I attended professional development (PD) this week entitled, Developing Proportional Thinking with Connections to Linear Algebra. The PD has inspired me to further explore the division of fractions. How, you might ask, is the division of fractions related to proportional thinking? Well the ratio is simply a secret division problem in disguise. The deeper your understanding of fractions (and the division of fractions in particular) the deeper you will be able to go with ratios and proportions.
Take the picture above as an example. What we are really trying to figure out here is how many one halves fit into three fourths. When we look at the fraction bars provided, we get a better sense of how many one half servings fit into our new total of three fourths. This visual model offers an alternative to the commonly used 'invert and multiply' trick. There are even more ways to divide fractions while maintaining a conceptual understanding of the mathematics involved. Take a look at the Math Playground video, How to Divide Fractions, for a glimpse into the common denominator method.
Our final resource for the week is homemade. When we were math students we learned a rather simple trick when dividing fractions. When given two fractions to divide, we were told to invert the second fraction and multiply straight across. I was never told WHY this trick works every time. Were you? The answer is likely no. For this reason, I felt inspired to create a ShowMe video entitled, Why Does the Invert and Multiply Trick Work? Check it out to see an algebraic proof of this childhood trick.