A first hurdle is to help students actually understand the number line graphs that they draw of solutions to inequalities. Many use middle school mnemonics about arrows pointing in the same direction as the inequality sign, and draw their graphs from these rules (and go wrong when the variable appears on the right hand sign of the inequality, of course). We've worked on listing a few actual numbers that are part of the solution, plotting these, and then drawing the graph afterward. That helps a good deal after a while.
The next hurdle is to understand the difference between AND and OR inequalities. The most effective approach so far has been a combination of the above mentioned insistence on a list of actual, specific numbers that satisfy the conditions, and lots of problem quartets like the following:
x > 2 and x < 5This way, we always get one inequality with no solution, one satisfied by all real numbers, and a couple plain vanilla and- and or-inequalities for the same pair of numbers. It's not magic, but it does seem to help to vary one thing at a time.
x > 2 or x < 5
x > 5 and x < 2
x > 5 or x < 2
Then enter the absolute value inequalities, and what a mess they are. There are so many different ways of solving them, and of talking about them, and I've made the mistake of covering several instead of sticking to one geometric approach and one algebraic approach. Now students are, quite predictably, using messy combinations of these.
Geometrically, the absolute value of x - 2 can be understood as the distance of x - 2 from zero, or the distance of x from 2. Last semester, with the Intermediate Algebra class, I relied on the former and had the students set up inequalities such as | x - 2 | > 3 by drawing a number line, placing "x - 2" more than three units away from zero on either side of zero, reading the resulting inequalities ( "x - 2 > 3" and "x - 2 < -3" ) from their sketch, and solving from there. It didn't really stick. I am not sure whether that was due to inadequate repetition or due to this approach being conceptually confusing.
Anyhow, with the Algebra 1 group this semester, I instead belabored the geometric interpretation of | x - 2 | as the distance between x and 2. I taped a large number line under the blackboard and we checked this definition by walking back and forth: -1 is three steps from 2, and sure enough, | -1 - 2 | = 3, and so forth. In order to solve inequalities such as | x - 2 | > 3 I had two students walk three steps from 2 on the number line in either direction, and we talked about what numbers were more than 3 units away from 2. This was difficult for many students (and not only the small group that always tunes out when I use any concrete representations because they think that's too middle school). My hunch is that there's some relation between this confusion and the difficulties Mr. K's students had with the meaning of "more."* Once students did pick up the idea it seemed to stick, but many never really got it. Maybe it's harder to ask when you're confused about what the walking up and down the number line is supposed to mean than when the material is more evidently academic.
I had first hoped to rely on this geometric approach to help students remember the direction of the inequality sign for the two linear inequalities in terms of which they will rewrite their absolute value inequalities, but gave up on that and introduced the approach that follows naturally from the algebraic definition of absolute value. If | x - 2 | > 3 then either x - 2 is greater than 3 or else the opposite of x - 2 is greater than 3.
Now I wish I'd used the geometric approach only for predicting and interpreting answers and stuck religiously to the algebraic approach to setting up the inequalities - because now students are using strange combinations of the two, such as x + 2 > -3 or -x + 2 > -3. In other words, complete confusion, with neither a clear concept nor a clear method to rely on. That's pretty discouraging even before thinking about the many students who did not even acknowledge the fact that there are two solutions to absolute value problems, that a distance can be in either of two directions... So, math teachers, what do I do now?
*We tend to assume too much about students' immediate grasp of the very idea of comparing quantities, let alone the isomorphism from this ranking of magnitudes to a spatial ordering along a line. Bob Moses, with his interest in pre-mathematical concepts that must be in place in order to succeed at Algebra, would presumably have a lot to say about this.
