Models — Core Maths Compendium

Build it up, step by step

Understanding models

Click each step below to reveal it — work through them in order the first time round.

Step 1 · What is a mathematical model?

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A mathematical model is a simplified representation of a real-world situation, built using equations, graphs or rules, used to make predictions or understand a system. Examples include a formula predicting population growth, or a straight-line graph modelling the cost of a taxi journey.

Step 2 · Assumptions in models

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Every model relies on simplifying assumptions to make it workable — e.g. assuming growth is constant, or that a relationship is exactly linear. These assumptions are rarely perfectly true in reality, but they make the model usable. You should always be able to identify the assumptions a given model makes.

Step 3 · Limitations of models

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Because models are simplifications, they have limitations: they may only be accurate within a certain range of values, and they may ignore real-world factors (e.g. a population model ignoring migration or disease). Extrapolating a model far beyond the data it was built from is especially risky.

Step 4 · Evaluating and refining a model

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A good exam answer critiques a model specifically: state an assumption, explain why it might not hold in reality, and describe the effect this would have on the model's predictions. Models can be refined by adding extra factors, or replacing an oversimplified assumption (e.g. using a curve instead of a straight line).

Worked example

A café models its monthly profit, P (£), using P = 1200 + 40n, where n is the number of extra customers per day. State one assumption this model makes, and explain a limitation.

Assumption: the model assumes profit increases by a constant £40 for every extra customer per day — a perfectly linear relationship.
Limitation: in reality this is unlikely to hold indefinitely — for very large n, the café might run out of seating/staff capacity, so profit per extra customer could fall, meaning the linear model would overestimate profit for large values of n.

Test yourself

Past-paper style question

A scientist models the height, h (cm), of a plant after t weeks using h = 3t + 5, based on data collected over the plant's first 6 weeks of growth.

(a) State one assumption made by this model.
(b) Explain why it would be unwise to use this model to predict the plant's height after 50 weeks. [4 marks]

Show the answer

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(a) The model assumes the plant grows at a constant rate (3 cm per week) — a perfectly linear relationship between height and time.

(b) Using the model to predict height after 50 weeks is extrapolation, going far beyond the range of data (6 weeks) the model was based on; in reality, plant growth typically slows or levels off as the plant matures, so the linear model would likely greatly overestimate the plant's actual height at 50 weeks.

Practice

Models worksheet

Five short questions on models. Work through them, then reveal the mark scheme to check.

Explain, in your own words, what is meant by a ‘mathematical model.’
A taxi fare is modelled as F = 3 + 2m, where m is the number of miles travelled. State one assumption this model makes about the cost of a taxi journey.
Explain the difference between interpolation and extrapolation when using a model to make a prediction.
Explain why extrapolating far beyond the range of the original data is generally considered risky.
A company uses a straight-line model to predict its revenue for the next 10 years, based on only 6 months of data. Explain why this could produce an unreliable prediction.

Mark scheme

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A mathematical model is a simplified representation of a real-world situation, built using equations, graphs or rules, used to describe, understand or make predictions about that situation.
It assumes the cost per mile is constant (£2 for every mile, regardless of traffic, distance already travelled, waiting time, etc.) — a perfectly linear relationship between cost and distance.
Interpolation means predicting a value that lies within the range of the original data used to build the model; extrapolation means predicting a value outside that range, which is generally far less reliable, as the model's assumptions may no longer hold.
Beyond the range of the original data, there's no evidence the same pattern/relationship continues to hold; real-world factors and trends can change, so predictions made this far outside the data are much less trustworthy.
Only 6 months of data is a very short and possibly unrepresentative sample (e.g. it might not capture seasonal variation), and predicting 10 years ahead is a huge extrapolation — the trend may not continue anywhere near that far into the future, making the prediction unreliable.

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