Locus

Locus is a fascinating topic, although not included in all Geometry curricula. The word locus can be interchanged with the words "set of points", so that the sentence "The set of points equidistant from the sides of an angle is the angle bisector." can be replaced by phrase "The locus of points equidistant from the sides of an angle is the angle bisector." in all cases.

At first, the concept of locus may seem complicated to students, as they have probably never heard the term before. In my classes, I often make a little joke of the explanation, showing them a picture of a locust (the insect) and explain that this is not a locus (the Geometry term).

Locust:

After we have gotten the name straight, I give the students some simple examples, using the words "set of points" and "locus". For example: "The locus of points equidistant from the parallel lines or "the set of points equidistant from parallel lines is a line, midway between the given lines, and parallel to both." At the same time, I draw the two given parallel lines on the board, and draw the locus as a dotted line, one point at a time.

The second example I give is a bit more complex, the locus of points equidistant from the sides of an angle. The complexity arises when we realize we need to define what we mean by the distance in this case. I draw an angle on the board, and a point inside the angle (see diagram below) and ask how we measure the distance from P, the point, to a side of the angle, ray AB.

A student says "Draw the distance with a ruler, from P to side AB." And I draw the following straight line, PE:

The class says "No, not like that, straight! At which point a lively discussion ensues as to what we mean by "distance" and why "straight" is not the word we should use. What finally results from the discussion is that the "distance" between a point and a line is defined and measured as the perpendicular from the point to the line. And so the need for clarity and precision in our mathematical terminolgy is reinforced; a mini-lesson within the lesson! And we end up with the following diagram:

Once we have that definition cleared up, I continue to draw points that are equally distant from ray AB and ray AC, and we end up concluding that the locus of points equidistant from the sides of an angle is a ray, the "angle bisector".

To reinforce the concept of locus, my students create "Flip Books". A "Flip Book" is an amusement that children often play with when they are young. An flip book is a book with a dozen or more pages having consecutive images of a sequence. The sequence could be a boy boucing a ball, or a snowman melting in the sunlinght, as shown in the child's flip book below. In this flip book, each page of the book shows a slightly different image of the snowman; in the first image he is a newly completed snowman, and in each successive image he has melted a bit more so that in the final image only his hat and broom remain.

Students can create their own "Flip Book" to illustrate a geometric theorem, by creating a series of geometric drawings, one on each page of a blank notepad. The notepad must have a glue binding; spiral or 3-ring notepads do not work as well. The example below shows the steps in creating a flipbook which demonstrates the locus definition of an angle bisector:

To create a "Geometry Flip Book", ask the students to bring a small pad of "sticky notes" to class, or just a small pad of paper from home (or the teacher can provide them). They should then draw a series of diagrams, such as the ones shown below. It is easier, by the way, to start at the back of the book with the last drawing, and than trace the "given" segments or circles on each sequential page, adding a point (or "dot") to the drawing on each page up through the stack which will then become the sequequental images in the resulting "movie".

Here is a brief example of the drawings that would be on 10 pages of a flipbook (one per page). It will take at least 20 pages to make a satisfactory flipbook. It is also helpful if the bottom 5 pages, and the top 5 pages are blank, to get the flipping motion started and ended.

The answer is: "The locus of points equidistant from the sides of an angle is a ray, the angle bisector."

Another interesting locus project is to create a "Locus Treasure Map". To introduce this project, let's take a look at a "real" treasure map, from the book Treasure Island, by Rober Louis Stevenson.

The geometric "treasure map" below is an example of how the idea of locus can be used creatively, in a geometry project for students. At first reading, this may not seem like a locus problem, but it certainly is! If we "translate" the words below into mathematical terminology, they would say "equidistant from two points (represented by Oak Tree 1 and Oak Tree 2) and 3 units from a given circle (represented by the Pond).

ANSWER: The locus of points equidistant from 2 points (the 2 Oak trees) is a line, the perpendicular bisector of the segment joining the 2 points. The locus of points a given distance from a circle is 2 circles, concentric with the given circle, one with radius the given distance inside the given circle, and the other with radius the given distance outside the given circle. The compound locus (the point or points that meet both conditions) is 4 points, ABC and D as shown.

Not only does this make the study of locus more interesting to the students, creating a locus treasure map gives them true "hands-on" experiences. And, long after they have forgotten much of the geometry that they studied in school, you can be sure that they will remember the fun they had designing their own locus treasure map! Many of my students have told me that this is one of the projects that they kept for many years, even after they had graduated from high school. And even those who did not keep it said that it was one of their fondest memories of math in high school.

"Enthusiasm is excitement with inspiration, motivation, and a pinch of creativity." Tom Krause