Dot card talk 5– number talk (visualising)
Stage 2 – A thinking mathematically targeted teaching opportunity focussed on exploring different ways to visualise a collection
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, 2023
You will need:
something to write on
something to write with
someone to talk to (if you can).
Watch Dot card talk 5 video (7:02).
(Duration: 7 minutes and 2 seconds)
[Text over a navy-blue background: Dot card talk 5. From Kazemi and Hintz. Small font text in the lower left-side corner of the screen NSW Mathematics Strategy Professional Learning team (NSWMS PL team). In the lower right-side corner of the screen is the red waratah and blue text of the NSW Government logo.]
Dot card talk 5, from Kazemi and Hintz.
[Text on a white background: You will need…
· something to write on
· something to write with
· someone you can talk to if you can.]
As well as having your mathematical eyeballs ready, you will need something to write on and write with to record your thinking, and if you can, someone to talk to, so you can share and compare your ideas.
[Text over a blue background: How many dots? How do you see them?]
OK, mathematicians. Eyeballs ready? I'm going to show you an arrangement of dots. How many dots can you see and how did you see them? Ready?
[Blue dots appear on a white background. Altogether, there are 8 groups of 4 dots. The smaller groups of dots are each arranged in a square-like pattern, with one dot in each corner. The overall pattern is also square like, and the groups of dots are arranged in 3 rows and columns. There are 3 groups of dots in both the top and bottom row. In the middle row, there is only 2 groups of dots, one on the left side and one on the right side. There is a blank, white space in the middle of the overall pattern. The pattern disappears from screen.]
OK. One more time.
[The pattern briefly reappears, then disappears again.]
Now, you may have noticed that there are 32 dots in total.
[Text: Over to you!]
Now take your pen and paper and record how you saw them. You might want to draw a few pictures and if you have more than one-way, record those too. If you do have someone with you, share with each other and see if you have the same way or a different way of seeing the dots. Push pause and come back once you've recorded how you saw the dots. Over to you!
[The dot pattern from earlier is printed on a white sheet of paper, which is laid out over a blue surface.]
Welcome back, mathematicians. I bet there was lots of interesting thinking going on out there. Now we have some other teachers joining us who would like to share their thinking. And as they do have a think about whether you saw the dots in the same way as them or in a different way. So first, we're joined by Tom. Hi, Tom.
Now, as Tom explains his thinking, I'm going to draw it on this piece of paper. So, Tom, how did you see the dots?
[As Tom talks, Sarah points to a few of the groups of 4 dots.]
Well, Sarah, I saw little chunks of four.
[The speaker draws a rectangle around the column of dots on the left side, which features 3 groups of 4 dots.]
You see, first I saw three fours there on the left.
[The speaker draws another rectangle around the column of dots on the right side, which also features 3 groups of 4 dots.]
And then three more fours over there on the right. I saw the same thing.
[The speaker draws a third rectangle around the column of dots in the middle, which features 2 groups of 4 dots.]
And then I saw the two fours in the middle. And then I added them all together.
OK, cool. So if we recorded that, it would look something like this…
[At the bottom of the page, the speaker writes “3 fours + 3 fours + 2 fours = 12 + 12 + 8”.]
Three fours plus three fours plus two fours. And this is equivalent to 12 plus 12 plus eight. And Tom, how did you find the total?
Well, Sarah, I doubled 12, which I know is 24, then added eight, and that is 32 dots.
[At the end of the equation, the speaker writes “= 32”.]
Thanks for sharing, Tom. Did I represent how you saw it? Did that match what happened in your brain?
Great. I wonder if any of you out there saw the dots like Tom did.
[Another piece of paper with the same pattern appears on screen.]
Now let's hear from another teacher. Hi, Mish. How did you see the dots?
Oh, hey, Sarah. I really liked this question. It did fun things inside my brain. So like Tom, the first thing I noticed was that there were clusters of fours, like on a dice pattern. And then what I noticed is that there's, like, a bigger structure of nine dots on dominoes. And so what I figured out from there was, I know nine fours is 36, and then I just took away one more four to get to 32 dots.
[The speaker places a cluster of 4 red dots in the centre of the pattern.]
So I guess I used my mathematical imagination to fill in the centre dot or the centre collection of dots, which would make it nine. And then I removed it mentally too.
Oh, cool. So I could represent that by putting these four counters in like this. That's really interesting thinking. So you visualised nine fours and then took one of those fours away.
[The speaker moves the 4 red dots in the middle of the pattern to the side of the page.]
So if we represented your thinking in an equation, it might look like this.
[At the bottom of the page, the speaker writes “9 x 4 – 4”.]
9 times 4, minus 4. Did I represent that accurately, Mish?
Oh yeah. I think the part with the counters is a great way of showing the imagination. But in my mind, when I think about the equation or the number sentence, I actually prefer to have it as nine fours, minus four. Cause I like being able to name the size of the unit, and in my head, that just clarifies everything more for me. So that would be the only revision I'd suggest.
Oh, I see. So more like this.
[To the equation, the speaker adds “= 9 fours – 4”.]
Nine fours minus four. Great. Another way of seeing the dots. So mathematicians out there, I wonder if your thinking was similar or different to Mish's.
[Another piece of paper with the same pattern appears on screen.]
And we have another teacher here to share their thinking. Hi, Penn. How did you see the dots?
Hi, Sarah. Well, I just heard Mish say that she imagined a change to the collection of dots. So I used her idea, but I imagined it changing in a different way.
[The speaker introduces 2 small squares of paper; one with a cluster of 4 dots, and one that is blank. She places the square of paper with 4 dots in the middle of the pattern, and then uses the blank square to cover the cluster in the middle row of the column on the right.]
I saw the chunk of four on the right moving into the middle, so it kind of looked like six on a dice.
[The speaker draws a large rectangle around the two full rows of clusters in the middle and on the left side of the pattern.]
And how did you find how many dots there were?
Well, I couldn't quite remember six fours. But I know that six of something is just five of something and one more. And I know that five fours is 20, and one more four is 24.
Nice way of using what you know, Penn.
And then I know two fours, which is double four, which is eight, and then 24 and eight more is 32.
Nice. So we could represent that like this.
[At the bottom of the page, the speaker writes “6 x 4 + 2 x 4 = 6 fours + 2 fours”.]
6 times 4, plus 2 times 4. Or 6 fours plus 2 fours. Did I capture how your brain saw the dots, Penn?
Wow. So many different ways of seeing the dots and working out how many there were just by looking and thinking. Cool.
[Text over a blue background: What’s (some of) the mathematics?]
So what's some of the mathematics?
[Text over a white background: What’s (some of) the mathematics? This collection has 32 dots…
Below, is an image of the pattern used in this video.
Text continues: … and people see different chunks inside the 32 dots.]
We notice that this collection has 32 dots and that people see different chunks inside the 32 dots.
[Three images appear, showing the workings of how Tom, Mish, Penn and Amanda all saw the pattern.]
We saw it here and here and here.
[Text: Mathematicians listen to and add on to the ideas of others. Text in a red speech bubble: I saw chunks of four, like Tom! Text in a blue speech bubble: I just heard Mish say that she imagined a change to the collection of dots, so I used her idea…]
We also saw that mathematicians listen to and add onto the ideas of others.
[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]
Tom, Mish and Pen all saw the 32 dots differently. How was your way of seeing the dots similar or different to theirs?
Create your own dot card number talk.
What are 3 different ways you can see the dots.
Use colour to show your thinking.