# Investigating ten-frames

A thinking mathematically targeted teaching opportunity exploring the features of a ten-frame.

## Syllabus

Syllabus outcomes and content descriptors from Mathematics K–10 Syllabus (2022) © NSW Education Standards Authority (NESA) for and on behalf of the Crown in right of the State of New South Wales, 2024.

## Outcomes

- MAO-WM-01
- MAE-RWN-01
- MAE-RWN-02
- MAE-CSQ-02

## Collect resources

You will need:

colour pencils or markers

something to write on.

## Investigating ten-frames

Watch Investigating ten-frames video (14:50).

### Speaker

Hello there little mathematicians welcome back

[Screen reads – you will need…a pencil and piece of paper and screen shows a pencil and some paper.]

Today we thought we'd spend some time exploring this little structure here, which we call a ten-frame. We use these a lot.

[Screen shows a ten-frame on a white piece of paper. The ten-frame shown is a rectangle with one vertical line and four horizontal lines, making 2 rows of 5 or a total of 10 boxes. The ten-frame is on top of a pink piece of A4 paper and a larger piece of navy paper is sitting above.]

They're really helpful in helping us understand and investigate and explore quantities and numbers.

And so, we thought it would be a really good time to really get in and analyse what exactly is this ten-frame all about.

And so, we thought that we might start off by actually trying to draw one.

And that's a really important idea, because mathematicians like writers know that when they really come to understand words or mathematical representations like a ten-frame, that they can also draw them too.

So, before I can draw it, I need to notice things about my ten-frame.

[Presenter picks up the ten-frame, moves the pink piece of paper to the side into a portrait position and then places the ten-frame paper back down.]

So, what are some things that you can tell me about what you see here?

Ah, ok, it has ten boxes. Let's check that. 1, 2, 3, 4, 5, 6, 7, 8, 9, 10. Ok, so something we notice.

[Presenter uses finger to point and count the boxes. They write ‘10 boxes’ on the pink paper.]

There are ten boxes. Something else that you notice?

Oh, yeah, around the outside, there's a big box isn't there? In fact, it's a big rectangle and there's one of them. So, there's one big rectangle.

[Presenter points to the rectangle on the white paper and outlines it with her finger. They write ‘1 big rectangle’ on the pink paper.]

Yeah, you're right, and the ten boxes sit inside the one big rectangle.

Don't they 'cause there's the big rectangle and there's one box and another box and a third box and the fourth box and a fifth box.

The sixth box and a seventh box and an eighth box and a ninth box and the tenth box. And they all sit inside.

[Presenter uses finger to trace the big rectangle and then each of the 10 boxes.]

Oh yeah, so there's this big rectangle and then there's other lines inside it that help partition it, don't they?

How many other lines are there? Yeah, there's one big line down the middle. One line down the the middle. You're right, it's one long line.

[Presenter points to the rectangle and traces the long horizontal line with her finger and writes ‘1 long line down the middle’ on the pink paper.]

Nice revising there mathematicians. One long line down the middle and then there's these shorter lines aren't there? And how many of those are there? Yeah, you could say there's eight 'cause you go 1, 2, 3, 4, 5, 6, 7, 8.

You're right, or if you drew the whole line, that would be one, wouldn't it? 1, 2, 3, 4 shorter lines.

[Presenter traces the short vertical lines either side of the long horizontal line with their finger and counts a total of 8. Then they trace the whole vertical lines with their finger and count to 4. They write ‘4 shorter lines’ on the pink paper.]

So actually, let's use our fingers and trace around. So, let's check we have ten boxes. 1, 2, 3, 4, 5, 6, 7, 8, 9, 10. One big rectangle. One long line down the middle. And four shorter lines.

[Presenter uses their finger and counts out the 10 boxes, touching their finger inside each box as they count. Then they trace 1 big rectangle, one long line down the middle and the 4 shorter vertical lines with their finger.]

Ok little mathematicians over to you, I'm going to put this representation of our ten-frame up here and now that we've thought about all the things that we can see over to you to draw one.

[Presenter moves the ten-frame on top of the navy paper and places the pink paper with writing on it underneath.]

So, pick up your pencil and get your paper. And get ready now to draw a ten-frame.

And while you do that, I'm going to do the same thing. How did you go? Does yours look a little bit like mine too? Ah.

[Presenter moves the pink paper with writing on it to the side and gets another piece of pink paper to draw on. They use a marker to draw a rectangle with a horizontal line in the middle and 4 vertical lines down, creating 10 boxes.]

Should we try to draw a ten-frame that has some things in it 'cause this one at the moment is representing zero. Because there's nothing inside the boxes.

[Presenter writes the word and numeral for zero underneath the ten-frame they just drew.]

Let's have a look at one that has some things inside it. What about this one? How many can you see here?

[Presenter removes the zero example, leaving a fresh piece of pink paper to draw on. They also remove the ten-frame from the top of the screen and replace it with a ten-frame showing a black dot in each of the first 4 boxes in the top row.]

Four that's right, it's representing four.

So little mathematicians I wonder now if you could, first of all, take a picture in your mind of what that looks like.

[Presenter mimics taking a picture of the ten-frame by bending their fingers and making a clicking noise.]

And then think about tracing around, so the one big rectangle with your finger.

The one line down the middle. And four shorter lines. And then we also need to represent four.

[Presenter traces around the one big rectangle, the line along the middle and the 4 shorter lines of the ten-frame with their finger.]

So, then I would colour in one dot. Colour in a second dot. Colour in a third dot. And colour in a fourth dot.

[Presenter points to each of the dots and then flips the ten-frame over face down.]

Now that you've got that picture in your mind. Pick up your pencil or your marker and draw your representation of four on a ten-frame.

Over to you, and I'm going to have a go at the same time, but this time I'm going to draw it myself underneath here so that you get a chance to think in your brain as well, ok?

You get started and I'll join you.

That's right, I'm thinking about the rectangle too. Like that. Then I'm thinking about the line down the middle. Then I'm thinking about four shorter lines that go inside my rectangle. 1, 2, 3, 4.

[Presenter holds a piece of paper up to hide their drawing and then moves it away to reveal after each step. They draw a rectangle, then a horizontal line across the middle, then 4 vertical lines, and finally colour in 4 dots in the first 4 boxes. The presenter removes the paper screen and shows their ten-frame.]

Then I was thinking about four. And I think it had 4 dots at the top because I remember the bottom row was empty and so was this space here. So, then I have to colour in my 4 dots.

OK, can you show me your drawing? Do ours look the same or similar?

And let's see if it looks like our representation of four. Yeah, I can see four here and I can see four here.

[Presenter flips ten-frame on white paper back over and points to the 4 depicted on the model and 4 depicted in their own drawing.]

I can still see the ten boxes look, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10. I can see the big rectangle. I can see the line down the middle, and I can see the four shorter lines, 1, 2, 3, 4.

[Presenter refers to the paper with the written points from earlier. They read the things a ten-frame needs and look at their drawing to check. They count the boxes with their finger, then they trace the big rectangle, line down the middle and 4 shorter lines with their finger.]

And look at this mathematicians, even if I turn it around, I can still see this.

Look, ten boxes. 1, 2, 3, 4, 5, 6, 7, 8, 9, 10. One big rectangle, one big rectangle with four sides. One long line down the middle like that. And four shorter lines, 1, 2, 3, 4.

[Presenter rotates the pink paper with the ten-frame drawing on it to the right. They count the 10 boxes, then trace the big rectangle with 4 sides, the long line down the middle, the 4 shorter lines and the 4 dots, counting as they go.]

And I still have 4 dots for this one too. 1, 2, 3, 4. Ok.

[Presenter rotates the model ten-frame on the white paper and places next to the ten-frame drawn on pink paper to compare them.]

I'm going to show you another representation. And this time I'd like you to take a picture in your mind's eye. Here it comes.

[Presenter removes the 4-dot example, leaving a fresh piece of pink paper to draw on. They also remove the model ten-frame and replace it with a ten-frame on white paper showing 6 black dots – a black dot in each of the first 3 boxes in the top and bottom rows.]

Ok, and over to you to draw. So, pick up your pencil's now. And draw the representation that you just saw.

[Presenter flips the ten-frame on the white paper over face down.]

Would you like to see it again to check? Here we go.

[Presenter flips the paper back over to show the ten-frame.]

Take a picture in your brain. And look at the details. How many dots are there? Where are they?

[Presenter flips it back over so the ten-frame is face down.]

Ok, have a go at drawing that from your mathematical imaginations. I'm gonna draw it too over here.

Ok. How did you go? Does your drawing look like this?

[Presenter draws their ten-frame off camera and then flips the paper over to show the ten-frame.]

This is what mine looks like.

There's my 6. Yeah, so I can see I have four shorter lines, 1, 2, 3, 4. One line in the middle, one big rectangle and ten boxes, 2, 4, 6, 8, 10.

[Presenter shows their ten-frame drawing, placing it in between the model ten-frame on the white paper and the written list of things a ten-frame has. The presenter uses their finger to trace over the 4 shorter lines, middle line, one big rectangle and 10 boxes in their drawing.]

I know I went backwards that time, didn't I?

And even if I turn it like this where it's really wonky. I can still see it. You're right, one big rectangle. One line down the middle and four smaller lines. That's right.

[Presenter turns their ten-frame drawing on the pink paper to the left slightly off centre and traces details again with their finger.]

Ok, I'm leaving you with one left to challenge you little mathematicians.

Here it comes.

[Presenter removes the 6-dot example, leaving a fresh piece of pink paper to draw on. They also remove the model ten-frame and replace it with a ten-frame on white paper showing 6 black dots – a black dot in each of the 5 boxes in the top row and one dot on the far left in the bottom row.]

Notice the features of the ten-frame and notice how many dots there are.

[Presenter flips the paper over face down so you cannot see the ten-frame.]

Uh-hm. Ok. You might like to draw it in the air and show what you saw in your brain. And then draw it on your piece of paper. I'm gonna draw one too.

[Presenter draws the ten-frame in the air with their finger.]

Ok, are you ready to have a look together?

[Presenter flips the paper back over to show the ten-frame.]

Ok, here was our ten-frame that we had. And how many dots is it showing us 6, Yeah, 'cause it's four empty spaces and 6 is 4 less than 10. And I also know it's 6, you're right, 'cause there's 5 on the top and one down the bottom.

[Presenter points to the 4 empty boxes on the model ten-frame.]

Here's my drawing of 6. How did yours go?

[Presenter shows their ten-frame drawing, placing it in underneath the model ten-frame on the white paper. They move the written list of things a ten-frame has off screen. Then they show the ten-frame example from earlier with 6 dots – 3 dots on the top row and 3 on the bottom row and place it next to the other example with 6 dots.]

Very nice and you know mathematicians. I noticed something, look. This one I did was 6.

Where's my marker? Look, 2, 4, 6.

[Presenter looks at the example with 2 rows of 3 and counts the dots by 2s using her fingers and then writes 6 underneath the ten-frame.]

This one was 6. Five, six. And they are 2 different ways of representing 6on the ten-frame.

[Presenter points to the dots on the example with a row of 5 and one more and writes 6 underneath the ten-frame.]

This one has one, 3 and a second 3 or 2, 2 and 2.

[On the example with 2 rows of 3, the presenter circles each row of dots with their finger. They then show how it could also be 3 columns of 2 by pointing to the groups of 2 on the ten-frame.]

And this one shows me 6 with one row of 5 and one more.

[On the example with 5 and one more, the presenter points to each of the six dots in ten-frame and circles the 5 dots on the top and one dot on the bottom with their finger.]

I wonder how many other ways that I could represent 6 dots on a ten-frame?

That sounds like an investigation to me mathematicians.

Over to you. Ok, press pause here little mathematicians, go investigate and come back and we'll show you what some other mathematicians came up with.

[Screen reads – pause.]

So firstly, what was some of the maths?

One is, we realise that when you use a ten-frame for example, you can represent the same quantity, the same amount like 6 in different ways.

[Screen shows 3 ten-frames one on the top and 2 underneath on different pieces of paper. The top example has 5 dots in the top row and one on the bottom row. The example to the left has been drawn earlier and has 5 dots in the top row and one on the bottom row. The third example has also been drawn earlier and has 3 dots in the top row and 3 dots in the bottom row.]

That's important cause it helps us see different things about those numbers. That mathematicians can record their ideas in different ways.

[Screen shows 2 ten-frames one on the top and one underneath on different pieces of paper. The top example has 4 dots in the top row. The one underneath is in portrait orientation and has 4 dots in the squares in the right column. The written points about what a ten-frame has is also shown to the side.]

Ten-frames is one of these. The ten-frames can be described as a mathematical pattern because they always have the same structure, and they always show us 10 boxes.

We also realise that when we draw representations, it helps us to make meaning from them, as we notice features.

So, let's look at what some other mathematicians discovered today.

Riley saw 6 as 4 and one and one.

[Screen shows a child’s drawn example of a ten-frame. In the top row, there are black dots in the first 2 boxes and then there is a blue dot in the fourth box. In the bottom row, there are black dots in the first 2 boxes and a red dot in the fifth box. Underneath the ten-frame it reads 4 + 1 + 1.]

And 6 as, 3 and 3.

[Screen shows a child’s drawn example of a ten-frame. In the top row, there are red dots in the first 2 boxes and then there is an orange dot in the fifth box. In the bottom row, there is a red dot in the first box and orange dots in the fourth and fifth boxes. Underneath the ten-frame it reads 6 is 3 and 3.]

Erica saw 6 as 5 and one more.

[Screen shows a child’s drawn example of a ten-frame. In the top row, there are 5 red dots in each of the boxes. In the bottom row, there is a green dot in the first box. Above the ten-frame it reads 5 and 1 is 6.]

And she saw it as 4 and one and one.

[Screen shows a child’s drawn example of a ten-frame. In the top row, there are red dots in the first 4 boxes and a blue dot in the fifth box. In the bottom row, there is a green dot in the first box. Above the ten-frame it reads 4 and 1 and 1 is 6.]

And Louise saw 6 as 2 and 2 and 2.

[Screen shows a child’s drawn example of a ten-frame. In the top and bottom row, there is an orange dot in the first box, a red dot in the third box and a blue dot in the fifth box. Underneath the ten-frame it reads 2 and 2 and 2.]

[End of transcript]

### Instructions

- Have a go at drawing your own ten-frames
- Draw 6 in a ten-frame in a few different ways