The topic of loci, traditionally seen as counterintuitive and visually dificult for high school students, can become a rich, illustrative tool for use in exploring the meaning of a definition and in developing mathematical reasoning. Through the advent of dynamic geometry software in the high school classroom, the notion of locus takes on meaning beyond simply an object to be mastered without connection to the rest of mathematics; locus becomes a means of illustrating, of exploring, of defining, and of providing proof.
The following unit was originally written for use with pre-service middle school teachers at the University of Georgia. However, it includes notions and concepts that can be developed in elementary schools as well as topics and problems that can be expanded and enhanced for successful use in the high school classroom.
One of the questions that we often deal with in mathematics is the question of how to define an object. Questions that come up include:
What is it about a circle that makes it a circle?
What do you know about a circle?
Perhaps the most helpful question: What things in life are circle-shaped?
LP vinyl record
bicycle wheel
the path of a tethered horse running around a pole
We can think of the shape of something in two dimensions by thinking
about the path that a ladybug takes when walking along this thing. Consider
a ladybug on the outer edge of a record. Can we find something that is true
about her path at every point?
Let's look at her path: Click Here:
We notice something about the distance between the ladybug and the center of the record no matter where she is on the path.
What?
Is this enough to define a circle?
How would you create a circle with a compass?
Does this definition fit with your experience with the compass?
Think of our ladybug as a moving point.
The path that this moving point draws, given specific conditions,
is called a locus.
The word locus is derived from the same root as the word location. What kind of connections can you make between the two words?
We can also define a locus as a set of points which satisfy a specific condition or conditions.
Now, we can formulate a defining characteristic of a circle:
Comments to Teachers:
Through this first activity on Geometer's Sketchpad, students have the opportunity to explore what it means to be a circle. Such an exploration makes the introduction of the word locus quite natural, for the students begin to see the need for such a word before it enters the discussion.
Further explorations can be done at this point, depending on the age and level of the students. The subtle contrast between the two definitions of locus provides an opportunity to discuss and "act out" these differences. For example, we can have 20 students stand at equal distances from a pole, denoting locus as a set of points, and then denote the path of a moving point by having one student walk around the pole keeping her distance from the pole constant.
Once the students are comfortable with what a locus is, we can move on to the following exercises. Each one is designed to develop different aspects of locus and to portray its different uses.
We have taken an object and found the condition of the path which draws the figure. Now, lets go the other way. We will start with a set of conditions which dictate how our point can move, and from there we will determine what shape the path is.
Notice that we are now concerned with the distance between a point and a line. Hmm. That brings up the question, how do we measure the distance from a point to a line? Can we measure like this?
On the next GSP sketch ,
there is a set of parallel lines and a point between them.
This next sketch gives you
the locus of points in the plane that fit the above condition. Run the animation.
We see that the moving point leaves a trail behind as it moves.
What do you think the locus of points equidistant from the ends of a
segment looks like? Let's construct this locus. Click here:
Here's another situation:
Let's relate this idea of loci to triangles.
Given the segment AB, we want to construct triangle ABC.
,
and explore this sketch.
Notice that we are now interested in the intersection of two loci.
Why? What does this intersection reveal about lengths of sides of triangles?
Does this relate to your previous experience? How?
Note to Teachers:
Notice that we have moved from thinking about loci as a separate topic to loci and the intersection of loci as a tool for understanding other areas of mathematics. In the above exploration, students are encouraged to discover relationships between the lengths of the sides of a triangle, such as the sum of two sides must be greater than the third side.
The next exploration takes the intersection of loci to the idea of illustrating a counterexample for a SSA congruence theorem. This example is rich with mathematics: the notion of counterexample, the use of loci as a tool for illustration and understanding, and the geometry of the triangle.
The intersection of loci can help us to solve problems and answer mathematical questions. Whether the intersection is a line, a single point, or more than one discrete point can help us illustrate things like the following:
Let's look at congruence theorems for triangles:
We know that SAS is a theorem for triangle congruence, but what about SSA? Is there a difference?
Look at the next sketch: .
One more situation:
Given any three points that do not all sit on the same line,
find a circle that intersects all three points:
Think about how to approach this.
If you get too frustrated, of if you find yourself going around in circles (no pun intended), press the HINT button on the GSP screen.
Final note:
I hope that this introduction to loci has sparked ideas about how to use dynamic software to teach geometry, and I hope that it has struck a chord (again, no pun intended) with your intuition about the imporance of seeing geometry as a tool to help us understand our world. You may take these explorations and embellish and adapt them to fit your own classroom. Happy teaching!!
