In this day and age of calculators, does anyone talk about slopes as
0.08333/1 (which is approximately 1/12 the maximum slope approved by ADA for
a short ramp), or 4781/1 which might represent an increase in sales per
month for some company over a particular time period? Although students
should no doubt start with slopes of 2/3, 1/2, 4, -3/2, I do not understand
why at any level we limit ourselves to "baby numbers." Introducing the
notion of the slope as a tangent certainly lends itself to a discussion of
slopes other than nice ratios. One of the questions I ask my students who
have had some algebra (even college algebra) is "Is a 1/2 slope very steep?"
Then I ask them if they could walk on a 1/2 slope. The first question, of
course is a judgement call and good for discussion. It is amazing how many
students who have done well in algebra classes think that in context of a
walking slope that 1/2 is not very steep. They are certainly surprised that
1/12 or (1/20 depending on the total rise) is the maximum legal slope for an
accessibility ramp. By the way, for a wheelchair parking space, the maximum
legal slope if 1/50.
Here are a couple of explorations using a scientific or graphics calculator.
1. Have students draw several angles, measure the angles with a protractor,
measure rise and run at several places and express each slope as a ratio to
1. Compare their different values of slopes with the tangent of the angle.
2. Have students use their calculator to find the tangent of an angle
(between 0 and 180 degrees). Based on the answer the students give, have
them draw the angle. This is how we can turn a calculator and ruler into a
protractor.
Martha
-----Original Message-----
From: Kirby Urner <urner@alumni.Princeton.EDU>
To: mathedcc@archives.math.utk.edu <mathedcc@archives.math.utk.edu>
Date: Wednesday, September 30, 1998 7:13 PM
Subject: Re: [MATHEDCC] slope: ratio or rate?
On Wed, 30 Sep 1998 13:54:34 -0500, Beth Hentges wrote:
<<SNIP>>
>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=(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.
>
>After this list discussion, I will make it a point to also build a 1 1/3 to
1
>stairway, also!
>
I agree with your stair-step approach, getting rise:run to mean
something visual (not algebraic). The right triangle approach
with TAN() = opp/adj segues nicely w/ 'tangent' as df(x)/dx or
f'(x).
As a newsgroup respondant pointed out, 'rate of change' is also
output:input -- what you get out per what you put in (a jeep
might give you a lot of 'rise' in a short 'run', while some
other vehicle might have a harder time getting you closer to
your goal).
Not sure how much weight to put on this 'ratio' versus 'rate'
distinction. I'm inclined to not worry if change_a/change_b
is a ratio with units, e.g. kgs/time or charge/cm^2. Maybe
it's a static condition (180 blades of grass per triangular
plot), in which case 'rate' seems a bit contrived -- ratio
OK, simply a translation of the word 'per' (x of something
'per' y of something else).
Kids sitting in class trying to bend their minds around this
stuff might be exposed to the concept of 'learning curve' -- it
it can be 'steep', which means you're trying to master
something difficult in a relatively short period of time. More
than a measure of 'work' (e.g. 'homework'), this is a measure
of 'power' -- because getting the homework done in an hour is
a 'steeper grade' than getting the same work done in a week.
Indeed, our talk of 'grades' and 'degrees' harks back to the
metaphor of a peak or summit, the top of some Tower of Babel,
which 'grad students' seek to ascend against the gradient, and
upon which 'degrees' are conferred as equi-altitude coutour
lines are crossed (i.e. as students get closer to the summit).
Students who just circle at a fixed altitude aren't doing any
work, vis-a-vis the gradient (grad dot F = 0), while those
who tackle the very steepest slopes (takes 'power' of
concentration) may risk burn out in the process.
Kirby
PS: this metaphoric use of everyday lingo as 'glue' for more
refined math-science concepts is an humanities approach to
mathematics -- and I push it further at my Oregon Curriculum
Network website.
PPS: I encourage students to 'eyeball' lots of web pages re
some math topic, looking for common threads, even if the content
is clearly 'above their heads' -- exposure to the 'look and
feel', without the grade pressure of thinking you'll
immediately be tested, can become almost a form of relaxation,
a vacation i.e. is what we mean by 'browsing'. Like, for
'gradient', here are some interesting tourist stops:
http://www.ma.iup.edu/projects/CalcDEMma/veccalc/veccalc9.html
http://www.ma.iup.edu/projects/CalcDEMma/veccalc/veccalc10.html
http://forum.swarthmore.edu/dr.math/problems/fresco6.3.97.html
http://www.math.iupui.edu/m261vis/tempgrad.html
http://www.geom.umn.edu/~fjw/Labs/Lab4/Summary.html
http://www.astro.virginia.edu/~eww6n/math/Gradient.html
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