My son's take on division of fractions

My son approached me last week to ask about dividing fractions. He presented me with the example

1/2÷1/3.

In school, he explained, they told the students to rewrite it by flipping the reciprocal:

1/2×3/1

then multiplying:

1/2×3/1=3/2.

That's right, I affirmed, that's the only way I've ever seen it done.

Well, I thought of this before she showed us, he said, and explained as he wrote, First, find the common denominator,

3/6÷2/6

...and then divide numerator and denominator:

3/2

because 6 divided by 6 gives 1 in the denominator.

I looked at it a moment and I finally exclaimed, That's brilliant!

I wish I had thought this up! The physical meaning is quite sensible, akin to adding or subtracting: Find the common denominator so that the two numerators represent the same number of units, then divide those units. I doubt that went through his head, but there's no question that this method works every time, either from the physical parallel or from the algebra.

(PHYSICAL PARALLEL) Division is precisely what the word indicates: take a quantity of (say) boxes, divide it into as many groups of boxes of the second quantity as possible, and report the number of groups. So 6 ÷ 2 = 3 because 6 can be divided into 3 groups of 2.



Suppose you want to divide instead 1/2 by 1/3. Looking at the picture, you might guess that 1/2 counts as a little more than one group of 1/3.



You might even guess that the remaining amount is half of 1/3, but you'd rather know than guess.

What makes a direct evaluation difficult? The boxes are different sizes. If we want to make an accurate comparison, we need to have the same box sizes.

That's not so hard, actually. Let's divide the whole into six parts, as in the diagram below. (Why 6? Because 2 × 3 = 6.) We see that 1/2 accounts for three of these six parts, so 1/2 = 3/6. Likewise, 1/3 accounts for two of these parts, so 1/3 = 2/6.



Now that all the boxes are the same size, we can compare the quantities directly. (This is the reason common denominators are so important with fractions.)



Thus 3/6 ÷ 2/6 = 3/2 (or one and a half, if you prefer). Since 3/6 = 1/2 and 2/6 = 1/3, we conclude that 1/2 ÷ 1/3 = 3/2.

This really oughtn't be so complicated as I'm making it out to be. I'll have to think about how to explain it some more, but I hope it gives you a feel for what's going on.

(ALGEBRA) For a,b,c,d∈Z (the set of integers), assume b,d≠0; then

(a/b)÷(c/d)=(ad/bd)÷(bc/bd)=ad/bc

which is exactly what one would expect from the traditional algorithm:

(a/b)÷(c/d)=(a/b)×(d/c)=ad/bc.

I have a physical explanation for the traditional method, but it's not as clear (to me) as the first method. The only reason that I can imagine that the second one is taught, rather than the first, is that it's easier than finding a common denominator.