Brackets decide which operation happens first

0:00Somebody hands you a sheet of working, and on it is a single line. Thirty, plus five, times four. What number is that? It is a fair question, and right now it does not have an answer. Not because the arithmetic is hard. The numbers are thirty, five and four. It is because that line can be read in two different ways, and the two ways give two different numbers.

0:27Ninety apart, as it happens. And here is the thing. Whoever wrote it down knew exactly which one they meant. They had a situation in front of them, and the situation said which operation came first. Then the situation got left behind, and the line went out into the world on its own. Before any of the maths at all, here is an ordinary English sentence with nothing mathematical in it.

0:57Ana sat next to Ben with the toys. Read that, and it is Ben who is holding the toys. Now the same words, with one comma dropped in. Ana sat next to Ben, with the toys. And now Ana is holding them. Nobody changed a word. The toys moved because of a comma. What the comma did was group things. It said that with the toys belongs to Ana, and not to Ben.

1:26Written maths has marks that do exactly that job, and they are called brackets. They do not calculate anything. They group. Back to our stranded line, then. Thirty plus five times four, with nothing around it to say what goes first. Here is the first way to read it. Straight across, left to right. Thirty plus five is thirty five. Thirty five times four is one hundred and forty. And here is the second way. Do the multiplication first.

2:01Five times four is twenty. Twenty plus thirty is fifty. One hundred and forty, and fifty. Same line, same three numbers. Now look hard at both of those ladders, and try to find the mistake in one of them. There is not one. Every single step in both columns is correct arithmetic. So we cannot settle this with more checking. We have to go back for the story. Here it is. Ana brought thirty loose marbles to the playground.

2:34Ben brought five small bags, with four marbles in each bag. How many marbles are on the playground? Count them. Four, eight, twelve, sixteen, twenty in the bags. And thirty already loose. Fifty. Not one hundred and forty. So the second reading is the one describing this playground, and the first one is not. But notice what the first reading is not. It is not a slip. One hundred and forty is four copies of everything they have between them, and that is a perfectly real thing to want.

3:11It is just not what happened here. The mistake was in the description, not in the adding. Which leaves us needing a way to write down which reading we meant. That is what brackets are for. Thirty, plus, open bracket, five times four, close bracket. The brackets say one thing out loud. Settle what is inside me first. And now the line cannot be read the other way. There is nothing left to guess.

3:41Notice what did not happen. No marble moved. There were fifty on the playground before we wrote the brackets, and there are fifty now. Brackets do not change what an expression is worth. They decide which of the two possible expressions you actually wrote down. And this is not some rare corner case, by the way. Take any line of that shape, with a plus and a times in it, and about nine times out of ten the two readings disagree.

4:14So, the rule for reading a bracket, in one sentence. Settle the inside, replace the bracket with what it came to, and carry on. On our line, the inside is five times four. That is twenty, so the whole bracket becomes twenty. Thirty plus twenty. Fifty. Two steps, and the bracket is gone after the first one. It has done its job. It held the multiplication together until the multiplication was settled.

4:45Write both steps down while you are learning this. The middle line is where mistakes get caught. You may have been handed a word to memorise for choosing an order. We are not going to use one, because a rule you can rebuild beats a word you can forget. Now a second situation, and this one has a trap in it worth walking straight into. You are at a shop. A pack of biscuits costs fifteen.

5:14A bag of rice costs fifty six. You hand over a hundred note, and you wait for your change. How much should come back? Let us build the line the way you would say it out loud. You start with the hundred you handed over. You take away fifteen for the biscuits. And you take away fifty six for the rice. Written down in a hurry, that comes out as a hundred, minus fifteen, plus fifty six.

5:43Two items, two amounts, in the order you picked them up. Look at it for a second before we work it out. Read it left to right, the way we read the first one. A hundred minus fifteen is eighty five. Eighty five plus fifty six is one hundred and forty one. So the change is one hundred and forty one. You handed over one hundred, and you are getting one hundred and forty one back.

6:13Forty one more than you walked in with, for buying two things. Nobody needs to check that arithmetic to know it is wrong. And that is the whole point of this bit. The answer was rejected by the situation, not by the sums. Both steps were right. Eighty five is right, and one hundred and forty one is right. The line itself is a fine line, too. It describes handing over a hundred, spending fifteen, and then being given fifty six. That was just not the shopping trip.

6:50So what did the line get wrong? It took the fifteen away, and then it added the rice on. But the rice was bought as well. It has to come off too. What we mean is both items together, taken away from the hundred. And both items together is a group, which is exactly the thing a bracket writes down. A hundred, minus, open bracket, fifteen plus fifty six, close bracket.

7:18Settle the inside. Fifteen and fifty six is seventy one. That is the shopping. A hundred minus seventy one is twenty nine. That is the change. And check it the way a shopkeeper would. Seventy one spent, twenty nine back, one hundred in total. The gap between the two answers is one hundred and twelve, which is fifty six twice over. The rice got added on instead of taken off, so it is out by that item twice.

7:52One last thing, about what brackets are for and what they are not for. Brackets are how you force a grouping, when the writing would otherwise leave it open. But most lines you will meet do not have any brackets in them at all. Thirty plus five times four, plain, with nothing around anything. That line is still perfectly readable, and there is one settled way to read it. It is just not left to right, and working out what it is comes next.

8:24So there are two tools here, not one. Brackets, for when the grouping has to be said out loud. And a way of reading the lines that carry none, which is where we are going. Both of them are doing the job that comma did. Making sure the number you meant is the number somebody else reads.