Introduction
Patterns are everywhere. We see them in nature, music, art, architecture, and mathematics. The arrangement of petals on a flower, the stripes on a zebra, the phases of the Moon, and even the rhythm of a song all follow patterns. In mathematics, recognising patterns allows us to make predictions, solve problems, and discover relationships between numbers and shapes.
Learning to recognise and describe patterns is one of the most important mathematical skills. By identifying how a pattern changes, we can predict future terms, write mathematical rules, and solve more complex problems in algebra and geometry.
What Is a Pattern?
A pattern is an arrangement of numbers, shapes, objects, or events that follows a predictable rule.
Patterns may:
- Repeat.
- Grow.
- Shrink.
- Change according to a rule.
Once the rule is known, future parts of the pattern can be predicted.
Figure 1. Patterns can be found in numbers, shapes, colours, and many everyday situations.
Number Patterns
A number pattern is a sequence of numbers that follows a rule.
Example 1
2, 4, 6, 8, 10, ...
Rule:
Add 2 each time.
Next terms:
12, 14, 16
Example 2
5, 10, 15, 20, ...
Rule:
Add 5 each time.
Next terms:
25, 30, 35
Example 3
100, 90, 80, 70, ...
Rule:
Subtract 10 each time.
Next terms:
60, 50, 40
Number patterns may involve:
- Addition
- Subtraction
- Multiplication
- Division
Shape Patterns
Patterns are not limited to numbers.
Shapes can also form patterns.
For example:
○ □ ○ □ ○ □ ...
Rule:
Circle, square, repeat.
Another example:
▲ ▲ ■ ▲ ▲ ■ ...
Rule:
Two triangles, then one square.
Shape patterns are often called repeating patterns.
Figure 2. Shape patterns repeat according to a predictable arrangement.
Growing Patterns
Some patterns become larger over time.
Example:
●
●●
●●●
●●●●
Each figure contains one more dot than the previous figure.
Rule:
Add one dot each time.
Growing patterns are common in mathematics because they help introduce algebra.
Describing a Pattern
A good mathematical description explains how the pattern changes.
Examples:
| Pattern |
Description |
| 3, 6, 9, 12 |
Add 3 each time |
| 50, 45, 40, 35. |
Subtract 5 each time |
| 4, 8, 16, 32 |
Multiply by 2 each time |
Using words helps explain the mathematical rule clearly.
Figure 3. Growing patterns increase according to a mathematical rule.
Finding the Next Term
To find the next term:
Step 1
Look for the pattern.
Step 2
Identify the rule.
Step 3
Apply the rule.
Example:
7, 10, 13, 16, ...
Difference:
+3
Next terms:
19, 22, 25
Example:
64, 32, 16, 8, ...
Rule:
÷2
Next terms:
4, 2, 1
Always check that the rule works for every term.
Extending Patterns
Once the rule is known, the pattern can continue.
Example:
1, 4, 7, 10, ...
Rule:
Add 3
20th term?
Continue adding 3 until the 20th number.
Mathematicians often develop formulas that allow large terms to be found without listing every value.
Patterns and Mathematical Rules
Many patterns can be described using mathematical rules.
Example:
2, 4, 6, 8, 10
Rule:
Multiply the term number by 2.
| Term Number. |
Value |
| 1 |
2 |
| 2 |
4 |
| 3 |
6 |
| 4 |
8 |
| 5 |
10 |
This rule can be written as:
Value = 2 × Term Number
This introduces the idea of algebraic relationships.
Figure 4. Patterns can often be represented using mathematical rules or formulas.
Patterns in Everyday Life
Patterns appear throughout the world.
Examples include:
- Traffic lights changing colour.
- Calendar dates.
- Music rhythms.
- Floor tiles.
- Brick walls.
- Flower petals.
- Animal markings.
- Seasons of the year.
Recognising these patterns helps us make predictions about future events.
Why Patterns Matter
Patterns help mathematicians:
- Make predictions.
- Solve problems.
- Discover formulas.
- Understand relationships.
- Develop algebra.
Many areas of mathematics begin with recognising simple patterns before developing more advanced ideas.
Figure 5. Natural patterns inspire many mathematical ideas and help scientists understand the world.
Worked Example
Question
Find the next three terms.
3, 7, 11, 15, ...
Solution
Step 1:
Find the difference.
7 − 3 = 4
11 − 7 = 4
15 − 11 = 4
Rule:
Add 4 each time.
Next three terms:
19, 23, 27
Real-World Connection
Patterns are essential in computer programming, engineering, architecture, and finance. Computer programs often search for patterns in data to recognise faces, predict weather, recommend videos, or detect fraud. Engineers use repeating patterns when designing bridges and buildings, while scientists study patterns in climate, genetics, and ecosystems to better understand the natural world.
Did You Know?
The seeds in a sunflower often form two sets of spirals that follow Fibonacci numbers (1, 1, 2, 3, 5, 8, 13, ...). This arrangement allows the seeds to be packed together very efficiently and is one of the most famous examples of mathematical patterns occurring in nature.
Key Terms
Growing pattern – A pattern that increases or decreases according to a mathematical rule.
Mathematical rule – A description or formula explaining how a pattern changes.
Pattern – A predictable arrangement of numbers, shapes, or objects.
Repeating pattern – A pattern in which the same arrangement occurs over and over.
Sequence – An ordered list of numbers or objects that follows a rule.
Term – Each individual number or object in a sequence.
Key Takeaways
- A pattern is a predictable arrangement that follows a rule.
- Patterns can involve numbers, shapes, diagrams, or real-world situations.
- Recognising the rule allows us to determine the next terms in a sequence.
- Growing patterns can often be described using mathematical rules or formulas.
- Patterns help mathematicians make predictions and discover relationships.
- Recognising patterns is an important foundation for learning algebra, geometry, and many other areas of mathematics.
