Patterns and Sequences | Fourth Grade
Complete Notes & Formulas
1. Complete an Increasing Number Pattern
Definition: An increasing number pattern (also called arithmetic sequence) is a pattern where each number increases by adding the same amount each time.
📐 Key Terms:
- Pattern/Sequence: A list of numbers following a rule
- Term: Each number in the sequence
- Common Difference: The amount added each time (must be positive for increasing)
- Rule: The mathematical operation used to get from one term to the next
📝 How to Find the Pattern:
- Step 1: Find the difference between consecutive terms
- Step 2: Check if the difference is the same for all pairs
- Step 3: That constant difference is your rule
- Step 4: Add the difference to find next terms
🔑 Key Formula:
Next Term = Current Term + Common Difference
Common Difference = Term₂ - Term₁
✏️ Examples:
Example 1: Simple Addition Pattern
Pattern: 5, 10, 15, 20, __, __, __
Solution:
Find difference: 10 - 5 = 5
Check: 15 - 10 = 5, 20 - 15 = 5 ✓
Rule: Add 5 each time
Next terms: 20 + 5 = 25, 25 + 5 = 30, 30 + 5 = 35
Answer: 5, 10, 15, 20, 25, 30, 35
Example 2: Larger Numbers
Pattern: 125, 150, 175, 200, __, __, __
Solution:
Find difference: 150 - 125 = 25
Rule: Add 25 each time
Next terms: 200 + 25 = 225, 225 + 25 = 250, 250 + 25 = 275
Answer: 125, 150, 175, 200, 225, 250, 275
💡 Pattern Types:
- Count by 2s: 2, 4, 6, 8, 10...
- Count by 5s: 5, 10, 15, 20, 25...
- Count by 10s: 10, 20, 30, 40, 50...
- Count by 100s: 100, 200, 300, 400...
- Any constant: Add the same number each time
2. Complete a Geometric Number Pattern
Definition: A geometric number pattern (also called geometric sequence) is a pattern where each number is found by multiplying or dividing by the same value each time.
📐 Key Terms:
- Geometric Pattern: Numbers multiply or divide by same amount
- Common Ratio: The number you multiply or divide by
- Growing Pattern: Multiplying (numbers get bigger)
- Shrinking Pattern: Dividing (numbers get smaller)
📝 How to Find the Pattern:
- Step 1: Divide the second term by the first term
- Step 2: Check if the ratio is the same for all pairs
- Step 3: That constant ratio is your rule
- Step 4: Multiply by the ratio to find next terms
🔑 Key Formulas:
Next Term = Current Term × Common Ratio
Common Ratio = Term₂ ÷ Term₁
✏️ Examples:
Example 1: Doubling Pattern (×2)
Pattern: 3, 6, 12, 24, __, __, __
Solution:
Find ratio: 6 ÷ 3 = 2
Check: 12 ÷ 6 = 2, 24 ÷ 12 = 2 ✓
Rule: Multiply by 2 each time
Next terms: 24 × 2 = 48, 48 × 2 = 96, 96 × 2 = 192
Answer: 3, 6, 12, 24, 48, 96, 192
Example 2: Tripling Pattern (×3)
Pattern: 2, 6, 18, 54, __, __
Solution:
Find ratio: 6 ÷ 2 = 3
Check: 18 ÷ 6 = 3, 54 ÷ 18 = 3 ✓
Rule: Multiply by 3 each time
Next terms: 54 × 3 = 162, 162 × 3 = 486
Answer: 2, 6, 18, 54, 162, 486
Example 3: Halving Pattern (÷2)
Pattern: 80, 40, 20, 10, __, __
Solution:
Find ratio: 40 ÷ 80 = 0.5 (or ÷ 2)
Rule: Divide by 2 each time (or multiply by ½)
Next terms: 10 ÷ 2 = 5, 5 ÷ 2 = 2.5
Answer: 80, 40, 20, 10, 5, 2.5
💡 Common Geometric Patterns:
- ×2 (Doubling): 1, 2, 4, 8, 16, 32...
- ×3 (Tripling): 1, 3, 9, 27, 81...
- ×10: 1, 10, 100, 1000, 10000...
- ÷2 (Halving): 100, 50, 25, 12.5...
3. Number Patterns: Word Problems
Definition: Real-world problems that involve identifying, extending, or applying number patterns to find solutions.
📝 Problem-Solving Steps:
- Step 1: Read the problem carefully and identify the pattern
- Step 2: Write out the sequence of numbers given
- Step 3: Find the rule (add, subtract, multiply, or divide?)
- Step 4: Extend the pattern to answer the question
- Step 5: Check that your answer makes sense
✏️ Example Word Problems:
Example 1: Saving Money (Increasing Pattern)
Problem: Maya saves money each week. Week 1: ₹50, Week 2: ₹75, Week 3: ₹100, Week 4: ₹125. If the pattern continues, how much will she save in Week 6?
Solution:
Pattern: 50, 75, 100, 125, __, __
Find difference: 75 - 50 = 25
Rule: Add ₹25 each week
Week 5: 125 + 25 = ₹150
Week 6: 150 + 25 = ₹175
Answer: ₹175 in Week 6
Example 2: Growing Bacteria (Geometric Pattern)
Problem: A bacteria colony doubles every hour. It starts with 5 bacteria. How many bacteria will there be after 4 hours?
Solution:
Hour 0: 5 bacteria
Hour 1: 5 × 2 = 10
Hour 2: 10 × 2 = 20
Hour 3: 20 × 2 = 40
Hour 4: 40 × 2 = 80
Answer: 80 bacteria after 4 hours
Example 3: Seating Arrangement
Problem: A theater has rows of seats. Row 1 has 20 seats, Row 2 has 24 seats, Row 3 has 28 seats. If the pattern continues, how many seats are in Row 7?
Solution:
Pattern: 20, 24, 28, __, __, __, __
Rule: Add 4 seats each row
Row 4: 28 + 4 = 32
Row 5: 32 + 4 = 36
Row 6: 36 + 4 = 40
Row 7: 40 + 4 = 44
Answer: 44 seats in Row 7
💡 Common Keywords:
- "Each time" → Look for repeated operation
- "Doubles" → Multiply by 2
- "Triples" → Multiply by 3
- "Increases by" → Addition pattern
- "Decreases by" → Subtraction pattern
- "Continues" → Extend the pattern
4. Number Patterns: Mixed Review
Definition: Practicing all types of number patterns together - increasing, decreasing, geometric, and special patterns.
📊 Types of Patterns Summary:
| Pattern Type | Rule | Example |
|---|---|---|
| Increasing (Add) | Add same number | 3, 7, 11, 15 (+4) |
| Decreasing (Subtract) | Subtract same number | 50, 42, 34, 26 (-8) |
| Geometric (Multiply) | Multiply same number | 2, 6, 18, 54 (×3) |
| Geometric (Divide) | Divide same number | 64, 32, 16, 8 (÷2) |
✏️ Mixed Practice Problems:
Problem 1: What's the pattern?
7, 14, 21, 28, 35, __
Solution:
Type: Increasing (Addition)
Rule: Add 7 (or multiples of 7)
Answer: 42
Problem 2: What's the pattern?
1, 4, 16, 64, __
Solution:
Type: Geometric (Multiplication)
Rule: Multiply by 4
Answer: 256
Problem 3: What's the pattern?
100, 90, 80, 70, __
Solution:
Type: Decreasing (Subtraction)
Rule: Subtract 10
Answer: 60
Problem 4: What's the pattern?
128, 64, 32, 16, __
Solution:
Type: Geometric (Division)
Rule: Divide by 2 (halving)
Answer: 8
📝 How to Identify Pattern Type:
- Check differences first: If same, it's addition or subtraction
- Check ratios next: If same, it's multiplication or division
- Growing numbers: Addition or multiplication
- Shrinking numbers: Subtraction or division
💡 Special Patterns to Know:
- Square Numbers: 1, 4, 9, 16, 25 (1², 2², 3², 4², 5²)
- Odd Numbers: 1, 3, 5, 7, 9, 11... (+2)
- Even Numbers: 2, 4, 6, 8, 10, 12... (+2)
- Powers of 2: 1, 2, 4, 8, 16, 32... (×2)
- Powers of 10: 1, 10, 100, 1000... (×10)
Patterns and Sequences Quick Reference Chart
| Pattern Type | Operation | Formula | Example |
|---|---|---|---|
| Increasing | Addition | Next = Current + d | 5, 10, 15, 20 |
| Decreasing | Subtraction | Next = Current - d | 20, 15, 10, 5 |
| Geometric Growing | Multiplication | Next = Current × r | 2, 6, 18, 54 |
| Geometric Shrinking | Division | Next = Current ÷ r | 80, 40, 20, 10 |
📐 Key Formulas to Remember:
Common Difference
d = Term₂ - Term₁
Common Ratio
r = Term₂ ÷ Term₁
📚 Fourth Grade Patterns and Sequences - Complete Study Guide
Master these pattern recognition skills for math excellence! ✨