# 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. …

## The rest of this article is only available to active members of Questia

Already a member? Log in now.

### Notes for this article

If you are trying to select text to create highlights or citations, remember that you must now click or tap on the first word, and then click or tap on the last word.
One moment ...
Default project is now your active project.
Project items
Notes
Cite this article

#### Cited article

Style
Citations are available only to our active members.
Buy instant access to cite pages or passages in MLA 8, MLA 7, APA and Chicago citation styles.

(Einhorn, 1992, p. 25)

(Einhorn 25)

(Einhorn 25)

1. Lois J. Einhorn, Abraham Lincoln, the Orator: Penetrating the Lincoln Legend (Westport, CT: Greenwood Press, 1992), 25, http://www.questia.com/read/27419298.

Note: primary sources have slightly different requirements for citation. Please see these guidelines for more information.

#### Cited article

Trigonometry from a Different Angle
Settings

#### Settings

Typeface
Text size Reset View mode
Search within

Look up

#### Look up a word

• Dictionary
• Thesaurus
Please submit a word or phrase above.
Print this page

#### Print this page

Why can't I print more than one page at a time?

Help
Full screen
Items saved from this article
• Highlights & Notes
• Citations
Some of your highlights are legacy items.

## Questia reader help

### How to highlight and cite specific passages

1. Click or tap the first word you want to select.
2. Click or tap the last word you want to select, and you’ll see everything in between get selected.
3. You’ll then get a menu of options like creating a highlight or a citation from that passage of text.

## Cited passage

Style
Citations are available only to our active members.
Buy instant access to cite pages or passages in MLA 8, MLA 7, APA and Chicago citation styles.

"Portraying himself as an honest, ordinary person helped Lincoln identify with his audiences." (Einhorn, 1992, p. 25).

"Portraying himself as an honest, ordinary person helped Lincoln identify with his audiences." (Einhorn 25)

"Portraying himself as an honest, ordinary person helped Lincoln identify with his audiences." (Einhorn 25)

"Portraying himself as an honest, ordinary person helped Lincoln identify with his audiences."1

1. Lois J. Einhorn, Abraham Lincoln, the Orator: Penetrating the Lincoln Legend (Westport, CT: Greenwood Press, 1992), 25, http://www.questia.com/read/27419298.

## Thanks for trying Questia!

Please continue trying out our research tools, but please note, full functionality is available only to our active members.

Your work will be lost once you leave this Web page.

Buy instant access to save your work.

Already a member? Log in now.

Search by...
Show...

### Oops!

An unknown error has occurred. Please click the button below to reload the page. If the problem persists, please try again in a little while.