Patterns They're Growing: Solved Exercises in Grade 2

Master growing patterns: counting sequences, shape patterns, and number relationships through these 3 detailed exercises.

Solution: Exercises 1 to 3
1 Counting by 2s
Exercise 1
Complete the growing pattern:
2, 4, 6, ___, ___, ___
Definition:

Growing pattern: A sequence where each number increases by the same amount

Pattern recognition method:
  1. Look at the first two numbers: 2 and 4
  2. Find the difference: 4 - 2 = 2
  3. Check if this difference is the same for next pairs
  4. Add the same difference to find the next numbers
Pattern
2, 4, 6, ?
Difference
+2 each time
Next
8, 10, 12
Step 1: Find the pattern rule

4 - 2 = 2, 6 - 4 = 2, so we add 2 each time

Step 2: Apply the rule

6 + 2 = 8, then 8 + 2 = 10, then 10 + 2 = 12

Step 3: Write the complete pattern

2, 4, 6, 8, 10, 12

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

2, 4, 6, 8, 10, 12

Applied rules:

• Counting by 2s: Add 2 to each number to get the next

• Pattern rule: Same difference between consecutive numbers

• Consistency: The same operation applies throughout

2 Shape pattern
Exercise 2
What comes next in this shape pattern?
Circle, Square, Triangle, Circle, Square, ___, ___
Definition:

Shape pattern: A repeating sequence of shapes following a specific order

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■
▲
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■
●
■
Sequence
Circle, Square, Triangle
Repeats
Circle, Square, Triangle
Step 1: Identify the repeating unit

The pattern repeats: Circle, Square, Triangle

Step 2: Continue the sequence

After Circle, Square, the next shape is Triangle

Step 3: Continue the pattern

After Triangle, the pattern starts over: Circle

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■
▲
●
■
▲
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Circle, Square, Triangle, Circle, Square, Triangle, Circle
Final answer:

The next two shapes are Triangle and Circle

Applied rules:

• Repeating pattern: Look for the smallest repeating unit

• Order matters: Shapes follow a specific sequence

• Continuation: After completing one cycle, start again

3 Number sequence
Exercise 3
Find the missing numbers:
10, 15, 20, ___, 30, 35, ___
Definition:

Number sequence: Numbers arranged in order with a consistent rule

Start
10, 15, 20
Rule
Add 5 each time
Complete
25, 30, 35, 40
Step 1: Find the difference between numbers

15 - 10 = 5, 20 - 15 = 5, so we add 5 each time

Step 2: Apply the rule to find missing numbers

20 + 5 = 25, 35 + 5 = 40

Step 3: Verify the pattern

10, 15, 20, 25, 30, 35, 40 (each number increases by 5)

10
15
20
25
30
35
40
10, 15, 20, 25, 30, 35, 40
Final answer:

The missing numbers are 25 and 40

Applied rules:

• Counting by 5s: Add 5 to each number to get the next

• Pattern consistency: The same difference applies throughout

• Verification: Check that the pattern holds for all numbers

Growing Pattern Rules and Methods
Start + Rule = Next Number
Basic Pattern Formula
Counting by 2s
+2 each time
Even numbers: 2, 4, 6, 8...
Counting by 5s
+5 each time
Ends in 5 or 0: 5, 10, 15...
Counting by 10s
+10 each time
Always ends in 0: 10, 20, 30...
Key definitions:

Growing Pattern: A sequence where each item follows a specific rule

Pattern Rule: The operation that transforms one item to the next

Repeating Pattern: A sequence of shapes or objects that repeats

Pattern Recognition Method:
  1. Observe: Look at the first few items in the sequence
  2. Find Difference: Calculate how much changes between items
  3. Identify Rule: Determine the pattern rule (add/subtract/multiply)
  4. Apply Rule: Use the rule to find missing items
  5. Verify: Check that the pattern works for all items
Tip 1: Count the difference between consecutive numbers to find the rule.
Tip 2: For shape patterns, identify the smallest repeating unit.
Tip 3: Check your answer by applying the rule to see if it continues the pattern.
Common patterns: Counting by 2s (even numbers), 5s, 10s, or any consistent addition.
Visual patterns: Shapes, colors, or objects that repeat in a specific order.
Growing Pattern Properties and Examples
Pattern Rule = Next - Previous
Finding the Pattern Rule
Types of Growing Patterns:

Number Patterns: Sequences of numbers that follow a rule

Shape Patterns: Sequences of shapes that repeat

Color Patterns: Sequences of colors that follow a rule

Advanced Pattern Techniques:
  1. Two-step patterns: Alternating between two different rules
  2. Multiplication patterns: Each number multiplied by a constant
  3. Visual patterns: Recognizing geometric growth
Memory aid: Counting by 10s always ends in 0.
Quick check: If adding the same number each time, it's a growing pattern.
Pattern clue: Numbers ending in 5 or 0 often indicate counting by 5s.
Shape tip: Draw out the pattern to visualize the repetition.
Pattern identification: Look for consistent changes between terms.
Verification method: Apply the rule to generate the next few terms.
Essential Pattern Rules:

• Counting by 2s: 2, 4, 6, 8, 10... (always even numbers)

• Counting by 5s: 5, 10, 15, 20, 25... (ends in 5 or 0)

• Counting by 10s: 10, 20, 30, 40... (always ends in 0)

• Pattern continuation: Same rule applies to all terms

Detailed Summary: Patterns They're Growing
Definitions of Key Concepts:
Growing Pattern: A sequence where each number or object increases according to a specific rule. The pattern "grows" because each element gets larger by following the same rule.
Pattern Rule: The mathematical operation (usually addition) that tells us how to get from one number to the next in the sequence.
Repeating Pattern: A sequence of shapes, colors, or objects that repeats in the same order over and over again.
Core Rules and Principles:
Consistency Rule: The same operation must be applied to every number in the sequence to maintain the pattern.
Difference Rule: In growing number patterns, find the difference between consecutive numbers to discover the pattern rule.
Repetition Rule: In shape patterns, look for the smallest group of items that repeats throughout the sequence.
Step-by-Step Methods:
For Number Patterns:
  1. Look at the first two numbers in the sequence
  2. Find the difference between them (subtract the smaller from the larger)
  3. Check if this difference is the same for the next pair of numbers
  4. If the difference is the same, that's your pattern rule
  5. Apply this rule to find missing numbers
For Shape Patterns:
  1. Identify the sequence of shapes: Circle, Square, Triangle
  2. Look for when the sequence starts over: Circle, Square, Triangle, Circle, Square
  3. Recognize that after Square, the pattern repeats with Triangle
  4. Continue the sequence following the established order
Examples (Simple → Advanced):
Simple: Counting by 2s: 2, 4, 6, 8, 10 (add 2 each time)
Medium: Counting by 5s: 5, 10, 15, 20, 25 (add 5 each time)
Advanced: Mixed patterns: 1, 3, 6, 10, 15 (the difference increases by 1 each time: +2, +3, +4, +5)
Tips, Tricks, and Common Pitfalls:
Tips:
  • Always check your work by applying the rule to see if it continues the pattern
  • Count by the pattern number to memorize: count by 2s, 5s, or 10s regularly
  • Draw out shape patterns if they're hard to visualize
Common Mistakes:
  • Forgetting that the same rule applies to ALL numbers in the sequence
  • Mixing up the difference calculation (subtracting the wrong way)
  • Not recognizing when a pattern changes rules
Key Notes for Memorization:
Counting by 2s: Always results in even numbers (2, 4, 6, 8, 10...)
Counting by 5s: Numbers always end in 5 or 0 (5, 10, 15, 20, 25...)
Counting by 10s: Numbers always end in 0 (10, 20, 30, 40, 50...)
Pattern Check: To verify your answer, apply the rule to the last known number to see if it produces the next number in the sequence.
Takeaways for Success:
• Look for the relationship between consecutive numbers
• Practice counting by 2s, 5s, and 10s to build number sense
• Remember that growing patterns always increase by the same amount
• Check your answers by continuing the pattern to ensure consistency

Questions & Answers

Question: I'm having trouble figuring out the pattern rule. In the sequence 3, 6, 9, 12, how do I know it's adding 3 each time and not multiplying?

Answer: Great question! To determine the pattern rule, you need to look at the difference between consecutive numbers:

  • 6 - 3 = 3
  • 9 - 6 = 3
  • 12 - 9 = 3

Since we're adding the same number (3) each time, it's an addition pattern. If it were multiplication, the differences would change:

For multiplication by 2: 3 × 2 = 6, 6 × 2 = 12, 12 × 2 = 24 (sequence: 3, 6, 12, 24)

Notice how the differences are different: 6-3=3, 12-6=6, 24-12=12. When differences are the same, it's addition; when ratios are the same, it's multiplication.

Question: What if the pattern doesn't seem to have the same difference? Like 2, 5, 9, 14. How do I figure that out?

Answer: That's a more advanced pattern! Let's look at the differences:

  • 5 - 2 = 3
  • 9 - 5 = 4
  • 14 - 9 = 5

The differences themselves form a pattern: 3, 4, 5. Each difference increases by 1. So the next difference would be 6, making the next number 14 + 6 = 20.

For now, focus on simpler patterns where the difference stays the same. As you advance, you'll learn about these more complex patterns too!

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

Answer: Here are some fun ways to practice patterns at home:

  • Household items: Create patterns with toys, blocks, or utensils (red cup, blue cup, red cup...)
  • Daily routines: Notice patterns in daily activities (breakfast, lunch, dinner)
  • Nature walks: Look for patterns in leaves, flowers, or tree rings
  • Art projects: Draw repeating color or shape patterns
  • Music: Clap out rhythmic patterns (clap, clap, pause, clap, clap, pause...)

The key is to make it fun and connect patterns to things they see every day!