How to Practise Counting Coins and Change

Counting a pile of coins and making a specific amount from coins look like the same skill. They are not, and children who are fluent at the first are often stuck on the second.

Adding a pile is arithmetic with one answer. Making 87p is a search problem with many valid answers and one shortest, and that is a different kind of thinking.

Three levels, in order

Total the pile. Straight addition, made harder than plain sums because the addends are unequal and unsorted. Sorting by denomination first is the strategy worth teaching, and it transfers to a lot of later work.

Make this amount. Given a target, choose coins. The child must work out which denominations get them there, which requires holding a running total and a remaining target at the same time.

Make it with the fewest coins. The same problem with an optimisation on top. This is where most of the mathematical interest lives.

Moving to level two before level one is secure produces confusion that looks like a money problem and is actually a working memory problem.

Where greedy works, and where it does not

The natural method for fewest coins is greedy: take the largest coin that fits, repeat.

For real currency systems this gives the minimum, and that is not an accident. Coin denominations are chosen so it works — the 1, 2, 5, 10, 20, 50 pattern common in many currencies has this property deliberately.

It is not true in general. With a hypothetical set of 1, 3 and 4, making 6 greedily gives 4+1+1, which is three coins. The optimum is 3+3, which is two.

That example is worth showing to older students. It demonstrates that a method which always works in practice can fail in principle, which is a genuinely useful idea and one that shows up again in algorithms.

Use the currency they will actually handle

Denominations differ enough between currencies that fluency does not transfer cleanly.

A child fluent with 1, 2, 5, 10, 20, 50 has internalised particular decompositions — that 70 is 50+20, that 8 is 5+2+1. Those specific facts do not help with a currency using 25 as a denomination, where making 75 is a different decomposition entirely.

Practise with the coins in the shop down the road. The transferable part is the strategy; the fluency is currency-specific.

Giving change is the hardest version

The skill shops actually need, and the one practice sets usually omit.

Giving change requires subtraction, then decomposition of the result into coins, held in working memory while a customer waits. It is two of the earlier skills combined under time pressure.

The method taught in shops is counting up rather than subtracting. From a 5.00 note for a 3.40 purchase: 3.50, 4.00, 5.00 — ten pence, fifty pence, a pound. No subtraction happens at all.

That technique is worth teaching explicitly, because it is genuinely easier and because children who learn only the subtraction method find giving change much harder than it needs to be.

Common mistakes to avoid

• Moving to making amounts before totalling piles is secure.
• Practising with a currency the child does not encounter.
• Teaching greedy as a rule without ever showing where it fails.
• Using only round targets, which avoids the harder decompositions entirely.
• Skipping physical coins. Handling them is doing real work for younger learners that a screen does not replicate.

How to do it with Coin Counting Practice

The Coin Counting Practice generates exercises in the currency you choose.

1. Choose the currency the learner actually uses.
2. Start with totalling piles, and move to making amounts once that is fluent.
3. Add the fewest-coins mode once making amounts is comfortable.
4. Work alongside physical coins for younger learners.

Other classroom tools are in the tools directory.

Frequently asked questions

Why is making an amount harder than counting one?

Counting is addition with a single answer. Making an amount requires choosing from many valid combinations while tracking a running total and a remaining target, which is a heavier working memory load.

Does taking the largest coin first always give the fewest?

For real currency systems, yes — denominations are chosen so it works. It is not true in general: with coins of 1, 3 and 4, making 6 greedily gives three coins where two suffice.

Which currency should we practise with?

The one the learner will use. The strategy transfers between currencies; the specific decompositions that make someone fluent do not.

Final thought

Get totalling secure before moving to making amounts. Most children who look stuck on money are being asked to do two new things at once.

Try the free Coin Counting Practice