Thankfully, I recently came across Sam Shah's method of exploring exponential functions using paper folding. I like this idea so much more than plain old point-plotting (though I ask the students to do that as well). Here's the worksheet I had my classes fill out before introducing the formal definition of an exponential function:

Paper Folding Exponential Functions

We spent 15-20 minutes on this worksheet in class before diving into the lecture. That's a lot of time for a college class, and I felt rushed the rest of period, which is never good. So, next semester I may assign it for homework the day before instead to save some time. I would have to give some kind of hint as to how to find

*f(x)*and

*g(x)*though, because the students were a bit unsure of the pattern going on. I don't think anyone got 2^x, but most did see that we were multiplying (or dividing) by two each time.

Once we got into the lecture part, I gave the students a great cheesy word problem:

One of my students noticed that it didn't take much time for the spending to increase significantly, which I was quite happy about. "So the moral of the story," I began to say...

"Is not to be exponential when you're shopping!" she finished for me.

Perfectly said.

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