The associative property states that (2 × 3) × 4 = 2 × (3 × 4). Grouping factors differently does not change the product. This permits flexible computation: 2 × 3 × 4 can be computed as (2 × 3) × 4 = 6 × 4 or as 2 × (3 × 4) = 2 × 12.
Use arrays and area models to show that regrouping factors rearranges the dimensions without changing the total. Provide three-factor problems and ask students to solve in two different groupings, verifying the answer matches.
You already know from the commutative property that the order of two factors doesn't change the product: 4 × 7 = 7 × 4. The associative property extends this freedom to three or more factors: when you multiply, it doesn't matter which pair you multiply first. Parentheses in a multiplication expression are just a suggestion about what to compute first — and you're free to choose a different order if it's easier.
Think about a problem like 2 × 3 × 4. You could do (2 × 3) × 4 = 6 × 4 = 24. Or you could do 2 × (3 × 4) = 2 × 12 = 24. Same answer either way. The parentheses don't change what's being multiplied — all three factors, 2, 3, and 4, are still in the problem. The only thing that changes is which multiplication you perform first.
The real power of this property shows up when you're trying to make a problem easier. Suppose you see 5 × 9 × 2. Your multiplication facts tell you 5 × 9 = 45, and then 45 × 2 = 90 — that works, but 45 × 2 is a bit of work. Alternatively, notice that 5 × 2 = 10, and 10 × 9 = 90 is trivial. Regrouping as (5 × 2) × 9 turns a harder problem into an easy one. The associative property gives you permission to do this.
This idea pairs with the commutative property to give you complete freedom when multiplying several numbers together: you can rearrange the factors in any order and group them any way you like. Together, these properties form the foundation for the more powerful distributive property and eventually for multi-digit multiplication, where you'll be breaking large products into pieces, computing each piece, and adding the parts — a strategy that only works because regrouping and reordering factors is always safe.