You are cycling up a hill and you spot a road sign: a red triangle with 1 in 5 printed inside. Your legs already know what it means — this is going to hurt. That little number is a gradient: it says the road climbs 1 metre upward for every 5 metres you travel along. A "1 in 20" sign would be a gentle slope you'd barely notice; "1 in 3" is a wall you'd struggle to walk up.

The gradient (or slope) of a straight line is exactly this idea, made precise: a single number that measures how steep the line is — how far it climbs for every step you take across it. It's the steepness of a wheelchair ramp, the pitch of a roof, the tilt of a ski run… and, as you'll see, the speed hidden inside a graph.

Rise over run

To pin down the steepness, take any two points on the line, (x_1, y_1) and (x_2, y_2). Walk from the first to the second. How far did you climb? That's the rise. How far did you move across? That's the run. The gradient m is the rise divided by the run:

A positive gradient means the line climbs as you read it from left to right; a negative gradient means it falls; and a gradient of zero means the line is perfectly flat. The bigger the number (ignoring its sign), the steeper the line.

The gradient in a nutshell

Reading the rise and the run

Here is the line through (1, 1) and (5, 4). Step the figure to see the run (how far you move across) and the rise (how far you climb). Going from the first point to the second, you move 4 across and 3 up, so

Worked example 1 — a clean positive gradient

Find the gradient of the line through (2, 1) and (6, 9).

Label the points: (x_1, y_1) = (2, 1) and (x_2, y_2) = (6, 9). Now subtract, keeping the y's on top and the x's on the bottom:

The gradient is 2: the line climbs 2 up for every 1 across. Positive, so it rises left to right — just as we expect, since the second point is higher and further right than the first.

Worked example 2 — a negative gradient

Find the gradient of the line through (1, 5) and (4, -1).

The gradient is -2. The minus sign is telling you something real: as you move to the right the line goes down, not up — it's a downhill slope, like a ski run or a slide. A negative gradient is not a mistake; it's a line that falls.

Check: it doesn't matter which point you call "first". Swap them and you get \frac{5 - (-1)}{1 - 4} = \frac{6}{-3} = -2 — the same answer, as long as the x's and y's are subtracted in the same order.

Worked example 3 — gradient is a rate of change

Here's where the gradient earns its keep. Draw a graph of distance (up the side) against time (across the bottom) for a runner. If she is at 20\text{ m} after 4\text{ s} and at 80\text{ m} after 16\text{ s}, the gradient of her distance–time line is

Look at the units: metres on top, seconds on the bottom — 5 metres per second. The gradient is her speed. That's the deep idea: whenever you plot one thing against another, the gradient tells you the rate at which one changes as the other does — metres per second, pounds per mile, litres per minute. Steepness and speed are the same measurement wearing different clothes.

Slide the gradient and watch

Pull the slider to change the gradient m of a line through the origin. Push it positive and the line climbs; push it negative and the line falls; slide it to 0 and the line lies flat. The further from zero, the steeper.