MBRA: Dividing fractions

One common technique taught to students for dividing fractions is to "invert and multiply." The idea here is that multiplying the numerator by the inverse of the denominator is the same as dividing, and multiplying fractions is fairly easy:
5
—
8		 = _____    becomes    5
—
8		 × 7
—
3		 = _____
3
—
7
The good news is that this approach works fine ... you get the correct answer. The bad news is that by teaching this as a "rule" the student usually doesn't know why it works. It is just a random rule ... probably one of many that the student learns for the test, but doesn't understand how/why it fits into any sort of unified whole.

We can do better. In fact, we can do better by NOT teaching "invert and multiply," but instead by teaching how to put together some fairly basic mathematical principles.

Example

We'll use the example from above:
5
—
8		 = _____
3
—
7

Key Pieces of Information

A) Any non-zero number divided by itself is one:    N
—
N		 = 1
B) Any number multiplied by one is itself:    N × 1 = 1
C) Any number multiplied by its reciprocol is one:    N
—
M		 × M
—
N		 = 1

Solution

Now we are ready to begin dividing our fraction.
1) We start by getting the reciprocol of the denominator: 7
—
3
2) We then use this reciprocol to construct a new fraction with a value of one: 7
—
3
7
—
3
3) We then multiply our problem fraction by our new fraction:
 5
—
8		 × 7
—
3
3
—
7		 7
—
3
4) This give us:
 35
—
24
21
—
21
5) Which is:
 35
—
24
1
6) Which is:
 35
—
24
7) Which can be written as a mixed number:
 1 11
—
24

Putting this all together in one line looks like this:

5
—
8		    =    5
—
8		 × 7
—
3		    =    35
—
24		    =    35
—
24		    =    35
—
24		    =    1 11
—
24
3
—
7		 3
—
7		 7
—
3		 21
—
21		 1