Learning and Teaching Mathematics  latest Issue
Volume 2017 Number 23, December 2017

From the editor
Author Duncan SamsonSource: Learning and Teaching Mathematics 2017, pp 2 –2 (2017)More LessIn the first article of LTM 23, Shula Weissman and Duncan Samson explore two further cases of curious calculations that make you look twice. In the second article in this issue, Debbie Stott and Pamela Vale present two activity sequences aimed at developing spatial reasoning in Foundation and Intermediate Phase learners. The third article, by Ashley Ah Goo and Franklyn Lau, explores an interesting numerical problem solving scenario, while in the fourth article Duncan Samson illustrates how openended geometry questions have the potential to nurture creative mathematical thinking. Craig Pournara then introduces an algebraic context and illustrates how keeping some aspects invariant while changing others provides an opportunity to generalise, while in the sixth article Michael de Villiers presents and comments on Thabit’s generalisation of the theorem of Pythagoras.
In the seventh article, Letuku Moses Makobe presents a useful trigonometric formula that can be used to find the height of a triangle given the base length along with the two base angles. The eighth article, by Moshe Stupel and Avi Sigler, explores a problem in probability using a geometric approach that ends with a surprising result. YiuKwong Man then uses one of the many visual proofs of the theorem of Pythagoras to prove the “Three Squares Problem”, after which Poobhalan Pillay presents a variety of solutions to a problem that appeared in the 2015 South African Team Competition. Alan Christison then explores the importance of prime numbers and prime factorisation in the expansion of a unit fraction into the sum of two unit fractions. The final article, by Jay Jahangiri and Moshe Stupel, is a charming “proof without words”.

Further curious calculations
Authors: Shula Weissman and Duncan SamsonSource: Learning and Teaching Mathematics 2017, pp 3 –5 (2017)More LessFollowing on from Duncan Samson, Benji Euvrard and Andrew Maffessanti’s article in LTM No. 22, Curious Calculations that Make You Look Twice, in this article we present a further two cases.

Developing spatial reasoning in foundation and intermediate phase learners
Authors: Debbie Stott and Pamela ValeSource: Learning and Teaching Mathematics 2017, pp 6 –11 (2017)More LessSpatial reasoning is a skill used in everyday life to solve problems using concepts of space, visualisation, and reasoning. As a cognitive skill, spatial thinking has been linked to high performance in both mathematics and science and is integral to the studies of engineering, geography, earth and environmental sciences.
The Foundation Phase CAPS document (2011a) specifically mentions spatial relationships (the position of two or more 3D objects in relation to the learner) and directionality (the ability to follow directions and to move/place oneself within a specific space) within the Space and Shape learning domain. Learners should be given opportunities to follow and give directions as well as describe their own positions and the positions of others, and other objects, in space using appropriate vocabulary.
In the Intermediate Phase CAPS document (2011b) it is outlined that learners should be able to recognise and describe shapes and objects in their environment that resemble mathematical objects and shapes, as well as explore properties of shapes by sorting, classifying, describing, drawing and interpreting as well as constructing and deconstructing models and objects. The learner’s experience of space and shape in this phase moves from recognition and simple description to classification and more detailed description of characteristics and properties of twodimensional shapes and threedimensional objects.

A square number puzzle
Authors: Ashley Ah Goo and Franklyn LauSource: Learning and Teaching Mathematics 2017, pp 12 –14 (2017)More LessAt a Teachers Across Borders workshop held in Mthatha we were given the following problem:

Euclidean geometry – nurturing multiple solutions
Author Duncan SamsonSource: Learning and Teaching Mathematics 2017, pp 15 –18 (2017)More LessIn a previous article (Samson, 2015) I highlighted the importance of providing pupils with opportunities to engage with geometric contexts in explorative and flexible ways. Such geometric scenarios should be posed with minimal directional guidance, thereby encouraging creative mathematical thinking. The beauty of such contexts is that they allow pupils to tackle the question in different ways.

Kevin’s question : an interesting discovery about squaring both sides of some equations
Author Craig PournaraSource: Learning and Teaching Mathematics 2017, pp 19 –21 (2017)More LessKevin’s question is based an important principle of learning that we work with in WMCS: What can come into focus if we keep some aspects invariant while varying others? In this instance Kevin kept the structure of the equation invariant and changed the value of the constant on both sides. This approach inevitably leads to some kind of investigation which then provides an opportunity to generalise – a fundamental goal of learning and doing algebra.

Thabit’s generalisation of the theorem of Pythagoras
Author Michael de VilliersSource: Learning and Teaching Mathematics 2017, pp 22 –23 (2017)More LessThere are many possible generalizations of the theorem of Pythagoras. Some of the most well known ones are the cosine formula, the distance in _{n} dimensions, and Ptolemy’s theorem. These three generalisations, along with seven others, are discussed in De Villiers (2009, pp. 6975). One of these other generalizations deserves to be better known and was apparently first proven by the Turkish scientist Thabit Ibn Qurra in approximately 900 AD.

Finding the height of a triangle given two angles and the included side
Author Letuku Moses MakobeSource: Learning and Teaching Mathematics 2017, pp 24 –26 (2017)More LessThis article presents a useful trigonometric formula that can be used to find the height of a triangle given the two base angles and the included side, i.e. the base length.

A problem in probability – its geometric solution and a surprising result
Authors: Moshe Stupel and Avi SiglerSource: Learning and Teaching Mathematics 2017, pp 27 –29 (2017)More LessThe broken stick problem is well known:
A stick is broken into three parts. What is the probability that a triangle can be constructed from the three parts?
The problem is solved using simple and beautiful geometrical considerations, and the result is 𝑝 = 0,25. In this article we use the same context to pose a slightly different question:
A stick is broken into three parts, 𝑥, 𝑦 and 𝑧. What is the probability that 𝑥𝑦 ≥ 𝑧^{2}?

A simple visual proof of two theorems in geometry
Author YiuKwong ManSource: Learning and Teaching Mathematics 2017, pp 30 –31 (2017)More LessThe Three Squares Problem represents an interesting geometrical result. If three congruent squares are placed side by side as shown in Figure 1, and three diagonals are drawn in as indicated, then the sum of the angles 𝛼_{1}, 𝛼_{2} and 𝛼_{3} is 90°. The reader is invited to explore this problem before reading on.

A multiple solution problem
Author Poobhalan PillaySource: Learning and Teaching Mathematics 2017, pp 32 –37 (2017)More LessLearners often tend to compartmentalize the different sections of the school Mathematics syllabus. One way to break down these perceived boundaries is to engage with problems that can lead to multiple and varied solutions. Exploration of such multiple solution problems encourages reflection and flexibility of thought, thereby enhancing one’s repertoire of problem solving skills and approaches for future challenges (De Villiers, 2016; Polya, 1945).
In this article I explore a variety of solutions to a problem that appeared in the Senior Individual paper of the 2015 South African Team Competition.

Investigating a unit fraction as the sum of two unit fractions
Author Alan ChristisonSource: Learning and Teaching Mathematics 2017, pp 38 –41 (2017)More LessWhile a unit fraction may be represented by the sum of an infinite number of unit fractions, this article considers an investigation related to the sum of two terms only. The impact and importance of prime numbers and prime factorisation in the expansion of a unit fraction into the sum of two unit fractions is highlighted. Only positive integers and sums are considered.

Proof without words
Authors: Jay Jahangiri and Moshe StupelSource: Learning and Teaching Mathematics 2017, pp 42 –42 (2017)More Less