Abstract: Yevdokimov, Oleksiy The purpose of the section is to supply teachers and students with a selection of interesting problems. In this issue we invite readers to deal with points and lines problems. For many of them symmetry plays a key role. The first problem below is one of the most famous examples from this topic. It is attributed to Sir Isaac Newton. All other problems have historical flavour too and all problems are from Dudeney (1958). Despite the simple problem statements they have, finding a direct and certain way towards a solution is often far from simple!

Abstract: Dion, Peter; Ho, Anthony An interesting mathematical problem, 'The Dog Problem', is easily stated. Four dogs each at the corners A, B, C and D of a large square of side length L start walking directly towards the dog on his right (Dog A to B, B to C, C to D, and D to A) all at the same speed. They do not to anticipate the target dog's movement and attempt a short cut. The dogs continually change direction, and we know by symmetry that they will spiral around each other and to meet in the middle of the square. What is the path length walked by each dog?

Abstract: Zelenskiy, Alexander S In a recent issue of Australian Senior Mathematics Journal there has been published an interesting article by Galbraith and Lockwood (2010). In that article the problem of finding the most favorable points for a shot at goal in Australian football is considered from different points of view. A similar problem was considered by Galbraith and Stillman (2006) in the context of soccer.

Abstract: Stoessiger, Rex When we examine numbers in the newspaper or magazines we might expect that their first digits are just as likely to be 8 or 9 as a 1 or a 2 and we might assume that each of the nine digits (zero is not used as a first digit of course) will occur 1/9 of the time. Unexpectedly this turns out to be untrue in many situations.

Abstract: Obara, Samuel The National Council of Teachers of Mathematics (NCTM, 1989, 2000) and the new Australian Curriculum: Mathematics for senior secondary (ACARA, 2010) highlight the importance of teaching spatial reasoning as early as preschool when mathematics is introduced. Studies have shown that there is a relationship between spatial abilities and mathematical achievement (Burnett, Lane, and Dratt, 1979; Casey, Nuttall, Pezaris, and Benbow, 1995; Geary, Saults, Liu, and Hoard, 2000). Activities that enhance spatial reasoning skills are invaluable to, and should be encouraged in, classroom instruction. Casey, Andrews, Schindler, Kersh, Samper, and Copley (2008) define spatial skills as "the ability to think and reason through the comparison, manipulation, and transformation of mental pictures" (p. 270). In this article I present an activity (Aichele and Wolfe, 2007, p. 11) to pre-service high school teachers. Though the activity seems simple, it can be challenging to students who have not been exposed to spatial tasks. The activity goal is to create a square from a given polygon by making one straight cut so that two pieces, when put together without overlap or gaps.

Abstract: Kenney, Rachael; Kastberg, Signe The beautiful history of the development of logarithms (Smith and Confrey, 1994), coupled with the power of the logarithmic function to model various situations and solve practical problems, makes the continued effort to support students' understanding of logarithms as critical today as it was when slide rules and logarithmic tables were commonly used for computation. They continue to play an important role despite the fact that calculators are now used for many computations involving logarithms: logarithmic scales can increase the range over which numbers can be viewed in a meaningful way. As described in the senior secondary curriculum, logarithmic scales are used regularly in astronomy, chemistry, acoustics, seismology, and engineering and students should be able to "identify contexts suitable for modelling by logarithmic functions" and be able to "use logarithmic functions to solve practical problems (Australian Curriculum and Assessment Reporting Authority [ACARA], 2012)".

Abstract: Foster, Colin A post-16 mathematics student, Peter (not his real name), asked me, "Does Pythagoras' theorem generalise to three dimensions?" I said, "Yes." But it turned out that we had different things in mind.

Abstract: Galbraith, Peter Recognition that real world problem solving expertise is a major educational goal continues to be reinforced internationally, at least officially, through documents that set specific goals for the learning of mathematics - as in the following: Mathematical literacy is defined in PISA as the capacity to identify, understand and engage in mathematics, and to make well-founded judgements about the role that mathematics plays in an individual's current and future private life, occupational life, social life with peers and relatives, and life as a constructive, concerned and reflective citizen. (OECD, 2001, p. 22).

Abstract: Padula, Janice When hoping to initiate or sustain students' interest in mathematics teachers should always consider relevance, relevance to students' lives and in the middle and later years of instruction in high school and university, accessibility. A topic such as the mathematics behind networks science, more specifically scale-free graphs, is up-to-date (think of the recent award-winning film, The Social Network, Spacey et al., 2010), highly relevant to students' lives, and accessible. This article illustrates how mathematicians and scientists work together collaboratively when applying mathematics and developing new scientific theories, and describes ways of teaching mathematics in a highly meaningful, real-world context with a topic which is widely acknowledged as being important mathematically and scientifically.

Abstract: Tay, Eng Guan; Leong, Yew Hoong We 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" (Lampert, 1990, p. 29). We should also be concerned about avoiding the scenario where instead of capitalising on such teaching opportunities, a wrong treatment of an unavoidable mathematical notion (such as a0) would leave an indelible impression on students that mathematics has strange inscrutable rules and devices or even more ironically mathematics is illogical.

Abstract: Bhattacharjee, Pramode Ranjan The theoretical study of Physical Science is based on many conventions. For a systematic study, one is to follow the normal conventions which have already earned international recognition. Now, what about those conventions which are not at all realistic and which have no resemblance with problems in real world' It is a high time to think of such conventions and to get rid of them with alternative replacements in compliance with real life situations or to deal with a problem where such a convention is used by alternative treatment so as to establish a bridge between theory and practice. Such attempts have been made in Bhattacharjee (2002, 2008, 2011, 2012). A means of getting rid of the most misleading sign convention of geometrical optics has been offered in Bhattacharjee (2002, 2012). Also considering the generalized equations of motion with the approach offered in Bhattacharjee (2008, 2011), one can easily do away with the need of using the ambiguous sign convention in regard to distance measurement which has been in regular use in solving typical problems of elementary mechanics for many years.

Abstract: Bardell, Nicholas S The roots of the general quadratic equation y = ax2 + 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 such as to render the discriminant negative. In this case, a simple two-dimensional x-y plot of the quadratic equation does not reveal the location of the complex conjugate roots, and the interested student might well be forgiven for asking, "Where exactly are the roots located and why can't I see them'" In the author's experience, this sort of question is hardly ever raised - or answered satisfactorily - in school Years 11 or 12, or in undergraduate mathematics courses. The purpose of this paper therefore is to provide a clear answer to this question by revealing the whereabouts of the complex roots and explaining the significance of the conjugate pairing.

Abstract: Yevdokimov, Oleksiy As usual, the purpose of this section is to supply teachers and students with a selection of interesting problems. In this issue we invite readers to deal with determinants that remain a core topic of the first course on linear algebra at the undergraduate level.

Abstract: Stupel, Moshe The notion of periodicity stands for regular recurrence of phenomena in a particular order in nature or in the actions of man, machine, etc. Many examples can be given from daily life featuring periodicity: day and night, the weekdays, the months of the year, the circulation of blood in our body, the function of the heart, the operation of a clock, the natural circulation of water, crop rotation, and tree crop rotation. In astronomy there are many periodical phenomena: the revolution of planets around the sun, the Solar Cycle and the Lunar Cycle, the cycle of intercalation in a period of 19 years: "Every 19 years of which 7 are intercalary and 12 are regular, is called a Period" (Maimonides, Sanctification of the Month 6). Mathematically the meaning of periodicity is that some value recurs with a constant frequency.

Abstract: Dion, Peter; Ho, Anthony For at least 2000 years people have been trying to calculate the value of pi, the ratio of the circumference to the diameter of a circle. We know that pi is an irrational number; its decimal representation goes on forever. Early methods were geometric, involving the use of inscribed and circumscribed polygons of a circle. However, real accuracy did not come until the use of infinite series techniques, in which one can, by calculating more and more terms, obtain smaller and smaller corrections all leading to a precise value. Such series go on forever, so the limitation on accuracy is how much time one is willing to devote to the task and how fast the computer is, but mainly how quickly your series converges.

Abstract: Bernhart, Frank R; Price, HLee Mack and Czernezkyj (2010) have given an interesting account of primitive Pythagorean triples (PPTs) from a geometrical perspective. We wish here to enlarge on the role of the equicircles (incircle and three excircles), and show there is yet another family tree in Pythagoras' garden.

Abstract: Balasooriya, Uditha; Li, Jackie; Low, Chan Kee For any density function (or probability function), there always corresponds a cumulative distribution function (cdf). It is a well-known mathematical fact that the cdf is more general than the density function, in the sense that for a given distribution the former may exist without the existence of the latter. Nevertheless, while the density function curve is frequently adopted as a graphical device in depicting the main attributes of the distribution it represents, the cdf curve is usually ignored in such practical analysis.

Abstract: Watson, Jane; Chance, Beth Formal inference, which makes theoretical assumptions about distributions and applies hypothesis testing procedures with null and alternative hypotheses, is notoriously difficult for tertiary students to master. The debate about whether this content should appear in Years 11 and 12 of the Australian Curriculum: Mathematics has gone on for several years. If formal inference is not included in Years 11 and 12, what statistical content, if any, should there be' Should students continue learning more data handling skills, which are a feature of the F-10 curriculum (Australian Curriculum, Assessment and Reporting Authority [ACARA], 2011)' Perhaps the focus should be on procedural aspects, such as correlation and lines of best fit, employing principles from calculus. Or perhaps the curriculum should drop statistics and focus on the more complex theoretical aspects of probability.

Abstract: Yevdokimov, Oleksiy The purpose of the section is to supply teachers and students with a selection of interesting problems. In this issue we invite readers to look back as far as more than one hundred years ago and work on a selection of problems that had been proposed to students and teachers at that time. The tradition of publishing problems in periodicals was well established in Europe by the beginning of the nineteenth century. The first two problems appeared in The Mathematical Visitor (Martin, 1881) that was published in the seventies of the nineteenth century in the US. The next two problems are from the Russian periodical Vestnik (Newsletter) of the Experimental Physics and Elementary Mathematics (Tsimmerman, 1886-1917). The final problem comes from the Italian mathematical magazine Il Pitagora where it appeared in 1897.

Abstract: Valahas, Theodoros; Boukas, Andreas 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 equations and relating these to horizontal axis intercepts of corresponding graphs. For example, given the quadratic expression x2 - 5x + 6 = 0, students would attempt to factorise by seeking two integers m and n such that m + n = -5 and m x n = 6, and thus express x2 - 5x + 6 = 0 in the form (x + n) (x + n).

Abstract: Staples, Ed The Catenary is the curve that an idealised hanging chain or cable assumes when supported at its ends and acted on only by its own weight... The word catenary is derived from the Latin word catena, which means "chain". Huygens first used the term catenaria in a letter to Leibniz in 1690 Hooke discovered that the catenary is the ideal curve for an arch of uniform density and thickness which supports only its own weight. (Wikipedia, catenary).

Abstract: Shriki, Atara 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. According to the intended curriculum in mathematics, in 9th grade, students should learn about quadratic functions, including simplification techniques. In the 10th grade, they are expected to solve a wide range of quadratic equations, construct graphs of parabolas, and connect algebraic and graphical representations of quadratic functions.

Abstract: Padula, Janice Have your students ever wondered what mathematics is, and exactly what it is that a mathematician does' In this paper different schools of thought are discussed and compared to encourage lively classroom discussion and interest in mathematics for high achieving Form 12 students and first (or higher) year university students enrolled in a mathematics degree program. (The topic also fits well under the rationale for Queensland Senior Mathematics B Syllabus, Queensland Studies Authority, 2008.) In particular the work and views of two mathematicians, Kurt G del (1931) and Ian Stewart (1996), mathematician and professor Reuben Hersh (1998) and university lecturer, researcher and writer Robyn Arianrhod (2003) are used to illustrate different views of mathematics. Two documentaries are suggested for viewing by students: Dangerous Knowledge, relating the work and place of Godel in the history and foundations of mathematics (Malone and Tanner, 2008), and How Kevin Bacon Cured Cancer (Jacques, 2008) which illustrates how mathematicians and scientists work together developing and applying mathematics.

Abstract: Fuentes, Sarah Quebec The promotion of proof as a process through which mathematics knowledge and understanding have been constructed will not necessarily motivate students, though, unless they believe that they are participating in meaningful mathematical discovery (Vincent, 2005, p. 94).

Abstract: Braiden, Doug 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 (Jordan and Smith, 2007, p. 2). Even if the solution can be found exactly it may be far too difficult to be clearly articulated such as those that form an infinite series. In either case, these real life problems are well beyond the scope of the secondary student to solve.

Abstract: Brown, Jill; Stillman, Gloria During July this year, the Australian Catholic University (ACU) hosted the Fifteenth Biennial Conference on the Teaching of Mathematical Modelling and Applications (ICTMA15). Teachers, mathematicians and mathematics educators from around the world descended on the Melbourne campus of ACU.

Abstract: Wongapiwatkul, Pimpalak; Laosinchai, Parames; Panijpan, Bhinyo An instruction method that uses Earth geometry and the great circle to enhance students' understanding of trigonometric ideas is described. Three visual aids to help students visualise the geometry of the Earth are presented, together with a method of calculating the shortest distance between two points on the surface of the Earth, that is, the great-circle distance.

Abstract: Sokolowski, Andrzej; Rackley, Robin An argument is presented that applying trigonometric functions to model harmonic motion offers a rich scientific context to exercise mathematical modelling through inductive inquiry in trigonometry classes as well. The paper describes an activity that uses a physics simulation called 'Wave on a String' created by the PhET Interactive Simulations Project at Colorado University at Boulder, and is available on the internet. The activity's cognitive learning objectives fit into the scope of the proposed Australian mathematics curriculum that highlights the development of the skills of mathematical modelling, data collection, and analysis.

Abstract: Padula, Janice The main elements of Kurt Godel's proof of the 'incompleteness' of a formal system such as Bertrand Russell and A.N. Whitehead's 'Principia Mathematica' are discussed together with ways to address potential difficulties for students. The article recommends the study of the logical-skeletal structure before students attempt the proof itself and describes how students may be introduced to the proof with a documentary highlighting its importance. In addition, the paper evaluates two books for the 'general reader', by E. Nagel and J.R. Newman and by Torkel Franzen, and a description of the proof's logical core written in clear English by Solomon Feferman.

Abstract: Griffiths, Martin A workshop for undergraduates and students in Years 11 and 12 on generating 'random' positive integers is described. The workshop explores a range of topics, from statistics to pure mathematics, including aspects of probability, random variables, and Fourier series. The teaching and learning that took place in the workshop is examined in the context of the 'Australian Senior Secondary Mathematics Curriculum'.

Abstract: Ghosh, Jonaki B A laboratory module on Fourier series and Gibbs phenomenon undertaken by Year 12 students using Mathematica is described. Paper and pencil methods were used to help students understand calculations while Mathematica added meaning to the calculations by providing graphical and numerical representations. Students were then able to focus on the behaviour of the graphs and the functions, which enabled them to visualize Gibbs phenomenon.

Abstract: Yevdokimov, Oleksiy Some interesting problems for teachers and students to solve are discussed. A few famous geometric inequalities for finding new proofs are highlighted.

Abstract: Mack, John; Czernezkyj, Vic The creation of a specific 'infinite ascent' from a single Primitive Pythagorean Triple (PPT) base and then a 'finite descent' from the PPT base is discussed. Some of the different challenges posed by Fermat as well as solutions for the same are highlighted.

Abstract: Hwang, Daniel The definition of 'twist', a transformation and its various applications are discussed. Some of the different classroom tasks to help illustrate the twist transformation are highlighted.

Abstract: Gough, John The need for proof to be and should be seen as a central component in school curriculum is discussed. Some of the various examples of proof and their application is highlighted.

Abstract: Galbraith, Peter; Lockwood, Terry The difficulty of a set shot in AFL football and how a set shot at goal varies with position on the field is discussed. A linking of the mathematics and modeling involved, to educational settings in terms of curriculum concerns and possible teaching approaches are highlighted.

Abstract: Kabael, Tangul Uygur The key aspects and features of the framework of 'three worlds of mathematics' that describes three levels of mathematics understanding are discussed. The teacher should design teaching activities that can help students relate prerequisite concepts with various chain rule applications.

Abstract: Farmer, Jim The key aspects and features of the Markov chain model used in an exercise to predict rainfall data for Darwin airport are discussed. The various factors that led to the failure of the model in such an exercise are highlighted.