Proof of the Week: Slicing brownies

BrowniesOver the Christmas break, Steven Strogatz Tweeted a nice problem:

“A puzzle for kids: Slice a rectangular brownie along its 2 diagonals. Prove all 4 pieces have the same amount of brownie.”

I have been trying to tackle a proof each week with one of my Y11 groups. I thought this would be very manageable for them, but it proved more challenging than I had expected – and provoked some interesting discussions.

There were two main difficulties. Firstly most of them labelled the sides of the Brownie as l (length) and w (width), but working out the area of a triangle which had a dimension of w/2 or l/2 posed problems for many who struggled to divide this by 2. Some interesting discussion about division, fractions and algebraic manipulation there. Secondly deciding at what point they had proved the result provoked some debate. Some students stopped once they had shown the areas, others wanted to find the volumes of the pieces.

The context of the problem also allowed us to talk about the assumptions they had made and the conditions under which the proof was valid.

All in all, a surprisingly satisfying problem. I’ll be using this one again. Thanks Steven!

 

Photo: Brownies by yum9me on Flickr. Used under Creative Commons Licence.

1 comment to Proof of the Week: Slicing brownies

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89 + = 93