McREL Standards Activity
Getting to the Bottom of the Pyramids
 Purpose:  As a result of this activity, students will be able to construct informal logical arguments to justify reasoning processes and conclusions.  Related Standard & Benchmarks:  Mathematics   Standard 1.  Uses a variety of strategies in the problemsolving process     Level III [Grade 68]     Benchmark 7. Constructs informal logical arguments to justify reasoning processes and methods of solutions to problems (i.e., uses informal deductive methods) 
 Mathematics   Standard 1.  Uses a variety of strategies in the problemsolving process     Level III [Grade 68]     Benchmark 5. Represents problem situations in and translates among oral, written, concrete, pictorial, and graphical forms 

 Student Product:  logical argument  Material & Resources:  No special resources required for this activity.  Teacher's Note:  Students work in pairs or triads to construct their arguments. Explain to the groups that their products should show their use of problem solving strategies as well as their final argument. A sample argument is provided below: Pyramid schemes depend on bringing in an exponentiallygrowing number of new participants. If you start with one person, who gets 10 people to join, and each of those people gets 10 more people to join, and so on, you have the total number of people growing by powers of 10. The total number of people involved grows to amazingly huge numbers without very many steps being required to reach these huge numbers. But these huge numbers create a problem. There are somewhere between five and six billion people in the world. Let’s suppose that every one of these people could be induced to join a particular pyramid scheme. For how many levels could this scheme run before it failed, for lack of new participants? You’ll be amazed when you see how quickly the number of required new participants grows to exceed the population. In the example above, I assumed that each person who joined would bring in ten new people. How many levels can be supported by a population of five to six billion? Let’s count them... 1.1 2.10 3.100 4.1,000 5.10,000 6.100,000 7.1,000,000 8.10,000,000 9.100,000,000 10.1,000,000,000 That’s ten levels, counting the one person at the top who started it. By the time these ten levels are filled, there will be a total of 1,111,111,111 participants. The eleventh level would require 10,000,000,000, or ten billion new participants to fill in. But there aren’t that many people in the world. There’s only between five and six billion, minus the somewhat over a billion who’ve already joined. Most of the billion people in the tenth level will not be able to get any new participants below them, and will therefore make no money at all. And of course, none of those who join in the eleventh level will get any new participants below them. There aren’t enough people to fill in the eleventh level, much less to start a twelfth level below that. At this point, the pyramid collapses. And when it does, a solid majority of those who had joined will not have made any return at all. They will have paid their money to get in, but the promise that they will profit as people join below them will never be fulfilled.  Activity  Your friend Darcy just received a chain letter in the mail that guarantees to make her some easy cash. The letter contains a list of two people and directs her to send $100 to the name at the top of the list. She is then supposed to remove the persons name from the top of the list, put her name at the bottom of the list and send it to 10 of her friends. The letter goes on to explain how she will end up receiving letters containing $100 from 100 people  netting her $10,000. This sounds like a great plan to Darcy because all her friends will make money, and their friends, and their friends... but will it be fair to everyone? At some point, there will be no one left to sell it to because everyone else is on the list. Use a diagram or manipulatives to represent the situation. Based on ideas from your representation, construct an informal logical argument to explain how the chain letter scheme will break down before everyone in the world gets $10,000.  
