> In a message dated 97-12-09 09:25:05 EST, you write:
>
> << WITH THE TI-83 OR 86 STUDENTS CAN DIVIDE A POLYNOMIAL
> BY A LINEAR BINOMIAL USING THE LIST FEATURE AND THE
> SYNTHETIC DIVISION ALGORITHM.
>
> ED >>
>
> Could you explain further, with the TI-83, how to do synthetic division?
>
> My students would be greatful.
Actually, _general_ polynomial division can be easily implemented on a TI-8x
calculator (not needed on the TI-92!), working with lists that represent the
coefficients of the dividend and divisor (numeric coefficients only, of
course). This is a nice little programming exercise, that's all. I suspect
hundreds upon hundreds of individuals have written such a program for their own
edification or use.
How such a program could be used effectively in an instructional setting is
another, and possibly more interesting, question. The old "Give a man a
fish..." line is very apt here -- I would myself see very little if any
long-term benefit to students simply providing them with a tool that would more
quickly knock off the sorts of problems that synthetic division, working with
pencil and paper, is well matched to. If any instructional "juice" is to be
gotten from working with such programs, I would see doing _more_ with them than
you could have done without them.
Getting used to the process of abstracting a polynomial as a (possibly sparse)
list can be in itself an enlightening step -- for example savoring the fact
that x^4 - 4*x really _means_ the five-element list {1,0,0,-4,0} is really
part of what we would like students to experience (or so it seems to me). I
like to do some direct work with lists in this way in front of the students
(and have them do some on homework too). The above list could be stored on the
TI-83 as P1, and it wouldn't be hard to calculate 2*P1, for starters (you would
need to insert the small-L list symbol the 83 uses to distinguish list names,
of course). To add two polynomials of different degrees you need to_augment_
the smaller list by prefixing zeros, but this is not too tough to do
informally.
Probably it would be the exceptional class that would benefit, as a whole, from
struggling with the niceties of the programming (but it never pays to underrate
students). Some of the ideas needed are dimL to determine the length of a
list, and a more formal approach to augmenting lists with prefixed or suffixed
zeros.
If I use a program as a "black box", without examining its inner workings, in
an educational setting, I always like to _push_ the program a bit, explore its
practical limitations, encourage the students to be _critical consumers_ of the
software. With a division program at hand, why stick to polynomials with
integer coefficients? Decimals, fractions? Are there questions of _precision_
that come up when we admit approximate numbers as coefficients? Just how high
a degree polynomial is it feasible to work with? Can the program be used to
divide x^50 -1 by x^2 -1 ? If not, can we use the program anyway to help us
figure out what the answer must be? What are some of the additional important
mathematical applications and insights that can be tackled if we can perform a
polynomial division in a matter of seconds? Etc. etc.
Working electronically one can _store_ results and feed them into further
calculations. If programs have also been developed for polynomial
multiplication and (simple) factoring, the relationships between these
operations can be explored in a new context -- the key to a deeper level of
understanding. The bottom line is that with different (and better) tools, one
should be doing different (and more challenging) tasks.
RWW Taylor
National Technical Institute for the Deaf
Rochester Institute of Technology
Rochester NY 14623
>>>> The plural of mongoose begins with p. <<<<
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