Intent

This activity is the second in the series in which students explore nonlinear relationships.

Mathematics

This activity asks students to make a prediction from two given data points. The most common approach, given the work students did in The Overland Trail, is to assume that the two points determine a linear function. In reality, an infinite number of functions can fit any two data points. Once a third data point is given, however, the nonlinear nature of the data becomes apparent. Even then, an infinite number of functions can be found that fit the three data points. Students are not expected to find an algebraic rule that fits the data in this activity, but they should be able to use graphs and tables to represent possible relationships.

Progression

Students will work on this activity in stages, with group work and whole-class discussion at each stage.

Approximate Time

30 minutes

Classroom Organization

Groups and whole-class discussion

Doing the Activity

Let students work for a few minutes on Questions 1 and 2, and then have them share their conclusions with the class. They will most likely have decided on an answer of 16 birdhouses for the 8 hours for Question 1 and concluded that the number of birdhouses that can be painted is double the number of hours.

Then have them continue with Question 3, emphasizing that you want them to think about ways to approach the task and that you don’t expect them to find an equation that represents the data. Give them about 10 minutes to work on this question.

Discussing and Debriefing the Activity

Ask groups to report on what they did. Here are some approaches they may have tried.

As you discuss the ideas, emphasize two key points: (1) that there may be more than one good answer and (2) even if they didn’t find an exact answer, looking for a different kind of rule is a good approach.

Point out that when there were only two pieces of information, other rules would have fit them. The rule y = (x - 1)² + 2 is one nonlinear example that fits the two initial pieces of information.

The same principle applies for the three points. Just because a rule fits the data doesn’t mean it must be the rule. People naturally think of the “two birdhouses per hour” rule for various reasons, including these: (1) It makes intuitive sense that Mia and her classmates paint the same number of birdhouses every hour. (2) That is the simplest rule that fits the information.