🌡️ Directed Numbers

Cambridge Lower Secondary · Grade 7 · Number & Calculation

Adding a Negative
5 + (−3) = 5 − 3 = 2
Adding a negative = subtracting
Subtracting a Negative
3 − (−2) = 3 + 2 = 5
Subtracting a negative = adding
Real-Life Context
Temperature: −8°C + 11°C = 3°C
Bank: −£40 − (−£15) = −£25

Watch the temperature change! 🌡️

−15°C
Start
→
+23°C change
−15 + 23
 
→
?°C
End

What you'll learn:

  • Placing negative numbers on a number line
  • Adding positive and negative integers
  • Subtracting positive and negative integers (e.g. 3 − (−2) = 5)
  • The two key rules: + (−) = − and − (−) = +
  • Real-life contexts: temperature, sea level, bank balance
  • Ordering and comparing negative numbers
Created by Mr Josh

📖 Learn: Directed Numbers

Part 1: The Number Line

Negative numbers are to the left of zero on the number line. The further left, the smaller the number.

−6 −5 −4 −3 −2 −1 0 1 2 3 4 5 6 ← Negative numbers Positive numbers →

Key facts:   −3 < −1  ·  −5 < 0  ·  2 > −10

Part 2: Adding Negative Numbers

When you add a negative number, you move left on the number line (just like subtracting).

📌 Rule: a + (−b) = a − b
Example:   5 + (−3) = 5 − 3 = 2   (start at 5, move 3 left)
Example:   2 + (−7) = 2 − 7 = −5   (start at 2, move 7 left, cross zero)
Example:   −4 + (−2) = −4 − 2 = −6   (start at −4, move 2 further left)
💡 Think of it as a temperature drop: −4°C dropping another 2°C → −6°C

Part 3: Subtracting Negative Numbers

When you subtract a negative number, you move right (it becomes addition!).

📌 Rule: a − (−b) = a + b
Example:   3 − (−2) = 3 + 2 = 5   (start at 3, move 2 right)
Example:   −1 − (−5) = −1 + 5 = 4   (start at −1, move 5 right)
Example:   −6 − (−6) = −6 + 6 = 0
💡 Think of debt: if you owe £6 and someone removes that debt, you gain £6!

Part 4: The Two Golden Rules

You See It Becomes Example
+ (−) − 8 + (−3) = 8 − 3 = 5
− (−) + 4 − (−5) = 4 + 5 = 9

Memory trick: "Two negatives make a positive" only when they are next to each other (same sign → +, different signs → −)

Part 5: Ordering Negative Numbers

To order negative numbers, think of the number line. The most negative is the smallest.

Order from smallest to largest: 3, −5, −1, 0, −8, 2
Answer: −8, −5, −1, 0, 2, 3
💡 Temperature analogy: −8°C is colder than −1°C, so −8 < −1
Created by Mr Josh

💡 Worked Examples

Example 1: Adding with a Negative Number

Calculate −3 + (−4)

Step 1: Recognise the rule: + (−) becomes −
Step 2: Rewrite: −3 + (−4) = −3 − 4
Step 3: Both negative: move 4 further left from −3
Answer: −7
−10 −9 −8 −7 −6 −5 −4 −3 −2 −1 0 1 2 start: −3 end: −7

Example 2: Subtracting a Negative Number

Calculate 3 − (−2)

Step 1: Apply the rule: − (−) becomes +
Step 2: Rewrite: 3 − (−2) = 3 + 2
Answer: 5   (start at 3, jump 2 to the right)
−2 −1 0 1 2 3 4 5 6 7 8 start: 3 end: 5
💡 Subtracting a negative ADDS — the submarine rises instead of sinking!

Example 3: Mixed Operations — Finding the Difference

The temperature in Moscow is −8°C. The temperature in Cairo is 15°C. What is the difference?

Step 1: Difference = Higher − Lower = 15 − (−8)
Step 2: Apply rule: 15 − (−8) = 15 + 8
Answer: 23°C difference
✈️ Think: to travel from −8°C to 15°C you go up 23 degrees.

Example 4: Multi-step with Negatives

Calculate −5 + (−3) − (−10)

Step 1: Replace signs: −5 − 3 + 10
Step 2: Work left to right: −5 − 3 = −8
Step 3: −8 + 10 = 2
Answer: 2
📋 Replace all + (−) with − and all − (−) with + first, then calculate.
Created by Mr Josh

🌡️ Number Line Jumper

Type any calculation involving negative numbers, then watch the jumper move on the number line!

Examples: 5 + (-3)  ·  -4 - (-2)  ·  -8 + 5  ·  3 - (-7)



Try these:

Created by Mr Josh

✏️ Exercise 1: Adding with Negative Numbers

Calculate each sum. Remember: a + (−b) = a − b

Created by Mr Josh

✏️ Exercise 2: Subtracting Negative Numbers

Calculate each difference. Remember: a − (−b) = a + b

Created by Mr Josh

✏️ Exercise 3: Mixed Operations

Simplify and calculate. Replace signs first, then evaluate.

Created by Mr Josh

🌍 Exercise 4: Temperature & Real-Life Problems

Use directed numbers in context. Type your answer (just the number).

Created by Mr Josh

📏 Exercise 5: Ordering Negative Numbers

Click the numbers in the pool to move them into the answer slots in order from smallest to largest. Click a placed chip to remove it.

Created by Mr Josh

📝 Practice Questions

  1. Calculate: 7 + (−3)
  2. Calculate: 2 + (−9)
  3. Calculate: −5 + (−4)
  4. Calculate: −8 + (−1)
  5. Calculate: 0 + (−6)
  6. Calculate: 10 − (−2)
  7. Calculate: 3 − (−8)
  8. Calculate: −4 − (−6)
  9. Calculate: −7 − (−7)
  10. Calculate: 0 − (−5)
  11. Calculate: −3 + (−2) − (−8)
  12. Calculate: 5 − (−3) + (−10)
  13. Calculate: −6 − (−4) + (−1)
  14. The temperature at midnight is −12°C. By noon it has risen by 18°C. What is the noon temperature?
  15. A submarine is at −45 m. It rises 28 m. What is its new depth?
  16. A bank account has a balance of −£80. A deposit of £55 is made. What is the new balance?
  17. The temperature in Oslo is −7°C and in Rome is 14°C. What is the difference in temperature?
  18. Write in ascending order: −3, 1, −7, 0, −1, 5
  19. Which is smaller: −4 or −9? Explain.
  20. A diver is at −18 m. A second diver is at −5 m. What is the difference in their depths?
  1. 7 + (−3) = 7 − 3 = 4
  2. 2 + (−9) = 2 − 9 = −7
  3. −5 + (−4) = −5 − 4 = −9
  4. −8 + (−1) = −8 − 1 = −9
  5. 0 + (−6) = 0 − 6 = −6
  6. 10 − (−2) = 10 + 2 = 12
  7. 3 − (−8) = 3 + 8 = 11
  8. −4 − (−6) = −4 + 6 = 2
  9. −7 − (−7) = −7 + 7 = 0
  10. 0 − (−5) = 0 + 5 = 5
  11. −3 − 2 + 8 = 3
  12. 5 + 3 − 10 = −2
  13. −6 + 4 − 1 = −3
  14. −12 + 18 = 6°C
  15. −45 + 28 = −17 m
  16. −80 + 55 = −£25
  17. 14 − (−7) = 14 + 7 = 21°C
  18. −7, −3, −1, 0, 1, 5
  19. −9 is smaller (further left on number line)
  20. −5 − (−18) = −5 + 18 = 13 m
Created by Mr Josh

🏆 Challenge: Multi-Step & Real-World Problems

  1. The temperature in Siberia on Monday is −23°C. On Tuesday it drops 8°C, and on Wednesday it rises 15°C. What is Wednesday's temperature?
  2. City A is at 250 m above sea level. City B is at −80 m (below sea level). What is the total difference in height between them?
  3. A company's profit over three months: January −£4 200, February −£1 800, March +£8 600. What is the overall profit/loss for the three months?
  4. A submarine starts at −120 m. It descends another 45 m, then rises 90 m. What is its final depth?
  5. The coldest temperature ever recorded in Antarctica is −89°C. The hottest recorded is 19°C. What is the range (difference) of these temperatures?
  6. Tom's bank balance is −£150. He receives his salary of £420 and pays a bill of £85. Then he withdraws £60 cash. What is his final balance?
  7. Four cities have these temperatures: Reykjavik −5°C, Moscow −12°C, London 3°C, and Dubai 28°C. (a) List them in ascending order. (b) Find the difference between the warmest and coldest.
  8. The Dead Sea shore is at −430 m. Mount Everest is at 8 849 m. A bird flies from the Dead Sea shore to the summit. How many metres does it travel vertically?
  1. Monday: −23°C. Tuesday: −23 − 8 = −31°C. Wednesday: −31 + 15 = −16°C
  2. 250 − (−80) = 250 + 80 = 330 m
  3. −4 200 + (−1 800) + 8 600 = −6 000 + 8 600 = +£2 600 profit
  4. −120 − 45 = −165; then −165 + 90 = −75 m
  5. 19 − (−89) = 19 + 89 = 108°C
  6. Start: −150. After salary: −150 + 420 = 270. After bill: 270 − 85 = 185. After withdrawal: 185 − 60 = £125
  7. (a) Moscow −12°C, Reykjavik −5°C, London 3°C, Dubai 28°C. (b) 28 − (−12) = 40°C
  8. 8 849 − (−430) = 8 849 + 430 = 9 279 m
Created by Mr Josh