• Quick retrieval from earlier learning (Find stationary points and decide their nature). Students answer on mini whiteboards.
• Show answers together and note who needs support in the main task.

1. Use calculus to find the stationary point of y = −2x² + 8x − 13. Type it like (2, −1). → (2, −5)Prompt: At a stationary point, dy/dx = 0.
2. Find the x-coordinates of the stationary points of y = x³ − 12x² + 36x + 4. Separate them with a comma. → x = 2 or x = 6Prompt: Differentiate and set dy/dx = 0.
3. y = −x³ + 6x² + 15x has a stationary point at x = −1. Is it a maximum or a minimum? (Maximum / Minimum) → MinimumPrompt: Find the second derivative d²y/dx².

• Show a distance-time graph of a journey and ask how fast it was going at one exact moment.
• Share the objective: Find where a function is increasing or decreasing. Read the success criteria together.

• Work through the example out loud, saying why you do each step, not just what you do.
• Example: For which values of x is f(x) = −x² − 12x + 3 increasing?
• Step 1: x < −6
• Answer: x < −6
• 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 which values of x is f(x) = −3x² − 6x + 3 increasing? (x < −1 / x > 1 / x < 1 / x > −1) → x < −1Prompt: Increasing means f′(x) > 0; decreasing means f′(x) < 0.
2. Where is f(x) = x³ − 3x² − 9x + 1 decreasing? (−3 < x < 1 / x < −1 or x > 3 / x > 3 / −1 < x < 3) → −1 < x < 3Prompt: Find where f′(x) < 0.

• 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 which values of x is f(x) = −3x² − 36x − 4 increasing? (x < −6 / x > −6 / x > 6 / x < 6) → x < −6Prompt: Increasing means f′(x) > 0; decreasing means f′(x) < 0.
2. Where is f(x) = x³ − 6x² + 1 decreasing? (x > 4 / 0 < x < 4 / x < 0 or x > 4 / −4 < x < 0) → 0 < x < 4Prompt: Find where f′(x) < 0.

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