You were told the commutative, associative and distributive laws in a single breath, and they blurred into one vague phrase: "you can rearrange stuff." But they do three very different things. Commutativity swaps two items. Associativity regroups the brackets. Distributivity spreads one factor across a sum. This satellite gives you one clean picture for each — so that when you use them in algebra, you know which rearrangement you actually performed.
All three laws — one live picture
Pick a law from the dropdown below, press Animate, and watch the actual rearrangement happen: a swap, a regrouping, or a spread. The numeric totals on either side stay equal throughout — the picture is doing the proof by showing that no quantity appeared or disappeared.
Commutative — swap the two inputs
Take two boxes holding a and b apples. Put box a on the left and box b on the right. Total apples: a + b. Now physically swap the boxes so that b is on the left and a on the right. Total apples: b + a. Same apples, same total.
Multiplication of real numbers works the same way: 3 \times 4 is three groups of four, 4 \times 3 is four groups of three, and a 3 \times 4 rectangular grid is the same grid whether you count rows first or columns first. Why: the grid has 12 dots no matter which direction you scan.
Associative — regroup the brackets
Now take three boxes a, b, c. The commutative law lets you swap two of them; the associative law lets you change which pair you add first. Either (a + b) + c — add the first two, then add the third — or a + (b + c) — hold onto the first, add the other two first, then combine. Same total.
Associativity and commutativity are independent: swapping the boxes is one kind of freedom, rebracketing them is another. The fact that addition enjoys both is what lets you write 3 + 5 + 7 + 2 with no brackets and no worry about the order — any rearrangement lands on the same total.
Distributive — spread one over two
Now the new shape. You have one factor a and a pair of things, b and c, that have been added together. Distributivity says you can multiply a into the bracketed sum by spreading — one copy of a lands on b, another copy lands on c, and then you add the two products.
The rectangle picture — visible in the canvas above when you pick "Distributive" — makes this unmistakeable. The tall rectangle has height a and total width b + c, so its total area is a(b + c). Split it vertically at width b: the left piece has area ab and the right piece has area ac. Same rectangle, two descriptions.
You are not swapping anything. You are not regrouping anything. You are taking a single factor and distributing it — handing a copy of a to each addend. That is why the law has its name.
Side by side — one table
Lined up, the three look like this. The difference is in what they rearrange.
| Law | Pattern | What moved |
|---|---|---|
| Commutative | a + b = b + a | the two items swapped places |
| Associative | (a + b) + c = a + (b + c) | the bracket slid from left pair to right pair |
| Distributive | a(b + c) = ab + ac | one factor spread across two addends |
Everything else — expanding 2(x + 3), collecting like terms in 4x + 7x, rearranging 5 + x + 3 into x + 8 — is a sequence of these three moves applied one after another. Naming the move you just made is the difference between algebra as rule-following and algebra as a language you can speak.
One warning
Distributivity is directional. Multiplication distributes over addition. Addition does not distribute over multiplication. Try it with numbers: 2 + (3 \times 4) = 14, but (2 + 3) \times (2 + 4) = 30. Nothing equal about those. Why: distributivity is saying "a rectangle of area a(b+c) splits into two rectangles." There is no analogous picture for "a number added to a product splits into two products" — because that statement is false.
If you ever catch yourself writing a + (b \cdot c) = (a + b)(a + c), pause. You have used a law that does not exist.
Related: Operations and Properties · Algebraic Identities · Real Numbers and Their Properties · Number Systems