Teaching Models

Understand Fractions

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

3 − 5 = 2

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

17 ÷ 3 = seventeen-thirds.

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

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 three-fourths can be modeled as follows.

Model of fraction

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 three-fourths can be identified as the shaded part of the region below.

Shaded square

A unit fraction is a fraction with a numerator of 1 (for example, one-half, one-third, one-fourth, one over n). The definition of a unit fraction, one over n, 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 one over n 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 one over n from 0 will be one over n.

Number line from 1

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

Number line from 2

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 = five over one = 5 × seven over one × 7 = thirty-five-sevenths, and thirty-five-sevenths equals 35 unit fractions of one-seventh. Just as 35 divided by 7 is 5, 35 unit fractions of one over n divided by 7 is 5 unit fractions of one over n. So 5 ÷ 7 = five-sevenths.

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 one-third 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.

Graphic of 1/3

Equivalent Fractions
Two fractions a over b and c over d are equivalent if there exists a number m such that m × a/m × b = c over d. For example, the fact that 2 × 3/2 × 4 = six-eighths implies that three-fourths is equivalent to six-eighths.

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 three-fourths is equivalent to six-eighths can be shown as follows.

Graphic of 6/8

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


Houghton Mifflin Math Grade 4