## Math 6 Chapter 2 Lesson 6: Properties of addition of integers

## 1. Summary of theory Tóm

### 1.1. Commutative property

The addition of integers is also commutative, that is:

a + b = b + a

### 1.2. Combined properties

Associative property of addition of integers

(a + b) + c = a + (b + c)

**Attention: **The above result is also called the sum of three numbers a, b, c and written a + b + c. Similarly, we can talk about the sum of four, five,…integer. When performing the addition of many numbers, we can change the order of terms arbitrarily, group the terms arbitrarily with signs (), [], {}.

### 1.3. Plus the number 0

a + 0 = 0 + a = a

### 1.4. Plus the number for

The opposite of integer a is denoted by -a. Then the opposite of (-a) is also a, that is: -(-a) = a. Clearly:

– If a is a positive integer, then -a is a negative integer, for example a = 3 then -a = -3.

– If a is a negative integer, then -a is a positive integer, for example a = -5 then -a = -(-5) = 5 (because 5 is the opposite of -5)

The opposite of 0 is still 0, so – 0 = 0

⇒ The sum of two opposite integers is always 0.

a + (-a) = 0

Conversely, if the sum of two integers is zero, then they are opposites:

If a + b = 0 then b = -a and a = -b.

## 2. Illustrated exercise

**Question 1:** Calculation: 248 + (-12) + 2064 + (-236)

__Solution guide__

248 + (-12) + 2064 + (-236)

= 248 + 2064 + [(-12) + (-236)]

= 248 + 2064 + (-248)

= [248 + (-248)] + 2064

= 2064

**Verse 2: **Find the sum of all integers x that satisfy: -6 < x < 5

__Solution guide__

x = -5, -4, -3, -2, -1, 0, 1, 2, 3, 4

Answer: -5 (because the remaining numbers are pairs of opposites and 0)

**Question 3:** Your kite Son flies at an altitude of 7 meters (above the ground). After a while, the height of the kite increased by 3 meters, and then decreased by 4 meters. What is the height of the kite (above the ground) after two changes of altitude?

__Solution guide__

7 + 3 + (-4) = 6 (meters)

we can also calculate: (7 + 3) – 4 = 6 meters

## 3. Practice

### 3.1. Essay exercises

**Question 1: **Calculate 5 + (-7) + 9 + (-11) + 13 + (-15)

**Verse 2: **Calculate the sums: (-17) + 5 + 8 +17

**Question 3:** Simplify the following expression: -11 + y + 7

### 3.2. Multiple choice exercises Bài

**Question 1: **Choose the incorrect statement in the following sentences?

A. The properties of the addition of integers include: the commutative property, the associative property, plus zero, plus the number of opposites.

B. a + 0 = 0

C. a + b = b + a

D. a + (-a) = 0

**Verse 2: **Sum of all integers x, knowing -4 < x < 4

A. 3

B. -1

C. 1

D. 0

**Question 3: **Choose the correct statement from the following statements:

A. If the sum of two natural numbers is 0, then both of those natural numbers are equal to 0

B. If the sum of two integers is 0, then both integers are equal to 0

C. The sum of many negative integers is also a negative integer whose absolute value is equal to the sum of their absolute values

D. The sum of two numbers is always 0

**Question 4: **Let x = 5, y = -7. Calculate |x+y| -x

A. -3

B. -4

C. 3

D. 4

**Question 5: **Calculate the value of the following expression:185 + (-300) +315

A. 800

B. -200

C. 200

D. 300

**Question 6: **The sum of all integers with absolute value less than 5 is

A. 1

B. -1

C. 2

D. 0

## 4. Conclusion

Through this lesson Add two integers with different signs, you need to complete some of the objectives given by the lesson, such as:

- Master the rule of adding two integers with different signs
- Apply the rule to do some exercises related to adding two integers with different signs

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