Mack and Czernezkyj (2010) have given an interesting account of primitive Pythagorean triples (PPTs) from a geometrical perspective. In this article, the authors wish to enlarge on the role of the equicircles (incircle and three excircles), and show there is yet another family tree in Pythagoras' garden. Where Mack and Czernezkyj (2010) begin with…

For at least 2000 years people have been trying to calculate the value of [pi], the ratio of the circumference to the diameter of a circle. People know that [pi] is an irrational number; its decimal representation goes on forever. Early methods were geometric, involving the use of inscribed and circumscribed polygons of a circle. However, real…

Trigonometry is a well known branch of Mathematics. The study of trigonometry is of great importance in surveying, astronomy, navigation, engineering, and in different branches of science. This paper reports on the discovery of flaws in the traditional definitions of trigonometric ratios of an angle, which (in most cases) make use of the most…

In high school geometry courses, students are often given a prepackaged statement that they are asked to prove. In these situations, the process of writing proofs is being abridged, if not misrepresented. To provide her students with a more authentic experience in writing a proof, the author provided them with a summative project for which they…

A parabola is an interesting curve. What makes it interesting at the secondary school level is the fact that this curve is presented in both its contexts: algebraic and geometric. Being one of Apollonius' conic sections, the parabola is basically a geometric entity. It is, however, typically known for its algebraic characteristics, in particular…

In this article, the authors investigate how the difficulty of a "set shot" at goal varies with position on the field. By "set shot" they mean a player kicking for goal following a mark or free kick. The analysis begins by defining the "angle of opportunity" as a measure of difficulty for kicks at a given distance and angle from goal, where the…

This geometrical account of primitive Pythagorean triples was stimulated by a remark of Douglas Rogers on a recent paper by Roger Alperin (Alperin, 2005). Rogers, in commenting on this paper, noted that Fermat in the 17th century had posed a challenge problem on Pythagorean triples that suggested he knew how to construct a sequence of them,…

Geometry is an area in which Australian students performed particularly poorly on the 2007 Trends in International Mathematics and Science Study (TIMSS). One innovative area of recreational geometry that has rich potential to engage and challenge a wide variety of students is "impossible geometry." An impossible geometric object is a…

As teachers of first-year college mathematics and science students, the authors are constantly on the lookout for simple classroom exercises that improve their students' analytical and computational skills. In this article, the authors outline a project entitled "Predicting Precipitation in Darwin." In this project, students: (1) analyze and…

Being a mathematician, the author started to wonder if there are any theorems in mathematics that seem very ordinary on the outside, but when applied, have surprisingly far reaching consequences. The author thought about this and came up with the following unlikely candidate which follows immediately from the definition of the area of a rectangle…

The number "e" is one of those fascinating numbers whose properties are of special interest to mathematicians. In this article, the author aims to provide a method of introducing a visual concept of the number "e". These ideas are suitable for secondary school and undergraduate tertiary students. The main concept involves areas under curves.…

An old chestnut goes something like this. The surface area of a pond in the form of an annulus is required, but the only measurement possible is the length of the chord across the outer circumference and tangent to the inner circumference. It is a beautiful example of invariance. Invariance in mathematics usually refers to a quantity that remains…

Perhaps a business colleague threw out a challenge. The year: around 1200. The place: Pisa. The challenge: Calculate how many pairs of rabbits will be produced in a year, beginning with a single pair, if in every month each pair bears a new pair which becomes productive from the second month on. The question and its solution found its way into the…

This article presents a guided investigation into the spacial relationships between the centres of the squares in a Fibonacci tiling. It is essentially a lesson in number pattern, but includes work with surds, coordinate geometry, and some elementary use of complex numbers. The investigation could be presented to students in a number of ways…

In this article, the author describes the Lutterloh method of making dress patterns, which was developed in Germany in the 1930s. The underlying principle involves the modification of basic designs and it is claimed that it provides better fits for women of different sizes and shapes than do other commercially available patterns. The method is…