Exploring Misconceptions: How Does a Change in Dimension Change the Area? Part One

How does the area of a square change if the side length is doubled?     I intended this assessment to draw out common misconceptions.   I expected students to either see it as doubling (disregarding the fact that the side length doubled in two dimensions) or to see it as quadrupled.   I also realized that some students would be stopped in their tracks by the fact that the dimension was a variable and decide there wasn’t enough information.  Hence, all three of those were among the choices on this multiple choice item.

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Most of the responses that I got were much as expected.   A few students were confused by the variable, a few students said that the area would double and most of them realized that it would quadruple.

As a follow-up question, I asked them to find the area of each square.   I expected this to be unnecessary, really, since they had largely determined that the area would quadruple.   I was really stunned by their responses:   “Four x”,   “Two x squared”,  “Four x squared”.     I was only a little surprised by the “four x”.   My students have experience with exponents but the majority of that experience is with numbers and not variables.   While we have done a little work with raising variables to powers, it is still a little bit novel for some of them so I should have anticipated this error.   However, how could they say that the area quadrupled and have a coefficient of 2?   I’ve spent a fair amount of time considering exactly where the error occurred here.   Was it carelessness and they just forgot about the 2 and then did not pause to consider whether the answer was reasonable?   Was it that they are not making the connection between the quadrupling and the coefficient?   Based on the debrief, I think it was the former but I still find myself wondering if that was all there was to it.   Hence, there will definitely be some sort of follow-up assessment to see if they have really “got it”.

My biggest take-away was a reminder to never assume too much.

When administering the assessment, I used it as a Commit and Toss assessment.   Students write their solution, crumple it in a ball and throw it around the room.  They keep throwing the balls until I tell them to stop so that no one knows whose assessment is whose. Students  examine the work on the assessment they have when time is called.   Then the class debriefs the assessment.

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