You know how you can show them this way, all justified and with lots of practice untilblueintheface:
But then a couple days later half of them will do this
and the other half will do this:
So, I stopped teaching it that way. I'm starting with something much like what most of us probably do:
Allison lives at 15 Sycamore Drive, and Sarah lives 8 houses away. Where does Sarah live?
But then, I'm sticking with that model for all kinds of problems.
Earlier in the lesson I made them write it out in words, i.e., "the distance from 200 to 3x is 896."
It was more of a pain initially, and not the most effervescent lesson I have ever delivered, but MAN, it did the trick. No more of that autopilot, forget to write two equations, forget that absolute value can't equal a negative nonsense.
This is an idea I stole wholesale from the article "A Conceptual Approach to Absolute Value Equations and Inequalities" by Mark W. Ellis and Janet L. Bryson, Mathematics Teacher April 2011, Volume 104, Issue 8, Page 592.
Inequalities are a natural extension of this concept. Where on the number line are all the values that are more than 896 away? That are less than 896 away?