A couple years ago, Kate Nowak asked us to ask our kids:
What is 1 Radian?” Try it. Dare ya. They’ll do a little better with: “What is 1 Degree?”
I really loved the question, and I did it last year with my precalculus kids, and then again this year. In fact, today I had a mini-assessment in precalculus which had the question:
What, conceptually, is 3 radians? Don’t convert to degrees — rather, I want you to explain radians on their own terms as if you don’t know about degrees. You may (and are encouraged to) draw pictures to help your explanation.
My kids did pretty well. They still were struggling with a bit of the writing aspect, but for the most part, they had the concept down. Why? It’s because my colleague and geogebra-amaze-face math teacher friend made this applet which I used in my class. Since this blog can’t embed geogebra fiels, I entreat you to go to the geogebratube page to check it out.
Although very simple, I dare anyone to leave the applet not understanding: “a radian is the angle subtended by the bit of a circumference of the circle that has 1 radius a circle that has a length of a single radius.” What makes it so powerful is that it shows radii being pulled out of the center of the circle, like a clown pulls colorful a neverending set of handkerchiefs out of his pocket.
If you want to see the applet work but are too lazy to go to the page, I have made a short video showing it work.
PS. Again, I did not make this applet. My awesome colleague did. And although there are other radian applets out there, there is something that is just perfect about this one.
Continuous Everywhere but Differentiable Nowhere 
“a radian is the angle subtended by the bit of a circumference of the circle that has 1 radius.”
I thought every circle has, depending on your point of view, “1 radius,” in the sense that whatever the length of its radius may be, that length is the same for all radii of that circle; or infinite radii, in that every circle does in fact have an infinite number of radii, all congruent to one another.
That said, I think a more accurate definition is “a radian is the angle subtended by an arc equal to its radius.” That is, a radian can be defined on ANY circle, the actual length of the radius being irrelevant in that a radian measures the same on ANY circle.
The one you give, aside from being awkward (would we say “radius equal to 1” or something like that rather than “1 radius”), leaves me with the impression that only a unit circle can be used to define/measure 1 radian, and that once we settle on what 1 unit of length is, we’ve somehow determined a measure of 1 radian (but that could vary depending upon the circle). I know you don’t think that, but that definition seems to leave that possibility open.
It took me forever to understand what your issue was with what I wrote — but I get it now.
You read it as “a radius equal to 1” instead of “the length of a single radius.” I also see what you mean about a kid who reads that who thinks “oh, there are an infinite number of radii in every circle.”
Your wording is definitely nicer.
Nice catch, Michael – I think your wording probably does clear it up a bit.
Sam – what a lovely applet your colleague created. I LOVE the colors, they definitely make the idea pop. I’ve linked to the applet on my own virtual filing cabinet. Thanks!
Love this applet. Just asked my precal kids on a quiz the other day: If a bug on this particular circle traveled one radian, how far in linear measurement did it travel? They all knew what the radius was, they had told me a hundred times what the definition of a radian was (in their own words), but only 2 of 23 could tell me how far that bug traveled!
That kind of thing is maddening, isn’t it? Feynmann tells a lovely story like this about a particular law of optic where students recite a law to him, he asks them to look at a lake and talk about refraction and they stare blankly at him. Understanding a fact and applying it in context are SO different. Maybe it is another ‘cognitive load’ issue?
Awesome demonstration of what exactly 1 radian is. I had never really taken time to think about this. Math is great.
Sam, I ordered Radian protractors from Jen Silverman. They absolutely had results in class. I think connecting radians to degrees and circles with real objects like a protractor made a huge difference in understanding.
@Glenn, thanks! @Tina, did you have them roll the protractors? @Sam, I have similar applets at proradian.net.
I love reading your blog because you are always asking, “Why are we teaching this?” and working to build conceptual understanding. I have been teaching Precalculus for 17 years, but with changing state standards and Common Core it is a bit confusing to know exactly what topics are considered “Precalculus” these days. If you have time, I’d love to have a list of your units (the topics you teach) to compare to mine. I am looking forward to using some of the ideas you have posted in your Precalc blogs. It is fun to see that you do some of the same things I do.
I just wanted to chime in my “Thanks for sharing!” I love this. Thank you also for the video as I initially fell into the “too lazy” category.
Awesome!