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Let’s take for granted that these students don’t have conceptual understanding of the Pythagorean Theorem, because if they did, then they wouldn’t make these mistakes. (I actually think that we need to be more careful with the ways that we toss around phrases like “conceptual understanding” but whatever.)

What do these mistakes reveal about how these kids think about right triangles and the Pythagorean Theorem in the absence of conceptual understanding? Why does this ever make sense to the student?

Thanks to Michael Fenton for the submission!