Home > #MTBoS, A Level, GCSE, Maths, Teaching > A nice area puzzle

A nice area puzzle

At some point over the last few days Danny Brown (@dannytybrown) tweeted this lovely puzzle out:

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While waiting in the car this afternoon I saw the screenshot I’d taken of it and had a think about it. Initially I’d made a daft mistake with the base of a triangle and got the wrong answer (5:6 if you were wondering), but after a rethink and some in head workings I got to an answer I’m now happy with. I have since written the solution down and want to write about it here.

Firstly, when attacking problems like this I like to sketch them out. This step is one many students are reluctant to take, and I try to drill it into them that sketching is always helpful in understanding problems. If I’d had a pen and paper handy when I’d attacked this problem originally I’d have sketched it and not made my daft error.

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From the sketch it is easy to see the area of shape A, this was easy to visualise.

In my working I then sketched shape B:

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When I visualised this I made my daft error, which was to assume the triangle had base x. From the sketch it’s easy to see that it’s not, and it will need calculating, for this I used Pythagoras’s Theorem:

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To calculate the area of the triangle I’d need an angle, so I thought about what I knew, the angle at E is made from a fold along EF so the angle must be equal to DEF.

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Using the tan ration of the right triangle I got tan (theta) to be rt3.

Then it was a case of calculating the height and then the area:

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The working written out properly looks a lot more thorough than my phone jottings:

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This is a nice puzzle that should be accessible to higher GCSE students and definitely A Level Students, but I worry that most would give up. We need to be giving our students the tools to unlock this sort of problem. I’m not sure how we can do that explicitly. I set these tasks, give them hints and then walk them through my thinking to model how I would attack them, and this has a great effect for many. But I wonder if this is enough.

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  1. December 31, 2014 at 12:22 pm

    I did it the same way (can’t see an alternative).
    One thing I’d add – as the final question wants the ratios, a variable is unnecessary; we can just pick an arbitrary length. I had width as 1 and length obviously root 3.
    Nice problem though.

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