Gradient between two points

Topic guide

What this worksheet practises

This worksheet provides practice on calculating the exact gradient (steepness) of a straight line connecting two coordinate points. This is identical to the "finding-the-gradient" topic, reinforcing the core coordinate geometry formula.

Key method

The gradient formula is: Gradient (m) = (Change in y) ÷ (Change in x).

• Identify the coordinates of your two points: Point A (x₁, y₁) and Point B (x₂, y₂).
• Calculate the difference between the y-coordinates. This gives you the vertical "rise".
• Calculate the difference between the x-coordinates. This gives you the horizontal "run".
• Divide the y-difference by the x-difference. Be extremely careful with negative numbers during the subtraction.

Worked example

Find the gradient of the line connecting A(3, −1) and B(5, 7).

Step 1: Calculate the change in y.

y of point B minus y of point A = 7 − (−1) = 7 + 1 = 8.

Step 2: Calculate the change in x. Because we started with point B for the y's, we must start with point B for the x's.

x of point B minus x of point A = 5 − 3 = 2.

Step 3: Calculate the gradient.

Gradient (m) = 8 ÷ 2 = 4.

Common mistakes to avoid

The two most common errors are: 1) Dividing the change in x by the change in y (putting the run on top of the rise), and 2) Mixing up the order of the points halfway through. If you calculate (y₂ − y₁), you absolutely must calculate (x₂ − x₁). If you switch to (x₁ − x₂), your final answer will have the wrong sign.

Things to remember

If your line goes downwards from left to right, your gradient must be a negative number. Always do a quick visual check (or sketch the points) to ensure the sign of your calculated gradient matches the physical reality of the line.