SQA Higher Applications of Mathematics

A mathematical model is a description of a real situation in mathematical terms: variables, relationships between them, and the assumptions that make the relationships usable. It is always a simplification, so the useful question is whether it is good enough for the purpose, not whether it is exactly true.

The modelling process

Modelling moves round a loop rather than along a line.

1. Decide what the model is for, and state the question it has to answer.
2. Identify the quantities involved and choose the variables.
3. State the assumptions that simplify the situation.
4. Choose a form for the relationship and write it as a formula, a table, a graph or a chart.
5. Use the model to calculate or predict.
6. Compare the output with what happens in reality or with data, and judge whether it is plausible.
7. Change the assumptions or the form if the output is not good enough, and go round again.

Variables and constants

A variable is a quantity that changes. The independent variable is the one chosen freely or the one that drives the change, often time or distance, and the dependent variable is the one whose value is worked out from it. A quantity that stays fixed in the situation is a constant or parameter, such as a fixed booking fee or a growth rate that is assumed not to change.

Suppose a taxi company charges a fixed £3.50 plus £1.80 for every mile. The independent variable is the distance $d$ in miles, the dependent variable is the fare $C$ in pounds, and £3.50 and £1.80 per mile are parameters. Choosing variables means deciding which way the dependence runs: the fare depends on the distance, not the other way round.

Some situations need more thought than that. For the height of a ball thrown upwards, suitable variables are the height above the ground and the time since it was thrown. For the number of customers in a café during the day, they are the number of customers and the time of day. Stating units when the variables are chosen (metres, seconds, pounds per month) prevents mistakes later, and the units form part of the model.

Forms of a model

The same relationship can be written in several ways, and each suits a different use.

A model for a situation in words often starts as a sentence. "The balance grows by 2% each month and £50 is added" describes a recurrence relationship. "The depth increases by 3 cm every minute" describes a linear one. Recognising which description goes with which form is the first skill, and the next two notes on relationships give the shapes to look for.

Assumptions

Every model leaves things out, and stating what was left out is part of the model. Typical assumptions are that a rate stays constant, that there are no other influences, that the quantities are measured exactly, and that the model is only used within a sensible range of the independent variable.

For the taxi fare, the assumptions are that the charge is exactly proportional to the distance after the booking fee, that there is no extra charge for waiting in traffic, and that nobody is charged for a part-mile differently. If a passenger reports a fare that does not match the formula, one of these assumptions is the place to look. The assumptions that fail tell you how to improve a model, which is covered further under choosing between models.

Estimating with a model

Some questions have no data and need an estimate built from assumptions you choose and state. The method is to break the quantity into parts that can each be estimated sensibly, multiply them, and round at every stage so that the working stays honest about how rough it is.

Estimate how many sandwiches a school canteen sells in a year.

1. Assume 1,200 pupils.
2. Assume about one in four buys a sandwich on a school day, so $1200 \times 0.25 = 300$ sandwiches a day.
3. Assume 190 school days in a year.
4. Then $300 \times 190 = 57\,000$ sandwiches.

The answer is given as "about 57,000", or better "roughly 60,000", because the inputs were only accurate to one or two figures. Changing an assumption changes the answer in a predictable way: if only one pupil in five buys a sandwich, the estimate falls to $1200 \times 0.2 \times 190 = 45\,600$. Showing how the result depends on the assumptions is what makes an estimate useful, because a reader can substitute their own figures.

Good estimates also check themselves for reasonableness. If the same canteen served 57,000 sandwiches and had 1,200 pupils, that is about 48 per pupil per year, which is plausible for a fifth of a school day each, and that agreement is some evidence the estimate is sensible.

Checking a model against the situation

Once a model gives an answer, three checks apply.

The answer should have the right units and size. A taxi fare of £0.03 for a ten-mile trip signals a slip somewhere. The behaviour should match what is known, for example a depth that increases should not be predicted to go negative. And the model should be used only over the range where the assumptions hold: a constant-rate model of a filling container stops being valid once the container is full.

If a check fails, the model is revised by altering an assumption or choosing a different form, for example replacing a constant rate by one that changes over time. The modelling loop then runs again from that step.