The activities in this lesson build intuition about the relationship between arc length and central angle, without yet naming the ratio of the arc length to radius as “radian measure.”

Students begin by using the example of a circular progress bar to make the observation that arc length for a particular central angle is dependent on the radius of the circle. Then students make initial observations about the relationships between arc lengths, radii, and angles, noting that the ratio of arc length to radius appears to be constant for the same angle measure across circles of different sizes. Finally students prove that the length of the arc intercepted by a central angle is proportional to the radius of the circle.

As students compare ratios of arc lengths to radii for congruent angles in circles with different radii, they are looking for and making use of structure (MP7).