We Distribute, Yet Things Multiply
Class 8 · Mathematics
4 ideas in Class 8 Mathematics, explained in plain words, with games and quizzes to practise.
Expanding Products with the Distributive Property
The distributive property says a(b + c) = ab + ac. If one number in a product goes up by 1, the product goes up by the other number. Multiplying two expressions with two terms each means multiplying every term of the first by every term of the second: (a + m)(b + n) is ab + mb + an + mn. Signs follow the rules for integers, so (a − u)(b + v) is ab + av − ub − uv.
Why it matters. This is the base for all algebra work. Children who understand it can expand brackets, spot the extra terms, and check their answers by putting in numbers.
Try it in two minutes
- Ask your child: 15 × 20 is 300, so what is 15 × 21? (add 15)
- Ask: what is 16 × 21? How does it compare with 15 × 20? (a + b + 1 more: 20 + 15 + 1 is 36 more)
- Draw a rectangle (a + m) by (b + n) split into four parts and name the area of each.
Try at home. Ask your child to write (x + 4)(x + 7) as four products and then add them, and to check with x is 1.
Squares of Sums and Differences
The square of a sum is (a + b)² = a² + 2ab + b² and the square of a difference is (a − b)² = a² − 2ab + b². The middle term 2ab is often forgotten: (a + b)² is not a² + b². These identities help us square numbers quickly, such as 65² = (60 + 5)² = 3600 + 600 + 25 = 4225, and 59² = (60 − 1)².
Why it matters. Children use these identities all through algebra, and they also give faster ways to square numbers. The most common mistake is to drop the middle term, which this topic corrects.
Try it in two minutes
- Draw a square of side 7 and split it as 5 + 2 on each side. Ask for the four areas (25, 10, 10, 4).
- Add the four areas to get 49 and match with 5² + 2 × 5 × 2 + 2².
- Ask your child to find 52² using (50 + 2)².
Try at home. Ask your child to square 31, 41 and 51 using (a + 1)² and look for a pattern.
Difference of Squares and Quick Multiplication
(a + b)(a − b) is a² − b², which makes products of numbers equally far from a middle number easy: 98 × 102 is 100² − 2², or 9996. The same idea gives squares as a² = (a + b)(a − b) + b². Multiplying by 11 means multiplying by 10 and adding the number once, so the digits are added in pairs. Multiplying by 101 shifts the number by two places and adds it.
Why it matters. Quick methods build number sense and reduce errors. Children also see the same algebra giving different shortcuts.
Try it in two minutes
- Ask your child to find 49 × 51 by writing it as (50 − 1)(50 + 1).
- Ask for 31² using (31 + 1)(31 − 1) + 1².
- Show that 2314 × 11 is 23140 + 2314 and add them.
Try at home. Ask your child to invent a pair of numbers around 100 and multiply them with (a + b)(a − b).
Patterns, Mistakes and Area Expressions
A pattern can be described in several ways, such as k(k + 2), (k + 1)² − 1 and k² + 2k, and expanding shows they are the same. Common mistakes are dropping the middle term of (a + b)², forgetting signs when multiplying negative terms, and not multiplying every term in a bracket. Areas can also be found in two ways, such as (m + n)² − 4mn and (n − m)², and the two must agree.
Why it matters. Checking that two expressions are equal, and spotting errors, are skills used throughout algebra. They help children trust their answers instead of relying on memory alone.
Try it in two minutes
- Make a pattern of tiles: Step 1 has 3 tiles, Step 2 has 8, Step 3 has 15. Ask for a rule (k² + 2k).
- Ask your child to describe the pattern in two different ways and show the two expressions are equal.
- Ask them to put numbers into both sides of a claim like (a + b)² is a² + b² to test it.
Try at home. Ask your child to check an expansion by putting in x is 1 and x is 2 on both sides.
What your child will practise
- Difference of Squares and Quick Multiplication: Space Quick Fire (Quick Fire)
- Expanding Products with the Distributive Property: Mission Control Quest (Match It Up)
- Patterns, Mistakes and Area Expressions: Space Puzzle (Clue Puzzle)
- Squares of Sums and Differences: Desert Quest (Flip and Match)
- Squares of Sums and Differences: Number Quest (Collect the Answer)
- Quiz: Difference of Squares and Quick Multiplication (8 questions)
- Quiz: Expanding Products with the Distributive Property (8 questions)
- Quiz: Patterns, Mistakes and Area Expressions (8 questions)
- Quiz: Squares of Sums and Differences (8 questions)