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Volume 2018 Number 24, June 2018

From the Editor
Author Duncan SamsonSource: Learning and Teaching Mathematics 2018, pp 2 –2 (2018)More LessIn the first article of LTM 24, Marc North encourages us to think more critically about the representations we make use of in the classroom. In the second article in this issue, Laura de Lange reflects on an openended problem solving activity she tried with one of her classes. The third article, by Moshe Stupel, Avi Sigler and Idan Tal, explores an interesting geometrical scenario, while in the fourth article Michael de Villiers and John Silvester illustrate the importance of ‘why not’ or ‘what if’ questions using a wellknown geometry theorem as a starting point. Duncan Samson then takes us on a short quadratic detour before Deepak Mavani and Andrew Maffessanti take us back to the future with multiple approaches to a finance question.

Which picture and why? being deliberate and explicit when using representations
Author Marc NorthSource: Learning and Teaching Mathematics 2018, pp 3 –8 (2018)More LessMULTITUDE OF REPRESENTATIONS
What is 12 × 7? The answer of course is 84. And that’s the end of the story … or is it? I recently posed the question in Figure 1 to a group of teachers.

Using Pokémon go to encourage patient problem solving
Author Laura de LangeSource: Learning and Teaching Mathematics 2018, pp 9 –12 (2018)More LessDan Meyer, in his TED talk entitled ‘Math Class Needs a Makeover’ 1, identifies a problem many of us face in our classroom – our students want to engage only with simple problems. When asked to engage with challenging, complicated or openended problems, students often show a lack of initiative in the problem solving process, and they tend to lack the perseverance and grit required to see the question through to a meaningful resolution. Meyer argues that any problem worth solving will either have a surplus of information that you have to sift through to find what is relevant, or insufficient information which forces you to go and find the information you need to solve the problem. However, this is seldom the type of problem that students encounter in the classroom, where the information provided is almost invariably exactly what is needed to solve the problem. What can we do to encourage patient problem solving in the classroom? How can we encourage students to overcome their impatience with irresolution, and to grapple meaningfully with complex and openended problems?

A dynamic investigation of conserved properties using a rotating “Pizza Slice”
Authors: Moshe Stupel, Avi Sigler and Idan TalSource: Learning and Teaching Mathematics 2018, pp 13 –16 (2018)More LessIn this article we describe an investigation that exploits the dynamic nature of geometry software environments such as GeoGebra. A wellknown initial scenario is posed which lends itself to dynamic exploration in such an environment. This basic idea, a square rotating about a vertex positioned at the centre of another square, is then extended to other regular polygons. The investigation leads to some interesting discoveries, and a number of generalisations can be made and justified.

A MerryGoRound the triangle
Authors: Michael de Villiers and John SilvesterSource: Learning and Teaching Mathematics 2018, pp 17 –20 (2018)More LessThe above quotations provide wry yet intriguing descriptions of ‘out of the box’ thinking, and how sometimes being ‘unreasonable’ can actually be productive. Asking the simple question ‘why not?’ in mathematics can often lead to the discovery and posing of new conjectures and problems. For example, suppose a particular result holds for an equilateral triangle. A first step may be to ask: ‘why not’ explore the same idea for a square, a regular pentagon, etc.? Similarly, ‘why not’ explore whether it also holds for an isosceles triangle or perhaps any triangle? ‘Why not’ explore the idea in 3D space? ‘Why not’ vary the idea itself? In a similar vein, Brown and Walter (1990) highlight the importance of ‘what if not?’ questions. What if it were not a triangle in the plane, but instead on a curved surface such as a sphere? This kind of thinking lies at the heart of a rather neglected area of mathematics education, namely the development of problem posing. In order to pose new problems it helps to be able to think divergently and ask questions that interrogate and vary the givens and conclusions of a result. The purpose of this article is to give an example of how this thinking led to the formulation of a new conjecture (for the authors at least) along with several associated results as well as their proof.

A short quadratic diversion
Author Duncan SamsonSource: Learning and Teaching Mathematics 2018, pp 21 –23 (2018)More LessIn the 1860s an Austrian engineer by the name of Eduard Lill devised a fascinating method for geometrically visualising the roots of polynomial equations. The technique, generally known as Lill’s Method, while perhaps a little obscure, nonetheless leads to the following fascinating result when applied to quadratic equations. Given the quadratic equation 𝑎𝑥2+𝑏𝑥+𝑐=0 then the roots of this equation are the 𝑥intercepts of the circle with diameter (0;1) and (−𝑏𝑎;𝑐𝑎).

Back to the future
Authors: Deepak Mavani and Andrew MaffessantiSource: Learning and Teaching Mathematics 2018, pp 24 –27 (2018)More LessThe purpose of this article is to highlight multiple approaches to solving the above question thereby providing different perspectives of hidden mathematical ideas within what appears to be a relatively straightforward scenario.

Isosceles trapezium with inscribed circle
Author Letuku Moses MakobeSource: Learning and Teaching Mathematics 2018, pp 28 –29 (2018)More LessThe diagram shows an isosceles trapezium with an inscribed circle. Devise a means for determining the perimeter of the trapezium if the only information known is the area of the trapezium and the base angle 𝜃.

Why does it Work? a mathematical explanation and further generalization of a card trick
Author Michael de VilliersSource: Learning and Teaching Mathematics 2018, pp 30 –31 (2018)More LessI was recently shown an interesting card trick by a young family relation who then specifically asked me if I could figure out why it worked. Of significance is that both she and her father (who originally showed her the trick) were entirely convinced of its validity. They had no doubt that it would always work, it was simply that they couldn’t figure out why it worked. This highlights the important, fundamental distinction I myself, and others, have consistently made between conviction and explanation in mathematics (De Villiers, 1990; Hanna, 1989; Harel, 2013). Unfortunately many curricular activities at school, and most textbooks, do not exploit this distinction to present proof as a meaningful experience to learners. Most often, doing a mathematical proof is only motivated as a means of gaining conviction, while its potential explanatory power is all but neglected.

On the height of a triangle with two known angles and the included side
Author YiuKwong ManSource: Learning and Teaching Mathematics 2018, pp 32 –33 (2018)More LessIn LTM No. 23, Makobe (2017) presented a formula for finding the height of a triangle with two known angles and the included side (Figure 1).

Functions having reciprocal roots
Authors: Alan Christison and Duncan SamsonSource: Learning and Teaching Mathematics 2018, pp 34 –37 (2018)More LessOne of us recently came across the following interesting statement in an article: “To find a polynomial whose roots are the reciprocals of the roots of a given polynomial, write the coefficients in reverse order.” In terms of the article the statement was simply used as an example of a mathematical algorithm. However, the statement itself grabbed our attention and some investigation ensued. In this article we will consider the statement more closely as it relates to linear, quadratic and cubic functions. In each case the relationship between the two functions with reciprocal roots is examined further. Note that roots of zero must be excluded since the reciprocal of zero is undefined. Thus, functions of the form 𝑓(𝑥)=𝑎𝑥𝑛+𝑏𝑥𝑚 and the like must first be written as 𝑓(𝑥)=𝑥𝑚(𝑎𝑥𝑛−𝑚+𝑏), and the algorithm applied to the portion within the brackets. In such cases, each of the two functions (the original as well as the one with reciprocal roots) will have a root of zero.

A surprising theorem with three different proofs
Authors: Moshe Stupel, Avi Sigler and Idan TalSource: Learning and Teaching Mathematics 2018, pp 38 –41 (2018)More LessMathematics involves expressing generality. Geometry, in particular, involves expressing generality about properties and relationships related to space and shape. As a shape or geometric construction is altered, which properties change, and which properties remain invariant? Dynamic Geometry Environments (DGEs) represent a powerful way of exploring such invariance by allowing the animation of mental imagery to be made visually explicit. In this article we highlight a surprising geometric theorem and prove it using three different approaches. Exposing pupils to different methods of solving a single problem is widely acknowledged as a powerful means of exploring the intertwined nature of mathematics, as well as expanding one’s knowledge base and ‘mathematical toolbox’.

Proof without words
Author Moshe StupelSource: Learning and Teaching Mathematics 2018, pp 42 –42 (2018)More LessConsider a parallelogram ABCD. Now construct quadrilateral KING with diagonals KN and GI. Points K and N are placed on sides AB and CD respectively such that KN is parallel to AD and BC. Points G and I are placed arbitrarily on sides AD and BC respectively.