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Volume 2019 Number 26, June 2019

From the editor
Author Duncan SamsonSource: Learning and Teaching Mathematics 2019, pp 2 –2 (2019)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.

Using different representations to compare and order fractions
Author Tom PenlingtonSource: Learning and Teaching Mathematics 2019, pp 3 –6 (2019)More LessWith reference to the CAPS document for the Intermediate Phase, one of the first aspects to consider when teaching fractions is that learners need to be able to compare and order common fractions with different denominators (up to at least eighths in Grade 4, and up to at least twelfths in Grade 5). In this article I discuss a number of different representations which I have found useful for learners as they engage with these types of fraction tasks.

A multiple solution task
Authors: Duncan Samson and Simon KroonSource: Learning and Teaching Mathematics 2019, pp 7 –11 (2019)More LessA number of recent articles in Learning and Teaching Mathematics have highlighted the value inherent in multiple solution tasks (e.g. De Villiers, 2016, 2017; Pillay, 2017; Samson, 2017; Stupel, Sigler & Tal, 2018, Wiggins 2018). Questions that lead to multiple and varied solutions allow for the exploration of the interconnected nature of mathematics. In addition, exploring and engaging with multiple solution tasks encourages reflection and flexibility of thought, two important mathematical habits of mind. In this article we present a carefully crafted question, adapted from a Cambridge A Level examination (9709/33 May/June 2017), which allows for a wide variety of solution approaches accessible to high school students. A number of these different approaches are illustrated.

The importance of structure for supporting children’s learning in mathematics
Author Marc NorthSource: Learning and Teaching Mathematics 2019, pp 12 –17 (2019)More LessComparison of two examples of children’s bookwork.

Relating volume to surface area
Author Letuku Moses MakobeSource: Learning and Teaching Mathematics 2019, pp 18 –19 (2019)More LessThere is an interesting relationship between the surface area and volume of a rectangular prism.

Visualising generality with regular arrays
Author Duncan SamsonSource: Learning and Teaching Mathematics 2019, pp 20 –25 (2019)More LessIt was the purpose of this article to show how appropriate pictorial contexts could be used to foster an authentic classroom experience of mathematical exploration through the investigation and articulation of expressions of generality. A number of examples are shown to illustrate possible subdivisions of the given pictorial terms into different component parts, but there are no doubt other potential deconstructions.

Isosceles trapezium with inscribed circle – further exploration
Author YiuKwong ManSource: Learning and Teaching Mathematics 2019, pp 26 –27 (2019)More LessIsosceles trapezium with inscribed circle – further exploration

An interesting collinearity
Authors: Michael de Villiers and Piet HumanSource: Learning and Teaching Mathematics 2019, pp 28 –30 (2019)More LessThe original problem is a pleasing, straightforward application of these two famous theorems, and would present a good practice challenge for learners preparing for the Third Round of the SA Mathematics Olympiad. It could also be used as enrichment to the high school geometry curriculum in a Mathematics Club. Due to the richness of the diagram, readers may even find additional interesting geometry properties.

Limitations in the value of sine and cosine
Author Alan ChristisonSource: Learning and Teaching Mathematics 2019, pp 31 –33 (2019)More LessTrigonometric functions are occasionally incorporated into otherwise algebraic equations in order to increase the complexity of the question. While this is a clever way of assessing algebraic as well as trigonometric knowledge within the same question, it is not without its complications. This article illustrates how limitations in the maximum and minimum values of the sine and cosine functions might impact on such questions. Two examples are presented and explored, and possibilities for extension activities are suggested.

On the area of an inscribed quadrilateral of a parallelogram
Author YiuKwong ManSource: Learning and Teaching Mathematics 2019, pp 34 –35 (2019)More LessIn LTM 24 Moshe Stupel presents an interesting proof without words. Starting with a parallelogram ABCD, construct quadrilateral KING with KN parallel to AB and CD, and G and I arbitrarily placed on sides AB and CD respectively. The area of quadrilateral KING is half the area of parallelogram ABCD.

Proofs without words
Authors: Moshe Stupel and Shula WeissmanSource: Learning and Teaching Mathematics 2019, pp 36 –37 (2019)More LessProofs without Words
Theorem 1 The angle bisectors of a quadrilateral form a cyclic quadrilateral.
Theorem 2 The exterior angle bisectors of a quadrilateral form a cyclic quadrilateral.