• 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. Type it like (2, −1). → (2, −8)Prompt: At a stationary point, dy/dx = 0.
2. Find the x-coordinates of the stationary points of y = x³ − 6x² + 7. Separate them with a comma. → x = 0 or x = 4Prompt: Differentiate and set dy/dx = 0.
3. y = −x³ + 27x has a stationary point at x = −3. 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: 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 84 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(84 − 2x). Find the maximum area in m². → 882Prompt: Expand, differentiate and set dA/dx = 0.
2. An open box is made from a 48 cm square of card by cutting a square of side x cm from each corner and folding up. V = x(48 − 2x)². Find the maximum volume in cm³. → 8192Prompt: 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 24 cm square of card by cutting a square of side x cm from each corner and folding up. V = x(24 − 2x)². Find the maximum volume in cm³. → 1024Prompt: Expand V, differentiate and solve dV/dx = 0.
2. A farmer has 108 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(108 − 2x). Find the maximum area in m². → 1458Prompt: Expand, differentiate and set dA/dx = 0.

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