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…

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I absolutely love point #2: there is no inverse function for absolute value. That seems to me to be the most crucial thing — and if I were to teach it again, I would highlight that.

I actually got pretty good at teaching them (with understanding). Here’s an awesome worksheet I made to get them thinking about the ideas but in a concrete way:

http://samjshah.com/2010/09/22/two-worksheets/

Also, some great T/F questions which revealed some amazing misconceptions even my strongest students had.

Sam

whoops, the T/F questions are: http://samjshah.com/2011/09/27/absolute-value/

I have told them that the absolute value bars are like a jail and the ‘a’ and ‘b’ are like guards. We have to get the guards away from the jail before we can break the x out of the jail. Kids seem to like the story.