String, scissors, pencils, paper.
“Which angle measurement are you more comfortable with, degrees or radians?” Allow answers, the majority will probably be degrees.
“Does anyone know where degrees come from? Why are there 360 in a circle?” Allow answers. Mention that 360 is a nice number. Babylonians liked it because there are approximately 360 days in a cycle of the sun, later peoples liked it because it has many factors, so they divided a circle up into 360 equal parts.
“And does anyone know where radians come from? Why do we use them at all?” Students generally don’t, but praise any who does. Mention that while degrees come from a convenient number for human use, radians come from inherent properties of a circle. That makes them more useful for certain calculations.
“That’s what we’ll show in todays’ class.”
- Define a radian, and its usefulness.
- Know the origin of radians.
- Identify various radian measures along the circumference of a circle.
Students will cut a length of string between 2 and 4 inches. They will anchor this string in the center of their paper, and use it to create a circle with that string’s length as the radius. Mark the center. Remind them to keep the string handy, as we’ll be using it.
I’ll draw my own circle larger circle on the board for reference.”
Make a mark at the right edge of the circle. This is the starting point, zero radians.
Place the length of string around the circumference of the circle, starting at the zero point, and mark where the string ends. Draw a line from this point to the center of the circle, and mark off the angle you have just drawn: You have now gone one radius length around the circumference of the circle, and the angle you’ve made is one radian.
Allow students to repeat this process on their own to mark off two radians, and three radians.
Point out that they are now almost halfway around the circle. Ask for an estimate of how many radius lengths would take you exactly halfway around the circle. Allow students to approximate. Answers will usually be around 3.1 or 3.2, or students might correctly estimate that it is pi radians. If not, lead them to it.
“Yes! I think it is around that many radians. Maybe around 3 point 1415926535897932384626…” Students will almost always get the picture before I reach the limits of what I have memorized. Stop when they do.
“That is why a 180 degrees—or a straight line—is equal to pi radians. Keep measuring radians from the 3rad point until you get the rest of the way around the circle.”
Allow students to continue until they reach the beginning point. Ask how many radians that took. Students should see that it is a bit over 6 radians. Lead the class, if necessary, toward realizing that the length is 2pi radians.
Checking for understanding:
“Now that we’ve established that half a circle equals pi radians, how could we convert between a given angle measured in degrees and that same angle measured in radians?” Lead students, if necessary, toward the conversion factor “pi radians / 180 degrees.” Give some examples of angles in degrees and ask students for their measures in radians. Try 90 degrees, 45 degrees, -30 degrees.
“And that, ladies and gentlemen, is where radians come from. They’re a bit cumbersome at first for human beings who are used to dealing with integers, but many types of calculations—most in Calculus, for instance—only make sense when we use this measurement, which is based on an actual property of circles rather than numbers we made up on purpose.”