Geometrical Formulations

"Tutor."

.

It's a role I've played for years in numerous guises. As a junior and senior in high school, most of my income came from tutoring. Pre-Algebra, Algebra, Geometry, Trig, Pre-Calculus, Biology, Chemistry, Latin, English: I not only helped with homework, but also re-taught material covered in class, helped prep for tests, and previewed upcoming concepts.

.

In college, I worked for a private tutoring center, expanding my repertoire to include Physics, Religion, SAT prep, and ACT prep. All of this to say, I think I've got a fair bit of tutoring experience under my belt... and it also probably demonstrates well that I enjoy tutoring!

.
Anyway, that's not what this blog is really about.
.
Actually, I wanted to share about one of the more exciting parts of today [warning: some of you will think me entirely bizarre for enjoying this, but...]: helping my fifteen year old sister with her Geometry homework after supper tonight.
.
As a high schooler, or actually, as a middle schooler, I hated Geometry. It was the first time I hadn't (intuitively or otherwise) understood a math class. I was still one of the more advanced students in the class, but, like Physics, it was too conceptual and spatial for me. I did obviously learn most of the concepts, enough that I could do well in the class and even tutor the subject a few years later, but I neither understood nor enjoyed Geometry. Math was always my favourite subject, but Geometry I certainly could have done without!
.
In college, years removed from the last math class I took (AP Calculus, as a junior), I found myself once again tutoring Geometry (and Physics...). As I studied the high school books and retaught the same concepts multiple times over the course of every week, Geometry finally clicked for me. Seven years after I took the class myself, I finally felt like I understood its most basic concepts.
.
And then, tonight, the moment that brought on all of this random reflection: I took a nap before dinner and came downstairs to find that my family had finished eating. As I warmed up some tomato soup and made myself a salad, I half-listened to Dad helping Rachel (my really-close-to-15-year-old sister) with her Geometry homework. A few seconds after I sat down across the table from them, Rachel read me the problem that was pestering them both [Dad would surely have figured it out quickly had he not been taking post-surgery pain medication...].
.

"Each figure above shows noncollinear rays with a common endpoint. Write a formula for the number of angles formed by n noncollinear rays with a common endpoint."

.

As Rachel quickly read the problem to me, all the words rushed over my head and my own high school feelings about Geometry quickly came back: ahhh! I pled the kind of confused tiredness that comes in the first few (or thirty, if you're me) minutes after waking up, and I started eating my supper.

.

.

But of course, like any other time that I get a problem into my head, my brain started processing it. I stole a look at the problem in the Geometry book, then started to doodle on a napkin. "What are you doing?," Rachel asked me. "Just trying to figure out the pattern," I helpfully explained. My napkin soon had columns of numbers like this:

.

1 = 0 = n - 1 = n - n

2 = 1 = n - 1 = n - (n - 1)

3 = 3 = n = n - 0

4 = 6 = n + 2 = n + 1/2n

5 = 10 = n + 5 = 2n

6 = 15 = n + 9 = n + 1.5n

.

Slowly, slowly, and as Rachel gave me numerous "you're so weird" glances, I started to see the pattern... and the formula emerged:

.

n - [(3-n)/2]n = no. of angles

.

So then, the easy part: explaining it in terms that a fifteen year old beginning Geometry student could understand! I showed her the formula, mostly explained how I got to it, let her copy it into her notebook, then made her "check" it against all five of the figures shown above. I also told her she should probably tell her teacher she had help; otherwise, she'd probably end up pegged as the "Geometry whiz kid." She told me not to worry: she was sure nobody else would solve the problem either and that her teacher would surely go over it in-depth. But, could she please take my napkin in to show her teacher?

.

.

It feels good to play with numbers again.