I-MATH
Triangle Properties
I-MATH
Triangle Properties
In many geometry books, the properties of
triangles, and their medians, angle bisectors, altitudes, and other
special lines in triangles, are listed as theorems. This topic
provides a wonderful opportunity for interactive, discovery
activities for students. In my classes, my students work in groups of
3 students per computer. They do each a number of activities,
exploring the properties of triangles using Sketchpad. If you don't
have sketchpad, there are many other possibilities for hands-on
investigations, using string, paper cutouts or "Tinker Toys". It is
best to have them to these activities before they read the theorems
in the textbook on these topics. This gives them opportunities for
hands-on experimentation with medians, angle bisectors.
I believe that when they discover the properties on their own, and help each other and discuss their discoveries, they understand the properties rather than just memorizing them, and not only does this help them to remember the properties, but they have a kind of "ownership" of the information, and their understanding is an active process.
Is the following statement always true, sometimes true, or never true? Explain your answer, and give examples in words and sketches. A median is perpendicular to a side of a triangle.
Student response: A median is sometimes perpendicular to a side of a triangle. It is not perpendicular to any side in a scalene triangle, the median to the base is perpendicular to the base in an isosceles triangle, and all 3 medians are perpendicular in an equilateral triangle, as seen in the following sketches:
Many students have told me that when they are trying to remember a property - for example, whether or not a median of a triangle is perpendicular to a side, and if so, in what kind of triangle - they picture themselves "dragging" a vertex of a triangle and visualize the median, as they did during their hands-on exploration with Geometer's Sketchpad. Other students did a quick series of paper and pencil sketches of triangles, and answered the question from their "doodles". This process is far superior to reading a theorem in a textbook and trying to just memorize it!
I developed a project that I call "Journey to the Center of a Triangle". My students did this project after they had explored triangles, but before they had completed their investigations. If they had already done all of the triangle activities, there would be nothing left to discover, and it would spoil this wonderful discovery activity!
Click on the next link, and it will take you to a printable page with the worksheet that accompanies this activity.
Journey to the Center of a Triangle has been such a successful project with my classes that I created a website for this project, to share our discoveries with other students and teachers. You should not direct your own students to this website until after they have completed their own project, or it will give them all the answers!