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Course Description:
A mathematical model is a simplification of some real-world phenomenon into a series of mathematical equations or structures. We create them in order to help make predictions, as well as to gain insights into what is happening in the real world. This class is oriented around a series of real-world questions; in each one we develop mathematical techniques appropriate to analyzing and answering the question mathematically, and then create models using our new-found techniques. This is (more or less) what a mathematician does when applying mathematics to the real world, so this class is essentially an introduction to how mathematics gets used to solve real-world problems. Unlike other math classes at Grauer, this one is heavily project-based, with most of the projects completed in groups of two to four.
The prerequisite for this class is completion of, or concurrent enrollment in, Pre-Calculus.
Math Modeling will typically be offered in alternating years with Statistics.
The textbook is:
No official textbook. All curriculum materials are pulled from various sources and curated/created by Peter.
The teacher is:
Detailed Course Topic List:
1. The mathematical modeling process
- Outline of the basic process, which includes posing a question, making mathematical assumptions, introducing variables, deriving relationships between the variables, testing the model, conducting a sensitivity analysis, and communicating results
- Example one: Renting vs. driving a car
- Example two: Analyzing the financial benefits and drawbacks of attending college
2. Case study: The mathematics of elections
- Possible election processes: first-past-the-post, Borda models, runoff models, approval voting
- Measures of fairness of election processes (Condorcet criterion)
- Voting paradoxes
- Concluding summative project: choice of research on gerrymandering of the Arrow voting theorem
3. Models of population growth
- Introduction / review of recursive sequences
- Models of population growth using sequences: Fibonacci model, chaotic models
- Introduction / review of matrices
- Leslie matrix model of population growth
- Concluding summative project: Choice of mathematical modeling problem asking one to model growth of a species in the environment
4. Fitting models to data empirically
- Review of basic function types: linear, quadratic, exponential, logarithmic
- Least-squares criterion of best-fitting curves
- Methods for linearizing data
- Concluding summative project: choices include modeling the growth of CO2 in the atmosphere
5. Graph theory
- Introduction to basic vocabulary of graphs
- Application: scheduling problems
- Euler/Hamilton circuits
- Concluding summative project: choices include researching the Traveling Salesman problem, and the 4-color theorem
6. Game theory
- Introduction: Prisoner's dilemma
- Basic concepts: dominated strategies, Nash equilibria, mixed strategies
- Application: models for predator/prey behavior and for males competing for mating rights in biology
- Application: determining the best pitch to throw in a given situation in baseball
7. Probability models
- Basic vocabulary of probability
- Markov chain models
- Application to random walk problems
- Simulation modeling (Monte Carlo simulations)
- Simulating queuing processes: the time it takes for a customer entering a line at a store to get served
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