Lesson Plan

• Describe how to translate a real‑world scenario into a mathematical function for optimisation.
• Apply the first‑ and second‑derivative tests to locate and classify stationary points.
• Solve optimisation problems (maximising profit, minimising material) and interpret the results in context.
• Communicate the complete solution process using appropriate mathematical notation.

• Projector or interactive whiteboard
• Printed worksheet with worked example and practice questions
• Graphing calculators or computer algebra system
• Whiteboard and markers
• Handout summarising first‑ and second‑derivative tests
• Rulers and graph paper (optional)

Start with a quick discussion of everyday decisions that involve getting the most profit or using the least material. Review that the derivative gives the rate of change and that stationary points occur where the derivative is zero. Explain that today students will model such situations, apply derivative tests, and clearly state the optimal solution.

1. Do‑now (5’): Short question on the highest point of a thrown ball; recall derivative concept.
2. Mini‑lecture (10’): Present the five‑step optimisation workflow and demonstrate the profit‑maximisation example with a graph.
3. Guided practice (15’): Work through the profit example together; students fill each step on the worksheet.
4. Collaborative activity (15’): Small groups solve the cylindrical‑can minimisation problem, use calculators, and present their classification of the stationary point.
5. Independent practice (10’): Students attempt two additional practice questions while the teacher circulates for support.
6. Exit ticket (5’): Write the five‑step procedure for solving any optimisation problem in their own words.

Recap the five‑step method and emphasise the importance of checking domain restrictions and interpreting results. Collect exit tickets to gauge understanding, and assign the remaining practice questions as homework, asking students to bring any uncertainties to the next lesson.