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.

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.”