Fractions Divided by Fractions

Someone told me, a couple of years ago, that you can divide across to divide fractions. And, I'm not ashamed to say, I was both in awe of my own ignorance at that moment and excited to make such a simple connection to something I learned years ago.

What helped make the moment especially poignant is that dividing across should be completely obvious when you know that multiplying across works, and that division is the inverse operation to multiplication. If you know, for example, that \[\mathtt{\frac{1}{3} \times \frac{2}{3} = \frac{1 \times 2}{3 \times 3} = \frac{2}{9}}\] then there is almost no excuse for not seeing that this implies that \[\mathtt{\frac{2}{9} \div \frac{2}{3} = \frac{2 \div 2}{9 \div 3} = \frac{1}{3}}\] But it turns out that I wasn't alone in missing that idea in school. I polled the G+ Mathematics Education community and found that 46% (of a very small sample, including me) learned about dividing across either at the moment of their reading the poll (38%) or at least as an adult (8%). I wouldn't be surprised if there are many adults reading this for whom this was their first exposure to the idea.

Modeling Division

I make sure to mention this method now—along with the method I was taught, multiplying by the reciprocal—almost every time I write instruction touching on fraction division.

Another method I really like, which I've included as an interactive 'conceptual calculator' module in the Fraction by Fraction Division lesson app (and embedded below), is the method of determining common denominators and then just dividing the numerators across.


With this model, the numerator is shown in the top bar, and the denominator in the bottom bar. The quotient turns out to be the number of shaded rectangles in the top bar divided by the number of shaded rectangles in the bottom bar. Thus, the division you see when this page loads is: \[\mathtt{\frac{4}{5} \div \frac{4}{1} = \frac{4}{5} \div \frac{20}{5} = \frac{4}{20}}\]Change the digits in the boxes to change the model of the quotient. You will slowly realize (if you haven't read this sentence) that the colors delineate "wholes" in the context of each division. In the lesson app, a drawing canvas lays on top of this model so that all kinds of drawing can happen.

I've loved this method for a long time—and still never put together that you can divide across!