A whole number times a fraction, read as repeated distance

0:00Aaron walks three kilometres in an hour. He walks for five hours. How far has he gone? Nobody hesitates over this one. Three, five times. Fifteen kilometres. You can even do it the slow way, an hour at a time, and watch it climb. Three. Then six. Then nine. Then twelve. Then fifteen. Same answer, and the multiplication said it in one step instead of five. That is what multiplying by a whole number is. You lay one copy down, over and over.

0:36Hold on to that picture of copies laid end to end. It is about to run out. Now the same sentence, with one number changed. A tortoise covers a quarter of a kilometre in an hour. It walks for three hours. How far? And here is the thing worth noticing: you did not have to stop and think. A quarter, and a quarter, and a quarter. Three quarters of a kilometre.

1:05The speed turned into a fraction, and nothing at all about the method changed. One copy, laid down three times. Which is exactly what Aaron's five hours were. So a fractional speed is not a special case, and it does not need a special rule. That sentence is the hinge of everything that follows. Keep it. Put the tortoise on a line, because this picture matters later. Zero at the left, and a tick mark every quarter of a kilometre.

1:39One hour of walking is one step, from zero to the first tick. The second hour is another step of exactly the same size. That lands on a half. The third lands here. Three quarters of a kilometre. Three steps of a quarter, and one long arrow underneath spanning all three. Now notice something, and notice that it does not bother you at all. Three hours of walking, and less than one kilometre covered. The tortoise is slow. That is the whole explanation.

2:15So the fraction can sit in the speed. Let us put it in the time instead. Back to Aaron, three kilometres in an hour. But today he only walks for a fifth of an hour. Twelve minutes. How far does he get? And this time, try to lay the copies down. You cannot. A fifth of a time is not a number of times. There is nothing to repeat, and no number of repeats to do it.

2:42The procedure has genuinely run out, and that leaves a choice. Either multiplication stops here, or it keeps the question and changes the method. And the question is still a perfectly sensible one. He covers three kilometres in a whole hour. A fifth of an hour is a fifth of that walking. So take the three kilometres and cut them into five equal parts. There they are. Each part is three fifths of a kilometre.

3:15A fifth of the time buys you exactly one of those parts. Three fifths of a kilometre. Six hundred metres. And it gets written down like this. One fifth, times three, is three fifths. No copies were laid down anywhere. The three got cut up instead. Two fifths of an hour now. Twenty-four minutes. The cutting is already done. Five parts, three fifths of a kilometre each. And two fifths of an hour is simply twice one fifth. So take two of the parts.

3:51Three fifths, and another three fifths. Six fifths of a kilometre. One whole kilometre, with a fifth of one left over. Two fifths, times three, is six fifths. So the new procedure is two moves. Cut, and then collect. And it answered exactly the question the old procedure answered. How far, in this much time. Something happened back there that is worth stopping on. One fifth, times three, gave three fifths.

4:25A three went in. Something smaller than three came out. Multiplying is supposed to make numbers bigger. Here it made one smaller. So I tried it one thousand seven hundred and twenty-eight times, with every multiplier I could build. Every multiplier below one shrank the number it multiplied. Every multiplier above one grew it. And a multiplier of exactly one left it precisely where it was. Not one exception in the whole sweep.

4:58Why it should work that way is another day's job. For now, just notice that it does. The two numbers in a product are doing different jobs, and the jobs have names. In two fifths times three, the two fifths is the multiplier. It is the one doing the multiplying, and it is the one that decides what happens. The three is the multiplicand. It is the one being multiplied, and it is the one that gets cut up.

5:30Now, those two names belong to the positions, not to the numbers. Write the same product the other way round, three times two fifths, and the roles swap over. Now the three is the multiplier, and it is the fraction that gets repeated. Two fifths, three times. Six fifths. Same two numbers, same answer, completely different procedure. That is worth sitting with for a moment. A fractional multiplier arrives in two parts, and each part drives one of the two moves.

6:07The number underneath says how many pieces to cut into. The number on top says how many of the pieces to keep. Two fifths. The five cuts. The two collects. In that order. It is worth getting it the wrong way round once, on purpose, to see what happens. Multiply the three by five and then halve it, and you get seven and a half. Against a right answer of six fifths. It is more than six times too big.

6:37I tried that backwards reading on one thousand four hundred and fifty-two products. It was not right once. Here is one where the fraction is the number being multiplied. Five grandchildren, and each of them inherits two thirds of a hectare. Five is a whole number, so copies can be laid down. Two thirds, five times over. Two thirds. Four thirds. And at the third one, exactly two whole hectares.

7:09Eight thirds. Ten thirds. Ten thirds of a hectare in all. Which is three whole hectares, and a third of one left over. Five plots, every one of them smaller than a hectare, and together they are more than three. One more, and this one needs a step before any multiplying starts. A tap fills eight litres in an hour. It is left running for an hour and a quarter.

7:37First, that hour and a quarter has to be written as a single fraction. One whole is four quarters, and one more quarter makes five. Five quarters of an hour. It does not mean one times a quarter. That would be a quarter of an hour, which is not what anybody said. Now the two moves. The four cuts: eight litres into four parts, two litres in each. The five collects: five of those parts. Ten litres.

8:08And notice the order. Cutting first kept the numbers small. Multiplying first would have taken you through forty on the way to the same ten. So there are two procedures now, with one question between them. A whole number in front: lay the copies down, one per unit. A fraction in front: cut the other number up, and collect what the top says. They look nothing alike, and both of them are called multiplication.

8:40That is the whole point. They are not two operations. They are one question, asked of two kinds of number. But look at what neither of them can reach yet. The tortoise again, a quarter of a kilometre in an hour, walking for a fifth of an hour. You cannot lay a quarter down a fifth of a time. And cutting a quarter into five pieces is not a move that has been made yet. The situation still has an answer. Finding it is what comes next.