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Counting Up: Subtraction as Making Change

Stop taking away and measure the gap instead. Counting up turns three-digit subtraction into two or three easy hops, and it is how change is made at a till.

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Key takeaways

  • Counting up replaces the take-away model of subtraction with a distance model: you measure the gap from the smaller number to the larger one.
  • Hopping to the next round number first, then across the hundreds, then to the target, turns a three-digit subtraction with borrowing into three easy additions.
  • The two models give the same answer because a difference is defined by the missing addend: if b plus d equals a, then a minus b equals d.
  • Counting up is strongest when the numbers are close together or the smaller one sits just below a benchmark, and weakest when the gap needs many hops.
  • Money is the native case, which is why the method is sometimes called the shopkeeper's method — the hops are the coins and notes handed back.

Stop taking away and measure the gap instead. To work out 512 − 386, start at 386 and count up: 14 gets you to 400, another 100 gets you to 500, another 12 gets you to 512. Add the hops — 14 + 100 + 12 — and the answer is 126. No borrowing, no column alignment, and every intermediate step is small enough to say out loud.

The move: a difference is a distance

The take-away model says subtraction removes one quantity from another. The distance model says a subtraction names the gap between two positions. Both are correct, and for mental work the second is far more useful, because measuring a gap can be done in pieces that you choose.

The equivalence is not a trick. Subtraction is formally the inverse of addition: a − b is the number d such that b + d = a. Research on how arithmetic strategies develop describes exactly this missing-addend counting strategy — given the whole and one part, children count from the known part up to the whole to find the unknown part — and places it among the routes that later give way to direct recall for small numbers. For large numbers, nobody ever recalls the answer, so the counting route stays useful for life.

The What Works Clearinghouse practice guide on mathematics intervention in the elementary grades makes the number line one of its six recommendations, for the related reason: representing quantities as positions makes the difference visible as a length rather than as an operation to be performed.

Choosing the hops

The method only pays if the hops are easy. The default sequence is: jump to the next round number, jump across the round numbers, then jump to the target.

1,204 − 968. From 968, hop 32 to reach 1,000. From 1,000, hop 204 to reach 1,204. Total: 32 + 204 = 236.

2,026 − 1,987. From 1,987, hop 13 to 2,000. From 2,000, hop 26 to 2,026. Total: 39.

803 − 467. From 467, hop 33 to 500. From 500, hop 300 to 800. From 800, hop 3. Total: 33 + 300 + 3 = 336.

The first hop is always a complement — how far to the next ten, hundred or thousand — which is why complements to 10 and 100 are the facts this method leans on hardest. If 467 to 500 is not instant, counting up is not yet fast.

Money is the native case

Handing back change is this method made physical. A £50 note against a £32.65 bill: from 32.65, 35p reaches £33; from £33, £17 reaches £50. The change is £17.35, and at a till it is handed over in exactly that order — the small coins that square the pennies, then the notes.

The same shape covers any decimal subtraction where one number sits just under a round figure. 100 − 62.40: 60p to 63, then 37 to 100, giving 37.60.

This is also why the method is sometimes called the shopkeeper's method. It survived in commerce because it needs no borrowing and each hop is independently verifiable while you are building it, which matters when someone is watching.

When counting up is the wrong choice

Every hop is another partial sum to hold, so the cost grows with the number of hops rather than with the size of the numbers.

7,412 − 1,689 takes three: 11 to reach 1,700, 5,300 to reach 7,000, 412 to reach 7,412, giving 11 + 5,300 + 412 = 5,723. That is correct, and it is more work than it needs to be, because holding three partial sums is exactly the load the method was supposed to remove.

Two rules of thumb. Prefer counting up when the numbers are close together, or when the smaller number sits just below a benchmark. Prefer something else when the gap spans several orders of magnitude and neither number is convenient.

The usual "something else" for a wide gap is to slide both numbers by the same amount so the smaller one becomes round — 7,412 − 1,689 becomes 7,723 − 2,000 = 5,723 in a single step. That method, and the reason it is safe, is covered in subtraction without borrowing. The other alternative is to round one number and correct afterwards, which is the compensation strategy — cheaper in hops, but it introduces a correction step whose direction is easy to reverse.

The mistakes worth naming

Counting down from the larger number. Starting at 512 and counting down to 386 gives the same distance, but the hops are subtractions again, and you have thrown away the reason for switching models.

Adding the hops to the wrong thing. The answer is the sum of the hops, not the last position reached. In 512 − 386 the answer is 126, not 512.

Losing a hop. Three hops is the practical ceiling for most people without writing anything down. If you need four, write the first one down or pick another method.

Forgetting to cross the benchmark cleanly. From 467 the hop to 500 is 33, not 43 — the complement to the next hundred, not to the next ten. Reading the tens digit and then stopping is the single most common slip.

Practising it

The two skills involved are complements and hop-tracking, and they improve separately. Complements respond to short, frequent drilling; hop-tracking responds to doing the same subtraction by two different routes and comparing.

IQ Test & Brain Training Games, our own app at Reign Creative, has a change-making mode that is this method in its natural setting — an amount tendered, an amount spent, and a 75-second clock — alongside Subtraction Sprint, which includes a borrowing-practice difficulty where counting up is usually the faster answer than the borrow it is named after. The Accuracy Trainer runs the same content with no timer, ten questions at a time, with an explanation available on a miss, which is the right setting while the hop sequence is still being learned. The app itself is described on the IQ Test & Brain Training Games app page.

One caution about practice design: a timer rewards whichever method you already have, not the one you are trying to install. Drill a new method untimed until it is the automatic choice, then put the clock back on. The wider ordering of methods, and which facts each one assumes, is set out in the mental math practice guide.

More of this material sits in the mental math topic hub and under education and brain training articles. England's national curriculum expects mental strategies to be built alongside written methods rather than after them, and counting up is the clearest case of why: the written algorithm and the mental one answer the same question by genuinely different routes, and having both means you can check one against the other.

Good to know

Frequently asked questions

What is the counting up method for subtraction?
Instead of removing the smaller number from the larger, you start at the smaller number and count up to the larger one in convenient jumps, then add the jumps together. For 512 minus 386 you go 386 to 400, 400 to 500, and 500 to 512, giving 14 plus 100 plus 12, which is 126.
Why does counting up give the same answer as taking away?
Because subtraction is defined as a missing addend. Asking what is left after removing 386 from 512 and asking what must be added to 386 to reach 512 are the same question written two ways, so the two procedures cannot disagree.
When is counting up a bad choice?
When the two numbers are far apart and neither sits near a benchmark, because each extra hop is another partial sum to hold. If the gap needs four or five jumps, sliding both numbers by the same amount, or the standard take-away method, will usually cost less.
Is this the same as the shopkeeper's method?
It is. Handing back change is counting up made physical: from the price you count coins and notes until you reach the amount tendered, and the change is the pile you built. The method survives because it requires no borrowing and each step is checkable as you go.

Sources

  1. A Framework for Remediating Number Combination Deficits (opens in a new tab)Fuchs, L. S. et al., Exceptional Children (2010) — accessed
  2. Assisting Students Struggling with Mathematics: Intervention in the Elementary Grades (opens in a new tab)What Works Clearinghouse, Institute of Education Sciences (2021) — accessed
  3. National curriculum in England: mathematics programmes of study (opens in a new tab)Department for Education (GOV.UK) — accessed

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