A whole can be divided into equal parts. Two equal parts make halves (each is 1/2); four equal parts make fourths (each is 1/4). The size of the parts matters: all parts of a whole must be the same size. Partitioning shapes and real objects (pizza, chocolate bars) makes this concrete.
You've already practiced cutting shapes into equal parts and naming those parts as halves and quarters. Now you're connecting that hands-on experience to the fraction — a number that names one of those equal parts. When you cut a pizza into 4 equal slices, each slice is one-fourth, written 1/4. The bottom number (the denominator) tells you how many equal parts the whole was cut into. The top number (the numerator) tells you how many of those parts you're talking about.
The most important idea at this stage is that the parts must be equal. This is stricter than it sounds. If you cut a circle into two pieces but one piece is bigger than the other, you do NOT have halves — you have two unequal pieces. One is bigger than half and one is smaller than half. This is a real distinction, not a technicality: the word "half" only applies when both pieces are the same size. The same rule applies to fourths: four pieces is not enough; they must be four *equal* pieces.
Think about sharing fairly. If you and a friend share a sandwich, a fair share means equal parts. Fractions are the math language for fair sharing and equal division — but the *equal* part is what makes the fraction valid. When you look at a picture of a shape divided into parts, always check: are the parts the same size? If not, the labels "half" and "fourth" don't apply.
Something that surprises many learners: a fourth (or quarter) is *smaller* than a half, even though 4 is bigger than 2. More cuts means smaller pieces. If you cut a candy bar into 4 equal parts, each piece is smaller than if you cut the same bar into only 2 equal parts. This is one of the first places where fraction intuition runs against whole-number intuition — a bigger number in the denominator does not mean a bigger piece; it means the whole was cut into more pieces, so each piece is smaller. Keeping this straight is the foundation of everything fractions will ask you to do next.