Solved Exercises on Multiplication and Division Fact Families in Grade 3

Master multiplication and division fact families: understanding relationships between multiplication and division facts, fact families, number bonds through these 5 detailed exercises.

Solution: Exercises 1 to 3
1 Basic fact family
Exercise 1
Write the multiplication and division fact family for the numbers 3, 4, and 12.
Definition:

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

Method for finding fact families:
  1. Identify the three numbers in the family
  2. Write two multiplication facts (switching order)
  3. Write two division facts (dividing the largest number)
  4. Verify that all four facts are related
Numbers
3, 4, 12
Multiplication
3×4=12, 4×3=12
Division
12÷3=4, 12÷4=3
Step 1: Identify the three numbers

We have 3, 4, and 12. The largest number (12) is the product of the other two.

Step 2: Write multiplication facts

3 × 4 = 12 and 4 × 3 = 12 (commutative property)

Step 3: Write division facts

12 ÷ 3 = 4 and 12 ÷ 4 = 3 (division reverses multiplication)

Step 4: Verify the relationship

Check: 3 × 4 = 12, so 12 ÷ 4 = 3 and 12 ÷ 3 = 4

Fact Family: 3×4=12, 4×3=12, 12÷3=4, 12÷4=3
Final answer:

Multiplication facts: 3×4=12, 4×3=12

Division facts: 12÷3=4, 12÷4=3

Applied rules:

• Commutative Property: 3×4 = 4×3

• Inverse Operations: Multiplication and division are inverse operations

• Fact Family Rule: Same 3 numbers create 4 related facts

2 Finding missing facts
Exercise 2
If 6×7=42, what are the other three facts in this fact family?
Definition:

Missing Facts: Given one fact in a family, find the other related facts using the same numbers.

Given
6×7=42
Find
7×6=?, 42÷6=?, 42÷7=?
Complete
4 facts
Step 1: Identify the numbers

From 6×7=42, the three numbers are 6, 7, and 42.

Step 2: Find the commutative multiplication fact

Using commutative property: 6×7=42 means 7×6=42

Step 3: Find the division facts

Division reverses multiplication: 42÷6=7 and 42÷7=6

Step 4: Verify all facts

Check: 6×7=42, 7×6=42, 42÷6=7, 42÷7=6

Fact Family: 6×7=42, 7×6=42, 42÷6=7, 42÷7=6
Final answer:

7×6=42, 42÷6=7, 42÷7=6

Applied rules:

• Commutative Property: Order doesn't matter in multiplication

• Inverse Relationship: Division undoes multiplication

• Fact Family Rule: 4 facts with same 3 numbers

3 Creating fact families
Exercise 3
Create the fact family for 8, 5, and 40.
Definition:

Creating Fact Families: Using three numbers where one is the product of the other two to write four related multiplication and division facts.

Numbers
8, 5, 40
Multiply
8×5=40
Divide
40÷8=5, 40÷5=8
Step 1: Check if numbers form a fact family

Verify: 8 × 5 = 40, so these numbers can form a fact family

Step 2: Write multiplication facts

8 × 5 = 40 and 5 × 8 = 40 (using commutative property)

Step 3: Write division facts

40 ÷ 8 = 5 and 40 ÷ 5 = 8 (division reverses multiplication)

Step 4: Verify the complete fact family

Check: 8×5=40, 5×8=40, 40÷8=5, 40÷5=8

Fact Family: 8×5=40, 5×8=40, 40÷8=5, 40÷5=8
Final answer:

8×5=40, 5×8=40, 40÷8=5, 40÷5=8

Applied rules:

• Fact Family Formation: One number equals product of other two

• Commutative Property: Order doesn't change product

• Division Inverse: Division reverses multiplication operation

Rules and methods, laws,...
3×4=12 → 4×3=12 → 12÷3=4 → 12÷4=3
Fact Family Relationships
Multiplication
a × b = c
Multiplication Fact
Commutative
b × a = c
Order Doesn't Matter
Division
c ÷ a = b
Division Reverses Multiplication
Key definitions:

Fact Family: A set of related multiplication and division facts that use the same numbers

Multiplication: Combining equal groups to find a total

Division: Splitting a total into equal groups

Complete methodology:
  1. Identify the numbers: Find the three numbers that make up the family
  2. Write multiplication facts: Use commutative property
  3. Write division facts: Reverse the multiplication
  4. Verify relationships: Check that all facts are consistent
Tip 1: The largest number in a fact family is always the product.
Tip 2: Division facts always start with the largest number.
Tip 3: Multiplication and division are opposite operations.
Key Properties: Commutative property of multiplication, inverse relationship between multiplication and division.
Memory Aid: Think of fact families like a triangle where each corner connects to the others.
Formulas to know by heart:

• If a × b = c, then b × a = c (commutative)

• If a × b = c, then c ÷ a = b (inverse)

• If a × b = c, then c ÷ b = a (inverse)

• Every fact family has exactly 4 related facts

Solution: Exercises 4 to 5
4 Missing number in fact family
Exercise 4
Complete the fact family: ?×6=30, 6×?=30, 30÷?=6, 30÷6=?
Definition:

Missing Number: Finding the unknown value in a fact family using known relationships.

Given
?×6=30
Find
?=30÷6=5
Complete
All 4 facts
Step 1: Find the missing factor

If ?×6=30, then ?=30÷6=5

Step 2: Complete multiplication facts

5×6=30 and 6×5=30

Step 3: Complete division facts

30÷5=6 and 30÷6=5

Step 4: Verify the complete fact family

Check: 5×6=30, 6×5=30, 30÷5=6, 30÷6=5

Fact Family: 5×6=30, 6×5=30, 30÷5=6, 30÷6=5
Final answer:

5×6=30, 6×5=30, 30÷5=6, 30÷6=5

Applied rules:

• Division Finds Factors: If a×b=c, then c÷b=a

• Commutative Property: Order doesn't change product

• Inverse Operations: Division reverses multiplication

5 Fact family verification
Exercise 5
Are 7, 8, and 54 a fact family? Explain why or why not.
Definition:

Fact Family Verification: Checking if three numbers can form a valid multiplication and division fact family.

Test
7×8=?
Result
7×8=56≠54
Conclusion
Not a fact family
Step 1: Test if 7×8=54

Calculate: 7×8=56, which is not equal to 54

Step 2: Test other combinations

7×54=378 and 8×54=432, neither equals 54

Step 3: Apply fact family rule

For a fact family, one number must be the product of the other two

Step 4: Draw conclusion

Since 7×8≠54, these numbers cannot form a fact family

7, 8, and 54 are NOT a fact family because 7×8≠54
Final answer:

No, 7, 8, and 54 are not a fact family because 7×8=56, not 54.

Applied rules:

• Fact Family Condition: One number must equal the product of the other two

• Verification: Always check if multiplication relationship holds

• Counterexample: If one relationship fails, no fact family exists

Multiplication and Division Fact Families Guide
a × b = c → b × a = c → c ÷ a = b → c ÷ b = a
Fact Family Structure
Key definitions:

Multiplication: Combining equal groups to find a total (repeated addition)

Division: Splitting a total into equal groups (repeated subtraction)

Fact Family: A group of 4 related multiplication and division facts using the same 3 numbers

Complete methodology:
  1. Analyze the numbers: Identify if one number is the product of the other two
  2. Write multiplication facts: Use both orders (commutative property)
  3. Write division facts: Divide the largest number by each smaller number
  4. Verify the family: Ensure all 4 facts are mathematically correct
Tip 1: The largest number in any fact family is always the product of the other two numbers.
Tip 2: Division facts always begin with the largest number (the product).
Tip 3: Multiplication and division are inverse operations - they undo each other.
Tip 4: Always check your work by verifying that multiplication and division facts correspond.
Commutative Property: In multiplication, a×b = b×a, so both forms appear in fact families.
Inverse Relationship: Multiplication and division are opposite operations that reverse each other.
Essential rules to remember:

• Every fact family consists of exactly 4 related facts

• Two multiplication facts and two division facts

• The largest number is always the product in multiplication facts

• The largest number is always the dividend in division facts

• If a×b=c, then c÷a=b and c÷b=a

Questions & Answers

Question: Why do we learn fact families? It seems like extra work to remember four facts instead of just one multiplication fact.

Answer: Great question! Fact families are incredibly important because they show the connection between multiplication and division:

  • Understanding Relationships: When you know 4×5=20, you automatically know 20÷4=5 and 20÷5=4
  • Better Problem Solving: If you know 3×7=21, you can quickly solve 21÷3=___ without having to figure it out from scratch
  • Building Mental Math: Fact families help you become faster at both multiplication and division
  • Foundation for Advanced Math: Understanding these relationships prepares you for algebra and more complex math

Think of it like learning a word family in reading - when you know "cat", you also understand "bat", "rat", "hat" share similar patterns. Similarly, 4×5=20, 5×4=20, 20÷4=5, and 20÷5=4 are all connected!

Question: My child is struggling to understand why 3×4=12 and 12÷3=4 are related. How can I help them visualize this connection?

Answer: Visual representations are perfect for this! Try these concrete approaches:

  • Array Model: Draw 3 rows of 4 dots each. Count all dots to get 12 (3×4=12). Then ask: "If I have 12 dots arranged in 3 rows, how many in each row?" (12÷3=4)
  • Grouping: Take 12 objects and divide them into 3 equal groups. Count how many are in each group (12÷3=4). Then show that 3 groups of 4 equals 12 (3×4=12)
  • Number Line: For 3×4, make 3 jumps of 4 units each to reach 12. For 12÷3, start at 12 and make 3 equal jumps back to 0, measuring each jump

The key insight: multiplication combines groups while division splits a total into groups. They're opposite actions!

Use manipulatives like blocks, counters, or even food items to make the concept tangible.

Question: How do I know if three numbers make a fact family? Like, do 4, 6, and 25 make a fact family?

Answer: Excellent question! To check if three numbers make a fact family, follow this simple test:

Step 1: Multiply the two smaller numbers together

For 4, 6, and 25: 4 × 6 = 24

Step 2: Compare the result to the largest number

Is 24 equal to 25? No, 24 ≠ 25

Step 3: Conclusion

Since 4×6=24 and not 25, these three numbers (4, 6, 25) do NOT make a fact family.

Let's try a real fact family: Numbers 3, 8, 24

  • 3 × 8 = 24 ✓ (equals the largest number)
  • So 3, 8, 24 DO make a fact family!
  • Fact family: 3×8=24, 8×3=24, 24÷3=8, 24÷8=3

Remember: For a fact family, the largest number must equal the product of the other two numbers!