Mathematical modelling with rational numbers
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 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.
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.
Markups and discounts are one model
A discount and a markup are the same calculation pointed in opposite directions. Both find a percentage of the base amount; a discount then subtracts it, while a markup adds it on. Taking 25% off 80 removes 20 to give 60, whereas adding 25% to 80 puts 20 on top to give 100. Seeing that a single model handles both is what lets you move comfortably between sale prices, price rises, tips and commission without learning a separate rule for each.
Adding tax to a price
Tax is a familiar add-on model. In Australia the goods and services tax adds 10% to most prices, so the total is the price plus a tenth of itself. A 60 dollar item carries 6 dollars of tax, making 66 dollars in all. Building the total as a base amount with a percentage slice laid on top is the same shape as interest on savings, and the same shape, reversed, as a discount. One chain of steps, reused across financial situations.
Comparing value with unit price
Modelling is not only about a single calculation; often it compares options. Faced with two pack sizes, the better buy is decided by price per unit, not the lower sticker price. A pack of 3 for 4 dollars 50 works out at 1 dollar 50 each, while 5 for 8 dollars is 1 dollar 60 each, so the smaller pack is actually the better value. Dividing price by quantity puts both choices on a common footing, which is exactly the kind of representation choice the modelling process asks you to justify.
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.