The Gauntlet

The Background
It is pretty common for Algebra 2 curricula to require binomial expansion in some capacity. In New York we have A2.A.36, Apply the Binomial Theorem to expand a binomial and determine a specific term of a binomial expansion.

Which on a standardized test yields this wonderful rare precious gem of a problem:

What is the fourth term in the expansion of (3x − 2)⁵?

(1) −720x²       (2) -240x       (3) 720x²       (4) 1,080x³

Which, come on. We don't even have to know anything. It's obviously (1). If they put practically the same answer twice it's got to be one of those, and there's a minus sign in the question, so we pick the negative one.

We include one 43-minute day on this in our Probability unit, I suppose because the most sensible application for us relates to Bernoulli trials. For example: A coin is tossed five times. Let H = the probability of heads and T = the probability of tails. Expand (H + T)⁵, and use it to determine the probability of 5 heads and 0 tails, 4 heads and 1 tails, etc etc.

The Issue
Presenting the binomial expansion in a way that makes any sense to a non-gifted regular-track eleventh grader is a (insert lots of swearing and edu-blog-inappropriate analogies here, and be creative because remember I was a sailor.) This year it so happens I will be out of school on Binomial Expansion Day, so I recorded a lesson for the sub to show. Well that's a lie. First I looked to see if anyone had already done a passable job at it, and as far as I can tell, no one has.

So I made my own. And I didn't slap it together, even. I spent hours on this thing. I didn't make the viewer wait while I very. slowly. wrote. x. plus. y. times. x. plus. y. times. x. plus. y...... zzzzzzzzz honk shuuuu, honk shuuuuu..... no! I typed it all out painstakingly ahead of time. I thought the highlighting of the groups starting at 5:00 was particularly inspired. It would have been better if they just kind of lit up, but I did the best I could with the tools I could bring to bear. And at first I was sure my NYSED = Death Star gag was comic gold but I'm pretty sure nobody gets it.



As you can see, it is really, really terrible. And I'm not even talking about the keyboard clicking noises and intermittent cat vocalizations, or the fact that I think you can see my mouse pointer when you can't. I could tweak this a little but the time investment makes me break into a cold sweat and at this point I'm going to have to leave this for the sub either way.

However...

The Challenge
Do it better. Some of you apparently think you got this. So bring it.

Format : Video, slides, slides with audio, smart notebook file, group activity, music video, mathematica slides, CAS investigation, whatever you like. As long as a teacher with a year or two under her belt could use it in a classroom with a reasonable amount of preparation. I'm not going to set a time limit but if I can't evaluate its worthiness in ~15 minutes its chances are bleak.

Level : Non-honors-math-taking, charming, fallible, fractions-are-dicey, probably-college-bound-but-mostly-not-future-engineers-scientists-mathematicians American high school juniors.

Objective : Present the binomial expansion in a way that makes sense. Bonus points if students are able to as a result completely expand a power of a binomial and find a specific term in an expansion.

Expect the students have at least minimal familiarity with : Basic probability, probability with "and" and "or", "how many ways" with combinations, "how many ways" with the fundamental counting principle, probability with combinations, Bernoulli experiments with "exactly", Bernoulli experiments with "at least" and "at most".

Deadline : Thursday, May 27, 2010, 10:00 PM. That's maybe too long but that's the next time I have any kind of break in my calendar, and this time of year is not leisurely for anyone, I know, so maybe more time is better.

Judges : Me (who will undoubtedly be influenced by comments from cohort and other readers) and possibly @samjshah if he was serious about this and possibly some students even.

How to Enter
Get it online somehow. Link to it in a comment on this post.

The Booty
Being featured in this very space right here, with its dozens of loyal readers : fame! And also, if you e-mail me a mailing address, this awesome and content-appropriate shirt in size XL which has only been worn once because it is way too big on me : fortune! In a universe where castoffs from my closet = fortune! Get crackin!

Binomial Theorem