Why one half and two quarters are the same

1. This is one whole bar. All of it is shaded. Nothing has been cut yet. Before any fraction makes sense you have to know what the WHOLE is. Everything else is a piece of this.
2. Cut it in half. One cut down the middle makes two pieces. They have to be the SAME SIZE, or they are not halves. One piece out of two is shaded: one half. A single piece like that has a name. It is a UNIT FRACTION. Every other fraction is built out of copies of one.
3. Now cut each half in two. Watch the shaded part while the cuts go in. It did not move. It did not change size. It is the same shaded part it always was. But there are four pieces now, and two of them are shaded. So we call it two quarters.
4. And again. Eight pieces, four of them shaded: four eighths. Look at where the shading stops. It is exactly the same place as before. Halfway along the bar, all three times.
5. Your turn. Shade one half. Tap the pieces to shade them. The bar is cut into quarters this time. So you will need more than one piece. Keep going until the counter says one half.
6. So those are three names for one amount. One half, two quarters and four eighths are the same amount of bar. Each time the pieces doubled, the shaded ones doubled too. Double the top and the bottom by the same number. The amount does not change. That is the whole rule.
7. Now watch something different. Back to one half: two pieces, one of them shaded. Keep your eye on HOW MUCH is shaded. This next bit is where everybody gets caught.
8. This time only ONE piece stays shaded. Eight pieces, and just one of them shaded: one eighth. The shaded part got SMALLER. Eight is a bigger number than two. But one eighth is a smaller piece than one half. Cutting the bar into more pieces makes every piece thinner.
9. Your turn again. Shade one half of this one. Same job, but the bar is cut into eighths now. Shade one half of it. You will need more pieces than last time. The shaded part will come out exactly the same size.
10. Quick check. One piece of this bar is shaded. Think about the size of the piece. Which is bigger, one half or one eighth? One half · One eighth, because 8 is bigger. Yes. More pieces means each piece is thinner, not bigger.

Double the top and the bottom together and the amount stays the same. One half, two quarters, four eighths. But a bigger bottom number on its own means SMALLER pieces. One eighth is much less than one half.