Intermediate Value Theorem & Elevators

I’ve used the elevator analogy with the intermediate value theorem before, but only after talking students through the intermediate value theorem first. This time, I took them through the following thought experiment first:

Step 1:

You enter the elevator on floor 2. You close your eyes and keep them closed until you arrive at floor 12, twenty seconds later.

Questions for discussion:

• At approximately what time was the elevator located at floor 7? How do you know? What assumptions are you making?
• Was there a time when the elevator was at floor 3? Floor 8? How do you know?
• Were you ever at floor 13? How do you know? Are you really sure?

Step 2:

Another day, you again enter the elevator on floor 2. You again keep your eyes closed, but another person gets on from some floor other than floor 2. You keep your eyes closed. The other person leaves the elevator at some point. After 60 seconds, you are on floor 12, and you open your eyes.

Questions:

• Was there a time at which the elevator was at floor 7? How do you know?
• Was there a time at which the elevator was at floor 13? How do you know?
• What was the highest floor at which you can guarantee the elevator was located during the minute long trip? The lowest floor?

Step 3

On yet another day, you are once again entering the elevator at floor 2 to go to floor 12. You close your eyes, same story as before. Another person gets on the elevator and leaves. This time, however, you open your eyes just long enough to see that the person leaves the elevator at floor 15. As before, the entire trip takes 60 seconds.

Questions:

• Was there a time at which the elevator was at floor 7?
• Was there a time at which the elevator was at floor 13? How do you know?
• Make a list of all of the floors that you can guarantee that the elevator could have stopped at during the 60 second trip.
• Can you guarantee that the elevator was never located at floor 17?

We then visited the driving principle to why we can do this thought experiment: why can we come to these conclusions without opening our eyes in the elevator? What is it about our experiences in elevators that makes this possible?

My students were primed to bring up continuity given that they worked through the concepts during the previous class. That said, there were quite a few lights that went on when I asked what it would be like to ride in a discontinuous elevator. Skipping floors, feeling the elevator move upwards and then arriving at a floor lower than where we started, or arriving at different floors just from closing or opening the doors.

Once we were comfortable with this, I threw the standard vocabulary of the intermediate value theorem:

Suppose f(x) has a maximum value M and a minimum value L over an interval [a,b]. There exists a value c in [a,b] such that L≤f(c)≤M as long as…

…and I left it there, hanging in the air until a student filled the silence with the condition of continuity over [a,b]. This was also a great time to introduce the idea of an existence theorem – it tells you that a mathematical object exists, and might give you some information on where to find it, but won’t definitively tell you exactly where it is located. Fun stuff.

We then talked about other examples of functions that are or are not continuous. Students brought up crashing into a wall after moving at a non-zero velocity. I also have this group of students the following period for physics, so I brought up what the velocity versus time graph actually looks at when you zoom in to the time of impact. (I like that this wasn’t a cognitive stretch for them given their experience zooming in on data on their calculators and graphs from Logger Pro.) The student that brought this up quickly argued himself back from saying that this was truly discontinuous.

This was a fun activity, and I’m glad I went through it. The concept of IVT is fairly intuitive, but we often present it in a way that doesn’t emphasize why it is special. In previous years, I started with the graph of a polynomial function bouncing up and down, asked students for the max/minimum value, and then asked them to identify whether they could do this for any value in the range between the maximum and minimum. They could, but never really saw the point of why that was special. Forcing them to imagine closing their eyes, limiting the information available to them, and then seeing how far they could take that limited knowledge made a difference in how this felt on the teaching end. I’ve seen some pretty good responses on my assessments of this concept as well, so it seems to have done some good for the students as well. (Phew!)