A shape problem to get your teeth into…

I’ve spent part of my day working on this problem, so thought I should blog about it.

A colleague had set an investigation for his Year 9 class to find the biggest tetrahedron that could be fit into a cube of side length 3cm.  In itself, this is a nice problem which pupils attempted initially via trial and improvement before realising that they could incorporate some Pythagoras’ calculations to help.  Pupils made models and hence practiced their skills of construction. Ace.  Fantastic.  Then another colleague who teaches the parallel Year 9 top set to me (NC levels 7a to 8a) began to think about how we could extend this task for our groups. That’s where it got more interesting…

At first, she was thinking about pupils thinking about a platonic solid that would fit within the tetrahedron.  We started to mull it over and discuss it:  The volume of the tetrahedron is a third of the volume of the cube.  Let’s get pupils to work out the fraction of the cube taken up by the tetrahedron and prove it algebraically.  Ace.  Fantastic.  Again.  This would hit lots of topics that pupils have covered this year; Pythagoras (both 2D and 3D), algebraic manipulation, surds…so we both grabbed pen and paper and set to work….

What we found out:  This problem is nowhere near as simple as it sounds it should be.  The proof led to all kinds of mathematical discussion and proofs within the main proof.  A tricky little problem which got two people with Maths degrees really scratching our heads and asking “Why?” and “How?” and occasionally “Huh?”

Where we’re at:  We got dangerously close to concluding the proof after school but I had to shoot off.  She’s since texted to tell me she’s completed it.  Tomorrow, I will do too.

Our conclusion:  This problem is way beyond the capabilities of our 9As (as wonderfully able and geeky and enthusiastic as they are) but could be utilised at a higher level.

Have a go. Leave a comment.

PS:  If anyone can think of a nice way to extend the original problem which would be suitable for 9A, I’d be really interested to hear them.


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