Assignment 4: Activity- Exploration of Medians
Many of us know that the medians of a given triangle can be found by connecting opposite vertices to the midpoint of each side of the triangle. This process can be achieved by using a compass, ruler, and pencil, but can also be achieved by using the mathematics program Geometers Sketchpad (GSP). Let's first begin our exploration of medians by walking through the steps necessary to construct the medians of any given triangle using GSP.
1. Place three points anywhere on the plane. These will be the vertices of your triangle.
2. Label these three points A, B, and C to identify your vertices.
3. Connect all three vertices to form the sides of your triangle, AB, BC, and AC.
4. Using the midpoint tool, under the Construct tab, find the midpoints of all three sides.
5. Now, connect all three midpoints to the opposite vertices of the triangle.
6. You can also label the three midpoints, using the text tool.
Through these easy steps, we have created the three medians of a given triangle. Looking at the last image, we can see that the three medians intersect at one given point inside the triangle. This point is called the Centroid and is usually labeled as G.
An exploration that would be extremely useful for students would be to figure out if the Centroid, the intersection of the triangle's three medians, can always be found inside the triangle. Students should be encouraged to explore all different kinds of triangles such as right triangles, isosceles triangles, and even equilateral triangles. Through this mini-exploration students will be able to discover more about the Centroid of triangles.
After allowing students to explore this question on their own or in groups, teachers could present the class with an illustration of a varying triangle and the location of the centroid as the triangle changes. On GSP teachers can easily vary the vertex A along a line to illustrate for students in one image the location of the Centroid as the triangle changes. Taking your already existing triangle ABC, teachers can merge the vertex A onto a line, l, and animate the vertex. This will automatically cause the vertex A to follow along the line, l, changing the shape of the triangle ABC. Students will be able to study the location of the Centroid, G, as the point varies. Screen shots of this animation can be seen below.
From the above snapshots of the animation on GSP students will be able to conclude that no matter the shape of the triangle, the Centroid will always be inside the triangle. As an additional exploration students could work on animating all three vertices on three different lines, l, m, and n to fully comprehend this mathematical principle about the Centroid. Merging vertex B onto a line m, and merging vertex C onto a line n, students could animate all three vertices. This exploration could be done after the teachers animation is shown, encouraging the students to try it on their own. Snapshots of what students might create can be seen below.
Link to GSP animation file
This activity/exploration is a great way to get students working hands-on to learn the mathematical concept of medians and the Centroid of a triangle. By working in small groups or pairs, students would even be able to learn from one another. In addition, through this exploration students would be required to discuss their ideas using proper mathematical terms, thus developing their mathematical vocabulary.