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…

Educators look for "teaching opportunities" within the curriculum to "bring the practice of knowing mathematics in school closer to what it means to know within the discipline". The need to emphasise disciplinarity--and the concomitant canons of logic, consistency, and connections--in the teaching of mathematics is in line with the proficiency…

Trigonometry is an integral part of the draft for the Senior Secondary Australian National Curriculum for Mathematics, as it is a topic in Unit 2 of both Specialist Mathematics and Mathematics Methods, and a reviewing topic in Unit 1, Topic 3: Measurement and Geometry of General Mathematics. However, learning trigonometric ideas is difficult for…

The senior school Mathematics syllabus is often restricted to the study of single variable differential equations of the first order. Unfortunately most real life examples do not follow such types of relations. In addition, very few differential equations in real life have exact solutions that can be expressed in finite terms. Even if the solution…

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…

The quest to find the equation of a catenary makes an ideal investigation for upper secondary students. In the modelling exercise that follows, no knowledge of calculus is required to gain a fairly good understanding of the nature of the curve. This investigation is best described as a scientific investigation--a "hands on" experience that…

In Years 9 and 10 of secondary schooling students are typically introduced to quadratic expressions and functions and related modelling, algebra, and graphing. This includes work on the expansion and factorisation of quadratic expressions (typically with integer values of coefficients), graphing quadratic functions, finding the roots of quadratic…

During the last decades of the twentieth century, changes in school curricula have resulted in proof-free or proof-lite curricula. In this article, the author argues that proof is, and should be seen to be, a central component in the school curriculum--from at least the middle of the primary years and upwards. He identifies proof with problem…

The transformations found in secondary mathematics curriculum are typically limited to stretches and translations (e.g., ACARA, 2010). Advanced students may find the transformation, twist, to be of further interest. As most available resources are written for professional-level readers, this article is intended to be an introduction accessible to…

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…

A problem given in the Australian Mathematics Competition for the Westpac Awards was stated as follows: With how many zeros does 2008! end? In this article, the author solves this problem, and provides further discussion on the related problems. These problems form a good model that helps students develop a logical thinking process toward problem…

Recently there have been several news items about possible cancer clusters in the Australian media. The term "cancer cluster" is used when an unusually large number of people in one geographic area, often a workplace, are diagnosed with cancer in a short space of time. In this paper the authors explore this important health issue using probability…

This article presents a mathematical solution to a motorway problem. The motorway problem is an excellent application in optimisation. As it integrates the concepts of trigonometric functions and differentiation, the motorway problem can be used quite effectively as the basis for an assessment tool in senior secondary mathematics subjects.…

The study of the number of intersection points of y = a[superscript x] and y = log[subscript a]x can be an interesting topic to present in a single-variable calculus class. In this article, the authors present a classroom presentation outline involving the basic algebra and the elementary calculus of the exponential and logarithmic functions. The…