I-MATH

Triangles

A triangle is the simplest polygon, with the least number of sides. Congruent triangles are triangles that are the same size and the same shape. This is easily seen when one looks at them from a transformational point of view. In the following diagrams, each pair of triangles is congruent and "connected" by a transformation.

When teaching congruent triangles, I find it helps students in finding the congruent triangles, and the correspondence, if they have some experience with transformations, especially visually and on GSP. I do not like to use the complicated transformation terminology, but that is a personal opinion.

An interesting note: when we get to similar triangles, we will see that the fourth transformation, "dilation" is a very useful concept to use in teaching similarity concepts and correspondence.

When we begin congruent triangles, most textbooks begin with the definition of congruent triangles and then go immediately to congruence postulates and theorems, SSS, SAS, ASA etc. This is what I did for many years. But one day a student asked me WHY? Why does SAS make two triangles congruent and SSA does not? I don't remember what I said, but it didn't satisfy Eric's curiosity. The next day, he came in to class with 4 pages of geometric constructions, and a fascinating exploration of the idea. I was amazed at what he had explored, constructed, and explained! He had really shed a light on this topic, and illuminated it for himself and for me!

It changed the way I teach congruent triangles.

And it was the beginning of many changes in the way that I teach mathematics.

That question, "why" has become the basis and the focus of my teaching. I love it when the students ask "why?" and if they don't, themselves, ask - then I ask them!

If you want to know what Eric wrote, and explored, click on the link below and print out the first worksheet. Use a compass and straightedge, and do the constructions. Then answer the questions on the second worksheet that follows (these are the worksheets that I use in my own classes).

Click on the link below to explore geometric constructions further. This link will take you to a fascinating website called "MathWorld". This website is "a comprehensive and interactive mathematics encyclopedia intended for students, educators, math enthusiasts, and researchers."

Triangles provide many fascinating explorations. Isosceles and equilateral triangles have special types of symmetry, and therefore students will find interesting relationships between the medians, angle bisectors, and other special segments in these triangles. Hands-on exploration of these triangles and special segments, using GSP, will reinforce these properties.

Right triangles are another set of triangles with interesting properties. There is some very interesting information about this famous theorem, and the Pythagorean inequality as well, for students (and teachers!) at the following web page: http://mathforum.org/dr.math/faq/faq.pythagorean.html

After an introduction to the Pythagorean Theorem (often called "the most famous theorem in mathematics"), students may be interested in exploring the internet to discover some of the 300 proofs that have been published related to the Pythagorean Theorem, or information about the mysterious "Pythagorean Society" of ancient Greece. Here are a few links to get them started: