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 Volume 2015, Issue 18, 2015
Learning and Teaching Mathematics  Volume 2015, Issue 18, January 2015
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Volume 2015, Issue 18, January 2015

From the Editors
Authors: Duncan. Samson, Marcus. Bizony and Lindiwe. TshabalalaSource: Learning and Teaching Mathematics 2015 (2015)More LessLTM 18 begins with a delightful article by James Metz who investigates what arrangement of three small, circular coffee filters would cover the largest circular area. In the second article in this issue, Poobhalan Pillay explores triangles that have one angle twice another. The third article, by Duncan Samson and Simon Kroon, explores three different visually engaging methods of carrying out long multiplication, while in the fourth article Marcus Bizony expands on Alan Christison's article on average gradients in LTM 17. Andrew Maffessanti and Duncan Samson then investigate the relationship between the angle subtended by a chord and the chord length, while in the sixth article Michael de Villiers explores Varignon's theorem and slays a geometrical 'monster'.

Keith's coffee filter problem
Author James. MetzSource: Learning and Teaching Mathematics 2015, pp 3 –6 (2015)More LessKeith posed the following problem :
"The coffee machine where I work requires a large circular filter, but the last one has just been used. However, there are some smaller filters left over from a previous machine. I decide to improvise by sticking three of the smaller filters together and cutting out as big a circle as possible to take the place of the large filter. What arrangement of the three smaller filters will cover the largest circular area?"

Triangles with one angle twice another.
Author Poobhalan. PillaySource: Learning and Teaching Mathematics 2015, pp 7 –9 (2015)More LessTriangles with One Angle Twice Another.

Methods of multiplication
Authors: Duncan Samson and Simon KroonSource: Learning and Teaching Mathematics 2015, pp 10 –15 (2015)More LessINTRODUCTION
In this article we explore three different visually engaging methods of carrying out long multiplication. The standard algorithmic approach to long multiplication, as typically taught at primary school level, is generally along the lines of that illustrated in Figure 1.

When a straight line meets a parabola
Author Marcus BizonySource: Learning and Teaching Mathematics 2015, pp 16 –17 (2015)More LessIn his article about the average gradient between two points in LTM No. 17 (pp. 1719), Alan Christison points out that when two lines are tangents to a parabola, the gradient at the point on the parabola midway between the two points of tangency is not only equal to the gradient of the straight line joining those points, but is also the average of the gradients at the two points of tangency. This scenario is illustrated in the following diagram.

Angles subtended by a chord.
Authors: Andrew Maffessanti and Duncan SamsonSource: Learning and Teaching Mathematics 2015, pp 18 –22 (2015)More LessINTRODUCTION
During a recent Grade 11 geometry lesson the class engaging with the theorem that angles subtended by a chord in the same segment of a circle are equal. As part of the discussion we touched on one of the corollaries of this theorem, namely that angles subtended by chords of equal length in a given circle are equal.

Slaying a geometrical 'Monster' : finding the area of a crossed Quadrilateral
Author Michael. De VilliersSource: Learning and Teaching Mathematics 2015, pp 23 –28 (2015)More LessINTRODUCTION
Varignon's theorem states that the midpoints of the sides of an arbitrary quadrilateral form a parallelogram. The parallelogram thus formed, known as the Varignon parallelogram, has the property that its area is half that of the original quadrilateral. In Figure 1 the Varignon parallelogram EFGH is formed by joining the midpoints of the sides of quadrilateral ABCD. Proving that the midpoints of an arbitrary quadrilateral doing deed form a parallelogram can readily be accomplished using the midpoint theorem. In Figure 2, one of the diagonals of quadrilateral ABCD has been drawn in. Since H and E are the midpoints of the sides of triangle ADB, it follows that HE is parallel to DB. Using a similar observation in triangle BCD we have FG parallel to BD from which it follows that HE is parallel to FG. Using AC, the other diagonal of quadrilateral ABCD, it can similarly be shown that EF is parallel to HG.

Surd equations and restrictions
Author Alan. ChristisonSource: Learning and Teaching Mathematics 2015, pp 29 –31 (2015)More LessINTRODUCTION
Restrictions form an important element of mathematics, and they arise in a number of different contexts. Restrictions are sometimes explicitly stated, for example a scenario where the solutions to a particular equation are restricted to natural numbers. There are also many situations where the restrictions are not explicitly stated but are mathematically embedded within the given algebraic equation or expression. Typical examples encountered in the school syllabus include algebraic fractions, logarithms, exponential equations, and equations involving surds. This article looks specifically at surd equations, and in particular how these embedded restrictions may be used to determine whether answers derived from solving such equations are indeed solutions of the original equation.

Finance mathematics  missed annuity payments
Author Andrew MaffessantiSource: Learning and Teaching Mathematics 2015, pp 32 –33 (2015)More LessA typical Advanced Programme Mathematics finance question asks learners to deal with a scenario where a series of annuity payments is missed from a loan amortization. By way of example, consider the following question:
Mario borrows R200 000 from a bank at an interest rate of 7% per annum, compounded monthly, for a period of 25 years. He makes his first repayment at the end of the first month. Due to unforeseen circumstances he is unable to make his loan repayments for the entire 11th year. However, Mario is determined to finish paying back the loan within the original agreed term (i.e. 25 years) and consequently needs to increase the repayment amount (starting in the 12th year) to take into account the twelve missed payments.

The grade 9 maths ANA  what can we see after three years?
Authors: Craig. Pournara, Sihlobosenkosi. Mpofu and Yvonne. SandersSource: Learning and Teaching Mathematics 2015, pp 34 –41 (2015)More LessINTRODUCTION
The Annual National Assessments (ANA) were introduced to address South Africa's poor performance in international mathematics assessments such as TIMSS and SACMEQ. The Grade 9 test was first written in 2012 with a national average of 12.7%. In 2013 the average increased to 13.9% but then dropped to 10.8% in 2014. The overall picture is one of very poor performance with the result that many questions are being asked about the value and purpose of the ANA.
The Department of Basic Education (DBE) identifies three purposes for the ANAs : (1) a measure for the state to gauge improvement in the education system, based on learner performance yearonyear; (2) a diagnostic assessment to identify areas of weakness in learners' performance; and (3) to provide a model of good assessment practice. While these are all important goals, they are not necessarily achievable through a single instrument. However, this is not the focus of our article. Our purpose here is to report on a content analysis of the three ANA papers which we undertook in the first quarter of 2015, paying particular attention to cognitive demand and levels of difficulty. We share our findings on these aspects, as well as on trends that arise as we look across the first three years of the Grade 9 Maths ANA.

Calculating variance using raw scores
Author Moses L. MakobeSource: Learning and Teaching Mathematics 2015, pp 42 –43 (2015)More LessVariance and standard deviation (the square root of the variance) are statistical measures that represent how spread out or dispersed a set of data points is. A small variance indicates that the data points are closely distributed around the mean, while a large variance indicates that the data points are much more spread out. Within the South African school syllabus, variance is usually calculated as the average of the squared differences from the mean.

Compound proportions
Author James. MetzSource: Learning and Teaching Mathematics 2015, pp 44 –45 (2015)More LessIn an old copy of An Advanced Arithmetic for High Schools, Normal Schools and Academies (Wentworth, 1898) I recently came across the following problem:
If 4 men mow 15 acres in 5 days of 14 hours, in how many days of 13 hours can 7 men mow 19.5 acres?
This problem is typical of a class of problems involving compound proportion in which more than two quantities are involved in two situations where one of the quantities is missing. The purpose of this paper is to revisit this class of problems and employ the tools of direct and inverse variation to find the solution. While this type of problem may have fallen out of fashion, there is still some valuable mathematics to be found within.

Effective interest rates : making sense or cents?
Author Craig PournaraSource: Learning and Teaching Mathematics 2015, pp 46 –50 (2015)More LessAn effective interest rate enables us to compare different interest rates over different periods, and with different compounding frequencies. Consequently, an understanding of effective rates is important for personal finance. In the SA school curriculum, effective interest rates are introduced in Grade 11 in both Mathematics and Mathematical Literacy. However, text books generally do not extend their discussion to engage with the relationship between effective rates and actual banking products. In this article I explore the notion of effective interest by focusing on its definition, its connection to percentage increase, and the derivation of the effective interest formula. I then discuss the interest rates of notice deposits as advertised by two big South African banks and show how effective rates are used in each case. Through this I show that the effective rate formula used at school is not the formula generally used by banks. Rather, banks quote an 'average effective annual rate' as this enables them to quote seemingly higher effective rates.