Should Slope-Intercept be King? A Case for Standard Form’s Ascension

  • Should Slope-Intercept be King? A Case for Standard Form’s Ascension

    By Levi Patrick, posted July 31, 2017 —

    There 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.

    Graphing 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 greater extent.

    2017_07_31_Patrick_Table1 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 and

    https://vimeo.com/217506476.)

    Graphing 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

    https://vimeo.com/217509353.)

    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.

    Standard 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 sense.

    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

    https://vimeo.com/217519757.)

    If 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?


    2017_07_31_Patrick_AuPicLevi 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.

     

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    Paul Hansen - 11/6/2018 3:11:50 PM

    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.


    Dylan Boone - 11/13/2017 2:27:22 PM

    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.