# Imagining dots (arrays)

Stage 2 and 3 – A thinking mathematically targeted teaching opportunity, focussed on using mathematical imagination to quantify a collection of dots

Adapted from Kling, G. & Bay-Williams, J. (2019). Math Fact Fluency: 60+ Games and Assessment Tools to Support Learning and Retention, Association for Supervision and Curriculum Development.

## 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

## Outcomes

- MAO-WM-01
- MA2-MR-01

- MAO-WM-01
- MA3-MR-01

## Collect resources

You will need:

- paper
- a pencil or a marker.

## Watch

Watch Imagining dots (arrays) part 1 video (6:25).

[White text on a navy-blue background reads ‘Imagining dots – Stage 2 and Stage 3 – Adapted from Kling and Bay-Williams (TCM, 2015)’. On the right, a blue half circle at the top and a red half circle at the bottom. In the middle bottom, a line of red dots forms another half circle. In the bottom left corner, a white NSW Government ‘waratah’ logo.]

#### Speaker

Today we're going to explore a task called imagining dots. It's adapted from an article written by Gina Kling and Jennifer Bay-Williams.

[Black text on a white background reads ‘You will need…’ Below, black text dot points (read by speaker).]

#### Speaker

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

Hey mathematicians. Today we're going to show you a collection of dots. I'm going to show you them for 3 seconds.

[White text on a blue background reads ‘Eye balls ready!’.]

#### Speaker

And it's your job to imagine how you see the dots on the page. You'll need to pay really close attention because I'm only going to show you the dots for 3 seconds. 3 seconds isn't very long. So are you ready? Are your eyeballs ready to take a picture?

[On a white background a large group of small blue circles, 6 high on the vertical and 9 long on the horizontal.]

#### Speaker

1, 2, 3. OK. 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 and help you to create a really good picture in your brain and where you saw the dots. We call this using our mathematical imagination. Right. Well, let's take another look, so you can check the picture you've been making in your mind.

Hold on to what you noticed. And let's take another look to check our ideas. Eyeballs ready.

[The group of blue circles from earlier.]

#### Speaker

1, 2, 3. You know, when I looked at the screen, there was so many dots. And I know that there's just too many for me to subitise. I can't instantly recognise how many dots there are. So, I might need to chunk the array into sections. How about we take a closer look?

[On a wooden desk, a sheet of white paper has the array of blue circles on it. The speaker uses a black marker to outline sections of the array (described when mentioned).]

#### Speaker

When I look closely at my array, my eye can see across the top here 4 in this top corner. So, this chunk here is 4. And actually, that's helped me to see how many there are in the rest of the row, because I can see another 4 and one more, which tells me that there are 5 here. So, now I know that my top row is made up of 9 dots.

Now, what about if we look down here? Can we use our subitising eyes to see some chunks coming down the column? Let's take a really close look. You know, when I look down here, if I halve the quantity in each of the columns by stretching my hand out on top here, can you see what I see?

Yeah. I put this line in halfway, and now I know there are 3. And if I uncover this, I can also use my subitising eyes to see another 3. Now I know coming down here, that there are 6 9s and I'm going to change these now to 9s. Because there's 9 in each row.

[Black text (read by speaker). Below the blue circle array has a pink arrow above it that points to the right. A pink speech bubble has white text that reads ‘There are 9 dots across here’. A second black speech bubble on the left has white text that reads ‘There are 6 dots along here’.]

#### Speaker

Now we know there are 9 dots across here and 6 going down here. We know that we're working with a 6-by-9 array and we can think about how we can work out 6 9s. Some of you might already know that 6 x 9 is 54. But what if you don't? What strategies could you use to efficiently work it out?

[On a wooden desk, a sheet of white paper with the blue circle array on it.]

#### Speaker

So, my friend Michael and I were talking and he said that when he looked at the array, he saw the 6 rows, and each of the row was composed of a 9. Now, Michael knew that 6 can be broken down into 3 and 3. And the reason Michael did this, because he knew 3 x 9 is 27. And the reason that helped Michael so much was he knows now that he has 27 here. And all of these are 27 as well. And when he combines them back together, they give him a total of 54.

So, I really liked Michael's idea of starting with what he knew. So, I had a really big think to myself, you know, what am I really confident in working with? And I'm really confident in working with 5s. I know that 5 of something and one of something will give me 6 of something. So, that means that if I know 5 x 9, which you can see here, there's my 5. And look here, here's my 5 x 9. And I can label that as 5 9s. And I know that 5 x 9 is 45. So, then I just need to think about adding on one more 9, which will then also give me a total of 54.

Then we thought about testing out one more strategy. We know that 9 is one less than 10. So, we wondered, could we use our knowledge of 10s? Let's test it out. Now we know that this array is made up of 6 x 9. But what if we were to imagine another column appearing over here?

[The speaker draws an extra column of circles on the right of the array.]

#### Speaker

And in that column, we can move from having 6 x 9 to 6 x 10. So, we know that all of these are 60, because we can use what we know about renaming.

We can rename 6 x 10 as 60. Then we can remove the 6 that we added on at the end, and we will also have a total of 54.

[White text on a blue background reads ‘Over to you…’ Below, further white text (read by speaker). At the bottom, three colour images of the 3 different arrays from earlier.]

#### Speaker

So over to you now, mathematicians. How could you use the same ways of thinking to solve 7 x 9, which we can also say is 7 9s?

[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]

## Instructions

- After watching the video, how could you use the same ways of thinking to solve 7 x 9? We can also call this 7 nines.
- How many ways can you think of to solve 7 x 9?

## Watch

Watch Imagining dots (arrays) part 2 video (5:51).

[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.]

#### Speaker

So while you were away, I was thinking that if I wanted to draw these dots, it would take me a really long time and would actually be a really inefficient way of recording my thinking. So it got me wondering what else do I have in my toolkit that could help me represent these ideas so that I could share my thinking with others?

Now mathematicians have this diagram they use called an area model, and it's a really efficient way of describing situations. Let's take a look.

[On a wooden desktop, a blank sheet of white paper on a larger sheet of blue cardboard. Above and to the left, an array of blue circles has been divided in half horizontally. There are 2 blue handwritten numerals ‘3’ on the vertical axis on the left and the word ‘nines’ is written at the top on the horizontal axis. On the right, further blue text reads ‘3 nines is 27’.]

#### Speaker

Let's start by looking at Michael's idea. So when we think about the area model, we can see up here on the array that it's actually shaped in a rectangle. So I'm gonna start by drawing that.

[The speaker uses a green marker to draw on the blank sheet of paper (explaining as she works).]

#### Speaker

Now, my rectangle doesn't need to be the exact same size as the array. It can just be the size it needs to be on the piece of paper. Now, I know that Michael used this idea of 9s. So I'm gonna write 9s up here because we can imagine the rows of 9s going across the page. Now, Michael used 3 and 3. Now we know that 3 is half of 6.

I need to have a look at this line here and imagine about halfway. So I'm gonna put a little mark just here. So I think that's about halfway. So to show that Michael thought about 3 9s, I'm gonna draw a line straight across here because Michael saw 3 9s and then another 3 9s repeated down below. Oh, look at that. Much more efficient way of recording these 54 dots. But we still get to see Michael's thinking. Let's have a look at one of the other strategies.

[On a wooden desktop, a second array of blue circles (as explained by speaker) and another blank sheet of white paper. The speaker uses a green marker to draw as she explains.]

#### Speaker

So using 5 9s instead of 6 9s was my idea because I was really comfortable in working with 5 of something. So when I look at this, I know I need to start with my rectangle first. Now let's have a look at how much space the 5 9s takes up. Now it almost comes right down the bottom. So I'm thinking about imagining the exact scene. And I have to leave enough space for just one row of dots here. So I think I'm gonna mark that about here. And then I can draw my line across, and I can label it as 5 9s and one 9. So here is 5 9s going across here and one more 9 down the bottom.

[On a wooden desktop, a third array of blue circles (as explained by speaker) and another blank sheet of white paper. The speaker uses a green marker to draw as she explains.]

#### Speaker

So let's take a look at these last one. Now this last one is just a little bit different because I actually drew on an extra column to make it 6 10s. So let's start with our rectangle. Now I drew this rectangle a little bit longer because I actually need to add this extra column here, and I think I'm gonna draw a line. I'm thinking about how much space these dots take up in my row across. And I come right to the end and I'm actually gonna draw a line down here.

So at the moment, I have 6 10s. But then I remembered that this was a line of imaginary dots that I'd drawn. And so, once I'd done my renaming 6 10s as 60, I knew I had to take these off. So I'm actually going to shade that off because 6 10s, which gives me 60. Then I needed to do 60-6=54 and I can even write 6x9 is equivalent in value to... 6x10-6. So mathematicians, take a look at how we've all represented our different ideas using the area model.

[In three still colour images, the 3 different area models and arrays from earlier.]

#### Speaker

What an efficient diagram to help us show our thinking.

[White text on a blue background reads ‘What’s (some of) the mathematics?’]

#### Speaker

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, a small colour image of the first array from earlier. A blue speech bubble on the left has white text that reads ‘Michael saw the chunks of three nesting inside of the 6 rows.’

#### Speaker

We can think about multiplicative situations with the same flexibility we use for whole numbers. We think about what we know about nine and six to see how we can be flexible. Michael saw the chunks of 3 nesting inside the 6 rows, so he thought he could use 3 9s and 3 9s to work out the total.

[Black text on a white background reads ‘What’s (some of) the mathematics?’. Below, further black text (read previously by speaker). Below, a small colour image of the second array from earlier. A pink speech bubble on the left has white text that reads ‘I was more confident in working with 5 nines. I thought 5 nines and 1 more nine.’

#### Speaker

I was more confident in working with 5 9s, so I thought five nines and one more nine would also give me the total.

[Black text on a white background reads ‘What’s (some of) the mathematics?’. Below, further black text (read by speaker). Below, 3 small colour images of the different area models/arrays from earlier. A blue speech bubble on the left has white text that reads ‘Representing our thinking using the area model was an efficient way to represent our ideas.’

#### Speaker

Mathematicians use diagrams to test out their ideas and to also help them communicate mathematical ideas to others. Representing our thinking using the area model was a really efficient way for us to represent our ideas without having to draw all the dots.

[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

- Michael and Penny both saw the arrays differently. How was your way of seeing the dots similar or different to their way of thinking?
- Try using the area model to represent the different ways you could break apart your array.