Operations with Rational Numbers - Seventh Grade
Addition, Subtraction, Multiplication & Division with Positive and Negative Numbers
1. Adding Positive and Negative Numbers
Sign Rules for Addition
Same Signs (++ or --):
• Positive + Positive → ADD and keep POSITIVE sign
• Negative + Negative → ADD and keep NEGATIVE sign
Different Signs (+- or -+):
• SUBTRACT the smaller absolute value from larger
• Use the sign of the number with LARGER absolute value
Examples
| Problem | Rule | Answer |
|---|---|---|
| 7 + 3 | Positive + Positive → Add | 10 |
| -7 + (-3) | Negative + Negative → Add, keep negative | -10 |
| 7 + (-3) | Different signs → Subtract 7-3 | 4 |
| -7 + 3 | Different signs → Subtract 7-3, keep negative | -4 |
Adding Decimals
Example 1: 5.6 + (-2.3)
Different signs → Subtract: 5.6 - 2.3 = 3.3
Answer: 3.3
Example 2: -3.7 + (-1.5)
Same signs → Add: 3.7 + 1.5 = 5.2, keep negative
Answer: -5.2
Adding Fractions
Example: -2/3 + 1/4
Find LCD: 12
-2/3 = -8/12, 1/4 = 3/12
-8/12 + 3/12 = -5/12
Answer: -5/12
2. Subtracting Positive and Negative Numbers
The Golden Rule
Keep - Change - Change (KCC)
KEEP the first number
CHANGE subtraction to addition
CHANGE the sign of the second number
Sign Rules for Subtraction
Two same signs: become POSITIVE (+)
• 5 - (+3) = 5 - 3 = 2
• 5 - (-3) = 5 + 3 = 8
Two different signs: become NEGATIVE (-)
• -5 - (+3) = -5 - 3 = -8
• 5 + (-3) = 5 - 3 = 2
Examples
| Problem | Change to Addition | Answer |
|---|---|---|
| 8 - 5 | 8 + (-5) | 3 |
| 8 - (-5) | 8 + 5 | 13 |
| -8 - 5 | -8 + (-5) | -13 |
| -8 - (-5) | -8 + 5 | -3 |
3. Multiplying Positive and Negative Numbers
Sign Rules for Multiplication
SAME SIGNS → POSITIVE (+)
• Positive × Positive = Positive
• Negative × Negative = Positive
DIFFERENT SIGNS → NEGATIVE (-)
• Positive × Negative = Negative
• Negative × Positive = Negative
(+) × (+) = +
(-) × (-) = +
(+) × (-) = -
(-) × (+) = -
Examples
Decimals:
3.5 × 2 = 7
(-3.5) × 2 = -7
(-3.5) × (-2) = 7
Fractions:
2/3 × 3/4 = 6/12 = 1/2
(-2/3) × (3/4) = -6/12 = -1/2
(-2/3) × (-3/4) = 6/12 = 1/2
4. Dividing Positive and Negative Numbers
Sign Rules for Division
SAME SIGNS → POSITIVE (+)
• Positive ÷ Positive = Positive
• Negative ÷ Negative = Positive
DIFFERENT SIGNS → NEGATIVE (-)
• Positive ÷ Negative = Negative
• Negative ÷ Positive = Negative
(+) ÷ (+) = +
(-) ÷ (-) = +
(+) ÷ (-) = -
(-) ÷ (+) = -
Examples
Decimals:
12 ÷ 4 = 3
(-12) ÷ 4 = -3
12 ÷ (-4) = -3
(-12) ÷ (-4) = 3
Fractions:
2/3 ÷ 1/2 = 2/3 × 2/1 = 4/3
(-2/3) ÷ 1/2 = (-2/3) × 2/1 = -4/3
(-2/3) ÷ (-1/2) = (-2/3) × (-2/1) = 4/3
5. Complete Sign Rules Summary
| Operation | Same Signs | Different Signs |
|---|---|---|
| Addition | Add, keep sign | Subtract, use larger sign |
| Subtraction | Change to addition (Keep-Change-Change) | |
| Multiplication | Result is POSITIVE | Result is NEGATIVE |
| Division | Result is POSITIVE | Result is NEGATIVE |
6. Multi-Step Word Problems
Steps to Solve
Step 1: READ the problem carefully
Step 2: IDENTIFY what operation(s) to use
Step 3: SET UP the expression/equation
Step 4: SOLVE using PEMDAS
Step 5: CHECK if answer makes sense
Example Problem
Problem: The temperature was 12°C. It dropped 5°C in the morning, then rose 3°C in the afternoon. What is the final temperature?
Step 1: Starting temp: 12°C
Step 2: Dropped 5°C: 12 + (-5) = 7°C
Step 3: Rose 3°C: 7 + 3 = 10°C
Or: 12 + (-5) + 3 = 10°C
Answer: 10°C
Quick Reference: Sign Rules
| Signs | Add/Subtract | Multiply/Divide |
|---|---|---|
| (+) and (+) | Add → Positive | Positive result |
| (-) and (-) | Add → Negative | Positive result |
| (+) and (-) | Subtract → Larger sign | Negative result |
💡 Important Tips to Remember
✓ Adding same signs: Add values, keep the sign
✓ Adding different signs: Subtract values, use sign of larger absolute value
✓ Subtracting: Change to addition (Keep-Change-Change)
✓ Multiplying/Dividing same signs: Result is POSITIVE
✓ Multiplying/Dividing different signs: Result is NEGATIVE
✓ Two negatives make a positive (only for × and ÷)
✓ Use PEMDAS for order of operations
✓ Always simplify final answers
✓ Check your work: Does the sign make sense?
✓ Practice makes perfect - work through many examples!
🧠 Memory Tricks & Strategies
Adding Same Signs:
"Same signs? Just add and keep the sign you had!"
Adding Different Signs:
"Different signs? Subtract and take the bigger one's sign!"
Subtracting:
"Subtracting a negative? Add it instead - it's positive ahead!"
"Keep-Change-Change: Keep first, change sign, change operation!"
Multiplying/Dividing:
"Same signs say YES to positive! Different signs? Negative is festive!"
"Two negatives make a positive - that's the rule, so be attentive!"
Two Signs Together:
"Same signs together? Plus! Different signs? It's minus for us!"
Overall Rule:
"Friends are the same, enemies are not - same signs positive, different are not!"
Master Operations with Rational Numbers! ➕ ➖ ✖️ ➗
Remember: Signs matter! Same signs = positive, different signs = negative (for × and ÷)