Chapter 15 - Locus

Locus is a very interesting topic in geometry, although not taught in all schools. A locus is defined as follows:

In mathematics, a locus (Latin for "place", plural loci) is a collection of points which share a property. The term 'locus' is usually used of a condition which defines a continuous figure or figures, that is, a curve. For example, a line is the locus of points equidistant from two fixed points or from two parallel lines. From Wikipedia, the free encyclopedia, at http://en.wikipedia.org/wiki/Locus_(mathematics)

Here are some examples: The locus of points equidistant from the sides of an angle is a ray, the angle bisector:

And the locus of points equidistant from two parallel lines is a third line, parallel to both given lines, and midway between them:

My students and I studied and constructed all the standard math problems on this topic. Then we did some very interesting locus projects. The students worked in groups of 2 or 3. The first project that I assigned was a "Treasure Map", a very interesting application of Loci. Here is the question that I asked in this assignment:

"You have found a scrap of paper which has clues describing where to find a buried treasure. The treasure map is torn, and parts of the clues are missing. Use your knowledge of locus and constructions to describe your work. If there is more than one possible place where the treasure might be buried, explain why and what the possible locations might be. If more information is needed, give an example of a clue that would make it possible to find the treasure more easily."

And here is the Treasure map:

In some years, when time allowed, I designed a treasure map that sent the students to different locations on campus for a "Locus Treasure Hunt", searching for a treasure that I had hidden. The instructions that I gave them to find the treasure were these:

"Begin your search at a point we'll call "point A", equidistant from the facing walls of Alexander Hall and Cooke Hall, and also equidistant from the two Monkeypod trees in front of Bingham Hall."

"From here, the next point (called point B) can be found 60 feet from point A and 120 feet from point A and 120 feet from the closest corner of Alexander Hall."

"The third point, C, is 120 feet from point B and equidistant from the facing walls or Pauhi Hall and Bingham Hall."

The students were intrigued, and enjoyed the opportunity to go outside instead of being stuck in the classroom!

In my writing-intensive math class, my students were expected to explain their work in writing. This is what Janine wrote: "We designed and drew this treasure map to be as realistic as possible. We think it is really cool looking, and we challenge you to find the treasure using our clues!"

Devon had this to say about the treasure map project: "This was the coolest project ever! I never would have thought that there could possibly be a connection at all between math and a treasure map! This was so cool, I even showed it to my Mom and Dad. Can we do more projects like this one, please?"

Another very interesting locus problem that we explored was one we called "Journey to the Center of a Triangle." There were 3 parts to this project, each one a different kind of "center". In this project, the students worked in groups of 3 or 4, and were asked the following question: "What is the center if a triangle?" Then 3 questions were asked, as follows:

"1) Imagine that you are a city planner. The 3 towns of Kai, Pua and Lani have pooled their funds and want to build a recreation center. Where would you build the rec centers so as to be fair to all 3 towns? Sketch, and explain your solution.

2) You are a sculptor, and have just completed a large metal mobile, in the shape of a triangle metal plate. The metal is the same thickness throughout. You are to hang this piece of artwork in the State Capitol so that it will be suspended with the triangular surface parallel to the ground. From what point should it hang? Sketch and explain your solution.

3) You are an architect. You have been asked to design an addition to a house for a client. Your client wants the addition to be circular. The site on which it is to be built is given. The house must fit within the setback lines and be attached to the existing house. Where would you place the center of the addition, to make it as large as possible?"

There were more details included in the project description, including diagrams. This project was very successful, and the students enjoyed working on it. This was a "group project", in which the students worked in groups of three or four. They each did the construction, made a conclusion, and then discussed their results with each other. They came up with very interesting conjectures as to what the center of a triangle was. They discovered that there is more than one type of center, and that each center has its own real-life application! To read more about this project and other projects too, please to the following web page, which I created on this topic: http://mathforum.org/~sanders/

I also asked the students to each keep a journal as they explored this project. In their journals, they should record their thoughts about the project, conjectures as to the solution (or solutions), and describe what ideas didn't work as well as those that did. In their reflections on this project, the students made very interesting comments. "When we started this project, we were looking for one definition of the center of a triangle. Whenever a point that might be considered the center fell outside we thought that it could not be the center. However, by the time we had finished experimenting we had found that the center of a triangle was different in different situations. It was true that if the center of a triangle fell outside they it could not work in some cases, like the sculptor, but in some cases (like the city planner) the center of the triangle might just be outside the triangle!" Lisa

Anthony made the following observation: "Even though it sure took a lot longer to find out what the center of a triangle is by all these constructions and exploring and guessing and arguing and finally agreeing, it actually was more interesting to figure it out for ourselves. And we sure learned more this way! I don't think I've ever spent so long on a math problem, but also I don't think I ever learned so much from one project either. If I remember nothing else from this year in geometry, I am sure I will remember the center of a triangle!"

The students really enjoyed the locus projects we did, and the Center of a Triangle project in particular. I am not sure whether or not this topic is covered in all geometry classes, but it certainly is an interesting one, and one of my favorites.

"We are usually convinced more easily by reasons we have found ourselves than by those which have occurred to others."

Blaise Pascal (1623-1662)