Why Does the Slope Formula Work?

Date: 10/29/2008 at 23:43:18
From: Paul =D
Subject: Why does slope work

To find the slope of a line, you do the formula y2-y1/x2-x1, when x1
is not equal to x2.  But why does this formula work?  I don't
understand the relationship of the subtraction and division with 
slope.  Is it simply an equation that creates different numbers
which signify different angles?

Date: 10/30/2008 at 10:37:32
From: Doctor Ian
Subject: Re: Why does slope work

Hi Paul,

In many cases, the best way to understand something is to re-invent it
for yourself. 

Suppose I have two lines, like

                 B                       C
              .                    .
           .                .
        .          .
     .     .
  A   

We would say that AB is "steeper" than AC, right?  But how could you
QUANTIFY that?  What kind of calculation could you do to attach a
number to the idea of "steepness"?  

That's what we're doing with slope.  We're trying to capture that
concept in a calculation. 

Now, in the case of AB and AC, they both have the same amount of
vertical travel ("rise").  But AB has less horizontal travel ("run").
If we divide rise by run, we get a larger number for AB, which is
nice, because it makes sense that a larger value should correspond to
a steeper line. 

Now, where does subtraction come in?  Well, suppose we think of each
line segment as the hypotenuse of a right triangle:

                 B                       C
              .  |                 .     |
           .     |          .            |
        .        | .                     |
     .     .     |                       |
  A______________|_______________________|   


If we know the coordinates of A, B, and C, can you see how we can use
subtraction to find the rise and run for each segment? 

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

Date: 10/30/2008 at 20:19:23
From: Paul =D
Subject: Thank you (Why does slope work)

Yay!  I get it now.  Rise and Run.  Thank you!