By using Ganita Prakash Book Class 6 Solutions and Chapter 1 Patterns in Mathematics Class 6 NCERT Solutions Question Answer, students can improve their problem-solving skills.
Class 6 Maths Chapter 1 Patterns in Mathematics Solutions
Patterns in Mathematics Class 6 Solutions Questions and Answers
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, tjicycles, trains, cars, planes, calendars, clocks, etc.)
Solution:
Mathematics is an essential part of our life and helps us to understand the world in different way. It teaches us to analyse data, identify patterns and develop creative solutions to complex problems. These skills are essential in todays fast paced, data driven world and are highly valued in many industries, including finances, healthcare and technology.
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, we can recognise the pattern in each of the sequence in table 1.
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:
- All l’s 1, 1, 1, 1,… (The next three numbers are 1, 1, 1. The rule is that every number is 1.)
- Counting numbers 1, 2, 3, 4, 5, 6, 7,… (The next three numbers are 8, 9,10. The rule is to add 1 to the previous number.)
- Odd numbers 1, 3, 5, 7, 9, 1,… (The next three numbers are 15, 17, 19. The rs to add 2 to the previous number.)n numbers 2, 4, 6, 8, 10, 12, 14,… (The next three is to add 2 to the previous number.)
- Triangular numbers 1, 3, 6,10,15, 21, 28,…
(The next three numbers are 36,45, 55. The rule is to add the next natural number is sequence.) - Squares 1, 4, 9, 16, 25, 36, 49,… (The next three numbers are 64, 81, 100. The rule is to square the next natural number.)
- Cubes 1, 8, 27, 64, 125, 216,… (The next three numbers are 343, 512, 729. The rule is to cube the next natural number.)
- Virahanka numbers 1, 2, 3, 5, 8, 13, 21,… (The next three numbers are 34, 55, 89. The rule is to add the last two numbers.)
- Powers of 2 1, 2, 4, 8, 16, 32, 64,… (The next three numbers are 128, 256, 512. The rule is to multiply the last number by 2.)
- Powers of 3 1, 3, 9, 27, 81, 243, 729,… (The next three numbers are 2187, 6561, 19683. The rule is to multiply the last number by 3.)
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:
1, 3, 6,10,15,… sequence forms a triangles, therefore it is called triangular numbers.
1,4, 9, 16, 25,… sequence forms a square, therefore, it is called square numbers.
1, 8, 27, 64, 125,… sequence forms a cube, therefore, it is called cubes.
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
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 is the sequence?
Solution:
Question 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.
Solution:
Yes
- Power of 2 We can visualize the powers of 2 as squares where each subsequent square has twice the number of smaller squares as the previous one.
- Power of 3 We can visualize the powers of 3 as cubes, where each subsequent cube has three times the number of smaller cubes as the previous one.
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:
Yes,
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
Thus, we see that adding up odd numbers gives square numbers.
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 l’s sequence up? What sequence do you get when you add the All l’s sequence up and down?
Solution:
Adding all l’s sequence up
1 = 1
1 + 1 = 2
1 + 1 + 1 = 3
1 + 1 + 1 + 1 = 4
…………
…………
…………
…………
and so on.
Here, we get a sequence of counting numbers
i. e. 1, 2, 3, 4,
Adding all l’s sequence up and down
1 = 1
1 + 1 + 1 = 3
1 + 1 + 1 + 1 + 1 = 5
…………
…………
…………
…………
and so on.
Here, we get a sequence of odd numbers.
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:
Adding up pairs of consecutive triangular numbers.
1 + 3 = 4
3 + 6 = 9
6 + 10 = 16
10 + 15 = 25
…………
…………
…………
…………
and so on.
Here, we get a sequence of squares.
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:
Adding up powers of 2 starting with 1.
1 = 1
1 + 2 = 3
1 + 2 + 4 = 7
1 + 2 + 4 + 8 = 15
1 + 2 + 4 + 8 + 16 = 31
…………
…………
…………
…………
and so on.
Adding 1 to each of the obtained numbers
1 + 1 = 2
3 + 1=4
7 + 1 = 8
15 + 1 = 16
31 + 1 = 32
…………
…………
…………
…………
and so on.
Here, we get a’ sequence in which next term is obtained by multiplying previous term by 2 or next term is double of the previous term.
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:
We have, triangular numbers.
1, 3, 6, 10, 15,…
On multiplying triangular numbers by 6 and add 1 to it, we get
1 × 6 + 1 = 7
3 × 6 + 1 = 19
6 × 6 + 1 = 37
10 × 6 + 1 = 61
15 × 6 + 1 = 91
…………
…………
…………
…………
and so on.
Hence, we get a sequence of hexagonal numbers.
The pictorial explanation of hexagonal numbers is given below.
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 recognise the pattern in each of the sequence in Table 3?
Solution:
Yes, we can recognise the pattern in each of the sequence in Table 3.
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 Polygon. Which number sequence do you get? What about the number of corners in each shape in the sequence of Regular Polygons? Do you get the same number sequence? Can you explain why this happens?
Solution:
| Shape | Number of sides | Number of corners |
| Triangle | 3 | 3 |
| Quadrilateral (Square) | 4 | 4 |
| Pentagon | 5 | 5 |
| Hexagon | 6 | 6 |
| Heptagon | 7 | 7 |
| Octagon | 8 | 8 |
| Nonagon | 9 | 9 |
| Decagon | 10 | 10 |
Sequence—3, 4, 5, 6, 7, 8, 9,10
We get, a sequence of counting numbers
Also, the number of corners in the shapes is same as 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?
Solution:
There are 1,4, 9, 16, 25 little squares, respectively in each of the sequence of stacked squares.
This gives a sequence of squares.
Reason
1 = 1
1 + 2 + 1 = 4
1 + 2 + 3 + 2 + 1 = 9
1 + 2 + 3 + 4 + 3+ 2 + 1 = 16
…………
…………
…………
…………
and so on.
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 by 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:
There are 3,12,48,192, 768 line segments in each shape of the Koch Snowflake.
The corresponding number sequence is 3, 12, 48, ….
i. e. 3 times the powers of 4.