Intent This activity introduces the idea of equivalent forms of linear functions by drawing on students’ ability to move among the various representations of a function.

Mathematics Students draw upon their understanding of the four representations of linear functions to informally become acquainted with the notion of equivalent equations. In particular, they graph an equation and then use what they have learned to determine a linear equation of the form y = ax + b from the graph. Students will then apply the distributive property as well as other basic symbol manipulation to demonstrate the equivalence of the two equations.

Progression Students work individually on the activity and then debrief their work as a class, making observations and conjectures about various symbolic forms of equivalent linear equations.

Approximate Time 5 minutes for introduction 30 minutes for activity (at home or in class) 15 minutes for discussion

Classroom Organization Individuals, followed by whole-class discussion

Doing the Activity Introduce this activity as an individual investigation. (Note that the activity makes no intent to define equivalent equations. A careful definition of equivalence—one that would involve sets of solutions—may detract from students bringing their own reasoning to the problems.)

Discussing and Debriefing the Activity Discussion can begin immediately with student presentations of Question 3a. Encourage discussion to clarify the presenters’ methods and reasoning. Also request presentations of Question 4a with the same goal—a discussion of a variety of approaches. Then ask for ideas about how to approach Questions 3b and 4b. Ask, What did you interpret the word equivalent to mean in Questions 3b and 4b? Where have you heard the term before? Students may observe that the ordered pairs that satisfy one equation also satisfy the other or that the related In-Out tables or graphs are the same. Others may say the equations are not equivalent because they don’t appear to be the same. Tell students that when two equations are equivalent, every pair of numbers that satisfies one equation also satisfies the other. With this definition, the two equations in Question 3 are equivalent, as are the two equations in Question 4. Let students know that they will return to this idea later in the unit

Supplemental Activities The Growth of Westville (extension) provides a western setting for examining situations that may appear to involve constant growth but do not lead to linear graphs, so this is a good follow-up to the series of activities. Westville Formulas (extension) is a follow-up to The Growth of Westville.