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.

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.

Two boxes swapping positions to illustrate commutativityA row with a box labelled a on the left and a box labelled b on the right, a curved double-headed arrow between them, and a second row showing the boxes swapped with b on the left and a on the right. Both rows end in the same total labelled a plus b. a + b = a + b b + a = a + b swap the two boxes
Commutativity is a swap. You interchange exactly two quantities. The total is unchanged because no apples were added, removed or broken apart — the only thing that changed is which box is where.

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.

Three boxes with different bracketings giving the same totalA diagram with three boxes labelled a b and c. In the top row the first two are enclosed by a bracket showing the pair a plus b added first, then c added next. In the bottom row the last two are enclosed by a bracket showing b plus c added first, then a added. Both rows give the same total a plus b plus c. a b c add first (a + b) + c = a + b + c a b c add first a + (b + c) = a + b + c
Associativity is a regrouping. The three boxes stay in the same order — $a, b, c$ — but the bracket slides from around the first pair to around the second pair. The total is unchanged because the bracket only decides the order in which you add, not which items are being added.

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.

a \times (b + c) = a \times b + a \times c

The rectangle picture makes this unmistakeable.

Distributive law as a rectangle split into two piecesA rectangle of height a and total width b plus c is split by a vertical line into a left piece of width b and a right piece of width c. The left piece's area is labelled a times b and the right piece's area is labelled a times c. Beneath the rectangle the equation a times the quantity b plus c equals a times b plus a times c is shown. a × b a × c b c a b + c a(b + c) = a·b + a·c
Distributivity is a spread, not a swap or a regrouping. One factor ($a$) is replicated across the terms of a sum ($b + c$), producing two products. The rectangle's total area is both $a(b + c)$ and $ab + ac$ — same area, 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