Multiplying Integers

Core Idea

Multiplying integers follows two sign rules: the product of two numbers with the same sign is positive, and the product of two numbers with different signs is negative. These rules can be derived from patterns (e.g., 3 × 2 = 6, 3 × 1 = 3, 3 × 0 = 0, 3 × (−1) = −3 — each step decreases by 3) or from the idea that multiplication by −1 reflects a number across zero on the number line. Mastery of integer multiplication is essential for simplifying expressions, working with exponents, and factoring polynomials.

How It's Best Learned

Show the pattern-based derivation so students see that the sign rules are logical consequences, not arbitrary. Use repeated addition to motivate: 3 × (−2) = (−2) + (−2) + (−2) = −6. For negative times negative, the pattern argument is most convincing. Practice with a mix of sign combinations and emphasize that counting the number of negative factors determines the product's sign.

Common Misconceptions

• Students may think negative × negative = negative, not recognizing the double-negative pattern.
• Confusing multiplication sign rules with addition sign rules (where same signs means add magnitudes).
• When multiplying more than two integers, students lose track of the sign — teach them to count the number of negatives (even = positive, odd = negative).

Explainer

You already know how to add integers. Multiplication is repeated addition, so it is natural to start there. What does 3 × (-2) mean? It means three groups of (-2): (-2) + (-2) + (-2) = -6. A positive times a negative is negative because you are repeating a negative quantity a positive number of times. This gives you the first rule without any memorization.

The harder case is negative times negative. The repeated-addition argument breaks down (what would -3 groups of something mean?), but the pattern argument makes it clear. Look at this column: 3×2=6, 3×1=3, 3×0=0, 3×(-1)=-3, 3×(-2)=-6. Each step down decreases by 3. Now do the same with -3: (-3)×2=-6, (-3)×1=-3, (-3)×0=0, (-3)×(-1)=?, (-3)×(-2)=?. Each step now *increases* by 3. The pattern forces (-3)×(-1)=3 and (-3)×(-2)=6. Negative times negative must be positive.

A more abstract way to see it: multiplying by -1 flips a number to its opposite on the number line. Multiplying by -1 twice flips twice — returning you to where you started. So (-1)×(-1) = 1, and by extension any negative times negative is positive.

When multiplying a chain of integers — like (-2)×(-3)×(-1) — you do not need to track running products. Just count the negative factors. An even number of negatives produces a positive result; an odd number produces a negative result. This "count the negatives" shortcut is essential for simplifying expressions with exponents later, where you will see things like (-2)⁵ (five negative factors → negative) versus (-2)⁴ (four negative factors → positive).