SWBAT to describe large quantities of digital storage through exponential notation

There is an incredible amount of storage space that we incorrectly quantify. The only way to understand these discrepancies is to use exponential notation.

15 minutes

I like to start off this investigation with a *demonstration* of the problem. (**Note**: I use a mac for this demonstration, which happens to be critical. If you have a PC, read through the Mac presentation and then check below for more information. This could also work really nicely with a PC. You just would change the presentation style a bit.) If you start with a Mac, try this presentation:

"So I created a video that is about 2 gigabytes (it is playing as students enter) and then I try to save it my 2 gigabyte flash drive. However, something doesn't match up. Why doesn't a video that is *just* under 2 gigabytes not fit?" I then open the Disk Utility App (all macs have this under Apps and Utilities) and erase the flash drive to "start over." Now before I load the video on, I click the image of flash drive and hit command+I to "get info." It tells me that about 789,000 KB are already used: Screen Shot 2013-09-02 at 2.48.06 PM.png

The video file size is here: Screen Shot 2013-09-02 at 2.48.27 PM.png

Together this is over 2 GB. I ask them how we can know this without finding the exact value.

So for some reason I can't upload the video on a Mac (maybe its the way the mac configures a flash drive). So I plug the flash drive into a PC and reformat it. But wait! Somehow the flash drive has a lot less space? Flash Drive Space.doc Now it only has 1.85 GB? How is this possible, and "what is a gigabyte?"

*If you don't have a Mac, you could pretty much reverse this presentation by plugging a 2 GB (or different) size flashdrive and showing how there isn't 2GB of space available. You could then use the screenshots below to show that if you plug it into a Mac it shows a larger amount space.*

What is a gigabyte? There are at least two standards for the definition of a gigabyte:

http://en.wikipedia.org/wiki/Gigabyte

Many hard drives and Macs use the "standard" definition and PC's use the binary definition. Does this difference matter? It seems to, there have been lawsuits over the discrepancy.

Lawsuit against Western Digital:

http://gizmodo.com/183980/western-digital-settles-hard-drive-capacity-lawsuit

Why do people care if these numbers are slightly off? Well first of all we have a lot of valuable things to save and we need to know if we are able to save them. Also, storage isn't cheap. Just look at the cost of gigabyte over time:

http://www.mkomo.com/cost-per-gigabyte

So how far off are the two systems?

20 minutes

Next I give students a chart of the two measurement systems and ask them to compare the number values without a calculator:

They can do this by looking at the KB as a reference. It is defined as 1000 in standard but 2^10 in binary. 2^10 = 1024 and 1024 > 1000. The next number comparison is 2^20 and 1000000, but 2^20 does not equal 2 * 2^10 (a mistake many students will make). However, by comparing

**2^20 as 2x2x2x2x2x2x2x2x2x2x2x2x2x2x2x2x2x2x2x2**

and

**2^10 as 2x2x2x2x2x2x2x2x2x2**

students will realize that 2^10 x 2^10 = 1024 x 1024, which much be larger than a million, which is larger than 1000x1000! They can extend this to all the other numbers.

About half way through, many students are ready for the next question. How far off are these measurements? This is an opportunity to show them **relative error**, a major algebra 1 topic. Here we treat the binary as the "actual" measurement. I model the first calculation (on the KB scale) and then give them this chart and ask them to verify that the other numbers are correct: Comparison of Decimal and Binary

To make these calculations, students simple find the "actual difference", which is the absolute value of the different between the two numbers, for KB that would be 2^10 - 1000 divided by the "observed value" which is 2^10.

I ask students to find the percents and find the actual byte differences. This will help them calculate if these differences matter.

20 minutes

To complete this lesson, we review our calculations and methods of comparing the binary and standard prefix systems. It is critical to review how they were able to compare all of these numbers without a calculator (see investigation for notes).

When we compare how far apart the two systems are, I like to use the "number of songs" difference as a reference. I ask "how many songs could fit in the missing space?" Assuming that an average song is about 6 MB

Then we look at the bigger picture and look at this problem on a worldwide scale. I show this NYT infographic about data flow on the internet:

We discuss some of the questions around the context of this graphic:

- Why do we think that traffic has increased so much over the past few years?
- Will it continue to increase?
- Will it ever stop increasing?
- Have you used more internet as you have gotten older?
- Who uses the internet the most?

Then we talk about the numbers of the graphic, because without an understanding of these numbers we *can't* understand this statistic. I want students to think about where we are with data usage and where we are going. This will stress the importance of having a standard system of measurement. We store valuable data and need to make sure we have enough space to hold that data.

Our investigation now looks into the future, how far off could our current data measurements be if we are dealing with such large numbers?