Equivalent fractions: many names, one number, and lowest terms

0:00Here is a paper strip, and we are calling its whole length one unit. Fold it in half and shade the left-hand part. That shaded stretch is one half. Now fold the same strip into four equal parts instead. The shading has not moved — but now it covers two of the four. Fold again, into eight. Still the same shaded stretch, and now it covers four of the eight.

0:27One half, two quarters, four eighths. Three different fractions, and one single length. Nothing was added and nothing was taken away. All that changed was the size of the step we measured with. That is what this video is about: one number, and the several names it answers to. Rather than fold one strip over and over, let us stack them. The top strip is one whole, undivided. Under it, the same length cut into two. Under that, into three. Then four, then five, then six.

1:05Every row is the same width, because every row is the same one unit. Only the number of cells changes. And every row closes exactly on the whole. Two halves, three thirds, six sixths — each one finishes flush with the right-hand edge. That is worth saying out loud, because it is the thing the wall is built on. A row of n cells, all n of them taken, is one.

1:35Now read the wall by looking down it rather than across. Find the right-hand edge of one half. Drop straight down. It lands exactly on the edge of three sixths. Not nearly — exactly, because both sit at the same distance from the left. So one half and three sixths are the same number, and you can see it without doing any arithmetic at all. Try another. The edge of two thirds drops onto the edge of four sixths.

2:06The wall answers counting questions too. How many sixths make one half? Line them up: three. How many sixths make one third? Two. The picture hands you the answer. Carry the wall further down — sevenths, eighths, ninths, tenths. One warning about that sevenths row before we use it. Its last cell is seven sevenths, and like every other row it closes on one whole. Now drop a line from the edge of three sixths again, and follow it all the way down.

2:41It passes through four eighths. It passes through five tenths. And of course it started at one half. Four names for one place. One half, three sixths, four eighths, five tenths. Try one yourself. Pause and write two fractions equal to two sixths — anything whose edge lines up with it. Four twelfths works. So does six eighteenths. There is no shortage of them. Here is the same idea arriving from somewhere completely different.

3:16One pancake, four children, shared equally. How much does each child get? Cut it into four equal pieces and hand one to each. Each child has one quarter. So far, so ordinary. But notice what that sentence actually did. It turned a division — one shared among four — into a fraction. One quarter. And that is not a coincidence or a trick of notation. The fraction one quarter simply is the answer to sharing one thing among four.

3:50Whenever you see a fraction, you may read it as a sharing if that helps. That one picture is holding three facts at once, and it is worth writing all three down. First, as a division. One pancake shared among four children gives one quarter each. Second, as a sum. The four shares put back together make the whole pancake — a quarter plus a quarter plus a quarter plus a quarter is one.

4:21Third, as a product. Four lots of one quarter is one. Three sentences, one picture. And the third is just the second written shorter. Now your turn with the same three. Three pancakes among four children — what does each get? Three quarters. And four children with three quarters each does rebuild the three pancakes. Here is where the sharing view earns its keep. Two cakes, five children. Each child gets two fifths.

4:55Now double everything. Four cakes, ten children. There is twice as much food, and twice as many mouths. What happens to one child's plate? Nothing. Each child still gets two fifths — and as a fraction of the food, four tenths. Two fifths and four tenths are the same share, because doubling both sides of a sharing changes nobody's portion. That is the whole reason multiplying the top and the bottom by the same number is allowed. It is not a rule someone invented. It is what fair sharing does.

5:35Let us watch that happen three times over. One pancake between two children: each gets one half. Two pancakes among four children: each still gets one half. Three pancakes among six: one half again. One half, two quarters, three sixths — every one of them the same plate. Now a chain that does not start at a half. Two pancakes among three children gives two thirds. Four among six gives two thirds. Six among nine, two thirds.

6:10Same three names, same one number, and the first of them is the shortest. Which raises the obvious question. If a number has many names, is there a best one? There is a shortest one, and finding it is just running the process backwards. Going up, you multiply both numbers by something. Coming down, you divide both by something. Take six ninths. Six and nine are both divisible by three, so divide both by three: two thirds.

6:44Can we go further? Two and three share nothing but one, so no. Two thirds is where it stops. That is the shortest name — the one you cannot shorten. And notice the number itself never moved. The strip is still shaded to the same place. Only the writing got tidier. Before we practise, a warning, because there is a very tempting mistake waiting here. You may multiply the top and bottom by the same number. You may not add the same number to both.

7:19Watch what adding does. Take one half, and add one to each number. You get two thirds. Mark them both on a line. One half sits here. Two thirds sits further along. They are not the same number. Adding changed the amount. Multiplying does not, because multiplying is scaling the whole situation up — more pieces, but proportionally smaller ones. The wall shows why. Doubling the number of cells halves the size of each cell, and those two effects cancel exactly.

7:55Adding one cell does not halve anything, so nothing cancels, and the length moves. A fraction is in its lowest terms when its two numbers share no factor above one. Take sixteen twentieths. Four divides both, so it is not in lowest terms yet. Divide both by four: four fifths. Now check. Four and five share nothing above one, so we are finished. And one more warning, this one about a habit rather than a rule.

8:28Some people talk about cancelling as though it were crossing out digits. It is not. It is dividing both numbers by a common factor. Here is why it matters. Take sixteen over sixty-two, and cross out the sixes. You would get one half. But sixteen over sixty-two is eight thirty-firsts, which is about one quarter. Nowhere near a half. Crossing out digits is not a method. One last thing, and it is a reassuring one. There is no single correct route down.

9:04Take thirty-six sixtieths. Halve both: eighteen thirtieths. Halve again: nine fifteenths. Now divide both by three: three fifths. Three small steps, and nothing divides three and five, so that is lowest terms. Now the other route. Notice that twelve divides both thirty-six and sixty. Divide once by twelve, and you land on three fifths immediately. Same destination. One route took three steps and one took one, and neither is more correct.

9:41So, three things to carry away. A number has many names, and each one records the step it was measured with. Multiplying top and bottom by the same number keeps the number still — because it is scaling a share, not changing it. Adding does not. And the shortest name is the one whose two numbers share nothing. Next time: putting two fractions side by side and deciding which is bigger.