Modelling · Topic 1 of 9
Introduction to Modelling
Theory
In the real world, we are often faced with problems involving variables that are difficult or entirely impossible to measure with absolute precision. To tackle this, we use mathematical modelling.
1. What is a Mathematical Model?
A mathematical model is a way to produce a reasonable, "ballpark" approximation to a complex problem.
- The primary goal is not to find one single "correct" or exact answer, but rather to calculate a highly educated and reasonable estimate.
2. The Role of Assumptions
To build a model, we must make certain assumptions about the variables involved.
- These assumptions are based on information that we already know, or can sensibly guess, about the world around us.
- When creating a simple mathematical model, it is standard practice to intentionally ignore minor details and special cases to keep the calculations manageable.
Note: As this introductory section focuses on the concept of modelling itself, the examples focus on identifying the sensible assumptions required before a calculation can take place.
Worked examples
Example 1
Example 1: The School Commute
A student wants to model the total time they spend travelling to and from school in a typical year. State three sensible assumptions they would need to make to build this mathematical model.
To build this model, the student must assume:
- The average time taken for a single, one-way journey to school (e.g., 20 minutes).
- The number of days they actually attend school in a typical year (e.g., 190 days).
- That they make exactly two journeys (one there, one back) on every single one of those days, ignoring any half-days, absences, or after-school clubs.
Example 2
Example 2: The Cinema Trip
A cinema manager wants to model the total volume of popcorn sold on a typical Saturday. State three assumptions they would need to make to estimate this figure.
To build this model, the manager must assume:
- The total number of customers who will visit the cinema on a typical Saturday.
- The proportion (or percentage) of those customers who will actually purchase popcorn.
- The average volume of a single portion of popcorn sold.
Example 3
Example 3: The Smartphone Battery
A tech reviewer is building a model to estimate how many times a smartphone is charged over its usable lifespan. State three assumptions required to construct this model.
To build this model, the reviewer must assume:
- The total usable lifespan of the smartphone in years (e.g., 3 years).
- The average number of times the phone is charged per week or per day.
- That the user's charging behaviour remains completely constant over the entire lifespan of the device, ignoring battery degradation over time.