My IB Mathematics course has started out well, with a successful introduction of functions, domain and range and composite functions. So maybe I started getting too cocky and didn't spend enough time on today's lesson - on inverses. Whatever the reason, it felt like hitting a wall. While students happily solved questions such as "what's k(x) if h(x) = 5x and h(k(x)) = x?", they could not even start on the very similar question "what's the inverse of h(x) if h(x) = 5x+3?" Even with me explicitly explaining the connection, by definition, to the previous question, students were dumbfounded by these developments.

Now I'm the one who's a bit stuck. Do I give an extra class on inverse functions - this would break my planning and mean I'm lowering my standards a bit. Or do I send them to the math study hall and go on with my planning, hopefully managing to fit in inverses now and then again? I'm leaning towards the latter but would really like to figure out what went wrong (and how to fix it). Maybe the language just got to abstract? Riley at Point of Inflection has a series of posts I probably should have read more carefully.

Ah well. "Where you stumble, there lies your treasure", is this nice saying I've always liked. Maybe there'll be something superbly useful in this mess somewhere.

I believe that the visual connections can be quite powerful in discussing inverses. The fact that log(x) 'undoes' e^x in the same way that x squared 'undoes' the square root of x is easier to understand. If you compose them graphically you can see the line y=x as the line of symmetry of the two graphs.

ReplyDeleteIt is a somewhat roundabout way of reintroducing inverses, but it could possibly serve as another outlet for their exploration/explanation.

Well, I did show then the reflection in the y=x line, but maybe you mean something else by "compose graphically"?

ReplyDeleteI wasn't sure that you had shown the students the possible visual representations. What really connected the ideas of inverses for me is when I realized that f(g(x)) = x when f and g are inverses, and that h(x) = x = f(g(x)) is the line of reflection. Essentially, there are three important functions involved in inverses - the two inverses and the identity function (y = x). I'm not sure where exactly I'm going with this... just trying to look at inverses in a different way.

ReplyDeleteYep, thanks, I think it's def the three function connections and inverse definition the students are struggling with. Then again, this is the first time I'm teaching inverses and I'm noticing new things about them every time I try to explain them to a student. Maybe it'll all come together to something decent next time. Tomorrow, though, we'll look at the definition again, work through a few exercises and move on.

ReplyDeleteSolved it! Mr Sam at http://samjshah.com/2009/04/14/composition-of-functions-and-their-inverses/ had a great idea which I copied. The thing that really did the trick was when the students saw how f and f-inverse undo each other step by step.

ReplyDeleteAwesome. I'm sure I will have to teach inverses soon. I definitely think I'll push to include something similar to this when that time comes. Thanks for sharing and I'm glad it worked out.

ReplyDelete