1. A Square and A Cube

Part I, printed Chapter 110 topics2 h

Chapter 1 of NCERT Mathematics for Class 8 (Part I, printed Chapter 1): A Square and A Cube. Three sections — Square numbers, Cube numbers and Where these ideas came from, and a puzzle. Ten videos, 2 h in all.

Worked answers to the 14 exercise questions in this chapter →

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Part 1

Square numbers

10 minWhat makes a number a perfect squareA hundred lockers, a hundred people, each changing every second, third, fourth. The doors left open at the end are exactly the perfect squares.10 min · 13 partsहिंदी10 minWhat a perfect square's last digits can and cannot beSquare every whole number and look only at the right-hand edge. Just six digits ever appear there, and four never do.10 min · 13 partsहिंदी11 minWhy the first n odd numbers add up to n²Subtract each square from the next and the odd numbers fall out in order, nothing skipped. Four cases is not yet a reason.11 min · 13 partsहिंदी10 minSquares hiding inside triangular numbersStack dots in rows of 1, 2, 3 and you get a staircase. Push two neighbouring staircases together and a perfect square appears.10 min · 13 partsहिंदी10 minSquare roots, and the prime-factor test for a perfect squareA square root is not a strange operation — it is a length. Give a square an area of 49 and its side is 7.10 min · 13 partsहिंदी

Part 2

Cube numbers

10 minWhat makes a number a perfect cube, and the three-identical-groups testA cube of edge 2 does not hold six unit cubes. Six is how many faces it has; it holds two floors of four.10 min · 13 partsहिंदी10 minTaxicab numbers: why 1729 is the number it isA mathematician called the number on a taxi dull. Ramanujan, ill in hospital, disagreed on the spot — and 1729 has been famous ever since.10 min · 13 partsहिंदी9 minCubes built from runs of consecutive odd numbersWrite out the odd numbers and take scissors to them: one, then two, then three. Every piece adds up to a cube.9 min · 13 partsहिंदी10 minCube roots, and what successive differences exposeSquaring throws things away — it folds −6 onto 6, so 36 has two roots. Cubing throws nothing away, and that changes everything.10 min · 13 partsहिंदी

Part 3

Where these ideas came from, and a puzzle

9 minThe oldest known tables of squares and cubesAround 1700 BCE somebody wrote out the squares. Not a rule for squaring — a list, pressed into wet clay and left to dry.9 min · 12 partsहिंदी

What you will be able to do

  • Explain why a locker's toggle count is exactly the count of factors of its number
  • Compute the squares of the first thirty natural numbers and tabulate them
  • Compute the differences between consecutive squares and identify them as the odd numbers in order
  • Recognise and extend the triangular numbers as dot staircases
  • Recover the sidelength of a square from its area
  • Count the unit cubes in a cube of a given edge, by layers
  • Verify that 1729 splits into two positive cubes in two distinct ways
  • Read the chapter's printed pattern and state how many terms each run has
  • Write and read the cube-root symbol, and state what it denotes
  • State when and where the earliest known lists of squares and cubes were made, and what they were written on

What this chapter assumes you already know

  • Finding all the factors of a number up to 100, and listing them in order
  • What "divides exactly" means, and that a factor leaves no remainder
  • What a prime number is
  • Area of a rectangle as length × breadth, and area measured in unit squares
  • Reading a fraction such as 3/5, and multiplying two fractions
  • Multiplying two decimals such as 2.5 × 2.5
  • Multiplying two two-digit and three-digit numbers by hand
  • Knowing that a number ending in 0 is a multiple of 10

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 square number is one you get by multiplying, and the odd factor count is a separate curiosity."
  • "Every number has an even number of factors."
  • "36 has the pair 6 × 6, so it has an odd factor count — done."
  • "Person d toggles locker n whenever d is less than n."
  • "1 is a special case that spoils the rule."
  • "Only whole numbers can be squared."
  • "A square is a shape; a square number is a number; the shared word is an accident."
  • "Lockers touched twice are the even-numbered ones."