In grades 5-8, the mathematics curriculum should include the continued development of number and number relationships so that students can--

• understand, represent, and use numbers in a variety of equivalent forms (integer, fraction, decimal, percent, exponential, and scientific notation) in real-world and mathematical problem situations;
• develop number sense for whole numbers, fractions, decimals, integers, and rational numbers;
• understand and apply ratios, proportions, and percents in a wide variety of situations;
• investigate relationships among fractions, decimals, and percents;
• represent numerical relationships in one- and two-dimensional graphs.

Focus

The use of concise symbols and language to represent numbers is a significant historical and practical development. In the middle school years, students come to recognize that numbers have multiple representations, so the development of concepts for fractions, ratios, decimals, and percents and the idea of multiple representations of these numbers need special attention and emphasis. The ability to generate, read, use, and appreciate multiple representations of the same quantity is a critical step in learning to understand and do mathematics.

As students progress through middle school, they build their knowledge of rational numbers as important both in their own right and as a foundation for rational forms in algebra. Ratio, proportion, and percent are introduced and developed in grades 5-8. In exploring these topics, students have many opportunities to develop their ability to reason proportionally.

To provide students with a lasting sense of number and number relationships, learning should be grounded in experience related to aspects of everyday life or to the use of concrete materials designed to reflect underlying mathematical ideas. Students should encounter number lines, area models, and graphs as well as representations of numbers that appear on calculators and computers (e.g., forms of scientific notation). Students should learn to identify equivalent forms of a number and understand why a particular representation is useful in a given setting.

Discussion

Understanding multiple representations for numbers is a crucial precursor to solving many of the problems students encounter. Toward this end, students can represent fractions, decimals, and percents in a variety of meaningful situations, thereby learning to move flexibly among concrete, pictorial, and abstract representations. Students should understand these numbers and their representations, the relationships among them, and the advantages and disadvantages of each.

Teachers should strive to make this process consistently positive; too often, students are taught that 2/4 = 1/2, only to be informed later that 2/4 is a "wrong answer" when the "correct" answer is 1/2. Discussing the appropriateness of certain representations in a given situation, such as the fact that it is better to write "68/100 dollars" on a check than reduce to "17/25 dollars," helps students recognize that there is no single, uniform way to represent a fraction but that the "best" way depends largely on the situation. Students learn, for example, that 15/100, 3/20, 0.15, and 15% are all representations of the same number, appropriate for a fraction of a dollar on a bank check, the probability of winning a game, the tax on a purchase of $2.98, and a discount, respectively. Similarly, they learn that +8, 8/1, and 8.0 are all appropriate representations of the same number, depending on whether they are subtracting integers, adding fractions, or labeling a coordinate axis with rational numbers.

Exponents and scientific notation give further examples of the elegance and complexities of concise notation. Numbers like 2⁽³⁰⁾ in the denominator of a fraction in a probability problem. 1.06⁽¹⁰⁾, as a factor in a population growth problem, 9.3 x 10⁸ in a science problem, and $3.46 million in a newspaper article are approximate representations, each of which is best suited to its own context. Calculators and computers can display forms like 7.23471 07, thus requiring that students learn other forms of scientific notation. Students also should recognize the difficulties inherent in various representations, such as the expression of 1/3 as a percent, 5.7 x 10⁹ as an ordinary decimal, or 5.999999999 on a calculator as 6. They also need to grasp some sense of the "infinite" quality of decimals.

Area models are especially helpful in visualizing numerical ideas from a geometric point of view. For example, area models can be used to show that 8/12 is equivalent to 2/3, that 1.2 x 1.3 = 1.56, and that 80% of 20 is 16. See figure 5.1.

Fig. 5.1. Area models for fractions, decimals, and percents

Later, students can extend area models to the study of algebra, probability, dimension analysis in measurement situations, and other more advanced subjects.

Concrete materials and representational models (pictures) should be used to continue the study of place value initiated in grades K-4. For example, students should see that the model in figure 5.2 represents 346 or 3.46, according to whether a small or a large square represents a unit.

Fig. 5.2. Base-ten blocks

Once students are familiar with a particular type of representation, it can be generalized and the equivalence of different representations for the same number can be established. This process should be approached conceptually before the computational techniques that convert one to another are developed.

In grades 5-8, number sense should be fostered through such questions as, How big is a million? or Could you carry a suitcase containing a million dollar bills? Operation sense should be expanded with such examples as, Is 2/3 x 5/4 more or less than 2/3? More or less than 5/4? Why is the product of a negative integer times a negative integer a positive integer?

Patterns that emerge when students examine terminating and repeating decimals are particularly appropriate for investigation in the middle grades. Questions about decimal expansions for fractions readily invite exploration: Which expansions are terminating, which are repeating, and which are nonrepeating? Which delay and then repeat? What is the relationship among expansions for families of fractions, such as 1/7, 2/7, 3/7, ..... , 6/7?

Just as K-4 students learn to represent single quantities with a number, middle school students should learn to represent comparison of quantities using ratios in various forms. Ratios should be introduced gradually through discussing the many situations in which they occur naturally. It takes little effort to relate these situations to students' interests: "If 245 of a company's 398 employees are women, how many of its 26 executives would you expect to be women?

Through these practical exercises, students should come to recognize that ratios are not directly measurable but that they contain two units and that the order of the items in the ratio pair in a proportion is critical. Thus, 23 persons per square mile is very different from 23 square miles per person. They need to understand the multiplicative nature of ratios to avoid such mistakes as "18 boys:15 girls :: 19 boys:16 girls" and to recognize nonexamples of ratios, for instance, doubling linear dimensions does not double area. They should begin to build other major ideas, including proportion, slope, and rational number. Ratios themselves can be extended to include more than two numbers, as in a recipe that has five ingredients. Spreadsheets provide an excellent means of working with ratios composed of more than two numbers, as illustrated in figure 5.3--students can see what happens to the amount of ingredients in a waffle recipe as the number of servings changes.

Fig. 5.3. Spreadsheet for waffle recipe

Graphs can be used to show relationships involving numbers, including number line graphs and two-dimensional graphs, which can be expanded over the middle school years from whole number coordinates to rational numbers. For example, the following graph (fig. 5.4) comparing rainfall in two Canadian cities can generate discussions about the best time to visit the cities and other matters of interest.

Fig. 5.4. Rainfall graph

Over grades 5-8, students should build a sense of number and of numerical relationships that gives them the flexibility to deal with numbers in many forms.