Show 3 × 4 = 4 × 3
Question
Verify the commutative property of multiplication using the example 3 × 4 = 4 × 3. Also explain why this property holds.
Solution — Step by Step
Step 1: Calculate 3 × 4.
3 × 4 means 3 groups of 4: 4 + 4 + 4 = 12
Step 2: Calculate 4 × 3.
4 × 3 means 4 groups of 3: 3 + 3 + 3 + 3 = 12
Step 3: Compare the results.
3 × 4 = 12 and 4 × 3 = 12
Both are equal: 3 × 4 = 4 × 3 = 12 ✓
This verifies the commutative property of multiplication.
Commutative Property of Multiplication
For any two whole numbers a and b: a × b = b × a
The product does not change when we swap the order of the factors.
3 × 4 = 4 × 3 = 12 7 × 5 = 5 × 7 = 35 25 × 4 = 4 × 25 = 100
Answer: 3 × 4 = 4 × 3 = 12. The commutative property of multiplication is verified.
Why This Works
The easiest way to understand this is with an arrangement of dots or objects.
Imagine 3 rows of 4 mangoes each:
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This is 3 × 4 = 12 mangoes.
Now tilt your head sideways (or rotate the arrangement 90°):
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Now we have 4 rows of 3 mangoes each — that's 4 × 3 = 12 mangoes.
Same mangoes, same total, just arranged differently. The number of mangoes did not change when we rotated the arrangement. This is exactly why multiplication is commutative.
💡 Expert Tip
This "rotate the grid" picture works for every multiplication. Whether it is 7 × 9 or 12 × 5, swapping the order gives the same product. This is why it doesn't matter whether we say "3 groups of 4" or "4 groups of 3."
Why This Is Useful
The commutative property lets us choose the easier order of multiplication.
For example, 4 × 25 might feel harder, but using the commutative property, we can think of it as 25 × 4 = 100 (since 25 × 4 is a nice round number from the multiplication table).
Or: 2 × 37 × 5 — we can rearrange as 2 × 5 × 37 = 10 × 37 = 370. Much easier!
📌 Note
The word "commutative" comes from the Latin word "commutare," meaning "to exchange." In mathematics, a commutative operation lets you exchange (swap) the order of the numbers without changing the result.
Common Mistake
⚠️ Common Mistake
Mistake: Applying the commutative property to subtraction or division.
Subtraction is NOT commutative: 8 − 3 = 5, but 3 − 8 = −5 (not the same!) Division is NOT commutative: 12 ÷ 4 = 3, but 4 ÷ 12 = 1/3 (not the same!)
Only addition and multiplication are commutative for whole numbers. Always check which operation you are working with before applying the property.