Operations with Integers · Lesson 3 of 6
Multiplication Patterns and Properties
“Extend times tables through zero and justify the sign and swapping rules.”
• Continue multiplication patterns into negative multipliers. • Use a sign table without losing the reason behind it. • Explain the roles of 1, −1 and zero in multiplication. • Justify commutativity for signed factors.
Continue the pattern across zero
In 4 × 3 = 12, 3 × 3 = 9, 2 × 3 = 6, each decrease of 1 in the first factor decreases the product by 3. Carry on: 1 × 3 = 3, 0 × 3 = 0, and (−1) × 3 = −3. For a second factor −3, the products −12, −9, −6, −3, 0, +3 rise by 3 each time the first factor drops by 1. The continuation explains the positive product of two negative factors.
| First factor | × 3 | × (−3) |
|---|---|---|
| 2 | 6 | −6 |
| 1 | 3 | −3 |
| 0 | 0 | 0 |
| −1 | −3 | 3 |
| −2 | −6 | 6 |
Problem
Continue 2 × (−3), 1 × (−3), 0 × (−3), (−1) × (−3).
- 1.The first products are −6, −3 and 0.
- 2.Each step lowers the first factor by 1 and raises the product by 3.
- 3.The next product is +3, so (−1) × (−3) = 3.
Changing signs does not change the product’s magnitude: |a × b| = |a| × |b|. Equal signs give a positive product and unlike signs a negative one, except that any product containing zero is zero. Multiplication by 1 leaves a number unchanged; multiplication by −1 gives its additive inverse.
Problem
Find (−1) × (−31) and 1 × (−31).
- 1.The additive inverse of −31 is +31, so multiplying it by −1 gives +31.
- 2.Multiplying by 1 leaves −31 unchanged.
- 3.The two answers are +31 and −31.
Swapping factors does not change the product
Compare 3 × (−4) and (−4) × 3: both have magnitude 12 and one negative factor, so both are −12. When both factors are negative, swapping also keeps the same magnitude and sign. This reasoning, combined with the familiar rule for positive factors, establishes commutativity for all integers.
Problem
Find (−15) × (−8) and (−8) × (−15).
- 1.Both factors are negative, so the result is positive.
- 2.15 × 8 = 120, regardless of order.
- 3.Both products equal 120.
The historical description of positive amounts as fortunes and negative amounts as debts offers another language for sign rules. It agrees with the token and pattern explanations: a debt multiplied by a positive count stays a debt, while reversing a debt gives a fortune.
A product with a zero factor is zero, even if the remaining factors have two negative signs. The positive/negative sign table applies to nonzero factors.
Quiz
What is (−1) × (−31)?
What is (−8) × 4?
Which expresses commutativity?
Continue 0 × (−3), (−1) × (−3), (−2) × (−3).
Which product is negative?
Practice Problems
- Extend the table 3 × (−4), 2 × (−4), 1 × (−4), 0 × (−4) to multipliers −1 and −2.
- Find (−9) × 10 and 10 × (−9); explain why they agree.
- Explain why 1 × a = a and (−1) × a = −a for a negative integer a.
- Complete (−30) × 12 and 12 × (−30), using magnitude and sign.
- Create a debt/fortune story for 4 × (−5) and its opposite.
- Explain why 0 × (−17) is zero rather than negative.
Key Takeaways
• Multiplication patterns continue consistently through zero. • Equal nonzero signs give a positive product; unlike signs give a negative product. • Product magnitude is the product of the magnitudes. • Multiplying by 1 preserves a number; multiplying by −1 gives its additive inverse. • Swapping two integer factors leaves their product unchanged.