ACARA v9 CONTENT DESCRIPTION “use mathematical modelling to solve practical problems involving additive and multiplicative situations including financial contexts; formulate the problems, choosing operations and efficient calculation strategies, using digital tools where appropriate; interpret and communicate solutions in terms of the situation”
Builds on the four operations, including the multiplication, division and estimation of the previous units. Year 5 brings these together as mathematical modelling: taking a real, practical problem, turning it into a number problem, solving it, and explaining the answer in terms of the original situation. Many everyday and financial problems are additive, solved by adding or subtracting, or multiplicative, solved by multiplying or dividing, and modelling is the skill of deciding which, working it out, and making sense of the result.
What mathematical modelling is
Mathematical modelling is a way of solving real problems with mathematics. It follows a cycle: formulate the problem by turning the situation into a number sentence, choose the operations and an efficient strategy, calculate the answer, then interpret and communicate it in terms of the situation. The mathematics is only part of the work; understanding the situation well enough to set it up, and explaining what the answer means, matter just as much. Modelling connects classroom calculation to the kinds of questions people really ask.
The modelling cycle
Five steps from a real problem to a clear answer.
Formulate: turn the situation into a number problem.
Choosing the operation
The first decision is which operation the situation needs. Additive situations combine or compare amounts and are solved by adding or subtracting: how much is left, how many altogether, how much more. Multiplicative situations involve equal groups, rates or sharing and are solved by multiplying or dividing: groups of a size, a total shared equally, a cost per item. Six boxes of eight pencils is six times eight, a multiplication, while forty-five dollars saved less eighteen spent is a subtraction. Recognising the type of situation points straight to the operation.
Choosing the operation
Additive or multiplicative? That picks the operation.
Decide whether the situation is additive or multiplicative.
From situation to number sentence
Formulating a problem means writing the situation as a number sentence. Four tables with six chairs each becomes four times six, which gives twenty-four chairs. Ninety apples packed into bags of six becomes ninety divided by six, which gives fifteen bags. The words of the problem are translated into numbers and an operation, and the answer is then read back with its units. Choosing an efficient strategy, and a calculator where the numbers are large, keeps the calculation quick and accurate once the problem is set up correctly.
From situation to number sentence
Translate the words into numbers and an operation.
Formulate: 4 tables, 6 chairs each. Chairs in all?
Modelling a money problem
Money problems are often modelled in more than one step. To buy three books at twelve dollars each and pay with fifty dollars, first multiply three by twelve to get thirty-six dollars, then subtract from fifty to find fourteen dollars change. Tickets at nine dollars each for four people plus six dollars parking is four times nine, then add six, giving forty-two dollars. Breaking a money problem into steps, each a single operation, makes a multi-step situation manageable, and an estimate along the way checks that each step is reasonable.
Modelling a money problem
Break a money problem into single-operation steps.
Model this money problem: Buy 3 books at $12 each, pay with $50.
Interpreting the solution
The last step is to interpret and communicate the answer in terms of the situation. A calculation gives a number, but the situation gives it meaning: twenty-three divided by four is five remainder three, which means five each with three left over, not simply five point seven five. The fourteen from fifty minus thirty-six is dollars of change, and the forty-eight from six times eight is the total number of pencils, not the number of boxes. A good solution states the answer with its units and relates it back to the question that was asked.
Interpreting the solution
A number means nothing until read back in context.
Relate the calculated number back to the situation, with units.
Modelling with confidence
Modelling with confidence means working through the whole cycle: read the situation, decide whether it is additive or multiplicative, write a number sentence, calculate with an efficient strategy, and then explain the answer in context with its units. Money problems may take several steps, each a single operation, with an estimate to check along the way. The mathematics and the meaning go together, and a clear answer always points back to the real question. With this habit a child can model many everyday and financial problems, ready for the larger investigations of later years.
Quick self-check
1. Mathematical modelling means...
2. 6 boxes of 8 pencils is best modelled by...
3. Ninety apples packed into bags of six is modelled by...
4. To buy 3 books at $12 and pay with $50, the change is found by...
5. 23 shared evenly among 4 gives 5 remainder 3. In the situation this means...