### Which is Larger, 1/2 or 1/3? An Introduction to Comparing Unit Fractions

PurposesStudent collaboration; Student engagement; Student model building; Reasoning; Representation
DescriptionThe data for this RUanalytic were gathered on September 24th, 1993, during Session 3 of the fourth grade fraction intervention and the focus is on the students as they first begin specifically to investigate ways to compare fractions based on models constructed with the Cuisenaire rods (Steencken, 2001; Yankelewitz, 2009). This RUanalytic captures the beginning of several “visits” that the children have with the comparison task: Which is greater, 1/2 or 1/3, and by how much? This RUanalytic is the first in a series of three related RUanalytics. The second RUanalytic in the series focuses on how much larger 1/2 is than 1/3 (See “Comparing 1/2 and 1/3: Confusion about the Unit”). The third RUanalytic in this series focuses on comparing the models that students created to compare fractions and analyzing students’ justification for their choice of unit (See “Comparing Models and Justifying the Choice of Unit”).

The investigation considered here builds directly from the preceding sessions of the intervention during which the students constructed Cuisenaire rod models to support their ideas about basic relationships among whole numbers and fractions. Determining number names for each of the ten Cuisenaire rods based on a specific rod, or train of rods, that was assigned as the “unit” and given the number name 1 has been fundamental to their reasoning throughout these sessions. (See RUanalytics: “Establishing Norms and Creating a Mathematical Community,” “Task Design Prompts Fourth Grade Students to Use Multiple Forms of Reasoning,” “Fourth Graders Design a New Rod,” “Fourth Graders Reason by Cases as They Explore Fraction Ideas,” and “Fourth Graders Build Toward Proportional Reasoning.”)

This particular exploration is framed contextually within a real-life situation presented to the class by the researcher. Toward the end of Session 3, as shown in Events 1, 2, and 3 of this RUanalytic, Researcher Carolyn Maher relates an actual situation in which she introduces chocolate candy bars as a metaphor for thinking about sharing and comparing particular fractions. Robert B. Davis refers to such metaphors as assimilation paradigms (1984).

The students recognize from Researcher Maher’s narrative how half of a smaller candy bar is smaller than the corresponding half of a larger bar and agree that for equitable sharing, it is important to work with “the same candy bar” and that this should be understood as a rule when comparing fractions. Immediately following this discussion, Researcher Maher poses the question: Which is larger 1/2 or 1/3? She asks the students to use the rods to support their conclusion and reminds them of the importance of keeping the unit constant, as illustrated by the candy bar.

During the remaining minutes of the session, as covered in this RUanalytic, the students are intent on developing a plausible solution to this problem. In Events 4 through 6, they begin a search to find a unit, referred to as “1” or at times, “the whole,” that will allow them to physically compare the lengths of the rods corresponding to the fractions 1/2 and 1/3. Students are looking for a particular rod, or train of rods, to represent the unit, for which there are shorter rods that can legally represent 1/2 and 1/3, such that a train made of two of the rods with the number name 1/2 is the same length as the unit and a train made of three of the rods with the number name 1/3 is the same length as the unit.

After considerable exploration of the rods, several students develop a model that uses a train of an orange rod and a red rod for the unit. In Event 7, Laura and Jessica share their solution with the class. They build their model at the overhead, naming the train of orange and red rods 1, and giving number names to the other rods based on this unit. They claim that, with the given unit, the purple rod has the number name 1/3 and the dark green rod has the number name 1/2. They use these rods to show that since the dark green rod is longer than the purple rod, 1/2 is greater than 1/3.

At the end of Event 7 and in Event 8, Researcher Maher asks students to determine how much bigger 1/2 is than 1/3. This leads to an investigation of another fraction concept, quantifying the difference between 1/2 and 1/3. Laura and Jessica use their model to claim that 1/2 is one red rod bigger than 1/3. Researcher Maher then asks students to consider the number name that they should give the red rod when the train of orange and red rods represents the unit.

Event 8 takes place at the very end of Session 3 and students can be seen packing up their belongings. Researcher Maher asks the students again to consider how much bigger 1/2 is than 1/3. Alan responds that it is 1/6 bigger: "Because I know already that … three reds would make a dark green and if there are two dark greens to make the orange and the red rod then it would take six red rods to make the orange and the red rod."

Students revisit this idea again in Session 4 (September 27), Session 5 (September, 29), and Session 6 (October, 1). To see a detailed picture of how students investigate these ideas in these sessions, viewers are invited to access the following RUanalytics: “Comparing 1/2 and 1/3: Confusion about the Unit” and “Comparing Models and Justifying the Choice of the Unit.”

References:
Davis, R. B. (1984). Learning mathematics: The cognitive science approach to mathematics education. Greenwood Publishing Group.

Gerstein, M. (2015) Establishing norms and creating a mathematical community. Retrieved from: http://dx.doi.org/doi:10.7282/T30C4XH9.

Steencken, E. (2001). Tracing the growth in understanding of fraction ideas: A fourth grade case study (Doctoral dissertation).

Van Ness, C. K. and Alston, A. (2015). Comparing 1/2 and 1/3: Confusion about the Unit.

Van Ness, C. K. and Alston, A. (2015). Comparing Models and Justifying the Choice of Unit.