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Topic: Simplifying Algebraic Expressions with Subtracted Expressions
Replies: 67   Last Post: Nov 25, 2013 12:57 PM

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 Louis Talman Posts: 5,090 Registered: 12/27/05
Re: Simplifying Algebraic Expressions with Subtracted Expressions
Posted: Nov 14, 2013 6:47 PM

On Thu, 14 Nov 2013 14:56:12 -0700, Joe Niederberger
<niederberger@comcast.net> wrote:

> I'm not arguing that your interpretations are not just fine -- I just
> wonder if that's what she meant. She seems to be saying there is no
> manipulation of "the unknown quantity" (x) in an algebraic fashion,
> either mentally or on paper. I'm skeptical.

I don't think "manipulation of 'the unknown quantity'" is what makes
algebra.

In Ma's solution, there's no *formal* manipulation of a symbol that used
consciously to represent an unknown number. It's the conscious formal,
symbolic representation of an unknown, together with its subsequent
manipulation, that turns the solution into an algebraic one.

Manipulating one or more unknowns, in and of itself, isn't algebra.
Representing the unknowns symbolically, and then manipulating those
symbols without having to think about what they represent is algebra.

Consider the problem: "A sundae with a cherry costs six dollars. The
sundae alone costs five dollars more than the cherry. What does the
cherry cost?"

I can say: The cost of the sundae alone is five dollars plus the cost of
the cherry, so the six dollar cost of the sundae with a cherry is five
dollars plus twice the cost of the cherry. Hence the cherry costs fifty
cents.

That wasn't algebra. We might call it pre-algebra, or even
proto-algebra. But it ain't algebra, because there's no conscious use of
symbols for unknowns, nor formal manipulation of such symbols. In fact,
I'd guess it was the need to simplify tortuous reasoning like that above
that ultimately led to algebra's invention.

Or I can say: Let x be the cost of the cherry and let y be the cost of
the sundae. Then y - x = 5, while y + x = 6. Now, without thinking about
the sundae or the cherry, I go on to say, so (y - x) + (y + x) = 11, or 2
y = 11. That means that y = 11/2, whence x = 1/2. So the cherry costs
half a dollar.

That was algebra.

I think that the non-algebraic solution shows deeper insight. Algebra is
for problems where insight isn't likely to be deep enough.

(I can also do what most beginning algebra students do: I can say "There
are two numbers in the problem, five and six. It's obvious that I want to
subtract the smaller from the larger. The cherry costs a dollar." This
solution points to a reading problem--not a mathematics problem.)

- --Lou Talman
Department of Mathematical * Computer Sciences
Metropolitan State University of Denver
<http://rowdy.msudenver.edu/~talmanl>

Date Subject Author
11/11/13 MVTutor
11/11/13 Robert Hansen
11/12/13 Bishop, Wayne
11/12/13 MVTutor
11/13/13 Jonathan Crabtree
11/13/13 Bishop, Wayne
11/14/13 Joe Niederberger
11/14/13 Robert Hansen
11/14/13 Joe Niederberger
11/14/13 Robert Hansen
11/14/13 Joe Niederberger
11/14/13 Robert Hansen
11/14/13 Louis Talman
11/14/13 Joe Niederberger
11/14/13 Robert Hansen
11/14/13 Robert Hansen
11/16/13 Bishop, Wayne
11/16/13 Robert Hansen
11/14/13 Joe Niederberger
11/14/13 Robert Hansen
11/16/13 Bishop, Wayne
11/14/13 Pam
11/15/13 Robert Hansen
11/15/13 Joe Niederberger
11/15/13 Robert Hansen
11/15/13 Joe Niederberger
11/15/13 Joe Niederberger
11/15/13 Joe Niederberger
11/15/13 Robert Hansen
11/16/13 Bishop, Wayne
11/16/13 Robert Hansen
11/17/13 Bishop, Wayne
11/17/13 Robert Hansen
11/17/13 Bishop, Wayne
11/15/13 Joe Niederberger
11/15/13 Robert Hansen
11/15/13 Pam
11/15/13 Robert Hansen
11/15/13 Pam
11/16/13 Robert Hansen
11/16/13 Robert Hansen
11/16/13 Joe Niederberger
11/16/13 Robert Hansen
11/18/13 Louis Talman
11/21/13 Robert Hansen
11/21/13 Louis Talman
11/16/13 Pam
11/16/13 Robert Hansen
11/16/13 Pam
11/16/13 Robert Hansen
11/18/13 GS Chandy
11/17/13 GS Chandy
11/17/13 Pam
11/17/13 Robert Hansen
11/17/13 Pam
11/17/13 Robert Hansen
11/17/13 Pam
11/18/13 Robert Hansen
11/18/13 Robert Hansen
11/18/13 Pam
11/18/13 Robert Hansen
11/25/13 Bishop, Wayne
11/25/13 Robert Hansen
11/22/13 Joe Niederberger
11/25/13 Bishop, Wayne
11/23/13 GS Chandy
11/24/13 Robert Hansen
11/25/13 Bishop, Wayne