Solved Exercises on Completing Number Patterns for Grade 3

Master completing number patterns: arithmetic sequences, geometric sequences, and pattern completion through these 5 detailed exercises.

Solution: Exercises 1 to 3
1 Basic arithmetic pattern
Exercise 1
Complete the pattern:
5, 10, 15, __, 25, 30
Definition:

Arithmetic pattern: A sequence where each term increases by the same amount (the common difference).

Pattern completion method:
  1. Look at the first two numbers: 5 and 10
  2. Find the difference: 10 - 5 = 5
  3. Check if the same difference applies to next pairs
  4. Apply the rule to find the missing number
Pattern
5, 10, 15, __, 25, 30
Rule
Add 5
Missing term
20
Step 1: Compare first two numbers

10 - 5 = 5, so we add 5 each time

Step 2: Check the rule with next pair

15 - 10 = 5 ✓ Same difference

Step 3: Apply the rule

15 + 5 = 20

Step 4: Verify with remaining numbers

20 + 5 = 25 ✓, and 25 + 5 = 30 ✓

Complete pattern: 5, 10, 15, 20, 25, 30
Final answer:

Rule: Add 5 to each number

Missing number: 20

Applied rules:

• Pattern Recognition: Look for consistent differences

• Arithmetic Sequence: Constant difference between consecutive terms

• Rule Application: Apply the identified rule to find missing terms

2 Decreasing pattern
Exercise 2
Complete the pattern:
40, 32, 24, __, 8, 0
Definition:

Decreasing arithmetic pattern: A sequence where each term decreases by the same amount.

Pattern
40, 32, 24, __, 8, 0
Rule
Subtract 8
Missing term
16
Step 1: Compare first two numbers

40 - 32 = 8, so we subtract 8 each time

Step 2: Check with next pair

32 - 24 = 8 ✓ Same difference

Step 3: Apply the rule

24 - 8 = 16

Step 4: Verify with remaining numbers

16 - 8 = 8 ✓, and 8 - 8 = 0 ✓

Complete pattern: 40, 32, 24, 16, 8, 0
Final answer:

Rule: Subtract 8 from each number

Missing number: 16

Applied rules:

• Decreasing Pattern: Same principle as increasing, but with subtraction

• Consistency Check: Verify the rule works for all consecutive pairs

• Negative Numbers: Understand that patterns can continue below zero

3 Geometric pattern
Exercise 3
Complete the pattern:
3, 6, 12, __, 48, 96
Definition:

Geometric pattern: A sequence where each term is multiplied by the same number.

Pattern
3, 6, 12, __, 48, 96
Rule
Multiply by 2
Missing term
24
Step 1: Compare first two numbers

6 ÷ 3 = 2, so we multiply by 2 each time

Step 2: Check with next pair

12 ÷ 6 = 2 ✓ Same multiplier

Step 3: Apply the rule

12 × 2 = 24

Step 4: Verify with remaining numbers

24 × 2 = 48 ✓, and 48 × 2 = 96 ✓

Complete pattern: 3, 6, 12, 24, 48, 96
Final answer:

Rule: Multiply each number by 2

Missing number: 24

Applied rules:

• Multiplication Pattern: When addition/subtraction doesn't work, try division/multiplication

• Geometric Sequence: Each term is multiplied by a constant factor

• Pattern Verification: Always verify the rule with multiple pairs

Completing Number Patterns: Summary
\(a_n = a_1 + (n-1)d\)
Arithmetic Sequence Formula
Arithmetic
Add/Subtract Same Number
5, 10, 15, 20, ... (add 5)
Geometric
Multiply/Divide Same Number
3, 6, 12, 24, ... (×2)
Alternating
Different Operations
2, 4, 3, 6, 5, 10, ...
Key definitions:

Number Pattern: A sequence of numbers that follows a specific rule

Rule: The mathematical operation that creates the pattern

Term: Each individual number in the sequence

Arithmetic Pattern: Each term changes by adding/subtracting the same number

Geometric Pattern: Each term changes by multiplying/dividing by the same number

Pattern Completion Method:
  1. Observe: Look at the given numbers and identify how many terms are provided
  2. Calculate Differences: Find the difference between consecutive terms
  3. Identify Pattern Type: Determine if it's arithmetic, geometric, or complex
  4. Test the Rule: Apply the rule to multiple pairs to verify consistency
  5. Apply to Missing Terms: Use the confirmed rule to find unknown numbers
  6. Verify Answer: Check that the completed sequence follows the rule consistently
Tip 1: Start by looking for addition or subtraction patterns first.
Tip 2: If adding/subtracting doesn't work, try multiplying/dividing.
Tip 3: Write down the differences between consecutive terms to see the pattern.
Tip 4: Always check your rule with at least 2-3 pairs of numbers.
Tip 5: Look for patterns in the differences themselves (e.g., differences might increase by 1 each time).
Common errors: Not checking enough pairs, mixing up operations, forgetting negative numbers, miscalculating differences.
Key strategies: Look for patterns in differences, check for multiplication/division, verify consistently, consider alternating operations.
Pattern Types and Rules:

• Arithmetic: Add or subtract the same number (e.g., 2, 5, 8, 11, ...)

• Geometric: Multiply or divide by the same number (e.g., 2, 6, 18, 54, ...)

• Alternating: Different rules for odd/even positions (e.g., ×2, -2, ×2, -2, ...)

• Complex: Relationship between multiple previous terms (e.g., Fibonacci)

• Quadratic: Differences themselves form an arithmetic pattern (e.g., 1, 4, 9, 16, ...)

Solution: Exercises 4 to 5
4 Alternating pattern
Exercise 4
Complete the pattern:
4, 8, 6, 12, 10, __, 18
Definition:

Alternating pattern: A sequence that uses different rules for odd and even positions.

Pattern
4, 8, 6, 12, 10, __, 18
Rules
×2, -2
Missing term
20
Step 1: Separate odd and even positions

Position 1: 4, Position 3: 6, Position 5: 10 (odd positions)

Position 2: 8, Position 4: 12, Position 6: ? (even positions)

Position 7: 18

Step 2: Analyze odd positions

4 → 6 (add 2), 6 → 10 (add 4), so add 6 to get 16

Step 3: Analyze even positions

8 → 12 (add 4), so add 6 to get 18, but position 7 is 18

Actually: 4×2=8, 8-2=6, 6×2=12, 12-2=10, 10×2=20, 20-2=18

Step 4: Apply the alternating rule

The pattern alternates: multiply by 2, then subtract 2

After 10, we multiply by 2: 10 × 2 = 20

Complete pattern: 4, 8, 6, 12, 10, 20, 18
Final answer:

Rule: Multiply by 2, then subtract 2, repeating

Missing number: 20

Applied rules:

• Alternating Operations: Different rules apply to consecutive terms

• Pattern Breaking: When simple arithmetic doesn't work, look for alternating rules

• Verification: Always test the identified rule throughout the sequence

5 Complex pattern
Exercise 5
Complete the pattern:
1, 2, 4, 7, 11, __, 22
Definition:

Increasing differences pattern: The differences between consecutive terms form their own pattern.

Pattern
1, 2, 4, 7, 11, __, 22
Differences
+1, +2, +3, +4, +5
Missing term
16
Step 1: Calculate the differences

2-1=1, 4-2=2, 7-4=3, 11-7=4

The differences are: 1, 2, 3, 4

Step 2: Identify the pattern in differences

The differences increase by 1 each time: +1, +1, +1, ...

Step 3: Continue the difference pattern

Next difference: 4 + 1 = 5

Step 4: Apply to find the missing number

11 + 5 = 16

Step 5: Verify with the final number

Next difference: 5 + 1 = 6

16 + 6 = 22 ✓

Complete pattern: 1, 2, 4, 7, 11, 16, 22
Final answer:

Rule: Add increasing numbers (1, 2, 3, 4, 5, 6, ...)

Missing number: 16

Applied rules:

• Complex Patterns: When simple rules don't work, look for patterns in the differences

• Second-order Differences: Differences of differences can form their own pattern

• Systematic Approach: Try different approaches when initial attempts fail

Advanced Pattern Completion Techniques
\(a_n = a_{n-1} + d_n\)
Variable Difference Formula
Advanced definitions:

Quadratic Pattern: A pattern where the differences between consecutive terms form an arithmetic sequence

Cubic Pattern: A pattern where the second differences form an arithmetic sequence

Fibonacci Sequence: Each term is the sum of the two preceding terms

Prime Number Pattern: A sequence using only prime numbers

Square/Cube Number Pattern: A sequence of perfect squares or cubes

Advanced Pattern Recognition Methodology:
  1. First-level differences: Calculate the difference between consecutive terms
  2. Second-level differences: Calculate the difference between the first differences
  3. Third-level differences: If needed, calculate differences of second differences
  4. Identify the pattern type: Based on which level of differences is constant
  5. Apply the appropriate formula: Linear, quadratic, or higher-order polynomial
Tip 1: For complex patterns, make a table showing the original sequence and its differences.
Tip 2: Look for special number sets: primes, squares (1, 4, 9, 16...), cubes (1, 8, 27...).
Tip 3: Remember that some patterns combine arithmetic and geometric elements.
Tip 4: If stuck, try reading the sequence backwards to see if that reveals a pattern.
Tip 5: Consider that the pattern might involve more than one mathematical operation per step.
Advanced errors: Assuming all patterns are simple arithmetic, not considering multiple levels of differences, overcomplicating simple patterns.
Advanced strategies: Using difference tables, recognizing special number sequences, combining multiple pattern types.
Advanced Pattern Types:

• Linear (Arithmetic): First differences constant (e.g., 2, 5, 8, 11...)

• Quadratic: Second differences constant (e.g., 1, 4, 9, 16, 25...)

• Cubic: Third differences constant (e.g., 1, 8, 27, 64...)

• Fibonacci-type: Each term depends on multiple previous terms

• Alternating: Different rules for odd/even positions

Questions & Answers

Question: I don't understand how to find the rule when the numbers seem random. How do I know if it's addition, subtraction, multiplication, or division?

Answer: Great question! Here's a systematic approach to determine the pattern type:

  • Start with addition/subtraction: Look at the difference between consecutive numbers
  • If the differences are the same, it's an arithmetic pattern (adding or subtracting the same number)
  • If differences change, try multiplication/division: divide each number by the previous one
  • If the ratios are the same, it's a geometric pattern (multiplying or dividing by the same number)

Example: In 2, 6, 18, 54...
Differences: 6-2=4, 18-6=12, 54-18=36 (not the same)
Ratios: 6÷2=3, 18÷6=3, 54÷18=3 (same ratio!)
So it's multiplication by 3.

Question: My child gets confused when the pattern involves decreasing numbers. How can I help them understand subtraction patterns?

Answer: Decreasing patterns can be tricky! Here are helpful strategies:

  • Use visual aids: Draw number lines or use manipulatives to show the decrease
  • Focus on the positive difference: Instead of saying "subtract 4," say "go back 4"
  • Connect to real-life: Temperature dropping, money being spent, etc.
  • Emphasize that subtraction patterns work just like addition patterns, but in reverse

Example: For 20, 16, 12, 8...
Show: 20-4=16, 16-4=12, 12-4=8
The pattern is "subtract 4" or "decrease by 4 each time."

Question: What happens when I get to really big numbers in patterns? How do I keep track?

Answer: Great question! As patterns continue, numbers can get large quickly. Here are strategies:

  • Always focus on the rule, not just the numbers
  • Write down the rule clearly: "multiply by 3" or "add 7"
  • Double-check your calculations as you go
  • Look for shortcuts: if multiplying by 10, just add a zero

Example: In the pattern 2, 6, 18, 54, 162...
Rule: multiply by 3
Next: 162 × 3 = 486
Even though 486 is large, the rule stays the same: multiply by 3.