How to Explain Number Patterns to Fourth Graders

"What comes next: 2, 6, 10, 14, ___?"

Patterns seem like simple puzzles, but they're actually the gateway to algebraic thinking. Fourth grade is when pattern recognition becomes formalized—students don't just continue patterns, they describe rules and make predictions.

Why Patterns Matter

Patterns are the foundation of mathematics:

• Multiplication tables are pattern collections
• Geometry is built on repeated shapes
• Algebra is the study of patterns expressed as equations
• Science uses patterns to predict phenomena

Students who see and express patterns develop the thinking skills that unlock higher mathematics.

Types of Number Patterns

Repeating Patterns

The same sequence cycles over and over:

2, 5, 8, 2, 5, 8, 2, 5, 8, ...
↑     ↑     ↑
Repeat unit: 2, 5, 8

To find what comes next: Identify the repeating unit and continue the cycle.

Growing (Arithmetic) Patterns

The same amount is added (or subtracted) each time:

3, 7, 11, 15, 19, ...
  +4  +4   +4   +4

Rule: Start at 3, add 4 each time.

Multiplicative Patterns

Each term is multiplied by the same number:

2, 6, 18, 54, 162, ...
 ×3  ×3   ×3   ×3

Rule: Start at 2, multiply by 3 each time.

Decreasing Patterns

Values go down instead of up:

50, 45, 40, 35, 30, ...
  -5   -5   -5   -5

Rule: Start at 50, subtract 5 each time.

Finding and Describing Pattern Rules

The Process

1. Look at consecutive terms: What changes between each pair?
2. Check if the change is consistent: Is it always the same?
3. Write the rule in words: "Start at ___, add/subtract/multiply ___"
4. Test the rule: Does it produce the pattern?

Example 1: Finding an Addition Rule

Pattern: 5, 9, 13, 17, 21, ...

Step 1: Find differences

• 9 - 5 = 4
• 13 - 9 = 4
• 17 - 13 = 4
• 21 - 17 = 4

Step 2: The change is consistent (+4)

Step 3: Rule: Start at 5, add 4 each time

Step 4: Test: 5 → 9 → 13 → 17 → 21 ✓

Example 2: Finding a Two-Step Pattern

Pattern: 1, 4, 7, 10, 13, ...

Differences: +3, +3, +3, +3

Rule: Start at 1, add 3 each time

What's the 10th term?

• Position 1: 1
• Position 2: 1 + 3 = 4
• Position 3: 4 + 3 = 7
• ...
• Shortcut: 1 + (9 × 3) = 1 + 27 = 28

The 10th term is 28.

Input-Output Tables

What is an Input-Output Table?

A table that shows the relationship between two sets of numbers:

Input (x)  | Output (y)
-----------|-----------
    1      |     5
    2      |     8
    3      |    11
    4      |    14

Finding the Rule

Look at how input relates to output:

• When input is 1, output is 5
• When input is 2, output is 8
• When input is 3, output is 11

What's happening?

• 1 → 5: Did we add 4? (1 + 4 = 5) ✓
• 2 → 8: Add 4? (2 + 4 = 6) ✗

Try: Multiply, then add

• 1 × 3 + 2 = 5 ✓
• 2 × 3 + 2 = 8 ✓
• 3 × 3 + 2 = 11 ✓
• 4 × 3 + 2 = 14 ✓

Rule: Multiply by 3, then add 2

Using the Rule

Once you know the rule, find any output:

• Input 5: 5 × 3 + 2 = 17
• Input 10: 10 × 3 + 2 = 32

Or find the input from an output:

• Output 23: What × 3 + 2 = 23?
• 23 - 2 = 21, 21 ÷ 3 = 7
• Input is 7

Patterns in Multiplication Tables

Skip Counting Patterns

The 4s row: 4, 8, 12, 16, 20, 24, 28, 32, 36, 40

• All even
• Alternate between "even + 4" pattern

The 9s pattern: 9, 18, 27, 36, 45, 54, 63, 72, 81, 90

• Tens digit goes up by 1
• Ones digit goes down by 1
• Digits always sum to 9!

Diagonal Patterns

    1  2  3  4  5
   ----------------
1 | 1  2  3  4  5
2 | 2  4  6  8 10
3 | 3  6  9 12 15
4 | 4  8 12 16 20
5 | 5 10 15 20 25

Main diagonal (1, 4, 9, 16, 25): Square numbers!

Factor Patterns

Numbers with many factors appear often in the table.
12 appears at: 1×12, 2×6, 3×4, 4×3, 6×2, 12×1

Shape Patterns

Growing Shape Patterns

Pattern 1:  ●

Pattern 2:  ●●
            ●

Pattern 3:  ●●●
            ●●
            ●

How many dots in each pattern?
Pattern 1: 1
Pattern 2: 3
Pattern 3: 6
Pattern 4: ?

The pattern: 1, 3, 6, 10, ...
Differences: +2, +3, +4, +5, ...

Pattern 4 would have: 6 + 4 = 10 dots

These are triangular numbers!

Pattern Block Sequences

Using pattern blocks:

Step 1: 🔷
Step 2: 🔷🔷🔷
Step 3: 🔷🔷🔷🔷🔷

Blocks: 1, 3, 5, 7, ...
Rule: Odd numbers! Add 2 each time.

Expressing Patterns Mathematically

Using Variables (Introduction)

Instead of words, we can use symbols:

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

• In words: "Multiply the position by 3"
• With a variable: "Position × 3" or "n × 3" or "3n"

Position (n) | Value (3 × n)
-------------|---------------
     1       |      3
     2       |      6
     3       |      9
     4       |      12

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

• In words: "Multiply position by 3, then subtract 1"
• Expression: 3n - 1

Position (n) | Value (3n - 1)
-------------|----------------
     1       | 3(1) - 1 = 2
     2       | 3(2) - 1 = 5
     3       | 3(3) - 1 = 8

Hands-On Activities

Pattern Block Building

Create a pattern with blocks:

• Build the first 3 steps
• Partner describes the rule
• Partner predicts step 4
• Check by building

Number Pattern Stories

Create contexts for patterns:

• "I save $5 each week. After week 1, I have $5. After week 2, I have $10..."
• "A tree grows 3 inches each year. It started at 12 inches..."

Function Machines

Draw a "machine" that follows a rule:

   INPUT
     ↓
  ┌─────┐
  │ +7  │
  └─────┘
     ↓
  OUTPUT

Put in 3 → Get out 10
Put in 5 → Get out 12
Partner guesses the rule!

Pattern Scavenger Hunt

Find patterns in:

• Phone numbers
• Addresses
• Tiled floors
• Nature (flower petals, pinecones)

Create Your Own Pattern

Make a number pattern. Give it to a partner. Can they:

• Continue it?
• Write the rule?
• Find the 10th term?

Common Mistakes and How to Fix Them

Mistake 1: Not Checking All Differences

Wrong: Assuming 2, 4, 7, 11 is "add 2" after seeing 2→4

Fix: Check ALL consecutive differences:

• 4 - 2 = 2
• 7 - 4 = 3
• 11 - 7 = 4

This is +2, +3, +4, ... (increasing differences)

Mistake 2: Starting Count at Wrong Position

Wrong: For pattern 5, 10, 15, 20... saying the "first term times 5" when position 1 should give 5

Fix: Check: Position 1 × 5 = 5 ✓, Position 2 × 5 = 10 ✓
The rule IS "multiply position by 5"

Mistake 3: Confusing the Rule with the Pattern

Wrong: "The rule is 3, 7, 11, 15"

Fix: The rule describes HOW to get the pattern.
Pattern: 3, 7, 11, 15
Rule: Start at 3, add 4 each time

Mistake 4: Only Looking for Addition

Wrong: Missing multiplication patterns like 2, 6, 18, 54

Fix: If addition doesn't show a constant difference, try multiplication (find ratios instead).

Building Algebraic Thinking

From Pattern to Expression

Pattern: The number of tiles in step n is n × (n + 1)

Step 1: 1 × 2 = 2 tiles
Step 2: 2 × 3 = 6 tiles
Step 3: 3 × 4 = 12 tiles
Step 4: 4 × 5 = 20 tiles

Predicting Any Term

Once you have a rule, you can find any term:

Pattern: 4, 7, 10, 13, ...
Rule: 3n + 1

What's the 100th term?
3(100) + 1 = 301

Connecting to Future Concepts

Patterns prepare students for:

Algebra

Variables and expressions grow directly from pattern rules.

Functions

Input-output tables become function tables.

Linear Equations

Arithmetic patterns become y = mx + b.

Sequences and Series

More complex patterns in higher math.

Practice Ideas for Home

Daily Number Challenges

"Continue this pattern 3 more terms: 1, 4, 9, 16, ..."
(Square numbers: 25, 36, 49)

Table Practice

Create input-output tables together:

• "I'll give you a number, you double it and add 1"
• Record in a table
• Let your child figure out your rule

Real-World Patterns

• Savings over time (weekly allowance)
• Plant growth (cm per week)
• Stacking blocks (1, 3, 6, 10...)

Pattern Games

Take turns:

1. Person A creates a pattern (gives first 4 terms)
2. Person B states the rule
3. Person B gives the next 3 terms
4. Swap roles

The Bottom Line

Patterns are not just puzzles—they're the language of mathematics. When fourth graders can look at a sequence of numbers and express the underlying rule, they're thinking algebraically.

The goal isn't just to find "what comes next." It's to understand:

• Why does this pattern work?
• How can I describe it in words?
• How can I predict any term?
• How might I express this symbolically?

This kind of thinking—looking beyond specific numbers to general relationships—is exactly what algebra requires. Pattern work in fourth grade isn't preparation for algebra; it IS algebra, in age-appropriate form.

Help your child see patterns everywhere, describe them precisely, and express them in different ways. That skill will serve them from fourth grade through calculus and beyond.