Okay, okay, I admit this isn’t suitable material for babies, nor for any normal kid whose age can be expressed with one digit! The “baby” phrase just popped into my head and I found it too cute to resist. A more accurate title might be “Joanie & Johnny’s First Infinite Series”, or “Kids’ First Infinite Series”.

I’m always thinking about ways to get those apathetic middle- and high-school students interested in math again by showing them something wild and crazy, yet simple and concrete enough (for *them*, of course) that it shouldn’t be too difficult to understand. My latest idea is an infinite series with a surprising and very simple visual “proof”. Can you guess what it is? Hint: the “proof” is in each of the figures above (surprise, surprise). What it proves is…

I’m sure each of these informal proofs — really variations on a single proof — has appeared in many places, but I ran across the one on the left first, in Roger Nelsen’s wonderful *Proofs Without Words.* (The one on the right is from the Wikipedia article “Geometric series”.) The two volumes of *Proofs Without Words*, published by MAA, contain dozens of marvelous and miraculous visual “proofs” on a wide variety of mathematical topics; I highly recommend browsing through them. Full disclosure, though: in my opinion, not many are as elegant or as easy to see as this one. Still, I believe that several of the proofs Nelsen has collected, certainly including this one, could be presented successfully on the middle-school level — though probably not with the above notation. Regardless of notation, of course, appreciating this proof requires some ability to do arithmetic with fractions.

I think most people with experience teaching high-school math would agree that (1) not many “apathetic” secondary-school students have much of an idea what a proof is (notwithstanding the emphasis on them in a typical geometry course), and (2) hardly any of them know why they or anyone should care! That may be a problem with my idea, but it strikes me more as an *opportunity*. I suspect that going through something like this proof would help with both problems; it would also expose students to the important idea of adding up infinitely many numbers and getting a finite sum.

Do you agree? Do you this could work (with a reasonable amount of scaffolding, naturally) as a discovery lesson for, say, 10th graders? If you think it’s worth trying, here’s an idea for “reasonable scaffolding”: Discuss the fact that 1/3 = 0.33333333…, with infinitely many 3’s, and ask the class what 0.33333333… with infinitely many 3’s actually means. Of course it’s a very simple infinite series, one with the same sum as the series I’ve been discussing:

Since both series have the same sum, an obvious question might be whether it’s possible to draw a picture to illustrate the above equation, and if so, what it would look like. I leave those questions as an exercise for the reader.