Optimising a Pen Design

Overview

Students investigate a realistic optimisation problem: designing a rectangular animal pen beside a straight river, using 40 m of fencing for three sides. They model the area as a function, sketch the graph, differentiate, and justify the maximum area in an annotated, NCEA-style response.

Learning intentions

Students will:

• Form a function to model a realistic optimisation problem.
• Sketch a quadratic function and identify its relevant features and domain.
• Find and use a derivative to locate a stationary point.
• Communicate a complete algebraic and graphical justification.

Success criteria

I can:

• Define variables and form an appropriate area function.
• Differentiate the function and solve (A'(x)=0).
• Explain why the stationary point gives a maximum.
• Present an annotated graph and relate my answer to the context.

Curriculum links

• Apply calculus methods in solving problems: derivatives, gradient functions, turning points, and communicating mathematical reasoning.
• Apply graphical methods in solving problems: sketching functions, identifying features, and connecting equations with graphs.
• Apply algebraic methods in solving problems: forming and solving a quadratic equation.
• NCEA relational and extended abstract thinking: selecting a logical strategy, connecting representations, and justifying a contextual conclusion.

Lesson structure (30 minutes)

1. 0–4 min · Hook and prediction. Teacher opens the optimisation hook and lesson sequence with the question: “Using 40 m of fencing for three sides of a pen beside a river, what dimensions create the greatest area?” Students sketch a possible design, estimate dimensions, and briefly share their prediction with a partner.

2. 4–9 min · Build the model. Teacher uses the labelled pen diagram and modelling prompts to establish (x) as the width perpendicular to the river and (y) as the fenced length parallel to the river. Guide students to write (2x+y=40), rearrange to (y=40-2x), and form [ A(x)=xy=x(40-2x)=40x-2x^2. ] Students copy the diagram, state the realistic domain (0<x<20), and explain what (A(x)) represents.

3. 9–15 min · Calculus demonstration. Teacher models the NCEA-style chain: [ A'(x)=40-4x,\qquad A'(x)=0 ] [ 40-4x=0\Rightarrow x=10. ] Use the derivative explanation and worked example to connect the derivative with gradient: at the maximum, the graph has a horizontal tangent. Students complete missing steps on the optimisation response worksheet and calculate (y=40-2(10)=20), then (A(10)=200\text{ m}^2).

4. 15–22 min · Graph and justify. Teacher displays the annotated graph instructions and asks students to sketch (A(x)=40x-2x^2) on suitable axes. Require labels for the intercepts ((0,0)) and ((20,0)), the vertex ((10,200)), the domain, and the contextual meaning of the vertex. Students annotate their worksheet graph and justify that the point is a maximum using either the downward-opening quadratic or a sign change in (A'(x)): positive before (x=10), negative after (x=10).

5. 22–27 min · NCEA response refinement. Teacher presents a model conclusion through the response checklist and peer-review prompt. Students swap work with a partner and check: variables defined, function formed, derivative shown, stationary point found, nature justified, units included, and conclusion linked to the pen. Students improve one line of algebra and one explanatory sentence.

6. 27–30 min · Exit check. Teacher uses the plenary and exit questions to ask: “Why is (x=10) not merely a possible width, but the width that maximises area?” Students complete the final question on the worksheet: “State the maximum area, the dimensions, and one reason the result is a maximum.” Collect responses to identify misconceptions.

Resources

• the optimisation hook and lesson sequence
• the optimisation response worksheet
• Whiteboard and markers
• Projector or interactive display
• Rulers and pencils
• Graphing technology, if routinely used by the class for checking

Assessment

• During modelling, check that students form (A(x)=40x-2x^2), rather than confusing area with perimeter.
• While circulating, question students about the domain, the meaning of (A'(x)), and why the stationary point is a maximum.
• Use the exit response to assess algebraic accuracy, graph annotations, contextual interpretation, and justification in the style of an NCEA response.

Differentiation

• Support students with the labelled diagram, a partially completed algebraic sequence, the sentence starters “Let (x) represent…” and “Since (A'(x)) changes from…”, and a displayed derivative rule.
• Pair students strategically for the modelling and peer-review stages; allow graphing technology to check the sketch after the algebra has been attempted.
• For students needing accessibility support, provide enlarged graph axes, reduced copying, and the option to explain the maximum verbally before writing.
• Extend confident students by asking them to justify the maximum using (A''(x)=-4), compare the result with endpoint areas, or generalise the model for (F) metres of fencing.