In this lesson, the students will explore the radian as a unit of measure. They will estimate the measure in radians of arcs, and also help us think about the connection between radians and degrees.
The students solidify how many radians are in various amounts of rotation and think about how that relates to degrees.
Mathematics in this Lesson
Targeted Understandings
This lesson can help students:
 Analyze the relationship between the measure of an angle in radians, the length of the arc it subtends, and the radius of the circle.
Explain why there are 2đťś‹Â radians in a full rotation.
Estimate the measure of an angle in radians, using the fact that about 6.28 radians are in a full rotation.
Evaluate the reasonableness of an angle measure in radians.
Explain why a circular radian protractor can be any size and still produce accurate angle measurements.
Solidify their understanding of the definition of a radian.
Common Core Math Standards
CCSS.MATH.CONTENT.HSF.TF. A.1. Understand radian measure of an angle as the length of the arc on the unit circle subtended by the angle.
Throughout this lesson, Claire and Mary utilize radians as a unit of measure. They find the measure of given angles and construct angles of a given size. They utilize various tools in their exploration that help them make sense of radians as the angle created by an arc equal to the length of the radius. These tools include patty paper, pipe cleaners, and transparency paper. For example, to find the measure of angles in radian, Mary and Claire create protractors using circular patty paper. They measure the radius of the paper using a pipe cleaner, which they then bend around the patty paper to mark off lengths equal to the radius.
CCSS.MATH.CONTENT.HSG. C.B.5. Derive using similarity the fact that the length of the arc intercepted by an angle is proportional to the radius, and define the radian measure of the angle as the constant of proportionality; derive the formula for the area of a sector.
Claire and Mary explore circles of varying sizes, analyzing the relationship between the radius and angles created by marking arcs with lengths equal to the radius.
According to the CCSSM, “Mathematically proficient students try to communicate precisely to others. They try to use clear definitions in discussion with others and in their own reasoning…They are careful about specifying units of measure… They calculate accurately and efficiently, express numerical answers with a degree of precision appropriate for the problem context…By the time they reach high school they have learned to examine claims and make explicit use of definitions.” Mary and Claire calculate and express angle measures in radians accurately, estimate radians effectively, and evaluate the reasonableness of their answers. For example, in Episode 1, they use pipe cleaners to first find the length of the radii of circles, and then to iteratively mark off arcs with that length [0:54]. Asked to approximate the number of radians, they use the “approximately equal” symbol and note that one angle is just under 2 radians and estimate it to be 1.8 radians. They use a similar process in Episode 2 to create angles with a given radian measure [e.g., 1:35]. In Episode 3, they use their methods again to measure given angles, claiming that one angle is between 1ÂĽ and 1â…“ radians [0:55], while another is over 2 radians, calling it 2.15 radians [3:05]. Finally, in Episode 4 they utilize an approximation of 6.28 radians for one full rotation while recognizing the exact amount is 2đťś‹. Precision is crucial when working with measurements and conversions between radians and degrees.Â
