Grade 3-5 Instruction
Through the study of various meanings and models for fractions – how fractions are related to each other and to the unit whole and how they are represented – students can gain facility in comparing fractions, often by using benchmarks such as 1/2 or 1. When students leave grade 5, they should understand the equivalence of fractions, decimals, and percents and the information each type of representation conveys. With these understandings and skills, they should be able to develop strategies for computing with familiar fractions and decimals.
During grades 3-5, students should built their understanding of fractions as parts of a whole and as division. They will need to see and explore a variety of models of fractions, focusing primarily on familiar fractions such as halves, thirds, fourths, fifths, sixths, eighths, and tenths. By using an area model in which part of a region is shaded, students can see how fractions are related to a unit whole, compare fractional parts of a whole, and find equivalent fractions. They should develop strategies for ordering and comparing fractions, often using benchmarks such as 1/2 and 1. For example, fifth graders can compare fractions such as 2/5 and 5/8 by comparing each with 1/2 – one is a little less than 1/2 and the other is a little more. By using parallel number lines, each showing a unit fraction and its multiples (see fig. 5.1), students can see fractions as numbers, note their relationship to 1, and see relationships among fractions, including equivalence. They should also be able to understand that between any two fractions, there is always another fraction.
Fig. 5.1: Parallel number lines with unit fractions and their multiples
Students in these grades should also investigate the relationship between fractions and decimals, focusing on equivalence. Through a variety of activities, they should understand that a fractions such as 1/2 is equivalent to 5/10 and that it has a decimal representation (0.5). As they encounter new meaning of a fraction – as a quotient of two whole numbers (1/2 = 1 divided by 2 = 0.5) – they can also see another way to arrive at this equivalence. By using the calculator to carry out the indicated division of familiar fractions like 1/4, 1/3, 2/5, 1/2, and 3/4, they can learn common fraction-decimal equivalence. They can also learn that some fractions can be expressed as terminating decimals but others cannot.
Students in grades 3-5 will need to be encouraged to routinely reflect on the size of an anticipated solution. Will 7 x 18 be smaller or larger than 100? If 3/8 of a cup of sugar is needed for a recipe and the recipe is doubled, will more or less than one cup of sugar be needed? Instructional attention and frequent modeling by the teacher can help students develop a range of computational estimation strategies including flexible rounding, the use of benchmarks, and front-end strategies. Students should be encouraged to frequently explain their thinking as they estimate.
A significant amount of instructional time should be devoted to rational numbers in grades 3-5. The focus should be on developing students’ conceptual understanding of fractions and decimals – what they are, how they are represented, and how they are related to whole numbers – rather than on developing computational fluency with rational numbers. Fluency in rational-number computation will be a major focus of grades 6-8.