72, 144, 216, 288, __, __
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
- Check for alternative representations
- Consider factor relationships and connections
We have: 72, 144, 216, 288
144 - 72 = 72, 216 - 144 = 72, 288 - 216 = 72
Each number increases by 72
288 + 72 = 360, 360 + 72 = 432
Check: 72 + 72 = 144, 144 + 72 = 216, 216 + 72 = 288, 288 + 72 = 360, 360 + 72 = 432
Pattern: 72n (where n = 1, 2, 3, 4, 5, 6)
72 = 2³ × 3², so this pattern connects to powers of 2 and 3
This is the 72 times table: 72×1, 72×2, 72×3, 72×4, 72×5, 72×6
72, 144, 216, 288, 360, 432
• 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 72n for term n
• Multiplication connection: These are multiples of 72
• Factor analysis: 72 = 8×9 = 2³×3², connects to multiple number systems
96, 192, 288, 384, __, __
Growing pattern: A sequence where each number increases by following a specific rule
We have: 96, 192, 288, 384
192 - 96 = 96, 288 - 192 = 96, 384 - 288 = 96
Each number increases by 96
384 + 96 = 480, 480 + 96 = 576
Check: 96 + 96 = 192, 192 + 96 = 288, 288 + 96 = 384, 384 + 96 = 480, 480 + 96 = 576
Pattern: 96n (where n = 1, 2, 3, 4, 5, 6)
96 = 2⁵ × 3, so this pattern connects to powers of 2 and multiples of 3
This is the 96 times table: 96×1, 96×2, 96×3, 96×4, 96×5, 96×6
96, 192, 288, 384, 480, 576
• 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 96 (96×1, 96×2, 96×3...)
• Algebraic representation: Express pattern as 96n for term n
• Factor analysis: 96 = 2⁵×3, connects to powers of 2 and multiples of 3
• Prime factorization: 96 = 2×2×2×2×2×3, showing exponential growth patterns
120, 240, 360, 480, __, __
Growing pattern: A sequence where each number increases by following a specific rule
We have: 120, 240, 360, 480
240 - 120 = 120, 360 - 240 = 120, 480 - 360 = 120
Each number increases by 120
480 + 120 = 600, 600 + 120 = 720
Check: 120 + 120 = 240, 240 + 120 = 360, 360 + 120 = 480, 480 + 120 = 600, 600 + 120 = 720
Pattern: 120n (where n = 1, 2, 3, 4, 5, 6)
120 = 2³ × 3 × 5, so this pattern connects to powers of 2, multiples of 3, and multiples of 5
This is the 120 times table: 120×1, 120×2, 120×3, 120×4, 120×5, 120×6
120, 240, 360, 480, 600, 720
• 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 120 (120×1, 120×2, 120×3...)
• Algebraic representation: Express pattern as 120n for term n
• Factor analysis: 120 = 2³×3×5, connects to multiple number systems
• Prime factorization: 120 = 2×2×2×3×5, showing complex exponential patterns
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
Factors: Numbers that divide evenly into another number
Prime Factorization: Breaking a number down into its prime number components
• 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
• Factor Analysis: Understanding prime factors reveals deeper pattern connections
• Multiple Representations: Patterns can be viewed arithmetically, algebraically, and geometrically
- 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
- Analyze: Consider factor relationships and connections to other patterns
- Represent: View the pattern from multiple perspectives (arithmetic, algebraic, geometric)
• 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)
• Factor analysis: Break down the pattern number into its prime factors
• Multiple approaches: Solve patterns using different methods to verify answers
• 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
• Overcomplicating: Missing simple patterns due to overthinking
• Ignoring factors: Not considering prime factorization connections
• 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"
• Factor: "Prime factorize to see deeper connections"
• Multiple: "Try different approaches to confirm answers"
Count by 120s: 120, 240, 360, 480, 600, 720...
Count by 144s: 144, 288, 432, 576, 720, 864...
Count by 180s: 180, 360, 540, 720, 900, 1080...
Count by 216s: 216, 432, 648, 864, 1080, 1296...
Count by 240s: 240, 480, 720, 960, 1200, 1440...
- 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
- Analyze factors: Consider prime factorization connections
- View multiple ways: Consider arithmetic, algebraic, and geometric perspectives
• 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
• Factor analysis: Consider prime factorization relationships
• Multiple perspectives: View patterns from different mathematical angles
108, 216, 324, 432, __, __
Rule: Add 108
Answer: 108, 216, 324, 432, 540, 648
168, 336, 504, 672, __, __
Rule: Add 168
Answer: 168, 336, 504, 672, 840, 1008
192, 384, 576, 768, __, __
Rule: Add 192
Answer: 192, 384, 576, 768, 960, 1152
252, 504, 756, 1008, __, __
Rule: Add 252
Answer: 252, 504, 756, 1008, 1260, 1512
Questions & Answers
Question: I noticed that 120 = 2³ × 3 × 5. How does this help with understanding the pattern?
Answer: Excellent observation! Prime factorization reveals deeper connections:
- 120 = 2³ × 3 × 5: So 120, 240, 360... connects to powers of 2, multiples of 3, and multiples of 5
- Each term: 120×1, 120×2, 120×3... = (2³×3×5)×1, (2³×3×5)×2, (2³×3×5)×3...
- Deeper patterns: You can think of this as 8×15, 16×15, 24×15... or 15×8, 15×16, 15×24...
- Connections: This pattern connects to the 2s, 3s, 4s, 5s, 6s, 8s, 10s, 12s, 15s, 20s, 24s, 30s, 40s, 60s, and 120s patterns!
Prime factorization helps you see how one pattern connects to multiple other patterns.
Question: What's the difference between arithmetic sequences and geometric sequences? What about quadratic sequences?
Answer: Great question! Here are the key differences:
- Arithmetic sequence: You add the same number each time (like 5, 10, 15, 20...)
- Geometric sequence: You multiply by the same number each time (like 2, 6, 18, 54...)
- Quadratic sequence: The difference between terms increases linearly (like 1, 4, 9, 16...)
- Our patterns: All the growing patterns we've seen are arithmetic sequences
- Formulas: Arithmetic uses addition (an + b), geometric uses multiplication (a × r^n), quadratic uses squares (an² + bn + c)
We'll study geometric and quadratic sequences in higher grades!
Question: How do patterns connect to algebra and more advanced math?
Answer: Patterns are the bridge to algebra and advanced mathematics! Here's how:
- Variables introduction: When we write patterns as 72n, n is a variable
- Functions: Patterns show input-output relationships (term number → value)
- Equations: Pattern formulas become equations we can solve
- Graphing: Arithmetic sequences graph as straight lines
- Real-world modeling: Many phenomena follow arithmetic patterns
- Calculus foundations: Patterns help understand rates of change
Mastering patterns now builds the foundation for algebra, functions, and calculus!