Understanding integers
Key Notes:
Understanding Integers through Real-Life Scenarios
| What Are Integers? |
- Definition:
Integers are whole numbers that can be positive, negative, or zero.
Example:... -3, -2, -1, 0, +1, +2, +3 ...
| Real-Life Scenarios Using Integers |
| Scenario | Integer Used | Explanation |
|---|---|---|
| Temperature | -5°C | 5 degrees below zero |
| Bank account balance | +500 or -200 | Credit (+) or debt (-) |
| Elevation | -50 m | 50 meters below sea level |
| Sports score (gain/loss) | +3, -2 | Gained 3 points, lost 2 points |
| Game levels | Level -3 to +5 | Below or above a starting level |
| Scenario 1: Financial Transactions |
Situation: Sarah has a bank account. She deposits 200 /- and then withdraws 50 /-.
Explanation:
- Deposit: +200/- (positive integer represents a gain)
- Withdrawal: -50/- (negative integer represents a loss)
- Net Amount: 200+(−50)=150
Result: Sarah has 150/- in her bank account.
| Scenario 2: Temperature Changes |
Situation: The temperature in a city changes from 5°C to -3°C overnight.
Explanation:
- Initial Temperature: +5°C
- Final Temperature: -3°C
- Change in Temperature: −3−5=−8
Result: The temperature dropped by 8°C.
| Scenario 3: Elevation Above and Below Sea Level |
Situation: A submarine dives 200 meters below sea level, then rises 150 meters.
Explanation:
- Initial Position: 0 meters (sea level)
- Dive: -200 meters
- Rise: +150 meters
- Final Position: −200+150 = −50
Result: The submarine is 50 meters below sea level.
| Scenario 4: Business Profits and Losses |
Situation: A business makes a profit of $1,000 in January but incurs a loss of $600 in February.
Explanation:
- January Profit: +1,000/-
- February Loss: -600 /-
- Net Profit/Loss: 1000+(−600)=400
Result: The business has a net profit of 400/- over the two months.
| Scenario 5: Points in a Game |
Situation: In a game, a player scores +10 points in the first round and loses 15 points in the second round.
Explanation:
- First Round: +10 points
- Second Round: -15 points
- Total Points: 10+(−15)=−5
Result: The player has a total of -5 points after two rounds.
| Scenario 6: Balancing a Checkbook |
Situation: John writes a check for $75 and later deposits $100 into his account.
Explanation:
- Initial Balance: $0
- Check Written: -$75
- Deposit: +$100
- Final Balance: −75+100=25
Result: John’s account balance is $25.
| Scenario 7: Stock Market |
Situation: A stock’s price increases by $5 on Monday and decreases by $8 on Tuesday.
Explanation:
- Monday Increase: +$5
- Tuesday Decrease: -$8
- Net Change: 5+(−8)=−3
Result: The stock’s price has a net decrease of $3.
| Scenario 8: Tracking Weight Loss and Gain |
Situation: Alex gains 3 kg over the holidays and loses 5 kg after exercising.
Explanation:
- Weight Gain: +3 kg
- Weight Loss: -5 kg
- Net Weight Change: 3+(−5)=−2
Result: Alex has a net weight loss of 2 kg.
| Scenario 9: Balancing an Equation |
Situation: To balance the equation x+5=2x + 5 = 2x+5=2, solve for xxx.
Explanation:
- Subtract 5 from both sides: x+5−5=2−5
- Simplify: x=−3
Result: xxx equals -3, indicating a value that makes the equation true.
| Scenario 10: Travel Distance |
Situation: A car travels 100 km north (positive direction) and then 150 km south (negative direction).
Explanation:
- Northward Travel: +100 km
- Southward Travel: -150 km
- Net Distance: 100+(−150)=−50
Result: The car is 50 km south of its starting point.
| ➕➖ Rules of Integer Operations with Real-Life Examples |
| A. Addition of Integers |
| Rule | Example | Real-Life Connection |
|---|---|---|
| Same signs → Add and keep the sign | -4 + (-3) = -7 | Two debts combined → More debt |
| Different signs → Subtract, take sign of larger | -6 + 4 = -2 | Owe 6 rupees, earn 4 rupees → Still owe 2 rupees |
| B. Subtraction of Integers |
| Rule | Example | Real-Life Connection |
|---|---|---|
| Subtracting an integer = Adding its opposite | 5 – (-2) = 5 + 2 = 7 | Removing a debt feels like gaining money |
| ✖️➗ Multiplication & Division of Integers |
| Rule | Example | Meaning |
|---|---|---|
| (+) × (+) or (–) × (–) = Positive | -2 × -3 = +6 | Two losses cancel each other |
| (+) × (–) or (–) × (+) = Negative | 4 × -2 = -8 | Positive times negative = loss |
🔁 Same rules apply to division
| 📊 Integers on the Number Line |
- Zero is the center.
- Positive integers go right →
- Negative integers go left ←
← -5 -4 -3 -2 -1 0 +1 +2 +3 +4 +5 →
Key Takeaways:
- Positive integers represent gains, increases, or movements in a positive direction.
- Negative integers represent losses, decreases, or movements in a negative direction.
- Understanding how to add, subtract, and interpret positive and negative integers helps in various real-life scenarios, from financial planning to tracking physical changes.
| Practice Scenarios (Word Problems) |
Temperature Change:
If it was 6°C in the morning and -2°C in the evening, what is the change in temperature?
➤ -2 – 6 = -8°C → Drop of 8°C
Bank Account:
Riya had ₹200. She spent ₹450. What is her balance now?
➤ 200 - 450 = -250 → She is ₹250 in debt
Elevator Problem:
An elevator goes from the 5th floor to the basement 3rd floor.
What is the change in levels?
➤ -3 – 5 = -8 → 8 floors down
Learn with an example
➡️ Which integer represents this scenario?
5 more centimetres of rain this year
-5 5
- Read the scenario:
- 5 more centimetres of rain this year
This scenario involves a gain or an increase. Use a positive integer to represent this scenario. - The integer 5 represents this scenario.
➡️ Which integer represents this scenario?
adding 6 stuffed animals to a collection
-6 6
- Read the scenario:
- adding 6 stuffed animals to a collection
This scenario involves a gain or an increase. Use a positive integer to represent this scenario. - The integer 6 represents this scenario.
➡️ Which integer represents this scenario?
9 degrees below zero
-9 9
- Read the scenario:
- 9 degrees below zero
This scenario involves a loss or a decrease. Use a negative integer to represent this scenario. - The integer -9 represents this scenario.
Lets practice !
