Regrouping is the step where two-digit arithmetic stops being "do each column on its own" and starts depending on place value. Children who learn it as a rule ("carry the one") tend to make the same few mistakes again and again. Children who see why it works usually stop making them.
What regrouping means
Our number system is built on tens. Ten ones make a ten; ten tens make a hundred. Regrouping is simply trading between those places:
- In addition, when a column adds up to 10 or more, ten of its ones are traded for one ten, which joins the next column. In 47 + 38, the ones make 15: that is 1 ten and 5 ones, so 5 stays and the 1 ten moves to the tens column.
- In subtraction, when the top digit in a column is smaller than the bottom digit, one ten is traded for ten ones. In 72 − 38, you cannot take 8 from 2, so one of the 7 tens becomes ten ones: 6 tens and 12 ones. Then 12 − 8 is 4, and 6 − 3 is 3.
"Carrying" and "borrowing" are older names for the same two moves.
Show it before you write it
Before the written method, let children regroup with real objects:
- Build the numbers with base-ten blocks, bundled straws, or coins (dimes and pennies work well).
- Combine or take away the ones first.
- Trade whenever you reach ten loose ones (or run short of them), and say it out loud: "Ten ones make one ten."
- Record what happened in the column method, one step at a time.
The little "1" written above the tens column then means something: it is the ten that was just made.
The most common mistakes
Subtracting the smaller digit from the larger. In 52 − 17, a child writes 7 − 2 = 5 in the ones column, giving 45 instead of 35. The fix is to always read the column top-down: "2 take away 7, can I do that?" If not, regroup.
Forgetting the carried ten. The ones are right, but the tens column is one short. Have children write the carried 1 in a fixed spot above the tens column and cross it off once it has been added.
Regrouping when it isn't needed. Once children learn to regroup, some do it in every problem. Mixing problems with and without regrouping makes them check each column first.
Losing track across a zero. In 403 − 167, there are no tens to trade from, so a hundred becomes ten tens first, and one of those becomes ten ones. Practice this case on its own after two-digit regrouping is secure.
A practice order that builds
Each step adds one new idea:
- Two-digit addition without regrouping, so the column method itself is familiar.
- Two-digit addition with regrouping.
- Two-digit subtraction without regrouping, then with it.
- A mix of problems that do and don't need regrouping.
- Three-digit addition and subtraction, including subtraction across zeros.
For addition and subtraction, the math generator can deal problems that all need regrouping, that never need it, or a mix, so you can stay on one step as long as you need.
Keep pages short
Ten to twenty problems, checked right away with the answer key, teaches more than a long page done without feedback. When a mistake shows up, ask the child to explain the problem with blocks or coins again before trying the next one.