Same and different (visual representations)

Stage 2 and 3 – A thinking mathematically targeted teaching opportunity exploring multiplicative strategies through the composition and partitioning of numbers

Syllabus

Syllabus outcomes and content descriptors from Mathematics K–10 Syllabus © NSW Education Standards Authority (NESA) for and on behalf of the Crown in right of the State of New South Wales, 2023

• MAO-WM-01
• MA2-MR-01

• MAO-WM-01
• MA3-MR-01

You will need:

• a pencil

Watch

Watch the Same and different (visual representations) video (5:30).

What is the same and different about these visual representations?

Transcript of Same and different (visual representations) video

[White text on a navy-blue background reads ‘Same… and different – With thanks to Maths for Love of youcubed’. On the right, a blue half circle at the top and a red half circle at the bottom. In the middle bottom, a line of red dots forms another half circle. In the bottom left corner, a white NSW Government ‘waratah’ logo.]

[White text on a blue background reads ‘OK, mathematician…it’s time to warm up our mathematical brains!]

Female speaker

OK mathematicians, it's time to warm up our amazingly mathematical brains.

[On a white background, various patterns of circles are arranged in number order from 1-35 in a 5 x 7 array. Black text in the top left reads ‘NSW Department of Education’ and in the bottom right ‘From: YouCubed’ and also a NSW Government red ‘waratah’ logo.]

Female speaker

Recently, we looked at this incredible visual of numbers from YouCubed. Yeah, and it gave us a really amazing insight into some different ways that you can think about quantities. And we asked you to play around with that and use colour to think about how you could represent how you see quantities. So here's a little insight into what I was noticing.

[Black text reads ‘What did you notice?’ Below, a section of the array from 1-14 is highlighted in different colours. Sections of the array appear below when mentioned by speaker.]

Female speaker

You might have noticed this, too, but 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 because I could see 2 4s inside of the 8. So I coloured the dots yellow because they're 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 4s. Yes, I can also see 8 2s and so I wrote that as well.

And then I noticed this with 3. See you have 3 takes the shape of a triangle here. So I notice this in 6, where I've got 2s, like I have for 2, which is why they're coloured in green, but those 2s take the shape of a triangle to me. And so, that helps me see that 6 is 3 2s. And look, I can see it here also in 12. I can see that triangle shape of structure, of the collections of 4 dots. And so, I can see 3 4s, which is 12.

[The earlier complete pattern array.]

Female speaker

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.

[On a dark grey background 20 circles are numbered from 1 to 20 in black on white. Inside each circle are various different combinations of coloured segments (as explained by speaker).]

Female speaker

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, one doesn't have any colour. Well, it's grey. And some of the other numbers have just one colour too, like one, 2, 3, 5, 7, 11, 13, 17 and 19. And it does make me wonder what might be special about those numbers and why some of them are red but some of them are not.

And then have a look at 10. I notice something about 10 with the colours inside of it. It's got the colours of 2 and the colours of 5. 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 made me start wondering about 20, because 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.

[The earlier pattern array from 1-20 is placed above the coloured 1-20 circle array. Further single images from each array appear as mentioned by speaker.]

Female speaker

We're interested in what you noticed here. And what we'd like you to think about is comparing actually what's the same in the... 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 head off. I found this really interesting. Here's one way of representing 6 from Dan. I need to have the colours beneath to think about the code. So the orange in the 6 represents the 2 and the green in the 6 represents 3. And that makes sense to me because 6 is 2 x 3 or 3 x 2.

And here's what this looks like on YouCubed. It's also showing me 3 2s, but it's showing me the chunks of the 2s inside the 6. I can see exactly how many things it is, whereas 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.

[White text on a blue background reads ‘Over to you!’ and then ‘ What’s (some of) the mathematics?’]

Female speaker

So what's some of the mathematics here?

[Black text on a white background reads ‘What’s (some of) the mathematics?’ Below, black text bullet points (read by speaker) and a colour image of the Dan Finkel 1-20 visual of numbers and the earlier pattern of numbers.]

Female speaker

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 x 2, that's multiplicative, whereas 5 plus one 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 that something can have the exact same value but look quite different. Just look at these two ways of representing 8.

[At the bottom, the 2 representations of 8 from the different arrays. The pattern array has 8 as 2 groups of 4 and the Dan Finkel version has 3 orange-coloured segments forming a circumference inside the ‘8’ circle.]

Female speaker

Alright mathematicians, you better go back to being creative and imagining, and we look forward to seeing you soon. Over to you.

[The NSW Government waratah logo turns briefly in the middle of various circles coloured blue, red, white and black. A copyright symbol and small blue text below it reads ‘State of New South Wales (Department of Education), 2021.’]

[End of transcript]

Instructions

• What’s the same in these visual representations of the numbers 1- 20? What's different?

• What are some things you find cool and/or curious?