Misdirection in Algebra
I’m back to Algebra I, first time since 2008-09, and loving it. I have three classes of ninth graders who already know how to use elephants for variables, know how to use transposition (as opposed to pendant subtraction) in solving equations, know that we pronounce neither T in Trenton with the sound they thought a T made, know a little something about the evils of testing, know that they are supposed to talk in class, know that perfect squares are never two more than a multiple of three, and are starting to get the swing of moving their desks between the three arrangements we use.
Soon they will learn about the holidays half way between the solstices and the equinoxes, other facts about the mean times we live in, that FOIL is no longer allowed (and some will hate me for that ), and that high school final exams are stressful, even in a “nice” school.
But last week they learned a little about exponents. My way.
Clearly demonstrate that is true.
Those were the instructions they saw on the board, and I got busy directing kids to put up homework or to review problems others had put up, but pushed every kid, slowly, to switch to the problem in front of them, and I could sense some discomfort.
Should we prove it?
Show it… You could try a proof, but that would be hard. And you could demonstrate with a few examples, but even a whole bunch of examples doesn’t show that it is always true.
Strange quiet as they worked. Or as a few worked. Most sort of just stared.
It doesn’t work!
No, I tried 3 and 2, it doesn’t work.
Well, wait, three plus two squared is twenty-five, and since we distribute the exponents, we get twenty-five on the other side. Work on fixing your mistake. Does someone else have something to add?
Four and six don’t work either!
Can someone help him with the arithmetic? Of course it works.
[I demonstrate how to distribute 2(a + b) and baldly assert the same thing happens with exponents]
Can we get a good explanation of why this is so?
Zero and one work.
Very nice, but is one example enough to show that this is true?
What can we do with one example?
We can have a counterexample.
Murmurs about distributing exponents. A girl starts to tell her neighbors that I am fooling around. The revolt is about to start.
What about ten and one? Don’t they work? Ten plus one is eleven, eleven squared is one hundred twenty-one, and ten squared plus one squared are also one hundred twenty one, right?
Lots of muttered yeses, and heads nodding yes, but finally a bunch of heads nodding no.
No, that’s one hundred one, it’s not one hundred twenty-one. It’s not true. These are counterexamples. These are all counterexamples!
I’m fighting to hold back the laughter, but they see. General chaos for a minute.
I share with them what other teachers say about exponents and kittens, and we vow never to distribute exponents ourselves. In one class (this story is a composite of the three classes, and quite a bit shortened), we went on to discover when is true. And in every class many kids are claiming that they knew from the beginning, but just didn’t say anything.
Some will still distribute an exponent here and there. But most won’t.