CBSE Important Questions Class 6 Maths Chapter 1 – Patterns in Mathematics

Class 6 Maths Chapter 1 Patterns in Mathematics is the first chapter of the new NCERT textbook Ganita Prakash. It teaches students to look for patterns in numbers and shapes. Students meet number sequences such as counting numbers, odd and even numbers, square numbers, cube numbers, triangular numbers, powers of 2 and 3, and Virahānka numbers, and they also study shape sequences like regular polygons, complete graphs, stacked squares and triangles and the Koch snowflake.

These CBSE important questions for Class 6 Maths Chapter 1 cover drawing the next shape in a sequence and describing its rule, showing with pictures why the sum of odd numbers is a square number, finding terms of number patterns, a sports-day case study on triangular, doubling and square patterns, and assertion-reason and multiple choice questions. Every question has a clear step-by-step answer in simple English, so students can practise on their own, parents can check the working, and teachers can use them for class tests. A free printable PDF of these important questions is also available on this page.

CBSE Important Questions Class 6 Maths Chapter 1 – Patterns in Mathematics

Patterns in Mathematics (New NCERT)

Medium

Q.

Can you recognize the pattern in each of the sequences in Table 1? 

1,1,1,1,1,1,1,…	(All 1'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 umbers)
1,4,9,16,25,36,49,…	(Squares)
1,8,27,64,125,216,…	(Cubes)
1,2,3,5,8,13,21,…	(Virahānka numbers)
1,2,4,8,16,32,64,…	(Powers of 2)
1,3,9,27,81,243,729,…	(Powers of 3)

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Patterns in Mathematics (New NCERT)

Medium

Q.

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?

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Patterns in Mathematics (New NCERT)

Medium

Q.

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?

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Patterns in Mathematics (New NCERT)

Medium

Q.

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?

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Patterns in Mathematics (New NCERT)

Medium

Q.

Count the number of lines in each shape in the sequence of Complete Graphs. Which number sequence do you get? Can you explain why?

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Patterns in Mathematics (New NCERT)

Medium

Q.

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.

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Patterns in Mathematics (New NCERT)

Medium

Q.

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

Examples of shape sequences 

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Patterns in Mathematics (New NCERT)

Medium

Q.

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?

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Patterns in Mathematics (New NCERT)

Medium

Q.

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?

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Patterns in Mathematics (New NCERT)

Medium

Q.

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?

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Patterns in Mathematics (New NCERT)

Medium

Q.

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.

1,1,1,1,1,1,1,…	(All 1'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,…	(Virahānka numbers)
1,2,4,8,16,32,64,…	(Powers of 2)
1,3,9,27,81,243,729,…	(Powers of 3)

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Patterns in Mathematics (New NCERT)

Medium

Q.

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

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Patterns in Mathematics (New NCERT)

Medium

Q.

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?

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Patterns in Mathematics (New NCERT)

Medium

Q.

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?

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Patterns in Mathematics (New NCERT)

Medium

Q.

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 :

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Patterns in Mathematics (New NCERT)

Medium

Q.

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?

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Patterns in Mathematics (New NCERT)

Medium

Q.

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!

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Patterns in Mathematics (New NCERT)

Medium

Q.

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?

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Patterns in Mathematics (New NCERT)

Medium

Q.

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

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Patterns in Mathematics (New NCERT)

Medium

Q.

To get from one shape to the next shape in the Koch Snowflake sequence, one replaces each line segment '__' by a 'speed bump' 

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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? 

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Q1. 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.

1. Regular Polygons – The shape shown is an 11-sided polygon (hendecagon). Yes, we can draw the next shape: a 12-sided polygon (dodecagon). Rule: each next polygon has one more side than the one before it.

2. Complete Graphs – The shape shown is the complete graph K7 (7 points, every point joined to every other point). Yes, we can draw the next shape, K8 with 8 points. Rule: each next graph has one more point (vertex) than the one before it, and all the points are joined to each other.

3. Stacked Squares – The shape shown is a 6 × 6 square made of 36 small squares. Yes, we can draw the next shape, a 7 × 7 square with 49 small squares. Rule: the side of the square grows by 1 each time, so the shapes show the square numbers 1, 4, 9, 16, 25, 36, 49, …

4. Stacked Triangles – The rows have 1, 3, 5, 7 and 9 small triangles, so the big triangle has 1 + 3 + 5 + 7 + 9 = 25 small triangles. Yes, we can draw the next shape by adding one more row of 11 triangles (25 + 11 = 36). Rule: each new row adds the next odd number, and the totals are square numbers (1, 4, 9, 16, 25, 36, …).

5. Koch Snowflake – No, it is very difficult to draw the next shape by hand. Rule: each time, every straight line segment is replaced by a 'speed bump' (a small triangle is added on the middle part of the segment). This is repeated again and again, so the boundary becomes more and more detailed.

The speed bump that replaces each line segment:

Q2. Use a pictorial representation to demonstrate that the sum of the first four odd numbers forms a square number.

Answer: The first 4 odd numbers are 1, 3, 5 and 7.
Start with 1 dot. Add 3 dots around it to make a 2 × 2 square (4 dots). Add 5 more dots to make a 3 × 3 square (9 dots). Add 7 more dots to make a 4 × 4 square (16 dots).
The picture shows that 1 + 3 + 5 + 7 = 16, which is 4 × 4, a square number.

Q3. From the given sequences, identify those that do not start with 1 as their first term.
(i) Even numbers (ii) Powers of 3 (iii) Cubes
(a) (i) and (ii)
(b) Only (i)
(c) (ii) and (iii)
(d) (i), (ii), and (iii)

Answer: (b) Only (i)
Even numbers: 2, 4, 6, 8, … – starts with 2.
Powers of 3: 1, 3, 9, 27, … – starts with 1.
Cubes: 1, 8, 27, 64, … – starts with 1.
So only the even numbers do not start with 1.

Q4. Which of the following sequences does not include the number 8?
(a) Virahānka numbers
(b) Even numbers
(c) Cubes
(d) Triangular numbers

Answer: (d) Triangular numbers
Virahānka numbers: 1, 2, 3, 5, 8, 13, … – has 8.
Even numbers: 2, 4, 6, 8, 10, … – has 8.
Cubes: 1, 8, 27, 64, … – has 8 (2 × 2 × 2 = 8).
Triangular numbers: 1, 3, 6, 10, 15, 21, … – 8 is not in this list.

Q5. Explain the Powers of 2 sequence. How can this sequence be represented visually using pictures or diagrams?

Answer: The Powers of 2 sequence is the list of numbers we get by starting from 1 and multiplying by 2 again and again: 1, 2, 4, 8, 16, 32, 64, …
We can picture it by counting the corner points (vertices) of shapes: a point has 1 vertex, a line segment has 2, a square has 4, a cube has 8, and the next shape (a hypercube) has 16. The number of vertices doubles each time.

Q6. During the Annual Sports Day at Green Valley School, the 6th grade students were divided into different sports activities, forming unique patterns in their group arrangements. The total number of students was 300, and each student participated in only one event.
The event organizers noticed the following patterns:
For the March Past, students stood in rows following a triangular number pattern: 1, 3, 6, 10, 15, …
For the Relay Race, students formed groups in a doubling sequence: 2, 4, 8, …
For Kho-Kho, students were lined up forming a square, ensuring that the number of rows and columns were equal.
The remaining students who did not participate in any event became spectators and watched the competitions.
Answer the following questions:
1. Find the 8th term in the triangular number sequence to determine how many students will be in the last row of the march past.
2. Predict the 5th group (last group) size in the relay race pattern.
3. If 81 students participated in Kho-Kho, how many rows are there in the arrangement?
4. How many students were spectators?

Answer: 1. Triangular numbers: 1, 3, 6, 10, 15, 21, 28, 36. The 8th triangular number is 36, so 36 students stand in the last row of the march past.
2. The group sizes double: 2, 4, 8, 16, 32. The 5th group has 32 students.
3. Square numbers: 1, 4, 9, 16, 25, 36, 49, 64, 81. Since 9 × 9 = 81, there are 9 rows.
4. March past: 1 + 3 + 6 + 10 + 15 + 21 + 28 + 36 = 120 students.
Relay race: 2 + 4 + 8 + 16 + 32 = 62 students.
Kho-Kho: 81 students.
Spectators = 300 − (120 + 62 + 81) = 300 − 263 = 37 students.

Q7. Pattern A: Start with 3 and add 7 successively.
Pattern B: Start with 74 and subtract 4 successively.
1. Identify the 12th term of Pattern A by continuing the sequence.
2. Find any two common terms in both patterns.
3. Compare the two patterns: At which term will Pattern A first exceed the value of Pattern B?

Answer: 1. Pattern A: 3, 10, 17, 24, 31, 38, 45, 52, 59, 66, 73, 80, … The 12th term of Pattern A is 80.
2. Pattern B: 74, 70, 66, 62, 58, 54, 50, 46, 42, 38, 34, 30, …
38 appears in Pattern A (6th term) and Pattern B (10th term).
66 appears in Pattern A (10th term) and Pattern B (3rd term).
Two common terms are 38 and 66.
3. Comparing term by term:

At the 8th term, Pattern A (52) is greater than Pattern B (46) for the first time. So Pattern A first exceeds Pattern B at the 8th term.

Q8. Study the following pattern carefully and answer the questions:
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. Express 225 in the same manner.
2. Find the value of 1 + 2 + 3 + … + 69 + 70 + 69 + … + 3 + 2 + 1?
3. Fill in the blank with the appropriate number.
1 + 2 + 3 + … + _____ + … + 3 + 2 + 1 = 99 × 99

Answer: In this pattern, the sum is always the square of the middle (highest) number: 1 = 1 × 1, 4 = 2 × 2, 9 = 3 × 3, 16 = 4 × 4, 25 = 5 × 5.
1. 225 = 15 × 15, so 225 = 1 + 2 + 3 + … + 14 + 15 + 14 + … + 3 + 2 + 1.
2. The highest number is 70, so the sum = 70 × 70 = 4900.
3. The right side is 99 × 99, so the highest number must be 99.
1 + 2 + 3 + … + 98 + 99 + 98 + … + 3 + 2 + 1 = 99 × 99. The missing number is 99.

Q9. The Virahānka numbers is a special sequence of numbers. The first few numbers of the Virahānka numbers are 1, 2, 3, 5, 8, 13, ... and so on. Describe how you can find the next number in the sequence of Virahānka numbers. Find the next five Virahānka numbers.

Answer: The first two Virahānka numbers are 1 and 2. From the third number onwards, each number is the sum of the two numbers just before it.
The next five numbers after 13 are:
8 + 13 = 21
13 + 21 = 34
21 + 34 = 55
34 + 55 = 89
55 + 89 = 144
So the next five Virahānka numbers are 21, 34, 55, 89 and 144.

Q10. Assertion (A): The next term in the sequence 1 × 1 = 1, 11 × 11 = 121, 111 × 111 = 12321 will be 1111 × 1111 = 1234321.
Reason (R): This pattern follows a sequence where each new term adds an extra 1 to the number, and the result forms a number that increases to the middle digit and then decreases in the same way.
Choose the correct option out of the choices given below.
(a) Both statements A and R are true, and R is the correct explanation of A.
(b) Both statements A and R are true, but R is not the correct explanation of A.
(c) Statement A is true, but statement R is false.
(d) Statement A is false, but statement R is true.

Answer: (a) Both statements A and R are true, and R is the correct explanation of A.
1 × 1 = 1, 11 × 11 = 121, 111 × 111 = 12321. Each time one more 1 is added, and the answer counts up to the middle digit and then back down. So 1111 × 1111 = 1234321. The Assertion is true, and the Reason correctly explains why.

Q11. Find the sum of 3rd and 5th hexagonal numbers.
(a) 44
(b) 56
(c) 80
(d) 117

Answer: (c) 80
In Ganita Prakash, the hexagonal numbers are 1, 7, 19, 37, 61, … (each new hexagon adds a ring of dots around the one before it).
3rd hexagonal number = 19
5th hexagonal number = 61
Sum = 19 + 61 = 80

Q12. Which of the following is both squared and cubed number?
(a) 125
(b) 27
(c) 36
(d) 64

Answer: (d) 64
125 = 5 × 5 × 5 (a cube, but not a square).
27 = 3 × 3 × 3 (a cube, but not a square).
36 = 6 × 6 (a square, but not a cube).
64 = 8 × 8 and also 64 = 4 × 4 × 4. So 64 is both a square number and a cube number.

Related Study Material on Extramarks

• CBSE Important Questions Class 6 Maths Chapter 2
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