# Finding halves

A thinking mathematically targeted teaching opportunity exploring examples and non-examples of halves

## 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
• MAE-FG-02
• MAE-GM-03
• MAE-2DS-02

## Collect resources

You will need:

• a few sheets of A4 paper

• scissors

• coloured markers/pencils

• an adult to help with the cutting.

## Watch

Watch Finding halves video (12:44) to learn how to play.

Find and explore halves around the environment.

### Transcript of Finding halves video

(Duration: 12 minutes and 44 seconds)

[Text over a navy-blue background: Finding halves. Small font text in the lower left-hand corner reads: NSW Mathematics Strategy Professional Learning Team (NSWMS PL team). In the lower right-hand corner is the red waratah of the NSW Government logo.

On a pink sheet, on the upper left-hand side, there are two rows of paper strips: the top row strip has been cut with a longer side on the left, the bottom row strip has been cut in half. On the right side of the paper strips, there are 2 small hexagons: the first hexagon has a line across its middle, the second has a line across near the bottom. Below the paper strips are 2 apples. Below the apples are 2 pig figures: the first pig has a rope going from its head to feet, the second one has a rope going from its shoulder to feet. Next to the apples and pigs are 3 plates of toasts: the first toast has been cut diagonally, the second has been cut down from the middle, and the last has been cut at an odd angle.]

### Speaker

Hello there, little mathematicians. I hope you're having a great day. I found this collection of objects around my house. What can you see? Uh-huh. Yes.

[She circles the toast with her fingers.]

### Speaker

These were some pieces of toast from my breakfast, and I'm looking forward to eating them. (LAUGHS) Yes. And one of my friends came and had a look at these objects, and they said, ooh, this is interesting, because I think that some of these objects are showing halves, but some are not halves.

[She places a blue sign saying ‘Halves’ on the top-left side of the sheet. She places a blue sign saying ‘Not Halves’ on the top-right side.]

### Speaker

And that got me thinking, I wonder if we could sort these objects into those that are halves and those that are not.

[She places her hands over the ‘Halves’ sign. Then over the ‘Not Halves’ sign.

### Speaker

Would you like to help me, mathematicians? Alright. Thank you. Let's get started.

[On top of the pink sheet are the signs ‘Halves’ and ‘Not Halves’. The speaker holds an apple over the middle of the sheet.]

### Speaker

Alright, let's start with this delicious looking apple. Now, this apple looks delicious-

[She holds two sides of the apple in each hand.]

### Speaker

-And it's been cut into one, two parts.

[She holds the two sides of the apple together in her left hand. She points to the signs with her right hand.]

### Speaker

Now, to help us figure out if this apple is showing halves or not, we need to think about what we know about halves. What do you know about halves? Ah, yes, I can hear you.

[She holds the apple side in her left hand over the ‘Halves’ sign. She holds the apples halves in front of each other, slightly apart. She pushes them together.]

### Speaker

And some of you are saying that halves need to be two parts of a whole object. So, do we have two parts of a whole object? One, two. Yeah, two parts of our whole apple.

[She points her right forefinger up.]

### Speaker

Yes, you're right, mathematicians-

[She points to the ‘Halves’ sign. She holds one apple side in each hand.]

### Speaker

But halves aren't just two parts of a whole, they have to be two equal parts of a whole.

[She holds one apple side in each hand – cut-side up.]

### Speaker

So, they need to be the same size.

[She turns the apple sides over and shows their bottom-side. Then she turns them to the cut-side. Then to the top. She holds them together. Then holds them apart.]

### Speaker

Let's have a look at our two pieces of apple. Do we think they're the same size? Yeah, I think you're right, mathematicians, because, look, when I put it back together, this apple lines up perfectly and both sides look exactly the same.

[She places the apple under the ‘Halves’ sign.]

### Speaker

So, we can say that this apple is showing halves.

[She brings another apple to the centre of the sheet.]

### Speaker

[She turns the apple around to each side.]

### Speaker

It looks like a whole apple, doesn't it? But if you look really carefully, can you see that?

[She twists the right side of the apple, and pulls the right side away.]

### Speaker

Yes. It's been cut into two pieces.

[She holds one side of the apple in each hand.]

### Speaker

So, again, we've got one, two parts, but we know that halves need to be equal parts.

[She turns the apple pieces around to each side.]

### Speaker

So, are these pieces the same size?

[She shows the top of the apple sides.]

### Speaker

No, they're not, are they?

[She shakes the apple piece in her right hand. Then she lifts up the piece in her left hand. She turns both of the pieces cut-side up.]

### Speaker

This one is much smaller than this one. So, we have two parts, but they're not the same size. So, this apple can go in the not halves section.

[She places the apple pieces under the ‘Not Halves’ sign.

The plates of toasts are in a row on the bottom of the sheet.]

### Speaker

Let's have a look at our pieces of toast.

[She points to the plate on the left with a toast that has been cut straight down the middle. She pushes the pieces apart.]

### Speaker

Let's start with this one, what do you notice? Yes, you're right, it has been cut into two pieces or parts. Yeah, so we know that halves need two parts, but we also know they need to be equal, don't we? So, how could we prove that from this piece of toast?

[She moves the plate aside and brings a piece of paper in its place. She places the toast on the paper. With a marker she traces around the toast. She lifts the toast to reveal the drawing. She puts the toast down.]

### Speaker

Maybe it would be easier if I put it on this piece of paper, and then if I trace around this piece of toast like this, uh-huh, see?

[She moves the right piece of the toast over the left piece. She traces the edges of the toast with her finger. She points to the corners. She flips the right piece back to its side.]

### Speaker

And then if we flip this piece of toast onto the other side, look at that, it's exactly the same size. Can you see how all of the edges line up perfectly, and the corners? And this is where it goes back to form the whole piece of toast. So, we can say that these are showing halves.

[She places the plate next to the ‘Halves’ sign. She takes the middle plate of toast.]

### Speaker

Alright, let's have a look at this one now. What do you think?

[She pushes the two pieces of toast apart.]

### Speaker

Yes, I can see that too. We've got our two pieces again, I wonder if they are the same size. Shall we use our same method?

[She moves the plate aside and brings a piece of paper in its place. She places the toast on the paper. With a marker she traces around the toast.]

### Speaker

Alright, put that on here. I trace around.

[She moves the right piece of the toast over the left piece. She flips the right piece back to its side. Then over the left piece again.]

### Speaker

OK. And then if I put one side onto the other, does it line up? Yes. And we can see that they are actually the same size. Let's put it back. Look at that, our whole piece of toast is showing halves.

[She places the plate under the ‘Halves’ sign.]

### Speaker

Alright, so we can put this one over here to show that it is also showing halves.

[She places the last plate of toast to the centre of the sheet.]

### Speaker

[She pushes the two pieces of toast apart.]

### Speaker

Yeah, I can see that, too. I can see that we've got two parts, but they don't look very equal to me.

(LAUGHS) Yeah, you too?

[She moves the right piece of the toast over the left piece. She moves it around. She flips the right piece back to its side.]

### Speaker

Yeah. And if we try to put this one on top of this one, oh, look at that. It's nowhere near big enough to cover the whole other side. So, we can say that this piece of toast is not showing halves.

[She places the plate under the ‘Not Halves’ sign.

She places 2 strips of paper on the sheet.]

### Speaker

OK, let's have a look at these two strips of rectangular paper.

[She points to the strip on the left.]

### Speaker

OK. Let's have a look at this one.

[She traces the length of the left-side of the strip. She points to the right side of the strip.]

### Speaker

I can see that there is one part here and one part here. So, that's one, two parts. But are they equal? Are they the same size?

[She places her right thumb and forefinger on the bottom corners of the smaller section of the strip. She pushes the strip up the sheet. She points to the left and longer side of the strip.]

### Speaker

Yeah, I agree, mathematicians, cause I can use my eye to really see that this part is much smaller than this part.

[She points to the smaller section, then the longer section. She circles the longer section with her finger.]

### Speaker

If I imagined moving this part over here, it wouldn't cover the whole part, would it?

[She folds the right section over to the left. She circles the longer section.]

### Speaker

I could do that by folding. It's nowhere near, there's all of this space left to cover.

[She unfolds the strip.]

### Speaker

Yeah, so, I'm thinking that too, we've got a not showing halves.

[She places the strip under the ‘Not Halves’ sign.

She moves the other strip of paper to the centre of the sheet.]

### Speaker

How about this one? Yeah. When I use my eye, you're right, I see the same thing -

[She points to the cut and the right edge of the paper. She circles the left section of the strip with her finger.]

### Speaker

-This part looks very similar to this part. So, maybe we do have halves. Let's try folding to see.

[She folds the strip.]

### Speaker

Ah, yes. Do you see that, mathematicians?

[She holds the paper up. She points to each corner. She traces the edge of the strip with her finger.]

### Speaker

Yes. See all of the corners and the edges line up perfectly. So, that tells me that these are two equal parts.

[She unfolds the strip and places it back on the sheet.]

### Speaker

So, we know where this is going. Over here to our halves.

[She places the strip under the ‘Halves’ sign.

Hexagons appear on bottom of the sheet. She moves them up slightly.]

### Speaker

And next I found these shapes. They're yellow, and we can call them hexagons because they have six sides.

[She moves the hexagon on the right to the edge of the sheet. She points to the hexagon on the left.]

### Speaker

[She points to the different sections of the hexagon.]

### Speaker

There are two parts, are they the same size?

[She points to the bottom part.]

### Speaker

Yeah, I'm imagining folding this shape over

[She points to the bottom right-hand corner of the hexagon. Then she points to its top right-hand corner. She touches the bottom left-hand corner of the hexagon. Then she points to the top left-hand corner.]

### Speaker

And I can see that this corner would touch that corner and this corner would meet that corner. And so, it's showing halves, two parts that are equal.

[She places the hexagon under the ‘Halves’ sign.

She moves the other hexagon to the centre of the sheet.]

### Speaker

[She points to the top section of the hexagon. Then the bottom part.]

This part here looks a lot bigger than this part, doesn't it? This is a teeny-tiny part. And if we visualised flipping it over, it wouldn't meet up. So, this is not showing halves.

[She places the hexagon under the ‘Not Halves’ sign.

2 pig figures appear on the bottom of the sheet.]

### Speaker

And finally, we've got my two little piggy friends.

[The speaker picks them up.]

### Speaker

They're really cute, aren't they?

[She points to the pig on the left and the red rope that runs from its head to feet. She points to the pig on the right and the red rope that runs from its shoulder to its feet.]

### Speaker

And on these two little piggies, do you notice that there's a red rope running from top to bottom, and a red rope running over this one's shoulder down to its feet?

[She points to the pig on the left. She points to its right side, then its left. Then the rope.]

### Speaker

Let's see, this one has one part here and one part here with the red rope dividing it. Do they look the same to you?

[She picks up the pig. She points to its right ear, and then its left ear. She points to its right arm, and then its left arm. She points to the semi-circle on the pig’s belly.]

### Speaker

Yeah, because on each side there is an ear, one ear there, one ear there, and there are some arms, one arm there, one arm there. And we can see that this shape has been split into two equal parts. So, this little pig is showing halves.

[She places the pig under the ‘Halves’ sign.

She picks up the other pig.]

### Speaker

[She points to the pig’s right side. She traces its left side with her finger.]

### Speaker

Yeah. If we look at this part and we compare it to this part, which one is bigger or smaller?

[She points to the pig’s left side.]

### Speaker

Yeah. This one is, isn't it?

[She points to the pig’s right side, and then its left side.

She places the pig under the ‘Not Halves’ sign.]

### Speaker

So, these are not equal in size. And so, this little pig is not showing halves.

Well, thank you, little mathematicians, you have been such a great help.

[The speaker places her hands over the ‘Halves’ section.]

### Speaker

We have now sorted our objects into halves-

[The speaker places her hands over the ‘Not Halves’ section.]

### Speaker

-And those that don't show halves. And this got me thinking, I wonder what other objects you could find or make that are showing halves and are showing not halves.

[Text over a navy-blue background: Over to you mathematicians…

1. Find and sort objects that show halves and those that don’t.

2. How do you know if they are showing halves or not.]

### Speaker

Alright, it's over to you, mathematicians. See how many different ways you can show halves. And how many different ways can you show parts that are not halves? Have fun, and remember to ask an adult to help if you need to use scissors.

[Text over a navy-blue background: What’s (some of) the mathematics?]

### Speaker

Just before you go, mathematicians, what's some of the mathematics we've learnt today?

A title on a white background reads: What's (some of) the mathematics?

A line below reads: We found some halves. We know they are two equal parts of a whole.

Below the line is an image of a plate of toast and an image of a paper strip.

### Speaker

Today we found some halves. We know they are two equal parts of a whole –

[The image of the toast with the right side placed over the left.]

### Speaker

-And we proved this with our piece of toast by placing one part on top of the other, showing that they were the same size.

[Below the image of the strip, an image of the strip being folded appears.]

### Speaker

With our rectangle, we folded one part on top of the other-

[Below the image of the strip, an image of the strip folded appears.]

### Speaker

-And there weren't any gaps or overlaps, and so we knew that these two parts were equal.

[A title on a white background reads: What's (some of) the mathematics?

A line below reads: Sometimes objects can have two parts that are not equal. These are not halves.

Below the line is an image of a plate of toast cut at an odd angle and an image of a paper strip cut with one side longer than the other.]

### Speaker

Sometimes objects can have two parts that are not equal. These are not halves.

[The image of the toast with the right side placed over the left.]

### Speaker

We proved this with our piece of toast by placing one part on top of the other, and we could see that it was much smaller than the one underneath.

[The image of the strip with the right sided folded over the left.]

### Speaker

We folded our rectangle and we could see that the corners and edges didn't line up, so these parts were not the same size.

[A title on a white background reads: What's (some of) the mathematics?

A line below reads: Halves can look different, but as long as they are two equal parts of a whole, they are still halves.

Below the line is a row of images: a toast cut in half, a toast cut diagonally, a paper strip cut in half.

Below this row is another set of images: two halves of an apple, a hexagon cut across the middle and a pig figure with a rope from its head to feet.]

### Speaker

We've also learned that halves can look really different, but as long as they are two equal parts of a whole, they are still halves. We sorted all of these objects as examples of halves because we could prove that the parts were the same size.

[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]

## Instructions

• Watch the video and see how we can sort objects into two groups. The objects can be grouped to show halves and objects not showing halves.

• Find objects around your home that are showing halves and some that are not.

• Sort the objects into two groups. You can label them as halves and not halves.