When it comes to the dividing fractions, I have to admit that I
graduated from the school of "Don't Ask Why, Just Invert and Multiply".
Were you taught something similar?

There are so many things wrong with that approach. It reinforces misconceptions that students may have about the mysterious and magical nature of math. Is dividing 1/4 by 1/3 really so incomprehensible that we shouldn't bother to explore its mind-boggling complexities? Fortunately for the current generation of students, that sort of teaching disappeared along with transistor radios and hoola hoops.

Or did it?

I had the opportunity to teach a 5th grade class recently. They had been immersed in a unit on fractions for several weeks and were just wrapping up a section on division. To find out what the students had learned so far, I placed the following problem on the board:

There are so many things wrong with that approach. It reinforces misconceptions that students may have about the mysterious and magical nature of math. Is dividing 1/4 by 1/3 really so incomprehensible that we shouldn't bother to explore its mind-boggling complexities? Fortunately for the current generation of students, that sort of teaching disappeared along with transistor radios and hoola hoops.

Or did it?

I had the opportunity to teach a 5th grade class recently. They had been immersed in a unit on fractions for several weeks and were just wrapping up a section on division. To find out what the students had learned so far, I placed the following problem on the board:

A lively conversation ensued.

Me: Does anyone know how to solve this problem?

Student1: Make it a multiplication problem.

Me: Like this?

Student1: No, you have to flip over the fractions.

Me: Oh, you mean like this?

Student1: No, they don't both get flipped. Only one of them.

Me: Which one?

Student1: I'm not sure. I forget which one.

Me(to myself):

*Uh huh. Where have I heard that before?*
Me: Does anyone have a different approach?

Student2: Well, half of 2/5 would be 1/5 so that's the answer.

Me(to myself):

*Ok, there's some number sense here but we're not quite there yet.*
Me: Do you think the problem is asking us to divide 2/5 into 2 equal parts?

Nearly the entire class nods a self assured yes.

Me: If that was true, wouldn't the problem read: 2/5 divided by 2?

The class expresses a collective look of bewilderment.

Me:
Actually, the problem asks you to figure out how many halves there are
in the fraction 2/5. It works just like whole numbers. When you are
given a problem such as 40 divided by 8, you are being asked to figure
out how many groups of 8 are in the number 40. So how many halves are in
the fraction 2/5?

Lots of blank stares. I better try another approach.

Me: Who thinks there's at least one 1/2 in 2/5?

Way
too many hands go up. Ok, we've got a number sense problem here. The
division lesson is over for now. Where's that giant number line? We've
got some work to do!

The old invert and multiply mentality is
alive and well today and continues to ensure that students have
absolutely no idea what is going on with division by fractions. While
invert and multiply is a tried and true method for dividing fractions
(provided one remembers which fraction to invert), it is not the only
way to divide fractions.

Suppose you were given the following problem:

You don't even need to invert and multiply in this case. You can
actually just divide the numerators and the denominators independently
and get a more direct answer.

How do you teach division of fractions to your students?

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