I needed a polar-coordinates-based assignment for my Calculus 2 students, so I pounced on it. The question they have been working on is, How long will it take to mow the lawn?

I read their work today. The following are some quotes from their writing.

“Establishing the polar function was difficult at first, until I thought about it as just a plain linear function.”

“I tried going on the treadmill to see what a comfortable walking speed for mowing would be.”

“Sorry for making this 13 pages. I really got into it.”

“Sometimes math needs a little touch up; this is when Photoshop is there to save the day.”

“The real real-world problem is how to convince your wife to upgrade mowers.”

“Rather than dealing with negatives and reciprocals, this paper will assume the lawnmower ‘un-mows’ the lawn from inside to out.”

“After realizing that the point on the outer ede of a circle has to cover more linear distance than a point near the center, angular velocity seems like it might have some flaws.”

I see in these excerpts students making mathematical connections that result from their struggles with the problem. I see them posing and refining mathematical models based on correspondence to the real world. I see them looking at this small slice of the world through a mathematical lens.

I am so proud of them.

NOTE: In original post, I did not know who had posted the video to 101qs. David Cox came through for me on Twitter. Credit given in revised post.

Sue, I learned a tremendous amount about my students’ thinking about polar coordinates from this task. The most stunning moment for me was when I realized that one student (ONE! Out of FORTY!) noticed that he could think of the radius as a linear function of theta. That speaks volumes about their ability to generalize properties of function types.

Conclusion-we get it all wrong…all so totally wrong when we approach polar coordinates from the perspective of the pretty flowers and other periodic shenanigans. No, we need to ask what “linear” functions look like in polar coordinates, and what “quadratics” look like, and “exponentials”. And we need to reinterpret the derivative using what we know from prior experience with these function families. Where is the derivative positive, negative and zero? And how can we SEE this in the polar graph? My students struggled mightily with that while the text was toying with periodic functions.

I really want to take your class! I feel like all I do is memorize formula after formula after formula (after all, I am using the book that blatantly states we should “pause to think” in between using formulas.) Please let me know if you end up teaching any online or distance classes :) I think it would be fun to write a 13 page paper on the mathematics of mowing the lawn (I’m not being facetious!)

This problem is quite rich with thinking and exploration. I’m curious how many different solution paths your students generated. I’m also curious about different assumptions students made.

I’m also curious about trying to answer other questions like, “How much of the mowing is is needless–i.e. how much time is spent moving grass that’s already been mowed over?” and … “What configurations guarantee that all the grass in the circle will get cut // what configuration will leave patches unmowed?”

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Thank you! We’ll be doing polar in a few days. Way cool!

Sue, I learned a tremendous amount about my students’ thinking about polar coordinates from this task. The most stunning moment for me was when I realized that one student (ONE! Out of FORTY!) noticed that he could think of the radius as a linear function of theta. That speaks volumes about their ability to generalize properties of function types.Conclusion-we get it all wrong…all so totally wrong when we approach polar coordinates from the perspective of the pretty flowers and other periodic shenanigans. No, we need to ask what “linear” functions look like in polar coordinates, and what “quadratics” look like, and “exponentials”. And we need to reinterpret the derivative using what we know from prior experience with these function families. Where is the derivative positive, negative and zero? And how can we SEE this in the polar graph? My students struggled mightily with that while the text was toying with periodic functions.

Hah, that is awesome, thank you for sharing.

I really want to take your class! I feel like all I do is memorize formula after formula after formula (after all, I am using the book that blatantly states we should “pause to think” in between using formulas.) Please let me know if you end up teaching any online or distance classes :) I think it would be fun to write a 13 page paper on the mathematics of mowing the lawn (I’m not being facetious!)

This problem is quite rich with thinking and exploration. I’m curious how many different solution paths your students generated. I’m also curious about different assumptions students made.

I’m also curious about trying to answer other questions like, “How much of the mowing is is needless–i.e. how much time is spent moving grass that’s already been mowed over?” and … “What configurations guarantee that all the grass in the circle will get cut // what configuration will leave patches unmowed?”