When a Fraction Means Whole: Dividing Fractions

If you have been around me long enough you have experienced one of these moments.  This particular Ah Ha moment came after teaching a lesson on dividing fractions. I felt like I had failed because I could not show how the algorithm worked. I was taught to change the division to multiplication and the divisor to it's reciprocal.
So 1/2  divided by 3/4, became 1/2 multiplied by 4/3. I had no problem showing this but really did not understand why. As a teacher, I wanted to be able to explain why we did that. I had to ask my students for a few days to investigate it further.
So talking it over with a few teacher friends, we began drawing a multitude of arrays trying to fit the pieces together and see if there was a visual pattern for the how the numbers came together. We continued to draw on the paper table cloths at the restaurant and it all clicked.

It was important to recognize that the divisor is the whole group you are trying to create.
So in 1/2 divided by 3/4. It means that I want a to see how many groups of 3/4 I can make with an amount of 1/2.
What I got out of this exploration was a clearer idea of using the language of whole number division in relation to the fractions.

*** Also, a teacher I worked with realized that we do not even need to multiple by the divisor. You can simply cross multiply and then simplify your answer.
For Example: 1/2 divided by 3/4= (4*1)/(2*3)=4/6=2/3...

A new way to teach an algorithm and a great visual to show the parts and whole.***