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Core + Extended Coordinate Geometry

Gradient of a Line for IGCSE Maths

Finding and interpreting the gradient between two points. This subtopic is part of Coordinate Geometry in the Cambridge IGCSE Mathematics 0580 syllabus (both Core and Extended tiers). Understanding gr

What You Need to Know

Finding and interpreting the gradient between two points. This subtopic is part of Coordinate Geometry in the Cambridge IGCSE Mathematics 0580 syllabus (both Core and Extended tiers). Understanding gradient of a line is essential for achieving a strong grade in your IGCSE Maths exam.

Understanding Gradient of a Line

The gradient of a straight line measures its steepness: gradient m = (y₂ − y₁)/(x₂ − x₁). A positive gradient slopes upward left-to-right; negative slopes downward; zero is horizontal; undefined is vertical. In IGCSE 0580, gradient questions appear on both Core and Extended papers and include calculating gradient from two coordinates, interpreting gradient in context (e.g. speed on a distance-time graph, rate of change), and recognising that parallel lines share the same gradient.

Step-by-Step Method

  1. 1

    Identify two points

    Read the coordinates of two clear points on the line from the graph or from the question. Label them (x₁, y₁) and (x₂, y₂).

  2. 2

    Apply the gradient formula

    m = (y₂ − y₁) / (x₂ − x₁). Subtract y-values over x-values — in that order. Never swap them.

  3. 3

    Simplify the fraction

    Cancel common factors to express m as a single fraction or integer. For example (6−2)/(5−1) = 4/4 = 1.

  4. 4

    Check the sign makes sense

    Look at the line: does it go up (positive) or down (negative) from left to right? Confirm your calculated sign matches.

  5. 5

    Interpret if needed

    In context: gradient of a distance-time graph = speed; gradient of a cost graph = cost per unit. Always state units when interpreting.

Worked Example

Question

Find the gradient of the line passing through the points A(−2, 5) and B(4, −1).

Solution

Step 1: Label the points. (x₁, y₁) = (−2, 5) and (x₂, y₂) = (4, −1). Step 2: Apply the formula. m = (y₂ − y₁) / (x₂ − x₁) m = (−1 − 5) / (4 − (−2)) m = −6 / 6 m = −1 Step 3: The negative gradient confirms the line slopes downward from left to right. Answer: Gradient = −1

Exam Tips for Gradient of a Line

  • Always subtract y-values in the numerator and x-values in the denominator — never the other way round.
  • Choose two points that lie exactly on grid intersections when reading from a graph — this minimises reading errors.
  • Gradient of −1/2 means: for every 2 units right, go 1 unit down. Use this to sketch lines quickly.
  • If the gradient comes out as a fraction, leave it as a fraction (e.g. 3/4) — don't round it to a decimal unless asked.

Practice Questions

Q1: Find the gradient of the line joining P(3, 7) and Q(8, 22).

Show hint

m = (22 − 7)/(8 − 3) = 15/5.

Q2: A line has gradient −3/4. Starting at (0, 5), find the coordinates of another point on the line.

Show hint

Move 4 right, 3 down: (4, 2). Or move 8 right, 6 down: (8, −1).

Q3: The graph of distance (km) against time (hours) is a straight line through (0, 0) and (3, 210). Find the speed.

Show hint

Speed = gradient = 210/3. Remember to state the units: km/h.

Frequently Asked Questions

What is gradient of a line in IGCSE Maths?

Finding and interpreting the gradient between two points.

Is gradient of a line in the Core or Extended syllabus?

Gradient of a Line is part of the Core and Extended syllabus for IGCSE Mathematics 0580.

How do I revise gradient of a line effectively?

Start with the revision notes to understand key concepts, then work through the worked examples step by step. Finally, practise past paper questions under timed conditions. Teacher Rig recommends spending focused revision sessions on gradient of a line rather than trying to cover everything at once.

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