Abstract: Merrotsy, Peter The product rule refers to the derivative of the product of two functions expressed in terms of the functions and their derivatives. This result first naturally appears in the subject Mathematical Methods in the senior secondary Australian Curriculum (Australian Curriculum, Assessment and Reporting Authority [ACARA], n.d.b). In the curriculum content, it is mentioned by name only (Unit 3, Topic 1, ACMMM104). Elsewhere (Glossary, p. 6), detail is given in the form.

Abstract: Turner, Paul 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 (Boyer and Merzbach, 1989, p. 509). Yet, it has resisted all attempts to establish it as a theorem.

Abstract: Teia, Luis The grand architecture of nature can be seen at play in a tree: no two are alike. Nevertheless, there is an inescapable similarity that makes us identify a tree anywhere in the world. Just saying "tree" recalls words like green, root, leaves, still, strong, branches. The tree of primitive Pythagorean triples is no different. It has a root, or a beginning. It is rooted not on earth, but on the soil of our mind. It has branches that spring from that root as it grows with the action of nature and time. In this case, it is not the proverbial Mother Nature, but the human nature - a nature formed by the human interpretation of reality. The Pythagoras' tree presented by Berggren in 1934 has stood still and strong for almost a century, but probably it is even older. Its leaves are triples, and they grow throughout its branches. Ultimately, when one looks at the Pythagoras' tree, one looks at a 'tree'. The root is the triple (3, 4, 5). All branches and leafs emerge from, and are dependent, of this root. Like any tree, all it requires is a seed and soil, and all develops automatically. The architecture that defines the tree is present throughout the tree and is a reflection of the beginning - the root. In other words, the root (3, 4, 5) plus the same movement repeated over and over again creates the tree. In this paper, we will look at how this basic geometrical and mathematical movement governs the birth and growth of the Pythagoras' tree. 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.

Abstract: Fitzherbert, John 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. Although not described by Ruffini, the process can be generalised to non-linear divisors. Ruffini's rule can be further generalised to evaluation of derivatives at a given point, does not require technology and, most importantly, it is not beyond the reach of high school mathematics students to prove why the rule works for polynomials of a specific degree.

Abstract: Bardell, Nicholas S The cubic polynomial with real coefficients y = ax3 + bx2 + cx + d in which a ≠ 0, has a rich and interesting history primarily associated with the endeavours of great mathematicians like del Ferro, Tartaglia, Cardano or Vieta who sought a solution for the roots (Katz, 1998; see Chapter 12.3: The Solution of the Cubic Equation). Suffice it to say that since the times of renaissance mathematics in Italy various techniques have been developed which yield the three roots of a general cubic equation. A 'cubic formula' does exist - much like the one for finding the two roots of a quadratic equation - but in the case of the cubic equation the formula is not easily memorised and the solution steps can get quite involved (Abramowitz and Stegun, 1970; see Chapter 3: Elementary Analytical Methods, 3.8.2 Solution of Cubic Equations). Hence it is not surprising that with the advent of the digital computer, numerical rootfinding algorithms such as those attributed to Newton-Raphson, Halley, and Householder have become the solution of choice (Weisstein, n.d.).

Abstract: Jourdan, Nicolas; Yevdokimov, Oleksiy Proof by contradiction is a very powerful mathematical technique. Indeed, remarkable results such as the fundamental theorem of arithmetic can be proved by contradiction (e.g., Grossman, 2009, p. 248). This method of proof is also one of the oldest types of proof early Greek mathematicians developed. More than two millennia ago two of the most famous results in mathematics: The irrationality of 2 (Heath, 1921, p. 155), and the infinitude of prime numbers (Euclid, Elements IX, 20) were obtained through reasoning by contradiction. This method of proof was so well established in Greek mathematics that many of Euclid's theorems and most of Archimedes' important results were proved by contradiction.

Abstract: Padula, Janice According to the latest news about declining standards in mathematics learning in Australia, boys, and girls, in particular, need to be more engaged in mathematics learning. Only 30% of mathematics students at university level in Australia are female (ABC News, 2014) although paradoxically it would seem, the majority of lawyers in Victoria are female, a profession which requires a good grasp of language, rhetoric and logic (Merritt, 2014). So why not engage girls - with one of their strengths in early childhood (and later) - language and its acquisition (Padula and Stacey, 1990) and at the same time assist boys in their teenage years when their language development has (usually) caught up with girls (Goodman, 2012; Gurian, Henley and Trueman, 2001)? One of the ways to do this would be to teach pure mathematics - in the form of proofs.

Abstract: Stupel, Moshe; Segal, Ruti; Oxman, Victor We present investigative tasks that concern loci, which integrate the use of dynamic geometry software (DGS) with mathematics for proving the obtained figures. Additional conditions were added to the loci: ellipse, parabola and circle, which result in the emergence of new loci, similar in form to the original loci. The mathematical relation between the parameters of the original and new loci was found by the learners. A mathematical explanation for the general case, using the 'surprising' results obtained in the investigative tasks, is presented. Integrating DGS in mathematics instruction fosters an improved teaching and learning process.

Abstract: Hassani, Mehdi; Kippen, Rebecca; Mills, Terence Life tables are mathematical tables that document probabilities of dying and life expectancies at different ages in a society. Thus, the life table contains some essential features of the health of a population. We will examine life tables from a mathematical point of view.

Abstract: Kissane, Barry Although the term is often used to denote electronic devices, the idea of a 'technology', with its origins in the Greek techne (art or skill), refers in its most general sense to a way of doing things. The development and availability of various technologies for computation over the past forty years or so have influenced what we regard as important in mathematics, and what we teach to students, given the inevitable time pressures on our curriculum. In this note, we compare and contrast current approaches to two important mathematical ideas, those of square roots and of integrals, and how these have changed (or not) over time.