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.
Three properties
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.
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.
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. \((10 - 4) - 2 \neq 10 - (4 - 2)\).
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.
| You change… | Property | Example |
|---|---|---|
| Order of two numbers | Commutative | \(4 + 7 = 7 + 4\) |
| Brackets with three numbers | Associative | \((1 + 2) + 3 = 1 + (2 + 3)\) |
| Multiply through a sum | Distributive | \(3(2 + 4) = 3 \times 2 + 3 \times 4\) |
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.