We’ve spent the last couple of chapters talking about dividing fractions: how to make sense of the operation, how to picture what’s going on, and how to do the computations.  But all of this kind of begs the question: When would you ever want to divide fractions, anyway?  How does that even come up?

It’s important that teachers are able to come up with situations and problems that model particular operations, which means you have to really understand what the operations mean and when they are used.

When to Multiply, When to Divide?

A common answer to

Come up with a situation where you would want to compute.

Is something like this:

My recipe calls for  cups of flour, but I only want to make half a recipe.  How much flour should I use?

But that problem doesn’t ask you to divide fractions.  It asks you to cut your recipe in half, which means dividing by 2 or multiplying by .

Why is it so hard to come up with division problems that use fractions? Maybe it’s because fractions are already the answer to a division problem, so you’re dividing and then dividing some more.  Maybe it’s because they just make it look so complicated.  In any case, it’s worth spending some time thinking about division problems that involve fractions and how to recognize and solve them.

One handy trick: Write a problem that involves division of whole numbers, and then see if you can change the numbers to fractions in a sensible way.

For each one of the fraction division questions, we can understand why it’s a division problem:

• I have feet of ribbon. How many 6-inch (that’s a foot) pieces can I cut from it? This means making equal groups of foot each and asking how many groups. That’s quotative division.
• My watch alarm goes off every half hour, and I don’t know how to shut it off. How many times will it go off during the hour movie? Again, we’re making equal groups of hour each, and asking how many groups.  Quotative division.
• My fish tank needs  gallons of water, and my bucket holds gallon. How many times will I need to fill my bucket in order to fill the tank? Once again: we’re making equal groups of gallon each, and asking how many groups (buckets).
• I want to measure cups of flour for a recipe, but I only have a cup measuring cup. How many times should I fill it? This is making equal groups of  cup and asking how many groups.
• I ran  miles before I twisted my ankle.  I only finished half the race.  How long was the race course? This one is a little different. This one is a little different. It’s the fraction version of partitive division.

Recall what partitive division asks:  For , we ask 20 is 4 groups of what size?

So for , we ask:  is half a group of what size?