How and When to Cancel Things in Fractions

One of the common errors in algebra is canceling bits in fractions that can’t be canceled. To remember the rule, though, you don’t have to memorize anything. You only have to think of a plain arithmetic problem to compare it.

To help you remember what’s allowed with , think about this simpler problem   .

Like this:

And just to keep you straight, here are the requirements (and a couple of examples) for the fancy <cue music> “distributive property”.

What do you think of this explanation? Post your thoughts in the comments. Need something else? Request it and I’ll get to it as soon as I can.

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5 Responses to How and When to Cancel Things in Fractions

  1. This tip works in many situations ( replacing the letters by some simple whole numbers such as 2, 3 and 5) and verify if it works.
    Because students have used many timesthe distributivity of the multiplicatipon over the addition, they MISTAKENLY believe that other operations are also distributive.
    The square root of a sum is NOT equal to the sum of the square roots of the 2 elements.
    The square of a sum is NOT equal to the sum of the squares of the 2 elements.
    a/b + c/d is NOT equal to a+c / b+d
    Generally speaking a function F{a+b) is NOT equal to F(a) + F(b)

  2. This is great. So simple an explanation. I’ll give kids something like this and as if it is true. I let them fool around a bit to see if they can discover the rule or property.

  3. I very much like rewriting 3x+4 / 2 as 1/2 * (3x+4) because now there is no need to justify/explain that each addend must be cut in half — as long as they grok the distributive property of multiplication over addition, of course.

    When I get around to my videos I am going do the same thing with division that we often do with subtraction: ban it! Just as we change subtraction to adding the additive inverse, we change division to the multiplicative inverse when we are not clear on the subtraction or division.

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