Division with fractions

This is going to be another of those rambling posts.

Homeschooling has given me a whole new perspective on grammar and elementary math. Before homeschooling I didn't think much about either subject. Now I adore these subjects. Too bad my kids don't share my fascination with either subject.

This morning I had a wonderful math moment.

To back up a bit. ....
I've been learning about math through teaching Sparkle and reading some math blogs. I read how treating multiplication like repeated addition falls apart when multiplying fractions. I've read elsewhere how many students (and many teachers) don't really understand dividing by a fraction. They might be able to do the procedure (don't question why, just invert and multiply), but cannot create an accurate word problem that reflects dividing by a fraction.

Most explanations of division of fractions that I've encountered explain division by a fraction with a lengthy discussion of division as the inverse of multiplication. It makes sense as I read it, but I didn't internalize it well enough to explain it to someone else.

Then a few days ago I was flipping through Sparkle's new math book. Dividing by a fraction is taught less than halfway through the new book. I was amazed. If you use this series of math books at the recommended levels, students would be encountering these problems halfway into the first semester of third grade! Sparkle will be older when she gets to it, but I still think that she will be on the young side for the concept. However, what really impressed me was that the concept was presented in a way that made total sense to me. I can really see a child can understand these concepts after working through these math books.

The problem (1/2) divided by (1/4) is expressed as the question "How may 1/4ths in 1/2?" It's the same way of phrasing (8 divided by 4) as "How many 4s in 8?"

My wonderful math moment was vaguely related.

Another tricky subject is subtraction of negative numbers.
I'm probably going to use incorrect terminology here, but I don't care.

Adding and subtracting are opposites.
A number plus its opposite equals the identity number for addition.
The identify number for addition is zero. (A number plus zero is itself.)
For example, the opposite of 5 is -5.
Subtracting a positive number is the same as adding the negative number.
5 - 5 = 5 + (-5)
The opposites cancel each other out. Swap subtracting with adding, Swap a number with its opposite.
So, subtracting a negative number is the same as adding the positive number because two opposites cancel each other out and everything stays balanced.

Similarly, multiplication and division are opposites.
The identity number for multiplication is one. A number times one is itself.
A number times its opposite equals one.
The opposite of a fraction would be the inverted fraction.
Inverting and multiplying to dividing by a fraction also involves two opposites. The inverted fraction is an opposite and changing dividing to multiplying is also an opposite.


Sorry if you've read this. It probably doesn't make much sense. I'm just typing this to try to capture my beautiful math moment for myself so I'll remember it later.