Teaching Models

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 = seventeen-thirds

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

seventeen-thirds = 5 + two-thirds

When the plus sign is omitted and seventeen-thirds is written as 5 two-thirds, 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 three-fourths we have this picture.

part of a whole

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.


A unit fraction is defined as a fraction with a numerator of 1 (for example, one-half, one-third, one-fourth, one over n). In a unit fraction, one over n, 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 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 to the right of 0 on the number line at a distance one over n from 0 will be one over n.

1/3 Number Line n = 3

The fraction m over n can represent the quotient of m and n, or m ÷ n. If the fraction is defined in terms of the unit fraction one over n, the fraction m over n means m unit fractions of 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 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.

1/5 Number Linem = 5, n = 6

Mixed Numbers
A fraction like five-sevenths is said to be proper because the numerator is less than the denominator. A fraction like four-thirds is said to be improper because the numerator is not less than the denominator. Whole numbers can also be written as improper fractions: 1 = five-fifths and 2 = four-halves. The improper fraction four-thirds can be written as the mixed number 1 one-third.

Teaching Model 18.1: Fractions and Regions

Houghton Mifflin Math Grade 3