Lesson Plan: Design a Medieval Fortress

Materials Needed

• Graph paper (1cm grid recommended)
• Pencil and eraser
• Ruler
• Scientific calculator
• Internet access for optional research on castle architecture

Learning Objectives

By the end of this lesson, you will be able to:

• Apply linear equations to manage a construction budget and material costs (ACMNA239).
• Use formulas for perimeter, area, and volume to design complex 3D structures (ACMMG242).
• Analyze parabolic motion using quadratic equations to simulate catapult trajectories (ACMNA241).
• Apply right-angled triangle trigonometry (SOH CAH TOA) to determine defensive sightlines (ACMMG245).
• Provide mathematical justification for engineering decisions.

1. Introduction: The Royal Engineer’s Quest

The Scenario: The year is 1324. The King has appointed you as the Royal Engineer. A rival kingdom is advancing, and you have been granted a 100m x 100m square plot of land to build a defensive fortress. However, the Royal Treasury is low. You have a strict budget of 20,000 gold pieces.

Construction Costs:

• Curtain Wall Stone: 10 gold per cubic meter ($m^3$).
• Tower Stone: 12 gold per cubic meter ($m^3$).

Success Criteria: Your fortress must be fully enclosed, include four towers, and stay under budget while proving its defensive capabilities through mathematics.

2. Body: Content and Practice

Phase 1: The Curtain Wall (I Do / We Do)

The curtain wall connects your towers. To maximize your internal space, you will build the wall along the perimeter of your 100m x 100m plot.

The Formulas:

• Perimeter ($P$): $P = 2L + 2W$
• Volume ($V_{wall}$): $P \times \text{height} \times \text{thickness}$ (Standard height: 8m, thickness: 3m)
• Cost ($C_{wall}$): $V_{wall} \times 10$ gold

Example Walkthrough: If we build a wall around the full 100m x 100m square:
$P = 2(100) + 2(100) = 400\text{m}$.
$V_{wall} = 400 \times 8 \times 3 = 9,600\text{m}^3$.
$C_{wall} = 9,600 \times 10 = 96,000$ gold.
Wait! That is way over budget! As the engineer, you must decide: will you make the fortress smaller, the walls thinner, or the walls shorter?

Phase 2: The Watchtowers (We Do / You Do)

You must place four identical cylindrical towers at the corners of your fortress. Each tower must be 15m high.

The Formula: $V_{tower} = \pi r^2 h$

The Task: Calculate the remaining budget after your wall design. Solve for the maximum radius ($r$) you can afford for your four towers. Remember, tower stone costs 12 gold per $m^3$.

Algebraic Scaffolding:
$C_{towers} = (4 \times \pi r^2 \times 15) \times 12$
$r = \sqrt{\frac{\text{Remaining Gold}}{4 \times \pi \times 15 \times 12}}$

Phase 3: Catapult Ballistics (You Do)

An enemy camp is spotted. Your catapult fires a stone following the path of the quadratic equation:
$y = -0.01x^2 + 0.9x + 1$
(Where $y$ is height in meters and $x$ is horizontal distance in meters).

• The Vertex: Find the maximum height of the projectile using $x = -b / 2a$.
• The Range: If the enemy camp is 80m away, will the stone hit them? (Evaluate for $x = 80$).
• The Impact: Solve for $y = 0$ using the quadratic formula to find exactly where the stone hits the ground.

Phase 4: Defensive Sightlines (You Do)

An archer stands atop a 15m tower. They spot an enemy at a horizontal distance of 40m from the base of the tower.

• Trigonometry Task: Use $\tan^{-1}(\frac{\text{opposite}}{\text{adjacent}})$ to find the angle of depression the archer must aim at to hit the target.

3. The Main Activity: The Design Blueprint

Using your graph paper, draw a scale bird's-eye view (1cm = 10m) of your fortress.

1. Layout: Draw your walls and four towers. Label all dimensions.
2. The Ledger: Provide a neatly written table showing your calculations for:
   • Total wall volume and cost.
   • Total tower volume and cost.
   • Combined total (Must be $\le$ 20,000 gold).
3. Tactical Report: Include your catapult analysis (vertex and impact point) and your archer’s angle of depression calculation.

4. Conclusion: Recap and Reflect

• Summary: Today we used linear equations to stay on budget, geometry to calculate material volume, and quadratics/trigonometry to plan our defense.
• Recap Question: If we doubled the radius of a tower, how many times more stone would we need? (Answer: 4 times, because $r$ is squared).
• Takeaway: Real-world engineering requires balancing creative design with the cold, hard limits of physics and finance.

5. Differentiation & Extensions

• Support: Provide a pre-filled spreadsheet or table where the student can "plug and play" different wall lengths to see the cost update automatically.
• Advanced Challenge 1 (Calculus/Optimization): Find the dimensions of a rectangular fortress that provide the maximum internal area for a fixed wall perimeter of 200m.
• Advanced Challenge 2 (Geometry): Calculate the volume of a "dry moat" (trapezoidal prism) dug around the fortress and determine how many laborers are needed if one person can move $2m^3$ of dirt per day.
• Creative Extension: Add a "Gatehouse" to your design. Research the shape of a portcullis and calculate the surface area of the iron bars needed to build it.

Assessment: Success Criteria