I’m cleaning out my email inbox and found this in an old, unread blog post:
(This is the blog post, by Dan Meyer. This is the original source of the image, by Rachel Kernodle.)
My efforts at teaching inverse functions this year were fun, but, alas, not very effective. I started with the “function box,” then added the “inverse function box” with the full range of appropriate sound effects, but the two “boxes” were just too similar. There wasn’t a strong enough visual signal of the opposite-ness of the two kinds of functions. Ms. Kernodle’s stapler/remover analogy could be the key to finally getting the message across.
The experiment was a success! Students walked into class today asking to know what the inverse trig functions do (they phrased it as, “What does the negative-one mean?”), two students had reasonable theories to explain it, and one hit it right on the nose.
Before going over the worksheet, I teamed up the students for an activity where one student would write a sine or cosine value on the board, indicating whether it were a sine or a cosine; the next student would name one or several angles with that sine/cosine value and draw the angle/s on the unit circle. They were performing the inverse sine and cosine functions without knowing it. Later when a student attempted to explain his theory for the meaning of sin−1, he suddenly came out with, “I think it’s what we were doing on the board!”
Today I’m conducting an experiment in forethought. For homework I gave my trig students a worksheet with problems they don’t know how to solve (arcsine, arccosine, and arctangent) mixed in with problems they do know how to solve (sine, cosine, and tangent), with the instruction that they were to complete as many of the problems as they knew how. At the top of the page is a list of twenty-six possible solutions, of which fourteen are used. (You can find the worksheet online here like I did.) I hypothesize that two results will come of this.
First, the students will be frustrated about not getting clearer instructions and therefore needing to figure out for themselves which problems to solve.
Second, after seeing sin−1 sandwiched between regular old sin, cos, and tan problems with a list of possible answers, the students will come to class tomorrow with ideas about what sin−1 means . Some may go beyond having an idea and may understand it quite well. If so, my experiment will have been a success by allowing a little forethought on the part of the students to ease the introduction of a new function.