# Lesson 7 Staking It Solidify Understanding

## Learning Focus

Calculate arc length for angles of rotation measured in radians.

Visualize the size of angles measured in radians, including radians given in decimal form.

How large is a radian? How can I estimate the size of angles measured in radians relative to degrees?

What good are radians? Are any calculations made easier when the angle is measured in radians? Are any contexts easier to describe using radians?

## Open Up the Math: Launch, Explore, Discuss

After considering different plans for laying out the archeological site described in *Diggin’ It*, Alyce, Javier, and Veronica have decided to make concentric circles at

### 1.

Veronica suggests they should locate the stakes around one circle and use those positions to mark where the stakes will go on all of the other circles. What do you think about Veronica’s idea?

#### a.

How will marking stake positions on one circle help them locate the positions of the stakes on all of the other circles?

#### b.

If there are

#### c.

How far apart are the stakes in radians?

Veronica has decided they should stake out the circle with a radius of

Veronica decides she will find the distance by setting up a proportion using degree measurements.

Alyce thinks they should find the distance by taking

of the circumference. Javier thinks they should use radian measurement in their calculation.

### 2.

Show how each team member will calculate this distance.

Veronica’s strategy:

Alyce’s strategy:

Javier’s strategy:

Javier has a different idea. He suggests they should figure out the locations of all of the stakes in Quadrant I first, and then it would be easy to find the locations of the stakes in all the other quadrants by using the Quadrant I locations.

### 3.

What do you think about Javier’s suggestion? How will marking the location of stakes in Quadrant I help them figure out the location of the stakes in other quadrants?

Javier has already started working on his strategy and has completed the calculations for several random points in Quadrant I (see Javier’s diagram).

Alyce has a concern: “Using the

Veronica has another idea: “I want to use a compass to orient the direction I should walk from the central tower, and then drive a stake every

### 4.

Develop a strategy to locate all of the other stakes in the first quadrant for these additional circles. Find the

### 5.

Javier has noticed that the farther away from the central tower, the farther apart the stakes appear to be. He suggests they should add radial lines to the diagram at angles of

### 6.

Alyce and Veronica love this idea and assign Javier the job of posting all of the stakes on the

Pause and Reflect

### 7.

Being able to convert quickly between degree measurement and radian measurement will help Javier a lot.

#### a.

Develop a strategy Javier can use consistently to convert angle measurements from degrees to radians.

#### b.

Develop a strategy Javier can use consistently to convert angle measurements from radians to degrees.

### 8.

Veronica has a different proposal for the soon-to-be-added

#### a.

Show how to calculate the arc length between stakes using

#### b.

Show how to calculate the arc length between stakes using radian measure.

## Ready for More?

Explain how the radian measure shows up in Alyce’s, Javier’s, and Veronica’s strategies for problem 2.

Veronica’s strategy:

Alyce’s strategy:

Javier’s strategy:

## Takeaways

If a circle has been divided into

I can find arc length for an angle of rotation measured in , by using the formula .

I can find arc length for an angle of rotation measured in , by using the formula .

I can convert degree measurement to radians by .

I can convert radian measurement to degrees by .

## Vocabulary

- arc length
**Bold**terms are new in this lesson.

## Lesson Summary

In this lesson, we continued to work with degree and radian measurement for angles of rotation. We found strategies for converting from one angle measurement to the other, and we saw that the formula for finding arc length for angles measured in radians was simpler than the formula for finding arc length for angles measured in degrees. This occurred because radian measure is defined as a ratio of arc length to radius.

### 1.

Find one point in each quadrant that lies on the circle

#### a.

Quadrant I:

#### b.

Quadrant II:

#### c.

Quadrant III:

#### d.

Quadrant IV:

### 2.

Label each point on the circle provided with the measure of the angle of rotation in standard position beginning with the rotation from

Point

Point

Point

Point

Point

Point

Point

Point