Click Back to return
THE MATHWRIGHT LIBRARY NEWSLETTER, March 2003, VOL
5, #2
A publication of Bluejay Lispware
James E. White, Editor
The official publication of the New Mathwright Library and Café:
In this issue:
Experimental geometry through logo: How to draw a star
Magnify your intuition about limits and derivatives
How
to draw a star
by Dan Kalman, The American
University, and Angela Hare, Messiah College
You
may know that the logo computer language was created by Seymour Papert
in order to place in students' hands a new medium that can support invention
and discovery in geometry. It is based on a philosophy of education that sees
playful exploration as a source of enduring knowledge. Mathwright contains
logo, and many of its interactive books and activities are written from this
viewpoint. How to draw a star is an excellent example, although it
requires no programming at all for the reader to explore its ideas. It is
written so that a High School student can derive many hours of creative entertainment
while reading the microworld either in her browser or off-line.
If
you draw a five pointed star on a sheet of paper, you will probably follow
a procedure that is very much like a logo program. The essential thing is
to decide on an angle through which to turn at each vertex of the star. If
you say to yourself right away that the angle must be 36 degrees, then you
are way ahead of the game. But there are still many interesting questions
that the microworld will pose, so read on. If, however, it is not clear why
that is so, then you can learn the answer to that question and many others
through experiment.
This
microworld artfully and unobtrusively visits many topics in geometry, including
the properties of regular polygons, inscribed and central angles in circles,
and it leads to the design of colorful symmetrical patterns that you can create,
and then print by taking snapshots of your screen.
The
Star Microworld provides many opportunities to explore the geometry of stars.
There are three activities pages. On the first you can explore the relationship
between the number of points in a star and the size of the angle formed at
each point. The second allows you to construct stars by drawing polygons
with vertices regularly spaced around a circle. The third automates this
process according to a user defined pattern for drawing the sides of the polygon.
Magnify!
by Dan Kalman, The American
University
One
of the first really new topics that a student encounters in calculus is the
idea of a limit. From a teacher's perspective, it presents a special
challenge -- and a great deal of chalk -- to make this seminal idea intuitive
and meaningful. It is especially important to do so, because almost every
major idea in the calculus (continuity, derivative, integral, and so on) depends
on it. But every teacher knows that this intuition comes slowly, and often
requires a certain amount of 'unlearning' in order for the idea to settle
in the student's mind.
Magnify
is a Microworld that starts with the basic idea, and makes it a visual and
interactive exploration. Essentially, the test for a limit is like using a
magnifying glass, or a microscope, to examine the properties of a function
more and more closely. Ideally, one examines them with infinite precision,
once the logic of the procedure is mastered. But before that, this microworld
allows the reader to arrive at that intuition in small, finite steps, with
its 'magnifying glass' that calculates and displays function values more and
more closely and displays them in both tabular and graphical form.
Calculations
like these would be tedious. The picture is often much more useful to spur
the imagination. This is so, especially because the reader may supply any
desired function and ask 'what if' questions.
On
another exploration page, the reader is led to the notion of derivative as
limiting slope. The idea here is that, on closer and closer inspection, a
well-behaved (differentiable) function resembles its tangent line more and
more closely. How do we see this? With our magnifying glass, of course! The
reader may guess the slope of the tangent line (the derivative) of a function
at a point and see for herself how good or poor a guess it is.