Questions: Associative Property of Multiplication

Question 1

A student needs to multiply 4 × 5 × 7. She groups it as (4 × 5) × 7 = 20 × 7 = 140, choosing to multiply 4 and 5 first because 20 is easy to work with. Which property justifies this choice?

• A. Commutative property — she swapped two of the numbers
• B. Associative property — she chose which pair of numbers to multiply first, without changing their positions
• C. Distributive property — she split one number into smaller parts
• D. Identity property — she multiplied by 1

Answer: (B)

The associative property says that the grouping of factors (which two you multiply first) does not change the product. She didn't swap any numbers' positions — 4 still comes before 5, which comes before 7. She only changed which multiplication to perform first by choosing the parentheses. That is exactly what associativity permits.

Question 2

What is the key difference between the commutative property and the associative property of multiplication?

• A. Commutative applies to addition; associative applies only to multiplication
• B. Commutative lets you reorder factors (swap their positions); associative lets you regroup factors (choose which to multiply first, without swapping)
• C. Commutative works with two factors; associative only works with four or more
• D. There is no real difference — both say you can rearrange numbers freely

Answer: (B)

Commutative: 3 × 4 = 4 × 3 — the numbers swap positions. Associative: (2 × 3) × 4 = 2 × (3 × 4) — the numbers stay in the same order, but the parentheses move to show a different pairing. Confusing them is the most common error with this property. In practice you often use both together, but they are logically distinct operations.

Question 3

The associative property of multiplication says that you can swap the positions of two factors without changing the product.

• T. True
• F. False

Answer: False

Swapping the positions of factors is the commutative property (3 × 4 = 4 × 3). The associative property is about regrouping — changing which pair of factors you multiply first — without moving any factor to a different position. (2 × 3) × 4 = 2 × (3 × 4): the order left-to-right is identical in both; only the parentheses (the grouping) changed.

Question 4

Using the associative property to compute 4 × 5 × 7 as (4 × 5) × 7 = 20 × 7 = 140 gives the same answer as (4 × 7) × 5 = 28 × 5 = 140.

• T. True
• F. False

Answer: True

The associative property guarantees that any grouping produces the same product. Both computations yield 140, even though (4 × 5) × 7 is typically easier because multiplying by 20 is simpler than multiplying by 28. The property gives you the freedom to pick the most convenient grouping — that freedom is its practical value.

Question 5

Explain in your own words how the associative property lets you choose the most convenient grouping when multiplying three numbers, and give an example where a strategic choice makes the calculation significantly easier.

Model answer: The associative property says that when multiplying three numbers, it doesn't matter which two you multiply first — the product will always be the same. This means you can look for a pair that produces a round number or a fact you know well. For example, to compute 2 × 7 × 5: multiplying left to right gives (2 × 7) × 5 = 14 × 5 = 70, which works but requires a two-digit multiplication. Instead, group the 2 and 5 first: 2 × (7 × 5) — wait, that's still 2 × 35. Better: (2 × 5) × 7 = 10 × 7 = 70. Multiplying by 10 is trivial, making the calculation much easier.

The strategic value of the associative property is that it turns a potentially hard calculation into an easy one by choosing a grouping that produces a round number (multiples of 10, 100, etc.). This is the engine behind many mental multiplication tricks: look for pairs that make friendly products, then multiply by the third factor.