## An interesting area problem

Here’s an interesting little question for you:

*Have you worked it out? How long did it take you to see it?*

It took me a few seconds at least, I had screenshotted the picture and was reaching for the pencil when the penny dropped, and that’s why I thought it was an interesting question.

The answer is, of course, 100pi. This follows easily from the information you have as the diagonal of the rectangle is clearly a radius – the top left is on the circumference and the bottom right is on the centre.

*So why didn’t I spot it immediately? *

I think the reason for me not spotting it instantly might be the misdirection in the question, the needless info that the height of the rectangle had me thinking about 6, 8, 10 triangles before I had even discovered what the question was.

I see this in students quite often at exam time, they can get confused about what they’re doing and it links to this piece I wrote earlier about analogy mistakes. The difference is I wasn’t constrained by my first instinct but all too often students can be and they can worry that it must be solved in the manner they first thought of.

Earlier today a student was working on an FP1 paper and he was struggling with a parabola question, he had done exactly this, he had assumed one thing which wasn’t the right way and got hung up in it. When he showed me the problem my instinct was the same as his, but when I hit the same dead end he had I stepped back and said “what else do we know”, then saw the right answer. I’m hoping that by seeing me do this he will realise that first instincts aren’t always correct.

I’m going to try this puzzle on all my classes tomorrow and Friday and see if they can manage it!

*How quickly did you see the answer? Do you experience this sort of thinking from your students? I’d love to hear any similar experiences.*

Cross-posted to Betterqs here.

Solved it quickly, but the direction displayed for the diagonal is a nice touch.

Aye, it adds to the misdirection I feel. I wonder what similar questions could be made? I might just show the picture at first and ask what they think the question is!

yes, very cool problem… looks like a great problem to share. Thanks!

The first time I saw this, I worked for a few minutes on the red herring 8 before the dawn came. I’ve definitely had more students than not, though, get frustrated about problems where there’s unneeded information, because they want to do something with every bit of information in the problem. For instance, consider a transversal crossing three lines, the outer two of which are parallel. The same side exterior angles of the parallel lines are marked 25x and 35x, while one of the angles (it doesn’t matter which) of the third intersection is marked 30x. The question asked what x was, and the most common answer I got was 2 (because 25x + 35x + 30x = 180 leads to 90x = 180, so x = 2). I had a devil of a time convincing students to ignore the 30x entirely. They didn’t want to believe me that it was irrelevant.

Aye, that certainly sounds a familiar tale.

This is actually an old problem, and one of my favorites — had a version of it in geometry class in 1967 and it was already classic then. Good to see it’s still going strong! ðŸ™‚

Aye it’s a lovely one isn’t it?