Multiplying and Dividing Integers

In Grade 6, you learned about integers and how to add and subtract them. Now it's time to master multiplication and division with positive and negative numbers!

---

Review: What Are Integers?

Integers are whole numbers and their opposites: ..., -3, -2, -1, 0, 1, 2, 3, ...

• Positive integers: 1, 2, 3, 4, ... (to the right of zero)
• Negative integers: -1, -2, -3, -4, ... (to the left of zero)
• Zero is neither positive nor negative

---

The Rules for Multiplying Integers

The rules for multiplying integers depend on the signs of the numbers.

Rule 1: Positive × Positive = Positive

When you multiply two positive numbers, the answer is positive.

Examples:

• 5 × 3 = 15
• 8 × 4 = 32
• 12 × 6 = 72

This is what you've always known!

Rule 2: Negative × Negative = Positive

When you multiply two negative numbers, the answer is positive.

Examples:

• (-5) × (-3) = 15
• (-8) × (-4) = 32
• (-12) × (-6) = 72

Why? Think of it this way: A negative times a negative "reverses the reverse," bringing you back to positive!

Memory trick: "Two negatives make a positive!"

Rule 3: Positive × Negative = Negative

When you multiply a positive and a negative number, the answer is negative.

Examples:

• 5 × (-3) = -15
• (-8) × 4 = -32
• 12 × (-6) = -72

Order doesn't matter: (-3) × 5 = 5 × (-3) = -15 (commutative property still works!)

Rule 4: Negative × Positive = Negative

Same as Rule 3! One positive and one negative always give a negative product.

Examples:

• (-7) × 2 = -14
• 9 × (-5) = -45
• (-10) × 8 = -80

---

Sign Rules Summary for Multiplication

Same signs → Positive product

• (+) × (+) = (+)
• (-) × (-) = (+)

Different signs → Negative product

• (+) × (-) = (-)
• (-) × (+) = (-)

Simple shortcut:

• Like signs (same) → Positive
• Unlike signs (different) → Negative

---

Multiplying with More Than Two Integers

When multiplying multiple integers, count the negative signs!

Rule:

• Even number of negatives → Positive result
• Odd number of negatives → Negative result

Examples:

Example 1: (-2) × (-3) × (-4)

• Three negatives (odd number)
• Answer will be negative
• Calculate: 2 × 3 × 4 = 24, make it negative
• Answer: -24

Example 2: (-2) × (-3) × 4

• Two negatives (even number)
• Answer will be positive
• Calculate: 2 × 3 × 4 = 24
• Answer: 24

Example 3: (-1) × (-1) × (-1) × (-1)

• Four negatives (even number)
• Answer will be positive
• Answer: 1

---

The Rules for Dividing Integers

Great news! Division uses the SAME sign rules as multiplication!

Rule 1: Positive ÷ Positive = Positive

Examples:

• 20 ÷ 4 = 5
• 36 ÷ 6 = 6
• 100 ÷ 25 = 4

Rule 2: Negative ÷ Negative = Positive

Examples:

• (-20) ÷ (-4) = 5
• (-36) ÷ (-6) = 6
• (-100) ÷ (-25) = 4

Rule 3: Positive ÷ Negative = Negative

Examples:

• 20 ÷ (-4) = -5
• 36 ÷ (-6) = -6
• 100 ÷ (-25) = -4

Rule 4: Negative ÷ Positive = Negative

Examples:

• (-20) ÷ 4 = -5
• (-36) ÷ 6 = -6
• (-100) ÷ 25 = -4

---

Sign Rules Summary for Division

Same signs → Positive quotient

• (+) ÷ (+) = (+)
• (-) ÷ (-) = (+)

Different signs → Negative quotient

• (+) ÷ (-) = (-)
• (-) ÷ (+) = (-)

Remember: Division and multiplication use the SAME sign rules!

---

Properties That Still Work with Integers

Commutative Property of Multiplication

Order doesn't matter when multiplying!

Examples:

• 5 × (-3) = (-3) × 5 = -15
• (-4) × (-7) = (-7) × (-4) = 28

Note: Division is NOT commutative!

• 12 ÷ 3 = 4, but 3 ÷ 12 = 0.25 (different!)

Associative Property of Multiplication

Grouping doesn't matter when multiplying!

Example:

• [(-2) × 3] × 4 = (-6) × 4 = -24
• (-2) × [3 × 4] = (-2) × 12 = -24
• Same answer!

Multiplication by Zero

Any number times zero equals zero!

Examples:

• 0 × 5 = 0
• (-7) × 0 = 0
• 0 × (-100) = 0

Multiplication by One

Any number times one equals itself!

Examples:

• 1 × 8 = 8
• (-6) × 1 = -6
• 1 × (-15) = -15

Multiplication by Negative One

Multiplying by -1 gives you the opposite!

Examples:

• (-1) × 5 = -5
• (-1) × (-8) = 8
• (-1) × 0 = 0

---

Real-World Applications

Temperature Changes

Problem: The temperature drops 3°F per hour for 4 hours. What's the total change?

Solution:

• Drop means negative: -3°F per hour
• For 4 hours: 4 × (-3) = -12°F
• Answer: The temperature dropped 12°F total

Banking (Debt)

Problem: You withdraw $25 from your account 3 times. What's the total change in your balance?

Solution:

• Withdrawal is negative: -$25
• Three times: 3 × (-25) = -$75
• Answer: Your balance decreased by $75

Elevators (Going Down)

Problem: An elevator descends 5 floors per minute for 3 minutes. Where is it relative to the starting floor?

Solution:

• Descending is negative: -5 floors
• For 3 minutes: 3 × (-5) = -15 floors
• Answer: 15 floors below the starting point

Sharing Debt

Problem: Four friends owe $60 total. If they split it equally, what does each person owe?

Solution:

• Debt is negative: -$60
• Divide by 4: (-60) ÷ 4 = -$15
• Answer: Each person owes $15 (or has -$15)

---

Order of Operations with Integers

Remember PEMDAS still applies!

Example: -3 × 5 - (-2) × 4

Step 1: Multiply first (left to right)

• -3 × 5 = -15
• (-2) × 4 = -8

Step 2: Rewrite

• -15 - (-8)

Step 3: Subtract (subtracting a negative = add)

• -15 + 8 = -7

Answer: -7

---

Common Mistakes to Avoid

❌ Mistake 1: Thinking negative × negative = negative

• Wrong: (-3) × (-4) = -12
• Right: (-3) × (-4) = 12 (two negatives make a positive!)

❌ Mistake 2: Forgetting the sign with zero

• Wrong: 0 ÷ (-5) = -0 (there's no negative zero!)
• Right: 0 ÷ (-5) = 0

❌ Mistake 3: Mixing up multiplication and addition rules

• Addition: -3 + (-4) = -7 (add and keep negative)
• Multiplication: (-3) × (-4) = 12 (different rule - makes positive!)

❌ Mistake 4: Not counting negatives in long problems

• Wrong: (-1) × (-2) × (-3) = 6
• Right: (-1) × (-2) × (-3) = -6 (three negatives = odd = negative!)

❌ Mistake 5: Forgetting parentheses with negatives

• 5 × -3 (ambiguous!)
• 5 × (-3) = -15 (clear!)

---

Practice Strategies

Strategy 1: Determine the sign first

1. Look at the signs (same or different?)
2. Decide if answer is positive or negative
3. Then multiply/divide the numbers normally
4. Apply the sign

Example: (-12) ÷ 4

1. Different signs → negative answer
2. 12 ÷ 4 = 3
3. Make it negative: -3

Strategy 2: Use patterns

• 5 × 3 = 15
• 5 × (-3) = -15
• (-5) × 3 = -15
• (-5) × (-3) = 15

See the pattern? Only the last one (two negatives) is positive!

Strategy 3: Think of real-world scenarios

• Debt (negative) shared among people (division)
• Temperature dropping (negative) over time (multiplication)
• These make the rules make sense!

---

Connection to Fractions

These same sign rules apply when multiplying and dividing fractions!

Examples:

• (-1/2) × (-1/3) = 1/6 (two negatives = positive)
• (3/4) ÷ (-1/2) = -3/2 (different signs = negative)

You'll use these rules throughout algebra!

---

Why These Rules Work

Multiplication by -1:

• Think of -1 × 5 as "the opposite of 5" = -5
• Then -1 × (-5) is "the opposite of the opposite of 5" = 5
• That's why negative × negative = positive!

Pattern thinking:

• 3 × 2 = 6
• 3 × 1 = 3
• 3 × 0 = 0
• 3 × (-1) = -3 (decreasing by 3 each time)
• 3 × (-2) = -6 (pattern continues!)

---

Quick Reference Chart

| Operation | Signs | Result | Example | |-----------|-------|--------|---------| | Multiply | Same | + | (-5) × (-3) = 15 | | Multiply | Different | - | 5 × (-3) = -15 | | Divide | Same | + | (-20) ÷ (-4) = 5 | | Divide | Different | - | 20 ÷ (-4) = -5 |

Master these rules and you'll be ready for algebra, equations, and advanced math!