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Learning and Teaching Mathematics
Learning and Teaching Mathematics is an official journal of AMESA  the Association for Mathematics Education of South Africa, http://www.amesa.org.za. It is aimed at mathematics educators at all levels of education, providing a medium for stimulating and challenging ideas, offering innovation and practice in all aspects of mathematics teaching and learning. It presents articles that describe or discuss mathematics teaching and learning from the perspective of the practitioner.
Publisher  Association for Mathematics Education of South Africa (AMESA) 

Frequency  Biannually 
Coverage  Issue 1 2004  current 
Language  English 
Journal Status  Active 
Collection(s) 


Exploring the minimum conditions for congruency of polygons
Polygons 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.



Camouflaged Functions
As 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.



Mathematics competitions  yearbased problems
Local 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.



A dynamic investigation of geometric properties with "proofs without words"
Dynamic 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.



Cautionary Tales of Geometric Converses
Within the South African school Mathematics syllabus, pupils are exposed to a number of geometric theorems along with their converses.



Finding the Greatest Common Divisor by Repeated Subtractions
The 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.

© Publisher: Association for Mathematics Education of South Africa (AMESA)
© Publisher: Association for Mathematics Education of South Africa (AMESA)