# Balancing numbers 1

A thinking mathematically targeted teaching opportunity focussed on exploring equivalence and counting principles using a balance arm

These videos are inspired by the work of Dan Meyer, Three-Act Tasks.

## 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-NSM-01

• MAO-WM-01
• MA1-NSM-01

## Collect resources

You will need:

• something to write on
• pencils or markers.

## Watch

Watch Balancing numbers 1 Part 1 video (0:41).

What do you notice about the balance scales?

### Transcript of Balancing numbers 1 Part 1

[No sound. Hulk is placed on one side of the scale and four bears are placed on the other side. The scale does not move.

What do you notice?

What do you wonder?]

[End of transcript]

## Reflection

• What do you notice?
• What do you wonder?

## Watch

Watch Balancing numbers 1 – Part 2 (1:19)

Investigate quantities needed to balance the scales.

### Transcript of Balancing numbers 1 Part 2

[No sound for the beginning of the video. The Hulk is placed on one side of the scale and 11 bears are placed one by one onto the other side. The scale makes a tiny wobble.]

### Speaker

So now I'm wondering how many more bears are needed to make the scale balance?

[Spoken words now appear on screen]

How many bears are equivalent in mass to the Hulk?

So write down in your notebooks, what's an estimate that you think is way too high? Write down an estimate that you think is too low and write down an estimate that you think is pretty reasonable.

Over to you mathematicians!

[End of transcript]

## Instructions

• How many more bears are needed to make the scales balance? (In other words, how many bears are equivalent in mass to the Hulk?)
• What's an estimate that is way too high?
• What's an estimate that is way too low?
• What's an estimate that you think is reasonable?

## Watch the last video

Watch Balancing numbers 1 – Part 3 (4:34)

Investigate quantities needed to balance the scales.

### Transcript of Balancing numbers 1 Part 3

[Screen reads – you will need:

• some pencils
• your workbook or some paper.

Screen shows a decorative picture of pencil and paper.]

[Screen reads – How many more bears are needed to make the scale balance? (How many bears are equivalent in mass to The Hulk?)]

[No sound for the beginning of the video. The Hulk is placed on one side of the scale and bears are placed on the other side one by one until there are enough to balance the scale.]

### Speaker

[Screen shows an aerial view of the balance scale. There are 15 small bear counters on the left-hand side of the scale and The Hulk on the right-hand side of the scale.]

So was your estimate reasonable? Hmm, I see you thinking.

So how many bears are needed to make the scale balance? In other words, how many bears are equivalent in mass to The Hulk? Hmm, let's have a look.

I wonder how I could work out how many without having to count everything?

[There are 15 bears and The Hulk displayed. The speaker arranges the bears into colours – green, blue, red and yellow.]

I wonder if I can see something inside of this collection? Oh, there's three, and three, and four.

[The speaker then arranges the bears into a ten frame by putting the 6 blue bears and 4 red bears together in 2 rows of 5.]

Oh, and that's making me think of three and three is six, and four more, and I can arrange them, uh-huh, into a ten frame, so I can use what I know about structures to work out how many by looking and thinking.

[The speaker puts the remaining 5 green and yellow bears underneath.]

And then I can see, look five, so one ten and five is fifteen.

Look, here's what I imagined in my brain. The rectangle around the outside, one long line down the middle and four internal lines and that makes ten.

[Speaker indicates with pencil by drawing a rectangle around the red and blue bears, then one long line across the middle and 4 internal lines to make a ten frame.]

Yeah, and here's where I can see five on my dice pattern, look. Mm-hmm. Like three on a dice and two on a dice and that's five all together. One ten and five. Fifteen.

[A square is drawn around the other 5 bears and dots are drawn under each bear indicating a dice pattern. Her hand sweeps the bears to the left to show that 10 and 5 makes 15, and she writes the numeral 15 next to her drawings.]

Fifteen bears and one amazing Hulk. Mm-hmm.

[The speaker circles the hulk with her pencil and pushes it above and writes 1 amazing hulk!]

Now I need to think about how I can record my findings as a mathematician. Here's one way. I could draw a picture.

[Screen shows a drawing with 6 blue bears, 4 red bears, 3 green bears and 2 orange bears on the left-hand side and The Hulk on the right. Next to the hulk is written “Grrr! I’m the toughest superhero there is.” A balance scale is drawn below the bears and The Hulk made up of a horizontal line and a triangle directly under the line in the centre. Underneath the scale “15 bears is equivalent to 1 Hulk” is written.]

Hmm and it's quite lifelike because look I can see the bears and I can see The Hulk, I can see his really cool hair, but I could also draw it a bit more like a mathematician, where I just put the most important mathematical information.

[At the top of the screen “15 bears is equivalent to 1 Hulk” is written. Underneath are 6 blue crosses, 4 red crosses, 3 green crosses and 2 orange crosses on the left-hand side and a green Hulk stick figure on the right-hand side. A balance scale is drawn below the crosses and stick figure made up of a horizontal line and a triangle directly under the line in the centre.]

Hmm, so over to you mathematicians. Create a drawing to represent the problem.

[Screen reads – Create a drawing to represent the problem: How many bears are equivalent in mass to The Hulk?]

What’s some of the mathematics here?

[Screen shows a photo of the bears and The Hulk balancing on the scale from earlier in the video.]

When both sides of the balance scale are level we can say they are equivalent. So, 15 bears is equivalent to 1 Hulk. Hmm and when we're thinking about representing mathematical ideas, like this situation with the balance scale, we can represent ideas in different ways. You can, you can create drawings that are lifelike, hmm, like I did here.

[Screen shows a photo of the drawing from earlier in the video with 6 blue bears, 4 red bears, 3 green bears and 2 orange bears on the left-hand side and the hulk on the right. Next to the hulk is written “Grrr! I’m the toughest superhero there is.” A balance scale is drawn below the bears and The Hulk made up of a horizontal line and a triangle directly under the line in the centre. Underneath the scale “15 bears is equivalent to 1 Hulk” is written.]

I can see their bears and I can see the Hulk, or I can do drawings that are more mathematical, where we just put the most useful information in there.

[Photo of the mathematical drawing from earlier in the video. At the top of the screen is written “15 bears is equivalent to 1 Hulk.” Underneath are 6 blue crosses, 4 red crosses, 3 green crosses and 2 orange crosses on the left-hand side and a green Hulk stick figure on the right-hand side. A balance scale is drawn below the crosses and stick figure made up of a horizontal line and a triangle directly under the line in the centre.]

Yes, so you can also create drawings that just include the most important mathematical information to represent situations and ideas.

Until next time mathematicians!

[End of transcript]