Measuring a length by choosing a smaller unit

0:00Here is a strip of paper. Nothing special about it. It is as long as it happens to be. And I am going to declare that this strip is one unit. One. The whole thing. That is not a discovery, it is a decision, and you are allowed to make it. Everything else in this video gets measured against this strip, and against nothing else. Now here is a shorter length. How long is it, in strips?

0:28Not one strip. Less than one strip. And whole numbers have just run out of things to say. So we do not abandon the measuring. We make the unit smaller until it fits. Fold the strip in half. Press the crease. Open it back up. One crease. Two parts. And they are equal, because folding is what makes them equal. Each of those parts is one half of the strip.

0:57So now I have a second measuring step available to me, smaller than the first. And notice the two numbers that just appeared. One crease, two parts. The number that matters is the two — the number of parts, not the number of folds. Two of these halves laid end to end give the whole strip back. That is what makes them halves. Fold it in half again, without opening it. Then open the whole thing out.

1:27Count the creases. Three. Count the parts. Four. So each part is one quarter of the strip. And this is worth saying out loud, because people expect thirds. Folding in half twice does not give thirds. It gives quarters. Two folds, four parts. Now build the run. One quarter is one step. Two of them is two quarters. Three of them is three quarters. And four of them is four quarters, which is the whole strip again, back where we started.

2:02One more fold. Three folds in total, and now open it right out. Seven creases. Eight parts. Each one is an eighth of the strip. The book prints this run and leaves most of it blank for you to fill in. So let us fill it in. One eighth. Two eighths. Three eighths. Four eighths — and look, four eighths reaches exactly halfway. Five eighths. Six eighths. Seven eighths, which is nearly the whole strip but not quite.

2:34And eight eighths. Every one of those is a count. Not a calculation. You are just laying down steps and counting them. Now that last line deserves a proper look, because it bothers people. Eight eighths. That is written like a fraction, so surely it must be less than one? Put the symbol down and look at the strip instead. Eight eighths is eight of those little pieces, laid end to end. And eight of them is the strip.

3:06So eight eighths is one. Not close to one — exactly one, and you can see it rather than believe it. The same thing happened with the quarters. Four quarters was the whole strip. Every run of a fractional unit closes back on one whole. That is what the number underneath is for. The book puts the point of this whole section in a box, and it is worth reading slowly.

3:35Once you have chosen a fractional unit, a fractional amount becomes measurable. That is a bigger claim than it looks. Read it backwards. The problem at the start was a length that would not fit into whole units. The fix was not a cleverer number. The fix was a smaller unit. Choose the step small enough and the awkward length becomes an ordinary count of steps. Which is the same move a ruler makes when it drops from centimetres to millimetres, and the same move the whole of measurement runs on.

4:12Now the section changes the object, on purpose, to show the strip was never the point. Forget paper. Here are pancakes, and each one is cut in half. One half is our fractional unit now. That is the step. Start collecting. One half. Add another. Two halves — and two halves is a whole pancake. Add another. Three halves. Then four halves, which is two whole pancakes. Then five halves. The step never changed. All that changed is how many you have.

4:47And I want to stop on the third column of that run, because something quietly important happened there. One half is less than a pancake. Two halves is exactly a pancake. Three halves is more than a pancake. So a fraction has just gone past one whole, and nothing broke. That matters, because people carry a rule around that a fraction is always less than one. It is not. Three halves is a perfectly good fraction. It just happens to be bigger than the whole you measured it with.

5:23The book does not stop to name that here, and neither will I. It comes back to it later in the chapter. For now, just notice that the run walked straight past one and kept going. The book then asks you to build the same run again with a different step, which is the real test of whether you have got it. So: quarters of a pancake. Same idea, smaller step.

5:51One quarter, two quarters, three quarters, four quarters — and four quarters is one whole pancake. Keep going. Five quarters. Draw it: one whole pancake, and one quarter left over. And nine quarters. Two whole pancakes, and one quarter left over. You did not divide anything. You counted quarter-pieces and then bundled every four of them into a pancake. Compare the two runs side by side and the pattern is the same. Only the size of the step is different.

6:27Back to the strip, because there is one fold we have not done. Every fold so far has been a halving, which is why we only ever got two, four and eight. You will hear people say a strip cannot be folded into three. It can. Curl it round and line the ends up. Three parts, and each one is a third of the strip. Now here is the nice bit. Fold that in half as well.

6:53Each third splits into two, so there are six parts now, and each one is a sixth. Half of a third is a sixth. You got a new fractional unit for free, out of one extra fold. Which brings us to the last idea, and it is about how you say these things in your head. Three quarters. You might read that as three over four, or three upon four, or three divided by four.

7:22None of those is wrong. But the book prefers a different reading, and I think it is right. Read it as three times one quarter. Three copies of the quarter-step. Here is why that reading is better. It tells you two separate things at once. The bottom number tells you how big each step is. The top number tells you how many steps you took. And that makes size visible without any arithmetic. Five sixths against five eighths — same count, and a sixth is a bigger step, so five sixths wins.

7:59Three eighths against five eighths — same step, more of them, so five eighths wins. No calculating. Just reading. And now, finally, the two numbers get their names, which the chapter has been holding back until you had something to attach them to. Take five sixths. The top number, five, is called the numerator. The bottom number, six, is called the denominator. Say what each one does, in one sentence, and never mix them up again.

8:33The denominator says how many of these steps make one whole. The numerator says how many you have got. So five sixths is: cut a whole into six, and take five of them. And none of this depended on paper. Do it with a circle, a square, a triangle — the argument is identical. Next time, that folded strip gets laid flat and turned into a number line, and every fraction gets a place to stand.