Counting complete subgraphs of three vertices in complete graphs, yields combinatorial arguments for identities for sums of squares of integers, odd integers, even integers and sums of the triangular numbers.

If you want your students to graph a cubic polynomial, it is best to give them one with rational roots and critical points. In this paper, we describe completely all such cubics and explain how to generate them.

The last ten years have seen an explosion of research in the zero-divisor graphs of commutative rings--by professional mathematicians "and" undergraduates. The objective is to find algebraic information within the geometry of these graphs. This topic is approachable by anyone with one or two semesters of abstract algebra. This article gives the…

We explore how curvature and torsion determine the shape of a curve via the Frenet-Serret formulas. The connection is made explicit using the existence of solutions to ordinary differential equations. We use a paperclip as a concrete, visual example and generate its graph in 3-space using a CAS. We also show how certain physical deformations to…

This article, which is organized around a single, well-known algorithm for root extraction, presents a way of incorporating dynamical systems into the teaching of mathematics. Included are sample exercises using complex numbers and the computer where students have the opportunity to do some analysis on this algorithm. (KR)

Presented are exercises that demonstrate the application of standard concepts in the design of algorithms for plotting certain fractals. The exercises can be used in any course that explains the concepts of bounded or unbounded planar sets and may serve as an application in a course on complex analysis. (KR)

Discussed is the process of translating situations involving changing quantities into mathematical relationships. This process, called dynamical modeling, allows students to learn new mathematics while sharpening their algebraic skills. A description of dynamical systems, problem-solving methods, a graphical analysis, and available classroom…