# Balancing numbers 3

A thinking mathematically targeted teaching opportunity focussed on exploring equivalence using different shaped objects and a balance arm

These videos are inspired by the work of Dan Meyer's Three-Act Tasks and Graham Fletcher's Equally balancing numbers.

## 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

• MAO-WM-01
• MA1-2DS-01
• MA1-NSM-01

## Collect resources

You will need:

• a pencil
• something to write on.

## Watch

Watch Balancing numbers 3 – part 1 video (0:54) to start thinking about equivalence.

Estimate equivalent quantities to balance a scale.

### Speaker

[No sound in the beginning. There is a balance scale shown. The presented places 3 yellow hexagon blocks on the left-hand side of the scale and 2 red trapezium blocks on right-hand side of the scale.]

So how many trapeziums of the red shapes are needed to balance the hexagons, the yellow shapes?

Write an estimate that's way too high, then write an estimate that's way too low, and then write an estimate that's reasonable.

Over to you.

[End of transcript]

## Reflection

How many trapeziums (red shapes) are needed to balance the hexagons (yellow shapes)?

• 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

Watch Balancing numbers 3 – part 2 video (1:39).

Investigate equivalent quantities on a balance scale.

### Speaker

[Screen reads – How many trapeziums (red shapes) are needed to balance the hexagons (yellow shapes)?

A balance scale has 3 yellow hexagon blocks added to the left-hand side and 6 red trapezium blocks are added to the right-hand side of the scale. The scale balances. Screen shows the view of the scales from above.]

Hmm, so three hexagons have the same mass as six trapeziums. What's another way I could prove they have equivalence? What do you notice here?

[Screens shows 3 yellow hexagon blocks joined together on the left and 6 red trapezium blocks in a pile to the right.]

Ah that looks like an interesting idea, for each hexagon I need 2 trapeziums. I think.

[Presenter places one red trapezium on top of half of one yellow hexagon.]

So, I think I need six trapeziums to cover the area of the three hexagons.

What do you think? Can you draw a picture to share your thinking?

Over to you.

[End of transcript]

## Instructions

• How many triangles are needed to cover the area of 3 hexagons?
• Michelle thinks she will need 6 trapeziums to cover the area of the 3 hexagons. What do you think?

• Draw a picture to share your thinking.

## Watch

Watch Balancing numbers 3 – part 3 video (1:32).

Investigate the concept of equivalence using shapes.

### Speaker

[Screen shows 3 yellow hexagon blocks laid flat on a table and 6 red trapezium blocks in a pile. The presenter places 2 red trapezium blocks on top of one yellow hexagon block. Then 2 more trapeziums are placed on top of another yellow hexagon and finally the last 2 trapeziums are placed on top of the last hexagon.]

Yeah, 6 trapeziums are needed to cover the area of 3 hexagons. For each hexagon, I need 2 trapeziums. Fancy a sweaty brain challenge? Of course, you do!

Keep watching.

[Presenter places 3 small green triangle blocks on top of one of the red trapezium blocks, showing 3 green triangles are needed to cover one red trapezium. Presenter then removes the trapeziums from the other hexagons and the half without the triangles on top and places to the side. They then remove the 3 small green triangles and the last trapezium and put them to the side. Presenter picks up 1 small green triangle and attempts to place it on top of the hexagons along different edges.]

So, how many triangles are needed to cover the area of three hexagons?

You might like to make a model, if you have some shapes, and definitely draw a picture to share your thinking.

Over to you mathematicians.

[End of transcript]

## Share or submit

• How many triangles are needed to cover the area of 3 hexagons?

• If you like, build the model and then draw a picture to share your thinking.