# Pentominoes

Stage 1 to 3 – A thinking mathematically targeted teaching opportunity focussed on investigating and creating pentominoes and the shapes they make

## 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
• MA1-2DS-01

• MAO-WM-01
• MA2-2DS-02

• MAO-WM-01
• MA3-2DS-03

You will need:

## Watch

Watch the Pentominoes part 1 video (2:41).

How many shapes can you make using 5 squares?

### Transcript of Pentominoes part 1 video

[A title over a navy-blue background: Pentominoes - part 1. Small font text in the upper left-hand corner reads: NSW Department of Education. In the lower left-hand corner is the white waratah of the NSW Government logo.

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

• Different coloured pens or pencils
• 5 paper squares
• Grid paper

Next to the first point is a blue pencil illustration. Next to the second point are 5 blue squares. Next to the third point is an image of a grid paper.

The speaker waves. On a wooden table are 5 white paper squares. On the right side of the table is a gridbook.

On the bottom right side of the screen, text appears: 5 paper squares.

The speaker moves the gridbook towards the middle of the screen. She pushes it back to the right.

She aligns 4 squares into a straight row. She lines 1 square directly below the first square on the left.

On the bottom right side of the screen, text appears: The edges have to line up perfectly.

She points to each square from bottom left to right.

In the gridbook, she draws a copy of her square arrangement.

On the bottom right side of the screen, text appears: Draw your pentomino.

A different text appears: Is there another pentomino that we could make?

The speaker moves the last square on the right below the third square. She puts her thumb up.

She moves the last square in between the second and third square. She waves her forefinger.

On the bottom left right of the screen, text appears: Remember, the edges have to line up perfectly

She moves the square back below the third square.

In the gridbook, using a pink marker, she draws a copy of her square arrangement.

On the bottom right side of the screen, text appears: Record your pentomino.

She puts her thumb up.

She moves the two squares on the bottom over the top of the squares that they were below.

On the bottom right side of the screen, text appears: Can I make this one as a new pentomino?

She puts her thumb up.

She takes the gridbook, flips it upside down, and points to the red drawing.

She points to the squares.

She flips the gridbook right-side up.

She flips the square arrangement.

On the bottom right side of the screen, text appears: No. It is the same shape just turned around.

She moves the last bottom square below the second top square.

She takes the top third square and places it above the second top square. She points to the square arrangement and the gridbook.

Text over a navy-blue background: Over to you!

Can you find all of the unique shapes you can make using all five squares? Don't forget to record your thinking.]

(NOT SPEAKING)

### Speaker

Over to you mathematicians! Can you find all of the unique shapes you can make using all five squares? Don't forget to record your thinking.

[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

• Find all the unique shapes you can make using all 5 squares.
• Remember, the edges of each square must join other squares perfectly and all of the shapes you make must be different. That means that 2 shapes are considered the same if one can be fitted exactly on top of the other, even if you have to turn it around or turn it over.
• Record the shapes you made.

## Watch

Watch the Pentominoes part 2 video (3:03).

Investigate organising pentominoes to create other shapes.

### Transcript of Pentominoes part 2 video

[A title over a navy-blue background: Pentominoes - part 2. Small font text in the upper left-hand corner reads: NSW Department of Education. In the lower left-hand corner is the white waratah of the NSW Government logo.

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

• a pencil or a pen
• some paper
• the pentominoes you drew

Next to the first point is a blue pencil illustration. Next to this is an illustration of a clipboard.

A gridpage with different coloured pentominoes.]

### Speaker

Alright, how did you go? A-ha, well, here are the ones that I found. I think I found 12 unique pentominoes. And this is a photocopy of all the pentominoes I thought I found. And now I'm gonna cut them out.

[Different coloured pieces of pentominoes are on a table.]

### Speaker

OK, so here they are, all of my little pieces. And let's check that we have 12, or that you have 12 as well. So I learned this strategy about rethinking the shapes as letters of the alphabet. So for example…

[She picks a pentomino that looks like a green ‘L’ and pushes up to the top of the sheet.]

### Speaker

…this one here looks a little bit like an L. And you might see this too…

[She picks a pentomino that looks like an orange ‘X’ and places it next to the L.]

### Speaker

…this one looks a bit like an X. And then this one's a bit like a T…

[She picks a pentomino that looks like a pink ‘T’ and places it next to the X.]

### Speaker

…and oh, a Z.

[She picks a pentomino that looks like a green ‘Z’ and places it next to the T.]

### Speaker

And oh, this one looks actually a bit like a V.

[She picks a pentomino that looks like a blue ‘V’ and places under the Z.]

### Speaker

I might put that one there. Oh look, and this one…

[She picks a pentomino that looks like a blue ‘P’ and places it under L.]

### Speaker

…if I turn it around is a P. And oh yeah, if I turn this one…

[She picks a pentomino that looks like a yellow ‘W’ and places it under X.]

### Speaker

…the yellow one looks like a W. Mm-hm, oh, and there's a U shape.

[She picks a pentomino that looks like a green ‘U’ and places it under T.]

### Speaker

A-ha, and that one looks like…

[She picks a pentomino that looks like a pink ‘I’ and places it under P.]

### Speaker

…an I, maybe a capital I, or as lowercase L. This one looks a little bit interesting…

[She picks a pentomino that looks like a purple ‘F’ and places it under W.]

### Speaker

…it's a bit like an F. Mm-hm. And, oh, if you turn this one around this way a little bit…

[She picks a pentomino that looks like an orange ‘N’.]

### Speaker

…looks like an N.

[She places it under U.]

### Speaker

And this one…

[She picks a pentomino that looks like a gray ‘Y’ and places it under V.]

### Speaker

…if you turn it like that, looks like a Y. Mm-hm. So they're our 12 pentomino pieces, and you might like to pause here and check that you have them as well. But now that we have all of our pentominoes, we can think about rearranging each of these shapes into different shapes. So for example…

[She picks the ‘I’ pentomino and pushes the other pentominos into a pile to the right. She turns I to the side.]

### Speaker

….move these guys out of the way, and then bring this to here.

[She picks the ‘L’ pentomino and places it over the I.]

### Speaker

And what if I use N maybe…

[She picks the ‘N’ pentomino.]

### Speaker

…and I'm actually aiming for a rectangle. I don't know why, I just decided that's what I was going to do.

[She puts the ‘N’ pentomino aside.]

### Speaker

But there's actually lots of different shapes that you can make with pentominoes. So mathematicians, over to you to go investigate.

[Text over a navy-blue background: What other shapes can you make by joining your pentominoes together?

• Can you make a rectangle?
• Can you make more than one?
• What else can you create?]

### Speaker

And so our challenge for you, mathematicians, is what are some other shapes you can make by joining your pentominoes together? Can you make a rectangle? Can you make more than one rectangle? And what else can you create? Over to you to get mathematical.

[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]

## What to do next

• Cut out pentominoes to investigate what rectangles you can make by joining your pentominoes together.
• Choose a challenge:
• Try making a rectangle with all of your pentominoes
• Challenge someone to a game of Penta Play from NRICH maths.
• Try making one of the animals from the images below. You can also find more pentominoes animals from Abaroth's World.