Category Archives: exponential growth and decay

If I had a million dollars…

During my final teaching internship, I taught one section of 8th grade Gifted Math/Algebra I.  These guys may have been my favorite class, even if they were the chattiest.  I also noticed they were VERY competitive.

I had to do an introductory exponential growth and decay lesson with them and remembered reading about a problem involving a guy walking into the classroom and offering either one million dollars or starting with one cent and doubling the money every day for 30 days to do a job.

*Cue in google.

I honestly do not remember where I found this problem, so I apologize for not giving credit.  However, this lesson (one that was very inquiry-based) worked out fantastic with this group of students.  (It’s based on the Conceptual Change Model, a model I learned about in my fantastic Math Methods course.)

I started with displaying the scenario on the SmartBoard:

You’re sitting in math class when in walks some rich and flashy guy and he has a job offer for you.  He doesn’t give too many details, hints about the possibility of danger.  He’s going to need you for 30 days, and you’ll have to miss school. (Won’t that just be awful?)  But do you ever sit up at the next thing he says.

You’ll have your choice of two payment options:

One cent that day, two cents on the next day, and double your salary every day thereafter for the thirty days; or

Exactly $1,000,000. (That’s one million dollars!)

I had the students write down which payment option they would prefer and why.  It’s important to only give them a minute to do this, otherwise many students will begin doing calculations.  I wanted them to commit to an answer prior to actually figuring it out mathematically.  They were then told to get into groups and share their thoughts.  One student from each group would then (anonymously) share the predictions.

Now is the time when I wanted them to figure out which payment option would be the most profitable.  I handed each group a partially filled out table and simply told them to figure it out in their groups. 😉

As they worked in their groups, I walked around to check on their progress, make sure they were on task, etc.  This class (as I mentioned earlier) was very competitive, so everyone was feverishly doubling the money in the right hand column.  When I heard complaints about it taking too long, I asked if there was an equation that would help them move things along faster.  When they heard this, all the groups started trying to figure out an equation.

I honestly do not recall how many groups figured out that it was 2^x, but I know it was well more than one group. Time was running away from us, so I didn’t get to have each group come up to explain their process in solving it (boo!). So, we moved on to discuss the results and what the graph of 2^x would look like, what the domain and range were, whether the graph of x^2 would increase faster or slower, etc. We then discussed what -2^x would look like and compared linear, quadratic and exponential functions.

All in all, it turned out to be a great lesson and students really seemed to have grasped the concept of exponential functions.

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