Date: Apr 25, 1995 3:10 PM
Author: Ronald A Ward
Subject: WLME--Calculus Sample
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