By Shelby P. Morge, posted November 9, 2015 –  

Classroom time constraints and desires for correct answers on grade-level assessments can often drive the instructional strategies that we use to teach students to compare fractions. When I ask the teachers I work with to compare two fractions, the most common strategies they use are finding common denominators, cross multiplying, and converting fractions to decimals. These strategies are fairly simple to teach, not time consuming, and do lead to correct answers; but they don’t encourage fraction number sense or thought about the relative size of the fractions (Van de Walle, Karp, and Bay-Williams 2013).

After the teachers solve one fraction-comparison task using their own strategies, I encourage them to act as though they have not learned those strategies and compare two new fractions using a reasoning approach that elementary school students might use (adapted from Van de Walle, Karp, and Bay-Williams 2013, p. 310). When they are encouraged to think more about what they know about the fraction concepts, teachers often come up with various strategies for determining the largest fraction. For example, when comparing 2/3 and 3/4, they may notice that both fractions are one unit fraction away from the whole and that 1/3 is a larger fraction than 1/4, so 3/4 is the larger fraction. Another comparison strategy might be to draw a picture of two same-size wholes and notice that fourths are smaller slices, so 3/4 is closer to 1.

Such activities as Fraction Comparisons on a Clothesline described in my previous post can support students’ understandings of fraction concepts and the relative size of fractions because students work with a number-line model and may create mental images to place their fraction on the clothesline. When placing the fraction, they must compare it with those around it, using strategies that make sense to them and that are similar to those described above. Additional conceptual strategies for fraction comparison that may be used include the following:

•     Fractions with a same-size whole or common denominators (such as 7/12 and 5/12)

•     Fractions with the same number of parts or common numerators (such as 3/4 and 3/5)

•     More than or less than 1/2 or 1 (such as 3/8 and 4/7), equivalent fraction concepts (Van de Walle, Karp, and Bay-Williams 2013)

•     Mentally breaking apart the number line into equal-size parts

The use of equivalent fractions can be a common strategy, but it can take a more conceptual approach when used to adjust how a fraction looks. As mentioned by Van de Walle and his colleagues (2013), Burns found that when comparing 6/8 to 4/5, students may change 4/5 to 8/10  so that both fractions are two parts away from the whole.

Avoiding teaching these strategies explicitly is important, because students will see them as something else they need to remember. In addition, students will be easily confused about when to use each strategy. Instead, these comparison strategies should come out naturally in conversations about how students decided to place their fractions on the clothesline. Teachers may select a fraction, such as 2/3 , and ask students to turn and talk to a partner about how they would decide where to place it on the clothesline (Chapin, O’Connor, and Anderson 2013). Then, teachers may ask students to share with the whole class how they decided to place the fraction where they did. These abilities to discuss with classmates, explain fraction reasoning, and visualize fractions on the number line will contribute to students’ long-term understanding of fractions and support learning in higher-level mathematics.

Your Turn

We want to hear from you. Please share your thoughts about using conceptual strategies for fraction comparison. Be sure to include any additional strategies that you have used with your students. Post your comments below or share your thoughts on Twitter @TCM_at_NCTM using #TCMtalk.

References

Chapin, Suzanne H., Catherine O’Connor, and Nancy Canavan Anderson. 2013. Classroom Discussions: A Teacher’s Guide for Using Talk Moves to Support the Common Core and More, Grades K–6. 3rd edition. Sausalito, CA: Math Solutions.

Van de Walle, John A., Karen S. Karp, and Jennifer M. Bay-Williams. 2013. Elementary and Middle School Mathematics: Teaching Developmentally. The professional development edition. Boston: Allyn and Bacon.

Shelby P. Morge, morges@uncw.edu, is an associate professor in the Department of Early Childhood, Elementary, Literacy, Middle Level, and Special Education at the University of North Carolina–Wilmington. She teaches mathematics education and field experience courses for middle level and elementary preservice and in-service teachers. Morge’s research focuses on mathematics-related beliefs, teacher and student understandings, and the use of assessment items for instruction and professional development. She is a former middle school and high school mathematics teacher.