📐 Gradient

Grade 8 · Cambridge Lower Secondary Stage 8

What you'll learn

  • Calculate gradient as rise ÷ run from a graph or two points
  • Interpret positive, negative, zero and undefined gradients
  • Link gradient to real-life rates (speed, cost per item)
  • Use gradient to compare steepness of lines
  • Apply gradient in practical distance–time and cost contexts

📖 Learn

1. What is Gradient?

Gradient measures how steep a line is — how much it goes UP (or DOWN) for every 1 unit it goes RIGHT.

Gradient = Rise ÷ Run = (change in y) ÷ (change in x)
Rise: vertical change (up is positive, down is negative)
Run: horizontal change (always go right, so always positive)
Formula using coordinates: m = (y₂ − y₁) ÷ (x₂ − x₁)
💡 Pick any TWO points on the line — the gradient is the same wherever you measure it.

2. Positive, Negative, Zero and Undefined

Positive gradient (m > 0): line slopes upward left to right ↗
Negative gradient (m < 0): line slopes downward left to right ↘
Zero gradient (m = 0): horizontal line → y = constant (e.g. y = 4)
Undefined gradient: vertical line → x = constant (e.g. x = 3). Cannot divide by zero.
💡 Steeper lines have larger absolute values of m. A gradient of 5 is steeper than 2.

3. Reading Gradient from a Graph

To find gradient from a graph:

Step 1: Choose two clear lattice points (where the line crosses grid intersections)
Step 2: Count squares up (rise) and squares right (run)
Step 3: Gradient = rise ÷ run (negative if the line goes down)
Example: If you go 4 squares up and 2 squares right → gradient = 4 ÷ 2 = 2
⚠️ If the line goes DOWN as you move right, the rise is NEGATIVE → negative gradient.

4. Gradient as a Rate of Change

In real life, gradient = the rate at which one quantity changes compared to another.

Distance–time graph: gradient = speed (km/h or m/s)
Cost–items graph: gradient = cost per item (£/item)
Temperature–time: gradient = rate of heating or cooling (°C per minute)
Example: A line from (0,0) to (5,150) on a cost graph → gradient = 150÷5 = £30 per item
💡 The units of gradient are: y-units per x-unit (e.g. km per hour)

5. Comparing Gradients

The steeper the line, the greater the absolute gradient.

m = 4 is steeper than m = 2 (both going up)
m = −5 is steeper than m = −2 (both going down; |−5| > |−2|)
m = 3 and m = −3 have the same steepness but opposite directions
Comparing speeds: on a d–t graph, steeper line = faster speed
💡 Use absolute value |m| to compare steepness regardless of direction.

✏️ Worked Examples

Example 1 – Gradient from Two Points

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

Step 1: m = (y₂ − y₁) ÷ (x₂ − x₁) = (9 − 1) ÷ (6 − 2) = 8 ÷ 4 = 2
Meaning: For every 1 unit right, the line goes 2 units up.

Example 2 – Negative Gradient

Find the gradient of the line through P(1, 8) and Q(4, 2).

Step 1: m = (2 − 8) ÷ (4 − 1) = −6 ÷ 3 = −2
Meaning: Line slopes downward — for every 1 right, it goes 2 down.

Example 3 – Gradient from a Real-life Graph

A taxi costs £2 flagfall + £3 per km. On a cost (£) vs distance (km) graph, what is the gradient?

Step 1: Two points: (0, 2) [start] and (4, 14) [4 km costs £2+£12=£14]
Step 2: m = (14 − 2) ÷ (4 − 0) = 12 ÷ 4 = 3
Meaning: Gradient = £3 per km (the cost rate)
💡 The y-intercept (2) gives the starting cost. The gradient gives the rate.

Example 4 – Steepest Line

Which is steepest: y = 3x + 1, y = −5x + 2, or y = 1.5x − 7?

Gradients: |3| = 3, |−5| = 5, |1.5| = 1.5
Answer: y = −5x + 2 is steepest (|m| = 5)

🎨 Visualizer

📐 Rise and Run Visualizer

Drag the sliders to change gradient. Watch the rise and run triangle.

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🖩 Gradient Calculator

📊 Gradient as Rate — Quick Quiz

🏔️ Steepness Ranker

Enter four gradients. They'll be ranked steepest to shallowest.

Exercise 1 – Gradient from Two Points

Calculate the gradient. Use negatives where needed.

Exercise 2 – Reading Gradients

Read the gradient from the equation y = mx + c. Enter the value of m.

Exercise 3 – Gradient as Rate of Change

Calculate the rate from the graph information given.

Exercise 4 – Comparing Steepness

Answer the question about which line is steepest or the gradient's absolute value.

Exercise 5 – Mixed Gradient Problems

📝 Practice – 20 Questions

🏆 Challenge – 8 Questions