# Creating balance

A thinking mathematically targeted teaching opportunity investigating equivalence using a balance arm.

## 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
- MA1-CSQ-01

## Collect resources

You will need:

- pencils or markers
- paper.

## Watch

These videos were created with Sally from Ellalong PS. Watch Creating balance part 1 video (5:08).

#### Speaker

Hey Mathematicians, let’s create balance!

For this task you will need something to write with and something to write on.

[Screen reads: Let’s balance!]

[Screen shows a number balance with 3 blue tags on one side of the balance. They are hanging on the numbers 5, 7 and 10. The left side of the number balance is covered by a blue piece of paper. Next to the balance is a yellow sheet of paper for notes.]

Hey there Mathematicians, how are you all going today?

I hope you’ve been working hard and been getting a really sweaty brain, and I hope you’re ready to get it even sweatier and do some investigating with me.

Here, I’ve got my number balance.

[Presenter points to the number balance.]

And I’m thinking that, actually, it’s not so balanced at the moment.

I was wondering, what do you notice about this side of the balance?

[Presenter points to the uncovered end of the number balance. She gestures to the 3 blue tags.]

Hm. Yeah, I noticed a few different things.

Firstly, I noticed that there were 3 blue tags.

[Presenter gestures to the 3 blue tags. On the yellow paper the presenter writes: 3 blue tags.]

I also noticed that the tags are hung on 5, 7 and 10.

[Presenter points to each number as she verbally says it. On the yellow paper the presenter writes: tags hung on 5, 7 and 10.]

I also noticed that there was some empty space between the tags.

[Presenter points to the empty spaces between the tags. On the yellow paper the presenter writes: empty spaces between tags.]

I noticed that between 5 and 7, there was one empty space.

[Presenter points to the space between 5 and 7.]

And between 7 and 10, there were 2 empty spaces.

[Presenter points to the 2 empty spaces between 7 and 10.]

I also noticed that 10 is the largest number, and 5 is the smallest.

[On the yellow paper, presenter writes: 10 is the largest and 5 is the smallest.]

And I noticed that this side was down, which makes me think that this side must be up.

[Presenter gestures to the uncovered side of the number balance, which is down, and then to the blue sheet of paper covering the left side of the number balance.]

I’m thinking that the arms aren’t balanced, because I know that for something to be balanced, it would have to have the same amount on each side. And I would expect my balance to be in a nice straight line.

[On the yellow piece of paper, the presenter writes: arms aren’t balanced. The presenter then gestures between the 2 sides of the number balance, demonstrating that an even number balance would have a straight line.]

I am also thinking that the sides aren’t equal, because if the sides were equal, I would, um, expect that they would have the same value on each side.

[On the yellow piece of paper, presenter writes: sides aren’t equal.]

So, mathematicians, let’s investigate our arm balance a little further.

[Presenter removes the blue sheet of paper that was covering the left side of the number balance. There are no blue tags on the left side.]

Hey, there are no tags on this side. Hm.

This makes me wonder.

What do I need to do to this side of the number balance, to make both sides equal?

[Presenter points to empty left side of the number balance.]

I know that I’ll need to add some tags.

I could just put the same numbers on. So, 10, 7 and 5 to make it balance, see?

[Presenter places blue tags on the left side of the number balance, placing them on the numbers 10, 7 and 5. This makes the number balance balance.]

But what if we were only allowed to keep one of the numbers on this side the same?

Oh, that’s tricky.

I’d have to think about which number would be the best to choose.

Would I want to keep the tag on 10?

[Presenter removes the blue tags from the numbers 5 and 7 on the left side of the number balance.]

Maybe on 5?

[Presenter removes the blue tag from the number 10 on the left side of the arm balance and places it on the number 5.]

Or would 7 be better?

[Presenter removes the blue tag from the number 5 on the left side of the arm balance and places it on the number 7.]

Over to you to find out how to make the arms balance.

Remember, you are only allowed to keep one number: 5, 7 or 10.

You might like to pause the video here and try to make the arms balance by only keeping one of the numbers the same.

Draw a picture to record your thinking.

How many different combinations can you find?

## Instructions

- Choose one of the numbers from the right side of the balance (10, 5 or 7).
- How could you use that number and some others to make the arms balance.
- How many different combinations can you find?
- Draw a picture to record your thinking.

## Watch

Watch Creating balance part 2 video (5:41) to see what to do next.

### Speaker

[Screen shows a number balance. On the left side of the balance there are no pegs, on the right side of the balance there are pegs on the numbers 5, 7 and 10. The balance is tipped towards the right side. A yellow piece of paper is above the balance with the names Kookaburra, T-rex and Elephant on it.]

Let’s have a look at how some students made the arms balance.

Now unfortunately, the students weren’t able to join me today.

But don’t worry, I’ve got the next best thing.

My friends Kookaburra, T-rex and Elephant.

[As the presenter introduces the special guests, a Kookaburra, a T-rex and an Elephant toy is shown on screen.]

Let’s start with Mr. Kookaburra’s thinking. He decided that he was going to keep his tag on 10 and use what he knows about 5 and 7 to make the arms balance.

[Presenter places the blue tag on the number 10 on the left side of the number balance, and then points to the numbers 5 and 7 on the right side of the number balance.]

He knows that 8 is one more than 7, so he’s going to hang one on 8.

[Presenter places a blue tag on the number 8 on the left side of the number balance.]

And because 8 is one more than 7, he thinks that he probably will need one less somewhere else.

He’s already placed the 10 and he’s got 8 here so that just leaves the 5. And he knows that one less than 5 is 4. So he thinks that 10, 8 and 4 will make the arms balance. Let’s see.

[Presenter places a blue tag on the number 4 on the left side of the number balance. The number balance shifts and becomes balanced on both sides.]

It does!

The arms are balanced.

Let’s record his thinking up here. So 10 and 8 and 4 will make the arms balance.

[On the yellow piece of paper, the presenter writes: 10 and 8 and 4.]

Let’s see if this one more one less strategy will work for other numbers. So he’s keeping his tag on 10, and he knows that 9 is one more than 8. And because he’s using the one more, one less strategy, he knows that he needs one less than 4 and 3 is one less than 4.

[Presenter places a blue tag on the numbers 9 and 3 on the left side of the number balance. The number balance shifts and becomes balanced on both sides.]

Let’s see if it balances again… and it does!

So 10 and 9 and 3 will also make the arms balance.

[On the yellow piece of paper, the presenter writes: 10 and 9 and 3.]

T-rex has also decided to put his first tag on 10.

[Presenter places the blue tag on the number 10 on the left side of the number balance.]

So he’s keeping the 10 on this side, and he knows that 5 and 7 is the same as or is equivalent to 12 and T-rex knows that double 6 is also the same as 12.

So, he’s going to put 2 tags on 6.

[Presenter places 2 blue tags on the number 6 on the left side of the number balance. The number balance shifts and becomes balanced on both sides.]

Let’s see if that makes the arms balance.

It does!

So 10 and 6 and 6 will also make the arms balance.

[On the yellow piece of paper, the presenter writes: 10 and 6 and 6.]

And because we know 6 and 6 is 12, we could say 10 and 12 would make the arms balance as well.

Now, Elephant wanted to be a bit different to Kookaburra and T-rex and he decided he was going to keep his tag on 5.

[Presenter places the blue tag on the number 5 on the left side of the number balance.]

And he’s going to use what he knows about 10 and 7. And he knows that one 10 and 7 more can be renamed as or is the same as 17, and he also knows that double 8 is 16 which is only one less than 17.

So he’s going to put 2 tags on 8 but because he needs one more than 16, he’s going to put a tag on one.

[Presenter places 2 blue tags on the number 8, and one blue tag on the number 1 on the left side of the number balance. The number balance shifts and becomes balanced on both sides.]

So he’s got 2 tags on 8, he kept a tag on 5 and one more tag on 1, and he has also made the arms balance.

Let’s record Elephant’s thinking as well. So he has 8 and 8 and 5 and 1.

[On the yellow piece of paper, the presenter writes: 8 and 8 and 5 and 1.]

So over to you mathematicians to find ways to make the arms balance using 4 pegs. You could draw another picture.

What’s some of the mathematics we’ve explored today?

Mathematicians think of equivalence as balance. When we first saw the number balance, one arm was up and the other was down, so we knew that the sides weren’t equivalent or balanced.

Numbers can be represented in different and equivalent ways. Kookaburra used what he knew about more and less to make the arms balance. He thought that if he kept the 10 the same, then he could think about what he knows about the numbers 5 and 7 to make the arms balance.

He started thinking about 7 and decided to put the tag on 8 because it was one more than 7. Then he knew he would just need to think of one less than 5 to make it balance with the other side.

Mathematicians use what they know to solve what they don’t know yet.

## Discuss

- Now that you've seen some different ways to make the arms balance, how could you make them balance using 4 tags.
- How could you be sure you have all the possible combinations?
- What other ways could you make the arms balance if you could use any numbers?
- Draw a number balance for someone else to solve.