## April 9, 2013

### Exponentials In Context

Between all the talk about noticing and wondering, taking a course on differentiation and doing fun projects with Geometry, I decided it was past time to do something exciting with my PreCalculus classes.  When students arrived in class they saw this on the board:

 Interest based differentiation!

In front of the board I pulled up a small table and spread out slips of paper (I cut up 3 copies of the document below so that there could be a maximum of three people in each group).

From there, I instructed students to start asking questions, as many as they could think of.  While most groups got started, I called up the students with a link instead of information on their paper to show them the corresponding Act 1 video.  Once everyone was started making a list of questions, I distributed the instructions and checklist (I hate rubrics- so many words! Everything was worth 1 point except the questions/answers were worth 4).  Some students had trouble coming up with interesting questions so I gave examples of "How long would it take to double?" and "What would happen after 2 years?" but recommended they pick something more significant than doubling/two years.  Mostly, though, they came up with interesting questions without assistance.  The sunglasses groups weren't confident in their answer, so we watched the end of the video to be extra sure, but everyone else was able to come up with an equation or interpret their equation without much difficulty. We had done a few examples of compound interest, but otherwise were working off of memories from last year.  I was impressed with how many people pulled out logs with little to no prompting when they needed to solve for an exponent (way to go Algebra 2 teachers!).

Grading these projects was surprisingly fun.  First, they were all different!

 Title of Report
Second, they came up with cute stories. They named characters, identified the source of bacteria (yogurt dropped in an agar tray by an unreliable intern, naturally) and one student even wrote a multi-page story. The premise of that one is a student who wants to work at a car dealership, but his parents are scientists so he needs to provide a proof this is a wise career choice. He gives his parents graphs of the exponential function and cites data. The concluding sentence is: “Now I know why I took PreCalculus, he thought.”

Another favorite sentence: “In the alleyway behind the snack-packing factory is a row of dumpsters where all the mispackaged or expired snacks go to die.” This obviously leads to a growth in the rat population. Another reason the rat population might grow is toxic waste:
At the local dump there are many rats. The rats love all the trash and debris that is so readily available to them. However, there is one thing that makes these rats grow and repopulate so quickly; it is the radioactive trash someone has been throwing away! At the current rate of P(t)=648e.016t the rat population will double in about 18 years! Not to mention the rats have radioactive superpowers.
Third, the questions they asked were truly interesting.  Groups thought to go forward and backwards in time, compare points, add complications (a fire meant all the library books were lost!) and change the rate.  Most kids asked different types of questions (after t years vs. when will it reach y value) but it may be worth specifying that the questions shouldn't all be the same type.

The group who looked at the photocopied dollar came up with this question, debated if it even made any sense, and then decided to go with it anyway because they found it interesting!
When shrunk two times how much will the dollar be worth in size?
15.5(.75)2=8.7 cm long
8.7/15.5=\$.56
Finally, the reflections were amusing.

One student wrote “our group is like a well oiled machine, and I was the on button.” When he shared this sentence with his group after school, the discussion became animated as they tried to figure out the type of machine they were:
If we’re a simple machine I can’t be the on button; there is no on button.
If we’re a car you’d be the key.
Would I be the driver?
No, we’re the machine. You can be the engine.
Ms. Cardone can be the driver.
So that makes (student 3)… everything else!
So, I’m the body of the car, and the air conditioning – in winter.
Everyone laughs.
End scene.