8th Grade - Gateway 1

The instructional materials for Big Ideas Math: Modeling Real Life Grade 8 partially meet expectations for the materials being consistent with the progressions in the Standards.

The materials concentrate on the mathematics of the grade, and are consistent with the progressions in the Standards. The publisher recommends using four resources together for a full explanation of the progression of skill and knowledge acquisition from previous grades to current grade to future grades. These resources include: “Laurie’s Notes”, “Chapter Overview”, “Progressions”, and “Learning Targets and Success Criteria”. For example:

• Laurie’s Notes, “Preparing to Teach” describe connections between content from prior grades and lessons to the current learning. In Chapter 4, Section 4, it states, “Students should know how to graph numbers on a number line and how to solve one-variable inequalities using whole numbers. In the exploration, students will be translating inequalities from verbal statements to graphical representations and symbolic sentences.”  Chapter Overviews describe connections between content from prior and future grades to the current learning, and the progression of learning that will occur. For example, In Chapter 5, in “Laurie’s Notes: Chapter Overview”, “The study of ratios and proportions in this chapter builds upon and connects to prior work with rates and ratios in the previous course.” (6.RP). In sections 5.1 and 5.2, students decide whether two quantities are in a proportional relationship using ratio tables (7.RP.2a) and use unit rates involving rational numbers. Then during the next three sections, students write, solve, and graph proportions (7.RP.2a-d, 7.RP.3). “Graphing proportional relationships enables students to see the connection between the constant of proportionality and equivalent ratios”, but the term, “slope” (8.EE.5-6), is not included. In the final lesson, students work with scale drawings (7.G.1).
• Each chapter’s Progressions page contains two charts. “Through the Grades”, lists the relevant portions of standards from prior and future grades (grades 6 and 7) that connect to the grade 8 standards addressed in that chapter.  This continues to show how grade-level content connects to future grades. For example, in Chapter 5, the table explains that the work in this chapter builds on 7th grade content around “write, graph, and solve one-step equations”(7.EE.A), and “solve two-step equations” (7.EE.B). It continues by explaining that this will go on to support high school around systems of linear equations (HSA.REI.C). 
• In Chapter 6, the Progressions page explains that the work in this chapter builds on the 7th grade content around using samples to draw inferences about populations and comparing two populations from random samples using measures of center and variability (7.SP). It continues by explaining that this will go on to support high school around classifying data as quantitative or qualitative  and making two-way tables to recognize associations in data (HSS.ID.B). 

Each lesson presents opportunities for students to work with grade-level problems. However, “Scaffolding Instruction” notes suggest assignments for students at different levels of proficiency (emergent, proficient, advanced). These levels are not defined, nor is there any tool used to determine which students fall into which level. In the Concepts, Skills, and Problem Solving section at the end of each lesson, problems are assigned based on these proficiencies, therefore, not all students have opportunities to engage with the full intent of grade-level standards. For example:

• In the Teacher Edition, Chapter 1, Section 1.4, page T-2, the assignments for proficient and advanced students include a reasoning task in which students solve a formula, that converts temperatures from degrees Fahrenheit to Kelvin, for F, and use the new formula to convert a temperature of 0.95 Kelvin to degrees Fahrenheit (page 30, number 29). This reasoning task is omitted from the assignments for emerging students, even though the learning target for this section is for students to “solve literal equations for given variables and convert temperatures.” (page T-25)
• In the Teacher Edition, Chapter 2, Section 2.5, page T-74, the assignments for proficient and advanced students include a critical thinking task in which students determine the transformations described using coordinate notation, for example (x,y) (2x + 4, 2y -3). This reasoning task is omitted from the assignments for emerging students, even though one of the success criteria for this section is, “Identify a dilation.”
• In Chapter 7, Section 7.3, Example 4, students determine which company charges more per cubic foot of mulch by comparing the slopes of two functions. One function is represented by a table and the other is represented by an equation. In practice problem, number 16, students determine which activity, kayaking or hiking, burns more calories per minute by comparing. One function is represented by an equation and the other is represented by a graph. Students compare slopes, but have no opportunities to compare other properties, such as y-intercepts, as stated in Standard 8.F.2.
• Each section within a chapter includes problems where the publisher states, “students encounter varying Depth of Knowledge levels, reaching higher cognitive demand and promoting student discourse.” For example, in Chapter 7, Section 7.4, students look for numeric patterns of falling objects in order to determine if the data represent a linear or nonlinear function. This supports Standard 8.F.3.
• In Chapter 7, Section 7.4, Exploration 1, students graph given equations representing the distance/time of a skydiver and a bowling ball. Students then decide if they represent linear or nonlinear functions. Finally, students compare the linear and nonlinear functions.
• In Chapter 7, Section 7.4, Problems 1 and 2 asks students to distinguish between linear and nonlinear equations, first with tables and then equations. They explain their thinking.
• In Chapter 4, Section 4.3, Example 3, students determine which ski lift is faster by identifying and comparing the slopes of the two proportional relationships. One relationship is represented by a graph and the other is represented by an equation.
• In Chapter 4, Section 4.3, Practice Problems 15 and 16, students identify, interpret, and compare slopes of proportional relationships represented by an equation and a graph, Problem 15, and by a narrative and a table, problem 16. Although there are no opportunities to compare y-intercepts, this is not specifically required by Standard 8.EE.5. Therefore, the materials meet the full depth of Standard 8.EE.5, which calls for comparisons of two different proportional relationships represented in different ways.
• Standard 8.EE.2, use square root and cube root systems to represent solutions to equations of the form $$x^2=p$$ and $$x^3=p$$, where p is a positive rational number.  Evaluate square roots of small perfect squares and cube roots of small perfect cubes.  Know that $$\sqrt{2}$$ is irrational. This standard is fully developed over the course of four sections in Chapter 9. For example:
• In Section 9.1, students use the radical sign to solve problems where ‘p’ is a rational number.  Chapter 9, Section 9.1, problem 5, students evaluate $$\sqrt{-81}$$.  This continues and students solve problems with the square root sign in example 5a: $$x^2=81$$. This is also embedded with work on perfect squares.
• In Section 9.2, students apply their knowledge of square roots to discovering the Pythagorean Theorem.
• In Section 9.3, students identify cubed roots and use approximate symbols to solve problems. Chapter 9, Section 9.3, Example 1: $$^3\sqrt{8}=?$$. Students identify perfect cubes and evaluate for them.
• In Section 9.5, Key Idea, students identify that $$\sqrt{2}$$ is an irrational number.

Materials explicitly relate grade-level concepts to prior knowledge from earlier grades. At the beginning of each section in Laurie’s Notes, there is a heading marked “Preparing to Teach” which includes a brief explanation of how work in prior courses relates to the work involved in that lesson. For example:

• In Chapter 1, Section 1.1, it explains that students worked with one and two-step equations, Standard 7.EE.A, and that they will build upon their understanding of solving linear equations with rational coefficients, Standard 7.EE.B.
• In the Teacher Edition, Chapter 2, Section 2.1, page T-43, the “Preparing to Teach” notes connect students’ prior work plotting points in the coordinate plane to the grade-level work of finding the coordinates of a translated figure and using coordinates to translate a figure. This supports Standards 8.G.1 and 8.G.3.
• In Chapter 2, Section 2.2, students use their understanding of lines of symmetry (4.G.3), and skill at plotting points in the coordinate plane (6.NS.8, 6.G.3), which supports the grade-level work of experimentally verifying properties of reflections (8.G.1). The Teacher Edition, “Preparing to Teach” notes, page T-49, make note that symmetry is content from a prior grade.
• In Chapter 3, Section 3.1, it states, “Students worked with transformations and congruent figures in the previous chapter, Standard 8.G.A 14. Now they will make conjectures about angles created by parallel lines and transversals, Standard 8.G.5.”
• Chapter 4, Laurie’s Notes on page T-141, states that “in previous courses, students used tables of values to show proportional relationships,” Domain 7.RP and that “in this lesson, they will use a table of values to create a graph of a linear equation, Standard 8.EE.5.” This follows the progression from the concept of proportional relationships to the concept of function.
• In Chapter 8, Section 8.1, it states, “Students should know how to raise a number to an exponent. The work in this section should be review for students, except powers with negative bases are now included.” In the Exploration 1, students complete a table that represents products from exponents and adds negative bases. This builds on their understanding of, Know and apply the properties of integer exponents (8.EE.1).
• In Teacher Edition, Chapter 9, Section 9.1, pg. T-373, the “Preparing to Teach” notes connect students’ prior work of find squares of numbers (6.EE.1), and areas of squares, (4.G.3 and 6.G.1), to the grade-level work of finding square roots, (8.EE.2).