# Part-whole triangles Stage 1

This is a thinking mathematically context for practise focused on building understanding of part-part-whole relationships.

Adapted from ‘Part-whole triangles’ by James Russo.

## 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.

## Outcome

- MAO-WM-01
- MA1-CSQ-01

## Collect resources

You will need:

- playing cards zero–13
- someone to play against
- pencils or markers
- something to write on.

## Part-whole triangles – Stage 1 part 1

Watch Part-whole triangles Stage 1 part 1 video (2:59).

[A title over a navy-blue background: Part-whole triangles. Below the title is text: Stage 1 (Part 1). Below this is in slightly smaller font text is: Adapted from James 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:

- playing cards zero - 13
- something to write with
- something to write on
- someone to play against.

On the right side of the points is an image of 2 hands of playing cards: the left hand is from zero - 8, the right hand is from 9 - 13.

On a table are 2 sets of 6 cards arranged in a triangular shape, with 1 card at the top, 2 below, and 3 at the bottom.]

### Speaker

OK, mathematicians! To play this game, you'll need playing cards from zero to 13, something to write with and something to write on, and someone to play against. Hi, mathematicians! Today, I wanna play a game with you that was shared by mathematician James Russo. And our game today is called Part-Whole Triangles. Here…

[The speaker traces the shape of each card set.]

### Speaker

…I have 2 triangles made using cards from a game I was just playing. What do you notice? If you're with someone today, you might like to share your thinking. Pause the video here and enjoy some thinking time.

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

### Speaker

Firstly, when I look at these 2 triangles, I notice that they're both made up using 6 cards. And that there are 3 layers of numbers in each triangle. The first layer…

[He points to the top cards.]

### Speaker

…the second layer…

[He points to the middle cards.]

### Speaker

…and the third layer.

[He points to the bottom cards.]

### Speaker

When I look at this example…

[He circles the left set.]

### Speaker

…I can see that at the top, I have 13.

[He points to the top card.]

### Speaker

And below…

[DESCRIPTION: He points to the middle cards.]

### Speaker

…I have two cards or the parts, that when added together is 13. In this example, I can see that five…

[He points to the left card in the middle row, which is 5.]

### Speaker

…and 8 more…

[He points to the right card in the middle row, which is 8.]

### Speaker

…give me a total of 13.

[He points to the top card.]

### Speaker

We sometimes also say that five and eight more has a sum of 13.

[He points to the bottom cards.]

### Speaker

Below my second layer, I can see that the…

[He points to the middle cards.]

### Speaker

…5 and the 8 have been re-partitioned into smaller chunks.

[He points to the bottom cards.]

### Speaker

Here, I can see the five has been partitioned into…

[He points to the first and middle cards in the bottom row, which are 1 and 4.]

### Speaker

…one and 4, and the 8 has been partitioned into…

[He points to the middle and last cards in the bottom row, which are 4 and 4.]

### Speaker

…2 fours. And you may also notice that this 4…

[He points to the 4 middle card.]

### Speaker

…is being used by both the five and the eight. Looking at the second triangle…

[He circles the card set on the right.]

### Speaker

…I can see that the 13…

[He points to the top card.]

### Speaker

…is partitioned into 5 and 8 more…

[He points to the middle cards.]

### Speaker

…that the 5…

[He points to the left card in the middle row, which is 5.]

### Speaker

…has then re-partitioned into…

[He points to the first and middle cards in the bottom row, which are 3 and 2.]

### Speaker

…3 and 2. And my 8…

[He points to the right card in the middle row, which is 8.]

### Speaker

…has been re-partitioned into…

[He points to the middle and last cards in the bottom row, which are 2 and 6.]

### Speaker

…2 and 6 more. When I arrange the cards in this way…

[The speaker traces the shape of each card set.]

### Speaker

…they're called part-whole triangles. And they're called part-whole triangles because the two cards below each number…

[He points to the middle cards of the right card set.]

### Speaker

…or the parts is the sum or total of the card above.

[He points to the top card.]

### Speaker

The whole. The aim of this game is to create as many different part-whole triangles as you can.

[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

- Each player gets 7 cards.
- Before starting the game, players attempt to make part-whole triangles from the cards they have been dealt.
- Players then take turns in trying to make part-whole triangles.
- Players may take a card from their opponent's unused pile or from the pile in the middle.

- Players then take turns in trying to make part-whole triangles.
- Players need 3 cards in their unused pile at all times.
- The winner is the person to make 6 part-whole triangles or the person with the largest number of triangles once all the cards have been used.

### Discuss

Can you create a different part-whole triangle for 13 using the numbers provided?

## Part-whole triangles – Stage 1 part 2

Watch Part-whole triangles – Stage 1 part 2 video (3:04).

[A title over a navy-blue background: Part-whole triangles. Below the title is text: Stage 1 (Part 2). Below this is in slightly smaller font text is: Adapted from James 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 clear table.]

### Speaker

Welcome back mathematicians. To help us learn how to play part-whole triangles today, I am joined by one of my favourite mathematicians, Caitlin. Hi, Catlin.

### Caitlin

Hi.

### Speaker

Caitlin, have you played part-whole triangles before?

### Caitlin

No, I haven't.

### Speaker

That's OK. I can explain as we go. Firstly, each player needs nine cards. Would you like to deal?

### Caitlin

Yeah, sure.

[In the centre of the table, Caitlin deals out 2 sets of 7 cards. She places the remaining cards between the sets.]

### Speaker

Now before we start playing, we have the opportunity to make any part-whole triangles that we can using the cards that we already have.

[The speaker points to the card sets. They pick up a set each and they lay them out at the bottom of the sheet over 2 rows. Caitlin places a 13 at the top of the table. Under this, she places a 5 and 8. Under 5, she places a 1 and 4. Under 8, she places another 4.]

### Speaker

OK Caitlin, I can see that you are actually able to make a part-whole triangle…

[The speaker traces the shape of Caitlin’s part-whole triangle set.]

### Speaker

…for the number 13. Unfortunately, however, though I couldn't. Now to play the game, we take turns in trying to make as many different part-whole triangles as we can. Now each time, I can ask you for one of the cards that you're not using…

[He points to Caitlin’s card set.]

### Speaker

…or I can take a card from the pile.

[He points to the remaining cards in the middle.]

### Speaker

When I have a look at my cards, I was also going to try and make a part-whole triangle for…

[He pushes up the second card from his top row, which is 13.]

### Speaker

…the number 13, and when I have a look at the cards you have…

[He circles Caitlin’s cards.]

### Speaker

…none of them will help me at this moment. So…

[He places 13 back in line.]

### Speaker

…what I'm going to do is I'm going to take a card from the pile.

[He takes a card from the pile, turns it over – reveals it’s a 6 - and places it on the right of the top row cards.]

### Speaker

And luckily, it's exactly the card that I needed.

[He pushes the ‘13’ card to the top of the table. He moves the first card in the bottom row, an ‘8’ below 13. He takes the second last card from the bottom row, which is a 5, and places it below 13. He takes the 6 and places it below 8. He takes the card that was next to 6, which was a 2 and places it next to 6. He takes a card from the bottom row, which is a 3 and places it next to 2.]

### Speaker

Now here I can see I've been able to now make a part-whole triangle…

[The speaker traces the shape of his part-whole triangle set.]

### Speaker

…for the number 13. My 13…

[He points to the top card.]

### Speaker

…has been partitioned into 5 and 8 more.

[He points to the middle cards. He points to the 5.]

### Speaker

My 5 has been repartitioned into…

[He points to the 3 and 2 cards.]

### Speaker

…3 and 2, and my 8…

[He points to 8.]

### Speaker

…has been repartitioned into 2 and 6.

[He points to the 2 and 6.]

### Speaker

And you'll notice here…

[He circles 2.]

### Speaker

…that the 2 is shared between the…

[He points to 5 and 8.]

### Speaker

…5 and the 8.

[A title on a white background reads: Let’s play!

Bullet points below read:

- Each player gets dealt 9 cards.
- Before starting the game, players attempt to make part-whole triangles from the cards they have been dealt.
- Payers then take turns in trying to make part-whole triangles.
- Players may take a card from their opponents unused pile or from the pile in the middle.
- Players need 3 cards in their unused pile at all times.
- The winner is the first person to make a 6 part-whole triangles or the person with the largest number of triangles once all the cards have been used.

Next to the points is an image of the 2 part-whole triangles. On the left of the triangles is a player’s card set. Above the triangles is the unused pile of cards.]

### Speaker

And now, young mathematicians, it's time for you to play part-whole triangles. The winner is the first person to make six part-whole triangles, or the person with the largest number of triangles once all the cards have been used.

[A title over a navy-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]

## Part-whole triangles – Stage 1 part 3

Watch Part-whole triangles – Stage 1 part 3 video (3:41).

[A title over a navy-blue background: Part-whole triangles. Below the title is text: Stage 1 (Part 3). Below this is in slightly smaller font text is: Adapted from James 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.

Text over a navy-blue background: A little while later…

On a table, over the left-hand side is a part-whole triangle with 6 cards. On the top right-hand side is a row of 5 cards. Under the row of cards is a blank paper.]

### Speaker

After playing a few more games of a part-whole triangle, it was actually Michelle that helped me realise that there may be more than one possible way to make a…

[The speaker traces the shape of the part-whole triangle, with his finger.]

### Speaker

…part-whole triangle using the numbers that I have…

[He traces the row of cards.]

### Speaker

…here. Before I show you another possibility, we know that mathematicians like to record their thinking to keep track of where they're up to. And to do that today, I'm going to be drawing a diagram. I'm going to do it like this.

[He points to the top card which is 13. He gets a marker and writes 13.]

### Speaker

At the top, I'm going to record my 13.

[He points to the top card and the cards beneath which are 5 and 8.]

### Speaker

When I move down a layer, I can see that I've partitioned my 13 into 5 and 8.

[He draws a line from 13 then writes 5 at the end of it. He draws another line and then writes 8 below.]

And as I move down another layer, I can see I've partitioned my…

[He points to the 5 and the cards beneath it which are 1 and 4.]

### Speaker

…5 into 1 and 4…

[He draws 2 lines from 5 then writes 1 and 4 below each.]

### Speaker

…and my 8…

[He points to the 8 and 4, and the card next to 4 which is a 4.]

### Speaker

…has been partitioned into two 4s.

[He points to the written 4 on the paper.]

### Speaker

Now I already have 1 four here…

[He draws a line from 8 then writes 4 below.]

### Speaker

…I just need to add my other four. Before I share another possibility with you, I wonder, can you rearrange our cards to make a different part-whole triangle for 13? You may like to use some of these cards as well.

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

Back to the table.]

### Speaker

To make another part-whole triangle today, I'm going to leave my first 2 layers and concentrate on this bottom layer.

[He removes the bottom layer, and he takes the 3 middle cards from the row which was 2, 3 and 5. He places these below the 5 and 8.

On the paper, on the right side of his previous text, he writes 13.]

### Speaker

13 is partitioned into…

[He draws 2 lines from 13 then writes 5 and 8 below each.]

### Speaker

…5 and 8. My 5…

[He points to the 5 and the cards beneath it which are 2 and 3.]

### Speaker

…is now partitioned into two and three.

[He draws 2 lines from 5 then writes 2 and 5 below each.]

### Speaker

And my 8…

[He points to the 8 and the cards beneath it which are 3 and 5.]

### Speaker

…is partitioned into three and five. And again, my middle three has been shared.

[He draws a line from 8 to 2. He then draws another line and writes 5 below.]

### Speaker

And now I can see that I have 2 different…

[He points to the 2 texts.]

### Speaker

…part-whole triangles, both for the number 13.

[A title on a white background reads: Let’s investigate

A bullet point below reads:

· Without moving the 13 and the eight, how many ways are there to complete this part-whole triangle?

On the right side is an image of a part-whole triangle with 6 cards. The top card, 13 and one of the cards below, 8 faces up, while the rest of the cards face down.]

### Speaker

And so, mathematicians, let's investigate a little further. Without moving the 13 and the eight, how many ways are there to complete this part-whole triangle?

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

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

Bullet points below read:

- This game helps us to see that smaller numbers can be found hiding inside of larger numbers. For example…
- 5, 4, 8 and 1 are all smaller than 13 and depending on how we break it apart, we can find all the numbers less than 13 hiding inside of it.

Between these points is a row of 3 part-whole triangle images with top card 13, and the second layer of 5 and 8 and a third layer of 3, 2, and 6. On the left image, 5 and 8 are outlined. Under the image is text: inside of 13, we can find 5 and 8. In the middle image: the 5, 2 and 6 are outlined. Under the image is text: inside of 13, we can find 5 and 2 and 6. On the right image, 8, 3 and 2 are outlined. Under the image is text: inside of 13, we can find 3 and 2 and 8.]

### Speaker

And so, what are some of the mathematics? This game helps us to see that smaller numbers can be found hiding inside of larger numbers. For example, inside of 13 we can see five and eight, or we can also see inside of 13 five and two and six. We can also see that inside of 13 we can find three and two and eight. Now, five, four, eight and one are all smaller than 13 and depending on how we break it apart, we can find all the numbers less than 13 hiding inside of it.

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

A bullet point below reads:

· Mathematicians record their thinking so that they can keep track of their ideas and to help them think strategically

Below the point are 2 images of part-whole triangles, with text on the right reflecting the breakdown. The left image is a part-whole triangle with top card 13, and the second layer of 5 and 8 and a third layer of 1, 4, and 4. The right image is a part-whole triangle with top card 13, and the second layer of 5 and 8 and a third layer of 3, 2, and 6.]

### Speaker

And we know that mathematicians record their thinking so that they can keep track of their ideas and to help them think strategically about solving problems.

[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]

### Discuss

Without moving the 13, how many part-whole triangles can you make? How will you know you have found them all?