Multiplication facts by understanding
Maths Multiplication Facts. Use what you know to work out what you don’t know.
Part 2. A Little More Introduction. A little more explanation.
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Here’s the challenge. Part 2.
In Part 1, I showed the 121 facts in a table. I want to take a more analytical look to get a more accurate picture of just how big the challenge is. The good news is that it’s not 121 facts for most people.
It’s a good approach before starting on any problem to work out just what that problem is, including, how big it is. And I kinda think it’s important to explain to you, and your pupils/children, the ‘Why?’ as well as the ‘How?’ and give you some credible evidence for both. I’ll get to ‘How?’ in Part 3. The ‘How?’ bit will be quite short and focused, once you know ‘Why?’
When 11 – 12 years old students came to my school (for dyslexics and dyscalculics) for an interview for a place, I’d ask them, ‘Which are the times table facts you know, the ones you’re really comfortable with?’ Inevitably the reply was, ‘I know the 2s, the 5s and the 10s.’ Some would add in the 0s and the 1s. That’s the reality. Over many years, I’ve asked hundreds of teachers at my training sessions about this and their experience is the same. So, work with what you have!
Let’s look at the impact of knowing those facts on the 121 facts (Fig. 1). I have shaded in the 0, 1, 2, 5 and 10 facts. See what’s left? And the really tricky ones are the sixteen with the white background. Just 16 facts. Children don’t need all those cartoons, songs (great tune, shame about the words!), one-fact mnemonics and dancing around for 121 facts. They need strategies for 16 facts. Well, even less than 16. And the bonus is they will learn some other maths, too. Read on.
Fig. 1
Why it’s even less than 16 facts to master. I’m puzzled by people promoting methods that have an overwhelming reliance on rote learning, relying only on memory. We should be teaching our pupils to understand, to know ‘Why?’ The times tables facts are a great topic for doing this, and they are an example of my other teaching mantra, ‘What else are you teaching?’
As two examples, let’s look at 3 x 4 and 4 x 3. That’s ‘3 lots of 4’ and ‘4 lots of 3’.
And let’s look at 2 x 5 and 5 x 2. That’s ‘2 lots of 5’ and ‘5 lots of 2’.
In both these examples, and for any two numbers multiplied together, the rule, the generalisation is:
1st number x 2nd number = 2nd number x 1st number
or number a x number b = number b x number a
or, using just the letters a and b as a way of representing ‘any number’, then it becomes
a x b = b x a
So, if you know one fact you get another one free! So, looking at the white square of numbers in Fig. 1, there are 4 facts which are a x a (6 x 6, 7 x 7, 8 x 8 and 9 x 9) and 6 facts that are a x b (for example 7 x 8) and 6 facts that are b x a (for example 8 x 7). Not 16 facts to learn, but 10.
And those 10 facts cause so much grief!
In case you needed to know: The maths name for this relationship is the ‘commutative property’.
Did you notice that I slipped in some algebra …. as an example of, ‘What else are you teaching?’ and to show how algebra is great for making generalisations.
In Part 3 I’m going to show you ‘How?’ to work out the ‘hard’ times tables facts by using the ‘easy’ times tables facts. I needed to show Parts 1 and 2 to show you ‘Why?’ otherwise I’d feel I wasn’t teaching an understanding of the maths.
3 x 4
= 12
4 x 3
= 12
2 x 5
= 10
5 x 2
= 10
3 x 4
= 12
4 x 3
= 12
2 x 5
= 10
5 x 2
= 10
3 x 4
= 12
4 x 3
= 12
2 x 5
= 10
5 x 2
= 10