I thought adding bracket is associative and removing bracket is distributive. thank you for your explanation.

Mechanically, the associative properties of addition and multiplication are about switching brackets around, not changing the number of brackets.

\(\displaystyle (a + b) + c \equiv a + (b + c)\) ASSOCIATIVE PROPERTY OF ADDITION.

Notice that no multiplication is involved and the number of brackets does not change.

\(\displaystyle (a * b) * c \equiv a * (b * c)\) ASSOCIATIVE PROPERTY OF MULTIPLICATION.

Notice that no addition is involved and the number of brackets does not change.

Essentially the associative property of addition says that if you need to add up

3, 7, and 11, it makes no difference if you add 3 and 7 to get 10 and then add 10 and 11 to get 21 or if instead you add 7 and 11 to get 18 and then add 18 and 3 to get 21 because the answer is the same either way.

Similarly, if you need to multiply 3, 7, and 11, you get the same answer of 231 whether you first multiply 3 and 7 to get 21 and then multiply 21 by 11 to get 231 or whether you first multiply 7 and 11 to get 77 and then multiply 77 by 3 to get 231.

As tkhunny has explained, the associative and commutative properties of addition do not have anything to do with multiplication, and the associative and commutative properties of multiplication do not have anything to do with addition. The distributive property is the only one that deals with both multiplication and addition at the same time.

\(\displaystyle a * (b + c) \equiv (a * b) + (b * c)\)

DISTRIBUTIVE PROPERTY OF MULTIPLICATION OVER ADDITION.

What this one means is that the product of 3 and the sum of 7 and 11 (meaning 3 times 18 or 54 in total) is the same as the sum of the products of 3 and 7 and of 3 and 11 (meaning the sum of 21 and 33 or 54 in total).

The arithmetic facts described by these properties are so obvious that students may think that they are missing some subtlety and get confused about merely naming things that they have known since second grade.