Introducing multiplication

Forget any idea you ever had that multiplication is difficult.

Unfortunately, some schools introduce this topic formally before children have grasped the relevant concepts, and then insist on pages of busy-work to try and get the technique implanted in their minds. Some will grasp it, and find the repetition tedious; others might master the technique, but with little understanding of what they’re doing. Still others will flounder, and may decide at the tender age of 6 or 7 that maths is far too difficult and should be avoided whenever possible. Sadly, very few children enjoy multiplication: the reason seems to be that they are not left to discover how it works when they are ready to do so.

When you’re home educating, you can choose the time to introduce your child to new concepts, and suit them to everyday life. Some children will see at quite a young age that there is a short-hand to repeated addition; others may not be ready or interested until they are almost teenagers. It doesn’t matter! Nevertheless, many children find that simple multiplication is easy to understand, when nobody has told them that it’s difficult. You don’t even need to tell them the word ‘multiplication’ when they first begin to use it.

Multiplying small numbers

When you think that your child might be able to understand simple multiplying, suggest an investigation for him: take three boxes, each with four chocolate bars, or something similar, and ask how many chocolate bars you have altogether. At first your child will want to count them. When he gets to twelve, tell him – casually – that he has the right answer. That’s the first lesson! Another day, try to use something quite different – for instance, giving three people each four cup-cakes. Ask how many you think you will need. Then count them together. Or find out how many legs three cats have.

You should be able to find several relevant ‘problems’ that your child can understand. This should take place over several days, but keep the same numbers to start with. Sooner or later, your child will say, ‘Twelve,’ without counting. This is a great discovery! When your child has truly understood that three lots of four ANYTHING make twelve, try playing with four lots of three. Give four people each three cookies. To adults it’s obvious that three lots of four is the same as four lots of three, but to a young child it’s not obvious at all. Stay patient. When your child sees this, he has begun an algebraic understanding that will be useful in any branch of mathematics that he ever takes.

To reinforce the concept, find three lego pieces that are four units long, and four lego pieces that are three units long. Line them up. Then – if your child is still interested – find two lego pieces that are six units long, and six lego pieces that are two units long. Ask the child what he expects. Will they be longer or shorter than the three lots of four units? If your child is ready for this stage of arithmetic, he will probably be quite excited, and experiment with other lego. Perhaps he has a piece that’s 12 units long. Or you can find twelve single-unit pieces. Encourage as much experimentation as your child wants, as he sees the relevance of numbers in his toys. On the other hand, if he’s not remotely interested and wonders what you’re talking about, put the idea aside, and build cars and trucks with his lego for a while.

Multiplication grid

But assuming there’s some interest, build up your child’s knowledge slowly, over several weeks or even months. Once he’s played around with the various options that multiply together to make 12, try some different experiments. How about five lots of four chocolate bars? How many legs do five dogs have? This time your child will probably realise much more quickly that there is a pattern, and that five lots of four ‘anything’ will come to 20. Continue to keep it low-key, but suggest building up a wall-chart so that he can see the pattern. Start with a 5×5 grid like this:

. 1 2 3 4 5
1 . . . . .
2 . . . . .
3 . . . . .
4 . . . . .
5 . . . . .

As your child ‘discovers’ a new multiplication fact, and becomes certain of it, add it in to the chart in the correct place, and keep it in a prominent place on the wall. You can ask him what he thinks should go in the column under ‘1’. If he doesn’t know, ask him to think about it for a while and tell you when he thinks he knows. The idea of multiplying by the number one to get the same number is another concept that is quite tricky to grasp sometimes. Eventually – and again this may take several months – you will have a complete chart like this:

. 1 2 3 4 5
1 1 2 3 4 5
2 2 4 6 8 10
3 3 6 9 12 15
4 4 8 12 16 20
5 5 10 15 20 25

This may be sufficient, or your child may ask about bigger numbers. If so, you could make a bigger 10×10 chart and fill in the results slowly. Show your child how to use a calculator, if he doesn’t know already, so he can check his answers, or discover patterns. Numbers can be fascinating! For instance, suggest that he keeps on adding up fives, and see when he can guess what the next one is going to be. Then try tens. It’s another big step forward in mathematical thinking when a child realises that multiplying by ten is, in effect, putting a zero after a number. This is the start of understanding ‘place value’, which is quite a stumbling block for many children (and adults!) To understand this concept – and other patterns that occur in multiplication – see the page on multiplication tricks.

Multiplication tables

What about learning tables? Traditionally, children chanted by rote, day after day, until the tables were ingrained in their minds. If you learned that way, you probably never forgot them, but can you remember something like 7×6 instantly, or do you need to chant your way mentally through the table first? In today’s age of calculators and computers, table learning is no longer necessary, but many people believe it’s still a good idea.

If you have a child who is good at learning by heart, then by all means introduce him to table chanting. There are cassette tapes available with table songs to listen to, if they appeal. Or be modern and try a ‘tables rap’, which I heard one enlightened classroom teacher using. It’s surprisingly easy to fit tables into a rap rhythm: three ONES are three, three TWOS are six; three THREES are nine, three FOURS are twelve….

On the other hand, if your child finds rote-learning confusing, and gets muddled when trying to sing or chant, don’t worry about it. It’s far more important to understand the concepts of multiplication than to know the facts by rote. The more familiar your child is with handling numbers and thinking about multiplying, the more he will remember ‘facts’ in his own time. If he can remember those up to 5 x 5 as on the grid above, he can always do the higher ones by other techniques – for instance as described in my article ‘multiplying on fingers’ for those between 6×6 and 10×10.

Alternatively, he may be interested in the patterns in tables, or have a quick grasp of the relationships between numbers. If he can double quickly, then any time he wants to know a multiple of 8, he need only remember the relevant multiple of 4, and double it. Knowing that 4 x 3 is 12, he can quickly work out that 8 x 3 is 24. Or perhaps he prefers to think of 8 as nearly 10 – in which case he could first work remember that 3 x 10 is 40, and then subtract 3 x 2, which is 6, to get 24 another way. There are lots of ways to multiply, and a child will have an excellent introduction to multiplication if he is encouraged to find his own ways, and to have a calculator always on hand for the times when he’s not sure, or can’t remember.

More articles about introducing children to maths informally:

Number bonds – beginning addition
Basic division

Algebra for six-year-olds
Prime numbers and factors

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