Dear Grade 9s: What algebraic equations are for

A friend of mine was making an algebra test for his junior high math class recently and asked me what I thought equations are for.  What he meant is that the students are required to write out an equation for a given word problem before solving the equation, but students are bound to ask what the point of writing an equation is if they can solve it without it.

For assessment purposes, of course teachers need to see students write out an equation to know that they can.  But for students, the problems are probably easy enough that they don't really need to write out an equation to solve the problem, and so they see no point in doing the extra algebraic work!  Of course, they have also been trained over the years to recognize nice, round numbers as "The Answer" to all types of questions, so it's hard for them to understand how an equation can be part of "The Answer."

So here's a simple reason that I might give to a hypothetical grade 9 class: the point of writing an equation is to communicate an argument, and to tell the reader what the writer thinks is in fact true reality.

As an example, consider . That tells the reader that the writer believes speed is distance travelled over time, and nothing more — it's not just a way to calculate, it tells us something about the true reality.

That's why is an important and celebrated equation in physics.  Einstein is telling us what he believes is the true reality: that matter is energy, and energy is matter; and in fact a certain amount of matter is precisely made up entirely of a certain amount of energy [1].

Of course, that's just a belief, but he had to communicate it with an equation — to be precise about what he's claiming about reality — or else other scientists couldn't perform experiments to prove or disprove it.  They did prove it experimentally, so now we know more about the true physical nature of reality, and it is described by a small equation.

So a student who provides just a numeric answer shows me he understands the situation described by the question well enough to do some easy calculations.  But a student who also provides a formula shows me she understands the situation well enough to describe and explain the true reality of the situation, or at least as it is claimed in the question, before doing the easy calculations.

[1] Mind you that's not exactly what Einstein is claiming, but hey, imagine this is said to a grade 9 math class, so it's close enough I say.