# Baby’s First Infinite Series

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.

# Gamow, Bread Rationing, and the Normal Distribution

You might be familiar with George Gamow, the mid-20th-century physicist who, with his student Ralph Alpher, came up with the Big Bang Theory long before there was any experimental evidence. Gamow is also the best writer for the layperson on science, and one of the best on math, I’ve ever read. In particular, his book One, Two, Three… Infinity is a masterpiece, jammed with fascinating ideas presented with absolute clarity. (Though much of it — mostly the non-math stuff — is kind of out-of-date now; the revised edition came out in 1961.)

A few years back, my friend Doug Hofstadter sent me a short article of Gamow’s about how the normal distribution was once used to expose a dishonest baker — apparently a true story. But true or not, it’s a fascinating story of mathematical probability in real life, one that I think would interest even a lot of apathetic middle-school and high-school students! A PDF of the story is available at

# Triangle Types, Quadrilateral Quotas, & Hexagon Hierarchy

Christopher Danielson’s great post on The Hierarchy of Hexagons begins with the observation that “School geometry seems to me one of the most lifeless topics in all of mathematics. And the worst of all? The hierarchy of quadrilaterals.” Then he describes his attempt at “breathing some life into this dead horse” by having students classify hexagons instead. Seems to me he was wildly successful: I never heard of future elementary teachers proving anything, but his class “proved that a Bob cannot be equilateral.” !! But what is a Bob, you say? Read his post — no, wait: read the rest of this post, then read his.

The chart above simply shows the relationship among all types of triangles in a Venn-diagram-like way that I think is exceptionally clear. I’ve never seen it done like this before, but if someone else has, please tell me.

All I’ve done with this chart myself is to show it to my geometry classes and ask them to draw examples of some of the types. But there other ways to use this idea, and one that seems much better to me now is: ask students to come up with their own graphical ways of showing the relationships among all types of triangle; then show some or all of them (and perhaps mine) to the class, and lead a “compare and contrast” discussion. Much better because much more likely to result in students really understanding! Comments, anyone?

For that matter, a classification like this begs the question of what types of figures there are of four or more sides, and of course the number of possibilities goes up rapidly with the number of sides. But how many are there with four sides, and how rapidly does the number go up? Well, what if a shape — say, a hexagon with five interior right angles — doesn’t have a name? (Actually, according to Christopher Danielson’s students, that’s a “Bob”.) Is a pentagon with two long sides and three short ones where the long sides are adjacent different from one where they’re not? What if the two long sides are parallel? What if the angles are all equal, or is that even possible? Etc. etc.
So a discussion of classifying triangles could lead to a discussion of partitions, of classification in general, of hierarchies, of (abstract) trees, and, no doubt, other topics.

# Zeno’s “Achilles and the Tortoise” Paradox and Geometric Series

When I decided to retrain as a secondary-school math teacher and applied for a Woodrow Wilson Indiana Teaching Fellowship a few years ago, I had to teach a 5-minute lesson. It could be on any subject I wanted, and I chose Zeno’s first paradox of motion, the “Achilles and the Tortoise” Paradox. As you may know, the Greek philosopher Zeno (5th century BCE) devised several “paradoxes of motion” that baffled all of his contemporaries. His first paradox involves a race between Achilles (a very fast runner, said to be “the fleetest of foot of all mortals”) and a turtle (a rather slow crawler), with the latter having a head start. Zeno gives a simple but surprisingly convincing argument that Achilles would never catch up: “In a race, the quickest runner can never overtake the slowest, since the pursuer must first reach the point whence the pursued started, so that the slower must always hold a lead.”

In my brief lesson, I explained that this argument assumes that the sum of an infinite geometric series cannot be finite, and showed why that’s not true, i.e., why such a series can converge to a finite value.

Why, in my interview lesson, did I teach something as advanced as convergence of infinite geometric series? In two words, intrinsic motivation. High-school Algebra II (at least in Indiana) covers geometric series and their sums, and infinite geometric series are very likely to be touched on. Still, students won’t be expected to understand why the sum of infinitely many numbers can be finite until they take calculus. But, while the details require methods of calculus, the concepts are simple; there’s no reason they can’t understand immediately. I’m fond of the infinite, that fruitful source of crazy ideas, and I’ve wanted from the beginning to expose students to the wildest and craziest ideas I could possibly get them to understand. And why did I want to do that? The main reason is that, based on my own experience, I believe that wild and crazy ideas can be a great source of intrinsic motivation for a lot of kids—certainly not the majority, but a lot. And intrinsic motivation is generally agreed to be far more reliable than external motivation. Along the same lines, Sawyer’s fine book Prelude to Mathematics comments “A non-mathematician learning mathematics…often has to plough through routine procedures, which can be extremely dull… An education should also contain elements that perform the functions of a cold bath—to provide a shock and keep one awake.”

I’ve given versions of this lesson three times now, including once to a high-school “math exploration” class, and I think it went over well each time. The latest version — available in narrative form — has two parts: “Resolving the Paradox” (roughly the original 5-minute form) and “Boring Details, A Related Example, and the General Situation” (about infinite geometric series in general and when they converge). The only technology needed is a yardstick and a greenboard drawing.

# Antidisestablishmentarianism the Dog

Warning: While I’m putting this in the “Lessons” category, I doubt if you can use it without major surgery!

I taught math at a small-town high school early this year, as a maternity leave replacement. I decided from the beginning to cultivate an eccentric and playful persona in front of my classes; this was easy to do because I really am both somewhat eccentric and decidedly playful! For example, my first day, I told them my family has a dog named “Antidisestablishmentarianism”: with 28 letters, that’s allegedly the longest word in the English language. (See photo. I’m pleased to say he’s registered with the American Kennel Club under that name, though we usually call him just “Tarry”.)

I explained where the name came from; the reason has minor mathematical implications. Specifically, I was inspired to call him that by the children’s song, “Bingo”. The first verse goes like this:
There was a farmer had a dog and Bingo was his name-o.
B, I, N G O;
B, I, N G O;
B, I, N G O,
and Bingo was his name-o.
The second verse is the same except that each “B” in the 2nd through 4th lines is replaced by a hand clap. And in each of the four remaining verses, one more letter in those lines is replaced by a hand clap, so that by the last verse those lines consists entirely of claps.
Thus, “Bingo” has one verse per letter of the dog’s name, and the length of each verse is also a function of the number of letters in the name. But I’ve always been amused by thinking about a version in which the dog’s name was Antidisestablishmentarianism! (And when my family gave me the opportunity to name our new puppy that, some years ago, I didn’t say no ☺ .)

This suggests the question I asked my students: how much longer would it take to sing if the dog’s name was much longer than “Bingo”—say, the 28-letter word “Antidisestablishmentarianism”? Removing the reference to a farmer to make it easier to squeeze all the syllables in, it might start like this:
There was a dog and Antidisestablishmentarianism was his name.
A N T I D I S E S T A B L I S H M E N T A R I A N I S M;
A N T I D I S E S T A B L I S H M E N T A R I A N I S M;
A N T I D I S E S T A B L I S H M E N T A R I A N I S M;
Antidisestablishmentarianism was his name-o.

There was a dog and Antidisestablishmentarianism was his name.
[Clap] N T I D I S E S T A B L I S H M E N T A R I A N I S M;
[Clap] N T I D I S E S T A B L I S H M E N T A R I A N I S M;
[Clap] N T I D I S E S T A B L I S H M E N T A R I A N I S M;
Antidisestablishmentarianism was his name-o.

There was a dog and Antidisestablishmentarianism was his name.
[Clap clap] T I D I S E S T A B L I S H M E N T A R I A N I S M;
[Clap clap] T I D I S E S T A B L I S H M E N T A R I A N I S M;
[Clap clap] T I D I S E S T A B L I S H M E N T A R I A N I S M;
Antidisestablishmentarianism was his name-o.

There was a dog and Antidisestablishmentarianism was his name.
[Clap clap clap] I D I S E S T A B L I S H M E N T A R I A N I S M;
[Clap clap clap] I D I S E S T A B L I S H M E N T A R I A N I S M;
[Clap clap clap] I D I S E S T A B L I S H M E N T A R I A N I S M;
Antidisestablishmentarianism was his name-o.

Good luck with the dozens of consecutive hand claps near the end… Anyway, most of my students gave the obvious but incorrect answer, 28/5 times as long. The next most obvious answer, (28/5)2 is closer, but also wrong. Of course the problem isn’t really well-defined, and a very interesting discussion of what the best answer is might have ensued, perhaps considering expressions involving the duration of a hand clap vs. saying the name of a letter, etc.; but I didn’t want to spend a lot of time on it. The main thing I wanted to accomplish with this was to connect with my new students, and I did!