SWBAT to use pattern and structure to compare large numbers

There is a shocking difference in size between a value and another value a thousand times larger.

15 minutes

This is a lesson about the patterns found in the way we name and use big numbers. Students need to understand that a million is a thousand times smaller than a billion and so forth. I want them to see this through their calculations and intuition. To get there, I like to start with a basic review of place value.

I ask them, "how high can you count?" This is accompanied by a list on the board. I usually stop around a trillion or so.

1

10

100

1000

1000000

1000000000

1000000000000

The class has about 2 minutes to write down the numbers and names of numbers along with any observations they make about this particular list. Since student might struggle in counting the zero's, I am very careful to color code an line of the table so that the number of zero's is easily decipherable. If my students need more support, I hand them typed copies of the table. Since students might not understand what I mean when I say, "write down anything you notice," I remind them of the fact that I *deliberately* chose these numbers. For example, "why did I go from 1 to 10 to 100? Aren't there numbers between 1 and 10 and 10 and 100? When you look at these numbers, you might notice that they *increase, *but give me a bit more and tell me in what *ways* they are increasing."

For the discussion, I like to show each number on a slide and ask for the name of the number, how they knew the name (in the context of place value or any other technique they used) and I ask them for the scientific representation of the number (this is something we have already begun to learn). Then we return to the entire list and discuss the patterns they notices. I am looking for them to notice that "the number of 0's increase each time" and that this represents a change in the powers of 10. Specifically, I like to focus on the differences between a million, billion and so forth. I believe it is imperative to recognize that these numbers are separated by a factor of 1000.

25 minutes

I start the investigation by briefly recapping the financial crisis and the case of the SEC against Fabulous Fab. I ask them the question, "does this win for SEC *matter*?"

They may give some general responses, but I ask them to support their answer with precise detail. The idea of the math practice here is for students to recognize that mathematics is way to solidify any argument. By bringing numbers into a debate, you can put your ideas into perspective. I ask, "what do you need to know in order to answer this question?" They know that they need the numbers. I am hoping they will ask two specific questions:

1.How much is he responsible for?

2.How much is the total loss from the financial crisis?

If they ask these questions, then I know my summary of the financial crisis and SEC case was effective. If they don't ask the questions, I will lead them there or just talk to them about why these questions might be the most fundamental. They might also want to know what the Fabulous Fab wil have to pay, but that is currently not available (although it certainly will be at some point).

I like to answer questions and get them hooked into the problem by showing this clip

Source: Colbert Report

After the clip, I hand them a print out of the two main figures:

1. Fabulous Fab responsible for 1 billion in losses

2. Financial Crisis 22 trillion in losses

There are also some key questions

1. Does this win *matter*? How do you know?

2. When you compare the numbers on your calculator, did you run out of room? Was anything strange about the number notation?

3. Do you need a calcualtor to solve this problem?

Extensions:

Mild:

wall street journal quote as reperations toward the crisis. How much does this *matter*? Use numbers to support your reasoning.

Medium:

This prosecution of the Fabulous Fab was relatively fast. If the SEC gets 1 billion in retorations from this 1 trial, how long would it take them to recuperate all the money? Lets assume that every trial gets 1 billion (on average).

I would give them the main dates of the trial on a handout.

There are more timeline shots on the NYT

Spicy:

Make a graph that demonstrates the answer to the "medium" question.

20 minutes

In the summary I give students a chance to debate if this SEC trial actually *matters?* I love to see the debate go here, because students can use their mathematics to argue either way. They can discuss the size of a billion in comparison to 22 trillion to point out that they have only dealth with 1/22000 of the problem. However, the opposing side could always show that a billion is *something*, which is certainly 1000 times larger than a million. The great thing here is that they will use their number sense to argue and to indirectly disprove the popular belief that math "has one answer." Here students show that mathematics goes beyond just a simple number answer. Instead, they are using the numbers to make meaning of the world around them.

Each question we focus on has value, but I think the discussion around the calculator is certainly pragmatic and relevant to their classroom experience. Since the calculator is their de facto technology, I use the TI-Smartview app to review what happens on the calculators and what calculators do when "they run out of room." The scientific notaiton is especially relevant in their calculations. It is also fun to show how we can convert between scientific notation and standard form on a graphing calculator.

However, it is critical to also discuss the intuition here. Students need to recognize that their is *no need* to use a calculator with such large numbers. That is the beauty of our number system. It is designed to work well with many types of both large and small numbers. I like to ask them, "what about these numbers makes them easy to think about?"

If there is time, I discuss the mild, medium and spicy extensions. The medium and spicy can be reviewed at once, since they are essentially the same question. I like to use programs like Desmos or Geogebra to model how the graphs can be set up to show the amazing time needed to recuperate all the losses. On some levels this is complex, but a major take away here needs to be that "since this try solved 1/22000 of the problem, we need to do this 21999 other times to finish." This would give me a chance to discuss fractions in the context of this problem. Any opportunity to discuss fractions is vital. Here we could use simple fractions to help them understand the ideas around the fraction 1/22000.