Any ideas on how to teach fractions?

I am doing some volunteer work, teaching fractions to an adult. She is not a slow learner, but has been away from the material for a long time and is struggling. We previously breezed through a review of whole number arithmetic.

I copied some online notes for teaching fractions that follow the usual progression – addition and subtraction and then multiplication and division.

I have been thinking that since addition and subtraction of fractions is so much more difficult, why not start with multiplication? It seems more natural. To multiply by 3, multiply the numerator by 3, which is following the general rule using 3/1. Dividing by 3 is a similar idea. Rectangles divided into regions could be used to illustrate the concepts. Once this is understood, multiplying by ⅔ can be seen as a two step process – first multiply by 2 and then divide by 3.

Before moving on to addition and subtraction, I was thinking of going over other ideas that would lead to a better feel for fractions. For example, being able to say that 4/13 is a little less than 4/12 = ⅓. Arrange the fractions 2/5, ¾ and 3/5 from smallest to largest.

When it comes to adding fractions, conversion of ⅓ to sixths can be explained as multiplying by 2/2 = 1, which will of course leave the value unchanged. This is a lot more logical than saying that you have to multiply numerator and denominator by the same value.

Does my radical approach have any merit? Do you have any other tips from learning or perhaps teaching fractions?