# Lesson 10

### Lesson Narrative

In this lesson, students experience a progression of learning that will build to a definition of radians in a subsequent lesson. The activities in this lesson build intuition about the relationship between arc length and central angle, without yet naming the arc length to radius ratio 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 analyze ratios in dilated circles, recalling that all circles are similar. Finally, they make initial observations about the relationships between arc lengths, radii, and angles, noting that the arc length to radius ratio appears to be constant for the same angle measure across circles of different sizes.

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

### Learning Goals

Teacher Facing

• Use the similarity of all circles to explore ratios in circles that are invariant under dilation.

### Student Facing

• Let’s analyze relationships between arc lengths, radii, and central angles.

### Student Facing

• I know that when a circle is dilated, some ratios, like the ratio of the circumference to the diameter, stay constant.

Building On