What is Slope?

Date: 9/2/96 at 17:33:48
From: Barbara Cummings
Subject: Slope

Dear Dr. Math,

My son is going into 10th grade. He is confused about slopes.  If 
he is given a slope = n equation, he doesn't understand how to apply 
that slope number to another graph problem.

Obviously Mom is at a loss also. Can you help?

Thanks, Barb and Ryan

Date: 9/3/96 at 18:46:4
From: Doctor Tom
Subject: Re: Slope

The slope is just a number that tells how steeply a line goes up or 
down.  If the line is perfectly level (it doesn't go up or down at 
all), the slope is 0 (zero).  If, as you go to the right, the line 
gets higher, we say it is sloping up, or has a positive slope.  If it 
goes down as we go to the right, we call that a negative slope.

So that's how to think of slopes.  Here's how to measure (or draw) 
them.  Let's first look at specific examples.

Suppose the slope is 1.  That means that if you go one inch to the 
right, the line goes up by 1 inch.  There's nothing special about 
"inch" - if you go 1 foot to the right, the line goes up 1 foot.  If 
you go 1 centimeter to the right, the line goes up 1 centimeter, and 
so on.  If you draw this line, you'll find it goes up at a 45 degree 
angle.

If the slope is 3, it goes up more steeply - in fact, if you go one 
unit to the right, it goes up 3 units, where "unit" can be inch, foot, 
centimeter, or whatever.

If the slope is 1/2, it means that if you go one unit to the right, 
the line goes up 1/2 unit.  If the slope is 1000, the line is very 
steep - going one unit to the right, the line rises by 1000 units, 
and so on.

If the line goes down to the right, it's the same thing, except with 
negative slopes.  A slope of -1 means that going 1 unit to the right, 
the line drops 1 unit.  A slope of -1/3 means for every inch the line 
goes to the right, it drops by 1/3 of an inch, et cetera.

There's one nasty problem, and that concerns lines that go straight up 
and down - you can't assign a sensible slope to them, because they 
never go to the right.

Often, you'll get problems like this:  What's the slope of a line that 
goes up 3 units for every 2 units it moves to the right?  Well, that 
means it goes up 1.5 units for each single unit to the right, so the 
slope is 1.5.

The "formula" for the slope of a line is simply to divide the motion 
up or down (positive or negative) by the motion to the right.  In all 
the original examples here, I used a motion of one unit to the right, 
so I'm dividing by 1 in my "formula."

Think about this.  Draw examples.  The idea of slope is a very key 
concept in algebra and beyond, and you'll have a great deal of trouble 
later if you don't understand it.

Basically, bigger slopes mean steeper lines.  Lines with positive 
slopes go up as you go right, and lines with negative slopes go down 
as the line goes to the right.  An exact, mathematical definition of 
slope is the amount of upward or downward movement the line makes as 
you move to the right.  In problems, of course, sometimes you don't 
have exactly this information and you have to work it out (as, for 
example, in my problem of a line going up 3 units as it moves to the 
right by 2 units).

Good luck.

-Doctor Tom,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/