Solved Exercises on Number Patterns in Grade 2

Master number patterns: skip counting, addition/subtraction patterns, geometric sequences with visual representations through these 3 detailed exercises.

Solution: Exercises 1 to 3
1 Addition Pattern
Exercise 1
Find the next three numbers in this pattern: 5, 8, 11, 14, __, __, __
Explain the rule and continue the pattern.
Definition:

Addition pattern: A sequence where each number is found by adding the same amount to the previous number.

Pattern finding method:
  1. Look at the first few numbers in the sequence
  2. Find the difference between consecutive numbers
  3. Identify the rule (add or subtract the same amount)
  4. Continue the pattern using the rule
Numbers
5, 8, 11, 14
Difference
8-5=3
Check
11-8=3
Check
14-11=3
5
→
8
→
11
→
14
→
17
→
20
→
23
5
8
11
14
17
20
23
Step 1: Find the difference between consecutive numbers

8 - 5 = 3, 11 - 8 = 3, 14 - 11 = 3

Step 2: Identify the rule

The rule is: Add 3 to each number to get the next number

Step 3: Continue the pattern

14 + 3 = 17, 17 + 3 = 20, 20 + 3 = 23

Rule: Add 3 to each number
Final answer:

The next three numbers are 17, 20, 23. The pattern continues by adding 3 each time.

Applied rules:

• Constant difference: Each consecutive pair has the same difference

• Pattern continuation: Apply the same operation to continue the sequence

• Arithmetic sequence: A sequence where each term increases by a constant amount

2 Subtraction Pattern
Exercise 2
Find the missing numbers in this pattern: 50, 45, 40, __, __, 25
Explain the rule and continue the pattern backward.
Definition:

Subtraction pattern: A sequence where each number is found by subtracting the same amount from the previous number.

50
→
45
→
40
→
35
→
30
→
25
Numbers
50, 45, 40
Difference
50-45=5
Check
45-40=5
50
45
40
35
30
25
Step 1: Find the difference between consecutive numbers

50 - 45 = 5, 45 - 40 = 5

Step 2: Identify the rule

The rule is: Subtract 5 from each number to get the next number

Step 3: Find the missing numbers

40 - 5 = 35, 35 - 5 = 30

Step 4: Verify the last number

30 - 5 = 25 ✓

Rule: Subtract 5 from each number
Final answer:

The missing numbers are 35 and 30. The pattern continues by subtracting 5 each time.

Applied rules:

• Constant difference: Each consecutive pair has the same difference

• Reverse pattern: Apply the inverse operation to go backward

• Decreasing sequence: A sequence where each term decreases by a constant amount

3 Repeating Pattern
Exercise 3
Find the next four numbers in this pattern: 2, 4, 6, 8, 2, 4, 6, 8, __, __, __, __
Explain the repeating unit and predict the 15th number in the sequence.
Definition:

Repeating pattern: A sequence where a group of numbers repeats over and over in the same order.

2
4
6
8
2
4
6
8
2
4
6
8
2
4
6
8
2
4
6
8
2
4
6
8
2
4
6
8
2
4
6
8
2
4
6
8
Step 1: Identify the repeating unit

The pattern "2, 4, 6, 8" repeats over and over

Step 2: Continue the pattern

After "2, 4, 6, 8", the next four numbers are "2, 4, 6, 8"

Step 3: Find the 15th number

The pattern has 4 numbers that repeat. 15 ÷ 4 = 3 remainder 3

So the 15th number is the 3rd number in the repeating unit: 6

Rule: Repeat "2, 4, 6, 8" continuously
Final answer:

The next four numbers are 2, 4, 6, 8. The 15th number in the sequence is 6.

Applied rules:

• Repeating unit: Find the group of numbers that repeats

• Modulo operation: Use division to find position within the repeating unit

• Pattern prediction: Extend the pattern based on the identified rule

Number Pattern Rules and Methods
Next = Previous ± Constant
Arithmetic Pattern
Addition Pattern
5, 8, 11, 14...
Add same number
Subtraction Pattern
50, 45, 40, 35...
Subtract same number
Repeating Pattern
2, 4, 6, 8, 2, 4...
Repeat unit
Key Rule: Find the difference between consecutive numbers to identify the pattern.
Pattern Logic: The same operation must apply to every consecutive pair.
Tip 1: Write down the differences between consecutive numbers to find the rule.
Tip 2: Look for repeating units in longer patterns.
Tip 3: Always check your answer by applying the rule to the last known number.
Key Rules to Remember:

• Constant difference: Each consecutive pair has the same difference

• Operation consistency: The same operation applies to every pair

• Pattern verification: Check your rule works for all known numbers

• Repeating units: Look for groups that repeat in complex patterns

aₙ = a₁ + (n-1)d
Arithmetic Sequence Formula
Pattern Type Example Rule Next Term Addition 3, 6, 9, 12 +3 15 Subtraction 20, 15, 10, 5 -5 0 Repeating 1, 3, 1, 3, 1 Repeat "1, 3" 3 Mixed 2, 4, 8, 16 ×2 32
Additional Practice
4 Skip Counting Pattern
Exercise 4
Complete the skip counting pattern by 5s: 5, 10, 15, __, __, __
Then start a new pattern counting by 10s from 20.
Definition:

Skip counting: Counting by a number other than 1 (like 2s, 5s, 10s).

5
→
10
→
15
→
20
→
25
→
30
Part 1: Skip counting by 5s

5, 10, 15, 20, 25, 30 (add 5 each time)

Part 2: Skip counting by 10s starting from 20

20, 30, 40, 50, 60, 70 (add 10 each time)

Final answers:

First pattern: 5, 10, 15, 20, 25, 30

Second pattern: 20, 30, 40, 50, 60, 70

Applied rules:

• Skip counting: Add the same number repeatedly

• Consistency: Apply the same operation to each term

• Pattern recognition: Identify the increment value

5 Complex Pattern
Exercise 5
Find the next three numbers in this pattern: 1, 3, 6, 10, 15, __, __, __
Explain the rule for this triangular number pattern.
Definition:

Triangular number pattern: A sequence where each number adds an increasing amount (1, 1+2, 1+2+3, etc.).

Numbers
1, 3, 6, 10, 15
Differences
+2, +3, +4, +5
1
3
6
10
15
Step 1: Find the differences between consecutive numbers

3-1=2, 6-3=3, 10-6=4, 15-10=5

Step 2: Identify the pattern in differences

The differences are increasing by 1 each time: +2, +3, +4, +5

Step 3: Continue the pattern

Next differences: +6, +7, +8

Next numbers: 15+6=21, 21+7=28, 28+8=36

Final answer:

The next three numbers are 21, 28, 36. This is a triangular number pattern where each number adds an increasing integer.

Applied rules:

• Increasing differences: Differences themselves form a pattern

• Complex pattern recognition: Look beyond simple arithmetic

• Triangular numbers: Each term adds the next integer (1, 1+2, 1+2+3, etc.)

Comprehensive Summary: Number Patterns
aₙ = a₁ + (n-1)d (arithmetic), aₙ = a₁×r^(n-1) (geometric)
Pattern Formulas
Key Definitions:

Number pattern: A sequence of numbers that follows a specific rule or pattern.

Arithmetic sequence: A sequence where each term is found by adding or subtracting the same number.

Geometric sequence: A sequence where each term is found by multiplying or dividing by the same number.

Repeating pattern: A sequence where a group of numbers repeats over and over.

Skip counting: Counting by numbers other than 1 (by 2s, 5s, 10s, etc.).

Triangular numbers: Numbers that form a triangular pattern (1, 3, 6, 10, 15, etc.).

Complete Methodology:
  1. Observe the sequence: Look at the first few numbers in the pattern
  2. Find the differences: Calculate the difference between consecutive numbers
  3. Identify the rule: Determine if it's addition, subtraction, multiplication, division, or repetition
  4. Test the rule: Verify it works for all known numbers in the sequence
  5. Apply the rule: Use it to find missing numbers or continue the pattern
Tip 1: Always calculate differences between consecutive numbers to identify the pattern.
Tip 2: Look for repeating units in longer sequences.
Tip 3: Check your work by applying the rule to the last known number.
Tip 4: Some patterns have increasing differences (like triangular numbers).
Common errors: Missing the pattern in differences, not checking the rule for all numbers, assuming simple arithmetic.
Key insights: Number patterns build foundational skills for algebra and problem-solving.
Core Rules and Principles:

• Consistency: The same rule must apply to every consecutive pair

• Difference analysis: Finding differences helps identify the pattern

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

• Pattern extension: Once identified, apply the rule to continue the sequence

• Complex patterns: Differences between differences may reveal the rule

Examples: 2,4,6,8 (add 2), 100,90,80 (subtract 10), 1,4,9,16 (square numbers)
Pattern Examples
Memory aid: "Look for the difference between neighbors" - if it's the same, it's arithmetic.
Success check: If 5, 8, 11... rule is add 3, then 11+3 should equal 14. ✓

Questions & Answers

Question: How do I know what number to add or subtract in a pattern? Sometimes I get confused.

Answer: Here's a foolproof method to find the pattern rule:

Step 1: Look at the first two numbers

  • Take the second number minus the first number
  • Example: In 5, 8, 11... → 8 - 5 = 3

Step 2: Check with the next pair

  • Take the third number minus the second number
  • Example: In 5, 8, 11... → 11 - 8 = 3

If the difference is the same, that's your rule! In this case, add 3 each time.

Always check at least two pairs to make sure the difference is consistent.

For decreasing patterns: If the numbers get smaller, you subtract. Example: 20, 15, 10... → 20-15=5, 15-10=5, so subtract 5.

Question: My child struggles with patterns that don't follow a simple addition or subtraction rule. How can I help them with more complex patterns?

Answer: For more complex patterns, use these strategies:

Strategy 1: Look at differences between differences

  • For 1, 3, 6, 10, 15...: differences are +2, +3, +4, +5
  • The differences themselves form a pattern!

Strategy 2: Look for repeating units

  • For 2, 4, 6, 8, 2, 4, 6, 8...: the unit "2, 4, 6, 8" repeats

Strategy 3: Try different operations

  • Multiplication patterns: 2, 4, 8, 16... (multiply by 2)
  • Square numbers: 1, 4, 9, 16... (1², 2², 3², 4²)

Encourage your child to write down the differences between consecutive numbers - this often reveals the hidden pattern!

Question: What if I see a pattern like 2, 4, 8, 16...? It's not adding the same number each time.

Answer: You've discovered a geometric pattern! Let's analyze it:

Looking at the differences:

  • 4 - 2 = 2
  • 8 - 4 = 4
  • 16 - 8 = 8

The differences aren't the same, so it's not an arithmetic pattern.

Looking at multiplication:

  • 2 × 2 = 4
  • 4 × 2 = 8
  • 8 × 2 = 16

The rule is: multiply by 2 each time!

This is called a geometric sequence where each term is found by multiplying the previous term by the same number (called the ratio). In this case, the ratio is 2.

The next numbers would be: 32, 64, 128...