Sometimes making teaching work is all about quality materials, innovative lessons, and all that jazz. Other times, it’s the small stuff that counts. In Algebra 2, I like to dive right into linear equations for the first few weeks, then skip back and start talking absolute value. This year, I did the same stuff I usually do, but with one tiny change in language, and the kids went from:
Here it is, and I realize probably everybody already does this, but I didn’t, and it is SO MUCH better.
Start with one-variable equations: |x + 5| = 8
I do everything on a number line and make students interpret the meaning of this as “the distance from x to -5 is equal to 8.” We solve some problems just like that, talking through them on a number line, but when I go to show the algebra “trick”, I teach them to drop the absolute value and set it equal to 8, or drop the absolute value and set it equal to … (here it comes)…. (seriously this is awesome)…. the OPPOSITE of 8.
So when the right-hand side has two terms, like |x – 2| = 2x – 12, the very first example of that type we look at, everyone already knows that either x – 2 = 2x + 12 or x – 2 = -2x + 12 and when I called on a student to explain why, she was like: “Duuuhhh, it’s the OPPOSITE.” Didn’t have to mention distributive property or anything and 100% of the class was ahead of me.
Then when we get to inequalities, it’s all very natural that something like |x – 2| > 7 will split into one branch of x – 2 > 7, and the other branch should be x – 2 < -7 because duuuhh… it’s the OPPOSITE!
And now we’re on to absolute value in two variables, and after some good old table of values graphing to convince them that they really didn’t want to have to do that all the time, I went back and talked about splitting up the absolute value equations into 2 linear equations, and wow… did not feel like I was earning my paycheck.
“So when we look at the second branch of y = |x + 4| – 2, how do you think we should write that? Should it be y = -x + 4 – 2 or y = -x – 4 – 2 or y = -x – 4 + 2?”
Usually questions like this spark an intense debate, or at least a lot of students with that scared “I ain’t picking anything” look.
But this year, all I got was eyerolls as 25 kids all picked the middle option. WHY?
“Because x +4 is in the absolute value, so you just take the opposite.”
“And why not take the opposite of -2?”
“Well, it’s not in the absolute value, so it doesn’t represent a distance where the answer could be in either direction. You’re just subtracting 2, like it says.”
So, 3 days of absolute value and we hit equations, inequalities, graphing two variables with tables and linear equation, including writing out interpretations referring back to the number line, no solution & all real number solution situations, and everyone’s on board. So if anyone out there is still teaching absolute value without using THE OPPOSITE, come give it a try. The most remarkable thing about this profession is how those tiny, tiny changes can have such an impact on students.