### Absolute Value – Making it Stick

I have seen two methods for teaching absolute value.  One embraces the notion of opposite:

Given |x-3| = 4 then the opposite is true, so |-x+3| = 4.  The problem with that method is students don’t remember what becomes negative and will write |x+3| = 4.

The other method involves distance from zero.  So if |x-3| = 4, then x-3 = 4 and x-3 = -4.  The main problem with that method is if the absolute value expression is not alone, such as 2+|x-3| = 4, students will write 2+x- 3 = 4 and  2+x- 3 = -4.   Also, since distributing a positive number into an absolute value doesn’t corrupt it, the students are completely confused when to have +/- a number.  While the complications of getting this other method straight exceed that of the first, with a little help from anchor theory (not tricks), I have had a good degree of success at the high school level teaching absolute value in two lessons.  I argue this method because of the solid connections with earlier grades’ understanding of absolute value as a distance from zero in the Common Core.

First I connect with 6.NS.7.c, positioning the absolute value expression, x, onto the number line, at a distance of one unit from zero. We move forward by discussing the scenario, “I’m thinking of a number that, if I add 3 to it, I am 6 units from zero.  Then we put that in symbols.  I slide clones of the expression to the number line.  Two numbers (or expressions) in the same place on the number line are equal (similar to 3.NF.3c). To solidify that understanding, I require students to show their expressions on number lines as they do their homework. That needs to soak in well before we deal with equations with absolute value expressions within them.  I argue that we don’t have an “absolute value” equation if there are additives and multipliers on the same side of the equation.  We could read the following equation as, “two added to twice the distance of (eight less than four times a number) is 30…yikes!

2 + 7|4x -8| = 30

I visualize that the absolute value expression has invaders.  To avoid my students asking the question, “When are we ever going to use that in real life?” I make this scenario interesting enough so they forget to ask.  In real life, does a king take down the fortress walls when there are invaders in his neighborhood?  Picture this scenario.

Seriously who would take down the walls at a time like that? The difference between using an anchor and a trick is that an anchor does not obscure the meaning of the math.  It simply makes it easier to remember how to think about it.  So we think about absolute value bars as cautionary:  This equation is at risk for being corrupted.  Those bars are dramatically different from parentheses.  While we could distribute the 7 inside the fortress and get correct answers (harmless), 7 would corrupt the equality if it were negative, so why take that chance?  It makes sense to say, “It is difficult to determine the distance with the 2 and 7 over there.”  We get the absolute value term alone by eliminating invaders properly and then we can write the expression as a distance from zero on the number line.

Then the value of x will survive and live happily ever after.