In grades K-4, the mathematics curriculum should include fractions and decimals so that students can--

• develop concepts of fractions, mixed numbers, and decimals;
• develop number sense for fractions and decimals;
• use models to relate fractions to decimals and to find equivalent fractions;
• use models to explore operations on fractions and decimals;
• apply fractions and decimals to problem situations.

Focus

Fractions and decimals represent a significant extension of children's knowledge about numbers. When children possess a sound understanding of fraction and decimal concepts, they can use this knowledge to describe real-world phenomena and apply it to problems involving measurement, probability, and statistics. An understanding of fractions and decimals broadens students' awareness of the usefulness and power of numbers and extends their knowledge of the number system. It is critical in grades K-4 to develop concepts and relationships that will serve as a foundation for more advanced concepts and skills.

The K-4 instruction should help students understand fractions and decimals, explore their relationship, and build initial concepts about order and equivalence. Because evidence suggests that children construct these ideas slowly, it is crucial that teachers use physical materials, diagrams, and real-world situations in conjunction with ongoing efforts to relate their learning experiences to oral language and symbols. This K-4 emphasis on basic ideas will reduce the amount of time currently spent in the upper grades in correcting students' misconceptions and procedural difficulties.

Discussion

All work at the K-4 level should involve fractions that are useful in everyday life, that is, fractions that can be easily modeled. Initial work with fractions should draw on children's experiences in sharing, such as asking four children to share a candy bar. The concept of a unit and its subdivision into equal parts is fundamental to understanding fractions and decimals, whether the quantity to be divided is a rectangular candy bar, a handful of jelly beans, or a piece of licorice. Initial instruction needs to emphasize oral language (one-fourth, two-thirds) and connect it to the models. Many productive activities can be used for initial instruction, such as folding paper strips into equal parts and describing the kind of parts (e.g., fifths) and the amount being considered (e.g., two-fifths).

In another activity, students construct a whole when given a part (fig. 12.1).

Fig. 12.1

Counting forward and backward by unit fractions (1/2, 1/3, 1/4, etc.) helps children build a strong awareness of fraction sequences and prepares them for both mental and paper-and-pencil computation. One relevant, thought-provoking activity appears in figure 12.2.

Fig. 12.2

Divide the class into two groups. Let one group be the "mixed" group and the other the "improper" group. Have each group count the number of thirds shown:

Fraction symbols, such as 1/4 and 3/2, should be introduced only after children have developed the concepts and oral language necessary for symbols to be meaningful and should be carefully connected to both the models and oral language.

An awareness of the relative size of fractions fosters number sense and enhances basic understandings. The following activity (see fig. 12.3) helps children think about the quantity represented by a fraction.

Fig. 12.3

Children need to use physical materials to explore equivalent fractions and compare fractions. For example, with folded paper strips, children can easily see that 1/2 is the same amount as 3/6 and that 2/3 is smaller than 3/4. (See fig. 12.3a)

Fig. 12.3a

Children also should use reasoning to determine that 1/5 is larger than 1/8 or 1/10 since fifths are larger than eights or tenths. Students should recognize that, for example, 3/4 is between 1/2 and 1 and that 1/3 is large compared to 1/10, about the same size as 1/4, and small compared to 5/6. They can also explore fractions that are close to 0, close to 1/2, or close to 1, as in figure 12.4. Experiences with the relative size of numbers promote the development of number sense.

Fig. 12.4

Physical materials should be used for exploratory work in adding and subtracting basic fractions, solving simple real-world problems, and partitioning sets of objects to find fractional parts of sets and relating this activity to division. For example, children learn that 1/3 of 30 is equivalent to "30 divided by 3," which helps them relate operations with fractions to earlier operations with whole numbers.

In grades K-4, children begin to encounter decimals in many situations--with calculators and metric measures, in tables of data, and in such daily activities as using a digital stopwatch. Thus, the curriculum needs to emphasize the development of decimal concepts.

The approach to decimals should be similar to work with fractions, namely, placing a strong and continued emphasis on models and oral language and then connecting this work with symbols. This is necessary if students are to make sense of decimals and use them insightfully. Exploring ideas of tenths and hundredths with models can include preliminary work with equivalent decimals (fig. 12.5), counting sequences, the comparing and ordering of decimals, and addition and subtraction.

Fig. 12.5

Decimal instruction should include informal experiences that relate fractions to decimals so that students begin to establish connections between the two systems. For example, if students recognize that 1/2 is the same amount as 0.5, they can use this relationship to determine that 0.4 and 0.45 are a little less than 1/2 and that 0.6 and 0.57 are a little more than 1/2. Such activities help children develop number sense for decimals.