PrepShorts · Study sheet · Class 7 Mathematics · Chapter 2, Operations with Integers
Chapter 2 · Operations with Integers
Negative numbers on the number line, and adding and subtracting them
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A signed number is a piece of compression, and a puck sliding on a straight track is the argument for paying its price.
The idea
A signed number is a piece of compression, and this section is the argument for paying its price. Struck in one direction only, a carrom coin needs one formula; allowed to travel both ways it appears to need four, one per combination of directions. The section refuses the four. It puts the direction inside the number, as a sign, and the single formula survives untouched. The same compression rescues subtraction: seven tokens cannot have eighteen taken from them, but pairs worth nothing can be added for free, and once that move is allowed every subtraction becomes an addition of an inverse and no subtraction is ever impossible again.
What you should be able to do
- Recover two integers from their sum and their difference by systematic trial, and say why the search always closes
- Explain why a movement with a direction is naturally modelled by a signed number rather than by a pair (size, direction)
- State the final-position formula for two carrom strikes and apply it when either or both strikes go leftward
- Read an arrow diagram on a number line and infer the signs of the two movements and which has the larger magnitude
- Separate the magnitude of an integer from its direction, and use the word correctly
- Carry out a subtraction that "runs out" of tokens by inserting zero pairs, and justify why inserting them changes nothing
- Rewrite any subtraction as an addition of an additive inverse, and read the notation −(−18) correctly
Words to know
| Term | Definition in one line | First introduced |
|---|---|---|
| integer | a whole number that may carry a minus sign, including zero | printed in the chapter title, p.24, and from p.25 onward in §2.1 |
| sum | the result of adding two numbers | printed in §2.1, p.24 |
| difference | first number minus second number, in this section's stated order | printed in §2.1, p.24, with the order fixed in the same passage |
| number line | the marked line on which integers are placed, negatives to the left of 0 | printed in "Carrom Coin Integers", p.26 |
| magnitude | how big a movement is, ignoring which way it goes | printed in bold, p.27 |
| final position | where the coin ends up after all its strikes, written P | printed in "Carrom Coin Integers", pp.25–26 |
| token | the counter used to stand for +1 or −1 | printed throughout p.28 |
| zero pair | one green and one red token together, worth nothing | printed on p.28 |
| additive inverse | the number that adds to a given number to give zero | printed on pp.28–29 |
| positive / negative | rightward / leftward, once the direction convention is fixed | printed throughout §2.1 |
| matching parity | sum and difference are both even or both odd | an added term; the chapter never names or states this condition |
| signed number | a number carrying its direction as a sign | the explanation's phrase; not printed in this chapter |
Page numbers in the provenance column are Part II's, printed pages 24–46.
One caution: This section fixes difference to mean first minus second, which is why a difference is allowed to be negative here at all. Say the order out loud when the word first appears; a student carrying "bigger minus smaller" from earlier classes cannot make sense of Rakesh's second puzzle.
Where people slip up
- "A difference cannot be negative." Rakesh's first trial row, (10, 15), produces −5 on the very first line of the chapter, and the second puzzle asks for −11 outright. The chapter's fixed order — first minus second — is what makes this coherent. Checked against p.24.
- "The minus sign means the number is small." The sign says which way; the magnitude says how far. Picture 1 on p.27 has the negative movement as the larger of the two. Make magnitude and direction two separate readings of one symbol.
- "Four direction cases means four rules to remember." The section sets the four cases up on p.26 precisely so it can throw them away. If the explanation lists the four cases and then teaches four rules, it has taught the opposite of the page.
- "Adding zero pairs is cheating — you are changing the number." A green and a red together are worth nothing, so the bag's value is untouched; only its contents change. Show the count of positives rising from 7 to 18 while the value stays 7.
- "7 − 18 cannot be done." It cannot be done by removal alone from seven tokens, which is exactly the difficulty the page stages. It can always be done once inverses are allowed. Nothing about integers is "impossible to subtract".
- "−(−18) is some new operation." It is the inverse of the inverse, and it lands back where it started. Tie it to the picture: turn round twice and you face the way you began.
- "The carrom coin's position is a distance, so it cannot be negative." The section's P is a position measured from 0, not a length. Its sign records which side of 0 the coin stopped on.
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Worked answers to this chapter’s exercises · this video explains Figure it Out · 2.1 Q1, Figure it Out · 5 Q12
Transcript1,292 words
Here is a puzzle. I am thinking of two numbers. They add up to twenty five. And their difference is eleven. Find them. But before you start, one thing has to be pinned down, or none of this works. Difference here means the first number minus the second one. In that order. Not the bigger minus the smaller. First minus second, whichever of them is bigger. Which means a difference is allowed to come out negative.
Hold on to that, because you are about to see one almost immediately. So guess. Ten and fifteen. They add to twenty five, which is right. And the difference? Ten minus fifteen. That is minus five. There it is. A negative difference, on the very first line we wrote. We wanted eleven and we got minus five, so the first number needs to be bigger. Try twenty and five. Sum twenty five, difference fifteen. Now we have gone too far.
Nineteen and six. Difference thirteen. Closer. Eighteen and seven. Difference eleven. That is the answer. And notice what really happened there. Every guess told you which way to move next. That is worth stopping on, because it was not random guessing. Keep the sum at twenty five, and push the first number up by one. The second one has to come down by one to keep the total. So the gap opens by two.
Every single time. Up one, gap opens two. It never varies and it never stalls. So the difference marches steadily, and you can walk straight to the answer. Now the second puzzle. Same sum of twenty five. But this time the difference is minus eleven. You do not need to start again. Minus eleven is eleven with its sign turned round. So turn the pair round. Seven and eighteen. The same two numbers, the other way about.
Here are six more, and some of them are strange enough to be worth naming. Sum twenty seven, difference nine. Sum four, difference twelve. Sum zero, difference ten. And sum zero, difference minus ten. Sum minus seven, difference minus one. Sum minus seven, difference minus thirteen. So two of those sums are zero. Two of them are negative. And three of the differences are negative. Three, not two. Those are different counts.
Every one of them closes. And here is why, though it is not usually said out loud. The sum and the difference always match. Both even, or both odd. Otherwise there is no answer at all. Now a completely different picture, and it is going to do a great deal of work. A puck on a straight track. It starts at zero, and you can strike it. For the moment, there is a restriction. You can only strike it one way. To the right.
Strike it four. It slides four along and stops. Strike it three more. It ends up at seven. So where does it finish? Add the two strikes together. That is the whole rule. First strike plus second strike gives you the final position. One line. Nothing to memorise. Enjoy it while it lasts. Because now we let go of the restriction. You can strike it whichever way you like. Right, or left. And immediately that one line is in trouble.
Both strikes rightward. Four then three. It finishes at seven, exactly as before. Both strikes leftward. Four then three. Now it finishes seven along, on the other side. Rightward four, then leftward seven. It goes back past the start and stops on the left. Leftward four, then rightward seven. That one stops on the right. Four different situations. And each of them seems to want a rule of its own.
Add them. Subtract them. Take the smaller from the bigger. Then decide which sign to keep. So do we now have four rules to learn instead of one? No. And this is the good part. Look at what is actually causing the trouble. We are carrying two separate things about every strike. How far it went, and which way it went. Two pieces of information, kept apart, so every combination of them has to be handled.
So stop keeping them apart. Put the direction inside the number itself. Draw the track as a number line, with zero somewhere in the middle. Rightward counts as positive. Leftward counts as negative. That is a choice, and we make it once. Now a strike is not a size with an arrow attached. It is a single signed number. Watch what that does. Strike the puck five, to the right.
Then strike it seven, to the left. That second one is minus seven. Now use the old rule. The one from when it could only go one way. Just add them. Five plus minus seven is minus two. And minus two is not a leftover. It is a position. Two units to the left of zero. Go and look at the track. That is exactly where the puck is sitting.
One formula. The same formula. It never needed replacing. The four cases did not turn into four rules. They got absorbed. So what did that cost, and what did it buy? It bought this. One symbol now carries two separate readings. Take minus seven. Its size is seven. That size has a name. It is the magnitude. And its sign says leftward. Two facts, living in one symbol. Which means you have to stop reading a minus sign as meaning small.
Minus seven is a smaller number than five. Nobody disputes that. But as a movement, minus seven is the bigger of the two. Seven beats five. Smaller number, larger movement. Both true at once, because they are different questions. Now run the whole thing backwards. Here are the arrows, with the numbers taken off. First arrow goes right. The second comes back further left. The puck ends up left of zero.
So the second movement must be negative, and it must be the bigger of the two. Second picture. Right, then back again, but a shorter way. It finishes right of zero. Same two signs, opposite conclusion. Here the first movement is the bigger one. Third picture. Left, then exactly as far back to the right. It lands on zero. Two questions to take away. The puck moved minus four and finished at five. What was the second move?
And if it moves one, then minus two, then three, then minus four, all the way to minus ten, where does it stop? Now subtraction, which is going to need a different picture again. Green tokens are worth plus one each. Red ones are worth minus one each. And there is one rule. A green and a red together are worth nothing at all. Put them in a bag. Seven greens. The bag is worth seven.
Now drop in one green and one red together. What is the bag worth now? Still seven. You added nothing, because that pair is worth nothing. The contents of the bag changed. Its value did not. That sounds like a technicality. It is about to be the entire trick. Because here is the problem. Seven, take away eighteen. You have seven green tokens, and I am asking you to hand me eighteen of them.
You cannot. There are only seven there. It looks impossible. So buy yourself some room. Drop in eleven of those pairs worth nothing. The bag now holds eighteen greens and eleven reds. And it is still worth seven. Now hand me the eighteen greens. What is left behind is eleven reds. Eleven reds is minus eleven. So seven take away eighteen is minus eleven. Taking a number away is just adding its opposite. Turn round twice, and you face the way you began.
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
- Writing a situation as an expression before computing itClass 7 · Ch 2, Arithmetic Expressions
Comes up again in
- Why a negative times a negative must be positiveClass 7 · Ch 2, Operations with Integers
- The sign rule for division, and why it follows from multiplicationClass 7 · Ch 2, Operations with Integers
- Commutative, associative, and distributive over the integersClass 7 · Ch 2, Operations with Integers
Either side of this one
- Why every equilateral triangle has three 60° anglesClass 7 · Ch 1, Geometric Twins