Radians and Degrees (Lesson 4.2)
Unit 4 - Day 3
Understand that a radian is an angle measure with an arc length of one radius
Use circumference to explain why 2pi radians corresponds to one full rotation
Use proportional reasoning to convert between angles measured in radians and degrees
Quick Lesson Plan
So far students have been working with angles in degrees but to transition to thinking about angles on a circle, they need a robust understanding of angles measured in radians. We love how this activity has them physically count radii around the circle and use proportional reasoning to split up the circle.
For this lesson, students will need a paper plate, Twizzlers, a protractor, scissors, and ruler. Many alternate materials are available. You can print circles onto pieces of paper and have them cut it out and Twizzlers can be replaced by string, floss, cooked spaghetti, anything that can bend and be cut!
Students start by thinking about partitions on the circle in terms of degrees and then establish the idea of 1 radian being the angle made whose arc length is 1 radius. Students estimate that there are roughly 3 radians in a semicircle and just over 6 radians in a whole circle. This answer becomes more specific as they think about the circumference of a circle in questions 10 and 11. Students end back where they began, by thinking about partitions of the circle but now in terms of radians. This lesson is a great way to review fractions and proportional reasoning.
It’s easy for students to confuse the terms radius and radian. We tell students that a radius is a length measured with a ruler and a radian is a measure for an angle, it is not the actual arc length. Students begin to associate angles with pi in them as being in radians, but it’s important that students realize pi is also a quantity telling you there are approximately 3.14 radii in half a circle.
You may note that we never formally teach the conversion formulas for switching from degrees to radians and vice versa. This is intentional, because we find that too often students resort to formulas and sidestep the mathematical ideas involved. Students in our class are expected to reason through the idea that pi/9 radians is 1/9 of the way to pi, and thus 1/9 of the way to 180˚. Similarly, 10˚ is 1/18 of 180˚ (it’s a half-circle cut into 18 equal chunks or slices), so the radian measure should be 1/18 of pi.
When filling out the angles of the unit circle for the first time, color coding can be very helpful. You may wish to show the unsimplified form of pi/3 as 2pi/6 so students see that all multiples of 30˚ are really multiples of pi/6 even if they can also be written as multiples of 60˚ or pi/3. In the Check Your Understanding, students will continue to think about equivalent forms of the angles on the circle and thus, which angles are multiples of one another. Make sure students see that each quadrant is not cut equally into thirds.