Slope

The slope of a line drawn on graph paper, or a road in real life, is a number that tells you how steeply the thing is slanted.

Slope in math

Here is a line drawn on graph paper.

To get the slope of the line from one point on the line to another, count squares as you go, then divide:

Questions

  1. What number did you get for the slope?
  2. Do you get a different number if you choose different starting and ending points?
  3. Do you get a different number if you don't go a whole number of squares?
  4. If this graph were the side view of a hill, do you think you could walk up it easily? Ride a bicycle up it easily?
Actually, there are a few more rules for measuring the slope of a line. A horizontal change is positive if you move from left to right, and negative if you move from right to left. A vertical change is positive if you move up, and negative if you move down.

Try computing the slope on the graph above by moving from right e on the graph above by moving from right to left. Do you get a different number?

Usually slope is measured by counting left to right. Depending on the line, you may go up or down as you go from left to right.

Get some graph paper and draw some lines with
  1. Slope 5
  2. Slope 2
  3. Slope 1
  4. Slope 1/2
  5. Slope 0
  6. Slope -1
  7. Slope -2
  8. Slope -5
Here is some online graph paper; you can draw lines.

If you don't see a piece of graph paper with some buttons at the top, your browser is not java-enabled. You'll just have to use some real graph paper and a pencil.

Another way to measure the slant of a line is to measure the angle it makes with a horizontal line.

AngleSlope
0o0
15o0.27
26.6o0.5
45o1
60o1.73
78.7o5
90oinfinity
This relationship between angles and slopes is called the tangent function. On a scientific calculatorhe tangent function. On a scientific calculator, if you put in the angle, then press the "tan" key, you should get the slope of the line. (This is a part of the mathematical subject trigonometry.)

Slope in real life

Slope in the real world is a little more complicated, because we live in 3 dimensions, not on a piece of graph paper. But the idea is the same if you think of a side view of the hill you're talking about as a line on graph paper.

First of all, the slope of a hill depends on the route you take. See the page on steepness of a hill if you haven't been there already.

If you have ever been hiking in the mountains or in the Grand Canyon, you probably noticed that the trails go back and forth across the hill instead of straight up. Why? (Sharp turns that make the trail go back and forth across the hill are called switchbacks.)

If you have been downhill skiing, you know that beginners are told to turn across the hill to slow down and stop. What happens if you point your skis straight down the hill?

Once you have chosen a route up the hill, you can measure the slope.

On highways with steep hills, sometimes there are signs warning truckers, sayiigns warning truckers, saying something like "9% grade; watch downhill speed." The number 9% is really a fraction, 9/100, so it is also the answer to the division 9 divided by 100. It is a slope where the vertical change is 9, and the horizontal change is 100.

Another way of giving a slope of 9% is to say "a gradient of 9 in 100", meaning that in every 100 horizontal units, the height increases by 9 units.

[This section Under construction]

Examples of slopes from real life