I express the second as "he change in y corresponding to a unit change in x.=
"
I begin by at least mentioning this possibility as one of many ways of
looking at the slope early on - in Elementary Algebra. I don't push it, but
mention that the fraction 3/4 and the complex fraction (3/4)/1 are
equivalent (and more generally, m =3D m/1), and remind them that we have
already seen that the usefulness of the slope is that it is a fixed ratio
associated with the given line which is independent of the two points used
to make the calculation - and that fact is tied to the facts about similar
triangles.
This arises when they know that the slope of some line is 3/4 and they want
to think about its graph - again the usual interpretation is that this is a
line that rises 3 units for each 4 that it moves to the right - and may be
the easier one to use in most instances, but I want them to see that the
two fractions above are equivalent and will produce the same graph.
I begin trying to get them to see this in El Alg and mention it again in
higher courses when the topic arises. Two of the benefits:
a. direct comparisons between lines - both numerically and graphically.
I ask them to imagine different lines running through a given point and
then if we move 1 unit to the right, the slope is the actual distance (in
those units) that the line will rise or fall.
b. in Business Calculus, when speaking of the marginal ____ (revenue, cost,
profit), it seems to me that this fact is the best way of explaining why
f'(a) represents the approximate change in ____ which corresponds to a unit
change in the independent variable (from a to a+1).
Bob Leibman
Austin Comm Coll
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