Optimisation and Modeling

Active learning builds deep understanding in optimisation because students must physically manipulate functions, debate constraints, and test predictions. This kinesthetic and social approach helps students confront misconceptions about stationary points and endpoints directly through concrete examples.

• Year 12
• Mathematics
• 4 activities
• 25 min–50 min

Learning Objectives

1. Analyze the relationship between the first derivative and the slope of a tangent line to identify stationary points.
2. Classify stationary points as local maxima, local minima, or points of inflection using the second derivative test.
3. Evaluate the function at critical points and endpoints to determine absolute extrema within a constrained domain.
4. Design a mathematical model to optimize a real-world scenario, such as maximizing the volume of a container or minimizing material cost.

Problem-Based Learning

⏱ 30 min·Pairs

Pairs: Derivative Card Sort

Provide cards with functions, graphs, first and second derivatives. Pairs match them to classify stationary points, then justify max/min using second derivative values. Conclude with a class share-out of mismatches.

Prepare & details

Explain how stationary points help us determine the most efficient design for a physical object.

Facilitation Tip: During Derivative Card Sort, circulate to clarify that stationary points are only candidates for extrema, not guarantees.

Setup: Groups at tables with access to research materials

Materials: Problem scenario document, KWL chart or inquiry framework, Resource library, Solution presentation template

• Analyze
• Evaluate
• Create
• Decision-Making
• Self-Management
• Relationship Skills

Problem-Based Learning

⏱ 50 min·Small Groups

Small Groups: Can Design Challenge

Groups receive fixed surface area specs to maximise volume of a cylindrical can using calculus. They derive the optimisation equation, test with second derivative, and build prototypes from paper. Compare designs.

Prepare & details

Assess the significance of the second derivative in determining the nature of an optimal solution.

Facilitation Tip: In the Can Design Challenge, ask guiding questions like 'What happens if you reduce the radius by 2 cm?' to press for constraint testing.

Setup: Groups at tables with access to research materials

Materials: Problem scenario document, KWL chart or inquiry framework, Resource library, Solution presentation template

• Analyze
• Evaluate
• Create
• Decision-Making
• Self-Management
• Relationship Skills

Problem-Based Learning

⏱ 40 min·Whole Class

Whole Class: Profit Maximisation Simulation

Project a cost/revenue function on screen. Class votes on production levels, calculates derivatives step-by-step, and discusses endpoint constraints like market limits. Adjust for new scenarios in real time.

Prepare & details

Justify how constraints on a domain change our approach to finding absolute extrema.

Facilitation Tip: In the Profit Maximisation Simulation, assign roles so students must explain their calculus steps to peers before moving to the next round.

Setup: Groups at tables with access to research materials

Materials: Problem scenario document, KWL chart or inquiry framework, Resource library, Solution presentation template

• Analyze
• Evaluate
• Create
• Decision-Making
• Self-Management
• Relationship Skills

Problem-Based Learning

⏱ 25 min·Individual

Individual: Graphing Software Exploration

Students input custom functions into Desmos or GeoGebra, locate extrema, and vary domains. They screenshot analyses for a portfolio, noting second derivative effects.

Prepare & details

Explain how stationary points help us determine the most efficient design for a physical object.

Facilitation Tip: For Graphing Software Exploration, provide a checklist of function features to investigate to keep students focused on key concepts.

Setup: Groups at tables with access to research materials

Materials: Problem scenario document, KWL chart or inquiry framework, Resource library, Solution presentation template

• Analyze
• Evaluate
• Create
• Decision-Making
• Self-Management
• Relationship Skills

Teaching This Topic

Teach this topic by starting with visual and tactile experiences before formal tests. Use graphing software to let students see concavity changes firsthand, then connect these observations to second derivative signs. Emphasize that calculus is a toolkit, not a single method; students should know when to use first derivatives, second derivatives, or endpoint evaluation.

What to Expect

Successful learning looks like students confidently identifying critical points, applying tests correctly, and justifying extrema choices with clear reasoning. They should articulate why endpoints matter and how concavity guides classification of stationary points.

Watch Out for These Misconceptions

Common Misconception

During Derivative Card Sort, watch for students who assume every zero of the first derivative is a max or min.

What to Teach Instead

Have pairs physically group functions by concavity before testing second derivatives, ensuring they link curvature changes to point classification.

Common Misconception

During Can Design Challenge, watch for students who ignore endpoints or constraints.

What to Teach Instead

Require teams to present their domain choices and explain why boundary values were evaluated, using the challenge’s material limits as a concrete anchor.

Common Misconception

During Graphing Software Exploration, watch for students who confuse second derivative zero with absence of an extremum.

What to Teach Instead

Ask students to adjust sliders until the second derivative is zero, then sketch the graph and test nearby points to observe behavior changes.

Assessment Ideas

Quick Check

After Derivative Card Sort, provide a function and interval. Ask students to list all critical points, endpoints, and classify extrema with justification using their sorted cards as reference.

Discussion Prompt

During Profit Maximisation Simulation, circulate and listen for students explaining how they set up the profit function, applied the first derivative test, and compared results to the budget cap.

Exit Ticket

After Graphing Software Exploration, give students a graph with labeled points. Ask them to classify each point and write a sentence explaining how the second derivative’s sign informed their choice.

Extensions & Scaffolding

• Challenge: Ask students to design a constraint scenario where the global maximum occurs at a stationary point and one where it occurs at an endpoint.
• Scaffolding: Provide partially completed sign charts or labeled graphs for students to fill in during the Card Sort activity.
• Deeper exploration: Introduce optimization with multiple constraints, such as Lagrange multipliers for advanced students.

Key Vocabulary

Stationary Point	A point on a curve where the gradient is zero, meaning the first derivative of the function is equal to zero at that point.
Second Derivative Test	A method using the second derivative of a function to determine whether a stationary point is a local maximum, local minimum, or a point of inflection.
Absolute Extrema	The maximum or minimum value of a function over a specific interval, which may occur at critical points or at the endpoints of the interval.
Constraint	A limitation or restriction placed on the variables in an optimization problem, such as a fixed amount of material or a maximum budget.