Complete Lesson on Multiples of 4 Patterns in Grade 4

Master multiples of 4: patterns, methods, examples, and applications with detailed explanations.

Multiples of 4: Definitions and Basic Examples
1 Understanding Multiples of 4
Definition:

Multiple of 4: A number that results from multiplying 4 by a whole number (1, 2, 3, ...)

How to find multiples of 4:
  1. Multiply 4 by 1, 2, 3, 4, 5, ...
  2. Each result is a multiple of 4
  3. Multiples continue infinitely
Base Number
4
Multiply by
1, 2, 3, 4, 5
Result
4, 8, 12, 16, 20
Step 1: Multiply by 1

4 × 1 = 4

Step 2: Multiply by 2

4 × 2 = 8

Step 3: Multiply by 3

4 × 3 = 12

Step 4: Multiply by 4

4 × 4 = 16

Step 5: Multiply by 5

4 × 5 = 20

The first 5 multiples of 4 are: 4, 8, 12, 16, 20
Key insight:

Multiples of 4 always end in 0, 2, 4, 6, or 8!

2 Identifying Multiples of 4
Definition:

Identifying multiples of 4: A number is a multiple of 4 if it divides evenly by 4 (no remainder)

Number to check
24
Divide by
4
Result
Yes (24 ÷ 4 = 6)
Step 1: Divide the number by 4

24 ÷ 4 = 6

Step 2: Check if the result is a whole number

6 is a whole number, so 24 is a multiple of 4

Step 3: Verify by multiplication

4 × 6 = 24 ✓

24 is a multiple of 4
Quick check:

Look at the last two digits: 24 ends in 24, and 24 ÷ 4 = 6, so it's a multiple of 4!

3 Pattern Recognition in Multiples of 4
Definition:

Pattern: Multiples of 4 increase by 4 each time

Sequence
4, 8, 12, 16, 20...
Pattern
Add 4 each time
Next 3
24, 28, 32
Step 1: Identify the pattern

Each number increases by 4

Step 2: Continue the pattern

20 + 4 = 24

24 + 4 = 28

28 + 4 = 32

Step 3: Verify with multiplication

4 × 6 = 24 ✓

4 × 7 = 28 ✓

4 × 8 = 32 ✓

Step 4: Describe the rule

Each multiple is 4 more than the previous one

Next 3 multiples of 4: 24, 28, 32
Pattern tip:

All multiples of 4 form an arithmetic sequence with a common difference of 4!

Multiples of 4: Core Rules and Methods
Multiple of 4 = 4 × k (where k = 1, 2, 3, ...)
Multiple of 4 Formula
Pattern 1
Always Even
Multiples of 4 are always even numbers
Pattern 2
Last Two Digits
If last two digits form a multiple of 4, the whole number is a multiple of 4
Pattern 3
Arithmetic Sequence
Add 4 each time to get the next multiple
Key definitions:

Multiple of 4: The product of 4 and any whole number

Pattern: A consistent mathematical relationship in a sequence

Arithmetic Sequence: A sequence where each term increases by the same amount

Finding multiples of 4 method:
  1. List method: Multiply 4 by 1, 2, 3, 4, 5...
  2. Pattern method: Add 4 to the previous multiple
  3. Division method: Check if division by 4 results in whole number
  4. Last two digits: If the last two digits form a multiple of 4, the whole number is a multiple of 4
Tip 1: Multiples of 4 always end in 0, 2, 4, 6, or 8.
Tip 2: Check last two digits: 124 → 24 ÷ 4 = 6, so 124 is a multiple of 4.
Tip 3: All multiples of 4 are also multiples of 2 (but not vice versa).
Tip 4: The pattern of multiples of 4 is always adding 4.
Important facts: Every multiple of 4 is even; 4 is the smallest positive multiple of 4; multiples of 4 form an infinite arithmetic sequence with common difference 4.
Memory aid: "4, 8, 12, 16, 20..." - count by 4s!
Essential rules:

• Every multiple of 4 is also a multiple of 2

• Multiples of 4 form an arithmetic sequence with common difference 4

• A number is a multiple of 4 if its last two digits form a multiple of 4

• Every multiple of 4 is even

• The smallest positive multiple of 4 is 4 itself

Advanced Examples and Applications
4 Word Problem Application
Real-world scenario:

Sarah puts 4 cookies in a jar every day. After how many days will she have put exactly 36 cookies in the jar?

Cookies per day
4
Total cookies
36
Days needed
36 ÷ 4 = 9 days
Step 1: Identify the pattern

Days: 1, 2, 3, 4, 5, 6, 7, 8, 9...

Cookies: 4, 8, 12, 16, 20, 24, 28, 32, 36...

Step 2: Find when total reaches 36

Looking at multiples of 4: 36 appears when day = 9

Step 3: Verify

4 × 9 = 36 ✓

Step 4: Answer

After 9 days, Sarah will have put exactly 36 cookies in the jar

Sarah will have 36 cookies after 9 days
Problem-solving tip:

Repeated addition of the same number creates multiples!

5 Advanced Pattern Recognition
Advanced pattern identification:

Recognizing multiples of 4 in larger numbers using the last two digits rule.

Large number
1,236
Last two digits
36
Is 36 ÷ 4 whole?
Yes (36 ÷ 4 = 9)
Step 1: Look at last two digits

For 1,236, the last two digits are 36

Step 2: Check if last two digits are divisible by 4

36 ÷ 4 = 9 (whole number)

Step 3: Apply the rule

If the last two digits form a multiple of 4, the entire number is a multiple of 4

Step 4: Verify with division

1,236 ÷ 4 = 309 ✓

1,236 is a multiple of 4
Advanced tip:

For large numbers, only check the last two digits to determine if it's a multiple of 4!

Complete Lesson Summary: Multiples of 4
Multiple of 4 = 4 × k (where k = 1, 2, 3, ...)
Core Formula
Key definitions:

Multiple of 4: A number that results from multiplying 4 by a whole number (1, 2, 3, ...)

Pattern: A consistent mathematical relationship in a sequence

Arithmetic Sequence: A sequence where each term increases by the same amount

Complete methodology:
  1. Find multiples: Multiply 4 by 1, 2, 3, 4, 5, ...
  2. Identify multiples: Check if a number divides evenly by 4
  3. Use pattern: Add 4 to get the next multiple
  4. Large numbers: Check if last two digits form a multiple of 4
  5. Verify results: Always check your answers by reversing the operation
Tip 1: Multiples of 4 always end in 0, 2, 4, 6, or 8.
Tip 2: For large numbers, check only the last two digits to determine if it's a multiple of 4.
Tip 3: Every multiple of 4 is also a multiple of 2.
Tip 4: The pattern of multiples of 4 is always adding 4.
Tip 5: The first multiple of 4 is 4 itself.
Key rules: Every multiple of 4 is even; multiples of 4 form an arithmetic sequence with common difference 4; the last two digits rule applies to all numbers.
Common mistakes: Forgetting that 4 is a multiple of 4; thinking odd numbers can be multiples of 4; not using the last two digits rule for large numbers.
Essential formulas and rules:

• Multiple of 4 = 4 × k (where k is any positive integer)

• To check if A is a multiple of 4: A ÷ 4 should result in a whole number

• Large number rule: If last two digits form a multiple of 4, the whole number is a multiple of 4

• Multiples of 4 form an infinite arithmetic sequence with common difference 4

• Every multiple of 4 is even and also a multiple of 2

Multiples of 4 Visualization: Patterns and Relationships
Visualizing Multiples of 4
The first 10 multiples of 4:
4, 8, 12, 16, 20, 24, 28, 32, 36, 40
Notice the consistent pattern of adding 4 each time.

Analysis: The chart shows how multiples of 4 form a perfectly linear pattern.

  • First 10 multiples of 4: 4, 8, 12, 16, 20, 24, 28, 32, 36, 40
  • Each multiple is exactly 4 more than the previous one
  • All multiples end in 0, 2, 4, 6, or 8
  • The pattern continues infinitely

Questions & Answers

Question: Why do multiples of 4 always end in 0, 2, 4, 6, or 8? Is there a reason for this pattern?

Answer: Yes, there's a mathematical reason for this pattern!

  • Since 4 is an even number, any multiple of 4 will also be even
  • Even numbers always end in 0, 2, 4, 6, or 8
  • Let's look at the pattern: 4×1=4, 4×2=8, 4×3=12, 4×4=16, 4×5=20, 4×6=24...
  • Notice how the last digits cycle through 4, 8, 2, 6, 0 and repeat

This happens because when we multiply 4 by numbers ending in 1, 2, 3, 4, 5, the last digits follow this pattern:

  • 4 × 1 = 4 (ends in 4)
  • 4 × 2 = 8 (ends in 8)
  • 4 × 3 = 12 (ends in 2)
  • 4 × 4 = 16 (ends in 6)
  • 4 × 5 = 20 (ends in 0)

This pattern repeats every 5 multiples!

Question: The rule says to check the last two digits for large numbers. Why two digits and not just one?

Answer: This is an excellent question! Let me explain why we need two digits:

  • Consider the number 14: The last digit is 4, but 14 ÷ 4 = 3.5, so 14 is NOT a multiple of 4
  • Now consider 24: The last digit is 4, and 24 ÷ 4 = 6, so 24 IS a multiple of 4
  • Looking at just the last digit isn't enough because it doesn't account for the tens place

Here's the mathematical reasoning:

  • Any number can be written as (100 × hundreds) + (last two digits)
  • Since 100 is a multiple of 4 (100 ÷ 4 = 25), the hundreds part is always divisible by 4
  • So we only need to check if the last two digits form a multiple of 4

Example: For 312 = (3 × 100) + 12 = 300 + 12. Since 300 is divisible by 4 and 12 is divisible by 4, then 312 is divisible by 4.

Question: Are all multiples of 4 also multiples of 2? And are all multiples of 2 also multiples of 4?

Answer: Great question! Let's break this down:

  • Yes, all multiples of 4 ARE multiples of 2 because 4 = 2 × 2
  • No, not all multiples of 2 are multiples of 4 (counter-example: 6 is a multiple of 2 but not 4)

Think of it this way:

  • Multiples of 4: 4, 8, 12, 16, 20, 24, 28, 32...
  • Multiples of 2: 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32...

As you can see, all multiples of 4 appear in the multiples of 2 list, but not all multiples of 2 appear in the multiples of 4 list.

This makes sense because if a number is divisible by 4, it's automatically divisible by 2 (since 4 is divisible by 2). But a number divisible by 2 might not be divisible by 4.