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 Volume 2013, Issue 15, 2013
Learning and Teaching Mathematics  Volume 2013, Issue 15, January 2013
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Volume 2013, Issue 15, January 2013

From the Editors
Authors: Duncan Samson, Marcus Bizony and Lindiwe TshabalalaSource: Learning and Teaching Mathematics 2013 (2013)More LessLTM15 begins with a classroom activity designed to engage learners in a meaningful handson investigation through the use of simple physical manipulatives. Wooden cubes are used to investigate the surface area and volume of rectangular prisms in terms of the effect of changing one, two or three dimensions by a specific factor. In the second article in this issue, Jim Metz investigates some interesting links between standardized rectangular paper sizes and the trigonometric ratios for special angles. The third article reports on teacher experiences at the Mathematics Education and Society Conference held in Cape Town in April 2013, while in the fourth article Shakespear Chiphambo discusses a learner's alternative approach to converting fractions into decimal form without a calculator. Tom Penlington then reflects briefly on aspects related to mental mathematics, while in the sixth article Ashley Ah Goo investigates a fascinating relationship between the diameter of a given circle and the dimensions of any inscribed triangle in the circle.

Investigating the surface area and volume of rectangular prisms
Authors: Senzeni Mbedzi and Duncan SamsonSource: Learning and Teaching Mathematics 2013, pp 3 –6 (2013)More LessThis paper describes a classroom investigation focusing on surface area and volume of rectangular prisms. The purpose of the activity is to investigate, with respect to the total surface area and volume, the effect of changing one, two or three dimensions of a rectangular prism by a given factor.

A4 rectangles and trigonometry
Author Jim MetzSource: Learning and Teaching Mathematics 2013, pp 7 –9 (2013)More LessWe are all familiar with standard A4 and A3 sheets of paper. Most of us are also probably aware that an A3 sheet can be cut in half to form two A4 sheets. A standard A4 sheet of paper measures 210 mm by 297 mm while a standard A3 sheet of paper measures 297 mm by 420 mm. From these dimensions we can see that A4 and A3 sheets of paper are similar since the ratio of the longer side to the shorter side is the same in both cases, i.e. 1,414.

Teacher experiences at the Mathematics Education and Society Conference
Author Mellony GravenSource: Learning and Teaching Mathematics 2013, pp 10 –14 (2013)More LessThe Mathematics Education and Society (MES) conference brings together mathematics educators from around the world and provides a forum for discussing the social, political, cultural and ethical dimensions of mathematics education. The conference also provides an arena in which to forge connections and to explore and establish future collaborative activity. The 7^{th} MES conference was hosted by South Africa and took place in Cape Town from 2  7 April 2013. 93 participants from 16 different countries attended, including Argentina, Australia, Brazil, Canada, the Czech Republic, Germany, Greece, Iceland, India, New Zealand, Norway, South Africa, Sweden, Switzerland, the United Kingdom, and the USA.

Converting fractions into decimal form without a calculator
Author Shakespear ChiphamboSource: Learning and Teaching Mathematics 2013, pp 15 –17 (2013)More LessThere are many approaches to converting a common fraction into decimal form without a calculator. For simple fractions the most direct approach is to convert the fraction into an equivalent form where the denominator is a power of 10 from which the decimal form can easily be determined.

Some thoughts on mental mathematics
Author Tom PenlingtonSource: Learning and Teaching Mathematics 2013 (2013)More LessMental calculation skills lie at the heart of numeracy. The word 'numeracy' has changed its meaning considerably over the years. Being numerate these days implies more than just developing the skills of computation but is linked to sense making and therefore understanding. According to Haylock (2001) the basis of being numerate lies in the ability to calculate mentally using a range of strategies.

Determining the diameter of a circle using any inscribed triangle
Author Ashley Ah GooSource: Learning and Teaching Mathematics 2013, pp 19 –21 (2013)More LessThis article investigates the relationship between the diameter of a given circle and the dimensions of any inscribed triangle in the circle. In addition, the relationship between different inscribed triangles in the same circle will be explored.

Exploring the difference of two squares
Author Duncan SamsonSource: Learning and Teaching Mathematics 2013, pp 22 –27 (2013)More LessOne of the aspects of mathematics that I particularly enjoy is how a simple idea can often lead to a wealth of mathematical exploration. One such idea is the classic 'difference of two squares'. Within the school curriculum the difference of two squares is usually introduced as a specific form of factorisation.

Some remarks about the form A cosx + B sinx
Author Marcus BizonySource: Learning and Teaching Mathematics 2013, pp 28 –30 (2013)More LessDeciding where a syllabus begins and ends is never an easy thing, and I suppose it will inevitably happen that every now and then a line is drawn across something that I personally see as being part of one big story. A good example concerns trigonometric equations. In our South African school syllabus we expect pupils to be able to solve quadratic equations in trigonometric functions, equations involving both double and single angles, and even rather arcane equations such as sin(x  20°) = cos(40°  2x).

Stomptapclapsnap : a game for promoting conceptual place value and listening skills
Author Debbie StottSource: Learning and Teaching Mathematics 2013, pp 31 –32 (2013)More LessAs part of our work with the South African Numeracy Chair Project we often engage with international academics and educators. We recently had the opportunity to spend time with Dr Ravi Subramaniam from the Homi Bhabha Centre for Science Education (HBCSE) in Mumbai (India) who shared this game with us. His version (included in his Grade 3 workbook) is called TAPCLAPSNAP and works with numbers up to 999. We immediately tried the game in some of our afterschool maths clubs. The game worked very well and I decided to extend it to include numbers in the thousands in order to give it more scope for different grades. In this article I share some of the ways of using this simple yet effective game in the classroom.

Circle mania
Author Tim MillsSource: Learning and Teaching Mathematics 2013, pp 33 –35 (2013)More LessWe have all no doubt come across the traditional analytical geometry question of proving that two circles either (a) touch one another, (b) intersect one another, or (c) don't intersect one another. A typical approach to this question is simply to compare the sum of the radii to the distance between the centres of the two circles.

A trisection concurrency : a variation on a median theme
Author Shunmugam PillaySource: Learning and Teaching Mathematics 2013, pp 36 –40 (2013)More LessEuclidean geometry is a potentially rich domain in which to provide learners with opportunities to engage in the processes of mathematical (re)discovery and (re)invention. To the extent that most scientific inquiry begins with inductive reasoning, usually leading to some unproved generalization (i.e. a conjecture), it is of general importance to engage learners in the process of inductive reasoning. This can be accomplished in a geometrical environment through the use of construction, measurement and observation, both by hand as well as with the aid of computing technology.

Dynamic geometry software as a dynamic tool for spatial exploration
Authors: Reinhard Holzl and Marc SchaferSource: Learning and Teaching Mathematics 2013, pp 41 –45 (2013)More LessDynamic Geometry Software (DGS) such as GeoGebra, Geometer's Sketchpad and Cabri Geometry offer a wealth of opportunities for an exploratory style of teaching and learning Mathematics, particularly in the exploration of space and shape. The new Curriculum and Assessment Policy Statement (CAPS) foregrounds the use of spatial skills and properties of shapes and objects "to identify, pose and solve problems creatively and critically" (South Africa. DBE, 2011, p. 9). Although the South African Mathematics curriculum no longer places importance on traditional Euclidean construction by means of straightedge and compass, DGS can nonetheless be used to engage with fundamental ideas relating to geometric shapes, symmetry and transformations. This was the impetus behind encouraging the use of GeoGebra in four township schools in the Grahamstown Education District whose Mathematics teachers participate in an ongoing inservice research and development programme hosted by Rhodes University.

Creating algebraic fractional sums that reduce further
Author Amur SinghSource: Learning and Teaching Mathematics 2013, pp 46 –50 (2013)More LessLet us begin by considering an example involving a sum of algebraic fractions where the denominators are of at most degree 2.