 “Learning to compare and order fractions is important to developing an understanding of fractions.” Marilyn Burns (About Teaching Mathematics, 2015, p 430)

“Helping students develop their ‘fraction sense’ is extremely important before they begin computing on and with fractions.” Julie McNamara and Meghan M. Shaughnessy
(Beyond Pizzas and Pies, 2015, p 132)

In this series of blog posts, I’m discussing five different strategies for comparing fractions. As background to this last strategy, I suggest you go back to the first blog post and read the other three posts in the series, which describe the first four strategies. You’ll also find activities you can download and use in your classroom.

By now, your students should have four solid comparison strategies in their toolbox–compare fractions with like denominators, compare fractions with like numerators, compare fractions one unit fraction from a whole, and use a benchmark of one half. All of these strategies are based on a solid understanding of fractions. This type of understanding develops only through the extensive use of concrete materials, pictorial representations, and lots of great math conversations! These strategies will be sufficient for handling many, but not all, fraction comparisons. Before you go on, look at each pair of fractions below and see if you can determine which would be the most helpful strategy for making each comparison. Think how you would explain your reasoning behind each comparison. You probably came up with something like this:

1. In the first pair 5/6 is greater than 3/4. They are both missing one piece (one unit fraction from a whole), and 1/6 is a smaller piece than 1/4, so 5/6 is closer to one whole. That makes it bigger than 3/4.
2. In the second pair, 3/5 is greater than 3/8. I’m getting 3 pieces in both fractions, but fifths are bigger pieces than eighths, so I’d rather have 3/5.
3. I know 6/8 is greater than 2/6, because 2/6 is less than 1/2 and 6/8 is greater than 1/2.
4. In the pair 2/6 and 4/6, the pieces are the same size–sixths–so of course I want more pieces. That means 4/6 is greater than 2/6.

When deciding which strategy to use, you might have cycled through the strategies, thinking of the most simple strategies first: Are the denominators the same? Are the numerators the same? Are they one unit fraction from a whole? How are they related to 1/2?

That’s really the process we want our students to go through as they compare fractions. Comparison of fractions should be based on an understanding of the fractions as numbers, not tricks.

But there will be some fractions that these first four strategies can’t help with. Consider comparing 8/12 and 3/4. Hmmmm, unlike numerators, unlike denominators, both greater than 1/2, one is a unit fraction from the whole but the other isn’t. Now what do I do? This is where an understanding of equivalency comes into play. We want students to notice a relationship between the denominators in this case and realize that fourths can be rewritten to twelfths. How do they develop that understanding? I think you know my answer to that–lots of experience with manipulatives, such as fraction tiles, exploring what fractions look like and represent. It’s pretty hard to look at these tiles and not see that 3/4 is equivalent to 9/12. And since 9/12 is greater than 8/12, that means that 3/4 is greater than 8/12. You can download printable fraction tiles here. There is also a multiplicative relationship between the denominators–4 and 12. Four times three is twelve, so it takes 3 times as many twelfths to have the same amount as fourths. In other words 1/4 is the same amount as 3/12. Pretty hard to miss that relationship looking at the tiles! After enough experience exploring these relationships with the manipulatives, it’s easy for students to understand why we can multiply the numerator and denominator by the same number (a fraction equivalent to 1) to find an equivalent fraction. So, yes, they learn a procedure for finding equivalent fractions, but it’s grounded in understanding. Often overlooked is that sometimes it makes more sense to find a common numerator, rather than a common denominator. Take, for example, the fractions 3/8 and 15/60. I cycle through the first four strategies, which are no help in this circumstance. Then I look for a relationship between the denominators and find none. So instead, I look for a relationship between the numerators, and I find a multiplicative relationship between 3 and 15–I know that 3 x 5 is 15. I multiply 3/8 by 5/5 (a version of 1) and find out that 3/8 is an amount equivalent to 15/40. Now I’m right back to one of my first strategies–comparing fractions with like numerators, 15/40 and 15/60. I know that fortieths are bigger than sixtieths, so I’d much rather have 15/40, which is the same as 3/8. Finally, what if none of the first four strategies work and there is no multiplicative relationship between either the numerator or denominator? Think, for example, of comparing 3/7 and 4/9. Unlike numerators, unlike denominators, neither is a unit fraction from the whole, both less than one half, no multiplicative relationship between the numerator or denominator. So do I have to find the least common denominator (LCD) in this case? Or, couldn’t I change them both to have a common numerator? When you truly understand fractions, the options are limitless! Now that your students know all the strategies and have had practice with each, they’ll need practice deciding which strategy is most efficient for a given pair of fractions. I created a set of cards with pairs of fractions that lend themselves to each strategy, and they’d make a great workstation. You don’t need a fancy recording sheet to use these cards. Have students choose a card and, in their math journal, record the correct comparison and explain which strategy they used to make the comparison. Remember, it’s the explanation that you’re really interested in to determine how well your students understand fractions. You can download the cards here.

One thing I feel the need to mention as I close this series. For your students to develop fraction sense, it’s important that you have a conceptual understanding of fractions. Many of us were not taught to deeply understand fractions. We learned procedures and tricks that did not help develop a conceptual understanding. It is our responsibility to pass on something better than that to our students. If you are still using tricks in your classroom, set a personal goal to develop yourself in this area. 