Radians (Lesson 9.7)
Understand how a radian can be used to measure an angle.
Convert angle measures from degrees to radians and vice versa.
For today’s lesson, students will be investigating a new unit of measurement: 1 Monkey String. Ok, well really it’s 1 radian but we’ll be referring to it throughout the experience as a piece of Monkey String. If your experience in algebra was anything like mine, you memorized the conversion for degrees to radians and that was the extent of your understanding. I truly had no clue what a radian actually was. We can’t let that happen for our students! So make sure you’ve got plenty of Monkey String on hand (this box will last a while!), scissors, and red pens.
A radian is really not that bad when you understand it’s just a different way to measure angles, just like kilometers is an alternative to miles when measuring distance. To get that idea across to students, they’ll be measuring out pieces of Monkey String that are the length of the radius of the circle on the handout. Have them cut all of these out in the beginning. Make sure to emphasize that they should all be the same length! No matter what, some students won’t and they will end up getting 6 pieces around the circle instead of 6.28 pieces.
I’d recommend modeling for students how they should be laying out their Monkey String. If they do this incorrectly, it’s tough to catch the concepts being highlighted. But once they’ve got the process started, they should work through the whole experience before you add the margin notes.
The circumference of a circle is found using C = 2πr
There’s actually not a lot of formalization in the activity today. Our main goal is to establish that a radian is simply a unit of measurement and you can use that to convert from one unit to another, just like we convert inches to feet and vice versa. Draw special attention to this by creating a conversion factor for degrees to radians in the margin notes.
In the Check Your Understanding, students will fill in a unit circle with both degrees and radians. Don’t rush this step! Help students think through each angle as a fraction of the whole circle. You might even want to do this as a Think Aloud with the class.
Possible Think Aloud:
Start with 0, π/2, π, 3π/2, 2π.
Then, try to get students to think of each angle in the unit circle as being a multiple of π/4 or π/6:
π/4, 2π/4, 3π/4, 4π/4, 5π/4, 6π/4, 7π/4, 8π/4
π/6, 2π/6, 3π/6, 4π/6, 5π/6, 6π/6, 7π/6, 8π/6, 9π/6, 10π/6, 11π/6, 12π/6