# Dot card talk 3: number talk (structures)

Stage 2 and 3 – A thinking mathematically targeted teaching opportunity focussed on using structures as a tool to quantify collections

## 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-RN-01

• MAO-WM-01
• MA1-RWN-01
• MA1-FG-01

## Watch

Watch Dot card talk 3 video (6:41).

Number talk investigating ways to estimate and quantify

### Speaker

[Screen reads – Okay mathematicians … what do you notice?]

Okay mathematicians, what do you notice? Now get your eyeballs ready. That's right, we're about to subitise. Here we go.

[Screen shows numerous scattered black dots varying in size.]

So, about how many dots did you see? Yes, there were a lot. Would you like to see it again? Here we go.

[Screen shows numerous scattered black dots varying in size.]

What do you think?

Yes, we asked the mathematicians, some students in our schools, and they, when they first saw this a block of or collection of dots like you, they said, well we think it's somewhere between 25 and 78 dots. And we said to them, how certain are you? And they said not at all.

[Screen shows the collection of dots hidden under the text What do you think? with a number line indicating a range from 25 to 78 and the text How certain are you? Not at all written underneath.]

So, here's the same collection of dots, just rearranged differently. You might have noticed that some of the dots were big, and some were medium size, and some were small, so we clumped them together.

[Screen shows numerous scattered black dots in 3 piles larger, medium, and small in a horizontal row.]

And now what do you think? How many dots are there altogether?

[Screen reads What do you think? with a number line indicating a range from 25 to 78.]

Yes, so our student mathematicians also said we don't really think that helped much. We still don't really know if it's between any less than 25 or any more than 78. We'd like some more information.

So here come the same collection of dots now just arranged in another different way.

[Screen shows numerous black dots in groups of large, medium and small in more structured arrays. There are 18 large black dots, 14 medium dots and 11 small dots.]

Aha and what do you think now, would you revise your estimate? Okay, so the students did too.

They were like, well, we think 25 is too small. We think there's more than that, so we think the smallest quantity there might be is 35 and we think 78 is too big.

So, we think the largest possibility is perhaps 55 and so our range has now changed between 35 and 55.

[Screen shows number line on the bottom of the screen indicating a range between 25 and 78. 35 and 55 have now been added to the number line to indicate that the range has now changed. Text underneath the number line reads How certain are you? We feel more confident…but we still don’t know!]

And we said to them, how certain are you? And they said we feel a bit more confident, but we still don't know. So ready, get ready to look again.

We're going to show you the same collection of dots just organised in a different way.

[Screen shows 4 ten-frames with either a large, medium or small black dot in each square and 3 dots underneath the ten-frames. The number line at the bottom of the screen still indicating a range between 25 and 78 with a smaller range of 35 and 55 also included on the number line.]

And now what do you think? Aha yes. Most of our mathematician said this too, arrgghh now we know, it's 43.

[Screen now shows the number 43 on the number line.]

Would you like to see it again ready? Oh, first of all, sorry, we said, How certain are you? And they said we're, almost completely certain.

And we said to them, why? Like what's helped you become far more confident in being able to offer a pretty good estimate and in fact being able to say, look, we're pretty sure it's 43.

[Screen shows the word Why?]

And this is what they said. It is that when you saw it like this, they could see the structure of the ten-frames. Can you see that?

[Screen shows 4 ten-frames with either a large, medium or small black dot scattered unevenly in each square and 3 dots underneath the ten-frames. The number line at the bottom of the screen indicates a range between 25 and 78 with a smaller range of 35 and 55 also included on the number line. The number 43 is in red and also appears on the number line.]

Yes, so it was the structure of the ten-frames that help them go oh there's just 4 ten-frames and 3 more which you call 43 in rename, with renaming. Yeah.

So, and they said it doesn't matter 'cause here they're wonky, look if we straighten them up a bit, that's what it looks like.

[Screen shows 4 ten-frames with either a large, medium or small black dot in each square and 3 dots underneath the ten-frames. The dots in and below the ten-frames are now straightened up so they are horizontally aligned. Below the ten-frames, the number line has been removed and we are left with the number 43.]

Straighter, yes, and we made it intentionally difficult with some of the dots different sizes, but it's the same representation. Here it is there.

[Screen shows 4 ten-frames with either a large, medium or small black dot scattered unevenly in each square and 3 dots underneath the ten-frames. The number 43 is written underneath the ten-frames.]

And all we did was delete the ten-frames underneath it.

[Screen shows 4 ten-frames with either a large, medium or small black dot scattered unevenly in each square. After a slight pause the ten-frames are deleted, leaving just the scattered dots on the page.]

Aha, and doesn't that change how you can quantify the collection. Because when we see it like this it doesn't look like we can see anything much of use, but when we have this structure underneath us, it, yeah, it really helps us see there's 4 tens and 3 leftover, so that's 43.

[Ten-frames are added back to the page underneath the scattered dots.

Screen reads: What some other mathematicians said…]

[Screen shows 3 groups of dots. The first group is a random group of dots inside a yellow box. The second group is a group of dots set up in arrays, being one array of large dots, one array of medium dots and one array of small dots. This group of dots is inside a blue box. The third group of dots are inside a green box and show 4 ten-frames with either a large, medium or small black dot scattered unevenly in each square and 3 dots underneath the ten-frames.

Screen reads: The random dots (the yellow box) was the hardest to estimate how many. Our range of estimates was from 25 to 78. The sorting and grouping of dots in arrays helped us revise our thinking. We realized that 25 was too small and that 78 was too big. Our range of thinking changed to estimate between 35-55 dots. We felt more confident with our estimation when there was some structure. When we saw a structure that we trust (a ten-frame) we could see it was 4 tens without having to calculate. This helped us revise our estimate even though the dots were messy in the ten-frame. Then, we could just use our knowledge of place value to rename the collection to know how many…43!]

They said the random dots, the one in the yellow box was the hardest to estimate how many. And their ranges went from 25 to 78. Would you agree with that?

That was the hardest one to see initially.

Uh-huh then they said the one that was the next, was helpful, was actually when it was set up or an attempting to set up in arrays.

So, even though it's not an array 'cause there weren't the same number of equal rows and columns, that helped them revise their thinking a little bit.

So, they said they thought 25 was too small and 78 was too big.

But that way of arranging the dots helped them see that was somewhere they thought between 35 and 55 and they could feel a bit more confident in their estimate.

Did you, did you feel the same with that?

Okay? And then when they saw this one, the green one, they said well, when we saw that structure, we know that we can trust a ten-frame and we could see that it was 4 tens without having to calculate.

And that this then helped us revise our estimate even though the dots were messy in the ten-frame. Then we could just use our knowledge of place value to rename the collection to know how many, 43.

[Screen reads: What’s (some of) the mathematics?

Screen shows 4 ten-frames with either a large, medium or small black dot scattered unevenly in each square. Screen reads: What was the mathematics? Familiar structures, like ten-frames, can help you determine how many there are in a collection without having to count them all. You can trust a structure even if the dots are a bit wonky.]

Aha, so what was the mathematics? Well, there's a couple of things in here, but two really big ideas. One is that familiar structures like ten-frames can help you determine how many there are in a collection without having to count them all.

And secondly that you can trust a structure even if the dots are a bit wonky. And if we combine this with something else that we also know is that you don't always need a structure to help you know how many either.

[Screen shows 5 scattered dots, 3 larger and 2 smaller.]

How many dots did you see?

[Screen shows 5 dots, 3 larger dots and 2 smaller dots placed within a ten-frame.]

How many dots did you see?

[Screen shows 5 scattered dots, 3 larger and 2 smaller to the left and to the right of the screen the 5 dots, 3 larger and 2 smaller placed within a ten-frame.]

Yes, so because the collections are small, they are in fact both 5, our brains can sort of see that we can see chunks like 4 and 1 or 2 and 3, and because we're really knowledgeable about the way that numbers are made up, we can say, well, I know 5 is 4 and 1 or when I see 3 and 2 then I just know that's 5, but structure is really useful when we've got big collections, look.

[Screen shows scattered dots of larger, medium, and small dots. Screen reads: …the collection has to be big enough for structure to help!]

Aha. Yes, and so the collection has to be big enough for structure to help us.

[Screen shows scattered dots of larger, medium and small dots placed within a ten-frame.]

Okay mathematicians, I look forward to you investigating ideas of structure. Until next time...

[End of transcript]

## Instructions

• Watch and follow along with the video, listening carefully and using your imagination skills.

• When prompted, try to work out how many dots you see on the screen.

• Continue to revise your thinking as the different structures are introduced.