
Re: Fractals
Posted:
May 7, 1995 6:23 AM


There appears to be two major points about wanting to introduce fractals into the early grades. Namely: 1) To get the students to become more familiar with 'infinity'. 2) To get the students to become more familiar with 'selfsimilarity'.
Let me suggest that both of these can be accomplished already in the 'traditional setting' without necessarily getting involved with 'fractals'.
a) Arithmetics. Consider the example of expressing the reciprocal of *one digit* numbers as decimals. (For the history buffs, one can discuss Egyptian unit fractions.)
1/9 = 0.111111...., 2/9 = 0.222222.... etc. The pattern is clear until one reaches:
1 = 9/9 = 0.999999....
To resolve this apparent puzzle, one may ask the student:
Do you know how to subtract one decimal from another? If so, subtract 0.999999.... from 1.
It clearly provides a heuristic feeling that somehow the *remaining digit* has receded to infinity. At the same time, the student may get a feeling of understanding something that is *infinitely small*.
In particular, the *smallness* is connected to the number of 0's needed before one can write down the first nonzero digit.
In this context, 1/7 = 0.142857142857...... is especially interesting. One can easily extend the discussion to more complicated fractions.
b) Geometry. Start with regular pentagon. Draw the five diagonals and one sees the appearance of a smaller regular pentagon on the inside. This clearly illustrates selfsimilarity as well as the infinite iteration process. The more advanced students may be asked to compute the side ratio for the two regular pentagon. As this point, one meets the Golden ratio.
In both cases, one does not have to face with the problem of confusing the issues because the more advanced techniques are not available.
Han Sah, sah@math.sunysb.edu

