SWBAT to compare lage numbers in certain contexts

Large numbers are best understood when we have visual and physical references.

15 minutes

I write the start up prompt on the board: **Today we are going to count the number of grains of sand on Earth and compare that to the stars in the universe. Which number do you think is larger?**

I circulate and see how they approach this question. I encourage them to explain their thinking and list out ideas they have for counting the total number of grains of sand on Earth. Then I show them three images of sand. How would you count the number of grains of sand in the picture?

As a class we discuss any ideas students suggest, ranging from counting individual grains, to counting an area, to counting a length. After we share their ideas, I give them 5 minutes to find the number of grains in each picture and determine the average number of grains in a cubic cm.

One key piece of information is that each photo represents a square centimeter.

Before we share results, we briefly review the discussion on the wide range of sand on Earth and the blurry lines that constitute the exact point at which sand becomes silt or gravel.

On average, students will get around 20 grains per centimeter and then 20^3 grains per cubic centimeter. I ask them to extend this to a cubic centimeter (and give them a minute on their own to try). Once we have established a reasonable number per cubic meter (20^3) x (100^3), we would discuss the next steps to solving the problem.

Essentially students need to find the volume of all the sand on Earth and divide it by the volume of a cubic meter (or whatever measurement is comfortable). We talk in depth about this division approach and I give students time to explain why this division would solve the problem.

At this point I would pull up two solutions to the number of grains on Earth. These sources are interesting because they both have clear flaws and radically different solutions.

Solution 1 is from the University of Hawaii: Solution 1

Solution 2 is from the Math Dude: Solution 2

In both cases, the researchers admit that they are simply guessing on the length, depth and width of coastlines, deserts, etc. Because of this, Solution 1 tells us there are 7.5 x 10^18 grains of sand and Solution 2 tells us that there 5.6 x 10^21 grains.

**I start the discussion by asking how many larger the Math Dude's estimation is.** Then we read through parts of each solution and I highlight their flawed reasoning, helping students identify and argue against the ways the made their calculations. I also ask for better approaches. For example, trying to look up a reliable source for the total length of the coastline.

I end this section by reminding students of our goal, "to find a better estimate of the number of grains on Earth."

20 minutes

We have a strong estimation process already because we are using three types of sand to deduce the average size of a grain. With this size they were already able to find an average volume of a cubic meter of sand. But now they need to find the volume of all the sand everywhere!

To help them in this daunting task, we give them the total area of all the deserts: Lists of Desserts by area

I put these into a table (which includes total area, but I like to leave that out and let students figure it out): desert size

I also give them a fact about deserts: only about 20% of deserts are covered in sand.

For the depth of a desert, there is no simple answer. Instead we give them some examples of average depths and extreme depths. This helps them make a reasonable average: desert size

Next we give them the length of all the coasts, from the CIA world fact book site:Coastline

Again, the dimensions of the coastline are in no way easy to simplify, but we can make some progress by looking at some averages and extremes: Beach depth and width

Interestingly, no estimate seems to try and count the sand we find elsewhere in rivers, oceans, mountains and dry land in general. To help students here, we give them total land and water areas on Earth. Total Land Area

The depth of sand in these areas can be something the speculate over. However, there are some general rules around depth here: Ultimate Depth

This should give enough for at least a *better* estimate than the ones we find online. However, the fun is arguing over the process used to estimate. Each group will have a reason as to how they estimated. They are rounding with intent. That is guaranteed to get some debate going.

Two other thoughts they might have.

First, we will create more sand as time goes on: Manufactured Sand

Second, they might want more data on the size of sand: Grain Size

I give students time to work with their partner and come up with the best estimate they can.

20 minutes

As students estimate the total grains of sand on Earth, I search for reasonable arguments that apply to any part of the problem solving process. Some groups may get through the whole process, but its more than likely that each group will have a piece of the puzzle solved. The summary is a time to compile all the solutions and form an answer that the class feels pretty good about.

During the estimation process, I make sure that students comment on each others reasoning and ask questions about the impact of each estimation process. For example, students might be surprised to find the range of estimates to fall within several magnitudes of each other. This shows that whatever the exact number of grains, our estimates will certainly show that they are above and below two numbers.

After about 15 minutes, I return to the question we had at the start of class. "Are their more grains of sand on Earth or stars in the universe." Instead of telling them, I let Carl Sagan do the talking. I show episode 8 of the Cosmost series, where Sagan tells us that there are *far* more stars in the universe than grains of sand on Earth.

A 2003 estimate tells us that there are 70 thousand million million million stars in the universe. I give the class two minutes to write this in scientific notation. After we agree on the answer, we finish class with an amazing observation: NPR

I end class by comparing these amazing numbers to something that *seems* small:

"There are the same number of molecules in 10 drops of water as there are stars in the universe." We take a few minutes to discuss how this could be possible. I leave it open ended though, because this is my transition into the world of the small.