Is that 3 supposed to be a 5, or am I just missing something?
Ignore me. I found the source of my error.
Thank you, I got as far as remembering x^2-25 is (x+5)(x-5) and then just forgot everything else. Now it’s all coming back to me. It actually is a relatively easy problem, if you could mine the math out of the deep recesses of your cranium.
For completeness, I left off another zero: x=-5.
Same as if someone can’t learn how to use a period, then.
Hm. I wonder how much trouble my dad had with calculus. Of course, he doesn’t exactly have a small nose.
I know exactly what you mean … in fact, you’ve described me to a tee with regard to English writing … first semester, first class, College English … two weeks later the teacher refused to even grade the essay without sitting me down and covering the paper with red ink … he said redo it and he’ll give me a D …
I had a choice, either drop out of college or buckle down and force myself to learn fundamental writing skills … never have I work so hard in a single class, and then just barely squeaking by with a B … now I read a lot of NWS copy and it’s completely obvious these writing skills are completely unnecessary … they taught me the wrong kind of written English for such a career … yeesh …
Algebra skills may never be used in Real Life, this is true, but there are other things the student of Algebra learns … deductive reasoning … starting with a simple truth and logically building on it to obtain a higher truth … there are very few matters in human endeavors where this problem solving tool can’t be used … Algebra makes for a great medium to teach these skills, where things are very much black and white …
California is saying that to be a college junior, you have to demonstrate the ability to perform logical deductive thinking by simplify this Algebraic statement … I see no harm in this, whereas I see plenty of harm from marital counselors who can’t deduce the source of a nasty spat … the grad school here has students take on real life cases and I could always deflect the kids by asking if they enjoyed taking calculus … [wolfish grin] …
Medical doctors knowing nothing of chemistry and prescribing drugs … then it becomes a cookbook profession; with these symptoms, take this drug … if it doesn’t work then the doctor is lost, unable to help his patient … whereas the doctor who understands organic chemistry sees the drug he’s using is the aldehyde form and maybe the alcohol form will be effective …
I’m old enough to remember doctors still performing forceps deliveries, because when they graduated med school C-sections were extremely dangerous … 30 years later C-sections are routine and safe but yahoo doctor is still clamping baby’s brain case and winching them through the birth canal … the medical doctor who won’t learn what he doesn’t want to is a danger to society … don’t want to learn chemistry, don’t be a doctor, good money to be made in the building trades after all …
Good thing math majors have to take a sociology class, or I would think that was a compliment … my brother had a type of stereo that would leave your ears ringing for days afterward …
I’m the male of my species, I don’t have periods … that’s why historians have to take a biology class …
Chasing down Wikipedia’s links brings me to this possibly outdated list of topics covered in Algebra I according to the California State Board of Education, and this list for Algebra II.
The Algebra I list includes “13.0 Students add, subtract, multiply, and divide rational expressions and functions. Students solve both computationally and conceptually challenging problems by using these techniques.” Dividing rational expressions (i.e. expressions that are ratios, or fractions) is what is going on in this problem, though this particular problem may or may not be more involved than the ones students would see in Algebra I. The Algebra II list includes “7.0 Students add, subtract, multiply, divide, reduce, and evaluate rational expressions with monomial and polynomial denominators and simplify complicated rational expressions, including those with negative exponents in the denominator.”
What if I proposed the problem “find all pairs (x,y) of real numbers satisfying xy-x-2y+2=x[sup]2[/sup]+xy-2x=0”? Too hard? At least it reads more like an actual algebra problem.
Like you said, no student’s fate will be decided by their ability to solve any single problem. Instead, (ideally) there is a battery of questions designed to tease out whether the student has indeed internalized some algebra. Along with English, and whatever else the California State Board has decided is indispensable.
I don’t know. That sounds to me a little like saying “Asking students to multiply two whole numbers together is not a good problem for testing students’ ability to use practical arithmetic skills, because most real world quantities are not whole numbers.”
It’s true that problems in algebra textbooks and tests tend to be less messy, more abstract or artificial, than those found in real life. This is not necessarily a bad thing. There are arguments both for and against abstractness, artificiality, and problems with neat and tidy answers.
This problem is one that I can’t imagine a direct practical application for, but it does test two crucial skills: (1) the ability to factor polynomials, and (2) the ability to work with (operate on and legally simplify) fractional expressions. Students who are not competent at such skills will have serious difficulty if and when they go on to take more advanced algebra, trigonometry, or calculus.
Thus, it is entirely reasonable to expect such students (i.e. the ones who will go on to take those kinds of math classes) to be able to solve such a problem. As the linked article notes, the debate comes in whether there should be “other possible ways of obtaining ‘college-level’ quantitative reasoning skills” besides these algebra-intensive classes.
** Chronos’** point is passing an Algebra test only measures the student’s ability to pass Algebra tests … not necessarily how much the student understands Algebra … military tests are famous for this … two of the four choices for answers were screw ball … then just randomly picking between the other two got you a passing 75% …
I’m not arguing against textbook problems being less messy. I’m arguing against textbook problems being less messy in a way that opens up solution techniques not available for more messy problems.
Do you mean setting problems not be arranged in such a way to have factors that divide out? (or cancel out, whatever you use. My Calc teacher in HS hated ‘cancel out’ so I’ve said ‘divide out’ ever since)
As far as I’m concerned, the difference between Algebra I and Algebra II is that imaginary/complex numbers. They don’t exist in Algebra I. And I can see someone saying that, outside a few particular fields, complex numbers are irrelevant to everyday life.
Sure, it opens up a whole host of mathematics, but it’s math that most people never actually get to nor need to use. That doesn’t mean it doesn’t need to be taught, but I do wonder if there’s too much focus on it.
I’ve heard of (but not verified) rumors that some are actually considering replacing Algebra II with a different form of math that is more useful. I can’t remember which ones specifically, but they’re all actually classes taught in college. I believe one of them is based on statistics.
It’s not a great medium if you’re a student with trouble understanding the abstraction of algebra and deductive reasoning can be learned in other ways. One might even put it the other way round, understanding deductive reasoning makes learning algebra trivial.
I mean that, if you’re going to use a single problem to test for algebra knowledge/skills, a factoring-and-simplifying problem is a poor choice for it. Of course, it’s also probably a poor choice to use any single problem: You should have an entire test of many questions, some of which are just factoring without any cancellation to give extra clues, and some of which are simplification problems for which the correct answer is “it’s already as simple as it can get”.
Sure, I agree with that. I wasn’t aware that the problem in the OP was a single problem used to determine eligibility for the degree or transfer. I thought it was just representative of the types of problem one would need to know?
And props for the obelus.
Thank-you. As a high school mathematics teacher, who on occasion had to teach students who had already failed Algebra I at least once (South Carolina required that you pass an end-of-course examination in Algebra I to graduate), this was my biggest unhappiness. I could easily (relatively speaking) teach them how to pass the EOC exam. But that did not mean that they really understood the basic concepts of Algebra I. Of course, neither did most of my pre-AP Geometry students (the equivalent of an honors course at my high school), and they usually passed Algebra I in a breeze as 8th graders.
If I had any one specific indictment of the general way high schools in America teach, it would be that they rarely take the effort to connect the subject matter to its actual usefulness in the real world. I cannot tell you how many Geometry textbooks I have seen where there is some Geometry principle theoretically demonstrated with a “real world” illustration, which has nothing to do with a real-world application of the principle. Instead, the graphic simply is a handy illustration of the principle. Real-world applications of, for example, the various rules regarding angles formed by a transversal across parallel lines are consistently absent. Which is stupid, because these rules are useful in a number of real-world contexts (for example, surveying). But most students will see them as nothin more than another set of Things To Memorize And Then Forget.
This really begins to show up with pre-Calculus classes. We spend hours teaching the students how to handle equation solving and expression simplification, without giving them any real-world examples showing how and why these skills might be used. And it’s important for them to understand that real-world equations and expressions are extremely messy, and it’s not easy to solve or simplify them. So because they don’t have a connection, they simply learn how to “game the system.” And sadly, this means that a lot of tests of their understanding of various function systems turns into nothing more than a form of intelligence test: can you spot the trick we want you to use to get to our pre-selected answer of choice? 
As for the implied underlying debate presented by the OP (and touched on by some posters), I will only say this: in my opinion, having taught all the various high school math courses, if the goal of forcing students to learn one specific math course is to teach them how to think logically, and how to sequence a series of steps to accomplish a goal (which is often the actual reason Algebra I is required), then Geometry is a much better class to require. If the goal is to teach them something really useful in the future, then frankly they should be required to take a probability and statistics course. The assertion that Algebra has some over-powering application to understanding future coursework in college is total bunk (Texas, for example, is in the process of completely re-thinking this at the high school and college levels). Yes, there are a number of college majors for which Algebra is a sine qua non. But there are far more majors for which it is not.
I agree with much of what you say here, and I’ll amplify by saying that a focus on algebra is a focus on calculus, and a focus on calculus is a focus on preparing students to be engineers, which is… let’s call it “premature optimization”, to borrow a concept from Computer Science, a STEM field which didn’t exist when calculus got raised to the highest heights of secondary and post-secondary math education. Computer Science doesn’t really require calculus, and I’ll whip any five of you who say it does. 
However, I have to differ by saying that probability and statistics would be a better place to learn logical thinking than geometry, because it goes from simple grounded-in-everyday-reality examples to fairly subtle logical arguments in a nicely gentle slope. For example, if it rained recently, you can conclude the sidewalk will be wet. However, if the sidewalk is wet, can you conclude it must have rained recently? What if it’s Missoula in August, when it’s dry and people water their laws and their sidewalks? Teaching people you can’t “flip the arrow” like that is very well-grounded in everyday experience and very important in terms of logical thinking, and it is precisely the kind of logic statistics touches on.
Geometry is useful, but it isn’t as massively necessary to being a functional adult, and, I would argue, doesn’t lead into that kind of “conversational” logic I mentioned above.