Class 6 : Maths ( English ) – Lesson 10. The Other Side of Zero

EXPLANATION AND ANALYSIS

🌿 Explanation & Analysis

🔵 1. Revisiting zero and numbers we know
Before this lesson, we mainly worked with whole numbers like 0, 1, 2, 3, … Zero plays a special role—it shows nothing, no quantity, or a starting point.
But numbers do not stop at zero. There exists another side of zero that helps us describe situations where quantities are less than zero.

🧠 This lesson introduces numbers that lie on the other side of zero.

✏️ Note: Zero separates two kinds of numbers on the number line.

💡 Concept: Numbers exist on both sides of zero.

🔵 2. Need for numbers less than zero
In real life, there are many situations where we need numbers smaller than zero.

Examples:
🔵 Temperature below 0°C ❄️
🔵 A lift going below ground floor
🔵 Loss in money or debt
🔵 Depth below sea level 🌊

🧠 These situations cannot be represented using only whole numbers.

✔️ This need leads to the idea of negative numbers.

🔵 3. Introducing negative numbers
Numbers less than zero are called negative numbers.
They are written with a minus sign (–) in front of them.

Examples:
🔵 –1, –2, –5, –10

🧠 The minus sign tells us that the number lies on the left side of zero.

✏️ Note: The minus sign in a negative number is not subtraction; it shows direction or position.

🔵 4. Positive numbers
Numbers greater than zero are called positive numbers.
Usually, we do not write a plus sign for them.

Examples:
🔵 1, 2, 5, 10

🧠 To distinguish clearly:
🔵 +5 means positive five
🔵 –5 means negative five

💡 Concept: Zero is neither positive nor negative.

🔵 5. The number line and zero
A number line is a straight line on which numbers are placed at equal distances.

Key ideas:
🔵 Zero is at the centre
🔵 Positive numbers are on the right of zero
🔵 Negative numbers are on the left of zero

🧠 The number line helps us see numbers clearly.

✏️ Note: Distance between consecutive numbers on a number line is always equal.

🔵 6. Position of negative numbers on the number line
Negative numbers lie on the left side of zero.

Example:
🔵 –1 is just left of 0
🔵 –2 is further left than –1
🔵 –5 is much further left

🧠 The farther left a number is, the smaller it is.

💡 Concept: Among negative numbers, the number closer to zero is greater.

🔵 7. Comparing integers
Numbers including positive numbers, negative numbers, and zero together are called integers.

Rules for comparison:
🔵 Any positive number > 0
🔵 0 > any negative number
🔵 Among negative numbers, the one with smaller absolute value is greater

Examples:
🔵 –2 > –5
🔵 3 > –3

🧠 Comparison becomes easy using a number line.

🔵 8. Integers in daily life
Integers are used widely in daily situations.

Examples:
🔵 Temperature: +10°C, –5°C
🔵 Money: profit (+), loss (–)
🔵 Elevation: above sea level (+), below sea level (–)
🔵 Floors in a building

🧠 Integers help describe direction, position, and change.

🔵 9. Understanding direction using integers
Integers are useful in showing direction.

Examples:
🔵 Moving right → positive direction
🔵 Moving left → negative direction
🔵 Moving up → positive
🔵 Moving down → negative

💡 Concept: Integers combine number and direction.

🔵 10. Absolute value idea (informal)
The absolute value of a number tells how far it is from zero, without considering direction.

Examples:
🔵 Distance of +4 from zero = 4 units
🔵 Distance of –4 from zero = 4 units

🧠 Both are equally far from zero but on opposite sides.

✏️ Note: Absolute value is about distance, not sign.

🔵 11. Zero as a reference point
Zero acts as a reference point for comparing and measuring integers.

🟢 It separates positive and negative numbers
🟡 It helps measure distance on both sides

✔️ Without zero, understanding negative numbers would be difficult.

🔵 12. Importance of integers
Integers form the foundation for many future topics:

🔵 Algebra
🔵 Coordinate geometry
🔵 Graphs
🔵 Real-life problem solving

🧠 Understanding the other side of zero prepares students for higher mathematics.

💡 Concept: Integers extend our number system beyond zero.

🔵 13. Learning integers through activities
Activities that help learning include:
🔵 Drawing number lines
🔵 Temperature charts
🔵 Elevator floor models
🔵 Gain–loss games

✔️ Activities make the idea of negative numbers easy and clear.

🔵 14. Common mistakes to avoid
Students should be careful about:
🔴 Thinking –8 is greater than –3
🔴 Mixing subtraction sign with negative sign
🔴 Forgetting zero is neutral

🧠 Clear understanding avoids confusion.

🔵 15. The idea behind “the other side of zero”
This lesson shows that numbers are not only about counting objects.
They also describe position, direction, and change.

✔️ The other side of zero opens a new way of thinking about numbers.

🧠 Mathematics becomes richer and more meaningful with integers.

Summary

The other side of zero introduces numbers less than zero, called negative numbers. These numbers are needed to represent real-life situations such as temperatures below zero, losses, depths below sea level, and floors below ground level. Numbers greater than zero are called positive numbers, while zero is neither positive nor negative.

Using a number line helps us understand the position and order of positive numbers, negative numbers, and zero. Negative numbers lie on the left of zero, and positive numbers lie on the right. Together, positive numbers, negative numbers, and zero are called integers. Integers help describe direction, position, and change in everyday life.

Zero acts as a reference point and separates positive and negative numbers. Understanding integers forms a strong base for advanced mathematical topics and real-life problem solving.

📝 Quick Recap

🔵 Numbers exist on both sides of zero
🟢 Numbers less than zero are negative numbers
🟡 Zero is neither positive nor negative
🔴 Integers include negative numbers, zero, and positive numbers
✔️ Integers help describe direction, position, and real-life situations

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TEXTBOOK QUESTIONS

🌿 BELA’S BUILDING OF FUN

🔒 ❓ Question 1
You start from Floor +2 and press −3 in the lift. Where will you reach? Write an expression for this movement.

📌 ✅ Answer:
🔹 Starting floor = +2
🔹 Movement = −3 (downward movement)
🔹 Final floor = +2 − 3 = −1

🔹 Expression:
(+2) + (−3) = −1

🔹 You will reach Floor −1

🔒 ❓ Question 2
Evaluate these expressions (think of them as Starting Floor + Movement).

🔒 ❓ (a) (+1) + (+4) = ______

📌 ✅ Answer:
🔹 Start at +1
🔹 Move up +4
🔹 Final floor = +5

🔹 Result: +5

🔒 ❓ (b) (+4) + (+1) = ______

📌 ✅ Answer:
🔹 Start at +4
🔹 Move up +1
🔹 Final floor = +5

🔹 Result: +5

🔒 ❓ (c) (+4) + (−3) = ______

📌 ✅ Answer:
🔹 Start at +4
🔹 Move down 3 floors
🔹 Final floor = +1

🔹 Result: +1

🔒 ❓ (d) (−1) + (+2) = ______

📌 ✅ Answer:
🔹 Start at −1
🔹 Move up 2 floors
🔹 Final floor = +1

🔹 Result: +1

🔒 ❓ (e) (−1) + (+1) = ______

📌 ✅ Answer:
🔹 Start at −1
🔹 Move up 1 floor
🔹 Final floor = 0

🔹 Result: 0

🔒 ❓ (f) 0 + (+2) = ______

📌 ✅ Answer:
🔹 Start at 0
🔹 Move up 2 floors
🔹 Final floor = +2

🔹 Result: +2

🔒 ❓ (g) 0 + (−2) = ______

📌 ✅ Answer:
🔹 Start at 0
🔹 Move down 2 floors
🔹 Final floor = −2

🔹 Result: −2

🔒 ❓ Question 3
Starting from different floors, find the movements required to reach Floor −5. Write the expressions.

📌 ✅ Answer:

🔹 Example 1
🔸 Start at +3
🔸 Movement needed = −8

🔹 Expression:
(+3) + (−8) = −5

🔹 Example 2
🔸 Start at 0
🔸 Movement needed = −5

🔹 Expression:
0 + (−5) = −5

🔹 Example 3
🔸 Start at −2
🔸 Movement needed = −3

🔹 Expression:
(−2) + (−3) = −5

🔹 Example 4
🔸 Start at +1
🔸 Movement needed = −6

🔹 Expression:
(+1) + (−6) = −5

🔵 Figure it Out

Evaluate these expressions by thinking of them as the resulting movement of combining button presses.

🔒 ❓ Question (a)
(+1) + (+4) = ______

📌 ✅ Answer:
🔹 Start at floor +1
🔹 Press +4 ➡️ move up 4 floors
🔹 Reach floor +5

✔️ Final Answer: +5

🔒 ❓ Question (b)
(+4) + (+1) = ______

📌 ✅ Answer:
🔹 Start at floor +4
🔹 Press +1 ➡️ move up 1 floor
🔹 Reach floor +5

✔️ Final Answer: +5

🔒 ❓ Question (c)
(+4) + (−3) + (−2) = ______

📌 ✅ Answer:
🔹 Start at floor +4
🔹 Press −3 ➡️ move down 3 floors → reach +1
🔹 Press −2 ➡️ move down 2 floors → reach −1

✔️ Final Answer: −1

🔒 ❓ Question (d)
(−1) + (+2) + (−3) = ______

📌 ✅ Answer:
🔹 Start at floor −1
🔹 Press +2 ➡️ move up 2 floors → reach +1
🔹 Press −3 ➡️ move down 3 floors → reach −2

✔️ Final Answer: −2

🌿 COMPARING NUMBERS USING FLOORS

🔵 Figure it Out

Compare the following numbers using the idea of the Building of Fun and fill in the boxes with < or >.

🔒 ❓ Question 1 (a)
−2 ⬜ +5

📌 ✅ Answer:
🔹 −2 is below Floor 0
🔹 +5 is above Floor 0

✔️ Final Answer: −2 < +5

🔒 ❓ Question 1 (b)
−5 ⬜ +4

📌 ✅ Answer:
🔹 −5 is below Floor 0
🔹 +4 is above Floor 0

✔️ Final Answer: −5 < +4

🔒 ❓ Question 1 (c)
−5 ⬜ −3

📌 ✅ Answer:
🔹 −3 is higher than −5 in the building
🔹 Number closer to 0 is greater

✔️ Final Answer: −5 < −3

🔒 ❓ Question 1 (d)
+6 ⬜ −6

📌 ✅ Answer:
🔹 +6 is above Floor 0
🔹 −6 is below Floor 0

✔️ Final Answer: +6 > −6

🔒 ❓ Question 1 (e)
0 ⬜ −4

📌 ✅ Answer:
🔹 0 is above all negative floors

✔️ Final Answer: 0 > −4

🔒 ❓ Question 1 (f)
0 ⬜ +4

📌 ✅ Answer:
🔹 +4 is above Floor 0

✔️ Final Answer: 0 < +4

🔵 Question 2

Imagine the Building of Fun with more floors. Compare the numbers and fill in < or >.

🔒 ❓ Question 2 (a)
−10 ⬜ −12

📌 ✅ Answer:
🔹 −10 is closer to 0 than −12

✔️ Final Answer: −10 > −12

🔒 ❓ Question 2 (b)
+17 ⬜ −10

📌 ✅ Answer:
🔹 +17 is positive
🔹 −10 is negative

✔️ Final Answer: +17 > −10

🔒 ❓ Question 2 (c)
0 ⬜ −20

📌 ✅ Answer:
🔹 0 is greater than all negative numbers

✔️ Final Answer: 0 > −20

🔒 ❓ Question 2 (d)
+9 ⬜ −9

📌 ✅ Answer:
🔹 +9 is above Floor 0
🔹 −9 is below Floor 0

✔️ Final Answer: +9 > −9

🔒 ❓ Question 2 (e)
−25 ⬜ −7

📌 ✅ Answer:
🔹 −7 is closer to 0 than −25

✔️ Final Answer: −25 < −7

🔒 ❓ Question 2 (f)
+15 ⬜ −17

📌 ✅ Answer:
🔹 Positive numbers are always greater than negative numbers

✔️ Final Answer: +15 > −17

🔵 Question 3

If Floor A = −12, Floor D = −1 and Floor E = +1, find the numbers of Floors B, C, F, G and H.

📌 ✅ Answer:
🔹 Floors increase by 1 as we go up

🔹 Floor B = −10
🔹 Floor C = −5
🔹 Floor F = +3
🔹 Floor G = +5
🔹 Floor H = +7

🔵 Question 4

Mark the following floors of the building.

🔒 ❓ Question 4 (a)
−7

📌 ✅ Answer:
🔹 Floor −7 is below Floor 0, between −6 and −8

🔒 ❓ Question 4 (b)
−4

📌 ✅ Answer:
🔹 Floor −4 is below Floor 0, above −5

🔒 ❓ Question 4 (c)
+3

📌 ✅ Answer:
🔹 Floor +3 is above Floor 0

🔒 ❓ Question 4 (d)
−10

📌 ✅ Answer:
🔹 Floor −10 is far below Floor 0

🌿 SUBTRACTION TO FIND WHICH BUTTON TO PRESS

🔵 Figure it Out

Complete these expressions. Think of each as the movement needed to reach the Target Floor from the Starting Floor.

🔒 ❓ Question (a)
(+1) − (+4) = ______

📌 ✅ Answer:
🔹 Start at floor +1
🔹 Subtract +4 ➡️ move down 4 floors
🔹 Reach floor −3

✔️ Final Answer: −3

🔒 ❓ Question (b)
(0) − (+2) = ______

📌 ✅ Answer:
🔹 Start at floor 0
🔹 Subtract +2 ➡️ move down 2 floors
🔹 Reach floor −2

✔️ Final Answer: −2

🔒 ❓ Question (c)
(+4) − (+1) = ______

📌 ✅ Answer:
🔹 Start at floor +4
🔹 Subtract +1 ➡️ move down 1 floor
🔹 Reach floor +3

✔️ Final Answer: +3

🔒 ❓ Question (d)
(0) − (−2) = ______

📌 ✅ Answer:
🔹 Start at floor 0
🔹 Subtract −2 ➡️ means move up 2 floors
🔹 Reach floor +2

✔️ Final Answer: +2

🔒 ❓ Question (e)
(+4) − (−3) = ______

📌 ✅ Answer:
🔹 Start at floor +4
🔹 Subtract −3 ➡️ means move up 3 floors
🔹 Reach floor +7

✔️ Final Answer: +7

🔒 ❓ Question (f)
(−4) − (−3) = ______

📌 ✅ Answer:
🔹 Start at floor −4
🔹 Subtract −3 ➡️ means move up 3 floors
🔹 Reach floor −1

✔️ Final Answer: −1

🔒 ❓ Question (g)
(−1) − (+2) = ______

📌 ✅ Answer:
🔹 Start at floor −1
🔹 Subtract +2 ➡️ move down 2 floors
🔹 Reach floor −3

✔️ Final Answer: −3

🔒 ❓ Question (h)
(−2) − (−2) = ______

📌 ✅ Answer:
🔹 Start at floor −2
🔹 Subtract −2 ➡️ move up 2 floors
🔹 Reach floor 0

✔️ Final Answer: 0

🔒 ❓ Question (i)
(−1) − (+1) = ______

📌 ✅ Answer:
🔹 Start at floor −1
🔹 Subtract +1 ➡️ move down 1 floor
🔹 Reach floor −2

✔️ Final Answer: −2

🔒 ❓ Question (j)
(+3) − (−3) = ______

📌 ✅ Answer:
🔹 Start at floor +3
🔹 Subtract −3 ➡️ means move up 3 floors
🔹 Reach floor +6

✔️ Final Answer: +6

🌿 ADDINGAND SUBTRACTING LARGER NUMBERS

🔵 Instruction Reminder (Visual Idea)
🔹 Think of numbers as movement in a mineshaft / lift
🔹 Addition (+) ➡️ move up
🔹 Subtraction (−) ➡️ move down

🔒 ❓ 1. Complete these expressions

🔒 ❓ (a) (+40) + ______ = +200
📌 ✅ Answer:
🔹 Start at +40
🔹 To reach +200, move up by +160
✔️ (+40) + (+160) = +200

🔒 ❓ (b) (+40) + ______ = −200
📌 ✅ Answer:
🔹 Start at +40
🔹 To reach −200, move down by −240
✔️ (+40) + (−240) = −200

🔒 ❓ (c) (−50) + ______ = +200
📌 ✅ Answer:
🔹 Start at −50
🔹 To reach +200, move up by +250
✔️ (−50) + (+250) = +200

🔒 ❓ (d) (−50) + ______ = −200
📌 ✅ Answer:
🔹 Start at −50
🔹 To reach −200, move down by −150
✔️ (−50) + (−150) = −200

🔒 ❓ (e) (−200) − (−40) = ______
📌 ✅ Answer:
🔹 Subtracting a negative means moving up
🔹 Move up by +40 from −200
✔️ −200 + 40 = −160

🔒 ❓ (f) (+200) − (+40) = ______
📌 ✅ Answer:
🔹 Subtracting a positive means moving down
🔹 Move down by 40 from +200
✔️ +200 − 40 = +160

🔒 ❓ (g) (−200) − (+40) = ______
📌 ✅ Answer:
🔹 Subtracting a positive means moving down
🔹 Move down by 40 from −200
✔️ −200 − 40 = −240

🔵 Final Visual Check (Mineshaft Logic)
🔹 All answers match correct upward/downward movement
🔹 Each result reaches the required target floor

🌿 BACK TO THE NUMBER LINE

🔒 ❓ Question 1
Mark 3 positive numbers and 3 negative numbers on the number line shown.

📌 ✅ Answer:

🔹 Positive numbers (numbers to the right of 0):
🔸 +2
🔸 +5
🔸 +8

🔹 Negative numbers (numbers to the left of 0):
🔸 −3
🔸 −6
🔸 −9

🔒 ❓ Question 2
Write down the above 3 marked negative numbers in the boxes.

📌 ✅ Answer:

🔹 −3
🔹 −6
🔹 −9

🔒 ❓ Question 3
Is 2 > −3? Why?
Is −2 < 3? Why?

📌 ✅ Answer:

🔹 2 > −3
🔸 On the number line, numbers increase as we move to the right
🔸 2 lies to the right of −3, so it is greater

🔹 −2 < 3
🔸 −2 lies to the left of 3 on the number line
🔸 Therefore, −2 is smaller than 3

🔒 ❓ Question 4
Find the following:

🔒 ❓ (a) −5 + 0

📌 ✅ Answer:
🔹 −5 + 0 = −5
🔸 Adding zero does not change the number

🔒 ❓ (b) 7 + (−7)

📌 ✅ Answer:
🔹 7 + (−7) = 0
🔸 A number and its opposite cancel each other

🔒 ❓ (c) −10 + 20

📌 ✅ Answer:
🔹 −10 + 20 = +10
🔸 From −10, move 20 steps to the right on the number line

🔒 ❓ (d) 10 − 20

📌 ✅ Answer:
🔹 10 − 20 = −10
🔸 From 10, move 20 steps to the left on the number line

🔒 ❓ (e) 7 − (−7)

📌 ✅ Answer:
🔹 7 − (−7) = 7 + 7 = 14
🔸 Subtracting a negative number means adding

🔒 ❓ (f) −8 − (−10)

📌 ✅ Answer:
🔹 −8 − (−10) = −8 + 10 = 2
🔸 Minus of minus becomes plus

🌿 THE TOKEN MODEL

📌USING TOKENS FOR ADDITON

🔒 ❓ Figure it Out

🔒 ❓ 1. Complete the additions using tokens.

📌 ✅ Answer:

🔹 a. (+6) + (+4)
🔸 Both are positive tokens
🔸 Total positive tokens = 6 + 4 = 10
🔸 Result = +10

🔹 b. (−3) + (−2)
🔸 Both are negative tokens
🔸 Total negative tokens = 3 + 2 = 5
🔸 Result = −5

🔹 c. (+5) + (−7)
🔸 Positive tokens = 5
🔸 Negative tokens = 7
🔸 Cancel 5 zero pairs
🔸 Remaining negative tokens = 2
🔸 Result = −2

🔹 d. (−2) + (+6)
🔸 Negative tokens = 2
🔸 Positive tokens = 6
🔸 Cancel 2 zero pairs
🔸 Remaining positive tokens = 4
🔸 Result = +4

🔒 ❓ 2. Cancel the zero pairs in the following two sets of tokens.
On what floor is the lift attendant in each case?
What is the corresponding addition statement in each case?

🔒 ❓ (a)

📌 ✅ Answer:

🔹 Green (+) tokens = 3
🔹 Orange (−) tokens = 5
🔹 Cancel 3 zero pairs
🔹 Remaining tokens = 2 negative tokens
🔹 Lift attendant is on Floor −2

🔹 Corresponding addition statement:
🔸 (+3) + (−5) = −2

🔒 ❓ (b)

📌 ✅ Answer:

🔹 Green (+) tokens = 6
🔹 Orange (−) tokens = 3
🔹 Cancel 3 zero pairs
🔹 Remaining tokens = 3 positive tokens
🔹 Lift attendant is on Floor +3

🔹 Corresponding addition statement:
🔸 (+6) + (−3) = +3

📌USING TOKENS FOR SUBTRACTION

🔒 ❓ Question 1
Evaluate the following differences using tokens. Check that you get the same result as with other methods.

📌 ✅ Answer:

🔹 (a) (+10) − (+7)
🔸 Start with +10 tokens
🔸 Remove 7 positive tokens
🔸 Remaining tokens = +3
➡️ Answer = +3

🔹 (b) (−8) − (−4)
🔸 Start with −8 tokens
🔸 Removing −4 means adding +4
🔸 −8 + 4 = −4
➡️ Answer = −4

🔹 (c) (−9) − (−4)
🔸 Start with −9 tokens
🔸 Removing −4 adds +4
🔸 −9 + 4 = −5
➡️ Answer = −5

🔹 (d) (+9) − (+12)
🔸 Start with +9 tokens
🔸 Need to remove 12 positive tokens
🔸 Add 3 zero pairs (+ and −)
🔸 Remaining = −3
➡️ Answer = −3

🔹 (e) (−5) − (−7)
🔸 Start with −5 tokens
🔸 Removing −7 adds +7
🔸 −5 + 7 = +2
➡️ Answer = +2

🔹 (f) (−2) − (−6)
🔸 Start with −2 tokens
🔸 Removing −6 adds +6
🔸 −2 + 6 = +4
➡️ Answer = +4

🔒 ❓ Question 2
Complete the subtractions.

📌 ✅ Answer:

🔹 (a) (−5) − (−7)
🔸 Removing −7 adds +7
🔸 −5 + 7 = +2
➡️ Answer = +2

🔹 (b) (+10) − (+13)
🔸 Remove 13 positive tokens
🔸 Add 3 zero pairs
🔸 Remaining = −3
➡️ Answer = −3

🔹 (c) (−7) − (−9)
🔸 Removing −9 adds +9
🔸 −7 + 9 = +2
➡️ Answer = +2

🔹 (d) (+3) − (+8)
🔸 Remove 8 positives
🔸 Add 5 zero pairs
🔸 Remaining = −5
➡️ Answer = −5

🔹 (e) (−2) − (−7)
🔸 Removing −7 adds +7
🔸 −2 + 7 = +5
➡️ Answer = +5

🔹 (f) (+3) − (+15)
🔸 Remove 15 positives
🔸 Add 12 zero pairs
🔸 Remaining = −12
➡️ Answer = −12

🟢 Concept Reminder (Visual Tip)
🔹 Subtracting a negative number means adding a positive number
🔹 Zero pairs (+1 and −1) do not change value
🔹 Token method helps “see” the answer clearly

🔒 ❓ Figure it Out

📌 Important:

🔒 ❓ 1. Try to subtract: −3 − (+5)
How many zero pairs will you have to put in? What is the result?

📌 ✅ Answer:
🔹 Start with −3 → means 3 negative tokens
🔹 We need to subtract +5 → but there are no positive tokens to remove
🔹 So, add zero pairs (each pair = +1 and −1)

🔹 To remove +5, we must add 5 zero pairs
🔸 This adds 5 positive and 5 negative tokens

🔹 Now remove the 5 positive tokens
🔹 Remaining tokens = original −3 and extra −5
🔹 Total negative tokens = −8

➡️ Result: −3 − (+5) = −8
➡️ Zero pairs added: 5

🔒 ❓ 2. Evaluate the following using tokens

🔒 ❓ (a) (−3) − (+10)
📌 ✅ Answer:
🔹 Start with −3 (3 negative tokens)
🔹 Need to remove +10 → add 10 zero pairs
🔹 Remove 10 positive tokens
🔹 Remaining negative tokens = −13

➡️ Result: −13

🔒 ❓ (b) (+8) − (−7)
📌 ✅ Answer:
🔹 Start with +8
🔹 Subtracting a negative means adding positives
🔹 So, +8 + 7

➡️ Result: +15

🔒 ❓ (c) (−5) − (+9)
📌 ✅ Answer:
🔹 Start with −5
🔹 Need to remove +9 → add 9 zero pairs
🔹 Remove 9 positive tokens
🔹 Remaining negatives = −14

➡️ Result: −14

🔒 ❓ (d) (−9) − (+10)
📌 ✅ Answer:
🔹 Start with −9
🔹 Add 10 zero pairs to remove +10
🔹 Remove 10 positive tokens
🔹 Remaining negatives = −19

➡️ Result: −19

🔒 ❓ (e) (+6) − (−4)
📌 ✅ Answer:
🔹 Subtracting a negative = adding a positive
🔹 +6 + 4

➡️ Result: +10

🔒 ❓ (f) (−2) − (+7)
📌 ✅ Answer:
🔹 Start with −2
🔹 Add 7 zero pairs
🔹 Remove 7 positive tokens
🔹 Remaining negatives = −9

➡️ Result: −9

✔️ Concept Check (Teacher Note):
🔹 Subtracting a positive → move more negative
🔹 Subtracting a negative → move positive
🔹 Zero pairs help when required tokens are missing

🌿 INTEGERS IN OTHER PLACES

📌CREDITS AND DEBITS

🔒 ❓ Question 1
Suppose you start with ₹0 in your bank account, and then you have credits of ₹30, ₹40, and ₹50, and debits of ₹40, ₹50, and ₹60. What is your bank account balance now?

📌 ✅ Answer:
🔹 Start balance = ₹0
🔹 Total credits = ₹30 + ₹40 + ₹50 = ₹120
🔹 Total debits = ₹40 + ₹50 + ₹60 = ₹150
🔹 Net balance = Total credits − Total debits
🔹 Net balance = ₹120 − ₹150
🔹 Net balance = −₹30

✔️ Final Answer: ₹−30 (negative balance)

🔒 ❓ Question 2
Suppose you start with ₹0 in your bank account, and then you have debits of ₹1, ₹2, ₹4, ₹8, ₹16, ₹32, ₹64, and ₹128, and then a single credit of ₹256. What is your bank account balance now?

📌 ✅ Answer:
🔹 Start balance = ₹0
🔹 Total debits = 1 + 2 + 4 + 8 + 16 + 32 + 64 + 128
🔹 Total debits = ₹255
🔹 Total credits = ₹256
🔹 Net balance = ₹256 − ₹255
🔹 Net balance = ₹1

✔️ Final Answer: ₹+1 (positive balance)

🔒 ❓ Question 3
Why is it generally better to try and maintain a positive balance in your bank account? What are circumstances under which it may be worthwhile to temporarily have a negative balance?

📌 ✅ Answer:
🔹 A positive balance means you have money available for daily needs
🔹 It helps avoid penalties, overdraft charges, and financial stress
🔹 It provides safety during emergencies

🔹 A temporary negative balance may be worthwhile when:
🔸 There is a genuine emergency (medical need, urgent repair)
🔸 Money is expected soon (salary, scholarship, refund)
🔸 Immediate spending is more important than waiting

✔️ Final Answer:
Maintaining a positive balance is generally safer, but a short-term negative balance can be acceptable in unavoidable situations with proper planning.

📌GEOGRAPHICAL CROSS SECTIONS

🔒 ❓ Question 1
Looking at the geographical cross section, fill in the respective heights.

📌 ✅ Answer:
🔹 a. Point A ➡️ +1500 m
🔹 b. Point B ➡️ −500 m
🔹 c. Point C ➡️ +300 m
🔹 d. Point D ➡️ −1200 m
🔹 e. Point E ➡️ +1200 m
🔹 f. Point F ➡️ −200 m
🔹 g. Point G ➡️ +100 m

🔒 ❓ Question 2
Which is the highest point in this geographical cross section?
Which is the lowest point?

📌 ✅ Answer:
🔹 Highest point ➡️ Point A (+1500 m)
🔹 Lowest point ➡️ Point D (−1200 m)

🔒 ❓ Question 3
Can you write the points A, B, …, G
(a) in decreasing order of heights
(b) in increasing order of heights?

📌 ✅ Answer:

🔹 Decreasing order (highest to lowest):
➡️ A, E, C, G, F, B, D

🔹 Increasing order (lowest to highest):
➡️ D, B, F, G, C, E, A

🔒 ❓ Question 4
What is the highest point above sea level on Earth? What is its height?

📌 ✅ Answer:
🔹 The highest point above sea level on Earth is Mount Everest
🔹 Height ➡️ +8848 m (approximately)

🔒 ❓ Question 5
What is the lowest point with respect to sea level on land or on the ocean floor? What is its height?

📌 ✅ Answer:
🔹 The lowest point is the Mariana Trench (Challenger Deep)
🔹 Height ➡️ approximately −11,000 m (negative because it is below sea level)

📌TEMPERATURE

🔒 ❓ Question 1
Do you know that there are some places in India where temperatures can go below 0°C? Find out the places in India where temperatures sometimes go below 0°C. What is common among these places? Why does it become colder there and not in other places?

📌 ✅ Answer:
🔹 Some places in India where temperatures sometimes go below 0°C are:
🔸 Leh and Kargil (Ladakh)
🔸 Drass (Ladakh)
🔸 Keylong (Himachal Pradesh)
🔸 Spiti Valley (Himachal Pradesh)
🔸 Gulmarg (Jammu & Kashmir)

🔹 What is common among these places?
🔸 All these places are located in very high mountainous regions.
🔸 They are far above sea level.

🔹 Why does it become colder there and not in other places?
🔸 As height above sea level increases, temperature decreases.
🔸 These regions receive less heat because the air is thinner at high altitudes.
🔸 Snow-covered land reflects heat instead of absorbing it.

🔒 ❓ Question 2
Leh in Ladakh gets very cold during the winter. The following is a table of temperature readings taken during different times of the day and night in Leh on a day in November. Match the temperature with the appropriate time of the day and night.

📌 ✅ Answer:

🔹 Daytime is warmer than night time.
🔹 Early morning hours are the coldest.

✔️ Correct matching:

🔸 14°C ➡️ 02:00 p.m.
🔸 8°C ➡️ 11:00 a.m.
🔸 −2°C ➡️ 11:00 p.m.
🔸 −4°C ➡️ 02:00 a.m.

🌿 EXPLORATIONS WITH INTEGERS

📌A HOLLOW INTEGER GRID

🔒 ❓ Question 1
Do the calculations for the second grid above and find the border sum.

📌 ✅ Answer:
🔹 The border sum means the sum of all numbers written in the outer boxes of the grid.
🔹 The centre box is not included in the border sum.
🔹 Add all numbers on
🔸 top row
🔸 bottom row
🔸 left column
🔸 right column
🔹 After adding all the border numbers of the second grid, we get the required border sum.

🔒 ❓ Question 2
Complete the grids to make the required border sum.

🔹 (a) Border sum = +4

📌 ✅ Answer:
🔹 Given border numbers are −10, −5 and 9.
🔹 Add the remaining border boxes so that the total becomes +4.
🔹 The centre box can have any number, as it does not affect the border sum.
🔹 One correct filling is possible by balancing positive and negative integers.

🔹 (b) Border sum = −2

📌 ✅ Answer:
🔹 Given border numbers are 6, 8, −5 and −2.
🔹 Add all known border numbers first.
🔹 Fill the empty border boxes with suitable integers so that the final total becomes −2.
🔹 More than one correct solution is possible.

🔹 (c) Border sum = −4

📌 ✅ Answer:
🔹 Given border numbers are 7 and −5.
🔹 The remaining border numbers must add up to −6 so that the total becomes −4.
🔹 There are many possible ways to fill the empty boxes.

🔒 ❓ Question 3
For the last grid above, find more than one way of filling the numbers to get border sum −4.

📌 ✅ Answer:
🔹 Yes, more than one way is possible.
🔹 Reason:
🔸 Different combinations of integers can give the same sum.
🔸 The centre box can be changed freely.
🔹 Hence, multiple correct answers exist.

🔒 ❓ Question 4
Which other grids can be filled in multiple ways? What could be the reason?

📌 ✅ Answer:
🔹 Any grid with empty border boxes can be filled in multiple ways.
🔹 Reason:
🔸 Integers have many combinations with the same sum.
🔸 The centre number does not affect the border sum.

🔒 ❓ Question 5
Make a border integer square puzzle and challenge your classmates.

📌 ✅ Answer:
🔹 Draw a 3×3 grid.
🔹 Fix a border sum, for example −6 or +10.
🔹 Fill some border boxes with integers.
🔹 Leave the rest blank.
🔹 Ask classmates to complete the grid so that the border sum is correct.

📌AN AMAZING GRID OF NUMBERS

🔒 ❓ 1. Try afresh, choose different numbers this time. What sum did you get? Was it different from the first time? Try a few more times.

📌 ✅ Answer:
🔹 When we choose different numbers and calculate the border sum, we again get the same total.
🔹 Even after trying many times with different choices, the sum remains unchanged.
🔹 This shows that the result depends on the structure of the grid, not on random choices.

🔒 ❓ 2. Play the same game with the grids below. What answer did you get?

📌 ✅ Answer:
🔹 For the first grid, adding all border numbers gives 0.
🔹 For the second grid also, the total border sum is 0.
🔹 Though numbers look different, positives and negatives cancel each other.

🔒 ❓ 3. What could be so special about these grids? Is the magic in the numbers or the way they are arranged or both? Can you make more such grids?

📌 ✅ Answer:
🔹 The special feature is both the numbers and their arrangement.
🔹 Each positive number has a matching negative number placed symmetrically.
🔹 Yes, many such grids can be made by balancing positive and negative integers.

🔒 ❓ Figure it Out

🔒 ❓ 1. Write all the integers between the given pairs, in increasing order.

📌 ✅ Answer:
🔹 a. Between 0 and −7
→ −6, −5, −4, −3, −2, −1

🔹 b. Between −4 and 4
→ −3, −2, −1, 0, 1, 2, 3

🔹 c. Between −8 and −15
→ −14, −13, −12, −11, −10, −9

🔹 d. Between −30 and −23
→ −29, −28, −27, −26, −25, −24

🔒 ❓ 2. Give three numbers such that their sum is −8.

📌 ✅ Answer:
🔹 Example: −3, −2, −3
🔹 −3 + (−2) + (−3) = −8

🔒 ❓ 3. Two dice have faces −1, 2, −3, 4, −5, 6.
Some numbers between −10 and +12 are not possible as sums. Find them.

📌 ✅ Answer:
🔹 Smallest sum = −5 + (−5) = −10
🔹 Largest sum = 6 + 6 = 12
🔹 All integers between −10 and 12 are not possible
🔹 Numbers like −9, −8, −7, 11 cannot be formed

🔒 ❓ 4. Solve these:

📌 ✅ Answer:
🔹 8 − 13 = −5
🔹 (−8) − 13 = −21
🔹 (−13) − (−8) = −5
🔹 (−13) + (−8) = −21

🔹 8 + (−13) = −5
🔹 (−8) − (−13) = 5
🔹 13 − 8 = 5
🔹 13 − (−8) = 21

🔒 ❓ 5. Find the years below (No year 0).

📌 ✅ Answer:
🔹 a. 150 years ago from present year
→ Present year − 150

🔹 b. 2200 years ago from present year
→ Present year − 2200

🔹 c. 320 years after 680 BCE
→ 680 − 320 = 360 BCE

🔒 ❓ 6. Complete the sequences:

📌 ✅ Answer:
🔹 a. −40, −34, −28, −22, −16, −10, −4

🔹 b. 3, 4, 2, 5, 1, 6, 0, 7, −1, 8

🔹 c. 15, 9, 12, 6, 1, −3, −6, −9, −12

🔒 ❓ 7. Make an expression closer to −30 using given cards.

📌 ✅ Answer:
🔹 Example:
(+1) − (+18) − (+7) − (−5) = −29
🔹 −29 is very close to −30

🔒 ❓ 8. Decide the sign of the result:

📌 ✅ Answer:
🔹 a. (positive) − (negative) → positive
🔹 b. (positive) + (negative) → can be positive or negative
🔹 c. (negative) + (negative) → negative
🔹 d. (negative) − (negative) → can be positive or negative
🔹 e. (negative) − (positive) → negative
🔹 f. (negative) + (positive) → can be positive or negative

🔒 ❓ 9. A string has 100 tokens arranged in a pattern. What is the value of the string?

📌 ✅ Answer:
🔹 Each + token cancels one − token
🔹 Total zero pairs are formed
🔹 Final value = 0

🌿 A PINCH OF HISTORY

🌟 Figure it Out

🔒 ❓ 1. Explain Brahmagupta’s rules using Bela’s Building of Fun or a number line.
📌 ✅ Answer:
🔹 Positive numbers mean moving upward / to the right.
🔹 Negative numbers mean moving downward / to the left.
🔹 Adding means moving in the same direction.
🔹 Subtracting a negative means reversing direction.
🔹 Zero means no movement at all.

🔒 ❓ 2. Give your own examples of each rule.
📌 ✅ Answer:
🔹 (+6) + (+4) = +10
🔹 (−7) + (−3) = −10
🔹 (+8) + (−5) = +3
🔹 9 + (−9) = 0
🔹 0 + (−6) = −6
🔹 5 − (−2) = 7
🔹 4 − 4 = 0

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OTHER IMPORTANT QUESTIONS

(CBSE MODEL QUESTION PAPER)

ESPECIALLY MADE FROM THIS CHAPTER ONLY

🔵 Section A — Very Short Answer

(Q1–Q6 | 1 × 6 = 6 marks)

🔵 Question
Q1. What is zero?

🟢 Answer
✔️ Zero represents no quantity and acts as a reference point on the number line.

🔵 Question
Q2. What are numbers less than zero called?

🟢 Answer
✔️ Numbers less than zero are called negative numbers.

🔵 Question
Q3. Write one negative number.

🟢 Answer
✔️ –3 is a negative number.

🔵 Question
Q4. Is zero a positive or a negative number?

🟢 Answer
✔️ Zero is neither positive nor negative.

🔵 Question
Q5. Where are negative numbers placed on the number line?

🟢 Answer
✔️ Negative numbers are placed on the left side of zero on the number line.

🔵 Question
Q6. What are integers?

🟢 Answer
✔️ Integers are numbers that include negative numbers, zero, and positive numbers.

🟢 Section B — Short Answer–I

(Q7–Q12 | 2 × 6 = 12 marks)

🟢 Question
Q7. Why do we need numbers less than zero? Give one example.

🟢 Answer
🔵 Numbers less than zero are needed to show situations like loss or temperature below zero.
🔵 Example: –5°C shows temperature below freezing point.

🟢 Question
Q8. Write the difference between positive and negative numbers.

🟢 Answer
🔵 Positive numbers are greater than zero.
🔵 Negative numbers are less than zero.

🟢 Question
Q9. Write any two real-life situations where negative numbers are used.

🟢 Answer
🔵 Temperature below 0°C.
🔵 Loss of money or debt.

🟢 Question
Q10. How does a number line help in understanding integers?

🟢 Answer
🔵 A number line shows the position of numbers clearly.
🔵 It helps in comparing positive numbers, negative numbers, and zero.

🟢 Question
Q11. Which is greater: –2 or –5? Give reason.

🟢 Answer
🔵 –2 is greater than –5.
🔵 It lies closer to zero on the number line.

🟢 Question
Q12. Write two examples of positive integers.

🟢 Answer
🔵 3
🔵 7

🟡 Section C — Short Answer–II

(Q13–Q22 | 3 × 10 = 30 marks)

🟡 Question
Q13. What are negative numbers? Write two examples.

🟢 Answer
🔵 Negative numbers are numbers less than zero.
🔵 They are written with a minus sign (–).
✔️ Examples: –3, –7

🟡 Question
Q14. Why is zero called a reference point on the number line?

🟢 Answer
🔵 Zero separates positive numbers and negative numbers.
🔵 It helps in comparing numbers on both sides of it.
✔️ Hence, zero is used as a reference point.

🟡 Question
Q15. Represent the integers –3, 0, and 4 on a number line (describe in words).

🟢 Answer
🔵 Draw a straight number line and mark zero at the centre.
🔵 Mark –3 three equal units to the left of zero.
🔵 Mark 4 four equal units to the right of zero.

🟡 Question
Q16. Which is greater: –1 or –6? Explain.

🟢 Answer
🔵 –1 lies closer to zero than –6 on the number line.
🔵 A negative number closer to zero is greater.
✔️ Therefore, –1 is greater than –6.

🟡 Question
Q17. Write three real-life situations where integers are used.

🟢 Answer
🔵 Temperature above or below 0°C.
🔵 Floors above and below ground level in a building.
🔵 Profit and loss in money.

🟡 Question
Q18. What do we mean by positive numbers? Give two examples.

🟢 Answer
🔵 Positive numbers are numbers greater than zero.
🔵 They are written without a minus sign.
✔️ Examples: 2, 9

🟡 Question
Q19. Compare –4 and 2. Which is smaller? Give reason.

🟢 Answer
🔵 –4 lies to the left of zero on the number line.
🔵 2 lies to the right of zero.
✔️ Therefore, –4 is smaller than 2.

🟡 Question
Q20. What are integers? Why are they useful?

🟢 Answer
🔵 Integers include negative numbers, zero, and positive numbers.
🔵 They are useful to show position, direction, gain, and loss in real life.

🟡 Question
Q21. Write two differences between positive numbers and negative numbers.

🟢 Answer
🔵 Positive numbers are greater than zero, while negative numbers are less than zero.
🔵 Positive numbers lie to the right of zero, while negative numbers lie to the left.

🟡 Question
Q22. Why are negative numbers important in daily life? Explain briefly.

🟢 Answer
🔵 Negative numbers help represent situations like loss, debt, and low temperature.
🔵 They help us understand values below a fixed level such as zero.

🔴 Section D — Long Answer

(Q23–Q30 | 4 × 8 = 32 marks)

🔴 Question
Q23. Explain why numbers less than zero are needed in mathematics. Give suitable examples.

🟢 Answer
🔵 In real life, some situations involve values less than zero.
🔵 Whole numbers cannot represent these situations properly.
🔵 Temperatures below 0°C are written using negative numbers, like –5°C.
🔵 Loss of money or debt is also represented by negative numbers.

✔️ Therefore, numbers less than zero are needed to describe real-life situations correctly.

🔴 Question
Q24. Explain the position of positive numbers, negative numbers, and zero on the number line.

🟢 Answer
🔵 Zero is placed at the centre of the number line.
🔵 Positive numbers are placed on the right side of zero.
🔵 Negative numbers are placed on the left side of zero.
🔵 Each number is placed at equal distance from the next number.

✔️ The number line helps in understanding the order and position of numbers.

🔴 Question
Q25. Compare –3 and –7 using the number line. Which is greater and why?

🟢 Answer
🔵 On the number line, –3 lies closer to zero than –7.
🔵 The number closer to zero on the left side is greater.
🔵 –7 lies farther to the left than –3.

✔️ Therefore, –3 is greater than –7.

🔴 Question
Q26. Explain with examples how integers are used to show direction.

🟢 Answer
🔵 Positive integers are used to show movement to the right or upward direction.
🔵 Negative integers are used to show movement to the left or downward direction.
🔵 For example, moving up 5 floors is written as +5, and moving down 3 floors is written as –3.

✔️ Integers help in showing both value and direction.

🔴 Question
Q27. What are integers? Explain their importance in daily life.

🟢 Answer
🔵 Integers include positive numbers, negative numbers, and zero.
🔵 They are used to show temperature, profit and loss, height, and depth.
🔵 Integers help in comparing values above and below a reference point.

✔️ Thus, integers are very important in daily life.

🔴 Question
Q28. Describe how zero helps in comparing integers.

🟢 Answer
🔵 Zero acts as a reference point on the number line.
🔵 Any number to the right of zero is greater than zero.
🔵 Any number to the left of zero is less than zero.
🔵 This helps in comparing positive and negative numbers easily.

✔️ Zero makes comparison of integers simple and clear.

🔴 Question
Q29. Explain the difference between –4 and 4 using the number line.

🟢 Answer
🔵 –4 lies four units to the left of zero on the number line.
🔵 4 lies four units to the right of zero.
🔵 Both are at the same distance from zero but in opposite directions.

✔️ –4 is a negative number and 4 is a positive number.

🔴 Question
Q30. Why is the number line useful in understanding integers? Give reasons.

🟢 Answer
🔵 It shows the exact position of integers.
🔵 It helps in comparing numbers easily.
🔵 It helps in understanding direction and distance from zero.
🔵 It makes learning about negative numbers simple.

✔️ Hence, the number line is very useful for understanding integers.

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