# Part-whole triangles Early Stage 1

This is a thinking mathematically context for practise focussed 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 © 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
• MAE-RWN-01
• MAE-RWN-02
• MAE-CSQ-01
• MAE-CSQ-02

## Collect resources

You will need:

• playing cards zero – 13

• someone to play against

• something to write on

• pencils or markers.

## Watch

Watch Part-whole triangles video part 1 (1:46).

Investigate the structure and rules of part-whole triangles.

### Transcript of Part-whole triangles video part 1

[A title over a navy-blue background: Part-whole triangles. Below the title is text: Early 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…

• 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 thirteen, something to write with and something to write on and someone to play against.

[In the middle of a table, are 2 sets of cards that each ha 1 card on top and 2 cards below.]

### 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.

[The speaker points to the left card set: first the card on top, then the bottom left and then bottom right. He repeats this with the right card set.]

### Speaker

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

[A title over a blue background: Over to you!]

### Speaker

When I look at this example…

[Back to the cards.

He points to the left card set.]

### Speaker

…I can see that at the top I have 6…

[He points to the top card]

### Speaker

…and below I have 2 cards…

[He points to the cards below.]

### Speaker

…or the parts, that when added together is 6.

[He points to 6.]

### Speaker

In this example, I can see that three and three more…

[He points to the ‘3’ cards.]

### Speaker

…give me a total of 6…

[He points to 6.]

### Speaker

…which we also sometimes say that 3 and 3 more gives the sum of 6. Looking at the second triangle…

[He points to the right card set.]

### Speaker

…I can see that nine…

[He points to the top card.]

### Speaker

…is partitioned into 6 and 3 more.

[He points to the cards below.]

### Speaker

When I arrange the cards in this way…

[He points to both card sets.]

### Speaker

…they're called a part-whole triangle, because the two numbers below…

[He points to the cards below.]

### Speaker

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

[He points to the top cards.]

### Speaker

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.]

## Watch

Watch Part-whole triangles video part 2 (3:13).

How to play the part-whole triangle card game.

### Transcript of Part-whole triangles video part 2

[A title over a navy-blue background: Part-whole triangles. Below the title is text: Early 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 white sheet of paper over a table.]

### Speaker

Welcome back, mathematicians. To help us learn how to play the game Part-Whole Triangles, I am joined by one of my favourite mathematicians, Kate. Hi, Katelyn.

Hi.

### Speaker

Katelyn, have you played Part-Whole Triangles before?

No, I haven't.

### Speaker

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

### Caitlin

Yeah, sure.

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

### Speaker

Now, before we start to play, we have the chance 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. They each place a card at the top of the sheet: 6 for Katelyn on the left, and 11 for the speaker on the right. Katelyn then places a 4 and 2 below her 6. The speaker places a 7 and 4 below his 11.]

### Speaker

So, now, we're ready to start. To play, we take turns trying to make more part-whole triangles.

[He points to the cards at the top.]

### Speaker

Each time, I can either take one of the cards that you're not using…

[He circles Katelyn’s cards]

### Speaker

…or I could turn over one of the cards from the middle pile.

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

### Speaker

Now, Katelyn, because you haven't played before, would you like to go first?

### Caitlin

I think you can go first.

### Speaker

OK, because this is your first time, I'm gonna share some of my thinking with you. Now, when I have a look at my cards…

[He circles his cards.]

### Speaker

[He circles Katelyn’s cards.]

### Speaker

…there are 2 possible things that I'm thinking of. The first one is I could do something like 8…

[He pushes the ‘8’ card, the card on the right on his set, and the another ‘8’ the card on the left.]

### Speaker

…is 8 and no more…

[He points to the ‘zero' card on Katelyn’s set, which is on the top right.]

### Speaker

…or I could do something…

[He places his cards back in line. He pushes out 4 and one, the 2 cards in the middle. He points to 5 on Katelyn’s set, which is on the top left.]

### Speaker

…like 5 is 4 and one more. In this case…

[He places his cards back in line.]

### Speaker

…I think I'm going to go with eight is eight and no more. So, can I please have your zero.

[Katelyn pushes her zero towards the speaker’s cards. The speaker pushes his 8 up, then pushes his other 8 below. He takes Katelyn’s zero and places it below the top 8. He moves the card set below the part-whole triangle at the top of the sheet.]

### Speaker

Using your zero I was able to make 8 is 8 and no more. Katelyn, it's your turn.

### Caitlin

[DESCRIPTION: The speaker pushes his 4 towards Katelyn’s cards.]

### Speaker

Oh, good one.

[Katelyn pushes her 5 card up, and then her 1 below it. She moves the 4 below the 5.]

### Caitlin

Now, I have 5 is one and 4 more.

### Speaker

We will now need to both draw cards from the pile to make sure we have three in front of us.

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

• Each player gets dealt 7 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 3 part-whole triangles in the middle of the sheet. Below the triangles is the player’s cards. Above the triangle 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 6 part-whole triangles 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]

## Instructions

• Each player receives 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 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 video part 3 (2:27)

Explore combinations using part-whole triangles.

### Transcript of Part-whole triangles video part 3

[A title over a navy-blue background: Part-whole triangles. Below the title is text: Early 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 blue background: Let’s investigate!

One a table, there are 2 sets of 3 cards, side by side. In each set, there is a one card at the top and 2 cards below.]

### Speaker

Partway through playing our game, both Caitlin and I notice that I have made a part-whole triangle for 12.

[The speaker circles the set on the left.]

### Speaker

And Caitlin has also made a part-whole triangle for 12.

[The speaker circles the set on the right.]

### Speaker

Have we noticed that they're made very differently? And this made us wonder.

[The speaker places a post-it note in between the card sets that reads: Without moving the 12, how many different part-whole triangles can you make? How will you know when you've found them all?]

### Speaker

Without moving the 12, how many different part-whole triangles can you make? And how will you know when you've found them all? Now, because we know mathematicians like to record their thinking to work systematically, today…

[He places a piece of paper on the bottom left side of the table.]

### Speaker

…I'm going to record my thinking like this. I'm going to write 12 is.

[On the piece of paper, he writes 12 is…]

### Speaker

And my triangle…

[Her circles the set on the left.]

### Speaker

…12 is 12 and zero.

[Under his handwriting, he writes a point: 12 and zero.]

### Speaker

On Caitlin's triangle…

[Her circles the set on the right.]

### Speaker

…it is one and 11.

[Under the previous point, he writes a point: one and 11.]

### Speaker

Over to you now mathematicians, how many different part-whole triangles can you make for 12?

[Text over a blue background: Over to you!

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

· This game helps to see that smaller numbers can be found hiding inside of larger numbers. For example…

Under the point are 2 images. On the left is 2 grids on top of each other, that each has 2 rows and 5 columns. The top grid has a red circle in each cell, the bottom grid has a red circle in the top left cell and a blue circle next to it.

On the right side are also 2 grids with 2 rows and 5 columns. The top grid’s top row has a red circle in each cell; the bottom row has 2 red circles then 3 blue circles. The bottom grid has a blue circle in the first 2 cells.

Below the left grids is text: inside of 12 we can find 11 and one. Below the right grids is text: inside of 12 we can find 7 and 5.

Below this is another point:

· 11, one, 7 and 5 are all smaller than 12 and depending on how we break it apart, we can find all the numbers less than 12 hiding inside of it.]

### Speaker

And so, what's some of the mathematics? This game helps to see that smaller numbers can be found hiding inside of larger numbers. For example, inside of 12, we can find 11 and one. But inside of 12, we can also find seven and five. 11, one, 7 and 5 are all smaller than 12. And depending on how we break it apart, we can find all the numbers less than 12 hiding inside of it.

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

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

Below the point is handwritten text and an image.
The text has a title: 12 is…
Below the title are points:

• 11 and one
• 10 and 2
• 9 and 3

The image is of 3 part-whole triangle sets for 12.]

### Speaker

We know that mathematicians like to record their thinking so they can keep a track of their ideas and to 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 12, how many part-whole triangles can you make?