This unit introduces students to nonproportional linear relationships by building on earlier work with rates and proportional relationships from grade 7, and on earlier grade 8 work around similarity and slope.

The unit begins by revisiting different representations of proportional relationships. Students create graphs, tables, and equations in order to interpret the constant of proportionality in a context. They see the constant of proportionality between two variables as the rate of change of one variable with respect to the other.

Next, students analyze a relationship that is linear but not proportional. In this context, students see that the rate of change has a numerical value that is the same as the slope of the line that represents the relationship. Students also view the graph of a line in the coordinate plane as the vertical translation of a proportional relationship.

In the following section, students are introduced to lines with non-positive slopes and vertical intercepts. They consider situations represented by linear relationships with negative rates of change and establish a way to compute the slope of a line from any two distinct points on the line. Students also write equations of horizontal and vertical lines.

In the last section, students consider what it means for a pair of values to be a solution to an equation and the correspondence between coordinates of points on a graph and solutions of an equation.

Progression of Disciplinary Language

In this unit, teachers can anticipate students using language for mathematical purposes, such as representing, generalizing, and explaining. Throughout the unit, students will benefit from routines designed to grow robust disciplinary language, both for their own sense-making and for building shared understanding with peers. Teachers can formatively assess how students are using language in these ways, particularly when students are using language to:

Represent

• Situations involving proportional relationships (Lesson 1).
• Constants of proportionality in different ways (Lesson 3).
• Slope using expressions (Lesson 10).
• Linear relationships using graphs, tables, equations, and verbal descriptions (Lesson 5).
• Situations using negative slopes and slopes of zero (Lesson 9).
• Situations by graphing lines and writing equations (Lesson 13).
• Situations involving linear relationships (Lesson 15).

Generalize

• Categories for graphs (Lesson 2).
• About equations and linear relationships (Lesson 7).
• In order to make predictions about the slope of lines (Lesson 10).

Explain

• How to graph proportional relationships (Lesson 3).
• How to use a graph to determine information about a linear situation (Lessons 5 and 6).
• How to graph linear relationships (Lesson 10 and 11).
• How slope relates to changes in a situation (Lesson 11).

In addition, students are expected to describe observations about the equation of a translated line. Students will also have opportunities to use language to interpret situations involving proportional relationships, interpret graphs using different scales, interpret slopes and intercepts of linear graphs, justify reasoning about linear relationships, justify correspondences between different representations, and justify which equations correspond to graphs of horizontal and vertical lines.

The table shows lessons where new terminology is first introduced in this course, including when students are expected to understand the word or phrase receptively and when students are expected to produce the word or phrase in their own speaking or writing. Terms that appear bolded are in the Glossary. Teachers should continue to support students’ use of a new term in the lessons that follow where it was first introduced.