Associative property of multiplication

### 1.1. Knowledge to remember

a) Calculate and compare the values of the two expressions:

(2 x 3) x 4 and 2 x (3 x 4)

We have: (2 x 3) x 4 = 6 x 4 = 24

2 x (3 x 4) = 2 x 12 = 24

So: (2 x 3) x 4 = 2 x (3 x 4).

b) Compare the values of the two expressions (a × b) × c and a × (b × c) in the following table:

We see that the values of (a × b) × c and a × (b × c) are always equal, we write:

(a ×b ) × c = a × (b × c)

*When we multiply a product of two numbers by a third, we can multiply the first number by the product of the second and the third.*

*Attention : *We can calculate the value of the expression a × b × c as follows:

a × b × c = (a × b) × c = a × (b × c)

### 1.2. Solving Textbook Exercises

**Lesson 1:** Calculate in 2 ways (according to the form)

*Template:* 2 × 5 × 4 = ?

Method 1: 2 × 5 × 4 = (2 × 5) × 4 = 10 × 4 = 40.

Method 2: 2 × 5 × 4 = 2 × (5 × 4) = 2 × 20 = 40.

a) 4 × 5 × 3 b) 5 × 2 × 7

3 × 5 × 6 3 × 4 × 5

__Solution guide:__

- Method 1: a × b × c = (a × b) × c.
- Method 2: a × b × c = a × (b × c).

a) 4 × 5 × 3 = ?

Method 1: 4 × 5 × 3 = (4 × 5) × 3 = 20 × 3 = 60.

Method 2: 4 × 5 × 3 = 4 × (5 × 3 ) = 4 × 15 = 60.

3 × 5 × 6 = ?

Method 1: 3 × 5 × 6 = (3 × 5) × 6 = 15 × 6 = 90.

Method 2: 3 × 5 × 6 = 3 × (5 × 6) = 3 × 30 = 90.

b) 5 × 2 × 7 = ?

Method 1: 5 × 2 × 7 = (5 × 2) × 7 = 10 × 7 = 70.

Method 2: 5 × 2 × 7 = 5 × (2 × 7) = 5 × 14 = 70.

3 × 4 × 5 = ?

Method 1: 3 × 4 × 5 = (3 × 4) × 5 = 12 × 5 = 60.

Method 2: 3 × 4 × 5 = 3 × (4 × 5) = 3 × 20 = 60.

**Lesson 2:** Calculate by the most convenient way

a) 13 × 5 × 2 b) 2 × 26 × 5

5 × 2 × 34 5 × 9 × 3 × 2

__Solution guide:__

- Apply the commutative and associative properties of multiplication to group numbers whose product is round ten, round hundred, … together.

a) 13 × 5 × 2 = 13 × (5 × 2) = 13 × 10 = 130.

5 × 2 × 34 = (5 × 2) × 34 = 10 × 34 = 340.

b) 2 × 26 × 5 = 26 × (2 × 5) = 26 × 10 = 260.

5 × 9 × 3 × 2 = (9 × 3) × (5 × 2) = 27 × 10 = 270.

**Lesson 3:** There are 8 classrooms, each room has 15 sets of tables and chairs with 2 students studying. How many students are studying in all?

__Solution guide:__

- Calculate the number of students in each classroom: 2 × 15 = 20 students.
- Calculating the number of students sitting in class, we multiply the number of students in each classroom by 8 (because there are 8 classrooms).

*Solution*

The number of students in each classroom is:

2 × 15 = 30 (students)

The number of students sitting in class is:

30 × 8 = 240 (students)

Answer: 240 students.

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