A Square and A Cube
Class 8 · Mathematics
4 ideas in Class 8 Mathematics, explained in plain words, with games and quizzes to practise.
Square Numbers and Their Patterns
A square number is a number multiplied by itself, like 6 × 6 which is 36. Squares have patterns: they end only in 0, 1, 4, 5, 6 or 9, they have an even number of zeros at the end, and each square is the sum of odd numbers starting from 1, such as 1 + 3 + 5 + 7 is 16.
Why it matters. Spotting these patterns lets a child check quickly whether a number can be a square and find the next square without long multiplication. Without them, big squares feel like random numbers.
Try it in two minutes
- Lay out 16 coins and ask your child to arrange them as a square (4 by 4).
- Remove one row and one column to leave a square of 9, then 4, then 1. Ask how many coins each step removed (7, 5, 3).
- Say that 1 + 3 + 5 + 7 is 16 and ask for 1 + 3 + 5 + 7 + 9.
Try at home. Ask your child to write the squares of 1 to 12 and circle the last digit of each. Which digits never appear?
Square Roots and Estimating Them
The square root of a number is the number which, multiplied by itself, gives it. Since 9 × 9 is 81, the square root of 81 is 9. To find a square root, write the prime factors in equal pairs: 324 is 2 × 2 × 3 × 3 × 3 × 3, so its root is 2 × 3 × 3, which is 18. If the factors cannot be paired, the number is not a perfect square.
Why it matters. Square roots are used to find the side of a square from its area. Pairing prime factors is a method that works for large numbers, and estimating helps when the answer is not a whole number.
Try it in two minutes
- Ask: a square garden has area 49 square metres, what is its side?
- Write 36 as 2 × 2 × 3 × 3 and circle the pairs. Ask what you get by taking one number from each pair.
- Try 50 as 2 × 5 × 5. Can you pair everything? What two squares does it lie between?
Try at home. Ask your child to find which two whole numbers the square root of 90 lies between, using squares they already know.
Cube Numbers and Their Patterns
A cube number is a number multiplied by itself three times, such as 4 × 4 × 4 which is 64. It also counts the small unit cubes needed to build a big cube: a cube with edge 4 uses 64 unit cubes. Cubes follow patterns too: for instance 7³ is the sum of 7 consecutive odd numbers, and a cube cannot end in exactly two zeros.
Why it matters. Cubes link numbers to solid shapes and volume. Knowing how cubes grow helps children estimate and avoid mistakes, such as confusing 5³ (125) with 5 × 3 (15).
Try it in two minutes
- Stack 8 small blocks or sugar cubes into a 2 by 2 by 2 cube. Ask how many layers and how many in each layer.
- Ask how many blocks a 3 by 3 by 3 cube needs (27) and how you could work it out.
- Ask why 5³ is not the same as 5 × 3.
Try at home. Ask your child to compute 1³, 2³, up to 6³ and then to say how many digits each cube has.
Cube Roots by Prime Factorisation
The cube root of a number is the number which, cubed, gives it. The cube root of 27 is 3, written ∛27 is 3. To find it, write the prime factors and group them into triplets: 3375 is 3 × 3 × 3 × 5 × 5 × 5, so its cube root is 3 × 5, which is 15. If a prime factor is left out of a triplet, the number is not a perfect cube.
Why it matters. Cube roots are used to find the edge of a cube from its volume. Grouping factors into triplets also shows how to adjust a number, by multiplying or dividing, so it becomes a perfect cube.
Try it in two minutes
- Write 216 as 2 × 2 × 2 × 3 × 3 × 3 and circle the groups of three.
- Take one number from each group and multiply: 2 × 3 is 6, so the cube root is 6.
- Try 500 as 2 × 2 × 5 × 5 × 5. Which prime is left without a full group?
Try at home. Ask your child to find the edge of a cube with volume 1000 cubic centimetres, and then explain how they know.
What your child will practise
- Cube Numbers and Their Patterns: Space Quick Fire (Quick Fire)
- Cube Roots by Prime Factorisation: Desert Puzzle (Clue Puzzle)
- Cube Roots by Prime Factorisation: Ocean Quick Fire (Quick Fire)
- Square Numbers and Their Patterns: Mission Control Sort (Sort It Out)
- Square Roots and Estimating Them: Lab Explorer Quest (Collect the Answer)
- Quiz: Cube Numbers and Their Patterns (8 questions)
- Quiz: Cube Roots by Prime Factorisation (8 questions)
- Quiz: Square Numbers and Their Patterns (8 questions)
- Quiz: Square Roots and Estimating Them (8 questions)