Class 6 NCERT Solutions (2024-2025) Ganita Prakash Maths Chapter 1 Patterns in Mathematics
1.1 What is Mathematics?
Mathematics is the study of numbers, quantities, shapes, patterns, and logical relationships. It is both a science and a language that helps us describe, analyze, and solve problems in the real world.
Key Aspects of Mathematics:
Numbers & Calculations – Arithmetic (addition, subtraction, multiplication, division), algebra, and number theory.
Shapes & Spaces – Geometry, trigonometry, and measurements (area, volume, angles).
Patterns & Relationships – Algebra, functions, and sequences.
Data & Chance – Statistics (averages, graphs) and probability (predicting outcomes).
Logic & Reasoning – Problem-solving, proofs, and critical thinking.
Why is Mathematics Important?
It helps in daily tasks (shopping, cooking, time management).
It is essential in science, engineering, technology, and finance.
It improves logical thinking and problem-solving skills.
Mathematics in Nature & Real Life:
Symmetry in flowers and snowflakes.
Patterns in weather, music, and art.
Measurements in construction, medicine, and sports.
In short, mathematics is the foundation of understanding and organizing the world around us!
For example, the understanding of patterns in the motion of stars,
planets, and their satellites led humankind to develop the theory of
gravitation, allowing us to launch our own satellites and send rockets
to the Moon and to Mars; similarly, understanding patterns in genomes
has helped in diagnosing and curing diseases—among thousands of
other such examples
Figure it Out Page number 2
1. Can you think of other examples where mathematics helps
us in our everyday lives?
Solution:
Mathematics plays a crucial role in our everyday lives in many ways.Examples of Mathematics in Daily Life:
Budgeting & Shopping:
Calculating expenses, discounts, and managing savings.
Comparing prices to find the best deals.
Time Management:
Cooking & Baking:
Travel & Navigation:
Home Improvement:
Measuring rooms for furniture, paint, or flooring.
Calculating areas and volumes for construction.
Banking & Finance:
Sports & Fitness:
Technology & Gadgets:
Health & Medicine:
Weather Forecasting:
Conclusion:
Mathematics plays a vital role in various professions, such as:
- Helping vegetable sellers calculate change accurately
- Enabling pot makers to craft pots of precise dimensions
- Assisting painters in estimating time and materials needed for a project
- Guiding masons in calculating the required number of bricks
- Aiding doctors in determining the correct dosage of medication
Mathematics is an essential tool in many real-world applications
Mathematics is everywhere—from simple daily tasks to complex decision-making. It helps us solve problems efficiently and make informed choices.
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2. How has mathematics helped propel humanity forward? (You
might think of examples involving: carrying out scientific
experiments; running our economy and democracy; building
bridges, houses or other complex structures; making TVs,
mobile phones, computers, bicycles, trains, cars, planes,
calendars, clocks, etc.).
Solution:
Mathematics has been the backbone of human progress, propelling advancements in science, technology, engineering, and society. Here are some key ways it has driven humanity forward:
1. Scientific Discovery & Experimentation
Physics & Astronomy: Mathematics enabled Newton’s laws of motion, Einstein’s theory of relativity, and space exploration (e.g., calculating rocket trajectories).
Medicine: Statistical models help in drug trials, epidemiology (e.g., predicting disease spread), and medical imaging (MRI, CT scans).
Chemistry: Equations predict reactions, helping design new materials, medicines, and energy solutions.
2. Engineering & Infrastructure
Bridges & Buildings: Calculus and geometry ensure stable structures (e.g., suspension bridges, skyscrapers).
Transportation: Algorithms optimize traffic flow; equations design safer cars, planes, and high-speed trains.
Electricity & Electronics: Maxwell’s equations underpin electrical engineering; binary math powers computers.
3. Technology & Communication
Computers & AI: Binary math, algorithms, and cryptography drive computing and machine learning.
Internet & Phones: Data compression (e.g., JPEG, MP3) and error-correcting codes enable fast, reliable communication.
GPS & Navigation: Relies on trigonometry and relativity to pinpoint locations accurately.
4. Economics & Governance
Finance & Markets: Probability and statistics guide investments, risk assessment, and economic policies.
Democracy & Voting Systems: Game theory helps design fair voting mechanisms and prevent manipulation.
Logistics & Supply Chains: Optimization math ensures efficient delivery of goods (e.g., Amazon, FedEx).
5. Everyday Life Innovations
Clocks & Calendars: Astronomy-based math keeps time accurate (leap years, time zones).
Consumer Tech: Math designs TVs (pixel algorithms), cameras (image processing), and even bicycles (gear ratios).
Weather Forecasting: Differential equations and supercomputers predict storms and climate trends.
Conclusion
Mathematics has helped propel humanity forward by Scientific Discovery, Technology, Economic and growth.
Mathematics is the "language of the universe," allowing us to model reality, solve problems, and innovate. Without it, modern civilization—from smartphones to space travel—would not exist.
1.2 Patterns in Numbers
Among the most basic patterns that occur in mathematics are
patterns of numbers, particularly patterns of whole numbers:
0, 1, 2, 3, 4, ...
The branch of Mathematics that studies patterns in whole
numbers is called number theory.
Number sequences are the most basic and among the most
fascinating types of patterns that mathematicians study.
Some key number sequences that are studied in
Mathematics.
1. All 1’s (1, 1, 1, 1, ...)
Q: What is the next term in the sequence: 1, 1, 1, 1, 1, ...?
A: 1
2. Counting Numbers (1, 2, 3, 4, ...)
Q: What is the 10th term in the sequence of counting numbers?
A: 10
3. Odd Numbers (1, 3, 5, 7, ...)
Q: What is the 6th odd number?
A: 11
4. Even Numbers (2, 4, 6, 8, ...)
Q: What is the sum of the first 3 even numbers?
A: 2 + 4 + 6 = 12
5. Triangular Numbers (1, 3, 6, 10, ...)
Q: How do you generate the next triangular number after 10?
A: Add the next counting number (5) → 10 + 5 = 15
6. Square Numbers (1, 4, 9, 16, ...)
Q: What is the 7th square number?
A: 7² = 49
7. Cube Numbers (1, 8, 27, 64, ...)
Q: What is the cube of 5?
A: 5³ = 125
8. Fibonacci (Virahฤnka) Numbers (1, 2, 3, 5, 8, 13, ...)
Q: What is the next Fibonacci number after 13?
A: 8 + 13 = 21
9. Powers of 2 (1, 2, 4, 8, 16, ...)
Q: What is 2 raised to the power of 6?
A: 64
10. Powers of 3 (1, 3, 9, 27, ...)
Q: What is 3⁵ (3 to the 5th power)?
A: 243
Bonus Challenge Questions:
Q: Which sequence has terms that are sums of the previous two numbers?
A: Fibonacci (Virahฤnka) numbers
Q: Which sequence represents numbers that can form equilateral triangles?
A: Triangular numbers
Q: What is the difference between consecutive square numbers? (e.g., 4 - 1 = 3, 9 - 4 = 5, etc.)
A: The differences are consecutive odd numbers.
Figure it Out Page Number 3
1. Can you recognize the pattern in each of the sequences
in Table 1?
Solution: Recognizing the Patterns in Each Sequence
| Sequence | Pattern (Rule) |
|---|
| 1, 1, 1, 1, 1, ... | All 1’s – Every term is 1. |
| 1, 2, 3, 4, 5, ... | Counting numbers – Start at 1, increase by 1 each time. |
| 1, 3, 5, 7, 9, ... | Odd numbers – Start at 1, increase by 2 each time. |
| 2, 4, 6, 8, 10, ... | Even numbers – Start at 2, increase by 2 each time. |
| 1, 3, 6, 10, 15, ... | Triangular numbers – Add the next counting number (e.g., 10 + 5 = 15). |
| 1, 4, 9, 16, 25, ... | Square numbers – Each term is n² (1², 2², 3², ...). |
| 1, 8, 27, 64, 125, ... | Cube numbers – Each term is n³ (1³, 2³, 3³, ...). |
| 1, 2, 3, 5, 8, 13, ... | Fibonacci (Virahฤnka) numbers – Each term is the sum of the two before it. |
| 1, 2, 4, 8, 16, 32, ... | Powers of 2 – Each term is 2โฟ⁻¹ (2⁰=1, 2¹=2, 2²=4, ...). |
| 1, 3, 9, 27, 81, 243, ... | Powers of 3 – Each term is 3โฟ⁻¹ (3⁰=1, 3¹=3, 3²=9, ...). |
2. Rewrite each sequence of Table 1 in your notebook, along
with the next three numbers in each sequence! After
each sequence, write in your own words what is the rule
for forming the numbers in the sequence.
Solution: Extending Each Sequence (Next 3 Numbers) + Rule in My Own Words
(a) All 1’s ( 1, 1, 1, 1, 1, 1, 1 ...)
(b) Counting Numbers ( 1, 2, 3, 4, 5, 6, 7 ...)
(c) Odd Numbers (1, 3, 5, 7, 9, 11 , 13 ...)
(d) Even Numbers ( 2,4, 6, 8, 10, 12, 14, .......)
(e) Triangular Numbers (1, 3, 6, 10, 15, 21, 28 ...)
Next 3 terms: 36, 45, 55
Rule: Start at 1, then add 2, then 3, then 4, and so on (each time adding the next counting number).
(f) Square Numbers ( 1, 4, 9, 16, 25,36,49,...)
Next 3 terms: 64, 81, 100
Rule: Each number is a perfect square (1×1, 2×2, 3×3, etc.).
(g) Cube Numbers (1, 8, 27, 64, 125, 216, 343, ...)
Next 3 terms: 512, 729,1000
Rule: Each number is a perfect cube (1×1×1, 2×2×2, 3×3×3, etc.).
(h) Fibonacci (Virahฤnka) Numbers (1, 2, 3, 5, 8, 13, 21,...)
(i) Powers of 2 (1, 2, 4, 8, 16, 32, 64, ...)
Next 3 terms: 128, 256, 512
Rule: Each number is double the previous one (or 2 raised to increasing powers).
(j) Powers of 3 (1, 3, 9, 27, 81, 243, 729, ...)
Next 3 terms: 2187, 6561, 19683
Rule: Each number is triple the previous one (or 3 raised to increasing powers).
1.3 Visualising Number Sequences
Pictorial representation of some number sequences:
(Pictorial Number Sequences)
1. All 1’s (1, 1, 1, 1, ...)
Q: If this sequence continues forever, will any term ever be different from 1?
A: No, every term is always 1.
2. Counting Numbers (1, 2, 3, 4, ...)
Q: How would the 5th term in this sequence be represented pictorially?
A: A group of 5 objects (e.g., 5 dots or 5 sticks).
3. Odd Numbers (1, 3, 5, 7, ...)
Q: If the sequence represents "dots in stacked L-shapes," how many dots form the 4th L-shape?
A: 7 dots (since the 4th odd number is 7).
4. Even Numbers (2, 4, 6, 8, ...)
Q: If these numbers represent pairs of shoes, how many shoes are there in the 5th term?
A: 10 shoes (5th even number = 10).
5. Triangular Numbers (1, 3, 6, 10, ...)
Q: If each term forms a triangle with dots, how many dots are added to go from the 3rd to the 4th triangle?
A: 4 dots (3rd term = 6 dots, 4th term = 10 dots → 10 – 6 = 4).
6. Square Numbers (1, 4, 9, 16, ...)
Q: If each square number is a grid of dots (e.g., 1×1, 2×2), how many dots form the 5th square?
A: 25 dots (5 × 5 grid).
7. Cube Numbers (1, 8, 27, 64, ...)
Q: If cubes are represented as 3D stacks of blocks, how many blocks make the 3rd cube?
A: 27 blocks (3 × 3 × 3).
Bonus Challenge Questions
Q: Which sequence would form a perfect staircase when drawn as dots?
A: Triangular numbers (each layer adds one more dot than the last).
Q: If "odd numbers" were represented as nested L-shapes, what would the 5th L-shape look like?
A: A larger L with 9 dots (5th odd number = 9).
Q: How are square numbers visually different from triangular numbers?
A: Squares form equal rows and columns (e.g., 4×4), while triangles form staircase layers (1, 2, 3, ... dots per row).
Figure it Out Page number 5
1. Copy the pictorial representations of the number sequences
in Table 2 in your notebook, and draw the next picture for
each sequence!
Solution:| Sequence | Visual Pattern | Next Term Drawing |
|---|
| All 1’s | ● (single dot) | ● |
| Counting numbers | ● → ●● → ●●● → ●●●● → ●●●●● | ●●●●●● (6 dots) |
| Odd numbers | L-shapes: 1, 3, 5, 7 ,9 dots | ●●●●● ●●●●●●
L with 11 dots (6th odd number) |
| Even numbers | Pairs: ●● → ●●●● → ●●●●●● → ●●●●●●●● | ●●●●●●●●●● (10 dots) |
| Triangular numbers | △ layers: 1, 3, 6, 10 dots | △ with 15 dots (add a row of 5) |
| Square numbers | 1×1, 2×2, 3×3, 4×4 grids | 5×5 grid (25 dots) |
| Cube numbers | 1³=1 block, 2³=8 blocks, 3³=27 blocks | 4×4×4 cube (64 blocks) |
2. Why are 1, 3, 6, 10, 15, … called triangular numbers? Why
are 1, 4, 9, 16, 25, … called square numbers or squares?
Why are 1, 8, 27, 64, 125, … called cubes?
Solution:
Triangular numbers: These numbers can be arranged in the shape of
a triangle by placing dots in increasing rows.
Square numbers: These numbers can be arranged in the shape of a
square by placing dots in equal rows and columns.
Cubes: These numbers can be arranged in the shape of a cube by
placing dots in three-dimensional layers
Triangular numbers (1, 3, 6, 10...):
Form equilateral triangles with dots:
● ● ● ●
● ● ● ● ● ●
● ● ● ● ● ●
● ● ● ●
Rule: Add a new row with +1 dot each time.
Square numbers (1, 4, 9, 16...):
Form perfect squares:
□ □□ □□□ □□□□
□□ □□□ □□□□
□□□ □□□□
□□□□
Rule: n2 (e.g., 32=9 dots).
Cube numbers (1, 8, 27, 64...):
Form 3D cubes with blocks:
1 block → 2×2×2 → 3×3×3 → ...
3. You will have noticed that 36 is both a triangular number and a square number! That is, 36 dots can be arranged perfectly both in a triangle and in a square. Make pictures in your notebook illustrating this! This shows that the same number can be represented differently, and play different roles, depending on the context. Try representing some other numbers pictorially in different ways!
3. 36 as Both Triangular and Square
Triangular: 8th triangular number = 36 dots:●
● ●
● ● ●
● ● ● ●
● ● ● ● ●
● ● ● ● ● ●
● ● ● ● ● ● ●
● ● ● ● ● ● ● ●
... (8 rows) Square: 6×6 grid = 36 dots.
● ● ● ● ● ●
● ● ● ● ● ● ● ● ● ● ● ●
● ● ● ● ● ●
● ● ● ● ● ●
● ● ● ● ● ●
Try with 1 or 1225 (next such number)!
4. What would you call the following sequence of numbers? That’s right, they are called hexagonal numbers! Draw these in
your notebook. What is the next number in the sequence?
4. Hexagonal Numbers (1, 7, 19, 37...)These are called hexagonal numbers because they can be arranged in a
hexagon shape.
Visual: Nested hexagons:1 dot → 6 dots around it → 12 more → 18 more → ...
5. Can you think of pictorial ways to visualise the sequence of
Powers of 2? Powers of 3?
Here is one possible way of thinking about Powers of 2:
5. Visualizing Powers of 2 and 3
Powers of 2: You can visualize the powers of 2 as squares where each
subsequent square has twice the number of smaller squares as the
previous one.
Powers of 3: You can visualize the powers of 3 as cubes, where each
subsequent cube has three times the number of smaller cubes as the
previous one.
● → ●● → ●●●● → ●●●●●●●● → ...
- Powers of 3 (1, 3, 9, 27...):
Triple each time:
● → △ of 3 → 3×3 grid → 3×3×3 cube → ...
Key Takeaways
Same number, different shapes: 36 can be a triangle and a square.
Patterns are everywhere: Hexagons, cubes, and grids reveal hidden math.
Try it! Draw your own sequences (e.g., pentagonal numbers).
Fun Challenge: Find the next number after 36 that’s both triangular and square! (Hint: It’s 1225.)
1. Square Grid Template
*(For square numbers, powers of 2, and 36-as-square)*
● ● ● ● ● ●
● ● ● ● ● ●
● ● ● ● ● ●
● ● ● ● ● ●
● ● ● ● ● ●
● ● ● ● ● ●
How to use:
Circle a 6×6 grid to show 36 as a square number.
Circle 1×1, 2×2, 3×3, etc. for square numbers.
For powers of 2: Start with 1 dot, then circle 2, 4, 8, etc.
2. Triangular Grid Template
*(For triangular numbers and 36-as-triangle)*
●
● ●
● ● ●
● ● ● ●
● ● ● ● ●
● ● ● ● ● ●
3. Hexagonal Grid Template
(For hexagonal numbers: 1, 7, 19, 37...)
● ● ●
● ● ● ●
● ● ● ● ●
● ● ● ●
● ● ●
How to use:
1 dot: Center dot.
7 dots: Center + 6 neighbors (first ring).
19 dots: Add 12 dots in the next ring (total = 7 + 12).
Next term (61): Add 24 dots in the outer ring (37 + 24).
4. Cube Template
(For cube numbers: 1, 8, 27...)
Front Layer Middle Layer Back Layer
● ● ● ● ● ● ● ● ●
● ● ● ● ● ● ● ● ●
● ● ● ● ● ● ● ● ●
How to use:
1³ = 1: One block in the center.
2³ = 8: 2×2×2 blocks (all layers).
3³ = 27: 3×3×3 blocks (fill all layers).
Activity Solutions
1. Drawing the Next Pictures
Triangular numbers: Add a row of 5 dots → 15 dots total.
Square numbers: Draw a 5×5 grid → 25 dots.
Cubes: Build a 4×4×4 cube → 64 blocks.
2. Why "Triangular," "Square," "Cubes"?
Triangular: Form perfect triangles (e.g., bowling pins).
Square: Form perfect squares (like a chessboard).
Cubes: Form 3D cubes (like Rubik’s cubes).
3. 36 as Triangle + Square
Triangle: 8 rows (1+2+3+...+8 = 36).
Square: 6×6 grid.
Triangular 36 Square 36
● ● ● ● ● ● ●
● ● ● ● ● ● ● ●
● ● ● ● ● ● ● ● ●
● ● ● ● ● ● ● ● ● ●
● ● ● ● ● ● ● ● ● ● ●
● ● ● ● ● ● ● ● ● ● ● ●
● ● ● ● ● ● ●
#### **4. Hexagonal Numbers**
- **Next number:** 61 (37 + 24).
- **Pattern:** Add multiples of 6 (6, 12, 18, 24...).
#### **5. Powers of 2/3 Visualization**
- **Powers of 2:** Binary splitting!
1 → ●
2 → ● ●
4 → ● ● ● ●
8 → ● ● ● ● ● ● ● ●
- **Powers of 3:** Ternary branching!
1 → ●
3 → ●─●─●
9 → 3 branches, each splitting into 3
### **Bonus Challenge**
Find the next number after 36 that’s both triangular *and* square!
**Hint:** It’s 1225 (triangular = 49th term, square = 35²).
1.4 Relations among Number Sequences
Example: What happens when we start adding up odd numbers?
1 = 1
1 + 3 = 4
1 + 3 + 5 = 9
1 + 3 + 5 + 7 = 16
1 + 3 + 5 + 7 + 9 = 25
1 + 3 + 5 + 7 + 9 + 11 = 36
.
.
.
This is a really beautiful pattern!
1 + 3 + 5 + 7 + 9 + 11 = 36.
Because such a picture can be made for a square of any size, this
explains why adding up odd numbers gives square numbers.
By drawing a similar picture, can you say what is the sum of the
first 10 odd numbers?
Now by imagining a similar picture, or by drawing it partially, as
needed, can you say what is the sum of the first 100 odd numbers?
Page 7
1. Can you find a similar pictorial explanation for why adding
counting numbers up and down gives square numbers?
Imagine arranging the numbers in rows, where each row adds one more
number and then subtracts one until it reaches back to 1.
When these numbers
are added up, they fill in the squares, forming perfect square grids.
2. By imagining a large version of your picture, or drawing it
partially, as needed, can you see what will be the value of 1 + 2 + 3
+ ... + 99 + 100 + 99 + ... + 3 + 2 + 1?
This question needs to be answered by students themselves. Here's an
explanation for better understanding:
When counting up to 100 and then back down, the total sum will form a square
number.
3. What sequence do you get when you start to add the All 1’s
sequence up? What sequence do you get when you add the All 1’s
sequence up and down?
Ans:
All 1’s sequence up: The result is the counting numbers (1, 2, 3,
4, ...).
All 1’s sequence up and down: The result is triangular numbers (1,
3, 6, 10, ...).
4. What sequence do you get when you start to add the Counting
numbers up? Can you give a smaller pictorial explanation?
Ans: When you start adding counting numbers, you get triangular numbers.
This question needs to be answered by students themselves.
5. What happens when you add up pairs of consecutive triangular
numbers?
Ans: Adding pairs of consecutive triangular numbers results in square
numbers.
6. What happens when you start to add up powers of 2 starting
with 1, and then add 1 to each of these numbers?
Ans: The sum of powers of 2 plus 1 results in the next power of 2, forming a
new sequence of powers of 2.
7. What happens when you multiply the triangular numbers by 6
and add 1? Can you explain it with a picture?
Ans: Multiplying triangular numbers by 6 and adding 1 results in a new
sequence.
8. What happens when you start to add up hexagonal
numbers? take 1, 1 + 7, 1 + 7 + 19, 1 + 7 + 19 + 37, ... ? Which
sequence do you
get? Can you explain it using a picture of a cube?
Adding up hexagonal numbers forms a new sequence.
9. Find your own patterns or relations in and among the
sequences in Table 1. Can you explain why they happen with a
picture or otherwise?
Various patterns can be discovered by exploring how sequences like triangular
numbers, squares, and cubes relate to each other. For example, triangular
numbers added together form square numbers, and visualizing this with
diagrams can help explain why this happens.
Page 10
1. Can you recognize the pattern in each of the sequences in Table
3?
This question encourages students to observe and identify the patterns
on their own. The patterns involve the growth and arrangement of shapes, such
as stacking triangles or squares and forming complete graphs. recognize how each sequence evolves and increases in complexity as more
shapes or connections are added.
2. How many little squares are there in each shape of the
sequence of Stacked Squares?
The number of little squares in each shape follows the pattern of square
numbers:
First shape: 1 square
Second shape: 4 squares (2x2)
Third shape: 9 squares (3x3)
Fourth shape: 16 squares (4x4)
Explanation: The number of little squares increases as the square of the
number of rows or columns in the shape, following the sequence 1, 4, 9, 16,
etc.
3. How many little triangles are there in each shape of the
sequence of Stacked Triangles?
The number of little triangles in each shape follows the pattern of
triangular numbers:
First shape: 1 triangle
Second shape: 3 triangles (1+2)
Third shape: 6 triangles (1+2+3)
Fourth shape: 10 triangles (1+2+3+4)
Explanation: The number of little triangles increases according to the
triangular number sequence, where each shape adds another row of triangles,
following the sequence 1, 3, 6, 10, etc.
4. How many total line segments are there in each shape of the
Koch Snowflake?
The number of line segments in each shape of the Koch Snowflake
increases as follows:
First shape: 3 line segments
Second shape: 12 line segments
Third shape: 48 line segments
Fourth shape: 192 line segments
Explanation: The number of line segments increases by multiplying by 4 each
time.
This results in the sequence 3, 12, 48, 192, which is 3 times the powers of
4 (i.e., 3 × 4^n)
