### 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:

1. With Don's help, a few students have written a program on a programmable calculator to change a fraction to its bimal. (Ch. 2)
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)
3. 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)
4. 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)
5. 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)
6. 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)
7. 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)
8. 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)