The purpose of this paper is to explore new geometrical relations in the Pythagorean theorem, and open doors for new interpretations of the physical world. Pythagoras is included in secondary education around the world including in Australian Curriculum (ACARA, n.d.), and hence this paper will be of interest to all.

Students are often asked to plot a generalised parent function from their knowledge of a parent function. One approach is to sketch the parent function, choose a few points on the parent function curve, transform and plot these points, and use the transformed points as a guide to sketching the generalised parent function. Another approach is to…

Discovery learning has long been a part of mathematics teaching in the elementary and middle grades. Since the 1960s and 1970s, based on the work of Jean Piaget, Jerome Bruner, and others, helping students 'discover' or 'construct' their own understandings of mathematical concepts through well-designed activities facilitated by a competent teacher…

This article summarises activities that happened during the first three weeks of a fictitious high-school-level linear algebra section that used magic squares as a teaching tool to inspire students to further investigate the topics. The author has been working with students from high school and college levels for years, and although this situation…

The radius of curvature formula is usually introduced in a university calculus course. Its proof is not included in most high school calculus courses and even some first-year university calculus courses because many students find the calculus used difficult (see Larson, Hostetler and Edwards, 2007, pp. 870- 872). Fortunately, there is an easier…

The sudden perception of a connection between ideas is exhilarating. We might call these moments 'Aha!' moments. The purpose of this paper is to demonstrate how several different ideas can come together in Year 12 mathematics. The subject Further Mathematics in the Victorian Certificate of Education (VCE) is the Victorian adaptation of General…

Jagadguru Shankaracharya Swami Bharati Krishna Tirtha (commonly abbreviated to Bharati Krishna) was a scholar who studied ancient Indian Veda texts and between 1911 and 1918 (vedicmaths.org, n.d.) and wrote a collection of 16 major rules and a number of minor rules which have collectively become known as the "sutras of Vedic…

In this paper, an investment problem is investigated in terms of elementary algebra, recurrence relations, functions, and calculus at high school level. The problem comes down to understanding the behaviour of a function associated with the problem and, in particular, to finding the zero of the function. A wider purpose is not only to formulate…

Paolo Ruffini (1765-1822) may be something of an unknown in high school mathematics; however his contributions to the world of mathematics are a rich source of inspiration. Ruffini's rule (often known as "synthetic division") is an efficient method of dividing a polynomial by a linear factor, with or without a remainder. The process can…

The opinion of the mathematician Christian Goldbach, stated in correspondence with Euler in 1742, that every even number greater than 2 can be expressed as the sum of two primes, seems to be true in the sense that no one has ever found a counterexample. Yet, it has resisted all attempts to establish it as a theorem. The discussion in this article…

This paper outlines an approach to definitively find the general term in a number pattern, of either a linear or quadratic form, by using the general equation of a linear or quadratic function. This approach is governed by four principles: (1) identifying the position of the term (input) and the term itself (output); (2) recognising that each…

Very much like today, the Old Babylonians (20th to 16th centuries BC) had the need to understand and use what is now called the Pythagoras' theorem x[superscript 2] + y[superscript 2] = z[superscript 2]. They applied it in very practical problems such as to determine how the height of a cane leaning against a wall changes with its inclination. In…

The leap into the wonderful world of differential calculus can be daunting for many students, and hence it is important to ensure that the landing is as gentle as possible. When the product rule, for example, is met in the "Australian Curriculum: Mathematics", sound pedagogy would suggest developing and presenting the result in a form…

This article aims to illustrate a process of making connections, not between mathematics and other activities, but within mathematics itself--between diverse parts of the subject. Novel connections are still possible in previously explored mathematics when the material happens to be unfamiliar, as may be the case for a learner at any career stage.…

In the "Australian Curriculum," the concept of mathematical induction is first met in the senior secondary subject Specialist Mathematics. This article details an example, the Tower of Hanoi problem, which provides an enactive introduction to the inductive process before moving to more abstract and cognitively demanding representations.…