Solved Exercises on Finding Rules in Number Patterns for Grade 3

Master number patterns: arithmetic sequences, pattern recognition, and rule identification through these 5 detailed exercises.

Solution: Exercises 1 to 3
1 Arithmetic sequence
Exercise 1
Find the rule and continue the pattern:
2, 5, 8, 11, __, __
Definition:

Arithmetic sequence: A sequence where each term increases or decreases by the same number (the common difference).

Pattern identification method:
  1. Look at the first two numbers: 2 and 5
  2. Find the difference: 5 - 2 = 3
  3. Check if the same difference applies to next pairs
  4. Apply the rule to find missing numbers
Pattern
2, 5, 8, 11
Rule
Add 3
Next terms
14, 17
Step 1: Compare first two numbers

5 - 2 = 3, so we add 3 each time

Step 2: Check the rule with next pair

8 - 5 = 3 ✓ Same difference

Step 3: Check with next pair

11 - 8 = 3 ✓ Same difference

Step 4: Apply the rule

11 + 3 = 14, then 14 + 3 = 17

Complete pattern: 2, 5, 8, 11, 14, 17
Final answer:

Rule: Add 3 to each number

Missing numbers: 14, 17

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
Find the rule and continue the pattern:
20, 16, 12, 8, __, __
Definition:

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

Pattern
20, 16, 12, 8
Rule
Subtract 4
Next terms
4, 0
Step 1: Compare first two numbers

20 - 16 = 4, so we subtract 4 each time

Step 2: Check with next pair

16 - 12 = 4 ✓ Same difference

Step 3: Check with next pair

12 - 8 = 4 ✓ Same difference

Step 4: Apply the rule

8 - 4 = 4, then 4 - 4 = 0

Complete pattern: 20, 16, 12, 8, 4, 0
Final answer:

Rule: Subtract 4 from each number

Missing numbers: 4, 0

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 Multiplication pattern
Exercise 3
Find the rule and continue the pattern:
2, 6, 18, 54, __, __
Definition:

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

Pattern
2, 6, 18, 54
Rule
Multiply by 3
Next terms
162, 486
Step 1: Compare first two numbers

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

Step 2: Check with next pair

18 ÷ 6 = 3 ✓ Same multiplier

Step 3: Check with next pair

54 ÷ 18 = 3 ✓ Same multiplier

Step 4: Apply the rule

54 × 3 = 162, then 162 × 3 = 486

Complete pattern: 2, 6, 18, 54, 162, 486
Final answer:

Rule: Multiply each number by 3

Missing numbers: 162, 486

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

Rules and methods, laws,...
\(a_n = a_1 + (n-1)d\)
Arithmetic Sequence Formula
Arithmetic
Add/Subtract Same Number
2, 5, 8, 11, ... (add 3)
Geometric
Multiply/Divide Same Number
2, 6, 18, 54, ... (×3)
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

Pattern Identification Method:
  1. Observe: Look at the first few numbers in the sequence
  2. Compare: Find the difference or ratio between consecutive terms
  3. Test: Verify the rule works for multiple pairs
  4. Apply: Use the rule to find missing terms
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.
Common errors: Not checking enough pairs, mixing up operations, forgetting negative numbers.
Key strategies: Look for patterns in differences, check for multiplication/division, verify consistently.
Formulas to know:

• Arithmetic: Next term = Previous term + Constant

• Geometric: Next term = Previous term × Constant

• Alternating: Different rules for odd/even positions

Solution: Exercises 4 to 5
4 Alternating pattern
Exercise 4
Find the rule and continue the pattern:
3, 6, 4, 8, 6, 12, __, __
Definition:

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

Pattern
3, 6, 4, 8, 6, 12
Rules
×2, -2
Next terms
10, 20
Step 1: Separate odd and even positions

Odd positions: 3, 4, 6 (positions 1, 3, 5)

Even positions: 6, 8, 12 (positions 2, 4, 6)

Step 2: Analyze odd positions

3 → 4 (add 1), 4 → 6 (add 2), so add 3 to get 9

Step 3: Analyze even positions

6 → 8 (add 2), 8 → 12 (add 4), so add 6 to get 18

Step 4: Notice the pattern

Actually, looking more closely: 3×2=6, 6-2=4, 4×2=8, 8-2=6, 6×2=12, 12-2=10, 10×2=20

Complete pattern: 3, 6, 4, 8, 6, 12, 10, 20
Final answer:

Rule: Multiply by 2, then subtract 2, repeating

Missing numbers: 10, 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
Find the rule and continue the pattern:
1, 1, 2, 3, 5, 8, __, __
Definition:

Fibonacci sequence: Each number is the sum of the two preceding numbers.

Pattern
1, 1, 2, 3, 5, 8
Rule
Add previous 2
Next terms
13, 21
Step 1: Look for simple arithmetic first

1→1: +0, 1→2: +1, 2→3: +1, 3→5: +2, 5→8: +3

This doesn't show a simple arithmetic pattern

Step 2: Try adding previous terms

1+1=2, 1+2=3, 2+3=5, 3+5=8 ✓

Step 3: Apply the rule

5+8=13, 8+13=21

Step 4: Verify the pattern

Each term equals the sum of the two previous terms

Complete pattern: 1, 1, 2, 3, 5, 8, 13, 21
Final answer:

Rule: Add the two previous numbers

Missing numbers: 13, 21

Applied rules:

• Complex Patterns: When simple rules don't work, look for relationships involving multiple previous terms

• Fibonacci Sequence: Special sequence where each term is sum of two preceding terms

• Systematic Approach: Try different approaches when initial attempts fail

Number Pattern Rules and Methods Guide
\(a_n = a_{n-1} + d\)
Arithmetic Sequence Rule
Key definitions:

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

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

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

Complete Pattern Recognition Methodology:
  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: When stuck, write out the differences between consecutive terms to spot the pattern.
Tip 2: For geometric patterns, divide consecutive terms to find the common ratio.
Tip 3: Remember that patterns can involve alternating operations or multiple rules.
Tip 4: Always double-check your answer by applying the rule to the entire sequence.
Common errors: Misidentifying the pattern type, not checking enough terms, calculation mistakes.
Success strategies: Systematic approach, verification of rules, practice with various pattern types.
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)

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.