Tuesday, April 10, 2012

Modeling Plane Mirror Images

Last post, students modeled both reflection (using tennis balls) and the length necessary for a mirror to be "full length."  We've done quite a bit of ray tracing in our work with shadows, so I had them try to prove, using their reflection model (the Law of Reflection) and ray tracing, that the necessary length for a FLM was indeed half of the subject's height.

This was tricky for them - I had several students successfully find an example ray diagram that seemed to fit the model, but an example isn't a general proof.  Most of those students did some guessing-and-checking (as did pretty much all of the stragglers that did it in groups on whiteboards at the beginning of class.  I wasn't altogether happy with how this went as a HW exercise.  Some more scaffolding might be needed, but mostly there wasn't enough perseverance by most.

As an aside, the large majority did indeed bring in something written down, but there was a significant fraction that brought in something like these:

So folks really didn't know whether they had "done it right" or not.  This is an overarching critical thinking skills problem, not a physics problem.  It's one that we rarely address anywhere, though.  The idea here is that we have two paradigms controlling the situation:

  • The angles of the incoming and reflected rays (preferably measured from the normal to the surface, but either will work at the moment) are equal
  • We see things when light goes into our eyes after bouncing off of the object.  To see something in a mirror, the light bounces off of the object, then the mirror, then our eyes.
The first drawing there doesn't involve the eyes at all - the head and feet are involved, but there's a lack of understanding of the mechanism of seeing and/or of what the heck what we're drawing means.  The second drawing has massively unequal angles for each ray, and so is impossible.

This is where students need to build up the amount of fluency with the mental models that we use to see that these are ridiculous on their face.  It's like we've said "OMG, she tweeted me this 6 page sob story it wow soooo boring," or "he won the game with a touchdown in extra innings." They have mental models of those situations that show these statements, without effort or analysis, to be absurd.  The same isn't true for most of their 'academic' models.  The same things true for looking at a force diagram and saying "oh, that's obviously unbalanced, but it's supposed to be moving with constant velocity, so something's wrong."  Part of it is our job to pay attention to that, require it, talk about it in class, etc. and to connect what we study to the real world, so that students really do think that "real life" and "school" aren't disjoint sets.

So... folks eventually (and mostly by trial and error) got to here:

At this point, we get to talk about example vs. proof and how to make an abstract argument here about the relationship between the length of the mirror and the height of the person.  This actually hinges upon something that the non-guessers figured out, and in connects the "real world" back to the light discussion.  Just have two folks stand and bounce a ball to each other.  They bounce it off of the point on the floor halfway between them.  "Of course!" they say.  It's great to recognize it after the fact, but that model's just not in there firmly enough if they don't see it the first time. This allows us to really easily find the places on the mirror where those rays reflect, and to make a congruent triangles argument to show that the model that we came up with experimentally follows naturally from the law of reflection.


OK, all of this guessing and checking is pain, and the midpoint trick only works if the observer and object are equidistant from the mirror.  We need a better way.  We did the classic "vertical piece of glass in a dark room with backlight mirror" trick - prop to glass on a sheet of paper, put a little lab weight in front of it, and maneuver an identical weight so that it always aligns with that image, when viewed from any angle.  Trace the object, image, and mirror a few times, and you can start to build a model to predict the size and location of an image.

The biggest idea that I'm pushing this year (so much so that I'm not having converging lenses, plane mirrors, etc. be separate standards) is the the location of the image is the point from which the light rays appear to diverge.  Everybody that sees the image points to it, and they only agree on one point.  It's a big picture that can get lost in all of the ray tracing rigamarole.

Each student's sheet (have each kid do one, it's quick) looks something like this:

The model follows pretty quickly at this point: the images are the same (perpendicular) distance behind the mirror that the objects are in front of it, and they're the same size as the objects. After making sure that they can do this for mirrors not aligned parallel to the paper's edges, we try to connect this again to the mechanism of seeing and the law of reflection.

First: have them pick one of the images and draw a couple of observers (these are supposed to be little eyes).  now, if we saw the image, and thought that it was there, what direction did the light rays have to come from?
 OK, good enough.  But... where did they really come from?
Hey - the law of reflection was upheld - automatically!  Look ma, no protractor!  Images are useful, quick, and easy.  Three great reasons for us to use them!

Let's bring it back to our ray tracing for the FLM.  It's sooooo much easier to just draw the image of the person (apologies, Randall) and work backwards to figure out where the rays came from, where they hit the mirror, etc.  This works great for figuring out the field of view of anyone, etc.  
The trickiest bit for the students is to see that this is exactly the same thing that we'll be doing with convex mirrors... and with concave mirrors... and with diverging lenses... and with converging lenses...  and with Coke glasses in restaurants... and with spearfishing... and with quarters in coffee cups.  Big picture, folks - keep their eyes on the prize.

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