Swap the numbers
Multiplying integers in either order gives the same product.
Same answer both ways?
3 × (−4): put 4 red tokens in the bag, 3 times. (−4) × 3: take 3 green tokens out, 4 times. These are different jobs.
Will the two products be equal?
What this lesson covers
The idea
For any two integers a and b, a × b = b × a, since swapping changes neither the magnitude nor the sign of the product.
Same answer both ways?
3 × (−4): put 4 red tokens in the bag, 3 times. (−4) × 3: take 3 green tokens out, 4 times. These are different jobs.
Will the two products be equal?
- Yes, they are equal
- Only their sizes are equal
- No, they are different
Predict, then turn the array
A product is an array of tokens. First predict the swapped product. Then turn the array on its side.
Turning changes nothing
Turn the array on its side: the number of tokens stays the same, so the size of the product does not change. The two numbers are the same two numbers, so like or unlike signs is also unchanged.
For any two integers a and b, a × b = b × a: swapping changes neither the magnitude nor the sign of the product. Multiplication is commutative.
- a × b | b × a
- 3 × (−4) = −12 | (−4) × 3 = −12
- (−15) × (−8) = 120 | (−8) × (−15) = 120
- 14 × (−5) = −70 | (−5) × 14 = −70
Notes
For any two integers a and b, a × b = b × a. Swapping changes neither the magnitude nor the sign of the product: multiplication is commutative.
Check yourself
14 × (−5) = −70. What number makes (−5) × ? = −70 true?
Answer: 14
(−5) × 14 = 14 × (−5) = −70, so the missing number is 14.
(−30) × 12 = −360. Without working it out again, what is 12 × (−30)?
Answer: -360
12 × (−30) = (−30) × 12 = −360.
Which is equal to (−15) × (−8)?
Does swapping two integers in a product ever change the sign of the product?
- (−8) × 15. That is −120. The sign changed because one of the numbers changed sign.
- (−8) × (−15) — correct. Yes! Swapping the two numbers gives the same product, 120.
- 8 × (−15). That is −120. Both numbers must stay as they were.
- Yes, when one number is negative. With one negative number the product is negative both before and after swapping.
- Yes, when both numbers are negative. With two negatives the product is positive both before and after swapping.
- No: the two numbers stay the same, so like or unlike signs stay the same — correct. Yes! Both before and after, the signs are the same pair.