But re: the expression vs equation mistake the student makes. I think of one issue I’ve had this year–I can tell students how to do x thing (solve for speed in the distance formula, simplify an exponential expression using division rules, place a square root between two consecutive whole numbers). I even bring in some investigation and open-ended tasks. They then do it, mostly correctly, and I feel good about myself. Then, I ask them again later. They try to use the simple interest formula for a distance problem, apply the product of powers rule when the expression is being divided, and try to evaluate a square root when I ask them to put it between whole numbers. Students are good at doing what we tell them to do, but less good at differentiating what we are telling them to do from that other thing we talked about that looks similar but I don’t remember and ughh math.
Algebra is usually taught with a big unit on expressions, then a big unit on equations. How much time is dedicated to understanding, in context, when an expression is the most valuable structure to answer a question, and when an equation will work better? That’s hard. Hard to know where students are, hard to come up with examples, hard to explain. But that’s the mistake this student makes, right? What’s the answer?