**The Penny Problem**

If you teach exponential functions, I have the perfect introductory activity for you! You may be familiar with the penny problem. Here’s how it goes…

“ Your friend is hiring you to help them with their business for 30 days. They offer you the choice of two different compensations. The first is to take $0.01 on the first day, and then double that amount each subsequent day. The second option is to take $1,000 on the first day, $2,000 on the second, $3,000 on the third, etc. for 30 days. Which option do you choose?”

**The Two Options**

I present this problem to my students. Some of them have heard the problem before and choose the penny. Some students are skeptical and choose option 2. I then put the students into groups of 2 or 3 and have them make a table of values for each option. I have them do this in the Desmos app, although they can also do it on paper or whiteboards. The benefit of Desmos comes when we analyze the graphs of the table.

**Option 1: **

**Option 2: **

After the students make tables for the two options, they can then start to analyze what they see. Ask questions like…

Which one is the better choice?

How much is total for option 1? Option 2?

If you were only working for 10 days, which one is the better choice? 20 days?

When does option 1 become a better choice?

I teach exponential functions after we’ve spend a LOT of time on linear functions, so I ask my students to name what kind of function they are seeing in option 2 (linear), and create an equation to model the table points. The students come up with y = 100x. **It's a good time to review that functions can be expressed as tables, graphs, and equations**! If you’re using Desmos this is also a good time to add this in to see the connected line on the graph (purple line below).

Finally, as an extension, I have my students try and come up with an equation that will connect the penny dots! This takes a little time and some of them get it, which then launches us into our lessons on the exponential function and what the variables represent.

Hope your students enjoy this intro activity as much as mine do! And when you're done, check out this mini project, real-life application of exponential functions!

"When will I ever use this?" The famous question asked by every Algebra student. Give your students a REAL application of exponential functions with this mini project.

**Profession:** Accountant

**Task:** Students will become accountants, learning how compound interest can affect the future value of an investment. They will be given several scenarios to analyze, in order to pick the best retirement account for their client.

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