Many ways exist to engage students without detracting from the mathematics. Certainly some are high-tech options, such as video games, online trivia sites, and PowerPoint® presentations that follow the same model as Jeopardy; but sometimes low-tech options can be just as powerful. One exciting way to connect with students is by incorporating…

The purpose of mathematics competitions, and in our case the South African Mathematics Olympiad (SAMO), is to promote problem solving skills and strategies, to generate interest and enthusiasm for mathematics and to identify the most talented mathematical minds. SAMO is organised in two divisions--a junior and a senior division--over three rounds.…

The proof is given that, if three equilateral triangles are constructed on the sides of a right triangle, then the sum of the areas on the sides equals the area on the hypotenuse. This is based on one of the hundreds of proofs that exist for the Pythogorean theorem. (MP)

On a triangular grid of equilateral triangles an analytical procedure is presented for solving puzzles that require one to measure out a specified amount of liquid from containers of various volumes. (JP)

Problems involving the generation and counting of isosceles triangles interior to a given isosceles triangle are described. (SD)

Some ways of thinking and acting geometrically are described which are related to the approach used by ancient humans. The focus is on intuitive geometric imagery, an attempt to resurrect a way of describing possible viewpoints of geometry outside of those commonly accepted. (MP)

Activities designed to lead pupils through the process of using the basic measuring and drawing devices of geometry are presented and move to the discovery of several surprising generalizations about arbitrary triangles. (MP)

The material examines areas generated by combinations of: (1) Circles and Triangles; (2) Closely Packed Circles; and (3) Overlapping Circles. The presentation looks at examples of certain areas and at logical ways to generate the necessary algebra to clarify the problems and solve general cases. Ideas for extension are provided. (MP)

People are noted as intrigued for centuries by interplay between algebra and geometry with many important advances viewed to have come down through some sort of linking of the two. Examples are given of advantages to learning and discovery that can be found in an investigative approach combining them. (Author/MP)

A problem involving the search for an equivalence class of triangles is viewed to provide several exciting and satisfying moments of insight. After solving the original problem, there is a brief discussion of a slight variation and several notes regarding related theorems and ideas. References for additional exploration are provided. (MP)

A mult tile is a set of polygons each of which can be dissected into smaller polygons similar to the original set of polygons. Using a recursive LOGO method that requires solutions to various geometry and trigonometry problems, dissections of mult tiles are carried out repeatedly to produce tile patterns. (MDH)

From the equality of the ratios of the surface areas and volumes of a sphere and its circumscribed cylinder, the exploration of theorems relating the ratios of surface areas and volumes of a sphere and other circumscribed solids in three dimensions, and analogous questions relating two-dimensional concepts of perimeter and area is recounted. (MDH)

A series of problems concerning a geoboard with "holes" is suggested. (SD)

Elucidated is the relationship among three threads of mathematical investigations: Kirkman's schoolgirl problems, finite geometries, and Euler's n-square officer problems. (JP)