I taught Absolute Value equations and inequalities for the first time in depth this year. There were the usual student struggles, but I kept being reminded of a conversation with Sam Shah that I had a year or more ago about this topic…He was frustrated that his students were more focused on memorizing “the procedure” rather than understanding how/why we solve absolute value problems the way we do. Now, I find myself with the same problem. Here’s my current idea for next year:
Day 1: Teach only problems of the form a|x| + b = c, where a,b,c are in R.
Day 2: Repeat/review this, or move onto inequalities of the form a|x|+b>c (or <c).
I want to make three big points in having only the variable inside the absolute value:
- It is important to isolate the absolute value.
- There is no “inverse” for an absolute value.
- When you get down to |x| = d, you HAVE TO THINK!
My hope is that when students get to the point of |x| = d or |x| > d, they stop to think what the possible answers are. There is no way to magically “cancel out” the absolute value. However, once we isolate that side of the equation, we can reason through the answer. I have been stressing in my class this year the idea that there are moments when you have to “put your hands in your lap and think” for a second. This would be one of those moments.
Only after students are comfortable with this, would I extend to problems of the form a|px+q|+b=c, etc. My hope is that they will get good at isolating the absolute value, and then THINKING about what makes sense next.
We will see how it goes next year…