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.

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?

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.