# Trigonometry Lesson 3 (Teachers)

### Investigating the Relationship between Arc Length and the Angle of Rotation

In this lesson, Mary and Claire investigate the relationship between an angle measure and the length of the arc that it cuts out in a circle

##### Episode 1: Making Sense

Mary and Claire think about whether or not you can use arc length to measure the angle of rotation.

##### Episode 2: Making Sense

Mary and Claire explore another student’s reasoning who thought that you could use arc length to determine the angle of rotation. They compare two arcs with different lengths.

##### Episode 3: Exploring

Mary and Claire find the distance traveled by the tips of the blades of two wind turbines. They relate those arc lengths to the angles of rotation made by the blades.

##### Episode 4: Repeating Your Reasoning

Mary and Claire explore the path traveled by two people who live in different locations, Michigan and the Caribbean, as the earth makes a full rotation. They figure out how much of a rotation the earth needs to make for each person to travel 1,000 miles.

##### Episode 5: Repeating Your Reasoning

The students use their conclusion from the previous episode to figure out how many degrees and gips the person in the Caribbean rotates as they travel 1,000 miles.

##### Episode 6: Reflecting

Mary and Claire revisit their thinking from the first episode. They consider what else you would need to know to determine the angle of rotation, if you know that length of the arc the object traces out while traveling along a circular path.

##### Episode 7: Reflecting

Claire and Mary create several circles of various sizes. On each of the circles, they trace out an arc whose length is the radius of the circle. They create angles at the centers of the circles that cut out these arcs. They compare those angles. A new unit of angle measure is defined: the radian.

### Mathematics in this Lesson

Targeted Understandings

This lesson can help students:

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Common Core Math Standards

Common Core Math Practices