Wave Slicing and Remainders: a cool way to find the period of a periodic function

Last month, in Falmouth High School in Maine, some Honors Physics students were estimating the period of a mass hanging from a spring. They used InquirySpace/Data Games software and Vernier motion sensors, and got data that looks like this (Reading is in meters; Time in seconds):

oscillating data

To do their investigations, they needed the period of this wave.

  • Some students found the peak of one wave, and subtracted its time from the peak of the next wave. This is the most straightforward and obvious. But if you do that, your period will always be a multiple of the time between points, in this case, 0.05 seconds. (This is part of what must have happened in the previous post.)
  • Some students—sometimes with prodding—would take the time difference across several waves, and divide by the number of periods. It’s not obvious to students that this technique gives a more precise measurement for the period. It’s interesting to think about how we know that this is so; for example, if you use five periods, it’s now possible to get any multiple of 0.01 seconds; but does that mean it’s actually more precise? (Yes.) This technique also gives students a chance to be off by one: do you count the peaks? No. You have to count the spaces between the peaks. (Getting students to explain why is illuminating.)
  • We could imagine trying to fit a sinusoid (and some students would, but it’s hard) or using a Fourier Transform (which is a black box for most students).

But this post is about an alternative to all of these techniques—one that uses all the data and gives a much more precise result than the first two.

You can read about this is excruciating detail in a paper I wrote. And I made one particularly careful group of students this (awkward and quickly-made) video describing the technique. So I will be brief here. Continue reading Wave Slicing and Remainders: a cool way to find the period of a periodic function

Strom’s Credibility Criterion

Long ago, way back when Dark Matter had not yet been inferred, I attended UC Berkeley. One day, a fellow astronomy grad student mentioned Strom’s Credibility Criterion in a (possibly beer-fueled) conversation—attributed to astronomer Stephen Strom, who was at U Mass Amherst at the time.

It went something like this:

Don’t believe any set of data unless at least one data point deviates dramatically from the model.

The principle has stuck with me over the years, bubbling in the background. It rose to the surface in a recent trip to the mysterious east (Maine) to visit a physics classroom field-testing materials from a project I advise called InquirySpace.

Background

There is a great deal to say about this trip, including some really great experiences using vague questions and prediction in the classroom, but this incident is about precision, data, and habits of mind. To get there, you need some background.

Students were investigating the motion of hanging springs with weights attached. (Vague question: how fast will the spring go up and down? Answer: it depends. Follow-up: depends on what? Answers: many, including weight and ‘how far you pull it,’ i.e., amplitude.)

So we make better definitions and better questions, get the equipment, and measure. In one phase of this multi-day investigation, students studied how the amplitude affected the period of this vertical spring thing.

If you remember your high-school physics, you may recall that amplitude has no (first-order) effect (just as weight has no effect in a pendulum). So it was interesting to have students make a pre-measurement prediction (often, that the relationship would be linear and increasing) and then turn them loose to discover that there is no effect and to try to explain why.

Enter Strom, after a fashion

Let us leave the issue of how the students measured period for another post. But one very capable and conscientious group found the following periods, in seconds, for four different amplitudes:

0.8, 0.8, 0.8, 0.8

Many of my colleagues in the project were happy with this result. The students found out—and commented—that their prediction had been wrong. So the main point of the lesson was achieved. But as a data guy, I heard the echo of Stephen Strom.

Continue reading Strom’s Credibility Criterion

Blocks and Shadows

Photo of blocks with their shadow
Four blocks and their shadow. I set them on graph paper just to make this shot.

Dan Meyer’s post today is lovely as usual, and mentions the tree/shadow problem (we math teachers make right triangles to help us figure things out because the “tree-ness and shadow-ness don’t matter”).

And that reminded me of a problem I gave teachers long ago in SEQUALS-land that (a) worked really well to get at what I was after and (b) could turn into a great modeling activity that could fit in to that first-year course my fellow revolutionaries and I are gradually getting serious about.

Here’s the idea: we want to be able to predict the length of the shadow of a pile of blocks. So we’re going to make piles of blocks and measure the shadows, which will lead us to make a graph, find a function, etc. etc.

The sneaky part is that we’re doing this in a classroom, so to make good shadows we bring in a floor lamp and turn the class lights off.

I will let you noble readers figure out why this messes things up in a really delicious way. Two delicious ways, actually. I’ll give away the second:

end of the shadow
The end of the shadow, closer up. Really: how long is it?

Of course we have all done height/shadow problems. But have you tried to measure a shadow lately? You have to make a lot of interesting decisions to measure a shadow; and a shadow from a pile of blocks made from a floor lamp exaggerates the problems, such as where do you measure from—the middle of the stack? The base on the shadow side? Where? And where do you measure to—where the fuzzy part of the shadow begins? Where it ends? And why is it fuzzy anyway?

This is why I love measurement as a strand so much. We always think of it as the weakling among content areas at the secondary level; it doesn’t have the intellectual heft of algebra or functions. But if you look closely (and go beyond the words in the standards) it’s a thing of beauty and (since we’re referencing Dan Meyer) perplexity. I did a chapel talk at Asilomar many Sundays ago in which I said that measurement was invented, inexact, and indirect. I still think that’s true, although as alliterative slogans go it’s hard to remember.

So: try this at home. Use Fathom if you have it. Come up with a function that models the shadow lengths. But don’t just figure it out like a math teacher—get the lamp, stack the blocks, and measure.