# Imagining dots (16)

A thinking mathematically targeted teaching resource focussed on using mathematical imagination to quantify a collection of dots

Adapted from Kazemi and Hintz – Intentional Talk, 2014

## 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-FG-01
• MAE-FG-02

## Collect resources

You will need:

• paper
• a pencil or a marker.

## Watch

Watch the Imagining dots Early Stage 1 part 1 video (2:05).

How do you see and imagine the collection of dots?

### Transcript of Imagining dots Early Stage 1 part 1

(Duration: 2 minutes and 5 seconds)

[A title over a navy-blue background: Imagining dots. Below the title is text in large font: Early Stage 1. Below this text is text that reads: Adapted from Kazemi and Hintz. Small font text in the lower left-hand corner reads: NSW Mathematics Strategy Professional Learning team (NSW MS PL team). In the lower right-hand corner is the red waratah of the NSW Government logo.]

### Speaker

Today, we're going to explore a task called Imagining Dots. It's adapted from Kazemi and Hintz.

[A title on a white background reads: You will need…

· eye balls

· something to write on and something to write with.]

### Speaker

For this task, you'll need your eyeballs ready, something to write on, and something to write with.

[The screen is cleared.]

### Speaker

Hi, mathematicians. Today, we're going to show you a collection of dots. I'm going to show you them for 3 seconds, and it's your job to imagine how you see the dots on the page.

[Text over a blue background: Eye balls ready!]

### Speaker

You'll need to pay really close attention because I'm only going to show you the dots for 3 seconds. It's not very long. So, are you ready? Are your eyeballs ready to take a picture?

[On the screen is 4 groups of blue dots arranged in a cross pattern: one group above , another below, another on the left and another on the right. Each group’s dots are set in a square.]

### Speaker

One, two, three.

[The screen is cleared.]

### Speaker

Oh! What did you see? And how did you see them? Can you close your eyes and imagine what you saw? We might take another look at the dots just to help you create a really good picture in your brain. We call this using our mathematical imagination. Right. Well, let's take another quick look so you can check the picture you've been making. Hold on to what you noticed, and let's take another look to check our ideas.

[Text over a blue background: Eye balls ready!]

### Speaker

[The same group of blue dots appear: 4 groups of blue dots arranged in a cross pattern: one group above, another below, another on the left and another on the right. Each group’s dots are set in a square.]

### Speaker

One, two, three.

[The screen is cleared.]

### Speaker

OK. So, have you created a picture of the dots using your mathematical imagination?

[Text over a blue background: Over to you…]

### Speaker

Now we're going to share some of the different ways that we could imagine the dots on the page. And sometimes it really helps us to draw a picture of what we saw.

[Below the text, 2 lines of text appears in slightly smaller font. The first line is: Draw a picture of how you saw the dots. The second line is: You might even like to draw a few pictures to show some of the different ways you saw the dots.]

### Speaker

So, grab a pencil and draw a picture of how you saw the dots. You might have seen the dots in different ways. So, you might even like to draw a few pictures. Push pause now, and come back once you've recorded how you saw the dots.

[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

• After watching the video, use your imagination to draw a picture of how you saw the dots.
• You might like to draw a few pictures to show some of the different ways you saw the dots.

## Watch

Watch the Imagining dots Early Stage 1 part 1 video (6:12).

Explore and record some ways dots could be seen.

### Transcript of Imagining dots Early Stage 1 part 2

(Duration: 6 minutes and 11 seconds)

[Text over a blue background: Welcome back…

Small font text in the upper left-hand corner reads: NSW Department of Education. In the lower right-hand corner is the white waratah of the NSW Government logo.]

### Penny

Welcome back, mathematicians.

[A large blue sheet of paper almost covers a table top. In the centre of the blue paper is a smaller piece of white paper.]

### Penny

So, let's have a look now at some of the ways that we saw the dots. I wonder if the ideas that we share here are the same or different to the way that you saw the dots. I'm going to describe the ways that I saw them to you. And I'm gonna try and explain how my mathematical imagination helped me to remember how I saw the dots. So, the first thing I noticed was all of the chunks looked like four on a dice pattern. So, I'm going to draw what one of those fours looked like.

[Penny points to the top middle section of the white paper.]

### Penny

Now, I know that there was one sitting right in here at the top. And that is going to be my first four.

[With a marker, she draws 4 circles set in a square.]

### Penny

Beautiful. Now, I'm gonna close my eyes again, and go back to my imagination. I'm gonna have another look at that picture I took in my brain, and I'm gonna think about what else that I see. That's right. All the fours look like four on a dice pattern. They were sitting inside the dots, and it was just a little bit wonky.

[She turns the white paper to the right until its top left corner points up.]

### Penny

I needed to turn my page so that I can see the four, and…

[She circles the 4 circles with her finger.]

### Penny

…this is my one four, there's gonna be another…

[On the left side of the circles, she draws another circle.]

### Penny

…four over here, another four over here.

[Below the circles, she draws another circle.]

### Speaker

And one more four…

[On the left side of the new circle, she draws another circle.]

### Penny

…or my fourth four is going to go over here. And then…

[She turns the paper straight up.]

### Penny

…I can turn my page back to fill them in.

[Next to the circle on the left, she draws 3 more circles so the 4 circles look like a square.]

### Penny

So, here's my second four…

[Next to the circle on the right, she draws 3 more circles so the 4 circles look like a square.]

### Penny

…oop, there we go and my last…

[Next to the circle on the bottom, she draws 3 more circles so the 4 circles look like a square.]

### Penny

…four down here. You can see…

[She turns the paper to the right.]

### Penny

…there's my four fours, like a dice pattern.

[She turns the paper straight up.]

### Penny

So, let's go back to the image and take a look and see how close I was.

[She moves the paper to the left of the blue paper.]

### Penny

I'm just gonna move that over to the side.

[On the right side of the blue paper, she places a sheet of paper with 4 groups of blue dots arranged in a cross pattern. Each group’s dots are set in a square.]

### Speaker

And here's a copy of the dots from the video. And I think I did a pretty good job using my imagination to draw the dots. And I'm actually going to write my name on this one because this was my idea.

[On the top left corner of the paper she drew on, Penny writes: Penny.]

### Penny

I'll just pop that to the side for a little while.

[She pushes her paper up. She takes the copy of the dots away. She places a white piece of paper on the blue paper.]

### Penny

Alright. Let's have a look at another way. Now, my friend Michael told me that he thought about the dots in a different way. He said he could see three fours like a triangle, and another four hanging down the bottom. So, let me think how I could draw that. First, I think I'll draw the triangle. And I might even draw the triangle in a different colour.

[With a purple marker, she draws a large triangle on the new piece of paper.]

### Penny

I'm gonna draw the triangle that will help me see the dots like Michael did.

And he said he saw four dots, like a dice pattern in each of the three corners of the triangle. So, let's put them in.

[In the bottom right-hand corner of the triangle, Penny draws 4 dots set in a square.]

### Penny

So, there's four dots in this corner of the triangle…

[At the top corner of the triangle, Penny draws 4 dots set in a square.]

### Penny

…four dots up the top corner and…

[In the bottom left-hand corner of the triangle, Penny draws 4 dots set in a square.]

### Penny

…four dots in the last corner. Here we go. Then Michael said something else. Now, I'm just gonna have another look back in my imagination. I have the three fours in the triangle, but Michael said he saw one more four hanging down the bottom.

[She points to the bottom middle section of the paper.]

### Penny

So, I better draw that one in down here.

[In the bottom middle section of the paper, Penny draws 4 dots set in a square.]

### Penny

And I'm gonna write Michael's name on this because this is Michael's thinking.

[On the top left corner of the paper she drew on, Penny writes: Michael.]

### Penny

Now, let's have a look.

[She moves the paper to the left. On the right side, she puts down the copy of the dots.]

### Penny

I move that one over. Yeah look…

[She traces the top groups of dots.]

### Penny

…there's Michael's triangle…

[She points to the top groups of dots.]

### Penny

…with fours in each corner, and that one…

[She points to the bottom group of dots.]

### Penny

…four hanging down the bottom.

[Penny’s paper is over the upper left side of the blue paper. Below it is Michael’s. On the right side is the copy of the dots.]

### Penny

Here we have two…

[She points to hers and Michael’s papers.]

### Penny

…different ways of imagining the dots. And this really helped me…

[She points to her paper.]

### Penny

…to draw a representation of my way that I saw the dots. But using my imagination also helped me to draw the way Michael saw the dots. I wonder if you can come up with any other ways to represent the dots?

Over to you mathematicians, to draw some other ways that you could represent the dots that you saw in this collection.

[Text over a blue background: What’s (some of) the mathematics?]

### Penny

What's some of the mathematics?

[A title on a white background reads: What’s (some of) the mathematics?

Text below reads: Mathematicians can use their imaginations to help them draw representations.

Below the text is a row of 3 images. On the left is an image of Penny’s drawing of the first group of dots. The middle image is her drawing circles where the other groups of dots were. The last image is her drawing the circles of each group of dots.]

### Penny

Mathematicians can use their imaginations to help them draw representations. When I imagined the dots, I saw them as four on a dice pattern, and I turned my piece of paper to the side so that I could easily see the four. When I turned my page back round, I was able to fill in the rest of the dots that showed me that I had four fours.

[A title on a white background reads: What’s (some of) the mathematics?

Text below reads: Mathematicians to the ideas of others.

Below the text, is a large blue speech bubble coming out of an image. Inside the speech bubble is text that reads: When Michael describes seeing the fours in the corners of the triangle, I could use my imagination to see what he was thinking. The image shows Penny’s and Michael’s papers next to the copy of the dots.]

### Penny

Mathematicians listen to the ideas of others. When Michael describes seeing the fours in the corners of the triangle, I could use my imagination to see what he was thinking. I closed my eyes and saw the triangle and knew that the fours would be nested in each of the corners.

[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]

## Discuss

• Can you draw some other ways to see the dots?
• Michael saw a triangle when he looked at the dots. Do you see any other shapes?
• Ask someone else how they saw the dots. They could describe it to you and see if you can draw a picture of their way of seeing the dots.
• Try drawing a different collection of dots. Show them to a partner and see if they can close their eyes and imagine the dots.