Directed Numbers

⚡ Do Now!

Try these before reading the lesson.

The temperature changes from 4°C to −2°C. Did it rise or fall?
A lift moves from floor 3 to basement level −2. How many floors does it travel?
Which number is greater: −3 or −8?
Calculate:
5 + (−7)

✅ Check the Do Now answers

It fell.
5 floors.
−3 is greater.
−2

🎯 By the end of this lesson...

You should be able to:

Explain what directed numbers are.
Identify positive and negative numbers.
Place integers correctly on a number line.
Compare and order directed numbers.
Add positive and negative numbers.
Subtract positive and negative numbers.
Multiply and divide directed numbers.
Use directed numbers in practical situations.
Check whether an answer is reasonable.

🤔 Think About This...

🌡️

A cold morning

At 6:00 a.m., the temperature is 3°C.

During the next few hours, it falls by 7°C.

What is the new temperature?

3 − 7 = −4

The new temperature is −4°C.

The negative sign tells us that the temperature is below zero. Directed numbers help us show both an amount and its direction.

1️⃣ What Are Directed Numbers?

Directed numbers are numbers that have a positive or negative sign to show direction from zero.

Positive direction

Positive numbers are greater than zero.

+1, +2, +3, +4, …

They usually represent movement upwards, forwards, gains, deposits or values above zero.

Negative direction

Negative numbers are less than zero.

−1, −2, −3, −4, …

They usually represent movement downwards, backwards, losses, debts or values below zero.

Zero is neither positive nor negative. It is the point separating the positive and negative directions.

🌍 Where Do We Use Directed Numbers?

🌡️

Temperature

−5°C means five degrees below zero.

🏦

Money

−$20 can represent a debt or an overdrawn balance.

🏢

Building floors

Floor −2 can represent a basement below ground level.

⛰️

Elevation

−30 m means thirty metres below sea level.

⚽

Goal difference

−4 means a team has conceded four more goals than it scored.

📈

Change

+8 shows an increase, while −8 shows a decrease.

2️⃣ The Number Line

A number line helps us see the position and direction of numbers.

◀

−5

−4

−3

−2

−1

0

1

2

3

4

5

▶

Numbers decrease Numbers increase

Moving to the right means the numbers are becoming greater.

Moving to the left means the numbers are becoming smaller.

3️⃣ Comparing Directed Numbers

The number farther to the right on a number line is always greater.

Example 1: Compare −3 and −7

−3 is closer to zero and lies farther to the right.

−3 > −7

−3 is greater than −7.

A negative number with a larger-looking digit is not necessarily greater.

For example, −10 is less than −2 because −10 lies farther left.

🧠 StudyNest Method: Comparing Negative Numbers

Step 1: Imagine or draw a number line.

Step 2: Locate both numbers.

Step 3: The number farther to the right is greater.

Step 4: For two negative numbers, the number closer to zero is greater.

−2 > −6

−8 < −3

4️⃣ Ordering Directed Numbers

Example 2: Arrange in ascending order

4, −2, 0, −7, 3

Ascending order means smallest to largest.

Imagine the numbers placed from left to right on a number line.

−7, −2, 0, 3, 4

Example 3: Arrange in descending order

−5, 6, −1, 2, 0

Descending order means largest to smallest.

6, 2, 0, −1, −5

5️⃣ Adding Directed Numbers

Addition can be shown as movement on a number line.

Add a positive number

Move to the right.

2 + 4 = 6

Add a negative number

Move to the left.

2 + (−4) = −2

Example 4: Calculate 3 + (−5)

Begin at 3.

Adding −5 means moving five spaces to the left.

Start: 3

← 5 spaces

Finish: −2

3 + (−5) = −2

🧠 Adding Numbers with the Same Sign

When both numbers have the same sign:

Add their numerical values.
Keep the common sign.

Example 5

(+4) + (+7) = +11

Example 6

(−4) + (−7) = −11

Same signs: add and keep the sign.

🧠 Adding Numbers with Different Signs

When the signs are different:

Subtract the smaller numerical value from the larger one.
Use the sign of the number with the larger numerical value.

Example 7: Calculate −9 + 4

9 − 4 = 5

The number with the larger numerical value is −9.

Therefore, the answer is negative.

−9 + 4 = −5

Example 8: Calculate −3 + 8

8 − 3 = 5

The number with the larger numerical value is +8.

−3 + 8 = 5

6️⃣ Subtracting Directed Numbers

Subtraction can be changed into addition by adding the opposite.

a − b = a + (−b)

💡 StudyNest Rule:

Keep the first number.

Change subtraction to addition.

Change the sign of the second number.

Example 9: Calculate 6 − 9

6 − 9

Change subtraction to addition and change +9 to −9.

6 + (−9)

6 − 9 = −3

Example 10: Calculate 4 − (−6)

Change subtraction to addition.

Change −6 to its opposite, +6.

4 + 6 = 10

4 − (−6) = 10

🔑 Why Does Subtracting a Negative Become Addition?

A negative represents movement in the opposite direction.

Subtracting means reversing that movement.

Reversing a negative direction produces a positive direction.

5 − (−3) = 5 + 3

Subtracting a negative number is the same as adding its positive opposite.

7️⃣ Multiplying Directed Numbers

For multiplication, calculate the numerical values first and then use the signs to determine the answer.

+ × +

+

+ × −

−

− × +

−

− × −

+

Same signs give a positive answer.

Different signs give a negative answer.

Example 11

(−4) × 6

The signs are different, so the answer is negative.

4 × 6 = 24

(−4) × 6 = −24

Example 12

(−7) × (−3)

The signs are the same, so the answer is positive.

(−7) × (−3) = 21

8️⃣ Dividing Directed Numbers

Division uses the same sign rules as multiplication.

+ ÷ +

+

+ ÷ −

−

− ÷ +

−

− ÷ −

+

Example 13

−36 ÷ 6

The signs are different.

−36 ÷ 6 = −6

Example 14

−48 ÷ (−8)

The signs are the same.

−48 ÷ (−8) = 6

🧠 StudyNest Thinking Method

Before calculating, identify the operation and signs.

Step 1: Look carefully at the operation sign.

Is the question asking you to add, subtract, multiply or divide?

Step 2: Identify whether each number is positive or negative.

Step 3: For addition:

Same signs → add and keep the sign.

Different signs → subtract and use the sign of the larger numerical value.

Step 4: For subtraction:

Change subtraction to addition and change the sign of the second number.

Step 5: For multiplication and division:

Same signs → positive.

Different signs → negative.

Step 6: Check whether the direction of your answer makes sense.

🌍 Real-Life Worked Examples

Example 15: Bank account

Tapiwa has $25 in his account.

A payment of $40 is deducted.

25 − 40 = −15

Tapiwa's balance is −$15. This means he owes the bank $15.

Example 16: Temperature change

The morning temperature is −3°C.

It rises by 8°C.

−3 + 8 = 5

The new temperature is 5°C.

Example 17: Lift movement

A lift begins at basement level −3 and moves up 7 floors.

−3 + 7 = 4

The lift stops at floor 4.

Example 18: Elevation

A diver is 18 metres below sea level.

The diver rises by 11 metres.

−18 + 11 = −7

The diver is still 7 metres below sea level.

⚠ Common Mistakes

Mistake 1:

Thinking −8 is greater than −3 because 8 is greater than 3.

On a number line, −8 lies farther left, so −8 < −3.

Mistake 2:

Treating subtraction and a negative sign as the same thing.

In 5 − (−2), the first sign is the operation and the second belongs to the number.

Mistake 3:

Forgetting to change the sign of the second number when changing subtraction into addition.

Mistake 4:

Using multiplication sign rules for addition.

The rule “two negatives make a positive” does not automatically apply to addition.

Mistake 5:

Saying:

−4 + (−3) = 7

The correct answer is −7 because both numbers are negative.

Mistake 6:

Giving a number without explaining its meaning in a real-life problem.

State whether it represents a debt, temperature, floor or direction.

🎮 Your Turn!

State which is greater: −4 or −9.
Arrange in ascending order:
3, −5, 0, −1, 7
Arrange in descending order:
−8, 4, −2, 1, 0
Calculate:
−7 + 12
Calculate:
−6 + (−9)
Calculate:
5 − 13
Calculate:
8 − (−4)
Calculate:
(−7) × 5
Calculate:
(−9) × (−4)
Calculate:
−56 ÷ 7
Calculate:
−72 ÷ (−8)
The temperature is −6°C and rises by 11°C. Find the new temperature.
A bank balance is $18. A payment of $30 is deducted. Find the new balance.
A lift begins on floor −2 and travels down 3 floors. Which floor does it reach?
A diver is 25 metres below sea level and rises 17 metres. State the diver's new position.

✅ Show Answers

−4
−5, −1, 0, 3, 7
4, 1, 0, −2, −8
5
−15
−8
12
−35
36
−8
9
5°C
−$12, meaning a debt of $12
Floor −5
−8 metres, or 8 metres below sea level

🧩 Sign Challenge

Without calculating the full answer, decide whether each result will be positive or negative.

(−8) × 7
(−6) × (−5)
48 ÷ (−6)
(−63) ÷ (−9)
(−4) + (−11)

✅ Show Challenge Answers

Negative
Positive
Negative
Positive
Negative

🧠 Remember This

Numbers increase as you move right on a number line.

Numbers decrease as you move left.

When adding the same signs, add and keep the sign.

When adding different signs, subtract and use the sign of the number with the larger numerical value.

When subtracting, add the opposite.

For multiplication and division:

same signs → positive

different signs → negative

🎯 Before Moving On...

✅ Can I explain what a directed number is?
✅ Can I place positive and negative integers on a number line?
✅ Can I compare two negative numbers correctly?
✅ Can I add directed numbers?
✅ Can I subtract directed numbers?
✅ Can I multiply and divide directed numbers?
✅ Can I apply directed numbers to practical situations?
✅ Can I explain what a negative answer means?

🎉 Well Done!

You can now compare, order and calculate with positive and negative numbers.

Remember:

Identify the operation → Check the signs → Apply the correct rule → Check the direction

📈 Your Progress

Lesson 4 of 17

Real Numbers Progress 24%

🎬 Move along the number line

⚡ Do Now!

🎯 By the end of this lesson...

🤔 Think About This...

A cold morning

1️⃣ What Are Directed Numbers?

Positive direction

Negative direction

🌍 Where Do We Use Directed Numbers?

Temperature

Money

Building floors

Elevation

Goal difference

Change

2️⃣ The Number Line

3️⃣ Comparing Directed Numbers

Example 1: Compare −3 and −7

🧠 StudyNest Method: Comparing Negative Numbers

4️⃣ Ordering Directed Numbers

Example 2: Arrange in ascending order

Example 3: Arrange in descending order

5️⃣ Adding Directed Numbers

Add a positive number

Add a negative number

Example 4: Calculate 3 + (−5)

🧠 Adding Numbers with the Same Sign

Example 5

Example 6

🧠 Adding Numbers with Different Signs

Example 7: Calculate −9 + 4

Example 8: Calculate −3 + 8

6️⃣ Subtracting Directed Numbers

Example 9: Calculate 6 − 9

Example 10: Calculate 4 − (−6)

🔑 Why Does Subtracting a Negative Become Addition?

7️⃣ Multiplying Directed Numbers

Example 11

Example 12

8️⃣ Dividing Directed Numbers

Example 13

Example 14

🧠 StudyNest Thinking Method

🌍 Real-Life Worked Examples

Example 15: Bank account

Example 16: Temperature change

Example 17: Lift movement

Example 18: Elevation

⚠ Common Mistakes

🎮 Your Turn!

🧩 Sign Challenge

🧠 Remember This

🎯 Before Moving On...

🎉 Well Done!

📈 Your Progress