Operations with Integers - Sixth Grade
Complete Notes & Formulas
1. Adding Integers
Rule 1: Same Signs
ADD the absolute values
KEEP the common sign
(+) + (+) = (+) → Positive + Positive = Positive
Example: 5 + 3 = 8
(−) + (−) = (−) → Negative + Negative = Negative
Example: −5 + (−3) = −8
Rule 2: Different Signs
SUBTRACT the smaller absolute value from larger
KEEP the sign of the number with larger absolute value
(+) + (−) → Subtract and use sign of bigger absolute value
Example: 8 + (−3) = 5 (positive because 8 > 3)
Example: 3 + (−8) = −5 (negative because 8 > 3)
Using Number Lines
• Start at the first number
• Move RIGHT for positive numbers
• Move LEFT for negative numbers
• Where you land is your answer
Example: −4 + 7
Step 1: Different signs, so subtract: 7 − 4 = 3
Step 2: Larger absolute value is 7 (positive)
Step 3: Answer is positive
Answer: 3
2. Subtracting Integers
The Golden Rule: Keep-Change-Change (KCC)
KEEP - CHANGE - CHANGE
KEEP the first number
CHANGE subtraction to addition
CHANGE the sign of second number
Formula
a − b = a + (−b)
Examples
Example 1: 7 − 3
7 − 3 = 7 + (−3) = 4
Answer: 4
Example 2: −5 − 3
−5 − 3 = −5 + (−3) = −8
Answer: −8
Example 3: 4 − (−6)
4 − (−6) = 4 + 6 = 10
Answer: 10
Example 4: −2 − (−7)
−2 − (−7) = −2 + 7 = 5
Answer: 5
3. Multiplying Integers
Sign Rules
SAME signs → POSITIVE (+)
DIFFERENT signs → NEGATIVE (−)
All Rules
| Operation | Sign of Result | Example |
|---|---|---|
| (+) × (+) | Positive (+) | 4 × 5 = 20 |
| (−) × (−) | Positive (+) | −4 × (−5) = 20 |
| (+) × (−) | Negative (−) | 4 × (−5) = −20 |
| (−) × (+) | Negative (−) | −4 × 5 = −20 |
Quick Tip:
• Even number of negatives → Positive result
• Odd number of negatives → Negative result
4. Dividing Integers
Sign Rules (Same as Multiplication!)
SAME signs → POSITIVE (+)
DIFFERENT signs → NEGATIVE (−)
All Rules
| Operation | Sign of Result | Example |
|---|---|---|
| (+) ÷ (+) | Positive (+) | 20 ÷ 4 = 5 |
| (−) ÷ (−) | Positive (+) | −20 ÷ (−4) = 5 |
| (+) ÷ (−) | Negative (−) | 20 ÷ (−4) = −5 |
| (−) ÷ (+) | Negative (−) | −20 ÷ 4 = −5 |
5. Mixed Operations with Integers (PEMDAS)
Order of Operations
PEMDAS
P - Parentheses
E - Exponents
M - Multiplication (left to right)
D - Division (left to right)
A - Addition (left to right)
S - Subtraction (left to right)
Example: −3 + 4 × (−2)
Step 1: No parentheses to simplify
Step 2: Multiply first: 4 × (−2) = −8
Expression becomes: −3 + (−8)
Step 3: Add: −3 + (−8) = −11
Answer: −11
Example: (−5 + 3) × (−2)
Step 1: Parentheses first: (−5 + 3) = −2
Step 2: Multiply: (−2) × (−2) = 4
Answer: 4
6. Adding Three or More Integers
Strategy
Method 1: Add from left to right
Method 2: Group positive numbers together, negative numbers together, then combine
Example: 5 + (−3) + 8 + (−7)
Method 2 (easier):
Positives: 5 + 8 = 13
Negatives: −3 + (−7) = −10
Combine: 13 + (−10) = 3
Answer: 3
7. Word Problems with Integers
Keywords
Addition: Increase, gain, deposit, rise, above
Subtraction: Decrease, loss, withdrawal, fall, below, difference
Positive: Profit, above sea level, temperature rise, credits
Negative: Debt, below sea level, temperature drop, debits
Example 1: Temperature
Problem: The temperature was −8°C in the morning. It increased by 12°C by noon. What is the temperature at noon?
Start: −8°C
Increase: +12°C
−8 + 12 = 4
Answer: 4°C
Example 2: Money
Problem: You have $50 in your bank account. You withdraw $30, then deposit $20. What is your balance?
Start: 50
Withdraw: −30
Deposit: +20
50 + (−30) + 20 = 40
Answer: $40
Example 3: Elevation
Problem: A submarine at −150 meters descends another 75 meters. What is its new depth?
Start: −150 meters
Descends (goes deeper): −75 meters
−150 + (−75) = −225
Answer: −225 meters
8. Input/Output Tables
How to Use
Step 1: Identify the rule (operation)
Step 2: Apply the rule to each input
Step 3: Write the output
Example: Rule: Add −5
| Input | Operation | Output |
|---|---|---|
| 3 | 3 + (−5) | −2 |
| 0 | 0 + (−5) | −5 |
| −4 | −4 + (−5) | −9 |
| 7 | 7 + (−5) | 2 |
Quick Reference: Integer Operations
| Operation | Key Rule |
|---|---|
| Addition (Same Signs) | Add, keep sign |
| Addition (Different Signs) | Subtract, larger sign wins |
| Subtraction | Keep-Change-Change |
| Multiplication | Same signs = (+), Different = (−) |
| Division | Same signs = (+), Different = (−) |
💡 Important Tips to Remember
✓ Addition same signs: Add and keep the sign
✓ Addition different signs: Subtract, use sign of larger absolute value
✓ Subtraction: Keep-Change-Change (change to addition)
✓ Multiplication/Division same signs: Result is positive
✓ Multiplication/Division different signs: Result is negative
✓ PEMDAS applies to integers just like other numbers
✓ Use parentheses to avoid confusion with signs
✓ Number line: Right = add positive, Left = add negative
✓ Zero is neutral: Adding 0 doesn't change the value
✓ Practice with real-world contexts (temperature, money, elevation)
🧠 Memory Tricks & Strategies
Adding Same Signs:
"Same signs add together, keep the sign forever!"
Adding Different Signs:
"Different signs subtract, bigger sign's where it's at!"
Subtraction:
"Keep it, change it, change it - KCC makes it easy!"
Multiplication/Division:
"Same signs positive, different negative - it's definitive!"
Negative Times Negative:
"Two wrongs make a right!" (−) × (−) = (+)
Number Line:
"Positive goes right, negative goes left, zero's in the middle - that's the test!"
Master Integer Operations! ➕ ➖ ✖️ ➗ 🎯
Remember: Same signs positive, different signs negative!