Main Question

A farmer has 100 meters of fencing to create a rectangular pen against the side of a barn (so one side doesn’t need fencing). What dimensions will give the maximum area?

More importantly, can you discover a general principle about how area and perimeter relate in optimization problems?

Key Concepts to Explore

• The trade-off between different dimensions when perimeter is fixed
• How to express area as a function of one variable
• The concept of optimization in geometric contexts
• Why certain shapes are “optimal” for given constraints

Expected Follow-up Questions

High-quality student engagement should include questions like:

• “What if we needed fencing on all four sides?”
• “Does this principle work for other shapes like triangles or circles?”
• “Why does the maximum occur at this specific point?”
• “How would additional constraints change the answer?”
• “What if we wanted to minimize perimeter for a given area instead?”
• “Can I visualize what’s happening as the dimensions change?”

Critical Thinking Indicators

Students demonstrate understanding when they:

• Question how the constraint affects the possible solutions
• Want to understand why the optimal solution has specific properties
• Explore what happens with different constraint values
• Connect this to real-world design problems
• Ask about generalizing to other shapes or conditions
• Investigate the mathematical reasoning behind optimization
• Show curiosity about related problems (circles, triangles, etc.)

Additional Exploration

• Compare rectangular, circular, and triangular pens with the same perimeter
• Investigate how the optimal ratio changes with different constraints
• Explore 3D versions (maximizing volume with surface area constraints)
• Connect to calculus concepts (derivatives and optimization)

Assessment Criteria

Strong responses should include:

• Clear setup of the constraint equation (perimeter = 100)
• Expression of area as a function of one variable
• Method for finding the maximum (completing the square or vertex form)
• Understanding that the optimal rectangle is a square when all sides need fencing
• Recognition that this is a special case due to the barn constraint
• Ability to generalize the approach to similar problems

Real-World Connections

• Architecture and space planning
• Agricultural field design
• Material efficiency in manufacturing
• Garden and landscape design
• Economic optimization in resource allocation