NCERT Solutions for Class 6 Maths Ganita Prakash Chapter 1 Patterns in Mathematics help students understand how mathematics is connected to everyday life through patterns and sequences.
The chapter begins by explaining the role of mathematics in daily activities like cooking, shopping, and traveling. It then introduces number patterns and sequences, encouraging students to identify rules behind them.
These solutions provide easy explanations, examples, and step-by-step answers to help students recognize patterns and think logically. It aligns with the latest CBSE Class 6 Maths Syllabus, helping students build strong logical and analytical thinking skills as per the syllabus guidelines.
NCERT Solutions for Class 6 Maths Ganita Prakash Chapter 1 Patterns in Mathematics
Mathematics is not just about numbers and formulas – it is all around us in our daily lives. In Chapter 1 of Ganita Prakash Class 6, titled "Patterns in Mathematics", students are introduced to the beauty of maths through patterns and sequences.
The chapter helps learners observe and understand how numbers and shapes follow certain rules or patterns. From natural things like petals on flowers to man-made things like calendars and tiles – patterns are everywhere!
This chapter encourages students to think logically, recognize repeated designs, and understand the order in numbers, helping build a strong foundation in mathematical thinking.
Class 6 Maths NCERT Solutions Ganita Prakash Chapter 1 Question Answers
Below are the detailed solutions for Patterns in Mathematics Class 6 solutions. These solutions are designed to help students understand how patterns work in numbers and shapes.
1.1 What is Mathematics? Figure it Out (Page No. 2)
Question 1. Can you think of other examples where mathematics helps us in our everyday lives?
Solution: Yes, maths helps us in many day-to-day things like managing money, cooking with correct measurements, calculating travel time and cost, decorating our homes, shopping, and even in playing games or keeping score.
Question 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 helped humanity grow in many ways. It is used in scientific experiments to understand the world better. It helps run our economy and support democracy by analyzing data and making fair systems.
Math is needed to build bridges, houses, and tall buildings safely. It also plays a key role in making TVs, mobile phones, computers, bicycles, trains, cars, planes, calendars, and clocks. Without math, it would be hard to design or improve these things. So, maths has made life easier, faster, and more advanced.
1.2 Patterns in Numbers Figure it Out (Page No. 3)
Question 1. Can you recognize the pattern in each of the sequences in Table 1?
Solution: Yes, the pattern in each of the sequences in Table 1 is recognizable.
Question 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:
1, 1, 1, 1, 1, 1, 1,… (All 1’s)
Next three numbers: 1, 1, 1
Rule: All 1s are to be written so that the next three numbers will also be 1.
1, 2, 3, 4, 5, 6, 7,….. (counting numbers)
Next three numbers: 8, 9, 10
Rule: Counting numbers increase by 1.
7 + 1 = 8
8 + 1 = 9
9 + 1 = 10
1, 3, 5, 7, 9, 11, 13,….. (odd numbers)
Next three numbers: 15, 17, 19
Rule: Odd numbers are not multiples of 2 and increase by 2.
13 + 2 = 15
15 + 2 = 17
17 + 2 = 19
2, 4, 6, 8, 10, 12, 14,….. (even numbers)
Next three numbers: 16, 18, 20
Rule: Even numbers are multiples of 2 and thus increase by 2.
14 + 2 = 16
16 + 2 = 18
18 + 2 = 20
1, 3, 6, 10, 15, 21, 28,….. (triangular numbers)
Next three numbers: 36, 45, 55
Rule: Two consecutive triangular numbers make a square number.
Here, 21 + 28 = 49
Now, next square number = 64, so 64 – 28 = 36
Next square number = 81, so 81 – 36 = 45
Next square number = 100, so 100 – 45 = 55
1, 4, 9, 16, 25, 36, 49,….. (squares)
Next three numbers: 64, 81, 100
Rule: 49 = 7 × 7
So 8 × 8 = 64
9 × 9 = 81
10 × 10 = 100
1, 8, 27, 64, 125, 216,….. (cubes)
Next three numbers: 343, 512, 729
Rule: 216 = 6 × 6 × 6
So 7 × 7 × 7 = 343
8 × 8 × 8 = 512
9 × 9 × 9 = 729
1, 2, 3, 5, 8, 13, 21… (Virahanka numbers)
Next three numbers: 34, 55, 89
Rule: The next Virahanka number is obtained by adding the previous 2 numbers.
13 + 21 = 34
34 + 21 = 55
55 + 34 = 89
1, 2, 4, 8, 16, 32, 64,….. (powers of 2)
Next three numbers = 128, 256, 512
Rule: 64 = 26
So 27 = 128
28 = 256
29 = 512
1, 3, 9, 27, 81, 243, 729,…… (powers of 3)
Next three numbers: 2,187, 6,561, 19,683
Rule: 729 = 36
So, 37 = 2,187
38 = 6,561
39 = 19,683
Question 1: 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:
-
The numbers 1, 3, 6, 10, 15, … are called triangular numbers because they can be represented by triangular arrangements of dots.
-
The numbers 1,4,9,16,25,… are called square numbers or squares because they can be represented by square arrangements of dots.
-
The numbers 1, 8, 27, 64, 125, … are called cubes, because they can be arranged in the form of cubes of unit blocks.
Question 2. 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!
Solution:
36 as a triangular number
or
36 as a square number
In the same way, number 9 can be represented in different ways as,
Similarly, number 10 can be represented as a rectangle and a triangle by arranging dots, as
There are many more numbers that can be represented in different ways.
Question 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?
Solution:
∵ 1,7, 19, 37 are hexagonal numbers as they follow a pattern. As
1
1 + 6 = 7
7 + 12 = 19
19 + 18 = 37
37 + 24 = 61
So, the next hexagonal number in the sequence is 61.
It can represented as
1.4 Relations Among Number Sequences Figure it Out (Page No. 8-9)
Question 1. Can you find a similar pictorial explanation for why adding counting numbers up and down, i.e., 1, 1 + 2 + 1, 1 + 2 + 3 + 2 + 1, …, gives square numbers?
Solution:
Question 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?
Solution:
1 = 1
1 + 2 + 1 = 4
1 + 2 + 3 + 2 + 1 = 9
1 + 2 + 3 + 4 + 3 + 2 + 1 = 16
1 + 2 + 3+ …+ 99 + 100 + 99 + … + 3 + 2 + 1 = 10000
Yes, we can see what will be the value of
1 + 2 + 3 +…+ 99 + 100 + 99 + … + 3 + 2 + 1
Question 3. Which 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?
Solution:
When we add all 1 ’s sequence up, we get the counting numbers, as
1 = 1,
1 + 1 = 2,
1 + 1 + 1 = 3,
1 + 1 + 1 + 1 = 4,…
When we add all 1 ’s sequence up and down, we get counting numbers depends upon number of times 1 occurs.
Question 4. Which sequence do you get when you start to add the counting numbers up? Can you give a smaller pictorial explanation?
Solution:
Adding counting numbers
1 = 1
1 + 2 = 3
1 + 2 + 3 = 6
1 + 2 + 3 + 4 = 10
1 + 2 + 3 + 4 + 5 = 15
…………
…………
…………
…………
and so on.
Here, we get sequence of triangular numbers.
A smaller pictorial explanation is
Question 5. What happens when you add up pairs of consecutive triangular numbers? That is, take 1 + 3, 3 + 6, 6 + 10, 10 + 15 …? Which sequence do you get? Why? Can you explain it with a picture?
Solution:
When we add up pairs of consecutive triangular numbers, we get the sequence of square numbers.
The sum of consecutive triangular numbers:
-
T1 + T2 = 1 + 3 = 4
-
T2 + T3 = 3 + 6 = 9
-
T3 + T4 = 6 + 10 = 16
-
T4 + T5 = 10 + 15 = 25
This sequence 4, 9, 16, 25, … is the sequence of square numbers 22, 32, 42, 52,…..
When we add two consecutive triangular numbers, the result is a square number because we are effectively completing a square shape. The smaller triangular number fits perfectly into the gap created by the larger triangular number, forming a complete square.
Question 6. What happens when you start to add up powers of 2 starting with 1, i.e., take 1, 1+2, 1+2 + 4, 1+2 + 4 + 8, … ? Now add 1 to each of these numbers — what numbers do you get? Why does this happen?
Solution:
On adding up powers of 2 starting with 1, as 1, 1 + 2 = 3, 1 + 2 + 4 = 7, 1 + 2 + 4 + 8 = 15, … we get a sequence like, 1 = 1, 1 × 2 + 1 = 3, 3 × 2 + 1 = 7, 7 × 2 + 1 = 15, 15 × 2 + 1 =31, …
Further adding up 1 to each number of the sequence obtained.
1 + 1 = 2, 3 + 1 = 4, 7 + 1 = 8, 15 + 1 = 16
Clearly, we get a number sequence of power of 2.
As, 2, 4, 8, 16, … .
Question 7. What happens when you multiply the triangular numbers by 6 and add 1? Which sequence do you get? Can you explain it with a picture?
Solution:
When we multiply triangular numbers by 6 and then add 1, we get a sequence of hexagonal numbers.
Triangular numbers are given by: Tn = [Math Processing Error]
Now, let’s multiply these triangular numbers by 6 and add 1:
First triangular number: T1 = 1
6 × 1 + 1 = 7
Second triangular number: T2 = 3
6 × 3 + 1 = 19
Third triangular number: T3 = 6
6 × 6 + 1 = 37
Sequences 7, 19, and 37 are part of the hexagonal number sequence and can be represented by constructing polygonal patterns within hexagons and their higher-order numbers.
Question 8. What happens when you start to add up hexagonal numbers, i.e.. 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?
Solution:
Adding up hexagonal numbers
1 = 1
1 + 7 = 8
1 + 7 + 19 = 27
1 + 7 + 19 + 37 = 64
1 + 7 + 19 + 37 + 61 = 125
…………
…………
…………
…………
and so on.
Here, we get a sequence of cubes.
Question 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?
Solution:
Given table is as
1,1, 1, 1, 1, 1,… (All l’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)
From the above table, we see that
On adding the counting numbers up and turn down
we get the square numbers.
1 = 1
1 + 2 + 1 = 4
1 + 2 + 3 + 2 + 1 = 9
1 + 2 + 3 + 4 + 3 + 2 + 1 = 16
1 + 2 + 3 + 4 + 5 + 4 + 3 + 2 + 1 = 25
1 + 2 + 3 + 4 + 5 + 6 + 5 + 4 + 3 + 2 + 1 = 36
…………
…………
…………
…………
and so on.
Thus, counting number is relate with square numbers. Also, we see triangular numbers added together for square numbers. „
1 + 3 = 4
3 + 6 = 9
6 + 10 = 16
10 + 15 = 25
10 + 15 = 25
…………
…………
…………
…………
and so on.
Also, we can look following patterns,
13 =12 =1
13 + 23 = (1 + 2)2 =9
13 + 23 + 33 =(1 + 2 + 3)2 = 36
13 + 23 + 33 +43 =(1 + 2 + 3 +4)2 = 100
…………
…………
…………
…………
and so on.
Here, we get a sequence of square of triangular numbers.
Thus, we can say that sequence relate to each other.
1.5 Patterns in Shapes Figure it Out (Page No. 11)
Question 1. Can you recognize the pattern in each of the sequences in Table 3?
Solution:
Yes, the patterns given in Table 3 are recognizable.
Regular polygons: Each shape sequence is obtained by adding 1 side to the previous polygon. The names of the polygons are thus as follows:
3 sides – triangle; 4 sides – quadrilateral; 5 sides – pentagon; 6 sides – hexagon and so on.
Complete graphs: A complete graph is a type of graph where every pair of vertices is connected by a unique edge.
For 3 vertices, K3 has 3 edges (like a triangle).
For 4 vertices, K4 has 6 edges.
For 5 vertices, K5 has 10 edges.
For 6 vertices, K6 has 15 edges.
The number of edges can be expressed by the formula: , where n is the number of vertices.
Stacked squares: Each new larger square is made up of a specific number of smaller squares arranged in a grid.
1 × 1 Square: Just 1 small square
2 × 2 Square: A 2 × 2 grid of small squares
Total = 2 × 2 = 4 small squares.
3 × 3 Square: A 3 × 3 grid of small squares
Total = 3 × 3 = 9 small squares.
4 × 4 Square: A 4 × 4 grid of small squares
Total = 4 × 4 = 16 small squares.
Stacked triangles: For a stacked triangle with n layers, the total number of small triangles can be calculated by summing the number of triangles in each layer.
For a stacked triangle with n layers, the total number of small triangles can be calculated by summing the number of triangles in each layer.
Total triangles , where n is the number of layers.
Koch snowflake: The Koch snowflake is a fractal curve known for its self-similarity and infinite complexity. It starts with an equilateral triangle and progressively adds smaller triangles to its side.
Each time you make a new layer on the Koch snowflake, it looks like the one before but with more details. The new triangles you add look like tiny versions of the big triangle, keeping the same shape, just smaller.
Question 2. Try and redraw each sequence in Table 3 in your notebook. Can you draw the next shape in each sequence? Why or why not? After each sequence, describe in your own words what is the rule or pattern for forming the shapes in the sequence.
Solution:
Regular Polygons: This sequence shows polygons that have equal sides and starts with a triangle having 3 sides and in each step in the sequence adds one more side to the previous shape.
Rule n – number of sides, n ∈ N
n + 1 = number of sides in the next shape
Complete Graphs This sequence shows that each point is connected to every other point and starts with two points connected by a line, three points form a triangle, four points form a square and so on. In each step, the number of lines increases.
Stacked Triangle: In this sequence, triangles are stacked to form a large triangle. In each step of the sequence starting with one triangle add more number of triangles to form a large triangle.
Stacked Triangle In this sequence, triangles are stacked to form a large triangle. In each step of the sequence starting with one triangle add more number of triangles to form a large triangle.
Koch Snowflakes In this sequence, starts with the triangle every side of the snowflake is replaced with 4 new sides. Each of these sides is a third of the length of the side, it is replacing or in this sequence, starts with triangle and in next step the line segment is replaced by a ‘speed bump’
In each step, the chances become tinier and tinier with very very small line.
1.6 Relation to Number Sequences Figure it Out (Page No. 11-12)
Question 1. Count the number of lines in each shape in the sequence of complete graphs. Which number sequence do you get? Can you explain why?
Solution:
There are 1, 3, 6,10,15 lines respectively in each shape in the sequence of complete graphs.
We get a sequence of triangular numbers.
Question 2. How many little squares are there in each shape of the sequence of stacked squares? Which number sequence does this give? Can you explain why?
Solution:
The number of squares in each shape of the sequence of stacked squares is: 1, 4, 9, 16,…..
The number sequence given by this is that of perfect square numbers.
Why does this happen? The number of squares is arranged in a grid-like pattern which again arrives at a square. Thus, a square number is obtained every time.
Question 3. How many little triangles are there in each shape of the sequence of Stacked Triangles? Which number sequence does this give? Can you explain why?
(Hint: In each shape in the sequence, how many triangles are there in each row?)
Solution:
There are 1,4, 9,16, 25, little triangles, respectively in each shape of the sequence of stacked triangles.
This gives a sequence of squares.
Reason In each shape, add the number of little triangles in each row
1 = 1
1 + 3 = 4
1 + 3 + 5 = 9
1 + 3 + 5 + 7 = 16
…………
…………
…………
…………
and so on.
Question 4. To get from one shape to the next shape in the Koch snowflake sequence, one replaces each line segment ‘—’ with a ‘speed bump’ _/\_. As one does this more and more times, the changes become tinier and tinier with very very small line segments. How many total line segments are there in each shape of the Koch snowflake? What is the corresponding number sequence? (The answer is 3, 12, 48 …, i.e., 3 times powers of 4; this sequence is not shown in Table 1.)
Solution:
In the Koch snowflake sequence, each iteration involves replacing each line segment with a ‘speed bump’, which creates 4 new segments for each original segment. Let’s break down how this affects the number of line segments and identify the sequence:
Koch snowflake iterations
1. Iteration 0 (starting shape):
2. Iteration 1:
3. Iteration 2:
4. Iteration 3:
-
Each of the 48 segments from Iteration 2 is replaced by 4 new segments.
-
Number of line segments: 48 × 4 = 192
-
The number of line segments in each iteration forms the sequence: 3, 12, 48, 192,…
-
This sequence can be described by the formula: 3 × 4n
Example: What happens when we start adding up odd numbers?
1 = 1 = square of 1
1 + 3 = 4 = square of 2
1 + 3 + 5 = 9 = square of 3
1 + 3 + 5 + 7 = 16 = square of 4
1 + 3 + 5 + 7 + 9 = 25 = square of 5
1 + 3 + 5 + 7 + 9 + 11 = 36 = square of 6
Why does this happen? Do you think it will happen forever? (Page 6)
Solution:
This happens because each odd number can be represented as (2n – 1), where (n) is a positive integer. When we sum the first (n) odd numbers, we get: 1 + 3 + 5 + ……. + (2n – 1) = n
This pattern will continue forever because it is a fundamental property of numbers. The sequence of odd numbers and their sums forming perfect squares is an inherent characteristic of the number system.
How can we partition the dots in a square grid into odd numbers of dots: 1, 3, 5, 7,…..? (Page 6)
Solution:
We can partition the dots in a square grid into odd numbers of dots: 1, 3, 5, 7, as follows:
By drawing a similar picture, can you say what is the sum of the first 10 odd numbers? (Page 7)
Solution:
Sum of first 10 odd numbers = 1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 + 17 + 19 = n2 = (10)2 = 10 × 10 = 100
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)
Solution:
The sum of the first 100 odd numbers = 1 + 3 + 5 + …………
= n2
= (100)2
= 100 × 100
= 10,000
NCERT Solutions for Class 6 Maths Ganita Prakash Chapter 1 PDF Download
Students can download the NCERT Solutions for Class 6 Maths Ganita Prakash Chapter 1 Patterns in Mathematics in PDF format from the link provided below.
These solutions are designed to help students explore the fascinating world of patterns, sequences, and number arrangements in a simple and engaging way. The answers include clear explanations for all textbook activities, pictorial patterns, number-based logic, and thinking-based questions.
