➗ Multiplying & Dividing Negative Numbers

Cambridge Lower Secondary Stage 7 · Directed Numbers

Positive × Positive
+ × + = +
e.g.   4 × 3 = 12
Positive × Negative
+ × − = −
e.g.   5 × (−2) = −10
Negative × Negative
− × − = +
e.g.   (−3) × (−4) = 12

Click a rule to see examples spark!

+ × + = +
+ × − = −
− × − = +
+ ÷ + = +
+ ÷ − = −
− ÷ − = +
⬆ Click any rule above to see examples animate in!

What you'll learn:

  • The four sign rules for multiplication and division
  • Multiplying integers including negative numbers
  • Dividing integers including negative numbers
  • Order of operations (BIDMAS) with negative numbers
  • Powers of negative numbers, e.g. (−2)³ = −8
  • Real-life problems: temperature drops, debts over weeks
Created by Mr Josh

📖 Learn: Multiplying & Dividing Negative Numbers

Part 1: The Four Sign Rules

When multiplying or dividing two numbers, the signs determine whether the answer is positive or negative.

Operation Rule Example
+ × + Positive 3 × 4 = 12
+ × − Negative 5 × (−2) = −10
− × + Negative (−3) × 4 = −12
− × − Positive (−3) × (−4) = 12
💡 Memory trick: Same signs → Positive. Different signs → Negative.
Think: "two negatives cancel each other out!"

Part 2: The Same Rules Apply to Division

The same sign rules apply for division:

Operation Rule Example
+ ÷ + Positive 12 ÷ 4 = 3
+ ÷ − Negative 12 ÷ (−4) = −3
− ÷ + Negative (−12) ÷ 4 = −3
− ÷ − Positive (−12) ÷ (−4) = 3

Part 3: Order of Operations (BIDMAS) with Negatives

BIDMAS still applies: Brackets → Indices → Division/Multiplication → Addition/Subtraction.

Example:   (−2) × 3 + 4 ÷ (−2)
Step 1:   Multiplication first: (−2) × 3 = −6
Step 2:   Division next: 4 ÷ (−2) = −2
Step 3:   Addition last: −6 + (−2) = −8
💡 Always deal with multiplications and divisions before additions and subtractions — even when negatives are involved!

Part 4: Powers of Negative Numbers

When you raise a negative number to a power, count how many negative signs multiply together.

Expression Expanded Answer Why?
(−2)² (−2) × (−2) +4 − × − = + (even power)
(−2)³ (−2) × (−2) × (−2) −8 +4 × (−2) = − (odd power)
(−3)² (−3) × (−3) +9 even power → positive
(−3)³ (−3) × (−3) × (−3) −27 odd power → negative
🔑 Key pattern: A negative number to an even power is always positive. To an odd power it is always negative.

Part 5: Pattern Spotter — See the Rule Emerge!

Watch what happens as we multiply 3 by decreasing numbers. Click each ? to reveal!

Created by Mr Josh

💡 Worked Examples

Example 1: Multiplying with Negative Numbers

Calculate:   (−7) × 4

Step 1: Ignore signs, multiply the numbers: 7 × 4 = 28
Step 2: Apply sign rule: (−) × (+) = (−)
Answer: −28
Different signs → negative answer.

Example 2: Dividing with Negative Numbers

Calculate:   (−36) ÷ (−9)

Step 1: Ignore signs, divide: 36 ÷ 9 = 4
Step 2: Apply sign rule: (−) ÷ (−) = (+)
Answer: +4
Same signs (both negative) → positive answer.

Example 3: BIDMAS with Negative Numbers

Calculate:   3 + (−4) × (−2) − 6 ÷ 3

Step 1: Multiply: (−4) × (−2) = +8
Step 2: Divide: 6 ÷ 3 = 2
Step 3: Now left to right: 3 + 8 − 2
Step 4: = 11 − 2 = 9
Always handle × and ÷ before + and − even with negatives!

Example 4: Powers of a Negative Number

Calculate:   (−5)³

Step 1: Write out the multiplication: (−5) × (−5) × (−5)
Step 2: First pair: (−5) × (−5) = +25
Step 3: Then: +25 × (−5) = −125
Answer: −125
Odd power → the negative survives. (−5)⁴ would give +625.
Created by Mr Josh

🔭 Multiplication Grid Visualizer

This grid shows multiplication extending into negative numbers. Green cells = positive, red cells = negative.

  

Notice the symmetry: every positive value in the top-right mirrors a positive in the bottom-left — because (−) × (−) = (+).

Created by Mr Josh

✏️ Exercise 1: Multiplying with Negative Numbers

Calculate each product. Include the sign in your answer (e.g. -12 or 20).

Created by Mr Josh

✏️ Exercise 2: Dividing with Negative Numbers

Calculate each quotient. Include the sign in your answer.

Created by Mr Josh

✏️ Exercise 3: BIDMAS with Negative Numbers

Use the correct order of operations. Show your working if needed.

Created by Mr Josh

✏️ Exercise 4: Powers of Negative Numbers

Calculate each power. Remember: even power = positive, odd power = negative.

Created by Mr Josh

📖 Exercise 5: Real-Life Worded Problems

Read each problem carefully. Write just the numerical answer (with sign).

Created by Mr Josh

📝 Practice Questions

  1. Calculate: 6 × (−3)
  2. Calculate: (−5) × 8
  3. Calculate: (−4) × (−7)
  4. Calculate: (−9) × 0
  5. Calculate: 12 ÷ (−4)
  6. Calculate: (−18) ÷ 6
  7. Calculate: (−20) ÷ (−5)
  8. Calculate: (−3) × (−3) × (−3)
  9. Calculate: (−2)⁴
  10. Calculate: (−5)²
  11. Calculate using BIDMAS: 2 + (−3) × 4
  12. Calculate using BIDMAS: (−2) × 5 − 6 ÷ (−3)
  13. Calculate using BIDMAS: (−4)² + 2 × (−3)
  14. Calculate using BIDMAS: 10 ÷ (−2) + (−3) × (−2)
  15. A submarine descends 15 m each hour. How far below the surface is it after 6 hours? (Give your answer as a negative number.)
  16. The temperature drops 4°C every hour. What is the change in temperature after 5 hours?
  17. Emma owes her friend £6 per week for 8 weeks. Write a multiplication to show her total debt and calculate it.
  18. A company loses £250 per day for 7 days. What is its total loss? (negative answer)
  19. The temperature is −24°C. It warms up equally over 8 hours to reach 0°C. What is the change per hour?
  20. If (−a) × (−b) = 36 and a = 4, find b.
  1. −18
  2. −40
  3. +28
  4. 0
  5. −3
  6. −3
  7. +4
  8. (−3)³ = −27
  9. (−2)⁴ = +16
  10. (−5)² = +25
  11. 2 + (−12) = −10
  12. −10 − (−2) = −10 + 2 = −8
  13. 16 + (−6) = 10
  14. −5 + 6 = 1
  15. 6 × (−15) = −90 m (90 m below surface)
  16. 5 × (−4) = −20°C change
  17. 8 × (−6) = −£48
  18. 7 × (−250) = −£1750
  19. (−24) ÷ 8 = −3°C per hour (wait, warming means +3°C/hour)
  20. (−4) × (−b) = 36 → 4b = 36 → b = 9
Created by Mr Josh

🏆 Challenge: Mixed Skills

  1. Calculate: (−3)² × (−2)³ ÷ (−6)
  2. Find all integer values of x such that (−2) × x = −14.
  3. A scientist records temperatures at midnight: −3°C, −7°C, −11°C, −15°C. The pattern continues. What is the temperature on the 8th night? Express the pattern as multiplication.
  4. Calculate: 5 − (−2)³ ÷ 4 × (−3) + 1. Show full BIDMAS working.
  5. The balance in a bank account changes by −£45 per week. Starting from £360, after how many weeks will the account first go below £0? Write and solve an inequality using multiplication.
  6. Is the statement true or false? "(−a)² is always positive for any non-zero integer a." Explain your reasoning.
  7. Calculate: [(−2)³ + (−3)²] × (−1)⁵
  8. A sequence is defined by: first term = 1, and each term is multiplied by −3 to get the next. Write the first five terms. What is the sign of the 10th term? Justify your answer.
  1. (−3)² = 9; (−2)³ = −8; 9 × (−8) = −72; (−72) ÷ (−6) = 12
  2. (−2) × x = −14 → x = (−14) ÷ (−2) = 7
  3. Pattern: starts at −3, decreases by 4 each night. Night n = −3 + (n−1) × (−4) = −3 − 4(n−1). Night 8 = −3 − 4×7 = −3 − 28 = −31°C
  4. (−2)³ = −8; −8 ÷ 4 = −2; −2 × (−3) = 6; then 5 − 6 + 1 = 0
  5. 360 + n×(−45) < 0 → 360 < 45n → n > 8. After 9 weeks it first goes below £0.
  6. True. (−a)² = (−a) × (−a) = a². Since a² ≥ 0 for all real a and is positive when a ≠ 0, the statement is true.
  7. (−2)³ = −8; (−3)² = 9; (−8 + 9) = 1; 1 × (−1)⁵ = 1 × (−1) = −1
  8. Terms: 1, −3, 9, −27, 81. The 10th term: multiplied by −3 nine times. (−3)⁹ is negative (odd power). So 10th term = 3⁹ = 19683, but sign is negative → −19683.
Created by Mr Josh