The roots of the general quadratic equation y = ax[superscript 2] + bx + c (real a, b, c) are known to occur in the following sets: (i) real and distinct; (ii) real and coincident; and (iii) a complex conjugate pair. Case (iii), which provides the focus for this investigation, can only occur when the values of the real coefficients a, b, and c are…

One of the author's undergraduate students recently asked him whether it was possible to generate a random positive integer. After some thought, the author realised that there were plenty of interesting mathematical ideas inherent in her question. So much so in fact, that the author decided to organise a workshop, open both to undergraduates and…

The study of Kurt Godel's proof of the "incompleteness" of a formal system such as "Principia Mathematica" is a great way to stimulate students' thinking and creative processes and interest in mathematics and its important developments. This article describes salient features of the proof together with ways to deal with potential difficulties for…

In this article, the authors present a lesson whose goal is to utilise a scientific environment to immerse a trigonometry student in the process of mathematical modelling. The scientific environment utilised during this activity is a physics simulation called "Wave on a String" created by the PhET Interactive Simulations Project at Colorado…

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 Volume 23(2) of the "Australian Senior Mathematics Journal," Boncek and Harden present an exercise in fitting a Markov chain model to rainfall data for Darwin Airport (Boncek & Harden, 2009). Days are subdivided into those with precipitation and precipitation-free days. The author abbreviates these labels to wet days and dry days. It is…

In this sequence 1/1, 7/5, 41/29, 239/169 and so on, Thomas notes that the sequence converges to square root of 2. By observation, the sequence of numbers in the numerator of the above sequence, have a pattern of generation which is the same as that in the denominator. That is, the next term is found by multiplying the previous term by six and…

The United States of America is a large and diverse country; nevertheless, high schools have a lot in common across the nation. In particular, mathematics teaching in American high schools, while not stereotyped, has some accepted norms which would be seen as unusual here in Australia. In order for the Australians to learn from the Americans, as…

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…

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.…

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…

Constructivist pedagogies are generally not considered to support the teaching of mathematics for externally assessed examination-based courses. In large part, teachers have believed that such approaches are inefficient in covering a set syllabus. This article summarises the author's learning journey in Year 12 mathematics in 2004 where attempts…

Many students experience considerable difficulties when they learn and then attempt to construct and communicate proofs of conjectures using the principle of mathematical induction. Although research on the pedagogy of mathematical induction has gained only occasional attention since the 1970s, there has been an increasing interest in this field…

In this note, we introduce and discuss convolutions of two series. The idea is simple and can be introduced to higher secondary school classes, and has the potential of providing a good background for the well known convolution of function.