A fraction as a part of one whole

0:00Last time, one-quarter meant a share. One thing, four children, and the piece you walked away with. This time the same symbol means something that sounds different, and turns out not to be. Here is a slab of toffee. Not shared with anyone — just sitting there, whole. Cut it into four equal pieces and pick one up. That piece is one-quarter of the slab. Nobody stood in a queue. Nothing was divided between people. And the name did not change.

0:33So one-quarter has two readings. Your share when four people split one thing, and one piece when one thing is split four ways. Same cut either way. The only difference is whether you are counting people or counting pieces. But before you name any piece there is a step people skip, and this section depends on it. You have to say what the whole is. Out loud. Because one-quarter is not an amount. It is an amount compared to something, and that something has to be pinned down.

1:10A quarter of this slab is a decent mouthful. A quarter of a crumb is nothing. So here is the rule for everything that follows. This intact slab, uncut, is the whole. Every piece we name gets measured against it, and against nothing else. A piece drawn off to the side starts to look like a whole in its own right. It is not. Keep the slab in view. Now snap the slab once, and not down the middle. Deliberately off-centre.

1:43Two pieces, one small and one large. Neither is a half, and we will come back to why that matters. The small one is easy to name, and the book names it for us — one-quarter of the slab. Which means something very specific. Four pieces that size, laid together, rebuild the whole slab exactly. That is what the four underneath tells you. Not four pieces on the table — there are only two. Four copies to make one whole slab: that is the test, and the only test.

2:18Now the large piece. Nobody has named that one, and here is where the chapter does something clever. You might expect the large piece to need a new name of its own. It does not. You already have a measuring stick — the small piece. So measure with it. Take the small piece and lay it inside the large one. Once. There is room for another. Twice. And once more. Three times, and the large piece is exactly used up.

2:52So the large piece is three of those quarter-pieces. Three quarters. Not a new idea, not a new symbol. A count, of a unit you already had. And check it. One quarter and three quarters — one piece plus three is four, which is the whole slab back. Right. Fresh slab, cut into six equal pieces. The obvious way is a grid. Two cuts down, one cut across. Count them. Six pieces, all the same size and shape.

3:25So each one is a sixth of the slab. Six of them rebuild it, so one over six. Pick one up and set it beside the slab. A neat little rectangle. Hold on to that shape, because we are about to cut the same slab six ways again and get something that looks nothing like it. Same slab again, still intact. Six equal pieces again — but cut differently. This time the cuts slant — a zigzag across the slab, with two straight cuts holding it together.

4:01Count what comes out. One, two, three, four, five, six. Six pieces, and every one of them is a triangle. Pull one out and set it beside the slab, exactly like before. Now put the two drawn-out pieces next to each other. The rectangle from the first cut, the triangle from this one. They look nothing alike. One is a stubby rectangle, the other a long slanted triangle. So the textbook stops here and asks the class a question. Stop with it.

4:37Cutting one whole into six equal parts in different ways gives pieces of different shapes. Are those pieces the same size? Pause if you like. Look at both, and commit to an answer. Because there is a real temptation here — the misconception this page exists to break. Different shape, so different fraction. The triangle is long and pointy, the rectangle squat — surely not the same amount of toffee.

5:07It is a very natural thought. And it is wrong. They are the same size. Both are a sixth of that slab, and the shapes are irrelevant. The first reason is already in front of you. Six equal pieces rebuilt that slab. Six equal pieces rebuilt this one. So each piece is one of six equal parts, whatever its outline. That argument finishes before you look at the shapes. But if you would rather see it — watch. I will turn the triangle into the rectangle without losing any toffee.

5:45Slide the triangle onto the rectangle. Most of it fits. A corner hangs over the top, and a notch is left empty. Cut the overhanging corner off. Swing it round, pivoting on the point where the slanted edge crossed the cut. And it drops into the notch exactly. Same toffee, rearranged. Nothing added, nothing thrown away. The shape was never what the name depended on. Only the number of equal pieces was.

6:18Now run the whole thing backwards, because you will need it that way round. Somebody hands you a piece. No slab, no cuts, no picture of where it came from. What fraction is it? Copy it. Lay another one down beside it. Two of them. Not a whole slab yet, so keep going. Three, and the slab closes up. Exactly, with no gap and no overlap. Three copies rebuild the whole, so the piece is one-third.

6:48Read that direction carefully. Nobody told you how the slab was cut. You worked the cut out from the piece. The number underneath answers one question. How many of these make one whole? So here is a set of eight pieces, all cut from the same slab, all unnamed. To make the counting honest, the slab is scored into a grid first. Six across, four down — twenty-four small blocks.

7:19Now every piece covers a whole number of blocks. Just count. This upright one covers two blocks. Twelve of those rebuild the slab, so it is one-twelfth. The big triangle covers six blocks — half of a four-by-three rectangle. Four sixes make twenty-four, so it is one-quarter. The tall thin strip is four blocks. Six of those rebuild the slab. One-sixth. The L-shaped piece is three blocks, and so is the smaller triangle. Both are one-eighth, though one has a corner cut into it and the other a slanted edge.

7:59The single square is one block out of twenty-four. And so is the narrow triangle — half of a two-block strip is one block. Every answer came from counting blocks. Not one came from looking at the shape. Which leaves one word to be careful about, and it is the word equal. Equal parts means parts of the same size. It does not mean parts of the same shape. A sixth can be a rectangle or a triangle or an L, as long as six of them rebuild the whole.

8:36But it does have to be the same size, and that half of the rule is not optional. Look at this. One slab, one cut, two pieces. So are they halves? No. The cut was off-centre. That piece is twice the other one — two-thirds and one-third. Two pieces is not enough. Two equal pieces makes halves, and at a glance these are not. And if you have ever snapped a real slab of toffee, you know the pieces come out ragged. Mathematics assumes the perfect cut; real toffee does not read the textbook.

9:14So, the whole of this in three lines. Say what the whole is, first, every time. A piece is named by how many copies of it rebuild that whole, not by what it looks like. And a bigger piece is measured by counting the named ones inside it. Three quarter-pieces is three quarters. One last thing. Everything today was a slab — a flat thing you cut up. But the same move works on something with no area at all.

9:45Take a length. A strip of ribbon, a stretch of road, a line on the page. Cut it into equal parts, ask how many copies rebuild it, and you have a fraction of a length. Which is where the chapter goes next — the most useful picture of the three.