Grades 9–12Applied MathematicsLesson available
Mathematical modelling with functions
The modelling cycle (simplify, formulate, solve, interpret, validate) with linear, quadratic and exponential models of real situations, and honest discussion of assumptions and limitations.
math.applied.modelling-with-functions
Builds on
Exponential and logarithmic functionsQuadratic equations and functionsRatios, rates and proportional reasoning
Required prerequisites, all the way down: 25
You will learn to
- Formulate linear, quadratic and exponential models from situations and data.
- Interpret the parameters of a model and use it to make predictions with stated assumptions.
- Validate a model against data and explain where and why it fails.
Definition
- A mathematical model is a simplified mathematical description of a real situation, together with its assumptions.
- Validation compares a model's predictions with observations.
Theorem
- A quantity growing continuously at rate r doubles every (ln 2)/r units of time.
- Every model leaves out features of reality; it is judged by whether it serves its purpose.
- Extrapolating far beyond the range of the data is risky.
Common misconceptions
- Treating a model's output as fact rather than as a consequence of its assumptions.
- Choosing the model that best fits past data regardless of whether it is plausible.
Optional discoveries
- Mathematical biology Models of populations and epidemics are built from differential equations.
Further reading
- Edward A. Bender, An Introduction to Mathematical Modeling (1978)
- COMAP and SIAM, Guidelines for Assessment and Instruction in Mathematical Modeling Education (GAIMME) (2016)