Helps you understand
Four operationsCommutative, associative and distributive properties are the shortcuts behind fast, accurate calculation.
Properties of numbers
Why does 7 × 8 equal 8 × 7, and how can such rules make mental maths easy?
The closure, commutative, associative and distributive properties, the special roles of 0 and 1, and how they turn hard calculations into easy ones.
5 depths≈ 215 min of reading85 practice questions
The ladder
Choose how deep to go
Every depth is open. Start where it feels interesting — you can always move back.
- Discoverabout 30 minRules that numbers always followMeet the properties of numbers through chairs, laddoos, kirana bills and socks: why 4 × 6 = 6 × 4, why you can add in any order, how breaking numbers apart makes sums easy, and what 0 and 1 do.Owner-accepted · not line-by-line reviewed
- Understandabout 40 minThe properties, preciselyState each property of whole numbers exactly, in words and with letters; see why it holds for + and × but fails for − and ÷; learn why division by zero is undefined; and use the properties for fast, reliable mental maths.Owner-accepted · not line-by-line reviewed
- Investigateabout 45 minAlways, sometimes or never?Test claims about whole numbers the way mathematicians do: predict, hunt for counterexamples, measure how badly subtraction and division fail to swap or regroup, discover patterns and shortcuts, and explain why the true ones must be true.Owner-accepted · not line-by-line reviewed
- Go deeperabout 50 minWhy the rules must be trueProve the commutative, associative and distributive laws for every whole number, see why long multiplication and divisibility tests work, show why division by zero would make 0 = 1, prove parity facts with letters, and meet the properties as the rules of algebra.Owner-accepted · not line-by-line reviewed
- Extendabout 50 minProperties beyond the whole numbersTake the properties into new worlds: integers and fractions that repair closure, clock arithmetic, non-commutative everyday actions, rounding inside computers, olympiad puzzles built on parity and the distributive law, projects to try and open questions like Goldbach.Owner-accepted · not line-by-line reviewed
Practise and apply
Question bank · 85 questions
8 sections from Foundation to Challenge, each with hints and a worked solution — or play a 10-question round for points and stars.
Foundation 21Core 35Stretch 19Challenge 10
The web
Connected ideas
Follow a connection when you’re curious — some lead into completely different areas.
Properties of numbers
Helps you understand
Order of operationsThe distributive property explains why multiplication is done before addition and how brackets change a result.
Related to
Number and shape patternsMany number patterns — like the sum of consecutive odd numbers — are properties of numbers in disguise.
Words to know
62 words are explained in these lessons
Natural numbersWhole numbersNumber lineSuccessorPredecessorPropertyCommutative propertyAssociative propertyDistributive propertyAdditive identityMultiplicative identityClosure+50 more