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 Grade K-4 Instruction 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 critical 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. 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 childrens' 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). Fig. 12.1: Students construct a whole when given a part . 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: Estimating the Color shaded part . Children need to use physical materials to explore equivalent fractions and compare fractions. For example, with folded paper strips as shown below, children can easily see that 1/2 is the same amount as 3/6 and that 2/3 is smaller than 3/4. . Children also should use reasoning to determine that 1/5 is larger than 1/8 or 1/10 since fifths are larger than eighths 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 5/8. They can also explore fractions that are close to 0, close to 1/2, or close to 1. Experiences with the relative size of numbers promote the development of number sense. Fig. 12.4: Sort the fractions into the regions below . 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.