StudyOSStudyOS CBSE Study Material, Class 5–12

Patterns in Mathematics

NCERT Class 6 · Mathematics Based on NCERT Class 6 Mathematics textbook · Free CBSE study kit

Everything you need to ace this chapter, in one place.
Notes · Mind Map · Quiz · Assertion-Reason · Case Studies · Question Bank · Past Papers

Get Full Study Kit →
Studying with friends? Send them this chapter.

Chapter Notes

Overview
This chapter introduces the idea that mathematics is fundamentally the search for patterns and the explanations for why those patterns exist. It explores number sequences like counting numbers, odd numbers, even numbers, triangular numbers, square numbers, cube numbers, Virahānka numbers, and powers of 2 and 3, as well as shape sequences like regular polygons, complete graphs, stacked triangles, stacked squares, and the Koch snowflake. The one big idea a student must walk away with is that patterns are everywhere in mathematics and nature, and understanding them helps us make sense of the world and solve problems.
Key Concepts (17)
Patterns
Regularities or repeated designs in numbers, shapes, or nature that can be observed and explained.
e.g. The pattern of odd numbers: 1, 3, 5, 7, 9, ... where each number is 2 more than the previous one.
Number Theory
The branch of mathematics that studies patterns in whole numbers.
e.g. Studying sequences like 1, 4, 9, 16, 25 (square numbers) is part of number theory.
Number Sequence
An ordered list of numbers that follow a specific rule or pattern.
e.g. 1, 2, 3, 4, 5, 6, 7, ... is the counting numbers sequence.
Counting Numbers
The sequence of numbers starting from 1 and increasing by 1 each time: 1, 2, 3, 4, 5, ...
e.g. When you count the number of students in a class: 1, 2, 3, 4, ...
Odd Numbers
Numbers that are not divisible by 2, ending in 1, 3, 5, 7, or 9.
e.g. 1, 3, 5, 7, 9, 11, 13, ...
Even Numbers
Numbers that are divisible by 2, ending in 0, 2, 4, 6, or 8.
e.g. 2, 4, 6, 8, 10, 12, 14, ...
Triangular Numbers
Numbers that can be arranged as a triangle of dots: 1, 3, 6, 10, 15, 21, 28, ...
e.g. 10 dots can be arranged in a triangle with 4 rows: 1 dot, 2 dots, 3 dots, 4 dots.
Square Numbers
Numbers that can be arranged as a square of dots: 1, 4, 9, 16, 25, 36, 49, ...
e.g. 16 dots can be arranged in a 4 by 4 square.
Cubes
Numbers that can be arranged as a cube of dots: 1, 8, 27, 64, 125, 216, ...
e.g. 27 dots can be arranged in a 3 by 3 by 3 cube.
Virahānka Numbers
A number sequence where each number is the sum of the two preceding numbers: 1, 2, 3, 5, 8, 13, 21, ...
e.g. 5 + 8 = 13, so 13 comes after 8.
Powers of 2
Numbers obtained by multiplying 2 by itself repeatedly: 1, 2, 4, 8, 16, 32, 64, ...
e.g. 2 × 2 × 2 = 8, so 8 is a power of 2.
Powers of 3
Numbers obtained by multiplying 3 by itself repeatedly: 1, 3, 9, 27, 81, 243, 729, ...
e.g. 3 × 3 × 3 = 27, so 27 is a power of 3.
Geometry
The branch of mathematics that studies patterns in shapes, including 1D, 2D, and 3D shapes.
e.g. Studying regular polygons like triangles, squares, and pentagons is part of geometry.
Regular Polygons
Shapes with equal-length sides and equal angles, such as regular triangle, square, pentagon, hexagon, etc.
e.g. A regular hexagon has 6 equal sides and 6 equal angles.
Complete Graphs
A shape sequence where each vertex is connected to every other vertex by a line.
e.g. K3 has 3 vertices and 3 lines connecting them.
Koch Snowflake
A shape sequence where each line segment is replaced by a 'speed bump' pattern, creating a fractal.
e.g. Starting with a triangle, each side is replaced by a bump, increasing the number of line segments.
Hexagonal Numbers
Numbers that can be arranged as a hexagon: 1, 7, 19, 37, 61, ...
e.g. 7 dots can be arranged in a hexagon shape.
Formulas & Equations (5)
1 + 3 + 5 + 7 + ... + (2n - 1) = n²
Reads as: The sum of the first n odd numbers equals n squared.
Variables: n = number of odd terms (positive integer); n² = square number
Use when: When asked to find the sum of odd numbers or to show that odd numbers add up to square numbers.
⚠ Students often write the sum as n instead of n², or forget that the last odd number is (2n - 1).
1 + 2 + 3 + ... + n + ... + 3 + 2 + 1 = n²
Reads as: Adding counting numbers up to n and back down to 1 gives n squared.
Variables: n = the largest counting number in the sequence; n² = square number
Use when: When asked to find the sum of a symmetric 'up and down' sequence of counting numbers.
⚠ Students think the sum is 2n or n(n+1)/2, forgetting the symmetric nature gives a perfect square.
1 + 2 + 4 + 8 + ... + 2^(n-1) = 2^n - 1
Reads as: The sum of powers of 2 from 1 to 2^(n-1) equals 2^n minus 1.
Variables: n = number of terms; 2^n = next power of 2
Use when: When asked to find the sum of a geometric series of powers of 2.
⚠ Students forget to subtract 1, writing 2^n instead of 2^n - 1.
Triangular number × 6 + 1 = Hexagonal number
Reads as: Multiplying a triangular number by 6 and adding 1 gives a hexagonal number.
Variables: Triangular number = 1, 3, 6, 10, 15, ...; Hexagonal number = 7, 19, 37, 61, ...
Use when: When asked to find hexagonal numbers from triangular numbers or to show the relationship.
⚠ Students multiply by 6 but forget to add 1, or use the wrong triangular number.
Number of line segments in Koch Snowflake = 3 × 4^(n-1)
Reads as: The number of line segments in the nth stage of the Koch snowflake is 3 times 4 raised to the power (n-1).
Variables: n = stage number (starting from 1); 3 = initial number of segments in triangle
Use when: When asked to find the number of line segments in a given stage of the Koch snowflake.
⚠ Students use 4^n instead of 4^(n-1), or forget to multiply by 3.
How It Works
Identifying a Number Sequence Pattern
  1. Step 1: Look at the given numbers in order.
  2. Step 2: Find the rule that connects one number to the next (e.g., add 2, multiply by 2, add the previous two numbers).
  3. Step 3: Use the rule to predict the next numbers in the sequence.
  4. Step 4: Verify the rule works for all given terms.
💡 Always check the rule for at least two consecutive pairs to ensure it is consistent.
Visualising Number Sequences with Pictures
  1. Step 1: Represent each number as a set of dots or shapes.
  2. Step 2: Arrange the dots in a pattern that matches the sequence (e.g., triangle for triangular numbers, square for square numbers).
  3. Step 3: Observe how the arrangement grows as the number increases.
  4. Step 4: Use the picture to understand why the sequence behaves the way it does.
💡 Pictures help explain why patterns work, not just what the pattern is.
Adding Odd Numbers to Get Square Numbers
  1. Step 1: Start with 1 (the first odd number).
  2. Step 2: Add the next odd number (3) to get 4, which is 2².
  3. Step 3: Continue adding the next odd number (5) to get 9, which is 3².
  4. Step 4: The sum of the first n odd numbers always equals n².
💡 This works because each odd number adds an L-shaped layer to a square, as shown in the picture.
Adding Counting Numbers Up and Down to Get Square Numbers
  1. Step 1: Start with 1.
  2. Step 2: Add 1 + 2 + 1 to get 4, which is 2².
  3. Step 3: Add 1 + 2 + 3 + 2 + 1 to get 9, which is 3².
  4. Step 4: The sum of counting numbers up to n and back down to 1 always equals n².
💡 The largest number in the middle is the side length of the square.
Adding Powers of 2 to Get One Less Than the Next Power
  1. Step 1: Start with 1 (2⁰).
  2. Step 2: Add 1 + 2 = 3, which is 4 - 1 (2² - 1).
  3. Step 3: Add 1 + 2 + 4 = 7, which is 8 - 1 (2³ - 1).
  4. Step 4: The sum of powers of 2 from 1 to 2^(n-1) equals 2^n - 1.
💡 Adding 1 to the sum gives the next power of 2.
Generating the Koch Snowflake Sequence
  1. Step 1: Start with an equilateral triangle (3 line segments).
  2. Step 2: Replace each line segment with a 'speed bump' shape (4 smaller segments).
  3. Step 3: Repeat the process for each new line segment.
  4. Step 4: The number of line segments multiplies by 4 at each stage.
💡 The number of segments at stage n is 3 × 4^(n-1).
Exam Traps — Don't Make These Mistakes
⚠ Confusing Triangular and Square Numbers
✗ Students write that 1, 4, 9, 16 are triangular numbers or that 1, 3, 6, 10 are square numbers.
→ Triangular numbers are arranged as triangles (1, 3, 6, 10, ...), while square numbers are arranged as squares (1, 4, 9, 16, ...). They are different sequences.
✓ Triangular numbers: 1, 3, 6, 10, 15, ... Square numbers: 1, 4, 9, 16, 25, ...
⚠ Mistaking Virahānka Numbers for Fibonacci Numbers
✗ Students write that the Virahānka sequence is 1, 1, 2, 3, 5, 8, ... (Fibonacci).
→ The Virahānka sequence in this chapter starts with 1, 2, 3, 5, 8, ... (not two 1's). The rule is the same (sum of previous two), but the starting numbers differ.
✓ Virahānka numbers: 1, 2, 3, 5, 8, 13, 21, ... (each term is the sum of the two preceding terms).
⚠ Forgetting to Add 1 to the Sum of Powers of 2
✗ Students write that 1 + 2 + 4 + 8 = 16 (instead of 15).
→ The sum of powers of 2 from 1 to 8 is 15, not 16. Adding 1 to the sum gives the next power of 2 (16).
✓ 1 + 2 + 4 + 8 = 15, and 15 + 1 = 16 (which is 2⁴).
⚠ Confusing Number of Sides and Number of Corners in Regular Polygons
✗ Students write that a pentagon has 5 sides and 4 corners.
→ In any closed polygon, the number of sides equals the number of corners (vertices). A pentagon has 5 sides and 5 corners.
✓ A regular pentagon has 5 sides and 5 corners. In any polygon, number of sides = number of vertices.
⚠ Miscounting Line Segments in Complete Graphs
✗ Students write that K4 has 4 lines (instead of 6).
→ In a complete graph, every vertex is connected to every other vertex. For 4 vertices, the number of lines is 3 + 2 + 1 = 6.
✓ K4 has 4 vertices and 6 lines (each vertex connects to the other 3, but each line is counted once).
⚠ Thinking 36 is Only a Square Number
✗ Students write that 36 is only a square number (6²).
→ 36 is both a square number (6²) and a triangular number (the 8th triangular number). It can be arranged as both a square and a triangle.
✓ 36 is both a square number (6 × 6) and a triangular number (1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 = 36).
Compare & Contrast
Divisibility by 2
Odd Numbers
Even Numbers
Odd numbers are not divisible by 2 (e.g., 1, 3, 5, 7), while even numbers are divisible by 2 (e.g., 2, 4, 6, 8).
Arrangement of dots
Triangular Numbers
Square Numbers
Triangular numbers (1, 3, 6, 10, ...) are arranged as triangles, while square numbers (1, 4, 9, 16, ...) are arranged as squares.
Inclusion of zero
Counting Numbers
Whole Numbers
Counting numbers start from 1 (1, 2, 3, ...), while whole numbers include 0 (0, 1, 2, 3, ...). The chapter focuses on counting numbers.
Base number
Powers of 2
Powers of 3
Powers of 2 are 1, 2, 4, 8, 16, ... (multiplying by 2 each time), while powers of 3 are 1, 3, 9, 27, 81, ... (multiplying by 3 each time).
Exam Focus
Marks pattern: 2 MCQs (1 mark each), 1 SA-I (2 marks), 1 SA-II (3 marks), 1 LA (5 marks) — total approximately 12 marks
Most asked: Identifying and extending number sequences (counting, odd, even, triangular, square numbers) · Sum of odd numbers equals square numbers · Visualising number sequences with pictures · Relations between number sequences (e.g., triangular and square numbers) · Regular polygons and their properties
Don't skip: Long answer question on adding odd numbers to get square numbers with pictorial explanation; Question on identifying and extending multiple number sequences from Table 1; Question on the relationship between triangular numbers and square numbers (e.g., 36 is both)
One-Line Revision (read aloud before the exam)
  1. Mathematics is the search for patterns and explanations for why those patterns exist.
  2. Number theory is the branch of mathematics that studies patterns in whole numbers.
  3. Counting numbers: 1, 2, 3, 4, 5, ...
  4. Odd numbers: 1, 3, 5, 7, 9, ...
  5. Even numbers: 2, 4, 6, 8, 10, ...
  6. Triangular numbers: 1, 3, 6, 10, 15, ...
  7. Square numbers: 1, 4, 9, 16, 25, ...
  8. Cubes: 1, 8, 27, 64, 125, ...
  9. Virahānka numbers: 1, 2, 3, 5, 8, 13, 21, ... (each term is sum of previous two).
  10. Powers of 2: 1, 2, 4, 8, 16, 32, 64, ...
  11. Powers of 3: 1, 3, 9, 27, 81, 243, 729, ...
  12. Adding odd numbers starting from 1 gives square numbers: 1 + 3 + 5 + 7 = 16 = 4².
  13. Adding counting numbers up and down gives square numbers: 1 + 2 + 3 + 2 + 1 = 9 = 3².
  14. Adding powers of 2 starting from 1 gives one less than the next power of 2.
  15. Geometry is the branch of mathematics that studies patterns in shapes.
  16. Regular polygons have equal-length sides and equal angles.
  17. The number of sides equals the number of corners in any polygon.
  18. 36 is both a triangular number and a square number.
  19. Hexagonal numbers: 1, 7, 19, 37, 61, ...
  20. The Koch snowflake sequence: 3, 12, 48, 192, ... (3 times powers of 4).

MCQs — 10 Questions with Answers

Q1. Which of the following is a square number?

  • A. 15
  • B. 25 ✓
  • C. 35
  • D. 45

Answer: B — 25 is a square number because 5 × 5 = 25, whereas 15, 35, and 45 are not perfect squares.

Q2. In the sequence of odd numbers 1, 3, 5, 7, 9, ..., what is the 10th odd number?

  • A. 17
  • B. 19 ✓
  • C. 21
  • D. 23

Answer: B — The 10th odd number is found by the pattern: odd number = 2n – 1, where n = 10, so 2(10) – 1 = 19.

Q3. What pattern do you observe when adding the first few odd numbers: 1 = 1, 1 + 3 = 4, 1 + 3 + 5 = 9?

  • A. You always get even numbers
  • B. You always get triangular numbers
  • C. You always get square numbers ✓
  • D. You always get cube numbers

Answer: C — The sum of consecutive odd numbers starting from 1 always produces square numbers: 1, 4, 9, 16, 25, etc.

Q4. Which number can be arranged both as a perfect square AND as a triangular number?

  • A. 16
  • B. 25
  • C. 36 ✓
  • D. 49

Answer: C — 36 is special because 36 = 6² (a square) and 36 = 1+2+3+4+5+6+7+8 (a triangular number).

Q5. In the sequence 2, 4, 6, 8, 10, ..., this sequence is called _____.

  • A. odd numbers
  • B. counting numbers
  • C. even numbers ✓
  • D. square numbers

Answer: C — This sequence contains only even numbers, which are numbers divisible by 2.

Q6. What is the next number in the sequence of powers of 2: 1, 2, 4, 8, 16, 32, ___?

  • A. 48
  • B. 56
  • C. 64 ✓
  • D. 72

Answer: C — In powers of 2, each number is multiplied by 2 to get the next, so 32 × 2 = 64.

Q7. A number sequence that can be arranged as dots in a triangular pattern is called a _____ number.

  • A. square
  • B. triangular ✓
  • C. even
  • D. odd

Answer: B — Triangular numbers (1, 3, 6, 10, 15...) get their name because the dots can be arranged in a triangle shape.

Q8. Looking at the pattern 1, 3, 6, 10, 15, 21, ..., which term comes next?

  • A. 25
  • B. 27
  • C. 28 ✓
  • D. 30

Answer: C — These are triangular numbers; each term adds the next counting number: 21 + 7 = 28.

Q9. In the sequence of cube numbers 1, 8, 27, 64, ..., the number 27 is 3 cubed. What does '3 cubed' mean?

  • A. 3 + 3 + 3
  • B. 3 × 3
  • C. 3 × 3 × 3 ✓
  • D. 3 to the power of 2

Answer: C — 3 cubed (3³) means 3 × 3 × 3 = 27, where a number is multiplied by itself three times.

Q10. Why do mathematicians study number patterns and sequences?

  • A. Only to pass exams
  • B. To find patterns and understand why they exist, which helps explain things in nature and science ✓
  • C. To confuse students
  • D. For no practical reason

Answer: B — Mathematicians study patterns to both discover what patterns exist and understand why they exist, which has real-world applications in science, nature, and technology.

Flashcards

What is a number sequence?

A number sequence is an ordered list of numbers that follow a specific pattern or rule.

What are odd numbers?

Odd numbers are numbers that cannot be divided equally by 2, like 1, 3, 5, 7, 9, 11, 13.

What are even numbers?

Even numbers are numbers that can be divided equally by 2, like 2, 4, 6, 8, 10, 12, 14.

What are triangular numbers?

Triangular numbers are 1, 3, 6, 10, 15, 21, 28... — numbers that can be arranged in a triangular shape with dots.

What are square numbers?

Square numbers are 1, 4, 9, 16, 25, 36... — numbers that result from multiplying a number by itself (1×1, 2×2, 3×3, etc.).

What are cube numbers?

Cube numbers are 1, 8, 27, 64, 125... — numbers that result from multiplying a number by itself three times (1×1×1, 2×2×2, etc.).

Why is 36 special in number patterns?

36 is special because it is both a square number (6×6) and a triangular number (1+2+3+4+5+6+7+8), so it can be arranged as both a square and a triangle.

What pattern do you get when you add odd numbers: 1 + 3 + 5 + 7 + 9 + 11?

When you add consecutive odd numbers starting from 1, you always get square numbers (1=1, 1+3=4, 1+3+5=9, 1+3+5+7=16, etc.).

What is the rule for the sequence 1, 2, 4, 8, 16, 32, 64?

This is the sequence of powers of 2, where each number is made by multiplying the previous number by 2.

Why do mathematicians study number patterns?

Mathematicians study patterns both to find what patterns exist and to understand why they exist, which helps explain many things in nature and science.

Important Board Questions

What is a number sequence? Give one example. [1 mark]

A number sequence is an ordered list of numbers following a pattern; examples include odd numbers (1, 3, 5, 7...), even numbers (2, 4, 6, 8...), or counting numbers (1, 2, 3, 4...).

Write the first 6 terms of the sequence of square numbers. How is each number formed? [2 marks]

Square numbers are 1, 4, 9, 16, 25, 36; each is formed by multiplying a counting number by itself (1×1, 2×2, 3×3, 4×4, 5×5, 6×6).

Add the first 5 odd numbers (1 + 3 + 5 + 7 + 9) and observe the pattern. Explain with a reason why this pattern occurs. Draw a diagram to show your explanation. [3 marks]

1 + 3 + 5 + 7 + 9 = 25 (which is 5²); draw a square grid and partition the dots into layers of odd numbers to show why the sum gives square numbers.

The sequence 1, 3, 6, 10, 15, 21, ... is called triangular numbers. (a) Why are they called triangular numbers? (b) Draw the first four triangular numbers using dots arranged in triangle shapes. (c) What is the next triangular number after 21? Explain the rule used to find it. [5 marks]

Triangular numbers are called that because dots can be arranged in triangle shapes; draw dots in triangular patterns (1 dot, 1+2 dots, 1+2+3 dots, etc.); the next term is found by adding the next counting number (21 + 7 = 28).

True or False: Every square number is also an odd number. Give a reason for your answer. [2 marks]

False — square numbers include 4, 16, 36, which are even; only odd-positioned squares like 1, 9, 25 are odd.

Assertion-Reason Questions for Patterns in Mathematics

Q1. Assertion (A):

The sequence 1, 4, 9, 16, 25 is called the sequence of square numbers.

Reason (R):

Because these numbers can be visualized by arranging dots in a square grid.

Answer: A — Both A and R are true, and R is the correct explanation of A — Both the assertion and reason are true. These are indeed square numbers (1², 2², 3², 4², 5²), and the reason correctly explains why they are called 'square' numbers—because they can be arranged in square patterns with dots.

Q2. Assertion (A):

Mathematics helps humanity by launching satellites and curing diseases.

Reason (R):

Because mathematicians understand the definitions of all numbers.

Answer: C — A is true but R is false — The assertion is true—the textbook gives these examples. However, the reason is false. These applications come from understanding patterns and WHY they exist, not just knowing definitions.

Q3. Assertion (A):

Adding consecutive odd numbers starting from 1 always gives square numbers.

Reason (R):

Because square numbers are always larger than odd numbers.

Answer: B — Both A and R are true, but R is NOT the correct explanation of A — The assertion is true (shown in the textbook with examples), but the reason is not the correct explanation. The reason why is because of how odd numbers can be layered around a square grid, not because of size.

Unlock more Assertion-Reason questions in Premium — unlock →

Case-Based Questions (CBQ) for Patterns in Mathematics

Case Study:

Priya is decorating her bedroom with a pattern of colored dots. She arranges them in rows for different designs: first design has 1 dot, second design has 3 dots, third design has 6 dots, and fourth design has 10 dots. She notices each new design has dots arranged in a triangular shape. Later, her brother shows her another pattern: 1, 4, 9, 16 dots arranged in square shapes for different designs.

Q1. What is the name of the sequence 1, 3, 6, 10 that Priya discovered?

  • A. Triangular numbers ✓
  • B. Square numbers
  • C. Odd numbers
  • D. Counting numbers

Answer: A

Q2. What is the next number in Priya's triangular pattern after 10?

  • A. 12
  • B. 13
  • C. 14
  • D. 15 ✓

Answer: D

Q3. Which number appears in BOTH Priya's triangular pattern and her brother's square pattern?

  • A. 9
  • B. 36 ✓
  • C. 10
  • D. 16

Answer: B

Unlock more case studies in Premium — unlock →

Next chapterLines and Angles →
Found this useful? Share it with your class group.

You've seen the free preview. Open the full interactive study kit — mind map, instant-graded quiz, progress tracking — free, no sign-in required.

Open Full Study Kit — Free →

Premium unlocks the full Question Bank, Daily Practice Papers, Chapter Quiz and Full Mock Papers for every chapter.