problem Solving, Item 9, students identify and interpret clusters, gaps, and outliers on a scatter plot, “The table shows the number of painters and sculptures enrolled in seven art schools. Jashar makes an incorrect scatter plot to represent the data. a. What error did Jashar likely make? b. Explain the relationship between the number of painters and sculptors enrolled in the art schools. c. Jashar’s scatter plot shows two possible outliers. Identify them and explain why they are outliers.” This question develops conceptual understanding and application of 8.SP.1, construct and interpret scatter plots for bivariate data to investigate patterns of association between two quantities.
• Lesson 4-4, Interpret Two-Way Frequency Tables, Do You Know How?, Item 4, students compare and make conjectures about data displayed in a two-way frequency table, “A basketball coach closely watches the shots of 60 players during basketball tryouts. Complete the two-way frequency table to show her observations” This question develops conceptual understanding and procedural skill of 8.SP.4, construct and interpret a two-way table summarizing data on two categorical variables collected from the same subjects.
• Lesson 6-3, Analyze Rotations, Practice & Problem Solving, Item12, students determine how a rotation affects a two-dimensional figure, “An architect is designing a new windmill with four sails. In her sketch, the sails’ center of rotation is the origin, (0,0), and the tip of one of the sails, Point Q, has coordinates (2,-3). She wants to make another sketch that shows the windmill after the sails have rotated 270° about the center of rotation. What would be the coordinates of ?” This question develops procedural skill and application of 8.G.3,describe the effect of dilations, translations, rotations, and reflections on two-dimensional figures using coordinates.

The instructional materials reviewed for enVision Mathematics Common Core 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, with few or no exceptions. Math Practices identification in this program according to the Teacher Resource Program Overview include:

• Materials provide a Math Practices and Problem Solving Handbook for students, “A great resource to help students build on and enhance their mathematical thinking and habits of mind.” This handbook explains math practices in student-friendly language and digital animation videos for each math practice are also available.
• Opportunities to apply math practices are found in the Explore It, Explain It, and Solve & Discuss It portions of the lesson. “The Solve & Discuss It calls on students to draw on nearly all of the math practices, but especially sense-making and solution formulation as well as abstract and quantitative reasoning. The Explore It focuses students on mathematical modeling, generalizations, and structure of mathematical models. The Explain It emphasizes mathematical reasoning and argumentation. Students construct arguments to defend a claim or critique an argument defending a claim.”
• The Math Practices and Problem Solving Handbook Teacher Pages, “provide overviews of the math practices, offer instructional strategies to help students refine and enhance their thinking habits, and include student behaviors to listen and look for for each standard.”
• Each Topic Overview contains Math Practices Teacher Pages which include, “Two highlighted math practices with student behaviors to look for, and questions to help students become more proficient with these thinking habits.” For example, in Topic 8, Model with mathematics suggested questions state, “What patterns do you recognize in the formulas for the volume of a rectangular prism and the volume of a cylinder? What patterns do you recognize in the formulas for the volume of a cylinder and the volume of a cone?”
• Math Practices boxes found in the student text provide, “Reminders to be thinking about the application of the math practices as they solve problems.”
• Math Practices Run-in Heads found in the Practice & Problem Solving questions, “Remind students to apply the math practices as they solve problems.”

The majority of the time the MPs are used to enrich the mathematical content and are not treated separately. Examples include:

• MP1: Make sense of problems and persevere in solving them. Lesson 8-2, Find Volume of Cylinders, Practice & Problem Solving, Item 12, students make sense of cylinders as they determine the greatest volume a cylinder can hold, “A rectangular piece of cardboard with dimensions 6 inches by 8 inches is used to make the curved side of a cylinder-shaped container. Using this cardboard, what is the greatest volume the cylinder can hold? Explain.” 
• MP2: Reason abstractly and quantitatively. Lesson 1-1, Rational Numbers as Decimals, Practice & Problem Solving, Item 17, students expand their knowledge about rational numbers and the relationships with decimals and fractions when explaining their answer,” When writing a repeating decimal as a fraction, why does the fraction always have only 9s or 9s and 0s as digits in the denominator?” 
• MP4: Model with mathematics. Lesson 2-1, Combine Like Terms to Solve Equations, Visual Learning, Item 18, students combine like terms on one side of the equation to use inverse operations to solve, “Nathan bought one notebook and one binder for each of his college classes. The total cost of the notebooks and binders was $27.08. Draw a bar diagram to represent the situation. How many classes is Nathan taking?”
• MP5: Use appropriate tools strategically. Lesson 5-4, Solve Systems by Elimination, Practice & Problem Solving, Item 13b, students use elimination to solve for a linear system of equations, “A deli offers two platters of sandwiches. Platter A has 2 roast beef sandwiches and 3 turkey sandwiches. Platter B has 3 roast beef sandwiches and 2 turkey sandwiches. What is the cost of each sandwich?” A picture of Platters A and B with total cost are provided.
• MP6: Attend to precision. Lesson 2-8, Understand the y-intercept of a Line, Practice & Problem Solving, Item 11, students use the relationship between variables in a problem situation to explain the meaning of the y-intercept, “The line models the temperature on a certain winter day since sunrise. What is the y-intercept of the line? What does the y-intercept represent?” 
• MP7: Look for and make use of structure. Lesson 7-4, Find Distance in the Coordinate Plane, Practice & Problem Solving, Item 12, students use the structure of ordered pairs and the Pythagorean Theorem to find distance between two points, “Point B has coordinates (2, 1). The x-coordinate of coordinate A is -10. The distance between point A and point B is 15 units. What are the possible coordinates of point A?”
• MP8: Look for and express regularity in repeated reasoning. Lesson 5-1, Estimate Solutions by Inspection, Practice & Problem Solving, Item 13, students compare the equations in a linear system to determine the number of solutions of the system, “Describe a situation that can be represented by using this system of equations. Inspect the system to determine the number of solutions and interpret the solution within the context of your solution. y = 2x + 10 and y = x + 15.”

The instructional materials reviewed for enVision Mathematics Common Core Grade 8 meet expectations that the instructional materials prompt students to construct viable arguments and analyze the arguments of others concerning key grade-level mathematics. 

Student materials consistently prompt students to construct viable arguments. These opportunities are found in the following activities: Solve & Discuss It!, Explain It!, Explore It!, Practice & Problem Solving, Do You Understand?, and Performance Tasks. Examples include:

• Lesson 1-5, Solve Equations Using Square Roots and Cube Roots, Do You Understand?, Item 3, students construct arguments related to the solutions of square and cube root equations. “There is an error in the work shown below. Explain the error and provide a correct solution. x$$^3$$ = 125.  $$\sqrt[3]{x^3}$$ = $$\sqrt[3]{125}$$. x = 5 and x = -5.”
• Lesson 2-5, Compare Proportional Relationships, Solve & Discuss It!, students use their understanding of proportional relationships to construct arguments and support their response, “Mei Li is going apple picking. She is choosing between two places. The cost of a crate of apples at each place is shown. Where should Mei Li go to pick her apples? Explain.”
• Lesson 3-1, Understand Relations and Functions, Practice & Problem Solving, Item 10, students use their understanding of relations and functions to justify their  arguments. “During a chemistry experiment, Sam records how the temperature changes over time using ordered pairs (time in minutes, temperature in). (0, 15), (5, 20), (10, 50), (15, 80), (20, 100), (25, 100). Is the relation a function? Explain.”
• Lesson 4-2, Analyze Linear Associations, Solve & Discuss It!, students construct arguments as they analyze bivariate data and connect it to linear associations. “What other factors should Angus also take into consideration to make a decision? Defend your response.”
• Lesson 6-2, Analyze Reflections, Practice & Problem Solving, Item 10, students use their understanding of reflections to construct arguments, “Your friend incorrectly says that the reflection of $$\vartriangle$$EFG to its image $$\vartriangle$$E’F’G’ is a reflection across the x-axis. a. What is your friend’s mistake? b. What is the correct description of the reflection?” 

Student materials consistently prompt students to analyze the arguments of others. These opportunities are found in the following activities: Solve & Discuss It!, Explain It!, Explore It!, Practice & Problem Solving, Do You Understand?, and Performance Tasks. Examples include:

• Lesson 1-4, Evaluate Square Roots and Cube Roots, Practice & Problem Solving, Item 15, students analyze the argument of others as they evaluate cubes and cube roots, “Diego says that if you cube the number 4 and then take the cube root of the result, you end up with 8. Is Diego correct? Explain.”
• Lesson 4-5, Interpret Two-Way Relative Frequency Tables, Do You Understand?, Item 3, students analyze the arguments of others as they find the relative frequency from a two-way table, “Maryann says that if 100 people are surveyed, the frequency table will provide the same information as a total relative frequency table. Do you agree? Explain why or why not.” 
• Lesson 5-3, Solve Systems by Substitution, Explain It!, students analyze the arguments of others as they graph a system of equations to determine the most cost effective cab company, “Jackson needs a taxi to take him to a destination that is a little over 4 miles away. He has a graph that shows the rates for two companies. Jackson says that because the slope of the line that represents the rates for On Time Cabs is less than the slope of the line that represents Speedy Cab Co., the cab ride from On Time Cabs will cost less. Do you agree with Jackson? Explain. Which taxi service company should Jackson call? Explain your reasoning.”
• Lesson 6-3, Analyze Rotations, Explore It!, students analyze the arguments of others as they explore a point on a circle after a rotation, “Maria boards a car at the bottom of the Ferris wheel. She rides to the top, where the car stops. Maria tells her friends that she completed $$\frac{1}{4}$$ turn before the car stopped. A. Do you agree with Maria? Explain. B. How could you use angle measures to describe the change in position of the car?”
• Lesson 8-1, Find Surface Area of Three-Dimensional Figures, Do You Understand?, Item 3, students analyze the arguments of others as they use formulas for polygons, “Aaron says that all cones with a base circumference of 8x inches will have the same surface area. Is Aaron correct? Explain.”

The instructional materials reviewed for enVision Mathematics Common Core Grade 8 meet expectations that the instructional materials assist teachers in engaging students to construct viable arguments and analyze the arguments of others concerning key grade-level mathematics.

Teacher materials assist teachers in engaging students in constructing viable arguments frequently throughout the program. Examples include:

• Lesson 1-6, Use Properties of Integer Exponents, Visual Learning, Example 4, Try It! Question 4d, “Write equivalent expressions using the properties of exponents. 8$$^9$$ ÷ 8$$^3$$” ETP (Effective Teaching Practices) Elicit and Use Evidence of Student Thinking teacher prompt states, “In part d, why not divide the exponent to get 8$$^3$$?”
• Lesson 2-2, Solve Equations with Variables on Both Sides, Visual Learning, Example 2, “Teresa earns a weekly salary of $925 and a 5% commission on her total sales. Ramon earns a weekly salary of $1,250 and a 3% commission on sales. What amount of sales, x, will result in them earning the same amount for the week?” ETP Pose Purposeful Questions teacher prompt states, “Could you solve this equation by subtracting 0.5x from both sides? Explain.” Within the example, the equation 0.5x + 925 = 0.3x + 1,250 is solved by first subtracting 0.3x from both sides.
• Lesson 6-1, Analyze Translations, Visual Learning, Example 3, “What is the rule that describes the translation that maps trapezoid PQRS onto trapezoid P’Q’R’S’?” ETP Pose Purposeful Questions teacher prompt states, “Suppose points P, Q, and R were translated 3 units right and 4 units up, but point S was not. Is this still a translation? Explain.” 

Teacher materials assist teachers in engaging students in analyzing the arguments of others frequently throughout the program. Examples include:

• Lesson 5-2, Solve Systems by Graphing, Explore It!, “Beth and Dante pass by the library as they walk home using separate straight paths. The point on the graph represents the location of the library. Draw and label lines on the graph to show each possible path to the library.” ETP After teacher prompt states, “Have students present their graphs and discuss how they chose and drew their lines. Ask students how they could have drawn infinitely many different pairs of lines that intersect at the same point; a line can be defined by two points. You can choose one of those points to always be the point of intersection. Then have them explain how they found the slope and y-intercepts of their lines. Have students discuss what a system of equations is and how they can use the equations of their two lines to write a system of equations to represent the situation; a system of equations is a set of two or more equations that have the same unknowns.”
• Lesson 6-8, Angles, Lines, and Transversals, Visual Learning, Example 1, Try It!, “Which angles are congruent to 8? Which angles are supplementary to ∠8?” A diagram of parallel lines intersected by the transversal with angles labeled is shown. ETP Elicit and Use Evidence of Student Thinking teacher prompt states, “Janine says that ∠8 and ∠2 are congruent because they are alternate interior angles. Do you agree? Explain.”
• Lesson 7-1, Understand the Pythagorean Theorem, Visual Learning, Example 1, an image of a triangle is provided. “is a right triangle with side lengths a, b, c. Construct a logical argument to show a$$^2$$+ b$$^2$$= c$$^2$$.” ETP Pose Purposeful Questions teacher prompt states, “How can you be certain that both larger squares are the same size? How can you justify that the areas of the smaller white squares inside the larger square are equal? What does the equation in the last section represent? Explain.” 

Teacher materials assist teachers in engaging students in both the construction of viable arguments and analyzing the arguments or reasoning of others frequently throughout the program. Each Topic Overview highlights specific Math Practices and suggests look fors in student behavior and provides questioning strategies. Examples include:

• Topic 6, Congruence and Similarity, Math Practices, look fors, “Mathematically proficient students: Justify their conclusions about reflections with mathematical ideas. Consider their reasoning about why they rotate figures a certain way or apply certain rules to the rotation of a figure. Make sense of accuracy in transformations and understand angles and congruence relationships. Use their understanding of the definition of the dilation to verify images that are the product of dilation.”
• Topic 6, Congruence and Similarity, Math Practices, questioning strategies, “What evidence did you use to support your solution? Are there similarities between reflections and rotations? Explain. How could you prove that figures are similar? How did you check whether your approach worked?”

The instructional materials reviewed for enVision Mathematics Common Core Grade 8 meet expectations that materials explicitly attend to the specialized language of mathematics.

The materials provide explicit instruction on how to communicate mathematical thinking using words, diagrams, and symbols. Each Topic Overview provides a chart in the Topic Planner that lists the vocabulary being introduced for each lesson in the Topic. As new words are introduced in a Lesson they are highlighted in yellow. Lesson practice includes questions to reinforce vocabulary comprehension and students write using math language to explain their thinking. Each Topic Review contains a Vocabulary Review section for students to review vocabulary taught in the Topic. Students have access to an Animated Glossary online in both English and Spanish. Examples include:

• Lesson 2-8, Understand the y-intercept of a Line, Visual Learning, Example 1, “The y-coordinate of the point where the line crosses the y-axis is the y-intercept.”
• Topic 4, Investigate Bivariate Data, Use Vocabulary in Writing, “Describe the scatter plot at the right. Use vocabulary terms in your description.” Students are provided a word bank containing, “categorical data, outlier(s), cluster(s), relative frequency, measurement data, and trend line.”
• Lesson 5-1, Estimate Solutions by Inspection, Visual Learning, Example 1, “A system of linear equations is formed by two or more linear equations that use the same variables. A solution to a system of linear equations is any ordered pair that makes all equations in the system true.”
• Lesson 6-3, Analyze Rotations, Visual Learning, Example 1, “A rotation is a transformation that turns a figure around a fixed point, called the center of rotation. The angle of rotation is the number of degrees the figure rotates. A positive angle of rotation turns a figure counterclockwise.”
• Topic 8, Solve Problems Involving Surface Area and Volume, Mid-Topic Checkpoint, Question 1, “Select all the statements that describe surface area and volume. Surface area is the sum of the areas of all the surfaces of a figure. Volume is the distance around a figure. Surface area is a three-dimensional measure. Volume is the amount of space a figure occupies. Volume is a three-dimensional measure.” 

The materials use precise and accurate terminology and definitions when describing mathematics, and support students in using them. A Vocabulary Glossary is provided in the back of Volume 1 and lists all the vocabulary terms and examples. Teacher side notes, Elicit and Use Evidence of Student Thinking and Pose Purposeful Questions, provide specific information about the use of vocabulary and math language. Examples include:

• Lesson 3-4, Construct Functions to Model Linear Relationships, Visual Learning, Example 3, Pose Purposeful Questions, “How is the initial value of a function represented in the equation y = mx + b? How is the constant rate of change represented in the equation y = mx + b?”
• Lesson 4-2, Analyze Linear Associations, Visual Learning, Example 1, Try It!, Elicit and Use Evidence of Student Thinking, “How did you decide where to draw your trend line? What type of relationship is it? Explain.”
• Lesson 6-7, Understand Similar Figures, Visual Learning, Example 1, Pose Purposeful Questions, “How do you know that the two trapezoids are facing in the opposite directions? Explain. How do you know the dilation of GHJK is a reflection of ABCD over the x-axis?”
• Lesson 7-1, Understand the Pythagorean Theorem, Visual Learning, Example 2, Pose Purposeful Questions, “Why is it possible to solve this problem using the Pythagorean Theorem? Would the Pythagorean Theorem still apply if one or more of the three side lengths were not a whole number?”
• Student Edition, Glossary, “irrational numbers: An irrational number is a number that cannot be written in the form $$\frac{a}{b}$$, where a and b are integers and b ≠ 0. In decimal form, an irrational number cannot be written as a terminating or repeating decimal. Example, The numbers $$\pi$$ and $$\sqrt2$$ are irrational numbers.”