Group Symmetries Connect Art and History with Mathematics
Throughout history, different cultures have produced designs to be used as ornamentation, as part of ceremonies, and as religious symbols. Many of these designs are mathematical in nature, and their bases are often the transformations of reflection and rotation in the plane. The images form groupings that appear to have an underlying unity. Thus, history and art merge to create a medium through which students can study the concrete operations of reflection and rotation in the plane, as well as the more abstract concept of symmetry groups. The resulting patterns give students a sense of the potential for creativity inherent in mathematics.
Sum of Songs: Making Mathematics Less Monotone!
Songs in the mathematics classroom can be fun and functional; they can supply motivation and mnemonics. Students and teachers with even minimal musicianship can enrich their connections to mathematics by using existing songs, writing raps, or writing new words for existing tunes.
Analyzing Unstable Systems of Linear Equations
The Curriculum and Evaluation Standards for School Mathematics (NCTM 1989, 146) states, "Students who are able to apply and translate among different representations of the same problem situation or of the same mathematical concept will have at once a powerful, flexible set of tools for solving problems and a deeper appreciation of the consistency and beauty of mathematics". The Standards also states, "The connections between algebra and geometry are among the most important in high school mathematics" (1989, 147). This article shows how the phenomenon of instability in the solution of a system of linear equations can be analyzed both algebraically and geometrically.
The Spaghetti Problem Problem
Who would think that all this mathematics could come from a strand of spaghetti? In attempting to solve this problem we used such tools as algebra, geometry, probability, statistics, computer simulations, and calculus.
Algebra in the Service of Geometry: Can Euler's Line Be Parallel to a Side of a Triangle?
This investigation of Euler's line has become a regular and valued unit in my honors-geometry syllabus. It originated with an intelligent question from a curious student. Its geometric foundation comprises sophisticated Euclidean triangle geometry. Its solution requires plentiful but not excessively complicated algebra. It culminates in the discovery of a conic locus that can be verified by construction on a computer screen.