I think students in Elementary Algebra through Precalculus have a pretty =
good understanding of slope as the ratio of the rise to the run, but I =
think they do not have an understanding of slope as a rate of change. =
This becomes quite a problem in Calculus when students need to think =
comparitively about slopes using the "racetrack principle." For example, =
my students have trouble answering the following question. If f' (x) < 3 =
on [5,9], and f(5) =3D 16, what can we say about f(9) ? They, of course, =
have even more trouble if they are given graphs of f' and of f and are =
asked to show that f(9) < 28. It took me quite a while to figure out the =
reason my students had trouble with these questions. I'm sure that part =
of the reason they have trouble with the question is that they are still =
learning the symbolism. Also, when I do try to write the algebra that =
shows why f(9) < 28, they see it as some sort of algebraic magic. =
However, I think their real difficulty is that they do not understand =
slope as rate of change. They do not understand the question above as a =
question about a rate of change. If they did, then the above question =
becomes "If the graph of f goes up no more than three units for every one =
unit it goes to the right, how far up does it go from sixteen if it goes =
four units to the right?" =20
As I was preparing a recent lesson for my Basic Mathematics class, I =
realized that my next lesson was the beginning of Calculus for them! The =
lesson was about ratios and rates. The textbook we use defines a ratio as =
a comparison of two quantities with the same units. I told my students =
that we express ratios as fractions. The book defines a rate as a =
comparison of two quantities with different units. I told my students =
that we think of the fraction as a division, and we do the division so =
that the denominator is one. We express rates as decimals. =20
When I first introduce the concept of slope in Elementary Algebra, I use a =
graphics calculator and graph various equations. I start with y =3D x, =
and I ask my students what they see. (I use a square graph.) They tell =
me that they see a straight line going through the origin that rises to =
the right at a forty-five degree angle. Of course, that takes some teeth =
pulling to get them to say that, and some of them aren't sure what a =
forty-five degree angle is.
Then I graph y =3D x and y =3D2x, sequentially on the same graph, and I =
ask my students what they see. Eventually they tell me that the graph of =
y=3D2x is the same as y=3Dx except that it is steeper. =20
We graph several more examples comparing them one to another. I always =
make sure that they compare the new graph to y=3Dx, first. So, y=3D3x is =
steeper that y=3Dx, but y =3D (1/3) x is flatter than y=3Dx. (Sometimes I =
type y=3D(1/3)x and sometimes I type 1x/3. I haven't yet decided which I =
like better for the purposes of my demonstration.) Then we compare y=3D3x =
to y=3D2x. Eventually we compare y=3D(4/3)x to y=3Dx and then to y=3D2x. =
I talk about the fact that (4/3)x is the same as y=3D(1 1/3)x. Therefore,=
the graph of y=3D(4/3)x should be steeper than y=3Dx and flatter than =
y=3D2x. I also play around with negative slopes. The graph of y=3D(-4/3)x=
has the same steepness as y=3D(4/3)x, but it falls to the right instead =
of rises.
After we have gone through this whole discussion, then I define slope as a =
measure of the steepness (and direction) of the line. We go back to the =
graph of y=3D(4/3)x, and I use the free-floating cursor to show that I can =
build a stairway for the graph where the rise is four and the run is =
three. =20
After this list discussion, I will make it a point to also build a 1 1/3 =
to 1 stairway, also!
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