How hard is graphing y > 2x + 1?

Coming up this week (for my Integrated Algebra students) is that algebraic mother lode: graphing linear inequalities (A.G.6: Graph linear inequalities, for the standards-aligned). This has been saved for the year's end since it is such a killer. The problem is, from the state's point of view, it should be easy for a ninth grade student. It should be, based on the state math performance indicators.

What do you have to do if you want to graph y > 2x + 1? (If you are a college-educated adult and have no idea what I'm talking about, this raises some other juicy questions about math education/retention/cultural significance.) Well, for starters, you have to recognize that you are going to need to graph the line y = 2x + 1. This sort of graphing is taught in seventh grade (7.A.7: Draw the graphic representation of a pattern from an equation or from a table of data) and again in eighth grade (8.G.17: Graph a line from an equation in slope-intercept form (y=mx+b)). This will no doubt call on you to plot points in the coordinate plane, something begun in fifth grade (5.G.12: Identify and plot points in the first quadrant).

You've also got to recognize also that since this is a strict inequality (>) you need to graph using a dashed line. (Why? Because the line does not include points in the solution set of the inequality.)

Finally, choose a point that is not on the line, plug in the numbers in the inequality (5.A.3: Substitute assigned values into variable expressions and evaluate using order of operations) and then evaluate the truth of the resulting inequality (4.A.3: Find the value or values that will make an open sentence true, if it contains <>) . If it is true, shade in that side of the line; if false, shade the other side.

And that's it. We'll let alone the case where y is not isolated (a further algebraic challenge) or the case of the system of linear inequalities (yup, two or more at once).

But for most students I teach, culling all of these skills at once is a lot to ask. If a student is weak in just one of these skills, and many are weak in several, the whole operation fails. Graphing a linear inequality becomes a labor that can last half a period, and most often contains at least one error. It can cause such frustration even though it is really just a combination of things my students already, supposedly, know.

So what's the problem? What happened to these students in fourth, fifth, seventh, and eighth grade? Were their teachers checked out? Did they leave these standards out of their instructional plans? Were their lessons poorly planned? It's possible, but I doubt this is a leading reason.

I believe that there were days in 2006 when my students were expert graphers. I think there was a day or two in 2004 when students were evaluating expressions with the best of them. Their teachers led them along through all the proper steps and understandings, and the students could do it. But this is not good enough today, in ninth grade algebra.

Based on my limited experience, I'm tempted to say the problem is the kind of understanding my students come to high school bearing. While students may have, at one time, under the right circumstances, with the proper scaffolding, etc. etc. known how to graph, plot points, evaluate expressions, and determine the truth of inequalities, they never truly understood the meaning of these skills. If my students are reaching into a vat of collected, memorized skills, and pulling them out, one at a time to graph y > 2x + 1, then of course it's hard. They could pull the wrong one. They could pull out a skill that is dusty, rusty and incompletely memorized. They are constantly in fear of "doing the wrong thing" or doing it "the wrong way".

If, on the other hand, they see a coordinate plane and recognize it as a sea of points, each of which registers true or false in the inequality; if plotting points is "second nature"; if the coordinate plane is readily divided by linear equations with differing slopes; in short, if students have a thorough understanding of what these various elements represent, then graphing inequalities will happen quickly and easily, conforming to a body of knowledge already obtained. The prerequisites need to be understood deeply and richly.

How do we instill this in children? I don't know. I'd like to know. I'm working on it.

Have students learned if they forget later?