What is a radian?

This is my second year of teaching precalculus and last year when I introduced radians I had students construct a radian “tape measure” and measure different distances around the unit circle. Some students left with a good understanding but many did not. This year I decided to take a different approach (admittedly less hands on) to introducing the topic. I’m making an effort to introduce topics in an inquiry fashion as much as possible so this is the activity that I used to introduce radians this year.

Intro to Radians Activity              Video to Accompany Activity

I handed out the activity to each student and told them that they shouldn’t talk or confer on the problems. I wanted their own individual observations. I then should the video to the entire class in its entirety. (The only thing I told them was that radians were another way to measure angles.) Once I played the video in its entirety, I played it again and paused it at each position as needed (For #1 I paused when r=1, etc.). Once we did #1-6 We went through and discussed their observations as a class. After a whole class discussion I asked them to answer #7 without collaborating or conferring. This gives me a good idea as to how well each student understands radians as a concept.

Although Algebra II touched on radians many students come to precalculus with a vague (if any) understanding of the concepts. I think it is valuable to understand why a radian is what it is and why it exists. Students had many “ah-ha” moments, especially in the whole class discussion.

Here are some student responses to #7. I tried to give a good sample of the class.

“One radian is equal to the radius along the edge of the circle. It also brings pi into the fold because when the radian count is equal to pi the point is at 180 degrees.”

“Imagine an x-axis. Draw a circle around the origin. A single radian is the distance around the circumference that is the same as the radius.”

“A radian is a way of describing degrees on a circle. It is closely related to pi in which half the circle is one pi radians.”

“It is the measure of an angle and its distance traveled along the unit circle in a counter clockwise direction.”

“One radian is one radius. It relates to pi is when it’s at 3.1 radian, and 2pi radians is the whole circle.”

“Radians are the angle that is formed by a radius. It’s related to pi because at half way around the circle the value is about the same value as pi.”

What I gathered from these responses is that next time (and tomorrow in class) I need to point out that the radian is the angle measure when the point travels the distance of one radian around the circle. Several students (as you can see above) think that the radian is the distance. Next year I may modify the activity and get the students in a computer lab so they can answer the questions at their own pace.

How could I improve this activity? What should I do different in the future? How do you introduce radians as a concept? Any feedback you can provide would be awesome!

(A huge thanks to Sam Shah and the Geogebra Applet that I recorded to make the video. Read his post that inspired my activity here.)