When I was introduced to radian measure in high school, I knew just one thing: How to convert between radians and degrees. Had you asked me to illustrate a radian on a circle or to explain why radian measure was useful, I would have been stumped.

In this post, I’ll describe a Web Sketchpad activity that helps students develop a deeper understanding of radians.

On page 1 of the Web Sketchpad model below (and here), drag point C. Notice that it is attached to segment AB. Now drag point B or press Animate Point B. What is the path of point C as point B moves around the circle? Web Sketchpad’s Trace widget allow your to see the path (The video at the end of this post demonstrates how to use the Trace widget and everything else described here.)

Moving on to page 2, press Animate Points. Pointst B and C are moving simultaneously at the same speed. As point B moves counterclockwise around the circle, point C moves back and forth along the segment. Observe point C and describe its path.

Point C seems to trace flower-like petals. How many petals does it trace for one full revolution of point B around the circle? Again, you can use the Trace widget to check your prediction. Does the number of petals seem especially significant in the context of circles?

How many petals do you think point C will trace if you keep the animation running? On page 3, everything is set up for you. Press Animate Points and observe the petals as they form. What is mathematically significant about the number of petals?  The video below demonstrates how you can build this construction from scratch on page 4 of the Web Sketchpad model.

The “pi-petal rose” was the basis for the logo of the late, great Key Curriculum Press. Read more about that logo here.