Actually, I don’t think we need to explain division of fractions. We just need to allow students to discover it for themselves.
We begin with multiplication of whole numbers by fractions, such as 15 * 2/3. There are many ways to model this problem. For example, divide 15 dots into 3 groups, and count the dots in 2 of the groups. One could also use a number line or fraction bars. At some point, students will realize that they can just multiply by the numerator and divide by the denominator.
Next, we ask a question such as “what number times 3/5 equals 24?” Again, there are many ways that one could figure out the answer, such as guess-and-check. Students will see that we are looking for a number which, when multiplied by 3 and divided by 5, gives 24. If they understand that multiplication and division are inverse operations, then the answer should be obvious. Multiply the 24 by 5 (to undo the division), and then divide the result by 3 (to undo the multiplication).
Now, another way to answer the question would be to divide 24 by 3/5. We just discovered that dividing by 3/5 is the same as multiplying by 5/3. I wonder if “invert and multiply” always works?