You should really read Kate's post because it's right on. I found myself laughing out loud as she explained the common mistakes students make regarding these equations and her snazzy way of trying out a different method (different to me, at least).
The idea is to emphasize that absolute value measures distance. I love this for two main reasons.
- It places value (no pun intended) on the topological property of absolute value as the Euclidean metric on the set of real numbers. If we see absolute value as just "making whatever's inside positive," we really lose the meaning of what absolute value can be used for in the topological sense. "Making everything positive" is a shallow definition (and I'm sad to say I'm guilty of using it).
- It eliminates the need for three cases. You know what I mean. "If it's equal to, set the inside equal to the positive and the negative; if it's less than, the inside goes between the negative and the positive; and if it's greatOR than, the inside is greater than the positive OR less than the negative." Oy. No matter how many times I'd justify those cases, and no matter how loudly I'd say "ORRR!" for the last case (shameful, I know), I would still get nonsense come test time.
To pound in the idea of what I've affectionately termed "Draw and Solve," we played a game the last twenty or so minutes of class. It was a version of Around the World, which was seriously one of my favorite games as a schoolgirl (and I don't just mean inside the classroom). Here's roughly how it went: I would pull up a slide with a problem. The first two students just had to draw the situation. Whoever drew it correctly first became the defender and stayed at the board and the next student in line would become the challenger. Now these two students had to solve the actual equation or inequality using the drawing the student before them had drawn. The winner of this round would stay, a new student would come up and challenge, and we'd start again with a fresh slide. So, the first slide looked something like this...
Thankfully, very few of my students are the "too cool for school" type, so they totally played along and seemed to enjoy themselves. I'm sure giving extra test credit to the winners was helpful also.
The real metric (see what I did there?) will be how they do on their upcoming test. I'm excited to see how it goes.
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