Learning and Teaching Mathematics  latest Issue
Volume 2016, Issue 21, January 2016

From the editors
Authors: Duncan Samson and Marcus BizonySource: Learning and Teaching Mathematics 2016 (2016)More LessDear LTM readers
In the first article of LTM 21, James Russo introduces a simple yet engaging mathematical board game to enhance learners' mental computation skills. In the second article in this issue, Debbie Stott builds on her experiences of using arrays to develop learners' conceptual understanding of multiplication and division. The third article, by Richard Kaufman, highlights an interesting relationship inherent in Pythagorean Triples, while in the fourth article Nicholas Kroon explores the beauty of mathematical proof. Michael de Villiers then presents two examples of appropriate generalisation activities that go beyond the normal curriculum, while in the sixth article YiuKwong Man illustrates an alternative method of finding the greatest common divisor of two or more numbers.

3inaRow Magic Dice
Author James RussoSource: Learning and Teaching Mathematics 2016, pp 3 –5 (2016)More Less'3inarow Magic Dice' is a simple, engaging and mathematically meaningful activity designed to reinforce counton and countback strategies, enhance pupils' mental computation skills and expose pupils to patterning and placevalue concepts as they navigate the 100's chart. The game is primarily aimed at pupils around 7 to 8 years of age. However, with minor modifications it can be played by younger or older pupils as well.

Using arrays for multiplication in the intermediate phase
Author Debbie StottSource: Learning and Teaching Mathematics 2016, pp 6 –11 (2016)More LessIn their book Maths for Mums and Dads, Rob Eastaway and Mike Askew (2010) list some common problems that children have with multiplication:
 "Making mistakes using techniques they have learnt mechanically, without understanding what they were doing
 Thinking that multiplication means multiple adding (when often it is about ratios)
 Assuming that multiplying always makes things bigger (so they are stumped when they discover that multiplying by 1/2 makes things smaller)" (p. 141)

QED  The Beauty of Mathematical Proof
Author Nicholas KroonSource: Learning and Teaching Mathematics 2016, pp 12 –16 (2016)More Less"Mathematics, rightly viewed, possesses not only truth, but supreme beautya beauty cold and austere, like that of sculpture, without appeal to any part of our weaker nature, without the gorgeous trappings of painting or music, yet sublimely pure, and capable of a stern perfection such as only the greatest art can show."

Generalising Some Geometrical Theorems and Objects
Author Michael De VilliersSource: Learning and Teaching Mathematics 2016, pp 17 –21 (2016)More LessGeneralisation is an important mathematical process, one that lies at the very heart of mathematical thinking. Not only does generalisation expand knowledge, but it often contributes to increasing our understanding by revealing relationships between different mathematical concepts and topics. With this in mind it is unfortunate that most formal school systems around the world tend to focus largely on 'routine' problems and the 'drill and practice' of mathematical techniques. Little opportunity is typically left to stretch and challenge more able learners. Given the pivotal role that generalisation plays in mathematics, it seems educationally necessary to design suitable activities for engaging talented mathematical learners (as well as prospective Mathematics teachers) in the process of generalisation that goes beyond the normal curriculum, and can possibly stimulate them to explore generalisations of their own. This article presents two possible examples of appropriate generalisation activities. The first is the generalisation of a familiar theorem for cyclic quadrilaterals to cyclic polygons, while the second is the generalisation of the concept of a rectangle to a higher order polygons.

Finding the Greatest Common Divisor by Repeated Subtractions
Author YiuKwong ManSource: Learning and Teaching Mathematics 2016, pp 22 –24 (2016)More LessThe greatest common divisor (GCD) of two or more positive integers is the largest integer that is a common divisor of the given integers  i.e. the largest integer that divides the given integers without leaving a remainder. There are many methods for finding the GCD of two or more integers. Some of the most commonly taught in schools are the methods of factor listing, prime factorisation, and common decomposition using the socalled ladder method. Each of these methods is briefly illustrated below for the pair of integers 18 and 24.

Cautionary Tales of Geometric Converses
Authors: Duncan Samson, Avi Sigler and Moshe StupelSource: Learning and Teaching Mathematics 2016, pp 25 –29 (2016)More LessWithin the South African school Mathematics syllabus, pupils are exposed to a number of geometric theorems along with their converses.

A dynamic investigation of geometric properties with "proofs without words"
Authors: Victor Oxman and Moshe StupelSource: Learning and Teaching Mathematics 2016, pp 30 –35 (2016)More LessDynamic geometry environments are a powerful way of engaging students in realtime mathematical exploration. Students are able to investigate mathematical properties through dynamic engagement by dragging objects and observing the effect immediately. Through this process it is possible not only to investigate geometric properties but to form conjectures and hypotheses relating to additional properties. Although it may be easy enough to establish that a conjecture is not true, we need to be a little more careful with establishing its veracity. Although the dynamic geometry environment can lead us to suspect that a conjecture is true, to verify that it is indeed true still requires a formal geometric proof. In this article we present a series of progressive tasks that are ideally suited to exploration in a dynamic geometry environment. The tasks are gradually developed through 'what if' question posing (Brown & Walter, 1990, 1993). Rather than getting students to attempt to prove various conjectures on their own (which they could of course do if they wanted), 'Proofs Without Words' (PWWs) are presented as a route to this verification process (Katz, Segal & Stupel, 2016; Nelsen, 2001; Sigler, Segal & Stupel, 2016). The idea is for students to engage with each PWW diagram, attempt to make sense of it, and then to articulate a formal geometric proof of the conjecture based on the PWW diagram.

Mathematics competitions  yearbased problems
Author Alan ChristisonSource: Learning and Teaching Mathematics 2016, pp 36 –38 (2016)More LessLocal and international mathematics Olympiads and Competitions often make use of problems that incorporate the year of the competition. I have used problems of this nature in the training of pupils to compete in the International Mathematics Competition (IMC), and have also devised and submitted a number of original questions to the IMC committee for possible inclusion in the competition itself. Problems for elementary level, i.e. primary school pupils who are 13 years or younger at the date of competition, must avoid high level mathematics, and should rather be based on the application of logic and general mathematical principles. This article looks at a number of problems I submitted to the 2015 IMC held in China, and comments on their adaptability for future years. The reader is encouraged to attempt these problems and use them in the classroom, with suitable adaptation as desired.

Camouflaged Functions
Author Duncan SamsonSource: Learning and Teaching Mathematics 2016, pp 39 –41 (2016)More LessAs part of the South African school Mathematics syllabus, pupils are exposed to a number of different functions and their graphical representations. Pupils generally become fairly adept at identifying what type of graph a given function represents when the function is given in one of its standard, and hence familiar, forms. However, when the function is presented in a nonstandard form pupils often struggle trying to make sense of the function.
As part of a 3day Matric revision camp at the start of our final term I presented a session entitled "Hidden Functions & Other Camouflage". My intention in the session was to motivate pupils to think more laterally and more flexibly when engaging with functions and their graphs, particularly in relation to functions given in nonstandard formats. In this article I share some of the ideas we explored.

Exploring the minimum conditions for congruency of polygons
Authors: Olga Plaksin and Dorit PatkinSource: Learning and Teaching Mathematics 2016, pp 42 –46 (2016)More LessPolygons form an important component of Euclidean geometry within most school curricula. In the earliest grades pupils observe, describe and sort polygons according to their general characteristics. This then develops into a more sophisticated classification system based on properties and definitions in which various 'families' of polygons emerge. The general idea of congruency is then introduced, and is formally explored with a specific focus on triangles.
The formal exploration of congruency in polygons with more than three sides is seldom dealt with at school level. It is our contention that exploring congruency in polygons other than triangles is likely to enhance a more meaningful appreciation for the concept of congruency, and may prevent pupils from making erroneous generalisations. By way of example, pupils are familiar with the idea that two triangles are congruent when their sides are correspondingly equal. Pupils might thus fall prey to the misconception that two quadrilaterals are also congruent if their sides are correspondingly equal.
The purpose of this article is to present an exploration of congruency in polygons other than triangles. We first explore the necessary conditions to establish congruency in quadrilaterals, and then use an inductive process to establish a broader generalisation of congruency in polygons. It is hoped that such an exploration in the classroom will deepen pupilsÃ¢?? appreciation for the concept of congruency.