PrepShorts · Teaching notes · Class 8 Mathematics · Chapter 1, A Square and A CubePrepShorts

Chapter 1 · A Square and A Cube

What makes a number a perfect square

Teaching notesNCERT10 min

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10 min.

What to assume they 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

What they should be able to do

  • Explain why a locker's toggle count is exactly the count of factors of its number
  • Pair off the factors of a given number into partner factors, and identify the number that has no distinct partner
  • Argue that a natural number has an odd count of factors exactly when it is a square, and use that to predict which lockers stay open
  • Identify the numbers whose factor count is exactly two, and name them as primes
  • Connect the sidelength of a square to its area, and read 1, 4, 9, 16, 25, 100 off that connection
  • Write and read the notation n² for n × n
  • Square a fraction and a decimal, not only a whole number
  • Distinguish a square number from a perfect square in the chapter's own sense

Where it usually goes wrong

  • "A square number is one you get by multiplying, and the odd factor count is a separate curiosity." They are the same statement. Squareness is the failure of the divisor pairing to pair everything off.
  • "Every number has an even number of factors." 1, 4 and 9 are the chapter's three counterexamples, printed side by side.
  • "36 has the pair 6 × 6, so it has an odd factor count — done." Not yet. You also need that no other factor of 36 is self-paired. The chapter asks for exactly that check and an explanation that skips it has skipped the proof.
  • **"Person d toggles locker n whenever d is less than n."** Only when d divides n. Person 7 never touches locker 12.
  • "1 is a special case that spoils the rule." 1 = 1 × 1 is the first square and the cleanest instance: its single factor is its own partner.
  • "Only whole numbers can be squared." The chapter squares 3/5 and 2.5 on the same page it defines the notation.
  • "A square is a shape; a square number is a number; the shared word is an accident." The chapter's own history section (Part I pp.15–16) shows the word carried both meanings from the start — see The oldest known tables of squares and cubes.
  • "Lockers touched twice are the even-numbered ones." Touched twice means exactly two factors, which is the definition of prime.

Questions to check understanding

  • List all factors of a given number and pair them up; say what the pairing shows
  • Decide, with a reason, whether a stated number has an odd or even factor count
  • Given a toggling rule, say which positions end in the changed state, and justify
  • "Which numbers have exactly three factors?" — the competency form, since these are the squares of primes
  • Complete a sidelength-and-area table, then read the notation n² off it
  • Square a given fraction and a given decimal
  • Explain in one or two sentences why a number with an odd factor count must be a square

Examples worth working on the board

Where a value is worked out here on the chapter's data rather than something printed, it says so.

  • The puzzle's set-up (Part I p.1). Queen Ratnamanjuri's will; her son Khoisnam plus 99 relatives, 100 people in all; a secret room with 100 lockers; each person numbered 1 to 100. The first goes down the row opening all of them; the second then flips every 2nd; the third flips the 3rd, 6th, 9th and onward; the fourth flips the 4th, 8th, 12th and onward; and the pattern continues through person 100. Whoever answers first keeps the whole inheritance; if all 100 answer together they share it. The chapter's hint tells the student to tally the toggles each locker receives.
  • Locker 6, worked on the page (Part I p.2). It is opened by person 1, shut by person 2, opened again by person 3 and shut by person 6. Its factors are 1, 2, 3, 6. Printed alongside in a green panel: 6: 1 × 6, 2 × 3, and the factor list.
  • Three more panels (Part I p.2), in violet, orange and blue: 1: 1 × 1, with the note that 1 is the only factor; 4: 1 × 4, 2 × 2, factors 1, 2, 4; 9: 1 × 9, 3 × 3, factors 1, 3, 9. Verified against the printed page — the numbers live inside the coloured panels, so anyone reading only extracted text will mis-set them.
  • 36, left to the student (Part I p.2). The chapter offers the pair 6 × 6 and asks the student to check that every other factor of 36 has a different partner. That check is the load-bearing step and the page does not do it. Working added here, to display: 36 = 1 × 36 = 2 × 18 = 3 × 12 = 4 × 9 = 6 × 6, so the factors are 1, 2, 3, 4, 6, 9, 12, 18, 36 — four couples and one loner.
  • The list the chapter draws the conclusion from (Part I p.2): 1 × 1, 2 × 2, 3 × 3, 4 × 4, …
  • How many lockers stay open. The chapter does not print the list, but its next line calls them "these 10 lockers", so the count is given on the page even though the roster is the exercise (Part I p.3). Leave the roster to the student.
  • The passcode clue (Part I p.3). The clue asks for the first five locker numbers touched exactly twice; the chapter reasons that a number touched twice has just two factors, so these are the primes, and prints the code 2-3-5-7-11. This one the book does answer, so the explanation may use it.
  • The area table (Part I p.3), six rows of data in two columns: sidelength 1 → 1 × 1 = 1 sq. unit; 2 → 2 × 2 = 4; 3 → 3 × 3 = 9; 4 → 4 × 4 = 16; 5 → 5 × 5 = 25; 10 → 10 × 10 = 100. An empty 5-by-5 grid of unit squares is drawn to the right of the table; it carries no numbers (verified on the printed page).
  • The notation lines (Part I p.3): 1 × 1 = 1² = 1; 2 × 2 = 2² = 4; 3 × 3 = 3² = 9; 4 × 4 = 4² = 16; 5 × 5 = 5² = 25, then a vertical ellipsis, then the general statement n × n = n².
  • Squares that are not whole numbers (Part I p.3). Sidelength 3/5 gives (3/5)² = (3/5) × (3/5) = 9/25; sidelength 2.5 gives (2.5)² = 6.25. The fraction is set as a stacked 3-over-5 in print, and the text layer flattens it to "35" — read from the printed page, not the extraction.

Figures to have open

  • The corridor of 100 lockers with a person walking it, stopping only at multiples. This is the chapter's own opening artwork (Part I p.1, a mansion interior with a row of lockers, the queen's portrait and two figures in the foreground); redraw it as a schematic strip of 100 numbered doors rather than reproducing the printed illustration.
  • The four coloured factor panels for 6, 1, 4 and 9 (Part I p.2). Standard schematic, but keep the numbers exactly as printed — they sit inside the panels.
  • A factor-pairing diagram: the factors of a number in a row with arcs joining partners, so the self-paired one shows as a loop. This is an added figure and it carries section 5.
  • The sidelength/area table with the empty 5-by-5 unit-square grid beside it (Part I p.3). Standard schematic.

Where this sits in the book

The book

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