MISSION · Algebra
Slope as Rise over Run
What the number in front of x is actually measuring
LEVEL 1 · EXPLORE
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LEVEL 2 · LEARN
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The idea in one sentence
Slope is one number that answers one question: when you move one step to the right, how far up or down do you go?
Rise over run
Pick any two points on a line. Count how far you move up to get from the first to the second — that’s the rise. Count how far you move right — that’s the run.
\[\text{slope} = \frac{\text{rise}}{\text{run}} = \frac{y_2 - y_1}{x_2 - x_1}\]The remarkable part is that the line doesn’t care which two points you picked. Take two points close together or two points far apart, and the fraction comes out the same. That constant ratio is what makes it a straight line.
What the sign and size tell you
| Slope | The line |
|---|---|
| Positive | Rises as you read left to right |
| Negative | Falls as you read left to right |
| Zero | Flat — no rise at all |
| Undefined | Vertical — the run is zero, and you can’t divide by zero |
A slope of 3 is steeper than a slope of \(\frac{1}{2}\), because three units up per step beats half a unit up per step. Compare the size to judge steepness, the sign to judge direction.
Slope is a rate
This is the part worth carrying past the graph. Slope is just a rate with the units left on:
- A line of cost against hours has a slope in dollars per hour — that’s the wage.
- A line of distance against time has a slope in miles per hour — that’s the speed.
- A line of water against minutes has a slope in gallons per minute — that’s the flow.
Every time you hear “per,” something has a slope.
Common mix-ups
- Flipping it to run over rise. The word order is the memory aid: rise first, because \(y\) comes second in \((x, y)\) and lands on top of the fraction.
- Subtracting the points in different orders. If you do \(y_2 - y_1\) on top, you must do \(x_2 - x_1\) on the bottom. Mixing the order flips the sign.
- Reading a steep-looking graph as a big slope when the axes use different scales. Check the numbers on the axes before trusting the picture.
- Calling a vertical line’s slope zero. Zero slope is flat. Vertical is undefined, which is a different thing entirely.
Check yourself
A line passes through \((2, 3)\) and \((6, 11)\). What is its slope? What happens to that answer if you subtract the points the other way round?
\(\frac{11 - 3}{6 - 2} = \frac{8}{4} = 2\). The other way round gives \(\frac{3 - 11}{2 - 6} = \frac{-8}{-4} = 2\) — the same, because both the top and the bottom changed sign.
LEVEL 3 · REFLECT
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