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Topic: Is "variable" confusing to students?

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Subject:   RE: Is 'variable' confusing to students?
Author: LFS
Date: May 12 2009
I know that I probably have a different point of view, but my college students
also have trouble with these things. So usually my first classes are dedicated
to this.

I tell them that a problem that says "simplify" is an expression and the answer
is a "snake" of equal signs, e.g.
Simplify 3x+2 - (4x+6)/3.
Answer:  3x+2 - (4x+6)/3=[3*(3x+2)-4x-6]/3=[9x+6-4x-6]/3=5x/3 .
Since it is a "snake" they cannot do anything to "both sides".

On the other hand, a problem that says "solve" is an equation and they need to
solve for the variable. Equations are sentences with 2 sides and the verb
"equals" and a "." at the end. We can do operations on both sides, BUT each time
we finish a sentence we must start a new sentence on a new line, e.g.
Solve: 3x+2=9x-1
Answer: 3x+2=9x-1  (subtract 3x and add 1)
           3=6x    (switch the equation around to get x on left)
          6x=3     (divide both sides by 6)
Now this is where I insist that they pay no attention to what they did in high
school and write another line.
I found if I insist they write another line here  - instead of what math
teachers frequently do: x=3/6=1/2 - they begin to understand the difference
between expressions and equations.
Of course, if possible I ALWAYS graph a SOLVE problem (using GeoGebra which is
the most excellent freeware) and then they can see the answer. Here this is not
possible, which is too bad. But they can substitute x=0.5 back into both sides
and check that both sides equal 3.5 (and then if they know and understand lines,
you can graph both sides as lines and they will intersect at (0.5,3.5)).

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