🔢 Number Patterns - Complete Notes & Formulae
1. Use a Rule to Complete a Number Pattern
Definition:
A number pattern is a sequence of numbers that follow a rule. The rule tells you how to get from one number to the next.
A number pattern is a sequence of numbers that follow a rule. The rule tells you how to get from one number to the next.
Common Pattern Rules
- Arithmetic (Add/Subtract): Add or subtract a fixed number each time.
nk = nk-1 + d - Geometric (Multiply/Divide): Multiply or divide by a fixed number.
nk = nk-1 × r - Increasing Patterns: The added difference increases each step.
nk = nk-1 + (difference increases) - Alternating Patterns: Alternate between two or more rules.
Add, then subtract, then add…
Example 1: 2, 5, 8, 11, ...
Rule: Add 3 each time
Example 2: 100, 50, 25, 12.5, ...Rule: Divide by 2 each time
2. Compare Patterns
Comparing Patterns: Look for similarities or differences in the rules used, starting numbers, or how quickly patterns grow.
Steps:
- Find the rule for each pattern
- Compare the rates of change (difference or ratio)
- Identify which pattern grows faster/slower
Example:
Pattern A: 5, 10, 15, 20, ... (+5 each time)
Pattern B: 5, 25, 125, 625, ... (×5 each time)
Observation: Pattern A increases steadily, Pattern B grows much faster.
Pattern A: 5, 10, 15, 20, ... (+5 each time)
Pattern B: 5, 25, 125, 625, ... (×5 each time)
Observation: Pattern A increases steadily, Pattern B grows much faster.
3. Complete an Increasing Number Pattern
Increasing Patterns:
- The amount added increases with each step (example: add 2, add 4, add 6...)
- Sometimes called arithmetic patterns with changing differences
Example: 1, 2, 4, 7, 11, 16, ...
Add 1, Add 2, Add 3, Add 4, ...
Next Term: 16 + 6 = 22
Example: 5, 6, 8, 11, 15, ...
Rule: Start at 5, Add 1, Add 2, Add 3, Add 4...
Next term: 15 + 5 = 204. Complete a Multiplication Number Pattern
Multiplication Patterns (Geometric Sequences)
- Each term is multiplied by the same number (the common ratio).
- Works for both increasing and decreasing patterns.
Example: 2, 4, 8, 16, 32, ...
Rule: Multiply by 2 each time
Next Term: 32 × 2 = 64
Formula:
If the first term is a and the common ratio is r: nk = a × rk-1
5. Number Patterns: Word Problems
Solving Pattern Word Problems: Read carefully, find the rule, use it to continue the pattern or answer the question.
Example 1: A plant grows 5 cm the first week, 8 cm the next, 11 cm the next … How tall after 5 weeks?
Pattern: +3 each week (5, 8, 11, 14, 17)
Final Height: 17 cm after 5 weeks.
Example 2: The number of chairs doubles each row. If the first row has 2, second has 4, how many in the 5th row?
Pattern: ×2 each time. Row 3: 8, Row 4: 16, Row 5: 32.
Answer: 32 chairs in the 5th row.6. Number Patterns: Mixed Review
Mixed Pattern Types:
- Simple Addition/Subtraction Patterns: 20, 18, 16, 14, ... (subtract 2)
- Multiplication/Division Patterns: 3, 9, 27, 81, ... (×3)
- Combination Patterns: Alternate adding/subtracting: 4, 8, 7, 11, 10...
- Special Patterns: Fibonacci: 0, 1, 1, 2, 3, 5, ... (add previous two terms)
Challenge Problem: Find the pattern and next term.
Pattern: 2, 6, 12, 20, 30, ...
Rule: Add 4, add 6, add 8, add 10, ...
Next term: 30 + 12 = 42
Pattern: 2, 6, 12, 20, 30, ...
Rule: Add 4, add 6, add 8, add 10, ...
Next term: 30 + 12 = 42
Tip: Always look for a rule: is it arithmetic (addition/subtraction), geometric (multiply/divide), alternating, or something special?