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Chapter 10 · The Other Side of Zero

Addition as movement: starting position plus movement gives target position

Teaching notesNCERT10 min

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10 min.

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What the video covers, what to say before it, where a class usually goes wrong, and what to set afterwards. An account is free, and it opens every chapter of every book.

What to assume they know

What they should be able to do

  • State the relation the chapter runs on: starting floor plus movement gives the target floor
  • Evaluate a sum of two signed numbers by locating a start and applying a move
  • Write an expression for a described journey, choosing which number is the position and which is the movement
  • Read the same expression a second way, as two movements combined into one
  • Predict when a sum of two signed numbers will be smaller than the number you started from
  • Show that swapping the two numbers in such a sum leaves the answer unchanged, and say why the picture makes that obvious
  • Evaluate a chain of three presses in order
  • Apply the same relation at a scale where no drawn building could hold it

Where it usually goes wrong

  • "Adding makes things bigger." (+ 4) + (– 3) is smaller than + 4. The picture kills the habit instantly: a movement can point down. Do not fix this with a rule about signs; fix it by moving the lift.
  • "You add the two floor numbers together." Only one of the two numbers in the sum is a floor. The other is a number of floors travelled. Confusing them is the commonest wrong reading of the whole section, and it is why the book keeps writing the relation out in words instead of in symbols.
  • "Gurmit ends up one floor below the ground." He ends one floor below the Toy Store. The combined movement is one floor down; the floor he reaches is a different number, and the chapter deliberately asks only about the movement.
  • "0 + (– 2) needs a special rule because you cannot take 2 from nothing." Nothing is being taken from anything. You stand at the entrance and go down two.
  • "A negative answer means I made an arithmetic slip." In this chapter a negative answer is a floor, and every floor is a real place with a shop on it.
  • "The mine is a different topic because it uses metres." It is the same relation with a different unit and a wider range. That is the whole point of putting it there.

Questions to check understanding

  • Evaluate a sum of two signed numbers and name the floor it lands on
  • Write an expression for a journey described in words
  • Given a start and a finish, state the press required
  • Evaluate a chain of three signed numbers
  • Decide whether a given sum will be larger or smaller than its first term, and justify it by direction rather than by size
  • Fill a missing addend in a stated sum
  • The solutions block bound with this chapter file answers the p.245, p.246 and p.251 exercises on its footer pages 1–2 and 4

Examples worth working on the board

  • The book's own first case (p.245). Start at the Food Court, which is + 1, and press + 2. The expression and the landing floor are both to be built.
  • Seven sums to run (p.245). (+ 1) + (+ 4); (+ 4) + (+ 1); (+ 4) + (– 3); (– 1) + (+ 2); (– 1) + (+ 1); 0 + (+ 2); 0 + (– 2). The first two exist to be compared with each other, and the last three exist to make 0 behave.
  • Reaching Floor – 5 from anywhere (p.245). The book supplies one worked start: from Floor + 2 the press needed is – 7. The task is to produce more such pairs. This is the first place in the chapter where a movement is the unknown, and it sets up Subtraction as the movement that gets you from start to target.
  • Gurmit (p.245). He is in the Toy Store and wants to drop two floors. He presses '+' twice by mistake, then '–' three times to fix it. The combined movement and the floor he ends on are different numbers, and the chapter states only the movement.
  • Three presses in a row (p.246). (+ 4) + (– 3) + (– 2) and (– 1) + (+ 2) + (– 3), alongside (+ 1) + (+ 4) and (+ 4) + (+ 1) again. The repeat of the first pair is deliberate — the same two expressions were read as position-plus-movement on p.245 and are read as movement-plus-movement here.
  • The mine (p.250). Checked against the printed page. A cutaway of a mine with a mineshaft down the middle; the levels the artwork labels are + 180 m, + 125 m, + 100 m, + 40 m, 0 m, – 50 m, – 80 m, – 125 m and – 175 m. The book works two additions at this scale: (+ 40) + (+ 60) and (– 90) + (– 55). The running text states the mine's range as – 200 to + 180, which is wider than the labels the picture carries.
  • The infinite lift (p.251). A vertical scale drawn down the outside edge of the page, marked every 100 from + 1000 to – 1000, with arrowheads at both ends. Four additions to run at this scale: – 125 + (– 30); + 105 + (+ 55); + 80 + (+ 150); – 99 + (+ 200). Checked against p.251.
  • Four missing addends (p.251). (+ 40) +? = + 200; (+ 40) +? = – 200; (– 50) +? = + 200; (– 50) +? = – 200. Worth staging here even though the unknown is a movement, because the second one is where the "adding makes it bigger" habit dies.

Figures to have open

  • The building from Numbering the floors below the ground: why zero needs another side, reused, with a movable pawn and a drawn arrow for the movement. The arrow must be a separate object from the pawn — that separation is section 1's argument.
  • A mine cutaway with a central shaft and labelled levels. Redraw; the content that must survive is that levels exist both above and below the 0 m mark and are labelled in metres.
  • A vertical scale with arrowheads at both ends, marked at regular intervals, standing free of any building. Reused in Laying the integers out in order, and why −8 is less than −2 when it is rotated into a number line.
  • 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.245 — the named sub-headings on addition as movement and on combining button presses, the worked Food Court case, the seven sums, the Floor – 5 task, and Gurmit
  • §10.1, p.246 — the four expressions read as combined movements
  • §10.1, p.250 — the mine, its two worked additions, and the range it is stated to cover
  • §10.1, p.251 — the infinite lift, the four missing addends, and the four additions to be run on it
  • Chapter Summary, p.269, for the general form of the relation once the building is dropped
  • Solutions block bound with this chapter file, footer pages 1–2 and 4

The book

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