# Balancing numbers (part-part-whole relationships)

A thinking mathematically targeted teaching opportunity exploring equivalence through part-part-whole relationships.

## 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.

## Outcomes

- MAO-WM-01
- MAE-RWN-01
- MAE-RWN-02
- MAE-CSQ-01
- MAE-CSQ-02

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

## Collect resources

You will need:

- something to write on
- pencils or markers.

## Balancing numbers – part 1

Watch Balancing numbers – part 1 video (4:41).

Text over a navy-blue background: Balancing Numbers. Talking about an equal-arm balance. 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 red waratah of the NSW Government logo.

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

Bullet points below read:

- Something to draw and write with
- Something to draw or write on.

An equal-arm balance sits on a table. It features a yellow beam, with numbers 1 to 10 written on each arm. On each arm, the number 1 is closest to the centre of the beam, while number 10 is closest to the end. Below each number is a yellow peg. On the right arm, a blue, plastic tag hangs from the peg below the number 7. The beam tilts to the right.

The speaker stands to the left of the frame and points to the equal-arm balance.

### Speaker

Hello, there, little mathematicians, we hope you are having a really good day today. We've got this great thing we love called an equal arm balance.

The speaker points to each of the numbers on both arms of the beam, each time starting from the centre then moving out to the end.

### Speaker

And look, if I start from the middle, I can see the numbers 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 along this side and I can see the numbers 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 along this side or this arm of the balance. I can also see that there's this blue peg hanging off of here, and I can see that my arms of my balance this side here, I think it's heavier than this side. So I was wondering what we might be able to do to make both of the arms balance out.

The speaker shows another blue tag to the camera. She puts it on the peg below the number 1 on the left side of the arm. The beam does not move.

### Speaker

So I've got some of the pegs here. Here's one of them. Let's see what happens if I put it on one. Oh, nope, that didn't make it balance out. Let's try two.

She moves the blue tag onto the peg below the number 2. The beam does not move.

### Speaker

So no, that didn't work yet. Let's try three.

She moves the tag to the peg below the number 3. The arm does not move.

Oh, still no success, but that's okay, I'm a mathematician, I'm going to keep trying and I'm going to keep going in order, I think. What about on four?

She moves the tag to the peg below the number 4. The beam does not move.

### Speaker

Nope, still not balanced out. What about on five?

She moves the tag to the peg below the number 5. The beam does not move.

### Speaker

Still not balanced. OK, what about six?

She moves the tag to the peg below the number 6. The beam moves slightly but returns to its original position.

Oh, looks like it's getting a bit closer. Let's see what happens if I put it on seven.

She places the tag on the peg below the number 7. The beam see-saws, tilting left and right, then eventually levels.

Oh, here we go. It looks like it's like a seesaw and oh, look at that now it's balanced out, and I can see on this side, I've got a peg on seven and on this side I've got a peg on seven and that made my arms go in a pretty straight line. What happens if I move it on to eight? Let's see what.

She removes the tag from the peg below the number 7 on the left arm. The beam tilts to the right. She puts the tag on the peg below the number 8 on the left arm. The beam tilts to the left. It bounces up to a level position, then eventually settles tilting heavily to the left.

### Speaker

Oh, now eight it's down the bottom. That must be because eight is bigger than seven. It's one bigger than seven. Yeah, because it's the next number in the counting words. What do you think will happen if I put it on nine? Oh, you think it'll bounce up like this because seven will be heavier and then it'll come back down? Let's see.

She removes the tag from the peg below the number 8. The beam tilts to the right.

### Speaker

Well, that bounce down and then.

She places the tag on the peg below the number 9. The beam crashes down to the left, bounces up slightly, then settles down on the left side.

Oh, yeah. When I put the peg on nine, it made the balance go down on this side now because nine is bigger than seven. What do you think will happen if I move my peg onto 10? Oh, you could get a like bounce off and then bounce down?

Let's try.

She removes the peg. The beam crashes to the right. She places the peg on the number 10. The beam crashes down to the left, bounces slightly, then settles.

### Speaker

Yes and yes.

She removes the tag on the left arm. The beam tilts to the right.

Wow, so what I'm wondering now, little mathematician's is I want to think of a way that I could get my arm to balance out, but I need to use two pegs.

The speaker holds two tags together. She places them both on the number 7 on the left side of the beam. The beam tilts to the left.

So if I put them both on seven two sevens, it's much heavier or bigger than one seven. I wonder what I could try. What if I put one peg on one and one peg on two?

She places one tag on the peg beneath the number 1 and one on the peg beneath the number 2. The beam does not move.

Oh, that didn't work. I know what if I keep one peg on two and one peg on three?

She takes the tag from the number 1 and places it on the peg below the number 3. The beam does not move.

### Speaker

That didn't work. What if I keep doing this like leapfrog and do three and four?

She takes the tag from the number 2 and places on the fourth peg. The beam levels out.

Oh, look at that little mathematicians, I can get my equal arm to balance with seven on one side and three and four on the other. And now I'm starting to wonder little mathematicians what are all the ways that I could get my equal arm to balance seven on one side using two pegs on this side? We know four and three. So over to you little mathematicians to find some other ways to balance seven on one side with two pegs on the other. And don't forget to draw your findings.

An image appears in the lower left-hand corner. It shows a hand drawing of a level equal-arm balance, with tags on the 3rd and 4th peg on the left side, and a tag on the 7th peg on the right side. Text below reads: 4 and 3 balance 7.

### Speaker

Here's how I drew this thinking.

Text over a blue background: Over to you!

OK, over to you.

The NSW Government logo flashes on screen. Text below reads: Copyright, State of New South Wales (Department of Education), 2021.

### Instruction

Write or draw the different ways we could arrange the pegs to make the equal-arm balance, balance!

## Balancing numbers – part 2

Watch Balancing numbers – part 2 (2:46)

### Speaker

Alright, how did you go there, little mathematicians?

[Screen shows an equal arm balance. It has 2 sides. On the left side of the equal arm balance is the number 10 which decreases all the way down to the number one. On the right side of the equal arm balance is the number one which increases all the way up to the number 10. There are pegs on the numbers 4 and 3 on the left side and a peg on the number 7 on the right side. The equal arm balance in balanced.]

We knew that 4 and 3 on one side would balance out 7 on the other.

Ah, yes. And I’m hearing that some of you came up with some different ways like…

[Screen shows the pegs on the left side being moved onto the numbers 1 and 6.]

One and 6. Oh, it’s always a bit fun waiting to see.

Yes! So one and 6 can balance 7. And…aha, some of you saw this one too, 5 and 2!

[Screen shows the pegs on the left side being moved onto the numbers 5 and 2.]

Woah, that’s pretty cool, little mathematicians.

So now this got us wondering. Someone said, well, what happens if we have 3 pegs?

[Speaker takes out a third peg.]

Oh my goodness. Well, what if we put this peg on one.

[Speaker places the peg on the left side of the equal arm balance, on the number one. The equal arm balance leans towards the left side.]

Now, 5 and 2 and 1- it’s heavier than 7 so we need to move something here.

What if I move my 2 and put it on 3?

[Speaker moves the peg on the left-side of the equal arm balance from the number 2 to the number 3. The equal arm balance leans towards the left side.]

Nope. That didn’t make it balance out and now this side is too heavy.

What if I move my 5 and put it on 2?

[Speaker moves the peg on the left-side of the equal arm balance from the number 5 to the number 2. The equal arm balance is almost balanced, but not quite.]

Ooh that looks like it’s getting closer, but not yet close enough. So what I might do little mathematicians, is move my 3 here just on 4. Just one more number to see what happens.

[Speaker moves the peg on the left-side of the equal arm balance from the number 3 to the number 4. The equal arm balance becomes balanced.]

Oh! And look! That balanced out!

Then I thought of something else we could’ve done. What if I still had this one here, and I moved 1 onto 2 so I have 2 pegs on 2.

[Speaker moves the peg on the left-side of the equal arm balance from the number 4 to the number 3. She then moves the peg from the number one to the number 2 so that there are 2 pegs on the number 2. The equal arm balance becomes balanced.]

Oh my goodness, and that worked too!

Alright, little mathematicians, here is your challenge. Imagine we’re starting from the beginning and we have a peg on the number 9. What are some different ways we could make our equal arm balance using 2 pegs and then you can challenge your little brains… or your big brains and think about what are some different ways we could balance 9 using 3 pegs?

[Speaker moves the peg on the right hand side to the number 9. She then picks up 2 pegs and then 3 pegs.]

Okay! Over to you!

[Screen reads: Draw or write your ideas.

What are some ways you can use 2 pegs to make this equal arm balance?

Screen shows an equal arm balance with a peg on 9.

What are some ways you can use 3 pegs to make this equal arm balance?

Screen shows an equal arm balance with a peg on 9.]

[End of transcript.]

### Instruction

Write or draw the different ways we could arrange the pegs to make the equal-arm balance, balance!