# How many unique characters? (exploring the 'for each' idea)

A thinking mathematically targeted teaching opportunity focussed on exploring multiplicative strategies including the cartesian product idea of 'for each'

## 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-MR-01

• MAO-WM-01
• MA3-MR-01

## Collect resources

You will need:

• pencils or markers

## Watch

Watch How many unique characters Part 1 video (2:00).

Explore how many minifigures can be made.

### Transcript of How many unique characters Part 1

(Duration: 2 minutes)

[White text on a navy-blue background reads ‘How many unique characters can you make?’ On the right, a blue half circle at the top and a red half circle at the bottom. In the middle bottom, a line of red dots forms another half circle. In the bottom left corner, a white NSW Government ‘waratah’ logo.]

[On a white desktop, a Lego ‘person’ dismantled into its 5 pieces.]

### Speaker

Hello there, mathematicians. We hope you're having a really lovely day today. This problem was inspired by a visit to the Lego shop. And when you visit the Lego shop, you can make your own minifigs. And we thought this was super cool.

[The speaker places down a yellow sticky note with ‘\$25’ written on it.]

### Speaker

And for \$25, you can make 3 minifigs. And each minifig can have one hat or hair, one face, one body. Oh, that face keeps rolling, one body, one pair of pants, and an accessory. So this could be one of the minifigs that we made if we join them all together.

[The speaker reassembles the Lego figure. The speaker moves aside the reassembled figure and then places 3 different Lego figures down on the desktop.]

### Speaker

And so when we went to the Lego shop, these were the 3 minifigs that we made. And we got three of them for \$25. But we started thinking there's a maths problem in here, because we could say that they're 3 minifigs, but what if we started to change their outfits and things? Actually, how many unique minifigs could we make for \$25 using one pair of pants, one shirt, one head, one piece of hair, and one accessory? Look, because we could swap pants, and now we could say this is now a different minifig because they have a different pair of pants. Yeah, or the robot could swap with the teddy bear and we could say that's a different minifig now because they have a different accessory. So using these minifigs, actually how many unique minifigs can we make for \$25? Over to you mathematicians.

[White text on a blue background reads ‘Over to you!’]

[The NSW Government waratah logo turns briefly in the middle of various circles coloured blue, red, white and black. A copyright symbol and small blue text below it reads ‘State of New South Wales (Department of Education), 2021.’]

[End of transcript]

## Instructions

• How many unique mini figs can we make for our \$25?

## Watch

You will explore a strategy that might help you think through this challenge. Watch How many unique characters Part 2 video (4:13).

Explore representing minifigure combinations.

### Transcript of How many unique characters Part 2

(Duration: 4 minutes and 13 seconds)

[Text over a blue background with a blue circular shape near the top right corner and a red one at the bottom right corner: How many unique characters can you make?

Below the text in smaller font is another line of text: 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.

On the left side of a large sheet of white paper, is a post-it note with text: \$25. Next to the note are 3 columns of Legoman figures with their pieces broken across 5 rows. On the right side is a notebook with a pen on top.]

### Speaker

[With her finger, she circles the first column of pieces.]

### Speaker

…if I joined these pieces together, there would be one unique minifig.

[She points the second column of pieces.]

### Speaker

And if I joined these ones together, that would be a second unique minifig.

[She points the third column of pieces.]

### Speaker

And if I joined these together, I'd have a third unique minifig. Aha. But just by changing, say for example, the pants.

[She swaps the pants in the fourth row of the second and third columns.]

### Speaker

Yes. I've now made two different, unique minifigs.

[She swaps the pants across all columns.]

### Speaker

What we're wondering is, how many different unique minifigs can I make just using this combination of pieces? And so a way to think about this is to think quite logically. So let's start at the top.

[She picks up the blue hat – the figure in the first row of the first column.]

### Speaker

I've got this blue hat, and the blue hat could be paired with the moustache face…

[She places the blue hat against each face – the figures in the second row.]

### Speaker

…the sunnies face, or the cool glasses face.

[She puts the blue hat back in place.]

### Speaker

And the afro…

[She picks up the afro – the figure in the first row of the second column. She places the afro against each face.]

### Speaker

…could be paired with the moustache face, the cool glasses face, the sunglasses face, or the cool glasses face.

[She puts the afro back in place.]

### Speaker

And the bun…

[She picks up the bun – the figure in the first row of the third column. She places the afro against each face.]

### Speaker

…could be paired with the moustache, the sunnies, or the cool glasses.

[She puts the bun back in place.]

### Speaker

[She picks up the blue hat.]

### Speaker

…there are three possible faces.

So let's draw this down.

[She picks up the pen.]

### Speaker

Let's write this down.

[On the notebook, a quarter down the page, she writes: blue hat.]

### Speaker

We might have the blue hat…

[Several lines from the previous text, she writes: afro.]

### Speaker

…the afro…

[Several lines from the previous text, she writes: bun.]

### Speaker

…or the button. And the blue hat…

[She picks up the blue hat.]

### Speaker

…might go with…

[She places the blue hat next to the face with the moustache. From the text ‘blue hat’, she draws a line and a moustache.]

### Speaker

...the moustache.

[She places the blue hat next to the face with the sunglasses. Under the previous line and moustache, she draws a line and sunglasses.]

### Speaker

It might go with the sunglasses.

[She places the blue hat next to the face with the cool glasses. Under the previous line and sunglasses, she draws a line and glasses.]

### Speaker

Or it could go with the cool glasses.

[She puts the blue hat back in place.]

### Speaker

And the same would happen for the afro…

[She picks up the afro, and places it next to all the faces. She puts it back in place.]

### Speaker

…and the bun.

[She picks up the bun, and places it next to all the faces. She puts it back in place.]

### Speaker

Yeah. So that could also have three options.

[From the text ‘afro’, she draws 3 forked lines. At the end of first line, she draws a moustache, at the second: sunglasses and the third: glasses.]

### Speaker

The moustache, the sunglasses face, or the cool glasses face. And the same with the bun.

[From the text ‘bun’, she draws forked 3 lines. At the end of first line, she draws a moustache, at the second: sunglasses and the third: glasses.]

### Speaker

Three options. So for each hat or hair, there are three options. I know, the moustache is now bigger than the sunglasses. That's how you know it's a cool moustache. (LAUGHS) OK, so for each hat, I've got three possibilities. So now I could think about for each of the faces. So now with the moustache…

[She picks up the moustache face.]

### Speaker

…that could be paired with…

[She places the moustache face against each top – the figures in the third row.]

### Speaker

…the blue shirt, the jacket, or the black shirt.

[She puts the moustache face back in place.]

### Speaker

So that means I could have…

[From the moustache drawing off ‘blue hat’, she draws 3 forked lines. At the end of first line, she writes: blue, the second: jacket and the third: black.]

### Speaker

…blue, jacket or black. And then what about the…

[She points to the sunglasses face.]

### Speaker

…sunglasses face with the blue hat?

[She picks up the blue hat and puts it onto the sunglasses face. She places the blue hat with the sunglasses face against each top.]

### Speaker

That could be with either the blue, the bomber, or the black.

[From the sunglasses drawing off ‘blue hat’, she draws 3 forked lines. At the end of first line, she writes: blue, the second: jacket and the third: black.]

### Speaker

Can you see a pattern emerging? I should call it jacket, black.

[She separates the blue hat from the sunglasses face and puts the sunglasses face back in place. She picks up the glasses face.]

### Speaker

[She puts the blue hat on the glasses face.]

### Speaker

And then I think about this... Ooh, it's quite nice in an angle, isn't it? It's very stylish. Yes. There'll be three options, right?

[She points to the tops.]

### Speaker

The blue, the jacket or the black?

[From the glasses drawing off ‘blue hat’, she draws a line. At the end of the line, she writes: blue, under this she writes: jacket and under jacket, she writes: black. She draws connecting lines from ‘jacket’ and ‘black’ to the glasses drawing.]

### Speaker

OK. OK.

[She puts the items back in place. She gets another pen.]

### Speaker

So I think now, I might use a different colour look.

[She draws a box around the drawings of the moustache, sunglasses and glasses off ‘blue hat’.]

### Speaker

So for each blue hat, there were three options.

[Above the box, she writes: 3 options.]

### Speaker

And then for each face, there's another three options.

[She draws a box around the texts ‘blue’, ‘jacket’ and ‘black’ off the moustache drawing. Above the box, she writes: 3 options.]

### Speaker

And look there's three…

[She points to the group of texts ‘blue’, ‘jacket’ and ‘black’ below.]

…here and there's three…

[She points to the group of texts ‘blue’, ‘jacket’ and ‘black’ below.]

### Speaker

…here. So, already with the afro…

[She picks up the afro, and places it next to the moustache face.]

### Speaker

…what do I already know? Ahh, I'm gonna leave that to you mathematicians.

[She puts the afro back in place.]

### Speaker

See if this strategy now will help you think about what are all the different unique minifigs. Over to you.

[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]

## Instructions

• Now, how many unique mini figs can we make for our \$25?