Back to Mathematical Modelling

Higher Applications of Mathematics

Linear models

Using straight-line models and interpreting gradients.

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 using constant rates of change, gradients and intercepts in practical models. 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

  • Linear model: y = mx + c
  • Gradient m is the change in output for one unit increase in input.
  • Intercept c is the starting value when the input is zero.

Worked examples

Modelling walkthrough 1

Build the model

A phone contract charges a fixed monthly fee plus a cost for extra data.

  1. Identify the fixed charge as the intercept.
  2. Identify the cost per unit as the gradient.
  3. Write the model in the form total cost = fixed charge + rate × units

A linear model is suitable when the rate stays constant.

Modelling walkthrough 2

Use the model

A phone contract charges a fixed monthly fee plus a cost for extra data.

  1. Substitute the number of extra GB into the model.
  2. Calculate the total cost.
  3. State the answer in pounds and explain what it represents.

The gradient and intercept have practical meanings, not just algebraic meanings.

Modelling walkthrough 3

Evaluate the model

A phone contract charges a fixed monthly fee plus a cost for extra data.

  1. Check whether the model is valid for the range given.
  2. Consider whether prices cap after a certain amount.
  3. Explain how that would limit the linear model.

A good answer says where the model may stop being reliable.

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: Scatter graphs and trendlinesStatistics: Linear regression

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