Multiplication: multiplication is repeated addition of the same number.

\[ a \times b = \underbrace{a + a + \cdots + a}_{b \text{ times}} \]

For example, \(4 \times 3\) means 4 taken 3 times: \(4 + 4 + 4 = 12\).

Properties of multiplication:

• Commutative: \(a \times b = b \times a\) (you can swap the order).
• Multiply by 1: \(a \times 1 = a\).
• Multiply by 0: \(a \times 0 = 0\).

Division: division means sharing into equal groups.

\[ a \div b = c \quad \Longleftrightarrow \quad b \times c = a \]

For example, \(20 \div 4 = 5\) because \(4 \times 5 = 20\).

Multiplication and division are inverse operations — you can always check a division result by multiplying.

Example 1: Multiplication — Repeated Addition

Problem: In a garden there are 5 flower beds with 6 flowers each. How many flowers are there altogether?

Solution:

1. 5 flower beds, 6 flowers each → multiplication: \( 5 \times 6 \).
2. As repeated addition: \( 6 + 6 + 6 + 6 + 6 = 30 \).
3. From the times table: \( 5 \times 6 = 30 \).

Answer: \( 30 \) flowers.

Example 2: Commutative Property — Swapping the Order

Problem: Calculate: \( 9 \times 3 \). Can you also calculate \( 3 \times 9 \)?

Solution:

1. \( 9 \times 3 \): 9 + 9 + 9 = 27.
2. \( 3 \times 9 \): 3 + 3 + 3 + 3 + 3 + 3 + 3 + 3 + 3 = 27.
3. Both give 27 — in multiplication you can always swap the order!

Answer: \( 9 \times 3 = 3 \times 9 = 27 \)

Example 3: Division — Sharing into Equal Groups

Problem: There are 36 biscuits to share equally among 4 children. How many does each child get?

Solution:

1. \( 36 \div 4 = ? \)
2. Ask: \( 4 \times ? = 36 \).
3. From the times table: \( 4 \times 9 = 36 \).
4. Check: \( 9 \times 4 = 36 \). Correct!

Answer: Each child gets \( 9 \) biscuits.

Example 4: The Link Between Multiplication and Division

Problem: If \( 7 \times 8 = 56 \), write two division facts.

Solution:

1. One multiplication fact gives two division facts.
2. \( 56 \div 7 = 8 \) (56 split into 7 groups = 8 in each group).
3. \( 56 \div 8 = 7 \) (56 split into 8 groups = 7 in each group).

Answer: \( 56 \div 7 = 8 \) and \( 56 \div 8 = 7 \).

Example 5: Word Problem — How Many Rows?

Problem: 42 students are sitting in rows of 7. How many rows are there?

Solution:

1. We are looking for the number of groups → division: \( 42 \div 7 = ? \)
2. \( 7 \times 6 = 42 \), so \( 42 \div 7 = 6 \).
3. Check: \( 6 \times 7 = 42 \). Correct!

Answer: There are \( 6 \) rows.

✗ Common mistake: Mixing up "how many groups" and "how many in each group" and using multiplication when division is needed.

✓ The correct way: If you know the total and want to find how many in each part — that's division. If you know the number of groups and the size of each — that's multiplication.

✗ Common mistake: Thinking the order matters in multiplication: \( 3 \times 8 \neq 8 \times 3 \).

✓ The correct way: In multiplication you can always swap the order! \( 3 \times 8 = 8 \times 3 = 24 \). Always choose the order that's easier for you.

✗ Common mistake: Not checking a division answer by multiplying.

✓ The correct way: After every division: multiply the quotient by the divisor. You should get the dividend back. \( 42 \div 7 = 6 \)? Check: \( 6 \times 7 = 42 \). Yes!

• Multiplication: \( a \times b \) = adding \(a\) a total of \(b\) times. Order does not matter.
• Division: \( a \div b = c \Leftrightarrow b \times c = a \). Always check by multiplying.
• Multiply by 0 = 0. Multiply by 1 = the same number. Multiply by 10 = add a 0.
• Multiplication and division are inverse operations — multiplication ≈ "repeated addition", division ≈ "repeated subtraction".