Sep. 28, 2018 - First Day of Class: Multivariable Functions (Stepan Paul)
Sep. 28, 2018 - First Day of Class: Multivariable Functions (Stepan Paul)
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On the first day of class we wanted to use several of the models to get students used to working with them. The learning objectives for the first day were for students to be introduced to functions of more than one variable, to understand how a function of two variables can be represented by either its graph or its contour plot, and for students to start using the idea of a trace (cross-section) of a surface in three dimensions.
At Harvard our classes are run in coordinated sections. This means that several instructors teach classes of about 30 students and all instructors use roughly the same lesson plan and worksheet each day. Below is an outline of how my section went.
After a brief introduction of myself and the course and an icebreaker activity, I started class by giving brief definitions of a contour plot and a graph of two variables using a weather map as motivation. We then proceeded straight to the worksheet problems.
The first problem of the worksheet is shown below. Students were given the function graph model, a set of dry erase markers, and a K'Nex “axes kit.” After some brief instruction about how to assemble to axes, students were set loose on Problem #1, working in small groups at the black boards.
My fear during Problem #1 was that students would become adept at relating the function graph and contour plot but not think about the underlying function itself. Because of this, my course assistants and I intervened to ask students to explain the connection of both their drawing and the model to the function they described.
The contours of the function were visible on the function graph model as an artifact of the 3D printing and thermoforming process so students who caught on to this fact were able to produce very nice contour plots in #1a. In #1b students were able to recognize that peaks were visible in the contour as concentric rings but when asked to imagine what the contours mirror a pit would look like, they had very strange misconceptions. Problem #1c was meant to foreshadow the idea that concept of derivative is more complicated for multivariable functions than for single variable functions.
The utility of the models was most apparent when students drew with the dry erase markers on them. In parts #1d and #1e below, students were able to sketch the cross sections and then orient them so that students could see the curve they drew as a cross section of the surface.
After a follow up discussion on Problem #1, we worked through Problem #2a together as a class. The idea here was for students to see how one could draw a contour plot for a function given with an explicit formula. In the last few minutes of class, students were given the 3D printed saddle model and I asked them to sketch the contours as cross sections of the function graph on the saddle.
When I do this same lesson next semester, I want to spend less time at the beginning talking myself and to give students more time to work on the rest of Problem #2. I may also change #1a to ask students to draw just one contour (e.g. where the function is equal to 2) to save time and to emphasize the connection to underlying function. To avoid the tangential conversations about the contour plot near a pit, I might eliminate #1b and save the discussion for later. I might also make Problem #1c part of Problem #2 instead to give students more exposure to the saddle model.
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