Number and Operations Standard for Grades 6–8

In grades 6–8, students should deepen their understanding of fractions, decimals, percents, and integers, and they should become proficient in using them to solve problems. By solving problems that require multiplicative comparisons (e.g., "How many times as many?" or "How many per?"), students will gain extensive experience with ratios, rates, and percents, which helps form a solid foundation for their understanding of, and facility with, proportionality. 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 lower grades and in everyday life. Students' facility with rational numbers and proportionality can be developed 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.

Understand numbers, ways of representing numbers, relationships among numbers, and number systems

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. For more discussion of useful representations for rational numbers, see the "Representation" section of this chapter.

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, 1010 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 also can 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 a 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—but also in new ways. For example, 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 operators (e.g., multiplying by 3/8 means generating a number that is 3/8 of the original number). In the middle grades, students also need to deepen their understanding of decimal numbers and extend the range of numbers and tasks with which they work. The foundation of students' work with decimal numbers must be an understanding of whole numbers and place value. In grades 3–5, students should have learned to think of decimal numbers as a natural extension of the base-ten place-value system to represent quantities less than 1. In grades 6–8, they should also understand decimals as fractions whose denominators are powers of 10. The absence of a solid conceptual foundation can greatly hinder students. Without a solid conceptual foundation, students often think about decimal numbers incorrectly; they may, for example, think that 3.75 is larger than 3.8 because 75 is more than 8 (Resnick et al. 1989). Students also need to interpret decimal numbers as they appear on calculator screens, where they may be truncated or rounded.

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 or 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 than 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

Percents, which can be thought about in ways that combine aspects of both fractions and decimals, offer students another useful form of rational number. Percents are particularly useful when comparing fractional parts of sets or numbers of unequal size, and they are also frequently encountered in problem-solving situations that arise in everyday life. As with fractions and decimals, conceptual difficulties need to be carefully addressed in instruction. In particular, percents less than 1 percent and greater than 100 percent are often challenging, and most students are likely to benefit from frequent encounters with problems involving percents of these magnitudes in order to develop a solid understanding.

Attention to developing flexibility in working with rational numbers contributes to students' understanding of, and facility with, proportionality. Facility with proportionality involves much more than setting two ratios equal and solving for a missing term. It involves recognizing quantities that are related proportionally and using numbers, tables, graphs, and equations to think about the quantities and their relationship. Proportionality is an important integrative thread that connects many of the mathematics topics studied in grades 6–8. Students encounter proportionality when they study linear functions of the form y = kx, when they consider the distance between points on a map drawn to scale and the actual distance between the corresponding locations in the world, when they use the relationship between the circumference of a circle and its diameter, and when they reason about data from a relative-frequency histogram.

In the middle grades, students should continue to work with whole numbers in a variety of problem-solving settings. They should develop a sense of the magnitude of very large numbers (millions and billions) and become proficient at reading and representing them. For example, they should recognize and represent 2 300 000 000 as 2.310⁹ in scientific notation and also as 2.3 billion. Contexts in which large numbers arise naturally are found in other school subjects as well as in everyday life. For example, a newspaper headline may proclaim, "Clean-Up Costs from Oil Spill Exceed $2 Billion!" or a science textbook may indicate that the number of red blood cells in the human body is about 1.910¹³. Students also need to understand various forms of notation and recognize, for instance, that the number 2.510¹¹ might appear on a calculator as 2.5E11 or 2.5 11, depending on the make and model of the machine. Students' experiences in working with very large numbers and in using the idea of orders of magnitude will also help build their facility with proportionality.

Students can also work with whole numbers in their study of number theory. Tasks, such as the following, involving factors, multiples, prime numbers, and divisibility, can afford opportunities for problem solving and reasoning.

1. Explain why the sum of the digits of any multiple of 3 is itself divisible by 3.
2. A number of the form abcabc always has several prime-number factors. Which prime numbers are always factors of a number of this form? Why?

Middle-grades students should also work with integers. In lower grades, students may have connected negative integers in appropriate ways to informal knowledge derived from everyday experiences, such as » below-zero winter temperatures or lost yards on football plays. In the middle grades, students should extend these initial understandings of integers. Positive and negative integers should be seen as useful for noting relative changes or values. Students can also appreciate the utility of negative integers when they work with equations whose solution requires them, such as 2x  + 7 = 1.

Dynamically Linking Area and Multiplication

Understand meanings of operations and how they relate to one another

In the middle grades, students should continue to refine their understandings of addition, subtraction, multiplication, and division as they use these operations with fractions, decimals, percents, and integers. Teachers need to be attentive to conceptual obstacles that many students encounter as they make the transition from operations with whole numbers.

Multiplying and dividing fractions and decimals can be challenging for many students because of problems that are primarily conceptual rather than procedural. From their 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 or decimals (Graeber and Tanenhaus 1993). Teachers should check to see if their students harbor this misconception and then take steps to build their understanding.

Figure 6.3 illustrates how students might use dynamic geometry software to examine how the product 3y is affected by the magnitude of y for nonnegative y-values. In this illustration, the product is represented as the area of a 3y rectangle. As a student changes the value of y by dragging a point along the vertical axis, the area of the rectangle changes. Referring to the area of the 31 rectangle, students can see that the area of the 3y rectangle is smaller when y is less than 1 and larger when y is greater than 1. That is, in contrast to the expectation that multiplication makes bigger, multiplying 3 by a number smaller than 1 results in a product that is less than 3.

Fig. 6.3. This dynamic area model shows the effect of multiplying by a number less than 1.

Fig. 6.4. Using benchmarks to estimate the results of a fraction computation

The division of fractions has traditionally been quite vexing for students. » Although "invert and multiply" has been a staple of conventional mathematics instruction and although it seems to be a simple way to remember how to divide fractions, students have for a long time had difficulty doing so. Some students forget which number is to be inverted, and others are confused about when it is appropriate to apply the procedure. A common way of formally justifying the "invert and multiply" procedure is to use sophisticated arguments involving the manipulation of algebraic rational expressions—arguments beyond the reach of many middle-grades students. This process can seem very remote and mysterious to many students. Lacking an understanding of the underlying rationale, many students are therefore unable to repair their errors and clear up their confusions about division of fractions on their own. An alternative approach involves helping students understand the division of fractions by building on what they know about the division of whole numbers. If students understand the meaning of division as repeated subtraction, they can recognize that 246 can be interpreted as "How many sets of 6 are there in a set of 24?" This view of division can also be applied to fractions, as seen in figure 6.5. To solve this problem, students can visualize repeatedly cutting off 3/4 yard of ribbon. The 5 yards of ribbon would provide enough for 6 complete bows, with a remainder of 2/4, or 1/2, yard of ribbon, which is enough for only 2/3 of a bow. Carefully sequenced experiences with problems such as these can help students build an understanding of division of fractions.

Fig. 6.5. Using the idea of division as repeated subtraction to solve a problem involving fractions

Students' understanding of operations with fractions, decimals, and integers can also be enhanced as they examine the validity and utility of properties of operations, such as the commutative and associative properties of addition and multiplication, with which they are familiar from their experiences with whole numbers. These properties can be used to simplify many computations with fractions, for example, 3(4/52/3) can be expressed as (32/3)4/5, which makes the calculation easier. The familiar distributive properties for whole-number operations can also be applied to fractions, decimals, and integers. Students already know that 326 can be computed by decomposing 26 and using the distributive property of multiplication over addition to get (320 + 36); in a similar fashion, they can compute 32 1/2 by expressing it as (32 + 31/2).

From earlier work with whole numbers, students should be familiar with the inverse relationship between the operation pairs of addition-subtraction and multiplication-division. In the middle grades, they can continue to apply this relationship as they work with fractions, decimals, » and integers. In the middle grades, students should also add another pair to their repertoire of inverse operations—squaring and taking square roots. In grades 6–8, students frequently encounter squares and square roots when they use the Pythagorean relationship. They can use the inverse relationship to determine the approximate location of square roots between whole numbers on a number line. Figure 6.6 illustrates this reasoning for and .

Fig. 6.6. Locating square roots on a number line

Compute fluently and make reasonable estimates

In grades 6–8, students should acquire computational fluency—the ability to compute efficiently and accurately—with fractions, decimals, and integers. Teachers should help students learn how to decide when an exact answer or an estimate would be more appropriate, how to choose the computational method that would be best to use, and how to evaluate the reasonableness of answers to computations. Most calculations should arise as students solve problems in context. Students should consider the features of the problem and the likely use of an answer to a calculation in deciding whether an exact answer or an estimate is needed, and then select an appropriate mode of calculation from among mental calculation, paper-and-pencil methods, or calculator use. For example, the cost of 1 1/4 pounds of cheese at $2.40 a pound can be found mentally, whereas the cost of 1.37 pounds of cheese at $2.95 a pound might be estimated, although a calculator would probably be the preferred tool if an exact answer were needed. Students should regularly analyze the answers to their calculations to evaluate their reasonableness.

Through teacher-orchestrated discussions of problems in context, students can develop useful methods to compute with fractions, decimals, percents, and integers in ways that make sense. Students' understanding of computation can be enhanced by developing their own methods and sharing them with one another, explaining why their methods work and are reasonable to use, and then comparing their methods with the algorithms traditionally taught in school. In this way, students can appreciate the power and efficiency of the traditional algorithms and also connect them to student-invented methods that may sometimes be less powerful or efficient but are often easier to understand.

Instruction in solving proportions should include methods that have a strong intuitive basis. The so-called cross-multiplication method can be developed meaningfully if it arises naturally in students' work, but it can also have unfortunate side effects when students do not adequately understand when the method is appropriate to use. Other approaches to solving proportions are often more intuitive and also quite powerful. For example, when trying to decide which is the better buy—12 tickets for $15.00 or 20 tickets for $23.00—students might choose to use a scaling strategy (finding the cost for a common number of tickets) or a unit-rate strategy (finding the cost for one ticket). (See fig. 6.7.)

Fig. 6.7. Two approaches to solving a problem involving proportions

As different ways to think about proportions are considered and discussed, teachers should help students recognize when and how various ways of reasoning about proportions might be appropriate to solve problems. Further discussion about ways to approach a contextualized problem involving a proportional relationship is found in the "Connections" section of this chapter.