Behind the Curtain of Dividing Fractions

Paula Beardell Krieg

Sep 2

The language of dividing fractions by fractions is so murky that it might as well be a lie. Not that I think anyone has set out to lie to me, but it sure does feel unfriendly.

Harkening back to my last post, I asked "Is half of two-thirds the same as two-thirds divided by one-half?" This is an uncomfortable question, designed to give you an idea of why this whole inquiry starts out feeling downright radioactive.

It's helpful to revisit what division asks. The question," what is 15 divided by 3" is actually asking: how many groups of 3 units fit into 15 units?

Did you get that? It's important that you get that. It's like saying, how many groups of 3 apples fill up the same space as 15 apples.

To generalize, how many of the divisors (in our example, the 3) fit into the dividend (the 15).

The statement above get makes me angry. Students are supposed to learn those words, divisor and dividend, which, come on, can so easily scramble the brain. Learning two words at the same time that kind of feel arbitrary and which sound so much alike is a recipe for failure. There's no way to accurately talk about division without choosing to either be overly wordy or using the correct language. I opt to, mostly, be overly wordy.

I apologize for being angry even before we get to dividing fractions. Channeling anger (and fear?) into clarifying the murk can lead to an illumination that is positively wonderous and joyful. That's my goal. Clarifying division of fractions in such a way it creates a deep sense of satisfaction and happiness.

Are you with me?

Here's what's next: Thinking about the original question at the top of this post,. What is two-thirds divided by one-half? It's time to draw a picture of two-thirds. Two-thirds of what, you may ask? Since this is mathematical thinking, the best default object for our inquiry is square.

I will explain this all in words and pictures, but if your interest and internet allows it, I recommend watching the following video.

The description of what goes on in the video is this:

*The question that is being asked is how many halves go into two-thirds*. Start by coloring two columns that represent two thirds of the square

If the equation is two-thirds divided by one-half the question being asked is how many halves go into two-thirds? All quantities refers to parts of the whole. Our whole is the whole square,

Keep in mind that we are not looking to find half of two-thirds, we are trying to find out how many halves fit into two-thirds.

Partitioning off a row that represents one-half of the square.

Above, the square is divided in half in a way that creates an array (grid) of smaller, countable units, and one-half of the square is shaded in. How many units fill one-half ? Three units fill one-half of the square (oh how I dislike math books that don't provide the answer to their questions. So glad I am not a math book.)

Now count how many groups of three units fit inside the 2/3rds part.

I can see that there is certainly one group of three in the 2/3rds part. But that leaves one unit unaccounted for. Or uncounted. Not sure which. But I digress.

There is one lonely unit that longs to be part of a group of three. Too bad. It's forever just one of three.

There you have it . There are one and one-third groups of on-half that go into two-thirds.

Eureka! Two-thirds divided by one-half is one and one-third.

Now take a look at the grid at the top of the page. Can you tell what division problem you can it's illustrating?

Before I go and tell you the division problem it's illustrating, I want to tell you that my plan for tomorrow is to examine what is half of 2/3rd. Or maybe what's one-fifth of two-thirds, and how does that differ from dividing two-thirds by a fifth, or a half.

The frame at the top can be interpreted as what is 5/8 divided by 3/5.

It has another interpretation as well, but leaving that until tomorrow.

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