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Estimated Time: 15–30 minutes
Students work in groups to interpret a contour diagram showing temperature as a function of both distance and time.
One possible introduction to this activity is to ask students whether they are familiar with topo maps for hiking. Or weather maps.
(This activity can be assigned or completed for homework.)
Some students have difficulty initially interpreting the description in words. How many walls are there? What are the different directions on the graph? Usually, at least one student in each group is able to put each group on track.
The first 3 questions are pretty straightforward; most students can determine where the window is, when it is open, and when the heat is on. This part of the activity already helps to interpret this basic representation of functions of 2 variables.
The graphing questions (4 and 5) emphasize the idea of “traces”, reducing functions of several variables to functions of one variable. Students usually have little difficulty with this part, but it can be viewed as optional depending on how much the instructor wishes to emphasizes this content.
The last 2 questions can be challenging, and warrant several minutes of group discussion.
A natural followup, either the same day or when introducing partial derivatives later in the course, is to ask how the temperature is changing at a given point, but without specifying with respect to distance or time. This forces students to confront the importance of specifying “with respect to what” (really “with what held constant”) when differentiating.