The stages of solving a problem — Mathematics, 11–13
Interpret, choose and carry out a strategy, then check the result in context — and see that one problem can be solved in several ways.
Three stages, every time
Solving a problem is not picking an operation straight away. There are three stages: first you interpret it (what is given? what is asked?), then you choose a strategy and carry it out, and finally you evaluate the result and check that it makes sense in the situation. You can also pose problems yourself: from the situation "24 eggs, 6 to a box" you invent the question — how many boxes are needed? (24 ÷ 6 = 4)
Many roads to the same answer
One problem can be attacked in different ways: run a simulation, work backwards, trial and error, solve a simpler problem first, try particular cases, or draw a diagram. A calculator or a spreadsheet can help, for example with trial and error. How many coins are in 8 piles of 6? Counting one by one, adding 6 + 6 + …, or doing 8 × 6 all give 48, but the last is the quickest. A strategy can be correct and still not efficient, so comparing them is how you learn.
Working backwards: grandma's biscuits
Grandma baked biscuits. João ate 1/4 of them; Pedro ate 1/3 of what was left; grandma ate half of the rest, leaving 3. How many were there? Work from the end. Before grandma: 3 is half, so there were 6. Before Pedro: 6 is 2/3, so 3 is 1/3 and there were 9. Before João: 9 is 3/4, so 3 is 1/4 and there were 12. Evaluate: 12 − 3 = 9; 9 − 3 = 6; 6 − 3 = 3 ✓. There were 12 biscuits.
Using every number in the text
Many people grab every number in the text and calculate with them. But some problems have too much data, others too little. "Rita, 11, bought 3 notebooks at 2 € each and a pen. How much did she pay for the notebooks?" Her age and the pen are extra: 3 × 2 = 6 €. But "Rita bought 3 notebooks; how much did she pay?" has too little data: the price is missing. Before calculating, ask: what is being asked, and which data do I need?
Where you see it: the party and the pizzas
18 friends come to a party and each eats 3 slices of pizza; each pizza has 8 slices. Interpret: 18 × 3 = 54 slices are needed. Carry out: 54 ÷ 8 = 6.75. Evaluate: nobody buys 6.75 pizzas, so it is 7 (with 7 × 8 − 54 = 2 slices left over). With the same data you can pose new problems: what if each ate only 2 slices? (18 × 2 = 36; 36 ÷ 8 = 4.5, so 5 pizzas).
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