Arithmetic with Polynomials and Rational Expressions Rational Expressions Uncategorized

Multiplying fractions

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?

3 replies on “Multiplying fractions”

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!

At least the student knows to multiply by a “special 1”.
My advice:
Put all loose parts in a baggie, just like when leftovers go in the fridge. Find factors first. Find special 1s – not “cancelling 🙂
Find anything that disappeared from the bottom, and make sure there are no disallowed values.
Go slow!!

I’m wondering if he is taking algebra 2, which is usually when kids learn about multiplying the conjugate, to clear denominators of irrational or complex numbers.

Not that it much matters if my first guess is correct, which is that this is the work of a kid who has absolutely no idea what’s going on and is just going through motions to get something that looks like an answer. He gives no evidence that he can multiply binomials. It also appears that he is using common factors of each single term, not understanding that he can’t “cancel” addends from both numerator and denominator (in this case, it results in the same answer, but he’s not using the right method).

So the next step would be to discover why he has no idea what’s going on. Did he flunk binomial multiplication, or did the complexity of the rational expressions confuse him? If he has no idea how to multiply binomials, then go back to that. If he does understand binomial multiplication, but lost it in the multiple steps of rational expression simplication, then that’s a good sign he just needs clear procedures. (I find this is often the case in algebra I and multi-step equations. Kids who can distribute and combine like terms in isolation go kerflooey when given a problem that asks them to do both, in order, like 2x -3(x+7) +5 = 17).

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