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. Students need to have multiple strategies for comparing fractions. I wrote a whole series of blog posts outlining the different strategies. You can check out the first post in that series and find links to the whole series here. I made a poster showing all the strategies in the order they should be considered.