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

Chapter 1 · A Square and A Cube

Cube roots, and what successive differences expose

Teaching notesNCERT10 min

LearnTeach

This video could not be loaded. Reload the page to try again.

Sign in with Google

10 min.

These teaching notes are for members

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.

Create a free accountSign in

What to assume they know

What they should be able to do

  • Write and read the cube-root symbol, and state what it denotes
  • Explain why a cube root has a single value where a square root has two
  • Find a cube root by dealing prime factors into three matching groups
  • State how a number's prime factorisation relates to that of its cube
  • Guess the cube root of a perfect cube without factorising, using the last digit and the leading part
  • Explain why the last digit determines a cube root's last digit but not a square root's
  • Build a table of successive differences for the squares and then for the cubes
  • State what the level of flattening tells you, and check the claim on both sequences
  • Compare the size of a difference of cubes with the corresponding difference of squares, and justify the comparison

Where it usually goes wrong

  • "Every cube has two cube roots, one of each sign, like squares." It does not. Cubing a negative gives a negative, so −8 has only −2 and 8 has only 2. The chapter's own (−6)³ = −216 on Part I p.13 is the evidence.
  • "The cube root of 1000 is 3." The chapter prints this and it is a slip. Say so if a student raises it; the value is 10, which the chapter's own general rule on the same line gives.
  • "The last digit of a square tells you the last digit of its root." It does not — an ending of 6 leaves two candidates. For cubes it does, and the contrast is the sharpest thing in this topic.
  • "Guessing a root is not real mathematics." The guess is forced, not free: the last digit and the leading part between them leave exactly one candidate. Show the forcing.
  • "Differences always go flat at the second level." That is what squares do. The level is the power, and the chapter's blank cube row is an invitation to discover it.
  • "The flat value for cubes must be 3, because the power is 3." This is the guess almost every student makes. Do not announce the correction — run the levels from the printed cubes and let the number appear.
  • "67³ − 66³ and 67² − 66² are roughly the same size." They are not remotely. The square difference is one odd number; the cube difference is a run of them.
  • "The three-groups test only tells you yes or no." It hands you the root at the same time, exactly as its two-group cousin did for squares.

Questions to check understanding

  • Find the cube root of a perfect cube by prime factorisation
  • Guess the cube root of a four- or five-digit perfect cube without factorising, and say what fixed each digit
  • State what number a given number must be multiplied by to become a cube
  • True or false, with reasons, on the last digits and parity of cubes
  • Build a difference table for a given sequence and say at which level it flattens
  • Compare 67³ − 66³ with 67² − 66² and justify which is greater without computing both in full
  • Explain why a cube root is unique where a square root is not

Examples worth working on the board

  • The definition (Part I p.14): 8 = 2³, so 2 is called the cube root of 8, written with the index 3 on the radical. The chapter then gives the general form algebraically — when one number is the cube of another, that other one is its cube root — and writes the cube root of 2³ as 2 and the cube root of n³ as n.
  • A misprint to correct silently. On Part I p.14 the chapter writes the cube root of 27 as the cube root of 3³, giving 3 — correct — and then in the same sentence writes the cube root of 1000 as the cube root of 10³ and prints the value as 3. It should read 10. Verified on that line. Do not show this as printed, and if the page is shown, cover or correct it.
  • The three-pile reading, backwards (Part I p.14): 3375 = 3 × 3 × 3 × 5 × 5 × 5, dealt as three groups of (3 × 5), so 3375 = (3 × 5)³ = 15³ and the cube root of 3375 is 15. Read the other way as triplets, 3375 = (3 × 3 × 3) × (5 × 5 × 5) = 3³ × 5³. The same page shows 500 = 2 × 2 × 5 × 5 × 5 failing the test.
  • The factorisation table (Part I p.15), four rows, with the number's factorisation on the left and its cube's on the right:
    • 4 = 2 × 2, and 4³ = 64 = 2 × 2 × 2 × 2 × 2 × 2 = 2³ × 2³
    • 6 = 2 × 3, and 6³ = 216 = 2 × 2 × 2 × 3 × 3 × 3 = 2³ × 3³
    • 15 = 3 × 5, and 15³ = 3375 = 3 × 3 × 3 × 5 × 5 × 5 = 3³ × 5³
    • 12 = 2 × 2 × 3, and 12³ = 1728 = 2 × 2 × 2 × 2 × 2 × 2 × 3 × 3 × 3 = 2³ × 2³ × 3³ The chapter's stated reading: every prime factor of the number turns up three times over in the cube's factorisation.
  • Left to the student (Part I p.15): the cube roots of 64, 512 and 729. Inputs only.
  • The successive-differences figure (Part I p.15). A row headed "Perfect Squares" carrying 1, 4, 9, 16, 25, 36 and an ellipsis; beneath it a row labelled Level 1 in red carrying 3, 5, 7, 9, 11; beneath that a row labelled Level 2 in blue carrying 2, 2, 2, 2. Looped arrows join each pair of entries to the entry below and between them. The chapter states that after two levels the differences have gone constant. Then a second figure headed "Perfect Cubes" carries 1, 8, 27, 64, 125, 216 with the same looped arrows drawn and no level rows printed at all — verified on the printed page. Building them is the exercise.
  • The argument for section 10, which the chapter does not give. Differencing drops the power by one: the gaps between consecutive squares vary like a first-power rule, and their gaps in turn are fixed, so squares flatten at level two. The gaps between consecutive cubes vary like a square rule, their gaps like a first-power rule, and only the next level is fixed — so cubes flatten one level lower down. The flat value is the product of the levels descended, which is why the squares' flat entry is 2 × 1 and the cubes' is 3 × 2 × 1.
  • Guessing a cube root (Part I p.16, item 4). The student is told 1331 is a perfect cube and asked to name its root without factorising, then to do the same for 4913, 12167 and 32768. The route, assembled from the chapter's own cube table on Part I p.12: the cubes of the ten digits end in ten different digits, so a cube's last digit names its root's last digit outright; and the part of the number above its last three digits is bracketed by the cubes of the small numbers, which fixes the root's tens digit. Give the route and the inputs; the four roots are the exercise.
  • The rest of §1.2 Figure it Out (Part I pp.16–17): item 1, the cube roots of 27000 and 10648; item 3(ii), whether any perfect cube ends in 8; item 5, which of 67³ − 66³, 43³ − 42³, 67² − 66² and 43² − 42² is greatest, with reasoning. Item 5 is section 11's material and it is a difference-table question in disguise — the two square differences are governed by the odd-number rule and the two cube differences by a rule that grows far faster.

Figures to have open

  • The chapter's successive-differences figure, squares complete and cubes blank (Part I p.15). Reproduce the blankness — anyone who "helpfully" fills the cube rows has removed the exercise and the discovery.
  • A digit map: 0 to 9 down one side and the last digit of each digit's cube down the other, with every arrow landing somewhere different. Set it beside the square version from What a perfect square's last digits can and cannot be, where arrows collide. An added figure, and it carries sections 5 and 6.
  • The three-tray dealing diagram for 3375, run in reverse so the root is lifted out. An added figure; reuse the asset from What makes a number a perfect cube, and the three-identical-groups test.
  • The chapter's four-row factorisation table (Part I p.15). Standard schematic.

Where this sits in the book

The book

Open in a new tab