I'm working an idea out in my head... I'm about to start discussing graph translations with my Precalculus students, which is a topic that I've never quite felt like I've really been able to guide my students to understanding. Instead, they memorize rules (this happens in all sorts of topics, of course, but I'll leave those for another day). In particular, I'm not satisfied with their understanding that when x is replaced by (x -2) in a function, the resulting graph has shifted by 2 in the positive direction, while (x + 2) would shift the graph in the negative direction. They seem to balk at first, and then just accept it as one of those weird math rules that I invented to make their lives a bit more complicated. They don't seem to reach the level of understanding that I want to see...
So here's my idea, though I'm changing details as I write: I want to line the students up (outside?) so that each of them represents a number on a number line (the x-axis). They will literally hold a function (colored ribbon at certain heights? a rigid model of a function?) in their hands. This way each student (x), will have a value for the function (y) in front of them. What I want them to see is that when I ask them to take the y from the person (x) who is two to their left as their own, the function moves to the right.
--will they have a solid object as the function? numbers to hold up? a ribbon? a lane line from the pool? does it matter?
--will all of the students participate, or will some watch? is watching or participating more effective?
--shall we record some video to look back on and share with the world?
--is this going to help them understand the concept?