NCERT Solutions for Class 6 Maths Chapter 1 Patterns in Mathematics

NCERT Solutions for Chapter 1 Class 6 Maths, "Patterns in Mathematics" covers the following six exercises:

Exercise 1.1: What is Mathematics?

This exercise introduces the concept of mathematics and its importance in daily life. It covers basic ideas and sets the foundation for understanding patterns. Students learn how math is used to solve real-world problems.

Exercise 1.2: Patterns in Numbers

This exercise focuses on identifying and analysing number patterns. Students practice recognizing sequences and understanding how numbers follow specific rules. It helps in developing skills for pattern recognition.

Exercise 1.3: Visualising Number Sequences

Here, students learn to visualise and interpret number sequences. The exercise emphasises understanding how sequences progress and how to predict subsequent numbers. It enhances student’s ability to work with visual patterns.

Exercise 1.4: Relations among Number Sequences

This exercise explores the relationships between different number sequences. Students learn how sequences are related and how to analyze these connections. It helps in understanding complex pattern relationships.

Exercise 1.5: Patterns in Shapes

This exercise involves recognizing and creating patterns using geometric shapes. Students practise identifying visual patterns and understanding how shapes form these patterns. It improves spatial reasoning and pattern-creation skills.

Exercise 1.6: Relation to Number Sequences

Students learn how number sequences can be related to geometric patterns. This exercise helps in understanding how sequences and shapes connect. It reinforces skills in pattern analysis and application.

Access NCERT Solutions for Class 6 Maths Chapter 1 Patterns in Mathematics

Exercise 1.1

1. Can you think of other examples where mathematics helps us in our everyday lives?

Ans: Here are a few examples of where mathematics helps us in our everyday lives:

1. Budgeting and Personal Finance: Maths helps manage money by creating budgets, tracking expenses, and planning savings.

2. Cooking and Baking: Recipes require measurements and proportions, and maths helps convert quantities and adjust servings.

3. Shopping: Maths is used to calculate discounts, compare prices, and determine the total cost of purchases.

4. Home Improvement: Maths is essential for measuring spaces, calculating areas, and determining quantities of materials needed for projects.

5. Travel Planning: Maths helps with calculating distances, estimating travel times, and managing expenses for trips.

6. Health and Fitness: Maths is used to track calorie intake, calculate body mass index (BMI), and monitor exercise routines.

7. Time Management: Maths helps in scheduling tasks, setting deadlines, and organising daily routines.

8. Gardening: Maths is used for spacing plants, calculating garden areas, and planning layouts.

These examples show how mathematics is a practical tool for solving everyday problems and making informed decisions.

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.)

Ans: Mathematics has been important in moving humanity forward in many ways:

1. Scientific Experiments: Maths help in planning and analysing experiments. For example, it is used to understand natural laws and forces in physics.

2. Economy and Democracy: Maths models financial systems and helps manage resources. It is also used in analysing election results and planning public policies.

3. Engineering and Architecture: Maths is used to design and build bridges, buildings, and other structures. Engineers use it to make sure these constructions are safe and stable.

4. Technology and Electronics: Maths is essential in making TVs, mobile phones, and computers. It is used in creating algorithms and digital signals.

5. Transportation: Maths helps design and operate bicycles, trains, cars, and planes. It is used to calculate speed, fuel efficiency, and directions.

6. Timekeeping: Maths is used to develop accurate calendars and clocks. It helps in making precise time measurements.

7. Medicine: Maths are important in medical imaging like MRI scans and in calculating medication dosages. It also helps model how diseases spread.

8. Astronomy and Space Exploration: Maths is key in calculating the paths of planets and stars. It is used in planning space missions and launching satellites.

Exercise 1.2

Table 1: Examples of number sequences

1. Can you recognise the pattern in each of the sequences in Table 1?

Ans: (a) 1, 1, 1, 1, 1, 1, 1, ……

The number ‘1’ is repeated indefinitely.

(b) 1, 2, 3, 4, 5, 6, 7, ……

Counting numbers starting from ‘1’. Each number increases by 1.

(c) 1, 3, 5, 7, 9, 11, 13, ……

Odd numbers. Start from 1 and keep adding 2.

(d) 2, 4, 6, 8, 10, 12, 14, ……

Even numbers. Start from 2 and keep adding 2.

(e) 1, 3, 6, 10, 15, 21, 28, ……

Triangular numbers. Each term is the sum of the natural numbers up to that term.

(f) 1, 4, 9, 16, 25, 36, 49, ……

Square numbers. Each term is the square of its position ($1^{2}$, $2^{2}$, $3^{2}$, ...).

(g) 1, 8, 27, 64, 125, 216, ……

Cube numbers. Each term is the cube of its position (($1^{2}$, $2^{2}$, $3^{2}$, ...).

(h) 1, 2, 3, 5, 8, 13, 21, ……

Virahānka numbers (Fibonacci sequence). Each number is the sum of the two preceding numbers.

(i) 1, 2, 4, 8, 16, 32, 64, ……

Powers of 2. Each term is raised to the power of its position ($2^{0}$, $2^{1}$, $2^{2}$, ...).

(j) 1, 3, 9, 27, 81, 243, 729, ……

Powers of 3. Each term is 3 raised to the power of its position ($3^{0}$, $3^{1}$, $3^{2}$, ...).

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.

Ans: (a) 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, …

The sequence consists of repeating the number 1. The next three numbers are also 1.

(b) 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, …

This is a sequence of consecutive counting numbers. The next three numbers are 8, 9, and 10.

(c) 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, …

These are consecutive odd numbers, increasing by 2. The next three numbers are 15, 17, and 19.

(d) 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, …

The sequence contains even numbers, increasing by 2. The next three numbers are 16, 18, and 20.

(e) 1, 3, 6, 10, 15, 21, 28, 36, 45, 55, …

A sequence of triangular numbers where each number is the sum of the natural numbers up to a certain point.
28 + 8 = 36
36 + 9 = 45
45 + 10 = 55

(f) 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, …

Square numbers are formed by multiplying a number by itself. The next three are 64, 81, and 100.
$8^{2}$ = 64
$9^{2}$ = 81
$10^{2}$ = 100

(g) 1, 8, 27, 64, 125, 216, 343, 512, 729, …

A sequence of cube numbers where each number is the cube of a natural number. The next three numbers are 343, 512, and 729.
$7^{3}$ = 343
$8^{3}$ = 512
$9^{3}$ = 729

(h) 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, …

This is the Fibonacci (Virahanka) sequence, where each number is the sum of the previous two. The next three numbers are 34, 55, and 89.
13 + 21 = 34
21 + 34 = 55
34 + 55 = 89

(i) 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, …

This shows powers of 2, where each number is multiplied by 2. The next three numbers are 128, 256, and 512.
64 × 2 = 128
128 × 2 = 256
256 × 2 = 512

(j) 1, 3, 9, 27, 81, 243, 729, 2187, 6561, 19683, …

This shows powers of 3, where each number is multiplied by 3. The next three numbers are 2187, 6561, and 19683.
729 × 3 = 2187
2187 × 3 = 6561
6561 × 3 = 19683

Exercise 1.3

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!

Ans: 

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? 

Ans: 1, 3, 6, 10, 15, … called triangular numbers: These numbers are called triangular numbers because they can be arranged in the shape of an equilateral triangle. For example, 3 can be arranged as a triangle with 2 dots in the base and 1 dot at the top. Each number in the sequence represents the total number of dots that form a triangle when arranged in increasing rows.

1, 4, 9, 16, 25, … called square numbers or squares: These numbers are called square numbers because they represent the area of a square. For example, 4 is the area of a square with sides of length 2 (2 × 2 = 4). Each number in this sequence is the product of a number multiplied by itself, which forms the area of a square.

1, 8, 27, 64, 125, … called cubes: These numbers are called cubes because they represent the volume of a cube. For example, 8 is the volume of a cube with each side of length 2 (2 × 2 × 2 = 8). Each number in the sequence is the result of multiplying a number by itself twice, which gives the volume of a cube.

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! 

Ans: Three other numbers that can be both triangular and square are 1, 1225, and 41616.

For instance, 1225 is the 49th triangular number, meaning it can be arranged in a triangular pattern with 49 rows. 

Additionally, 1225 can also be arranged in a square, with each side containing 35 dots (since 35 × 35 = 1225). 

This shows how certain numbers can be represented in multiple ways, demonstrating their versatility in both triangular and square formations. 

The fact that a number like 1225 fits perfectly into both shapes highlights the fascinating connections between different geometric and mathematical properties.

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? 

Ans: Let's break down the pattern in this sequence:

1st number = 1
2nd number = 1 + 6 = 7 (This is found by adding 6 × 1 to the 1st number)
3rd number = 7 + 12 = 19 (This is found by adding 6 × 2 to the 2nd number)
4th number = 19 + 18 = 37 (This is found by adding 6 × 3 to the 3rd number)
5th number = 37 + 24 = 61 (This is found by adding 6 × 4 to the 4th number)

Thus, the pattern involves adding multiples of 6, with each multiple increasing by 6. So, to find the next number in the sequence, you would add 6 × 5 = 30 to the 5th number (61). Therefore, the next number in the sequence would be 61 + 30 = 91.

This sequence demonstrates a regular pattern of growth, with each term increasing by a progressively larger multiple of 6.

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:

Ans: Pictorial Representation for powers of 3:

Exercise 1.4

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? 

Ans: 

1. The first term is 1 (which is $1^{2}$).

2. The second term is 1 + 2 + 1 = 4 (which is $2^{2}$).

3. The third term is 1 + 2 + 3 + 2 + 1 = 9 (which is $3^{2}$).

Let's find the next three terms:

4. The fourth term is 1 + 2 + 3 + 4 + 3 + 2 + 1 = 16 (which is $4^{2}$).

5. The fifth term is 1 + 2 + 3 + 4 + 5 + 4 + 3 + 2 + 1 = 25 (which is $5^{2}$).

6. The sixth term is 1 + 2 + 3 + 4 + 5 + 6 + 5 + 4 + 3 + 2 + 1 = 36 (which is $6^{2}$).

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? 

Ans: 

1. Understand the Pattern: The sequence starts with increasing numbers from 1 up to 100 and then decreases back to 1. It forms a symmetrical pattern around the number 100.

2. The sum of Increasing Sequence:
   $\text{Sum}_{1 \text{ to } 100} = \frac{100 \times (100 + 1)}{2} = \frac{100 \times 101}{2} = 5050$

3. The sum of Decreasing Sequence:
   $\text{Sum}_{1 \text{ to } 99} = \frac{99 \times (99 + 1)}{2} = \frac{99 \times 100}{2} = 4950$

4. Add the Two Sums:
   $\text{Total Sum} = 5050 + 100 + 4950 = 10000$

So, the value of 1 + 2 + 3 + ... + 99 + 100 + 99 + … + 3 + 2 + 1 is 10,000.

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?

Ans: When adding up a sequence of 1’s, such as 1 + 1 + 1 + 1, the result is 4. 

Similarly, when adding down a sequence of 1’s, such as 1 + 1 + 1 + 1 the sum remains 4. This shows that whether you add the 1’s up or down, the total sum is the same in both cases.

4. Which sequence do you get when you start to add the counting numbers up? Can you give a smaller pictorial explanation?

Ans: 

When you add up the counting numbers sequentially, you get the following sequence:

1. Start with 1.

2. Next, add 1 + 2 to get 3.

3. Then, add 1 + 2 + 3 to get 6.

4. Finally, add 1 + 2 + 3 + 4 to get 10.

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?

Ans: Identify the Triangular Numbers: Triangular numbers are: 1,3,6,10,15,…

Add the First Pair:
1 + 3 = 4
This is the first pentagonal number.

Add the Second Pair:
3 + 6 = 9
This is the second pentagonal number.

Add the Third Pair:
6 + 10 = 16
This is the third pentagonal number.

Add the Fourth Pair:
10 + 15 = 25
This is the fourth pentagonal number.

Each sum of consecutive triangular numbers forms a pentagonal number because the total number of dots creates a pentagon shape. This pattern reflects the arrangement of dots in a pentagonal figure.

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?

Ans: When you start adding up powers of 2, you get a sequence of numbers that are one less than the next power of 2. Here's a detailed explanation:

Adding Powers of 2:

1. Start with 1: 1

2. Add the next power of 2 (2): 1 + 2 = 3

3. Add the next power of 2 (4):1 + 2 + 4 = 7

4. Add the next power of 2 (8): 1 + 2 + 4 + 8 = 15

5. Add the next power of 2 (16): 1 + 2 + 4 + 8 + 16 = 31

Adding 1 to Each Number:

1. Add 1 to 1: 1 + 1 = 2

2. Add 1 to 3: 3 + 1 = 4

3. Add 1 to 7: 7 + 1 = 8

4. Add 1 to 15: 15 + 1 = 16

5. Add 1 to 31: 31 + 1 = 32

Explanation:

When you add up powers of 2, the sum is always one less than the next power of 2. This happens because:

• The sum of the first n powers of 2 is $2^{0}$ + $2^{1}$ + $2^{2}$ + ... + $2^{n-1}$.

• This sum can be simplified using the formula for a geometric series: $S_{n} = 2^{n} - 1$.

• Therefore, when you add 1 to this sum, you get $2^{n}$, which is the next power of 2.

So, the sequence you get after adding 1 to each of these sums is 2, 4, 8, 16, 32,…

This is the sequence of powers of 2.

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? 

Ans: Triangular numbers follow the sequence: 1, 3, 6, 10, 15, 21, etc. When you multiply each triangular number by 6 and add 1, you get a new sequence:

• $1 \times 6 + 1 = 7$

• $3 \times 6 + 1 = 19$ (increase of 12)

• $6 \times 6 + 1 = 37$ (increase of 18)

• $10 \times 6 + 1 = 61$ (increase of 24)

• $15 \times 6 + 1 = 91$ (increase of 30)

Thus, the sequence becomes 7, 19, 37, 61, 91, and so on. This pattern shows that each term increases by 6 more than the previous increase.

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?

Ans: Hexagonal numbers follow the sequence: 1, 7, 19, 37, and so on. When you sum these numbers sequentially, you observe the following results:

• The sum of the first hexagonal number is 1, which equals $1^{3}$ (the cube of 1).

• Adding the second hexagonal number 7 gives 1 + 7 = 8, which is $2^{3}$ (the cube of 2).

• Adding the third hexagonal number 19 results in 1 + 7 + 19 = 27, which is 333^333 (the cube of 3).

• Adding the fourth hexagonal number 37 yields 1 + 7 + 19 + 37 = 64, which is $4^{3}$ (the cube of 4).

• Adding the fifth hexagonal number 616161 gives 1 + 7 + 19 + 37 + 61 = 125, which is $5^{3}$ (the cube of 5).

This pattern illustrates that the cumulative sum of the first n hexagonal numbers equals $n^{3}$, demonstrating an interesting relationship between hexagonal numbers and perfect cubes.

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?

Ans: Here are two simple patterns:

1. Multiples of 3: The sequence 3,6,9,12,15,18,… includes numbers that are multiples of 3. Each number is 3 more than the previous one.

2. Starting at 10 and Increasing by 5: The sequence 10,15,20,25,… starts at 10, with each number increasing by 5.

In the first sequence, each term is 3 times a whole number. In the second sequence, each term starts at 10 and adds 5 each time. Both sequences show how regular patterns can be created with simple rules.

Exercise 1.5

Table 3: Examples of shape sequences

1. Can you recognise the pattern in each of the sequences in Table 3? 

Ans: (a) Regular Polygons: Examples include triangle, quadrilateral, pentagon, and hexagon. In these shapes, the number of sides increases by 1 with each step, starting from 3. This forms a continuous number sequence where each polygon has one more side than the previous one.

(b) Complete Graphs

The number of lines in the sequence is as follows:

For K2 = 1, K3 = 3, K4 = 6, K5 = 10, and K6 = 15. The resulting series is 1, 3, 6, 10, 15,…. This forms a triangular number sequence, where each term represents the total number of lines that can form a triangle. Triangular numbers are generated by adding consecutive natural numbers, making this sequence grow in a predictable pattern.

(c) Stacked Squares: The number of small squares in each layer follows the pattern: 1, 4, 9, 16, 25, and so on. This sequence represents square numbers, where each term is the result of squaring a natural number ($1^{2}$, $2^{2}$, $3^{2}$, etc.). 

The arrangement visually forms a perfect square, showing how the number of small squares increases as the layers grow, perfectly fitting into a square grid. Hence, it’s a clear representation of a square number sequence.

(d) Stacked Triangles: The number of small triangles in each layer follows the pattern:

Thus, it is also a square number sequence represented through triangular formations.

(e) Koch Snowflake: Number of sides in each becomes 4 times.

The number of sides in each increases by a factor of 4.

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.

Ans: (a) Regular Polygon

A polygon with 11 sides, is known as a hendecagon.

(b) K6

The image shows a complete graph with 7 vertices (K7), where every point is connected to every other point with straight lines.

(c) Stacked Squares

The total number of squares is 6 × 6 = 36. This calculation represents a perfect square, showing how the number of small squares forms a 6x6 grid.

(d) Stacked Triangles

The total number of triangles is 1 + 3 + 5 + 7 + 9 + 11 = 36. This sum represents the sequential addition of odd numbers, resulting in the total number of triangles in the arrangement.

(e) Koch Snowflake

The shape is a Koch snowflake, created by repeatedly adding triangular bumps to each side of an equilateral triangle.

Exercise 1.6

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? 

Ans: 

Both sequences are the same because, in a regular polygon, the number of sides equals the number of vertices.

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?

Ans: 

Therefore, the sequence is 1, 3, 6, 10, 15, and so on. This is known as a triangular number sequence.

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? 

Ans: 

We obtain the sequence 1, 4, 9, 16, 25, 36, and so on. This sequence represents square numbers.

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?) 

Ans: 

Sequences 1, 4, 9, 16, 25, 36, and 49 represent square numbers. By adding a stacked triangle at the bottom, we can determine the next number in this square sequence.

5. To get from one shape to the next shape in the Koch Snowflake sequence, one replaces each line segment ‘—’ by a ‘speed bump’ __ⴷ__. 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.)

Ans: 

The sequence 3, 12, 48, 192, 768, and so on begins with 3. Each subsequent term is found by multiplying the previous term by 4. Similarly, in the next Koch Snowflake iteration, four new lines are added for each line of the previous shape. 

Benefits of NCERT Solutions for Class 6 Maths Chapter 1 Patterns in Mathematics

• Provides easy-to-follow explanations about number patterns and sequences, helping students understand how numbers follow specific rules and form predictable patterns.

• Offers clear steps and methods for identifying and extending patterns, making it simpler for students to learn and apply these concepts effectively.

• Helps students understand the basics of number patterns, which is important for grasping more complex mathematical topics.

• Includes various practice problems to help students improve their ability to recognise and work with different patterns, enhancing their problem-solving skills.

• The FREE PDF download allows students to study and practice at their own pace, making learning more convenient and adaptable.

Important Study Material Links for Class 6 Maths Chapter 1 - Patterns in Mathematics

Conclusion 

NCERT Solutions for Class 6 Maths Chapter 1, Exercise 1.6, "Relation to Number Sequences," helps students understand how number sequences relate to patterns. The straightforward explanations make it easier to identify, analyze, and extend these sequences. Practising these exercises enhances students' ability to work with number patterns, which is essential for grasping more advanced math concepts. With the solutions available as a FREE PDF download, students can study and review at their own pace, making learning both flexible and effective.

Chapter-wise NCERT Solutions Class 6 Maths 

The chapter-wise NCERT Solutions for Class 6 Maths are given below. Go through these chapter-wise solutions to be thoroughly familiar with the concepts.

Related Important Links for Maths Class 6

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