Dissecting Computer: Building a Hovercraft

electronics-dissection_4749 — Extracting the hard drive from an old computer.

Our school was recycling some old computers, so my students convinced me that it would be worthwhile o dissect a few of them to see if there was anything worth saving. It was quite remarkable to see just how interested they were in examining the insides of the machines — a few desktop computers and a monitor — but I guess I shouldn’t have been surprised. After all, it’s getting harder and harder to open up their iPods and other electronics, and even more difficult to repair and repurpose them, so I can see why students would jump at the chance of looking inside a device. Also, they tend to like to break things.

electronics-dissection_4763 — Pulling apart a monitor.

To get them to think a little more about what they were seeing, I got a couple students to draw a scale diagram of one of the motherboards, and write up a report on what they’d done.

mapping-motherboard_4835 — Diagramming a motherboard.

Some of the other students spent their time trying to make all the motors, LED’s, and lasers work by hooking them up to 9-Volt batteries. Then they found the fans… and someone had the brilliant idea that they could use it to make a hovercraft. Using a gallon sized ziplock bag and some red duct tape, a prototype was constructed.

The fan would inflate the bag which would then let air out the bottom through small holes. I convinced them to try to quantify the effectiveness of their fans before they put the holes in by hooking the bag up to one of our Vernier pressure sensors that plug into their calculators. Unfortunately, the sensor was not quite sensitive enough.

hovercraft_4756 — Attempting to measure the hovercraft’s bag pressure using a gas pressure sensor connected to a calculator.

This was not how I had planned spending those days during the interim, but the pull of following the students’ interests was just too strong.

Ski Trip (to Hidden Valley)

skiing_4829 — Approaching a change in slope.

We took a school trip to the ski slopes in Hidden Valley. It was the interim, and it was a day dedicated to taking a break. However, it would have been a great place to talk about gradients, changes in slopes, and first and second differentials. The physics of mass, acceleration, and friction would have been interesting topics as well.

skiing_4817 — Calculus student about to take the second differential.

This year has been cooler than last year, but they’ve still struggled a bit to keep snow on the slopes. They make the snow on colder nights, and hope it lasts during the warmer spells. The thermodynamics of ice formation would fit in nicely into physics and discussion of weather, while the impact of a warming climate on the economy is a topic we’ve broached in environmental science already.

skiing-snow-making_4811 — The blue cannon launches water into the air, where, if it’s cold enough, it crystallizes into artificial snow. The water is pumped up from a lake at the bottom of the ski slopes.

Wiggle Matching: Sorting out the Global Warming Curve

To figure out if the climate is actually warming we need to extract from the global temperature curve all the wiggles caused by other things, like volcanic eruptions and El Nino/La Nina events. The resulting trend is quite striking.

I’m teaching pre-Calculus using a graphical approach, and my students’ latest project is to model the trends in the rising carbon dioxide record in a similar way. They’re matching curves (exponential, parabolic, sinusoidal) to the data and subtracting them till they get down to the background noise.

mona-loa-data — Carbon dioxide concentration (ppm) measured at the Mona Loa observatory in Hawaii shows exponential growth and a periodic annual variation.

Regression with Gnumeric

Finally, I’ve found a spreadsheet application (Gnumeric) with a reliable Solver for doing regressions. And it’s free. The only tricky part is that there’s no native port for Macs; you have to use a command line package manager to install it (I used Fink).

Gnumeric, however, seems to be an excellent tool for data analysis.

Warming of the West Antarctic Ice Sheet

… a breakup of the ice sheet, … could raise global sea levels by 10 feet, possibly more.

— Gillis (2012): Scientists Report Faster Warming in Antarctica in The New York Times.

In an excellent article, Justin Gillis highlights a new paper that shows the West Antarctic Ice sheet to be one of the fastest warming places on Earth.

antarctica-melting-days — The black star shows the Byrd Station. The colors show the number of melting days over Antarctica in January 2005. This number increases with warming temperatures (image from supplementary material in Bromwich et al., 2012).

Note to math students: The scientists use linear regression to get the rate of temperature increase.

The record reveals a linear increase in annual temperature between 1958 and 2010 by 2.4±1.2 °C, establishing central West Antarctica as one of the fastest-warming regions globally.

— Bromwich et al., (2012): Central West Antarctica among the most rapidly warming regions on Earth in Nature.

Analyzing the 20th Century Carbon Dioxide Rise: A pre-calculus assignment

The carbon dioxide concentration record from Mona Loa in Hawaii is an excellent data set to work with in high-school mathematics classes for two key reasons.

The first has to do with the spark-the-imagination excitement that comes from being able to work with a live, real, scientific record (updated every month) that is so easy to grab (from Scrippts), and is extremely relavant given all the issues we’re dealing with regarding global climate change.

The second is that the data is very clearly the sum of two different types of functions. The exponential growth of CO₂ concentration in the atmosphere over the last 60 years dominates, but does not swamp, the annual sinusoidal variability as local plants respond to the seasons.

Assignment

So here’s the assignment using the dataset (mona-loa-2012.xls or plain text mona-loa-2012.csv):

1. Identify the exponential growth function:

Add an exponential curve trendline in a spreadsheet program or manual regression. If using the regression (which I’ve found gives the best match) your equation should have the form:

$y = a b^{cx} + d$

while the built-in exponential trendline will usually give something simpler like:

$y = a e^{bx}$

2. Subtract the exponential function.

Put the exponential function (model) into your spreadsheet program and subtract it from data set. The result should be just the annual sinusoidal function.

minus-exp — Dataset with the exponential curve subtracted.

If you look carefully you might also see what looks like a longer wavelength periodicity overlain on top of the annual cycle. You can attempt to extract if you wish.

3. Decipher the annual sinusoidal function

Try to match the stripped dataset with a sinusoidal function of the form:

$y = a \sin (bx+c) + d$

A good place to start at finding the best-fit coefficients is by recognizing that:

a = amplitude;
b = frequency (which is the inverse of the wavelength;
c = phase (to shift the curve left or right); and
d = vertical offset (this sets the baseline of the curve.

Wrap up

Now you have a model for carbon dioxide concentration, so you should be able to predict, for example, what the concentration will be for each month in the years 2020, 2050 and 2100 if the trends continue as they have for the last 60 years. This is the first step in predicting annual temperatures based on increasing CO₂ concentrations.

Exponential Growth of Cells

Today I grew, and then killed off, a bunch of bacteria using the VAMP exponential growth model to talk about exponential and logarithmic functions in pre-Calculus. I also took the opportunity to use an exponential decay model to talk about the development of drug resistance in bacteria.

exp-growth — Two cells are reproducing (yellow) during a run of the exponential growth model.

Students had already worked on, and presented to each other, a few bacterial growth problems but the sound and the animation helped give a better conceptual understanding of what was going on.

After watching and listening to the simulation I asked, “What happens to the doubling time?” and one student answered, “It gets shorter,” which seems reasonable but is incorrect. I was able to explain that the doubling time stays the same even though the rate of reproduction (the number of new cells per second) increases rapidly.

exp-growth-graph — Graph showing how the number of cells increases over time.

exp-decay-model — Switching from growth to decay (half-life of 50 sec).

Then I changed the model from growth to decay by changing the doubling time to a half-life. Essentially this changes the coefficient in the exponent of the growth equation from positive to negative. The growth rate’s doubling time was 100 seconds, but I used a half life of 50 seconds for decay to accelerate things a bit, but still show the persistence of the last of the bugs.

exp-decay-graph — Exponential decrease in cell population/biomass.

The cells died really fast in the beginning, and while there was just one cell was left at the very end, it was pretty clear just how persistent that last cell was; cells were dieing so slowly at the end.

This is similar to what happens when someone takes antibiotics. The typical course lasts for 10 days, but you’ve killed enough of the bacteria to loose the symptoms of sickness after two or three. Those final few that remain are the most resistant to the antibiotic, and if you don’t kill them then, once you stop taking the antibiotic, they’ll start to grow and replicate and you’ll end up sick again with a new, antibiotic-resistant population of bacteria.

I thought that using the VAMP model for the demonstration worked very well. The sound of the cells popping up faster and faster with exponential growth seemed to help amplify the visual effect, and make the whole thing more real. And during the decay phase, having that last cell hang on, seemingly forever, really helped convey the idea that bacteria can be extremely persistent.

Geometry at the City Museum

city-museum-geom-2298 — Ms. Wilson believes that the City Museum makes a great field trip for her geometry class. I think she has a point.

I had my pre-Calculus students take pictures of curves at the City Museum. Ms. Wilson’s geometry students had to photograph shapes and angles instead. Then they had to put together a slideshow of what they found, which, from what I heard, went very well.