# Dot card talk 2 – number talk (5 sevens)

Stage 2 – A thinking mathematically targeted teaching opportunity focussed on using multiplicative situations flexibly to represent quantities

## 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-MR-01

## Collect resources

You will need:

• pencils or markers

• your maths workbook.

## Watch

Be ready to look and think like a mathematician. Watch Dot card talk 2 video (6:36).

Number talk investigating different ways to quantify collections.

### Speaker

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

We are about to do some subitising or technically subitising and visual recognition.

So, I am about to show you a collection of dots and I'd like you to see if you can see how, you see the dots. Think about how you see the dots and how many there are in total.

Okay, let's see what you notice. Here we go.

[Screen shows 5 large circles, 2 circles in the top row and 3 in the bottom row with blue outline. Each circle has 7 pink dots with 6 in a dice pattern and the seventh dot to the right in the middle row. To the right of the images there are the words ‘How do you see the dots’ and ‘And how many are there in total? After a brief pause, the images are removed from the screen with only the text remaining.]

Oh, I know I could hear you groaning.

It's meant to be fast mmm because in this case we are really trying to get inside of how your brain sees collections.

Uh-huh, so we're using our subitising and visual recognition skills. So, try to recreate what you think you saw at least in your mind.

So, for me I saw a lot some large blue circles and inside those circles were some smaller pink dots.

So, I'm trying to work out how many large blue circles I saw and how many small pink ones were inside.

Okay, would you like to have another look to check? Here we go.

[Screen shows 5 large circles, 2 circles in the top row and 3 in the bottom row with blue outline. Each circle has 7 pink dots with 6 in a dice pattern and the seventh dot to the right in the middle row. To the right of the images there are the words ‘How do you see the dots’ and ‘And how many are there in total? After a brief pause, the images are removed from the screen with only the text remaining.]

Okay, so what I'm really trying to get you to think about is how do you see the dots, because that will help you answer this question of 'And how many are there in total?'

Okay, ready? Here's what it looked like.

[Screen shows 5 large circles, 2 circles in the top row and 3 in the bottom row with blue outline. Each circle has 7 pink dots with 6 in a dice pattern and the seventh dot to the right in the middle row. To the right of the images there are the words ‘How do you see the dots’ and ‘And how many are there in total?]

Yeah, and now let's have a think about how different people saw these dots.

So, this, this is what Michael was thinking. So, when he saw this collection of dots, he could see the 5 groups um because he saw the 2 and the 3 more and he knew that was 5, and then he knows 5 sevens is 35, so he could actually see that quantity as a whole.

But Sharon, when she saw it, she was looking she actually saw some chunks.

So inside of the 5 groups where the 7 dots are, she saw the 7 as 5 and 2 more.

So, she said what she knew in her head was that 5 fives is 25 and then 5 twos is 10 and she was able to join 25 and 10 together to get 35.

But Lucy thought about it differently. She saw four groups like on a dice pattern and one more group of 7.

So, she said she worked out 4 sevens, which is 28, and then added another 7 dots to get to 35.

Yeah, and Millsy thought about it like this. She saw the 5 big groups altogether and inside of them she saw six like on a dice pattern. Yeah, and the one more dot. So, she worked out 5 sixes and then 5 ones.

[Screen shows the page set out in quarters showing examples of how different people saw the dots.

In the first example, it shows 5 large circles, 2 circles in the top row and 3 in the bottom row with blue outline. Each circle has 7 pink dots with 6 in a dice pattern and the seventh dot to the right in the middle row. The text below the image reads ‘Michael: I know 35 is 5 sevens’.

In the second example, it shows 5 large circles, 2 circles in the top row and 3 in the bottom row with blue outline. All circles show the first 5 dots being pink and the bottom 2 dots in each of the circles being orange and the screen reads Sharon: I saw 5 and 2. So, I know 5 fives is 25. Then, 5 twos is 10. 25 and 10 is 35.

The third example shows 5 large circles, 2 circles in the top row and 3 in the bottom row with blue outline. The first 4 circles all have 7 pink dots and the fifth circle has all orange dots. The screen reads Lucy: I saw 4 sevens and 1 seven. I know 4 sevens is 28. Then I added another 7 to get 35 dots.

The fourth example shows 5 large circles, 2 circles in the top row and 3 in the bottom row with blue outline. In all circles the first 6 dots are pink and the seventh dot, in the second row to the right is orange. The screen reads Millsy: I saw 5 sixes and 5 ones. I know 5 sixes is 30. I know 5 ones is 5. 30 + 5 is 35.]

Yes, and so what's really amazing about doing dot card talks like this is that we can see that lots of different people can think about quantities differently, and so this got me thinking about I wonder how we could rearrange these dots into different structures so that we can see different ways of thinking.

[Screen reads – How could I rearrange these ways of thinking into different structures?]

And here I'll show you what happened in my imagination.

When Millsy said this, that she saw 5 sixes and 5 ones, this made me think about what happens if I take out those boundaries of the group like this and rearrange them so I can still see the 5 sixes and the 5 ones, but now it looks like an array.

[Screen shows 5 large circles, 2 circles in the top row and 3 in the bottom row with blue outline. Each circle has 6 pink dots in a dice pattern and the seventh dot to the right in the middle row is orange. Screen reads – Millsy: I saw 5 sixes and 5 ones. I know 5 sixes is 30. I know 5 ones is 5. 30 plus 5 is 35.

The outline is then removed around all of the circles and the dots are moved into an array. The array displays 5 rows of 7 dots with the first 6 dots in each row being pink and the seventh dot in each row being orange. The screen reads 5 sixes plus 5 ones.]

Yes, so if I come back to all of these different ways of thinking, here is Millsey's original way of thinking. And here it is just rearranged in an array structure.

[Screen again shows the page set out in quarters showing the examples of how different people saw the dots with the presenter highlighting Millsy’s original way of thinking and then showing the example rearranged in an array structure.

Screen reads – Imagine the other collections reforming into arrays … draw what they would look like]

So, your job mathematicians is, 'Can you imagine the other collections reforming into arrays and draw what they would look like?’

Okay, I'll put them back up on the screen for you. It would be a good idea to pause the video here.

[Screen again shows the page set out in quarters showing the examples of how different people saw the dots. Millsy’s example is still arranged in the array structure.]

Okay, shall we have a look together after you've had a chance to draw? Okay, so here's what I left you to think about.

[Screen reads – Let’s have a look together! Screen again shows the page set out in quarters showing the examples of how different people saw the dots. Millsy’s example is still arranged in the array structure.]

Let's have a look at what Sharon did in her thinking, so here was how she chunked the 7 together into fives and twos and here's what that would look like as an array.

[Screen shows 5 large circles, 2 circles in the top row and 3 in the bottom row with blue outline. All circles show the first 5 dots being pink and the bottom 2 dots in each of the circles being orange and the screen reads Sharon: 1 saw 5 and 2. So, I know 5 fives is 25. Then, 5 twos is 10. 25 and 10 is 35. To the right, indicated by a blue arrow pointing right, is an array of 35 dots, with 5 rows of 7 dots. The first 5 dots in each row are pink and the last 2 dots in each row are orange. The text under the array of dots reads 5 fives plus 5 twos.]

Is that similar to yours? Yeah, 5 fives and 5 twos and yes, I can see that too.

What I can see now that inside of 5 sevens where there's 5 fives is a square number.

Yeah, I've never thought about that before that inside of 5 sevens is 5 fives, so inside of 5 sevens is a square number, and now I wonder actually how many other square numbers are in there. Oh, okay, you can go off and investigate that if you're curious also.

Let's have a look at what Lucy's might look like.

[Screen shows a group of 5 large circles, 2 circles in the top row and 3 in the bottom row with blue outline. The first 4 circles have 7 pink dots, and the last circle has 7 orange dots. The screen reads Lucy: I saw 4 sevens and 1 seven. I know 4 sevens is 28. Then I added another 7 to get 35 dots. To the right, indicated by a blue arrow pointing right, is an array of 35 dots, with 5 rows of 7 dots. The first 4 rows contain 7 pink dots and the last row contains 7 orange dots. Screen under array reads 5 sevens plus 1 seven.]

Yes, so you can see that too. So, the 4 sevens are in one colour and the 1 seven more is in the other. And so, where Lucy partitioned her array in this way where she partitioned the 5 into four and one, the 5 groups into four and one, Sharon partitioned it in different way where she kept the 5 groups, but she partitioned the part inside the group, the 7 into 5 and 2 more.

[Screen shows a group of 5 larger circles, 2 circles in the top row and 3 in the bottom row with blue outline. All circles show the first 5 dots being pink and the bottom 2 dots in each of the circles being orange and the screen reads Sharon: I saw 5 and 2. So, I know 5 fives is 25. Then, 5 twos is 10. 25 and 10 is 35. To the right, indicated by a blue arrow pointing right, is an array of 35 dots, with 5 rows of 7 dots. The first 5 dots in each row are pink and the last 2 dots in each row are orange. The text under the array of dots reads 5 fives plus 5 twos.]

Alright, and Michael’s yes, would look like this.

[Screen shows a group of 5 large circles, 2 circles in the top row and 3 in the bottom row with blue outline. Each circle has 7 pink dots with 6 in a dice pattern and the seventh dot to the right in the middle row. Above the image, the screen reads Michael: I know 35 is 5 sevens. To the right, indicated by a blue arrow pointing right, is an array of 35 dots, with 5 rows of 7 dots. All dots in each row are pink. The text under the array of dots reads 5 sevens.]

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

Alright mathematicians, so let's talk about what the maths was in there that we saw. Or some of it anyway, 'cause it's quite a lot.

One thing is that it helped us see that quantities can quite different but have the same value and that we can represent quantities in different ways and that also we can think about multiplicative situations with the same flexibility we use for whole numbers, and this is really helpful for us as we come to start using really efficient strategies in multiplication and division.

[Screen reads – What was the mathematics? Quantities can look different but have the same value, we can represent quantities in different ways and we can think about multiplicative situations with the same flexibility we use for whole numbers.

To the right of the text are 2 images. The first is an image set out in quarters with examples of how different people saw the dots.

The first example shows 5 large circles, 2 circles in the top row and 3 in the bottom row with blue outline. Each circle has 7 pink dots with 6 in a dice pattern and the seventh dot to the right in the middle row. The text below the image reads ‘Michael: I know 35 is 5 sevens’.

In the second example, it shows 5 large circles, 2 circles in the top row and 3 in the bottom row with blue outline. All circles show the first 5 dots being pink and the bottom 2 dots in each of the circles being orange and the screen reads Sharon: I saw 5 and 2. So, I know 5 fives is 25. Then, 5 twos is 10. 25 and 10 is 35.

The third example shows 5 large circles, 2 circles in the top row and 3 in the bottom row with blue outline. The first 4 circles all have 7 pink dots and the fifth circle has all orange dots. The screen reads Lucy: I saw 4 sevens and 1 seven. I know 4 sevens is 28. Then I added another 7 to get 35 dots.

The fourth example shows 5 large circles, 2 circles in the top row and 3 in the bottom row with blue outline. In all circles the first 6 dots are pink and the seventh dot, in the second row to the right is orange. The screen reads Millsy: I saw 5 sixes and 5 ones. I know 5 sixes is 30. I know 5 ones is 5. 30 + 5 is 35.

The second image is an image set out in quarters with examples of 4 different dot arrays.

The first example is an array of 35 dots, with 5 rows of 7 dots. All dots in each row are pink. The text under the array of dots reads ‘Michael: I know 35 is 5 sevens’.

The second example is an array of 35 dots, with 5 rows of 7 dots. The first 5 dots in each row are pink and the last 2 dots in each row are orange. The text under the array of dots reads ‘Sharon: I saw 5 and 2. So, I know 5 fives is 25. Then, 2 fives is 10. 25 and 10 is 35’.

The third example is an array of 35 dots, with 5 rows of 7 dots. The first 4 rows contain 7 pink dots and the last row contains 7 orange dots. The text under the array of dots reads ‘Lucy: I saw 4 sevens and 1 seven. I know 4 sevens is 28. Then I added another 7 to get 35 dots’.

The fourth example is an array of 35 dots, with 5 rows of 7 dots. The array displays 5 rows of 7 dots with the first 6 dots in each row being pink and the seventh dot in each row being orange. The text under the array of dots reads ‘Millsy: I saw 5 sixes and 5 ones. I know 5 sixes is 30. I know 5 ones is 5. 30 plus five is 35’.

Alright mathematicians, until next time. See you then.

[End of transcript]

## Instructions

• Imagining the other collections reforming into arrays, draw what they would look like.

• Record your thinking in your notebook.

• How do people in your family see the collection?