A tenth part

0:00Here is a pencil, lying against a ruler whose every unit is cut into ten equal parts. Start at zero and travel along it. One whole unit, two whole units, three whole units. And then four more of the small parts. So the pencil is three units and four tenths. But there is a second way of saying exactly the same thing. Forget the whole units for a moment, and count nothing but the small parts, all the way from zero.

0:32There are thirty four of them. Thirty four tenths. The same pencil, the same length, and two different names for it. Those two names only work together because of one fact, and it is the only new thing in this whole idea. Ten tenths make one unit. Watch it happen. Here is one tenth. Another, and another, until ten of them are laid end to end. And they reach exactly as far as one whole unit does. Not nearly. Exactly.

1:06So any time you have ten tenths sitting together, you are allowed to trade them in for a single unit. And any time you find you need tenths, you are allowed to break a unit back into ten of them. That trade is not new. It is the same one that turns ten ones into a ten, and ten tens into a hundred. The arithmetic you already know simply reaches further down than you thought it did.

1:35Let us use that on the pencil, and walk from one name to the other on purpose. Thirty four tenths. Take ten of them and trade them in. That is one unit. Take another ten. Two units. And another ten. Three units. Four tenths are left over, with no ten left to make. Three units and four tenths. Now go the other way, which is easier. Three units, each one worth ten tenths, is thirty tenths.

2:06Plus the four that were already there. Thirty four. One length. One bar. Chopped two different ways, and it never changed size while we did it. Now four lengths built out of the same two digits, and they are worth being slow about. Four and one tenth. Four tenths. Forty one tenths. And forty one and one tenth. Turn each one into a plain count of tenths, and see what you get.

2:35Four and one tenth is forty one tenths. Four tenths is four tenths. Forty one tenths is, well, forty one tenths. So two of those four are the very same length, written two different ways. And forty one and one tenth is four hundred and eleven tenths, which is nowhere near any of them. It sits thirty seven whole units away from forty one tenths. These are not near misses. Read them slowly.

3:06Here is a trap that catches people, and it is worth seeing once. Two objects, each one measured against its own ruler. The first is eight tenths long. And so is the second. The same count. The same number of parts. But now put the two rulers side by side. The unit on the second ruler is two and a half times as long as the unit on the first.

3:33So its tenths are two and a half times as long as well, and its eight tenths reach two and a half times as far. A count of tenths is not a length. It only becomes a length once you know what one unit is. Which makes the first question to ask about any scale: what is one part of this one worth? Here are eight lengths. Some are written as units and tenths, and some as a plain count.

4:02Nine tenths. One and seven tenths. A hundred and thirty tenths. Thirteen and one tenth. Ten and five tenths. Seven and six tenths. Six and seven tenths. Four tenths. Put them in order, smallest first. Trying to do that by eye is close to hopeless. So put every one of them into the same form first. A plain count of tenths. And now they are just whole numbers, and putting whole numbers in order is something you can already do.

4:35Four, nine, seventeen, sixty seven, seventy six, a hundred and five, a hundred and thirty, a hundred and thirty one. Look at the last two. A hundred and thirty tenths is exactly thirteen units, and thirteen and one tenth is one tenth more than that. The one with the bigger looking number written on it is the smaller of the two, by the narrowest margin on the list. Now adding. Somebody measures their lower arm at two units and seven tenths.

5:10And their upper arm at three units and six tenths. How long is the whole arm? Add the parts that match each other. Units with units, first. Two units and three units is five units. Then tenths with tenths. Seven tenths and six tenths. Thirteen tenths. So the answer is five units, and thirteen tenths. Which is a completely true statement, and a slightly odd looking one. Thirteen tenths? Thirteen tenths is a perfectly good count. There is nothing illegal about it.

5:48It is just not the tidiest way to leave an answer, because it has a whole unit hiding inside it. Ten of those thirteen tenths make one unit. So hand them over. One unit crosses to the units side, and three tenths stay behind. Five units, plus the one that just arrived, is six. Six units and three tenths. And if that trade felt familiar, it should have. It is exactly what you do when a column of ones adds up past nine, and you carry one across.

6:22The tenths carry at ten for the same reason the ones do. Ten of anything here makes one of the next thing up. Now do that same sum a completely different way, and watch what happens. Turn both arms into plain counts of tenths before you start anything. Two units and seven tenths is twenty seven tenths. Three units and six tenths is thirty six tenths. And now simply add the two counts. Twenty seven and thirty six.

6:55Sixty three tenths. And sixty three tenths is six units and three tenths. The same answer as before. Not by luck, and not because one of the two routes is the correct one. There is no correct one. Both of them are describing the same pile of units and tenths, so they were never going to disagree. Subtraction now, and the one place where it gets interesting. A hand measures twelve units and four tenths, from wrist to fingertip.

7:29The palm on its own is six units and seven tenths. How long is the middle finger? Start with the tenths, the way you would with any subtraction. Four tenths, take away seven tenths. You cannot. There are not enough of them there. So go and get some. Take one whole unit off the twelve, and break it into ten tenths. That leaves eleven units and fourteen tenths, which is the same hand, written a different way.

7:59Fourteen take away seven is seven tenths. Eleven take away six is five units. Five units and seven tenths. One last thing, and it is a good way of practising all of this at once. A sequence in tenths. Four. Four and three tenths. Four and six tenths. What is the step? Turn them into counts. Forty, forty three, forty six. Three tenths each time. So the next one is forty nine tenths, which is four and nine tenths.

8:35Here is another. Seven and six tenths, then eight and seven tenths. That step is eleven tenths, which is more than a whole unit. Steps are allowed to be. And they do not have to climb. Thirteen and five tenths, then thirteen, then twelve and five tenths. Down five tenths each time, so the next one is twelve units exactly. In every one of them the trick is the same. Put everything into tenths, and it becomes a run of whole numbers you already know how to handle.