Understand Fractions

Fractions
Fractions are numbers that are needed to solve certain kinds of division problems. Much as the subtraction problem

creates a need for numbers that are not positive, certain division problems create a need for numbers that are not integers. For example, fractions allow the solution to 17 ÷ 3 to be written as

When a and b are integers and b ≠ 0, then the solution to the division problem a ÷ b can be expressed as a fraction, .

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. The whole may be a group consisting of n objects where part of the group consists of k objects and k < n. The fraction can be modeled as follows.

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 a fraction with a numerator of 1 (for example, , , , ). The definition of a unit fraction, , is to take one unit and divide it 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 located 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 . 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 at a distance m × from 0. The numerator of the fraction tells how many segments. The denominator tells the size of each segment.

A straightforward way to show that fractions represent a solution to a division problem is by using equivalent fractions. What is 35 ÷ 7? It is 5 because 35 equals 7 × 5. What is 5 ÷ 7? This is more difficult because 5 is not a multiple of 7. However, 5 = = 5 × × 7 = , and equals 35 unit fractions of . Just as 35 divided by 7 is 5, 35 unit fractions of divided by 7 is 5 unit fractions of . So 5 ÷ 7 = .

Finding a Fractional Part of a Number
The word of is often used to pose problems involving the multiplication of a whole number by a fraction. At this level, students have not yet learned to multiply fractions. The problem of finding of 6 can be modeled in terms of a group of 6 objects that has been separated into 3 smaller groups, each of which has 2 objects.

Equivalent Fractions
Two fractions and are equivalent if there exists a number m such that m × × b = . For example, the fact that 2 × × 4 = implies that is equivalent to .

Geometrically, this concept can be conveyed in terms of a picture in which there are two ways of representing the same part of the whole. The fact that is equivalent to can be shown as follows.

Because equivalent fractions represent the same number, they are referred to as equal.

A fraction is in simplest form if the numerators and denominators are as small as possible. A more formal way of stating this is to say that in a simplest form fraction, the numerator and denominator have no common factors other than 1.

Teaching Model 19.4: Compare and Order Fractions