Chapter 10: Why do we Invert and Multiply?

When dividing two fractions, we are taught to "invert and multiply" (and don't ask why). It is the rule that all math students inevitably learn, relatively few challenge, and fewer can rationalize. It is like wetting your toothbrush before adding toothpaste and brushing...you don't quite know why you do it, you just know it works.

This week in class, we were challenged to rationalize the "invert and multiply" rule. Why does it work? How can we explain why it works? How can we show that it works? Below is my thought process on why we are able to "invert and multiply" two fractions that are being divided:

Let's use the following equation:

1/2 ÷ 3/4

Since these fractions are being divided, we could rewrite the equation as:

1÷3 
2÷4

Now we have a fraction in the numerator (1/3) and a fraction in the denominator (2/4). We can now multiply the entire equation by the fraction of 3/3 (to get rid of the 3 in the numerator fraction) and 4/4 (to get rid of the 4 in the denominator fraction)...excuse my terminology!:

[(1/3) ÷ (2/4)] x (3/3) x (4/4)

which can also look like:


(1÷3)(3)(4)
(2÷4)(3)(4)

As you can see, multiplying then dividing by 3 in the numerator and 4 in the denominator would cancel the respective terms, leaving:

(1) x (4)     
(2) x (3)

 If we separate the terms, we see our rule... invert and multiply!

(1/2) x (4/3)

https://www.pinterest.com/explore/dividing-fractions/

The mathematics applied to this solution is not necessarily complex, and I am sure there are a million other more effective and simple ways to solve it (see the picture to the left). The point I am trying to make, is that, using either numbers, visuals, or manipulatives, mathematics educators should work with students to facilitate the derivation of the "invert and multiply" (and other!) math rules. An unexplained/non-derived formula or rule is a difficult concept to grasp, and passing it along without question promotes a passive style of learning which resonates with primitive methods of teaching. I truly believe that something as simple as collaboratively deriving the "invert and multiply" rule could drastically affect the confidence, and therefore academic ability, of all learners within a mathematics classroom.




Thanks!


UPDATE: Here is a link to essentially the same proof as the one shown in the blog, but shown in a much clearer manner! It also shows other ways of proving the "invert and multiply rule"!

http://www.mikesmathclub.org/div_fractions.pdf