What’s the student thinking, and how can we help?

Rational_expressions
  • John

    They seem to be applying the rule that whenever you add fractions, you need to find a common denominator, so they start by multiplying both top and bottom by the common denominator, 2d, and they multiply that across the numerator in both of the fractions. However, they then seem to think that this will also create a common denominator for both the top and the bottom fractions, and it does not. But what they seem to be missing is a sense of order in how to approach the problem. I think I would ask them to tell me where to start with this problem. I might ask them to draw in some parenthesis around the top and bottom fractions to help them see that they must simplify those first. If that doesn’t help, I might just rewrite the top fraction separately and simply only that. This was really helpful for me to think through. As someone who sees these errors almost instantly in physics, but has never taught math before, I found this to be quite challenging, and I’m sure I’ve missed a lot. It’s going to be an interesting year teaching algebra II for sure.

  • Michael P

    One thing that I’d like to add is that the student is confusing two techniques for handling a complex fraction. One option, as John said, is for the student to add the fractions in both the numerator and the denominator. Another option, though, is to eliminate the denominators by multiplying by a common factor of ALL the denominators of ALL the smaller fractions. My sense is that this is the strategy that he begins with, and then he reverts back to that initial strategy of combining the fractions.So, I’m not sure if the student is substantively confused about how to pursue either strategy. I wonder if the student is just confusing two things that they know how to do. I think that I would ask the student to explain why they multiplied both parts of the larger fraction by “2d/1”. If the student explains that it’s to “kill the denominators” then I would ask why the denominators are still present in the next step.