• Quick retrieval from earlier learning (Average rate of change over an interval). Students answer on mini whiteboards.
• Show answers together and note who needs support in the main task.

1. x: 0, 1, 2, 3, 4 f(x): −3, −4, −1, 6, 17 Find the average rate of change of f from x = 2 to x = 4. → 9Prompt: Average rate = change in f(x) ÷ change in x.
2. f(x) = x² + 2x − 1. Find the average rate of change of f between x = 0 and x = 1. → 3Prompt: Evaluate f at both ends of the interval.
3. A ball is thrown upwards. Its height in metres after t seconds is h(t) = 17t − 5t². Find its average speed upwards (rate of change of height, in m/s) from t = 0 to t = 2. A negative answer means it is coming down. → 7Prompt: Find the height at each time.

• Show a distance-time graph of a journey and ask how fast it was going at one exact moment.
• Share the objective: Estimate the gradient of a curve with chords. Read the success criteria together.

• Work through the example out loud, saying why you do each step, not just what you do.
• Example: For y = x², find the gradient of the chord from x = −3 to x = −2.9.
• Step 1: (8.41 − 9) ÷ 0.1
• Step 2: = −5.9
• Answer: -5.9
• Pause after each step and ask a student to say what comes next.

• Do these together. Students try each step on whiteboards before you reveal it.
• Use the hints as prompts rather than giving the answer.

1. For y = x², find the gradient of the chord from x = −2 to x = −1.9. → -3.9Prompt: Gradient = change in y ÷ change in x.
2. For y = x², chords from x = 3 to x = 3 + h have gradients: h = 0.1: 6.1 h = 0.01: 6.01 h = 0.001: 6.001 What is the gradient of the curve at x = 3? → 6Prompt: As h shrinks, the chord gets closer to the tangent.

• Students work on the student sheet. Circulate and check the first two answers of every student early.
• Students who finish: write a question of their own for a partner, with the answer.

The 6 questions are on the student sheet below, with answers in the key.

• Two questions on their own. Sort answers into got it / nearly / not yet to plan the next lesson.

1. For y = x², chords from x = −1 to x = −1 + h have gradients: h = 0.1: −1.9 h = 0.01: −1.99 h = 0.001: −1.999 What is the gradient of the curve at x = −1? → -2Prompt: As h shrinks, the chord gets closer to the tangent.
2. For y = x³, find the gradient of the chord from x = 1 to x = 1.1. Give 2 decimal places. → 3.31Prompt: Work out both y-values.

Free online practice with hints and printable worksheets for this skill: eurekami.com/math/grade-12.