Locus

Introduction

A locus is the set of all points that satisfy a given condition or conditions. The plural of locus is loci (LOW-sigh). Let’s investigate this idea further.

Task

Using cardboard, string, colored labeling dots, 2 yardsticks, masking tape, and scissors, investigate the definition of a locus of points. Present your findings using a 3-dimensional model.

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string

Knot

Process

Complete the investigation:

1. Given a point A, what is the locus of points in a plane that are 2 feet from point A?

a. Cut a square piece of cardboard with side length of 3 inches. Cut a piece of string about 30 inches in length. Poke a small hole in the center of the cardboard. Label the small hole A. Thread the string through the hole. Tie a knot in the string and cu it so that it measures 2 fee from point A to the end of the string.

b. Tape the cardboard square securely to the floor with the knot under the cardboard. Pull the string to its full length along the floor. Mark the floor at the end of the string with a colored dot. The dot will represent a point that is 2 feet from point A.

c. Pick the string up and extend it in another direction. Mark the floor at the end of the string with another colored dot. Repeat this process until you have about 20 dots on the floor.

d. Imagine placing more and more dots on the floor. Describe the set of points.

e. Describe the locus of points in a plane at a given distance from a fixed point on the plane.

2. Given two parallel lines, what is the locus of points in a plane that are equidistant from the two parallel lines?

a. Tape two meter (or yard) sticks to the floor so that they are parallel and at least one foot apart. Make sure that the ends of the meter sticks are even.

b. Cut a piece of string the length of the distance between the meter sticks. To be accurate, stretch the string between the same markings on the sticks, say between 10 centimeters on each stick.

c. Find and mark the midpoint of the piece of string. Lay the string between the same markings on the two sticks. Place a colored dot on the floor at the midpoint of the string. Pick up the string and mark about 10 additional points with colored dots.

d. Imagine placing more and more dots. Describe this set of points.

e. Describe the locus of points in a plane that are equidistant from the two parallel lines.

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Resources

Basics 1

Basics 2

Basics 3

Loci

Compound Locus

Locus of Perpendicular Bisectors

Locus of Concurency

Complete the following tasks:

1. Identify the locus of points that satisfies each condition.

a. all points in a plane that are 2 feet from a given line

b. all points in a plane that are equidistant from two given points

c. all points in a plane that are equidistant from the sides of a given angle

d. all points in a plane that are equidistant from two intersecting lines

2. A compound locus is the intersection of loci that satisfies two or more conditions. Identify the compound locus of points that satisfies each set of conditions.

a. all points in a plane that are equidistant from the sides of a given 90˚ angle and 4 feet from the vertex of the angle.

b. all points in a plane that are equidistant from two given points and 10 centimeters from the line containing the two points.

c. All points on a coordinate plane that are 2 units from P(1, 3) and 1 unit from Q(1,6)

3. If the phrase “in a plane” was deleted from the two problems in this investigation, what would the locus be for those two problems? Provide a sketch

4. Teach the class some of the basics of what you have learned and design a quick problem for a volunteer to practice (Don’t pick a goof ball). DO NOT READ BORING PARAGRAPHS TO THE CLASS! Discuss it like you know what you are talking about! Use several visual aids to help the class understand the topic

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Evaluation

Web Quest Evaluation Rubric

Presentation
	70	80	90	100
Were the presenters enthusiastic about this presentation?	Presenters may as well been asleep they were so boring.	Presenters were faking their enthusiasm or could have been more lively.	Presenters were reasonably excited about their topic.	Presenters were excited enough to make the audience want to know more.
Did the presenters make eye contact or just read to the class?	Read to the class long boring paragraphs	Some reading with a small bit of eye contact	Maintained eye contact throughout at least half the presentation	Great eye contact for most of the presentation. Made the listener feel that they were part of the presentation.
Did the presentation flow well or were there long pauses of silence?	Several long pauses of silence, very unorganized	Some pauses of silence, unorganized in parts	Very few pauses of silence, 1 part unorganized	No pauses; presentation flowed well
Did all team members have a role in delivering the presentation?	One person gave the entire presentation.	One person gave most of the presentation.	Both persons had equal speaking parts but they did not seem to function as a team.	Both persons had equal speaking parts and functioned well as a team.

Project Design 70 80 90 100

Did the project address all necessary information?	Left off several items	Left off a 2-3 of items	Left off one item	All required items included
Did the project have an interesting look?	Plain, no color, 1 picture	A little color, 2-3 small pictures	Colorful, a few pictures, but not very pleasing	Very interesting, pleasing look, good pictures
Did the visual aids offer variety and maintain interest?	Aids were uninteresting and did not pertain to subject.	Aids were too small to see but were appropriate.	Aids were interesting but were all the same type.	Interesting visual aids that were large enough to see.
How much effort seemed to be put into this project?	Looked and sounded like it was all done last night.	A little effort was put into the poster but none into the presentation (or vise-versa)	A respectable amount of work was obvious.	An outstanding job that Mrs. Papizan will show for future reference.

Presentation Avg: Project Avg: Overall Score:

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Conclusion

Now that you have finished your quest on loci, hopefully you have a basic understanding. If you enjoyed this exploration, further research may lead you into a career in science using loci.