How well do you understand fractions?

I have been tutoring math to a young woman who is taking several courses in order to pass the GED high school equivalency test. Since she understands the basic arithmetic operations, I figured the next step would be to teach fractions.

It was surprising to see how little she understood them. Some things that seem transparently obvious were beyond her understanding, like 2/7 + 3/7 = 5/7, 1/12 is less than 1/10, and ⅔ = 4/6. Do you find these to be obvious also? Do you know anyone who would have trouble with this? I found out that my SIL also has difficulty with these ideas.

I did not want to just teach rules, but to provide a basic understanding. It has been helpful to talk in terms of cutting a pizza into equal portions. If you cut it into 7 equal slices then each slice is 1/7. 2/7 + 3/7 is equivalent to combining 2 slices with 3 others. Each of 12 slices is less than each of 10 slices. And dividing a pizza in 3 and taking 2 of the 3 slices (⅔) is the same as dividing each of the thirds in half and taking what is now 4 of the 6 slices. Do any of you have any other ideas? I want to teach cross multiplication to show that two fractions are equivalent, but I can’t think of a simple way of explaining it.