This is more of a How Can You Do This, because I bombed. Students were baffled. I’m not sure how to work the scaffolding on this one, even though I got the concept from a textbook. Any help appreciated. ADD: Suggestions to fix this lesson are in the comments.

Trailer for THEM!

Followed (after discussion) by:

Now, you may have some student give a clever answer using evolutionary biology, but at some point establish this is a Super-Science sort of situation, leading to:

Suppose you had a super-enlarger ray. How large could you make the ant so it still survives?

And here’s the textbook version (McDougal Littell, Algebra 2):

A common ant absorbs oxygen at a rate of about 6.2 milliliters per second per square centimeter of exoskeleton. It needs about 24 millilitres of oxygen per second per cubic centimeter of its body. An ant is basically cylindrical in shape, so its surface area S and volume V can be approximated by the formulas for the surface area and volume of a cylinder:

a. Approximate the surface area and volume of an ant that is 8 millimeters long and has a radius of 1.5 millimeters. Would this ant have a surface area large enough to meet its oxygen needs?

b. Consider a “giant” ant that is 8 meters long and has a radius of 1.5 meters. Would this ant have a surface area large enough to meet its oxygen needs?

So: no dice, even after simplifying the cylinder to a rectangular prism and exploring the 1x1x1 cube with the sides doubled first. (I could have led in with the Athenian Plague problem, which is roughly equivalent.)

In any case, students were paralyzed with indecision and I nearly had to do the problem for them to wrap it up. So, how could this work?

I think the movie was really cool, and the problem seems to be an interesting one. But where exactly did they get stuck? It seems like calculating the volume and surface area for both parts should’ve been straightforward – just plug and chug.

But the 2nd part of both questions – can the ant meet its oxygen needs? is pretty tricky. Maybe I’m just a little slow in the morning, but I think I’m missing the crucial piece of the puzzle to finish the problem… What aspect of that part tripped up your students? the whole thing?

The Athenian Plague problem seemed more obvious to me (maybe b/c there was less lead-in?) and you could’ve used blocks or something to get them figure out how to double the size of the temple on their own, and then got them to think through why the other equations worked as well.

Discrete examples (like in the textbook) are easy. The problem is to figure out the exact size when the ant goes from surviving to not surviving. They weren’t able to generalize.

Athenian Plague does seem like a good lead-in problem (previous day likely), but I didn’t remember it until after everything was done.

I’m pretty sure I didn’t clue to the whole creature scaling issue until I was maybe in my senior year in high school. It’s not immediately intuitive based on what we can see in nature.

Respiration and Heat exchange are the two really most critical issues, but hardest to visualize. Structural strength is less critical, but easier to examine.

I’d start by having them build a fairly simple structure and then rebuilding it at a much larger scale. Ideally, it’d be an origami ant, but that might be overwhelming, so maybe something with pasta, aluminum foil, or balsa wood could be done.. I’m hoping that the falling apart of the second one would lead to an investigation of weight vs cross section of the limbs.

I’m also not sure this would do much better than what you’ve got.

I was giving them the oxygen statistics above; the science wasn’t really the problem.

You may be right something like a model might be necessary for them to visualize this fully.

I could just stick with the two textbook problems and call it a wrap, it’s just I feel like the _interesting_ question in the WCYDWT sense is exactly at what point does the ant go boom.

Perhaps one solution would be to bypass the silly cylinder/prism approximation. Give them the surface are & volume (or mass?) of an 5 mm ant. Then have them figure out from scaling properties what it would be for a 10 (or 8) mm ant. That should be enough to let them find the pattern, and devise a solution.

I can totally see them getting hung up on dealing with pi.

I’m thinking that an activity BEFORE this problem building rectangular prisms with the centimeter cubes and making tables of side length/surface area/volume to look at the patterns (and possibly graph the patterns & find equations for them) might be helpful.

Once they have that big idea in their heads, then the ant problem is still engaging and challenging, but more manageable. Then adding that idea of when does it become to big to support itself (oxygen) is a great extension (would that happen when the graphs intersected?).

The trailer mentions they multiply faster than they can be killed. I have created an excel spreadsheet exam question based on the rate of repopulation of an alien as a % vs the number killed every day and see how long it would take to get the population down to a manageable size and nuke the last 100.

Basically a loan type question using exponential regression increase subtracting the number of ants killed to find the balance each day until the balance reaches a target. If you want a sample of the question and sample spreadsheet, please feel free to email me.

[…] all my lessons that have worked beautifully, an equal number have failed. Fine, so you tried something creative and it didn’t work: get out there and try it […]

m, on October 9, 2009 at 9:52 am said:I think the movie was really cool, and the problem seems to be an interesting one. But where exactly did they get stuck? It seems like calculating the volume and surface area for both parts should’ve been straightforward – just plug and chug.

But the 2nd part of both questions – can the ant meet its oxygen needs? is pretty tricky. Maybe I’m just a little slow in the morning, but I think I’m missing the crucial piece of the puzzle to finish the problem… What aspect of that part tripped up your students? the whole thing?

The Athenian Plague problem seemed more obvious to me (maybe b/c there was less lead-in?) and you could’ve used blocks or something to get them figure out how to double the size of the temple on their own, and then got them to think through why the other equations worked as well.

Jason Dyer, on October 9, 2009 at 1:10 pm said:Discrete examples (like in the textbook) are easy. The problem is to figure out the exact size when the ant goes from surviving to not surviving. They weren’t able to generalize.

Athenian Plague does seem like a good lead-in problem (previous day likely), but I didn’t remember it until after everything was done.

Mr. K, on October 9, 2009 at 2:16 pm said:I’m pretty sure I didn’t clue to the whole creature scaling issue until I was maybe in my senior year in high school. It’s not immediately intuitive based on what we can see in nature.

Respiration and Heat exchange are the two really most critical issues, but hardest to visualize. Structural strength is less critical, but easier to examine.

I’d start by having them build a fairly simple structure and then rebuilding it at a much larger scale. Ideally, it’d be an origami ant, but that might be overwhelming, so maybe something with pasta, aluminum foil, or balsa wood could be done.. I’m hoping that the falling apart of the second one would lead to an investigation of weight vs cross section of the limbs.

I’m also not sure this would do much better than what you’ve got.

Jason Dyer, on October 9, 2009 at 3:27 pm said:I was giving them the oxygen statistics above; the science wasn’t really the problem.

You may be right something like a model might be necessary for them to visualize this fully.

I could just stick with the two textbook problems and call it a wrap, it’s just I feel like the _interesting_ question in the WCYDWT sense is exactly at what point does the ant go boom.

Mr. K, on October 9, 2009 at 4:00 pm said:Perhaps one solution would be to bypass the silly cylinder/prism approximation. Give them the surface are & volume (or mass?) of an 5 mm ant. Then have them figure out from scaling properties what it would be for a 10 (or 8) mm ant. That should be enough to let them find the pattern, and devise a solution.

I can totally see them getting hung up on dealing with pi.

Mrs. Fuller, on October 10, 2009 at 10:54 am said:Wow, I love the idea!

I’m thinking that an activity BEFORE this problem building rectangular prisms with the centimeter cubes and making tables of side length/surface area/volume to look at the patterns (and possibly graph the patterns & find equations for them) might be helpful.

Once they have that big idea in their heads, then the ant problem is still engaging and challenging, but more manageable. Then adding that idea of when does it become to big to support itself (oxygen) is a great extension (would that happen when the graphs intersected?).

What do you think?

Jason Dyer, on October 13, 2009 at 8:38 am said:Would that happen when the graphs intersected?Yes.

I like all the suggestions. I’m still mulling and may compile a “fixed” version of this lesson, but for now I’ll just point to these comments.

Peter, on July 7, 2010 at 12:32 pm said:The trailer mentions they multiply faster than they can be killed. I have created an excel spreadsheet exam question based on the rate of repopulation of an alien as a % vs the number killed every day and see how long it would take to get the population down to a manageable size and nuke the last 100.

Basically a loan type question using exponential regression increase subtracting the number of ants killed to find the balance each day until the balance reaches a target. If you want a sample of the question and sample spreadsheet, please feel free to email me.

How to be creative « The Number Warrior, on July 28, 2010 at 8:03 am said:[…] all my lessons that have worked beautifully, an equal number have failed. Fine, so you tried something creative and it didn’t work: get out there and try it […]