## Fraction Concepts

Certain division problems create a need for numbers that are not integers. For example, fractions make it possible to write the solution to 17 ÷ 3 as

17 ÷ 3 =

When **a** and **b** are integers and b ≠ 0, then the solution of the division problem a ÷ b can be expressed as a fraction, . If 0 ≤ a ≤ b, and b ≠ 0, the fraction is called a **proper fraction.** If a ≥ b ≥ 0, and b ≠ 0, then is an **improper fraction.** Improper fractions can also be written as the sum of a whole number and a proper fraction. For example,

= 5 +

When the plus sign is omitted and is written as 5 , it is called a **mixed number.**

At this grade level, students should learn to identify fractions with models that convey their properties. Proper fractions can be modeled in terms of a “part of a whole.” Here the “whole” may be a group consisting of n objects where “part” of the group consists of k objects and k < n. In the case of we have this picture.

Equivalently, the whole may consist of a region that is divided into n congruent parts, k of which belong to a subregion. For example, the fraction can be identified as the shaded part of the region below.

A unit fraction is defined as a fraction with a numerator of 1 (for example, , , , ). In a unit fraction, , one whole unit is divided into n equal parts. One of these smaller parts is the amount represented by the unit fraction. On the number line, the unit fraction represents the length of a segment when a unit interval on the number line is divided into n equal segments. The point to the right of 0 on the number line at a distance from 0 will be .

The fraction can represent the quotient of m and n, or m ÷ n. If the fraction is defined in terms of the unit fraction , the fraction means m unit fractions of . In terms of distance along the number line, the fraction means the length of m abutting segments each of length . The point is located to the right of 0 a distance m × from 0. The numerator of the fraction tells how many segments. The denominator tells the size of each segment.

**Mixed Numbers**

A fraction like is said to be proper because the numerator is less than the denominator. A fraction like is said to be improper because the numerator is not less than the denominator. Whole numbers can also be written as improper fractions: 1 = and 2 = . The improper fraction can be written as the mixed number 1 .

**Teaching Model 18.1:** Fractions and Regions