# Lesson 6 Diggin’ It Develop Understanding

## Learning Focus

Locate points in a plane using coordinates based on horizontal and vertical movements or based on circles and angles.

Use degrees and radians to measure angles.

Are there other ways to describe the location of a point in the plane other than by giving its

What proportionality relationships can I find between corresponding points and arc lengths of concentric circles? How can I justify why those proportionality relationships exist?

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

Alyce, Javier, and Veronica are responsible for preparing a recently discovered archeological site. The people who used to inhabit this site built their city around a central tower. The first job of the planning team is to mark the site using stakes so they can keep track of where each discovered item was located.

### 1.

Alyce suggests that the team place stakes in a circle around the tower, with the distance between the markers on each circle being equal to the radius of the circle. Javier likes this idea but says that by using this strategy, the number of markers needed would depend on how far away the circle is from the center tower. Do you agree or disagree with Javier’s statement? Explain.

### 2.

Show where the stakes would be located using Alyce’s method if one set of markers were to be placed on a circle

### 3.

After looking at the model, Veronica says they need to have more stakes if they intend to be specific with the location of the artifacts. Since most archeological sites use a grid to mark off sections, Veronica suggests evenly spacing

Javier suggests they record the location of each stake and its distance around the circle for the set of stakes on each circle. Veronica suggests it might also be interesting to record the ratio of the arc length to the radius for each circle.

### 4.

Help Javier and Veronica complete this table.

Inner Circle: | Outer Circle: | |||||

Location | Dist. from | Ratio of arc length to radius | Location | Dist. from | Ratio of arc length to radius | |

Stakes 0, 12 | ||||||

Stake 1 | ||||||

Stake 2 | ||||||

Stake 3 | ||||||

Stake 4 | ||||||

Stake 5 | ||||||

Stake 6 | ||||||

Stake 7 | ||||||

Stake 8 | ||||||

Stake 9 | ||||||

Stake 10 | ||||||

Stake 11 |

### 5.

What patterns might Alyce, Javier, and Veronica notice in their work and their table? Summarize any things you have noticed.

## Ready for More?

Justify why the patterns Alyce, Javier, and Veronica might observe on a

## Takeaways

Angles can be measured in degrees or radians.

The radian measure of an angle is:

We can locate points in the plane by providing two different types of coordinates:

Rectangular coordinates:

Polar coordinates:

## Vocabulary

- polar coordinates
- radian
- rectangular coordinate system
**Bold**terms are new in this lesson.

## Lesson Summary

In this lesson, we learned how to locate points in a plane using either rectangular or polar coordinates. We also revisited the definition of the radian measurement of an angle.

### 1.

Find the **exact** length of

#### a.

#### b.

### 2.

Write an equivalent expression by dividing out all common factors.