Properties of operations

You can already read number sentences from basic math notation and combine numbers from counting and numbers. This chapter names three patterns that keep showing up: rules about order, grouping, and splitting a sum before you multiply. They do not replace careful arithmetic — they explain why different written forms can still mean the same amount.

Same story, three rules

Imagine three friends. Friend A gets \$2 each day for 3 days and \$2 each day for 5 more days. You can count the money in different ways:

Distribute is about multiplying through a bracket. Commute is about order. Associate is about which pair you combine first when there are three or more numbers.

Commutative property

Commute means to move or swap. For addition and multiplication of whole numbers, changing the order does not change the result.

Example: \(3 + 5 = 5 + 3 = 8\). Three dots then five dots is the same total as five dots then three dots.

Subtraction and division are not commutative. \(5 - 3 \neq 3 - 5\) and \(6 \div 2 \neq 2 \div 6\). Order matters there.

Associative property

When you add or multiply three or more numbers, brackets tell you which pair to do first. The associative property says different groupings give the same answer (for + and × only).

Example: \((2 + 3) + 5 = 5 + 5 = 10\) and \(2 + (3 + 5) = 2 + 8 = 10\). You may combine \(2 + 3\) first or \(3 + 5\) first.

Again, subtraction and division are not associative. \((10 - 4) - 2 \neq 10 - (4 - 2)\).

Distributive property

The distributive property links multiplication to addition. Multiplying a sum is the same as multiplying each addend and then adding — “distribute” the multiply across the bracket.

Example: \(2(3 + 5) = 2 \times 8 = 16\). Split the width into \(3\) and \(5\): \(2 \times 3 = 6\) and \(2 \times 5 = 10\), then \(6 + 10 = 16\). The area model below shows one tall rectangle split into two parts.

Later, in written calculation, partial products in multiplication use this idea: you multiply by each digit of the bottom number separately, then add the rows.

Which property is it?

What comes next

In written calculation you will stack digits and use carry and borrow. Knowing these properties helps you check work (e.g. swap order to verify an addition) and understand why multiplication splits into several rows before you add them.

History

Long before algebra used letters, merchants and accountants rearranged piles and rows of goods. Indian and Islamic mathematicians in the Middle Ages wrote rules for how sums and products behave. The words commutative, associative, and distributive became standard in European algebra books in the 1800s, when school math settled on one notation for all.

The distributive rule is ancient geometry in disguise: a large rectangle split into two smaller ones has total area equal to the sum of the parts. That picture is still one of the clearest explanations today.

Checkpoints

1. Which property lets you write \(7 + 4\) as \(4 + 7\) without changing the total?
2. Is \((5 \times 2) \times 3\) the same as \(5 \times (2 \times 3)\)? Which property says so?
3. Rewrite \(4(6 + 1)\) as a sum of two products.
4. Why is \(8 - 3\) not commutative? Give two different answers for \(3 - 8\) and \(8 - 3\).
5. A rectangle is 5 units tall and \((2 + 3)\) units wide. How does the distributive property describe its area?

Applications

Mental math

Distribute to make friendly numbers: \(7 \times 98 = 7 \times (100 - 2) = 700 - 14 = 686\). Commute to add easier first: \(6 + 47 + 4 = (6 + 4) + 47 = 57\).

Shopping and sharing

Buying 3 packs of \((2 + 5)\) items is the same as \(3 \times 2\) items plus \(3 \times 5\) items. Swapping the order you scan barcodes does not change how many items you have (commute on addition when you merge carts).

Later algebra

Expanding \(a(b + c)\) in polynomials and factorising common factors both rely on the distributive property. You will meet them again in polynomials: expansion & factorisation.

Question bank

Easy

Intermediate

Difficult