Visual RepresentationsIn my Fraction Arithmetic page, I present a visual strategy for figuring out how to add, subtract, and multiply fractions. (There is also a discussion of this strategy's implications for division, but I suspect that is unlikely to find broad acceptance, as it is somewhat complicated.)
The core idea of the strategy is the use of rectangles on grid paper, and the first step involves choosing the right dimensions for the rectangle that represents 1. What I like about this is that it is directly connected to very basic understandings about fractions, and about operations. For example, if you understand the meaning of 2/3, the meaning of 1/5, and the meaning of multiplication, it is not hard to see how this image leads you to figuring out the product of the two fractions:
On the other hand, if a student doesn't understand the meaning of the fractions, or the area model of multiplication, it is entirely pointless to try to teach them a multiplication trick (multiply the tops, and multiply the bottoms, say). Learning such a trick may help them to get answers in the short run, but it may soon be forgotten, not to mention that it will bring with it some misconceptions, such as "add the tops and add the bottoms" for addition.
Presenting this visual strategy to students who don't understand the basic underlying concepts forces us to shift the conversation to those concepts, which in the end is the only way to make fraction arithmetic meaningful.
Working with this representation also provides an environment to get to the tricks (e.g. using a common denominator) with understanding. In other words, even if you think it is essential to teach those tricks, you'll do it more effectively by investing some time up front with this representation.
One implication of this approach is that it is a good idea to use grid-paper rectangles early on when introducing fractions, as those become a powerful tool further down the line.
This does not mean throwing away the "pies" representation. That way of looking at fractions is not as powerful for the purposes of arithmetic, but it is great for making connections with other ideas: angles, time, money, and of course percents and decimals.
See my Slices page for several worksheets you could use for this, and some notes on making the connections. While I believe those have a lot of potential, I have not yet really explored it. I hope some of you will come up with some good activities using these, and let me know about them!
Low Threshold, High Ceiling, etc.
Finally, check out the Egyptian Fractions Challenge. I would encourage to work through it yourself, and you'll see that it has the key characteristic of a great activity: it is interesting to the teacher as well as to the student! In fact, it satisfies all the criteria I once listed for a rich curricular activity, and goes further: it is based on the Erdos-Straus conjecture, which number theorists have yet to prove.