Some of my precalculus students were staying after school to (gasp) study for a test coming up in a couple of days (new behavior). Anyway, they were working through logarithm and natural log problems, and one of the students muttered, "what are these used for (good for) anyway?". Then another students quickly pipes up with, "shhhhh, she'll assign us a project!".
This made me laugh. Throughout the year, I've given them various small "projects" to research things like:
* how do the conic sections show up in real life (what is it about their properties that lends them to the use).
* who uses imaginary numbers
* how is trigonometry used and find one fact you find interesting about it.
* what are fractals and what's one use of them.
These are usually small posters (8.5 x 11) or just handed in on paper. They have to cite their sources and have a good presentation. This way, I make them do the work, and sometimes, I have nice posters to hang around my room.
The other funny incident happened while they were reacquainting themselves with log and ln and "e" and their calculators. After a bit, I asked them to look on their calculators and notice the positioning as a pair of ln & e ... and log & 10^x ... and linked it to the positioning of x^2 & sqrt(x), and cos x & arccos x. I made the connection that they are all inverse operations of each other, and that's why they're placed that way. Then a student comes up with another example of inverse operations: "oh like the positioning of the ON & OFF keys".