Mathematical Modelling and Problem Solving: From Words to Equations
Core concepts
Modelling means translating a described situation into symbols, solving, then interpreting the result in context. Candidates should be able to define variables clearly, build linear and simple quadratic models, identify constraints, and judge whether an answer is reasonable. Units, scale, and assumptions all belong in the answer.
A reliable modelling cycle
- Define the variables and their units.
- Formulate the relationship, stating assumptions.
- Solve with the appropriate algebra.
- Interpret the result and check it against the original wording.
- Reflect on whether the model's limits matter.
Traps to watch
- Answering the algebra rather than the question - a negative time or a fractional number of people signals a modelling error, not a clever answer.
- Ignoring stated constraints such as a fixed budget or maximum capacity.
- Mixing units (minutes with hours, cents with dollars) inside one calculation.
- Over-generalising beyond the data range used to build the model.
- Skipping the assumption statement, which is often worth marks on its own.
Worked example
A printer charges a fixed $20 plus $0.05 per page. A budget is $45. Model: 20 + 0.05p = 45, so 0.05p = 25 and p = 500 pages. Interpreting: 500 pages is the maximum, and any non-integer result would need rounding down, because you cannot buy part of a page within a budget.
Rule of thumb
Write the units beside every number as you go. If the units do not cancel to the units the question asks for, the model is wrong before the arithmetic is.