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Volume 2019 Number 27, December 2019

From the editor
Author Duncan SamsonSource: Learning and Teaching Mathematics 2019, pp 2 –2 (2019)More LessDear 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.

Recognising patterns using the 100 grid
Author Tom PenlingtonSource: Learning and Teaching Mathematics 2019, pp 3 –7 (2019)More LessIntroductionExploring 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.

The Girl Who Saved the King of Sweden
Author Duncan SamsonSource: Learning and Teaching Mathematics 2019, pp 8 –9 (2019)More LessThe Girl Who Saved the King of SwedenHi Duncan
I’m having great difficulty trying to work out a maths problem in this book.

On the areas of triangles bounded by nonparallel sides and diagonals of a trapezium
Author YiuKwong ManSource: Learning and Teaching Mathematics 2019, pp 10 –11 (2019)More LessIntroductionConsider 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.

Introduction into QR coding into the mathematics classroom
Authors: Michal Seri and Hanna SavionSource: Learning and Teaching Mathematics 2019, pp 12 –16 (2019)More LessIntroductionEducationally 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.

Using real world contexts to explore fundamental mathematical ideas
Author Eric A. PandiscioSource: Learning and Teaching Mathematics 2019, pp 17 –22 (2019)More LessIntroductionThere 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 nontrivial 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.

An elementary proof of Ptolemy’s theorem
Author YiuKwong ManSource: Learning and Teaching Mathematics 2019, pp 23 –25 (2019)More LessIntroductionPtolemy’s theorem is a wellknown 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 = ef
In 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).

A look back : the International Mathematics Olympiad, 1959
Author Alan ChristisonSource: Learning and Teaching Mathematics 2019, pp 26 –29 (2019)More LessIntroductionThe 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 60^{th} 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.

More area, perimeter and other properties of circumscribed isosceles trapeziums and cyclic kites
Author Michael de VilliersSource: Learning and Teaching Mathematics 2019, pp 30 –33 (2019)More LessIntroductionIn 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.

Proof without words : parallel lines passing through the points of intersection of two circles
Authors: Moshe Stupel and Victor OxmanSource: Learning and Teaching Mathematics 2019, pp 34 –34 (2019)More LessCD 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.