Whenever I finish a new post I will tweet out a message about it and I often encourage people to drop on by and share some wisdom. I definitely need some tonight.

Had a great conversation with a colleague today about his Algebra II Honors class. They are examining exponential functions and are ready to talk about logs. He came by with what seemed like a straightforward question – but it no longer feels like it is. He sketched the graph off y = 2 ^ x and marked pi on the x-axis. He talked about working his kiddos through the argument that there must be some power of 2 that yields pi as the answer. He talked about a method of exhaustion making better and better guesses to get closer and closer. We talked about how this conversation could be approached as carefully as possible. I talked about the intermediate value theorem but advised that it not be named yet. I talked about temperature during a day and speed on a car’s speedometer. But as he pushed me I realized that all of these arguments rely on comfortably knowing that this function is continuous and that if 2 ^ 1 = 2 and 2 ^ 2 = 4 that there MUST be some value of x between 1 and 2 so that 2 ^ x = pi. We ended the conversation – because I had a committee to run to – with this questions: How do we convince Algebra II students that this function actually does have to have an input x that yields every output y in a region? How do we recognize a function as continuous? What are the markers? This feels like a question that I should have had a better answer for and maybe in October when my brain is smarter I might have. So, I ask you out there – how can we convince Algebra II students that there is some real x so that 2 ^ x = pi?

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Would they have this same problem if it were a linear function of some sort? Would they require convincing that there exists a real number x such that 5x-3 = pi?

Interesting question. This assumption of continuity might have even deeper roots than I was thinking of. I am not so sure that my colleague feels that they need this convincing so much as he worries about the implications of not being able to provide this argument. Does that make sense?

Right. I doubt they NEED to be convinced of this. I find that HS students are willing to believe quite a lot without solid proof. Normally that’s something I want to break them of, but in this case, I’d take it as a blessing!

I would step back to the domain and range of the function. Do the students believe that the domain is R and the range is (0,infty)? From there maybe it’s easier to convince themselves that it must attain that particular height.

I think you have to approach it with an application problem that is continuous. Unfortunately most exponential application problems aren’t really continuous: things lile insect populations are actually discrete since the population at any moment is always integral. Even radioactive decay isn’t truly continuous since the mass changes in jumps (of an admittedly tiny size).

Perhaps you make something up: a balloon is being blown up such that it’s volume increases exponentially. Volume is truly continuous (Okay yes, air molecules are discretely volumetric, but I think we can get close enough with a tiny bit of handwavium). At some point, the volume equals pi. Maybe?

That is the direction that my mind was going but my colleague was pressing me more in the direction of what structural properties are inherent in some of these functions that tip me off to the continuous nature of the function. I agree that a model like the one you propose will go a loooong way toward convincing the kiddos that these statements about continuity are true. I am just trying to answer a question that is a bit deeper but might ultimately be not terribly important.

Do students believe that is between 6 and 7? That may be another convincer.

Yeah, they do and we have invoked this sort of reasoning since before Algebra II in some cases. What this says to me is that even the students are developing a nice sense of continuity and its implications. I guess what I am trying to get at is some profound statement to make to the students about when/why they can safely make these assumptions. I know we get lazy about it in Calculus and invoke continuity at times without explicitly acknowledging it – or we make an assumption about continuity when we have no real right to do so.

This is fun to think about and I appreciate all those jumping in on the conversation.