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 $x$ - and $y$ -coordinates?

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 $12 meters$ from the center and a second set on a circle $18 meters$ from the center.

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 $12 stakes$ around each circle and using the coordinate grid to label the location of these stakes. The central tower is located at the origin and the first of each set of $12 stakes$ for the inner and outer circles is placed at the points $(12, 0)$ and $(18, 0)$ , respectively. Alyce also wants to make sure they record the distance around the circle to each new stake from these initial stakes. Your job is to determine the $x$ and $y$ -coordinates for each of the remaining stakes on each circle, as well as the arc length from the points $(12, 0)$ or $(18, 0)$ , depending on which circle the stake is located. Keep track of the method(s) you use to find these values.

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: $r = 12 meters$			Outer Circle: $r = 18 meters$
	Location	Dist. from $(12, 0)$ along circular path	Ratio of arc length to radius	Location	Dist. from $(18, 0)$ along circular path	Ratio of arc length to radius
Stakes 0, 12	$(12, 0)$	$0$	$0$	$(18, 0)$	$0$	$0$
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 $12 –m$ and $18 –m$ circle should also be true for any circle that is concentric with these two given circles.

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.

Retrieval

1.

Find the exact length of $A B$ in each right triangle.

a.

b.

2.

Write an equivalent expression by dividing out all common factors.

a.

$\frac{x^{2} - 6 x - 16}{8 - x}$

b.

$\frac{x^{2} - 2 x - 3}{1 - x}$