Math Background

Addition and Subtraction of Fractions: Overview

If a candy bar is broken into two halves to share, it is not accurate to say, “I want the bigger half.” If it is truly broken into two halves, then both parts are equal. One half of the candy bar combined with the other half would equal the whole candy bar. The fraction one-half means one of two equal parts of the whole. The 1 is the numerator, and the 2 is the denominator of the fraction.

numerator over denominator

Before computing with fractions, it's important to understand the meanings of the numerator and denominator of a fraction. Figure 1 shows a rectangle divided into four equal parts.

rectangle divided into four equal parts
Figure 1

Figure 2 displays three of the four parts shaded. The fraction three-fourths represents three of four equal parts. The 3, or numerator, tells you how many parts are shaded, while the 4, or denominator, shows how many equal parts the whole is divided into.

rectangle divided into four equal parts - three parts are shaded
Figure 2

When adding fractions such as two-sixths + three-sixths , you add only the numerators, while keeping the same denominator ( two-sixths + three-sixths = five-sixths ). A pictorial demonstration of addition of fractions with like denominators provides a much clearer explanation of why we do not add the denominators. Figure 3 shows a rectangle divided into six equal parts.

a rectangle divided into six equal parts
Figure 3

Figure 4 shows the same rectangle with two of six equal parts shaded.
Figure 5 shows three parts of the whole (six parts) shaded. When adding two-sixths + three-sixths, the result is five-sixths , which means five of six equal parts are shaded, as shown in Figure 6.

a rectangle divided into six equal parts, with two of six equal parts shaded
Figure 4

a rectangle divided into six equal parts, with three of six equal parts shaded
Figure 5

a rectangle divided into six equal parts, with five of six equal parts shaded
Figure 6

It is important to notice that three-sixths + two-sixths = five-sixths and five-sixths is a fraction with a denominator of 6. Since you have added 2 of six equal parts to 3 of six equal parts your result is 5 of six equal parts. Similarly, in subtraction, if you were to start with 5 parts shaded, then erase the shading from 2 parts, 3 shaded parts would remain. Thus, five-sixthstwo-sixths = three-sixths.


Houghton Mifflin Math Grade 3