Challenge

Jason Merrill recently posted a fun geometry puzzle inspired by his work on the Lawnmower Math activity for Desmos.  Here’s my paraphrase of the puzzle:

Suppose a lawnmower is tethered to a circular peg in the middle of the lawn. As the lawnmower moves along its spiral path, the rope shortens as its winds around the peg. At the moment the lawnmower contacts the peg, how much rope remains uncoiled?

When I first considered this problem it seemed hard.  After some thought, it seemed obvious.  Then, after some more thought, it seemed hard again.  That’s the sign of a compelling problem!

I enjoyed working out a solution, the heart of which I’ve included below.  Jason graciously included my solution in his post sharing his own, and he also does a wonderful job describing the journey of making simplifying assumptions, both mathematical and physical, that allow us to start moving toward a solution.  It’s the kind of work that often goes unmentioned in problem solving, especially in school mathematics, and this puzzle provides a nice opportunity to make that thinking transparent.

I highly recommend reading the puzzle and his solution at his blog.  Thanks for the fun problem, Jason!

I recently profiled an erroneous high-stakes math exam question that had two correct answers.

Here, it is possible to map AB onto A’B’ using either a glide reflection or a rotation.

It’s interesting to note that there are actually two distinct rotations that map AB onto A’B’, as demonstrated below.

This raises an interesting question:  given two congruent objects, under what circumstances will two distinct rotations exist that map one onto the other?

In a comment on the original post, Joshua Greene offered another interesting follow-up question:

Under what circumstance, if any, are two line segments of equal length not images of each other under rotation? In which of those cases, if any, are the two line segments images of each other under glide reflection?

With enough work, even erroneous exam questions are redeemable!

In class, and on Twitter, I posed a question that led to lots of great conversation.

To increase the volume of a cylinder, is it better to increase the radius or the height? Great conversation starter today.#math #mathchat

— Patrick Honner (@MrHonner) January 9, 2015

There are many reasons I love this particular question. It’s familiar, accessible, and usually counter-intuitive. And it bridges algebra and geometry in a very natural way.

A reasonable response is to argue that an increase in radius will be better. The volume of a cylinder is , so an increase in radius seems to have a squared effect on the volume, while the effect of an increase in height is only linear.

If you imagine the grey cylinder shown below, this argument seems to make sense. Increasing the height a little bit adds a small blue disk of volume to the top, but increasing the radius a little bit adds a large blue shell. The additional volume of the shell clearly appears to be more than that of the disk.

However, this argument is a lot less convincing if you imagine a different cylinder.

It’s not obvious which additional volume here is larger, which suggests some further thinking is in order. At this point, some multidimensional extreme-case thinking usually leads to an appropriate conclusion: Namely, that the answer depends on the dimensions of the cylinder.

This problem is my standard introduction to partial derivatives. It creates great context for computing and comparing

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But its versatility is another reason I like this problem so much. Geometry and Calculus students can both engage in this problem in a meaningful way, using the tools available to them to analyze the situation. And it always results in great conversations!