Authors:Duncan Samson Abstract: Dear LTM readersIn the first article of LTM 27, Tom Penlington shares various activities which introduce learners to new mathematical words as they look for patterns and describe relationships between numbers in the 100 grid. In the second article in this issue, Duncan Samson sheds light on a clever bit of mental arithmetic which a reader found in the novel The Girl Who Saved the King of Sweden. PubDate: 2019-12-01T00:00:00Z

Authors:Tom Penlington Abstract: IntroductionExploring patterns is the essence of mathematics. However, learners in the Intermediate Phase often find it challenging to describe patterns. The 100 grid is a useful and versatile tool that can be used in Foundation and Intermediate Phase classrooms to teach and consolidate a number of mathematical concepts while engaging with the notion of pattern. PubDate: 2019-12-01T00:00:00Z

Authors:Duncan Samson Abstract: The Girl Who Saved the King of SwedenHi DuncanI’m having great difficulty trying to work out a maths problem in this book. PubDate: 2019-12-01T00:00:00Z

Authors:Yiu-Kwong Man Abstract: IntroductionConsider a trapezium ACDB (not necessarily isosceles) with diagonals AD and BC intersecting at E, as illustrated in Figure 1. Since ACD and BCD have equal bases and equal heights, their areas are the same. Now, since CDE is a common portion of these two triangles, it follows that the area of ACE is equal to the area of BDE. PubDate: 2019-12-01T00:00:00Z

Authors:Michal Seri; Hanna Savion Abstract: IntroductionEducationally relevant technology has long been used as an aid to teaching, and the richness and sophistication of such technology continues to increase. In addition, smartphones have become part of our lives. This is particularly true of our students who have been born into a world of mobile technology, instant information, and immediate communication. Incorporating the use of Quick Response (QR) codes in the mathematics classroom is one of many ways to capitalize on this technology in an educationally meaningful way. After installing the appropriate application on a smartphone, scanning a QR code links one directly to the desired content, be it animations, video clips, sound clips, images, or online content. This article describes an innovative activity that capitalizes on the mobility of the smartphone through the use of QR scanning, creating an engaging and motivating learning experience. PubDate: 2019-12-01T00:00:00Z

Authors:Eric A. Pandiscio Abstract: IntroductionThere is a solid rationale for infusing the middle level and high school Mathematics curriculum with robust mathematical ideas derived from real world contexts (see e.g. NCTM 2000; Koedinger & Nathan, 2004). Using this rationale as a base, this article poses non-trivial mathematical problems that originated in a context outside the classroom, and analyses the fundamental mathematical content of the problems. The core concepts relate to ratio and proportional reasoning as well as the interpretation of graphical information and possible misconceptions related thereto. PubDate: 2019-12-01T00:00:00Z

Authors:Yiu-Kwong Man Abstract: IntroductionPtolemy’s theorem is a well-known result in plane geometry and can be stated as follows:If ABCD is a cyclic quadrilateral with sides a, b, c, d and diagonals e, f, then ac +bd = efIn this article we present an elementary proof of Ptolemy’s theorem based on basic trigonometry, circle geometry and transformation geometry. The proof is different to those described elsewhere – see for example Johnson (1929), Coxeter & Greitzer (1967), Ostermann & Wanner (2012) and Miculita (2017). PubDate: 2019-12-01T00:00:00Z

Authors:Alan Christison Abstract: IntroductionThe International Mathematics Olympiad (IMO) is the World Championship Mathematics Competition for High School students. The first IMO was held in Romania in 1959, with seven countries competing. 2019 marks the 60th anniversary of the event. The 2019 IMO was held in Bath in the United Kingdom. The Olympiad currently has over one hundred competing countries from across five continents. South Africa has competed each year since 1992. This article considers Question 2 from the 1959 IMO. PubDate: 2019-12-01T00:00:00Z

Authors:Michael de Villiers Abstract: IntroductionIn two recent issues of the Learning & Teaching Mathematics journal, the following interesting area and perimeter formulae for an isosceles trapezium with an inscribed circle were derived and proven by the authors cited below:In this article further interesting properties of an isosceles trapezium with an inscribed circle, as well as for a kite that is cyclic, will be derived. These two types of quadrilaterals have respectively been called ‘circumscribed isosceles trapeziums’ and ‘right kites’ in De Villiers (2009), and the same terminology will be used here. PubDate: 2019-12-01T00:00:00Z

Authors:Moshe Stupel; Victor Oxman Abstract: CD and EF are parallel lines that pass through the points of intersection of two circles, A and B. With such a configuration it will always be true that CD = EF, since quadrilateral DCEF is a parallelogram. PubDate: 2019-12-01T00:00:00Z

Authors:Duncan Samson Abstract: We hope you enjoy the wonderfully diverse array of articles in this issue, and remind you that we are always eager to receive submissions. Suggestions to authors, as well as a breakdown of the different types of article you could consider, can be found at the end of this journal. PubDate: 2019-06-01T00:00:00Z

Authors:Tom Penlington Abstract: With reference to the CAPS document for the Intermediate Phase, one of the first aspects to consider when teaching fractions is that learners need to be able to compare and order common fractions with different denominators (up to at least eighths in Grade 4, and up to at least twelfths in Grade 5). In this article I discuss a number of different representations which I have found useful for learners as they engage with these types of fraction tasks. PubDate: 2019-06-01T00:00:00Z

Authors:Duncan Samson; Simon Kroon Abstract: A number of recent articles in Learning and Teaching Mathematics have highlighted the value inherent in multiple solution tasks (e.g. De Villiers, 2016, 2017; Pillay, 2017; Samson, 2017; Stupel, Sigler & Tal, 2018, Wiggins 2018). Questions that lead to multiple and varied solutions allow for the exploration of the interconnected nature of mathematics. In addition, exploring and engaging with multiple solution tasks encourages reflection and flexibility of thought, two important mathematical habits of mind. In this article we present a carefully crafted question, adapted from a Cambridge A Level examination (9709/33 May/June 2017), which allows for a wide variety of solution approaches accessible to high school students. A number of these different approaches are illustrated. PubDate: 2019-06-01T00:00:00Z

Authors:Letuku Moses Makobe Abstract: There is an interesting relationship between the surface area and volume of a rectangular prism. PubDate: 2019-06-01T00:00:00Z

Authors:Duncan Samson Abstract: It was the purpose of this article to show how appropriate pictorial contexts could be used to foster an authentic classroom experience of mathematical exploration through the investigation and articulation of expressions of generality. A number of examples are shown to illustrate possible subdivisions of the given pictorial terms into different component parts, but there are no doubt other potential deconstructions. PubDate: 2019-06-01T00:00:00Z

Authors:Michael de Villiers; Piet Human Abstract: The original problem is a pleasing, straightforward application of these two famous theorems, and would present a good practice challenge for learners preparing for the Third Round of the SA Mathematics Olympiad. It could also be used as enrichment to the high school geometry curriculum in a Mathematics Club. Due to the richness of the diagram, readers may even find additional interesting geometry properties.

Authors:Alan Christison Abstract: Trigonometric functions are occasionally incorporated into otherwise algebraic equations in order to increase the complexity of the question. While this is a clever way of assessing algebraic as well as trigonometric knowledge within the same question, it is not without its complications. This article illustrates how limitations in the maximum and minimum values of the sine and cosine functions might impact on such questions. Two examples are presented and explored, and possibilities for extension activities are suggested. PubDate: 2019-06-01T00:00:00Z

Authors:Yiu-Kwong Man Abstract: In LTM 24 Moshe Stupel presents an interesting proof without words. Starting with a parallelogram ABCD, construct quadrilateral KING with KN parallel to AB and CD, and G and I arbitrarily placed on sides AB and CD respectively. The area of quadrilateral KING is half the area of parallelogram ABCD. PubDate: 2019-06-01T00:00:00Z

Authors:Moshe Stupel; Shula Weissman Abstract: Proofs without Words Theorem 1 The angle bisectors of a quadrilateral form a cyclic quadrilateral. Theorem 2 The exterior angle bisectors of a quadrilateral form a cyclic quadrilateral.