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 five 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 five different strategies to solve our problem. 23 minus 19. Now I know what you're thinking, ha, 23 minus 19. This is not much of a brain sweat for me yet. Stick with me. Your challenge is coming.

Okay, So what I'd like you to think about is what is one strategy that you could use to solve this problem? Okay, and once you 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, [hold closed fist out] like this means I'm still thinking. This means, [shows thumbs up hand signal] I have one possible strategy of thinking through this problem. This means [shows two fingers] I have another strategy and so on.

Okay, 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 minus 19 as 23 minus 10. And they said that was 13 and then 13 minus 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 3 here is what this number represents in the in the number.

And I know these are 10 '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 subertise 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 two 10s, which is what this shows me and my 3 ones.

And we're going to represent their thinking using a number line and we'll 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 that they did was, the first thing was they got rid of 1 jump of 10 so.

So I'm now thinking about where my 10 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? Okay. 1, 2, 3, whoops. 4. 9, which leaves? 4. So the 13 minus 9 is 4 and so what we have here is the one 10 and the 9 more of 19 and I can record the Strong man's team's thinking over here as 23 minus 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 reassembled 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're 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'll call them, thought, really 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 and 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. 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 a 19 to work out the space between the difference. And they said, well, since we know this is one 10 and this is another 10, 19. must be here because 19 is one less than 20. That's right, and then they added one. 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 plus 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 minus 19 because 19 is not a landmark number. So in actual fact I can say this 23 minus 19 is equivalent in value to 24 minus 20. And they said and I immediately just know it in my head that that's a difference of 4. And we're 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 strong man 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 10s, 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, wooh, minus 20 and as you'll see it leaves 4.

So we thought this was really interesting the, the Strongman, the robot guys and the Flamingo team had come up with three different ways or different strategies to think about 23 minus 19 .

And my challenge for you now mathematicians is how could you use these different strategies? The blue strategy, the purplestrategy and the green strategy, or the green strategy to think about this problem instead. 2 and three tenths minus 1 and nine tenths. Over to you mathematicians. Plus, remember George Polya, five ways to solve one problem. You can try these three strategies, but you need another two over to you. Alright, welcome back mathematicians.

How did you go? I posed this problem to you is how could you use these strategies to think of solving this problem? And then I said as well, can you think of two other strategies to solve it? Alright, so let's talk about how we can use thinking about 23 minus 19 to help us solve 2 and three tenths minus 1 and nine tenths.

And the first thing that we need to think about for a moment is place value. So I've got this little chart here to help us to start with. Well, I think what you might have like a strategy that you might have heard or someone might have said she before. Is this idea of well just move the decimal point. And so the first thing to talk about is the fact that this decimal point never actually ever moves. So in mathematics the decimal point, this thing here, that separates the whole numbers from the fractional numbers is always between the ones and the tens, always and it's always there because it represents the shift from working with whole quantities to now having to work with fractional quantities.

So for example, if I wanted to represent two and three tenths, I have a bit of a problem in that my paddle pop sticks are hard to re-partition into tenths, and so I'd like to invite you into my mathematical imagination and here's the one that I need to partition into 10 smaller parts or to tenth it. So I'm going to make it bigger so that you can see what's happening, and then what I do is I imagine tenthing it. So for me, the first thing I do is half. And then for each half. Half I've made. I then fifth each one and that now gives me 10 equal size parts, so I've 10th it and I only need 3 of them for the number that I'm making and I'll shrink them back down to size and I can put them now back into my place value chart. And I can see here now that I have three tenths and my number is 2 and three tenths.

And now what I want to do is actually multiply each of those by 10 so that I can get to a whole number. And I like whole numbers'cause my brain feels more comfortable working with them. So if I multiply each of those collections or each of those sections by 10. I end up with two tens and three ones, otherwise known as 23, and you'll notice that the decimal point has not moved and so from this collection now I know that I need to remove 19 and so if I have two bundles of 10, that's 20 and if I get rid of 19. It means I just have one paddle pop stick left, which means I have 4 in my collection and back into my place value chart.

I now have 4 ones and what I need to do is divide each of those by 10. And what that leaves me with is 4 tenths, and as you'll notice, the decimal point has not shifted at all. But what I've done is multiplied my collection to work with them, and then whenI finished doing my operating, I've divided the remaining quantity by 10 to make sure that I'm keeping on the same quantity. And it's a fair outcome.So that's one way that you can use something like 23 minus 19 to help you workout 2 and three tenths - 1 and nine tenths.

And now what I wonder is how could you use those other strategies? And remember you still need two more. So back to you mathematicians.