This material shows how to use basic techniques, principles of counting, and geometry to count squares on geoboards. The methods are elementary in that the proofs are easily conceptualized. A discussion of other approaches illustrates that easily stated problems may lead to very difficult and sophisticated methods. (MP)

Three examples are presented which involve constructions employing a compass, a centimeter ruler, and a calculator. Employing mathematical tools to support understanding and discovery of mathematical principles is viewed as being clearly a desire of mathematics educators, but it is felt such tools are not frequently used to their fullest. (MP)

A possible logical flaw based on similar triangles is discussed with the Sherlock Holmes mystery, "The Muskgrave Ritual." The possible flaw has to do with the need for two trees to have equal growth rates over a 250-year period in order for the solution presented to work. (MP)

Geoboards are viewed as excellent vehicles for developing intuitive understandings of various topics in transformational geometry. The activities described are designed to show pupils that when a triangle or parallelogram is subjected to a transformation called a shear, its area is not changed. Ways for finding areas are detailed. (MP)

A complete characterization of minimum conditions for congruence of quadrilaterals is presented. Convex quadrilaterals are treated first, then concave quadrilaterals are considered. A study of such minimum conditions is seen to provide some interesting and important activities for students. Only background in triangle congruence is necessary. (MP)

Ways of working with mirrors to explore the reflections of various objects are noted. Angles and formulas helpful in constructing interesting "reflection chambers" are provided. (MP)

Presented is an edited transcript of an interview with Dr. Caleb Gattegno exploring the teaching approaches used in a film of the same name. (MP)

Worksheets for duplication concerned with classification and measure of various geometric shapes are given. (MK)

This module, designed for use at the high school level as a four- to eight-hour topic, includes: the geometry of a sphere, the coordinate system used to describe points on the earth's surface, parallel and meridian sailing, and the solution of right spherical triangles. (Author/MK)

Although traditional instructional methods and textbooks mitigate against it, investigation can be a viable alternative strategy in the geometry classroom. Many theorems in geometry, such as Euclidean space, lend themselves easily to student experimentation. (Author/SJL)