Forwards or Backwards: Choosing a Strategy
Two problems can look identical and need opposite methods. Telling them apart takes about four seconds — once you know what to look at.
Show me the tellWorked example — Forwards or Backwards: Choosing a Strategy
One problem, solved out loud
After spending a third of her money and then 20 riyals on lunch, Sara has 60 riyals left. How much did she start with?
- Notice what is given: the END amount, not the start.
This is the whole decision, and it is made before any arithmetic. What you know tells you which end to stand at.
- Start at 60 and undo the last thing that happened: 60 + 20 = 80.
Undo means the opposite operation, in the opposite order. She spent, so you add.
- Undo the third: 80 is two thirds, so a third is 40, and the whole is 120.
Spending a third leaves two thirds. Undoing a fraction means asking what the remainder is a fraction OF.
- Check it forwards: 120 − 40 = 80, 80 − 20 = 60. ✓
Working backwards always gives you a free check, because the forward path is the problem itself.
Answer120 riyals
Forwards or Backwards: Choosing a Strategy — explained
That method has a name, and naming it is what turns it from something you watched into something you can reach for. It is WORKING BACKWARDS: start from the known end state and undo each step, opposite operation, opposite order. Its partner is WORKING FORWARDS: start from the known beginning and do each step as it comes. Neither is better. The choice is decided by one question — which end of the story do you actually know? If you are given the start, go forwards. If you are given the end and asked for the start, go backwards. Almost every learner who gets stuck on a problem like Sara's is not bad at arithmetic; they are standing at the wrong end of it, trying to guess the number that goes at the front.
The idea — Forwards or Backwards: Choosing a Strategy
Before solving, ask which end of the story you were given. That answer chooses the method for you.
You practised choosing, not just calculating — which is why a problem about aeroplanes felt like a problem about money.
Questions about Forwards or Backwards: Choosing a Strategy
How do you know whether to work forwards or backwards in a math problem?
Ask which end of the story you were given: if you know the start, go forwards; if you know the end and need the start, go backwards.
What does 'working backwards' mean in problem solving?
It means starting from the known end state and undoing each step using the opposite operation in the opposite order.
How much money did Sara start with if she had 60 riyals left?
She started with 120 riyals: undoing the lunch spending gives 80, and undoing the third spent gives 120.
Why does working backwards give you a free check of your answer?
Because the forward path is the problem itself, so redoing it forwards from your answer confirms whether it matches the given end.
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