Same... and different

Ok, mathematicians, it's time to warm up our amazingly mathematical brains. Recently we looked at this incredible visual of numbers from YouCubed.

Yeah, and it gave us and really amazing insight into some different ways that you can think about quantities. And we asked you to play around with it and use colour and, yeah, use colour to think about how you could represent how you see quantities and the chunks inside that.

So here's a little insight into what I was noticing. You might have noticed this too. That I started to see some really important relationships I thought between the quantities. I'll give you an example of what I mean. So here is 4 and for me I saw 4 like on a dice pattern, but when I saw 8 I knew it was 8 'cause I could see 2 fours inside of the 8. So I coloured the dots yellow because they are the 4 and I did a green highlighter around the outside to show that that was related to 2, and that's how I could see 2 fours.

Yes I can also see 8 twos and so I wrote that as well. And then I notice this with 3. See how 3 takes the shape of a triangle here. So I notice this in 6 where I've got twos like I have for 2, which is why they're coloured in green, but those twos take the shape of a triangle to me, and so that helps me see that 6 is 3 twos and look I can see it here also in 12. I can see that triangle orient... triangle shape, or structure of the collections of four dots, and so I can see 3 fours which is 12.

So this way from YouCubed is one way of thinking about and imagining quantities, but we'd like to have a look at another way today, and this one comes from Dan Finkel. Yes, and it's really different isn't it? We can still see it's numbers 1 to 20 here to start with. What are some things that you're noticing?

Yes, 1 doesn't have any colour. Well, it's grey. And some of the other numbers have just one colour too like 1, 2, 3, 5, 7, 9, 11, 13 and 17 and 19. Yeah, and some of those ones are red and some of them are different colours. And then have a look at 10. I notice something about 10 with the colours inside of it. It's got the colours of two and the colours of five, and that made me think about. If you multiply 2 by 5, you get 10, or if you multiply 5 by 2, you also get 10. And then that may be start wondering about 20, 'cause it has the same colours.

But it also has the same number of orange sections as 4. Oh, I know it's so interesting, isn't it? And so it's now over to you mathematicians. We're interested in what you notice here and what we'd like you to think about is comparing, actually, what's the same in the visuals of the numbers in the way that YouCubed thought about it and the way that Dan Finkel thinks about it in math for love? What's different? And also what are just some really cool, curious, interesting things that you notice?

I'll share with you one before you can go. Before you head off. Here's, I found this really interesting. Here is one way of representing 6 from Dan and for me to know that it's, to think, I need to have the colours beneath to think about the code. So the orange in the 6 represents a 2 and a 3 in the 6 represents threes and that makes sense to me because 6 is 2 threes or 3 twos. And here's what this looks like on YouCube. It's also showing me 3 twos, but it's showing me the chunks of the 6 inside the 6. I can see exactly how many things it is. Where as with Dan's I can't actually see the quantity, but I can see still some really important relationships.

So over to you mathematicians to notice and explore and wonder and test out and share your amazing ideas. Over to you.

So what's some of the mathematics here? Some really important ideas in mathematics is that bigger numbers are made up of smaller numbers, and these visuals help us see the composition of numbers. It's also really important that we know that you can decompose or partition numbers in ways that help us see multiplicative situations, as well as additive ones. So for example, when I see 6 as 3 twos that's multiplicative, whereas 5 plus 1 is an additive way of thinking. Some numbers can be partitioned into equal groups in different ways, and other numbers can't be partitioned into equal groups at all.

That's really interesting, and what I find so fascinating is it something can have the exact same value, but look quite different. Just look at these two ways of representing 8. Alright mathematicians, you better go back to being creative and imagining and we look forward to seeing you soon. Over to you.