ZOOMING IN ON THE LEARNING PROGRESSIONS: FRACTIONS
Developing fraction sense is critical to the number sense in middle-grade mathematics. Skip Fennell, mathematics educator and past president of NCTM, stated in the December 2007 NCTM News Bulletin entitled Fractions is Foundational,
As students develop a sense of fractions, they will also recognize that they must approach the ordering of a set of fractions such as 7/8, 3/8, 5/8, and 9/8 differently from a set such as 3/5, 3/7, 3/4, and 3/8. Such experiences provide students with the background that they need to begin finding common denominators, creating equivalent fractions, and adding and subtracting fractions. Students also need to understand what really happens when they multiply and divide fractions. Far too many students are adept at carrying out these procedures without understanding that products typically get smaller when they multiply fractions and that quotients get larger when they divide them. Experiences with rate and proportion provide middle-grade students with everyday situations that involve fractions as well as contextual links to algebra.
This paragraph elicits important ideas about student learning of fractions and pedagogical implications for teachers. For example, understanding the meaning of fraction and the magnitude of fractions is critical to knowing how to order fractions. As Fennell mentions in his newsletter, ordering of a set of fractions such as 7/8, 3/8, 5/8, and 9/8 is different than a set such as 3/5, 3/7, 3/4, and 3/8 because it requires students to reason through the relative magnitude of these fractions. The first set all have the denominator of eighth so they can think about 1/8 as the unit fraction and see that they can order starting with 3/8 because it is 3 parts the unit fraction 1/8. This corresponds to the Common Core Math Standard:
Develop understanding of fractions as numbers.
Understand a fraction 1/b as the quantity formed by 1 part when a whole is partitioned into b equal parts; understand a fraction a/b as the quantity formed by a parts of size 1/b.
In addition, as students order the rest of the fractions, they can visualize the other fractions being a long a number line starting with 3/8, 5/8, 7/8, and then 9/8 hopefully also noticing that each fraction is 2/8 away from each other and that 9/8 is a fraction greater than a whole. This relates to the third-grade Common Core Math Standard:
Understand a fraction as a number on the number line; represent fractions on a number line diagram.
The second set of fractions is more complex because they do not have the same partitions or common denominators as the first set did. Yet, the numerators are all the same. Using that piece of information, students who have a strong integrated understanding of fraction will be able to reason through the ordering of the fraction. By examining and visualizing 3/5, 3/7, 3/4, and 3/8, they might think, “If I had to have 3 pieces of a chocolate bar and they were cut into fourth, fifth, seventh, and eighth, which piece would be the largest and the smallest?” They would see visually that % is the largest and 3/8 is the smallest size. Using that reasoning, they can order the fractions from smallest to largest starting from 3/8, 3/7, 3/5, and 3/4. They may be reasoning through benchmark fractions such as xh to make sense of the relative size. For example, they can use equivalent fractions and benchmark fractions to reason to easily see that 3/8
is less than У2 since 4/8 would be equivalent to Уг and % is greater than Уг because Уг is the same as 2/4 and so % is greater. This relates to the fourth-grade Common Core Math Standard:
Compare two fractions with different numerators and different denominators, e.g., by creating common denominators or numerators, or by comparing to a benchmark fraction such as 1/2. Recognize that comparisons are valid only when the two fractions refer to the same whole. Record the results of comparisons with symbols >, =, or <, and justify the conclusions, e.g., by using a visual fraction model.
One of the most common interpretations for rational numbers is the part to whole relationship (Barnett-Clarke, Fisher, Marks, & Ross, 2011). This interpretation can be visually represented with a continuous region model or a discrete set model. For example, XA can be represented as 1 out of 4 parts or can be represented as 1 out 4 discrete objects. For example, the fraction XA is read “sa boon-eh il” which means “out of 4 parts 1 part” which explicitly states the part to whole relationship. Partitioning a whole into equal parts is an important prerequisite to fractions. In this example, a student shows interesting and creative ways to cut the cake into 1/8. Although the area is not exact, these figures demonstrate an understanding of equal areas.
The second interpretation of a rational number is as a measure and can describe the amount of something (like a distance, an area, a capacity or volume, or a duration in time) in relation to the size of a unit, which is considered equal to 1 (Barnett-Clarke, Fisher, Marks, & Ross, 2011).
A number line is a mathematical model that provides a rich environment for understanding and reasoning about rational numbers (Moss and Case, 1999). The number line is a model that has generality in that it can easily represent the magnitude of numbers and also show the density of fractions. The third interpretation of a rational number is as a quotient and can indicate a division operation: That is, 3/4 can be seen as equivalent to the arithmetic expression 3 ^ 4 (Barnett-Clarke, Fisher, Marks, & Ross, 2011). This interpretation is illustrated in the research lesson highlighted below called the Sharing Brownie task. Two other interpretations include rational number as a ratio and as an operator, which we will focus in the next chapter.