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

Fill in the blank using a property: \(9 + 6 = 6 + \underline{\hspace{1.5em}}\).

Answer: \(9\) — commutative property of addition.

Compute: \(2(4 + 3)\).

Answer: \(14\). Either \(2 \times 7 = 14\), or \(2 \times 4 + 2 \times 3 = 8 + 6 = 14\) (distributive).

Intermediate

Evaluate both sides and compare: \((3 + 4) + 5\) and \(3 + (4 + 5)\).

Answer: Both equal \(12\) — associative property of addition.

Use distribution to find \(5 \times 102\).

Answer: \(510\). Steps: \(5 \times (100 + 2) = 500 + 10 = 510\).

Difficult

Explain why \(12 \div (3 \div 2)\) is not the same as \((12 \div 3) \div 2\).

Answer: Left: \(12 \div 1.5 = 8\). Right: \(4 \div 2 = 2\). Division is not associative; grouping changes the meaning.

A garden bed is 4 m wide and \((2 + 3)\) m long. Write two ways to find its area and show they match.

Answer: \(20\) m². One rectangle: \(4 \times 5 = 20\). Split length: \(4 \times 2 + 4 \times 3 = 8 + 12 = 20\) (distributive).