## Key Concepts

- Commutative property of multiplication
- Partitioning through place value and applying distributive property

**Introduction:**

- Understand the commutative property of multiplication.
- Understand the place values.
- Understand the distributive property for multiplication.
- Understand the partitioning through place value and applying distributive property.

**3.5 Mental math strategies for multiplication**

Mental math strategies help in multiplying large and small numbers in your head. Learning mental math techniques helps you to convert complex multiplication into simpler multiplications and use addition and subtraction to save time.

For doing mental math multiplication, we also use some multiplication properties as follows.

Commutative property

E.g., 3×4 = 4× 3=12

Distributive property

E.g., 3×(2+ 3) = (3 × 2) +( 3× 3) = 6 + 9 =15

Associative property

E.g., (6×5)× 7= 6 x (5× 7) = 210

**3.5.1 Commutative property of multiplication **

As per the commutative property of multiplication, when we multiply two integers, the answer we get after multiplication (the product) will remain the same, even if the position of the integers is interchanged.

Let A and B be the two integers, then,

A × B = B × A

Examples of commutative property of multiplication

- 1 × 2 = 2 × 1 = 2

- 3 × 8 = 8 × 3 = 24

### Example: 1

- Find the value of 3 × 12

**Step 1:** Multiply 3 × 12 = 36

**Step 2:** Multiply 12 × 3 = 36

=> 3 × 12 = 12 × 3 = 36

(Commutative Property)

**Example:2**

Find the value of 6×20 using commutative property.

**Step1: **Multiply 6×20 = 120

**Step2: **Multiply 20×6 = 120

**Step3:** 6 x 20 = 20×6 = 120

**3.5.2 Partitioning through place value and applying distributive property**

**What is a Place Value?**** **

Place value is the value of each digit in a number. The value of every digit in a number is different based on its position in the number.

Place value is the value of a digit according to its position in the number such as ones, tens, hundreds, and so on.

Following image shows the place value:

**Example:**

Find the value of 4 5 6 7

4567 = Four thousand, five hundred and sixty-seven

= (4×1000) + (5× 100) + (6× 10) + (7× 1)

= 4000 + 500 + 60 + 7

= 4567

### Distributive property of multiplication

According to the distributive property, multiplying the sum of two or more addends by a number will give the same result as multiplying each addend individually by the number, and then adding the products together.

Multiply 3×(3 + 4).

3×(3 + 4) = (3× 3) + (3× 4)

3×(7) = 9 + 12

21 = 12

This below image indicates the distributive property of multiplication.

**Example 1:**

Multiply 8×27.

**Solution:**

Calculating 8 × 27 can be made easier by breaking down 27 as 20 + 7

The distributive property of multiplication

8 × (20 + 7 )

= 8 × 20 + 8 × 7

= 160 + 56

= 216

**Example 2:**

Multiply 6×34

= 6×34

= (3×2)× 34

=2×(3× 34)

=2×(102)

= 204 (Associative property)

### Use multiplication facts and place value to multiply by multiples.

**Example 1**:

Find the value of 9×80.

9×80 = 9× 8 tens

= 72 tens

= 720

9×80 = 720

**Example 2:**

Expand the multiplier and distribute the multiplicand to each place value:

Multiply 3×1847

9(1,000) + 9(800) + 9(40) + 9(7) =?

9,000 + 7,200 + 360 + 63 =?

Associate (group) addends for easier mental addition:

(9,000 + 7,200) + (360 + 63) =?

16,200 + 423 = 16,623

### Break Apart and Use Addition (Mental math strategy)

Multiply mentally to find the product. Explain which strategy you used to multiply 8 x 903.

**Solution: **

8×903.

903 is close to 900.

Find 8×900 and adjust the answer.

8×900 = 7200

8 × 903 = 8(900 + 3)

Here, 8 × 900 = 7200 and 8 × 3 = 24

Adjust the answer by adding 24 to 7200.

7200 + 24 = 7224

7224 is the answer

Here, we use distributive property.

## Exercise

- Multiply mentally to find each product. Explain which strategy has been used.

a) 3 x 898 b) 34 x 6 c) 4 x 87 - Multiply 5 x 4,567.
- Write the property of each one

9 x 2 = 2 x 9

8 x (20 + 4) = (8 x 20) + (8 x 4)

4 x (5 x 8) = (4 x 5) x 8 - Write the answer to 2 (4 + 3)
- (72 x 12) + (72 x 57) = 72(12 + ____)

_______________ Property - Use place value and the distributive property to find the product of 548 and 5.
- What is 2 × 16 × 5?
- Do multiplication using distributive property.

18 x 5 =?

44 x 5 =?

503 x 8 =?

890 x 5 =? - Find the value of 3 x (5 + 2).
- Find the value of A

3 x (5 + A) = 45

### Concept map:

### What have we learned:

- Understand Commutative property of multiplication
- Understand Place value
- Understand distributive property
- Understand the Partitioning through place value and applying distributive property