1. What a Fraction Means: Equal Parts, Numerator, Denominator

A fraction names part of a whole. The most important word is equal: the whole must be cut into equal-size pieces.

In \(\frac{3}{8}\):

• The denominator \(8\) tells how many equal parts make one whole.
• The numerator \(3\) tells how many of those parts we are using.

Think in three pictures:

• Area model: 3 of 8 equal pizza slices.
• Set model: 3 of 8 students wearing blue shirts.
• Number line model: a point three steps out of eight from 0 to 1.

Worked example: A pan of cornbread is cut into 12 equal pieces. A family eats 5 pieces. The fraction eaten is \(\frac{5}{12}\). The fraction left is \(\frac{7}{12}\), because \(12-5=7\).

Common mistake: writing the bigger number on bottom automatically. The denominator is not "the bigger number"; it is the number of equal parts in the whole.

Case study - cafeteria tray: A tray has 10 equal brownie squares. If 4 are sold before lunch, the sold amount is \(\frac{4}{10}\). The unsold amount is \(\frac{6}{10}\). Both fractions describe the same tray, just different parts of it.