• 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 = 3x² + 18x + 29. Type it like (2, −1). → (−3, 2)Prompt: At a stationary point, dy/dx = 0.
2. Find the x-coordinates of the stationary points of y = x³ − 12x² + 36x + 1. Separate them with a comma. → x = 2 or x = 6Prompt: Differentiate and set dy/dx = 0.
3. y = x³ − 3x² − 24x has a stationary point at x = −2. Is it a maximum or a minimum? (Maximum / Minimum) → MaximumPrompt: 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: Solve optimisation problems with calculus. Read the success criteria together.

• Work through the example out loud, saying why you do each step, not just what you do.
• Example: A farmer has 80 m of fencing to make a rectangular pen against a straight wall (the wall is one side). If the two sides at right angles to the wall are x m, the area is A = x(80 − 2x). Find the maximum area in m².
• Step 1: A = 80x − 2x²
• Step 2: dA/dx = 80 − 4x = 0 → x = 20
• Step 3: A = 20 × 40 = 800 m²
• Answer: 800
• 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. A farmer has 92 m of fencing to make a rectangular pen against a straight wall (the wall is one side). If the two sides at right angles to the wall are x m, the area is A = x(92 − 2x). Find the maximum area in m². → 1058Prompt: Expand, differentiate and set dA/dx = 0.
2. An open box is made from a 42 cm square of card by cutting a square of side x cm from each corner and folding up. V = x(42 − 2x)². Find the maximum volume in cm³. → 5488Prompt: Expand V, differentiate and solve dV/dx = 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 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. An open box is made from a 30 cm square of card by cutting a square of side x cm from each corner and folding up. V = x(30 − 2x)². Find the maximum volume in cm³. → 2000Prompt: Expand V, differentiate and solve dV/dx = 0.
2. A farmer has 28 m of fencing to make a rectangular pen against a straight wall (the wall is one side). If the two sides at right angles to the wall are x m, the area is A = x(28 − 2x). Find the maximum area in m². → 98Prompt: Expand, differentiate and set dA/dx = 0.

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