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 Volume 2014, Issue 16, 2014
Learning and Teaching Mathematics  Volume 2014, Issue 16, June 2014
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Volume 2014, Issue 16, June 2014

From the Editors
Authors: Duncan Samson, Marcus Bizony and Lindiwe TshabalalaSource: Learning and Teaching Mathematics 2014 (2014)More LessLTM16 begins with an article by Tom Penlington who presents a variety of ideas and classroom strategies for developing multiplicative reasoning and understanding. In the second article in this issue, Dorit Patkin and Ilana Levenberg present a visually striking and handson approach to finding the sum of the interior angles of convex polygons. The third article, by Amur Singh, explores a fascinating number link between Euler's formula and the golden ratio, while in the fourth article Donald Flint considers the distinction between drawing and constructing tangents. Marcus Bizony then presents a quick method for finding rational approximations to square roots, while in the sixth article Duncan Samson and Peter Breetzke delve into the beauty and intrigue of cyclic numbers.

Developing basic multiplication skills for understanding
Author Tom PenlingtonSource: Learning and Teaching Mathematics 2014, pp 3 –6 (2014)More LessFollowing on from my brief article in LTM 15 (December 2013) regarding mental mathematics, the purpose of this article is to suggest a variety of strategies or ideas for developing multiplicative reasoning and understanding.

Investigating the sum of the interior angles of convex polygons using paper and scissors
Authors: Dorit Patkin and Ilana LevenbergSource: Learning and Teaching Mathematics 2014, pp 7 –11 (2014)More LessThis article explores an inductive process for finding the sum of the interior angles of convex polygons through the use of paper and scissors. The activities presented make use of drawing and cutting as a simple but experientially meaningful way to explore geometric concepts. The didactic and pedagogical functions of the activities presented in this paper aim to develop the learning of mathematics based on engagement and reasoning as well as to provide a creative and interesting mathematical experience.

Complex numbers and the golden ratio : an interesting number link
Author Amur SinghSource: Learning and Teaching Mathematics 2014, pp 12 –16 (2014)More LessThe article begins by briefly introducing the idea of the golden ratio. Euler's formula is then explored for different angle measurements and the link between Euler's formula and the golden ratio is established and discussed. Some concluding comments and observations are then made.

Off on a tangent
Author Donald FlintSource: Learning and Teaching Mathematics 2014, pp 17 –20 (2014)More LessThe process of drawing a tangent to a circle seems like a simple enough procedure. If we wanted to draw a tangent to a specific point on the circumference of a circle all we would need to do is place a straight edge against the specified point and then gradually rotate the straight edge until our visual acuity judged it to be tangential. We could then draw the desired tangent. If the situation was somewhat different and we were required to draw a tangent to a circle from a specified point outside the circle, we could similarly rely on our visual judgement. This time we would need to place our straight edge against the external point and then gradually manoeuvre the straight edge towards the circle, getting things close as best we could with our eyesight. We could then draw the desired tangent.

Finding rational approximations to square roots
Author Marcus BizonySource: Learning and Teaching Mathematics 2014, pp 21 –23 (2014)More LessSuppose that n is a natural number, and that we have integers l and h so that l is an underestimate of √n while h is an overestimate for √n. This situation is illustrated in Figure 1 where the curve shown represents y = x^{2}. We can now use linear interpolation to calculate a new estimate, e, for √n as illustrated in Figure 2.

The beauty of cyclic numbers
Authors: Duncan Samson and Peter BreetzkeSource: Learning and Teaching Mathematics 2014, pp 24 –29 (2014)More LessCyclic numbers stemming from the decimal expansions of certain fractions provide a fascinating context for mathematical exploration and investigation. The purpose of this article was not to engage too heavily with the theory behind cyclic numbers as this has been done extensively elsewhere through the use of the algorithmic process of long division, number theory, modular arithmetic, and group theory. Rather, what we hope we have accomplished in this article is to show how a simple idea can be developed into an ever expanding investigation resulting in intriguing discoveries which hopefully spark a mathematical desire to explore both further and deeper.

Investigating factors
Author Alan ChristisonSource: Learning and Teaching Mathematics 2014, pp 30 –31 (2014)More LessIn 2007 the following question appeared in the Third Round of the Junior Section of the South African Mathematics Olympiad:
"The sum of the factors of 120 is 360. Find the sum of the reciprocals of the factors of 120."

Why does the 'tipandtimes' rule work?
Author Faaiz GierdienSource: Learning and Teaching Mathematics 2014, pp 32 –36 (2014)More LessThis article is about understanding why the 'tipandtimes' or 'invertandmultiply' rule for dividing by a fraction works. I will illustrate the reasoning behind this rule using proper and improper fractions as well as a practical example using ribbon material. The reasoning entails using multiplying or dividing by 1 and equivalence (Askew, 2008, 2013) or equivalent ways to rewrite fractions or whole numbers. In the case of the practical example using ribbon material, a central idea is changing or switching the measuring unit. The two examples used come from primary school preservice teachers (PSTs) whom I teach.

Snakes and ladders for integer consolidation
Author Andrew HalleySource: Learning and Teaching Mathematics 2014, pp 37 –39 (2014)More LessLearners' difficulties in operating with integers are wellknown, and their errors can have far reaching implications. If learners struggle with integers, the associated problems may haunt them throughout their secondary school career. While working in the Wits Maths Connect Secondary Project we identified the need for learners to practice operating with integers without simply carrying out tedious and repetitive calculations. In this article I share a game that I developed which provides learners with a meaningful context in which to practice the addition and subtraction of integers.

Over and over again : two geometric iterations with triangles
Author Michael De VilliersSource: Learning and Teaching Mathematics 2014, pp 40 –45 (2014)More LessIteration in mathematics  doing the same procedure over and over, with each successive step using the results of the previous step  is a fundamental concept and procedure in mathematics dating back to ancient times.

Solving cubic equations numerically
Author Marcus BizonySource: Learning and Teaching Mathematics 2014, pp 46 –48 (2014)More LessFor learners who are familiar with the formula for solving quadratic equations, it is greatly disappointing that there isn't a similar method for solving cubic equations. There are of course standard techniques, but these are rather complex and generally not accessible at school level. The types of cubic equations we solve at school typically have at least one integer solution, and this solution (often found by setting up a table of values or using a "guess and check" approach) is used as the basis to finding the other two solutions. This presents the somewhat paradoxical situation that in order to solve a cubic equation you have to know one of the solutions already. But what if the cubic equation doesn't have an integer, or even rational, solution? How would one go about solving such an equation? One would hope that this is exactly the sort of question an engaged student would ask!

Solids, positions and rotation axes
Author Dorit PatkinSource: Learning and Teaching Mathematics 2014, pp 49 –53 (2014)More LessThreedimensional objects such as cylinders, cones and spheres are in fact solids of revolution which are formed by rotating a plane shape about an axis that lies in the same plane. These solids are usually engaged with in the classroom with reference to the attributes that characterise them and in terms of volume and surface area calculations, but this is generally done without reference to such objects being solids of revolution.
This paper presents an approach for engaging with these solids from a different perspective, that of manipulating (rotating) a 2dimensional shape in order to obtain a 3dimensional solid, carried out through the use of mental imagery. The activities in this paper focus on solids of revolution through the use of mental imagery and are suitable for learners in the upper grades of elementary school as well as for pre and inservice mathematics teachers.