Class 7 Maths (Part II) Chapter 2 : Operations with Integers | Number Puzzles, Token Model for Multiplication, Division Rules & Practical Problems
Complete Chapter 2 (Part II) guide: Rakesh's number puzzles (sum and difference), modelling integer movement on a line (carrom strikes, direction), using colored tokens (positive/negative) to understand addition, subtraction and multiplication, rules for multiplying and dividing integers (signs logic), properties like commutativity, associativity and distributivity, Collatz conjecture with integers, temperature drops, plus solved examples and practice questions for CBSE Class 7 Maths
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Operations with Integers
Class 7 Mathematics Chapter 2 | Complete Guide | Operations on positive and negative numbers 2025
Chapter at a Glance – Operations with Integers
This chapter recaps integers and explores operations like addition, subtraction, multiplication, and division using models like tokens and number lines.
Main Topics Covered
- Recap of integers using puzzles and carrom coin model
- Addition and subtraction with token model
- Multiplication of integers using token model for all sign combinations
- Patterns in integer multiplication
- Division of integers and rules for signs
- Properties: commutative, associative, distributive
- Expressions with integers and pick the pattern
- Applications in exams, elevators, temperature, profit/loss
- Magic grids and Terhüchü game
Key Takeaways for Exams
Sign Rules Multiplication
(+) × (+) = (+), (-) × (-) = (+), (+) × (-) = (-)
Sign Rules Division
Same as multiplication for quotient signs
Additive Inverse
-a for a, used in subtraction as addition of inverse
Commutative
a × b = b × a
Associative
(a × b) × c = a × (b × c)
Distributive
a × (b + c) = a × b + a × c
Identity
1 × a = a, -1 × a = -a
Token Model
Green +, red -, zero pairs for operations
Key Rules & Properties – Operations with Integers
Important rules for signs and properties of operations.
Multiplication Sign Rules
| Multiplier | Multiplicand | Product Sign |
|---|---|---|
| Positive | Positive | Positive |
| Positive | Negative | Negative |
| Negative | Positive | Negative |
| Negative | Negative | Positive |
Division Sign Rules
| Dividend | Divisor | Quotient Sign |
|---|---|---|
| Positive | Positive | Positive |
| Positive | Negative | Negative |
| Negative | Positive | Negative |
| Negative | Negative | Positive |
Properties
- Commutative: a × b = b × a
- Associative: (a × b) × c = a × (b × c)
- Distributive: a × (b + c) = a × b + a × c
- Identity: 1 × a = a
- Inverse: -1 × a = -a
- Subtraction: a - b = a + (-b)
Concept Cards – Quick Explanations
Integer Recap
Positive, negative, zero. Modeled on number line or tokens.
Addition/Subtraction Tokens
Green +, red -. Zero pairs to enable subtraction.
Multiplication Positive × Positive
Add positives multiple times = positive.
Multiplication Positive × Negative
Add negatives multiple times = negative.
Multiplication Negative × Positive
Remove positives (using zero pairs) = negative.
Multiplication Negative × Negative
Remove negatives (using zero pairs) = positive.
Patterns in Multiplication
Product decreases/increases by multiplicand as multiplier changes.
Division as Inverse
Convert to multiplication: what × divisor = dividend.
Brahmagupta's Rules
Fortune (positive), debt (negative) for signs in mul/div.
Distributive Property
Visual with rectangular tokens.
Magic Grid
Product of circled numbers always same.
Examples + Solutions
Example 1: Exam Marks
Problem: 50 MCQs, +5 correct, -2 wrong. Mala: 30 correct, 20 wrong. Total marks?
Solution: 30 × 5 + 20 × (-2) = 150 - 40 = 110
Explanation: Positive for correct, negative for wrong.
Example 2: Elevator Position
Problem: Descends 3m/min from 0. Position after 1 hour?
Solution: 60 × (-3) = -180 (180m below ground)
Explanation: Negative for down.
Example 2(b): From 15m Above
Problem: From 15m above, after 45 min?
Solution: 15 + 45 × (-3) = 15 - 135 = -120
Explanation: Starting positive, add negative movement.
Figure it Out Solutions (All Solved)
Page 8
1(a) 3 × (-2)
-6
1(b) (-5) × (-2)
10
1(c) (-4) × (-1)
4
1(d) (-7) × 3
-21
2(a) (-123) × 456
-56088
2(b) (-123) × (-456)
56088
2(c) 123 × (-456)
-56088
3. Simple rule for multiplying two integers
Multiply magnitudes, sign positive if same signs, negative if different.
Page 10
(a) 4 × (-3)
-12
(b) (-6) × (-3)
18
(c) (-5) × (-1)
5
(d) (-8) × 4
-32
(e) (-9) × 10
-90
(f) 10 × (-17)
-170
Page 15
1(a) 14 × (-15)
-210
1(b) -16 × (-5)
80
1(c) 36 ÷ (-18)
-2
1(d) (-46) ÷ (-23)
2
2. Temperature lowering from 32°C at 5°C/hour. After 10 hours?
32 + 10 × (-5) = 32 - 50 = -18°C
3(a) Profit/loss: 3000 white +8, 5000 grey -5
3000×8 + 5000×(-5) = 24000 - 25000 = -1000 (loss)
3(b) Grey 6400 -5, white x +8, zero profit
8x - 5×6400 = 0 → 8x = 32000 → x=4000
4(a) (-3) × _ = 27
-9
4(b) 5 × _ = -35
-7
4(c) _ × (-8) = -56
7
4(d) _ × (-12) = 132
-11
4(e) _ ÷ (-8) = 7
-56
4(f) _ ÷ 12 = -11
-132
Page 19
1(a) (-5) × (18 + (-3))
-75
1(b) (-7) × 4 × (-1)
28
1(c) (-2) × (-1) × (-5) × (-3)
-30
2(a) (-27) ÷ 9
-3
2(b) 84 ÷ (-4)
-21
2(c) (-56) ÷ (-2)
28
3(a) _ × (-1) = 27
-27
3(b) _ × (-1) = -31
31
3(c) _ × (-1) = -1
1
3(d) _ × (-1) = 1
-1
3(e) _ × (-1) = 0
0
4. Given expression = -4, find negative of terms.
-47 + 56 - 14 + 8 - 2 + 8 - 5 = 4
5. Modified Collatz with integers.
Patterns: Cycles or to 1, observe for given starts.
6(a) Anita 40 marks, 15 correct, all answered.
Incorrect: x, 15×4 + x×(-2) = 40, 60 - 2x = 40, x=10. Total questions 25.
6(b) Anil -10, 5 correct.
5×4 + x×(-2) = -10, 20 - 2x = -10, x=15 incorrect. Unanswered? Assume all answered.
7. Pick the pattern machine.
Analyze inputs/outputs for operation, e.g., a + b - c.
8. Temperature 8°C, drops 5°C/hour, after 4 hours.
8 + 4 × (-5) = 8 - 20 = -12°C
9(a) 3 consecutive product -6
-3, -2, -1? Product -6 no. -1,0,1=0. Try -3,-2,1=6. -2,-1,3=-6 yes.
9(b) 120
4,5,6=120
10. Pibs +13, -9 to +85
10×13 + 5×(-9) = 130 - 45 = 85
10(a) +20
5×13 + 5×(-9) = 65 - 45 = 20
10(b) +40
10×13 + 10×(-9) = 130 - 90 = 40
10(c) -50
5×(-9) - 5×13 = -45 - 65 = -110 no. Solve 13x - 9y = -50
Extra Practice Questions (Exam-Ready) – Chapter 2
25+ Questions • Categorized by Marks • With Detailed Solutions • Difficulty Tags
1-Mark Questions (Very Short Answer)
1. Additive inverse of -7.
2. 5 × (-3)
3. (-8) ÷ 2
4. Sign of (-) × (-)
5. 1 × a = ?
2-Mark Questions (Short Answer)
6. Explain token model for (-4) × 2.
7. Why is multiplication commutative?
8. (-10) + 5 - (-3)
9. Brahmagupta's rule for debt × fortune.
10. Distributive property example.
3-Mark Questions (Reasoning)
11. Patterns when multiplicand negative.
12. Why division converts to multiplication.
13. Token for 3 × (-4).
14. Associative check: (2 × -3) × 4 vs 2 × (-3 × 4)
15. Magic grid product always same?
4–5 Mark Questions (Application)
16. Temperature 20°C drops 4°C/hour, after 6 hours.
17. Profit +10/bag A, -6/bag B. 200A, 300B total.
18. Find x: (-5) × x = 35
19. Expression: (-2) × (3 + (-4))
20. Collatz starting -5.
Challenge Questions (6+ Marks)
21. Prove distributive for negatives.
22. Machine pattern: inputs to output.
23. Pibs equation 13x - 9y = 10.
24. Arrange expressions increasing order.
25. Given product, find nearby.
Common Mistakes & How to Avoid
Mistake 1: Sign Errors in Multiplication
Forgetting (-) × (-) = (+).
Avoid: Remember same signs positive, different negative.
Mistake 2: Subtraction Without Inverse
Not converting to addition of inverse.
Avoid: Always a - b = a + (-b).
Mistake 3: Division Sign Confusion
Wrong quotient sign.
Avoid: Same as multiplication signs.
Mistake 4: Associative Grouping Wrong
Incorrect order in expressions.
Avoid: Property holds, but check signs.
Mistake 5: Token Model Zero Pairs Forget
Not adding zero pairs for removal.
Avoid: Always use for negative multipliers.
Mistake 6: Distributive Over Subtraction
Forgetting negative distribution.
Avoid: a × (b - c) = a×b - a×c.
History & Fun Facts
Ancient Origins
Brahmagupta (628 CE) first stated rules for positive/negative multiplication/division using fortune/debt.
Real-Life Applications
- Temperature: Negative for below zero.
- Profit/Loss: Positive profit, negative loss.
- Elevators: Above/below ground.
- Stock Market: Gains/losses.
Fun Facts
- Integers crucial in programming for loops, indices.
- Negative numbers invented for debt accounting.
- Zero invented in India, key for negatives.
- Collatz conjecture unsolved for positives, modified for integers.
Did You Know?
Chinese used red positive, black negative in ancient math.
Quick Revision One-Pager
Sign Rules
| Operation | Same Signs | Different Signs |
|---|---|---|
| Mul/Div | Positive | Negative |
Quick Rules
- ✓ Subtraction: Add inverse
- ✓ Multiplication: Magnitude ×, sign by rules
- ✓ Division: Convert to mul
- ✓ Commutative: Order swap ok
- ✓ Associative: Grouping ok
- ✓ Distributive: Over add/sub
Mind Map
Central: Integers Operations
- Models: Number line, tokens
- Multiplication:
- Signs, patterns
- Properties
- Division: Inverse mul
- Applications: Real-life scenarios
Exam Tips
Before Solving
Identify signs, use rules
During Solving
Step by step, check inverse
After Solving
Verify with model if unsure
Time-Savers
Memorize sign rules
Interactive Quiz – 15 Questions
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