By Shelby P. Morge, posted October 26, 2015 –

One of the most common strategies for comparing fractions is cross multiplication, which may lead to correct answers but does not help students make sense of the relative size of fractions or develop fraction number sense (Van de Walle, Karp, and Bay-Williams 2013). In the Common Core State Standards for Mathematics (CCSSM) (CCSSI 2010), first-grade students begin working with halves and fourths by dividing shapes into equal parts. By third grade, students are placing unit fractions on a number line, so they must develop appropriate strategies for comparing and ordering fractions. I like to encourage the preservice and in-service teachers that I work with to use conceptual strategies for fraction comparison. The following Fraction Clothesline activity, adapted from the Illuminations Fractional Clothesline lesson (NCTM 2015) is an example of an engaging activity that requires thought and communication to compare fractions, not just procedures.

Required Materials

• About 25 feet of rope
  
• Clothespins
  
• Fraction cards—Make a set of fraction cards that are appropriate for your students. For example, younger students may work with halves, thirds, and fourths; older students may work with mixed numbers and improper fractions. Be sure to include 0 (or the smallest number) and 1 (or the largest number that you want to be on the opposite end of your number line). Also include some equivalent fractions, such as 1/3 and 2/6.

Activity Description

Place small groups of the fraction cards with small groups of students. Ask them to put their cards in order from least to greatest. This will help them get a sense of the numbers they have and activate their prior knowledge for comparing fractions. Ask students to explain how they decided which fraction was the smallest. Which was the largest (etc.)? Then ask the whole class, Who has the largest number? Who has the smallest number? Ask students to place those numbers on the clothesline, using the clothespins.

Then ask which fractions would be easiest to place next. Students often mention halves and fourths. Have students place those on the number line. Then ask the class if everyone agrees with their placement. Make sure that students pay attention to the spacing, not just the order. For example, 1/4 should be halfway between 0 and 1/2. Once the easiest fractions are placed and the class agrees that they are placed correctly, allow all students to place a fraction from their group on the clothesline.

After all fractions are placed, ask the class to look at the clothesline and determine whether students agree with the placement of the fractions. Select two fractions that you feel are too close together or too far apart and ask about their placement. Select individual fractions and ask students to explain their strategy for placing the fraction where they did. This conversation may encourage students to visually divide the number line into equal-size parts or to use alternative strategies to compare fractions, such as closeness to 1/2 or 1.

Extensions

This is just one example of an activity to encourage the development of students’ fraction number sense. This activity can be adapted by using simpler fractions, such as halves, thirds, and fourths. It can also be extended by including mixed numbers and improper fractions. In one classroom (Annenberg Learner 1997), a teacher kept the clothesline up in her classroom and added equivalent fractions, decimals, and percents as they came up in class conversations throughout the year.

What activities do you use to develop students’ conceptual understanding for fraction comparison? I invite you to share, in the comments section below, your students’ work on this activity or other activities that you use to encourage conceptual understanding of fraction comparison. I look forward to hearing about what is happening in your classrooms.

Your Turn

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References

Annenberg Learner. 1997. Teaching Math: A Video Library, 5–8. Fraction Tracks [DVD]. http://www.learner.org/resources/series33.html

Common Core State Standards Initiative (CCSSI). 2010. Common Core State Standards for Mathematics (CCSSM). Washington, DC: National Governors Association Center for Best Practices and the Council of Chief State School Officers. http://www.corestandards.org/wp-content/uploads/Math_Standards.pdf

National Council of Teachers of Mathematics (NCTM). 2015. Illuminations Resources for Teaching Math. “Fractional Clothesline.” http://illuminations.nctm.org/Lesson.aspx?id=2867

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.