Squares and Square Roots
Class 8 · Mathematics
5 ideas in Class 8 Mathematics, explained in plain words, with games and quizzes to practise.
Square Numbers and Their Endings
Part of: Square numbers
A square number is a number you get by multiplying a number by itself, like 49 which is 7 × 7. Square numbers end only in 0, 1, 4, 5, 6 or 9, so a number ending in 2, 3, 7 or 8 is never a perfect square. The square of an even number is even, and of an odd number is odd.
Why it matters. Knowing the endings of squares gives a fast check before any calculation. It also builds number sense: squares grow quickly, and patterns in their last digits help with larger numbers.
Try it in two minutes
- Ask your child to square the numbers 1 to 10 and write the units digits in a row.
- Ask: which digits never appear? What does this tell about 1237?
- Try 20, 200 and 2000 and count the zeros at the end of each square.
Try at home. Ask your child to find five numbers that are surely not squares just by looking at their last digit.
Patterns in Square Numbers
Part of: Square numbers
Square numbers hide patterns. The sum of the first n odd numbers is n²: 1 + 3 + 5 + 7 = 16. Between n² and (n+1)² there are exactly 2n non-square numbers. The square of an odd number is the sum of two consecutive numbers, and (a - 1)(a + 1) is a² - 1.
Why it matters. Spotting patterns is a core maths skill. These patterns give quick mental routes: instead of multiplying 47 × 49, a child can use 48² - 1.
Try it in two minutes
- Write 1, 1 + 3, 1 + 3 + 5 on paper and ask your child what each sum is.
- Ask: what will the sum of the first 10 odd numbers be? Check by adding.
- Ask how many numbers lie between 5² and 6².
Try at home. Ask your child to find 19 × 21 using 20² - 1.
Quick Squares and Pythagorean Triplets
Part of: Square numbers
You can square a number quickly by splitting it, like 23² = (20 + 3)² = 400 + 120 + 9 = 529. For a number ending in 5, such as 35, write (3 × 4) hundreds + 25 = 1225. Three numbers a, b, c with a² + b² = c² are called a Pythagorean triplet, like 3, 4, 5.
Why it matters. Short methods build confidence with larger numbers and show that maths has structure. Pythagorean triplets also appear in building and measuring, since 3, 4, 5 gives a right angle.
Try it in two minutes
- Ask your child to find 25², 35² and 45² using the rule for numbers ending in 5.
- Check one with long multiplication.
- Check that 3² + 4² equals 5², then try 6, 8 and 10.
Try at home. Ask your child to find 65² and 95² with the rule for numbers ending in 5.
Square Roots by Factorisation
Part of: Square roots
Finding a square root is undoing a square. One way is to break the number into prime factors and pair them: 144 = 2 × 2 × 2 × 2 × 3 × 3, so its square root is 2 × 2 × 3 = 12. If a prime has no partner, the number is not a perfect square. Another way is to subtract 1, 3, 5, ... again and again until you reach 0.
Why it matters. Factorisation shows why a number is or is not a square, and prepares for algebra and for working with areas. The pairing idea also tells us what to multiply or divide by to make a square.
Try it in two minutes
- Write 36 as a product of primes: 2 × 2 × 3 × 3. Circle the pairs.
- Take one from each pair: 2 × 3 = 6. That is the square root.
- Try 72. Which factor has no pair?
Try at home. Ask your child to find the square root of 400 by prime factors and by subtracting odd numbers.
Square Roots by Division and Decimals
Part of: Square roots
The long division method finds square roots of large numbers. Put a bar over each pair of digits from the right, and each bar gives one digit of the root. For decimals, bars go in pairs from the decimal point in both directions. If a number is not a perfect square, the remainder tells how much to subtract or add to reach one.
Why it matters. Square roots appear when we know an area and want a side length. The long division method also strengthens division, estimation and checking skills.
Try it in two minutes
- Ask your child: a square tile has area 144 square cm. What is its side?
- Check the guess by multiplying.
- Ask how many digits the root of 4096 has, using pairs of digits.
Try at home. Ask your child to estimate the square root of 70 and say which two whole numbers it lies between.
What your child will practise
- Patterns in Square Numbers: Lab Explorer Quest (Collect the Answer)
- Quick Squares and Pythagorean Triplets: Space Quick Fire (Quick Fire)
- Square Numbers and Their Endings: Mission Control Quest (Match It Up)
- Square Roots by Division and Decimals: Ocean Puzzle (Clue Puzzle)
- Square Roots by Factorisation: Jungle Puzzle (Clue Puzzle)
- Quiz: Patterns in Square Numbers (8 questions)
- Quiz: Quick Squares and Pythagorean Triplets (8 questions)
- Quiz: Square Numbers and Their Endings (8 questions)
- Quiz: Square Roots by Division and Decimals (8 questions)