If you teach algebra, you've most likely seen the error $(x-5)^2=x^2+25.$ Previously, in explaining this error, I would resort to discussions about what "squaring" means and the value of the distributive property. This year has been different, though.
Since the beginning of the year, I've been teaching students that they can check their work simplifying expressions by plugging in a number into the original and the simplified versions and seeing if they get the same result. We emphasized this approach throughout our review of linear expressions.
Today in class, we had exactly the problem of simplifying the expression $(x-5)^2$. Some students gave the incorrect response of $x^2+25$, but as I listened to their group conversations, I could hear their discussions "I plugged in 2 to both expressions. $(x-5)^2$ gave me 9, but $x^2+25$ gave me 29." This was music to my ears. Rather than simplification being a set of rules to remember and apply, simplification becomes attached to the real experience of ensuring that I get the same answer each time.
I started this blog to share my transformation from math nerd to math nerd who loves to share math with young people. I teach high school in Philadelphia, Pennsylvania. Your comments are always welcome.