Before reading this article, you might like to look at ‘Introducing multiplication‘ if you haven’t already done so: a page which gives an introduction to this topic and shows how you can gently show your child what multiplication is without any drill or boredom.
If your child is interested in the topic, or if you decide to use some kind of curriculum or workbooks that have written arithmetic, it may be tempting to think that a load of number facts or tables may need to be learned before any progress can be made.
Some children do like rote-learning, in which case they may find it easy to chant tables; others may intuitively understand how multiplication works. Some kinaesthetic learners may like to multiply on their fingers, and still others may want to use a calculator every time. There are no right or wrong methods, and no real reason why anyone needs to be able to do multiplication without a calculator, unless it interests them.
However, for those who find numbers fascinating – and those who want quick short-cuts – there are several number patterns that show up in multiplication, sometimes known as ‘tricks’.
Multiplication tricks and short-cuts
The most obvious example is multiplying by ten. Nobody needs to learn the ten times table: all they need is to understand what the decimal system of counting is and what we mean by a zero. Then it’s clear that ten times two is twenty; ten times three is thirty, and so on.
Try not to think of it as ‘adding a zero’ to the number, although that’s what it looks like. But you’re not actually adding a zero (two added to zero is, after all, still two) – you’re moving the digit one place to the left and then, since you can’t leave a space, putting a zero to mark the ‘units’ place. This is important to understand, otherwise when your child wants to multiply (say) two-and-a-half by ten and realises that two-and-a-half is 2.5 he might assume the answer is 2.50 whereas it is, of course, 25. No zero necessary, because the five is in the ‘units’ place.
To show your child how to multiply by ten, if he has not already grasped the concept intuitively, show him the table:
1×10=10
2×10=20
3×10=30
4×10=40
.. and so on, and ask if he sees the pattern. He may see it instantly, or it may take some time. Patterns are not always obvious to young children.
Multiplying by eleven is the next easiest. You multiply by ten, and then add the number you’re multiplying. So for numbers up to nine there’s a very clear pattern:
1×11=11
1×11=22
3×11=33
4×11=44
.. and so on. If your child has a basic understanding of simple equations, then he may see that this is an example of a(b+1) = ab + a where b, in the eleven times table, is 10.
For two-digit numbers greater than nine, the pattern isn’t quite so obvious but it’s still easy to multiply by eleven. For ten times, of course, the answer is 110. For others, the algebraic equation still holds, but a number has to be ‘carried’. For example,
15×11 = (15×10) + (15×1) which is 150 + 15, or 165
35×11 = (35×10) + (35×1) which is 350 + 35, or 385
There is still a pattern, or trick: you leave the units digit in the same place, move the tens digit to the hundreds place, and then add the digits together to give the tens digit. (In those examples, 6=1+5, and 8=3+5). Why does this work? If you lay the sums out as long multiplication, you should be able to see it. Of course, if you’re multiplying 11 by a number whose digits add up to ten or more, you’ll have to do a bit more carrying. For instance:
29×11 = (29×10) + (29×1) = 290 + 29 = 319
This isn’t quite so neat, but you can still think of it in the same way: move the two to the hundreds place, add two and nine together to get 11, put the 1 in the tens place, and then increase the two to three.
Those who know the two times table well (and this is one which is well worth being able to know without thinking about it) will quickly realise that multiplying by 12 uses the same kind of algebraic equation: in other words, multiply by ten, multiply by two, and add the results. No neat patterns, but a quick way of doing this in your head.
Multiplying by lower numbers
Multiplying by five is straightforward: multiply by ten and then halve the result. For example:
4×5 = half of 4×10
4×10=40, so 4×5=20
If you’re multiplying five by an odd number such as 5 or 7, the result will end in 5 rather than zero. For example:
7×5 = half of 7×10
7×10=70, so 7×5=35
Multiplying by two and three are worth understanding so well that your child can do them in his head, as they’re the basis for so many other multiplication tables. There aren’t any easy tricks to these, but there are checks you can do having done the sum.
Anything multiplied by two must give an even number: in other words, the last digit must be 2, 4, 6, 8 or 0.
Anything multiplied by three must give a number whose digits add up to three, six or nine. For example:
3 x 12 = 36. (3+6=9)
3 x 23 = 69 (6+9=15 which is still two digits, so add again: 1+5=6)
Multiplying by four isn’t too difficult if you know the two times table well and can double numbers in your head: just multiply by two, and then multiply by two again! For example:
4×6 = 2 x (2×6)
2×6=12, so 4×6=24
Multiplying by eight is then another double. So:
8×6 = 2 x (4×6) = 48
Multiplying by nine can be done in various ways. You can treat it similarly to multiplying by 11, but subtracting rather than adding. In other words, multiply by ten and then take away the number you’re multiplying by. For example:
9×12 = (10×12) – 12
= 120 – 12
= 108
Alternatively you can multiply by three and then multiply by three again. For multiplying nine by numbers less than ten, there’s quite a nice pattern:
9×2=18
9×3=27
9×4=36
9×5=45
… and so on. The tens increase, the units decrease, and the sum of the two digits is nine. Indeed, the sum of the digits of any result from multiplying by nine will be 9, so that’s a check that can be done at the end, similar to the check on multiplying by three.
Multiplying by six and seven are often thought of as the most difficult, as there are no easy patterns or quick techniques. Still, it’s not difficult to see that multiplying by six is the same as multiplying by two and then by three (or by three and then by two – the result will be the same). Or you could multiply by five and then add the number you’re multiplying by. For sevens, you could multiply by five and then double the number you’re multiplying and add that… but the more computations you have to do, the more likelihood there is of making mistakes.
So… the easiest way to multiply anything between six and nine, if you can easily do multiplication of anything up to five, is to multiply on your fingers – an ancient technique that’s rarely taught in schools.
More articles about introducing children to maths informally:
Algebra for six-year-olds
Prime numbers and factors
Geometrical shapes
Writing fractions