Counting up treats subtraction as a distance question. Rather than asking what is left after taking 293 away from 471, it asks how far it is from 293 up to 471. The two questions have the same answer, and the second one is often easier to do in your head.
Anyone who has been handed change in a shop has seen this method. A cashier working out change from a twenty rarely subtracts. They count up from the price to the note.
How the method works
- Start at the smaller number.
- Jump up to the next friendly number, usually the next multiple of ten.
- Keep jumping in easy amounts, hundreds and tens, until you reach the larger number.
- Add all the jumps. That total is the difference.
Friendly means whatever is easy for that student. Bigger jumps mean fewer steps but demand more confidence. Smaller jumps are slower and safer. Both give the same answer.
A worked example
471 minus 293, counted up from 293
293 -> 300 jump of 7 300 -> 400 jump of 100 400 -> 471 jump of 71 ------------ total 178
The money version
Counting up is the natural way to work out change, and it is worth practising in that form because the numbers mean something.
Change from 20 dollars for a bill of 13 dollars 65 cents
13.65 -> 13.70 5 cents 13.70 -> 14.00 30 cents 14.00 -> 20.00 6 dollars ------------ 6 dollars 35 cents
The first jump is five cents, the second is thirty cents, and the last is six dollars. Together that is 6.35, which checks out because 13.65 + 6.35 = 20.00.
Why it works
Subtraction has two meanings that give the same number. One is take away: start with 471 and remove 293. The other is difference: how far apart are 293 and 471 on the number line. Counting up measures that distance directly, in convenient pieces. Because the pieces cover the gap exactly once, with no overlap and no gap left over, their total is the whole distance.
Common mistakes
Losing a jump
Three or four jumps are easy to hold in your head, until one of them is not. Write each jump down as you make it. The arithmetic in this method is easy, so almost every wrong answer is a lost jump rather than a bad calculation.
Jumping past the target
Going from 400 straight to 500 when the target is 471 means the last step has to come back down, and the subtraction the method was supposed to avoid has reappeared. Stop at the largest friendly number that is still below the target.
Counting down instead
Counting down works, but it is harder to keep straight and gives away the advantage. Always start at the smaller number and go up.
Check yourself
624 - 358
358 -> 360 2 360 -> 400 40 400 -> 600 200 600 -> 624 24 --- 266
1,000 - 673
673 -> 700 27 700 -> 1000 300 --- 327
Change from 50 dollars for a bill of 31 dollars 20 cents
31.20 -> 32.00 80 cents 32.00 -> 50.00 18 dollars 18 dollars 80 cents
Check: 31.20 + 18.80 = 50.00.
Would 293 to 471 still give 178 with different jumps?
Yes. Try 293 to 393 as a jump of one hundred, then 393 to 471 as a jump of seventy eight. That is 100 + 78 = 178 again. The route does not matter, only the distance.
See also
Sources
- Counting up, sometimes called the shopkeeper method or the counting-up algorithm, is taught as a mental subtraction strategy in Everyday Mathematics and in many other elementary programmes. Examples on this page are our own.