Multiplication today

When we were in school the way we were taught was basic, we were given a method and we learned it by rote; it worked so we kept doing it regardless of the fact we had no real connection as to why it worked or how, we just did it. That’s fine until we need to adapt the way we do the same thing with something different such as decimals, then we had to use a completely new method to deal with them. We became a series of methods but there was no real glue of “why” holding them all together.

Back in the 90s it was finally accepted there was a better way of teaching and that was to give the children methods with explanations as to how they worked and although some of the ‘how’ may not be totally clear when they first came across them, they would as they got older and more confident in using the skill. Multiplication is one of them and the methods they use these days for long multiplication may seem cumbersome and strange but in fact they rely on methods which we employed even if we didn’t realise at the time.

It all comes down to place value

Let’s imagine we have a sum; let’s say 25 x 7 there are several ways we could approach this. 

1.  Think of it as 4 lots of 25 make 100, so that would be 175 [4x25 =100 and 3x25=75]
2.  20x7 and 5x7, by splitting up the 25
3.  

25

x7

175

3

      

The ‘old fashioned way’ of doing it which most of us are familiar with.

The first is a way we could do it in our heads quite quickly if we can ‘see’ the concept of 4 lots of 25 is 100 and most children by the time they are in year 5 have little problem with this. The second way, by splitting the number into its tens and units is also a mental maths way of dealing with it as long as they have developed the skill to be able to ‘hold the figures’ in their minds. Many do and they find this a logical way forward because the place value is maintained – the 2 remains a 20 and not, as we would say with the third method, a 2.

All three methods are valid and should be methods used by a child in year 5 and 6. As quick methods I would still use them with children in the secondary school and into adulthood, they do not lose heir validity just because we get older. J

Ok so we have three ways of looking at simple multiplication but what about the one time dreaded long multiplication? Modern methods favour a much more gentle transition from short to long; they are after all an extension of one another not an invention of a brand new concept as we would have experienced it. Let’s face it the only difference is we have to wield more numbers.

There are 3 main ways of looking at it and I will show you all three.

Extra Long Multiplication
42
× 37
14	7×2
280	7×40
60	30×2
1200	30×40
1554	42×37

This method is an extension of the one I showed you earlier and it takes each of the numbers and multiplies them. The sums they are doing are shown down the side and the whole thing comes together at the bottom. Effectively it is the same as we used to do but with a bit more added. Many of you will be saying this is very confusing but in fact if you look at it, is very logical; the two numbers are multiplied by the 7 first and then by the 30.

	Using a grid
×	40	2	Total
30	1200	60	1260
7	280	14	294
	37 × 42	Total	1554

This is the method most children in year 5 are familiar with and will be using with a degree of confidence by now. It tends to be taught BEFORE the example above and is very popular with the children. It is logical and can be used to teach decimal multiplication as well so it is multi-functional. It lets the child ‘see’ place value on each of the numbers being held and is more an exercise in addition and times tables than anything else.

The down side to this method is the need to be accurate in addition and of course to know the times tables, but that does apply to all these methods and if your child is still poor at those, it will hinder their understanding of these methods.

The final method is our most familiar and it is when most parents sigh with relief because they are on ‘home ground’ one more, the old fashioned long multiplication.

Long Multiplication
42
× 37
294	7×42
1260	30×42
1554	42×37
This is the shortest!

Yes, this is the quickest method and there will be some year 5 children who will get this quite quickly and go from the grid to this, but they will have unconsciously taken on board the place value of each of the numbers and will understand when their answers go wrong. I will give you an example; the most common thing to forget when doing this method is that zero which has to go in ‘because you are multiplying by a tens number’. In years gone by if I had said to a child, “that cannot be right because you are multiplying a 40-something by a 20-something so its got to be 800-something” they would not have understood my logic. These days they will, and they will make the noises of, “of course” and then re-think their logic. They have kept place value, something we were not even aware of most of the time.

I am often asked which is the ‘best method’ to begin with, and I would say the grid is the best. It makes sense, it relies on a few skills and teaches them how to multiply by 10s, 100s and 1000s without them realising. It is transferable and to a lesser degree can support mental math development.