How to Teach Carrying and Borrowing (Regrouping)

In the United States, the Common Core standards ask first graders to add and subtract within 20, and they name the strategies to use: “making ten (e.g., 8 + 6 = 8 + 2 + 4 = 10 + 4 = 14)” and “decomposing a number leading to a ten (e.g., 13 − 4 = 13 − 3 − 1 = 10 − 1 = 9)” (Common Core State Standards, Grade 1, Operations and Algebraic Thinking). The written method for multi-digit numbers, the standard algorithm, follows over the next few grades. Japan teaches the single-digit part, sums like 8 + 5 that pass 10 and the subtractions that undo them, in the first year of primary school.

Many teachers now say “regrouping” rather than “carrying” and “borrowing”, partly because “borrowing” suggests something will be given back, which it never is. I use the older words in this post because most parents learned them, but they mean the same thing.

This is a point where many children stumble. Until now, counting on fingers was enough. With 8 + 5, there are no fingers left. Below I go through the order in which Japanese schools teach it, because I find that order clear, and what parents can do at home.

The base: ten as one unit

Carrying and borrowing work the same way. You treat ten as one bundle. That is all.

So before carrying, a child needs the pairs that make ten. Asked “how many more to make 10 from 8?”, they should say “2” without counting. 1 and 9, 2 and 8, 3 and 7, 4 and 6, 5 and 5. These five pairs appear in every carrying and borrowing problem within 20.

When a child stumbles on 8 + 5, I think the real gap is often here. If so, it is quicker to pause carrying and firm up the pairs first. Lay out 10 counters, cover some with a hand, and ask how many are hidden. That is plenty of practice.

Adding with carrying

Take 8 + 5. Japanese textbooks usually think it through like this:

1. 8 needs 2 more to make 10.
2. Split 5 into 2 and 3.
3. 8 and 2 make 10.
4. 10 and 3 make 13.

In Japan the split is often drawn as two branches under the 5. The shape looks like a pair of cherries on a stem, so it is sometimes called sakuranbo keisan, cherry calculation.

 8 + 5
    / \
   2   3

 8 + 2 = 10    10 + 3 = 13

The useful habit is to make the bigger number into 10. For 8 + 5, filling up the 8 takes fewer steps than filling up the 5. For 3 + 9, fill up the 9: split 3 into 1 and 2, 9 and 1 make 10, 10 and 2 make 12. Seeing that either number can be made into 10 is part of this stage.

Subtracting with borrowing

Take 13 − 8. The method Japanese first graders learn first is this:

1. Split 13 into 10 and 3.
2. 8 cannot come out of 3, so take 8 from the 10, which leaves 2.
3. 2 and 3 make 5.

You subtract from the ten and then add the rest back. In Japanese this is called genkahō, the subtract-then-add method. In English it is sometimes described as “subtracting from ten”.

The other method suits problems like 12 − 3, where the number taken away is small:

1. Split 3 into 2 and 1.
2. 12 take away 2 is 10.
3. 10 take away 1 is 9.

You subtract twice, so it is called gengenhō, the subtract-subtract method. It is the “decomposing a number leading to a ten” in the Common Core example above. Japanese textbooks usually teach subtracting from ten first and this one after.

Taking 8 from 10 is just one of the pairs that make ten. The base is the same.

Which is harder?

I believe borrowing trips children up more than carrying.

Carrying is “add, then add”, always in one direction. Subtracting from ten is “split, subtract, add”. There is an addition in the middle of a subtraction. Why add when you are taking away? I think this switch of direction is the confusing part.

The other reason is that seeing 13 as “10 and 3” takes practice in itself. A child who only sees thirteen as one number will not think of splitting it. Here, real objects help most. Use a bundle of ten sticks and three loose ones, or a ten bar and three unit cubes, and lay out 13 as one ten and three ones.

Carrying and borrowing in column arithmetic

With two-digit numbers and the written column method, the same idea appears in place value.

When the ones add up to 10 or more, send one ten over to the tens column. That is carrying. When the ones cannot be subtracted, take one ten from the tens column and put it in the ones as ten ones. That is borrowing.

If a child understands “ten as one unit” from the single-digit stage, the column method only changes how things are written down. And if a child is lost in column borrowing, going back to 13 − 8 usually sorts it out.

Where to go back when a child is stuck, and what not to say

Counting on fingers. Not wrong. But counting up from 8 one by one is slow and invites miscounts. Go back to the pairs that make ten.

Right answers, but cannot explain the split. The child may be doing it in their head. I don’t think they need to draw cherries every time. If they are quick and right without drawing, move on.

Adding when they should subtract. In subtracting from ten, the child is unsure what gets added to what. Act it out with counters: take from the ten, then gather what is left with the loose ones.

Things not to say. “Why can’t you do this?” and “We just did this” are easy to say, but in my view they push children toward guessing. So does “hurry up”. Speed can wait until the answers are reliably right.

Where MathCal fits

MathCal is a math drill app I make, where children write their answers on the screen by hand.

To see ten as one unit, there are bead lessons. In Addition Beads, when ten unit beads gather, the child taps to exchange them for one ten bar. In Subtraction Beads it goes the other way: when there are not enough ones, one ten bar is broken into ten unit beads. Both start with problems that need no regrouping and move on to ones that do. In Subtraction Beads, borrowing comes up in two-digit minus two-digit problems. For the written method, there are Column Addition and Column Subtraction lessons too.

For practice, use the Basic Calculation Drill. Turn on No Subtraction, No Multiplication and No Division, and only single-digit addition remains. You can start with No Carrying or Borrowing switched on, so all answers stay at 9 or below, and then switch it off to mix in problems like 8 + 5. Enable Dots on Number shows dots for each number, so a child can see the quantities while thinking. Note that subtraction in this drill is single digit minus single digit only, so it does not cover problems like 13 − 8.

For borrowing and two-digit work, there is the Column Arithmetic Drill. It gives column addition and subtraction from 1 to 5 digits, answered by hand one digit per cell. Set X digits to 2 and Y digits to 1 for two-digit and one-digit problems, with carrying and borrowing mixed in.

Today’s Drill, Today’s Challenge and Today’s Lesson on the home screen are free every day. If no birthday is set, Today’s Lesson is chosen from Addition Beads and Subtraction Beads. Choosing any drill or lesson from the full lists requires a one-time purchase.

For times tables, see How to Learn Times Tables: Say Them, Write Them, Use Them. For daily routines, see Ideas for Getting More out of MathCal.

Carrying and borrowing are both about ten as one unit. When they look hard, going back to the pairs that make ten is usually the shortest way forward.