SQA Higher Applications of Mathematics Mathematical Modelling: formulae, graphs, units, accuracy and spreadsheets

What Mathematical Modelling actually demands

Mathematical modelling is the thread that runs through the whole course: every finance, statistics and planning question begins by turning a described situation into mathematics. The examiners test whether you can choose the right model, keep your units and accuracy honest, and use a spreadsheet to do the heavy calculation. This guide ties together the three dot-point pages of the module; each has its own worked examples and practice.

Building models with formulae and graphs

The starting skill is to define variables, with units, and write a formula. A linear model $y = mx + c$ suits a fixed start plus a constant rate, such as a standing charge plus a unit price. A piecewise linear model uses different rules over different intervals, such as a tariff that changes after a threshold, and you state the interval beside each rule. An exponential model $y = a \times b^{t}$ multiplies by a fixed factor each period, modelling growth when $b > 1$ and decay when $0 < b < 1$. The test for which family fits is simple: constant differences point to linear, constant ratios point to exponential.

Units, accuracy and tolerance

A model is only trustworthy if its units are consistent and its accuracy is honest. Convert every quantity to one system before calculating, and report a result no more precisely than the data justify. Two error measures matter: the absolute error $|\text{measured} - \text{true}|$ in the original units, and the percentage error $\dfrac{\text{absolute error}}{\text{true value}} \times 100$, which divides by the true value and lets you compare errors on different-sized quantities. A tolerance such as $50 \pm 0.4$ mm gives an acceptable range from $49.6$ mm to $50.4$ mm; a value passes if it lies inside the range.

Spreadsheets for modelling

The course requires you to use a spreadsheet, and printed output is submitted with the question paper. A formula begins with = and refers to cells; filling it down copies the calculation. A relative reference (B2) shifts when filled, so it tracks each row, while an absolute reference ($E$1) is locked, so a shared rate stays fixed. Functions such as SUM, AVERAGE and STDEV calculate over a range, and goal seek finds the input that produces a target output, which solves break-even and savings-target problems without rearranging an equation.

How Mathematical Modelling is examined

A typical SQA profile for this area:

• Setting up a model. Defining variables and writing a linear, piecewise or exponential formula from a description.
• Predicting and evaluating. Using the model to predict, and commenting on its limitations.
• Units and error. Converting units, rounding sensibly, and reporting absolute or percentage error against a tolerance.
• Technology. Building the model in a spreadsheet with correct references, functions and goal seek.

Check your knowledge

A mix of recall and method questions covering the module. Attempt them, then check against the solutions.

1. A model has constant differences over equal steps. Linear or exponential? (1 mark)
2. Write a formula for a cost of a $\pounds 6$ fee plus $\pounds 2$ per unit, for $u$ units. (1 mark)
3. A part is specified as $30 \pm 0.5$ mm. State the acceptable range. (2 marks)
4. A model gives $84$ but the true value is $80$. Find the percentage error. (2 marks)
5. Explain what a relative cell reference does when a formula is filled down. (1 mark)