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### What is Slope?

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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
```

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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."

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/
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Associated Topics:
High School Basic Algebra
High School Definitions
Middle School Algebra
Middle School Definitions

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