A fraction is, fundamentally, an equivalence class of ratios, though teaching
this idea using this language is not appropriate in lower-level courses. But it
helps to understand what is really going on when you add fractions, for
example, if you keep this in mind. This is also what the idea of a rational
number really amounts to, if you confine the terms of the ratio to integers.
A _rate_, on the other hand, is a single number. For example, _speed_ is a rate
-- we always think of it as a single number. Any ratio whose second term is
not zero has an associated rate, which can be obtained by division. Any two
equivalent ratios correspond to the same rate. To any given rate correspond
any number of (equivalent) ratios.
Any line in the plane has a slope ratio, basically expressing "rise over run",
or rise _compared to_ run. For any line that is non-vertical, there is an
associated rate, the _slope_. Given a slope, we can develop a convenient
expression for a slope ratio, which allows for finding a new point on the line,
etc. This idea then scales upward as needed when talking about secants and
tangents and...
You won't find the above definitions in any mathematical dictionary! We looked
around for a long time before we decided that we were not going to find an
adequate definition of ratio, and would have to conceptualize it ourselves.
If anyone is aware of similar definitions appearing in a published source,
however, I would be very interested to learn about it.
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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