Combat Models Project Introduction

The purpose of this investigation is for students to see the real power of calculus to describe and explain the world around them. In this project, students will use calculus to understand and explore a basic model for conventional combat and use this model to learn about the Battle of Trafalgar. From there, students can explore different combat models or apply the conventional combat model to other historic battles. Students will need to bring together topics and concepts from across the course and put them together in ways they may not have seen as a part of the standard first course in calculus.

This investigation has 4 essential components. The first component, the Battle of Trafalgar, is essential to the project and must be completed first. The three extensions are independent of each other and build from the background of this first component. Teachers can use some or all of them, depending on the background of their students and the amount of time they would like to commit to the project. We generally try to give at least two weeks for projects like this one. Student material and teacher material, most with solutions, are provided so teachers can see what content is involved in each section.

In the first component of the project, students will model conventional (direct fire) combat, where combat takes place in the open with both sides exposed to their opponent. Students learn about the Lanchester model of conventional combat, a deterministic model in which the rate at which one side loses troops depends on the size of the opposing side and the effectiveness of the opposition fire. Students will apply this model to the Battle of Trafalgar, the famous 1805 naval battle in which Admiral Lord Nelson abandoned orthodox tactics and led his British fleet to victory over a larger force of French and Spanish ships. They will first use Euler's method to explore the solution numerically, then find analytic solutions.

Once students understand this basic model, there are several possible extensions.

• Battle of the Alamo: Students can apply the methods they learned in the Battle of Trafalgar investigation to another historic battle. This is the simplest of the three extensions.
• Guerilla warfare: Students can learn how the model for conventional combat can be modified to create a model for guerilla (area fire) warfare, where each side is not exposed to its opponents. This change in tactics reduces the effectiveness of each side, and it makes the size of both forces factors in the rate at which an army loses troops. Students can apply this model to the Vietnam War.
• Stochastic model: Students will interpret the coefficients of fighting effectiveness probabilistically, and develop a stochastic model that will present a different perspective on what might have happened during a battle. This section is appropriate for strong BC students.

Links to Files for Handouts and numerical solutions in Excel:

• Background info on Euler's method - intro to EM and Excel spreadsheet
• Battle of Trafalgar - student handout, teacher handout, and Excel spreadsheet
• Battle of the Alamo - student handout, teacher handout, and Excel spreadsheet
• Guerilla Warfare - student handout and solutions
• Stochastic model - student handout, teacher handout, and supplemental teacher handout, and integrating factor handout
• And finally, a TI-calculator program, written by Paul Thompson. (Requires TI-Connect. Save a local copy first by right-clicking link, then select "save as...")

If you choose to have your students work through this investigation and find errors or have questions, please e-mail one of us at: "belledin at ncssm dot edu" or "teague at ncssm dot edu." One of us (or our colleagues) will get back to you with support.

Note: Teacher handouts are password-protected. Please feel free to email one of us in the NCSSM math dept to receive the login credentials.