Let's talk 2

Hello there mathematicians, welcome back.

Today I've got a problem to get you thinking hard and this is it. How would you solve 15 x 9 or 15 nines?

Yes, so what we're thinking about here is what are the different strategies that you could come up with to solve this problem? So it's exactly like a number talk that you might be familiar with where we're thinking about, at the moment I'm thinking about one different strategy that I could use to solve this problem. And if you have one strategy it be saying, yeah, I have one way of thinking it through. Yeah, and then I have a second way of thinking it through and you would keep going.

Aha ok so, I think everybody's got one strategy now, so so let's share some thinking together. So we asked this question of some students and they can't be here with me today, so I'm using these minifigs to represent their thinking. So one way of thinking this team over here.

The thing that they started with was this idea of they know that they can partition numbers, and in this case they were interested in partitioning 15. And they said what we know about 15 is that it can be composed of 10 and 5 more. And they said we think we can use this to help us because what that means is that 15 nines is equivalent to 10 nines and 5 more nines. And they said this is really helpful for them because what they know about 10 nines is that if they use the commutative property they can say that's 9 tens. And then they can just use what they know about place value to say that's 90. And then they said with 5 nines it's actually a number fact that they know, and they know that 5 nines is 45.

And so then all they needed to do was combine these two quantities so they had 90 and 45 more and they rethought about that actually as 100 + 35. So they've moved one of the 10s from this number across to 90 to reform 10 tens as 100 and then they had 35. And they said they could actually just rename that using place value as 135.And so that was one way of thinking about this problem.

But the, this team here, they look like a foreman, the foreman's team we'll represent them in red. They said we're going to think about 9, and so they said, we're going to think about this that 15 x 9 is pretty close to 15 x 10. And then they would just need to get rid of 15 ones that they added on. So what they're saying is that 15 tens minus 15 ones is a way that they could rethink 15 x 9. And so they said what they then knew was that 15 tens, they could rename that using their knowledge of place value to be 150. And they know 15 ones is 15, so all they had to do is then subtract 100, subtract from 150, 15, which they said leaves 135, so that was a second strategy.

But of course, because you know we're trying to embrace our inner George Polya, we were thinking well, what's another way that we could think about this? And this team represented by this minifig had a different strategy where they also looked at 8, and this idea of partitioning it and they said, well, we think that you could think about 15 nines as 15 eights plus 15 ones. So it would look like this using symbols 15 x 8 + 15 x 1. Yes, because what they did to the 9 here was partition the 9 into 8 and 1 more. Uh huh. And so then they said with 15 x 8 we can use a strategy called doubling and halving. So 15 x 8 is equivalent to 30 x 4 which is equivalent to 60 x 2 which is 120, they said, And then they just needed to work out 15 ones and they said 15 ones are 15 and then they added 120 + 15 to get 135.

So that's three different strategies that we were thinking about that we could use to solve 15 x 9. I wonder for you mathematicians, if you can think of another two strategies that you could use and also could you use any of these strategies to solve this problem here for you? 16 x 25, 16 twenty-fives.

So over to you mathematicians.

Ok, so what's some of the mathematics here? So a couple of really important things. One that as a mathematician you can think flexibly about numbers and situations. So when you see 15 nines, you can think about 15 tens minus 15 ones. You could think about 15 eights plus 15 ones. Or you could think about 10 nines and 5 nines more.

So we also found three different strategies to solve the same problem and this is where we said back to you mathematicians. This is a really important mathematical skill. Cathy Fosnot said that to be mathematical is when we can look to the context of the problem and make decisions about what to do.

So back to you.