In grades 5-8, the mathematics curriculum should include the study of number systems and number theory so that students can--

• understand and appreciate the need for numbers beyond the whole numbers;
• develop and use order relations for whole numbers, fractions, decimals, integers, and rational numbers;
• extend their understanding of whole number operations to fractions, decimals, integers, and rational numbers;
• understand how the basic arithmetic operations are related to one another;
• develop and apply number theory concepts (e.g., primes, factors, and multiples) in real-world and mathematical problem situations.

Focus

The central theme of this standard is the underlying structure of mathematics, which bonds its many individual facets into a useful, interesting, and logical whole. Instruction in grades 5-8 typically devotes a great deal of time to helping students master a myriad of details but pays scant attention to how these individual facets fit together. It is the intent of this standard that students should come to understand and appreciate mathematics as a coherent body of knowledge rather than a vast, perhaps bewildering, collection of isolated facts and rules. Understanding this structure promotes students' efficiency in investigating the arithmetic of fractions, decimals, integers, and rationals through the unity of common ideas. It also offers insights into how the whole number system is extended to the rational number system and beyond. It improves problem-solving capability by providing a better perspective of arithmetic operations.

Instruction that facilitates students' understanding of the underlying structure of arithmetic should employ informal explorations and emphasize the reasons why various kinds of numbers occur, commonalities among various arithmetic processes, and relationships between number systems. Steen (1986, p. 6), past president of the Mathematical Association of America, articulated the spirit of this standard when he noted that "above all else, it [the mathematics curriculum] must not give the impression that mathematical and quantitative ideas are the product of authority or wizardry."

Number theory offers many rich opportunities for explorations that are interesting, enjoyable, and useful. These explorations have payoffs in problem solving, in understanding and developing other mathematical concepts, in illustrating the beauty of mathematics, and in understanding the human aspects of the historical development of number.

Discussion

As students reach grade 5, they begin to recognize--in both arithmetic and geometric settings--the need for numbers beyond whole numbers.

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. Such numbers as the 2 and are needed to describe the length of the diagonal of a square or the circumference of a circle. Encountering irrationals as nonexamples helps students appreciate rationals.

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 to 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. Yet in comparing -2 and -5 on a number line or a thermometer, students see that -2 is greater than -5, so they learn that comparing negative integers is different from comparing whole numbers.

Students should extend their knowledge of whole number operations other systems; for example, they need to see how 1.4 + 6.7 is related to 14 + 67 and how dividing - 12 by - 3 is related to dividing 12 by 3. Concrete materials and representational models, such as area models, should be used to provide a firm basis for understanding such ideas. The transition from whole numbers to fractions and decimals can be difficult for students. Although they may multiply the numerators and then the denominators, for example, they often do not understand why a similar procedure does not work in adding fractions. Concrete or representational models can help students clarify these anomalies.

Students should understand how analogies among structures can give a clearer picture of mathematics. For example, in contrasting the missing-addend interpretation of subtraction with the missing-factor interpretation of division, students learn that

and

This suggests that subtraction can be considered in terms of addition and that division can be thought of in terms of multiplication, illustrating the analogy that division is to multiplication as subtraction is to addition.

Relationships among operations are developed over many grades. In K-4, multiplication can be viewed as repeated addition and division can be considered repeated subtraction. However, because the multiplication of fractions and decimals is not repeated addition and division is not always repeated subtraction, students should be exposed to other interpretations of multiplication and division as well. For example, as shown in figure 5.1, 1.2 x 1.3 can be represented by an area model. In the later grades, students learn that the subtraction of integers is the same as adding the opposite and the division of fractions is the same as multiplying by the reciprocal.

Another goal of this standard is to offer meaningful answers to such questions as, Why can't we divide by zero? Is there a smallest number? A largest number?

Challenging but accessible problems from number theory can be easily formulated and explored by students. For example, building rectangular arrays with a set of tiles can stimulate questions about divisibility and prime, composite, square, even, and odd numbers. (See fig. 6.1.)

Fig. 6.1. Tile explorations

This activity and others can be extended to investigate other interesting topics, such as abundant, deficient, or perfect numbers; triangular and square numbers; cubes; palindromes; factorials; and Fibonacci numbers. The development of various procedures for finding the greatest common factor of two numbers can foreshadow important topics in the 9-12 curriculum, as students compare the advantages, disadvantages, and efficiency of various algorithms. String art and explorations with star polygons can relate number theory to geometry.

Another example from number theory involves making connections between the prime structure of a number and the number of its factors:

Find five examples of numbers that have exactly three factors. Repeat for four factors, then five factors. What can you say about the numbers in each of your lists?

Students might give 4, 9, 25, 49, and 121 as examples of numbers with exactly three factors. Each of these numbers is the square of a prime.

Without an understanding of number systems and number theory, mathematics is a mysterious collection of facts. With such an understanding, mathematics is seen as a beautiful, cohesive whole.