# Number visuals

Number visuals is a thinking mathematically targeted teaching opportunity focussed on investigating and recording the ways we see visual representations 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
• MAE-RWN-01
• MAE-RWN-02
• MAE-CSQ-01
• MAE-CSQ-02
• MAE-FG-01
• MAE-FG-02

• MAO-WM-01
• MA1-RWN-01
• MA1-RWN-02
• MA1-CSQ-01
• MA1-FG-01

You will need:

## Watch

Watch Number visuals video (5:21).

Investigate and compare how we see number visuals.

### Transcript of Number visuals video

[A title over a navy-blue background: Youcubed Number Visuals. Below the title is text: From: https://www.youcubed.org/wp-content/uploads/2017/07/WIM-day-2-gr-5-9-vF.pdf. Small font text in the upper left-hand corner reads: NSW Department of Education. In the lower left-hand corner is the white waratah of the NSW Government logo.

A sheet of paper with circles of different sizes and arrangement.]

### Speaker

Hello, everybody. This is a task from Youcubed at Stanford University. And it's all about thinking and representing numbers in different ways. So, the first thing I'd like you to think about is what do you notice when you see this?

[The speaker gestures at the sheet.]
OK, and I'm gonna move this over a little bit here…

[She pushes the sheet to the left. In her right hand is a piece of paper.]

…aha, and I'm gonna record down some of the things that you can see.

[She puts the paper down.]
Yeah, so I can see that too, the numbers increase in size.

[She points to each circle set, dragging her finger across each row.]

They get bigger.

[She points to each circle set in the top row.]

Look one dot, two dots, three dots, four dots, five dots. Mm-hmm. So, oh...well, that's something that we could notice.

[On the paper on the right, she writes: smallest quantity is 1.]

The smallest... ..quantity is one, and the largest quantity?

[She points to the last circle set in the bottom row. This set has 5 groups of 4 dots.]

How many are there that you see?

[Below the text, she writes: largest.]

Yeah, I see that too. In fact, what I see here are fours.

[She colours in the group of dots at the top of the set.]

Look, one four, mm-hmm, a second four.

[Using a green marker, she colours in the group of dots at the top right of the set.]

Yeah, you can see it too, a third four…

[Using an orange marker, she colours in the group of dots at the bottom right of the set.]

…a fourth four

[Using a black marker, she colours in the group of dots at the bottom left of the set.]

…and a fifth four.

[Using a red marker, she colours in the group of dots at the top left of the set.]

Oops. And so, that's 20 dots, yeah. So, the largest quantity on this sheet... some of you are using a different one. The largest quantity here is 20.

[On the paper on the right, next to ‘largest’, she writes: quantity is 20.]

Yeah, and that's actually an important thing to notice that one…

[She points to the first circle in the top row.]

…is one dot, but 20...

[She points to the last circle set in the bottom row.

Below the text she’s written, she writes: inside of 20, we can see 5 fours.]

…so inside of 20, we can see 5 fours.

Ah, yes, I can see you're thinking there, too, that if I partitioned these more…

[In the last circle set, she draws dotted lines in between each group of dots.]

…now you wouldn't see...oops, you wouldn't see fours, but twos. Aha, and there would be 10 twos, yeah.

[Below the text she’s written, she writes: 10 twos.]

Oh...yeah, and some of you are now saying that if I rethink this as four again…

[In the last circle set, she outlines the bottom right group of dots.]

…as a collection of four, it looks like this four.

[She points to fourth circle set in the top row. It has four large circles set in 2 columns of 2 circles. She outlines the set.]

Yeah, like four on a dice pattern. Aha, yes. And the five, look…

[She outlines the fifth circle set in the top row.]

…it looks like this five.

[She outlines the last circle set in the bottom row.]

See that it's the same...yeah. So, mmm, how are we gonna say that, that inside of the bigger numbers…

[She points to the last circle set in the bottom row.]

…we see the same spatial structures or patterns of the smaller numbers?

[She points to the fourth and last circle set in the top row.]

Yeah, and can we see that…

[She points to the last circle set in the bottom row.]

…anywhere else, anything else that has a five in it? Where can you see it? There's a five here…

[She points to the last circle set in the top row.]

…and a five there.

[She points to the last circle set in the bottom row.]

Oh! Up here, look.

[She points to the last circle set in the third row. It has 5 groups of 3 dots.]

Yeah, which is 15, 'cause it's 5 threes.

[She points to each group of dots.]

Aha. So, what we're saying here is that when we have five of something, it follows the structure like a hexagon.

[In last circle set in the top row, she points to each circle.]

So, five is represented like a...

[Below the text she’s written, she writes: 5 is represented like a...]

…oh, not a hexagon, a pentagon.

[She continues writing: pentagon.]

Thank you for correcting. Yes, look, like, five…

[Below the text she’s written, she draws 5 dots in a pentagon shape.]

…and then there's 5 threes... whoops, mm-hmm.

[Next to the dots, she draws 5 groups of 3 dots in a pentagon shape.

Below the first group of dots, she writes: 5 ones. Below the second group of dots, she writes: 5 threes.]

5 ones, 5 threes, and 5 fours.

[She draws 5 groups of 4 dots in a pentagon shape. Below this, she writes: 5 fours.]

Ah, so that's interesting, isn't it? And the fours…

[She points to the fourth circle set in the top row.]

…I think are always like squares, are they? 4 fours…

[She points to the first circle set in the bottom row.]

3 fours…

[She points to the second circle set in the third row.]

2 fours.

[She points to the third circle set in the second row.]

Oh my gosh, look at that. Did you see that pattern?

[She points to fourth circle set in the top row.]

Look, 1 four…

[She outlines the third circle set in the second row.]

2 fours…

[She outlines the second circle set in the third row.]

3 fours…

[She outlines the first circle set in the bottom row.]

4 fours. Wow! It increases by the number of fours on the diagonal. OK, mathematicians, it is over to you to have a look at, what do you notice, and how could you use colour to help you capture some of the things that you are seeing? OK, over to you to have fun with these representations.

[A title over a navy-blue background: Over to you!

Over a grey background, the red waratah of the NSW Government logo appears amongst red, white and blue circles. Text: Copyright State of New South Wales (Department of Education), 2021.]

[End of transcript]

## Instructions

• Explore the number visuals and record the different ways you see each number visual made up of other numbers.

• Record your thinking using the visuals worksheet in your student workbook.