Fractions: Divide and Conquer
A lot of students begin by finding a common denominator for the dividend and divisor when dividing by a fraction. And a lot of teachers intervene by saying, "Remember, you only need a common denominator for addition and subtraction. For division, just flip and multiply."
Technically these teachers are right: you don't need a common denominator when dividing by a fraction. At the same time, the common denominator approach is in fact a viable alternative to flipping and multiplying. Take, for example, the expression 1/2 ÷ 1/8, for which flipping and multiplying yields 1/2 x 8/1, or 8/2, or 4. Simplifying the same expression using the common denominator approach results in 4/8 ÷ 1/8, which is also 4, since there are indeed four eighths in four-eighths.
To generalize, once you've converted the dividend and divisor to fractions with the same denominator, all you need to do to find the quotient is divide the numerator of the dividend by the numerator of the divisor. In other words, disregard the common denominator, and divide the numerators.
For many students, the common denominator approach is more straightforward and intuitive than flipping and multiplying. Does this mean you should scrap the flip and multiply approach? Not necessarily, since it's often more efficient than the common denominator approach. But it can also lead to more errors, including those caused by students flipping the dividend rather than the divisor.
A key to preventing such errors is to teach students the flip and multiply approach before they learn to divide fractions rather than as something unique to fractions. The point being that the same rules apply whether you're dividing fractions or whole numbers. It only makes sense, then, to stress when teaching students to divide whole numbers that dividing by any number is the same as multiplying by that number's reciprocal. Example: 24 ÷ 6 = 24 x 1/6.
Do this, and students will be more likely to get the mathematical basis for flipping and multiplying when dividing fractions, and be more proficient with it as a result. Still, be sure to also teach students the common denominator approach, since they don't need to flip and multiply fractions in order to divide and conquer them.
Image provided by Phillip Martin with permission
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