So here’s the question: should we teach students that the slope-intercept form of the equation of a line is y = mx + b ? Or should we teach it as y = a + bx ? And does it matter?
I contend that it does matter, and that there are several good reasons for using the second form. Furthermore, I contend that the arguments for using the first form are not particularly convincing.
Let’s look first at the arguments for y = mx + b. Why m for slope? More importantly, why does the growth term, mx, come before the initial value, b ? Defenders of y = mx + b will say that m is traditional, and that the x term comes before the constant because that is standard form for polynomials: list the terms in descending order by the power of the variable. Surely a cubic is ax3 + bx2 + cx +d, so why shouldn’t a linear form have the first-power term, mx, before the constant term, b ?
I’m not thrilled about using m just because it’s traditional, since nobody knows why m was chosen. There are plenty of theories, but the only remotely plausible one is that it stands for the French word monter, and even that hypothesis doesn’t have much evidence to support it. Even if there is a good reason for m, then surely there’s no good reason for b to stand for y-intercept. If you’re going to call the second parameter b just because it’s second, then at least call the first parameter a because it’s first, instead of using a mysterious m of unknown origin.
Having disposed of the names of the parameters, let’s turn to the more important question: the order of the terms. The argument by analogy with polynomials is a fine one for those who are enamored with the patterns found in theoretical mathematics. I’m often in that camp, but as a teacher I am more swayed by the uses to which we put our math. Students learn about linear functions (and exponential functions, as we’ll see) before they learn about cubics. Algebraic functions ought to mean something, at least some of the time. In particular, linear functions should demonstrate constant additive growth: start with an initial value, then keep adding the same quantity. The first number, our initial value, is the y-intercept, since it’s the value when x is zero. The second number must be the slope, since it is the amount by which y increases each time whenever x rises by one. Now we’re completely making sense of the nature of a linear function.
As an extra bonus, note the parallel to the exponential function: start with an initial value, then keep multiplying by the same quantity each time. Everybody agrees that the simplest form of the exponential function is y = a(bx). If we use the y = mx + b form of the linear function, we now have perfect parallelism: each starts with an initial value and then repeatedly applies the same operation. Left to right for exponentials we have initial value followed by repeatedly multiplying by b x times, and repeated multiplication is exponentiation. So left to right for linears we have initial value followed by repeatedly adding b x times, and repeated addition is multiplication. Voilà!