Multiplying two-digit numbers

You can introduce your child to the general concepts of multiplication at quite a young age – see my article ‘Introducing multiplication’ for some ideas about how to do this. You can then reinforce the concept by helping him understand various multiplication tricks and patterns, and show how how to multiply on his fingers. But moving from the practical application to written arithmetic can cause difficulties for many children.

All too often, text-books present multiplication as a formula to be learned, without full and complete explanation as to why it works. Children may follow examples, and even work some correctly on their own. But if they don’t know why the technique is successful, they are likely to get it wrong – perhaps missing out a line, or forgetting to multiply by tens or hundreds as appropriate.

No amount of drill or busy-work will help a child to understand something which isn’t explained properly in the first place. So if you have any doubts about your child’s understanding of the basic concept of multiplication, don’t introduce it on paper just yet!

When you’re sure your child is familiar with the principles, you can see if he’s ready to think about learning about multiplying two-digit numbers, and the shorthand techniques for doing so.

Firstly, however, introduce the idea of simple multiplication in terms of carpet tiles on a floor. If you want to multiply 4 by 6, for instance, you could represent it like this:

Assume that there are 6 cm (or inches, if you prefer) along the top, and four down the side. The total, which can be counted – or checked on a multiplication grid – is 24.

It’s also easy to multiply by 10 or by multiples of 10. Again, if your child is not familiar with doing this, don’t continue any further with this topic until he is. The page Multiplication tricks may help. He should be able to see for himself how to multiply by 10, and also by 100, and have a good understanding of place value, before he can fully understand written multiplication.

So, to check that he understands about multiplying by ten, ask what would happen if the grid above were 6 by 40 cm instead of 6 by 4. He should be able to tell you that there would be 240 squares in all. If it were 60 by 40, there would be 2400. Before doing any long multiplication, it’s crucial to understand about place value, and how to multiply and divide by 10. If your child has any difficulty with these concepts, leave long multiplication alone for a while.

Assuming he’s happy with these ideas, what happens if you want to multiply 6 by a larger number such as 64? You’re unlikely to want to create a tables chart that goes up that far. Of course, you could use a calculator. If that’s your child’s immediate choice, encourage it. But offer to show him how to do it on paper as well.

If he has been introduced to multiplication in ways that are enjoyable, he will probably want to know how he can multiply larger numbers without the calculator. So draw another diagram. Rather than dividing into all the little squares, here’s a simple rectangle:

Imagine that the top is 64 cm and the side is 6 cm high. To a child who has only done simple multiplication, this could look quite daunting. But ask your child how you might split the rectangle up into countable sections.

For instance, he might suggest marking the top in tens, so that you would have six lots of 10 x 6 rectangles, and a 4 x 6 rectangle at the end. This would be an excellent way of finding out the answer, and a child is most likely to remember things like this which he has discovered himself.

Once your child has tried various methods, show him (if he hasn’t thought of it for himself) how you could divide it into two rectangles, one of 60 x 6 and the other 4 x 6, like this:

The larger, yellow section represents 60 x 6, the smaller, blue section at the right represents 4 x 6. Since 6 x 6 = 36, your child knows that 6 x 60 = 360. He also knows that 4 x 6 = 24. Adding them together, you get a total of 384.

It’s a good idea to check this kind of answer on a calculator, to be sure you haven’t made any mistakes. It’s also fun for a child to see that he has found the right result! If he likes this challenge, try some other, similar problems which he can do by splitting into two simple multiplication problems, and then adding them together.

When your child is familiar with this kind of problem solving – and this may happen in half an hour when he’s in the mood for concentrating on new ideas, or it may take place over several weeks, or even months – then is the time to show how he can write out a multiplication problem on paper, as a shorthand for a diagram and repeated addition:

x 6

Explain that the first thing to do is to multiply the 6 by the 4 of 64, which gives 24, just as it did when using the diagram technique. If your child is not yet familiar with ‘carrying’ tens, simply write the 24 down:

x 6

Now explain that the 6 must also be multiplied by the 60 part of 64. Don’t say ‘by the 6’ because many children become confused at that. The ‘6’ in ’64’ is not representing the number 6, but the number 60. When we multiply 6 by 60, we get 360. So put that underneath the 24:

x 6

Then add 24 to 360 – by whatever method your child likes to add – and you will get the answer 384.If your child is confident with carrying units, and fairly complex addition, you can explain that, when you multiply the 6 by the 4 and get 24, you usually just write down the ‘4’, as that’s the ‘units’ part of the answer, and remember that there are an extra 20 units waiting. This is often written as a little ‘2’ under the tens column, like this:

x 6

Then when you multiply 6 by 60, and get 360, you must add in the extra 20 units which are waiting, to get 380. As you already have the units part of the answer, add 380 to 4 and again you will have 384.

This is a complicated process for a child to grasp, and needs a great deal of practice. Not busywork or drill, which generally just make a child bored with the whole subject, but examples in real life, and problems to solve at odd moments. Keep returning to the idea that multiplication is repeated addition, and counting out lego pieces or carpet tiles or measuring anything appropriate that shows in practical terms what multiplication is.

If your child enjoys working out new methods, and finds the techniques fun, suggest that you could give him a new problem each week, and challenge him to solve it in three different ways each time. If he finds this easy, you could see if he can work out how to multiply a three-digit number by a single-digit number. Start with lower numbers, to avoid ‘carrying’, such as 213 x 3.

Always encourage your child to check his workings on the calculator. This teaches him to check his work, and also gives immediate feedback which is non-emotive. He may surprise you by trying much higher numbers and developing his own techniques. That’s fine: there are usually several possible methods to solve any kind of mathematical problem, and none is inhernently better than any other.

Eventually your child will want to know the quickest techniques of solving problems on paper, but in the meantime he will enjoy maths much more if he has as much time and space as he likes to work out number relationships in his own way.

For more articles about teaching basic maths, see:

Number bonds – beginning addition
Introducing fractions
Introducing algebra

Prime numbers and factors

If you have older children, and are concerned about teaching them maths, see:

Maths and the home educated teen Understanding anglesSimultaneous equations