Proving the Distributive Law

What is the "Distributive Law"?

You will no doubt remember your old maths teacher telling you this one:

57 x 32 = (50 x 30) + (50 x 2) + (7 x 30) + (7 x 2)

or even more simply

5 x 12 = (5 x 10) + (5 x 2)  

or   (5 x 6) + (5 x 6)   

or   (5 x 8) + (5 x 4) etc

I thought it might be good to get the kids to show this one visually and to have them prove the Distributive Law.

Let's see what they came up with...

Alright - this is 6 x 5 = 30...........................and this shows (6 x 3) + (6 x 2) = 30

And here we have 8 x 3 = 24............which can be shown to be (5 x 3) + (3 x 3) = 24

and finally 7 x 4 = 28.......................which can be seen to be the same as (5 x 4) + (2 x 4) = 28

Nice work class!

The conversation continued....

Having had a chance to model these facts and use the Distributive Law a bit, we then sat down to talk it through.

"So, the 12 times tables is the same as the 10 times tables plus the 2 times tables!"

"Yes because 12 x 3 is 36 and 10 x 3 plus 2 x 3 is 36 too!?

"So the 15 timnes tables is...?"

"Oh, that would be the 10 times tables plus the 5 times tables."

"Right! And the 17 times tables....?"

"It's the 10 times tables plus the 7 times tables."

"Good work. Then how about the 26 times tables?"

"Hmm, that would be the 2 times tables plus the 6 times tables."

WHAT?! Fantastic - an opportunity to learn!

And so we continued the conversation.

Why is multiplying by 26 NOT the same as multiplying by 2 and multiplying by 6? And why WOULD multiplying by 26 be the SAME as multiplying by 20 (not 2) and multiplying by 6?

Place value, Distributive Law, times tables facts, arrays, modelling - a lot of interesting stuff in what could have been a bit of a tedious, repetitive activity involving number.