Adding and Subtracting With Regrouping
Carrying and borrowing are the same move in opposite directions: ten of one column is worth one of the next. Everything else is bookkeeping.
What regrouping means
Ten of any column is worth exactly one of the column to its left.
Ten ones make a ten. Ten tens make a hundred. That is the whole of regrouping, and both carrying and borrowing are just that fact used in one direction or the other. A child who has been told to "carry the one" without ever being told what the one is has been given a ritual, and rituals fall apart the moment a question looks slightly unfamiliar.
The same move, both ways
Adding: a column holds too much, so ten of it is swapped for one of the next column along. That is carrying.
Subtracting: a column does not hold enough, so one from the next column along is swapped back into ten. That is borrowing.
Carrying, in addition
Start at the right, always, because that is the only direction in which a carry can be dealt with as soon as it appears. Add each column, and if the total reaches ten or more, write the ones digit and carry the tens digit into the next column.
Work out 3,487 + 2,695.
- Ones: 7 + 5 = 12. Write 2, carry 1 into the tens.
- Tens: 8 + 9 + 1 = 18. Write 8, carry 1 into the hundreds.
- Hundreds: 4 + 6 + 1 = 11. Write 1, carry 1 into the thousands.
- Thousands: 3 + 2 + 1 = 6.
6,182
Borrowing, in subtraction
Again start at the right. If the top digit is too small to take the bottom one from, go next door and take one, which arrives as ten. The column you took from is now worth one less, and that is the part children forget to write down.
Borrowing is a bad name
Nothing is given back. It is an exchange: one ten is swapped for ten ones, and the total has not changed at all. Calling it exchanging rather than borrowing removes a surprising amount of confusion, because children keep waiting for the repayment step.
Subtracting across zeros
This is the case that stops children, and it is worth its own section. In 6,002 minus 1,847 the ones column needs to take 7 from 2, so it goes next door for a ten. But the tens column is a zero, and so is the hundreds. There is nothing there to take.
So you keep going left until you find a column that has something, and then exchange down through each empty column on the way back. It is not a special rule. It is the same exchange happening three times in a row.
Work out 6,002 - 1,847.
- Ones need a ten. Tens and hundreds are both zero, so go to the thousands.
- Exchange: 6 thousands becomes 5 thousands and 10 hundreds. Then 10 hundreds becomes 9 hundreds and 10 tens. Then 10 tens becomes 9 tens and 10 ones.
- Now the ones column has 12. 12 - 7 = 5.
- Tens: 9 - 4 = 5. Hundreds: 9 - 8 = 1. Thousands: 5 - 1 = 4.
4,155
Common mistakes
1. Taking the smaller digit from the larger one, whichever is on top
Seeing 2 above 7 and writing 5, because 7 take 2 is 5. It is the single most common subtraction error there is, and it is not carelessness: the child has decided subtraction means "find the difference between these two digits". The top number is the one being reduced, so the direction is fixed.
2. Carrying but not adding the carry in
Writing the small 1 above the next column and then adding that column without it. The carry is not a note about what happened, it is a number that has to join the next sum. Reading each column out loud including the carry catches this instantly.
3. Not lining the columns up
Writing 348 above 27 with the 2 under the 3, because both numbers were pushed to the left. Ones must sit under ones. Squared paper fixes this permanently, and it fixes it faster than any amount of explaining.
Practice questions
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What to learn next
Column addition and subtraction are the machinery that long multiplication and long division are built on, so it is worth having them solid before either.
For parents
If your child is stuck on subtraction, give them coins rather than a worksheet. Ask them to pay 47 cents from two twenty-cent pieces and a ten, and they will exchange without being told to, because the coins make the exchange physical. Then show them that the written method is a record of exactly what their hands just did.
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