In order to graduate from a California Community College and transfer to a four year university, one has to solve this problem.
Anyone know the answer? Easy or Hard?
x(2) – 25xy + 6x – 5y - 30
------------÷---------------------
X(2) + 5x5x - 15
Solve for what? It isn’t an equation, it’s an expression: Nothing is being equated, it’s just a mathematical relationship between numbers and variables which could potentially be simplified, but without telling what the expression is equal to, none of the variables can be solved for. It’s also very inconsistent: Is x a variable or a function? Why is one term written “5x5x” instead of 25x[sup]2[/sup], or is that part of what they’re actually testing for? Was this thing horribly, grotesquely mistranscribed and mutilated at some point?
I do like the little obelus in the middle of the bar. That’s cute.
Your formatting is hard to make out, but I clicked on the link and there it shows up clearly.
I don’t see any algebra problem at all, just a rational expression containing a bunch of common factors which the student is presumably supposed to cancel out. Answer: 5(x-3)/x(y+6). Difficulty: trivial, assuming the student has taken an introduction to algebra course in high school.
I only skimmed the article, but in my view the point is not how much mathematics the student may end up studying at university, rather that if he or she cannot take an introductory course (algebra in this case) and answer a basic no-brain question, then he or she has zero chance of passing any class in any subject. That is, the student (possibly through no fault of their own) is simply not ready to graduate.
OK, I checked the LA Times article. Here’s the expression:
x² - 25 xy + 6x - 5y - 30
--------- ÷ --------------------
x² + 5x 5x - 15
And, yes, it’s trivial. A computer can do it. Morally, a computer should do it.
That isn’t true and shouldn’t be promoted as a stereotype. I went to a prestigious university for undergraduate education and an Ivy League graduate school. I got the highest grades in all of my statistics classes and analytical computer programming classes. I work in high level Information Technology now. I simply don’t do Algebra except organically and never understood the relevance to it for anything whatsoever.
We have had this debate before in other threads. There are some people that “get” algebra and others that simply don’t. I might be able to if I tried hard enough but Algebra II might be the most useless course that you can take. I doubt I would do any better now than when I was 16 and forced to take it even though I have over 20 years of experience in IT and engineering. It has no predictive, practical or real-world value. My position is that it should be replaced by almost anything in high schools whether that is folk banjo playing or just extended recess.
The real-world implications are that you have kids that are dropping out of high school because they can’t pass a course that should only exist as an elective and not a requirement. There are few people in the world that have actually used any form of Algebra once they finished the class and almost immediately forgot every useless thing they were force-fed because of early 20th century educational conventions.
That isn’t true and shouldn’t be promoted as a stereotype.
I see this is in General Questions. If and when it gets moved, I may have more to say, but for now I will confine myself to the following relatively factual remarks:
The OP’s formatting is a bit confusing, but the problem as stated in the link is clear.
DPRK gave the correct answer.
It’s not a particularly difficult nor nonstandard problem for the average student who has recently been studying algebra and has been taught to do these kinds of problems. Many, though not all, high school students regularly solve problems considerably more advanced than this. In fact, I believe that this sort of problem would be the kind of thing included in the math that some high schools require for graduation.
Although, the thread title notwithstanding, I doubt that any high school or college would make solving any one particular problem a graduation requirement; rather, the student would have to get a passing grade on a test or in a class that included such problems.
If someone can’t learn Algebra, and there’s many people who can’t, then that someone can’t be a college junior … seems fair to me …
Let’s look at the other tennis shoe …
If someone can’t learn to write standard English, and there’s many people who can’t, then that someone can’t be a college junior … seems just as fair to me …
A university degree is supposed to mean a universal education … sure, I might know a bunch about zoology, but I also know something about history, literature, math, physics, underwater basket weaving, political science, and so on … I understand there’s no Algebra in 14th Century French poetry, but if you want to be a college graduate, you’re going to have to learn Algebra …
These days, seems like all the kids have to do is show up to class every day to get a high school diploma … you’re guarantied at least a B because we don’t want to damage your self-esteem … are our junior colleges to be the same, here’s your AA degree in attendance? … eventually we’ll be giving Masters degrees in civil engineering to dogs …
Life’s a bitch … sometimes we have to learn something that we really really really hate … I could have died a happy man if I never knew anything about septic tanks … but then I’d have unplugged the clean out and walked away, out of sight out of mind as long as the toilet flushes … but then where would we be? … standing in a big pile of shit is where we’d be …
I think calculus should be required of all college students … not because it’s useful … rather it’s a good way for math majors to make some extra money tutoring … especially cute pre-nursing students with small noses …
I was not even claiming that Algebra is or is not a useful subject. If the quiz were on second-century Chinese literature, and the student cited Shakespeare as a prominent poet of the Song Dynasty, I would also regard that as problematic. You might argue that Chinese poetry is useless as well, but the student is being tested for graduation, so presumably is supposed to have passed some class in some subject. The OP asked whether the linked problem was an easy or hard algebra problem, and as has been pointed out it is not much of a puzzle to anyone who was awake during class; in fact it is just a mechanical exercise.
Here’s an interesting blog post: "Should Doctors Have to Take Physics and Chemistry?’ – Chad Orzel (ScienceBlogs) – July 30th, 2010 … the author presents both sides and it’s an interesting question … with the shortage of MD’s maybe we should start letting people be doctors without understanding “hard science” …
I don’t want to insult anyone but do we really want to let people be doctors who are too stupid to learn chemistry … it’s not that godawful hard … and the only way to see if they can learn is to make them pass the class … I’m just having a hard time believing medical school is easier than first year calculus …
I teach high school math to highly motivated students at varying levels of math proficiency. There are many paths to the state of being a student unable to simplify that expression, and few of those paths should prevent you from graduating and going on to study something that doesn’t involve algebra.
At best such a requirement wastes students time by requiring rote learning of algorithms they don’t have the mathematical foundations to grasp the meaning of, at worst it prevents students from getting degrees society needs.
Now I personally consider it possible that every student with the ability to learn other subjects at a college level could learn this much math if they had being instructed in an appropriate manner at lower levels, but it is obvious that math education fails to do that for a lot of students, and the time to remedy that is not at the end of college.
And it’s also completely possible that the particular form of abstraction is not possible for all students, even those quite capable in other ways, just as I suspect not everyone has the aptitude to play the piano properly. (I’ll be happy to concede the point if someone presents a successful piano lesson study with randomization.)
It’s not an established fact that intelligence is this one dimensional. By the time students get to senior high they have spent a decade absorbing and organizing knowledge, not only in the class room, but in their daily lives, and that process is not uniform or perfect, so they will have a range of starting points, and some students who are very bright in all other ways may simply have a starting point for chemistry so poor and/or different from the rest that it isn’t cost efficient for the student and the school system to teach them chemistry unless it’s actually indispensable to their desired career path.
According to the article, the task is to simplify it, not solve it. And the simplification is trivial.
It’s an easy problem, but it’s not a good problem for testing students’ ability to use practical algebra skills.
Part of the difficulty is that simplification problems are inherently open-ended. Sometimes when you get one, the answer is “This cannot be simplified; it’s already in its simplest possible form”. Sometimes you can do one simplification, but don’t know if there are any others. And sometimes an expression can be expressed in two different ways, but it’s a matter of taste which one is considered simplest.
And part of it is the techniques used in the simplification. The first part of the problem, to the left of the division sign, is super-easy. The polynomial on the top of the right-hand side, however, takes a modicum more effort. But factoring is always easier when you have a good guess as to what one of the factors will be, which I did: Since I knew that it was a simplification problem, I expected that one of the factors would cancel out with the left portion, which it did.
But really, that’s gaming the system. That’s a technique for solving textbook problems, but real-world problems almost never work out that nicely. This problem isn’t so much testing how good students are at algebra (an immensely practical and useful real-world skill), as it is at testing how good they are at faking algebra skill (an impractical and useless real-world skill).
Is this problem something we’d typically bin (in the US) into Algebra I or II? I’m pulling up a table of contents but I don’t remember the labels we apply to different topics/expressions/equations.
Christ, I guess it was over 20 years ago, but I passed through AP Calc with a breeze and even took an extra Calc course in college, and I haven’t the faintest memory of how to simplify the equation in the OP. Anyone care to walk me through it?
That isn’t true, and shouldn’t be promoted as a stereotype.
I learned to do this sort of stuff at age 11 or 12.
So can one of you geniuses please simplify it and walk me through it? I’m not even sure where to begin. I can’t cancel the x sqauareds in the first part there from what I remember. Am I just missing something really obvious?
We start with the expression:
((x^2 - 25) / (x^2 + 5x)) / ((xy + 6x -5y - 30) / (5*x - 15))
Then factor the polynomials to get:
((x+5)(x-5) / (x(x+5))) / ((x-5)(y+6) / (5(x-3)))
This is the “simplest” general form of the expression. Note that the expression is undefined if x=0 or x=3 or x=5 or y=-6. Depending on what the expression is being used for, this can be critically important or totally irrelevant.
If we assume the expression has a definite value, we can simplify further by moving factors to the numerator or denominator. (That is, 1/(1/x) = x if x!=0.)
((x+5)(x-5)5(x-3)) / (x(x+5)(x-5)(y+6))
And then cancel factors that are in both numerator and denominator to get:
(5*(x-3)) / (x*(y+6))