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Volume 2018 Number 25, December 2018

From the editor
Author Duncan SamsonSource: Learning and Teaching Mathematics 2018, pp 2 –2 (2018)More LessWe 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. If you have an idea but aren’t sure how to structure it into an article, you are welcome to email the editor directly – we’d be happy to engage with you about turning your idea into a printed article.

When our intuition is challenged
Author Duncan SamsonSource: Learning and Teaching Mathematics 2018, pp 3 –5 (2018)More LessThe above ‘definition’ of logic, while playfully tongueincheek, is nonetheless a humbling reminder of the fallibility of the human mind. I am sure we have all encountered situations where the answer to a problem is unexpected or even somewhat counterintuitive. In a short article entitled “When our Intuition Lets us Down”, Marcus Bizony (2006) briefly highlights a number of scenarios where our intuition might be a little misleading. He makes the important point that as educators we should be on the lookout for such scenarios, where we can use the power of mathematics to challenge our intuition. In this article I explore three such contexts that have the potential to surprise and intrigue, and thus provoke rich mathematical discussions.

Tiling with a trilateral trapezium and Penrose tiles
Author Michael de VilliersSource: Learning and Teaching Mathematics 2018, pp 6 –10 (2018)More LessTilings have been around in human culture for hundreds if not thousands of years, mainly for decorative, religious and artistic purposes. However, tiling also has a functional component – consider for example a tiled kitchen or bathroom where it is far easier to replace a single broken tile than it would be to repair a wall or floor covered by one large sheet. In the past century or so tilings have been more rigorously investigated mathematically, but there are still a number of unanswered questions.

Strategies for understanding subtraction
Author Tom PenlingtonSource: Learning and Teaching Mathematics 2018, pp 11 –14 (2018)More LessSubtraction, along with addition, multiplication and division, represent the four basic operations. Flexibility in performing these operations is important, and as educators we should be aware of a variety of approaches that could potentially resonate differently with different learners. In this article I explore various strategies for carrying out subtraction.

Convergence with respect to triangle shapes – elementary geometric iterations
Authors: Hans Humenberger and Franz EmbacherSource: Learning and Teaching Mathematics 2018, pp 15 –19 (2018)More LessWhen thinking about ‘convergence’ one typically considers number sequences or contexts arising from calculus. However, there are also interesting phenomena of convergence arising from elementary geometry. Cases of convergence in the realm of geometry are useful in that they can readily be visualised using Dynamic Geometry Software (DGS) environments. In this article we explore a number of convergence scenarios that could be used in the school classroom in order to promote conjecturing, exploration, reasoning and proof.

The parallelogram identity
Author Letuku Moses MakobeSource: Learning and Teaching Mathematics 2018, pp 20 –21 (2018)More LessThe parallelogram identity

Makhubo constructs a 45° angle
Authors: Ntaopane Joseph Makhubo and Brad UySource: Learning and Teaching Mathematics 2018, pp 22 –25 (2018)More LessMathematics is filled with wonderful surprises. This article describes an account of one such surprise from a recent workshop conducted by Teachers Across Borders – Southern Africa (TABSA) in Bloemfontein. TABSA is an international nongovernmental organization. The weeklong workshop in Bloemfontein was attended by local classroom teachers. Ntaopane Joseph Makhubo (the first author) was one of these local teachers, while Brad Uy (the second author) was one of the workshop facilitators.

A diagonal property of a rhombus constructed from a rectangle
Author Michael de VilliersSource: Learning and Teaching Mathematics 2018, pp 26 –27 (2018)More LessFor any given rectangle EBFD, a rhombus ABCD can be constructed (as shown in Figure 1) by ensuring that AB = AD. The easiest way to do this is to construct the perpendicular bisector of diagonal BD, since a rhombus has diagonals that bisect at right angles. The points A and C are located respectively where this perpendicular bisector cuts ED and BF.

Perpendicular lines : four proofs of the negative reciprocal relationship
Author Harry WigginsSource: Learning and Teaching Mathematics 2018, pp 28 –31 (2018)More LessOne of the great joys of mathematics is finding multiple ways of arriving at a solution or proving a result. Some approaches can be messy and longwinded, while others can be elegant and succinct. But the joy lies in arriving at a common endpoint, however circuitous the route may have been. In this article we explore four different ways of proving the negative reciprocal relationship between the gradients of perpendicular lines. Each proof uses elementary ideas from other topics encountered in the high school Mathematics curriculum.

Right quadrilaterals
Authors: James Metz and Duncan SamsonSource: Learning and Teaching Mathematics 2018, pp 32 –35 (2018)More LessLet us define a right quadrilateral as any quadrilateral containing at least one pair of opposite angles that are right angles (see De Villiers, 2009, pp. 7374). Figure 1 shows right quadrilateral ABCD with right angles at B and D.

Exploring unit fractions
Author Alan ChristisonSource: Learning and Teaching Mathematics 2018, pp 36 –39 (2018)More LessIn their article “The Beauty of Cyclic Numbers”, Duncan Samson and Peter Breetzke (2014) explored, among other things, unit fractions and their decimal expansions. They showed that for a unit fraction to terminate, its denominator must be expressible in the form 2.5 where and are nonnegative integers, both not simultaneously equal to zero. In this article I take the above observation as a starting point and explore it further.

A geometric proof of Heron’s formula
Author YiuKwong ManSource: Learning and Teaching Mathematics 2018, pp 40 –41 (2018)More LessA geometric proof of Heron’s formula

Proof without words
Author Moshe StupelSource: Learning and Teaching Mathematics 2018, pp 42 –42 (2018)More LessConsider a square ABCD. From vertex A, a straight line segment is drawn that intersects side DC at an arbitrary point E. The angle bisector of ∠ is then drawn, intersecting side BC at point F.
Prove that BF + DE = AE.