The
Use of Calculators and Computers in Don's Books and
Tapes
Don refuses to let his students use a
calculator to do simple arithmetic. Much of what he does involves doing
the arithmetic in your head and looking for patterns. The arithmetic,
algebra, and patterns of
infinite series and infinite sequences, however, make this a worthwhile
use of
calculators and computers. Appendix 3 in Don's worksheet book describes
how to write programs to do these jobs, and shows what programs are in
each chapter. Below are some examples:
- With Don's help, a few students have written a program on a
programmable calculator to change a fraction to its bimal. (Ch. 2)
- A student will iterate, by hand, a function like 5 + x/2, starting with
say 1.
This gives an infinite sequence, which on a calculator shows that it gets
to 10. Then using Mathematica, the student can take the sequence to
100
decimal places, with 150 iterations and see that it still doesn't get to 10.
(Ch. 8)
- Most students are asked to solve quadratic equations
first by guessing and eventually finding the secrets of the
sum and product of the roots. Then when they work on the
quadratic equation x^2 - x - 1 = 0 by trying numbers with
pencil and paper and then a calculator. They get sequences
of numbers too big and too small, leading to a 5 decimal
approximation of the Golden Mean. Then they can use Derive
to find a quick solution and see the exact (although
irrational) answer, as well as see approximations with as
many places as they like. These are two of about ten ways
Don gets kids to solve quadratic equations. (Ch.8)
-
Students, after doing it with a diagram on graph
paper and finding patterns, will write a program on a calculator or
computer to get the sum of the infinite series 2/5 + (2/5)^2 +
...(Ch.1)
- A 7th-grader while trying things on a calculator, took
repeated square roots of a number. He found that no matter what number
he started with he would always get to 1. (Ch.10)
- While Ian was
playing with powers of powers on his calculator during Physics class, he
came across a function which goes to e as x goes to infinity. (Ch.11)
-
Don wrote a program on a
programmable calculator that finds the area under a curve by plotting
points under the curve, counting these points, then finding
the ratio of the number plotted to the number filling a 1x1 square.
(Ch.13)
- Don helped Khaki use
Derive to plot a graph, then zoom in on a point until
the curve looks like a straight line. She found the slope of this line.
Keeping track of this slope for a few points, she figured out a rule to
find the slope at a given x-coordinate, arriving at the derivative! (Ch.14)
To order Don's
materials
Mathman home
Mathematica
Inverse Symbolic
Calculator
Computer Algebra
Information Network (from the Netherlands)