ACARA v9 CONTENT DESCRIPTION “use mathematical modelling to solve practical problems, involving rational numbers and percentages, including financial contexts; formulate problems, choosing representations and efficient calculation strategies, using digital tools as appropriate; interpret and communicate solutions in terms of the situation, justifying choices made about the representation”
Mathematics earns its keep when it solves real problems. Working out a sale price, splitting a bill, calculating interest on savings: these are everyday situations where the numbers you have been learning about come together. Mathematical modelling is the skill of taking a real problem, turning it into mathematics, solving it, and translating the answer back into the real world. It is where the whole year of number work pays off.
This year you learn to model practical problems involving rational numbers and percentages, especially in financial contexts like discounts, savings and sharing costs. The aim is not just to get an answer, but to choose a sensible approach, carry it out efficiently, and explain what the answer means for the original situation.
The modelling cycle
Modelling follows a cycle. You begin with a real problem, often described in words. You then translate it into mathematics, deciding which numbers and operations capture the situation. Next you solve the maths, and finally you interpret the result, reading it back into the real context and asking whether it makes sense. If a sale price comes out higher than the original, something has gone wrong, and the interpretation step is where you catch it.
The modelling cycle
Turn a real problem into maths, solve it, then read the answer back into the real situation.
Mathematical modelling is a cycle. You start with a real problem, translate it into mathematics, solve the maths, then interpret the answer back in the real situation, checking it makes sense. The maths is a tool in the service of a real question.
The translation step is the one that takes practice. A problem about three friends sharing a 90 dollar bill equally becomes the calculation 90 divided by 3, giving 30 dollars each. A problem about a 25 percent discount becomes a percentage calculation followed by a subtraction. Learning to spot which mathematics a situation calls for is the core of modelling, and it draws on every idea about fractions, percentages and ratios you have met.
Modelling money problems
Financial problems are among the most common models, and most reduce to a short chain of percentage steps. To find the sale price of an 80 dollar jacket reduced by 25 percent, you find 25 percent of 80, which is 20 dollars, then subtract it to get 60 dollars. Interest works the same way but adds rather than subtracts: a 50 dollar deposit earning 10 percent gains 5 dollars, growing to 55 dollars.
A discount as a model
A real money problem becomes a short chain of percentage steps.
A money problem is modelled as a chain of steps. To find a sale price, take the original 80 dollars, find 25 percent of it, which is 20 dollars, and subtract to reach 60 dollars. The percentage and ratio skills from earlier units do the work.
Notice how the same handful of skills, finding a percentage and then adding or subtracting, models a wide range of real situations. This is the power of proportional reasoning: once you can scale a quantity up or down by a percentage or a ratio, you can model discounts, markups, interest, tax and currency conversion with the same approach. The final and often forgotten step is to communicate the answer clearly in terms of the situation, saying the jacket costs 60 dollars rather than just writing 60, so that the mathematics actually answers the question that was asked.
Teaching tip: real receipts, catalogues and bank statements are ideal modelling material. Ask the student to work out a discounted price from a real sale, or how a savings balance grows with interest, so the maths attaches to something they recognise. The habit of stating the answer in full, with its dollar sign and its meaning, is worth reinforcing every time.
Watch for answers left as bare numbers with no interpretation. A model is not finished at the calculation; encourage the student to write a sentence saying what the number means, since this is exactly the communicating step the curriculum asks for and the one most easily skipped.
Builds on: Ratios (AC9M7N08). That unit built ratio reasoning; this unit applies it to model real problems with percentages and money.
Quick self-check
1. What is the first step in mathematical modelling?
2. A jacket costs $80 and is reduced by 25%. What is the sale price?
3. A phone bill of $90 is split between 3 friends in the ratio 1 : 1 : 1. How much does each pay?
4. After solving the maths in a model, what should you always do?
5. A $50 deposit earns 10% interest in a year. How much is in the account after one year?