Solved Exercises on Multiplication Fact Families in Grade 3

Master multiplication fact families: understanding relationships between factors and products through these 5 detailed exercises with visual aids and step-by-step solutions.

Solution: Exercises 1 to 3
1 Basic fact family
Exercise 1
Complete the fact family for 3 × 4 = 12:
3 × 4 = 12
4 × 3 = ?
12 ÷ 3 = ?
12 ÷ 4 = ?
Definition:

Multiplication fact family: A group of related multiplication and division facts that use the same numbers.

For example: 3, 4, and 12 form a fact family because they can make: 3×4=12, 4×3=12, 12÷3=4, 12÷4=3

Fact family method:
  1. Identify the three numbers in the fact (factors and product)
  2. Write both multiplication facts (order doesn't matter)
  3. Write both division facts (product divided by each factor)
  4. Verify that all facts are true
Starting fact
3 × 4 = 12
Complete family
4 × 3 = 12
12 ÷ 3 = 4
12 ÷ 4 = 3
Step 1: Identify the numbers in the fact

We have 3, 4, and 12 from the given fact: 3 × 4 = 12

Step 2: Write the commutative multiplication fact

When we multiply the same two numbers in reverse order: 4 × 3 = 12

Step 3: Write the division facts

Divide the product by each factor: 12 ÷ 3 = 4 and 12 ÷ 4 = 3

Step 4: Verify all facts

3 × 4 = 12 ✓, 4 × 3 = 12 ✓, 12 ÷ 3 = 4 ✓, 12 ÷ 4 = 3 ✓

Complete fact family: 3 × 4 = 12, 4 × 3 = 12, 12 ÷ 3 = 4, 12 ÷ 4 = 3
Final answer:

4 × 3 = 12, 12 ÷ 3 = 4, 12 ÷ 4 = 3

Applied rules:

• Commutative Property: Order doesn't matter in multiplication (a×b = b×a)

• Multiplication-Division Relationship: Division "undoes" multiplication

• Fact Family Rule: Use the same 3 numbers in all 4 operations

2 Finding missing facts
Exercise 2
If 5 × 6 = 30, what are the other three facts in this fact family?
5 × 6 = 30
? × ? = ?
? ÷ ? = ?
? ÷ ? = ?
Definition:

Fact family members: The same 3 numbers used in 2 multiplication and 2 division facts.

Factors × Factors = Product, Product ÷ Factor = Other Factor

Given fact
5 × 6 = 30
Complete family
6 × 5 = 30
30 ÷ 5 = 6
30 ÷ 6 = 5
Step 1: Identify the three numbers

From 5 × 6 = 30, we have the numbers 5, 6, and 30

Step 2: Write the commutative multiplication fact

Switch the order of the factors: 6 × 5 = 30

Step 3: Write the first division fact

Divide the product by the first factor: 30 ÷ 5 = 6

Step 4: Write the second division fact

Divide the product by the second factor: 30 ÷ 6 = 5

Step 5: Verify all facts

5 × 6 = 30 ✓, 6 × 5 = 30 ✓, 30 ÷ 5 = 6 ✓, 30 ÷ 6 = 5 ✓

Complete fact family: 5 × 6 = 30, 6 × 5 = 30, 30 ÷ 5 = 6, 30 ÷ 6 = 5
Final answer:

6 × 5 = 30, 30 ÷ 5 = 6, 30 ÷ 6 = 5

Applied rules:

• Commutative Property: Multiplication is commutative (a×b = b×a)

• Division-Multiplication Inverse: Division reverses multiplication

• Fact Family Pattern: Always 2 multiplication and 2 division facts

3 Using arrays to visualize
Exercise 3
Draw an array for 2 × 7 = 14 and write the complete fact family.
Show how the array represents all four facts.
Definition:

Array model: A visual representation of multiplication using rows and columns.

For 2 × 7 = 14: 2 rows with 7 items in each row, totaling 14 items.

Array
2 rows of 7
Fact family
2×7=14, 7×2=14
14÷2=7, 14÷7=2
Step 1: Draw the array for 2 × 7

Draw 2 rows with 7 dots in each row (total 14 dots)

Step 2: Identify the numbers

From the array, we see 2, 7, and 14

Step 3: Write multiplication facts

Looking at rows: 2 × 7 = 14, Looking at columns: 7 × 2 = 14

Step 4: Write division facts

14 total items ÷ 2 rows = 7 items per row, 14 total items ÷ 7 items per row = 2 rows

Step 5: Verify with array

The array shows 2 groups of 7, 7 groups of 2, 14 split into 2 equal groups of 7, 14 split into 7 equal groups of 2

Array: 2 rows × 7 columns = 14 total items
Fact family: 2×7=14, 7×2=14, 14÷2=7, 14÷7=2
Final answer:

2×7=14, 7×2=14, 14÷2=7, 14÷7=2

Applied rules:

• Array Model: Rows × Columns = Total items

• Commutative Property: Same array viewed differently

• Part-Whole Relationship: Total can be divided by parts

Rules and methods, laws,...
\(a × b = c\)
Basic Multiplication
Multiplication Facts
a × b = c
and b × a = c
Division Facts
c ÷ a = b
and c ÷ b = a
Fact Family
4 related facts
using same 3 numbers
Key definitions:

Factor: Numbers being multiplied together (a and b in a×b=c)

Product: Result of multiplication (c in a×b=c)

Quotient: Result of division (result of c÷a=b)

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

Complete methodology:
  1. Identify the three numbers: Two factors and one product
  2. Write multiplication facts: Both orders (a×b=c and b×a=c)
  3. Write division facts: Product divided by each factor
  4. Verify relationships: Check that multiplication and division are inverse operations
Tip 1: Use arrays or groups to visualize multiplication facts.
Tip 2: Remember that multiplication is commutative (order doesn't matter).
Tip 3: Division "undoes" multiplication - they are inverse operations.
Tip 4: Always check that your division facts work by multiplying back.
Common errors: Mixing up factors and products, forgetting commutative property, incorrect division facts.
Memorization trick: Think of fact families as "fact triangles" where the top number is the product.
Key relationships to know:

• Commutative Property: a×b = b×a

• Multiplication-Division Inverse: If a×b=c, then c÷a=b and c÷b=a

• Fact Family Structure: 2 multiplication facts + 2 division facts = 4 related facts

• Part-Whole Relationship: Product contains the factors as parts

Questions & Answers

Question: I don't understand why 2 × 7 = 14, 7 × 2 = 14, 14 ÷ 2 = 7, and 14 ÷ 7 = 2 are all connected. Can you explain how they relate to each other?

Answer: These facts are connected because they all involve the same three numbers: 2, 7, and 14!

Think of it like this:

  • 2 × 7 = 14: If you have 2 groups with 7 items in each group, you have 14 total items
  • 7 × 2 = 14: If you have 7 groups with 2 items in each group, you still have 14 total items
  • 14 ÷ 2 = 7: If you have 14 total items and divide them into 2 equal groups, each group has 7 items
  • 14 ÷ 7 = 2: If you have 14 total items and put 7 items in each group, you get 2 groups

They're like a family of facts because they all use the same numbers in different ways. Multiplication combines things together, while division takes things apart!

Question: My child is struggling to remember fact families. Are there any visual tools or games that could help them understand better?

Answer: Yes! Here are some effective visual tools and activities:

  • Array models: Use objects like counters, blocks, or draw dots in rows and columns
  • Fact family triangles: Draw a triangle with the product at the top and factors at the bottom corners
  • Number bonds: Show how the three numbers connect in circles
  • Manipulatives: Use actual objects to physically group and separate

Games ideas:

  • Create fact family houses with the three numbers on top and four facts inside
  • Play matching games where children pair multiplication facts with their division partners
  • Use flashcards showing just one fact and ask for the other three in the family

The key is connecting concrete manipulatives to abstract symbols gradually!

Question: Why do we need to learn fact families? Can't I just memorize my multiplication tables?

Answer: Great question! Learning fact families does more than just help with memorization:

  • It builds understanding: You see how multiplication and division are related, not just separate topics
  • It helps with division: If you know 4 × 5 = 20, you automatically know 20 ÷ 4 = 5 and 20 ÷ 5 = 4
  • It improves problem-solving: When solving word problems, you'll better understand what operation to use
  • It creates connections: Instead of memorizing many separate facts, you learn patterns that connect them

Think of fact families like learning a whole family tree instead of just individual people - you understand how everything connects!

Solution: Exercises 4 to 5
4 Word problem fact family
Exercise 4
Sarah has 3 bags with 8 apples in each bag. She has 24 apples total.
Write the complete fact family for this situation.
Then solve: If she wants to put the apples in 4 bags equally, how many apples per bag?
Definition:

Word problem fact family: Translating a real-world situation into mathematical facts.

3 bags × 8 apples per bag = 24 total apples

Original facts
3 × 8 = 24
Complete family
8 × 3 = 24
24 ÷ 3 = 8
24 ÷ 8 = 3
New problem
24 ÷ 4 = 6
Step 1: Identify the multiplication fact from the problem

3 bags × 8 apples per bag = 24 total apples, so 3 × 8 = 24

Step 2: Write the commutative multiplication fact

8 × 3 = 24 (same total with different grouping)

Step 3: Write the division facts

24 total apples ÷ 3 bags = 8 apples per bag, so 24 ÷ 3 = 8

24 total apples ÷ 8 apples per bag = 3 bags, so 24 ÷ 8 = 3

Step 4: Solve the new problem

If 24 apples are divided into 4 bags equally: 24 ÷ 4 = 6 apples per bag

Step 5: Verify all facts

3 × 8 = 24 ✓, 8 × 3 = 24 ✓, 24 ÷ 3 = 8 ✓, 24 ÷ 8 = 3 ✓

Fact family: 3×8=24, 8×3=24, 24÷3=8, 24÷8=3
New problem: 24÷4=6 apples per bag
Final answer:

Complete fact family: 3×8=24, 8×3=24, 24÷3=8, 24÷8=3

New problem: 6 apples per bag

Applied rules:

• Real-world connection: Translate words into mathematical operations

• Fact family completion: Use same 3 numbers in 4 related facts

• Problem extension: Apply understanding to new scenarios

5 Mixed operations
Exercise 5
Given that 9 × 2 = 18, find:
a) The complete fact family
b) If 18 stickers are shared among 6 friends equally, how many does each friend get?
c) If each friend gets 3 stickers, how many friends can share 18 stickers?
Definition:

Mixed operations: Combining fact families with new division scenarios.

Using known facts to solve related problems.

Given fact
9 × 2 = 18
Fact family
2 × 9 = 18
18 ÷ 9 = 2
18 ÷ 2 = 9
New problems
18 ÷ 6 = 3
18 ÷ 3 = 6
Step 1: Write the complete fact family for 9 × 2 = 18

Numbers involved: 9, 2, 18

Multiplication: 9 × 2 = 18, 2 × 9 = 18

Division: 18 ÷ 9 = 2, 18 ÷ 2 = 9

Step 2: Solve part b - 18 stickers shared among 6 friends

This is division: 18 ÷ 6 = 3 stickers per friend

Step 3: Solve part c - 18 stickers with 3 per friend

This is division: 18 ÷ 3 = 6 friends

Step 4: Verify all calculations

9 × 2 = 18 ✓, 2 × 9 = 18 ✓, 18 ÷ 9 = 2 ✓, 18 ÷ 2 = 9 ✓

18 ÷ 6 = 3 ✓, 18 ÷ 3 = 6 ✓

Step 5: Check using multiplication

3 stickers × 6 friends = 18 stickers ✓, 3 stickers × 6 friends = 18 stickers ✓

a) Fact family: 9×2=18, 2×9=18, 18÷9=2, 18÷2=9
b) 3 stickers per friend
c) 6 friends
Final answer:

a) 9×2=18, 2×9=18, 18÷9=2, 18÷2=9

b) 3 stickers per friend

c) 6 friends

Applied rules:

• Fact family extension: Use known facts to solve new problems

• Division interpretation: Sharing equally or finding group size

• Verification: Check answers using inverse operations

Multiplication Fact Family Fundamentals and Practice
\(a × b = c \Rightarrow b × a = c, c ÷ a = b, c ÷ b = a\)
Fact Family Relationship
Key definitions:

Multiplication: Repeated addition (a×b means a groups of b items)

Division: Splitting into equal groups (c÷a means c items split into a groups)

Fact Family: Four related facts using the same three numbers

Commutative Property: Order doesn't matter in multiplication

Complete methodology:
  1. Identify components: Recognize factors and product in the given fact
  2. Apply commutativity: Write the multiplication fact with factors reversed
  3. Derive divisions: Write both division facts using product and factors
  4. Verify relationships: Confirm all facts are mathematically correct
  5. Extend applications: Use fact families to solve related problems
Tip 1: Use arrays or grouping objects to visualize multiplication facts.
Tip 2: Remember that multiplication and division are inverse operations.
Tip 3: Always verify division facts by multiplying back to check.
Tip 4: Look for patterns in fact families to aid memorization.
Common errors: Confusing factors with products, forgetting commutative property, incorrect division facts.
Memorization strategies: Use fact triangles, visual arrays, and repeated practice.
Essential formulas and relationships:

• Commutative Property: a×b = b×a

• Inverse Relationship: If a×b=c, then c÷a=b and c÷b=a

• Fact Family Structure: 2 multiplication + 2 division = 4 related facts

• Verification Method: Multiply division results to check accuracy

Visual Fact Family Patterns

📊
Fact Family Triangle

Top: Product

Bottom: Two Factors

Connects all 4 operations

Array Visualization

Rows × Columns = Total

Shows commutative property

Connects to division naturally

Operation Relationships

Multiplication combines groups

Division separates groups

They are inverse operations

Memory Strategy

Learn families, not isolated facts

Connect multiplication to division

Use visual representations

Advanced Applications:

• Solving word problems using fact families

• Understanding fraction relationships

• Preparing for more complex arithmetic

• Building foundation for algebraic thinking