Pretty Functions

For the last few months, my son Eric has been spending an amazing amount of time hunched over a new calculator--his TI-83 Plus. He had to get it for his seventh-grade algebra course, and now he's writing little programs for it, playing games (passed from one classmate to another), and even plotting data and various functions.

At first glance, his graphing calculator looked quite intimidating to me--all those mysteriously labeled keys (accompanied inevitably by cryptic error messages whenever I tried to do something and failed to adapt sufficiently quickly to the machine's predilections). It didn't take long, however, for me to appreciate the remarkable things the calculator could do. I certainly wished that I had had such a tool when I was in middle or high school and especially in college. My student days date back to a dim, prehistoric past of slide rules and punched cards.

My father had worked in an engineering office, so when the first microchip calculators from Hewlett Packard and Texas Instruments appeared, I learned about them very quickly from him. Despite the high price, I got one of these early models. That antique now looks very primitive next to a TI-83 Plus.

I took a definite step upward in 1988 when I obtained the HP-28S. This hand-held calculating machine was capable of handling not just numbers and simple operations, such as multiplication or computing square roots, but also algebraic expressions and complex tasks, such as taking derivatives and plotting graphs. I could enter the expression "sin x," press a few keys to take its derivative and see the answer "cos x" on the calculator's four-line liquid crystal display, then use a few more keystrokes to generate a graph of the function.

The following years saw the rapid development of graphing calculators and their gradual deployment in the classroom (see Curves and Lying Calculators, February 17, 1997).

Lately, I've been playing with a somewhat different sort of graphing calculator. Graphing Calculator happens to be the name of a computer program for quickly visualizing two- and three-dimensional mathematical objects. Developed largely by Ron Avitzur of Pacific Tech in San Carlos, Calif., it's surprisingly easy to learn and use and wonderfully free of glitches, a rarity in the software world these days.

In fact, owners of most Macintosh computers already have a version of this software on their machines, though few realize how powerful and versatile a tool it really is. It serves as another reminder of how Apple has long neglected to promote and take advantage of the Macintosh's superior mathematical capabilities. A Windows version of Graphing Calculator (once called NuCalc) is now available, as are enhanced editions for the Macintosh (http://www.nucalc.com/).

One of the first things I did was to go to a math textbook called Functions and Graphs by I.M. Gel'fand of Rutgers University in New Brunswick, N.J., and his colleagues. This slim, rather stark volume reminded me a little of the math textbooks that I had encountered in school. It goes right to the point with no fancy graphics or bright colors to entertain the eye--just the mathematical facts, presented cleanly and logically.

Many of the book's exercises involve plotting a given function, then seeing what happens when you modify the function in various ways. I can still remember the great swaths of time that I spent painstakingly plotting functions by hand on paper not only in high school but also in college calculus. Doing this with the Graphing Calculator program was much faster, and, using the software's animation capabilities, I could really see how functions worked under different conditions and better understand relationships between them.

Variations of the function 1/(x² + 1), shown in purple.

Yet, there is something to be said for plotting functions by hand, point by point, at least some of the time. One reason that I could appreciate what the plots on the Graphing Calculator were showing me was that I had, at one time, plotted graphs manually--something that forced me to think about what I was doing, what the points and resulting curves meant, and where they came from.

So, I think you need both--hands-on experience in the world of functions, then a spiffy high-tech graphing calculator to explore that realm and go beyond.

References:

Erickson, T. 1995. Introducing Dynamic Algebra with NuCalc: Investigating Symbols, Functions, and Graphs. Berkeley, Calif.: Key Curriculum Press.

Gel'fand, I.M., E.G. Glagoleva, and E.E. Shnol. 1990. Functions and Graphs. Boston: Birkhäuser.

Peterson, I. 1988. Calculus in the palm of your hand. Science News 133(Jan. 23):62.

Information about Pacific Tech's Graphing Calculator (NuCalc) program for visualizing functions is available at http://www.nucalc.com/.

Comments are welcome. Please send messages to Ivars Peterson at ipeterson@maa.org.