Back to Mathematical Modelling

Higher Applications of Mathematics

Variables, formulae and models

Choosing variables and writing useful relationships.

Modelling referenceModelling mixed quiz

Modelling reference

Check variables, formulae, model types, units, errors and assumptions.

Mixed revision quiz

Review Mathematical Modelling with adaptive feedback.

Before you start

  • Be confident substituting numbers into simple formulae.
  • Check the units in the question before calculating.
  • Be ready to explain what an answer means in the real situation.

Method helper

Which model or method do I use?

Does the output change by a fixed amount each step?

Use a linear model.

Is there a starting fee plus a rate per unit?

Use y = mx + c and interpret the intercept.

Does the situation involve area, a turning point or a squared variable?

Use a quadratic model.

Is the amount increasing or decreasing by a percentage repeatedly?

Use an exponential model.

Have you been given a formula and values?

Substitute into the formula carefully.

Do you need to make one variable the subject?

Rearrange the formula one step at a time.

Are measurements rounded or allowed within a range?

Calculate tolerance or error.

Do the units look inconsistent?

Convert units and check dimensional sense.

Are you testing several assumptions or scenarios?

Use technology to compare model outputs.

Does the prediction seem unrealistic?

Comment on model limitations.

Modelling lesson

Key idea

  • This topic focuses on identifying variables, forming useful formulae and explaining model assumptions. Mathematical modelling is about using maths to represent a real situation well enough to support a decision.
  • A good model identifies the variables, uses a sensible formula or graph, and states the assumptions being made.
  • For Higher Applications, the interpretation matters. You should explain whether the output is realistic, what it means in context, and what might make the model less reliable.

Key formulae, definitions and methods

  • Define each variable with units before using it.
  • Independent variable = input; dependent variable = output
  • A model is useful only if its assumptions are reasonable for the context.

Worked examples

Modelling walkthrough 1

Build the model

A school minibus trip is modelled using distance, fuel use and ticket income.

  1. List the quantities that can change, such as distance, passengers and fuel price.
  2. Choose clear variable names and units.
  3. Decide which variable is the input and which is the output.

The model is clearer because every variable has a meaning and unit.

Modelling walkthrough 2

Use the model

A school minibus trip is modelled using distance, fuel use and ticket income.

  1. Write a formula from the words in the problem.
  2. Substitute known values carefully.
  3. Round the result to a sensible level for the context.

The formula gives a useful estimate, but the output still needs interpretation.

Modelling walkthrough 3

Evaluate the model

A school minibus trip is modelled using distance, fuel use and ticket income.

  1. State one assumption, such as constant fuel use.
  2. Explain how that assumption could affect the answer.
  3. Suggest one improvement, such as using actual fuel data from previous trips.

A limitation comment shows whether the estimate should be trusted.

Watch out

  • Using a model without defining the variables and units.
  • Choosing a linear model when the rate of change is not constant.
  • Treating a model prediction as an exact fact rather than an estimate.
  • Forgetting to convert units before substituting values.
  • Giving a calculation without commenting on assumptions or limitations.

Technology and data connection

Related Higher Applications topics

Spreadsheet: Formulae and cell references

Next step

Move into practice

Use the learning notes to identify variables, assumptions and units, then try varied formulas, model types and reasonableness checks.

Modelling mixed quiz