Introduction
Patterns are sequences that follow a specific rule or relationship. In mathematics, we encounter patterns in numbers, shapes, and symbols. Being able to identify the rule in a pattern helps us predict what comes next.
The next number is 10 (rule: add 2)
Understanding Patterns
A pattern is a sequence where elements are arranged according to a specific rule. We can identify two main types:
- Numerical Patterns: Sequences of numbers following a mathematical rule
- Symbolic Patterns: Sequences of shapes, letters, or symbols following a visual rule
Example: △, ○, △, ○, △, ○,... (Alternating triangle and circle)
Symbolic Patterns
Identifying Symbolic Patterns
To determine the rule in a symbolic pattern:
- Examine the sequence carefully
- Look for repeating elements or cycles
- Note any changes in position, rotation, or size
- Identify the relationship between consecutive elements
Example 1: Shape Sequence
Pattern: □, ○, △, □, ○, △, □, ○, △,...
Rule: The sequence repeats square, circle, triangle
Next element: □
Example 2: Letter Pattern
Pattern: A, C, E, G, I,...
Rule: Skip one letter in the alphabet (A, skip B, C, skip D, etc.)
Next element: K
Position and Rotation Patterns
Some patterns involve changes in position or rotation:
Pattern: ↑, →, ↓, ←, ↑, →, ↓, ←,...
Rule: Rotating 90° clockwise each time
Next element: ↑
Symbolic Pattern Table
| Pattern | Rule | Next Element |
|---|---|---|
| ♠, ♥, ♦, ♣, ♠, ♥,... | Repeating suits in order | ♦ |
| ▲, ▼, ▲, ▼,... | Alternating up and down | ▲ |
| 1, *, 2, *, 3, *,... | Alternating number and asterisk | 4 |
Numerical Patterns
Identifying Numerical Patterns
To find the rule in a numerical pattern:
- Look at the differences between consecutive numbers
- Check if numbers are multiplied by a certain factor
- Look for alternating patterns
- Check for more complex relationships (squares, cubes, etc.)
Example 1: Adding Pattern
Pattern: 5, 8, 11, 14, 17,...
Rule: Add 3 to the previous number
Next element: 20
Example 2: Multiplying Pattern
Pattern: 3, 6, 12, 24, 48,...
Rule: Multiply previous number by 2
Next element: 96
Example 3: Alternating Pattern
Pattern: 1, 3, 2, 4, 3, 5, 4, 6,...
Rule: Alternating between +2 and -1
Next element: 5
Complex Numerical Patterns
Some patterns involve more complex mathematical operations:
Example 1: 1, 4, 9, 16, 25,... (Square numbers)
Example 2: 2, 5, 10, 17, 26,... (n² + 1)
Example 3: 1, 1, 2, 3, 5, 8,... (Fibonacci sequence)
Numerical Pattern Table
| Pattern | Rule | Next Element |
|---|---|---|
| 10, 20, 30, 40,... | Add 10 | 50 |
| 1, 4, 7, 10,... | Add 3 | 13 |
| 5, 10, 20, 40,... | Multiply by 2 | 80 |
| 1, 3, 6, 10, 15,... | Add increasing numbers (1+2=3, 3+3=6, etc.) | 21 |
Applying Rules
Using Rules to Find Elements
When given the rule, we can find specific elements in a sequence:
Example 1: Given Rule "Add 5"
Starting with 3, find the 4th term:
1st: 3
2nd: 3 + 5 = 8
3rd: 8 + 5 = 13
4th: 13 + 5 = 18
Example 2: Given Rule "Multiply by 3 then subtract 1"
Starting with 2, find the 3rd term:
1st: 2
2nd: (2 × 3) - 1 = 5
3rd: (5 × 3) - 1 = 14
Example 3: Given Position-to-Term Rule "2n + 1"
Find the 5th term:
n = 5: 2(5) + 1 = 11
Finding Rules from Positions
We can find general rules for patterns by examining the position of each term:
Sequence: 3, 5, 7, 9, 11,...
Position (n): 1, 2, 3, 4, 5,...
Term: 3, 5, 7, 9, 11,...
Rule: term = 2n + 1
Practice Exercise
Question 1
Find the next element in this pattern: □, △, ○, □, △, ○, □, △, ___
Pattern repeats □, △, ○. Next element is ○
Question 2
What is the rule for this pattern: 4, 7, 10, 13, 16,...?
Add 3 to the previous number
Question 3
Find the 6th term in this pattern if the rule is "multiply by 2": 3, 6, 12, 24,...
5th term: 48
6th term: 96
Question 4
What comes next in this pattern: A, D, G, J, ___?
Rule: Skip 2 letters (A, skip B,C → D, etc.)
Next letter: M
Question 5
Find the rule for this pattern: 1, 4, 9, 16, 25,...
Square numbers (n²)
Question 6
Given the rule "add 4 then subtract 1" starting with 5, find the 4th term
1st: 5
2nd: 5 + 4 - 1 = 8
3rd: 8 + 4 - 1 = 11
4th: 11 + 4 - 1 = 14
Ready for more challenges?
Full Practice SetQuick Test
You have 60 minutes to complete the test. Try to work on your speed as you prepare towards the final exam.
Good luck and remember to check your answers with the solutions provided. If you have any questions, feel free to ask your teacher or refer to the video lessons for more help.