• Quick retrieval from earlier learning (Find the gradient of a curve at a point). Students answer on mini whiteboards.
• Show answers together and note who needs support in the main task.

1. Find the gradient of y = −x² + 2x + 9 at x = −2. → 6Prompt: Differentiate first.
2. Find the gradient of y = x² + 3/x at x = −1. → -5Prompt: 3/x = 3x⁻¹, which differentiates to −3x⁻².
3. At what x is the gradient of y = x² + 4x − 7 equal to 14? → x = 5Prompt: Differentiate, then set dy/dx equal to the gradient.

• Show a distance-time graph of a journey and ask how fast it was going at one exact moment.
• Share the objective: Interpret a derivative as a rate of change in context. Read the success criteria together.

• Work through the example out loud, saying why you do each step, not just what you do.
• Example: V(t) is the number of users of an app after t days. V′(15) = 32. What does this mean?
• Step 1: On day 15, users are increasing at about 32 per day
• Answer: On day 15, users are increasing at about 32 per day
• 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. V(t) is the number of users of an app after t days. V′(5) = 36. What does this mean? (The total change since the start / On day 5, users are increasing at about 36 per day / The average rate over the whole period / There are 36 users on day 5) → On day 5, users are increasing at about 36 per dayPrompt: A derivative is a rate: units of output per unit of input.
2. The cost (in dollars) of making x phone cases is C(x) = 3x² + 28x + 200. Use C′(8) to estimate the extra cost of making one more case after 8. → 76Prompt: The derivative gives the cost per extra case at that point (marginal cost).

• 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 8 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. The cost (in dollars) of making x phone cases is C(x) = 4x² + 54x + 200. Use C′(17) to estimate the extra cost of making one more case after 17. → 190Prompt: The derivative gives the cost per extra case at that point (marginal cost).
2. T(m) is the temperature (°C) of a drink m minutes after it is made. T′(22) = −1. What does this mean? (The total change since the start / The drink is at 1 °C after 22 minutes / The average rate over the whole period / At 22 minutes, the drink is cooling at about 1 °C per minute) → At 22 minutes, the drink is cooling at about 1 °C per minutePrompt: A derivative is a rate: units of output per unit of input.

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