Patterns in Mathematics
Mathematics is often called the science of patterns. This opening Class 6 Ganita Prakash chapter looks at patterns in numbers and shapes — counting, odd and even numbers, triangular and square numbers, and growing shape sequences — and shows how spotting the rule behind a pattern lets us predict what comes next.
Learning objectives
- Recognise and extend number patterns.
- Identify odd, even, triangular and square numbers.
- Describe patterns made by shapes.
- Find the rule that generates a pattern.
Key concepts
Number sequences
A sequence is a list of numbers arranged by a rule. The counting numbers 1, 2, 3, 4… go up by 1, the even numbers 2, 4, 6, 8… go up by 2, and the odd numbers 1, 3, 5, 7… also go up by 2. Once we know the rule, we can continue the sequence as far as we like.
Triangular and square numbers
Some numbers make neat shapes with dots. Triangular numbers 1, 3, 6, 10… can be drawn as triangles, and square numbers 1, 4, 9, 16… can be drawn as squares. A square number is a number multiplied by itself, so 4 = 2 × 2 and 9 = 3 × 3.
Patterns with shapes
Patterns also appear in shapes that grow step by step, such as a row of squares or an L-shape that adds one more square each time. Counting the squares at each step often reveals a number pattern hiding inside the shape pattern.
Finding the rule
To extend a pattern, we look for the rule: how each term is made from the one before, or from its position. The sum of the first few odd numbers gives square numbers (1 + 3 = 4, 1 + 3 + 5 = 9), a surprising link between two patterns.
Important formulas
Even number
even number = 2 × a whole number
Square number
square number = n × n
Odd-number sum
1 + 3 + 5 + … (n terms) = n × n
Key definitions
- Pattern
- An arrangement of numbers or shapes that follows a rule.
- Sequence
- An ordered list of numbers formed by a rule.
- Triangular number
- A number of dots that can be arranged in a triangle, like 1, 3, 6, 10.
- Square number
- A number that is a whole number multiplied by itself, like 4, 9, 16.
Solved examples
Q1. Write the next two numbers: 2, 4, 6, 8, ___, ___.
Solution: The rule is 'add 2', so the next numbers are 10 and 12.
Q2. Is 16 a square number? Why?
Solution: Yes, because 16 = 4 × 4, so it can be drawn as a 4-by-4 square of dots.
Q3. Find the sum 1 + 3 + 5 + 7.
Solution: The sum of the first 4 odd numbers is 4 × 4 = 16.
Common mistakes to avoid
- Guessing the next term without first finding the rule.
- Confusing triangular numbers with square numbers.
- Thinking a square number means doubling (4 squared is 16, not 8).
- Assuming every increasing list is a pattern, even with no fixed rule.
Patterns in Mathematics — MCQ Quiz
10 questions with instant feedback. Use number keys 1–4 to answer.
The next number in 1, 3, 5, 7, ___ is:
Practice questions
Short answer
What is a number sequence?
An ordered list of numbers formed by following a rule.
Give an example of a triangular number.
6 (it can be drawn as a triangle of dots: 1 + 2 + 3).
What do you get by adding the first four odd numbers?
1 + 3 + 5 + 7 = 16, the square number 4 × 4.
Long answer
Explain how to find and extend the rule of a pattern, using 2, 5, 8, 11 as an example.
To find a rule, compare each term with the one before it. In 2, 5, 8, 11 each term is 3 more than the previous one, so the rule is 'add 3'. Using this rule we extend the pattern: 11 + 3 = 14, then 14 + 3 = 17, and so on. Knowing the rule lets us predict any later term without drawing the whole sequence.
Describe triangular and square numbers and one link between number patterns.
Triangular numbers (1, 3, 6, 10…) are dots arranged as triangles, each formed by adding the next counting number (1, 1+2, 1+2+3…). Square numbers (1, 4, 9, 16…) are dots arranged as squares, formed by n × n. A neat link is that adding consecutive odd numbers gives square numbers: 1 = 1, 1 + 3 = 4, 1 + 3 + 5 = 9 — showing how one pattern is built from another.
HOTS (Higher Order Thinking)
Without adding all of them, find 1 + 3 + 5 + 7 + 9 + 11.
These are the first 6 odd numbers, and their sum is 6 × 6 = 36.
The 10th triangular number is 55. What is the 11th, and why?
Each triangular number adds the next counting number, so the 11th is 55 + 11 = 66.
Quick revision
Revision notes
- A pattern follows a rule; find the rule to extend it.
- Even: 2,4,6…; Odd: 1,3,5…; both step by 2.
- Triangular numbers 1,3,6,10…; square numbers 1,4,9,16… (n × n).
- Sum of the first n odd numbers = n × n.
Key takeaways
- Maths is the study of patterns in numbers and shapes.
- Finding the rule lets you predict what comes next.
- Different patterns can be surprisingly linked.
Frequently asked questions
What is a pattern in mathematics?
An arrangement of numbers or shapes that follows a definite rule.
What is a square number?
A whole number multiplied by itself, such as 9 = 3 × 3.
How are odd numbers linked to square numbers?
Adding consecutive odd numbers from 1 gives square numbers (1, 4, 9, 16…).