Great Balls of Fire

After learning about gravity and taking the midmorning break,  the Peregrine 3-4 graders and I worked on understanding nuclear fusion in the core of the Sun and where elements come from.

I started by setting the context.  The students had studied atoms and molecules the previous year so I started by drawing a molecule of water (two hydrogen atoms and one oxygen atom) and reminding them of the evidence for atoms and molecules.  Then we zoomed in to one hydrogen atom and discussed the Rutherford experiment showing that atoms are very fluffy; most of their volume is nearly empty while nearly all their mass is concentrated in a tiny volume in the center (nucleus).  Then we zoomed in further by a factor of 10,000 to the nucleus.  For a hydrogen atom, the nucleus is a single positively charged particle called a proton.  I held up a ping-pong ball as a proton and said that if protons really were that size, the atom would have to be the size of South Davis.

To reinforce the sense of scale, I showed the movie Powers of Ten.  This classic ten-minute movie should be seen by anyone wanting to understand the universe.  I also took the time to answer questions about it.

The basic rules of nuclear physics are actually understandable by anyone. Last year we investigated the effects of electrical charge, and concluded that like charges repel while opposite charges attract. Atoms beyond hydrogen in the periodic table have more protons.  But why do the protons stick together if they repel each other? There must be some form of glue.  I demonstrated two magnets  which repelled each other.  They were "donut" magnets threaded onto a rod so they didn't flop around and the repulsion was clear.  But when I turned the rod vertically and one magnet fell with enough speed onto the other one, they touched briefly.  That was enough for the velcro on their surfaces to attach and keep them together.  The velcro is a short-range force, like the strong nuclear force which keeps a nucleus together.

But protons alone can't generate sufficient strong nuclear force to keep nuclei together.  Another type of particle, with similar mass but no charge and called a neutron, provides the glue.  Nuclei need roughly equal amounts of protons and neutrons to be stable.  I modeled this with a bunch of ping-pong balls I had wrapped with velcro.  The "protons" had velcro hooks and the "neutrons" had velcro loops, so that you needed roughly equal numbers of each to build up a large nucleus.  (The different types were also different colors to make the idea plainly visible.)  Adding a neutron to a nucleus adds mass, but doesn't otherwise change the properties of the atom.  For example, a proton plus a neutron is still hydrogen, but we call it a different isotope of hydrogen.  Similarly, carbon-12 (usually written with a superscript 12 on the left) and carbon-14 are different isotopes of carbon which differ by two neutrons.

With that in mind, we can start building up more complicated elements from hydrogen.  Element number 2 (two protons) is helium, and we need two neutrons to provide the glue so the most common isotope of helium is helium-4.   The protons have to be smashed together at very high speed if they are to ever get close enough for the "velcro" of the strong nuclear force to make them stick, so we need very high temperatures to make this fusion process happen. (High temperature means the individual microscopic particles are wiggling or bounding around at high speed.)  We find it difficult to make these high temperatures on Earth, but the core of the Sun is 15 million degrees (Celsius; tens of millions of degrees if you think in Fahrenheit) and this happens quite routinely.  In fact, most stars turn hydrogen into helium in their cores.

Fusing helium into even heavier elements is harder, but most stars will do that as well by the ends of their lives.   It turns out that crashing two heliums together results in an unstable isotope of element 4 (beryllium), which quickly decays back into two helium-4 nuclei. But if you manage to crash a third helium into the two heliums before the two-helium complex has a chance to decay, you make carbon-12 (the most common isotope of element 6, carbon; again, equal amounts of protons and neutrons).  Then, if you crash another helium into that, you get element number 8: oxygen. Another helium into that produces element 10, argon.  These helium capture reactions are common in massive stars (substantially more massive than the Sun), and they create more of the even-numbered elements than the odd-numbered elements (nitrogen, fluorine, etc).  They go all the way up to iron (element 26).  I modeled all this with the velcro-covered ping-pong balls.

Have you noticed what we've done here?  We've explained the origin of the elements using basic, well-understood physical processes. That's pretty cool! Here's a graph of the observed abundances:

You can see that hydrogen is the most abundant, followed by helium, then the even-numbered elements carbon, oxygen....through iron.  But why are there elements beyond iron if stars only make up to iron? Well, stars make up to iron when they are in equilibrium.  But when they explode (a supernova), so much energy is released that even more complicated nuclei can be made.  I won't explain the details here, but the abundances of all the elements beyond iron are well understood as consequences of supernovae.  That we can understand all the features of the above plot is, to me, one of the most amazing things in all of science.

The supernova explosions are also what throw the newly-manufactured elements back into space, where they can mix into gas clouds that eventually collapse to form new stars. That means that the atoms in your body were once inside another star.  (Not from the Sun, because new atoms made in the Sun won't escape until the end of its life.)

Supernovae make some unstable elements, like uranium.  The most common type of decay for a heavy element is to violently eject a "bullet"  made of two protons and two neutrons, in other words a helium nucleus (again I modeled this with the ping-pong balls).  This is why there is helium on Earth; our gravity is too weak to hold on to helium gas, but helium produced by radioactive decays is trapped in rocks underground.  When we drill for natural gas, we can capture some of this helium and eventually use it to fill balloons.  When it escapes from the balloon, it eventually escapes into space.

Big Bang 

I left out one detail in the story above: most of the helium in the plot was actually made in the Big Bang.  Some of the kids had expressed interest in the Big Bang previously, so I used the remaining time to talk about that.  I used the usual balloon-with-stickers demo, and I also showed this interactive tool made by an undergraduate student of mine. The point of the tool is to show that although we see all galaxies moving away from us, observers in all other galaxies also see all galaxies moving away from them. So we are not at the center of anything. If we think back in time, all galaxies were closer to each other, so the universe was denser (and hotter).  Far enough back in time, the universe was so hot (everywhere) that a fair amount of hydrogen fused into helium.  This is called Big Bang nucleosynthesis (BBN). We can look at the abundance of various  BBN byproducts, like hydrogen-2 (aka deuterium) and confirm that this really happened.

Wrapping up

Most of this trimester we worked on understanding the immense size of space.  If this makes you feel insignificant, remember that you are made of atoms from another star.  You are a part of the universe which can actually understand itself. 

There is a season

At the end of last Friday's session with the 3-4 graders we discussed
the seasons. It's natural to think that winter is when we are farther
from the Sun, but is that true?  What evidence can we reason with
here?

Well, for one, we know that when it's winter in the northern
hemisphere, it's summer in the southern hemisphere.  That's a pretty
good clue that the seasons are not caused by the Earth getting closer
to and farther from the Sun.  What else?  One boy, to my amazement,
specifically mentioned that if/when we are closer to the Sun, it
should appear to be bigger, and it doesn't appear to be bigger in
(northern) winter.  I was pretty surprised, but then I recalled that
he had recently seen the movie Agora.  This is a trenchant
observation: the Sun actually looks smallest (as seen from anywhere on
Earth) in January, indicating that January is when Earth is furthest
from the Sun. (You can't see this just by looking, because the Sun is
so blindingly bright and it's a fairly small change, but you can use a
pinhole camera to do it.)


Most people (these kids included) "know" that seasons are caused by
"tilt," but what does that really mean?  It's very useful to take a
globe (on a standard stand tilted by the right amount) and make it
orbit a light source to see how this plays out.  The tilt actually
points to the same place in the sky (judging by the stars) all the
time, but since Earth goes around Sun (or vice versa, it doesn't
matter for this point), for part of the orbit the tilt points the
north pole toward the Sun.  This is northern summer, when there is
midnight Sun near the north pole, and longer daylight hours in
northern latitudes generally.  (You can see this by spinning the globe
in the aforementioned model. Make sure to keep the north pole always
pointing toward the same place in the room, like the clock on the wall,
even as it goes around the "Sun.") Six months later, the Sun is on the
opposite side of Earth (relative to the stars) so it's the south's
turn to have summer and the north pole has constant darkness.  In
between those two extremes, neither hemisphere is favored.

So we are NOT tilted closer to the Sun.  The tilt merely allows the

Sun to shine a greater or lesser fraction of the time on a certain

hemisphere, depending on where the Sun is relative to this constant

tilt.  (This fits the evidence that whatever hemisphere has summer, it

does not see the Sun as closer and bigger.)  Translated into what we

see on the ground, it means that summer is when the Sun rises earlier

and sets later, and goes higher in the sky at noon.  You can see all

these things with the globe and light setup.  I had had the kids keep

a sketch of the position and time of sunset or sunrise each week for

the past six weeks, and these data match the model I just described.

I'll analyze that in a bit more detail, and explore the Moon's orbit

around Earth, in my next visit to the school.

Turn! Turn! Turn!

This spring I am assigned to work with the Peregrine School 3-4
graders on astronomy, and last Friday was my first day, so I started
with basics like how we know the Earth spins.  We tend to feel
superior to people in the past who believed that the Sun went around
the Earth but, really, how can you use basic observations to show that
it doesn't?  I suspect that most people on the street would be stumped
by this if I didn't allow "satellites" or "NASA" as an answer.

If we only had the observation that the Sun rises and sets every 24
hours, we wouldn't be able to conclude anything.  Each star also rises
and sets in roughly (later we'll see why I say roughly) 24 hours, so
based on pure majority rule, it might be easy to attribute the
apparent motion of the Sun and stars to Earth spinning.  This model
invokes only one thing (Earth) moving, vs the other model invokes a
grand conspiracy of everything else in the universe circling us at an
agreed-upon rate of once every 24 hours.  Sounds like a no-brainer,
but why don't we feel Earth moving?

The kids had lots of ideas in response to this question. It moves so
slowly we can't feel it? No, its circumference is about 24,000 miles
so if it spins in 24 hours its equator must move 1,000 mph.  It moves
so quickly we can't feel it? Gravity?  Centrifugal force? It's so big
we don't feel it move?  There were so many ideas about this that I
decided to explore Galilean relativity: if you are in a laboratory
moving at constant velocity, there is no experiment you can do to
prove you are not actually stationary.  Think about a smooth flight in
an airplane.  If you drop something does it fly backward, indicating
that you are actually traveling at 500 mph?  No, it falls straight
down.  Unless you look out the window, you can't tell that you're
moving---there is NO experiment which will tell you this.  If you do
look out the window, all you can conclude is that you are moving
relative to Earth...there is no experiment you can do that says Earth
is stationary and you are the one who is moving.

This was a pretty new and shocking idea for the kids, so we spent a
long time discussing it.   I remembered a great video I had seen
demonstrating one aspect of this, so I sketched it out and asked them
to predict what would happen.  Say you have a pitching machine which
shoots baseballs at 100 mph, but you mount this machine in the back of
a pickup truck which goes 100 mph the other way.  What will the ball's
speed be, relative to people on the ground? 200 mph? 100 mph?  Zero?
We analyzed this until break time, then after break I showed them the
video.  Mythbusters also did a similar thing, which you can see much more clearly.

The bottom line is that velocities are relative.  So if everything in
your vicinity is moving together at the same speed, it can all be
considered stationary.  This applies to your vicinity on Earth:
although Earth's rotation causes different parts of Earth to move in
different directions and speeds, the part that you experience at any
one time is so small that to high precision it's all moving at the
same speed in the same direction. (When winds move air over hundreds
of miles, that air does eventually feel the effect of Earth's
rotation, the Coriolis effect.)

So not feeling Earth's spin is not a valid argument that it must be
still.  But how do we prove it spins? The Coriolis effect is one way,
but I considered that too advanced for this audience.  Instead, I
explained the Foucault pendulum, which many of them had seen but
probably didn't realize the significance at the time.

Next, we tackled Earth's motion through space.  Spinning is not enough
to explain all our observations, because the stars rise and set every
day slightly faster than the Sun, which means that over time the Sun
loses more and more ground to the stars, and over the course of a year
we see the Sun make one complete circle around the sky relative to the
stars.  What could explain this?  Well, maybe the Sun does go around
Earth, in addition to Earth spinning. But maybe the Earth goes around
the Sun. How could we tell the difference?  I'll get to the bottom of
that next time I visit the school, but for the time being I wanted to
focus on other changes throughout the year and tie them all together
into a coherent model.  I'll post that part of our discussion soon.

Cold and Clammy

After spending most of the morning studying the dynamics of sand along California's beaches,  I had about 30 minutes left to tie up some loose ends I had left on my last visit.  On that visit, I had promised that I could catastrophically crush an aluminum soda can using just heat and cold, but it didn't work.  As soon as I left the school that day, I realized what I had done wrong, but instead of just explaining what I did wrong, I planned some activities to build up to an explanation.  The first was measuring the dew point in the room. (Note: there are lots of dew-point activities written up on the web; I'm just linking to a random one here out of laziness.  In particular, I saved time compared to the activity in this link by starting with cool rather than warm water.)  The dew point was about 10 C, in a room with a temperature of about 20 C.  I also had them answer some questions related to dew point, such as: Which city would you rather travel to, one where the dew point is 50 F or one where it is 80 F?  Explain why, and suggest a plausible location for each city.

Then we related dew point to relative humidity. I wanted to make a graph of amount of water that air holds, vs temperature of the air.  At any temperature, there is a maximum amount it can hold, so I can sketch this maximum amount as a curve which changes with temperature.  I elicited from them how I should sketch it: the warmer the air, the more water it can hold. On that same graph, how would we represent the air in this room? We know it's 20 C, and we know the amount of water in the air is substantially less than the maximum---if it were close to the max we would have seen condensation very quickly as soon as we began to cool the glass.  So I made a mark indicating that conceptually.  As we cooled the glass, we lowered its temp, so I drew a line going leftward from that point.  When it hits the max curve, it condenses.

So the dew point is an indication of how much water is in the air, but what we feel as humidity is really how much water is in the air relative to the maximum it could hold at that temperature. This is called relative humidity.  For example, the dew point was about 10 C, or about 50 F, and in a 70 F room that doesn't feel humid.  But in a 52 F room, that would feel clammy as well as cool.  So I asked the kids to brainstorm how they could build a device to measure relative humidity.  To my surprise (because I was hearing some whining) someone came up really quickly with the idea of a wet thermometer.  I said "Brilliant!" and tried to elicit more details.  Why is being wet important?  Because then there will be evaporation.  OK, how will evaporation change your thermometer reading?  There was much discussion of this, with about half the class leaning toward warmer and half toward colder, but eventually I steered them toward thinking about getting out of a swimming pool and feeling cold as all those little water drops on your skin evaporate.  The thermometer will definitely read a colder temperature! So how does this help you determine humidity? Well, if the air is very humid already, there won't be much evaporation, so the wet thermometer won't read much colder than a dry thermometer.  If the air is very dry, there will be a lot of evaporation and the wet thermometer will read much colder than a dry thermometer.  So we did the experiment, and we found about 16 C (61 F) for the wet one and 20 C (70 F) for the dry one.  Then we find a table which tells us the relative humidity as a function of dry-bulb temp and the temperature difference between dry and wet bulbs.

Now for the grand finale. I reminded them how much a substance expands when going from liquid to gas.  Similarly, when a gas condenses to liquid, it occupies much less volume.  So I put a small amount of water into an empty aluminum soda can, heat the can with a torch so that the gas in the can is mostly water vapor, then plunge the can upside-down into an ice bath.  The water vapor in the can condenses quickly.  Suddenly, there's a lot of empty space in the can, and it collapses catastrophically because the pressure on the outside of the can (standard atmospheric pressure) is so much greater than the pressure on the outside of the can (very little because the gas is gone).  When I tried to do this demo previously, I was not cognizant of the key role of condensation and I put very little ice into a giant pail of water, virtually guaranteeing that I would not get condensation.  You can see a video of this kind of demo here.  It was a satisfying conclusion.  Three kids wanted to take crushed cans home for keepsakes.

A River of Sand

Today was my last day with the 5-7 graders.  We spent most of the time learning about beaches: how sand gets there and how sand moves once it's there (California grade 6 standard 2c).  It's a lot more interesting than you might think, and it's explained well in this video.  Normally I show just short clips of videos, 30 seconds or a minute here and there to support whatever I want to talk about; a lot of "educational" videos have a lot of fluff surrounding the critical part(s).  But I found this video to be packed full of good visualizations of what's going on with beaches, far better than I could set up myself, and very little fluff.  So we watched all 20 minutes (ok, I skipped the fluffy first 80 seconds), and I highly recommend it for parents too.  Aside: It's from the 1960's, and told in the "voice of God" style strongly reminiscent of the films I was shown when I was in elementary school.  Science videos today are quite different, typically with a friendly host just like us who wants to take part in experiments just like we do.  That's probably an improvement on average, but I experienced some nostalgia for the "voice of God" style as I watched it. 

After the video, I set the kids to work on the "Rollin' Down the Sand Highway" activity from this packet (the last activity in the packet).  I didn't provide maps, but just looked them up online as needed.  Some kids had never seen a topo map before, so I explained that in context.  But for most of the time most of the kids were stuck on the math, which surprised me because the math is pretty simple.  I guess it's a question of applying math outside of math class!  It's always easier to apply a concept when you've just learned it and you know that the problem you've been given can be solved using that concept.

More specifically, the students did not have a clear idea of how to go about converting cubic yards of sand per year to dump trucks per minute.  I led them through  the easy step of converting cubic yards per year to dump trucks per year, and I thought this would give them the boost they needed to complete it on their own, but I was wrong.  As I circulated around the room helping students, it came out that we would need to know how many minutes per year, and the students were able to come up with that number (although they may have Googled it on a mobile device behind my back): 525,600.  But there was a huge amount of confusion regarding whether they multiply or divide by this number, and whether the result would be dump trucks per minute or minutes per dump truck.  I walked them through how I think about it, and they stared at me totally lost; I stared back wondering how they could not have seen this before.  So I backed up and (much wailing and gnashing of teeth omitted here) found a way to get it across. 

Here's what worked: let's say that you are asked to compute 3 times 4, divided by 7 times 3:

3x4
----  = ?
7x3

The kids universally said the following: multiply across the top and also across the bottom:

3x4     12
----  = ---
7x3     21

This surprised me because it's not what I would do, but once I figured out that one kid was thinking like this, I repeated it for all the kids.  Although the answer surprised me, it's not wrong, so let's continue along these lines and see what happens.  The natural next step is to simplify the fraction 12/21: is this its simplest form?  The typical answer from a student was: ...um...well, I don't see any common factors.  And of course it's hard to see the common factors when you're staring at 12/21.  But if instead you look at

3x4    
----  =?
7x3    
the common factor of 3 is jumping up and down screaming "I'm a common factor!"  So cancel the 3's and you immediately get 4/7.  This is not only much less work than writing 12/21 and then trying to simplify; it avoids the potential for a lot of mistakes.  Although this kind of thing is second nature to me, it was not natural for the kids, who were intent on following the specific rules they had learned about multiplying fractions.

I had to go through all this just to get to the main idea: we can do the same kind of thing with items like dump trucks and minutes instead of specific numbers.  We are given dump trucks per year and we want to get dump trucks per minute, so we can represent the problem like this:

dump trucks      ?         dump trucks
--------------- x   --   =   ---------------
year                     ?           minute


We have to get rid of years and introduce minutes, so if we put years per minute in the question marks, we get:

dump trucks      year           dump trucks
--------------- x   -----     =   ---------------
year                   minute           minute

The years on the left cancel each other, leaving dump trucks in the numerator and minutes in the denominator.  If we had instead tried:

dump trucks      minute           dump trucks
--------------- x   ---------    =   ---------------
year                      year                 minute

this equation is manifestly false; the right hand side should contain dump truck minutes on the top and years squared on the bottom. This kind of thinking seemed to be new to the 6th graders, and I'm glad I did it because it's really important.  It provides a system for making sure you do the right thing.  Don't know whether to multiply by 525,600 or divide by 525,600?  One system popular among the students was to just try one approach, and then if the teacher says it's wrong, just do the other!  But here's a system which makes clear that we have to multiply by years/minute, or 1/525600. And not sure if the resulting number represents dump trucks per minute or minutes per dump truck? Again, the system makes clear that the result is dump trucks per minute.

Another thing the kids need to internalize much better is sanity checking.  If you multiply 722,222 cubic yards by the $5 per cubic yard it costs to remove, you should get a number bigger than 722,222, not less than 722,222.  The kids didn't apply this kind of sanity checking to any of their results, and therefore didn't catch any of their mistakes before showing their answers to me.  This was the first math-based activity I had really done with the upper-graders, and I was probably naive to expect that they could apply math outside the context of a math class.  I should have given a little primer on how to estimate before calculating, how to check that your answer is right after calculating, etc.  This is not really math; it's metacognition in a math context, and I'm now kicking myself for not emphasizing metacognition throughout this trimester with the upper graders.

In any case, we spent a lot of time on this activity: 1 hour, including the movie, before break; then maybe another 20 minutes after break.  It was worth it to work through these issues, but then I did have to cut down on my planned post-break activity.  I'll dedicate the next blog post to the humidity-related activities we did in the last 30 minutes of the morning.

Making Clouds in a Bottle

As our final activity in our March 12 physics-behind-the-weather extravaganza, we made clouds in a bottle, pretty much as shown in this video.  This experiment shows that to form clouds we need three things: humid air, a change in temperature/pressure which brings the air below the dew point, and condensation nuclei, which are provided by particles in the smoke we waft into the bottle.  Without these, water droplets tend not to form; see the Wikipedia article on supersaturation.

After many minutes of squeezing to see the clouds form and releasing to see the clouds disappear, they tended to stop forming.  I think that must be due to the particles being driven into the liquid water rather than floating in the air.

This was a relatively quick (~15 minutes) activity, but I was surprised at how the kids had problems getting the smoke into the bottle.  They didn't think about how the bottle had to exhale before it could inhale the smoke.  I didn't trust the kids with matches, so I was quite busy servicing the kids as they each tried to get smoke multiple times.  My advice to teachers is to have several adults help if you do this with a larger number of kids.

Balloon in a bottle

This is a great demo for making air pressure dramatically visible.
The "bottle" is a special flask with a hole in the bottom. Stuff most
of a limp balloon into the neck of the flask, mount the balloon's neck
onto the flask's neck, and inflate the balloon.  When you take your
mouth off the balloon, it deflates, of course.  Ask your audience to
predict what will happen if you put the stopper in before taking your
mouth off.  Now inflate the balloon again, insert the stopper in the
hole at the bottom of the flask, take your mouth off, and TA-DA!  The
balloon does not deflate, despite having its mouth held wide open by
the neck of the flask!  This is really an astounding demo, and people
are delighted every time I do it.

The key to understanding this is that inflating the balloon pushes air
out of the flask through the hole in its bottom.  With no stopper, air
rushes back in through the same hole when the balloon deflates. But
with a stopper, pulling air back in to the flask is not possible. As
soon as the balloon deflates just a tiny bit, the same amount of air
in the flask must occupy a larger volume, which means it lowers its
pressure.  Each square inch of latex in the balloon's surface now
starts to feel a higher pressure from the inside of the balloon than
from the outside of the balloon, so it can't deflate any more.
(Experts: I am purposely omitting surface tension to keep it
simple for a young audience.)  If you now remove the stopper, it
quickly deflates by drawing air into the flask through the newly made
hole.

There are further variations such as pouring water in the balloon
before removing the stopper (which creates a nice squirt of water when
you do remove the stopper).

If you can't find this specialized flask, you may be able to do a
similar demo with a regular bottle by using a straw in parallel with
the balloon to vent the bottle as you inflate the balloon.  From the
videos I've seen, it takes some dexterity and practice to do this and
remove the straw to prevent further air flow at the critical moment,
but it is doable.

I did this Tuesday with the upper-graders at Peregrine School to
introduce pressure to our study of weather.  Pressure is related to
temperature; pumping up a bike tire shows that compressing a gas
raises its temperature, and there are demos I will describe in a
future post which show how cooling a gas makes its pressure drop.  So
hot air masses are associated with high pressure, and cool air masses
(and storms) are associated with low pressure.  In low-pressure storms
like hurricanes, air is pushed from higher-pressure regions on the
periphery toward the center.  Combining this with what we had just
learned about the Coriolis effect, we see that in the northern
hemisphere air will be deflected to its right, making a
counterclockwise circulation which is easily seen in satellite images:

In the southern hemisphere, air is deflected to its left as it tries
to go from outskirts to center, thus creating clockwise circulation.
We saw in our previous activity that the Coriolis effect cannot
determine the circulation in toilet bowls because a few inches' travel
is too small to be affected by riding on the 12,000-mile
merry-go-round we call Earth.  But over hundreds of miles, the
Coriolis effect does build up and cause these wind patterns.

The next and last activity in our  physics-behind-the-weather extravaganza was making clouds.

Coriolis Effect

After learning how convection makes air "want" to circulate from
equator to pole at high altitudes and back from pole to equator at low
altitudes, we learned how that idealized picture of air flow is
modified by the Coriolis effect.

I won't explain the effect here, because there are plenty of good Web
resources on that, with MIT providing perhaps the best demo.  As a
teacher, my quandary was whether to simply show the video of this
demo, or to try to actually do it in class. I took a risk on the
latter, and it worked out amazingly well.  I built something like the
contraption shown in the MIT video, by borrowing a sturdy turntable (a
low one, much lower than a pottery turntable, so that a fall from it
would not hurt) and clamping a 2"x12"x8' plank to it.  I put a kid on
each end, asked them to take some practice throws, and the spun it and
had them throw at will.  They had a blast! Everyone insisted on taking
a turn---multiple turns---so we had plenty of chances to explore the
effects of different variables, such as direction of rotation, speed
of rotation, and distance over which the ball travels.  When break
time arrived, the kids just kept playing with it, and kids from other
grades also came in the room and demanded their turn.  I can honestly
say this was the most fun science activity I have done with kids.
They begged me to leave the device at school, but I did not leave it
because it would require constant adult supervision to keep it safe.
I strongly urge anyone teaching the Coriolis effect to do this
activity. The activity described in this post took about an hour, but
kids could play with this device for many hours.  If you don't think
you can build one, a merry-go-round should provide a good substitute.

Back in discussion mode, we took some time to understand that Earth's
northern hemisphere is like a merry-go-round spinning
counterclockwise, so when low-altitude air tries to circulate from
pole to equator it gets deflected to the right and becomes a wind from
the northwest, whereas when high-altitude air tries to circulate from
equator to pole it gets deflected to the right and becomes a wind from
the southwest.  This means that one giant circulation cell from
equator to pole and back is not feasible; it gets broken up into three
cells, from equator to about 30 degrees latitude, from 30-60 degrees
latitude, and from latitude 60 to the pole:

(Image from Wikipedia)

The sourthern hemisphere is like a merry-go-round spinning clockwise,
so its air circulation cells (called Hadley cells) are a mirror image
of the north's.  On a faster-spinning planet such as Jupiter, the Coriolis effect
is much stronger and therefore the equator-to-pole tendency is broken up
into more cells.  Each cloud band in this photo represents a cell:

An important thing to note about the Coriolis effect is that it does
not determine the direction of toilet flushes.  Our experiment showed
that when traveling distances which are short compared to the diameter
of the merry-go-round, the ball is not noticeably deflected.  Water
traveling a few inches from rim to center of a toilet ball is
traveling only a tiny, tiny fraction of the Earth's diameter, so the
Coriolis effect cannot be important.  What is important is how water
is injected into the bowl.

I reinforced that idea by showing this video from a travel documentary,
whose producers and stars are completely taken in by locals living on the
equator purporting to demonstrate that water drains in different
directions just north of the equator vs just south of the equator.  As
you watch this video, can you spot a major mistake the locals make?
Also, can you explain how the locals got the water to drain in
different directions at will?

In the next activity, we explored how the Coriolis effect also explains why storms like hurricanes circulate counterclockwise in the northern hemisphere and clockwise in the southern hemisphere.

Convection and Weather

This morning I guided the upper-graders at Peregrine School through a
set of weather-related activities.

First, we reviewed what we had learned previously about energy in the
Earth-Sun system.  Briefly, although the Earth's core is very hot and
slowly cooling off due to heat flowing outward through the surface,
the vast majority of heat that we experience comes from the Sun.  How
could we figure that out from everyday observations? Julia nailed it:
the surface temperature varies quite a bit from pole to equator and
from day to night, which is indicative of the Sun rather than of a
constant flow of heat outward in all directions from the Earth's core.
We also reviewed how the Sun heats the ground, not the air (because
air is transparent to visible light); air near the ground is heated by
the ground and that heat then gets mixed throughout the atmosphere.
Whenever something is heated from the bottom, as our atmosphere is,
you get convection (one of the three forms of heat transport we had
discussed earlier).

Convection is the reason we have weather. Hot air rises, cool air
sinks, and so air is always in motion.  To assess the solidity of
their understanding of convection, I immersed a shot-glass full of hot
water (dyed red) into a big container of cool water (dyed ever so
slightly yellow to provide contrast), but first, I asked them to make
predictions about what would happen.  This is a really nice, really
simple experiment or demo.  You can see the hot fluid rising in wisps;
eventually all the red collects on the top half of the large
container.  But the true test of understanding is predicting what
happens when I immerse a shot-glass full of cool water into a big
container of warm water.  The kids showed a good understanding by
predicting that the cool water would not rise at all, and just stay in
the shot-glass (and the bit of cool water which might spill in the
process of setting the shot glass down in the large container would
also settle on the bottom of the large container).  In weather, this
is called an inversion: if cold air gets under a layer of warm air, it
is trapped there, and among other things air pollution can build up in
a city where there's an inversion. (The Wikipedia article on inversion
has some decent pictures, and a Google image search on "weather
inversion" also yields some nice pics.)

Convection transports heat in the oceans as well as in the atmosphere.
There are ocean currents which circulate warm equatorial water toward
northern regions and bring cold water from the north back down toward
the equator to get warmed up again.  The sea off California's coast is
rather cold because the current here comes from Alaska.

But the takeaway message of this part of the day (which took probably
only 20-25 minutes) is that the Sun provides the energy for moving air
around, which makes weather happen.  Because of the way the Sun's
energy hits Earth, hot air must rise from equatorial regions and cold
air must sink near the poles.  But the only way for this to be
sustainable is with a "conveyor belt": hot air which rises from
equatorial regions moves toward the poles, where it cools, sinks, and
moves along the surface back toward the equator.  This creates wind
and weather patterns.  Our next activity was designed to add more
nuance to this general idea.

Before proceeding to the next activity, I presented the class with a
lava lamp for long-term loan.  This will constantly remind them of
convection even when I'm not there!

Climate Change

Yesterday we tied together California (Grade 6) Science Standards 6
(resources), 3 (Heat), and 4 (Energy in the Earth System).  We'd
already done quite a bit of 3 and 4, so we started with a discussion
of resources.  The consequences of using resources (6a) led naturally to
the greenhouse effect, which builds on our previous understanding of
heat flow in the Earth-Sun system.  We had previously calculated a
rough temperature that Earth "should" be at, ie the stable temperature
at which Earth should radiate just as much heat into space (in the
form of infrared light) as it gets from the Sun (mostly in the form of
visible light).  This temperature was just below freezing, and it
turns out that a natural greenhouse effect makes Earth livable.

We started with this video, which is a nice short demo of how carbon
dioxide absorbs infrared light.  C02 is by no means the only
greenhouse gas; water vapor is also very important, and methane
absorbs much more infrared light on a gram-for-gram basis, but there
is not enough methane in the atmosphere to make it the most important
greenhouse gas overall.  We also watched a short clip of another
video, which demonstrated how the temperature of a bottle of carbon
dioxide increased more than a bottle of air when both were heated by a
lamp.  This latter experiment requires only basic equipment and a
teacher might consider having the kids do the experiment, but I
suspect the experiment could be finicky in real life: you will have to
make sure there are no leaks in the C02 bottle, etc.

The kids were ahead of me on this one. They had already made the leap
to climate change, but I wanted to do at least a quick review to fill
in the logic.  The atmosphere is basically transparent to visible
light, the form in which we get energy from the Sun; if it's not
transparent to infrared light, the form in which Earth gets rid of its
heat, then Earth must heat up.  As stated above, we need a certain
amount of natural greenhouse effect to avoid freezing over, but there
can be too much of a good thing.  We spent the rest of the time in
small groups, playing with a computer simulation of all this. This
simulation is really good, so I encourage you to click Run Now (it
takes a minute to load and start).  You can adjust the level of
greenhouse gases from none (to see our previous calculation in action)
to lots.  As I circulated around the groups, we discussed the effect
of clouds (keep us cooler during the day but warmer at night) vs
greenhouse gases (always keep us warmer).  We also looked at the
Photon Absorption tab, which shows what's going on microscopically.
You can shoot visible or infrared photons (the smallest unit of light)
at a variety of molecules to see which are greenhouse gases.  In the
main (Greenhouse Effect) tab, the view is too zoomed out to see what
the photons are interacting with when they bounce around.  This was a
successful activity: students learned something as they explored, and
some students worked into their recess break to finish answering the
questions on the worksheet.

(Maven alert: it's common to say that greenhouse gases "trap" heat,
but this is not technically correct. It's more accurate to say that
they impede the flow of energy.  I didn't correct the kids when they
said "trap", but teachers should be aware of this.  Saying "trap" as a
teacher leaves you open to refutation.)

After the recess break, we discussed feedback loops and the
physics/engineering definition of positive and negative feedback
(which have nothing to do with psychological concepts such as negative
reinforcement or positive attitude).  I asked them to classify 11
different situations as positive or negative feedback (eg, foxes
provide negative feedback on the rabbit population), and they did very
well, so the concept is possibly less challenging than I imagined.  We
briefly discussed how confusing it is to have delayed feedback (eg
Alice says something to Bob and three days later he raises his voice).
Psychological experiments have shown that when feedback is delayed a
long time, people get very confused as to what causes what: they think
their actions have no effect, or the opposite effect.  (For more on
this, I recommend the book The Logic of Failure.)

So it is with climate change.  Scientists knew of CO2's heat
"trapping" properties more than a century ago and predicted rising
temperatures as we dumped more CO2 into the atmosphere, but it takes
so long for the heat to build up that it's easy to ignore.  By the
time we really see the temperature rise in a very convincing way, we
have dumped so much C02 into the atmosphere that temperatures will
rise much more even if we take immediate action.  Compounding this is
variability: if you just pay attention to the temperatures in your
neighborhood, there is so much variability from day to day and season
to season that it's impossible to notice a change in the average
temperature.  To see the change, you have to average together many
thousands of temperature measurements.

Even after getting people to accept that line of reasoning, they will be
unimpressed by the global average change so far: 1.4 degrees Fahrenheit.
What's a degree or two between friends?  But the change has been much
larger in some regions (the Arctic) and even 1.4 degrees results in a lot of
dislocation and expense: species have to adjust their ranges all over the world,
malaria may be able to move further from tropical regions, etc.  Won't Canada
and the northern US be happy to be a little bit warmer? Maybe, but it's not that
simple. Rain patterns may shift, so farmers in Canada may not be so happy after
all.  And northern forests are being destroyed at a rapid rate now that certain
kinds of beetles can survive the winter further north; beetles are mobile, but trees
are not, and the northern trees will be destroyed before they have time to
adapt to the beetle.  And areas which do gain from climate change may
be overrun with refugees from areas which lose big-time.

Anyway, the delayed-feedback idea led into the carbon cycle.  Over
tens of thousands of years the carbon cycle will remove excess carbon
from the atmosphere, so the Earth will not get hotter without limit
(thus answering an earlier question from a student). 

Our final activity was looking at this interactive flood map.  Seas
rise because the ocean heats up and expands (a very slow process) and
because of melting glaciers (not as slow, but still not easy to
predict).  The standard prediction for the year 2100 (when these
students will be old, but quite possibly still alive) is about 1 meter
of sea level rise, so I asked the students to dial in 1 meter and
answer a few questions about impacts on their house and on nearby
areas.  But the slowness of the ocean expansion means that the impact
of the current amount of carbon is further down the road, and has been
estimated to be 21 meters.  So I asked the students to dial in 21
meters and answer a few more questions.  This was another successful
activity combining student exploration with learning; I urge readers
of this blog to try the interactive flood map as well. Twenty-one
meters seems insane, so some kids need to be reassured that it will be
slow, over hundreds of years and perhaps a thousand years, so people
will have time to evacuate and adjust.  Still, evacuation and
adjustment are costly financially and emotionally so it may be better
to prevent the need for so much evacuation and adjustment in the first
place.

I didn't have time for a few things I wanted to show, but I can link to them here.
First, a quick Google image search for "glacier comparison" shows how fast most
glaciers are melting.  It is astounding*.  Second 30 seconds from this story about the
documentary Chasing Ice provide another dramatic look at glacier melting. (Sorry,
you will probably have to watch an ad to see this, but I couldn't find a better link.)

P.S.: Another important point for teachers of this subject is to emphasize that
"global warming" doesn't mean "every part of the Earth warms all of the time."
There is a model behind the predictions, a model with moving parts which affect
each other so that the predictions are richer than a novice imagines. For example,
a warmer atmosphere will also be a more humid atmosphere, so many areas will
get more precipitation and more intense storms.  If you live in a place where it's
cool enough to snow occasionally, then yes, global warming predicts that you can
get more snow.  People who think a big snowstorm contradicts predictions of
climate models simply haven't taken the time to get familiar with what climate
models really predict.  A scientific model should make a rich set of nuanced
predictions: that makes it easier to set up stringent experimental tests of the model.
This nuance does mean that scientists must work harder to educate the public.  If
any scientists are reading this, I plead with you to put in that hard work.  Society
needs you.

*Climate change deniers have recently made a big deal about a study showing that glaciers in some parts of the Himalayas are actually growing.  Note the qualified phrase "some parts of the Himalayas."  This is NOT what's happening to most glaciers around the world.  As noted above, climate change may have some "winners" as well as losers.  But I doubt the "winners" will feel very secure with so much dislocation in the world.

The Dating Game

Our third activity last Friday was on radioisotope dating.  (Everyone
has heard of carbon dating but carbon is just one of many radioactive
isotopes used for dating, and not even the most useful one for
geology, as we'll see below.)  This tied in with the other two
activities, because a certain age pattern in seafloor rocks was a
prediction generated by students' model of continental motions.  So
how do we measure the ages of rocks?

I prepared a whole bunch of small pieces of paper which were purple on
one side and white on the other.  Each group got a handful and put
them all purple-side up.  These represent potassium-40 atoms.  Each
group started constructing a graph of the number of potassium-40 atoms
vs time.  Let's say we start with 100 such atoms at a time we call
zero (we'll see later what this really means).  Over 1.25 billion
years, half of the potassium-40 atoms decay into argon-40 atoms.  The
students represent this by flipping over half of the "atoms" so the
white side is up.  Once flipped over, it can never flip back.  One
analogy is that once a ball rolls downhill, it's not going to roll
back up; similarly the argon-40 atom is in a lower-energy state.  (The
word "decay" evokes an irreversible process, for good reason.) So now
the student plots 50 atoms at a time of 1.25 billion years.  After
another 1.25 billion years, half of the remaining potassium-40 atoms
decay, so we now have 25 left and we plot that.  We may also want to
keep track of the number of argon-40 atoms, so (in a different color
pencil) we put zero of those atoms at time zero, 50 at 1.25 billion
years, and 75 at 2.5 billion years.

Keep going with this process.  In another time step, the 25
potassium-40 atoms decay into 12 or 13 argon-40 atoms. An atom can't
be half-decayed, and there is an element of randomness in this
process, so you can flip a coin or just decide randomly if it's 12 or
13.  (Aficionados will recognize that there is some probability of 11
or 14 as well, but that's beyond the scope here.)  Keep going until
you run out of graph paper, then connect the dots.

Now, how can we tell how old a rock is?  Look at the ratio of potassium-40 to
argon-40 atoms: 1:0 at the start, 1:1 at t=1.25 billion years, 1:3
at 2.5 billion years, 1:7 at 3.75 billion years, etc.  Measuring this
ratio provides an unambiguous estimate of the age of the rock.  I had
the kids field a few practice questions where I would give an age and
they would give a ratio or percentage, or vice versa. (If you're more comfortable
with percentages, as a percentage of the total potassium-40+argon-40 atoms, it's
100% potassium-40 at the start, 50% at 1.25 billion years, 25% at 2.5 billion
years, 12.5% at 3.75 billion years, etc.)

But what do we mean by the "age" of a rock? This is really key to
understanding the whole thing.  Argon is a gas, so when a rock is
molten the argon will just bubble out.  So when a rock solidifies, it
has no argon atoms, and a 1:0 ratio of potassium-40 to argon-40 atoms.
So t=0 corresponds to the last time that the rock solidified, which is
exactly the tool we need to date the age of new ocean floor oozing out
of the mid-Atlantic ridge!

The half-life of potassium-40 is well suited to dating rocks because
so many of them are so old.  What if we wanted to date something
younger, like a human skeleton from an archaeological dig?  Even if it
was 4000 years old, the ratio of potassium-40 to argon-40 would be so
close to 1:0 that we wouldn't be able to tell.  We need something with
a shorter half-life, like carbon-14, which decays to carbon-12 with a
half-life of only 5730 years.  Carbon-14 is great for dating
skeletons, but if we tried to use it to date a rock, we would most
likely find zero carbon-14 left so we would only be able to say that
it's many half-lives old.  To say that a rock is at least 10 of those
half-lives, or about 60,000 years old, is not very useful.

(Teacher warning: a lot of the implementation details are different
for carbon-14 dating.  For instance, there is never a 1:0 ratio of
carbon-14 to carbon-12, not even close. So adapting this exercise to
carbon dating would be tricky.)

This was a good activity for the 25 minutes we had left.  I wouldn't
try to squeeze this activity into any less time, but we definitely
could have used more time.  For example, we could have plotted the
ratio; used our computer skills to make plots, including of the ratio;
brought in some algebra to calculate things quickly through an
equation rather than graphically; etc.

Trembling in our Books

Yesterday we did three activities related to plate tectonics: making a
model of continental motion and generating predictions from it;
locating earthquakes; and radioisotope dating of rocks.  The second
activity followed roughly the reasoning outlined here.  However, I
didn't want to get into S and P waves, so instead of measuring the
distance from the epicenter to the seismograph by analyzing the wave
form, I decided to "simplify" and give students the time of arrival at
the three seismographs.  Only after we started the activity did I
realize that although the timing information I gave was sufficient,
some serious algebra would be required to solve the problem with just
that information.  So I ended up giving them the distance from the
earthquake to one of the seismographs, just to get them started.

Using timing information to solve for a location is an important
problem with many real-world aspects.  For example, GPS uses exactly
the kind of reasoning shown in the last figure of the page linked to
above, but in full 3-d with satellites distributed around the Earth,
to solve for your full 3-d location.  So I like the pure-timing aspect
of my version of the activity, but I have to find a way to make
workable for 5-7 graders.

Still, I don't think the kids noticed all this scrambling going on
behind the scenes.  They got the main ideas: the intersections of two
circles are the candidate epicenters based on two seismographs, and a
third seismograph can be used to resolve the ambiguity.  And they had
fun finding the mystery location of the epicenter.  I think we took
about 40 minutes on this activity, including a 5-minute opening
discussion on the link between earthquakes and our previous activity.

Get My Drift?

Yesterday we did three activities related to plate tectonics: making a
model of continental motion and generating predictions from it;
locating earthquakes; and radioisotope dating of rocks.

In the first activity, I gave students cutouts of the continents.
(The best way to find these is by googling terms related to this
activity; you can't just print a world map because of the distortion
inherent in most projections.)  The cutouts were on their desks as the
students filed in, so it was interesting to see what the students did
without any instructions: mostly arrange them as they are now rather
than try to put them together like a puzzle.  But it only took a small
hint to get them assembling the puzzle.  Once each group settled on a
way of fitting the continents together, I had them glue the model to
one side of a handout I had prepared.  On the other side they were
instructed to make four specific predictions about what would be
observable if this model were true.  I had to drop some major hints,
but the groups did eventually come up with the same four major
categories: (1) fossils on once-adjacent pieces of land should be the
same even though they are now very far apart; (2) living creatures on
once-adjacent pieces of land should be similar (making allowance for
evolutionary changes and for especially mobile animals such as birds
to be excluded from this analysis; (3) an expanding ocean floor should
be young in the middle where it spreads apart, and progressively older
near the continents (some groups put more emphasis on finding an
identifiable mid-ocean feature, but it's basically the same idea); (4)
once-adjacent pieces of land should have very similar older rock
layers even though they are now very far apart.  One thing no one got
even though I mentioned GPS is that we should be able to measure the
distance between, say, North America and Europe increasing very
slightly each year (it is, by a few centimeters per year).

I had planned for this to be iterative.  In my original plan the
groups were to make a very specific prediction such as "fossils found
in this part of Antarctica match the fossils in this part of
Australia", and then I would look that up quickly (to prevent
computers from being a distraction), and then after seeing how all
four predictions went they would make a better model on a new sheet of
paper (I brought lots of continent cutouts).  But the initial puzzle
assembling took much, much longer than I anticipated.  Some groups
took a lot of time to trim their rough-cut continent cutouts in
exquisite detail; others rearranged theirs many times; others just
didn't focus as much as I would have liked.  So we didn't go through
another iteration.  But one lesson that was clear to me at least is
that although South America fits nicely into Africa, almost nothing
else matches that clearly.  At some point you have to guess (this is
clear when comparing the different guesses of the different groups),
and at that point you have to look for fossil evidence to verify or
falsify your guess.  That whole process is what science is really all
about!

In the time left before break, I asked the students to guess why the
continents move.  They had a lot of crazy theories, but I steered it
back to what we had learned last week: the core of the Earth is hot,
heat flows to areas of lower temperature, and it can flow through
radiation, conduction, and/or convection.  We talked about how each of
these might or might not apply in this case, and figured out that
convection is well suited to transporting heat through the mantle,
which is fluid although not really molten.  Once we got this all into
a diagram with convection loops in the mantle, it was clear that this
was a very plausible mechanism for making continents move. 

This whole activity took 45 minutes, and as I mentioned I probably
should have budgeted much longer, and/or come up with ways to save
lots of time on the puzzle-assembly.  Devoting time to verify or
falsify specific predictions and come up with a better model would
have been a great illustration of the process of science.  Maybe it
should be a homework.  But, apart from this reservation, I think it's
a great activity.

Heat, Earth, and Sun

Friday I started earth science with the 5-7 graders at Peregrine
School. We started half an hour late because of the all-school
discussion of the meteor strike over Russia.  So I squeezed a lot into
35 minutes before a shortened recess break.  We reviewed the structure
of the Earth and then we talked about the three different ways heat
flows: conduction, convection, and radiation (which in this context is
just another word for light; it does not mean ionizing radiation,
which is what you need to protect your DNA from).  I brought a torch
and a saucepan to make the discussion of conduction more concrete:
cookware designers want the bottom to conduct heat very well so that
the food is heated evenly, but they want the handle to conduct heat
poorly so that you don't burn yourself.  Then I added water to segue
to convection.  Because hot fluids rise, convection occurs whenever a
fluid is heated from below, which occurs in very diverse contexts:
boiling water on the stove, fluid rock in Earth's mantle, and the
movement of air in the atmosphere.

Next, I drew a Sun far from our diagram of Earth, and I asked how heat
gets from the Sun to the Earth.  It can't be conduction or convection,
because empty space can't do either of these.  It's radiation (light).
So we observed thermal radiation (the light emitted by an object by
virtue of its temperature), noting the brightness and color of a light
bulb at different temperatures (achieved by changing the voltage).  We
analyzed the color in detail by looking through diffraction gratings
to make rainbows from the white light, and noting which color in the
rainbow was brightest.  The pattern that emerges is: raising the
temperature makes the light bluer, and makes it much brighter.  We
think of red hot as being about the hottest temperature we ever
encounter, but really white hot is even hotter (the light is a mixture
of red, green, and blue), and blue hot is even hotter than that.  (The
ocean and sky are blue because they scatter the blue light from the
Sun, not because they are emitting light.)  Even objects at room
temperature emit thermal radiation, but that light is "redder than
red" or infrared.  These kids had played with an infrared camera
before, so I didn't bring one, but we discussed their IR camera
experience in this new light.  (Read this post to get the basics of
the IR camera experience.)

The last point I made before recess break: Earth's temperature is a
balance between the energy it gets from the Sun and the energy
(infrared light) it emits into space.  To maintain a roughly stable
temperature, it must emit as much as it gets.  We would examine that
balance in more detail after the break.  During the break, I had a
trick to keep them thinking about this subject: I brought a parabolic
mirror, pointed it at the Sun, and we entertained ourselves setting
things on fire.

After the break, before moving on, I felt they needed more practice with
conduction, convection, and radiation, so I had them work in groups to design
thermoses.  We put together ideas from the different groups to arrive at a
consensus design which minimizes conduction, convection, and radiation.

Back to the main thread: I noted how the parabolic mirror gathered energy from
the Sun over a largish area and concentrated it on a small area.  If we
measured the power (energy per second) falling over one square meter
(about twice the area of the mirror), we would find that it's about
one kilowatt, or 1 kW.  I brought a 1 kW hair dryer to make that more
concrete.  We then talked about night vs day, and how the Sun is
fairly low in the sky during part of the day, and concluded that the
average power from the Sun on 1 square meter of Earth would be more
like 300 W.  So each square meter of Earth should emit about 300 W of
infrared light in order to maintain a stable temperature.

Recall that power emitted ("brightness") increases strongly as the
temperature of an object increases.  So if the temperature of that
square meter of Earth is low, it will emit less than it absorbs, and
that will raise its temperature.  But if the temperature goes up very
high, it will emit more than it absorbs, and the temp will come down.
We ought to be able to calculate the temp which is just right so that
it emits exactly 300 W.  This is where we returned to the computer
programming that the kids are loving so much.  Most of these kids are
not familiar with algebra, but they can (with lots of guidance from
me) write a loop over a range of plausible temperatures and print out
the power emitted at each temperature.

To do this, I had to give them the equation for power (in watts)
emitted as a function of temperature: 0.0000000567 T⁴, where T is in
Kelvins.  That led to a discussion of Fahrenheit vs Celsius vs Kelvin.
Fahrenheit is defined so that water freezes at 32 degrees and boils at
212, a 180-degree difference; Celsius is defined so that water freezes
at 0 degrees and boils at 100.  Therefore, each Celsius degree is
"bigger" by 180/100 or 9/5.  Therefore Fahrenheit = 9/5 Celsius + 32.
Kelvin = Celsius + 273 (I explained about absolute zero), so
Fahrenheit = 9/5 (Kelvin-273) + 32.  Admittedly, most students didn't
follow all these steps, but at least one did, and I told the others to
just use this to convert while focusing on the logical steps needed to
carry out their program.

So each group wrote a Python script to check from 1 to 1000 Kelvins,
at each step printing out the power emitted and the Fahrenheit
temperature.  It turns out that 26 F is the right temperature for 300
W.  Is this a reasonable answer?  We discussed the approximations
involved (primarily albedo, using snow as an example).  Then we tried
representing this information graphically.  Instead of scanning a list
of numbers to find the right temperature, I taught them how to make a
graph of power emitted vs temperature.  We then added a horizontal
line at 300 W, and the temp at which the line intersects the curve is
the "right" temp.  I really want to work on graph-making and
-interpreting skills, so we discussed the labels we should put on each
axis, and how to summarize the plot in words.

As a teaser for next week, a slightly more rigorous calculation shows
that Earth's global average temperature should be even colder than 26
F.  The reason we are not in fact that cold is that our atmosphere
intercepts some of the outgoing infrared light and turns it back to
the surface: the greenhouse effect.  There is a natural greenhouse
effect which makes our planet livable.  The kids had of course heard
of the greenhouse effect and global warming, so they were able to see
right away that the problem is not the greenhouse effect per se; it is
that we are adding to the natural greenhouse effect, resulting in too
much of a good thing.  More on that next week.

The original plot we made:

and a zoom in to the important part: