The square root function returns the positive value. Otherwise, it would be multi-valued and therefore not a function.
I saw this thread fall off with no replies so didn’t bother checking it over the weekend. Thanks for the replies, though it doesn’t seem anyone disagrees with the OP. I expected some disagreement.
To address a few comments:
First off, the reason I pick on Algebra is the nature of Algebra itself. Algebra is THE main tool of science and mathematics as well as used by many other disciplines. However, it is terribly not useful on its own. Look at the word problems of a Algebra textbook. They are usually very silly. Now look at a Calculus book. Those are legitimate word problems. Algebra is essential for many things. However, Algebra by itself has limited use.
Because Algebra is essential for so many diciplines, I still think a year of Algebra should be required. However, Algebra II should be an elective IMO. There are other Maths that can be taught for students not interested in Algebra-intensive disciplines (math, science and others)
This is a good idea but usually not ‘politically’ feasible. You see, education is dominated by ‘socialistic’ thinking. I mean that much money is spent on slower kids and smarter kids are thought to be doing fine on their own. In addition, teachers are taught and expected to use the smarter kids to help teach the slower ones. Smarter kids 'don’t need the help while slower ones do’mentality. Anything that deals with separating the faster and slower students is thought ‘elitist’ and is not looked upon favorably by many. In addition, we have to worry about the ego of the slower students…To avoid the slower students feeling like they are slower by having accelerated classes which they are not apart it is perfectly acceptable to sacrifice the smarter students education.
You see? Very ‘socialistic’. I’m sure there are parts of the country where this is not true but it was VERY true where I taught and was true of nearly every teacher I met and this topic was brought up with.
You need to teach. I say this because every new teacher goes through a huge shock when they start because they believe as you do. The problem is politics. When you teach college, if a student fails it is their fault. In HS, if a student fails, it is not thier fault, IT IS YOUR FAULT! THAT is the mentality you will deal with. YOU are a bad teacher because they failed.
Now, most students will not study, nor do homework very much. Some won’t even pay attention in class. Keep in mind that if these students fail, it isn’t their fault, it is yours. Most teachers quickly realize not to give bad grades. My community accepted D’s as passing so my lowest grade was a D. However, the administration greatly frowned on more than 10-15% D’s. Some teacher friends were forced to give C’s since D’s reflected badly on them. In fact, there were many high schools out there that would not even ALLOW their teachers to give F’s.
In addition, there is a stress and wear-and-tear issue on teachers. Giving poorer grades causes the community and administration (your bosses) to feel you are not a good teacher. Your life is much more negative and stressed. To stick to your guns involves much sacrifice. It is no wonder many decide to not make this sacrifice.
This is the reality of many schools and teachers.
Cont…
Ahhhhhh! That explains it. I have taught 4 years at the HS level and 6 years at College level. As I aluded to above, teaching at the College level is a completely different job that only looks similar to teaching HS. There are about as similar as an accountant is to airline pilot. (ok bad example but they are very, very different jobs)
Salute Moejuck! My hat is off to you. Now, go kiss your administration (boss). It is because of them you are still teaching. In many districts, you would have been fired after your first or second year. Serious.
New licenses and standards for teachers will not help this probem. It is not a problem with the teachers but a problem with the community that has a ‘kids should never fail’ mentality.
This is a major pet peeve of mine. I was a darned good math teacher if I do say so myself. I still have a drawer full of letters from former students saying so. One even arrived 6 years after I taught her (and she hated me at the time 
) thanking me.
I always hammered the ‘why’ over the ‘how’. I can’t see doing otherwise. I firmly believed this approach worked very well. If you emphasize ‘how to do this’ over ‘why to do this’, then Algebra degenerates into memorizing what to do when in confronted with this type of problem. This is not learning and will dissipate quickly after the class is over.
I think the reason why math teachers don’t do this is because many math teachers are not really firm with their math. They do not really know why things are done. In fact, many teachers teaching math are not math majors – either a minor or less or the dreaded ‘Math education major’ (shudder - but that is another topic).
cont…
I’m mixed on this. Of course, you’re right. Schools should teach students skills they will need in life. I am for classes like this separate from the Algebra/Trig/Calc sequence (don’t need those more watered down)
However…To use English education as an example…
When will most students ever read Shakespeare in their adult life? Maybe HS is the place where students are exposed to subjects/ideas/experiences that will be DIFFERENT than the rest of their existance?
I’m torn. I see the practical side but a part of me feels that students should be immersed in the different. 
Need to go. I will check back later
Here’s a cite regarding the square roots:
http://mathworld.wolfram.com/SquareRoot.html
(Parenthetical comment mine).
Wake up call, in your original post regarding this, you mention “Take square root of both sides”. This implies you are using the square root as a function, by defining square roots to be nonnegative numbers (as is the standard). The fact that we can define square root as a function this way is what allows us to “Take square root of both sides”. This is very commonly done, in fact. In your next step, however, you then abandon using the square root as a function, by stating that the square root of (4 - 9/2)[sup]2[/sup] is 4 - 9/2 (a negative number). This is inconsistent, and incorrect.
Good job!
The math on the SAT would have to be revamped, wouldn’t it? If algebra is optional, then not all college-bound students would take it and they would not be able to do well on the math section of the SAT.
If students are presented with logic and statistics questions on the exam, then students opting for more traditional HS courses would not do well.
What could be done to fix this kind of situation, besides making certain math courses mandatory (as they are now) or doing away with math on the SAT all together (not gonna happen)?
To run with your English education example, what’s happening in schools today is kind of like starting your junior high school students out on Shakespeare, without stopping by Judy Blume or S.E. Hinton. Sure, there may be some kids who learn an appreciation of English literature that way, but for most contemporary kids, it’ll leave them frustrated and without an understanding of how learning this stuff applies to them. Worse, if you spent their whole school careers teaching Chaucer and Shakespeare, how would they cope with the driver’s ed manual, written in plain but, to them, foreign English?
I agree, real life math skills should be taught in classes separate from Algebra, Calculus, and Geometry. But not teaching them seems downright cruel.
Cabbage, thank you for the cite. I apologize to the audience for what appears to be a highjack of the OP. However, being a diehard, I am going to give it one more shot before I concede. Let’s forget about taking square roots of both sides.
How about simple factorization like a[sup]2[/sup] - b[sup]2[/sup] = (a - b)(a + b)
So, (4 - 9/2)[sup]2[/sup] - (5 - 9/2)[sup]2[/sup] = 0
Factorizing the above, we get:
(4 - 9/2 - 5 + 9/2)(4 - 9/2 + 5 - 9/2) = 0
Therefore, one solution is that (4 - 9/2 - 5 + 9/2) = 0
Which leads to 2 x 2 = 5
Now, what rule / cite are you going to use to refute this one?
Once you answer this, then I will explain the reason for this highjack, and how it relates to the OP’s conclusion.
andymurph64
Seems like a fault of the textbooks rather than Algebra. Algrbra is very useful. The basic concept of algebra is to perform mathematical operations on a quantity while deferring the question of the value of that quantity, and then going backwards, undoing the operations to find what the original quantity was. This is an extremely valuable skill. I can’t see anyone doing any sort of “white collar” job without doing this in some form or another. Here’s an example of a simple algebra problem: you see a shirt on sale for $20. It’s marked as 20% off, and the sale ends today. If you come back tomorrow, what will its price be?
Monstro:
I don’t see the problem. Those that don’t take algebra won’t do well on the SAT. Lesson in responsibility: you can skip the classes, but there will be consequences.
Wake up Call:
I have the feeling that you already know that this isn’t valid, and are just teasing us, but I’ll bite.
Your proof can be rewritted as:
1*0=0
Therefore 1=0 and 0=0
4=4
Adding an equal number to both sides,
4=5
You proof is based on a fallacy called “affirming the consequent”. You see, we know if((4 - 9/2 - 5 + 9/2) = 0) then (((4 - 9/2 - 5 + 9/2)(4 - 9/2 + 5 - 9/2) = 0).
From this you conclude that if((4 - 9/2 - 5 + 9/2)(4 - 9/2 + 5 - 9/2) = 0) then ((4 - 9/2 - 5 + 9/2) = 0)).
But that’s not valid, because you’re switching around the if-then statement.
Wake Up Call, yeah, I agree with The Ryan. That’s similar to a common technique to solve an equation such as:
(x-2)(x+3) = 0
We know that if two real numbers have a product of zero, then one of them must be zero. So the solutions to the above are x=2, x=-3.
But if we don’t have any variables to solve for (i.e., if we’re just dealing with expressions of real numbers, no variables), there’s nothing to “solve” for. If we have
(4 - 9/2 - 5 + 9/2)(4 - 9/2 + 5 - 9/2) = 0
we know (4 - 9/2 - 5 + 9/2)=0 or (4 - 9/2 + 5 - 9/2) = 0. The latter is true, while the former is clearly false.
Bingo. Which brings us to the last sentence of the OP:
Have students take more Math. However, there is math out there besides Algebra/Trig/Calc that will better serve students not wanting to be there.
As we saw above, Cabbage had to resort to the rule of “using square root as a function”, and The Ryan had to resort to the rule of “fallacy of affirming the consequent” simply to show that 4 cannot equal 5. So, when we come to a kid that has to learn more Algebra/Trig/Calc, we are forcing her/him to learn a lot of “rules” and exceptions – for what purpose? I suppose the OP is saying “why are we doing this?”
There are indeed different kinds of math to teach. Some are down to earth math as MrVisible alluded. Others are based on Topology, Set Theory and Group Theory which use minimal axioms and “rules” to stimulate logical thinking.
So, my conclusion is: Yes to more math, but make it interesting and stimulating to the kids. As for Algebra/Trig/Calc, either make it less, or turn it into an elective for those that are interested in the language of sciences.
He’s going to use the fact that (4 - 9/2 + 5 - 9/2) = 0, and so anything multiplied by that is 0, and that any real number multiplied by 0 is 0. Furthermore, he’ll point out that the assertion that (4 - 9/2 - 5 + 9/2)(4 - 9/2 + 5 - 9/2) = 0 implies (4 - 9/2 - 5 + 9/2) = 0 is false, as the hypothesis is true and the conclusion is false.
I think you’re out of line here. You submit a “proof” that 4=5, I point out what’s wrong with it, and from this you come to some conclusion about the pointlessness of learning a lot of rules? That’s quite a leap; in fact, I don’t even see the connection. How else is someone supposed to point out a flaw in your proof? Can you give me a desired response to your proof? One that, in your opinion, wouldn’t have demonstrated whatever point it is that you are trying to make?
Now, this is hysterically funny.
The OP asks about better ways to teach mathematics to kids. The mathematicians wander off to talk about math, while the rest of the thread tries desperately to keep the subject about education, and ultimately, kids.
This is part of what turned me off to mathematics when I was a kid. It’s not that I wasn’t interested in math; I actually enjoyed the hell out of geometry. But it seemed, ultimately, pointless and needlessly obscure. I only found out later exactly how important it really was.
I don’t see teaching basic bookkeeping and financial management as a distraction from higher mathematics. Instead, consider the possibility that they may serve as a gateway drug to the purer forms of math. If a student’s interest is piqued by some practical self-interest, they may end up being better mathematicians altogether. And if not, we’ll end up with adults who understand that a credit card with 21% interest is a bad idea, no matter how high the credit limit or how nice the complimentary toaster.
Wake up call
What exceptions? If (x-a)(x-b)=0, then x=a or b. You were trying to argue that since x=a or b, then x=a. That’s simply not valid logic. There’s nothing special to algebra there. Even if a student doesn’t learn algebra, they should at the very least understand why “x=a” does not follow from “x=a or b”.
The fallacy of affirming the consequence is also a basic logical idea. Everyone that graduates from high school should understand what’s wrong with it, even if they don’t know the official name. These aren’t esoteric concepts; they are crucial to functioning in this world.
MrVisible
The combination of “nice” and “complimentary” gave me an interesting mental image. “Wow, you’re really good at picking out bread. I really like how you put those slices in me. You’re such a wonderful person, you deserve to have perferctly toasted bread”.
MrVisible,
You liked Geometry but didn’t like other math?
Allow me to ramble some more
Mathematics suffers from a large flaw. The flaw is that in most subjects what you initally see is what you get. So, if you like Literature, you will continue to like literature.
The problem with mathematics is that you don’t really ‘see what you get’. Therefore, College math departments attract people to get degrees who THINK they like math and don’t get people who think they don’t like math.
The problem is that high level math is not what many people think it is – including the Math majors themselves. When many Math majors initially run into higher level mathematics they are repulsed and chalk up the course as an aberation. In fact, many math majors really have no idea what most higher level math is.
The problem really hits when they go on to Grad school when they find the ‘aberation’ is actually the reality.
It is a crude example but ‘mathematics’ is much, much more like your Geometry class than it is your Algebra class.
As a fun exercise, a fellow graduate student in math did a survey. He asked people what they thought higher level of Math students did. They responded that they thought they did ‘very complicated Algebra problems’.
That is so completely different of reality
andymurph64, I find that fascinating. If we’re spending that much time teaching students the rudiments of higher mathematics, and even the ones who are most intrigued by it are completely unaware of its actual nature, don’t you think something’s wrong with the way it’s being taught?
I enjoyed Geometry immensely, after struggling hard to even stay awake in Algebra. I still do geometric constructions for fun, these 20 years later. But I thought of it as something fun to do, a class I could ace easily, and not as a useful skill. Certainly not as a career option.
Anyway, the level of abstraction in teaching math is what I have a problem with. It’s taught as if it were a philosophy, instead of it being an essential life skill, and a possible career door-opener. Sure, there will be some students who are interested enough in the subject for its own sake to pursue it, but unless you invoke a student’s self interest, the odds are in favor of them barely getting up the energy to turn in their homework.
It seems like a logical progression to me: teach practical math. From there, teach advanced practical math, and some higher level math with immediately obvious practical applications. Then, move on to the more advanced math, with an eye towards its usefulness in science and engineering applications. Then start pushing the hard stuff; if they can be, they’re already hooked.
The end result is that even the least successful students end up knowing enough to successfully navigate real life, and the most successful students still go on to be mathematicians and engineers, but with a solid foundation for real-life applications as well.
Of course, it’s just a theory.
So if we assume that algebra is required to do well on the SAT, then it would be prudent to make it mandatory. If it’s a requirement for the SAT, then it’s a requirement for going to college. Making it optional for college-bound students is asking for a lot of unnecessary pain and headache.
Perhaps the OP is talking about making it optional for non-college-bound students?
Mr. Visible,
‘Practical Math’ and ‘Advanced practical Math’ are not really Math You see, when most people say ‘practical math’ they mean balancing checkbooks, calculating sales prices, tax returns and so on in that vein. These use numbers and involve calculation but most math people do not consider it math. They consider it arithmetic at best. {To tell a math teacher that teaching arithmetic is teaching math would be like telling an author that teaching first graders how to write the alphabet is teaching writing }
It is also not a progression from practical math to Alegbra and beyond. ‘Practical’ math is an offshoot and a dead end one at that.
I have a hunch that business educators (and such) consider ‘practical’ math to be math and so do not try to teach it. This leaves this area an orphan.
I think there should be a class where many of these topics are combined under a ‘nuts and bolts of being an adult’ class. The problem is that HS curriculums are jammed full now. Adding another required class just adds to the problem.
You wouldn’t believe the hoops you need to jump through to get a course added to the curriculum. If you want it required as well…multiply it by 100. I’ve done this at the college level where I created a course to substitute for the required College Algebra. It was designed for people where the one required course was all the math they would take. It took many, many hours and years of time.