Evaluating expressions that mix integers and brackets

0:00Here is a habit worth more than any single technique in this topic. When a situation turns into arithmetic, write the whole plan down first, and work it out second. That sounds like a stylistic preference. It is not. Once the plan is written, every sign in it has already been decided. You never reach a step where you have to stop and ask whether this part gets added or taken away.

0:28The sign bookkeeping stops being a series of small decisions and becomes one piece of arithmetic. We will do it twice, in two quite different situations. And then we will run the whole idea backwards, and try to recover a plan from nothing but its answers. A test. Fifty multiple choice questions. A right answer earns five marks. A wrong answer carries a penalty worth two. Mala answers thirty of them correctly and twenty of them wrongly.

1:03The tempting move is to work out the marks she gained, then the marks she lost, and then remember to take the second away from the first. Instead, settle the two per-question values first, and put the penalty inside the number. A right answer is worth five. A wrong answer is worth minus two. Now the whole test is a single expression. Thirty times five, plus twenty times minus two.

1:33And every operation in it is an addition. Nothing is left to decide later. So work the plan. The two multiplications happen before the addition. That is a convention rather than a law, and it is doing real work here, so let us have it out loud. Thirty times five is one hundred and fifty. Twenty times minus two is minus forty. And one hundred and fifty plus minus forty is one hundred and ten.

2:06Notice that nobody ever decided to subtract anything. The minus was in the plan from the beginning, sitting inside the value of a wrong answer, which is where it belongs. Here is a question the setup raises and almost nobody asks. What is the best score this paper can give, and what is the worst? The best is every question right. Fifty times five is two hundred and fifty. The worst is every question wrong. Fifty times minus two is minus one hundred.

2:43And you might wonder whether leaving questions blank moves either of those. It cannot. A blank scores nothing at all. Nothing is better than a penalty, so turning any wrong answer into a blank always raises the total and never lowers it. Which means the worst possible paper is the one where you answered every single question, and got every single one of them wrong. Second situation, and it comes with a much better picture.

3:16A mine shaft, with galleries running off it at different depths. Ground level counts as zero. Above it is positive, below it is negative. It is a number line stood on its end. A lift travels in this shaft at three metres a minute. It starts at ground level, and it goes down for one hour. Where does it end up? There are two ways to answer that, and they are not the same way written twice.

3:46First method. Work out how far it travelled, then take that away. Sixty minutes, at three metres a minute. Sixty times three is one hundred and eighty metres of shaft. It went downwards, so take that off the starting level. Zero minus one hundred and eighty is minus one hundred and eighty. The lift is a hundred and eighty metres below ground, and that is correct. But look carefully at where the thinking happened.

4:18The direction never entered the arithmetic. It stayed in your head, and it was spent at the moment you chose to subtract. Second method. Put the direction into the number instead. The lift is going downwards, so call its speed minus three metres a minute. That is a modelling choice and it is worth naming as one. A single number is now carrying both how fast and which way, which is a compression, and compressions are always worth noticing.

4:52Then the whole journey is one product. Minus three, times sixty. Minus one hundred and eighty. The same answer, obviously. But nothing was decided halfway through this time. The minus went in at the start, and the arithmetic carried it the rest of the way. Which looks like a matter of taste, right up until the situation changes. So change it. The lift now begins fifteen metres above ground, and descends for forty five minutes.

5:27Watch what that does to each method. The second one barely notices. Starting position, plus rate times time. Fifteen, plus minus three times forty five. Minus three times forty five is minus one hundred and thirty five, and fifteen plus that is minus one hundred and twenty. One extra term, and the shape of the plan did not change at all. The first method has to be argued again. You still get a hundred and thirty five metres of travel, but you are subtracting it from fifteen now instead of from zero, and what that means has to be thought through a second time.

6:11Now run the whole idea backwards. Here is a machine. Three numbers go in, and one comes out. You are not told what it does. You only get to see what it did. Five, eight and three gave ten. Ten, eleven and twelve gave nine. Five, eight and minus three gave sixteen. Compare the first row with the third. Only one input changed, from three to minus three, and the output went from ten to sixteen.

6:43The input dropped by six and the answer rose by six, so the machine is subtracting that one. First plus second, minus third. Check it against the other rows and it holds throughout. Here is a harder machine, and it is the one worth your time. Four, eight and minus three gave twenty eight. Six, nine and six gave minus forty eight. Two, three and minus two gave eight. Try adding and subtracting the three inputs in every combination there is, and not one of the twenty seven possibilities fits.

7:22The reason is that a multiplication is buried inside this one. Take the first input, and subtract the product of the other two. Four, minus eight times minus three. Eight times minus three is minus twenty four. And four minus minus twenty four is twenty eight. Which only works because the multiplication happened first. That rule fits all five rows, so that is the answer. But it is worth ten honest seconds on what has actually been established.

7:59Five rows do not determine a rule. They recommend one. I can write down a different rule that agrees with every row you have been shown, and then disagrees with the simple one by more than a million on the very next. It is an ugly rule and nobody would ever propose it, and it fits the evidence exactly as well. That is not a reason to distrust the simple answer. Simplicity is a genuinely good reason to prefer a rule.

8:32It is a reason to know what sort of thing you are holding: the best explanation of the evidence, rather than a proof. Which is the same distinction that separates a real argument from a handful of checked examples. Last thing, and it is about the brackets. It is tempting to read them as tidiness. They are not tidiness. They are the plan. Four, minus the product of eight and minus three, is twenty eight.

9:03Now move one bracket. Four minus eight, all of it times minus three, is twelve. Same three numbers, same order, same operations, and two different answers. But do not over-learn that, because brackets do not always bite. Five times minus three, times four, is minus sixty, and it stays minus sixty whichever pair you bracket, because everything in sight is a multiplication. Brackets change the answer when a multiplication sits beside an addition or a subtraction. That is the whole condition, and it is worth knowing exactly rather than as a superstition.