A fraction as an equal share, and why one-ninth is less than one-fifth

0:00Here is one pancake, and here are two children who both want it. Split it fairly. One cut, straight down the middle, and each child takes a piece. You already have a word for what each child got. A half. And you already have a way of writing it. One over two. Now look hard at that two, because everything in this video turns on what it is counting. It is not counting pieces the child is holding — the child is holding one piece.

0:32It is counting the children. Two people shared it, so there is a two underneath. Now the same pancake, and four children instead of two. Same rule. Split it fairly, everyone gets the same. Two more cuts, and each child takes a piece. That piece is a quarter — one over four. And the four underneath is the number of children again. Not the pieces you hold. The people you shared with.

1:02Now put the two pictures side by side, drawn the same size, so only the pieces differ. Which child got more — the one who shared with one other person, or the one who shared with three? Nobody needs mathematics for that. The two-child share is visibly bigger. One-half beats one-quarter. So now the book asks a harder pair, and it does something I think is excellent. It does not warn you about the mistake. It has a character make the mistake, out loud.

1:38The question is: which is bigger, one-fifth or one-ninth? And the character answers straight away. Nine is bigger than five, so one-ninth is bigger. Now — that is wrong. But do not just dismiss it, because it is a completely reasonable thing to think. You have spent five years learning that a bigger number written down means a bigger amount. Nine beats five, every time, and it always has. The instinct is sound. It is being applied to the wrong thing.

2:14So the second character does not argue about the numbers at all. They go back to the pancake. One-fifth is not the numbers one and five. It is an instruction. Take one pancake. Find five children. Split it so that everybody gets the same. Your share is one-fifth. And one-ninth says: take one pancake, find nine children, split it fairly. Your share is one-ninth. Say those two sentences out loud and the comparison is no longer about the digits.

2:49It is about a queue of five people, and a queue of nine people, standing around one pancake. That is the whole trick of this section. Read the number underneath as people, not as a number. So which queue would you rather be in? Five children round one pancake. Or nine children round the same pancake. Five, obviously. More people means the same amount of food is being split further, so everybody's piece is smaller.

3:21So one-fifth is bigger than one-ninth. The character's guess was backwards. And notice both statements are true at the same time, for the same reason. Nine is more than five. One-ninth is less than one-fifth. They are not in conflict. The second one is what the first one does to a fixed amount of food. More sharers, smaller share. That is the rule, and it never fails. The book then pushes the same reasoning until it becomes obvious, which is a good habit.

3:58Which is bigger — one hundredth, or one two-hundredth? Do not look at the digits. Picture the queue. A hundred children round one pancake. That is already a miserable sliver each. Now two hundred children round the same pancake. Every sliver is cut in half again. One hundredth is bigger. And you did not calculate anything — you imagined a longer queue. Which means you can now order any two of these on sight. Bigger number underneath, smaller share. Always.

4:34These things have a name, and the book gives it two. One of the equal parts that a single whole splits into is called a fractional unit. It is also called a unit fraction — same object, two names, and you will meet both. One half, one third, one quarter, one fifth, one tenth, one fiftieth, one hundredth. Every one of them is a fractional unit. And the list does not stop. There is a one two-hundredth, and a one-thousandth, and there is no last one.

5:08Every one has a one on top, because that is what it means: your share, when the whole is cut that many ways. So do not memorise the list. Memorise the sentence that makes it. Now the book sets some sharing questions, and they get quietly harder. Three apples, all about the same size, weigh one kilogram between them. How much does one apple weigh? One kilogram split three ways. A third of a kilogram.

5:41Next: one kilogram of rice, packed into four equal packets. How much in each? One kilogram split four ways. A quarter of a kilogram each. Both of those follow the same sentence. One whole, split so many ways, and you take one part. But the next one breaks that pattern on purpose, and it is the most important question in the set. Four friends want to share three glasses of orange juice, equally.

6:13Now look at what is different. Every question so far had one thing being shared. This has three. So do the obvious thing. Take each glass and split it four ways, one part to each friend. From the first glass, everybody gets a quarter. From the second, another quarter. From the third, another quarter. So each friend ends up holding three quarter-glasses. Three-quarters of a glass. And that is worth pausing on, because a share is not always one single piece.

6:48Here it is three of them. The fractional unit was still a quarter — you just got three of those units instead of one. One more, and this one puts two different fractions together. A big fish weighs half a kilogram. A small one weighs a quarter of a kilogram. What do they weigh together? Now you cannot just add the numbers underneath — a two and a four do not make anything sensible.

7:16So go back to the pieces. A half is two quarters. You can see that on the pancake: cut a half in two and you have two quarters. So the big fish is two quarters, and the small one is one quarter. Two quarters and one quarter is three quarters. Three-quarters of a kilogram. And notice what made that work. You made both amounts out of the same size of piece, and then you counted pieces.

7:45Hold on to that, because it is the whole of adding fractions, and it comes back later in this chapter. Now something the chapter includes that I am glad it does. The way we write fractions — one number above another, with a line between — is comparatively recent. The ideas are far older, and they lived in ordinary words long before anyone drew that line. The Rig Veda, thousands of years old, has a word for three-quarters. Tri-pada.

8:18The book traces that word forward into modern Indian languages — teen paav in Hindi, mukkaal in Tamil. And then it asks you to do something better than reading about it. Go and find the fraction words used in your own home, in whatever language you speak there. Almost every language has them, and most of them are older than the notation. People needed to talk about half a thing long before anybody needed to write it down.

8:52So finish with words rather than symbols. Here are six of them, jumbled. One and a half. One quarter. Two and a half. Three quarters. One half. One and a quarter. Put them in order, smallest first — and do it by thinking about the amounts, not by translating into digits. One quarter is the smallest — one thing, split four ways. Then one half. Then three quarters, which is still under a whole thing.

9:22Then one and a quarter, then one and a half — both past a whole now. And two and a half is the largest. So: a fraction is a share, the number underneath counts the sharers, and more sharers means less each. Next time, the same fractions seen a different way — not as a share between people, but as a part of one whole thing.