8th Grade - Gateway 2

The instructional materials for Big Ideas Math: Modeling Real Life Grade 8 partially meet expectations that the materials develop conceptual understanding of key mathematical concepts, especially where called for in specific standards or cluster headings. The instructional materials do not always provide students opportunities to independently demonstrate conceptual understanding throughout the grade-level. 

Each lesson begins with an Exploration section where students develop conceptual understanding of key mathematical concepts through teacher-led activities. For example:

• In Chapter 9, Lesson 1, Exploration 1, students are given the area of squares and asked to find the side lengths introducing square roots. The remaining work of the lesson is procedural with the exception of three questions (7,8 and 9) in Concepts, Skills, and Problem Solving in which students apply the understanding of square root to the area of a square. (8.EE.2)
• In Chapter 2, Lesson 1, Exploration 1, students create a figure on the coordinate plane. They copy this figure onto transparent paper and move it around the coordinate plane describing its location in comparison to the original location. The lesson continues on with figures, examples, illustrations and models to explain and facilitate the understanding of translations conceptually. (8.G.1)

The instructional materials do not always provide students opportunities to independently demonstrate conceptual understanding throughout the grade-level. During the “Example” sections, the focus is on explaining procedures. For example:

• In Chapter 10, Lesson 1, students find the volume of cylinders. In Exploration 2, students discover the formula for volume through an investigation using nets. In the remainder of the lesson gives and applies the formula for volume procedurally. 
• Chapter 1, Lesson 3, students solve equations with variables on both sides. In Exploration 1, students conceptually solving for both variables using perimeter and area, however, the remaining work of the lesson is procedural as students solve and check their equations.

The instructional materials for Big Ideas Math: Modeling Real Life Grade 8 partially meet expectations that the materials are designed so that teachers and students spend sufficient time working with engaging applications of mathematics. 

The instructional materials present opportunities for students to engage in application of grade-level mathematics; however, the problems are scaffolded through teacher-led questions and procedural explanation. The last example of each lesson is titled, “Modeling Real Life,” which provides a real-life problem involving the key standards addressed for each lesson. This section provides a step-by-step solution for the problem; therefore, students do not fully engage in application. For example:

• Chapter 2, Lesson 3, Example 4, Modeling Real Life, “A carousel is represented in a coordinate plane with the center of the carousel at the origin. You and three friends sit at A(−4, −4), B(−3, 0), C(−1, −2), and D(−2, −3). At the end of the ride, your positions have rotated 270° clockwise about the center of the carousel. What are your locations at the end of the ride? A rotation of 270° clockwise about the origin is the same as a rotation of 90° counterclockwise about the origin. Use coordinate rules to find the locations after a rotation of 90° counterclockwise about the origin.” [Coordinate plane with original and rotated positions included.] The example provides step-by-step instructions on how to solve the problem. “A point (x, y) rotated 90° counterclockwise about the origin results in an image with coordinates (2y, x). [Each origin point with solution included] Your locations at the end of the ride are A′(4, -4), B′(0, -3), C′(2, -1), and D′(3, -2).” (8.G.2, 8.G.3)
• Chapter 4, Lesson 7, Example 3, Modeling Real Life, “You finish parasailing and are being pulled back to the boat. After 2 seconds, you are 25 feet above the boat. At what height were you parasailing? You are 25 feet above the boat after 2 seconds, which can be represented by the point (2, 25). You are being pulled down at a rate of 10 feet per second. So, the slope is −10. Because you know a point and the slope, use point-slope form to write an equation that represents your height y (in feet) above the boat after x seconds.” [Step-by-Step solution provided.] “The height at which you were parasailing is represented by the y-intercept. So, you were parasailing at a height of 45 feet.” (8.EE.6)

Throughout the series, there are examples of routine application problems that require both single and multi-step processes; however, there are limited opportunities to engage in non-routine problems. For example:

• Chapter 7, Lesson 3, Problem 15, Dig Deeper, “You and a friend race each other. You give your friend a 50-foot head start. The distance y (in feet) your friend runs after x seconds is represented by the linear function y=14x+50. The table shows your distance at various times throughout the race. For what distances will you win the race? Explain. [Table provided.]” (8.F.3, 8.F.2, multi-step, routine)
• Chapter 9, Lesson 2, Problem 14 Dig Deeper, “Objects detected by radar are plotted in a coordinate plane where each unit represents 1 mile. The point (0, 0) represents the location of a shipyard. A cargo ship is traveling at a constant speed and in a constant direction parallel to the coastline. At 9 a.m., the radar shows the cargo ship at (0, 15). At 10 a.m., the radar shows the cargo ship at (16, 15). How far is the cargo ship from the shipyard at 4 p.m.? Explain." (8.G.7, multi-step, routine)
• Chapter 3, Lesson 4, Problem 19, “A map shows the number of steps you must take to get to a treasure. However, the map is old, and the last dimension is unreadable. Explain why the triangles are similar. How many steps do you take from the pyramids to the treasure?" (8.G.6, multi-step, routine)

The instructional materials for Big Ideas Math: Modeling Real Life Grade 8 partially meet expectations that the three aspects of rigor are not always treated together and are not always treated separately. 

The instructional materials present opportunities in most lessons for students to engage in each aspect of rigor, however, these are often treated together. There is an over-emphasis on procedural skill and fluency. For example:

• Chapter 2, Lesson 5, Dilations, begins with a visual to represent the meaning of the word ‘dilate’. In Exploration 1, students conceptualize what it means to dilate a polygon. The lesson provides five examples showing step by step procedures for dilating a figure using scale factor. During Concepts, Skills and Problem Solving, students independently solve problems procedurally. For example, in Problems 9-14, students “Tell whether the blue figure is a dilation of the red figure” with figures given for each problem. In Problems 15-21, students dilate and identify a figure based on given vertices and scale factors.  
• Chapter 4, Lesson 4, Graphing Linear Equations in Slope-Intercept Form, Exploration 1, Deriving an Equation, builds on the prior learning of graphing proportional relationships. The students conceptualize and understand graphing linear equations using slope intercept. The lesson continues with two examples that show step by step procedures for identifying slopes and y-intercepts and graphing linear equations. For example, Example 1 “Find the slope and the y-intercept of the graph of each linear equation. y = -4x - 2”. Example 2, “Graphing a Linear Equation in Slope Intercept Form. Graph y = -3x + 3. Identify the x-intercept." The lesson continues with independent, procedural practice. For example, in the Concepts, Skills and Problem Solving section, Problems 14-22, students find the slope and the y-intercept of the graph of the linear equation.
• Chapter 9, Lesson 2, The Pythagorean Theorem, Exploration 1, Discovering the Pythagorean Theorem, students work with a partner to conceptually explore an informal proof. The lesson continues with four examples that model and provide step by step procedures for: “Finding the Length of a Hypotenuse”, “Finding the Length of the Leg,” Finding the Length of the Three-Dimensional Figure” and “Finding a Distance in a Coordinate Plane”. During independent practice in Concepts, Skills and Problem Solving section, Problems 7-12, students find the missing length of a triangle.

The instructional materials reviewed for Big Ideas Math: Modeling Real Life Grade 8 partially meet expectations that the instructional materials prompt students to construct viable arguments and analyze the arguments of others concerning key grade-level mathematics. 

“You be the Teacher” found in many lessons, presents opportunities for students to critique the reasoning of others, and construct arguments. For example: 

• Chapter 1, Lesson 1, Problem 31, You Be the Teacher, “Your friend solves the equation. Is your friend correct? Explain your reasoning.” The student work is provided to examine. 
• Chapter 2, Lesson 3, Problem 12, You Be the Teacher, “Your friend describes a sequence of rigid motions between the figures. Is your friend correct? Explain your reasoning.” The student work is provided to examine. 
• Chapter 5, Lesson 3, Problem 15, You Be the Teacher, “Your friend solves the system. Is your friend correct? Explain your reasoning.” The student work is provided to examine. 

The Student Edition labels Standards of Mathematical Practices with “MP Construct Arguments,” however, these activities do not always require students to construct arguments or analyze arguments of others. In the Student Edition “Construct Arguments” was labeled once for students and “Build Arguments” was labeled once for students.  Examples of missed opportunities include the following:

• Chapter 4, Lesson 2, Exploration 1, Construct Arguments is identified in the Math Practice blue box with the following question, “Do your answers to parts (b) and (c) change when you draw △DEF in a different location in part (a)? Explain.” 
• Chapter 7, Lesson 1, Exploration 2, Construct Arguments is identified in the Math Practice blue box with the following question, “How does the graph help you determine whether the statement is true?” 

Chapter 8, Lesson 1, Exploration 1, Build Arguments is identified in the Math Practice blue box with the following questions: “When is the value of (-3)n positive? Negative?”

The instructional materials reviewed for Big Ideas Math: Modeling Real Life Grade 8 partially 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. 

There are some missed opportunities where the materials could assist teachers in engaging students in both constructing viable arguments and analyzing the arguments of others. For example:

• In Chapter 2, Lesson 5, Example 2 students are shown how to dilate a figure and identify it. Laurie’s Notes identifies MP3, prompting teachers to ask, “How do you think the perimeters of the two triangles compare? Explain.” “How do you think the areas of the two triangles compare? Explain.” Neither of these questions allow students to construct viable arguments or critique the reasoning of others because the work and explanations are given to the students. 
• In Chapter 4, Chapter Exploration, Laurie’s notes, identifies MP3 and prompts teachers to “listen to and discuss students' responses to the generalizations in parts (e) and (g)”. 
• Chapter 8, Lesson 7, Example 2, students multiply using scientific notation. Laurie’s Notes identifies MP3 and  prompts teachers to “Make sure that students realize that the Commutative and Associative Properties allow this to happen. The Product of Powers Property is used to multiply the powers of 10.”