Quandaries and Queries
 


Name: Arthur

Who is asking: Other
Level: Secondary

Question:
Note: This is sort of a Pre-Algebra or just an Algebra question so this inquiry should be directed accordingly. Thank you. :)

Description of Problems (mandatory reading):
When complicated expressions exist on both sides of a fraction, and both these expressions contain like and unlike terms with "literal" coefficients, the procedural precedence suddenly gets extremely confusing and seems impossible to simplify. This seems confusing and impossible to simplify because there is a HUGE contradiction in the way Procedural Precedence and the Distributive property deal with Parentheses. The Procedure says you MUST deal with what's INSIDE the parenthesis FIRST, BEFORE dealing with ANYTHING on the outside where as the Distributive property contradicts that by saying to go ahead and ignore the Precedence and use a factor on the outside of the Parenthesis.

Rhetoric and Commentary (optional reading):
Itís an obvious direct and blatant contradiction that seems like it must be hurting in more long term sociological ways than any of us might think of at first. How am I supposed to know which way I'm supposed to do what if things are getting explained this way in the books? <-- That's a semi-rhetorical unclear question expressing my uncontrollable frustration on the matter and doesn't necessarily require a semi-rhetorical answer unless it can break up some of the tension and frustration here. My apologies for the rhetoric and commentary.

"QUESTIONS" related to Problems (mandatory reading):
Could you please explain why there is such a huge contradiction in the way Procedural Precedence and the Distributive property deal with Parentheses? Or at least why the material is presented that way? Also how can I tell which method (Procedural or Distributive) I should be using when confronted with simplifying an expression containing Parenthesis? MOST IMPORTANTLY; How am I supposed "simplify" complicated expressions with like and unlike terms containing parenthesis on both sides of a fraction? In other words when you have a complicated expression in both the numerator and another one in the denominator.

Final Comments (optional reading):
As you can see, the text books have left me severely confused. My confusion plus a complicated fractional expression equal a total and complete inhalation of any kind of inspirations I used to have about learning math which in turn leave me extremely depressed because I then start to believe that I have no future to look forward to which in turn leads to the inevitable. Itís a wonder to me that any one ever makes any progress in school at all ever. Please HELP ME, I've been brainwashed by the school system and bad text books!!!! Also my utmost apologies for not being able to clearly express my questions and the problems I'm having with Pre-Algebra in a short and to the point way. I just cant see a way to explain it in short form that would help you understand how to pinpoint my confusion on the matter. After all if you don't understand what I'm asking, then how could I possibly understand your answer? <-- Again, really sorry for the rhetoric. I've got to do something! about that.

Sincerely and completely confusterated n' complexified, -Arthur.

P.S. To answer your question, I'm out of college but trying to relearn Algebra on my own from books so whether or not I can be considered a student is up to your definition of student. Other wise just mark me down as "Other". ;)



Hi Arthur,

To simplify a fraction, it is best to (try to) factor out the numerator and the denominator. But in order to do so, it is sometimes necessary to expand the summands in each expression as in the case of

[(9-x) + (x-3)(x+3)]/[(x+3)2 - 9] which can be simplified by expanding the numerator and the denominator: (9+x) + (x-3)(x+3) = 9 - x + x2 - 9 = x2 - x,

(x+3)2 - 9 = x2 + 6x + 9 - 9 = x2 + 6x,
and then factoring them: x2 - x = x(x-1), x2 + 6x = x(x+6), to get [(9-x) + (x-3)(x+3)]/[(x+3)2 - 9] = [x(x-1)]/[x(x+6)] = (x-1)/(x+6) I think that this is the essence of your ``mandatory reading''.

At this point I note that the identity

[(9-x) + (x-3)(x+3)]/[(x+3)2 - 9] = (x-1)/(x+6) itself becomes quite intricate when written out as a text:

"The quotient whose numerator is the sum of two terms, namely nine diminished by an unknown fixed quantity and the product of this unknown quantity diminished by three and the same unknown quantity augmented by three, and whose denominator is nine subtracted from the square of this unknown quantity augmented by three, is equal to the ratio of the unknown quantity diminished by one and the unknown quantity augmented by six.''

It is hard enough to understand the sentence, let alone to do the algebra in a similar text. This shows how powerful a tool the modern algebraic notation is. However, like most tools, it also has its drawbacks, and our schools often produce students who can compute 20 - 1.5 correctly, yet cannot figure out the amount of gas to charge to a customer who wants to buy a $1.50 pizza slice and spend the remainder of his $20.00 bill in gas. Hence there is now a tendency to do more `"word problems'' in math classes, and bring back the textual meaning in algebraic computations. Sometimes the results are good, and sometimes they are bad, but I think that overall it is a good idea to use mathematics as well to help students to appreciate the value of language.

Claude


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