What are these kids struggling with? What are they getting right? Are there interesting differences between the three different students whose work is shown?

Thanks again to Ms Miles (twitter / blog) for the submission!

Holy cow. I think that the first thing I would have to do is sit down and ask these students if they had ever seen anyone do these things before. Hopefully, they would admit that the answer is no and then we could talk about why. The biggest issue – I think it is clear – is that the absolute value expression does not fit neatly into any order of operations pnemonic that the student has been developing. If the absolute value sign is simply seen as some impenetrable super parentheses of sorts, then many of the mistakes above would not happen. There is some struggle between zero and one when dividing a number by itself. This probably comes from hearing someone say that items ‘cancel’ when simplifying fractions. It is awfully hard to distinguish between 0 and 1 in that case. It’s depressing that each student started off with 9 + 3x = 12x essentially.
To fix this moving forward I might urge the teacher to make a habit of simply replacing the entire absolute value expression with some dummy variable, solve for that variable then work back from there.