In case you’re having trouble reading the kid’s work, and because the top of the question is cut off, I’m going to reproduce the problem in text below the image:

Question: What is the product of $\frac{x^2-1}{x+1}$ and $\frac{x+3}{3x+3}$?

Answer: $\frac{3x^2-3}{3x+3}*\frac{x+3}{3x-3}$

$\frac{3x^2+9}{9x+9} \rightarrow \frac{1x^2+3}{3x+3}$

The usual: What does he know, what doesn’t he know, and what would you do next?

• Not sure why the student multiplied the first fraction by 3/3? Maybe they thought they needed to find a “common denominator” (even though they are still not common.)

The student DOES know that one can multiply fractions by multiplying the numerators and multiplying the denominators. Let’s go back to that, using just numbers, but some in which the product can be simplified and some where it cannot. How can you “predict” whether or not you will be able to simplify “before you even start?” I don’t like to teach “cross-canceling” but rather recognizing that a common factor in the numerator and denominator will result in a fraction that CAN be simplified. This is especially important when working with rational expressions because even if the student DID multiply the binomials correctly (another concept for remediation) they would then have to go back to the factored form to simplify. Just like with plain fractions, student needs to able to think about factored forms of numerator and denominator (so they will need to be able to factor rather simple polynomials, in this case) and whether there will be a common factor.

Next up: back-tracking to multiplying binomials, but that’s going to definitely take some more conceptual development!

• louise

At least the student knows to multiply by a “special 1”.