Stories from Middle School: Partitions and the Quadratic Equation

The instructions  from our coach were simply to “review the problems we did yesterday.”  To our Mathcounts team, this came as a surprise. We had sacrificed nearly a month of our lunches to work on math problems, and we were used to reviewing each set after we finished.  But an entire lunch period for reviewing? That worked too.

The four of us, three girls and a guy, grabbed our papers, sat down at random desks in the room, and began going over the problems we had missed. However, this didn’t take very long, since most mistakes in Mathcounts occurred because of two reasons: 1) carelessness, or 2) a failure to recognize one clue that would unlock the entire question.

After a bunch of “OHHH”s and “NOOOO”s (or maybe they were all from me) we had finished “reviewing,” except for one question that none of us knew how to solve involving something called partitions.

Usually, when this happened, our coach would pull out the solutions manual, show us their brilliant way of solving the problem, and we would all be enlightened.

This time though, their solution was to make a  list. The brute force method that was always looked down upon. Not that we never used it, but in an official solution? No.  We weren’t satisfied. (or maybe it was just me again)

So we set out to find a pattern instead and crowded around the chalkboard as one of us began listing partitions. (The only way to find the intelligent method was by using the stupid method first, right?)

A few minutes later, we had filled up the entire board with partitions and supposedly spotted a pattern. Now we just had to keep testing it.

“Let’s go list the partitions for 8!” I exclaimed, perhaps a little too enthusiastically.

“Yeah!” my friend replied. (a little  overly eager as well) She grabbed a sheet of paper and a marker, and we sat down and began listing.

 “Ugh, you guys are such nerds!” one of our team members told us as we methodically said partitions and wrote them down.

“Dude, you can’t say that since we’re probably all nerds,” our other team member pointed out.  He picked up the chalkboard eraser and started erasing our beautiful list of partitions.

“I’m going to derive the quadratic formula.”

 I’m not sure if that was supposed to come as a surprise or not.  I mean, I know middle schoolers typically don’t derive long algebraic formulas in their spare time, but my friend and I were the ones listing partitions. Neither of us were really better off.

So we stuck to our list as he wrote ax^2+bx+c=0 on the board.

Our team member who had called my friend and I nerds walked over to watch him, and I couldn’t resist calling out, “Who are you calling a nerd now!” Which maybe was slightly misdirected, but it  felt nice to say.

Anyways, our “pattern” for partitions ended up not working, and there wasn’t much we could do about it since finding a new pattern would be too difficult . We decided to leave our partitions and go watch the derivation of the quadratic equation that was taking place a few meters away.

At this point, I had no idea what was going on, but I watched as the left side of the equation gradually got smaller and smaller and the right side got more and more complex. And soon, the left side of the equation was just x and the right side was negative b plus or minus the square root of b squared minus 4c. All over 2a. We had the quadratic equation!!!

No wait. We didn’t. We almost had the quadratic equation. An “a” was missing in the middle.

incorrect derivation

Because I can, and because my handwriting is kind of horrible

For some reason, this cracked all of us up. Here we had this brilliant board of algebra with beautiful process and beautiful everything (ok maybe the handwriting was a little questionable, but still) and all of it was invalid because of a puny little “a” lost somewhere. Oh the irony.

“Where’s the ‘a’???”

“How do you forget an a?”

“Where’s the ‘a’???”

“I don’t know! We messed up somewhere!”

“Where?”

“WHERE’S THE A???”

Anyone walking in on us these next few minutes would have seen four hysterical kids chanting “WHERE’S THE A???” every few seconds in front of an algebra-filled board that obviously had many a’s written on it.

And that’s exactly what happened when the lunch bell rang and next period’s math class started coming in. They probably thought we were maniacs. But we didn’t really care. We just had to find that “a” (if that weren’t obvious enough already)

Our coach came up behind us and said “I see where the mistake is…”

“Where?” I immediately asked. “Wait no don’t tell us!” I wanted to find it ourselves.

But it was ridiculously hard to find a single mistake in a huge sea of algebra, (especially when we were laughing our heads off)  and our coach finally had to point out the line where we messed up.

We spent the next 30 seconds staring at the middle of the board, still saying “Where’s the A???” and still being partially hyper.

Then…I saw it.

“I FOUND THE MISTAKE!!!” I gasped in between bouts of hysteria. ” Right there! If you multiply the bottom by this then the whole fraction’s supposed to be changed because of that rule, you know, the –THERE! That’s where the a is!”

No one could understand me. They went back to searching.

A couple more seconds passed. The late bell rang. None of us moved.

Then…

“I SEE IT!” the original person doing the deriving exclaimed.

He managed to sanely explain the problem to everyone else and changed the string of mistakes it had led to until the “a” was restored to the last line. And we had it– a step by step derivation of the quadratic equation!

“YES! I FINALLY GOT IT BY MYSELF! I have to write this down!” He did deserve most of the credit. The rest of us were just watching him and getting lost. (or was it just me?)

That didn’t stop us from telling each other and the class looking at us weirdly, “WE FOUND THE A!!!”

Except there was another problem with our success. We were late to our next class. So as the beautiful derivation was being copied, my friend and I hurriedly ran out of the room, hysterically laughing, and hoping that our next teacher wouldn’t mind too much.

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8 thoughts on “Stories from Middle School: Partitions and the Quadratic Equation

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