GCSE MATHS • HIGHER TIER

Gradient Between Two Points Made Simple

Learn how to calculate the gradient of a straight line directly from two coordinate pairs using a clear, reliable GCSE method.

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Clear method

Exam practice

Worked proofs

Gradient Between Two Points in Four Moves

1

Identify

Write down the two coordinate pairs clearly.

2

Find the Rise

Calculate the change in y using the same coordinate order.

3

Find the Run

Calculate the change in x using the same coordinate order.

4

Divide and Check

Divide change in y by change in x, simplify and check the sign.

What You’ll Learn

Understand what gradient measures.

Use two coordinate points to calculate gradient.

Apply the gradient formula accurately.

Calculate change in y and change in x.

Recognise positive and negative gradients.

Understand horizontal and vertical special cases.

Use gradient skills in wider coordinate geometry.

Key Rules

Write the Points Clearly

(x₁, y₁) and (x₂, y₂)

Find the Change in y

y₂ − y₁

Find the Change in x

x₂ − x₁

Divide Rise by Run

Gradient = change in y ÷ change in x

Keep the Same Point Order

Use the same order for both subtractions

Simplify the Gradient

Reduce the final answer where possible

Worked Proof: Gradient Between Two Points

Step-by-Step Method

  • 1. Identify the two coordinate pairs.
  • 2. Find the change in y.
  • 3. Find the change in x.
  • 4. Divide change in y by change in x.
  • 5. Simplify the answer.
  • 6. Check whether the gradient is positive or negative.

More Worked Examples

1

Find the gradient between (1, 4) and (5, 12).

Change in y = 8

Change in x = 4
Gradient = 2
Answer: 2

2

Find the gradient between (3, 8) and (7, 4).

Change in y = 4 − 8 = −4

Change in x = 4
Gradient = −1
Answer: −1

3

Find the gradient between (0, 5) and (4, 13).

Change in y = 8

Change in x = 4
Gradient = 2
Answer: 2

4

Find the gradient between (−2, 1) and (4, 7).

Change in y = 6

Change in x = 6
Gradient = 1
Answer: 1
  • Common GCSE Mistakes
  • Subtracting the coordinates in different orders.
  • Mixing up x-values and y-values.
  • Forgetting negative signs.
  • Dividing change in x by change in y.
  • Not simplifying the final gradient.
  • Exam Tips
  • Write the two coordinate pairs clearly before calculating.
  • Use the same point order for both subtractions.
  • Calculate change in y before change in x.
  • Remember: gradient = change in y ÷ change in x.
  • Keep negative signs throughout the calculation.
  • Check whether the final sign matches the direction of the line.

Skills Used in Gradient Between Two Points

Coordinates

Gradient

Change in y

Change in x

Rates of Change

Coordinate Geometry

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Frequently Asked Questions

Gradient measures the steepness and direction of a straight line.

Subtract the y-values and divide by the corresponding difference between the x-values.

Using the same order keeps the signs of the vertical and horizontal changes consistent.

It means the line falls from left to right.

A horizontal line has gradient 0.

Its change in x is zero, and division by zero is impossible.