I really wanted to be involved in the MTBOS (Matho Twitto Blogo Sphere) challenges this year – I was a bit lame last year and never got off my bum to do anything about it. I’ve been a bit late on last week’s Exploring the MTBOS challenge but I thought I could catch up today.

I wanted to talk about a problem that my Year 8s have been working on this past week – The Painted Cube. Now, this used to be one option for GCSE coursework and I’ve used it for that in the past. I think it is a pretty good example of a ‘Low Threshold, High Ceiling’ task – the problem is easy to pose, everyone can do something and the students can run with it, extending it to the limit of the their Mathematical ability.

Here’s the problem:

Imagine a large cube made up from 27 small blackcubes.

Imagine dipping the large cube into a pot of yellow paint so the whole outer surface is covered, and then breaking the cube up into its small cubes.

How many of the small cubes will have yellow paint on their faces?

Will they all look the same?

nRich have a nice animation to help with the launch here.

There were a number of skills that I wanted to be used during the few hours that we were looking at this task. I wanted students to collect results systematically, I wanted them to look for and spot patterns, to come up with general rules, draw isometrically, consider the volume of cubes and cuboids and to relate their rules to the physical reality of the problem. I knew I could encourage some to look at graphs to summarise and present their results, to begin thinking about algebraic proof, expansion of quadratic and cubic brackets and, potentially, propose results in 4D (sadly, none of my 12 year olds got that far this time round).

I had students working in pairs or individually – I gave them access to some cubes but encouraged them to move towards visualising using diagrams because we didn’t have enough cubes for a 10x10x10 example. Lots of discussion and lots of excellent thinking ensured. The quality of the questions was superb and the student support for each other was excellent. Some students weren’t clear on drawing isometrically, so a student expert demonstrated with the visualiser while another narrated and answered questions. There was some tussling over the multilink cubes but that calmed down after a while.

We spent two hours of lesson time looking at this problem. We looked at finding nth terms earlier in the term and are about the begin a section on volume, so I thought this was a good place to drop this problem in.

I’ve asked every student to spend another hour on their own for homework to take the problem further, so I’m looking forward to see what they bring back after the weekend.

I love that from a simple problem we can get so much Maths that connects to everything else – when they come back on Monday we’re going to mindmap all of the Maths we used in this one simple problem and marvel at the wonder of Mathematics.