Park City Mathematics Institute 2006
Bowen Kerins and Ben Sinwell
Developing Mathematics: Some Applications of Geometric Thinking: While geometry is a wonderful field in its own right, geometric ideas are also useful tools for studying and understanding all kinds of phenomena inside and outside mathematics. Teachers in this course will look at some basic geometric habits of mind like studying continuous change and looking for things that don't change, and they'll apply these habits to a wide variety of situations. For example, we may look at
- Geometric optimization: what are some general-purpose tools for minimizing paths or maximizing area?
- Geometric invariants: what attributes of a figure stay the same when the figure is changed in some regular way?
- The fundamental theorem of algebra: what does it say, why is it fundamental, and why is it true?
- Fixed points: If you pick a positive number, take its square root, take the square root of that, and keep up the good work, what happens and why?
- What geometric thinking is common to all these and similar questions?
No background with any of these questions will be assumed. One of the tools we may use in the course is a dynamic geometry environment (like The Geometer's Sketchpad® or Cabri). Here, too, we'll start from scratch. By the end of week three, we promise some beautiful and exciting applications of geometry.
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