Real problems do not arrive with a formula attached. "Will my phone battery last the bus ride to the tournament if I stream videos the whole way?" There is no equation in the question, no numbers, and no list of what matters. Mathematical modelling is the process of turning a messy real question like this into mathematics, solving the mathematics, and then checking whether the answer makes sense back in the real world. Engineers, doctors, economists and climate scientists all work this way. The process is a cycle, not a straight line: you often discover halfway through that an assumption was wrong and go back to fix it.
What a model is
A mathematical model is a simplified description of a real situation using mathematics: an equation, a graph, a table or a set of rules. It is simplified on purpose. A model of a phone battery ignores the colour of the case and the song you are listening to, and keeps only what affects the answer most. Every simplification is an assumption, a statement you accept as true for the model even though it may not be exactly true. The statistician George Box is known for the saying that all models are wrong, but some are useful. A good model is not perfect; it is accurate enough to answer the question, and honest about where it stops working.
- Ask a question. Make it specific and measurable: not "how long do batteries last?" but "how many hours of video streaming before my phone drops from 100% to 20%?"
- Make assumptions and choose variables. Decide what you will measure (the variables), what you will keep the same, and what you will ignore.
- Plan and collect data. Decide how, how often and how many times you will measure, then collect the data carefully and record units.
- Build the model. Graph the data, look at its shape, and fit a linear, quadratic or exponential model (by hand or with technology).
- Test the model. Compare the model's predictions with data you did not use to build it. If the fit is poor, go back and revise your assumptions or your model.
- Report. Answer the original question in context, say how well the model fits, and state its limits.
A complete model: the phone battery
| Stage | For the battery question |
|---|---|
| Question | How many hours of video streaming before the battery falls from 100% to 20%? |
| Variables | Independent: time streaming t (hours). Dependent: battery level B (percent). |
| Assumptions | Same phone, same video app, same brightness and volume, on Wi-Fi, room temperature, no charging, no other apps running. |
| Data plan | Start at 100%, record the battery level every 30 minutes for 3 hours. |
| Time t (h) | 0 | 0.5 | 1 | 1.5 | 2 | 2.5 | 3 |
|---|---|---|---|---|---|---|---|
| Battery B (%) | 100 | 91 | 83 | 74 | 66 | 57 | 49 |
| Change since previous row | −9 | −8 | −9 | −8 | −9 | −8 |
Build. The battery drops by 8 or 9 percentage points every half hour, an almost constant rate, so a linear model fits. Over the 3 hours it fell from 100 to 49, a rate of percentage points per hour, and it started at 100. (Technology gives with , essentially the same line.) Now solve the model for the question's target.
Test. On another day, under the same conditions, the battery read 30% after 4 hours. The model predicts , so it was off by 2 percentage points: a good fit, slightly optimistic. Report. Under these conditions the battery should reach 20% after about 4.7 hours (about 4 hours 42 minutes), and since the test suggests the model runs a little high, planning for about 4.5 hours is safer.
When a model fails the test
Testing sometimes shows that a model is wrong in shape, not just slightly off. Suppose (invented data) a cup of tea in a 21 °C room measures 90, 72, 59, 49 and 42 °C at 0, 5, 10, 15 and 20 minutes. A straight line through the first and last points has slope °C per minute and predicts °C at 40 minutes: colder than the room, and below freezing. That is impossible, so the assumption "the tea cools at a constant rate" must be wrong. The drops, 18, 13, 10 and 7 degrees, keep getting smaller. Back in the cycle, a better assumption is that the tea's temperature gap above the room, , shrinks by about the same factor (about ) every 5 minutes, which is an exponential pattern that levels off at room temperature.