One of the strengths of Peregrine School is that learning is
integrated. Instead of offering isolated concepts and facts, we try
to design activities which build on and reinforce each other. The
past week or two, the elementary students have been reviewing how to
measure. They have measured their bedrooms and mapped their rooms on
pieces of graph paper, for example. So I thought I would follow that
up by specifically relating length with area and volume and leading
into one of the most underappreciated aspects of science: scaling
relations.
(By the way, I didn't get time to talk about a lot of these ideas with
each group, so this post may be a good resource for parents who would
like to follow up the activity and investigate more.)
Beforehand, I printed out some graph paper with 3/4" squares to match
the unifix cubes we had on hand. This way, when we shifted from
measuring area (counting squares of graph paper) to measuring volume
(counting cubes), the kids would not be distracted by a change in the
size of the measuring unit. To people who understand area and volume,
a change in grid size might be an irrelevant detail, but to those who
don't, it could be the start of a long and unnecessary detour.
First, I asked them to draw a map of a room which was 1x1, and measure
the area by counting the squares inside (1, obviously). This should
be really boring and easy, but it took a surprisingly long time to get
everybody clear on what I was asking. Maybe next time, to speed
things up, I would start by drawing the 1x1 square so they would
instantly see my intention rather than having to parse my words.
Next, I asked them to draw a 2x2 square, predict the area, and then
measure it. Over the course of 3 groups, I learned to be stricter
about making the prediction. Once kids understand what the task is,
they are quite happy to do bigger and bigger squares and get answers
without making any predictions. But they don't learn anything this
way; only if they see that their prediction was wrong will they take
the trouble to figure out what's really going on. (In principle, the
discomfort of realizing that reality conflicts with his/her beliefs
("cognitive dissonance") will spur the student to make the effort to
really understand. In practice, it doesn't always work that
way...perhaps the subject of another post. But the fact that some
students won't care when their predictions are wrong should not stop
us from trying; they are even less likely to care if they never commit
to a prediction. And if we get even a bare majority of the class
modifying their beliefs based on evidence, we are doing a great
thing!)
So, in the spirit of being strict about making predictions, I ask you
to make a prediction. If you were to double the length and the width
of the 1x1 square, what would happen to the area? What if you were to
triple the length and width? Quadruple? Make your predictions before
reading on!
With the kids telling me the results of their experiments, I compiled
the following chart:
length area
---------------
1 1
2 4
3 9
4 16
5 25
Note that very few of them made correct predictions of these results!
I also made a graph of this, to show them the pattern in different
ways. The graph shows a very rapidly rising curve. The area
increases much faster than the length!
Why should we care? None of them knew. And I think this is a weakness
with traditional math education. These concepts are taught in a
vacuum and never made relevant. (A recent op-ed in the New York Times
agrees with me on this.) So I gave two examples:
(1) if you wanted a room twice as long and twice as wide, how many
times more carpet would you need to cover the floor? Amazingly, no
one got this right even after I pointed to the chart, with numerous
hints! The most popular answer was double. They were not able to
transfer the idea from the chart to a new situation, so they had not
really learned. Good thing I asked them instead of just moving on!
So we discussed this a bit before moving on.
(2) Suppose you have a small room full of people, like at a party.
There's a door in each wall so if there's a fire, they can easily
escape. But now imagine a room 3x longer and 3x wider. It can fit 9x
the people, so now there will be long lines at the doors when the fire
alarm goes off. People could die if architects didn't think about
these things! We'll need a lot more doors for rooms which, if we
thought only about length, might not seem so much bigger. Or you
could think about evacuating a city. A city 10x the size in each
direction has 100x the people, but nowhere near 100x the number of
escape routes, so it may be simply impossible to evacuate in a short
amount of time.
(3) I didn't have time to show that surface area behaves the same way
as area, but it does. That is, if we wanted to carpet ALL the
surfaces of a room, a room twice as big in each dimension would still
take only 4x the carpet of the smaller room. (In either case, the
total surface area is 6x the area of the floor.) Furthermore, the
same is still true of an irregular surface such as the surface of your
body. A dog twice as large in each dimension will have quadruple the
surface area you need to brush!
Ok, now we moved on to volume. I asked them to model a 1x1x1 room
with cubes and tell me the volume, then predict the volume of a 2x2x2
room. Again, no one got it right! We quickly built up a new column
in our chart:
length area volume
---------------------------
1 1 1
2 4 8
3 9 27
4 16 64
5 25 125
(actually, I filled in the 64 and 125 due to lack of time). Again,
why do we care? Well, the volume increases much faster than even
the area does, so if you wanted a swimming pool twice as long,
twice as wide, and twice as deep, it would have quadruple the surface area but
you should plan to pay for eight times the water to fill it up!
And here's where I finally got to the applications I thought would
interest the kids. If you have a dog twice as long, twice as tall,
and twice as wide as another dog, it will have 8x the volume and thus
8x the weight. You will have to feed that dog 8x the amount of energy
if it's going to have the same activity level as the smaller dog (thus
explaining why larger dogs generally do NOT have the same activity
level as larger dogs). If you have a dinosaur four times the size of
a rhino in each dimension, it will weigh as much as 64 rhinos!
This actually explains why there is a limit to the size of animals.
If you double the size of an animal in each dimension, it has eight
times the weight to support. But the strength of its bones grows only
as their cross-sectional area (4x in this example). So the bones will
not be able to support the weight. To make that bigger animal, we
will need to change the plan and not just scale everything up. This
explains why we don't see mice the size of elephants. To go from
mouse-size to elephant size requires a drastic redesign and
proportionally much thicker bones. And there's a limit to how thick
you can make the bones, so there's a limit to the redesign strategy.
So why are the largest whales so much bigger than elephants? Because
whales don't really support their weight; the ocean does.
And why were the largest dinosaurs so much bigger than elephants? I'm
guessing that it's due to a more subtle limit on the elephant. If you
double each dimension of an elephant so it has 8x the weight, then it
also produces 8x the heat because it is warm-blooded. But this heat
can only escape through the surface area, which is 4x the original surface area. So the 2x-scale elephant would overheat! Dinosaurs (at least the biggest
ones, as far as we know) were cold-blooded, so they didn't have this
issue. (It was also thought at one time that the largest dinosaurs
were swamp-dwellers so that water supported some of their weight, but
this seems to be unsupported by evidence.)
(Why don't blue whales overheat? I'm guessing it's because
water conducts heat away much better, and because they are very long
and skinny, thus maximizing their surface area; they are NOT 2x-scale
copies of smaller whales.)
This heat consideration explains a couple of other things. Larger
mammals have slower metabolisms than smaller mammals, to avoid
overheating. (Slower metabolisms, by the way, enable longer life
spans. Ever notice how the life span of a mammal correlates roughly
with its size?) And elephants have enormous ears to help them get rid
of heat.
Finally, let's think about insects. Insects get oxygen by absorbing
it through their "skin", so if we make a 2x-scale insect, we can get
oxygen in at 4x the rate, but we have 8x the amount of flesh which
needs oxygen. That's not gonna work. So we have a limit on the size
of insects. But why were there 2.5-foot dragonflies hundreds of
millions of years ago? Turns out that there was much more oxygen in
the atmosphere at that time, so each bit of surface area could take in
oxygen more effectively, and the area/volume tradeoff could be pushed
further towards more volume.
Now, I didn't get time to talk about many of these ideas, at least in
any detail. We will expand my science slot to 30 minutes with each
group. But here are some resources for parents to follow up:
Wikipedia article on the square-cube law: this repeats much of what I said about volume vs area, but has links to related articles in more depth. In particular, if you like to think about animals, you might want to check out....
Wikipedia article on allometry: allometry is the fact that most larger animals are NOT scaled-up smaller animals, but have changes in body plan.
Wikipedia article on Meganeura: this is the 2.5-foot dragonfly. It has a link to the 1999 Nature article linking insect gigantism to oxygen supply.
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