In grades 68, 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.
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.
At the heart of flexibility in working with rational numbers is a solid understanding of different representations for fractions, decimals, and percents. In grades 35, 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 gradesas measures, quantities,
parts of a whole, locations on a number line, and indicated divisionsbut
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 35,
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 68, 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. »
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 68. 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.
- Explain why the sum of the digits of any multiple of 3 is itself divisible by 3.
- 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.
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.
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 expressionsarguments 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.
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 operationssquaring and taking square roots. In grades 68, 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 .
In grades 68, students
should acquire computational fluencythe ability to compute efficiently
and accuratelywith 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
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 buy12 tickets for $15.00 or 20 tickets for $23.00students 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.)
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