The sum of the first consecutive odd numbers is a square number.
Why? What do perfect squares have to do with odd numbers? At first glance, these are two seemingly unrelated types of numbers.
Some of us (okay, me) may have presented something like this:
1 + 3 + 5 + … + (2n – 1)
(2n – 1) + … + 5 + 3 + 1
The sum of each column is 2n. We have n columns. The total is then n × 2n = 2n². We added the sum twice so 2n² ÷ 2 = n².
Can you see what perfect squares have to do with odd numbers? Me neither.
Compare that with the following explanation¹ given in Paul Lockhart’s “A Mathematician’s Lament”.
Inspired by this pictorial representation, I created this poster below.
¹ Lockhart might say it’s not the fact that perfect squares are made up of odd numbers which can be represented as L-shapes. What matters is the idea of chopping the square into these nested shapes.
Chris Hunter 
Vancouver residents may recognize some of the numbers. The photographer, mag3737 (flickr), sent me the following:
1(leaf): On a car at an auto-show in Yaletown a few years ago
3: The “Qube”, 1333 W. Georgia
5: Shop on S. Granville in Marpole
7: Utility/manhole cover, probably somewhere downtown. Actually, I have a
vague feeling this one might be from the corner of Smithe & Howe, but I
wouldn’t swear to it.
9: Somewhere around the armoury at Georgia & Beatty, possibly even on one
of those tanks/artillery things
11: Probably Sacramento again.
15: Save-On Foods on Cambie just south of False Creek
17: Somewhere in London, England
19: The main Post Office downtown
Chris,
This might be just the jump start I need to keep a talk I’m doing on visuals that scream sequences/series. May I borrow your graph of cool numbers, ahem, odd numbers? I would like to use it and, a la Dan Meyer’s #anyqs ask “Is this always true?” first.
Then proceed with the proof. Any thoughts you have on similar items would be awesome too.
Steal away! Let me know how it worked. I’ve been working on a couple more posters and I’ll be posting them soon.
May I ask a critical question that also pertains to the Running with Scissors post?
What is the purpose of the numerals in the boxes? They are beautiful, to be sure. But I wonder whether they add meaningfully to the information in the diagram, or whether they decorate it.
I think I’ve decorated Lockhart’s diagram.
My idea was that students (teachers?) would recognize the numerals as being odd, see the squares in the diagram (or maybe be encouraged to find them by the n^2 at the bottom), and make connections. By including the nine 9’s instead of nine dots as in Lockhart’s diagram, I thought it would make it more obvious that we are talking about odd numbers here. Lockhart posed the problem in a preceding page. Without the numerals, would students/teachers ask “What do odd numbers have to do with perfect squares?”. Without the numerals, would students/teachers see nested L’s as odd numbers? Probably not (at least at first), otherwise Lockhart couldn’t have written about that Eureka moment.
Numerals or not, I thought Lockhart’s diagram of squares broken into nested L-shapes was a brilliant example of how a pictorial explanation can be more powerful than a symbolic one. Sometimes, I hear from teachers that pictorial representations are for younger students (“Drawing pictures is great in elementary, but in secondary students should be able to use variables”) or struggling students (“Drawing pictures is great for those students who just don’t get it using variables”).
As for the Running with Scissors post… looks like decoration. Thanks again for giving me something to think about.