Why Is Slope Rise Over Run and Not Run Over Rise?

Date: 03/05/2005 at 21:27:04
From: kristin
Subject: Why rise over run???

Someone asked our math teacher yesterday, why is it rise over run and
not run over rise... I was wondering since our teacher didn't know the
answer, if you might.  If you could answer this that would be so great!!!

I think that maybe it could be both ways bcause like for instance:  
(I'm going to show my work here)

(4,2)(3,-7)
2-(-7) / 4-3 = 9/1
y=mx+b
1/9 + 2
y=9/1x + 2

and when you graph it you put the dot at the 2 place and go up 9 
lines and go out one line and put a dot (then connect those dots) 
BUT you can also put a dot at the 2 go out one, and up 9 and put a 
dot there (then connect) and it's the same thing.  Do you think rise 
over run can also be run over rise???

Date: 03/05/2005 at 23:04:40
From: Doctor Peterson
Subject: Re: Why rise over run???

Hi, Kristin.

There are at least two questions here: why do we define the word 
"slope" to mean the ratio of "rise" to "run", and why is that the 
right number to use in the slope-intercept form of a line?

The answer to the first question is that any number we assign to a 
"slope" ought to be bigger when the line is sloped more steeply. We
want the "slope" to tell us how much the line is sloped. A steeper 
line goes up more in the same distance:

        o
       /
      /
     /            o
    /           /
   /          /
  o         o
  steep    less steep

If we used the "run" over the "rise", then the less steep line would
have a greater slope, which wouldn't make sense:

        o
       /|
      / |
     /  |          o
    /   |        / |
   /    |      /   |
  o-----+    o-----+
    6/6        3/6      <-- actual slope: steeper has greater slope
    6/6        6/3      <-- run/rise: steeper has smaller slope!

How about the question of what goes in the "m" spot in the equation?
Let's look at the intercept and one other point on the line:

        |       /
        |     o (x,y)
        |   / |
        | /   |
  (0,b) o-----+
        |
        |
        +---------------

What is the slope of that line?

      y-b   y-b
  m = --- = ---
      x-0    x

If we multiply both sides of the equation by x, we get

  mx = y - b

and adding b to both sides gives

  mx + b = y

So if we define slope as rise/run, then this is the equation that any
point on the line has to fit. If you defined slope differently, you
would get a different equation.

So here's how it works: we define slope in a way that makes sense
based on what the word "slope" means; then we find that we can use
that slope value in an equation that describes any point on the line.
One could use a different formula for something like "slope", and get
a different equation; but this one gives us a reasonable definition
AND a nice little equation to use it in.

If you have any further questions, feel free to write back.

- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/