Solved Exercises on Using Patterns and Mental Math to Divide in Grade 4

Master division using patterns and mental math: division patterns, mental division strategies, and pattern-based division through these 5 detailed exercises.

Solution: Exercises 1 to 3
1 Division Pattern with Tens
Exercise 1
Use patterns to divide:
400 ÷ 4, 4000 ÷ 4, 40000 ÷ 4
Definition:

Division pattern with tens: When dividing numbers ending in zeros, divide the non-zero part first, then add back the zeros

Mental division method:
  1. Divide the basic fact (without zeros)
  2. Count the zeros in the dividend
  3. Add the same number of zeros to the quotient
Basic fact
4 ÷ 4 = 1
Pattern
Add zeros
Results
100, 1000, 10000
Step 1: Identify the basic division fact

4 ÷ 4 = 1 (the basic division fact)

Step 2: Apply to 400 ÷ 4

400 has 2 zeros after 4

So 400 ÷ 4 = 1 with 2 zeros added = 100

Step 3: Apply to 4000 ÷ 4

4000 has 3 zeros after 4

So 4000 ÷ 4 = 1 with 3 zeros added = 1000

Step 4: Apply to 40000 ÷ 4

40000 has 4 zeros after 4

So 40000 ÷ 4 = 1 with 4 zeros added = 10000

400 ÷ 4 = 100
4000 ÷ 4 = 1000
40000 ÷ 4 = 10000
Final answer:

400 ÷ 4 = 100, 4000 ÷ 4 = 1000, 40000 ÷ 4 = 10000

Applied rules:

• Zero pattern: Divide the non-zero digits first, then add back the zeros

• Place value: Each zero represents a power of 10

• Consistency: The same basic fact applies to all variations

2 Mental Division with Multiples of 10
Exercise 2
Use mental math to divide:
240 ÷ 6, 2400 ÷ 6, 24000 ÷ 6
Definition:

Mental division with multiples of 10: Use basic division facts and add zeros based on place value

Basic fact
24 ÷ 6 = 4
Pattern
Add zeros
Results
40, 400, 4000
Step 1: Find the basic division fact

24 ÷ 6 = 4 (the basic division fact)

Step 2: Apply to 240 ÷ 6

240 has 1 zero after 24

So 240 ÷ 6 = 4 with 1 zero added = 40

Step 3: Apply to 2400 ÷ 6

2400 has 2 zeros after 24

So 2400 ÷ 6 = 4 with 2 zeros added = 400

Step 4: Apply to 24000 ÷ 6

24000 has 3 zeros after 24

So 24000 ÷ 6 = 4 with 3 zeros added = 4000

Step 5: Verify the pattern

Each time we add a zero to the dividend, we add a zero to the quotient

240 ÷ 6 = 40
2400 ÷ 6 = 400
24000 ÷ 6 = 4000
Final answer:

240 ÷ 6 = 40, 2400 ÷ 6 = 400, 24000 ÷ 6 = 4000

Applied rules:

• Basic facts: Know your multiplication/division facts

• Place value: Understand how zeros affect the result

• Pattern recognition: Same basic fact applies to all variations

3 Division Using Fact Families
Exercise 3
Use fact families to divide:
36 ÷ 4, 360 ÷ 4, 3600 ÷ 4
Definition:

Fact families: Related multiplication and division facts that use the same numbers

Fact family
4 × 9 = 36
Division
36 ÷ 4 = 9
Pattern
Add zeros
Step 1: Identify the fact family

4 × 9 = 36, so 36 ÷ 4 = 9

Step 2: Apply to 36 ÷ 4

Using the fact family: 36 ÷ 4 = 9

Step 3: Apply to 360 ÷ 4

360 has 1 zero after 36

So 360 ÷ 4 = 9 with 1 zero added = 90

Step 4: Apply to 3600 ÷ 4

3600 has 2 zeros after 36

So 3600 ÷ 4 = 9 with 2 zeros added = 900

Step 5: Verify using multiplication

9 × 4 = 36 ✓, 90 × 4 = 360 ✓, 900 × 4 = 3600 ✓

36 ÷ 4 = 9
360 ÷ 4 = 90
3600 ÷ 4 = 900
Final answer:

36 ÷ 4 = 9, 360 ÷ 4 = 90, 3600 ÷ 4 = 900

Applied rules:

• Fact families: Multiplication and division are inverse operations

• Pattern extension: Same basic fact applies to multiples of 10

• Verification: Use multiplication to check division

Rules and methods, laws,...
\(a \div b = c \text{ means } b \times c = a\)
Division Relationship
Zero Pattern
Divide basic fact, add zeros
200 ÷ 5 = 40 (2 ÷ 5 = 0.4, add 2 zeros)
Fact Family
Inverse operations
6 × 7 = 42, so 42 ÷ 6 = 7
Mental Division
Break apart numbers
84 ÷ 4 = (80 ÷ 4) + (4 ÷ 4) = 20 + 1 = 21
Key definitions:

Division: The operation of splitting a number into equal parts

Quotient: The result of a division problem

Dividend: The number being divided

Divisor: The number that divides the dividend

Fact family: A set of related multiplication and division facts

Complete methodology:
  1. Identify the basic fact: Find the division fact without zeros
  2. Count zeros: Count how many zeros are in the dividend
  3. Apply the pattern: Use the basic fact and add zeros to the quotient
  4. Verify: Check your answer by multiplying
Tip 1: Always start with the basic division fact before adding zeros.
Tip 2: Use fact families to connect multiplication and division.
Tip 3: Break larger numbers into smaller parts for easier mental division.
Tip 4: Always verify your division by multiplying the quotient by the divisor.
Common errors: Forgetting to add zeros to the quotient, miscounting zeros, not verifying answers.
Mental math strategies: Use known facts, break apart numbers, recognize patterns.
Formulas to know by heart:

• Basic division facts: Know all facts up to 12 ÷ 12

• Zero pattern: Divide the non-zero part, then add back zeros

• Fact families: If a × b = c, then c ÷ a = b and c ÷ b = a

• Verification: Quotient × Divisor = Dividend

Solution: Exercises 4 to 5
4 Mental Division with Breaking Apart
Exercise 4
Use mental math to divide:
72 ÷ 6, 720 ÷ 6, 7200 ÷ 6
Definition:

Mental division by breaking apart: Split the dividend into parts that are easily divisible by the divisor

Breaking apart
72 = 60 + 12
Division
(60 ÷ 6) + (12 ÷ 6)
Result
10 + 2 = 12
Step 1: Break apart the dividend

72 = 60 + 12 (both parts are divisible by 6)

Step 2: Divide each part separately

60 ÷ 6 = 10

12 ÷ 6 = 2

Step 3: Add the quotients

10 + 2 = 12

Step 4: Apply to 720 ÷ 6

720 has 1 zero after 72

So 720 ÷ 6 = 12 with 1 zero added = 120

Step 5: Apply to 7200 ÷ 6

7200 has 2 zeros after 72

So 7200 ÷ 6 = 12 with 2 zeros added = 1200

72 ÷ 6 = 12
720 ÷ 6 = 120
7200 ÷ 6 = 1200
Final answer:

72 ÷ 6 = 12, 720 ÷ 6 = 120, 7200 ÷ 6 = 1200

Applied rules:

• Breaking apart: Split numbers into parts that are easily divisible

• Distributive property: (a + b) ÷ c = (a ÷ c) + (b ÷ c)

• Zero pattern: Same basic fact applies to multiples of 10

5 Pattern-Based Division Challenge
Exercise 5
Use patterns to divide:
54 ÷ 9, 540 ÷ 9, 5400 ÷ 9, 54000 ÷ 9
Definition:

Pattern-based division: Recognize and extend division patterns to solve related problems

Basic fact
54 ÷ 9 = 6
Pattern
Add zeros
Results
6, 60, 600, 6000
Step 1: Identify the basic division fact

54 ÷ 9 = 6 (the basic division fact)

Step 2: Apply to 540 ÷ 9

540 has 1 zero after 54

So 540 ÷ 9 = 6 with 1 zero added = 60

Step 3: Apply to 5400 ÷ 9

5400 has 2 zeros after 54

So 5400 ÷ 9 = 6 with 2 zeros added = 600

Step 4: Apply to 54000 ÷ 9

54000 has 3 zeros after 54

So 54000 ÷ 9 = 6 with 3 zeros added = 6000

Step 5: Verify all answers

6 × 9 = 54 ✓, 60 × 9 = 540 ✓, 600 × 9 = 5400 ✓, 6000 × 9 = 54000 ✓

54 ÷ 9 = 6
540 ÷ 9 = 60
5400 ÷ 9 = 600
54000 ÷ 9 = 6000
Final answer:

54 ÷ 9 = 6, 540 ÷ 9 = 60, 5400 ÷ 9 = 600, 54000 ÷ 9 = 6000

Applied rules:

• Pattern recognition: Same basic fact applies to all multiples of 10

• Zero counting: Number of zeros in dividend equals zeros in quotient

• Verification: Always check with multiplication

Key Concepts: Division Patterns and Mental Math Strategies
\(a \div b = c \text{ means } b \times c = a\)
Division Relationship
Key definitions:

Division: The operation of splitting a number into equal parts

Quotient: The result of a division problem

Dividend: The number being divided

Divisor: The number that divides the dividend

Fact family: A set of related multiplication and division facts

Complete methodology:
  1. Identify the basic fact: Find the division fact without zeros
  2. Count zeros: Count how many zeros are in the dividend
  3. Apply the pattern: Use the basic fact and add zeros to the quotient
  4. Verify: Check your answer by multiplying
Tip 1: Always start with the basic division fact before adding zeros.
Tip 2: Use fact families to connect multiplication and division.
Tip 3: Break larger numbers into smaller parts for easier mental division.
Tip 4: Always verify your division by multiplying the quotient by the divisor.
Tip 5: Practice with basic division facts to build confidence.
Common errors: Forgetting to add zeros to the quotient, miscounting zeros, not verifying answers.
Mental math strategies: Use known facts, break apart numbers, recognize patterns.
Pattern recognition: Look for relationships between related division problems.
Formulas to know by heart:

• Basic division facts: Know all facts up to 12 ÷ 12

• Zero pattern: Divide the non-zero part, then add back zeros

• Fact families: If a × b = c, then c ÷ a = b and c ÷ b = a

• Verification: Quotient × Divisor = Dividend

Questions & Answers

Question: I don't understand why we can just add zeros when dividing numbers with zeros at the end. Can you explain?

Answer: Great question! This works because of place value. Here's why:

  • When you see 400 ÷ 4, think of it as 4 hundreds ÷ 4 = 1 hundred = 100
  • Each zero represents a power of 10 (10, 100, 1000, etc.)
  • So 400 ÷ 4 = (4 × 100) ÷ 4 = 4 ÷ 4 × 100 = 1 × 100 = 100

The zero pattern is just a shortcut for understanding place value relationships!

Question: How do I know which numbers to break apart when using the breaking apart strategy for division?

Answer: Choose numbers that are easily divisible by your divisor! Here's how:

  1. Look for the largest multiple of your divisor that's close to your dividend
  2. For 72 ÷ 6, think: What's the closest multiple of 6? 60 (since 6 × 10 = 60)
  3. Then break 72 into 60 + 12, since both 60 and 12 are divisible by 6

Practice recognizing these relationships, and it will become easier!

Question: What if I forget my basic division facts? How can I still use patterns?

Answer: If you forget a basic fact, you can use multiplication to figure it out!

For example, if you forget 24 ÷ 6:

  • Think: "What number times 6 equals 24?"
  • Try: 6 × 4 = 24, so 24 ÷ 6 = 4

Once you have the basic fact, you can apply the zero pattern to solve 240 ÷ 6, 2400 ÷ 6, etc.

Complete Summary: Use Patterns and Mental Math to Divide
Key Definitions:

Division: The operation of splitting a number into equal parts. It's the opposite of multiplication.

Quotient: The result you get when you divide one number by another.

Dividend: The number you're dividing up (the big number).

Divisor: The number you're dividing by (the number outside).

Fact Family: A group of related multiplication and division facts using the same numbers.

Core Rules and Principles:

• Zero Pattern: When dividing numbers with trailing zeros, divide the basic fact first, then add the zeros back.

• Fact Families: Division and multiplication are inverse operations (if a × b = c, then c ÷ a = b).

• Breaking Apart: Split difficult dividends into easier parts to divide separately.

• Verification: Always check your division by multiplying the quotient by the divisor.

Step-by-Step Method:
  1. Identify the basic division fact: Find the division without considering zeros.
  2. Count the zeros: Count how many zeros are at the end of the dividend.
  3. Apply the pattern: Use the basic fact and add the same number of zeros to the quotient.
  4. Verify: Multiply the quotient by the divisor to check your answer.
Simple Examples

Zero Pattern: 50 ÷ 5 = 10 (since 5 ÷ 5 = 1, add one zero)

Fact Family: If 4 × 7 = 28, then 28 ÷ 4 = 7 and 28 ÷ 7 = 4

Breaking Apart: 48 ÷ 4 = (40 ÷ 4) + (8 ÷ 4) = 10 + 2 = 12

Advanced Examples

Multiple Zeros: 3000 ÷ 5 = 600 (since 30 ÷ 5 = 6, add 2 zeros)

Complex Breaking: 96 ÷ 8 = (80 ÷ 8) + (16 ÷ 8) = 10 + 2 = 12

Pattern Recognition: 720 ÷ 9 = 80 (since 72 ÷ 9 = 8, add 1 zero)

Tips: Always start with the basic division fact. Know your multiplication tables well - they help with division too!
Common Pitfalls: Don't forget to add zeros to your answer. Don't confuse the dividend and divisor.
Memorization Tips: Practice basic division facts daily. Use fact families to connect multiplication and division.
Key Takeaways:

• Division patterns help you solve problems faster using mental math.

• The zero pattern: Divide the basic fact, then add back the zeros.

• Fact families connect multiplication and division facts.

• Breaking apart numbers makes complex division easier.

• Always verify your answers by multiplying back.