Intent

In this summative activity, students describe, in writing, methods for moving from one representation of a linear function to another.

Mathematics

This activity draws upon and strengthens the connections among the four representations of a function, with a particular focus on linear functions. Students are encouraged to work from rate of change and starting value in identifying these connections. They learn that linear has an algebraic as well as a geometric meaning and are introduced to a common form of the linear function, y = ax + b.

Progression

Students work on this activity in groups, perhaps submitting a first draft submitted for feedback, with a final draft due at a later date.

Approximate Time

30 minutes

Classroom Organization

Groups

Materials

Students’ work on the activities in Traveling at a Constant Rate

Doing the Activity

Remind students that the term linear was initially used to refer to a function or situation that led to a straight-line graph. (See the introductory comments on Wagon Train Sketches and Situations.) Then ask, How can you tell from looking at a rule whether its graph is a straight line?

The goal here is for students to recognize that rules for straight lines have a specific algebraic form. Students might reply with something like y = s + rx, where s represents the starting value and r represents the rate, likely suggesting numbers in place of the variables s and r. Help students recognize that the s- and r-values can be positive, negative, or zero. (See the “Mathematics” section in the introduction to Traveling at a Constant Rate for a relevant discussion.)

Ask, Why do situations with constant rates have rules of this special type? Students may offer a variety of responses to this question.

Explain that rules of this type—namely, of the form y = s + rx—are called linear functions and that this term has both a geometric and an algebraic meaning.

Also explain that a common way to write a rule for a linear function is y = ax + b. Ask students to identify the meaning of the a and b terms. Also state that any rule equivalent to a rule of this form is linear as well.

As groups work, help students to recall relevant activities from the unit. Do you remember an activity from this unit in which you had to create an In-Out table from a graph?

Discussing and Debriefing the Activity

No formal debriefing of this activity is necessary.