Why is this mistake attractive to the student? How could you make it unattractive?

Thanks to Tina Cardone for the submission.

  • What’s the significance of the two erasers?
    Depending on the student… I point out that going up or forward is “positive,” remind that “neg divided by neg is positive,” show that when the x & y both go the same direction, that’s a positive relationship/slope, but if one’s getting bigger while the other’s getting smaller (this works better if the slope exercise includes something with context and meaning)… and the thing that works most consistently is that visual-kinesthetic: “If you signed your name on this line and it’s going up, it’s a positive slope…”
    Now, the perpendicular thingy… some of my folks benefit from understanding what perpendicular means (helps w/ horiz and vert lines) … my procedure-focused folks benefit from “it’s got to pass two tests” (“upside down and backwards” appeals to me, but not to them) — you have to flip the fraction *and* switch the sign.

  • I think the erasers are obscuring the student’s name, but I could be wrong.

    The most basic error took me the longest to notice: both the numerator and the denominator have negative signs. So there’s a fundamental misconception about fractions there, with regards to a negative sign being multiplication by -1 and canceling -1/-1 and so on.

    The next error is that the answer given looks like run/rise instead of rise/run. Then, the -3 means the student is either moving up the y axis backwards or getting flipped around somehow.

    It’s tempting to write this off as a failure of rote skill, but it’s not. I’d go back to basics and talk about double negatives, negatives in fractions, and finding the rise and run. Then I’d make sure they understand “sliding” a line up and down to create a family of parallel lines. Ultimately, knowing that distinct parallel lines have the same slope and different y-intercepts is to understand local/global and absolute/relative change, two very abstract yet fundamental concepts.