

Children’s Strategies in Ordering Rational Numbers  
Edited
by Glenda Lappan, Michigan State University, East Lansing. MI 48823 Prepared
by Thomas Post and Kathleen Cramer, 

Our
comments here are based on interviews from two teaching experiments with
fourth and fifthgrade children. The students were taught many aspects of
fractions using a variety of manipulative materials including circular and
rectangular pieces, Cuisenaire rods, number lines, and chips. Each student
had his or her own materials. Each worked independently, in small groups,
and as a whole class, spending much time talking about and performing a
wide variety of fractionrelated tasks.
One of the more fundamental rationalnumber notions that we explored with students was that of fraction order and equivalence. The following comments reflect the nature of children’s thinking as they grapple with these concepts. For all children, their previous wholenumber schemas have influenced their ability to reason about the order relation for fractions. Children reasoned that onethird is greater than onehalf because three is greater than two. Children often regressed to additive or subtractive strategies when comparing fractions. Twofifths equaled fiveeighths because 2 + 3 = 5 and 5 + 3 = 8. They considered threefourths and twothirds to be equal because the difference between the numerator and denominator in each fraction was one. For some children the influence of wholenumber ideas on their thinking about rational numbers was persistent. Perhaps the fourth and fifth graders we worked with were hesitant to use multiplicative strategies because the concept of multiplication had not yet been fully developed. In one sense it seems logical that they would use the arithmetic ideas most familiar to them. We believe many experiences with physical models of fractions are essential to overcome the influence that wholenumber schemas have on their rationalnumber reasoning. Although one strategy is sufficient to order whole numbers, multiple strategies (depending on whether fractions have the same numerators, the same denominators, or nothing the same) are needed to order fractions. This idea was new to children and caused considerable difficulty. In the identical numerator situation (2/3, 2/5) the ordering decision is made by comparing the size of pieces. (Thirds are larger than fifths so 2/3 > 2/5.) In the identical denominator situation (4/11, 7/11) the ordering decision is made by comparing the number of pieces. (The pieces are the same size, and four pieces are fewer than seven pieces. Therefore, 4/11 < 7/11.) Children initially confused these issues. When asked to compare 3/9 and 4/9, some argued, "3/9 is greater than 4/9 because in fourths the pieces are smaller and it would take more of them to equal the whole unit," or similarly, "because in 3/9 the pieces are larger because there are fewer of them." The words more and greater can lead to misunderstandings. More can mean more pieces in the partitioned whole, or more can mean more area covered by each part. Greater can mean a greater number of parts in the partitioned whole or a greater fraction size. When asking a student to select the greater of two fractions (or which of two fractions is more), the correctness of the answer will depend on how the student interprets the words greater or more. For example, many children responded that 1/3 is greater than 1/2 because you have more pieces when you divide the whole into thirds than when you divide the whole into halves. Children often followed up our ordering questions with their own question: "Do you mean size of piece or number of pieces?" They needed clarification of the word more before they could give an answer. Focusing on the size of piece, 1/2 is greater than 1/3; if the number of pieces is the variable considered, then 1/3 is greater than 1/2. Students with strong mental images developed through extensive experiences with concrete aids employed ordering strategies that had not been taught. An example of a studentcreated strategy that we later named the residual strategy is the following: When comparing 5/6 and 7/8 the student observed that both fractions have one piece left over (to make a whole). Since 1/6 is greater than 1/8, more is left over for 5/6; thus, it is the smaller fraction. Students also created what we called the transitive strategy: "3/7 is less than 5/9 because 3/7 doesn't cover half the unit and 5/9 covers over half." Students who created their own strategies for comparing fractions and those who mastered the multiple strategies for comparing fractions demonstrated a quantitative understanding of fractions that would enable them to think reasonably about them. These students would not make the common error of adding numerators and denominators in, for example, the problem 1/2 + 1/3. They could reason that a sum of 2/5 would be unreasonable because it is less than 1/2, and the answer must be greater than 1/2 because one of the addends is 1/2. Thinking quantitatively about fractions depends on the internal images children have of fractions and their ideas of order and equivalence. Our experience has shown that extended instruction with a variety of manipulative materials can help children overcome the initial influence that wholenumber ideas have on their thinking and enable them to think quantitatively about fractions. 

Activities
for Ordering Rational Numbers
The following samples of activities from our teaching experiments are designed to develop order and equivalence concepts using fraction circles similar to the ones shown in figure 1. Fraction circles can be easily constructed by duplicating the pattern on construction paper, laminating the paper for durability, and then cutting them apart. 







To
compare the size of fractions, such as 1/2 and 1/3 or 3/6 and 3/9, students
need to understand the inverse relationship between the number of pieces
into which the whole is divided and the size of the resulting pieces. By
using the fraction circles as a frame of reference, onethird can be shown
to be less than onehalf because when a circle is divided into three equal
parts, the size of each piece is smaller than either of the two pieces that
will result when the same size circle is divided into two equal parts. With
fractions, "more" pieces does mean "less" size.
Children can be led to discover this relationship. Figure 2 presents an activity to help children see this relationship and to come to understand how to order fractions with identical numerators. In the first example, the brown and red pieces are compared. Three brown pieces are needed to cover the whole circular area, whereas twelve red pieces are needed to cover the same circular area. More red pieces are needed than brown ones, and the red pieces are smaller in size. This information is recorded on the chart. Three more comparisons are shown in figure 2. Notice the last question in this activity. After several identicalnumerator comparisons, students are asked to make an identicaldenominator comparison. Children need to realize that different ordering strategies are used for different types of fractions. They often want to apply the same strategy in all situations. For this reason, it is helpful to include identicaldenominator comparisons among identicalnumerator comparisons in all ordering activities that you do with children. 

Fig. 2




The next activity introduces children to the idea of equivalent fractions. By using fraction circles, equivalent fractions are initially defined as fractions that take up the same amount of area. For example, by superimposing two blue pieces over one yellow piece, children can see that 2/4 is the same amount as 1/2. By "creating" the equivalence chart (fig. 3) cooperatively, children will construct for themselves many of the common equivalences.  


Fig. 3 Equivalence chart






The activity should start by finding fractions equal to 6/12. Start by defining the whole circular area as the unit. Ask children to show 6/12 of the whole circular area with their pieces. Now ask if they can find any other ways, using pieces of another color, to cover those six reds exactly. Students may try to cover 6/12 with the whites and see that it cannot be done. They will easily see that one yellow covers 6/12, and by exploring they will find that four grays and two blues also do the job. Ask students to name the fractions for these pieces and record that information on the chart. The rest of the chart can be completed in a similar manner, perhaps starting with 12/12 next. Once the chart is completed, a large classroom model of it can be posted and children can refer to it when the need arises. Using just these two fractioncircle ideas, teachers can create a variety of activities that use these circles to teach fraction ideas. Many other physical models are very useful, such as fraction bars and Cuisenaire rods. What is important is that initial fraction concepts be introduced concretely so children will be able to operate meaningfully on fractions represented abstractly. References Behr, Merlyn J., Ipke Wachsmuth, Thomas R. Post, and Richard Lesh. "Order and Equivalence of Rational Numbers: A Clinical Teaching Experiment." Journal for Research in Mathematics Education 15 (November 1984):32341. Post, Thomas R., Merlyn J. Behr, and Richard Lesh. "Researchbased Observations about Children’s Learning of Rational Number Concepts." FOCUS: On Learning Problems in Mathematics 8 (Winter, 1986):3948. Post, Thomas R., Ipke Wachsmuth, Richard Lesh, and Merlyn J. Behr. "Order and Equivalence of Rational Numbers: A Cognitive Analysis." Journal for Research in Mathematics Education 16 (January 1985): 1836. 
