Role #6: They facilitate applications. See 21 Feb blog.
If the model for a “word” problem is the equation it should be recognized as a linear equation in one variable (because it matches the definition of a linear equation in one variable) then any one of the general properties known about linear equation is one variable is applicable (A nice illustration of deductive reasoning).
Me: What kind of creature is ?
Class: Linear equation in one variable.
Me: What do you know about a linear equation in one variable?
I would expect conclusions to vary to match other objects when they occurred. I like to believe that as soon as they recognized the model to be a linear equation in one variable, they felt that, except for a few boring computation details, the problem is solved, by using deductive reasoning.
Unfortunately, many students do not know the definition of linear equation in one variable and therefore are deprived of that level of confidence.
An example from my consulting experience. A client asked if I could solve (for x of course) a proprietary formula which they used with one of their products. I noticed that the formula was an exponential equation with x in the exponent. Because I knew the definition of the type of equation, I knew I could easily solve the equation. I accepted their challenge as a paid activity and profited handsomely.
Clearly if you don’t know what kind of mathematics creature you are working with, it is impossible to use deductive reasoning to apply everything you know about such a creature.
Knowing the definition of graph of an equation makes it possible to extract information from any such graph. Constructing such a graph is easier if the definition of graph is known.