Algebraic Expressions vs Algebraic Equations

Give the algebraic expression that represents the perimeter of the following polygon. 
001 Perimeter Diagram

Over the last five years working as a tutor I have seen many students struggle with a problem like the one above. To answer it we must remember that the perimeter of any shape can be found by adding up its side-lengths. With side-lengths that are numbers this is straightforward, but with side-lengths that have variables and numbers it can be tricky—you need to know how to identify like-terms and how to sum them correctly.

001 Perimeter Expression

Therefore, the perimeter of the polygon is 13x-34 units.

But getting an answer such as 13x-34 is not necessarily what students have in mind as an answer. Often, they expect and, perhaps, want a number. There are likely many ways of generating an incorrect numerical solution to the problem, but one common way is to try to solve for x. For example, suppose a student sets the sum of the sides equal to 6, i.e. the number of sides of the polygon.

001-student-err-v2

The big mistake here is that students invent and solve an algebraic equation rather than find, as the question asks, an algebraic expression. Equations and expressions both contain variables and/or numbers, but you can only solve for x in the former and not the latter.*† A good way to tell the difference is that algebraic equations always have an equal sign and an algebraic expression on at least one side of the equal sign.

001 Alg Eq Alg Expr

Algebraic expressions, however,  whether they’re alone or in equations,  can be simplified. That is essentially the core exercise of the question above: each of the shape’s side-lengths is an algebraic expression that when added with all the other side-lengths gives a simplified algebraic expression representing the perimeter.

Algebraic expressions can also be evaluated, as shown above when the hypothetical student substituted his solution for x back into 13x-34.  Here’s another example:

001 Eval Alg Expr

Tomato paste. Tomato sauce. Expression. Equation. Whether you’re cooking pasta or writing a math test, the meanings of words matter.
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*  I recently read in a textbook an algebraic expression being defined as a monomial or the sum of monomials. This seems a fitting definition for polynomials, but too restrictive for algebraic expressions. Based on my understanding that polynomials are composed of terms that can’t have a variable as a denominator, the said textbook definition would exclude rational expressions as algebraic expressions, which seems strange to me. 1/x may not be a monomial, but is, in my opinion,  a valid algebraic expression.
† The “and/or” business in this sentence is just to make sure that numbers by themselves are not to be forgotten as algebraic expressions, as weird as that is. For example, “8” written alone is technically an algebraic expression. The logic here is that numbers alone are considered to be monomials (e.g. 8x^0 = 8) and monomials are considered to be algebraic expressions. This brings up a problem in my thinking that algebraic equations must contain at least one algebraic expression and that all algebraic equations are solvable (or at least be shown to have no solution). Thus, according to my post, “8=8” would be an algebraic equation, but clearly nothing can be solved for here. All this is very confusing and life would probably be simpler if numbers were not regarded as monomials and so algebraic expressions.
‡ If you have read the previous footnotes, then it’s pretty clear that the didacticism of the concluding zinger is quite a bit diluted by the fact that words can have multiple, contradictory meanings and that one or more those meanings can be ambiguous. In the end, perhaps, it’s all about what you need to get done. If a particular word’s meaning has just a low-enough level of ambiguity to get you justifiably out of a jam, then it’s probably good enough.

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