I have just returned from an inspiring morning at #mathsmeetglyn organised by @mathsjem watching Don Steward give a fantastic, brain-stretching whistle-stop tour of some of the great problems on his site.
The theme was “Generalising” – he started off by saying that his current mission is to get some generalising into every single lesson because without it, well, you’re not doing real maths.
I filled 10 pages of notes, some of which I’ll come back to, but wanted to share how he took this Edexcel Exemplar Foundation GCSE question and “generalised” it.
Now, first up, I don’t mind admitting that after working at it for about 3-4 minutes I was completely stumped. Don had added “You can’t cut the tiles” to the question which was essential in my view, but didn’t actually help me. I was completely fixated on the fact that all tiles have to be in the same orientation. They don’t.
I think it’s a pretty poor question, probably mainly because I couldn’t do it. But there is a serious point around what questions like this are really testing. If there is a simple “trick” you need to get, is that fair?
A debate to be had there, I’m sure, but more interesting was what Don did with it next.
What other areas can you fit 40 x 30cm “carpet tiles” into and how many do you need? Start with:
120cm x 60cm 120cm x 70cm 120cm x 80cm 120cm x 90cm
Do you need to go any further? Can you write a general statement from this? i.e. can you prove that all multiples of 10 will work if the width is 120cm? What other widths does this work for and why? And then, of course, what happens if you try different sized carpet tiles.
It feels a bit like one of those Maths GCSE coursework questions that were set in the days before I was a teacher. But I really like the idea of taking what is a pretty bad question and turning into some interesting maths.
mho maths 
What an interesting way to extend the question! I’ll certainly be using that activity soon.
As for the initial question, I agree that if there’s a trick that’s the key to unlocking a problem, then it’s probably not a good classroom task in most cases.
In this case I think there is some mathematcial reasoning that could help: 100 (cm) isn’t a multple of either 40 or 30, therefore the tiles cannot possibly have the same orientation; 120 is a multiple of both 30 and 40, therefore a row of tiles in the same orientation, whichever it s, will always fit this dimesnion exactly; can we make 100 by adding a combinations of 30s and 40s?
I think that approach to solving the problem could be useful in helping wth the extension task too.
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