An important part of being a flexible mathematician is knowing that one size does not fit all. In other words, mathematicians use different strategies depending on the situation. A good example is comparing fractions. I will go on the record and emphatically state that kiddos should not be cross-multiplying to compare fractions. Yes, I know it's fast. Yes, I know it works. But it's critical that our kiddos understand fractions, and cross-multiplying is not a means to that end. So how should we compare fractions?
First, students need to have good fraction sense. That is, they need to deeply understand what a fraction like 1/8 means. That takes lots of concrete experiences with fractions. Check out this blog post for more on fraction number sense.
Next, it totally depends on the fractions being compared. I have a four-step approach for comparing fractions, as shown in the poster below. I'm BIG on #2--relate to 1/2. You simply can't do enough to develop benchmarks of 0, 1/2, and 1. Here's another blog post with more information on that.
Click here to grab a copy of the poster. And if you want some fractions cards to practice, check out this blog post.