Problem-solving method
Making a systematic list, explained for parents
For problem sums that ask how many different ways. Here is why the order matters more than the list, and how your child can know they have found every one.
- Used from
- P3
- Taught at
- P3–P6
What this method is
When a question asks how many different ways, the answer is to write them all out. In an order. A child listing at random will find most of them and have no way of knowing which one they dropped.
Fix one thing, vary the rest, then move the fixed thing on. Done that way, the list can be pointed at as proof. The skill being taught is not listing, it is knowing when you have finished.
When to reach for it
These are the words in a question that point at this method. You do not need to be able to solve it to spot them.
“How many different ways”
The answer is a count of possibilities, which means the possibilities have to exist on paper first.
“all the possible”
Completeness is being asked for, not an example. That is what forces an order rather than a scattering.
“using the digits 2, 4 and 6”
A small fixed set to arrange. This is the classic shape, and it is nearly always a list question.
“at least one of each”
A constraint that removes rows. List everything first, then cross out, rather than trying to hold the rule in your head while listing.
“in how many ways can she pay”
Coins and notes. Fix the largest denomination, work down, and the order takes care of itself.
Watch it work
Using the digits 2, 4 and 6, and using each digit at most once, how many different 2-digit numbers can be made?
| Tens digit | Numbers made |
|---|---|
| 2 | 24, 26 |
| 4 | 42, 46 |
| 6 | 62, 64 |
Choose what to hold fixed
tens digit first, starting at 2Any choice works as long as it is a choice. Deciding this before writing anything is what separates a list from a scattering.
List every number that starts with it
24, 26Two numbers, and then stop. Knowing there are no others starting with 2 is a small moment of certainty, and the method is built out of those.
Move the fixed digit on and repeat
starting with 4 → 42, 46 · starting with 6 → 62, 64The same two steps again, twice. Nothing new is being decided, which is why this method works under exam pressure.
Count the rows
3 tens digits × 2 each = 6 numbersThe count comes off the shape of the list rather than off the fingers. A child who can explain why it is six has the method, whatever number they wrote.
The one it gets confused with
Both of these also write things down as they go. What differs is what makes you stop writing.
Are you hunting for one answer that fits, or counting every possibility? Guess and check stops when it finds. A list stops when it is complete.
Does the question give you an end result? Then there is one answer to reverse to, not a set to enumerate.
How can you tell it has clicked?
Ask the only question that matters here: how do you know that is all of them? A child with the method points at the order. A child without one counts again and hopes for the same number.
A list written in the order the ideas arrived is the thing to watch for. It is often complete by luck, which hides the gap until a harder question turns up.
One thing to try tonight
Next time you are ordering food, ask how many different combinations there are from two mains and three drinks.
Then ask the harder one: how do you know you got them all? If they cannot answer that, it is the thing to practise, not the counting.
Where your child will meet it
Want the bigger picture on how these methods fit together? Maths heuristics, explained for parents
See which methods your child reaches for
StudyLah names the method in every worked solution, so you can see whether the problem was the method or the arithmetic.