Chapter 3 of 14
Chapter 3: Patterns Around Us
Growing Patterns, Repeating Patterns, Rotational Turns & Symmetrical Borders | CBSE Session 2026–27
Core Pedagogical Competencies & Concepts
- Identifying repeating units (motifs) and generating geometric patterns.
- Understanding rotational turns: Half Turn (180°), Quarter Turn (90°), and Three-Quarter Turn (270°).
- Analyzing growing number sequences involving addition, multiplication, and geometric growth.
- Recognizing patterns in border designs, rangolis, and fabrics.
All Textbook Exercises & In-Text Tasks (15 Questions)
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Q1
Number Sequence Pattern Rules
Find the rule and write the next three terms for each pattern: (a) 7, 14, 21, 28, __, __, __; (b) 2, 4, 8, 16, __, __, __; (c) 1, 4, 9, 16, 25, __, __, __; (d) 100, 92, 84, 76, __, __, __.
[4 Marks]
✅ Step-by-Step Solution & Working
(a) 7, 14, 21, 28, ...
• Rule: Add 7 to each term (Multiples of 7).
• Next three terms: 28 + 7 = 35, 35 + 7 = 42, 42 + 7 = 49.
(b) 2, 4, 8, 16, ...
• Rule: Multiply each term by 2 (Doubling rule).
• Next three terms: 16 × 2 = 32, 32 × 2 = 64, 64 × 2 = 128.
(c) 1, 4, 9, 16, 25, ...
• Rule: Square numbers (1×1, 2×2, 3×3, 4×4, 5×5) or consecutive odd addition (+3, +5, +7, +9, +11...).
• Next three terms: 6×6 = 36, 7×7 = 49, 8×8 = 64.
(d) 100, 92, 84, 76, ...
• Rule: Subtract 8 from each term.
• Next three terms: 76 - 8 = 68, 68 - 8 = 60, 60 - 8 = 52.
• Rule: Add 7 to each term (Multiples of 7).
• Next three terms: 28 + 7 = 35, 35 + 7 = 42, 42 + 7 = 49.
(b) 2, 4, 8, 16, ...
• Rule: Multiply each term by 2 (Doubling rule).
• Next three terms: 16 × 2 = 32, 32 × 2 = 64, 64 × 2 = 128.
(c) 1, 4, 9, 16, 25, ...
• Rule: Square numbers (1×1, 2×2, 3×3, 4×4, 5×5) or consecutive odd addition (+3, +5, +7, +9, +11...).
• Next three terms: 6×6 = 36, 7×7 = 49, 8×8 = 64.
(d) 100, 92, 84, 76, ...
• Rule: Subtract 8 from each term.
• Next three terms: 76 - 8 = 68, 68 - 8 = 60, 60 - 8 = 52.
CBSE Step-by-Step Marking Scheme
Correct rule and terms for (a) & (b)
2.0 M
Correct rule and terms for (c) & (d)
2.0 M
Q2
Rotational Symmetry & Letter Turns
Which English capital letters look exactly the same after a HALF TURN (180° rotation)? Test the letters: A, H, I, M, N, O, S, X, Z.
[3 Marks]
✅ Step-by-Step Solution & Working
Let us test each letter by rotating it upside down by 180°:
• H: Rotates to H (Same) ✅
• I: Rotates to I (Same) ✅
• N: Rotates to N (Same) ✅
• O: Rotates to O (Same) ✅
• S: Rotates to S (Same) ✅
• X: Rotates to X (Same) ✅
• Z: Rotates to Z (Same) ✅
Letters that change:
• A becomes upside down (∀) ❌
• M rotates to W ❌
Conclusion: The letters that look identical after a half turn are H, I, N, O, S, X, and Z.
• H: Rotates to H (Same) ✅
• I: Rotates to I (Same) ✅
• N: Rotates to N (Same) ✅
• O: Rotates to O (Same) ✅
• S: Rotates to S (Same) ✅
• X: Rotates to X (Same) ✅
• Z: Rotates to Z (Same) ✅
Letters that change:
• A becomes upside down (∀) ❌
• M rotates to W ❌
Conclusion: The letters that look identical after a half turn are H, I, N, O, S, X, and Z.
CBSE Step-by-Step Marking Scheme
Correctly identifies all 7 valid letters
2.0 M
Explains why A and M fail
1.0 M
Q3
Number Patterns • Square Numbers
Observe the pattern of dot arrays: 1, 4, 9, 16... What are the next two numbers in this sequence? Explain the rule.
[3 Marks]
✅ Step-by-Step Solution & Working
Rule and Next Terms:
1. Rule: These are Square Numbers, formed by multiplying a whole number by itself (or by arranging dots in perfect squares):
• 1 × 1 = 1
• 2 × 2 = 4 (difference +3)
• 3 × 3 = 9 (difference +5)
• 4 × 4 = 16 (difference +7)
2. Next Two Numbers:
• 5 × 5 = 25 (16 + 9)
• 6 × 6 = 36 (25 + 11)
The next two numbers in the sequence are 25 and 36.
1. Rule: These are Square Numbers, formed by multiplying a whole number by itself (or by arranging dots in perfect squares):
• 1 × 1 = 1
• 2 × 2 = 4 (difference +3)
• 3 × 3 = 9 (difference +5)
• 4 × 4 = 16 (difference +7)
2. Next Two Numbers:
• 5 × 5 = 25 (16 + 9)
• 6 × 6 = 36 (25 + 11)
The next two numbers in the sequence are 25 and 36.
CBSE Step-by-Step Marking Scheme
Stating the multiplication rule or consecutive odd additions
1.5 M
Correctly identifying 25 and 36
1.5 M
Q4
Secret Codes • Decoding Patterns
If the word 'MATH' is coded as 'N B U I' by shifting each letter forward by one alphabet (A→B, B→C), what is the secret word coded as 'C B M M'?
[2 Marks]
✅ Step-by-Step Solution & Working
Decoding the Message:
The encoding rule adds +1 letter. To decode, we reverse the rule by shifting each letter backward by one alphabet (-1):
• C - 1 = B
• B - 1 = A
• M - 1 = L
• M - 1 = L
The original secret word is BALL.
The encoding rule adds +1 letter. To decode, we reverse the rule by shifting each letter backward by one alphabet (-1):
• C - 1 = B
• B - 1 = A
• M - 1 = L
• M - 1 = L
The original secret word is BALL.
CBSE Step-by-Step Marking Scheme
Reversing the +1 rule to -1 letter shift
1.0 M
Accurately decoding to 'BALL'
1.0 M
Q5
Textbook Exercise • Real-World Context
In Chapter 3 ('Patterns Around Us'), a classroom project investigates real-world mathematics: If a school bus travels 48 kilometres on 4 litres of diesel, what is the mileage (kilometres per litre) of the bus? How much diesel is needed for a 120-kilometre journey?
[3 Marks]
✅ Step-by-Step Solution & Working
Step-by-Step Solution:
1. Mileage calculation:
• Distance = 48 km, Diesel used = 4 L.
• Mileage = 48 ÷ 4 = 12 km per litre.
2. Diesel for 120 km:
• Required diesel = Total distance ÷ Mileage = 120 ÷ 12 = 10 litres.
1. Mileage calculation:
• Distance = 48 km, Diesel used = 4 L.
• Mileage = 48 ÷ 4 = 12 km per litre.
2. Diesel for 120 km:
• Required diesel = Total distance ÷ Mileage = 120 ÷ 12 = 10 litres.
CBSE Step-by-Step Marking Scheme
Mileage = 12 km/L
1.5 M
Diesel needed = 10 L
1.5 M
Q6
Practice Time • Measurement & Conversions
Convert the following measurements into standard units:
(a) 4 metres 65 centimetres into centimetres.
(b) 3500 millilitres into litres and millilitres.
(c) 2 kilograms 750 grams into grams.
[3 Marks]
(a) 4 metres 65 centimetres into centimetres.
(b) 3500 millilitres into litres and millilitres.
(c) 2 kilograms 750 grams into grams.
✅ Step-by-Step Solution & Working
Conversions:
• (a) 1 m = 100 cm → (4 × 100) + 65 = 465 cm.
• (b) 1000 mL = 1 L → 3500 mL = 3 L 500 mL.
• (c) 1 kg = 1000 g → (2 × 1000) + 750 = 2750 g.
• (a) 1 m = 100 cm → (4 × 100) + 65 = 465 cm.
• (b) 1000 mL = 1 L → 3500 mL = 3 L 500 mL.
• (c) 1 kg = 1000 g → (2 × 1000) + 750 = 2750 g.
CBSE Step-by-Step Marking Scheme
465 cm
1 M
3 L 500 mL
1 M
2750 g
1 M
Q7
Try These • Geometry and Shapes
Look at a solid brick:
(a) How many faces does a brick have?
(b) What is the shape of each face?
(c) How many straight edges and how many corners (vertices) does a brick have?
[3 Marks]
(a) How many faces does a brick have?
(b) What is the shape of each face?
(c) How many straight edges and how many corners (vertices) does a brick have?
✅ Step-by-Step Solution & Working
Properties of a Brick (Cuboid):
• (a) Faces: A brick has 6 flat faces.
• (b) Shape of faces: Each face is a rectangle.
• (c) Edges and Corners: It has 12 straight edges and 8 corners (vertices).
• (a) Faces: A brick has 6 flat faces.
• (b) Shape of faces: Each face is a rectangle.
• (c) Edges and Corners: It has 12 straight edges and 8 corners (vertices).
CBSE Step-by-Step Marking Scheme
6 faces
1 M
Rectangular faces
1 M
12 edges and 8 corners
1 M
Q8
Money & Accounts • Rate List Calculation
At a recycling shop, the rate list shows: Waste paper = ₹12 per kg, Plastic = ₹18 per kg, Iron = ₹25 per kg.
Ramesh sells 10 kg of waste paper, 5 kg of plastic, and 4 kg of iron.
(a) Find the cost of each item sold.
(b) What is the total money Ramesh earns?
[3 Marks]
Ramesh sells 10 kg of waste paper, 5 kg of plastic, and 4 kg of iron.
(a) Find the cost of each item sold.
(b) What is the total money Ramesh earns?
✅ Step-by-Step Solution & Working
Calculations:
• Waste paper: 10 × ₹12 = ₹120.
• Plastic: 5 × ₹18 = ₹90.
• Iron: 4 × ₹25 = ₹100.
• Total Earnings: ₹120 + ₹90 + ₹100 = ₹310.
• Waste paper: 10 × ₹12 = ₹120.
• Plastic: 5 × ₹18 = ₹90.
• Iron: 4 × ₹25 = ₹100.
• Total Earnings: ₹120 + ₹90 + ₹100 = ₹310.
CBSE Step-by-Step Marking Scheme
Individual costs (₹120, ₹90, ₹100)
1.5 M
Total = ₹310
1.5 M
Q9
Fractions Exploration • Parts of a Whole
A rectangular chocolate bar has 16 equal pieces. Reena gives 1/4 of the chocolate to her brother and 1/2 of it to her friend.
(a) How many pieces does her brother receive?
(b) How many pieces does her friend receive?
(c) How many pieces are left for Reena?
[3 Marks]
(a) How many pieces does her brother receive?
(b) How many pieces does her friend receive?
(c) How many pieces are left for Reena?
✅ Step-by-Step Solution & Working
Fractional Breakdown:
• Total pieces = 16.
• Brother's share (1/4): 16 ÷ 4 = 4 pieces.
• Friend's share (1/2): 16 ÷ 2 = 8 pieces.
• Remaining for Reena: 16 - (4 + 8) = 16 - 12 = 4 pieces (which is 1/4 of the bar).
• Total pieces = 16.
• Brother's share (1/4): 16 ÷ 4 = 4 pieces.
• Friend's share (1/2): 16 ÷ 2 = 8 pieces.
• Remaining for Reena: 16 - (4 + 8) = 16 - 12 = 4 pieces (which is 1/4 of the bar).
CBSE Step-by-Step Marking Scheme
Brother = 4 pieces
1 M
Friend = 8 pieces
1 M
Reena = 4 pieces
1 M
Q10
Time & Clocks • Elapsed Time
A school cricket match began at 9:30 AM and concluded at 1:15 PM.
(a) Convert 1:15 PM into 24-hour railway clock time.
(b) How long did the cricket match last?
[3 Marks]
(a) Convert 1:15 PM into 24-hour railway clock time.
(b) How long did the cricket match last?
✅ Step-by-Step Solution & Working
Time Calculations:
• (a) 24-hour time: 1:15 PM = 12:00 + 1:15 = 13:15 hours.
• (b) Duration:
• From 9:30 AM to 12:30 PM = 3 hours.
• From 12:30 PM to 1:15 PM = 45 minutes.
• Total duration = 3 hours and 45 minutes.
• (a) 24-hour time: 1:15 PM = 12:00 + 1:15 = 13:15 hours.
• (b) Duration:
• From 9:30 AM to 12:30 PM = 3 hours.
• From 12:30 PM to 1:15 PM = 45 minutes.
• Total duration = 3 hours and 45 minutes.
CBSE Step-by-Step Marking Scheme
13:15 hours in 24-hr clock
1 M
Duration = 3 hours 45 minutes with steps
2 M
Q11
Perimeter & Area • Fencing a Field
A square vegetable garden has a side length of 15 metres.
(a) What is the total length of wire fence needed to go around the garden once?
(b) If fencing costs ₹20 per metre, find the total cost of fencing.
[3 Marks]
(a) What is the total length of wire fence needed to go around the garden once?
(b) If fencing costs ₹20 per metre, find the total cost of fencing.
✅ Step-by-Step Solution & Working
Perimeter & Cost:
• (a) Perimeter of square: 4 × Side = 4 × 15 m = 60 metres.
• (b) Total Cost: Perimeter × Rate = 60 m × ₹20/m = ₹1,200.
• (a) Perimeter of square: 4 × Side = 4 × 15 m = 60 metres.
• (b) Total Cost: Perimeter × Rate = 60 m × ₹20/m = ₹1,200.
CBSE Step-by-Step Marking Scheme
Perimeter = 60 m
1.5 M
Cost = ₹1,200
1.5 M
Q12
Mental Math • Division and Multiplication
Solve mentally using number breakdown:
(a) 72 ÷ 6 = ?
(b) 15 × 8 = ?
(c) 25 × 40 = ?
(d) 96 ÷ 8 = ?
[2 Marks]
(a) 72 ÷ 6 = ?
(b) 15 × 8 = ?
(c) 25 × 40 = ?
(d) 96 ÷ 8 = ?
✅ Step-by-Step Solution & Working
Solutions:
• (a) 72 ÷ 6 = (60 ÷ 6) + (12 ÷ 6) = 10 + 2 = 12.
• (b) 15 × 8 = (10 × 8) + (5 × 8) = 80 + 40 = 120.
• (c) 25 × 40 = (25 × 4) × 10 = 100 × 10 = 1000.
• (d) 96 ÷ 8 = (80 ÷ 8) + (16 ÷ 8) = 10 + 2 = 12.
• (a) 72 ÷ 6 = (60 ÷ 6) + (12 ÷ 6) = 10 + 2 = 12.
• (b) 15 × 8 = (10 × 8) + (5 × 8) = 80 + 40 = 120.
• (c) 25 × 40 = (25 × 4) × 10 = 100 × 10 = 1000.
• (d) 96 ÷ 8 = (80 ÷ 8) + (16 ÷ 8) = 10 + 2 = 12.
CBSE Step-by-Step Marking Scheme
All 4 mental math facts correct
2 M
Q13
Patterns & Symmetries • Rotational Turns
Look at the letter 'H' and the letter 'N':
(a) What does each look like after a half-turn (180° rotation)?
(b) What does each look like after a quarter-turn (90° rotation)?
[2 Marks]
(a) What does each look like after a half-turn (180° rotation)?
(b) What does each look like after a quarter-turn (90° rotation)?
✅ Step-by-Step Solution & Working
Rotational Symmetry:
• (a) Half-turn (180°): Both 'H' and 'N' look exactly the same after a half-turn.
• (b) Quarter-turn (90°): 'H' rotates to look like 'I' with sleeping bars, and 'N' rotates to look like 'Z'.
• (a) Half-turn (180°): Both 'H' and 'N' look exactly the same after a half-turn.
• (b) Quarter-turn (90°): 'H' rotates to look like 'I' with sleeping bars, and 'N' rotates to look like 'Z'.
CBSE Step-by-Step Marking Scheme
Half-turn explanation (same)
1 M
Quarter-turn explanation
1 M
Q14
Smart Charts • Data Interpretation
In a Class 4 survey of 40 children, 20 like Cricket, 10 like Football, and 10 like Badminton.
(a) What fraction of children like Cricket?
(b) Draw a simple circular pie chart (Chapati Chart) representing this data.
[3 Marks]
(a) What fraction of children like Cricket?
(b) Draw a simple circular pie chart (Chapati Chart) representing this data.
✅ Step-by-Step Solution & Working
Data & Chapati Chart:
• (a) Fraction: 20 out of 40 = 20/40 = 1/2 (half of the class).
• (b) Chapati Chart: A circle divided in half represents Cricket (1/2). The other half is split equally into two quarters (1/4 for Football and 1/4 for Badminton).
• (a) Fraction: 20 out of 40 = 20/40 = 1/2 (half of the class).
• (b) Chapati Chart: A circle divided in half represents Cricket (1/2). The other half is split equally into two quarters (1/4 for Football and 1/4 for Badminton).
CBSE Step-by-Step Marking Scheme
Fraction = 1/2 for cricket
1.5 M
Pie chart sectors explained (1/2, 1/4, 1/4)
1.5 M
Q15
Chapter Review • True or False
State True or False:
(a) A circle has no corners and no straight edges.
(b) 1 litre is equal to 100 millilitres.
(c) The perimeter of any rectangle is 2 × (Length + Breadth).
[3 Marks]
(a) A circle has no corners and no straight edges.
(b) 1 litre is equal to 100 millilitres.
(c) The perimeter of any rectangle is 2 × (Length + Breadth).
✅ Step-by-Step Solution & Working
Answers:
• (a) True: A circle is a smooth curve with 0 corners.
• (b) False: 1 Litre is equal to 1000 millilitres.
• (c) True: Perimeter = 2 × (L + B).
• (a) True: A circle is a smooth curve with 0 corners.
• (b) False: 1 Litre is equal to 1000 millilitres.
• (c) True: Perimeter = 2 × (L + B).
CBSE Step-by-Step Marking Scheme
True, False, True with explanations
3 M
🌟 NEP 2020 Competency-Based Case Study
Secret Spy Code Pattern
Rohan receives an encrypted message: 'GDKKN VNQKC'. The encryption rule shifts every English letter 1 position backward in the alphabet (e.g., B becomes A, C becomes B).
Analytical Task: Decrypt Rohan's secret message using the reverse pattern rule.
Evaluation & Model Answer:
Decrypted Message: 'HELLO WORLD'.
Working: To reverse the encryption, we shift each letter 1 step forward in the alphabet:
• G ➔ H, D ➔ E, K ➔ L, K ➔ L, N ➔ O (HELLO)
• V ➔ W, N ➔ O, Q ➔ R, K ➔ L, C ➔ D (WORLD).
Working: To reverse the encryption, we shift each letter 1 step forward in the alphabet:
• G ➔ H, D ➔ E, K ➔ L, K ➔ L, N ➔ O (HELLO)
• V ➔ W, N ➔ O, Q ➔ R, K ➔ L, C ➔ D (WORLD).
Rubric: 1 Mark for reverse shift rule; 1 Mark for correct decrypted phrase.