The purpose of this discussion is to connect each location on the circle with the arc length and the central angle. Display a circle with a marked center for all to see. Invite students to add each of the following to the display:

• The point where Tyler entered the path
• The point on the path where the sign indicates half a mile
• The point on the path that is northwest of the fountain
• The point on the path where there is a duck statue

As students add each point, ask the class, “How far has Tyler traveled?” Continue calling on students until they've shared a distance in miles and an angle in degrees.

If students are confused by the fact that their radian measures of angles are the same as the arc lengths they’ve calculated, remind them that the radius of the circle is 1 unit, so it makes sense that when we calculate the ratio of arc length to radius for angles in these circles, the answer is the arc length divided by 1—that is, it’s simply the arc length.

The goal is to discuss the idea that, by relating arc length to a circle’s radius, the radian measure of an angle shows how many radii make up the arc defined by a central angle.

Ask students to share their strategies for the last problem. If possible, select one student who calculated that the circumference of the circle in the activity is 2 units, and another who reasoned that, because 360 degrees is 2 times 180 degrees, the radian measure of a 360-degree angle must be twice that of a 180-degree angle. Then ask students:

• “Does the word ‘radian’ seem to relate to a particular part of a circle?” (The word “radian” sounds like “radius.”)
• “How is a radian connected to the radius?” (The radian measure of an angle is the ratio of the arc length to the radius, so it tells how many radii fit along the arc defined by the angle.)
• “How did your answers to the second and third questions connect?” (The string measurement showed that the arc length was about 3 units in length. The radian measure of the angle ended up being or 3.14 units, which matches our approximate string measurements.)
• “What if the circle had a radius of 2 units instead of 1? How would the radian measures change?” (They wouldn’t change. The ratio of arc length to radius is constant no matter the size of the circle.)
• “We defined the radian measure of an angle as . How does this relate to the idea that the arc length and radius are proportional?” (We can think of the angle measure as being the constant of proportionality.)