A half is one of two equal parts of a whole; a quarter is one of four equal parts. Understanding these equal divisions is foundational for all fraction concepts.
Use physical objects — fold paper, cut sandwiches, divide a group of blocks — and always check whether the parts are equal by comparing them directly. Seeing that unequal pieces are NOT halves or quarters is as important as seeing what equal pieces look like.
You already know how to split a shape into parts — you practiced decomposing shapes by cutting them apart or breaking them into pieces. Now the key idea is that the parts must be *equal* in size. That is what makes a half or a quarter different from just any two pieces of something.
A half means one out of two equal pieces. If you cut a sandwich right down the middle so both pieces are exactly the same size, each piece is a half. If one piece is bigger than the other, neither piece is a half — they have to match. Together, two halves make one whole. You can check your work by putting the pieces back together: do they fit perfectly with no gaps or extra bits?
A quarter means one out of four equal pieces. If you take a square piece of paper and fold it in half, then fold it in half again the same way, you get four equal pieces. Each one is a quarter. Four quarters make one whole — the same whole you started with. You might already know the word "quarter" from coins: a quarter coin is worth one out of four equal parts of a dollar, which is exactly the same idea.
The most important thing to remember is the word equal. Whether you are making halves or quarters, the pieces must all be the same size. This idea — that equal parts fit back together perfectly into the whole with no gaps or overlaps — is the foundation for all fraction work you will do later, where every fraction describes a certain number of equal parts out of a whole.