Solved Exercises on Understanding Division in Grade 4

Master division concepts: sharing equally, grouping, relationship with multiplication, division facts, and word problems through these 5 detailed exercises.

Solution: Exercises 1 to 3
1 Sharing Equally
Exercise 1
Sarah has 12 apples and wants to share them equally among 4 friends. How many apples does each friend get?
Definition:

Division: Splitting a number into equal groups or parts. The operation of finding how many times one number goes into another.

Sharing Method:
  1. Identify the total amount (dividend)
  2. Identify the number of groups (divisor)
  3. Distribute equally to find how many in each group (quotient)
Total Apples
12
Number of Friends
4
Apples per Friend
?
Step 1: Set up the division problem

We need to divide 12 apples among 4 friends: 12 ÷ 4

Step 2: Use sharing strategy

Imagine distributing 12 apples equally to 4 people:
1st round: Each gets 1 apple → 4 apples distributed, 8 remaining
2nd round: Each gets 1 more → 8 apples distributed, 4 remaining
3rd round: Each gets 1 more → 12 apples distributed, 0 remaining

Step 3: Count the result

Each friend receives 3 apples after 3 rounds of distribution

Step 4: Verify with multiplication

Check: 3 × 4 = 12 ✓

12 ÷ 4 = 3
Final answer:

Each friend gets 3 apples.

Applied rules:

• Division as sharing: Distribute items equally among groups

• Relationship with multiplication: Division is the inverse of multiplication

• Verification: Multiply quotient by divisor to check answer

2 Grouping Strategy
Exercise 2
There are 15 candies. If we make groups of 3 candies each, how many groups can we make?
Definition:

Division as grouping: Finding how many groups of a specific size can be made from a total amount.

Total Candies
15
Group Size
3
Number of Groups
?
Step 1: Set up the division problem

We need to find how many groups of 3 can be made from 15: 15 ÷ 3

Step 2: Use grouping strategy

Create groups of 3 from 15 items:
Group 1: 3 candies → 12 remaining
Group 2: 3 candies → 9 remaining
Group 3: 3 candies → 6 remaining
Group 4: 3 candies → 3 remaining
Group 5: 3 candies → 0 remaining

Step 3: Count the result

We made 5 groups of 3 candies each

Step 4: Verify with multiplication

Check: 5 × 3 = 15 ✓

15 ÷ 3 = 5
Final answer:

We can make 5 groups of 3 candies each.

Applied rules:

• Division as grouping: Determine how many sets of a given size fit into the total

• Counting strategy: Repeatedly subtract the group size until none remain

• Verification: Multiply number of groups by group size to check answer

3 Relationship with Multiplication
Exercise 3
Find the missing number: ? ÷ 6 = 4. Use the relationship between division and multiplication.
Definition:

Inverse operations: Division and multiplication are inverse operations. If a ÷ b = c, then b × c = a.

Unknown ÷ Divisor
? ÷ 6
Equals
=
Quotient
4
Step 1: Understand the relationship

If ? ÷ 6 = 4, then ? = 6 × 4 (using inverse relationship)

Step 2: Use multiplication to solve

? = 6 × 4 = 24

Step 3: Verify the solution

Check: 24 ÷ 6 = 4 ✓

Step 4: Confirm with multiplication fact

Since 6 × 4 = 24, then 24 ÷ 6 = 4 ✓

24 ÷ 6 = 4
Final answer:

The missing number is 24.

Applied rules:

• Inverse relationship: Division undoes multiplication and vice versa

• Multiplication facts: Use known multiplication facts to solve division problems

• Verification: Always check your answer using the inverse operation

Division Facts, Properties, and Methods
Dividend ÷ Divisor = Quotient
Division Components
Division Fact
12 ÷ 3 = 4
because 3 × 4 = 12
Division by 1
Any number ÷ 1 = itself
15 ÷ 1 = 15
Division by Self
Any number ÷ itself = 1
8 ÷ 8 = 1
Key definitions:

Dividend: The number being divided

Divisor: The number dividing the dividend

Quotient: The result of division

Remainder: What's left over after division (not covered in basic understanding)

Division methods:
  1. Sharing equally: Distribute items into equal groups
  2. Repeated subtraction: Keep subtracting the divisor until reaching zero
  3. Using multiplication facts: Think of related multiplication facts
  4. Grouping: Make equal-sized groups from the total
Tip 1: Division is the opposite of multiplication. Use multiplication facts to solve division problems.
Tip 2: When dividing by 1, the answer is always the same as the number you started with.
Tip 3: When dividing a number by itself, the answer is always 1.
Tip 4: Always verify your division by multiplying the quotient by the divisor.
Division properties: Division is not commutative (a÷b ≠ b÷a) and not associative ((a÷b)÷c ≠ a÷(b÷c)).
Special cases: Any number divided by itself equals 1; any number divided by 1 equals itself; division by zero is undefined.
Important division facts to remember:

• Basic division facts up to 100 (related to multiplication tables)

• Division by 1: n ÷ 1 = n

• Division by self: n ÷ n = 1 (where n ≠ 0)

• Zero divided by any number: 0 ÷ n = 0

• Division by zero: n ÷ 0 is undefined

Visualizing Division Relationships
Division Patterns
Visualizing division facts and relationships between numbers.

Analysis: The chart shows various division relationships and patterns that help understand division concepts.

  • Division as repeated subtraction: 12 ÷ 3 means subtracting 3 from 12 four times
  • Division as grouping: 12 items grouped in sets of 3 gives 4 groups
  • Division as sharing: 12 items shared equally among 3 groups gives 4 items per group
Solution: Exercises 4 to 5
4 Word Problem Application
Exercise 4
A pizza restaurant made 24 pizzas today. They want to deliver these equally to 8 stores. How many pizzas will each store receive?
Definition:

Word problems: Real-world situations that require mathematical operations to solve. Identify key information and determine which operation to use.

Total Pizzas
24
Number of Stores
8
Pizzas per Store
?
Step 1: Identify key information

Total pizzas = 24, Number of stores = 8, Need to find: Pizzas per store

Step 2: Determine the operation

"Equally distribute" indicates division: 24 ÷ 8

Step 3: Solve using known facts

Think: What number multiplied by 8 gives 24?
8 × 3 = 24, so 24 ÷ 8 = 3

Step 4: Verify the answer

Check: 3 pizzas × 8 stores = 24 pizzas ✓

24 ÷ 8 = 3
Final answer:

Each store will receive 3 pizzas.

Applied rules:

• Problem-solving strategy: Identify total amount, number of groups, and unknown

• Keywords for division: "each," "per," "equally," "shared among," "split into"

• Verification: Multiply to ensure the answer makes sense in context

5 Division Patterns and Facts
Exercise 5
Complete the pattern: 36 ÷ 4 = 9, 36 ÷ 9 = ?, 36 ÷ 6 = 6, 36 ÷ ? = 4. Explain the relationship.
Definition:

Division patterns: Recognizing relationships between related division facts based on multiplication tables.

Given Fact
36 ÷ 4 = 9
Find
36 ÷ 9 = ?
Find
36 ÷ ? = 4
Step 1: Analyze the first fact

36 ÷ 4 = 9 means 4 × 9 = 36

Step 2: Find 36 ÷ 9

Since 9 × 4 = 36, then 36 ÷ 9 = 4

Step 3: Verify the third fact

36 ÷ 6 = 6 because 6 × 6 = 36

Step 4: Find the missing divisor

If 36 ÷ ? = 4, then ? × 4 = 36, so ? = 9

Step 5: Complete the pattern

36 ÷ 4 = 9, 36 ÷ 9 = 4, 36 ÷ 6 = 6, 36 ÷ 9 = 4

36 ÷ 9 = 4 and 36 ÷ 9 = 4
Final answer:

The missing values are 4 and 9. These facts are related through the multiplication table of 36: 4×9=36, 6×6=36, 9×4=36.

Applied rules:

• Fact families: Related multiplication and division facts using the same numbers

• Commutative property: If 4×9=36, then 9×4=36

• Pattern recognition: Identifying relationships between related facts

Division Laws, Methods, Rules, and Definitions
Dividend ÷ Divisor = Quotient
Division Formula
Key definitions:

Division: The process of splitting a number into equal parts or groups.

Dividend: The number being divided (the whole).

Divisor: The number by which the dividend is divided (the size of each group).

Quotient: The result of division (the number of equal groups).

Complete methodology:
  1. Read the problem carefully: Identify what is being divided and how many groups or parts.
  2. Determine the operation: Look for keywords like "share," "divide," "each," "per," "equal groups."
  3. Set up the division sentence: Write the dividend ÷ divisor = ?
  4. Choose the strategy: Sharing, repeated subtraction, or use multiplication facts.
  5. Solve the problem: Perform the calculation.
  6. Verify the answer: Multiply the quotient by the divisor to check.
Tip 1: Use multiplication facts to solve division problems. If you know 6×7=42, then 42÷6=7.
Tip 2: Remember that division is the inverse operation of multiplication.
Tip 3: Practice division facts regularly to build fluency.
Tip 4: Draw pictures or use manipulatives to visualize division problems.
Common errors: Confusing dividend and divisor, forgetting to verify answers, mixing up division and multiplication facts.
Exam preparation: Master division facts up to 100, practice word problems, understand the relationship between multiplication and division.
Division facts to know by heart:

• Division facts for tables 1-10 (related to multiplication tables)

• Any number divided by 1 equals itself: n ÷ 1 = n

• Any number divided by itself equals 1: n ÷ n = 1 (where n ≠ 0)

• Zero divided by any number equals 0: 0 ÷ n = 0

• Division by zero is undefined: n ÷ 0 = undefined

Exercise with Visualization: Division Concepts
Exercise 6: Division Patterns
Consider the following division patterns:
f₁(x) = 24 ÷ x (for x = 1,2,3,4,6,8,12,24)
f₂(x) = 36 ÷ x (for x = 1,2,3,4,6,9,12,18,36)
f₃(x) = 48 ÷ x (for x = 1,2,3,4,6,8,12,16,24,48)

Analysis: The chart shows how division results change as divisors increase.

  • As the divisor increases, the quotient decreases
  • Division facts form patterns based on multiplication tables
  • Only divisors that evenly divide the dividend result in whole numbers

Questions & Answers

Question: I don't understand the difference between sharing equally and grouping in division. Can you explain when to use each method?

Answer: Great question! Both methods represent division but focus on different aspects:

  • Sharing equally: You know the total and the number of groups, find how many in each group
    Example: 12 cookies shared among 4 children → 12 ÷ 4 = 3 cookies per child
  • Grouping: You know the total and the size of each group, find how many groups
    Example: 12 cookies, put 3 in each bag → 12 ÷ 3 = 4 bags

Both give the same answer but model different real-world situations. Choose the method that matches the problem's context.

Question: How do I know if my division answer is correct? Sometimes I'm not sure if I divided correctly.

Answer: The best way to check division is to multiply your answer by the divisor:

  • If you calculated 24 ÷ 6 = 4, check: 4 × 6 = 24 ✓
  • This works because division and multiplication are inverse operations
  • You can also draw a picture or use counters to verify your answer

Always perform this verification step, especially when solving word problems!

Question: My child is struggling with division facts. Are there any strategies to help them memorize these better?

Answer: Yes! Here are effective strategies for learning division facts:

  • Connect to multiplication: If they know 7 × 8 = 56, then 56 ÷ 8 = 7
  • Start with easier facts (÷1, ÷2, ÷10) and build up
  • Use visual aids like arrays, groups of objects, or drawings
  • Practice with games, flashcards, or apps that make practice fun
  • Focus on patterns: dividing by 10 moves the decimal, dividing by 5 is half of dividing by 10

Consistent practice for just 10 minutes daily is more effective than longer, infrequent sessions.

Question: Why can't we divide by zero? My teacher said it's impossible but didn't explain why.

Answer: Great thinking! Division by zero is undefined because it creates logical contradictions:

If 10 ÷ 0 = ? then ? × 0 should equal 10, but anything times 0 equals 0, not 10!

Also, as we divide by smaller and smaller numbers, the result gets larger and larger:

  • 10 ÷ 1 = 10
  • 10 ÷ 0.1 = 100
  • 10 ÷ 0.01 = 1000

As the divisor approaches zero, the result approaches infinity, which isn't a specific number. That's why mathematicians say division by zero is undefined.

Question: What are some keywords in word problems that tell me to use division?

Answer: Here are common keywords that indicate division is needed:

  • Sharing equally: "shared among," "divided equally," "split between," "distributed among"
  • Groups of: "groups of," "sets of," "packs of," "rows of," "arrays of"
  • Rate or unit: "per," "each," "every," "for every," "at each"
  • How many times: "how many times does...go into," "how many...fit into"

Example: "There are 24 students in 4 equal groups" → 24 ÷ 4 = 6 students per group