## Wednesday, December 1, 2010

### Thin Lens Equation/Rational Equations

This lesson went well with my honors students this week, and I'll see how it goes with my lower track students next week. The idea is that the only legitimate context for much of the material in Algebra II is its scientific context, and I used this to give my students this context while practicing solving rational equations.

In my mind this is a step up from the "average rate" questions that dominate most of the applications of rational equations that I've seen. This is also easier for students (and this teacher) to understand and explain than resistor and circuit problems.

Here's my power point presentation, which I used to tease them ("Ever wonder why objects in your sideview mirror are closer than they appear?"), leading into the worksheet.

They split up into pairs and worked on the worksheet together.

Thin Lens Worksheet

1. Do all mirrors have a particular focal length? How do they decide what focal length to use for those mirrors? Do my glasses have a focal length? (Do they distort distances?)

2. Gah! I just wrote a long comment and its disappeared. That means I'm going to write a shorter, worse one.

First of all, here are four diagrams that show what the focal length is.
http://en.wikipedia.org/wiki/File:Focal-length.svg

Let's start with this: light from a light bulb fills out a room easily because the light leaves the bulb in every which direction. It's not all moving in a straight line. Contrast this with a laser light, which barely scatters, and instead travels straight forward. This straight kind of light is called collimated light.

Now, send collimated light through a converging lens, and the light will converge around a single point (the lens will bend the light inwardly). This point is called the focal point. There's a distance in between the lens and the focal point. This is the focal distance, and you can see this in the first figure above.

But, wait, not every lens bends light inwardly. Some lenses are diverging lenses, and the spread light out instead of pushing it inward. Here the light will never focus on a single point. Still, we define a focal point using the extension of lines, but we change the sign of the focal length to indicate that the lens is diverging. This can be seen in the second diagram.

So, yeah, this is true of every lens and mirror. (Mirrors are another small step away from lenses, but the way that we define focal length is quite similar and can be seen in the third and fourth diagrams above.)

Now, what about glasses? These lenses also bend light, and so they also have a focal length. Reading glasses are easier to deal with, since they make things larger using the same principles as a magnifying glass. They have a focal length that makes small objects appear larger than they actually are by distorting their apparent distance.

Thinking about it now, here's how I might do this lesson. I would give kids spoons, and have them play with them, asking them to figure out whether their images are upside down or right side up. I'll challenge them to make an image that is right side up. Eventually, they'll discover that right-side up is possible, but only if the spoon is very close. Then maybe I'll give them another lens (I would need another converging lens, like the thing they use in gas stations) to figure out when the image is flipped right side up in that case. Then we could spill the beans with a diagram, and then use an equation to get really precise info from that diagram (i.e. the distances that will actually flip the image.)

3. There must be some physics teacher who has done this work already and made an awesome optics lesson. If only I had somewhere where I could publicly post my question! (Shameless plug: http://area51.stackexchange.com/proposals/29616/teaching-and-tutoring-math-and-science)