Solved Exercises on Identifying Arithmetic Patterns in Grade 3

Master arithmetic patterns: sequences, skip counting, and number relationships through these 5 detailed exercises.

Solution: Exercises 1 to 3
1 Skip Counting Pattern
Exercise 1
Complete the pattern: 2, 4, 6, 8, __, __, __. Explain the pattern rule.
Definition:

Arithmetic Pattern: A sequence of numbers where each number is found by adding the same amount to the previous number. The constant amount being added is called the common difference.

Pattern Recognition Method:
  1. Examine the sequence: Look at the first few numbers
  2. Find the difference: Subtract each number from the next
  3. Identify the pattern: Determine the constant amount being added
  4. Apply the rule: Add the constant amount to continue the sequence
  5. Verify: Check that the pattern continues consistently
2
→ +2
4
→ +2
6
→ +2
8
→ +2
10
→ +2
12
→ +2
14
Rule: Add 2 to each number to get the next number
Position Number Pattern Rule 1 2 2 +2 2 4 2 + 2 +2 3 6 4 + 2 +2 4 8 6 + 2 +2 5 10 8 + 2 +2 6 12 10 + 2 +2 7 14 12 + 2 +2
Starting number
2
Common difference
+2
Rule
Add 2
Step 1: Examine the sequence

Look at the given numbers: 2, 4, 6, 8

Step 2: Find the difference

4 - 2 = 2, 6 - 4 = 2, 8 - 6 = 2

Step 3: Identify the pattern

Each number is 2 more than the previous number

Step 4: Apply the rule

8 + 2 = 10, 10 + 2 = 12, 12 + 2 = 14

Step 5: Verify the pattern

Check: 10 + 2 = 12, 12 + 2 = 14 ✓

2, 4, 6, 8, 10, 12, 14
Final answer:

The complete pattern is: 2, 4, 6, 8, 10, 12, 14. The rule is to add 2 to each number.

Applied rules:

• Arithmetic Sequence: Each term increases by a constant difference

• Pattern Recognition: Identify the consistent change between terms

• Continuation Rule: Apply the identified rule to extend the sequence

2 Increasing Pattern
Exercise 2
Find the next three numbers in the pattern: 5, 10, 15, 20, __, __, __. What is the pattern rule?
Definition:

Increasing Arithmetic Pattern: A sequence where each number is greater than the previous number by a constant amount. This is also called skip counting.

5
→
10
→
15
→
20
→
25
→
30
→
35
Rule: Add 5 to each number to get the next number
Starting number
5
Common difference
+5
Next numbers
25, 30, 35
Step 1: Identify the starting number

The sequence starts with 5

Step 2: Find the pattern difference

10 - 5 = 5, 15 - 10 = 5, 20 - 15 = 5

Step 3: Apply the rule

20 + 5 = 25, 25 + 5 = 30, 30 + 5 = 35

Step 4: Verify the continuation

Check: 25 + 5 = 30, 30 + 5 = 35 ✓

5, 10, 15, 20, 25, 30, 35
Final answer:

The next three numbers are 25, 30, 35. The pattern rule is to add 5 to each number.

Applied rules:

• Skip Counting: Counting by a specific number repeatedly

• Constant Difference: The same amount is added each time

• Pattern Continuation: Extend the sequence using the identified rule

3 Pattern Table
Exercise 3
Complete the pattern: 1, 4, 7, 10, __, __, __. Show the pattern in a table.
Definition:

Pattern Table: A visual representation showing how numbers in a sequence relate to their position in the pattern, helping to identify the underlying rule.

1
4
7
10
13
16
19
Position Number Pattern Difference 1 1 1 +3 2 4 1 + 3 +3 3 7 4 + 3 +3 4 10 7 + 3 +3 5 13 10 + 3 +3 6 16 13 + 3 +3 7 19 16 + 3 +3
Rule: Add 3 to each number to get the next number
Starting number
1
Common difference
+3
Next numbers
13, 16, 19
Step 1: Analyze the sequence

Look at: 1, 4, 7, 10

Step 2: Calculate the differences

4 - 1 = 3, 7 - 4 = 3, 10 - 7 = 3

Step 3: Identify the pattern rule

Add 3 to each number to get the next

Step 4: Continue the pattern

10 + 3 = 13, 13 + 3 = 16, 16 + 3 = 19

Step 5: Complete the sequence

1, 4, 7, 10, 13, 16, 19

1, 4, 7, 10, 13, 16, 19
Final answer:

The completed pattern is: 1, 4, 7, 10, 13, 16, 19. The rule is to add 3 to each number.

Applied rules:

• Table Organization: Organize numbers by position to identify patterns

• Constant Difference: The same amount is added each time

• Sequential Addition: Each number depends on the previous number

Arithmetic Pattern Recognition
Next Term = Previous Term + Common Difference
Arithmetic Pattern Formula
Pattern 1
Add 2
2, 4, 6, 8, 10...
Pattern 2
Add 3
1, 4, 7, 10, 13...
Pattern 3
Add 5
5, 10, 15, 20, 25...
Pattern 4
Add 10
10, 20, 30, 40, 50...
Key definitions:

Pattern: A sequence of numbers that follow a specific rule or order

Arithmetic Pattern: A sequence where each number increases or decreases by the same amount

Common Difference: The constant amount added to get from one number to the next

Sequence: A list of numbers in a specific order

Term: Each number in the sequence

Pattern Recognition Steps:
  1. Observe the sequence: Look at the given numbers
  2. Find the difference: Subtract consecutive numbers
  3. Identify the rule: Determine the constant difference
  4. Apply the rule: Add the difference to continue the pattern
  5. Verify: Check that the pattern continues consistently
Key Principle: Arithmetic patterns have a constant difference between consecutive terms.
Pattern Recognition: Look for the consistent change between numbers.
Tip 1: Always subtract to find the pattern difference.
Tip 2: Check your rule works for all given numbers.
Tip 3: Use tables to organize complex patterns.
Tip 4: Patterns can increase or decrease by a constant amount.
Solution: Exercises 4 to 5
4 Decreasing Pattern
Exercise 4
Find the next three numbers in the pattern: 20, 17, 14, 11, __, __, __. What is the pattern rule?
Definition:

Decreasing Arithmetic Pattern: A sequence where each number is less than the previous number by a constant amount. The common difference is negative.

20
→ -3
17
→ -3
14
→ -3
11
→ -3
8
→ -3
5
→ -3
2
Position Number Pattern Difference 1 20 20 -3 2 17 20 - 3 -3 3 14 17 - 3 -3 4 11 14 - 3 -3 5 8 11 - 3 -3 6 5 8 - 3 -3 7 2 5 - 3 -3
Rule: Subtract 3 from each number to get the next number
Starting number
20
Common difference
-3
Next numbers
8, 5, 2
Step 1: Analyze the sequence

Look at: 20, 17, 14, 11

Step 2: Find the differences

17 - 20 = -3, 14 - 17 = -3, 11 - 14 = -3

Step 3: Identify the pattern rule

Subtract 3 from each number to get the next

Step 4: Continue the pattern

11 - 3 = 8, 8 - 3 = 5, 5 - 3 = 2

Step 5: Verify the pattern

Check: 8 - 3 = 5, 5 - 3 = 2 ✓

20, 17, 14, 11, 8, 5, 2
Final answer:

The next three numbers are 8, 5, 2. The pattern rule is to subtract 3 from each number.

Applied rules:

• Decreasing Pattern: Each term decreases by a constant amount

• Negative Difference: The common difference is negative

• Consistency: The same operation is applied to each term

5 Mixed Practice
Exercise 5
Identify the pattern rule and find the next three numbers for each: (a) 3, 6, 9, 12, __, __, __ (b) 15, 12, 9, 6, __, __, __
Definition:

Mixed Pattern Practice: Working with both increasing and decreasing arithmetic patterns to strengthen pattern recognition skills.

Part (a) Analysis

Sequence: 3, 6, 9, 12

Differences: 6-3=3, 9-6=3, 12-9=3

Rule: Add 3

Part (b) Analysis

Sequence: 15, 12, 9, 6

Differences: 12-15=-3, 9-12=-3, 6-9=-3

Rule: Subtract 3

Part (a)
3, 6, 9, 12, 15, 18, 21
Part (b)
15, 12, 9, 6, 3, 0, -3
Rules
Add 3, Subtract 3
Step 1: Analyze part (a)

3, 6, 9, 12: Each number increases by 3

Step 2: Continue part (a)

12 + 3 = 15, 15 + 3 = 18, 18 + 3 = 21

Step 3: Analyze part (b)

15, 12, 9, 6: Each number decreases by 3

Step 4: Continue part (b)

6 - 3 = 3, 3 - 3 = 0, 0 - 3 = -3

(a) 3, 6, 9, 12, 15, 18, 21 (Add 3)
(b) 15, 12, 9, 6, 3, 0, -3 (Subtract 3)
Final answer:

(a) 15, 18, 21 (add 3 each time), (b) 3, 0, -3 (subtract 3 each time)

Applied rules:

• Pattern Comparison: Increasing and decreasing patterns both have constant differences

• Rule Identification: Determine whether to add or subtract

• Extension Practice: Apply the rule to continue the sequence

Advanced Pattern Concepts
a_n = a_1 + (n-1)d
General Term Formula
Key definitions:

Arithmetic Sequence: A sequence where each term differs from the previous term by a constant amount

Common Difference: The constant amount added to get from one term to the next

Term: Each number in the sequence

Position: The order number of a term in the sequence

Advanced Pattern Strategies:
  1. Look for the difference: Always subtract consecutive terms to find the pattern
  2. Check all differences: Verify the same difference applies to all pairs
  3. Consider direction: Patterns can increase or decrease
  4. Visualize: Draw or represent patterns to understand them better
  5. Connect to multiplication: Skip counting is repeated addition
Tip 1: Arithmetic patterns always have a constant difference between terms.
Tip 2: The difference can be positive (increasing) or negative (decreasing).
Tip 3: Use tables to organize complex patterns with many terms.
Tip 4: Practice with different starting numbers to strengthen pattern recognition.
Pattern Consistency: The same operation is applied to every term.
Foundation for Algebra: These patterns prepare for more advanced math concepts.
Advanced rules:

• Constant Difference: In arithmetic patterns, the difference between consecutive terms is always the same

• Direction Matters: Positive differences create increasing patterns, negative differences create decreasing patterns

• Verification: Always check that your pattern rule works for all given terms

• Extension: Once you know the rule, you can find any term in the sequence

Example
Pattern
Rule
Simple
2, 4, 6, 8
Add 2
Medium
5, 10, 15, 20
Add 5
Advanced
20, 17, 14, 11
Subtract 3

Questions & Answers

Question: Why do we need to learn patterns? Can't we just memorize the answers?

Answer: Learning patterns is much more powerful than memorizing answers! Here's why:

  • Understands the concept: You learn WHY the numbers work the way they do
  • Applies to new problems: Patterns help you solve problems you've never seen before
  • Builds problem-solving skills: You develop critical thinking abilities
  • Prepares for advanced math: Patterns are the foundation for algebra
  • Real-world connections: Many situations follow patterns

Patterns teach you to think like a mathematician!

Question: How can I help my child practice pattern recognition at home?

Answer: Here are fun ways to practice at home:

  • Number sequences: "What comes next in 2, 4, 6, 8...?"
  • Real-world patterns: Count stairs, fence posts, or tiles
  • Calendar patterns: Look for weekly and monthly patterns
  • Music patterns: Rhythms and beats follow patterns
  • Art patterns: Create repeating designs with colors or shapes
  • Games: Pattern-based board games or card games

The key is making it engaging and connecting to everyday experiences!

Question: What happens if the difference changes? Like in the sequence 2, 4, 7, 11...

Answer: That's a great observation! In the sequence 2, 4, 7, 11...

  • The differences are: 4-2=2, 7-4=3, 11-7=4
  • The differences themselves form a pattern: 2, 3, 4...

This is called a "quadratic pattern" where the differences change by a constant amount. For now, focus on arithmetic patterns where the difference stays the same, but it's excellent that you're noticing other types of patterns!