As I write this, I've graded the final exams for one of my three Algebra 2 classes. I'm going to schedule this post to show up on the blog as my last Algebra 2 class starts its final tomorrow, so that none of my students could possibly stumble upon this post and have an advantage when taking the test.
My final consists of a department-wide 15-question multiple choice section, as well as a Darren-made 15-question short answer section. Students did the multiple guess part first, then started the short answer part.
The second question on the short answer section (so there's no excuse for brains having already been turned to mush) was something like this: |something|=-6x . It doesn't matter what was in the absolute value sign, but it was something simple and linear like x-5. Since you, my reader, don't know what was in the absolute value sign, you can't solve the problem, but I ask you this: what can you tell me about the answer x?
Look at the problem and think for a moment.
I understand it's been awhile since many of you have worked with absolute values, so here's what I'm looking for: since the absolute value of a real number is never negative, then x must be negative because -6 times a negative number would be positive.
Solving the problem would give two answers, one positive and one negative--the positive one is extraneous, meaning that when you substitute it back into the original equation it doesn't work.
This was one of the easiest problems on the test, and yet only 3 of my 30 or so students wrote down the correct answer (which is just the negative answer). Most did all this work and then didn't even take the time (and they had plenty of time) to substitute their answers back in to check them. To say I'm disappointed would be an understatement.
7 comments:
That is scary. But what frightens me more is the new education philosophy which will blame you for the students' lack of performance, despite the fact that you probably taught them the concept multiple times.
I was going to say that x has to be negative, but then I see that you answered that in the body of your post.
Maybe you can have a math post similar to the trivia posts, where you don't post the answer until a day late.
--> THINKING <-- about a problem first can very often help solve it. To paraphrase one of my math teachers, "None of these problems require heavy calculation. If you wind up with a page full of equations, stop, you're doing it wrong!"
I polled the Bradley kids.
Both the 8th grader and the 6th grader knew x was negative. They think that a high proportion of their classmates would know it.
I know the right answer now, but I'm pretty certain that I missed it on a test in my high school math class. I learned from my mistake and eventually got a BS in Math. So, don't lose heart. Kids have to learn to check their work at some point. Maybe some of your students just hit that point.
Sometimes I try to remember what I was like as a mark student in high school, and I'm quite certain I would have missed that problem. But, I kept studying and working, and now I'm pretty good at math.
Ahhh yes the ole "extraneous" solution. It gets most of my students too. I don't make it as obvious, but I tell them pretty straight forward that if there is a variable outside the absolute value, then you need to check it because most likely one won't work. And yet...
I will say that I am not sure how my students would have answered your question. Some would have probably said no solution right away because the absolute value equation was equal to a negative. The minds of high schoolers :-)
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