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Chapter 2 · Arithmetic Expressions
Removing a bracket after a minus sign flips every term inside
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The sign flip is not a rule to memorise. It is forced by what taking away means.
The idea
The sign flip is not a rule to be memorised — it is forced by what taking away means. Removing a whole is the same as removing each part in turn; and removing too much obliges you to hand some back. That is the entire justification, it covers the awkward case where a negative inside comes out positive, and it also explains why a bracket sitting behind a plus comes out untouched. The chapter says as much in a boxed aside: work it out from the meaning instead of remembering when to change signs.
What you should be able to do
- Rewrite an expression of the shape "a number minus a bracketed total" without its bracket, changing the sign of every term that comes out
- Explain that change from the meaning of subtraction rather than by quoting a rule
- Handle the case where a negative term inside the bracket emerges positive
- State that a bracket not preceded by a minus leaves its terms' signs alone, and say why
- Check a proposed bracket removal by evaluating both forms and comparing
- Predict how the value of an expression moves when one of its terms is nudged up or down, without recomputing
- Recognise that raising a term that is itself negative makes the total larger, not smaller
Words to know
| Term | Definition in one line | First introduced |
|---|---|---|
| removing brackets | rewriting an expression so that a bracketed part is no longer bracketed | this topic; printed pp.35–36 (Part I, §2.2) |
| negative sign | the minus standing in front of a bracket, which decides whether the signs inside change | this topic; printed pp.35–36 (Part I, §2.2) |
| term (of an expression) | one of the pieces a sum falls into | Every expression can be rewritten as a sum of terms; printed p.28 (Part I, §2.2) |
| inverse | the number with the same size and the opposite sign | Every expression can be rewritten as a sum of terms; printed p.28 (Part I, §2.2) |
| brackets | the paired symbols that force one part to be settled first | Brackets decide which operation happens first; printed p.27 (Part I, §2.2) |
| value | the single number an expression stands for | Writing a situation as an expression before computing it; printed p.24 (Part I, §2.1) |
| nudging a term | changing one term slightly and reading off the effect on the total without recomputing | an added phrase; the chapter runs the exercise under a printed subheading of its own but names no such operation |
Where people slip up
- "Removing a bracket means rubbing it out." The expression that survives is a different string of symbols.
200 – (40 + 3)and200 – 40 + 3are not the same number, and the chapter makes that comparison explicitly on p.35. - "Only the first term inside changes." Both 40 and 3 flip. This is the single most common slip, and it is why the chapter uses a two-term bracket first.
- "Everything inside becomes negative." No — every term reverses. A negative term inside emerges positive, which is precisely what happens to the −100 in Example 13. A student who converts rather than reverses will get 150 instead of 350 and will not know why.
- "A bracket after a plus needs the same treatment." Example 14 exists to say no, and to explain the difference rather than legislate it.
- "−15 must be less than −16, because 15 is less than 16." The chapter builds the whole first column of the tinker grid to catch this, then asks the question outright. Raising a negative term raises the total.
- "If a term goes up by one, the total goes up by one — always." True when the term goes up. But an exercise on p.38 changes the subtracted amount to keep a difference fixed, and there the two adjustments must move the same way, not opposite ways.
- "Two changes always cancel." Column three of the tinker grid contains a pair that cancels and a pair that does not, side by side, on purpose.
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Worked answers to this chapter’s exercises · this video explains Figure it Out · 3 Q1, Figure it Out · 3 Q2, Figure it Out · 3 Q3, Figure it Out · 3 Q5, Figure it Out · 3 Q6, Figure it Out · 3 Q7, Figure it Out · 3 Q8, Figure it Out · 3 Q9
Transcript1,444 words
Two hundred, minus a bracket, and inside the bracket, forty plus three. There are two honest ways to work that out, and they had better agree. Route one. Settle the bracket first, because that is what a bracket is for. Forty plus three is forty three. So we want two hundred, take away forty three. One hundred and fifty seven. Route two. Take the forty off first, and land on one hundred and sixty.
Then take the three off that as well. One hundred and fifty seven again. The same answer, and it is not luck. Taking away a total is taking away each part of it in turn. So the bracket can come off. The only question is what happens to the three on the way out. Here is what almost everybody writes when that bracket comes off. Two hundred, minus forty, plus three.
It looks harmless. Every symbol is still there, in the same order. But work it out. Two hundred take away forty is one hundred and sixty, and plus three is one hundred and sixty three. One hundred and sixty three, against one hundred and fifty seven. Those are six apart. And six is exactly two threes. That is the fingerprint of this mistake, and it is worth knowing. The three was taken away in one version and handed over in the other, so the gap is two of them, never one.
And the wrong line is not meaningless. It is the correct answer to a different question. Take away forty, then be given three back. Which is not what anybody asked for. Let us put that somewhere you can feel which answer is right. You buy two things. One costs fifteen. The other costs fifty six. You hand over a note worth one hundred. What comes back? One way is to add the bill up first. Fifteen and fifty six is seventy one.
One hundred take away seventy one. Twenty nine comes back. The other way is to pay for them one at a time. One hundred take away fifteen leaves eighty five. And eighty five is not just a step. It is the change you would have got if you had bought only the first thing. Then take the fifty six off that, and there is the twenty nine again. And the tempting version claims one hundred, minus fifteen, plus fifty six, which is one hundred and forty one. More than the note you handed over.
Now the case that makes people give up on this. Five hundred, minus a bracket, and this time there is a subtraction inside it. Two hundred and fifty, minus one hundred. Settle it first and there is no trouble at all. Two hundred and fifty take away one hundred is one hundred and fifty. Five hundred take away one hundred and fifty is three hundred and fifty. Now take the bracket off, and watch what is forced.
One hundred and fifty is what we are removing. Suppose we start by removing the whole two hundred and fifty. We have gone too far. We have taken off one hundred more than we were meant to. So one hundred has to come back. Five hundred, minus two hundred and fifty, plus one hundred. Three hundred and fifty. The minus one hundred came out as a plus one hundred, and nobody had to remember a rule to see why.
The other candidate is sitting right there, and it deserves killing properly. Five hundred, minus two hundred and fifty, minus one hundred. That is what you get if you make everything inside the bracket negative, instead of reversing each term. Work it out. Five hundred take away two hundred and fifty is two hundred and fifty. Take away one hundred is one hundred and fifty. One hundred and fifty. And now look at where that number came from.
One hundred and fifty is what was inside the bracket in the first place. So the wrong answer is a number already in front of you, which is exactly why it feels right. Three hundred and fifty against one hundred and fifty. Two hundred apart, which is twice that hundred, as before. Every term inside reverses. A plus becomes a minus, and a minus becomes a plus. They do not all become negative. Reversed, not converted.
Every bracket so far has had a minus in front of it. What if there is a plus? Somebody keeps a coin collection in two bags. Twenty eight in the first, thirty five in the second. They give away ten coins out of the second bag. Twenty eight, plus a bracket, thirty five minus ten. The second bag has twenty five in it now, and twenty five with twenty eight is fifty three.
Take the bracket off and nothing changes. Twenty eight plus thirty five minus ten. Still fifty three. The thirty five stayed positive and the ten stayed negative. And that is not a second rule to learn alongside the first. There is nothing in front of that bracket asking for anything to be reversed, so nothing is reversed. A bracket sitting behind a plus is a bracket you never needed. This is the moment to say how to carry all of this around with you.
You could learn it as a rule. Minus in front, flip everything inside. Plus in front, flip nothing. And that serves you right up until the day you cannot remember which way round it goes. There is a better thing to keep, and it fits in one sentence. Taking away a whole is taking away each of its parts. And if you take away too much, you hand the extra back.
That is all of it. That one idea produces every case we have done today. Two hundred minus a bracket holding forty and three: take off both of them. Five hundred minus a bracket that had already lost one hundred: take off too much, then give the hundred back. Work it out from what it means, and you will never need to remember which sign does what. Now something a bit different, and it is a habit worth building.
Fifty three, plus negative sixteen. That comes to thirty seven. Now change one thing. Fifty four, plus negative sixteen. You could add it all up again. Or you could notice that fifty four is one more than fifty three. One more going in, one more coming out. Thirty eight. You did not compute that. You traced it. It works on the other term too. Fifty two plus negative sixteen is one less than thirty seven, so thirty six.
And it works for any size of nudge, not just one. Move a term up by five and the total goes up by five, whichever term you moved. This is not a trick for people who dislike arithmetic. It is what being a sum of terms means. Here is where that habit gets tested properly. Fifty three plus negative sixteen is thirty seven. Now change the second term to negative fifteen.
Answer before you work it out. Does the total go up, or does it go down? Almost everybody says down, because fifteen is smaller than sixteen. So let us stop arguing and put the two numbers on a line. Negative sixteen sits here. Negative fifteen sits here, to its right. Further right means larger. Negative fifteen is larger than negative sixteen. So the term went up by one, and the total goes up by one with it. Thirty eight.
Fifteen is smaller than sixteen. Negative fifteen is larger than negative sixteen. Both of those are true. The size of a number and its value are two different questions, and a line keeps them apart. One last thing. What happens when two terms move at the same time? Start with negative eighty seven, plus negative sixteen. That is negative one hundred and three. Now try negative eighty eight, plus negative fifteen.
The first term went down by one. The second term went up by one. Down one and up one. They cancel, and the total does not move at all. Negative one hundred and three again. Now negative eighty six, plus negative eighteen. The first went up by one. The second went down by two. Those do not cancel. Up one and down two leaves down one. Negative one hundred and four.
And negative ninety seven with negative twenty six. Down ten and down ten, both the same way, so down twenty. Negative one hundred and twenty three. So never ask whether two changes cancel. Add them together, with their signs, and the answer falls straight out.
Where this fits
Taken from the notes each video was made from, not from the reading order — these are the ideas this one rests on and the ones that later rest on it.
Builds on
- Every expression can be rewritten as a sum of termsClass 7 · Ch 2, Arithmetic Expressions
- Commutative and associative: why order and grouping are freeClass 7 · Ch 2, Arithmetic Expressions
Comes up again in
- The distributive property, and using it to compute fasterClass 7 · Ch 2, Arithmetic Expressions