Tag Archives: volume

Exothermic fauna: A surface area poem

Two animals

Are both close to a cube

In shape.

The height of the first animal

Is 4 cm

The height of the second animal

Is 6 cm

Which animal

Would experience the greater


Of heat loss?


Thanks to Cathy Yenca and Chris Robinson for the find, whose source is evidently a 2007 U.S. textbook.

mhampton‘s reformulation of my poem (in the comments below) led to some fun over on Twitter, which I collected on Storify.


More junk food math

You’ve seen the billboards for these, right?

Basically, they’re mini McNuggets.

McNuggets have a lot of fat. Most of it (as with all fried foods with the possible exception of cheese curds) is in the outer coating.

So what happens when we shrink the McNugget?

We could model the chicken by volume and the outer coating by surface area. In this case, shrinking the McNugget by a factor 1/2 would yield a doubling of the ratio of coating to chicken. As a result we predict a substantial increase in fat as a percentage of weight.

But that’s a crude model-the coating isn’t infinitely thin.

A more realistic model might treat the coating as having a thickness, and thus a volume. When we shrink the McNugget to make a McBite, do we shrink the thickness of the coating proportionately? This seems unlikely. But even if we do, the ratio of coating to chicken increases.

More likely is that the coating is of constant thickness. In other words, McBites are probably smaller chicken bits covered with the same coating as McNuggets. Once again, we increase the ratio of coating to chicken, and we predict a substantial increase in fat as a percentage of weight.

Heading over to McDonald’s online Nutrition Information, we learn that McNuggets have about 12 grams of fat for a 65 gram serving. That’s about 18% fat by weight. This figure is consistent across serving sizes.

Mc Bites? 19 grams of fat for an 85 gram serving. That’s about 22% fat by weight. Also consistent across serving sizes.

How different is 22% fat from 18% fat, you ask? Very different. They’re both bad news, but 22% fat is (coincidentally) 22% more fat than 18% is. So take a serving of McNuggets, which has a lot of fat. And put almost a quarter more fat in. That’s McBites. Oh, and also increase the portion size from 97 grams (6 piece McNuggets) to 128 grams (Regular size McBites).

Middle school math will get you far, kids.

Now if we can get an estimate of the McNugget to McBite scale factor, we can set up a system of equations and figure out just how much of that fat is in the coating.

Salt, continued

Callum writes:

Filling a container with salt, sorry but…is this kind of thing inspiring to students learning maths? Isn’t this all rather pointless – you filled a container with salt – well done!


But can we agree that my salt problem is no worse than any of the following?

These were gleaned from a quick sample of middle-school, remedial college and mathematics for elementary teachers textbooks on my shelf. I grabbed four books off my shelf that I thought might have volume problems in them and found a problem for the gallery in each of them. No cherry-picking here. And I swear I didn’t leave out any compelling applications of volume of a cylinder.

But even if we agree it’s no worse, that’s not a very strong argument in favor of the problem.

So is there any aspect in which it might be better? I think there are several.


The initial question-will it fill or spill? admits student guesses. Students will have a hunch about the answer, and an intuitive sense of why it will or will not fill or spill.

This contrasts with these other problems. Students are asked to find a volume for the sole purpose of finding a volume. Not in order to answer anything some more meaningful question. And not even I have an intuitive sense of the volume of 678 flapjacks.

Plus, the question can come from the students. They can ask whether it will fill or spill; I don’t think I’ll have to. And then we’ll need to find some volumes in order to make a good prediction.


These are real containers. Perhaps not very compelling containers (although I’m a big fan of vintage Tupperware). But real containers nonetheless. Unlike anything in the problems above, these are objects in their daily lives.

Perhaps this is a sign of my hopeless math geekdom, but I am pleasantly surprised every time I refill my salt container that it fits perfectly. No leftover salt; no space left in the container. A perfect fit. I imagine some of that enthusiasm will be contagious in the classroom. And perhaps inspire some students to look at the containers in their own homes a little bit differently and wonder which ones are “bigger” than others.

the answer

Did I mention that the salt fills the container perfectly? And that we can see it happen before our eyes?

I’m not looking to draw eyes away from the Super Bowl with this problem, nor to cause students to switch their major. But I hope they’ll be a bit more invested in the outcome than they are in the textbook problems above.

intuition again

Here’s an interesting task from the math for elementary teachers book.

from Beckmann, S. (2010). Mathematics for Elementary Teachers. Boston: Pearson.

My instinct is that, at middle school, where the salt task would be appropriate, this will still be part of some students’ intuition. It is much more abstract to run the calculations and see that they are very, very close than to run them and then see that closeness play out in the physical world.

I’m not hoping to draw students to mathematics with this problem; I’m hoping to get them engaged for a lesson on volume.

But we’ll see. I’ll be using the problem with my future elementary teachers in a few weeks. This is not a population that is already sold on math (although by this late in the semester, I’ve reeled them in pretty well). I’ll report back.

And I welcome further critiques.

How do we make volume compelling?

Math 2.0: A newbie tries his hand

Will it fill? Will it spill?

THe question[s]

Two questions: Will the salt fill the Tupperware? and How long will it take to empty the package?

Some useful information


This is how much was rattling in the bottom of the container at the end.