# Part-whole triangles Stage 2 and 3

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

Adapted from ‘Part-whole triangles’ by James Russo

## 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-AR-01
• MA2-AR-02
• MA2-MR-01

• MAO-WM-01
• MA3-AR-01
• MA3-MR-01
• MA3-MR-02

## Collect resources

You will need:

• playing cards zero – 13

• someone to play against

• something to write on

• pencils and markers.

## Watch

Watch Part-whole triangles Stages 2 and 3 part 1 video (2:59).

Investigate the structure of part-whole triangles.

### Transcript of Part-whole triangles Stages 2 and 3 part 1 video

[A title over a navy-blue background: Part-whole triangles. Below the title is text: Stage 2 and Stage 3 (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…

• playing cards 0–13
• 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 0-8, the right hand is from 9–13.]

### 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.

[On a table is 10 cards arranged in a triangular shape, with 1 card at the top, 2 in the second row, 3 in the third row and 4 at the bottom.]

### Speaker

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 the card set.]

### Speaker

…I have a triangle made using cards from a game when I was playing. What do you notice? And what do you wonder? If you have someone with you 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!

Back to the card set.]

### Speaker

When I look at this triangle, there are a few things that I notice. First of all, I know notice that it's made up using ten cards. I also notice that it is arranged into…

[With his finger, he crosses each card row.]

### Speaker

…four very distinct layers. In this example, 13…

[He points to the top card.]

### Speaker

…is at the top of my triangle. And then I have two cards…

[He circles the second row.]

### Speaker

…in the next layer or parts. In this example, I have…

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

### Speaker

…five and eight.

[He points to the card next to 5, which is 8.]

### Speaker

And when I take five and eight more, it gives me a total of 13 or the whole. And we sometimes also say that five and eight more has a sum of 13.

When I move into my…

[He crosses the third card row.]

### Speaker

…third layer of the triangle, I can see that five has been partitioned into…

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

### Speaker

…three and two more. And eight has been partitioned into…

[He points to 2 and the card next to it, which is 6.]

### Speaker

…two and six more. And you may notice that this two…

[He circles 2.]

### Speaker

…is being shared by both the five and eight. When I move down to my…

[He crosses the bottom card row.]

### Speaker

…fourth layer of the triangle, I can see that three…

[He points to 3.]

### Speaker

…has been partitioned into two and one more…

[He points to the first and second cards in the bottom row, which are 2 and 1.]

### Speaker

…two…

[He points to 2.]

### Speaker

…has been partitioned into one and one more…

[He points to the second and third cards in the bottom row, which are 1 and 1.]

### Speaker

…and six…

[He points to 6.]

### Speaker

…has been partitioned into one and five more.

[He points to the third and last cards in the bottom row, which are 1 and 5.]

### Speaker

You may also notice that this one…

[He points to first 1 in the bottom row.]

### Speaker

…is being shared by three and two.

[He points to the cards directly above.]

### Speaker

And this one…

[He points to second 1 in the bottom row.]

### Speaker

…is being shared by two and six.

[He points to the cards directly above.]

### Speaker

When my cards are arranged in this way…

[He traces the shape of the card set.]

### Speaker

…it's what we call a part-whole triangle, where the parts underneath…

[He circles the second row cards.]

### Speaker

…add together to give you the whole.

[He points to the top card.]

### Speaker

Here, I can also see the parts…

[He circles the first and second cards in the third row.]

### Speaker

…or three and two add together to give me the whole…

[He points to the card directly above.]

### Speaker

…of five.

[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 given.

• Players 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 person to make 6 part-whole triangles or the person with the largest number of triangles once all the cards have been used.

## Watch

Watch Part-whole triangles Stages 2 and 3 part 2 video (3:41).

How to play the part-whole triangle card game.

### Transcript of Part-whole triangles Stages 2 and 3 part 2

[A title over a navy-blue background: Part-whole triangles. Below the title is text: Stage 2 and 3 (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.

#### Speaker

Welcome back mathematicians!

[A large blank sheet of paper over a table.]

#### Speaker

To help us learn how to play the game part-whole triangles, I am joined by 1 of my favourite mathematicians Caitlin. Hi Caitlin.

Hi.

#### Speaker

Caitlin, have you played the game part-whole triangles before?

No, I haven't.

#### Speaker

That's OK. I can explain as we go. Firstly each player needs 12 cards. So can you give us 12 cards each please?

#### Caitlin

Yeah sure.

[In the centre of the table, Caitlin deals out 2 sets of 12 cards. She places the remaining cards between the sets. The speaker and Caitlin take their sets and lay them out in rows at the bottom of the sheet.]

#### Speaker

Before we start the game, we get the opportunity to make any part-whole triangles we can using the cards that we have been dealt.

[Caitlin takes card 13 and places it at the top right side of the sheet. The speaker takes 7 and places it in the middle-left side of the side, and places 5 below. He puts them back in the rows. Caitlin creates a second layer of 7 and 6; third layer of 4, 3, and 3; fourth layer of 2, 2, 1, leaving a space for another card. The speaker places 4 in the middle, then 0 and 4 below it. He puts them back in the rows.]

#### Speaker

And so Caitlin I can see that you were so close to being able to make a part-whole triangle that has four layers. You're actually just missing…

[He points to the empty space next to 1 in Caitlin’s part-whole triangle.]

#### Speaker

…1 more number. So now is probably where I tell you how to play. What we are going to do is we're going to take turns trying to either complete or make our part-whole triangles. Now, each time you can take a card from…

[The speaker outlines his row of cards with his finger.]

#### Speaker

…my unused pile, and this is your unused part here…

[He outlines Caitlin’s row of cards with his finger.]

#### Speaker

…or you can take a card from the middle. So Caitlin, over to you.

#### Caitlin

Well, I know that 3 is 1 and 2 more, so I know I need the number 2. So I'm going to draw a card…

[Caitlin takes a card from the pile in the middle of the sheet. It’s a 2.]

#### Speaker

…and I got the 1 I need.

[She places the 2 next to the 1 in her set.]

#### Speaker

And you know what? I think I've been actually quite lucky today because I also had a 2.

[The speakers points to the 2 in his row.]

#### Speaker

You just didn't notice that I had the 2 over here. And what I'm going to try is I'm going to try and start my part-whole triangle with a 9.

[He pushes 9 up towards the top of the sheet.]

#### Speaker

And this is something that I did before but I just didn't have the right numbers to just keep going. So what I now notice is that 9 can be partitioned in lots and lots of different things. It can be partitioned into 5 and 4, it can be partitioned into 7 and 2, and both of these combinations I've tried already. A combination that I didn't try was 3 and 6 more. And I noticed that you have a 6.

[He pushes the 3 below 9.]

#### Speaker

So Caitlin, can I have your 6?

#### Caitlin

Sure.

[Caitlin hands him the 6. He places it below the 9.]

#### Speaker

And now I can see that I have nine has been partitioned into 3 and six more, and now I can keep going and building on that.

[He builds a third layer of 2, 1, and 5 and a fourth layer of 1, 1, 0, and 5.]

#### Speaker

And with a little bit of luck I have also now managed to make a part-whole triangle for nine. We will need to make sure that we both have 3 cards in our unused pile. If you haven't got 3 cards, draw 1 from the pile in the middle.

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

• Each player gets dealt 12 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 opp1nts 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 12 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 a 12 part-whole triangle. In the image, on the left is a pile of unused cards. Below the part-whole triangle is a row of 3 cards.]

#### Speaker

It's now time for you to play part-whole triangles. The winner is the first person to make 2 part-whole triangles that have four layers, or the person with the largest number of triangles once all the cards have been used.

[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.]

## Watch

Watch Part-whole triangles Stages 2 and 3 part 3 video (4:02).

Investigate and record relationships within part-whole triangles.

### Transcript of Part-whole triangles Stages 2 and 3 part 3

[A title over a navy-blue background: Part-whole triangles. Below the title is text: Stage 2 and 3 (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, on the left-side is a 4-layer part-whole triangle. On the top left-side is a row of 2 cards. Below the cards is a blank piece of paper.]

#### Speaker

After playing a few more games of part-whole triangle, it was actually Michelle that helped me discover that there's more than 1 possible way to make…

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

#### Speaker

…a part-whole triangle for the number 13.

[He points to the top card which is 13.]

#### Speaker

Now before I make another part-whole triangle for 13, I'm going to record my current triangle on this piece of paper, and I'm going to do it like this. Going to start with my 13 at the top.

[On the paper, he writes: 13.]

#### Speaker

I can then see that 13 has been partitioned into 5 and 8.

[He points to the cards beneath 13 which are 5 and 8. On the paper, below 13, he writes 5 and 8. He draws 2 lines from 13 to the 5 and 8.]

#### Speaker

If I have a look at my 5…

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

#### Speaker

…I can see it has been partitioned into 3 and 2.

[Below 5, he writes 3 and 2 below each. He draws 2 lines to connect them to 5].

#### Speaker

My 8 has been partitioned into 2 and 6. My 2 is already here.

[He draws a line from 8 to 2. He draws a line from 8 then he writes 6. He points to 2 and the cards beneath which are 2 and 1.]

#### Speaker

My 3 has been partitioned into 2 and 1 more.

[He draws 2 lines from 3 then writes 2 and 1 below each.]

#### Speaker

My 2 has been partitioned into 2 1s. I already have 1, 1.

[He draws a line from 2 to 1. He draws a line from 2 then he writes 1.]

#### Speaker

And my 6 has been partitioned into 1 and 5 more.

[He draws a line from 6 to 1. He draws a line from 6 then he writes 5.]

#### Speaker

Before I show you another possible way to make a part-whole triangle for 13, I wonder can you rearrange these…

[He traces the part-whole triangle.]

#### Speaker

…cards to make a different part-whole triangle? You may also like to use some of the cards that I have up here.

[He points to the row of cards.

Text over a blue background: Over to you!

Back to the part-whole triangle.]

#### Speaker

I want to share with you now another possible way that I discovered to make a part-whole triangle for the number 13.

[He removes the third and 4th layers of the part-whole triangle. He builds a third layer of 2, 3, and 5 and a 4th layer of 1, 1, 2, and 3.

On the paper, under the black text, in red he records the new part-whole triangle breakdown.]

#### Speaker

When I look at these 2…

[On the paper, he points to the breakdown at the top then the 1 below.]

#### Speaker

…part-whole triangles that I've made side by side, I can notice that both triangles have 13 as the top, and both triangles partition 13 into 5 and 8.

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

• Without moving the 13, 5 or 8 how many ways can you complete this part-whole triangle?
• What is the tallest part-whole triangle (largest number of layers) you can create using the numbers 0 to 13?

Next to the points is an image of a part-whole triangle. Only the cards at the top (13) and second layer (5 and 8) are facing up.]

#### Speaker

And so now, mathematicians, let's investigate a little further. I wonder without moving the 13, 5, or 8, how many ways can you complete this part-whole triangle? I'm also wondering, what is the tallest possible part-whole triangle with the largest number of layers you can create using the numbers 0 to 13?

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

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

• This game helps us to see that smaller numbers can be found hiding inside of larger numbers. For example…
• 6, 3, 2 and 1 are all smaller than 9 and depending on how we break it apart, we can find all the numbers less than 9 hiding inside of it.

Between these points is a row of 3 part-whole triangle images with top card 9, a second layer of 3 and 6, third layer of 2, 1 and 5, and 4th layer of 1, 1, 0 and 5. On the left image, the second layer of 3 and 6 are outlined. Under the image is text: inside of 9, we can find 6 and 3. In the middle image: the 6 in the second layer, and the 2 and 1 of the third layer are outlined. Under the image is text: inside of 9, we can find 6 and 2 and 1. On the right image, the 3 of the second layer, the 5 of the third layer, and the first 1 of the 4th layer are outlined. Under the image is text: inside of 9, we can find 3 and 5 and 1.]

#### Speaker

What's some of the mathematics? This game helps us to see that numbers can be found hiding inside of larger numbers. For example, inside of 9, we can find 6 and 3. But we can also find 6, and 2, and 1, or 3, and 5, and 1. 6, 3, 2, 1, and 5 are all smaller than 9, and depending on how we break it apart, we can find all the numbers less than 9 hiding inside of it.

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

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

Below the point is an images of a part-whole triangle, with text on the right reflecting the breakdown.]

#### Speaker

We also know that mathematicians record their thinking so that they can keep track of their ideas and help them think strategically when 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, 5 or 8 how many ways can you complete this part-whole triangle?
• What is the tallest part-whole triangle, the largest number of layers, you can create using the numbers 0 to 13?