The additive inverse, and how it turns every subtraction into an addition

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What to assume they know

Addition as movement: starting position plus movement gives target position: starting position plus movement gives the target
Subtraction as the movement that gets you from start to target: target minus start gives the movement needed
Laying the integers out in order, and why −8 is less than −2: the number line, with 0 in the middle of the picture
The idea, from earlier classes, of an operation being undone

What they should be able to do

Write the inverse of a given integer, including 0
State that a number and its inverse add to 0, and use that as the definition
Explain why the inverse relation runs both ways
Explain why 0 is its own inverse without appealing to a special case
Re-route a journey through 0 and write the two legs as an addition
Convert any subtraction into an addition, and any addition into a subtraction
Predict the sign of a difference before computing it, by looking at the inverse
Use the phrase the book's summary uses — additive inverse — and connect it to the word used earlier in the chapter

Where it usually goes wrong

"The inverse of a number is the number with a minus sign in front." That works for + 3 and fails for – 3, whose inverse has a plus sign. The definition is about what the pair adds to, not about what the symbol looks like.
"0 has no inverse" or "the inverse of 0 is undefined." Add 0 to 0 and you get 0, which is exactly what the definition asks for. It is the only number that is its own partner, and that is worth a section rather than a footnote.
"Two minus signs cancel, so – (– 3) is 3 by a rule about signs." The rule is a consequence, not a starting point. Derive it by walking the journey; a student who has only the sign rule cannot say why (– 3) – (+ 8) does not turn into an addition of + 8.
"Converting to addition is just an alternative method, for people who like addition." It is the reason the sign rules can be stated at all, and it is what §10.5 says Brahmagupta's rules amount to.
"Every subtraction becomes an addition of the same number." It becomes an addition of the inverse. The number that changes is the second one, and only the second one.
"Once you convert, the answer changes." Nothing about the journey changes; only the way it is written down. Run both routes on the same line and land on the same point.

Questions to check understanding

Write the inverse of a stated integer
State what a number and its inverse add up to
Rewrite a given subtraction as an addition
Rewrite a given addition as a subtraction
Evaluate a difference by first converting it
Say which number is its own inverse, and justify it
Given a sum that comes to 0, name the missing term
The solutions block bound with this chapter file answers the p.246 and p.255 exercises on its footer pages 2 and 6

Examples worth working on the board

Basant's mistake (p.246). Standing at the entrance, he presses + 3 by accident and wants to stay where he is. What he presses to fix it, and the expression that records the pair, are the content of sections 1 and 2. The book then asks the same question for + 4 followed by – 4.
Six inverses to write (p.246). + 4, – 4, – 3, 0, + 2, – 1. The 0 in the middle of that list is not padding; it is section 5.
The matching task (p.246). Two rows of four numbers to be joined by lines. The top row reads + 5, – 7, – 8, + 9 and the row beneath it reads – 9, + 8, – 5, + 7. None of the four pairs sits directly above its partner, which is what makes the task worth doing.
The second reading of inverse (p.246). The book gives it twice over: from Floor + 4 press – 4 and you are back at the entrance; from Floor – 2 press + 2 and you are back at the entrance.
The clue (pp.251–252). The long subtraction from – 200 up to + 2000 came out as + 2200, and so does the corresponding addition. The book records the coincidence on p.251 and generalises it in the paragraph at the top of p.252. Treat it as evidence, not yet as proof.
The two methods (p.255). Going from 2 to – 3. The first method reads the movement straight off the line. The second breaks the journey at 0: first the leg from 2 down to the origin, then the leg from the origin down to – 3, and adds the two legs. The book prints the two resulting expressions in colour to make the point that one of them has no subtraction sign in it at all.
Four conversions (p.255). (+ 7) – (+ 5); (– 3) – (+ 8); (+ 8) – (– 2); (+ 6) – (– 9). Each is to be rewritten as an addition. The third and fourth are where the surprise from section 7 finally gets its reason.
The Summary's two examples (p.269). The inverse of 7, and the inverse of – 543. The second is chosen to be a number nobody would picture on a line, which is the point: by then the relation is arithmetic, not geography.

Figures to have open

A number line with a pawn and two arrows that can be laid head to tail, one of them growing out of 0. Reused from Laying the integers out in order, and why −8 is less than −2.
A two-leg journey drawn on one line with the break at 0 marked, so that the two legs can be seen to be movements in the same direction. This is the figure the whole topic turns on and it must be built, not shown finished.
A rewrite movement in which only the sign of the second term moves. Static before-and-after panels will not carry section 9.
No photograph or textbook data table is needed.

Where this sits in the book

NCERT Ganita Prakash, Class 6, Chapter 10 "The Other Side of Zero", §10.1 "Bela's Building of Fun", p.246 — the named sub-heading on returning to zero, Basant, the definition of inverse, the six inverses and the matching task
§10.1, pp.251–252 — the long subtraction whose answer matched an addition, and the paragraph that generalises the observation
§10.1, p.255 — the named sub-heading on converting between the two operations, the two methods, and the four conversions
Chapter Summary, p.269, for the fuller name and its two examples
§10.5, p.268, for the same fact stated as one of Brahmagupta's rules for subtraction — the forward pointer this topic should end on
Solutions block bound with this chapter file, footer pages 2 and 6