The difference between a tautology and a justification

Here's an optics homework question I asked the other day:

A concave mirror has a focal length of 10 cm.  It is desired that an image of an object be created that is upright and magnified three times.  (a) Should the object be placed greater or less than 10 cm from the mirror?  Justify your answer.

Now, in and out of class, we have reviewed this type of situation ad nauseum.  We have showed with equations, experiment, and ray diagrams that an object closer to the concave mirror than a focal length produces an enlarged virtual image; any object placed beyond a focal length from the mirror will produce a real image.  My students hopefully KNOW these mirror facts as well as they know that a field goal is worth three points.

Consider this justifcation, then:

The object should be placed less than 10 cm from the mirror, because that produces an enlarged, virtual image.

Such a justification earns no credit!  This response is a tautology, circular reasoning that essentially says "it is because it is."  I mean, *I* know that this student is reasoning from personal experience or memorized facts, but the student could just as easily be guessing and restating the conditions from the question stem.

A correct justification here must go beyond a restatement of the facts, no matter how simple the reasoning might be.  Possible correct justifications:

*... because an object placed inside the focal point of a concave mirror will *always* produce a virtual, enlarged image.  That's a property of concave mirrors.

*... look at my ray diagram here.  You see that the image is virtual and enlarged, and the object is inside the focal point.

* ... look at this calculation here.  I get an object distance of nearly 7 cm, which is clearly less than 10 cm.

* ... in my class we looked at images in concave mirrors, and we saw that only objects close to the mirror produced upright images.

Don't fall into the tautology trap.

A different use of a clicker quiz -- snell's law

Springtime for seniors brings a competition as to who can do the least amount of work.  That's certainly not true for all seniors, but if you listen to any high school faculty this time of year, you'd think that this year's crop of 17 year olds were the passive-aggressive scourge of Satan.  In a previous note, I explain my use of the "exemption" as one prong in my defense against the senior slide.

The gist of the exemption:  Students who put in particularly strong effort on out-of-class assignments earn the right to skip a future assignment of their choice.  The first exemption of the year in general physics is awarded to the student with the highest homework average from the previous trimester.  From here on, exemptions will be given for things like maintaining an A homework average over the course of a full week, writing a perfect quiz when that quiz is based on a homework assignment, or a perfect fundamentals quiz.

The homework problem for today involved refraction in a triangular block of glass.  I've introduced Snell's Law already, and we've done several basic problems with simple geometry.  Today's problem was more complicated, because the normal was NOT straight up and down the page.  Take a look at the diagram to the right, which is slightly editied from a problem in the Glencoe text.  The homework problem did not label the 49 degree angle; instead, it gave θ_i as 45 degrees, and asked to find angles A and B, and θ_exit

I warned everyone ahead of time that angles A and B are NOT 60 degree angles.  That doesn't prevent half the class from making that assumption anyway, but it sets up for success those students who pay attention.

For today's quiz, I gave the diagram shown, with the 49 degree angle.  I asked the six questions at the end of this post, to be answered on the classroom response system (the clickers).

Now, I am careful NOT to use the clickers for quiz purposes through most of the year.  At first, I absolutely do not want students seeing how their peers did.  I don't want boasting about good scores, I don't want the sour-grapes rationalization that inevitably follows when someone gets a 25%.  Over the course of the year, 1/4 on a multiple choice quiz is a drop in the bucket.  My guys can deal with that.  What they CAN'T deal with at first is the idea that the average score on the quiz might only be 50%. 

In the spring, though, I use the clickers because I WANT score distributions to be public.  Everyone needs to see that yes, people are getting 6 correct answers on this quiz.  The clickers' instant feedback allows me to pinpoint the folks who missed a straightforward question, and make sure they know why they got it wrong.  And most importantly... when the quiz is over, I can IMMEDIATELY and publicly award an exemption to everyone who got a perfect score. 

GCJ



1. What is the angle of incidence at the left edge, labeled θ_i in the diagram?

2. What is the angle of refraction at the left edge, labeled r in the diagram?

3. What is angle A?

4. What is angle B?

5. What is the angle of incidence at the right edge?

6. What is θ_exit?

 

Using Red AND Green Lasers

As soon as we get back from spring break, I will be discussing "physical optics" -- double slits, diffraction gratings, thin films, and so on.  I always start with the double slit -- why light through a double slit produces a diffraction pattern, and the equation for the location of the minima and maxima. 

Here the best demonstration, I think, is to shine a red laser through a diffraction grating.  (No, I don't start with a real double slit... a diffraction grating produces sharp maxima which don't invite arguments about whether we're really seeing bright and dark spots; and, the mathematics for diffraction gratings are identical to those of double slits.  Later, after the class is comfortable predicting the location of the bright spots, I explain the qualitative differences between diffraction gratings, double slits, and single slits.)

This demonstration can start off quantitatively.  Using the known slit spacing of the diffraction grating, the wavelength of the laser (printed on there somewhere), and the distance to the screen, predict the distance between the first two bright spots using dsinθ = mλ. 

Next, ask the qualitative question: if the red laser is replaced by a green laser, what happens to the distance between bright spots?  Let the class argue it out amongst themselves:  the wavelength of green light is smaller than the wavelength of red light.  (Yes, you have to know that fact for the AP exam.)  By the above equation, reducing the wavelength while keeping d and m constant reduces sin θ as well.  As the sine of an angle decreases, the angle itself decreases.  So, the angle to the bright spot decreases, and the bright spots will be closer together. 

Note that it is legitimate to argue using the small angle approximation equation x = mλL/d.  You'd find that reducing the wavelength reduces x, the distance from the central maximum to the mth spot.

Then, if you have different diffraction gratings, you can ask how the distance between spots will change using these other gratings.

"But I don't have a green laser," you say.  I bought one a few years ago for $200.  But now both green and red lasers have come way down in price.  Search around.  I just bought a pack of twelve keychain red lasers for about $2 each; I found green lasers for about $10 each.  Considering its other uses in geometric optics (for ray tracing in an aquarium, say) or in an astronomy unit (to point out stars in a nighttime observing session), the $10 is worth it.

“Recurrent” labs: image distance for a convex lens

I’ve been reading The Physics Teacher journal for over a decade. Every issue contains at least one interesting idea that’s somewhat new to me. I encourage you not to be put off by the frequent buzzword-heavy piece by someone trying to show “scientifically” that his pet new teaching method works… mine each issue for the experiments, demonstrations, and new ways of thinking about old topics.

This month’s issue (May 2009) contains what, to me, is the most revolutionary article I’ve ever read in TPT. Mikhail Agrest, of the College of Charleston, writes about his approach to introductory physics labs, which he calls the “Recurrent” method. Agrest presents a fully developed method that includes pieces of things that I have done, but never completely in the way he suggests. I’m going to try a Recurrent lab tomorrow.

According to Agrest, a Recurrent lab consists of three separate stages. First, an essentially traditional lab is conducted in which a parameter (like the focal length of a lens) is measured. Next, students are asked to use that parameter to predict the results of a slightly different experiment – for example, use the measured focal length to predict the location of an image given an object distance. Finally, students must perform that very experiment in front of the teacher to verify their prediction. Students’ grades are based in part on the accuracy of the prediction.

I’ve done similar experiments in the past, in which students predict an unknown quantity for a grade. The major inspiration provided by Agrest is to let the students develop their experimental method first, before challenging them to make a high-stakes prediction. My own contribution is to make the final prediction into a sort of competitive game: the lower the uncertainty in the prediction, the more credit the lab group can earn.

Stage I: We conduct a standard laboratory exercise with a convex lens. Students project the image of a candle onto a screen, and measure image and object distances. I ask each partnership to set up a graph of 1/do vs. 1/di before they start collecting data – each data point is to be graphed immediately. This way the students better see the relationship between the physical measurements they make and the graph… if they just make a table and graph it later, the lab becomes an exercise in arithmetic manipulation. A substantial part of their grade will be earned for the quality of the graph’s presentation.

Stage II: Once a lab group and I agree that they have investigated a reasonable range of object and image distances, I give them a new object distance: 5 meters. They are asked to use their graph to predict an image distance, including an uncertainty. They will do some calculation, and discover that they’re really looking for the x-intercept of their graph. (Tomorrow, I’ll explain that they’ve found the focal length of their lens.)

I give guidance as to the format of the image distance prediction (i.e. “30 +/- 2 cm), but I let them estimate the uncertainty in any way they please. The rules for stage III will guide their determination of uncertainty.

Stage III: I will compare their measured focal length to the value stated on the box. Eight of twenty points for the lab will come from the accuracy of their measurement. I set up a system of rewards for these eight points:

0 points are earned if the box’s focal length does not fall within the stated uncertainty.
4 points are earned if the measurement matches the box’s focal length, no matter how large or crazy the uncertainty.
7 points are earned if the measurement matches the box’s focal length, and the uncertainty is 10% or less of the measured value.

Then, for all groups whose measurement matches the box’s focal length, bonus points are awarded: everyone gets one point for each group with a larger uncertainty.

I suggest the students imagine that I have hired them to predict the image distance… it is most important that they be RIGHT. After that, the more precise the prediction, the better. My own thought is that this kind of game teaches the deep meaning of experimental uncertainty better than any mathematical exercise. Much credit to Mr. Agrest for the inspiration to refine the experimental approach described here.

(And yes, folks, I'm aware that the picture at the top of the post is emphatically NOT a convex lens. Please feel free to explain how I know that in your comment.)

GCJ