Imagining dots (18)
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 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, 2021
You will need:
- something to write on
- pencils or markers.
Watch Imagining dots (Stage 1) part 1 video (2:12).
[A title over a navy-blue background: Imagining dots. Below the title is text in large font: 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.
Today we're going to explore the task, imagining dots. It's adapted from Kazemi and Hintz.
[A title on a white background reads: You will need…
Bullet points below read:
· eye balls
· something to write on and something to write with.]
For this task, you will need your eyeballs ready, something to write on, and something to write with.
[The screen is cleared.]
Hi mathematicians, today we're going to show you a collection of 18 dots. I'm going to show you them for three seconds and it's your job to imagine how you see them on the page. You'll need to pay really close attention because I'm only going to show you the dots for three seconds.
[Text over a blue background: Eye balls ready!]
Three seconds isn't very long, so are you ready? Are your eyeballs ready to take a picture?
[On the screen is 2 rows of groups of blue dots. Each row has 3 columns. In the top row, each group has 2 dots side-by-side. In the bottom row, each group has 2 dots over 2 dots, making them look set in a square.]
One, two, three.
[The screen is cleared.]
So, 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 to help you create a really good picture in your brain. We call this, using our mathematical imagination. Right, well let's take another look so we can check the picture you've been making in your imagination. Hold on to what you noticed and let's take another look to check our ideas.
[The dots appear again: 2 rows of groups of blue dots. Each row has 3 columns. In the top row, each group has 2 dots side-by-side. In the bottom row, each group has 2 dots over 2 dots, making them look set in a square.]
One, two, three.
[Text over a blue background: Over to you…
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.]
OK. So now you've created a picture of dots using your mathematical imagination. We're going to share some of the different ways that we could imagine the dots on the page. Sometimes it can help us to draw a picture of what we saw. Now, let's get our pencils and draw a picture of how we saw the dots. You might have seen the dots in different ways so you might like to draw a few pictures. If you have someone with you, you can share your ideas and see if you have the same or different ways of seeing the 18 dots. Push pause here 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]
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 Imagining dots (Stage 1) part 2 video (9:02).
[White text on a blue background reads ‘Welcome back…’. Further white text in the top left reads ‘NSW Department of Education’. In the top right, a blue half circle and a curved line formed by red dots. In the bottom right corner, a white NSW Government ‘waratah’ logo.]
[On a wooden desktop, a sheet of white paper on a larger sheet of blue cardboard. On the left, a chunky black marker pen, a purple marker pen and a blue marker pen.]
Welcome back mathematicians. When I first saw the dots, the first thing that I noticed was that all the chunks across the bottom looked like 4 on a dice pattern. So now I'm going to draw what that 4 looked like. Now I know there was one right over here on the side of the page.
[The speaker uses the chunky black marker to draw 4 circles in the bottom left corner of the white paper.]
So I'm going to draw my first 4 over here and I can see that 4 just like a dice pattern. Now I'm going to close my eyes again and go back into my imagination and that picture I took in my brain and I'm going to think what else I saw. Oh, that's right. There was one 4 here, another 4 here.
[The speaker draws 2 more patterns of 4 circles to the right of the first one.]
So now I've got 2 4s. And when I close my eyes, I remember seeing one more 4 over here. And I see those 4s going in a straight line across the page. But I know that's not all the dots. So when I close my eyes again, I can actually visualise the dots that I saw moving to make another 4 up here. I know there was 4 because I saw 2 and 2. And I know that 1 x 2, 2 x 2, are 4. I might imagine these 2 dots moving across to create another 4.
[The speaker draws 2 lots of 2 circles above and uses the purple marker to draw an arrow that points to the right. She then draws 2 more circles in purple marker above the 2 black circles on the right. She proceeds to put a purple line through the 2 black circles on the left.]
And so now I can cross these ones off, because I imagine them moving over to here to create one more 4. And then when I close my eyes one more time, I've seen my 3 4s down the bottom, 4 up here. And then I remember there were 2 more dots just above this 4 over here.
[The speaker uses the black marker to draw 2 circles on the left of the white paper. She then writes with the marker at the top of the page. She then circles each group of 4 circles with the blue marker pen.]
So then I can write, 18 is 4 4s. And we can check that. One 4, 2 4s, 3 4s and 4 4s. So there are 4 4s and 2 more. So let's go back to the image again and let's take a look at how close I was using my imagination to draw the 18 dots.
[The speaker places a white piece of paper on the desktop. It has 18 blue circle stickers on it.]
Now here's the 18 dots. How do you think I went? Yeah, we can see one four, 2 4s, 3 4s and there's my fourth four where I imagine the dots moving. And just like here, there's my 2 more over there. I think closing my eyes and using my imagination was a really great strategy to show how the 18 dots were arranged on the page.
My friend Michael though, he told me he saw it another way. He told me that he saw 3 6s going across the page. Oh, I'm definitely going to have to close my eyes and imagine where those sixes were. So I'm gonna use my purple Texta here to show that Michael saw one 6, 2 6s and 3 6s.
[The speaker uses the purple marker to draw 3 large circles.]
The first 6 that he saw over here, the first thing he saw was the 4 dots down the bottom, just like I did.
[The speaker uses the black marker to draw 4 small circles inside the left-hand large purple circle. She then draws 2 further circles above them.]
Now, Michael used his imagination too, but his was a little bit different to mine. So he saw his 4 and his 2 up the top, and he used his imagination to imagine these dots travelling down the page.
[The speaker uses a green marker pen to draw two arrows that point down from the top 2 circles to the group of 4 below. She then draws two small circles above the group of 4 circles.]
Can you see how the 4 of the dice patterns and the 2 up the top have now combined to make a 6? And we know that it's 6 because we see this pattern all the time on a dice as well. So because Michael imagined these 2 dots travelling down to make a 6, I'm actually going to cross these 2 out, because they're now down here.
[The speaker puts a line through the 2 circles at the top.]
So now that I know that Michael could see one 6 here, he told me that when he closed his eyes and used his imagination, he could see another 6 here and another 6 here.
[The speaker draws 6 small circles inside the remaining 2 large purple circles.]
So I can draw them in and I can draw them like a 6 on a dice because that's how Michael imagined them. So then Michael has noticed that 18 is 3 6s.
[The speaker writes ’18 is 3 sixes’ at the top of the white paper. She proceeds to outline the 3 groups in green marker pen.]
And we know 18 is 3 6s, because here is one 6, all of these are 2 6s and all of these combine here to make 3 6s. So, let's now take a look at the 18 dots we saw on the screen and let's see how Michael went using his imagination.
[The speaker places down the white sheet of paper with the 18 blue circular stickers from earlier.]
Can we still see that 18 is 3 6s? Here's the 2 dots that Michael imagined moving down the page. And here is one 6, 2 6s, 3 6s. Oh, great job using your imagination Michael.
[The 2 drawings from earlier side by side. Above, the sheet of paper with the 18 blue circle stickers from earlier.]
Now that we've drawn out how Michael and I saw the dots, we can use our drawings to compare the different ways that we saw the 18 dots. Both Michael and I imagined some dots moving to create a new chunk. I used mine over here to imagine these 2 dots adding to create another 4 over here. Where Michael imagined these 2 dots at the top travelling down to make 3 6s. Imagining the dots moving really helped both Michael and I to remember how we saw the dots.
[White text on a blue background reads ‘What’s (some of) the mathematics?’]
What's some of the mathematics?
[Black text on a white background reads ‘What’s (some of) the mathematics?’. Below, further black text (read by speaker). Below, 2 small colour images of the 2 drawings from earlier, each one alongside the white paper with 18 blue circle stickers on it.]
Mathematicians can use their imaginations to help them draw representations. Michael and I closed our eyes and imagined how we saw the dots. We used our imaginations to imagine the dots moving and this helped us to remember the chunks of dots.
[Black text on a white background reads ‘What’s (some of) the mathematics?’. Below, further black text (read by speaker). Below, a small colour image of ‘Michael’s method’ of drawing the groups of circles from earlier alongside the white paper with 18 blue circle stickers on it. On the left a blue speech bubble has white text inside (read by speaker).]
Mathematicians listen to the ideas of others. When Michael described seeing the 2 dots travelling down the page to make a 6, I could use my imagination to see what he was thinking. I saw the 2 dots from the top move down the page to meet up with the 4 dots that were already there. And then I instantly recognised what Michael saw which was a dice pattern of 6.
[Black text on a white background reads ‘What’s (some of) the mathematics?’. Below, further black text (read by speaker). Below, a small colour image of both earlier methods of drawing the groups of circles alongside the white paper with 18 blue circle stickers on it. On the left a blue speech bubble has white text inside (read by speaker).]
Listening to the ideas of others also helped us to compare the different ways that we can see the 18 dots. Michael and I both used our imaginations, but we created different chunks. I used the chunks of 4 to help me and Michael used chunks of 6.
[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]
Discuss and reflect
Can you draw some other ways to see the dots?
Michael saw chunks of six and Penny saw chunks of four when they looked at the dots. Do you see any other chunks hidden in the dots?
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.