Patterns and Rules | JHS Mathematics | The Let's Learn Project

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.

\[ \text{Example: } 2, 4, 6, 8, \underline{\quad} \]

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 Set

Math Challenge

Timed Test in 7 Levels

Test your skills with identifying patterns and predicting elements