Stage 3 – A thinking mathematically targeted teaching opportunity focussed on using reasoning and problem solving to explore the surface area of objects

Adapted from NRICH Maths


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 
  • MA2-MR-01 
  • MA2-3DS-01
  • MAO-WM-01 
  • MA3-2DS-02 
  • MA3-3DS-01

Collect resources

You will need:

  • a pencil

  • your student workbook

  • 5 cubes (if you'd like).


Watch Brushloads video (3:38).

Arrange the cubes to make as little or as many brushloads.


Hi there mathematicians, we have a challenge for you today.

It's called brush strokes and it comes from NRICH maths.

[Screen shows a piece of paper, 5 yellow cubes and a larger blue cube, which the presenter picks up and rolls around in her hand before placing to the side.]

Now to do this it would be good if you had some cubes or something to help you visualise.

So, you might have had the cubes that you made, and you could use them for multiple purposes including this task. You need 5 of them. Or you could use other cubes.

I'm using these ones today. So, the challenge is when the painters come along to paint whatever it is that you construct, that for each face they need 1 brush load of paint.

[Presenter points to each of the faces of the yellow cubes.]

So, the challenge is to arrange the cube so that they need the least amount of paint.

So, for example, if I arrange my cubes like this to be painted as a structure, but I'll lay it down so you can see it.

[Presenter places cubes in tower formation.]

There would be 5 faces on this side that need painting. And then 5 faces on this side.

5 faces on this side. 5 faces on this side and one for each of the top. Yes, so there will be 4 fives plus 2 ones.

[Presenter traces her finger over the 5 front cubes of the tower, the faces on its side, and then the top and bottom.]

Yes, which is 20 plus 2, which is 22.

[Presenter now writes on paper 4 fives plus 2 ones equals 20 plus 2 equals 22.]

So, it would need 22 loads of paint. And we don't have to worry about the ones on the inside 'cause that's inside our construction.

Yes, now when I'm thinking about what's the most efficient structure that I could build a couple of rules apply.

I can't do any crazy things like this.

[Presenter moves top cube so that the 2 corners of each cube are touching. She then moves top cube down to join face to face with the fourth cube in the tower.]

They have to join, the faces have to join perfectly, and I could think about things like - I'm laying them down for you to see.

But if I make a structure like this and put it down.

[Presenter has 5 yellow cubes, and she lays 3 cubes down and puts 2 cubes on top, and then rolls the structure back to lay flat.]

Now I might think about, so I still have five faces on the front and five faces umm- this is tricky to pick up, underneath that need paint. So that's 2 fives?

[Presenter points to the 5 faces on the front, then picks up cubes and indicates 5 faces on the back.]

And then on the bottom now and then I'm going to go around the edges.

[Presenter now points to the 10 edges.]

So, under here there's 3, 4, 5, 6, 7, 8, 9, 10. Plus 10 ones and two fives is 10. And 10 more is 20.

[Presenter writes on paper 2 fives plus 10 ones equals 10 plus 10 equals 20.]

Oh, so this structure is more efficient than the first one.

And what I should do, mathematicians, very good pickup, is record my thinking.

So, the first one I'll draw it as a sideways tower. Five and this one looks like this.

[Presenter now draws the 2 figures. The top figure contains 5 cubes joined together horizontally. The second figure under the writing contains a row of 2 squares joined horizontally on top, with 3 squares joined horizontally underneath.]

Okay, so over to you mathematicians to find the most efficient or and or the least efficient construction.

Have fun!

[Screen reads: Your challenge!

Can you find ways of arranging 5 cubes so that:

· You need as few BL’s (brushloads) as possible?

· You need as many BL’s (brushloads) as possible?]

Over to you, mathematicians!

Okay mathematicians, what's some of the maths here?

Yes, so you really need to tap into your reasoning and problem-solving skills with this task as you try to work out the smallest number of brush loads or some people are calling them brush strokes and the largest.

You'll also be exploring ideas of surface area with different objects, and this will really help you build your mathematical imaginations as you're trying to work out which services haven't been painted and need to be painted.


[End of transcript]


  • Can you find ways of arranging 5 cubes so that:
  • you need as few brushloads as possible?

  • you need as many brushloads as possible?

  • Record your thinking in your student workbook.
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