Comparing Fractions

By Cassandra Hatfield, RME Assessment Coordinator

4 × 5 is 20. 8 × 1 is 8.
So, ⁵⁄₈ is the greater fraction.

As a middle school math teacher, I found it interesting that when I would ask students to compare two fractions, many were quick to give me an answer, but when I asked them to order fractions it took those same quick students quite a bit more time and they often got the wrong answer. Intrigued by this I began to ask my students “how do you know?” I quickly found out that many students were comparing fractions using the “butterfly method” or cross multiplication. In this example, because the larger cross product is on the left, the larger fraction is on the left.

This “trick” will not take a student very far in their journey of mathematics. Although a student can arrive at the correct answer, the “trick” does not require any thought about the relative size of the fractions. If a student does not understand anything about the relative size of the fractions, how would the student order three or more fractions or think conceptually about this word problem: Max paid $12 for his portion of dinner. This was one-third of the total bill. How much was the total bill?

Instead of using the “trick” above, students should be given opportunities to compare fractions that have been intentionally selected by the teacher for investigation. This is not to say, that this should be a lesson titled “Many Ways to Compare Fractions.” Students should be given the opportunity to compare the fractions and share strategies with their classmates in a discussion. The teacher should listen for students who use specific strategies and poster those strategies in the classroom.

Below are some fractions I have intentionally selected for comparison. I’ve shared one strategy that could be used for comparison.

(Van De Walle, Karp, Bay-Williams, 2013, p. 310-311)

Summing it All Up
For students that are struggling with the abstractness of the above strategies, the use of number lines could be helpful. The IES Practice Guide, which is supported by research evidence, states, “conceptually, number lines and number paths show magnitude and allow for explicit instruction on magnitude comparisons" (IES Practice Guide, 2010).

First, see if the student understands how to partition a number line into equal parts and identify distances on the number line. If the student does, then models like the one shown below might be helpful for students to visually see the comparison strategy. 

Model of comparing ⁴⁄₅ and ⁹⁄₁₀ to a whole.

Now it’s your turn. Share with us a different strategy you would have used to compare the fractions above.

Van De Walle, J.A., Karp, K.S., & Bay-Williams, J.M. (2013) Elementary and middle school mathematics: Teaching developmentally. Upper Saddle River, New Jersey: Pearson.

Siegler, R., Carpenter, T., Fennel, F., Geary, D., Lewis, J., Okamoto, Y., Thompson, L., Wray, J. (2010) Developing effective fractions instruction for kindergarten through 8th grade: A practice guide. Washington, DC: National Center for Education Evaluation and Regional Assistance, Institute of Education Sciences, U.S. Department of Education. Retrieved from http://ies.ed.gov/ncee/wwc/publications/practiceguides/.

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