# Paper planes

Stage 2 and 3 – a thinking mathematically targeted teaching opportunity investigating testing and recording conjectures around symmetrical and asymmetrical objects.

## 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
- MA2-DATA-01
- MA2-DATA-02

- MAO-WM-01
- MA3-DATA-02

## Collect resources

You will need:

- paper for folding paper planes
- recording materials.

## Paper planes – part 1

Watch the video Paper planes – part 1 (6:34) to learn how to create your paper planes!

The videos were created with Olivia from Summer Hill PS, Michelle from Lake Heights PS, Kim from Mullumbimby PS and Kelly from Keiraville PS.

[A title over a navy-blue background: Paper Planes part 1. Below the title is text: MathXplosion – Go Fly a Kite. 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.

A title on a white background reads: You will need…

Bullet points below read:

- Paper (for example A4 paper or newspaper)
- Recording materials (paper and a pencil, pen or marker)

Below the points is a row of 3 images: on the left is an illustration of a piece of paper, in the middle is an illustration of a newspaper and on the right is an image of writing instruments – pencil, pen, markers.]

### Speaker

For today's activity, you will need a device to watch Go Fly a Kite, some paper for folding like A4 paper or newspaper, and also some recording materials like paper and a pencil.

[Text over a blue background: Let’s play!]

### Speaker

Let's play!

[The speaker is holding a green paper plane.]

### Speaker

Hi mathematicians. I've been thinking about how symmetry helps things fly and I was wondering if this would apply to a paper aeroplane. So, I thought a really fun task today would be to make a symmetrical paper plane and an asymmetrical paper plane and then see what happens when we fly them.

[On a table are 2 pieces of white paper.]

### Speaker

Hi, mathematicians. For today's activity, you will need some paper for folding. I have just taken some A4 paper and cut it in half.

[The speaker sets aside the paper on the right, pushing the one on the left to the middle.]

### Speaker

I'm going to begin by making my symmetrical paper plane.

[She folds the piece of paper into half; aligning the 2 long edges.]

### Speaker

I'm going to follow this rectangular piece of paper in half to form my line of symmetry.

[She unfolds the paper. She lays it out with the fold going down the middle. She points to the left side of the paper.]

### Speaker

Every time I do something to this side…

[She points to the right side of the paper.]

### Speaker

…I must repeat the same on the other so that it looks like a mirror image. And look…

[She turns the paper long-side up.]

…if I place a mirror along the line of symmetry...

[She places a mirror on the fold.]

### Speaker

I should see that when I lift it up, the other side is identical. Let's check. Is it symmetrical?

[She lifts the mirror off the fold. She places it back.]

### Speaker

It is. It's a mirror image.

[She takes the mirror away.]

### Speaker

OK, let's keep going.

[She turns the paper, so the fold is down the middle.]

### Speaker

I'm next going to fold the nose of my plane.

[She folds the left corner of the paper, aligning it to the middle fold.]

### Speaker

I'm going to take this corner and form a triangle.

[She presses the bend down. She folds the right corner of the paper, aligning it to the middle fold.]

### Speaker

I'm going to repeat the same process on the other side because I want it to be a mirror image.

[She turns the paper long-side up, the pointy end pointing to the left.]

### Speaker

Let's get my mirror again to check.

[She places a mirror on the fold.]

### Speaker

Placing it along my line of symmetry. Is it the same?

[She takes the mirror away.]

### Speaker

Yes, it is.

[She places a mirror on the fold.]

### Speaker

I can keep checking…

[She takes the mirror away.]

### Speaker

…and I can see that it is identical.

[She folds the paper in half.]

### Speaker

I'm next going to make my wings.

[She folds the diagonal side of the wing aligning it with the bottom edge.]

### Speaker

Now, you can make a paper plane in a different way. This is just the way that I've been taught.

[She folds the straight part of the paper aligning it with the bottom edge.]

### Speaker

I'm folding my wings down twice and repeating the same on the other side.

[She turns the paper over. She folds the diagonal side of the wing aligning it with the bottom edge. She folds the straight part of the paper aligning it with the bottom edge.]

### Speaker

Folding down once and twice.

[She holds the paper up.]

### Speaker

I have now made a symmetrical paper plane. And my next task is to make a plane that is not symmetrical, an asymmetrical plane.

[A piece of paper is on the table long side up.]

### Speaker

I'm going to make one side of my plane exactly like I made my first plane so I have a fair experiment.

[She folds the piece of paper into half; aligning the 2 long edges.]

### Speaker

So, now when I fold this rectangular piece of paper in half to form my line of symmetry…

[She lays it out with the fold going down the middle.]

### Speaker

…I want each side to be different.

[She folds the left corner of the paper, aligning it to the middle fold.]

### Speaker

As I fold down the nose on this side, I can make a triangle. And over here, I can also fold down…

[She folds the right corner of the paper, leaving space between the middle fold and the ‘triangle’s’ side.]

### Speaker

…the corner of the paper to make another triangle but they are not identical. This means they are not symmetrical. I can further check this…

[She turns the paper long-side up, the folded end pointing to the left. She places a mirror on the fold.]

### Speaker

….by getting my mirror placing it along my line of symmetry. Let's see if the other side is symmetrical.

[She takes the mirror away.]

### Speaker

Is it a mirror image?

[She places a mirror on the fold.]

### Speaker

No, it's not. It is not identical nor is it a mirror image. Therefore, it is not symmetrical. OK, let's keep going.

[She folds the paper in the middle, leaving the bigger corner fold on top.]

### Speaker

This is the side that I'm keeping the same as my symmetrical plane, my first plane.

[She folds the diagonal side of the wing aligning it with the bottom edge.]

### Speaker

So, I need to create the wings the same by folding the paper down twice.

[She folds the straight part of the wing aligning it with the bottom edge. She turns the paper over.]

### Speaker

On this side, I do need to form a wing, but I can make it however I like, so I might fold it this way.

[She folds the straight part of the wing aligning it with the bottom edge.]

### Speaker

Kind of like a rectangular wing.

[She folds the straight part of the wing again aligning it with the bottom edge.]

### Speaker

OK, well, so now I have two paper planes.

[She holds the symmetrical plane in her left hand and the asymmetrical plane is on the table on the right-hand side. She raises the plane in her left hand.]

### Speaker

This one is symmetrical, each side is a mirror image of each other. And this one…

[She picks up the asymmetrical plane.]

### Speaker

… well, this is asymmetrical. It is not identical to each side. OK, over to you mathematicians to make two paper planes, one that is symmetrical and one that is asymmetrical.

Then come on back and we'll talk about what's next.

[Text over a blue background: Over to you!]

### Speaker

Over to you, mathematicians.

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

### Speaker

So, what's some of the mathematics we explored today?

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

Bullet points below read:

· We can describe objects as symmetrical or asymmetrical.

· Symmetrical means that one side is a mirror image of the other.

· Asymmetrical means that one side is not a mirror image of the other side.

Below the second point on the right side, is an image of the plane with a mirror on the fold. Below the third point on the right side, is an image of the asymmetrical plane with 2 different wing folds.]

### Speaker

We can describe objects as symmetrical or asymmetrical. Symmetrical means that one side is a mirror image of the other. Asymmetrical means that one side is not a mirror image of the other side.

[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

- Create your symmetrical paper plane.
- To do this fold your piece of paper in half to create a line of symmetry.

- To keep our planes symmetrical, each side of the centre fold line needs to be a mirror image. That means that any fold that we make on the right-hand side of the centre fold line needs to be mirrored on the left-hand side.
- Create your asymmetrical paper plane.
- To do this fold you piece of paper in half to create a line of symmetry. Fold one half of your paper in the exact same way as you did to create your symmetrical plane.

- Fold the other half of your paper in a different way (so the two sides don’t match).

## Paper planes – part 2

When you're ready, watch the next video Paper Planes – part 2 (5:05).

[A title over a navy-blue background: Paper Planes part 2. Below the title is text: MathXplosion – Go Fly a Kite. 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.

A title on a white background reads: Where the planes landed. Below is an image of 2 paper planes in a carpeted hallway. On the left side is a text box that reads: Asymmetrical plane. On the right is a text box that reads: Symmetrical plane. At the bottom left corner is a text box that reads: Starting spot. Next to this text box is a green ribbon.]

### Speaker

Welcome back mathematicians. I hope you had fun making your symmetrical and asymmetrical paper planes. Now I'm just going to fly my planes. And here is a picture of where they landed. As you can see, my symmetrical plane flew further than my asymmetrical plane. But it got me wondering. Just flying the planes once really confirmed my conjecture that symmetrical planes fly further than asymmetrical planes. So I thought a fair test would be to fly the planes ten times, and then I'll be able to either confirm or reject the idea that symmetry is important to help things fly.

[On a table is a large sheet of paper. A title is on the paper: Our conjecture is that the symmetrical plane will fly further. Below this is a smaller blank paper.]

[DESCRIPTION:

I've written my conjecture at the top of the page. Our conjecture is that the symmetrical plane will fly further. I now need to draw a table to record my observations. I'm going to call this table…

[She writes on the top of the smaller paper: Plane that flew further. She underlines it.]

### Speaker

… 'plane that flew further'. Next, I need to draw a large rectangle.

[She draws a rectangle that fills the space below the title.]

### Speaker

And then I'm going to make a skinny rectangle by drawing a vertical line on the left-hand side to record the times I fly the plane.

[In the rectangle, allowing a small space from the left, she draws a line from its top to its bottom.]

### Speaker

Then I'm going to think about this section…

[Within the rectangle, with her finger, she traces the larger rectangle on the right.]

### Speaker

…that is left and get my eye to imagine where about half is.

[She marks a dot in the middle of the top side of the right rectangle. Then she draws a line down the middle to the bottom.]

### Speaker

Drawing a line across the top…

[Within the full rectangle, allowing a small space from the top-left corner down, she draws a line from left to right.]

### Speaker

…so that I can record a column for the symmetrical…

[In the second column of the rectangle, she writes: Symmetrical.]

### Speaker

…plane and the asymmetrical.

[In the third column of the rectangle, she writes: Asymmetrical.]

### Speaker

In the rest of this table…

[Within the full rectangle, she traces the area below the headings.]

### Speaker

…I need to make ten equal parts, which we also call tens. I can get my eye and again and think about where half is…

[She traces the left side of the rectangle.]

### Speaker

…along here because half of ten is five.

[She marks a dot halfway down the left side of the rectangle.]

### Speaker

And if I made ten tens or ten equal parts, I know that in the first half I need to have five fifths and in the bottom half I need to have five fifths. And one thing I know about this is that it's a little less than a quarter. So if I get my eye again and imagine where I think about a quarter is.

[She points to a spot a few centimetres down the left side of the rectangle.]

### Speaker

And then thinking that the notion of one fifth is a little less. I can make my first row.

[She draws a line across from left to right. In the first cell, she writes: 1.]

### Speaker

Now we know that in this part here…

[With her right forefinger and thumb, she traces the area below the row and halfway down.]

### Speaker

…I need to have four equal parts, which I can think about like quarters. And what I know about quarters is that it is half of half. So if I halve…

[She marks a dot halfway down the first row and the halfway down the rectangle.]

### Speaker

…the next part here, and then I halve it again.

[She marks two more halfway points between the other first row and halfway down the rectangle.]

### Speaker

I should have four equal parts.

[She draws a line across from each dot on the left, to right.]

### Speaker

And when I combine them with my first part or my first fifth, I should have five-fifths.

[In the first cell of each subsequent row, she writes 2-5.]

### Speaker

I'm now going to repeat that process in my second half. I'm going to get my eye and imagine where about one quarter is.

[She points to a spot a few centimetres below the last row.]

### Speaker

And then one-fiffth is a little less. So I can make my first row.

[She draws a line across from left to right.

### Speaker

Now in this part…

[With her left forefinger and thumb, she points to the line she drew and the bottom of the rectangle.]

### Speaker

…I need to make four equal sections, which I can think about like quarters. So like I did before imagining…

[She points to halfway down the left side between the last row and bottom side.]

### Speaker

…where around half this..

[She marks this with a dot.]

### Speaker

…and then halving each half to make quarters…

[She marks two more halfway points between the last row and bottom side.]

### Speaker

…of four equal parts.

[At each dot, she draws a line across from left to right. In the first cell of each new row, she writes 6-10.]

### Speaker

I now have made a table with ten rows to record the times I fly the plane. Alright, mathematicians, over to you now to go and draw your own table and then take your planes and fly them ten times, recording in the table. Which plane flew the furthest each time. Then come back and we will look at your results.

[Text over a blue background: Over to you!

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]

## Paper planes – part 3

When you're ready, watch the last video Paper planes – part 3 (1:23).

[A title over a navy-blue background: Paper Planes part 3. Below the title is text: MathXplosion – Go Fly a Kite. 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.

From Paper planes – part 2, the table has been filled in with ticks.]

### Speaker

Hi, mathematicians! I've just flown my planes ten times, and as you can see, my symmetrical plane flew the furthest eight times, and my asymmetrical plane only flew the furthest twice. This tells me that today, I can confirm my conjecture is true. That the symmetrical plane did fly further. And I'm wondering, what did your data show?

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

### Speaker

So what's some of the mathematics we explored today?

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

Bullet points below read:

· Mathematicians raise conjectures.

· They then gather evidence to either confirm or reject this.

· We can use a table to collect and organise our observations.

On the right side of each point is an image. The top image is of the handwritten conjecture. The middle image is of the two paper planes, the testing area and where they have landed, and the bottom image is of the table.]

### Speaker

Mathematicians raise conjectures. We did that today when we wrote our conjecture that a symmetrical plane will fly further. They then gather evidence to either confirm or reject this. We carried out this today through our fair experiment by throwing an asymmetrical and symmetrical plane ten times. Finally, we can use a table to collect and organise our observations. Well, mathematicians. I hope you enjoyed making paper planes today. And remember, flying paper planes isn't magic, it's maths.

[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

- What do you notice in the data you collected? Did one plane fly further more often than the other plane? Did you notice anything interesting? Record you thinking in your workbook.
- Does the conjecture that symmetry is important to help things fly hold true?