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Mathematics › 🔄 Grade 7 › Multiplying and Dividing Integers: Cracking the Sign Code
🔄 Grade 7 · Lesson 3 of 14

Multiplying and Dividing Integers: Cracking the Sign Code

Two negatives really do make a positive — and here's the pattern that proves it.

Grade 7Middle School
Multiplying and Dividing Integers: Cracking the Sign Code — illustration
💡
The big idea: Sign rules for multiplication and division follow directly from extending the patterns you already know. Three positive times three negative is negative. Three negative times three negative must extend the pattern — and that extension forces the answer to be positive. Understanding why eliminates the need to memorize.
🎯 By the end, you'll be able to
  • Determine the sign of a product or quotient using the same-sign/different-sign rules
  • Derive the sign rules from a shrinking-pattern table rather than memorizing them
  • Multiply or divide two or more integers correctly including three-or-more-factor chains
  • Count the number of negative factors to predict the sign of a product
📎 You should already know
  • What negative integers are (integer intro)
  • Whole-number multiplication and division

Warm up: a pattern you can trust

Look at this column of products — each line multiplies 3 by a decreasing number:

3 × 3 = 9
3 × 2 = 6
3 × 1 = 3
3 × 0 = 0
3 × (−1) = ?
3 × (−2) = ?

The products decrease by 3 each step. After 0, continuing the pattern gives −3, −6 … So 3 × (−1) = −3. A positive times a negative is negative — the pattern demands it.

Now watch what happens with two negatives

Start a new column, now with the first factor decreasing:

3 × (−3) = −9
2 × (−3) = −6
1 × (−3) = −3
0 × (−3) = 0
(−1) × (−3) = ?
(−2) × (−3) = ?

The products increase by 3 each step. After 0, the pattern continues: +3, +6 … So (−1) × (−3) = +3. A negative times a negative is positive. Not because of a rule you memorized — because the pattern requires it.

🔑 The two sign rules

Same signs → Positive: (+)(+) = (+)    and    (−)(−) = (+)

Different signs → Negative: (+)(−) = (−)    and    (−)(+) = (−)

These rules work for both multiplication and division — the sign of the result depends only on whether the signs of the two numbers match.

🎮 Integer Walker LIVE
Set the hop size (first factor) and the number of hops (second factor). A negative hop size means moving left; a negative number of hops reverses direction. Watch where you land.
✨ Three or more factors? Just count the negatives.

For chains like (−2) × (−3) × (−1):

  • Count the negative factors: three negatives.
  • Each pair of negatives cancels to a positive. Odd count of negatives → negative product. Even count → positive product.
  • Here: three negatives (odd) → negative. |2 × 3 × 1| = 6, so the answer is −6.
📝 Worked example: Find (−4) × (−7).
  1. Signs: both negative — same sign → positive result.
  2. Multiply the absolute values: 4 × 7 = 28.
  3. Apply the positive sign.
✓ (−4) × (−7) = <strong>+28</strong>.
📝 Worked example: Find (−36) ÷ 9.
  1. Signs: negative ÷ positive — different signs → negative result.
  2. Divide the absolute values: 36 ÷ 9 = 4.
  3. Apply the negative sign.
✓ (−36) ÷ 9 = <strong>−4</strong>.
📝 Worked example: Find (−2) × 5 × (−3).
  1. Count negative factors: two (−2 and −3). Even count → positive product.
  2. Multiply the absolute values: 2 × 5 × 3 = 30.
  3. Apply the positive sign.
✓ (−2) × 5 × (−3) = <strong>+30</strong>.

Check your understanding

1. What is (−6) × (−4)?
Both signs are negative — same signs → positive. 6 × 4 = 24, so the answer is +24.
2. What is (−5) × 7?
Different signs (negative × positive) → negative. 5 × 7 = 35, so the answer is −35.
3. What is (−48) ÷ (−6)?
Same signs (both negative) → positive. 48 ÷ 6 = 8, so the answer is +8.
4. A submarine descends 15 metres per minute for 4 minutes. What is the depth change as a signed number?
Each minute is −15 m (descending). 4 groups of −15: 4 × (−15) = −60 m. The submarine is 60 metres deeper.
5. What is (−2) × (−3) × (−1)?
Three negative factors — odd count → negative. Absolute value: 2 × 3 × 1 = 6. Answer: −6.
✅ Key takeaways
  • The sign rule for multiplication/division: same signs → positive; different signs → negative.
  • Derive this from the shrinking-pattern table — don't just memorize it.
  • For chains of factors, count the negatives: odd count → negative product; even count → positive.
  • The magnitude of the result is always |factor1| × |factor2| (or ÷) regardless of signs.
  • The same sign rules apply to division as to multiplication.