...or more formally known as "Completing the Factoring by Using the Multiplicative Property of 2." We're going to release it under Creative Commons, so feel free to use it in your classes too. Just make sure you give credit where credit is due.
Here it is. Bask in the glory!
Beautiful isn't it? So elegant. So simple. So WRONG!
I've been teaching this stuff for a while and haven't seen this misconception before. I thanked the student for providing us with the best wrong answer I've ever seen. He looked at me kinda funny then realized I was dead serious. I loved this.
What an opportunity to discuss why we do what we do. This student had the process for solving a quadratic equation by factoring cold. But he didn't understand why we set the equation equal to zero. I don't know about your students, but mine do pretty well with the What. It's the Why that drives this class. I was a bit surprised at how long it took for a student to step up and explain the error in a way that wasn't, "if you plug in the numbers they don't work."
I do my best to make my students think, but they still try to become good little algorithm followers.
The fight goes on.
Wednesday, February 10, 2010
The 2 Product Property
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I have the same problem all the time. I try to go over a quick proof of why "given xy = 0, then either x=0 or y =0." They seem to get it then, but they constantly try to use quad formula or factor as your kids did with it not equal to zero.
Yeah, the mistake in the last step is pretty classic.
(x+1)(x+4) = 2 is fine.
But then you should say
x+1 = 2/(4+x),
x = -1 + 2/(4+x),
and then of course substitute for x on the RHS,
x = -1 + 2/(4 + -1 + 2/(4+x))
x = -1 + 2/(3 + 2/(4+x))
and iterate as many times as you want...
x = -1 + 2/(3 + 2/(3 + 2/(3 + ...)))
You get x = about -0.438 or thereabouts,
and the actual solutions to the original quadratic are -0.43844 or so and ... wait, another solution? Where did that come from?
It's a fun puzzle to ask students at what point in this process you lost the second solution.
And it's fun to generate nice rational approximations to the square roots by using this kind of process too.
Several nice things here. One is that I recently thanked a student in an algebra 1 class where I am the math coach (one of seven Detroit small high schools I currently work with) and was teaching a guest lesson on problem solving for coming up with the best wrong answer to a problem. In this case, too, there was a very logical error that I had hoped someone would come up with, but hadn't seen in other classes in Detroit where I'd done the same problem (unfortunately, not enough kids engaged or engaged deeply enough to make the error).
This student was crestfallen for a second until I made him realize that I was very happy that he'd made his hypothesis. The danger of sincere talk that SEEMS like sarcasm to kids is that they've been battered by too much negative talk (what Marshall Rosenberg, the author of NONVIOLENT COMMUNICATION, TEACHING CHILDREN COMPASSIONATELY, and other excellent books)calls "jackal language," be it at home, from peers, or from teachers and other authority figures in school, to accept a compliment unless it's glaringly obvious. Luckily, the student in question quickly accepted my assurances that I was sincere.
My other observation here is the idea that a quadratic equation that isn't factorable over the integers doesn't magically become so "ultimately" through manipulating it. Either the solutions are integers and the quadratic is factorable in Z or they aren't and it isn't. ;)
I wonder, though, if this student's mistake comes from a misunderstanding of how completing the square SEEMS to be a case of making something that isn't factorable in Z suddenly just that. Of course, the solutions don't turn out to be integers unless completing the square wasn't actually necessary to begin with. ;)
Finally, nice stuff, and nice addition from Josh.
It seems to me that the error was simply misapplying an algorithm. We get (x-a)(x-b)=0 then x-a=0 or x-b=0. This student figured that if it works for 0, then it works for 2. Apparently the zero product property hasn't made a huge impression on him yet.
where do you teach in porterville? I have family that teaches at both Monache and Porterville High. Amazing how small the world really is.
BTW, I teach high school math in Walnut...suburb of Los Angeles.
As to the topic at hand, no matter how many different ways I teach this (and I have tried about 4 different ways in 7 years) a student will always do what you did.
I teach at Sequoia Middle School but taught at Monache for 11 years. Who do you know? Small world indeed.
David Koontz, he is my uncle. He is still there and works with the at risk students, runs ASB, and has been coaching softball there for years now. My parents both grew up there and my grandmothers both still live there.
My family and I try and visit a few times a year. I spent many many weeks/months up there at a time growing up. I learned a lot of my core values from my grandfather who was a farmer up there off of 144. Let me tell you, once you do some manual labor like farming (which I did)it makes you want to get an education so you won't have to live your life like that.
Did you grow up there?
That's crazy. I know Dave very well. I was the head baseball coach while at Monache, so he and I worked together quite often. Good man.
I moved to Porterville when I was in the 8th grade, but had lived in the valley and have been around agriculture for most of my life.
what a small world.
I found this page because of meta musings and their topic of standards based grading. Then I caught the statement at the bottom of your page about not speaking for Porterville Unified and I figured there can't be more than one Porterville in CA.
Was the move to the middle school due to budget cuts or just cause? I hear you guys are going through a lot of crap right now.
I think I will check back because I use examview as well. Although I have just started the basics. We don't have the whole network ability to us yet...not sure if there is a plan to do it.
Maybe you know the answer to this one, does the school have to have a special version to let you add your own questions? I can't remember what the specific warning is when I add a new question. But it says something like it won't be added to the questions bank. Is there a way to add them in?
Nice blog and a lot of good information here.
Yeah, pretty sure there's only one Porterville. I moved to the middle school because I was given the opportunity to teach 3 block periods of advanced students. I have quite a bit of autonomy because I'm the only one who teaches my sections. Bonus: school is about 1 mile from home.
You can add the questions with whichever version you have. I get the same message. There is a way to build questions on a question bank editor which allows you to import the questions into future tests. The tests I'm currently making are very skill specific so I can export the test as a question bank and them it will be available for import on later tests.
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