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Re: Fractals
Posted:
May 7, 1995 6:23 AM
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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
'self-similarity'.
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 non-zero 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 self-similarity 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
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