Today in my beginning calculus class, I tried a combination physical/outdoor/modeling activity to explore velocity. I divided the class into five small groups of six. Each group was given a large orange, a blank transparency, and overhead marker.
Within a given group, one student threw the orange vertically upward, a second student counted off seconds of flight out loud, other students were responsible for observing where the orange was located at each second of flight until it "splatted" on the ground. Each group was encouraged to come to some agreement before conducting the experiement as to the elevation of certain horizontal marks on a nearby building, used as background for estimating the height of the orange. Each group made a table of its data, then drew a model (in this case, a graph) of the height above the ground versus the elapsed time in seconds.
When we returned to the classroom, each group presented its model. There were many similarities, including the length of flight (3.5 to 5 seconds), the maximum altitude reached (30 to 45 feet), the general shape of the curve, and so forth. But there were also differences, including the starting altitude (3.5 to 7.5 feet) as well as specific differences cited earlier.
Using one of the models, we then discussed "average velocity over an interval," its relation to the slope of a secant line, what the secant line itself represented, as well as a process for determining the "instantaneous velocity at a particular time," and its relationship to the slope of the curve at a point, the slope of the tangent line to the curve, what happens when you zoom-in on a graphing calculator repeatedly on a curve at a point, and so forth.
I felt that it was a good way to "celebrate" mathematics, to employ the Way of Archimedes, and to view a concept from several perspectives (tabular or numerical and geometric--we didn't have time to also explore the concept from an algebraic point of view). Of course, when you run these experiments, it's something like Forrest Gump's box of chocolates--you never know what you're going to get! I did take photographs to record the event, and we'll mount those on oaktag, along with the small-group models.
Ron Ward/Western Washington U/Bellingham, WA 98225 firstname.lastname@example.org