“If I had an hour to save the world I would spend 59 minutes defining the problem and one minute finding solutions”
- Al Einstein

## Tuesday, November 9, 2010

### Brainstorming: Graph Translation

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.

Still TBD:
--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?

1. The idea sounds interesting. I think it's worth a shot.

Here is how I understand the question. When x is replaced by x-2, we can say that the graph shifts two units to the right. But another point of view is that the graph stays in the same position, and the *coordinate system* moves two units to the left.

There is a scene from the 1984 comedy movie "Top Secret!" that illustrates the concept. The protagonist is on a train that appears to be departing from the station. Then it is revealed that the train is not moving, but the *station* is moving backwards. http://www.youtube.com/watch?v=AgPFhLFSkaI

2. Thanks, David... I didn't have the idea of the coordinate system moving, even for myself. Plus, any excuse to incorporate classic cinema into the classroom!

3. This sounds cool - some kinesthetic learning too. Can you film it so other learners can view it later to see it. It can be tricky to see the overall picture when you are a part of a ribbon.
You need a set of axes - ones that you can shift for the point above. Maybe heavy rope, something like that. If not, use lines on a gym floor or an outside area with lines on it. If you do film it, post it on your blog so we can see how it went.

4. I think it's a great idea. As a student (and a secondary ed math major), I definitely learn better through examples such as this one. My AP Calc teacher in high school taught us this way and brought a whole new level of understanding to the class. I say go for it. I don't really think the objects matter, but I would go with numbers or even something like shapes (circle, square, star, triangle, and whatnot). I also think that the whole class should participate so everyone gets an opportunity to be part of the example and yes, record it so not only can you share it with future students, but the current class can see what's going on, as well.

5. I'm appreciating the encouragement. I really need to get myself pretty well scripted here, since I've managed to confuse several of my colleagues already. That doesn't bode too well. I'm not going to chicken out, though... I don't think I can really damage their understanding, here.