The author of this article, while recently working through some problem sets on determining volumes by triple integrals in cylindrical and spherical coordinate systems, realized that, although the textbook he was using included many interesting problems involving spheres, cylinders and cones and the increasingly complex solids that arose from the…

Direct sum decomposition of Abelian groups appears in almost all textbooks on algebra for undergraduate students. This concept plays an important role in group theory. One simple example of this decomposition is obtained by using the kernel and range of a projection map on an Abelian group. The aim in this pedagogical note is to establish a direct…

A "convex" polygon is one with no re-entrant angles. Alternatively one can use the standard convexity definition, asserting that for any two points of the convex polygon, the line segment joining them is contained completely within the polygon. In this article, the author provides a solution to a problem involving convex lattice polygons.

An exploration of nonconvex, crossed, and degenerate polygons (NCCDPs) are described with the help of examples with pedagogical tips and recommendations that are found useful when teaching the mathematical process of extending geometric patterns to NCCDPs. The study concludes that investigating such extensions with interactive geometry software…

Rene Descartes lived from 1596 to 1650. His contributions to geometry are still remembered today in the terminology "Descartes' plane". This paper discusses a simple theorem of Descartes, which enables students to easily determine the number of vertices of almost every polyhedron. (Contains 1 table and 2 figures.)

Prediction is a great skill to have in any walk of life: it can, in fact, save lives at times. While the two investigations posed in this column may not be that dramatic, they might just increase one's appreciation of some important connections between grids and rectangles and the divisors of numbers that appear in the dimensions of those…

This article represents an attempt to reconcile discussions of aspects of educational research with recent developments in complexity science. It is argued that current characterizations of and distinctions among research methodologies in education are potentially counterproductive, in large part because they tend to be defined against or in terms…

This study examined the effectiveness of different types of art activities in the reduction of anxiety. After undergoing a brief anxiety-induction, 84 undergraduate students were randomly assigned to color a mandala, to color a plaid form, or to color on a blank piece of paper. Results demonstrated that anxiety levels declined approximately the…

This paper discusses an investigative approach to problems concerned with approximations. Two problems are analysed in detail and both are inspired by their historical origins. The first examines an ancient Chinese formula for calculating the area of a segment. The second looks at Durer's construction for the trisection of an angle. It is argued…

By stretching the area under the curve x[superscript [alpha]] it is shown to be of the form x[superscript [alpha] + 1] p([alpha]). Geometry is then used to prove p([alpha]) = 1/([alpha] + 1).

Hexaflexagons (or flexagons) have a magic that has enthralled some of the greatest minds for over half a century. Their name comes from "flexible hexagon", which is the form in which they were first discovered by their inventor, Arthur Stone, an English graduate student at Princeton in 1939. In this article, the author talks about using flexagons…

In this note, we recall the convex (or barycentric) coordinates of the points of a closed triangular region. We relate the convex and trilinear coordinates of the interior points of the triangular region. We use the relationship between convex and trilinear coordinates to calculate the convex coordinates of the symmedian point of the triangular…

These materials are intended to provide meaningful mathematical experiences for pre-algebra students. These experiences emphasize the development of computational skills, mathematical concepts, and problem-solving techniques. This bulletin may be used as the basis for the second term of a one-year course, or for the second year of a two-year…

This is one of a series of geometry modules developed for use by secondary students in a laboratory setting. The thrust of this module is to introduce the student to transformations by having the student physically transform sets of points. Heavy use of manipulatives is made to aid the student in the transforming activity. Individual sections…

Presents a geometry exercise which is designed to help students create three-dimensional clay figures, think in three dimensions, and develop vocabulary. Reproducible worksheets are included. (PK)