I’ve been doing Algebra with Leo for some time, and every once in a while I actually amaze myself at just how cool Algebra is. Leo and I have in mind writing a book called “Algebra is Magic” which would just be a bunch of short solutions to what look like difficult problems, but which just “magically” drop out through Algebra.
Whenever we do any math, I’ve been trying to engrain that the first question is always “What’s the model?”, that is, what is the formal ‘equational’/’mathematical’/’algebraic’ form of whatever problem we’re doing?
Now, usually I know where we’re going, because I can do most of the things that a K-12 student would ever encounter more-or-less in my head, or at least I can tell right away what the algebraic form is going to be. So, together we write down the model (which I already know … or am able to work out without thinking too hard, so I lead him through this), and then we push through the algebra, and “voila!” the answer. (We always, of course, check the answer, at least for making sense, and usually for being correct by pushing it back through the original equation.)
Now, usually, the “voila!” moment is unsurprising to me, because I knew it was going to work, because I could see that we did it right, and where it was going. But, as I said at the outset, sometimes the problem is hard enough that even I’m not sure whether we’re right, and even I am surprised when it works!
There have been many of these, and our plan is to write these up, first in this blog, and eventually in a book.
Here’s just the latest one, as an example:
We’re starting on Calculus, and the book that I’m using (Stewart’s Calculus [8th ed.] from Engage Learning) has a diagnostic test in the preface. For the most part these are pretty simple high school Algebra problems, and even Leo can do most of them … maybe with a little guidance from me … and I can, of course, do all of them. But there are a couple that are even tricky for me.
One such is this:
So, first off, we know “SOHCAHTOA”, and have used it a lot, but I had to recall what secant was. (I know its either inverse sine or cosine, but the way I remember which one is that I remember that it’s as confusing as possible, so to be as confusing as possible secant is inverse cosine, and cosecant is inverse sine!)
Okay, fine, but on the face of it I had no sense that this was going to be anything but a mess to prove. But we decided to use the “Algebra is Magic” mantra:
“Get a model and just push it through!”
Okay, so even I don’t know where this was going, but went with it, but here goes:
(I’m using ‘p’ here instead of ‘o’ to mean ‘opposite’, to avoid confusion with zero)
Blindly substitute into:
Next, blindly push through the algebra, not really knowing where this is going:
Multiply through by a:
Multiply through by h:
And then … Whoa! What The…. That’s The Pythagorean Theorem!
We weren’t headed there at all; we didn’t actually know where we were headed; Algebra brought us to The Pythagorean Theorem … by Magic!
And if we take The Pythagorean Theorem as axiomatic (which, of course we do!), then we just proved the target expression!
QEfingD … MAGIC … BANG!
(This, of course, led into into long discussions of the difference between theorems and axioms, including studying a bunch of proofs of The Pythagorean Theorem itself. But none of that was as magical as running headlong into it without knowing where we were headed!)