When we get to rational expressions in intermediate algebra classes, I have my students fill out this survey.
I let the students work for five minutes or so and then we discuss the answers. There are a lot of enlightened oh’s and ah’s. From then on in class I need only remind them that “canceling” only works with factors. I conclude the survey by discussing the necessity of distinguishing between adding and multiplying and more generally the necessity of looking carefully at all algebra problems.
I don’t think I do this type of exercise often enough. By “this type of exercise” I mean a set of questions that force my students to review what they know in a compare and contrast format. I am going to try this method for other concepts. I will be publishing the worksheets as I develop them.