10. The Other Side of Zero

11 topics2 h

Chapter 10 of NCERT Mathematics for Class 6: The Other Side of Zero. Four sections — Numbers below zero, Integers on the number line, The token model and Integers in the world, in puzzles, and in history. 11 videos, 2 h in all.

Worked answers to the 49 exercise questions in this chapter →

0 of 11 done — sign in to track this chapter

Part 1

Numbers below zero

9 minNumbering the floors below the ground: why zero needs another sideNumbering the floors below the ground is a naming problem before it is a rule. Zero turns out to need another side.9 min · 12 partsहिंदी10 minAddition as movement: starting position plus movement gives target positionAdding a negative number needs no rule about signs — only a change in what addition means. Start, move, arrive.10 min · 12 partsहिंदी9 minSubtraction as the movement that gets you from start to targetSubtraction has two meanings, and with negatives you have to choose between them every single step.9 min · 12 partsहिंदी

Part 2

Integers on the number line

9 minLaying the integers out in order, and why −8 is less than −2Why is −8 less than −2, when 8 is bigger than 2? Because “less than” was always about position and never about size.9 min · 12 partsहिंदी9 minThe additive inverse, and how it turns every subtraction into an additionThe additive inverse is the end of subtraction, not a rule about signs. Anyone leaving with “two minuses make a plus” has missed it.9 min · 12 partsहिंदी

Part 3

The token model

9 minZero pairs: why a positive and a negative token cancelOne green token and one red token cancel — and that single fact is why addition cannot depend on the order you do it in.9 min · 12 partsहिंदी9 minSubtracting with tokens by first putting zero pairs inTake away four red tokens when there are none there. Put zero pairs in first and suddenly there are.9 min · 12 partsहिंदी

Part 4

Integers in the world, in puzzles, and in history

9 minCredits, debits, and what a negative balance actually meansA negative balance is not a smaller number. It is a direction — and the bank means something specific by it.9 min · 12 partsहिंदी10 minSea level and freezing point: zero as a chosen reference, not an absenceSea level is not where the sea is. Zero is a chosen reference here, not an absence of anything.10 min · 12 partsहिंदी10 minInteger grids, and why the total comes out the same every timeThe total in an integer grid is decided before you play. Circle whatever you like and it comes out the same.10 min · 12 partsहिंदी9 minBrahmagupta's rules, and how long it took the world to accept themBrahmagupta wrote the rules for negative numbers down. The world took a thousand years to accept them — and that was not a delay in discovering them.9 min · 12 partsहिंदी

What you will be able to do

  • State the chapter's opening question: whether anything lies to the left of 0
  • State the relation the chapter runs on: starting floor plus movement gives the target floor
  • Recall both earlier meanings of subtraction and say which of the two the chapter keeps
  • Compare two floor numbers by saying which floor is lower
  • Write the inverse of a given integer, including 0
  • Lay out any given integer as a collection of tokens of a single colour
  • Subtract by removing tokens, in the case where enough of them are present
  • Identify a credit as a positive movement and a debit as a negative one
  • State what heights on a geographical cross section are measured from
  • Compute the sum of a row and of a column in a bordered grid and compare them
  • State where and in what activity negative numbers were first written down

What this chapter assumes you already know

  • Counting numbers, and zero as the number that comes before 1
  • Reading a simple scale where the marks are evenly spaced
  • Addition of whole numbers, and the fact that order does not change a sum
  • Reading a vertical scale where equal gaps mean equal steps
  • Subtraction of whole numbers as taking away, and as finding what is missing
  • The habit, from earlier classes, of checking a subtraction by adding back
  • Comparing whole numbers with < and >, and reading those symbols aloud
  • Reading an evenly spaced scale and interpolating between two marked ticks

It builds directly on Every number sequence is a rule, not a list, Line and ray: what changes when you refuse to stop and Why adding the odd numbers gives the squares.

Where people usually slip up

Sentences students actually say, taken from the notes the videos were made from. Each one is answered on the page of the video it belongs to.

  • "A minus sign means subtract."
  • "Negative numbers are numbers you cannot have."
  • "The floor below the ground should be Floor 1, then Floor 2 going down."
  • "Zero is positive"
  • "The ground floor is 0 because there is nothing there."
  • "– 3 is bigger than – 2, because 3 is bigger than 2."
  • "Adding makes things bigger."
  • "You add the two floor numbers together."