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 everyday life. Some such examples are:
- Calculating time duration.
- Estimating different measures.
- Budgeting to balance income and home expenses.
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,
| Patterns | Recognization of patterns |
| 1, 1, 1, 1, 1, 1, 1,… (All l’s) | Sequence of all 1 ’s |
| 1, 2, 3, 4, 5, 6, 7, … (Counting numbers) | A sequence of consecutive counting numbers starting from 1 |
| 1, 3, 5, 7, 9, 11, 13, … (Odd numbers) | A sequence of consecutive odd numbers starting from 1 |
| 2, 4, 6, 8, 10, 12, 14, … (Even numbers) | A sequence of consecutive even numbers starting from 2 |
| 1, 3, 6, 10, 15, 21, 28, … (Triangular numbers) | In the sequence, each term is the sum of first n consecutive counting numbers |
| 1,4, 9, 16, 25,36, 49, … (Squares) | In the sequence, each term is the product of counting number by itself starting from 1 |
| 1, 8, 27, 64, 125, 216, … (Cubes) | In the sequence, each term is the product of counting number by itself thrice starting from 1 |
| 1, 2, 3, 5, 8, 13, 21, … (Virahanka numbers) | In the sequence, each term (starting from third term) is the sum of previous two terms |
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, 1,1,1,1 | Sequence of all 1 ’s |
| 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 | A sequence of consecutive counting numbers starting from 1, adding 1 to the previous term to get the next term, as 1, 1 + 1 = 2, 2 + 1 = 3, 3 + 1 = 4, … |
| 1,3, 5, 7, 9, 11, 13, 15, 17, 19 | A sequence of consecutive odd numbers starting from 1, adding 2 to the previous term to get the next term, as 1, 1 + 2 = 3, 3 + 2 = 5, 5 + 2 = 7, 7 + 2 = 9, … |
| 2, 4, 6, 8, 10, 12, 14, 16, 18, 20 | A sequence of consecutive even numbers starting from 2, adding 2 to the previous term to get the next term, as 2, 2 + 2 = 4, 4 + 2 = 6, 6 + 2 = 8, 8 + 2= 10, … |
| 1, 3, 6, 10, 15, 21, 28, 36, 45, 55 | In the sequence, each term is the sum of first n consecutive counting numbers, as 1 = 1; 1 + 2 = 3; 1 + 2 + 3 = 6, 1 + 2 + 3 + 4 = 10; 1 + 2 + 3 + 4 + 5 = 15; 1 + 2 + 3 + 4 + 5 + 6 = 21; … |
| 1, 4, 9, 16, 25, 36, 49, 64, 81, 100 | In the sequence, each term is the product of counting number by itself starting from 1, as 1 × 1 = 1, 2 × 2 = 4, 3 × 3 = 9, 4 × 4 = 16, 5 × 5 = 25, … |
| 1, 8, 27, 64, 125, 216, 343, 512, 729 | In the sequence, each term is the product of counting number by itself thrice starting from 1, as 1 × 1 × 1 = 1, 2 × 2 × 2 = 8, 3 × 3 × 3 = 27, 4 × 4 × 4 = 64, 5 × 5 × 5 = 125, 6 × 6 × 6 = 216,… |
| 1,2,3,5,8,13, 21,34, 55, 89 | In the sequence, each term (starting from third term) is the sum of previous two terms, as 1, 2, 1 + 2 = 3, 2 + 3 = 5, 3 + 5 = 8, 5 + 8 = 13, 8 + 13 = 21, … |
| 1, 2, 4, 8, 16, 32, 64, 128, 256, 512 | In the sequence, next term is the double of previous term, as 1, 1 × 2 = 2, 2 × 2 = 4, 4 × 2 = 8, 8 × 2 = 16, 16 × 2 = 32,… |
| 1,3, 9, 27,81, 243,729, 2187, 6561, 19683 | In the sequence, ne×t term is the thrice of previous term, as 1, 1 × 3 = 3, 3 × 3 = 9, 9 × 3 = 27, 27 × 3 = 81, 81 × 3 = 243,… |
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?
Answer:
Triangular Numbers: Each number in this sequence can be arranged in the shape of an equilateral triangle. For example:
- 1 dot forms a single point (a triangle with one dot).
- 3 dots can be arranged into a triangle (two dots at the base one above).
- 6 dots form a larger triangle (three at the base, two in the middle, and one on top,) and so on.
Formula: The nth triangular number is given by \(\frac{n(n+1)}{2}\)
Square numbers: Each number in this sequence can be arranged into a square shape. For example:
- 1 dot forms a 1 × 1 square.
- 4 dots can be arranged into a 2 × 2 square.
- 9 dots form a 3 × 3 square and so on.
Formula: The nth square number is n2.
Cubes: Each number in this sequence can be visualized as a cube. For example:
- 1 dot represents a 1 × 1 × 1 cube.
- 8 dots can be arranged into a 2 × 2 × 2 cube.
- 27 dots form a 3 × 3 × 3 cube and so on.
Formula: The nth cube is n3.
Question 3.
You will have noticed that 36 is a triangular and square number! That is, 36 dots can be arranged perfectly both in a triangle and 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:
Three other such numbers are 1, 1225, and 41616.
1225 is the 49th triangular number.
Also, 1225 can be represented by a square having 36 dots along its side.
Question 4.
What would you call the following sequence or numbers?
That’s right, they are called hexagonal numbers! Draw these in your notebook. What is the next number in the sequence?
Answer:
The next number in the sequence will be 61.
The hexagonal numbers are numbers that can be arranged in the form of a hexagon. The sequence here is as follows:
1 + (6 × 1) = 7
7 + (6 × 2) = 19
19 + (6 × 3) = 37
37 + (6 × 4) = 61
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?
Answer:
Yes, similar pictorial explanation can be made for why adding counting numbers up and down gives square numbers.
It can be observed that these figures form perfect square numbers by following the ‘adding counting numbers up and down’ method. This happens because each of these sequences is to be thought of as forming a symmetric pattern around a central line. This symmetry allows us to ‘fold’ the pattern along this central line.
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 + 2 + 3 + …… + 100 + ……. + 3 + 2 + 1 = 100 × 100 = 10000
Hence, it is 1002 = 100 × 100 = 10,000
Question 3.
Which sequence do you get when you start to add the all 1s sequence up? What sequence do you get when you add the all 1s sequence up and down?
Answer:
Adding the All 1’s sequence up:
The All 1’s sequence’ is a sequence of ones: 1, 1, 1, 1, 1,…..
When we add this sequence cumulatively, we get the sequence of counting numbers:
Starting sequence: 1, 1, 1, 1, 1,…..
Cumulative sum:
1
1 + 1 = 2
1 + 1 + 1 = 3
1 + 1 + 1 + 1 = 4
1 + 1 + 1 + 1 + 1 = 5 And so on.
This gives us the counting numbers sequence:
Sequence: 1, 2, 3, 4, 5, 6,…..
Adding the all 1’s sequence up and down:
When you add all 1’s sequence both up and down, you create a symmetric sequence centered around the middle:
Starting sequence: 1, 1, 1, 1, 1,…..
Up and down cumulative sum:
For n = 1: 1
For n = 2: 1 + 1 + 1 = 3
For n = 3: 1 + 1 + 1 + 1 + 1 = 5
For n = 4: 1 + 1 + 1 + 1 + 1 + 1 + 1 = 7 And so on.
This gives us the sequence of odd numbers:
Sequence: 1, 3, 5, 7, 9, 11,…..
Question 4.
Which sequence do you get when you start to add the Counting numbers up? Can you give a smaller pictorial explanation?
Solution:
Counting numbers adding up:
1, 1 + 2, 1 + 2 + 3, 1 + 2 + 3 + 4,
Or 1, 3, 6, 10
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:
After adding up pairs of consecutive triangular numbers, we get square numbers. As
1 + 3 = 4, 3 + 6 = 9, 6 + 10 = 16, 10 + 15 = 25, 15 + 21 = 36,… Pictorial representation. As
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?
Answer:
Adding powers of 2:
The powers of 2 sequences is:
20 = 1
21 = 2
22 = 4
23 = 8
24 = 16 And so on
Now, let’s add them cumulatively:
1. Sum of first power: 1
2. Sum of first two powers: 1 + 2 = 3
3. Sum of first three powers: 1 + 2 + 4 = 7
4. Sum of first four powers: 1 + 2 + 4 + 8 = 15
5. Sum of first five powers: 1 + 2 + 4 + 8 + 16 = 31
This sequence is 1, 3, 7, 15, 31,…..
Now, add 1 to each of these sums:
1. 1 + 1 = 2
2. 3 + 1 = 4
3. 7 + 1 = 8
4. 15 + 1 = 16
5. 31 + 1 = 32
This gives us the sequence 2, 4, 8, 16, 32,…..
The sequence 2, 4, 8, 16, 32 … is the powers of 2 again!
This happens because the cumulative sum of powers of 2 up to 2n is always one less than the next power of 2, so adding 1 brings us to that next power of 2.
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:
Hexagonal No. are: 1, 7, 19, 37,……….
Let us add them:
1 = 13 (third power of 1)
1 + 7 = 8 = 2 × 2 × 2 = 23 (third power of 2)
1 + 7 + 19 = 27 = 3 × 3 × 3 = 33 (third power of 3)
1 + 7 + 19 + 37 = 64 = 4 × 4 × 4 = 43 (third power of 4)
1 + 7 + 19 + 37 + 61 = 125 = 5 × 5 × 5 = 53 (third power of 5)
Question 9.
Find your patterns or relations in and among the sequences in Table 1. Can you explain why they happen with a picture or otherwise?
Solution:
3, 6, 9, 12, 15, 18,……… (consecutive multiples of 3).
10, 15, 20, 25,…….. (first number is 10. Then increase of 5 in each term)
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:
Regular polygon: In the sequence, next polygon is obtained by increasing the number of sides by 1. Complete Graph: In the sequence, next shape is obtained by increasing the number of vertices by 1.
Stacked Squares: In the sequence, next bigger square is representing the square numbers.
Stacked Triangles: In the sequence, next bigger triangle is representing the sum of consecutive odd numbers starting from 1.
Koch Snowflake: In the sequence, each time each line segment ‘—’ is replaced by a speed bump
As one does this more and more times, the changes become tinier and tinier with very-very small line segments.
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:
(a) Regular Polygon: Polygon of 11 sides. (hendecagon)
(b) K7
(c) Stacked Squares
Total number of squares = 6 × 6 = 36
(d) Stacked Triangles
Total number of triangles = 1 + 3 + 5 + 7 + 9 + 11 = 36
(e) Koch Snowflake
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 corners in each shape in the sequence of regular polygons? Do you get the same number sequence? Can you explain why this happens?
Answer:
The first polygon has 3 sides – Triangle
The second polygon has 4 sides – Square
The third polygon has 5 sides – Pentagon
And so on.
3, 4, 5, ….. – This forms a sequence of counting numbers.
The same happens for the number of vertices/corners in each shape.
In a regular polygon, the number of sides is always equal to the number of corners. This is because each corner is formed by two sides meeting, and the number of these meeting points is equal to 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:
Hence, we get the sequence: 1, 3, 6, 10, 15,…… It is a triangular number sequence.
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:
Clearly, we get a sequence of square numbers 1,4,9,16, 25, … Because we count the number of little squares in each shape and get each one is the product of counting number by itself.
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:
We get the sequence: 1, 4, 9, 16, 25, 36, 49, This is the sequence of square numbers.
Hence by adding a Stacked Triangle at the bottom, the next number in the square sequence will come.
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;+his sequence is not shown in Table 1).
Solution:
By counting line segments, we get the number sequence: 3, 12, 48, 192, 768, …
In the sequence, each term is thrice of power of 4, as
3 × 1 = 3
3 × 4 = 12
3 × (4 × 4) = 48
3 × (4 × 4 × 4) = 192