This section introduces students to functions and develops the idea of a function as a rule that assigns to each allowable input exactly one output.

A focus of this section is on examples of “input–output rules,” such as “Divide by 3” or “If even, then . . . . If odd,
then . . . .” In the starting examples, the inputs are (implicitly) numbers, but students note that some inputs are not allowable for some rules. For example,  is not even or odd. Students then practice identifying statements that describe functions and use language such as “[The output] is a function of [the input]” and “[The output] depends on [the input].”

In this section, students use linear and piecewise linear functions to model relationships between quantities in real-world situations, interpreting information from graphs and other representations in terms of the situations. The lessons provide an opportunity for students to coordinate and synthesize their understanding of new and old terms that describe aspects of linear and piecewise linear functions.

Students begin the section making connections between linear relationships and functions. This work builds on their previous experiences in grade 7 with proportional relationships and the constant of proportionality and in grade 8 connecting the slope of a line to the rate of change and making sense of the vertical intercept in context. Students also consider when a linear function is a good model for a situation, when it’s not a good model, and when it could work for part of a situation, leading to piecewise linear functions.

Scatterplot, horizontal, time in hours after midnight, 0 to 12 by ones, vertical, temperature in degrees Fahrenheit. Fifty points approximate a straight line from point 2 5 comma 50 increasing to 5 point 75 comma 59 and then decreasing from there to 12 comma 52 point 5.

The lesson in this section is optional because it offers additional opportunities to practice standards that are not a focus of the grade.

In this section, students connect the terms “independent variable” and “dependent variable” (which they learned in grade 6) with the inputs and outputs of a function. They use equations to express a dependent variable as a function of an independent variable, viewing formulas from earlier grades (for example, ) as determining functions. They work with tables, graphs, and equations for functions, learning the convention that the independent variable is generally shown on the horizontal axis. They also work with verbal descriptions of a function arising from a real-world situation, identifying tables, equations, and graphs that represent the function, and interpreting information from these representations in terms of the real-world situation.

A graph in a coordinate plane, horizontal axis, distance in meters, 0 to 24 by threes, vertical axis, time in seconds, 0 to 10 by ones. Graph begins at the origin and moves steadily upward and to the right, passes through ( 3 comma 1)  and ( 18 comma 6 ).

In this section, students extend their understanding of volume from right prisms to right cylinders and right cones.

Students begin by investigating the volume of water in a graduated cylinder as a function of the height of the water, and vice versa. They examine depictions of cylinders, prisms, spheres, and cones in order to develop their abilities to identify radii, bases, and heights of these objects.

Next, students use these abilities together with geometric knowledge developed in earlier grades to perceive similar structure in the formulas for the volume of a rectangular prism and the volume of a cylinder—both are the product of base and height. After gaining familiarity with a formula for the volume of a cylinder by using it to solve problems, students perceive similar structure in a formula for the volume of a cone.

The focus of this section is establishing the formula for the volume of a sphere. The first lesson invites students to use equations and graphs of proportional relationships to make sense of what happens to the volumes of familiar shapes when one dimension changes. A goal of this work is for students to reason about volume using equations, which will continue throughout the section.

The second lesson in this section is optional, as it contains work beyond the scope of the grade: nonlinear functions. If time allows, consider using this lesson to give students experience working with nonlinear functions in the context of three-dimensional shapes.

The following two lessons see students reasoning first about the volume of hemispheres and how it compares to the volume of prisms, cylinders, and cones of similar dimensions. Building on the foundation of these connections, the formula for the volume of a sphere is established.

The section concludes with a lesson using the Information Gap routine. This lesson offers students opportunities to practice using what they have learned about the volume of a sphere and how that volume relates to the volume of cones and cylinders.