Today we blasted off from our Earth-Moon base and explored the other planets. I started with this image of the terrestrial planets, which accurately depicts their relative sizes but not their distances. I brought in a big yoga ball to represent the Sun and we went in order from the Sun (ie from the left in that image). For each planet I elicited what they already knew or thought they knew about each planet, and then enriched it as best I could. For example, they knew Mercury is hot because it's close to the Sun...but what about the side away from the Sun (ie the night side, which is not always the same side)? It is actually very cold; why would that be? To put it another way, why is the day/night temperature variation on Earth not very extreme? That led to a discussion of atmospheres, which further led to a discussion of cratering, which further led to comparisons between Mercury and our Moon (similar size, both airless and cratered, extreme day/night temperature variation). I won't try to document each planet's discussion here, but 45 minutes flew by. (Here are links to a similar image comparing some asteroids in the asteroid belt, one comparing the gas giant planets (aka Jovian planets) and an image comparing the dwarf planets outside Neptune's orbit.) As we went, I filled in a table of planet sizes (diameters) and distances from the Sun, for later reference. I rounded the numbers quite a bit so kids would more easily see the comparisons. For example, rounding the Sun's diameter to 800,000 miles and Earth's to 8,000 we easily see that the Sun is about 100 times bigger across. This is way easier to understand than listing the exact numbers and doing the exact computation to find that it is 109 times bigger across.
Just before the break, I addressed why Pluto is no longer considered a planet. Short answer: it became clear that Pluto was just one of many smallish iceballs which are very unlike terrestrial planets and also very unlike Jovian planets, so they deserve their own class. When Pluto was the only known example, it didn't occur to anyone to put it in its own class. A nice example of how the way we classify things can change as we get more data.
After the break, we worked on understanding the distances and sizes by building scale models. First, we did the pocket solar system to understand the relative distances. It's quite amazing to see how relatively jam-packed the inner solar system is compared to the outer solar system, yet even in the inner solar system there are many tens of millions of miles between planets.
Next, the sizes. With the 65-cm-diameter yoga ball as the Sun, I pulled balls of various sizes out of my box: softball, baseball, tennis ball, ping-pong ball, etc. Because I had two ping-pong balls, students suggested they could be Earth and Venus, which are nearly the same size. Does this accurately depict how much smaller than the Sun these two planets are? Well, Earth is 100 times smaller than the Sun, so on this scale it should be 0.65 cm across, or 6.5mm (1/4 inch). That's way smaller than a ping-pong ball, so I had to rummage around in my kit, where I found some allspice. Allspice varies in size, but we did find some which were 6mm across. That's right, if the Sun is a yoga ball, Earth is the size of an allspice!
Whenever we do a calculation, we have to double-check it. I held up the yoga ball and the allspice and asked the kids if they thought 100 allspice would fit across the yoga ball. Yes, it looks about right. Out of curiosity, how many would fit in the yoga ball? Some of them guessed 100x100, because the yoga ball is 100x bigger in each of the two dimensions which are easily seen. But the yoga ball is also 100x bigger in the third dimension, so its volume is 100x100x100 or 1,000,000 (a million) times bigger. One million Earths could fit into the volume of the Sun. (The Sun's density is a bit less than Earth's, so the Sun's mass is "only" 318,000 times bigger than Earth's. For older kids, adding density and mass to this whole discussion might make sense.)
OK, so now we have Earth and Venus. What about Jupiter? Using the same reasoning, we found a ball about Jupiter's size (a small whiffleball, not much bigger than a ping-pong ball), and Saturn is just a bit smaller. Uranus and Neptune could be represented by small marbles. Mars could be a small allspice or an average peppercorn, and Mercury could be a mustard seed. Amazing! (If you're a teacher who would like to do this kind of activity, check out the peppercorn Earth website for some supporting materials.)
Finally, if these are the sizes of the planets in our scale model, what are the distances between planets? The Earth-Sun distance is about 100 Sun diameters, so we need 65 meters or about 200 feet. That's about the distance from our classroom to the far side of the playground. Jupiter is 5 times farther, so maybe we could put it at the KFC a block or so away. Pluto is 40 times further than Earth from the Sun, so that would be 8,000 feet or 1.6 miles, the distance from school to home for some of the kids. Imagine...all that space in between would be empty. Even Mercury, closest to the Sun, would be about 80 feet away and the size of a mustard seed!
At the end, I asked the students to choose a favorite planet or moon, learn more about it, and make a poster over the next two weeks. We'll put the posters up all over school at the appropriate distances to make a scale model. At the center of each poster will be a small object size to match the scale model. To fit the scale model into the school, some of them will have to be very small objects, like a grain of sand. Teacher Brittany will work with the students on the math for that, and I'll report back on the scale model in a few weeks.
Showing posts with label scaling relations. Show all posts
Showing posts with label scaling relations. Show all posts
Friday, May 10, 2013
Monday, February 20, 2012
100 Days
Primaria has been counting the days of school since it started in
September, to give the kids practice counting. Last week they got up
to 100, which was a big milestone they celebrated. So I sneaked in an
extra science activity when they did PE. The Sun is about 100 times
bigger than the Earth, which I thought made a nice connection between
science and what they were celebrating. I brought a 12-inch (1-foot)
globe to represent the Earth, and marked out a 100-foot diameter
(actually, 109 feet, to be more accurate) circle in the park to
represent the Sun. I marked it out with about a dozen cones, and to
help visualize a real circle I draped a 50-foot rope around the edges
of the first 2-3 cones; more would have been better.
It's one thing to say the the Sun is 109 times bigger than the Earth,
but another to see it! If you are familiar with the park, imagine the
Sun covering all the grass in the narrower east-west direction, and
most of the grass in the north-south direction. Compare that to
little old Earth, the 12-inch globe in my hand! This is the beginning
of a scale model, but to be true to the scale model we would have to
put Earth a few miles away, like in downtown Davis! The Sun looks
small only because it is really far away.
Part of the reason that saying "the Sun is 109 times bigger than the
Earth" has less impact than it should is that we are comparing
diameters, not volumes. The large area of grass I marked is like a
cross-section of the Sun, 109 times Earth's size in one dimension AND
109 times Earth's size in the second dimension as well, for over
10,000 times the area. But we should really try to visualize a sphere
109 feet tall as well, which would make for a volume over a million
times the volume of the globe in my hand. So we could just as easily
say that the Sun is over a million times Earth's size, if we are
talking about volume. See my post(s) on scaling relations
for more on this. I didn't try to get the Primaria kids to think
abstractly about this, but I did ask them to imagine the circle
representing the Sun as extending 100 feet high. Then we ran around
the Sun a few times.
September, to give the kids practice counting. Last week they got up
to 100, which was a big milestone they celebrated. So I sneaked in an
extra science activity when they did PE. The Sun is about 100 times
bigger than the Earth, which I thought made a nice connection between
science and what they were celebrating. I brought a 12-inch (1-foot)
globe to represent the Earth, and marked out a 100-foot diameter
(actually, 109 feet, to be more accurate) circle in the park to
represent the Sun. I marked it out with about a dozen cones, and to
help visualize a real circle I draped a 50-foot rope around the edges
of the first 2-3 cones; more would have been better.
It's one thing to say the the Sun is 109 times bigger than the Earth,
but another to see it! If you are familiar with the park, imagine the
Sun covering all the grass in the narrower east-west direction, and
most of the grass in the north-south direction. Compare that to
little old Earth, the 12-inch globe in my hand! This is the beginning
of a scale model, but to be true to the scale model we would have to
put Earth a few miles away, like in downtown Davis! The Sun looks
small only because it is really far away.
Part of the reason that saying "the Sun is 109 times bigger than the
Earth" has less impact than it should is that we are comparing
diameters, not volumes. The large area of grass I marked is like a
cross-section of the Sun, 109 times Earth's size in one dimension AND
109 times Earth's size in the second dimension as well, for over
10,000 times the area. But we should really try to visualize a sphere
109 feet tall as well, which would make for a volume over a million
times the volume of the globe in my hand. So we could just as easily
say that the Sun is over a million times Earth's size, if we are
talking about volume. See my post(s) on scaling relations
for more on this. I didn't try to get the Primaria kids to think
abstractly about this, but I did ask them to imagine the circle
representing the Sun as extending 100 feet high. Then we ran around
the Sun a few times.
Friday, September 23, 2011
Length, Area, Volume, Dinosaurs and Giant Insects
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.
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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