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What math teacher hasn't seen a student try to simplify a fraction by canceling the denominator with a single term of a multi-term numerator? The students have never learned any algebraic property that makes this a legitimate choice, and they've been told repeatedly that they can't do it, but still they persist.

One technique that I've found useful in helping students avoid this pitfall involves introducing the idea of multi-term numerators before we even start talking about algebraic fractions.

I'll put up a fraction like this on the board, and ask the students to simplify it:

Students will typically say, "Add the 12 and 8 to make 20, and then simplify by canceling out a four, leaving ."

I tell them that's correct, and then ask them if they can find a different way to simplify it. Eventually someone will come up with the idea of factoring a 4 out of the numerator:

= and then canceling the four out, leaving = .

This may seem like a roundabout way of simpifying the fractions, but I tell them it's a very important idea, because someday soon they'll have fractions that the terms in the numerator aren't like terms, so they won't be able to combine them, and this is the technique they'll have to use.

I give them a couple more examples that we work through together, and then I set them loose to try some on their own. A worksheet is provided below which gives examples for students to work through. Insist that students simplify the fractions in the manner shown on the board, and optionally tell them to check their answers by combining the terms in the numerator and evaluating the fraction that way. Note that the last few problems involve algebraic fractions, so they cannot be checked as described above.

Lesson by Mr. Twitchell

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