Teaching Column Addition and Subtraction at Home: Start by Lining Up the Digits

In Japan, children learn column addition and subtraction in grade 2, around age 7 or 8. The national course of study for grade 2 says children should be able to add and subtract two-digit numbers and understand how to do it in columns, with simple three-digit cases too. In grade 3 it extends to three and four digits (MEXT, Course of Study for Elementary Schools, 2017, in Japanese). England is similar: the national curriculum asks Year 3 pupils to “add and subtract numbers with up to 3 digits, using formal written methods of columnar addition and subtraction” (National curriculum in England: mathematics programmes of study).

The column method is less a way of calculating than a set of rules for writing. If you write by the rules, the same steps give the answer however many digits there are. If the writing goes wrong, the answer goes wrong even when each small sum is right. In this post I go through how to teach it at home, starting with lining up the digits.

Different ways of writing it

Before the steps, a word about style. The column method looks slightly different from country to country and even from school to school. In Japan the carried 1 is written small near the tens digit, above it or just above the line depending on the textbook, and in subtraction the digit you borrow from is crossed out with the new value written above it. In many UK classrooms the carried digit goes small below the answer line, and borrowing is called “exchanging”. In many US books the carried digit sits above the top number, and the whole idea is called “regrouping”.

None of these is more correct than the others. The steps are the same. If your child’s school teaches one style, I think it is best to use the same style at home.

1. Line up the place values

The first and most important thing, I believe, is to line up the digits by place value. Ones under ones, tens under tens. Line them up on the right.

Here is my own example. When writing 34 + 5 in columns, a child may put the 5 under the 3.

Wrong

   3 4
 + 5
 _____
   8 4

Right

   3 4
 +   5
 _____
   3 9

On the left, the 5 has been added to the tens, so the answer comes out as 84. The 5 is a ones digit, so it belongs under the 4, and the answer is 39.

Children write from left to right, so they want to line numbers up on the left too. A two-digit number with a one-digit number is where this happens most. I think it helps to agree in words, right at the start, that numbers line up on the right edge.

2. Use squared paper

For lining up digits, squared paper helps more than anything, I think. If each square holds one digit, the columns become the place values.

Without squares, digits vary in size and spacing, and the columns drift a little more with every line. With few digits the answer still comes out, but at three or four digits it becomes unclear which digit belongs to which place.

How long to use squared paper is up to the child. Once they can keep the columns straight without it, I think it is fine to move to plain paper.

3. Draw the line and start from the ones

After writing the numbers in columns, draw a line under them. The answer goes below the line.

Start from the ones, working right to left. This is the opposite of the reading direction, so children sometimes want to start with the tens. Without carrying, the answer still comes out right. With carrying, they end up correcting the tens after writing them. I think it is better to make “start from the ones” a habit from the beginning.

4. Write the carried 1

My own example: 47 + 38.

   1
   4 7
 + 3 8
 _____
   8 5

1. Ones: 7 + 8 = 15. Write 5 in the ones and carry 1 to the tens.
2. Tens: 4 + 3 = 7, plus the carried 1 makes 8.
3. The answer is 85.

As I said above, where to write the carried 1 depends on the style. I think writing it in the same place every time matters more than which place you choose.

The most common mistake is forgetting to add it. In the example above, forgetting gives 75. If the child points at the carried 1 before adding the tens, it is harder to forget.

5. In subtraction, borrow from the next place in the top number

The most common mistake in column subtraction, I think, is swapping top and bottom.

My own example: 52 − 37.

Wrong

   5 2
 − 3 7
 _____
   2 5

Right

   4
   5 2
 − 3 7
 _____
   1 5

In the ones, 2 − 7 does not work, so a child may do 7 − 2 = 5 instead. In column subtraction you always take the bottom number from the top number.

The right steps:

1. Ones: you cannot take 7 from 2, so take 10 from the tens. Cross out the 5 in the tens and write 4.
2. The ones become 12. 12 − 7 = 5.
3. Tens: 4 − 3 = 1.
4. The answer is 15.

The best way to catch a swapped subtraction, I think, is to estimate first. 52 − 37 is roughly 50 − 40, so the answer should be a little over 10. A child who knows that can see that 25 is wrong.

Later come problems like 302 − 127, where the tens digit is 0 and there is nothing to borrow. You borrow from the hundreds into the tens, then from the tens into the ones, twice in a row. If a child gets stuck here, going back to two-digit borrowing usually sorts it out. The idea behind carrying and borrowing is in How to Teach Carrying and Borrowing (Regrouping).

6. Check the answer

There are two ways to check.

One is to compare with the estimate. The other, for subtraction, is to add back. If 52 − 37 = 15, then 15 + 37 should make 52.

I do not think every answer needs checking. But when mistakes keep happening, these two checks make it easier to see where things went wrong.

An order for teaching at home

Putting it together, this is the order I think works:

1. Two-digit plus one-digit and two-digit plus two-digit without carrying, practising only the writing
2. Addition with carrying
3. Subtraction without borrowing
4. Subtraction with borrowing

At stage 1, it is fine if the child could do the sums in their head. The point is to learn the writing. Move on to carrying and borrowing once the writing is solid. If the child does not see why you “take 10”, go back to real objects: lay out 52 with five 10-coins and two 1-coins, and change one 10-coin for ten 1-coins.

The Column Arithmetic Drill in MathCal

MathCal is an arithmetic drill app where children write their answers by hand on the screen. I make it.

The Column Arithmetic Drill gives column addition and subtraction from one to five digits, already lined up by place value. Answers are handwritten one digit per box. You choose the number of digits for X and Y, so a grade 2 child can start with two digits and one, or two and two. Each recognised digit is shown small at the edge of its box, and boxes that could not be read change colour. Check answer cannot be pressed until every box has been read, so nothing gets left blank. Besides the answer boxes, the child can write freely over the problem, for example to note a carried 1.

The drill is used in portrait. My reason is simple: column arithmetic is written downwards, so the screen has to be tall. I have also adjusted how the boxes are read and checked by watching my daughter use it and fixing what went wrong.

To see the steps, there are Column Addition and Column Subtraction lessons. The problem is lined up on the right, the column being worked on lights up, and each tap reveals the answer for one column. The carried 1 and the borrowing marks are drawn small. I think understanding and practice should turn together, like two wheels, so the idea is to watch the steps in a lesson and then use your hands in the drill.

Today’s Drill, Today’s Challenge and Today’s Lesson on the home screen are free every day. Choosing any drill or lesson from the full lists requires a one-time purchase. There are no ads. An overview of arithmetic from grade 1 to 6 is in What Kids Learn in Arithmetic, Grade by Grade.