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Monday, November 7, 2011

This Is a Fun Question to Ask Your Calculus Students

"What is 1 Radian?" Try it. Dare ya. They'll do a little better with: "What is 1 Degree?"

9 comments:

  1. Mine get it because I taught them in precal. =P But when I begin it in precal, they really struggle with what 1 degree is. They even just struggle trying to define an angle even though it's clear they know what it is. Once you explain it, though, they seem to love that it makes sense (unlike most of the math they've done to this point, reportedly).

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  2. I ask this to my Pre-Calc students every year...it's literally an hour of them having circular discussions, resulting in me assigning the question to them as homework for the next class. It's awesome to sit back and watch them try and figure it out together.

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  3. :) Good question.

    I'm wondering how you describe the fundamental difference in thinking between measuring with degrees vs. radians.

    Here's how I think of it, and how I try to explain the difference - I'd be curious to hear other ways of thinking about it and explaining it.

    The degree is an arbitrary way of splitting up a circle, I think based originally on the Babalonian calendar. But the idea behind the degree is pretty simple: I want to cut the circle into a certain number of equal wedges (in this case, 360 of them). The radian approach is different in that it doesn't simply come from picking a different number of wedges to split the circle into. The radian approach says: I don't care how many wedges I end up with, I just want to cut out the perfect wedge - the wedge whose arc is the same as the radius (it turns out that you get 2pi of these). The pizza analogy goes something like: the degree is the size of the slice you end up with when you need to share pizza equally with 359 of your friends, the radian is what you end up with when you are cutting the perfect slice for yourself, with no thought about how the rest of the pizza will be cut.

    Another thing that puzzles students is that degrees have units, but radians don't. You can explain this in a similar way, but again, I'm wondering how (if?) students best understand this.

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  4. So, it's definitely true that 360 degrees in a circle is arbitrary, we have the Sumerians and Babylonians to thank, where 360 was probably the natural intersection of using a base 60 number system, and their being 365 days in a year, so that the angle the sun moves in one day on the sky is about 1 degree.

    But radians are quite natural, in fact they are the most natural angular measure, and I think of them as just measuring the length of the arc of a circle. For a unit circle, if you go all the way around, the arc (the circumference) has a length of 2pi, if you go some fraction of the way around the circle, you are marking the angle you make by the ratio of the length of the arc you've described to the full arc of the circle. That's why all of the trigonometric functions are so clean in radians, radians is a measure of arc length.

    I think its missing a deeper beauty to suggest that radians is the natural unit for 'perfect' slices, since your choice of a 'perfect' slice is quite arbitrary in itself.

    From a dimensional analysis point of view, radians are the best dimensionless measure of angle.

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  5. What's it say of my style that my kids would tell you "approximately 57 degrees"? Some might say it's circular thinking, but then I'd just laugh at them.

    On a more serious note, I love the idea of asking a question that sounds so simple yet is so difficult for kids to answer. For my kids, I often ask "what is x?" And the frequency with which they answer "1" is scary.

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  6. I think the answers I would like are:

    a little less than 60 degrees (since 2π of them, about 6.3, make a circle)

    and

    one radian is the ratio between a half-circumference and the radius - of any arbitrary circle.

    It's an annoying question, I'll give you that.

    Jonathan

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  7. The question is great in that it stops kids cold, even after several days of work with the unit circle. It reveals a lot about how kids approach open questions like this vs the usual process exercises. I like how some find the equivalent degree measure, some wonder where pi went to, and then there's "This is a trick question, right?" But after discussion, and demonstrations at the board, they settle on an arc that is the same length as the radius. No one disagreed. Definitely worth the time. Thanks, Kate, and thanks to commenters, too.

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  8. As the years passed I found that it worked better for me to start by saying it is about 57 degrees & then get into the discussion of why such a seemingly obsucre value was chosen. I don't think they follow the development well at all before getting a real concrete sense that it is just another unit of measure - like 12 inches is a foot.

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  9. We define a radian in precalc as "how far Pacman would need to open his mouth to eat an arc one radius long"

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