The Sand Reckoner

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SWBAT use powers of 10 to estimate extremely large quantities

Big Idea

Although we will most likely never be able to estimate the exact number of grains of sand in the universe, we can make a reasonable estimate.

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15 minutes

I like to start our investigation of scientific notation with a question that demonstrates the usefulness of scientific notation. "How many grains of sand are on Earth?" I supplement this question with two other questions:

1. What is daunting about this question?

2. How would you start to answer this question?

Students might a have a few comments and I make sure to let them share their ideas. This is a daunting task. I want students to listen to each other and develop their ability to take on questions that seem impossible. Once a few students have shared ideas, I give the class a few minutes to record their individual thoughts. We do a group share after. I like to record students' ideas on the board.

Common responses around why this is so daunting:

"It seems like sand is everywhere. How do we know where it is and where it isn't?"

"All sand is different, what do we mean by sand?"

"Is there an average size to sand? How accurate is this average?"

"How do deal with the sand under ground and under oceans?"

"What about sand in the atmosphere?"

Common responses around how we would start to solve this problem:

"We would look at the surface area of all the beachs and deserts."

"We would look at an average size for a grain of sand."

"Look at the depth and volume of beaches and deserts."

After we share I ask, "could we ever find an exact answer?" I give students a chance to answer, explain and debate. I supplement the conversation with images that show how varied beaches and deserts really are. I show the size of a grain and charts that measure sizes of different types of sand. Then I show images that help students understand how complex it would be to find an exact distance of coastline and exact size to the deserts on Earth. 

I tell students that I am asking a question that they could never answer exactly. However, would could find ways to give reasonable estimates. The idea is that in the real world we constantly estimate using mathematics. Real problems very rarely have exact answers. Instead, we model an answer that seems reasonable and then debate the validity of our answer. Here we could try and give upper and lower parameters to our estimate. I ask them, "how could we find an estimate that is almost guaranteed to be a little bit above and another that is a little bit below the exact value?"

I like to chart their responses for "over estimates" and "under estimates."

Common over estimation strategies:

"Assume the entire surface of the Earth is covered with sand."

"Assume all sand is very fine."

"Round the surface area and depths of coasts and deserts up."

Common under estimation strategies:

"Assume that only deserts and beaches have sand."

"Assume sand is coarse."

For each response, I ask them to tell me what why their idea would give a larger or smaller estimate. I also ask if a strategy would separate our estimate by a little or a lot from the exact value.

With these ideas, I like to graph a line for our overestimate and for our underestimate. This is a general graph with no actual numbers. The idea is for students to see that whatever the exact value is, it must be sandwiched between the over and under estimates (possible a great way to mention limits!) When we estimate, our goal is to get as close as possible to an exact value. If this gap is too large then we can not make sense of our question.

After I show this idea, I move toward the investigation. "We are actually looking to answer a question much bigger than this. We are looking to find the number of grains that fit into the universe.

Coda: Students often want to know the answer. I reply, "no one knows. But we will make some progress towards a reasonable estimate." We do come back to this in a future lesson and leave it on hold for now since it requires us to manipulate numbers in scientific notation."


25 minutes

I like to tell a story around this investigation.

"This calculation is based on the assumptions from Archimedes, a mathematician from ancient Greece. What is interesting is that he seems to have made a pretty reasonable estimation. 

He overestimated some values (he thought the circumference of the Earth = 555,000 km when it is about 40,000 km) and underestimated others (he thought the Sun was no more than 30 times larger than the moon when the Sun has a diameter more than 400 times that of the moon). He also imagined that the size of the universe could be imagined through a simple ratio. Imagine the ratio of the size of the Earth's orbit to the size of the Earth. Archimedes believe that this ratio equaled the ratio of the size of the universe to the size of Earth's orbit around the sun. This gives us a tiny universe (with a diameter of 2 light years, where our current observations show at least 91 billion light years).

However, in Archimdes work, it is amazing that the number he reached for a value larger than all the grains that can fit in the universe is comparable to our modern day upper estimate, even though his perception of the universe was radically different. In this lesson we look at the numbers he needed to invent to even think about these estimations."

I give the students the myrian hand out (from my class blog) and have them work through each question with a partner. 

In the third century B.C., the Greek Mathematician Archimedes wrote a book called The Sand Reckoner. Writing to Gelon, King of Syracuse, he wrote:

Many people believe, King Gelon, that the grains of sand are without number. Others think that although their number is not without limit, no number can ever be named which will be greater than the number of grains of sand. But I shall try to prove to you that among the numbers which I have named are those which exceed the number of grains in a heap of sand the size not only of the Earth but even of the universe.

Think about this, the number of grains of sand that would fill the entire universe. Could such a number exist? Archimedes had to invent new numbers to describe this quantity. He used the largest Greek number at the time, a myriad, which we now call "ten thousand."

1. Write a myriad as a power of 10

2. Archimedes began his investigation by thinking of a "myriad of myriads." This is like saying ten thousand times ten thousand. What do we call a "myriad of myriads"? Write this as a power of ten.

3. He continued his investigation by examining numbers of different order. The first order included all numbers from 1 to 100,000,000. Numbers in the second order ranged from 100,000,000 to (100,000,000 x 100,000,000). Write 100,000,000 times 100,000,000 as a power of ten.

4. The third order continued with this pattern and reached 100,000,000 cubed. Write this as a power of 10.

5. The fourth order ended with 100,000,000 to the fourth power. Write this as a power of 10.

6. Archimedes followed this pattern until he reached the "100,000,000th order." Write the highest number of this order as a power of 10. How many digits would this number contain?

7. Archimedes believed that the number of grains needed to fill the universe would be about 10 to the sixty-third power. How many times smaller is this that the number you analyzed in question 6?


20 minutes

For the summary, I discuss the answers to question that students seemed to master during the investigation, but ask students to demonstrate the last two questions in detail. I look for students who used the laws of exponents to explain the size comparison between 10^63 and the 100 millionth order. I believe it is imperative to spiral back to the logic of this comparison and show the magnificent size different between such large numbers. I want students to walk away from this lesson thinking that you can have two enormous numbers and still have an enormous different between them. As we work towards smaller numbers and the very tiny, I want them to return to this idea and think, "wow you can have two big numbers and two small numbers and still have enormous distances between them. I wonder if this is true for all numbers."