25, 50, 75, 100, __, __
Growing pattern: A sequence where each number increases by following a specific rule
- Examine the first few numbers in the sequence
- Calculate the difference between consecutive numbers
- Determine the consistent rule that generates the pattern
- Apply the same rule to find missing numbers
- Verify the rule applies to all given terms
- Express the pattern algebraically when possible
We have: 25, 50, 75, 100
50 - 25 = 25, 75 - 50 = 25, 100 - 75 = 25
Each number increases by 25
100 + 25 = 125, 125 + 25 = 150
Check: 25 + 25 = 50, 50 + 25 = 75, 75 + 25 = 100, 100 + 25 = 125, 125 + 25 = 150
Pattern: 25n (where n = 1, 2, 3, 4, 5, 6)
25, 50, 75, 100, 125, 150
• Consistent difference: The same number is added to each term
• Sequential progression: Each term follows the previous by the same rule
• Pattern verification: Check that the rule applies to all given terms
• Place value understanding: Recognize how digits change in each position
• Algebraic representation: Express pattern as 25n for term n
12, 24, 36, 48, __, __
Growing pattern: A sequence where each number increases by following a specific rule
We have: 12, 24, 36, 48
24 - 12 = 12, 36 - 24 = 12, 48 - 36 = 12
Each number increases by 12
48 + 12 = 60, 60 + 12 = 72
Check: 12 + 12 = 24, 24 + 12 = 36, 36 + 12 = 48, 48 + 12 = 60, 60 + 12 = 72
Pattern: 12n (where n = 1, 2, 3, 4, 5, 6)
12, 24, 36, 48, 60, 72
• Consistent difference: The same number is added to each term
• Sequential progression: Each term follows the previous by the same rule
• Pattern verification: Check that the rule applies to all given terms
• Multiplication connection: These are multiples of 12 (12×1, 12×2, 12×3...)
• Algebraic representation: Express pattern as 12n for term n
35, 70, 105, 140, __, __
Growing pattern: A sequence where each number increases by following a specific rule
We have: 35, 70, 105, 140
70 - 35 = 35, 105 - 70 = 35, 140 - 105 = 35
Each number increases by 35
140 + 35 = 175, 175 + 35 = 210
Check: 35 + 35 = 70, 70 + 35 = 105, 105 + 35 = 140, 140 + 35 = 175, 175 + 35 = 210
Pattern: 35n (where n = 1, 2, 3, 4, 5, 6)
35, 70, 105, 140, 175, 210
• Consistent difference: The same number is added to each term
• Sequential progression: Each term follows the previous by the same rule
• Pattern verification: Check that the rule applies to all given terms
• Multiplication connection: These are multiples of 35 (35×1, 35×2, 35×3...)
• Algebraic representation: Express pattern as 35n for term n
Growing Pattern: A sequence of numbers that increases following a specific rule
Pattern Rule: The mathematical operation that creates the next number in the sequence
Term: Each individual number in the pattern sequence
Difference: The amount added or subtracted to get from one term to the next
Sequence: An ordered list of numbers following a pattern
Algebraic Expression: A mathematical statement that describes the pattern rule
• Consistency Rule: The same operation must be applied to every term
• Constant Difference: In growing patterns, the difference between consecutive terms remains the same
• Sequential Order: Each term follows the previous one in a predictable manner
• Repetition Principle: The pattern rule repeats throughout the entire sequence
• Verification Rule: Always check that your rule applies to all given terms
• Algebraic Connection: Patterns can be expressed using variables and formulas
- Observe: Carefully examine the given numbers in the pattern
- Compare: Calculate the difference between the first two numbers
- Verify: Confirm that the same difference applies to subsequent pairs
- Identify: Determine the consistent pattern rule
- Apply: Use the rule to find missing numbers in the sequence
- Check: Verify that your answers maintain the established pattern
- Validate: Ensure the rule works for the entire sequence
- Express: Represent the pattern algebraically when possible
• Subtract consecutively: Find the difference by subtracting each number from the next
• Double-check consistency: Ensure the rule works for all given terms
• Look for multiples: Many patterns involve counting by larger numbers
• Use mental math: Practice quick addition to speed up pattern completion
• Notice place value: Pay attention to how digits change in each position
• Connect to multiplication: Recognize patterns as multiplication tables
• Express algebraically: Practice writing patterns as formulas (an + b)
• Inconsistent rule: Changing the pattern rule partway through
• Wrong difference: Calculating incorrect differences between terms
• Skipping verification: Not checking that the rule applies to all terms
• Arithmetic errors: Making calculation mistakes when continuing the pattern
• Place value confusion: Misunderstanding how digits change in different positions
• Algebraic errors: Incorrectly expressing the pattern as a formula
• Rhyme: "Find the gap, keep the trap, follow the path!"
• Memory trick: Think of climbing stairs with equal steps
• Visual cue: Imagine a number line with equal jumps
• Phrase: "Same jump, same fun - that's how patterns are done!"
• Strategy: "Check, check, check - always verify!"
• Algebraic: "Look for the multiplier and express as an + b"
Count by 12s: 12, 24, 36, 48, 60, 72...
Count by 15s: 15, 30, 45, 60, 75, 90...
Count by 20s: 20, 40, 60, 80, 100, 120...
Count by 25s: 25, 50, 75, 100, 125, 150...
Count by 50s: 50, 100, 150, 200, 250, 300...
- Read carefully: Understand what the worksheet asks
- Find the rule: Calculate the difference between given numbers
- Apply consistently: Use the same rule throughout the sequence
- Check work: Verify each number follows the pattern
- Write clearly: Present answers in an organized manner
- Review: Double-check your answers before submitting
- Express algebraically: Write the pattern as a formula when possible
• Take time: Don't rush through problems
• Show work: Write down calculations to avoid errors
• Double-check: Review answers before moving on
• Ask for help: If stuck, seek assistance
• Connect concepts: Relate patterns to multiplication tables
• Algebraic thinking: Express patterns using variables
18, 36, 54, 72, __, __
Rule: Add 18
Answer: 18, 36, 54, 72, 90, 108
28, 56, 84, 112, __, __
Rule: Add 28
Answer: 28, 56, 84, 112, 140, 168
42, 84, 126, 168, __, __
Rule: Add 42
Answer: 42, 84, 126, 168, 210, 252
36, 72, 108, 144, __, __
Rule: Add 36
Answer: 36, 72, 108, 144, 180, 216
Questions & Answers
Question: I see that some patterns are like multiplication tables. How do I write that as a formula?
Answer: Great observation! When a pattern starts at the beginning of a multiplication table, you can write it as:
- For 12, 24, 36, 48...: This is 12×1, 12×2, 12×3, 12×4..., so the formula is 12n
- For 25, 50, 75, 100...: This is 25×1, 25×2, 25×3, 25×4..., so the formula is 25n
- General form: If the pattern starts at the first multiple of a number, the formula is (that number) × n
Where n represents the position in the sequence (1st term, 2nd term, etc.).
Question: What if a pattern doesn't start at the beginning of a multiplication table?
Answer: If a pattern doesn't start at the beginning, you need to adjust the formula:
- For 15, 25, 35, 45...: This increases by 10 each time
- Start with the rule: Each term adds 10
- Find the starting point: The first term is 15
- Formula: 10n + 5 (because 10×1 + 5 = 15, 10×2 + 5 = 25, etc.)
The general form is: (difference) × n + (adjustment to match first term).
Question: Why do we need to learn to write patterns as formulas? Isn't just knowing the rule enough?
Answer: Writing patterns as formulas is very powerful because:
- Quick calculation: You can find any term without listing all the terms before it
- Higher math: This prepares you for algebra and more advanced math
- Problem solving: Formulas help solve complex problems more efficiently
- Pattern analysis: You can compare different patterns and see relationships
- Real-world applications: Scientists and engineers use formulas to model patterns in nature
Learning formulas now builds a strong foundation for all future math success!