Re: [MATHEDCC]: Why Simplify Fractions
Kathy Burgis (kburgis@ALPHA.LANSING.CC.MI.US)
Tue, 07 Oct 1997 17:30:24 -0400
msnyder@tiac.net wrote:
>
> I originally wrote this on Monday, but it has not appeared in my mailbox,
> so I guess it didn't get distributed, so I am resending it to the list. My
> apologies if it did in fact get sent out to the list.
>
> Kathy Burgis wrote:
>
> >This will probably cause another flood of mail, but I guess I have to
> >admit that I do not think simplifying fractions is a worthwhile activity
> >either. In fact, I have problems with the concept of "simplified form"
> >in general.
> >
> >Cheers,
> >Kathy Burgis
>
> Here are some typical "Simplify" problems. I would be interested in why
> you object to them (if you do):
>
> 1. Simplify: 3x + 2(x-1) -4(-x+(2(1-x)))
>
> 2. Simplify: (x^2 + 2x + 2)(x^2 - 2x + 2)/(4 + x^4)
>
> 3. Simplify: (x^(4/5))^(5/16)
>
> 4. Simplify: (ln 16)/(ln 4)
>
> As for "simplifying" fractions (i.e., reducing them to lowest terms), I
> don't see what the BFD is all about: you find the prime factorization of
> the top and the bottom, then cancel common factors. This is hardly rocket
> science.
>
> And there *is* a reason to reduce fractions to lowest terms: that makes
> the fraction more understandable. If you survey 1309 people and find that
> 561 believe a certain thing, then that belief is held by 561 out of 1309,
> so the fraction of people that believe it is 561/1309. Reducing that to
> 3/7 means that 3 out of 7 people believe that thing. The latter is much
> easier to use and to comprehend. Of course, you could use a calculator to
> find the decimal equivalent 0.428571... Depending on the situation, that
> might be more useful than 3/7. But I think students should know both how
> to reduce a fraction to lowest terms, and know that a/b means a divided by
> b. Each has its own uses. If I had 28 people, then I would expect that 12
> of them held that belief (using multiplication of fractions--the *reduced*
> one times 28, or equality of ratios, but *not* a calculator), whereas if I
> had 362 people, then I would expect that about 155 of them would hold that
> belief (multiplying 0.428571... times 362, but *not* multiplying fractions
> or using ratios).
>
> And, of course, reducing fractions to lowest terms always gives you an
> exact result, whereas a calculator only gives you something approximate.
> The approximate result might be good enough, or even more desirable than
> the reduced fraction, of course, but that depends on the situation.
>
> mark snyder
Mark,
I certainly do not object to any of your simplify examples. I do think
it is important that students learn to manipulate symbols. What I do
object to is conveying the idea that there is only one correct form. So
I would prefer, for example, to give students groups of expressions, and
ask "Which of these of expressions are equivalent?"
It seems to me that, in your discussion about reducing fractions, you
are making a plea for students' learning of proportional reasoning. If
so, I agree completely. What I do not agree with is, again, using our
classrooms to promote the idea that there is only one "correct" way to
express a ratio or fraction.
Cheers,
Kathy Burgis
Lansing Community College
Lansing, Michigan
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