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Subject:   RE: Formulas
Author: jazy
Date: Apr 7 2005
Thank you very much David.
On Apr  6 2005, Mathman wrote:
> On Apr  6 2005, jazy wrote:
> David, I am one of those students who
> knows to invert and multiply
> when when working with a problem
> that deals with the division of a
> fraction.  Could you please
> explain the principle on which this is
> based on.

Ingrid, I am
> referring to equivalence of fractions, related also to equality of
> ratio and all other such relationships.  That is:

ma/mb <==> a/b
> I also refer to the equivalence x/x=1 and x/1 = x.

So, when you
> have as a specific example (2/3)/(5/7), and if applying this
> principle, you would multiply numerator and denominator by the same
> quantity.  That can be any quantity, but we choose the inverse of
> the lower fraction, (7/5).  This is done to force the denominator to
> be a 1.

The result would be [(2/3)(7/5)]/[(5/7)(7/5)].  The
> second equivalence makes this result in [2/3)(7/5)]/[1], and again
> simply (2/3)(7/5).

We avoid all of this writing, although the
> process of "invert and multiply"  is based upon it, and write simply
> the end result (2/3)(7/5) from the initial question, and carry on
> from there.

This is my point with transforming formulas.  The
> fact is that we "do the same to both sides", but having done so many
> times we see that we don't need to, and can simply transpose.  I
> believe that many students get frustrated by having so much baggage
> to carry around wit hall of that writing, and that it makes them
> more prone to error and disappointment.  A few examples would do to
> set the principles, then lots of graded [very easy to difficult]
> questions for practice applying the learned principles.  That is
> just my opinion and experience, and I do not challenge in any way
> that of others who prefer to keep on writing additional terms on
> both sides at every step.  However, I would suggest that it is not
> done in higher education where it would be far too much of a burden.
> But techniques learned with simpler problems would still be
> applicable.  To me, it is in a sense learning to "read" mathematics.
> So, after some preliminary work establishing the rules, I would
> concentrate on learning to resolve formulas and equations by
> transposing terms.  It is in fact what I did, and it worked well
> over the years.


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