At TLC middle school, we set aside an hour a day for PSST (Problem Solving and Strategic Thinking). It’s a hybrid of math, science, and problem solving, and it can seem abstract; so here’s an example of a PSST session that focuses on the sun — an appropriate topic for a hot summer day in Durham, NC. This particular problem about the sun originated from a book about solar power — as you can see in the excerpt below, it claims that more than a million Earths would fit in the sun.

I was skeptical that a *million* Earths would fit in the sun, and became curious to learn out exactly how many Earths would fit in the sun, given what we know about the size of each.

Turns out it’s not so easy to find the size of either the sun or the Earth. I started by looking for the diameter of the sun — here are some figures I found:

Source #1: diameter of the sun is 865,374 miles

Source #2: radius of the sun is 435,000 miles, so the diameter would be 870,000 radius

Source #3: radius of sun is 432,450,

so presumably the diameter would be double that, although this source inexplicably says, and I quote, “The mean radius of the sun is 432,450 miles, which makes its diameter about 864,938 miles…” Ummm… where did the extra 38 miles come from there?

In any case, let’s go with 865,000 miles for now — if we were doing this for real, I’d have a few students take on the task of investigating exactly how scientists measure the sun (it can’t be with a big tape measure) as well as the range of values reputable scientists come up with.

For the Earth, which is not a sphere, the diameter depends on where you measure — the equatorial diameter is longer than the polar diameter (with students, we’d make a rough model of the Earth so they could see this in action).

A quick Google search says this:

But another source told me this:

Since the Earth is not a perfect sphere, there are three numbers to know when answering questions about the diameter of Earth. The rotation of the planet has slightly flattened it out, so it has a larger diameter at the equator than at the poles. The equatorial diameter of Earth is 12,756 km, its polar diameter is 12,713 km, and its average diameter, which is referred to in common usage, is 12,742 km. For our friends who are not using the metric system, that translates to 7,926 miles.

[Pasted from http://www.universetoday.com/15055/diameter-of-earth/ ]

So Earth’s diameter is somewhere between 7,926 and 7,918 miles should do it — let’s go with 7,920 for a rough calculation. Again, if TLC were in session, a pair of students would investigate how people have tried to measure the Earth’s size, going back to the Greek scientist Eratosthenes (click for a 6-minute Carl Sagan video).

So the sun’s diameter is 865,500 miles, and the Earth’s is 7,920 miles — when we divide, we get 109.2803 — so you could line up 109 earths along the diameter of the sun.

But does that mean we could fit a *million* Earths inside the sun???

Well, let’s review (or look up, depending on where we are in math) the formula for finding the volume of a sphere:

V stands for volume of a sphere, pi is 3.14159…, and r is the radius of the sphere, raised to the third power in this equation.

Okay, let’s plug in some numbers:

V of Sun = 4/3 (pi) (432,750)^3

4/3*pi*(432,750)^3=3.39468779490276E17

For those of you not up on your scientific notation as expressed by a computer with an “E,” the number after the “E” is the number of zeroes to add after the decimal place, so that’s the same as 3.394… x 10^17, or roughly 339,468,779,000,000,000 cubic miles. Spoken, that’s 339 quadrillion, 468 trillion, etc…

[At TLC, we would take the time to introduce/review the concepts of exponents and scientific notation]

V of Earth = 4/3 (pi) (3960)^3

4/3*pi*(3960)^3=2.60120252602498E11

That works out to 260,100,000,000, or “just” 260 billion cubic miles. So as we anticipated, Earth is way smaller. But how much bigger is 339 quadrillion (the sun) than 260 billion (the Earth)?

Well, a quadrillion is a thousand times bigger than a trillion, so it’s a thousand thousand times bigger than a billion — and that’s a million. So that books was right. Sort of…

Let’s go ahead and divide the actual numbers to see how many Earths fit in the sun:

3.39468779490276E17/2.60120252602498E11=1.30504555525338E6

If our numbers are right, slightly more than 1.3 millions Earths fit inside the sun. That’s 30% more than “just” a million. Wow!

One of the things we will do at TLC is work to expand our thinking when we work with something like a sphere, so I thought it would be interesting to consider the volume of a sphere with a diameter of one mile.

When I did that, I made a mistake — **and that’s actually a good thing**, because it’s by playing around and making mistakes that we solidify concepts in our minds. Students at TLC will blog twice a week about math, and in those blogs, they will often describe the mistakes they make and/or the misconceptions they once held, but that they now understand. They will also write about concepts or topics they don’t understand; that’s a great opportunity for a teacher to have a conversation with a student about that topic/concept.

Let’s get to my mistake — here’s the wrong diagram I made:

*(My son frighteningly knew pi to 11 digits after the 3 — I had written 3.14 and he corrected me — that’s why I have that cross-out)*

So what’s wrong with that equation?

Well, one of the first tip-offs for me is that the result is more than 4 cubic miles, which seemed odd, because the sphere in question would actually *fit* *inside* a cubic mile, so how could it be more than four cubic miles?

[Again, at TLC we would demonstrate this by putting something like a golf ball or a ping-pong ball inside a square box that it fit into tightly]

Looking at the equation above, do you see where I messed up? I cubed the diameter of the sphere, rather than its radius. If we re-do the equation, here’s what we get:

So a mile-long sphere contains about half the volume of a cubic mile. Interesting… that means a baseball packed in a box takes up only about 52% of the volume of the box — that’s a lot of wasted space.

Moving back to the Earth and the sun, I once read somewhere that it takes light 8 minutes to get to the Earth. Let’s check that quickly — it’s about 93 million miles from the Earth to the Sun (how do we *know *that? we would go through the same drill as with measuring the diameter of the sun) and light travels at roughly 186,000 miles per second (how do we know *that*?). So when we divide 93,000,000 by 186,000, we get 500 seconds, which works out to 8.333 minutes. So yes, sunlight takes a little more than 8 minutes to get from the sun to the Earth.

One final connection I’d make: some time ago, I had Tweeted about how much (or really, how little) water there is on the Earth. I was able to find that long-ago Tweet using a tool called All My Tweets. By searching for the word “water,” I found this Tweet from May 2012:

And by following that link, I was re-acquainted with this very cool image that I had not seen for more than a year, but that connects nicely with this PSST session:

## Spheres representing all of Earth’s water, Earth’s liquid fresh water, and water in lakes and rivers

The largest sphere represents all of Earth’s water, and its diameter is about 860 miles (the distance from Salt Lake City, Utah, to Topeka, Kansas). It would have a volume of about 332,500,000 cubic miles (mi

^{3}) (1,386,000,000 cubic kilometers (km^{3})). The sphere includes all the water in the oceans, ice caps, lakes, and rivers, as well as groundwater, atmospheric water, and even the water in you, your dog, and your tomato plant.source: http://ga.water.usgs.gov/edu/2010/gallery/global-water-volume.html

There are all sorts of directions we could take that one… global warming comes to mind as we might think about how much of our frozen water is melting these days…

And of course, teachers at TLC would also encourage students to make connections of their own. The whole idea behind PSST is that the problem — in this case “how big is the sun?” — can lead to all sorts of scientific and mathematical inquiries.

So let’s review all the concepts we might have learned about in this PSST session:

Exponents, division, multiplication, the volume of spheres, the speed of light, global warming, and a review of large numbers such as quadrillion and trillion.

Note that at times, we might use our PSST time to work individually with students on math, wherever they are in a traditional curriculum. Our goal at TLC is to learn as much math as we can in three years, and each student will typically cover a traditional 6-7-8 math curriculum in about a year and a half. I’ll explain how students can do that by moving at their own pace in a future post that details the math curriculum at TLC.