Grade 5-8 Instruction
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 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.
Understanding multiple representations for numbers is a crucial precursor to solving many of the problems students encounter. Toward this end, students can represent fractions, decimal, 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, 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.
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
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?
Arithmetically, the fraction 2/3 becomes necessary as the only solution to the whole number problem 2 ÷ 3, such as in the real world situation in which two pizzas are divided among three people. The integer –1 becomes necessary so that the problem 2 – 3 has a solution, such as when a player loses three points in a game when he or she has only two points. The need to measure more precisely than to the nearest inch gives rise to numbers like 3 5/8 inches.
As students expand their mathematical horizons to include fractions, decimals, integers, and rational numbers, as well as the basic operations for each, they need to understand both the common ideas underlying these number systems and the differences among them. For example, to compare 2/3 and 3/4, students can use concrete materials to represent them as 8/12 and 9/12, respectively, and then conclude that 8/12 is less than 9/12, since 8 is less than 9. Thus, they learn that comparing fractions is like comparing whole numbers once common denominators have been identified.