(These materials were created, gathered, and developed by Sadeka Harris, an Instructional Facilitator at the Bronx Adult Learning Center and a NYSED Teacher Leader).

WARM-UP

A good way to begin building/drawing out background knowledge for the pyramid scheme problem is with the warm-up below (from the website Open Middle).

The answer here is 4⁵ + 2 x 3 (or 4⁵ + 3 x 2), which equals 1030, but many of us (teachers and students alike) might think it is 5⁴ + 2 x 3, which only equals 631.

If any of your students come up with 631, ask if anyone else got a higher number. If no one has, tell them that you created an expression that equals 1030 and ask them to try to figure out how you placed your digits. The goal of this warm-up is to get students thinking about the power of exponents (pun intended).

Follow up this math warm-up with a visual to begin to make sense of how pyramid schemes work. Have students do a notice wonder in response to this image:

There are lots of things students may notice and wonder about this image, including that there are more new investors, the money flows up, and that there is a single person at the top.

HISTORICAL INFORMATION ON PYRAMID & PONZI SCHEMES

A scheme is an elaborate plan or program of action that is usually devious in nature.  

A simple definition of a Pyramid scheme is a system of making money based on recruiting a never-ending number of “investors.” The initial promoters recruit investors, who in turn recruit more investors, and so on.

Ponzi schemes take their name from Charles Ponzi, who ran one in 1919. Mr. Ponzi promised investors outrageous returns in a short period of time by claiming he could take advantage of the difference in currency values through buying and selling international mail coupons. 

Initial investors usually get their returns but more and more investors are needed to keep the fraud going and, eventually, it crumbles revealing the plot and some investors make no money or lose money.

Next hand out the central situation they will be exploring (the pdf link at the top of this post) and ask students what they notice and what they wonder. Teachers can use this opportunity to ask multiple students to explain the situation in their own words to help everyone make sense of how it works.

HELPING STUDENTS UNDERSTAND THE PROCESS (Optional)

If students are still confused about how the process described on the email works, it may help to act out a smaller version of the situation. You’ll need materials to represent letters and materials to represent money. For the letters, blank sticky notes work fine – just make them a different color if you are also using sticky notes to represent money. For the money, you can use things like chips, prop money, green sticky notes, etc.

• Explain that instead of 8 names on the list, 5 new investors each round, and $5, we are going to simplify things to 3 names on the list, 2 new promoters each round and $2.
• Ask for a student volunteer to be a the promoter.
• Write three names on the board, with the promoter’s name in the third spot. This represents the names on the letter.
• The promoter “sends a letter” to 2 investors. Two investors (other students in the class) each receive a letter. (The promoter is in the third spot)
• The two investors each send the letter to two new investors (four people). The promoter moves to position #2.
• The four new investors from the last round each send letters to two new investors. The promoter moves to position #1 on the list. Those 8 people each send $2 to the promoter.
• The promoter receives $16.

THE CENTRAL TASK

Once students can explain how it works, have them start working on the central problem:

If the process goes as planned, how much money would be sent to Max? Show your calculations.

Before you let students start working, ask for a few guesses.

Give students some time to work independently before giving them the option of exploring further in groups. It is highly recommended that have calculators available to students and/or make sure they know they can use the calculators on their phones.

SAMPLES OF STUDENT REASONING

Here are some strategies you might see. (Please note that some of these strategies do not result in the correct answer. They are included here to help teachers think about what types are student thinking they may need to build off of):

Click here to get a closer look at these sample strategies

MAKING CONNECTIONS TO THE MATH

It is likely that many if not all of your students who do the repeated multiplication, will not necessarily make the connection to exponents. After students share their strategies, if exponents don’t come up, you can ask students if there are any other ways to write 5x5x5x5x5x5x5x5. If no one knows, you can show them the exponent notation 5⁹ (or 5⁸ people multiplied by $5)

MAKING CONNECTIONS TO THE REAL WORLD

Here are some questions for further student discussions:

• Will Max really earn that much money? What could possibly go wrong?
• Why do you think Ponzi schemes like this are illegal?

TEACHING SLIDES

For any teachers for whom it is helpful, we created these slides to be projected in a classroom or used for remote/virtual instruction: PYRAMID PROBLEM SLIDES