Should Slope-Intercept be King? A Case for Standard Form’s Ascension
Levi Patrick, posted July 31, 2017 —
must be a better way. If
you’ve ever taught prealgebra or algebra 1, you know what I know: There is
nothing about teaching the slope-intercept form of a line that is as easy as it
should be. I have wondered why the current learning progression prioritizes the
slope-intercept form (y = mx
+ b). It seems to me that we jump too quickly into the slope-intercept
form in the hopes that we can leverage (or establish) a sense of what we meant
in seventh grade by unit rate.
We often prioritize the simplicity of the y-intercept and rate-of-change
combination because we believe it helps
students more easily relate symbolic, graphic, and tabular representations.
Shouldn’t our students be able to immediately connect direct variation to lines
that are shifted vertically by b? My experience indicates that this is
not as seamless as we might hope. Students make the leap to slope intercepts
before having a sense of how lines actually work, which leads to instructional
time spent plugging and chugging to find the slope and y-intercept
rather than solidifying understandings around covariation and rate of change.
To kick off my series on disagreeing, I would like to
make the case that standard form deserves a great deal more of our attention,
if not real estate, in the middle school curriculum.
number bonds. Although
many of my students
were bewildered by slope and y-intercepts, they seemed to easily grasp
things like number bonds by
this point quite sufficiently, and I’m certain that can be leveraged to a
Start by taking any number (let’s say, 12) and consider
all the pairs of numbers that add to 12. A few of those are shown in the table.
I start with 0 and then list multiples of 3.
Think about how these pairs of numbers have to add to
12; we can write a very straightforward equation: x + y = 12. The graph of this
equation includes not only all those points we just mentioned but also infinitely
many more if you consider all rational numbers. (You can see it in action at
the following two sites: https://www.desmos.com/calculator/tcuaxpsiwh
perimeter. Once we have a sense of what number
bonds look like graphed, I think it is interesting to start thinking about what
other similar graphs look like. Let’s leverage a fourth-grade idea: perimeter. The equation for an x
by y rectangle is 2x + 2y
= k, where k is
the perimeter. Check out the number bonds graphed along with the perimeter
graph here. (Look at the connections and similarities at the following two
sites: https://www.desmos.com/calculator/aq0fredv1t and
What I start wondering is whether there is much more
that we should be doing to address the idea of covariation before we move to
slope and intercept. Aren’t we seeing some interesting relationships in the
numbers here? I would argue that they are much more apparent than those in a
traditional prealgebra experience.
form, intercepts, and factor families. The final
convincing piece for me is the idea that we can come to appreciate how the
factors for x and y are
related to the constant, k. If you think about how we look at factors
and roots later on, introducing standard form earlier could make a lot of
For the equation 3x
+ 4y = 24, we can imagine x = 0 and see that y must equal 6 for the equation to be
true. Thus, we have the y-intercept (0, 6). Likewise, we can find the x-intercept
of (8, 0). We can even begin to appreciate the idea that if the coefficient of x
(currently 3) was instead 12, we can imagine the x-intercept shrinking
down from 8 to 2 since 2 • 12 = 24. (See this https://www.desmos.com/calculator/i963mn0x7a and
not king, is there room? Ultimately, I argue that there’s an
important component of covariational thinking that has been glossed over and is
too often addressed as an algebraic manipulation exercise in prealgebra and algebra
1 rather than a meaningful experience based in a graphic representation. I
believe a renewed focus on standard form will allow us to highlight key
mathematical relationships that help students advance their thinking about the
structure of equations and functions.
I’d love to hear your thoughts on this. Have I
convinced you to disagree with the current prominence of the slope-intercept form? Is there room
for spending more time on the standard
form in the prealgebra (or earlier) curriculum? Should there be? What do
you like about this idea, and what can or should be improved?
Levi Patrick serves Oklahoma as the director of computer
science and secondary mathematics education. He is the vice president of
program for the Association of State Supervisors of Mathematics and serves NCTM
as chair of the Professional Development Services Committee. Patrick taught eighth
grade, algebra 1, and geometry in Oklahoma City and in the Putnam City Public
Schools, developed curriculum and mentoring programs as a mathematics specialist
at the K20 Center for Education and Community Renewal at the University of
Oklahoma, and has been involved in the development of the #OKMath community
(http://OKMathTeachers.com) and the Oklahoma Mathematics Alliance for the past
few years. He and his wife, Roslyn, also an educator, live in Oklahoma City
with their Jack Russell “Terror,” Piper.
Thanks to Levi Patrick for introducing this issue. I agree that the standard or general form of linear equations should be the starting point. Other forms have more specialized uses, and they either separate the variables, with parametric or Cartesian forms, or unbalance the variables, in the case of slope-intercept. That is, Y is given as a function of X; in that sense the variables are treated differently.
Standard form Ax + By + C = 0 is perfectly balanced; more importantly, it is simple and extremely useful. Starting with 2 points p0 and p1, the A coefficient is simply p1y - p0y, and B is p0x - p1x. They are differentials. And C is the cross product p1x * p0y - p0x * p1y, which leads students directly to the 2x2 matrix determinant with those 4 numbers. No need to go through the slope/intercept form, as almost everyone does.
We know that if you plug in values for X and Y and the standard form evaluates to zero, then the point is on the line. However, that has limited usefulness since most points are not on the line. What we don't teach (and thus few people know) is that if the point is not on the line, then the non-zero result is negative when the point is on the left side of the line, and positive on the right side! (when viewed from p0 to p1) This becomes very useful when marching around a polygon and trying to determine if the point is inside or outside the polygon.
Furthermore, not only is the result negative or positive, but its magnitude, when normalized by dividing it by the quantity (A*A + B*B), is precisely the perpendicular distance from the point to line! This is very useful for putting a "halo" around a polygon, or determining if a pick-point is close enough to the line to register a pick.
That's not all -- we used a cross product to determine the C coefficient; how about the dot product? If you find the differentials dx = px - p0x and dy = py - p0y and then find dot = (dx * B + dy * A), then this product is 0 when point p is "lined up" with p0 (on either side), negative when "behind" p, between 0.0 and 1.0 when "between" p0 and p1, 1.0 when lined up with p1, and greater than 1.0 when "beyond" p1! This, combined with the perpendicular distance away from the line, means that the standard form is actually a linear "local" coordinate system determined by p0 and p1!
How beautiful is all of this? And how exciting for students? Not only that, but it is a direct lead-in to 3D figures. In 3D the line segment becomes an arbitrary triangle with 3 points p0, p1 and p2, and completely analogous things happen in 3D space! 3D is supposed to be advanced stuff with most students never see, but it's the most fun of all, and the most important, since we live in a 3D world.
I agree standard form does make it easier to see the x-intercept and y-intercepts for students at first. I like the idea of showing students standard form earlier as well because students can see the relationship betweent the y's coefficient and the x's coeffient to the rate of change in a table. It could also lead to more of a smooth transition to finding slope because students can more easily see the differences between x and y in a table and where they are represented in an equation that is in standard form. The leap to slope-intercept becomes much smaller because you can introduce slope through solving standard form into slope-intercept which shows students algebracally why slope is change in y / change in x. Then showing students the procedures for finding slope from two points becomes more meaningful for students.