The importance of mathematical reasoning is unquestioned and providing opportunities for students to become involved in mathematical reasoning is paramount. The open-ended tasks presented incorporate mathematical content explored through the contexts of problem solving and reasoning. This article presents a number of simple tasks that may be…

A Steiner chain is defined as the sequence of n circles that are all tangent to two given non-intersecting circles. A closed chain, in particular, is one in which every circle in the sequence is tangent to the previous and next circles of the chain. In a closed Steiner chain the first and the "n"th circles of the chain are also tangent…

In line with the latest positions of Gottlob Frege, this article puts forward the hypothesis that the cognitive bases of mathematics are geometric in nature. Starting from the geometry axioms of the "Elements" of Euclid, we introduce a geometric theory of proportions along the lines of the one introduced by Grassmann in…

This article shows how simple instruments may be used to construct angles of certain measures and applies this procedure to more detailed problems. A proof of the Pythagorean Theorem is given using these procedures. (CT)

This article, written by a high school junior, shows that there can never be more than two isosceles triangles having the same perimeter and area. (MK)

Establishes that the congruence criteria for polygons with more than three sides (such as ASASA for Quadrilaterals) are easily proved within the scope of the standard high school geometry course. Also argues that elegant applications of these criteria are more easily found once these new criteria are known. (PK)