Let’s talk 1

Welcome back mathematicians. We hope you're having a really lovely day today. Today we thought we would embrace our inner George Polya, who was a really famous mathematician who also once famously said this. That it's better to solve one problem in 5 different ways than to solve 5 different problems, and so, to Georges' point, we're going to think about how many different ways, in fact, can we think of 5 different strategies to solve our problem, 23 - 19.

Now I know what you thinking, oh, 23 - 19. This is not much of a brain sweat for me yet. Stick with me. Your challenge is coming. OK, so what I'd like you to think about is what is one strategy that you could use to solve this problem?

OK, and once you've thought of one strategy you might, you know, can you think of a second strategy that you could use? Yeah, and for those of you that are familiar, we're sort of doing a number talk. Aren't we? Where in a classroom, we might use hand signals. Like this [shows fist] means I'm thinking. [Shows thumbs up] This means I have one possible strategy of thinking through this problem. [Holds out two fingers] This means I have another strategy and so on.

OK, so hopefully you've got one way of thinking about this. We thought about this with some students too, they can't be here with us today, so we're going to represent their thinking. So the team represented by the strong man suggested, well, you could think about 19 and partition it, into its parts, so to break it apart. And they said, really 19 is made up of 10 and 9.

So we could think of 23 -19 as 23 - 10. And they said that was 13 and then 13 - 9 and they said that what they would do is subtract the ones by using the jump strategy. So let's have a look at what that looks like on a number line. And we've been playing around with this idea of, you know, how do we record number lines and get our eye in to make them proportional. So we'll share with you a strategy that we've been using with these guys today.

And the first thing is, we've modeled our quantity so we have 23. The two long sticks here are each 10. That is what this number here represents and the three here is what this number represents in the, in the numbers. And I know these are ten 'cause I made them, but we could, I could prove to you it's 10 by snapping them in half. And what I know is that my brain and your brain has this capacity to subitise quantities, so without having to count, I can actually see this chunk of 3 and this chunk of 2, and I know 3 and 2 together is 5 and double 5 is 10.

So that has to be 10 bricks high, and if I line that up. That's also 10, so now I have my 2 tens, which is what this shows me and my 3 ones. And we're going to represent their thinking using a number line and will use blue for the strong man. And yeah, we've been using them almost like a measure, and if I come here and carefully mark the end, that's where 23 goes. And actually my number line could keep going if I wanted. And this is where zero would be and also, my number line would keep going in the other direction, and what the strongman team said they did, was the first thing was, they got rid of 1 jump of 10. So, so I'm now thinking about where my ten is and I know there's 3 here. So if I go with the 3 left behind strategy, that will be a jump of 10 and I can prove that by using direct comparison.

And then they said, now we would count back by ones 9 times. So can you help me keep track of the count? 1. 2. 3, whoops. 4. [Continues removing blocks one by one - no sound] 9, which leaves? 4, so the 13 - 9 is 4 and so what we have here is the 1 ten and the 9 more of 19 and I can record the strong man's team's thinking over here as 23 - 19 is equivalent in value to 4.

So like George Polya, though we're like, well, let's see what other strategies that we can come up with. And so, as I reassemble these blocks, someone else in our group had a really interesting idea and they were thinking about, well, I know something about addition and subtraction and that is that they are related, and so I can use addition to solve subtraction problems. So enter in fancy robot dancing man, that's what we decided to call him. And this team, the green team we will call them, thought, re-thought, about the problem and they said.

Well, actually, when you're solving subtraction, you can just think addition. So what I know is that 19 plus something is equivalent in value to 23 and we need to work out what the difference is. They said then what they would do is 19 plus 1 is 20, because that gets them to a landmark number and then they said from 20, they know that just to add, 3 more is 23 because they would rename it. And what we wondered about, is how we could record that on a number line.

So this is what we came up with. We said, well, we could use our 23. And I'm going to try to line them up so that you can see them. And here's my number line. There's 23 with my arrow 'cause that it extends in that direction and zero and my arrow. And what they were saying is that what we what we know is that 23 is here and we need to find 19 to work out the space between the difference and they said, well, since we know this is 1 ten and this is another ten. 19 must be here because 19 is 1 less than 20, that's right. And then they added 1. And then they added 3 more.

Yeah, so they still have, if I take this section of brick off, it's still a difference of 4. But they just thought about the problem differently, so in this case what they thought about was 19 plus something is 23 and they worked out that that means 19 + 4 is 23. That was their solution and then we were having a really interesting conversation about how you can use addition to solve subtraction and in fact subtraction to solve addition, when along came the Flamingo team, and the Flamingos were like, well, hold on a second.

We've got another way that we could think about this problem, and they said we would just rethink the problem altogether, where I don't want to deal with 23 - 19 because 19 is not a landmark number. So, in actual fact, I can say this, 23 - 19 is equivalent in value to 24 - 20 and they said, and I immediately just know it in my head that that's a difference of 4.

And we were like, wow, can you explain your thinking more please? It was a bit like this. Can you explain your thinking more please Flamingo? Of course I can Robot. So this is what happened. Because, because what the Robot team and the Strongman team were wondering about is that if this is 23, and if I now make a collection of 24, you know this, this tower is one block more than this one, so how does this work? So let's have a look, so we'll use the 24 and I'll line this up as best as I can to create our number line.

And this time we're starting at 24, but again, our number line can continue in this direction. And this is where zero is. And it continues in this direction. And the first thing they did was to take a big jump to subtract 20. So to work out 20 what I'm going to think about is this section here. There's 4 more than the number of tens, and so I'm going to leave the same quantity behind, so that will give me 10. And I can check by measuring.

And I'm going to do the same thing where there's 4 extra, so I'm going to do the 4 left behind strategy. And that's going to give me a really big mega jump of minus 20. And as you'll see, it leaves 4. So we thought this was really interesting. The Strongman, the Robot guys and the Flamingo team had come up with 3 different ways, or different strategies, to think about 23 - 19 and my challenge for you now mathematicians is how could you use these different strategies?

The blue strategy, the purple strategy and the green strategy, or the green strategy to think about this problem instead. Ah, told you it was gonna get a bit more sweaty!

Over to you to think about that, mathematicians.

Welcome back mathematicians. How did you go? OK, so let's debrief this idea of how we can use what we solved here with these guys, I'll just move them out of the way. What we did with the blue team, the green team, oh sorry, the Strongman, the Robot and the Flamingo. And how we could use those strategies in this context, to help us think about this context. And it really comes down to this idea of renaming numbers. So I know some of you will be looking at this and you'll maybe have thought about a trick that you might have been told once about adding zeros and subtracting zeroes. So let's clarify that for a moment.

Here I've got some paddle pop sticks. I think you can see that. I'll put them on there so it's easier. OK, I've got 3 paddle pop sticks. OK, now add zero more paddle pop sticks to my 3. Yes, I still have 3. Now, take, take away zero paddle pop sticks. This is a really cool thing about mathematics. This law actually, that when you add zero or subtract zero it doesn't change what happens. So when we learn this trick of adding a zero, it's mathematically incorrect.

What actually you are using here, is this knowledge of place value and renaming. So let's have a look at that.

I've got this little part portion odds chart for you to look at. And look if I have 3 ones over here, we would write the number 3. 1, 2, 3, so we would write 3 here. If I move them into here, I actually still only have 3 ones and if I add zero or take zero away, I still just have 3 ones. What I know is that if I want to make them move across into tens where a zero will appear when I write it, like this, then what I need to do to each of these is multiply them by 10, which means these 3 ones become 3 tens.

And they would then go into here and so I still have a 3 in my tens column and now a zero in my ones place because the zeros letting me know hey, these 3 are worth 3 tens, not just 3 ones. OK, so you hopefully were using some sort of strategy around renaming to help you here, and we could do the same with each of these strategies. So let's have a look at the first one.

Because what you might have then thought about is, we had these blocks and each one of these blocks was representing 1. You know, we had 23 and we have 23 blocks. What we can think of is that maybe each one of these blocks instead of now being worth 1 is actually worth a 10. Which means these are also all worth 10s. There's 1 of them, 2 of them, a third 10, a fourth 10, a fifth 10, a sixth 10, a 7th 10, an 8th 10, a 9th 10 and a tenth 10. And we do this really cool thing with place value that when we get to 10 tens, we regroup, and we rename it. And so actually, this becomes 100 or is representing 100 and the same, yeah, is happening over here.

And so now what I'm thinking about is 230 minus 190 or 23 tens minus 19 tens, and we're going to use the same strategy. So the first thing these guys did was think about, you could think about, was instead of 23 - 10. It's now 22 tens minus 10 tens, which is equivalent to 100. And we could still use the same left behind strategy. With my 3 left behind, but I'm just now saying that's 100 gone and so actually this is now 23 tens, and that's 10 tens, which is 100 and that's gone. And then the next thing they did was go by ones, which in this case is 10. So that's 1 ten, another 10, a 3rd ten and now I have to, yeah, repartition my 100, a 3rd ten, a 4th ten, a 5th ten, a 6th ten, a 7th ten, an 8th ten and a 9th ten and because these are all worth ten, that means this is worth 4 tens, which means 23 minus 19 equals 4.

I could say 23 tens minus 19 tens is 4 tens. Which is 40 and we just rename it. So that's one way that we could think about this problem. All right back over to you mathematicians, to think about how you can use renaming to adjust, or think about, your other strategies. Back to you.