NCERT Textbook: Patterns in Mathematics | Mathematics (Maths) Class 6 PDF Download

 Page 1


PATTERNS IN
 MATHEMATICS
1
1.1 What is Mathematics?
Mathematics is, in large part, the search for patterns, and for 
the explanations as to why those patterns exist.
Such patterns indeed exist all around us — in nature, in 
our homes and schools, and in the motion of the sun, moon, 
and stars. They occur in everything that we do and see, from 
shopping and cooking, to throwing a ball and playing games, to 
understanding weather patterns and using technology.
The search for patterns and their explanations can be a fun 
and creative endeavour. It is for this reason that mathematicians 
think of mathematics both as an art and as a science. This year, we 
hope that you will get a chance to see the creativity and artistry 
involved in discovering and understanding mathematical 
patterns.
It is important to keep in mind that mathematics 
aims to not just find out what patterns exist, but also the 
explanations for why they exist. Such explanations can 
often then be used in applications well beyond the context in 
which they were discovered, which can then help to propel  
humanity forward.
Chapter 1_Patterns in Mathematics.indd   1 10-08-2024   11:55:05
Page 2


PATTERNS IN
 MATHEMATICS
1
1.1 What is Mathematics?
Mathematics is, in large part, the search for patterns, and for 
the explanations as to why those patterns exist.
Such patterns indeed exist all around us — in nature, in 
our homes and schools, and in the motion of the sun, moon, 
and stars. They occur in everything that we do and see, from 
shopping and cooking, to throwing a ball and playing games, to 
understanding weather patterns and using technology.
The search for patterns and their explanations can be a fun 
and creative endeavour. It is for this reason that mathematicians 
think of mathematics both as an art and as a science. This year, we 
hope that you will get a chance to see the creativity and artistry 
involved in discovering and understanding mathematical 
patterns.
It is important to keep in mind that mathematics 
aims to not just find out what patterns exist, but also the 
explanations for why they exist. Such explanations can 
often then be used in applications well beyond the context in 
which they were discovered, which can then help to propel  
humanity forward.
Chapter 1_Patterns in Mathematics.indd   1 10-08-2024   11:55:05
Ganita Prakash | Grade 6
2
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
1. Can you think of other examples where mathematics helps 
us in our everyday lives?
2. How has mathematics helped propel humanity forward? (Y ou 
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.)
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. 
Table 1 shows some key number sequences that are studied in 
Mathematics.
Math 
Talk
Chapter 1_Patterns in Mathematics.indd   2 10-08-2024   11:55:05
Page 3


PATTERNS IN
 MATHEMATICS
1
1.1 What is Mathematics?
Mathematics is, in large part, the search for patterns, and for 
the explanations as to why those patterns exist.
Such patterns indeed exist all around us — in nature, in 
our homes and schools, and in the motion of the sun, moon, 
and stars. They occur in everything that we do and see, from 
shopping and cooking, to throwing a ball and playing games, to 
understanding weather patterns and using technology.
The search for patterns and their explanations can be a fun 
and creative endeavour. It is for this reason that mathematicians 
think of mathematics both as an art and as a science. This year, we 
hope that you will get a chance to see the creativity and artistry 
involved in discovering and understanding mathematical 
patterns.
It is important to keep in mind that mathematics 
aims to not just find out what patterns exist, but also the 
explanations for why they exist. Such explanations can 
often then be used in applications well beyond the context in 
which they were discovered, which can then help to propel  
humanity forward.
Chapter 1_Patterns in Mathematics.indd   1 10-08-2024   11:55:05
Ganita Prakash | Grade 6
2
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
1. Can you think of other examples where mathematics helps 
us in our everyday lives?
2. How has mathematics helped propel humanity forward? (Y ou 
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.)
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. 
Table 1 shows some key number sequences that are studied in 
Mathematics.
Math 
Talk
Chapter 1_Patterns in Mathematics.indd   2 10-08-2024   11:55:05
Patterns in Mathematics
3
Table 1 Examples of number sequences
 1, 1, 1, 1, 1, 1, 1, ... (All 1’s)
 1, 2, 3, 4, 5, 6, 7, ... (Counting numbers)
 1, 3, 5, 7, 9, 11, 13, ...  (Odd numbers)
 2, 4, 6, 8, 10, 12, 14, ... (Even numbers)
 1, 3, 6, 10, 15, 21, 28, ... (Triangular numbers)
 1, 4, 9, 16, 25, 36, 49, ...  (Squares)
 1, 8, 27, 64, 125, 216, ...  (Cubes)
 1, 2, 3, 5, 8, 13, 21, ...  (Virahanka numbers)
 1, 2, 4, 8, 16, 32, 64, ...  (Powers of 2)
 1, 3, 9, 27, 81, 243, 729, ...  (Powers of 3)
 Figure it Out
1.  Can you recognize the pattern in each of the sequences 
in Table 1?
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.
1.3  Visualising Number Sequences
Many number sequences can be visualised using pictures. 
Visualising mathematical objects through pictures or diagrams can 
be a very fruitful way to understand mathematical patterns and 
concepts. 
Let us represent the first seven sequences in Table 1 using the 
following pictures.
Math 
Talk
Chapter 1_Patterns in Mathematics.indd   3 10-08-2024   11:55:05
Page 4


PATTERNS IN
 MATHEMATICS
1
1.1 What is Mathematics?
Mathematics is, in large part, the search for patterns, and for 
the explanations as to why those patterns exist.
Such patterns indeed exist all around us — in nature, in 
our homes and schools, and in the motion of the sun, moon, 
and stars. They occur in everything that we do and see, from 
shopping and cooking, to throwing a ball and playing games, to 
understanding weather patterns and using technology.
The search for patterns and their explanations can be a fun 
and creative endeavour. It is for this reason that mathematicians 
think of mathematics both as an art and as a science. This year, we 
hope that you will get a chance to see the creativity and artistry 
involved in discovering and understanding mathematical 
patterns.
It is important to keep in mind that mathematics 
aims to not just find out what patterns exist, but also the 
explanations for why they exist. Such explanations can 
often then be used in applications well beyond the context in 
which they were discovered, which can then help to propel  
humanity forward.
Chapter 1_Patterns in Mathematics.indd   1 10-08-2024   11:55:05
Ganita Prakash | Grade 6
2
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
1. Can you think of other examples where mathematics helps 
us in our everyday lives?
2. How has mathematics helped propel humanity forward? (Y ou 
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.)
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. 
Table 1 shows some key number sequences that are studied in 
Mathematics.
Math 
Talk
Chapter 1_Patterns in Mathematics.indd   2 10-08-2024   11:55:05
Patterns in Mathematics
3
Table 1 Examples of number sequences
 1, 1, 1, 1, 1, 1, 1, ... (All 1’s)
 1, 2, 3, 4, 5, 6, 7, ... (Counting numbers)
 1, 3, 5, 7, 9, 11, 13, ...  (Odd numbers)
 2, 4, 6, 8, 10, 12, 14, ... (Even numbers)
 1, 3, 6, 10, 15, 21, 28, ... (Triangular numbers)
 1, 4, 9, 16, 25, 36, 49, ...  (Squares)
 1, 8, 27, 64, 125, 216, ...  (Cubes)
 1, 2, 3, 5, 8, 13, 21, ...  (Virahanka numbers)
 1, 2, 4, 8, 16, 32, 64, ...  (Powers of 2)
 1, 3, 9, 27, 81, 243, 729, ...  (Powers of 3)
 Figure it Out
1.  Can you recognize the pattern in each of the sequences 
in Table 1?
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.
1.3  Visualising Number Sequences
Many number sequences can be visualised using pictures. 
Visualising mathematical objects through pictures or diagrams can 
be a very fruitful way to understand mathematical patterns and 
concepts. 
Let us represent the first seven sequences in Table 1 using the 
following pictures.
Math 
Talk
Chapter 1_Patterns in Mathematics.indd   3 10-08-2024   11:55:05
Ganita Prakash | Grade 6
4
Table 2 Pictorial representation of some number sequences 
Squares
1 4 9 16
25
Triangular 
numbers
6 3 1 10 15
Cubes
27 8 1 64 125
           
All 1’s
 1  1  1  1  1  
          
Counting
 
 1  2   3  4  5 
numbers
           
Odd
 1  3  5  7  9  
numbers
        
Even
 2  4  6  8  10  
numbers
Chapter 1_Patterns in Mathematics.indd   4 10-08-2024   11:55:06
Page 5


PATTERNS IN
 MATHEMATICS
1
1.1 What is Mathematics?
Mathematics is, in large part, the search for patterns, and for 
the explanations as to why those patterns exist.
Such patterns indeed exist all around us — in nature, in 
our homes and schools, and in the motion of the sun, moon, 
and stars. They occur in everything that we do and see, from 
shopping and cooking, to throwing a ball and playing games, to 
understanding weather patterns and using technology.
The search for patterns and their explanations can be a fun 
and creative endeavour. It is for this reason that mathematicians 
think of mathematics both as an art and as a science. This year, we 
hope that you will get a chance to see the creativity and artistry 
involved in discovering and understanding mathematical 
patterns.
It is important to keep in mind that mathematics 
aims to not just find out what patterns exist, but also the 
explanations for why they exist. Such explanations can 
often then be used in applications well beyond the context in 
which they were discovered, which can then help to propel  
humanity forward.
Chapter 1_Patterns in Mathematics.indd   1 10-08-2024   11:55:05
Ganita Prakash | Grade 6
2
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
1. Can you think of other examples where mathematics helps 
us in our everyday lives?
2. How has mathematics helped propel humanity forward? (Y ou 
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.)
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. 
Table 1 shows some key number sequences that are studied in 
Mathematics.
Math 
Talk
Chapter 1_Patterns in Mathematics.indd   2 10-08-2024   11:55:05
Patterns in Mathematics
3
Table 1 Examples of number sequences
 1, 1, 1, 1, 1, 1, 1, ... (All 1’s)
 1, 2, 3, 4, 5, 6, 7, ... (Counting numbers)
 1, 3, 5, 7, 9, 11, 13, ...  (Odd numbers)
 2, 4, 6, 8, 10, 12, 14, ... (Even numbers)
 1, 3, 6, 10, 15, 21, 28, ... (Triangular numbers)
 1, 4, 9, 16, 25, 36, 49, ...  (Squares)
 1, 8, 27, 64, 125, 216, ...  (Cubes)
 1, 2, 3, 5, 8, 13, 21, ...  (Virahanka numbers)
 1, 2, 4, 8, 16, 32, 64, ...  (Powers of 2)
 1, 3, 9, 27, 81, 243, 729, ...  (Powers of 3)
 Figure it Out
1.  Can you recognize the pattern in each of the sequences 
in Table 1?
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.
1.3  Visualising Number Sequences
Many number sequences can be visualised using pictures. 
Visualising mathematical objects through pictures or diagrams can 
be a very fruitful way to understand mathematical patterns and 
concepts. 
Let us represent the first seven sequences in Table 1 using the 
following pictures.
Math 
Talk
Chapter 1_Patterns in Mathematics.indd   3 10-08-2024   11:55:05
Ganita Prakash | Grade 6
4
Table 2 Pictorial representation of some number sequences 
Squares
1 4 9 16
25
Triangular 
numbers
6 3 1 10 15
Cubes
27 8 1 64 125
           
All 1’s
 1  1  1  1  1  
          
Counting
 
 1  2   3  4  5 
numbers
           
Odd
 1  3  5  7  9  
numbers
        
Even
 2  4  6  8  10  
numbers
Chapter 1_Patterns in Mathematics.indd   4 10-08-2024   11:55:06
Patterns in Mathematics
5
 Figure it Out 
1. Copy the pictorial representations of the number sequences 
in Table 2 in your notebook, and draw the next picture for 
each sequence!
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?
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! 
4. What would you call the following sequence of numbers? 
 
1 7 19 37
 That’s right, they are called hexagonal numbers! Draw these in 
your notebook. What is the next number in the sequence?
5. Can you think of pictorial ways to visualise the sequence of 
Powers of 2? Powers of 3?
Math 
Talk
Chapter 1_Patterns in Mathematics.indd   5 10-08-2024   11:55:06