Number Patterns

For Elementary Students

What Is a Number Pattern?

A number pattern is a list of numbers that follows a rule.

Think about it like this: If you can figure out the rule, you can predict what number comes next — like a detective solving a mystery!

Simple Patterns: Adding the Same Number

The easiest patterns add the same number each time.

Example: 3, 6, 9, 12, 15, ...

What's the pattern?

3 → +3 → 6 → +3 → 9 → +3 → 12 → +3 → 15

Rule: Add 3 each time

Next number: 15 + 3 = 18

Another Adding Pattern

Example: 10, 15, 20, 25, ...

What do we add each time?

• 10 to 15 → add 5
• 15 to 20 → add 5
• 20 to 25 → add 5

Rule: Add 5

Next number: 25 + 5 = 30

Patterns That Subtract

Example: 50, 45, 40, 35, ...

What's happening?

• 50 to 45 → subtract 5
• 45 to 40 → subtract 5
• 40 to 35 → subtract 5

Rule: Subtract 5

Next number: 35 − 5 = 30

Skip Counting Is a Pattern!

When you skip count, you're making a pattern!

Count by 2s: 2, 4, 6, 8, 10, ... (add 2 each time)

Count by 5s: 5, 10, 15, 20, 25, ... (add 5 each time)

Count by 10s: 10, 20, 30, 40, 50, ... (add 10 each time)

Finding the Rule

Steps:

1. Look at the first two numbers
2. What do you add or subtract to get from one to the next?
3. Check if it's the same for the next pair
4. That's your rule!

For Junior High Students

Types of Number Patterns

1. Arithmetic patterns — add or subtract the same number

2. Geometric patterns — multiply or divide by the same number

3. Special patterns — follow unique rules

Arithmetic Patterns (Adding/Subtracting)

Definition: Each term changes by a constant difference.

Example: 2, 5, 8, 11, 14, ...

Find the pattern:

• Difference: 5 − 2 = 3
• Check: 8 − 5 = 3, 11 − 8 = 3, 14 − 11 = 3 ✓

Rule: Add 3 each time

Next term: 14 + 3 = 17

General form: Each term = previous term + d (where d is the constant difference)

Subtracting Patterns

Example: 100, 93, 86, 79, ...

Differences: 93 − 100 = −7, 86 − 93 = −7

Rule: Subtract 7 (or add −7)

Next term: 79 − 7 = 72

Geometric Patterns (Multiplying/Dividing)

Definition: Each term is multiplied by a constant ratio.

Example: 3, 6, 12, 24, 48, ...

Find the pattern:

• Ratio: 6 ÷ 3 = 2
• Check: 12 ÷ 6 = 2, 24 ÷ 12 = 2, 48 ÷ 24 = 2 ✓

Rule: Multiply by 2

Next term: 48 × 2 = 96

General form: Each term = previous term × r (where r is the constant ratio)

Dividing Patterns

Example: 1000, 100, 10, 1, ...

Ratios: 100 ÷ 1000 = 0.1, 10 ÷ 100 = 0.1

Rule: Divide by 10 (or multiply by 0.1)

Next term: 1 ÷ 10 = 0.1

How to Identify the Pattern Type

Step 1: Find the differences between consecutive terms

If differences are constant → Arithmetic (add/subtract)

If differences are NOT constant → Check ratios

If ratios are constant → Geometric (multiply/divide)

If neither → Special pattern

Example 1: 5, 8, 11, 14

Differences: 3, 3, 3 (constant) → Arithmetic, add 3

Example 2: 2, 6, 18, 54

Differences: 4, 12, 36 (not constant)

Ratios: 6÷2=3, 18÷6=3, 54÷18=3 → Geometric, multiply by 3

Special Patterns

Square Numbers: 1, 4, 9, 16, 25, 36, ...

Rule: n² (1², 2², 3², 4², 5², 6², ...)

Triangular Numbers: 1, 3, 6, 10, 15, 21, ...

Rule: Add 2, then 3, then 4, then 5, ... (increasing differences)

1
1 + 2 = 3
3 + 3 = 6
6 + 4 = 10
10 + 5 = 15

Fibonacci-like: 1, 1, 2, 3, 5, 8, 13, ...

Rule: Each term is the sum of the two before it

• 1 + 1 = 2
• 1 + 2 = 3
• 2 + 3 = 5
• 3 + 5 = 8
• 5 + 8 = 13

Finding the nth Term (Arithmetic)

Formula for arithmetic pattern: aₙ = a₁ + (n − 1)d

Where:

• aₙ = the nth term
• a₁ = first term
• n = position number
• d = common difference

Example: Find the 20th term of: 7, 11, 15, 19, ...

• a₁ = 7
• d = 4
• n = 20

Calculate: a₂₀ = 7 + (20 − 1) × 4 = 7 + 19 × 4 = 7 + 76 = 83

Finding the nth Term (Geometric)

Formula: aₙ = a₁ × r^(n−1)

Where r = common ratio

Example: Find the 6th term of: 3, 6, 12, 24, ...

• a₁ = 3
• r = 2
• n = 6

Calculate: a₆ = 3 × 2⁵ = 3 × 32 = 96

Real-Life Patterns

Savings: Deposit $10 each week → 10, 20, 30, 40, ... (arithmetic)

Population growth: Doubles each year → 100, 200, 400, 800, ... (geometric)

Stacking blocks: Rows of 1, 3, 6, 10, ... (triangular)

Practice