One of the things most high school students don’t learn about in their science classes is quantitative error analysis. (The closest they usually get to the topic is significant figures.) This year, I started off by teaching my students to estimate uncertainties and propagate the uncertainties through their calculations. Quantifying their estimated uncertainty will continue to be a requirement on lab write-ups throughout the year.
I had them do a couple of activities relating to uncertainty, including the thickness of aluminum foil lab I described in my previous post, entitled Learning by Screwing Up. I did a couple of additional activities with them that are worth mentioning.
I drew a line on the white board with a marker. I held up a meter stick with no markings on it and asked them to estimate the length of the line (from their seats) to within 0.1 meter, which they were able to do correctly. Then I held up a meter stick marked only in 0.1 m increments. They successfully improved their estimate to within 1 cm. Then I held up a meter stick marked only in 1 cm increments, and they improved their estimate to within ±2 mm, despite the fact that the closest student was looking from 2-3 m away. I did this activity in each of my five classes, with the same result.
Then I had them measure the length of the hallway outside my room, based on the concrete tiles that it’s made of. The tiles appear to have been hand cut–they’re about 75 cm long, but this varies by 1-2 cm. Thus the accumulated error caused by using the tiles to measure the hallway (about 37 m long) was about 1 meter! I also had them measure the width. Then, I gave them the dimensions of US paper currency (15.6 cm long × 6.6 cm wide) and asked them to calculate how many dollars it would take to paper the entire hallway (roughly $12,850, assuming one-dollar bills) , and to estimate the uncertainty in dollars (roughly ±$350).
Both results are fun, and more than a little surprising. The hope is that I can turn out a crop of students who have a fairly good intuitive understanding of uncertainty and how to estimate it.
I wish I had learned propagation of uncertainty in high school. It’s an incredibly useful tool if you know how to use it, and it would have been nice to have had it in the metaphorical toolbox earlier than my Junior year of college.
It would also have been nice if we’d gotten more instruction on it than three slides full of partial derivatives, but that’s another story.
What I’m teaching them is condensed from the tutorial created by the Columbia University physics department at http://phys.columbia.edu/~tutorial/.