Cancel Out “Cancel Out”

It’s important when demonstrating mathematical procedures that you stress the mathematics behind those procedures (i.e., the “why” behind the “what” or “how”), since failing to do so can cause or reinforce students’ misconceptions. Take, for example, multiplying fractions, as in the case of 3/5 x 5/8. Commonly—and correctly—teachers note that “the fives cancel out,” leaving 3/8 as the product.

The problem is that unless students understand why the fives cancel out, they see it as some mechanical (if not magical) step in the process. A common mistake, among others, I’ve seen students make as a result has been to also cancel out when adding or subtracting fractions—such that the sum of 3/5 and 5/8 would then equal the product of 3/5 and 5/8. To prevent this and other related errors, explain the mathematical basis for cancelling—i.e., you’re not really cancelling common factors, but rather dividing them to yield a quotient of 1 (represented as 1/1). Then show students (or better yet, have them discover themselves) that while this works when multiplying fractions, it doesn’t work when adding or subtracting them.

Is it ever ok to mutter “cancel out the 5s” as you work through a problem or just execute the cancellation without a word like your teachers may have done? Sure, once you’ve prevented or alleviated any misconceptions related to this. But until then, cancel out “cancel out,” and instead talk to students about the math behind it.

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3 Responses to Cancel Out “Cancel Out”

  1. Jim Cavallero says:

    Once again Dave you have a made a great point. The “details” are so important.

    Jim

  2. Shelley says:

    Thanks, for the really important and easy to do tips for Math teachers. I share with the teachers who are involved in our Math Coaching program (grades 7,8,9) and they really appreciate the help.

  3. colintgraham says:

    In this situation, where the concept is maybe being introduced, I think you could use either the distributive law or the commutative law to help out.

    Using distributive law (good for manipulatives):
    3/5 x 5/8 = 3/5 x (1/8+1/8+1/8+1/8+1/8)
    As there are now five separate eighths and you want 3/5 of five, that gives you 3 eighths (1/8+1/8+1/8)

    Your approach seems to be more like the commutative law, applied either to the numerator or to the denominator, but not both:
    3×5 = 5×3, so 3/5 x 5/8 = 5/5 x 3/8
    5×8 = 8×5, so 3/5 x 5/8 = 3/8 x 5/5

    If students are happy with simplifying fractions, then having them carry out the multiplications and ‘reduce to lowest terms’ (simplify is much better!) can work too, particularly if you focus on the division by 5 in the simplification and go back and ask the students where 5 was in the original ‘problem’ (one of them on the top and one on the bottom).
    (3×5)/(5×8) = 15/40 = 3/8
    This can be reinforced by having a series of similar problems and getting students to notice or write down the common factors in the simplification, which can help some notice the pattern for themselves without you having to prompt.
    You could also ‘sneak’ in more challenging problems such as 5/6 x 12/17 to show that common factors can be ‘cancelled’ too!

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