Mixed fractions: naming what lies past 1

0:00Last time we put fractions on a number line and found that some of them land past the one. Here are the ones we drew. One half, two thirds, two fifths, four fifths — and then three halves, six fifths, seven fifths, eight fifths, nine fifths. Sort them by a single question: does this bar reach past one whole unit, or does it stop short? Four of them stop short. Five of them carry on past.

0:30Nothing sits in the middle. Every fraction is on one side or the other, because a bar either reaches the one mark or it does not. That sort is the whole subject of this video. The ones on the right need a second way of being written, and by the end you will have it. Look only at the column that went past. Three halves. Six fifths. Seven fifths. Eight fifths. Nine fifths.

1:01Something is true of every one of them, and it is not true of anything in the other column. The top number is bigger than the bottom number. Three beats two. Six, seven, eight and nine all beat five. Now check the left column. One half — one is smaller than two. Two thirds, two fifths, four fifths — smaller every time. So there is a test sitting here. If the top number is bigger, the bar goes past one whole.

1:32But a pattern you have noticed is not yet a reason. Before we use it, let us see why it has to be true. Ask a simpler question first. How many steps does one whole unit take? If the unit is cut into five, then five of those steps bring you exactly to one. Not four, not six. Five. Cut it into three, and three steps close it. Cut it into eleven, and eleven do.

2:02So the bottom number is not just a name. It is the number of steps that one whole costs. Now the test explains itself. The top number is how many steps you took. The bottom is how many one whole needs. Take more steps than one whole needs, and of course you end up past it. Take fewer, and of course you fall short. That is the reason. The rule was only ever a description of it.

2:33Now we can ask a better question than which side of the one it lands on. How many complete whole units are inside it? Take three halves. Lay the halves out one at a time. First half. Second half — and those two close a whole. Third half. There is nothing to pair it with, so it sits outside on its own. Three halves is one whole unit, and one half left over.

2:59Do it again with five halves. Two of them make a whole. Two more make a second whole. The fifth is left. Two whole units, and one half over. The number has not changed. We have only stopped counting halves and started counting wholes. The same story with a different step, because this is where it becomes obvious. Cut the unit into thirds. One third, two thirds — still short. Three thirds, and the whole closes exactly.

3:31There is the fact we needed. Three thirds is one, precisely. So what does a fourth third do? It cannot fit inside a whole that is already full. It has to spill over the edge. Four thirds is one whole unit and one third past it. And notice you did not need the test to see that. You watched the whole fill up and watched the next piece land outside.

3:58Your turn. Pause if you want to. How many whole units are inside seven halves? Two halves per whole. Two, four, six — that is three wholes — and a seventh half left over. Three whole units, and a half. Next. Four thirds. Three thirds make the whole, so it is one whole and one third. And seven thirds. Three, then three more — two wholes — with a seventh third over. Two wholes and one third.

4:27Same move every time. Group the pieces into complete sets, count the sets, and keep what will not make another. Let us do one carefully and write it down properly. Eight thirds. Three thirds is a whole. Three more is a second whole. That uses six of them. Two are left, and two thirds is not enough for a third whole. So eight thirds is two whole units plus two thirds.

4:56And here is the notation. Write the two, and write the two thirds beside it — no symbol between them. Two and two thirds. It says the same thing the sentence said, in less space. The number on the line has not moved. Eight thirds and two and two thirds are the same place. This form has a name. It is called a mixed number, or a mixed fraction — both names mean this.

5:25It has two parts, and they have names too. The two in front is the whole part. The two thirds is the fractional part. The fractional part is always less than one whole. That is what makes it a leftover — if it reached a whole, you would have taken that whole out. Now the warning, and it matters. There is no symbol at all between the two parts, and that is genuinely confusing.

5:53Two and two thirds does not mean two multiplied by two thirds. Two times two thirds would be four thirds, which is a different number entirely. The missing symbol is a plus. It is two plus two thirds. Say the plus out loud in your head every time you read one of these. Here is a question worth stopping on. Every fraction bigger than one — can all of them be written this way?

6:23Try four halves. It is bigger than one, so it belongs in the right-hand column. Two halves make a whole. Two more make a second whole. And now stop, because there is nothing in your hand. Four halves is two. Exactly two. There is no leftover to write beside it. So the answer is no. A mixed number needs a fractional part, and a fraction that comes out to an exact whole has not got one.

6:52Eight quarters does the same thing, and nine thirds, and ten fifths. When the bottom number divides the top exactly, you get a whole number and nothing else. Everything so far ran one way. Now run it backwards. Take three and three quarters, and write it as a single fraction. Do not reach for a rule. Just undo what we did — open each whole back up into quarters. One whole is four quarters. So the first whole opens into four, the second into four, the third into four.

7:29Four plus four plus four, and then the three quarters that were already there. Count them all: fifteen quarters. Now look at what you just did. Three wholes, four quarters each — that is three times four. Then add the three. That is the rule people memorise, and here it is arriving as a shorthand for counting you have already done. Both directions, quickly, so the move is yours. Nine halves. Two per whole, so four wholes and a half. Four and a half.

8:03Nineteen sixths. Six per whole. Twelve is two wholes, eighteen is three — and one over. Three and one sixth. Now the other way. Seven and two thirds. Seven wholes at three thirds each is twenty-one, plus two, is twenty-three thirds. Two and three elevenths. Two elevens is twenty-two, plus three, is twenty-five elevenths. And every one of those pairs is a single place on the number line, written two ways. The spelling changed. The number never did.

8:41Three things to carry away from this. The bottom number tells you what one whole costs in steps. Take more steps than that and you have passed it — that is the whole of the size test. The mixed form is a regrouping, not a new number. Complete sets pulled out, remainder left behind, same place on the line. And there is a plus sign hiding in the notation. Read it every time, until you stop needing to.

9:11One caution before we finish. Neither spelling is the correct one. Fifteen quarters is not untidy, and three and three quarters is not more advanced. You will meet answers written both ways, sometimes on the same page, and both will be right. Next time: comparing two fractions when the bottom numbers do not match — which is a harder question than it sounds.