Kelli Prine submits the above, and asks “This student keeps factoring the terms and canceling.  What can I do to clarify the process?”

I expect some excellent comments on this one. Don’t disappoint.

• Wow. It’s so tough, watching this student turn addition and subtraction into multiplication with those little noted signs floating above. How do we get them to see that this problem is totally different from the one above it because multiplication is not the same as addition?

Maybe it’s time for some alternate representations, like algebra tiles, or area problems, or something where the variable comes with some units or real-world meaning that show clearly that these expressions with multiplication are almost totally unlike the expressions they came from in the previous step.

• Oh, I meant to note: Usually I have the problem that students “cancel” when it’s not division undoing multiplication, but rather when there’s still addition in it. Here I see maybe a tiny hint of that in (7) where the “cross-canceling” seems to have had some kind of glitch to it, and no factoring prior to it, but at least in the marked problem the “canceling” is fine, it’s the magical conversion of addition to multiplication that’s the problem. That’s an unusual one!