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.

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.

One Tenth of A Picture Is Worth A Hundred Words (WIL #3)

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This gallery contains 3 photos.

The old saw, “a picture is worth a thousand words”, applies to teaching mathematics and the other STEM disciplines as much as anything, and I think most STEM teachers are reasonably aware of that. But there’s a discipline called graphic … Continue reading

Syntax of Math Notation: the Anatomy of a Term

At every level I’ve taught at, I’ve found a lot of students are confused about the syntax of algebraic notation. Here’s a handout I gave to my middle-school, high-school precalculus, and college calculus students, showing the coefficient, a single variable, and an exponent, and saying what the defaults are if anything is missing.

I think many of my students found this helpful, but it could be used as the basis for something more engaging, maybe even perplexing, to use Dan Meyer’s word (Ten Design Principles For Engaging Math Tasks). Let’s challenge students to think about what the defaults should be — no, what they must be — if parts of the term are missing, as they often are. If there’s no coefficient, you must assume it’s 1. Why? Because multiplying by 1 doesn’t change the value; multiplying by anything else does. And so on.

Feel free to use these however you like (though I’d appreciate it if you’d give me credit). Higher-resolution versions are available of both the “challenge” version and the “reference” version.