Multiply Signed Changes
Apply multiplication signs and interpret repeated changes.
Supplied materials
- Paper or a note app.
- Supplied pretend change: −3 units each interval. Products to examine: 4 × (−3), 0 × (−3), and (−4) × (−3).
Learn the idea
Four copies of a negative change make a negative total: 4 × (−3) means −3 + −3 + −3 + −3. This gives −12. One positive factor and one negative factor produce a negative product.
Two negative factors produce a positive product. This is consistent with distributing multiplication over addition: [4 + (−4)] × (−3) = 0. Since the 4 × (−3) part is −12, the other part must be +12 to sum to zero. A negative count is an algebraic extension, not a literal negative number of objects.
- Add four copies of −3 to find −12.
- Write 4 × (−3) + (−4) × (−3) = 0.
- Use the missing opposite +12 to explain the two-negative product, then calculate (−5) × (−3).
See the method in action
4 × (−3) = −12 and (−4) × (−3) = +12. Their sum is 0, agreeing with [4 + (−4)] × (−3) = 0.
Check your reasoning
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See the answer and explanation
15
Both factors are negative, so the product is positive. Their magnitudes multiply to 5 × 3 = 15.
Take it further
Change the first factor to +5. Now (+5) × (−3) = −15. The magnitude stays fifteen while the sign changes because only one factor is negative.
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