Multiply and Divide Within 100 Using Strategies and Properties

Master multiplication and division within 100 through visual examples, step-by-step strategies, and practice exercises.

Multiplication and Division Strategies
\(a \times b = c \Rightarrow c \div b = a\)
Inverse Relationship
Key Definitions:

Multiplication: Repeated addition of equal groups

Division: Splitting into equal groups or finding how many groups

Factors: Numbers being multiplied together

Product: Result of multiplication

Dividend: Number being divided

Divisor: Number you divide by

Quotient: Result of division

Fact Family
\(4 \times 5 = 20\)
\(5 \times 4 = 20\)
\(20 \div 4 = 5\)
\(20 \div 5 = 4\)
All related facts
Distributive Property
\(7 \times 8 = 7 \times (5 + 3) = (7 \times 5) + (7 \times 3)\)
Break apart numbers
Commutative Property
\(6 \times 7 = 7 \times 6\)
Order doesn't matter
Tip 1: Use known facts to solve unknown ones (7×8 = 5×8 + 2×8 = 40+16 = 56).
Tip 2: Think of division as "how many groups?" or "how many in each group?"
Tip 3: Use arrays and area models to visualize multiplication and division.
Key Note: Multiplication and division are inverse operations - they undo each other!
Exercise Solutions: 1 to 3
1 Using Known Facts Strategy
Exercise 1
Solve 8 × 7 using the distributive property by breaking apart 7 into 5 + 2.
Then verify your answer using division.
Definition:

Distributive Property: \(a \times (b + c) = (a \times b) + (a \times c)\)

Method:
  1. Break the larger number into easier parts
  2. Apply the distributive property
  3. Calculate each part separately
  4. Add the partial products
  5. Verify using inverse operation (division)
Original Problem
\(8 \times 7\)
Break Apart
\(8 \times (5 + 2)\)
Result
56
Step 1: Break 7 into 5 + 2

8 × 7 = 8 × (5 + 2)

Step 2: Apply distributive property

8 × (5 + 2) = (8 × 5) + (8 × 2)

Step 3: Calculate each part

8 × 5 = 40 and 8 × 2 = 16

Step 4: Add partial products

40 + 16 = 56

Step 5: Verify with division

56 ÷ 8 = 7 ✓ and 56 ÷ 7 = 8 ✓

\(8 \times 7 = 56\)
Final answer:

8 × 7 = 56, calculated using 8 × (5 + 2) = (8 × 5) + (8 × 2) = 40 + 16 = 56

Applied rules:

• Distributive Property: Break apart one factor to make calculation easier

• Inverse Operations: Use division to verify multiplication results

• Partial Products: Add individual products to get final answer

2 Fact Families Application
Exercise 2
If 6 × 9 = 54, write all related multiplication and division facts.
Then solve: 54 ÷ 6 = ? and 54 ÷ 9 = ?
Definition:

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

Given Fact
\(6 \times 9 = 54\)
Related Facts
\(9 \times 6 = 54\)
Division Facts
\(54 \div 6 = 9\), \(54 \div 9 = 6\)
Step 1: Write the commutative fact

6 × 9 = 54, so 9 × 6 = 54

Step 2: Write the division facts

Since 6 × 9 = 54, then 54 ÷ 6 = 9 and 54 ÷ 9 = 6

Step 3: Solve the division problems

54 ÷ 6 = 9 and 54 ÷ 9 = 6

Step 4: Verify relationships

6 × 9 = 54, 9 × 6 = 54, 54 ÷ 6 = 9, 54 ÷ 9 = 6

\(6 \times 9 = 54\)
\(9 \times 6 = 54\)
\(54 \div 6 = 9\)
\(54 \div 9 = 6\)
Final answer:

54 ÷ 6 = 9 and 54 ÷ 9 = 6

Applied rules:

• Fact Families: Four related facts using the same three numbers

• Inverse Operations: Multiplication and division undo each other

• Commutative Property: Order doesn't matter in multiplication

3 Word Problem Application
Exercise 3
Emma has 48 stickers. She wants to put them equally on 6 pages of her sticker book.
How many stickers will go on each page? Use multiplication to verify your answer.
Definition:

Division as Equal Sharing: Splitting a total into equal groups

Total Stickers
48
Number of Pages
6
Stickers Per Page
8
Step 1: Set up the division problem

48 stickers ÷ 6 pages = ? stickers per page

Step 2: Think of related multiplication fact

What number times 6 equals 48? 6 × ? = 48

Step 3: Use known facts

6 × 8 = 48, so 48 ÷ 6 = 8

Step 4: Verify with multiplication

8 stickers per page × 6 pages = 48 stickers ✓

\(48 \div 6 = 8\)
Final answer:

Emma will put 8 stickers on each page.

Applied rules:

• Division as Equal Sharing: Split total into equal groups

• Inverse Operations: Use multiplication to check division

• Fact Families: Connect multiplication and division facts

Exercise Solutions: 4 to 5
4 Distributive Property Strategy
Exercise 4
Use the distributive property to solve 9 × 7 by breaking apart 9 into 5 + 4.
Show all steps and verify your answer.
Definition:

Distributive Property: \(a \times (b + c) = (a \times b) + (a \times c)\)

Original Problem
\(9 \times 7\)
Break Apart
\((5 + 4) \times 7\)
Result
63
Step 1: Break 9 into 5 + 4

9 × 7 = (5 + 4) × 7

Step 2: Apply distributive property

(5 + 4) × 7 = (5 × 7) + (4 × 7)

Step 3: Calculate each part

5 × 7 = 35 and 4 × 7 = 28

Step 4: Add partial products

35 + 28 = 63

Step 5: Verify with related division

63 ÷ 9 = 7 ✓ and 63 ÷ 7 = 9 ✓

\(9 \times 7 = 63\)
Final answer:

9 × 7 = 63, calculated using (5 + 4) × 7 = (5 × 7) + (4 × 7) = 35 + 28 = 63

Applied rules:

• Distributive Property: Break apart one factor to simplify calculation

• Partial Products: Calculate each part separately, then add

• Verification: Use inverse operation to confirm accuracy

5 Mixed Operations Strategy
Exercise 5
Complete these equations by finding the missing numbers:
a) 8 × ___ = 72
b) 56 ÷ ___ = 8
c) ___ × 7 = 49
Explain your thinking for each.
Definition:

Inverse Operations: Multiplication and division are opposite operations

Part A
\(8 \times ? = 72\)
Part B
\(56 \div ? = 8\)
Part C
\(? \times 7 = 49\)
Step 1: Solve part a) 8 × ___ = 72

Think: 8 times what number equals 72? 8 × 9 = 72, so answer is 9

Step 2: Solve part b) 56 ÷ ___ = 8

Think: 56 divided by what number equals 8? 56 ÷ 7 = 8, so answer is 7

Step 3: Solve part c) ___ × 7 = 49

Think: What number times 7 equals 49? 7 × 7 = 49, so answer is 7

Step 4: Verify all answers

8 × 9 = 72 ✓, 56 ÷ 7 = 8 ✓, 7 × 7 = 49 ✓

a) 9, b) 7, c) 7
Final answer:

a) 8 × 9 = 72, b) 56 ÷ 7 = 8, c) 7 × 7 = 49

Applied rules:

• Inverse Operations: Use multiplication to solve division problems and vice versa

• Fact Families: Connect related multiplication and division facts

• Number Sense: Think about what operation will give the desired result

Multiplication and Division Properties & Methods
\(a \times b = c \Rightarrow c \div b = a\)
Inverse Relationship
Key definitions:

Multiplication: Repeated addition of equal groups

Division: Splitting into equal groups or finding how many groups

Fact Family: Set of related multiplication and division facts

Distributive Property: Breaking apart numbers to make calculations easier

Commutative Property: Order doesn't matter in multiplication

Complete methodology:
  1. Identify the operation needed: Determine if it's multiplication or division
  2. Choose the strategy: Use known facts, distributive property, or fact families
  3. Apply the method: Follow the steps of the chosen technique
  4. Verify the result: Use inverse operation to check your answer
Tip 1: Use doubles facts (like 6×6=36) to find nearby products (6×7=36+6=42).
Tip 2: For division, think "what number times the divisor equals the dividend?"
Tip 3: Draw arrays or equal groups to visualize multiplication and division.
Tip 4: Always verify division by multiplying to check your answer.
Common errors: Confusing multiplication and division, forgetting to verify answers, misapplying properties.
Exam preparation: Practice fact families, memorize basic facts, understand inverse relationship.
Properties to know:

• Commutative: \(a \times b = b \times a\)

• Distributive: \(a \times (b + c) = (a \times b) + (a \times c)\)

• Identity: \(a \times 1 = a\), \(a \div 1 = a\)

• Zero: \(a \times 0 = 0\), \(0 \div a = 0\)

• Inverse: \(a \times b = c \Rightarrow c \div b = a\)

Questions & Answers

Question: Why do we need to learn strategies for multiplication and division? Can't I just memorize the facts?

Answer: Great question! While memorizing facts is helpful, strategies are essential for several reasons:

  • Understanding: Strategies help you understand WHY multiplication and division work, not just memorize answers
  • Problem-solving: When you encounter a fact you haven't memorized, strategies help you figure it out
  • Flexibility: You can choose the easiest approach for each problem
  • Checking work: Strategies help you verify if your memorized answer is correct

For example, if you don't know 8×7, you can use 8×5 + 8×2 = 40 + 16 = 56. This builds number sense!

Question: My child is struggling with division. How can I help them understand the connection to multiplication?

Answer: The connection between multiplication and division is crucial! Try these approaches:

  • Use fact families: Show how 6×4=24, 4×6=24, 24÷6=4, and 24÷4=6 are all connected
  • Frame division as "undoing" multiplication: "If 6×4=24, then 24÷4 must equal 6"
  • Visual representations: Draw 24 dots arranged in 6 rows of 4, then show how dividing means counting the rows or columns
  • Real-world problems: "If you have 24 cookies to share among 6 friends, how many does each get?" connects to 24÷6=4

The key is helping your child see that multiplication and division are opposite operations that work together!

Question: What's the distributive property? It sounds complicated!

Answer: Don't worry - it's simpler than it sounds! The distributive property is just breaking apart numbers to make multiplication easier:

  • Instead of: 7×8 (which might be hard to remember)
  • You can break 8 into: 5 + 3
  • Then do: 7×(5+3) = (7×5) + (7×3) = 35 + 21 = 56

It's like taking a big problem and splitting it into smaller, easier problems! Think of it as a way to use facts you DO know to solve facts you don't know yet.