Assignment #7

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.