A Tangent to Conics
From a discrete mathematicians perspective, mathematics could be described as a connected graph. No matter what branch you are currently studying, it can be connected to another branch of mathematics. High school mathematics is presented to students as a tree. Students often see math as four branches that are connected, but there is no cycle. This investigation is an example of how teachers can help students see mathematics as a connected science and not as four separate branches of the “math tree”. This investigation provides discovery of those common threads in mathematics by bringing together the topics of triangles, circles, tangency, and conic sections.
Tangency is a concept that is covered lightly or not at all in high school geometry. This activity is a great way to teach this concept. Discovering that tangency can be found using the properties of triangles is a great way to intertwine some basic ideas of geometry like triangles and circles. GSP allows students the freedom to work backwards and discover the usefulness of the properties of isosceles triangles in order to create the tangent circles. The discovery and exploration that GSP allow are very important to improving the depth at which students can study geometry in high school. Most of the labor involved in using the compass and straightedge are gone and replaced with the labor of thought.
The introduction of conic sections through geometry is also an interesting perspective that this investigation brings to light. Often, conic sections are taught as a separate branch and students and teachers often miss the connection between geometry and conics. Another difficult concept for students to understand is a locus of points. This investigation uses both these difficult concepts and ties them to concepts that are easier for students to understand. Using prior knowledge to develop new knowledge is essential in giving students an appropriate foundation for higher mathematics.