Patterns in Mathematics Class 6 NCERT Solutions Ganita Prakash Maths Chapter 1

Get the simplified Class 6 Maths NCERT Solutions of Ganita Prakash Chapter 1 Patterns in Mathematics textbook exercise questions with complete explanation.

Ganita Prakash Class 6 Maths Chapter 1 Solutions Patterns in Mathematics

NCERT Solutions for Class 6 Maths Ganita Prakash Chapter 1 Patterns in Mathematics

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:
Mathematics helps us in managing money, preparing food, figuring out distance, time and cost of travel, baking, home decorating etc.

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.)
Answer:
Mathematics drives progress in various fields. For example, it aids in predicting weather patterns, designing video games, improving medical imaging, and analyzing social media trends. It has helped enhance delivery services and make new gadgets, showing how math is crucial for progress everywhere.

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?
Answer:
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.
Answer:
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

1.3 Visualising Number Sequences Figure it Out (Page No. 5-6)

Question 1.
Copy the pictorial representations of the number sequences in Table 2 in your notebook, and draw the next picture for each sequence!
Solution:

Question 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:

• 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 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!
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

Question 5.
Can you think of pictorial ways to visualize the sequence of powers of 2 and powers of 3?
Here is one possible way of thinking about the powers of 2: ____________

Answer:
Different ways to pictorially represent powers of 2 and powers of 3 are:
(a) Binary tree method (for powers of 2): 1, 2, 4, 8, 16,…..

(b) Triangles (for powers of 3): 1, 3, 9, 27,…..

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?
Answer:
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?
Answer:
When we multiply triangular numbers by 6 and then add 1, we get a sequence of hexagonal numbers.
Triangular numbers are given by: T_n = \(\frac{n(n+1)}{2}\)
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?
Answer:
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: \(\frac{n(n+1)}{2}\), 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 = \(\frac{n(n+1)}{2}\), 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 sides in each shape in the sequence of Regular Polygons. Which number sequence do you get? What about the number of comers in each shape in the sequence of Regular Polygons? Do you get the same number sequence? Can you explain why this happens?
Solution:

The number sequence we get 3,4,5,6,7, 8,9,10, i.e., the counting numbers starting from 3, in both cases: number of sides and number of comers.
This happens as the number of comers depends upon the number of sides.

Question 2.
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 3.
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?
Answer:
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 4.
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 5.
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.)
Answer:
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):

• Initial shape: An equilateral triangle
• Number of line segments: 3

2. Iteration 1:

• Each of the 3 line segments is replaced by 4 new segments (creating a ‘speed bump’).
• Number of line segments: 3 × 4 = 12

3. Iteration 2:

• Each of the 12 segments from Iteration 1 is replaced by 4 new segments.
• Number of line segments: 12 × 4 = 48

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

Intext Questions

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