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 Volume 2015, Issue 19, 2015
Learning and Teaching Mathematics  Volume 2015, Issue 19, December 2015
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Volume 2015, Issue 19, December 2015

From the Editors
Authors: Duncan Samson, Marcus Bizony and Lindiwe TshabalalaSource: Learning and Teaching Mathematics 2015 (2015)More LessIn the first article of LTM 19, Mellony Graven and Debbie Stott share their experiences of enthusing mathematical passion through organising Family Maths Events. In the second article in this issue, Tom Penlington discusses a variety of numerical strategies that could be useful when engaging with division in the classroom. The third article, by Ronit BassanCincinatus and Dorit Patkin, outlines a sequence of activities aimed at developing conceptual understanding of the lowest common denominator, while in the fourth article Duncan Samson reflects on the value of geometric contexts that allow for explorative and flexible approaches. Alan Christison and Tim Mills then explore a specific method of solving linear simultaneous equations, while in the sixth article Moses Makobe presents an alternative formula for determining the least squares line of best fit.

Families enjoying Maths together  organising a family Maths event
Authors: Mellony Graven and Debbie StottSource: Learning and Teaching Mathematics 2015, pp 3 –6 (2015)More LessAs early as 197 4, Bronfenbrenner wrote about the importance of family involvement in child development, particularly with regard to the success of intervention programmes. Evidence indicates that the family is the most effective and economical system for fostering and sustaining the development of a child. The evidence indicates further that the involvement of the child's family as an active participant is critical to the success of any intervention programme. Without such family involvement, any effects of intervention, at least in the cognitive sphere, appear to erode fairly rapidly once the programme ends (Bronfenbrenner, 1974).

The complexity of teaching division
Author Tom PenlingtonSource: Learning and Teaching Mathematics 2015, pp 7 –9 (2015)More LessIn my experience in working with teachers, division (along with the division algorithm) is often considered the most difficult of the four basic operations for learners to master conceptually. The traditional division algorithm is an abstract procedure that does not reflect the way most learners think. Long division is challenging, and as the process involves multiple steps it makes several demands on the learner. If we consider the various skills needed in order to master the process of long division, we would need to include, amongst others, (i) an understanding that the computation begins from the left as opposed to the right, (ii) multiples of numbers and the multiplication tables, (iii) estimation skills to assess the reasonableness of numbers calculated, (iv) sequencing skills, (v) subtraction, and (vi) carefully structured organisation of the written calculation.

Determining the lowest common denominator
Authors: Ronit BassanCincinatus and Dorit PatkinSource: Learning and Teaching Mathematics 2015, pp 10 –12 (2015)More LessDetermining the lowest common denominator is a fundamental skill required when comparing, adding or subtracting fractions  whether numeric or algebraic. This paper outlines a sequence of activities that aims to develop conceptual understanding of this important skill. The basic concept of the lowest common multiple is first introduced. This is then extended to the notion of a lowest common denominator within the context of numeric fractions and finally in relation to algebraic fractions.

Devising explorative Euclidean Geometry questions
Author Duncan SamsonSource: Learning and Teaching Mathematics 2015, pp 13 –16 (2015)More LessEspecially now that Euclidean Geometry is once again a component of the Grade 11 and 12 curriculum, it is important for us to provide pupils with opportunities to engage with suitable geometric contexts in explorative and flexible ways. We should be exposing our pupils to scenarios that require them to think creatively, and with minimal directional guidance. The beauty of such scenarios is that students are exposed to a variety of approaches and strategies, as different students are likely to tackle such questions in different ways. A single simple scenario can often lead to a wealth of exploration and generalisation.

Linear simultaneous equations
Authors: Alan Christison and Tim MillsSource: Learning and Teaching Mathematics 2015, pp 17 –19 (2015)More LessThe following method of solving linear simultaneous equations was shown to Mike Werth (Hilton College Mathematics Department) by a pupil in his class, and subsequently featured in an online FET mathematics discussion group.

Determining the least squares regression line
Author L. Moses MakobeSource: Learning and Teaching Mathematics 2015, pp 20 –22 (2015)More LessA scatterplot is a useful means of visually engaging with bivariate data. The scatterplot provides a graphical representation of the data, and allows for the visual detection of relationships such as a linear trend. The numerical method usually used for determining the equation of the line of best fit is that of least squares regression. Conceptually, the least squares line of best fit is the straight line that minimises the sum of the squares of the residuals, where the residual for each data point is the difference in the actual yvalue and the predicted yvalue (illustrated by the dotted vertical line segments in Figure 1).

Tangents to cubic functions
Author Andrew MaffessantiSource: Learning and Teaching Mathematics 2015, pp 23 –25 (2015)More LessI recently challenged my Grade 12 class with the following:
"Take any cubic function. Determine the equation of the tangent at point P and determine Q, the point where the tangent and the function intersect again."

Flashback to the past : a 1949 Matric Geometry question
Author Michael De VilliersSource: Learning and Teaching Mathematics 2015, pp 26 –31 (2015)More LessRecently I was paging through a copy of the August 1996 issue of the unfortunately now defunct journal Spectrum. I was drawn to a section on old Mathematics Papers where the following problem, from Paper 2 of the 1949 National Senior Examinations for the Union of South Africa, caught my attention.

Insights into teaching geometry
Author Eric A. PandiscioSource: Learning and Teaching Mathematics 2015, pp 32 –34 (2015)More LessTeaching geometry at secondary school level is a vital yet complex task, while constantly improving our instruction is an essential and demanding component of the education profession. Taking these premises as a backdrop, this article offers insights into effective pedagogy to stimulate discussion and reflection. The ideas stem from a combination of experiences: observing a multitude of geometry classrooms in both secondary and university settings; working with teachers in a professional development capacity; and teaching geometry myself while being observed by talented and capable mentors. I offer the distillation of these experiences by describing a "model" Euclidean Geometry course  an amalgam of successful characteristics  that blends significant content with knowledgeable pedagogy to foster critical thinking, logical reasoning skills, and a thorough understanding of geometric concepts at a sophisticated level.

Calculating quartiles for boxandwhisker plots
Author Duncan SamsonSource: Learning and Teaching Mathematics 2015, pp 35 –38 (2015)More LessQuartiles are useful for describing the distribution of a data set as they split the distribution into four equal parts, each part containing one quarter of the total. As such, quartiles form a critical component of the "fivenumber summary" used for drawing boxandwhisker plots. Calculating the upper and lower quartiles of a data set should be a relatively straightforward procedure. However, while the basic concept of a quartile is simple enough, there are many different accepted methods for their calculation, with different methods producing slightly different numerical values. In this article I explore those particular methods that I have seen employed at school level in South Africa.

Investigating x^{n} in terms of the sum of its lesser powers
Author Alan ChristisonSource: Learning and Teaching Mathematics 2015, pp 39 –42 (2015)More LessA problem in a recent international mathematics competition for primary school level was stated as follows:
Every positive integer can be expressed as a sum of distinct powers of 2.
How many threedigit numbers are sums of exactly 9 distinct powers of 2?