ACARA v9 CONTENT DESCRIPTION “use mathematical modelling to solve practical problems involving rational numbers and percentages, including financial contexts; formulate problems, choosing efficient calculation strategies and using digital tools where appropriate; interpret and communicate solutions in terms of the situation, reviewing the appropriateness of the model”
What mathematical modelling means
Mathematical modelling is the habit of turning a real situation into a calculation, working out the answer, and then reading that answer back into the situation. Most of the numbers we meet outside the classroom arrive wrapped in a story: a price with a discount, a recipe shared between friends, two products of different sizes to compare. Modelling is the bridge between the story and the arithmetic. You decide what the situation is really asking, strip it down to the numbers that matter, do the sum, and then say what the result means in plain words. The mathematics here is the rational-number and percentage work students already know; what is new is the discipline of moving carefully between the world and the calculation, in both directions.
From a situation to a number and back
A shopping problem becomes a calculation, then a worded answer.
modelling means turning a real situation into a calculation, solving it, then reading the result back into the situation.
The modelling cycle
It helps to picture modelling as a short cycle with four stages. First you formulate the problem: you decide exactly what you are trying to find, and you note the numbers and the assumptions you will rely on. Next you compute: you choose an efficient strategy, carry out the arithmetic, and reach for a calculator when the values are awkward. Then you interpret: you translate the bare number back into the situation and say what it means, with the right units and a sensible level of rounding. Finally you review: you check whether your assumptions were fair and whether the answer actually fits the question. If the review uncovers a problem, you loop back to the start and adjust, so the cycle can run more than once before you are satisfied.
The modelling cycle
Formulate, compute, interpret, review, then loop back if needed.
the modelling cycle: formulate the problem, compute a solution, interpret it in context, then review whether the model was appropriate.
Working with percentages
Three percentage moves cover most everyday problems. The first is finding a percentage of a quantity: 20% of $80 is 0.2 times 80, which is $16. The second is a percentage increase or decrease: to increase $50 by 10% you find 10% of $50, which is $5, and add it to get $55, while a 15% decrease works the same way but takes the slice off instead. The third is expressing one quantity as a percentage of another: 12 out of 48 is 12/48, which simplifies to 1/4, or 25%. It is worth holding the three forms of a percentage together in your mind, because 20% is the same number as the decimal 0.2 and the fraction 1/5, and you can switch to whichever form makes a particular calculation easiest.
Three percentage moves
A percentage of an amount, a percentage change, and one amount as a percentage of another.
find a percentage of an amount, increase or decrease by a percentage, or write one amount as a percentage of another.
A worked financial example
A single worked example shows the cycle in action. A jacket is priced at $80 and the shop is offering 15% off; the question is what you will actually pay. Formulating the problem, you decide you need the sale price, which is the original price with the discount removed. Computing, you find 15% of 80, which is 12, and subtract it: 80 minus 12 is 68. Interpreting the result, the jacket will cost $68 at the till. Every step stays in ordinary arithmetic, and the numbers are kept simple on purpose, so the structure of the reasoning is easy to see and to copy onto a new problem of the same shape.
A 15% discount, worked through
Take 15% of $80 off the original price to find what you pay.
a 15% discount on $80 removes $12, leaving a sale price of $68.
Interpreting and reviewing the answer
The last two stages are the ones students most often skip, and they are what make an answer trustworthy. Interpreting means stating the result in the words of the situation rather than leaving a bare number on the page: not 68, but a sale price of $68. Checking that it is reasonable is a quick guard against slips, since a discount should always give a smaller price, and $68 sitting a little below $80 passes that test at a glance. Reviewing means looking back at the assumptions: did we ignore a delivery fee, a rounding rule, or a tax that applies at the register? If one of those matters, the model needs adjusting; if not, we can trust it for the decision at hand.
Interpret, then review
State the answer in words, check it is reasonable, and review the assumptions.
state the answer in the words of the situation, check it is reasonable, and review the assumptions the model relied on.
Why this matters
Modelling is how mathematics earns its keep in daily life. Budgeting, shopping, planning a trip and comparing two deals are all modelling tasks, and the same four-stage habit of formulate, compute, interpret and review keeps the answers meaningful rather than merely correct on paper. Strong rational-number and percentage skills feed straight into these everyday financial decisions, which is why they are worth practising until they are automatic. This unit stays with simple, realistic numbers and ordinary arithmetic; compound interest and the formal formulae of financial mathematics belong to later years, and the goal here is the clear thinking that makes any later technique reliable.
Quick self-check
1. What is 20% of $80?
2. A $50 item increases by 10%. What is the new price?
3. A $80 jacket is 15% off. What is the sale price?
4. 12 out of 48 students walk to school. What percentage is that?
5. In the modelling cycle, what do you do AFTER computing a solution?