Trigonometry from a Different Angle

By Cavanagh, Michael | Australian Mathematics Teacher, Spring 2008 | Go to article overview

Trigonometry from a Different Angle


Cavanagh, Michael, Australian Mathematics Teacher


I read with interest the article on teaching trigonometry recently published in The Australian Mathematics Teacher (Quinlan, 2004). The article reports on a lesson given by a student-teacher in which the pupils were involved in a practical activity designed to introduce the tangent ratio and demonstrate its utility in some real-life contexts. Quinlan (2004) concludes with some general principles for introducing new mathematical concepts, ideas which he was fortunate enough to have learned when he completed his teacher training in the 1950s. The author also suggests that teachers begin by allowing students to explore concrete examples of a concept before presenting its definition, and that the formal terminology and symbolism associated with the concept should be introduced much later, after students have developed a sound grasp of the basic ideas.

Re-thinking classroom practices

My recollections of the mathematics methodology subjects I undertook in the early 1980s are quite different. I remember being encouraged to adopt a very expository style of teaching in which each new concept is introduced by its formal definition. The teacher should then explain a few carefully chosen examples for students to copy into their books, and then provide plenty of graded practice exercises from the textbook for students to complete. It is what Mitchelmore (2000) calls the ABC approach: where abstract definitions are taught before any concrete examples are considered. So, for many years, my teaching of trigonometry in Year 9 began with exercises in identifying opposite and adjacent sides in right-angled triangles, definitions of the trigonometric ratios and the mnemonic SOHCAHTOA, then lots of work on calculating unknown sides and angles, all devoid of any realistic context. Finally, right at the end of the topic, I gave the class some word problems involving applications like angles of elevation and compass bearings.

It was only when I undertook further study some years later and was exposed to alternative ways of thinking about the nature of mathematics and its pedagogy that I began to reassess my classroom practice. There was no blinding light or sudden conversion but, over time, I did make some significant changes in my teaching. In my trigonometry lessons this meant not following the textbook so slavishly, changing the order in which students tackled the basic ideas associated with right-angled triangles, and reconsidering the kinds of classroom activities I provided for students. I was also mindful of the Standards for Excellence in Teaching Mathematics in Australian Schools (AAMT, 2002) and the advice on professional practice in Domain 3. In particular, I wanted to use a variety of teaching strategies and try to take account of students' prior mathematical knowledge. The purpose of this article is to outline briefly some of the elements of my new approach and how I developed them.

Introducing the ratios

First, I thought it important for my Year 9 students to understand that "sine", "cosine" and "tangent" are ratios whose value depends on the relative size of the sides in a right triangle. I used a diagram like Figure 1, found in many textbooks, and asked the students to measure BF, CG, DH, and EI, the lengths of the sides opposite the marked acute angle, [theta]. Then the students measured AF, AG, AH and AI, the lengths of the hypotenuse in each triangle. Finally, I asked the students to divide the values for each of the opposite sides by the hypotenuse in [DELTA]ABF, [DELTA]ACG and so on, until they obtained approximately the same value in each case, and so I was able to explain that they had found the sine ratio! This was not a very auspicious beginning at all and the students were unconvinced by my explanation but they accepted it and we moved on to repeat the process for the two remaining ratios. In hindsight, this approach was still too abstract and provided no rationale for measuring those particular sides to obtain the three ratios. …

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