Grade 6-8 Instruction
The study of rational numbers in the middle grades should build on students’ prior knowledge of whole-number concepts and skills and their encounters with fractions, decimals, and percents in the lower grades and in everyday life. Students’ facility with rational numbers and proportionality can develop in concert with their study of many topics in the middle grades curriculum. For example, students can use fractions and decimals to report measurements, to compare survey responses from samples of unequal size, to express probabilities, to indicate scale factors for similarity, and to represent constant rate of change in a problem or slope in a graph of a linear function.
In the middle grades students should become facile in working with fractions, decimals, and percents. Teachers can help students deepen their understanding of rational numbers by presenting problems, such as those in figure 6.1 that call for flexible thinking.
Fig. 6.1: Problems that require students to think flexibly about rational numbers
At the heart of flexibility in working with rational numbers is a solid understanding of different representations for fractions, decimals, and percents. In grades 3-5, students should have learned to generate and recognize equivalent forms of fractions, decimals, and percents, at least in some simple cases. In the middle grades, students should build on and extend this experience to become facile in using fractions, decimals, and percents meaningfully. Students can develop a deep understanding of rational numbers through experiences with a variety of models, such as fraction strips, number lines, 10 x 10 grids, area models, and objects. These models offer students concrete representations of abstract ideas and support students’ meaningful use of representations and their flexible movement among them to solve problems.
As they solve problems in context, students can also consider the advantages and disadvantages of various representations of quantities. For example, students should understand not only that 15/100, 3/20, 0.15, and 15 percent are all representations of the same number but also that these representations may not be equally suitable to use in a particular context. For example, it is typical to represent a sales discount as 15%, the probability of winning a game as 3/20, a fraction of a dollar in writing a check as 15/100, and the amount of the 5 percent tax added to the purchase of $2.98 as $0.15.
In the middle grades, students should expand their repertoire of meanings, representations, and uses for nonnegative rational numbers. They should recognize and use fractions not only in the ways they have in lower grades – as measures, quantities, parts of a whole, locations on a number line, and indicated divisions, they should encounter problems involving ratios (e.g., 3 adult chaperones for every 8 students), rates (e.g., scoring a soccer goal on 3 of every 8 penalty kicks), and operations (e.g., multiplying by 3/8 means generating a number that is 3/8 of the original number).
In the lower grades, students should have had experience in comparing fractions between 0 and 1 in relation to such benchmarks as 0, 1/4, 1/2, 3/4, and 1. In the middle grades, students should extend this experience to tasks in which they order and compare fractions, which many students find difficult. For example, fewer than one-third of the thirteen-year-old U.S. students tested in the National Assessment of Educational Progress (NAEP) in 1988 correctly chose the largest number from 3/4, 9/16, 5/8, and 2/3 (Kouba, Carpenter, and Swafford 1989). Students’ difficulties with comparison of fractions have also been documented in more recent NAEP administrations (Kouba, Zawojewski, and Strutchens 1997). Visual images of fractions as fraction strips should help many students think flexibly in comparing fractions. As shown in figure 6.2, a student might conclude that 7/8 is greater than 2/3 because each fraction is exactly "one piece" smaller that 1 and the missing 1/8 piece is smaller than the missing 1/3 piece. Students may also be helped by thinking about the relative locations of fractions and decimals on a number line.
Fig. 6.2: A student's reasoning about the sizes of rational numbers
Multiplying and dividing fractions and decimals can be challenging for many students because of problems that are primarily conceptual rather than procedural. From experience with whole numbers, many students appear to develop a belief that "multiplication makes bigger and division makes smaller". When students solve problems in which they need to decide whether to multiply or divide fractions or decimals, this belief has negative consequences that have been well researched (Greer 1992). Also, a mistaken expectation about the magnitude of a computational result is likely to interfere with students’ making sense of multiplication and division of fractions and decimals (Graeber and Tanenhaus 1993). Teachers should check to see if their students harbor this misconception and then take steps to build their understanding.
Teachers can also help students add and subtract fractions correctly by helping them develop meaning for numerator, denominator, and equivalence and by encouraging them to use benchmarks and estimation (see fig. 6.4). Students who have a solid conceptual foundation in fractions should be less prone to committing computational errors than students who do not have such a foundation.
Fig. 6.4: Using benchmarks to estimate the results of a fraction computation
Fig. 6.5: Using the idea of division as repeated subtraction to solve a problem involving fractions
Students should also develop and adapt procedures for mental calculations and computational estimation with fractions, decimals, and percents. Mental computation and estimation are also useful in many calculations involving percents. Because these methods often require flexibility in moving from one representation to another, they are useful in deepening students’ understanding of rational numbers and helping them think flexibly about these numbers.