NSW Education LIVE – Eddie Woo
This page is a transcript of the video NSW Education LIVE with Eddie Woo.
Duration – 19:37
Good morning everyone, I'm Eddie Woo. I'm a mathematics teacher at Cherrybrook Technology High School and welcome to Morning Maths with Mr. Woo, thanks so much for tuning in. Today, I hope we learn something together and even have a little bit of fun.
Now, I want to acknowledge country, right now, I'm on Darug land and Cherrybrook technology High School owes its existence to these people who the traditional custodians of the land have cared for this environment over thousands of years and for all the lands that you are tuning in from, I hope that you pay respect to the elders past, present and emerging who are the reason why we can meet and live and learn on these wonderful lands.
Now, today, we're going to be doing a bit of learning and I wanna say to all the parents out there, thanks for getting your kids logged in. You're gonna need some materials to do the learning today, you'll need some paper like, you know, a spare sheet or a book, something that you can write and draw on and you'll also need something to write with but as an added bonus, if you just got like a pencil or a pen, I'd love it if you could try and get some different colours like, if you got a coloured pencil set or even some markers, anything you can use to show some difference there. If you do have just a lead pencil, we can make it work, I'll show you how later but we'll come to that.
Now, what we're going to be doing today is having you think about how mathematics is all around us and I'm gonna give you one particular example. Now, I have something here, a little bit of a prop to help me illustrate. This is a map of the world as you can see, I picked this particular one because, well, there's actually lots of maths that you can see and I'm gonna bring it just a little bit closer so you can see a bit more clearly. It has all these different countries on it and before I explain the particular maths we're going to have a think about here.
I'd love you at home to talk to the person next to you if you've got someone nearby and think about what kind of maths could you spot in a map like this? We could do so many different explorations, maybe you're thinking about world population, maybe you're thinking about the area of these different countries. I think it's appropriate at a time like this that we think globally. Everyone around the world is in the same kind of situation, we are trying our best to be responsible and stay healthy and well. So, I think it's fitting that we have a look at the entire world map here. You can see, let me move around to the other side here, you've got kind of the American continents over here. Now, what is the particular bit of maths that I want to draw out of this map for you? Well, you might have had a bit of a clue before when I told you the materials that we needed for this lesson. It's something that you might not think is very mathematical and that is colour.
There are lots of different colours on this map. I've seen lots of black and white maps before and I'm sure you have too and they're useful but colour maps we can see much more clearly 'cause we can tell the countries apart. Now, let me bring this even closer let me come back around to this side. If I show you say, here we go, you can see Europe in there, hopefully, it's big enough that you can see it reasonably clearly and if you have a close look, I want you to see how many colours you can count on this map and I really mean, all of the colours you can see. For instance, you know, the water here is in blue, you can see Norway in green, Sweden in yellow, Finland in purple, this is the Russian Federation over here in this kind of salmon pink colour. I'm counting something like 1, 2, 3, 4, 5, 6, 7, at least 7 colours, maybe 1 or 2 more that I haven't spotted there. At least 7 colours are on this map and I guess that makes sense because, there are hundreds of countries on this map and we wanna make sure that we can tell them all apart but my question to all of you is, how many colours do we actually need to colour this map? Did we need 7? Can we actually do it with fewer? Now, this is where your materials are gonna come in handy. If you've got a blank piece of paper there or a blank page in your workbook underneath your heading, I want you to draw with me some imaginary maps. So, if you are not very good like I am at drawing like the exact things real like I'm really terrible at drawing a map of Australia 'cause it's like a weird, funny, awkward shape, then don't worry about that we're just gonna do some imaginary maps, okay? So, firstly, let's just kind of like our big map that I was showing you before, let's just have an imaginary map here that's gonna include our world, okay? Now, in our world, I want us to draw some imaginary countries with imaginary borders. Now, before we get too crazy and complicated, let's do a simple one to start with. I'm gonna challenge you to draw a map, any map with just full lines, okay? They don't have to be straight lines, they can be wiggly and all that kind of thing but let's just use 4 to start with so this doesn't go too complicated on us too fast so if I draw something like this, let's try, there's 1 line, there's 2, there's 3 and let me put in a fourth one over here, all right.
Now, you can imagine this might be like what I did when I showed you the map close up and I held it close to the camera, this might be a zoomed-in portion of a map and that's fine. Now, my question to you is, how many different colours do we need to colour this map? Now, we used 4 lines, right? And maybe your map looks very similar to mine or maybe it looks completely different, it's one of the cool things about mathematics, we can explore all different kinds of situations. In this map, I can count, there are a different number of, well, I guess we could call them countries, different regions on the map that is separated by borders. I'm counting, let's do a count here. I'll stay with the black here. I count one up here, 2 over here, 3, 4, and then 5. Now, there are 5 sections on the map so I guess I could use 5 different colours, one colour for each section and then they would definitely will be different and we wouldn't have any sections of the map that are next to each other which are the same colour but it actually turns out, I don't need 5 different colours even though there are 5 sections on the map and I'll show you and I want you to see if you can do this on your own piece of paper as well.
Let's pick a colour like say, I've got red here, okay? If I take, let's do these in order, I'll start at the top over here, If I take the first section and colour this in red, I'm just gonna do some lines like this so I don't use too much ink, here's my first section. Now, before I jump on to the second one here, I know it's gonna be a different colour but before I do that, I notice that there are other sections on the map that I could colour with the same colour and still not have 2 sections that are next to each other which are identical, right? you can see, have a look on mine and perhaps on yours you've got another section here. I could do either section 3 or country 3, I guess, or country 4, I could do either them in red, and I would still be okay, they wouldn't be up against country 1 and so everything will still look all right. So, I'm gonna do country number 3, let's do that guy. Okay, so, so far, I've already used 1 colour to do 2 different sections so I already know I need less than 5, how much less than 5? Now, if you are unable to find some colour pencils at home or colour textas, that's okay. Instead of using a new colour for the next thing, maybe you want to have lines that go in a different direction or you could put dots or any kind of pattern so you can distinguish these sections from each other. Let's keep on going around the map, I'm up to section 2 now so here's my blue marker, I'm gonna do this in this direction over this way.
Now, as I colour this in and perhaps as you look at your own map, you can see just like before, there's another section of the map that I could colour with the same colour and still not have it touching my country number 2 over here, which one is it? Well, in this case, it's country number 4, right? It doesn't touch country number 2 so I can use the same colour and everything will be fine, let's colour this guy in blue. Okay, fantastic, now, I've got one last section to do and I can't do it in the same colours that I've already used, red and blue because, country number 5, it's a bit unusual actually, country number 5, it touches every other country so therefore, I definitely have to use a different colour for it so thankfully, I've got an orange marker here and I'm going to colour it in horizontally like so and hopefully, you can see what I've got now is a finished map, I've used all of the different colours that I needed but it wasn't 5, it was only 3 and maybe when you've had a go with your map as well, you can say that you didn't need as many colours as you had countries, you could use fewer of them.
Now, it turns out, we actually never really need 5 colours to do a map and we can prove this, I mean, no matter how crazy you make your map look, let's do a more complicated one now I'm drawing over here. Ok? I limited you to 4 lines last time, this time let's just go to town, let's put as many lines on there as we like. The only thing that I would suggest is make sure that you actually make the countries big enough to colour, okay? 'Cause otherwise, this would be very, very difficult for you to see whether it's going to work, okay? Let's have a go at putting a bunch of crazy lines on here so I'm gonna go put one right across the middle there, one up there, I'm gonna put a country inside a country that's a bit weird but it does happen, put one there, another line here, let's do something weird, let's put a bunch of countries here in a row, put another one there and then I'll put one here which kind of touches that, all right. Now, you may still be drawing your map, that's totally fine, take as much time as you like, this can go weird and crazy and you can put lots of lines here, I mean, you might think, wow, that is weird. Is there ever a country inside another country? The answer is, yes, there are and even if you're not doing countries, you can think a little more locally in Australia, we have the ACT, if you're from the Australian Capital Territory, hello, the ACT of course, is completely enclosed by NSW so these kinds of things happen in that all the time. Now, as we have a look here, boy, I should have checked this beforehand. There's so many different regions, I have lost count, actually. Let's have a quick look and see if we can do this, I've got 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, oh my gosh, 12 countries, how am I going to colour this?
Now, before we have a go at colouring this, I want you to notice because, unfortunately, I can't see all of the maps that you guys have been creating but I can guess that some of you have encountered this problem. As you go about colouring, you might start to use one colour, then another one and then realise, Oh, no, I'm starting to use lots and lots of different colours here and, you know, maybe I do need, you know, like, 7 colours like I did on here. Well, I'm gonna show you, we only need the number of colours that I have here in my hand I've got, let me come a bit closer, I've got 4 markers here and in fact, I don't really even need or 4 because, I can use white as one of my colours. I'm gonna show you that we only need 4 colours to do any crazy different map, let me just demonstrate on this one, okay. Now, to work out which colour to start with, I should say which region in which country to start with, rather than doing it like this and just going 1, 2, 3, 4, 5, I'm actually gonna suggest to you to look for a particular kind of place. I want you to start with the kind of country that touches the most other countries. Let me say that again, what you want to search for is the country, let's use a darker colour for writing some words there, there we go, look for the country that touches the most or the highest number of other countries. Sorry, this is a bit messy. Now, why would we start with this? Hopefully, you'll start to realise why as we go. Now, when I have a look here, I'm just doing this very quickly I didn't plan this map or anything like that so I'm guessing it's gonna be a country somewhere in the middle, right? Like country number 5, you can see it touches 1, 2, 3, 4, 5, 6 different countries so that's a pretty good candidate, down here this one touches a lot as well but I don't think it's many, 1, 2, 3, 4, okay, so I'm going to start with number 5 so let's have a go with this one.
Like before, I'll start with red, okay? Now, once you've got this guy coloured in, okay? I again wanna suggest to you, don't just do this in random order, let's see if there's any other countries we can colour with this one before we have to give up, okay? Which ones can we do? Well, apart from all those ones that touch I can do the other countries so long as they're not bordering this one. Let's start with number 1 over here, that landlocked one that looks a bit like the ACT. Let's colour this one in, I can use red there, okay? That one's all good and you can see I can continue this process, I can keep searching for countries that are in colour with this, that's, you know, mean, I can still use red and minimise the number of colours that I'm using so I'm gonna give you a second to do that while I catch up and have a go over here and while I do this and continue colouring, for all the people at home who are a little bit older, perhaps you're in your 10, or you're 11, or you're 12, I want you to think about, does this sound like a complicated bit of math? Does it look very, you know, intricate, and like a difficult bit of calculus or trigonometry? Doesn't look all that complicated, right? But there's actually some deep mathematics in here which we'll reveal shortly, all right.
Now, I'm searching for my next one which touches lots of other countries. I said this one touches 1, 2, 3, 4, this one, oh, this one touches quite a few, one, 2, 3, 4, 5, okay, let's do this one, 6, all right. So hopefully, you're making some progress and my challenge to you like I've showed you before, is see if you can do this with just 4 colours or fewer, okay? I can do that one, which means 8, okay, there we go. 3 is okay. Like so, where else can I go? All right, 12, that doesn't touch anything, and I think I might be done for this colour, I think that's finished, okay, let's use green this time, all right, where am I gonna go? Okay, now, I think I'm actually going to go up here, let's go to 11 and I'm being a bit sneaky like I said before even though I do have a full colour here I've got an orange marker, I'm actually gonna use white as my final colour because, what that means is, at this point, I think I'm actually finished.
I think I have done every little area here and I've used, as I mentioned before, a grand total of 4 colours. Now, this is not an accident, I didn't bring full whiteboard markers with me unintentionally, I knew that I could only use 4 colours and the reason why is this wonderful piece of maths called the four colour theorem. In maths, when we say theorem, what we mean is actually something that we have proven to be true no matter which way you try something, you know, you might have heard of Pythagoras' theorem, that's to do with right angle triangles and no matter what right-angled triangle you have, Pythagoras' theorem, for the parents in the room who are remembering this kicking around in their memory, a2 + b2 =, which are the 2 shorter sides squared, If you add them together, you get c2 no matter what right angles triangle you start with. Well, the four colour theorem, it says that no matter what map you start with, you could have a fairly simple one, a complicated one, even one as crazy as our actual world that we live in, you only need 4 colours and so I said before if you're a bit older, well, what does this have to do with some of the more advanced mathematics that I've been learning? Well, if you're a Year 11 or 12 student who studies mathematics standard and there are several thousand of you around NSW and they're equivalent ones in different jurisdictions.
Then I want you to think about a part of mathematics called networks. Networks is a name for a part of mathematics that is also called graph theory and if you think about every map that you've drawn as a network, you can start to see this is a graph theory problem. This map over here that we created for example, 1, 2, 3, 4, 5, I could say each country represents a node on the map, let's draw it up like this 1, 2, 3, 4 and then 5 and then I can show the borders between the different countries as the different links between them as the different pods that I can take through this network. 1, for example, it connects to 5, 1 also connects to 2 and these are the only 2 countries that it borders so I can finish with all of those different branches there. Let's have a look at 2, 2 connects to one, like I said, it connects to 5 and it connects to 3, 3, I've already done the connection to 2, it connects to 5 and also connects to 4, 4 connects to 5 as well and then that is all of its connections and as you can see, 5 connects to every other one like we showed before and what we're showing here is we can understand, all right, so long as 2 vertices on here, 2 of the nodes are neighbouring so long as they have different colours, we will be able to create a map that's equivalent to that 'cause really, all these wiggly, wobbly borders don't matter, it's just whether they share a connection or not. So, as a bit of challenge for all of you, I'd love you to go away and try and draw some more maps, we know according to the 4 colour theorem, you only need 4 colours no matter what map you create but can you create a map that requires just 2 colours? What about a map that you can create that requires exactly 3 like this one over here?
Have fun with that, good luck exploring mathematics and have a great day everyone, stay safe at home, bye.
End of transcript.