What’s the fastest way to help this student?

Do you have a good way to help students understand the rigor needed in these sort of proofs?

Thanks to Jonathan Newman for the submission.

  • It seems like they have the factoring idea and are just stumbling around forgetting what is actually equal to what, and what is being substituted for what.

    I would suggest having them write down the Pythagorean identity that they are trying to use; specifically I’d have them go back to a two-column-ish format where they give a reason for their progress from one expression or equation to the next. “Factoring” and “Because cos^2 t + sin^2 t = 1” seem like what they’re using or trying to use.

    I would also get them to box or circle or something each expression that they’re substituting; in other words, if on one line they have cos^2 t + sin^2 t then they can circle it and on the next line put a circled 1 in its place.

  • I can’t tell if they incorrectly remembered the Pythagorean identity or if they tried to memorize this proof and got it messed up in the middle. Regardless they lose me entirely on the last line where they try to fake it on the left and lose their 2 on the right. Were they in a hurry?

    However, this looks like it’s on a whiteboard- did they explain what was going on?

  • It looks like they copied this out a notebook without paying attention to where there were positives and negatives.