http://serc.carleton.edu/spaceboston/2010activities/46734.html#discussion http://serc.carleton.edu/spaceboston/2010activities/46734.html#post12756
Some random thoughts from a mathematician as I read through things:

1. SUMMARY: Is it worth making the sentence "... glass spheres of a different size but the same total mass" a little clearer? That is, you are willing to put many small spheres into the same cup to match the mass of the one big sphere in the other cup. (I assume this is what you mean? Or do you mean you are going to have single small spheres, one in a cup, but made of a denser material so that the mass of the small sphere is the same as the mass of a large sphere?

2. BACKGROUND: Writing math in an e-mail is indeed a pain. People write: k^2 and k^3 for k squared and k cubed.

I completely ignored the issue of how much food a polar bear needs to eat in order to maintain its muscle mass, c.f. a black bear. If you get into this dicussion with kids, they will bring this up. So being big in a cold climate means you get to "keep" more of the energy you produce, but you have to find a lot more to eat to be that efficient. It's a trade-off that you will need to be prepared to think about with kids.

3. I think all looks good - in this writing stage of things. Of course, I am interested to see if reality matches the ideal here. Let me know how it all turns out when you test-run all this for yourself.

Fun stuff!

- J



]]> James Tanton 1277647080 http://serc.carleton.edu/spaceboston/2010activities/46734.html#post12756 http://serc.carleton.edu/spaceboston/2010activities/46734.html#post12783
Really neat idea, cleverly designed. I'll be very interested to see if the students understand the planetary importance; could be a real winner!

A few content notes: Most petrologists would put 1200C at the low end of upper mantle temperatures. A good range is 1200 - 1400C, with an accepted characteristic temperature of 1300 or 1350C.

"The date of the formation of the Earth is difficult to define but it is now generally agreed that the Earth was nearly fully formed by 4.567 billion years ago." This is actually not right - that age, 4.567 Ga, is the age of the first tiny spherical solids, the CAIs. The planetesimals that later assembled to form the Earth were just beginning to grow at that time. The protoEarth itself, the body that received the Moon-forming impact, was not assembled until at least 30 million or perhaps 60 - 90 million years later (4.53 to 4.47 Ga).

There's new evidence that Mercury's core may still be partly molten...not much more I can say about that. Also, Europa is thought to have a subsurface ocean, and something is melting the interior of Enceladus. Might be important to say something at the end about other influences?

I can now articulate more about your statement "...not all of the residual heat energy can be accounted for in this way. Instead, the Earth is still so hot inside because its surface is too small, relative to its volume, for this heat to escape." The surface-to-volume argument applies to all sources of heat energy, whether they are "fossil" accretionary heat or ongoing radiogenic heat. Your statement makes me, at least, think that radiogenic heat is a separate problem from surface heat loss. It may be better to say that there are two major sources of heat in planetary interior: accretionary heat not yet lost to space, and radiogenic heat, also dwindling with time. So why is the interior of the Earth still hot where Mars and the Moon are colder?

Little clean-up note: "As a result they have 23 = 8 times as much volume..." OK, I know you know 23 does not equal 8 ...can you write this out more?

How do you know polar bears produce eight times as much heat in their bodies? Not necessarily true. And do the two kinds of bears have the same normal body core temperature? I'd take care not to push this analogy too far. Also depends upon food sources...

Can't wait to hear how this turns out! I'd love to see the data.

all the best -

Lindy]]> Lindy Elkins-Tanton 1277732460 http://serc.carleton.edu/spaceboston/2010activities/46734.html#post12783 http://serc.carleton.edu/spaceboston/2010activities/46734.html#post12811 Thanks very much for your valuable comments. I was particularly glad to get the scientific details correct. I hope I have changed things with regard to those details so that they are more accurate. I hesitated to put in too much about the outer planets' moons because I thought it might muddy the waters. On the other hand, now that I've written it in it may be that it could lead to fruitful discussion.

You're right about the polar bear analogy, both of you. I added a sentence about it being 'only an analogy' so that readers will, I hope, not take it too far. I am making a mental note for questions about food: certainly the heat-efficiency is paid for with a food inefficiency.

I've fixed all the mathematical typos: I thought they were all set and didn't check them when I pasted the work into the site from the word processor. I had typed all the exponents as superscripts and foolishly assumed they had pasted that way. Now I believe they are formatted correctly so that I won't have to use the '^' key.

As for whether reality will match the ideal...a lot depends on what materials I can get to use for the glass spheres. If I can get two different sizes of pyrex glass spheres then I expect I can get decent data. If not, well, we'll see. As you may recall, I hesitated to even write this activity because I was so unsure about whether it would be practical in the lab. I'll certainly be testing things out myself before I set students loose on it. I am interested to see how it turns out, though, and I really expect that it will spark some interesting discussion in class.

Thanks again for your thoughts. This was a very stimulating class and my thoughts are still whirling as a result. If only I could teach an entire course about planetary science! I almost think we covered enough this week to fill it.

I'll be in touch once I've had a chance to run some tests and put it past my class(es). Nothing like this is ever really 'live' until you try it. Even so, I'm content for this activity to made live on the SERC site. Maybe someone else will get to it before I do and try it out!

Take care,

Aaron]]> Aaron Keller 1277750880 http://serc.carleton.edu/spaceboston/2010activities/46734.html#post12811 http://serc.carleton.edu/spaceboston/2010activities/46734.html#post12867
I, too, find this an interesting activity and want to know what happens when you actually try it out. I’ll be interested to hear how the kids connect it to planet cooling, and also wonder what connections kids will make to related daily-life situations. (Perhaps they have been surprised by the amount of time it took for a pot heated on the stove to cool, or something like that.)

The level of detail you’ve provided throughout this write-up will be helpful to other teachers who consider it. It has also enabled you to continue developing your ideas about the science involved as others react to specifics.

About the bears: If you present this a little differently, you can avoid the problematic aspects of comparing different species, or bears with planets. You could just note that some scientists have observed that within a species, individuals in colder parts of the range tend to be larger. Others have noted that high surface area/volume parts like ears tend to be smaller in cold-dwelling critters (again, within a species) – larger in warmer parts of the range. I’ll bet there are exceptions to the rule, and it would be interesting for kids to think about.

Regarding the Standards: Eventually, you may opt to link your activity to the National Science Education Standards as that could be useful to other teachers who are considering it. Obviously not a high priority. I'm glad you have a good degree of freedom in designing your program.

Ellen
]]> Ellen Doris 1277868360 http://serc.carleton.edu/spaceboston/2010activities/46734.html#post12867