# Playing with tessellations

Stage 2 and 3 – A thinking mathematically targeted teaching opportunity investigating tesselating patterns using modified 2D shapes

## 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
• MA2-2DS-02

• MAO-WM-01
• MA3-2DS-03

## Collect resources

You will need:

• a few sheets of paper

• scissors

• sticky tape

• pencils or markers.

## Watch

Watch the Playing with tessellations video (13:34).

Explore how shapes create tessellation.

### Transcript of Playing with tessellations

[A title over a navy-blue background: Playing with tessellations. Below the title is text: ‘…It’s a Metamorphosis’ follow up Russo. 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 white waratah of the NSW Government logo.

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

Bullet points below read:

• A few sheets of paper
• Scissors
• Sticky tape
• Pencils/markers

On the right side is an image of a table with green paper on the left. On the right side is a red scissors, pencil, marker and a sticky tape.

The speaker waves her hands.

A large white sheet of paper is over a wooden table.]

### Speaker

Welcome back, mathematicians. I hope you're having a really great day. Now, you may have been exploring tessellating patterns and shapes recently, and we've discovered that we can actually start with a square and alter its shape and still create a tessellating design, which is really cool. And this got me thinking, I wonder if we could do the same starting with a different shape.

[She places a green triangle in the middle of the sheet.]

### Speaker

So, I started thinking about what I already know about tessellating shapes, and I know that triangles can tessellate…

[She places another green triangle on the left of the other triangle.]

### Speaker

…because I've been playing with them and exploring…

[She places another green triangle on the right of the other triangle.]

### Speaker

…and I can prove that they're tessellating…

[She places an upside-down green triangle between the first and middle triangles.]

### Speaker

…because as I line up each of their sides…

[She places an upside-down green triangle between the last and middle triangles.]

### Speaker

…there are no gaps or spaces. They line up perfectly, like a puzzle piece. So, this made me think…

[She holds onto the triangle on the right, and pushes away the rest to the left.]

### Speaker

…if I know that a triangle can tessellate just like a square can tessellate, what would happen if I…

[She places a green curvy triangular shape on the right of the other triangle.]

### Speaker

…altered the shape of the triangle by cutting a curved section out and attaching it to the other side? Would this new irregular shape still tessellate? What do you think?

What's your conjecture? Now, remember, a conjecture is a mathematical idea that we can prove or disprove. So, mathematicians, what's your conjecture? Will our new irregular shape still tessellate? Uh-huh, you are right, mathematicians. No matter what we're thinking, we need to be able to prove and show our thinking. So, that sounds like the cue for a really excellent mathematical investigation. Are you ready?

[She pushes the shapes to the top of the sheet.]

### Speaker

OK. For our investigation today, you're gonna need some light coloured paper to draw on.

[She places a marker, pencil, scissors, sticky tape and green paper on the sheet.]

### Speaker

You're going to need some pencils or markers, some scissors, sticky tape or masking tape, and some other paper to create our stencil.

[She picks up a green triangle.]

### Speaker

And the very first thing we're going to do is construct our triangle.

[Text over a blue background: Let’s create!

The speaker holds up the green paper.]

### Speaker

So, to make my triangle, I took an A4 piece of paper and I folded it in half.

[She folds the paper in half. She opens it up.]

### Speaker

Great, and I can see that's half because it's been divided into two equal pieces.

[She folds the paper in half, and half again.]

### Speaker

And then I wanted to create quarters. So, I know that if I half my half again, that I will partition…

[She opens the paper up.]

### Speaker

…my whole into four pieces or quarters.

Now, I'm only going to need one of these rectangles, so I'm gonna cut it out.

[She cuts along the paper folds from the bottom. She reaches the middle fold and she turns the paper to side and cuts up. She holds the cut up piece.]

### Speaker

OK, so the triangle I want to create today has…

[She picks up the triangle.]

…all equal sides and angles, and we call it an equilateral triangle.

[She puts down the triangle.]

### Speaker

And to create it, I'm going to fold…

[She folds the rectangle paper in haf.]

### Speaker

…my rectangle in half vertically. Beautiful. Like this.

[She opens it up.]

### Speaker

And then I'm going to take this left vertex…

[She points to the bottom left corner of the rectangle.]

### Speaker

…or corner, and bring it up to the middle fold.

[She aligns the corner to the middle fold.]

### Speaker

But I want to make sure that the right vertex is also folded right down the middle like this. See that? I'm gonna use this to trace one side of my triangle.

[She traces the side with a pencil. She opens the fold.]

### Speaker

Aha, alright. And to create the other side, we're gonna do exactly the same thing but on the other side.

[She aligns the corner to the middle fold.]

### Speaker

So, we're going to take our right vertex and bring it to the middle fold, making sure that the left vertex or corner is also folded down the middle here, like a little pinch. Have to be quite careful and slow doing this. Alright, and I'm gonna trace that line as well.

[She traces the side. She opens the fold.]

### Speaker

And when I open up, aha, I have the outline of my equilateral triangle, which I'm gonna cut out.

[She cuts at the drawn lines.]

### Speaker

Great.

[She holds the cut pieces together and apart.]

### Speaker

I can see that those two shapes also tessellate, don't they?

[The sheet is cleared. The triangle is the centre of the sheet, the scissors on the left. The speaker holds the pencil.]

### Speaker

Now that we've got our equilateral triangle…

[She faintly draws a curvy line along one side of the triangle.]

### Speaker

...we're going to draw one continuous curved line on the inside that's exactly the same length as one of the sides. So, you can get a bit creative here, and I think mine is gonna look like this.

[She draws a curvy line along one side of the triangle.]

### Speaker

There. Beautiful. Alright, and now we're gonna cut out along that line as carefully as we can…

[She cuts at the drawn line.]

### Speaker

…so we can create the base of our hopefully tessellating design. Alright, beautiful. So, that's the part that I've cut out of my triangle.

[She puts the cut-up triangle down. She points to the smaller piece.]

### Speaker

And now we want to attach this part to one of the other sides of our triangle. So, I wonder how I can move this…

[She holds the smaller piece.]

### Speaker

…so that it lines up with one of the other sides? I could translate it, slide it…

[She moves it over to the right side.]

### Speaker

….but it's not quite lining up there, is it? So, what do you think I need to do? Yes, I need to rotate it…

[She rotates the pieces to the left.]

### Speaker

…don't I? I'm gonna rotate it in an anti-clockwise direction so that the two straight sides line up like that. Let's see that again. So, I started here…

[She puts the piece back where she cut it.

…and I translated it or slid it across, and then I rotated it. She moves it to the right. Turns it to left, until the 2 straight sides meet.]

### Speaker

And so, it lines up. Now, I'm gonna get my masking tape and very carefully secure the two pieces together.

[She places tape over the two pieces.]

### Speaker

This bit is a bit fiddly. Need to make sure that the corners match up, uh-huh, as best we can, right. Now, we've created our stencil, it's time for us to explore whether we can create a repeat tessellating pattern on our sheet of paper.

[She aligns the bottom of the piece to the bottom of the sheet.]

### Speaker

So, I'm gonna begin by lining up this side, this straight side, with the bottom of the sheet and carefully tracing around.

[She traces the shape.]

### Speaker

It's quite hard to make sure it doesn't move. Alright.

[She removes the piece to reveal the outline.]

### Speaker

And so here, I have the outline of my stencil, of my shape. Now, I'm wanting to make this shape…

[She holds up the cut piece.]

### Speaker

…tessellate with this shape.

[She points to the outline.]

### Speaker

Mathematicians, can you help me out?

[She moves the cut piece around.]

### Speaker

Can you see how I might be able to move this stencil so that it connects and there aren't any spaces? I mean, I could slide it or translate it up…

[She moves the cut piece over the outline.]

### Speaker

…but then it definitely isn't connecting. Uh-huh, yep, I could reflect it, I could flip it.

[She places the piece over the outline, then, lifts it from the right, and she places it down.]

### Speaker

No, because then I've got that space in there, don't I?

[She points to the empty space between the piece and outline.]

### Speaker

Let's try another movement and I'm going to rotate it.

[She aligns the top vertexes, and she rotates the piece slightly towards the upper left until the piece shape matched the outline.]

### Speaker

Uh-huh, see that, it's kind of like a puzzle, isn't it?

Yes, that's lining up beautifully. Alright.

[She traces the piece.]

### Speaker

I'm gonna trace around this one. Look at that, beautiful. So, so far, I'm thinking that we're proving our conjecture that we can actually create a tessellating design from our irregular shape. Let's keep going.

[She aligns the top vertexes, and she rotates the piece slightly towards the upper left until the piece shape matched the outline.]

### Speaker

So, we rotated it clockwise to tessellate that way. I wonder if we continue the same pattern, if we tessellate it again, if we rotate it, sorry, again. Yes, clockwise. Can you see that? It fits. Alright, I'm gonna trace around this one.

[She traces the piece.]

### Speaker

Huh, I wonder if I continue to rotate it...

[She aligns the top vertexes, and she rotates the piece slightly clockwise until the piece shape matched the outline.]

### Speaker

Yes, look at this.

[She continues to rotate and trace the piece under she has completely drawn all over the paper.]

### Speaker

And there we go. We've proved our conjecture that even if we draw a curved line inside our triangle and remove that piece, we can still create a tessellating design. As I was looking at my design, I noticed all sorts of things. I noticed the shape that the lines were making as they move towards the centre and how they all joined in almost a swirling pattern. Then I also noticed that in my pattern, I seemed to have created some other shapes.

Can you see that too? How about if I do this? I might make it easier to see.

[She takes a ruler and places it along a straight line near the centre of the paper. She marks the line.]

### Speaker

Each of these straight lines…

[She places the ruler along the next straight line, she marks the line and continues until she comes back to the original line.]

### Speaker

…straight sides of my original triangle, are actually connecting and they're forming another shape. Uh-huh. Can you see what I'm seeing? What shape are they making? Yes, a hexagon, isn't it?

[She points to the lines she drew.]

### Speaker

Because it has one, two, three, four, five, six sides. And we know that a six sided 2D shape is called a hexagon because hex means six. Now, this got me thinking, why has my squiggly curved sided triangle created this hexagon design? But then I remembered something.

[She brings over a couple of triangles. She places the bottom of a triangle along a hexagon side.]

### Speaker

That inside of shapes…

[She places other triangles along the hexagon sides, filling the hexagon with triangles.]

### Speaker

…are other shapes hiding. And when I tessellate each of my equilateral triangles, can you see? They create a hexagon.

[She picks up the curvy cut piece.]

### Speaker

And so, even though we've changed the shape of our original triangle, the pieces all connect and tessellate to create the outline of a hexagon. And that got me thinking, mathematicians, I wonder what other shapes we could use to start off with to create another tessellating design. And I wonder whether they will make any different shapes when they are combined?

So, over to you, mathematicians. Have fun exploring, playing, and getting creative with maths.

[A title over a blue background: Over to you, mathematicians…

Below the title is a numbered list:

1. Create your own tessellating design using a triangle.
2. What other shapes can you create a tessellating design with?
3. What shapes can you find ‘hiding’ in your pattern?]

### Speaker

So, over to you, mathematicians. Start by creating your own tessellating design using a triangle, and then investigate what other shapes you can start with to create a different tessellating design. Finally, what shapes can you find hiding in your pattern?

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

### Speaker

So, what's some of the mathematics?

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

A bullet point below reads:

• As mathematicians, we can use what we already know to help us solve something we don't know yet. For example, we used our knowledge of tessellating shapes to make a conjecture about whether another shape will tessellate or not.]

### Speaker

Well, as mathematicians, we can use what we already know to help us solve something we don't know yet. For example, we used our knowledge of tessellating shapes…

[Below the point, an image of tessellated triangles appear.]

### Speaker

…in this case the triangle, to make a conjecture about whether another shape…

[Next to the image, an image of the cut-up curvy triangle next to a triangle appears.]

### Speaker

…will tessellate or not.

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

A bullet point below reads:

• We can also alter a shape's orientation without changing the shape itself.]

### Speaker

We can also alter a shape's orientation without changing the shape itself.

[Below the point, an image of the curvy triangle over its outline appears. Above it is text: Translate (slide).]

### Speaker

We can translate it or slide.

[Next to the image, an image of the curvy triangle on the side of its outline appears. Above it is text: Reflect (flip).]

### Speaker

We can reflect or flip and…

[Next to the image, an image of the curvy triangle on an angle to its outline appears. Above it is text: Rotate (turn).]

### Speaker

…we can also rotate or turn.

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

Bullet points below reads:

• You can combine two dimensional shapes to form other shapes. This helps us to see that inside bigger shapes are smaller shapes, just like inside bigger numbers are smaller numbers.
• There is beauty and patterning within all of mathematics.

On the right side of the points is an image of the paper drawn with tessellations and a hexagon filled by tessellated triangles.]

### Speaker

You can combine two dimensional shapes to form other shapes, and this helps us to see that inside bigger shapes are smaller shapes, just like inside bigger numbers are smaller numbers. And of course, there is beauty and patterning within all of mathematics.

[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 own tessellating design using an equilateral triangle.

• What other shapes can you create a tessellating design with?

• What shapes can you find ‘hiding’ in your pattern?

## Other ways to play

• What happens if you create a tessellating design with different kinds of triangles (scalene or isosceles)?
• Why do you think this happens?

## Discuss/reflect

• How do we know that these shapes tesselate?

• What evidence can you use to prove something tesselates or doesn’t tesselate?

• Describe the way you are moving your stencil to create your tessellating design? (Are you rotating, translating, reflecting or a combination?)

• Are there any shapes you can think of that you think might be difficult to create tessellating designs? Which ones? Why?