5.+Trig+Ratios+-+Teachers

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Teaching Ideas Student Misconceptions Textbook Analysis

Teaching Ideas

 * Exploring the Unit Circle**
 * The unit circle as a diagram of the night sky - trigonometry as a way to do astrology/astronomy.
 * Use the idea of the Carousel (see Useful Analogies below)
 * Consider introducing the idea of squine and cosquine http://www.youtube.com/watch?v=yk-QCbTGNIY


 * Really hammer home the idea of functions**
 * Write sin(x) not sin x.
 * Show function machines.


 * Use lots of graphs**
 * It's astounding how few textbook show the graphs of the trig functions in the context of each problem.
 * Help explain the difference between sin x and sin 2x using graphs. This explains why you can't simplify sin(2x)/2 by dividing the 2x by 2.
 * When looking at trig equations, show the graphics - this helps explain why, for example, sin 3x = 2 has so many solutions for just x in the range 0 <= x <= 360
 * When looking at the Auxiliary Angle Transform a sin x + b cos x, show the graphs. It's clear the result is another sin (or cos) function of a different amplitude, offset by an angle.


 * Useful Analogies**
 * Rotating angles around the unit circle is like riding a carousel - round and round we go. Negative angles: go the other direction

Flickr:jarefman

 * Make the difference clear between an equation and an identity:
 * Trig identities are like Superman and Clark Kent. The LHS is the same as the RHS - they are just wearing different costumes. Superman looks all powerful, but sometimes it's easier to grab hold of Clark Kent.
 * The t-transformation and the auxiliary angle transformation are just mechanisms to switch between forms:
 * Trig equations are when we hit specific values - when our carousel reaches a specific point.

**Textbook Analysis**

 * Summary**
 * Pender moves some Prelim material to Year 12 - allowing him to focus on essentials and do the harder trig topics using radian measure.
 * Fitzpatrick also introduces radian measure early.


 * Detailed Analysis**


 * **Unit Circle**
 * Fitzpatrick 2Unit has good diagrams and explanations of reflection properties that lead to identities using reference angles.


 * **Equations**
 * Pender gives very clear breakdown of concepts and techniques for solving trig equations.
 * Grove section is weak - no clear methodology, most exercise question are too easy for Extension 1 students.


 * **t-method and Auxiliary angle method**
 * Grove combines these into one exercise - probably not enough exposition. Pender and Fitzpatrick treat these separately.
 * Only Pender shows a graph of the summation of the two trig functions - seems like a major omission in the other books.
 * Note: Pender & Fitzpatrick are working in radians when they cover this work.


 * **Harder trig problems (including 3D)**
 * Most of the questions in Grove are too easy. Although looking at past HSC papers, the examiners appear to have been kind and provided diagrams for the harder questions.