# Well behaved …

Mea culpa.

In my time, I may have made a few things up. I’ve “created” a new meaning for the verb: “to pythagorize”, and I regularly tell my students of the “Forgotten Formula”

(The first: to pythagorize means to “to philosophize in the manner of the Pythagoreans”. In my classroom it means to use or apply Pythagoras’ theorem. For example, to find the magnitude of a vector I may tell my students to “pythagorize the i and j (and k if working in three dimensions) components.” I think it may be stretch to suggest we are philosophizing in the manner of the Pythagoreans when doing this.

The second is a bit of cod psychology. The most useful item not included on the A level formula sheet is, imho, the identity:

sec2x = tan2x + 1

yet my experience has taught me that students often grind to a halt in a question when remembering this identity would unlock the problem for them. Hence I have christened it “The Forgotten Formula”, have it labelled as such on my wall, and refer to it by this name. On more than one occasion, I have had to explain to my students that this is one of my little (many?) quirks and if they start talking of “the forgotten formula” with other teachers and mathematicians they will, at best, look back at them blankly. But if it helps my students remember, than I’m happy)

So it should have come as no surprise to me when introducing my students to “well behaved functions” they thought it was more of my make-believe (although they loved the concept, and I may have embellished my teacher talk with tales of mis-behaving functions having to spend Thursday lunchtime in detention.) So I had to convince them it was “a thing” and that we describe a function as well behaved if it is continuous and all of its derivatives are defined and continuous. For our purposes, in A level maths, we were exploring the sign change method to find a root between a pair of values, and discussing when this method may not work (if the function is not well-behaved!)

I’ve taught this class for two years, I hope I’ve managed to teach them something. I suspect that, in ten years time, they may not remember much maths, but probably will remember when we talked of well behaved functions!