Solved Exercises on Multiplication Fact Families in Grade 3

Master multiplication fact families: related multiplication and division facts, fact triangles, and inverse relationships through these 3 detailed exercises.

Solution: Exercises 1 to 3
1 Creating a Fact Family
Exercise 1
Create a fact family for the numbers 4, 5, and 20.
Write all related multiplication and division facts.
Definition:

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

Fact family method:
  1. Identify the three numbers in the family (two factors and one product)
  2. Write two multiplication facts using the factors
  3. Write two division facts using the product and factors
  4. Verify that all facts are related
20
4
5
4 × 5 = 20
5 × 4 = 20
20 ÷ 4 = 5
20 ÷ 5 = 4
Step 1: Identify the three numbers

We have 4, 5, and 20. The product (20) is the largest number.

Step 2: Write multiplication facts

4 × 5 = 20 and 5 × 4 = 20 (commutative property)

Step 3: Write division facts

20 ÷ 4 = 5 and 20 ÷ 5 = 4 (division is the inverse of multiplication)

Step 4: Verify all facts are related

All facts use the same three numbers: 4, 5, and 20

Final answer:

Fact family for 4, 5, 20: 4×5=20, 5×4=20, 20÷4=5, 20÷5=4

Applied rules:

• Commutative property: Order doesn't matter in multiplication (4×5 = 5×4)

• Inverse relationship: Multiplication and division are opposite operations

• Fact family structure: Two multiplication facts and two division facts using the same numbers

2 Finding Missing Facts
Exercise 2
Complete the fact family: 6 × 7 = 42, 7 × 6 = 42, ____, ____
Explain the relationship between multiplication and division.
Definition:

Inverse operations: Multiplication and division are opposite operations that undo each other.

6 × 7 = 42
7 × 6 = 42
42 ÷ 6 = 7
42 ÷ 7 = 6
Multiplication
↔
Division
Step 1: Identify the known facts

We have: 6 × 7 = 42 and 7 × 6 = 42 (multiplication facts)

Step 2: Determine the missing facts

Using the inverse relationship: 42 ÷ 6 = 7 and 42 ÷ 7 = 6

Step 3: Verify the relationship

If 6 × 7 = 42, then 42 ÷ 6 = 7 and 42 ÷ 7 = 6

Step 4: Check the answers

6 × 7 = 42 ✓, 7 × 6 = 42 ✓, 42 ÷ 6 = 7 ✓, 42 ÷ 7 = 6 ✓

Final answer:

The missing facts are: 42 ÷ 6 = 7 and 42 ÷ 7 = 6

Applied rules:

• Inverse operations: Division undoes multiplication

• Fact family completeness: Every family has 2 multiplication and 2 division facts

• Relationship verification: Products divided by factors give the other factor

3 Solving Problems with Fact Families
Exercise 3
If 8 × 9 = 72, what are the other three facts in this fact family?
Use the fact family to solve: 72 ÷ 8 = ? and 72 ÷ 9 = ?
Definition:

Problem solving with fact families: Using known facts to find unknown related facts and solve problems.

72
8
9
8 × 9 = 72
9 × 8 = 72
72 ÷ 8 = 9
72 ÷ 9 = 8
Step 1: Identify the known fact

Given: 8 × 9 = 72

Step 2: Find the commutative fact

9 × 8 = 72 (same factors, different order)

Step 3: Find the division facts

72 ÷ 8 = 9 (product divided by first factor gives second factor)

72 ÷ 9 = 8 (product divided by second factor gives first factor)

Step 4: Solve the problems

72 ÷ 8 = 9 and 72 ÷ 9 = 8

Final answer:

The complete fact family: 8×9=72, 9×8=72, 72÷8=9, 72÷9=8

Solutions: 72 ÷ 8 = 9 and 72 ÷ 9 = 8

Applied rules:

• Commutative property: Factors can switch positions in multiplication

• Inverse relationship: Division reverses multiplication

• Problem solving: Use known facts to find unknown related facts

Fact Family Rules and Methods
a × b = c → c ÷ a = b, c ÷ b = a
Inverse Relationship
Multiplication Facts
3×4=12, 4×3=12
Commutative property
Division Facts
12÷3=4, 12÷4=3
Inverse operations
Fact Family
3, 4, 12
4 related facts
Key Rule: Every fact family has exactly 4 related facts (2 multiplication, 2 division).
Relationship Logic: Multiplication and division are inverse operations.
Tip 1: Use arrays or groups to visualize multiplication facts.
Tip 2: Remember that division facts always start with the largest number.
Tip 3: If you know one fact, you can find all others in the family.
Key Rules to Remember:

• Fact family structure: 2 multiplication facts and 2 division facts

• Inverse operations: Multiplication and division undo each other

• Commutative property: Order doesn't matter in multiplication

• Three numbers: Each family uses the same three numbers

{a, b, a×b} → {a×b=b×a, a×b=a×b, (a×b)÷a=b, (a×b)÷b=a}
General Fact Family
Fact Family Mult. Facts Div. Facts Numbers Used 2, 3, 6 2×3=6, 3×2=6 6÷2=3, 6÷3=2 2, 3, 6 4, 5, 20 4×5=20, 5×4=20 20÷4=5, 20÷5=4 4, 5, 20 6, 7, 42 6×7=42, 7×6=42 42÷6=7, 42÷7=6 6, 7, 42 8, 9, 72 8×9=72, 9×8=72 72÷8=9, 72÷9=8 8, 9, 72
Additional Practice
4 Mixed Fact Family Practice
Exercise 4
Complete the following fact families:
a) 3 × 6 = 18, ___, ___, ___
b) ___, 9 × 8 = 72, ___, ___
c) 5 × 7 = 35, ___, ___, ___
Definition:

Mixed practice: Combining different fact families to strengthen understanding of the relationships.

Part a: Fact family for 3, 6, 18

3 × 6 = 18, 6 × 3 = 18, 18 ÷ 3 = 6, 18 ÷ 6 = 3

Part b: Fact family for 8, 9, 72

8 × 9 = 72, 9 × 8 = 72, 72 ÷ 8 = 9, 72 ÷ 9 = 8

Part c: Fact family for 5, 7, 35

5 × 7 = 35, 7 × 5 = 35, 35 ÷ 5 = 7, 35 ÷ 7 = 5

Final answers:

a) 3×6=18, 6×3=18, 18÷3=6, 18÷6=3

b) 8×9=72, 9×8=72, 72÷8=9, 72÷9=8

c) 5×7=35, 7×5=35, 35÷5=7, 35÷7=5

Applied rules:

• Consistency: Apply the same method to all parts of the problem

• Fact family structure: Always 2 multiplication and 2 division facts

• Verification: Check that all facts use the same three numbers

5 Word Problem Application
Exercise 5
Emma has 24 cookies. She wants to put them in bags with 4 cookies in each bag. How many bags will she need? Use the fact family for 4, 6, and 24 to solve.
Definition:

Word problem application: Using fact families to solve real-world problems involving multiplication and division.

24
4
6
4 × 6 = 24
6 × 4 = 24
24 ÷ 4 = 6
24 ÷ 6 = 4
Step 1: Identify the problem

Emma has 24 cookies, 4 per bag, find number of bags

Step 2: Set up the division

24 ÷ 4 = ? (total cookies ÷ cookies per bag = number of bags)

Step 3: Use the fact family

From the 4, 6, 24 family: 24 ÷ 4 = 6

Step 4: Verify with multiplication

4 × 6 = 24 (4 bags with 6 cookies each = 24 total)

Final answer:

Emma needs 6 bags. The fact family shows that 24 ÷ 4 = 6.

Applied rules:

• Problem translation: Convert word problems to mathematical operations

• Fact family application: Use known facts to solve related problems

• Verification: Check answers using inverse operations

Comprehensive Summary: Multiplication Fact Families
{a, b, a×b} ↔ {a×b=b×a, a×b=a×b, (a×b)÷a=b, (a×b)÷b=a}
Fact Family Formula
Key Definitions:

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

Factors: Numbers being multiplied together (the smaller numbers in a fact family).

Product: The result of multiplication (the largest number in a fact family).

Inverse operations: Operations that undo each other (multiplication and division).

Commutative property: The order of factors doesn't change the product (a×b = b×a).

Related facts: Facts that use the same numbers in different arrangements.

Complete Methodology:
  1. Identify the numbers: Find the two factors and their product
  2. Write multiplication facts: Create two facts using the commutative property
  3. Write division facts: Create two facts using the inverse relationship
  4. Verify completeness: Ensure you have exactly 4 related facts
  5. Apply to problems: Use fact families to solve related problems
Tip 1: The largest number is always the product in multiplication facts.
Tip 2: Division facts always start with the largest number (the product).
Tip 3: If you know one fact, you can find all others in the family.
Tip 4: Use arrays or visual models to understand the relationships.
Common errors: Forgetting the commutative property, mixing up multiplication and division, missing facts in the family.
Key insights: Fact families connect multiplication and division, building foundational understanding for algebra.
Core Rules and Principles:

• Family structure: Every fact family has exactly 4 related facts (2 multiplication, 2 division)

• Three numbers: Each family uses the same three numbers consistently

• Inverse relationship: Multiplication and division are opposite operations

• Commutative property: Factors can be multiplied in any order

• Verification principle: Products divided by factors give the other factor

Examples: {2,3,6}, {4,5,20}, {6,7,42}, {8,9,72}
Sample Fact Families
Memory aid: "Big number divided by small number gives the other small number."
Success check: If 4×5=20, then 20÷4 should equal 5 and 20÷5 should equal 4. ✓

Questions & Answers

Question: Why do we need to learn fact families? Why not just memorize the multiplication tables?

Answer: Fact families are incredibly important for several reasons:

Connection between operations:

  • They show how multiplication and division are related
  • If you know 6 × 7 = 42, you automatically know 42 ÷ 6 = 7

Better problem-solving:

  • When you encounter a division problem like 42 ÷ 6, you can think "what times 6 equals 42?"
  • This connects to your knowledge of multiplication facts

Deeper understanding:

  • Instead of memorizing separate tables, you see the relationships between numbers
  • This builds a stronger foundation for more advanced math

Fact families help you understand math concepts rather than just memorize them!

Question: My child is struggling with the concept of inverse operations. How can I help them understand that multiplication and division are opposites?

Answer: Here are effective ways to teach inverse operations:

Use concrete examples:

  • "If you have 3 bags with 4 apples each, you have 3 × 4 = 12 apples"
  • "If you divide 12 apples into 3 bags, each bag has 12 ÷ 3 = 4 apples"

Visual representations:

  • Draw arrays: 3 rows of 4 dots = 12 total dots
  • Then group the 12 dots into 3 groups to show division

Fact family triangles:

  • Draw a triangle with 12 at the top, 3 and 4 at the bottom
  • Show how you can multiply the bottom numbers to get the top
  • And divide the top by either bottom number to get the other bottom number

The key is showing that multiplication "builds up" groups while division "breaks down" groups.

Question: What happens if I only know one fact in a fact family? Can I figure out the rest?

Answer: Absolutely! If you know one fact in a family, you can find all the others:

Example with multiplication:

  • If you know: 5 × 6 = 30
  • Use commutative property: 6 × 5 = 30
  • Use inverse relationship: 30 ÷ 5 = 6
  • Use inverse relationship: 30 ÷ 6 = 5

Example with division:

  • If you know: 24 ÷ 4 = 6
  • Use inverse relationship: 6 × 4 = 24
  • Use commutative property: 4 × 6 = 24
  • Use inverse relationship: 24 ÷ 6 = 4

This is why fact families are so powerful - knowing just one fact gives you access to three more related facts!

Once you know the three numbers in the family (like 4, 6, 24), you can write all four facts.