variable: A symbol used to represent one or more numbers.
High school students are justifiably confused by the two distinct concepts:
- a variable as something that “varies” in an expression, such as the h in the expression $4.50\cdot h$; and
- an unknown quantity that is a "specific unknown” in an equation
The definition of a variable as being a symbol used to represent both of these cases explicitly states this as: “a symbol used to represent one or more numbers.” Where the “one number” case is the “specific unknown” in a simple equation, such as $18.00=4.50\cdot h\;$ where the $h$ can only represent the one number $4$ to turn the open sentence into a true statement.
The more than one number case being the letter h standing for values in a table such as: $1, 2, 3,\>$ or $4$ being substituted into an equation to form a pattern such as $C=4.50\cdot h$, the definition of a variable now being interpreted as a symbol, h, used to represent one number when that number is substituted in for it and a symbol used to represent more than one number when other numbers are substituted in for it. Thus generating a table of values of $C$ such as: $4.50, 9.00, 13.50,\>$ and $18.00$.
Another example, the variable, say x, in the quadratic equation represents a parabola when it "varies" over a given domain. But the variable in the equation $0=x^2+2x+1$ is "an unknown quantity. It does not vary." Thus, in this case the vari-able has lost its ability to "vary." Yet in both situations they are referred to as a variable, and this duality is embodied in the definition of a variable as "a symbol used to represent one or more numbers." Could the motivation behind this definition be such that we don't have to make the distinction between an "unknown specific quantity" and a "varying" quantity?
Does anyone agree that the above argument has been logically developed or is there some flaw in my reasoning? Thank you.