One sandwich, two people
You and a friend share a rectangular sandwich. A cut down the middle can give each of you an equal amount. Each share is one half of the sandwich.
Now imagine a cut very close to one edge. There are still two pieces, but one is much larger. Calling either piece “a half” would hide that difference. Fractions such as halves and quarters describe equal parts of a chosen whole.
Try it yourself
Draw two identical strips. Divide the first into two equal parts and the second into eight. Shade one part of each strip and compare their sizes.
Look at one piece when there are two parts, then one piece when there are eight. More equal parts mean each individual part is smaller, as long as the whole stays the same size.
One of four equal pieces is called one quarter. The pieces must be equal: a tiny corner and a huge leftover piece would not be two equal shares.
Always name the whole
Half a small sandwich and half a large sandwich are both halves, but they may contain different amounts of food. “A half” describes a relationship to a whole, not a fixed size by itself.
When comparing fractions, use strips of the same length. That makes comparing their shaded fractions fair. If the wholes changed size without warning, the picture could be misleading.
Make your own example
Fold a strip of paper into two equal sections, then fold it again to make four. Unfold it and check the sections. Two of the four sections cover the same length as one of the two original sections.
Do not worry about the written fraction yet. First get comfortable describing the picture: how many equal parts make the whole, and how many of those parts are selected? The next lesson gives those two counts their names.
For the full picture, explore seven friendly starting points for basic arithmetic.