Lessons · Teaching · adding and subtracting, and what regrouping really is
Regrouping is trading ten of one place for one of the next
Carrying and borrowing are the same move in two directions: ten ones are worth one ten, so ten of anything in one place can be traded for one of the place to its left. Adding trades upward when a place overflows past nine; subtracting trades downward when a place has too little to take from.
Hone is a place to practise a career, one idea a day. This is one of its lessons, written out in full and free to read without an account.
In beta. This lesson was written for Hone and has not yet been checked by a state certification officer. Practice material, not professional advice. What that means.
What it is for
'Borrowing' is a bad name and it teaches a bad idea, because nothing is borrowed and nothing is given back. A child told to 'borrow one from the neighbour' will happily borrow from a zero. Told that they are exchanging one hundred for ten tens, they can see why 300 minus 7 needs two exchanges and not one.
How to think about it
Line the places up, never the digits. Work right to left. When a place goes over nine, write the ones digit and carry the ten to the next place. When a place is too small to subtract from, take one from the place to the left and add ten to this one, and if that place is a zero, go left again until you find something to exchange.
Worked example
47 + 38: seven ones plus eight ones is fifteen onesFifteen is one ten and five ones, so the five stays and the ten moves left.
Write 5, carry 1: four tens plus three tens plus the carried ten is eight tens, so 85The carry is not a mark, it is a ten that changed address.
300 - 7: the ones place has nothing, and so does the tensGo left until there is something to exchange. This is the case that defeats 'borrow from the neighbour'.
Exchange one hundred for ten tens, then one ten for ten ones: 2 hundreds, 9 tens, 10 ones, minus 7 is 293Two exchanges, both of them the same trade.
Your turn
Name how many of one place are traded for a single one of the place to its left.
One of any place is worth of the place to its right
Solve one, graded on the server
The trap
Subtracting the smaller digit from the larger out of habit. Asked for 62 minus 38, a pupil does 8 minus 2 in the ones column because it is easier, gets 36, and has answered a different question. The order in a subtraction is not negotiable.