Questions: Commutative Property of Multiplication

Question 1

You know that 7 × 9 = 63. Using only the commutative property, which other multiplication fact do you automatically know?

• A. 7 × 10 = 70, because 9 and 10 are close
• B. 9 × 7 = 63, because order of factors does not change the product
• C. 63 ÷ 7 = 9, because multiplication and division are related
• D. 14 × 9 = 126, because doubling one factor doubles the product

Answer: (B)

The commutative property states that a × b = b × a — switching the order of factors gives the same product. So knowing 7 × 9 = 63 immediately and automatically tells you 9 × 7 = 63. Option C is true but requires the inverse relationship between multiplication and division — a different property, not the commutative property alone.

Question 2

A word problem says: 'There are 5 shelves, each holding 8 books.' A student writes 8 × 5 = 40 instead of 5 × 8 = 40. Is the student's answer correct?

• A. No — the problem says 5 shelves with 8 books each, so only 5 × 8 is valid
• B. Yes — the commutative property guarantees the product is the same regardless of order
• C. No — 8 × 5 means 8 shelves with 5 books, which is a different total
• D. Only if the student explains why the order was switched

Answer: (B)

The commutative property guarantees 5 × 8 = 8 × 5 = 40, so the numerical answer is correct either way. The physical setup described in the problem (5 groups of 8) is different from 8 groups of 5, but the total count is identical. The commutative property applies to the numerical product — the answer 40 is correct regardless of which order is written.

Question 3

The commutative property of multiplication means that 3 × 4 and 4 × 3 describe the same physical situation.

• T. True
• F. False

Answer: False

The commutative property guarantees the same *product* (3 × 4 = 4 × 3 = 12), but the two expressions can describe different physical situations. Three groups of four is a different arrangement than four groups of three — even though both total 12. In a word problem, the order of factors carries meaning about the real-world setup. The property is about numerical equality, not situational identity.

Question 4

Because of the commutative property, a student who knows 8 × 6 = 48 automatically knows 6 × 8 = 48 without any extra work.

• T. True
• F. False

Answer: True

This is precisely what the commutative property delivers. Every fact in the multiplication table appears twice — once as a × b and once as b × a — but both give the same product. Knowing one immediately gives you the other for free. This is why the commutative property effectively cuts the number of unique multiplication facts in half.

Question 5

Why does the commutative property cut the number of multiplication facts you need to memorize roughly in half?

Model answer: Because a × b = b × a, every fact in the times table appears twice — for example, 3 × 7 and 7 × 3 are the same fact written in a different order. Once you know one, you know the other automatically. So the multiplication table is symmetric: every entry above the diagonal mirrors one below it, and you only need to learn one of each pair.

This halving is why the commutative property is one of the most practically useful properties in arithmetic. A student who grasps this can approach an unfamiliar fact (like 9 × 4) by recalling the more familiar version (4 × 9 = 36) — then flip it. The property isn't just a rule to recite; it's a memory shortcut with real daily value.