This paper is presented in two parts. Through an example the first part takes up the issue of applying mathematics to situations that form part of the life context of students--the priority expressed in three curriculum statements presented. Then, noting the particular point in time--development of a National Curriculum for Mathematics--the second…

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…

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…

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…

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,…

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…

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…

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…

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…

Making connections between various representations is important in mathematics. In this article, the authors discuss the numeric, algebraic, and graphical representations of sums of absolute values of linear functions. The initial explanations are accessible to all students who have experience graphing and who understand that absolute value simply…

During the period 1729-1826 Bernoulli, Euler, Goldbach and Legendre developed expressions for defining and evaluating "n"! and the related gamma function. Expressions related to "n"! and the gamma function are a common feature in computer science and engineering applications. In the modern computer age people live in now, two common tests to…

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…