Lesson Objectives

- Demonstrate an understanding of integers
- Demonstrate an understanding of how to find the absolute value of a number
- Learn how to add two or more integers using a number line
- Learn how to add two integers with the same sign
- Learn how to add two integers with different signs
- Learn how to add more than two integers

## How to Add Integers

### Adding Integers on a Number Line

Up to this point, we have only performed addition with whole numbers. Now that we have a good understanding of the integers and the absolute value operation, we are ready to tackle integer addition. In our number line addition lesson, we showed how to add two whole numbers using a number line.

Adding two Whole Numbers on a Number Line

When we add two integers using the number line, the process is very similar. When we add two non-zero whole numbers, we always move to the right. This is due to the fact that adding a positive number results in a larger number, and numbers increase as we move to the right on the number line. When we deal with negative numbers, the opposite is true. If we add a negative number, the result will be a number that is smaller. We know that numbers decrease as we move left on the number line, therefore adding a negative number results in a movement left.

Adding two Integers on a Number Line

Example 2: Add (-4) + (-3) using a number line

Example 3: Add (-7) + 12 using a number line

(+) + (+) = +

+4 + (+9) = +13

We should already know this addition fact from our studies with whole numbers. We see a similar result when adding two or more negative numbers. When we add two or more negative numbers, we obtain a negative result.

(-) + (-) = -

(-4) + (-9) = -13

In each case, we can add the absolute values and attaching the common sign to the answer. Let's look at a few examples.

Example 4: Add (-9) + (-11)

Example 5: Add (-2) + (-5) + (-22)

For this problem, we will start at 0 on the number line, and then move right by 6 units (+6), followed by a movement left of 11 units (-11): We can see from the above image that (-11) has the larger absolute value, and the sign of the sum ends up being negative. Once we know the sign, we can think about the number part. Notice how the two numbers work against each other. One is moving us to the right and the other to the left. This means the net movement right or left can be calculated by subtracting the larger absolute value minus the smaller. In this case, we can subtract 11 - 6 = 5. Then we can attach the known sign, which is "-". This leads us to an answer of -5.

6 + (-11) = -5

Adding two Integers with Different Signs

Example 7: Add 34 + (-51)

Example 8: Add 22 + (-13) + 19 + 7 + (-30)

Adding two Whole Numbers on a Number Line

- Start on the number line at the leftmost number of the addition problem
- Move to the right by the number of units being added

- Start at 6 on the number line
- Move 9 units right and arrive at 15, our answer

When we add two integers using the number line, the process is very similar. When we add two non-zero whole numbers, we always move to the right. This is due to the fact that adding a positive number results in a larger number, and numbers increase as we move to the right on the number line. When we deal with negative numbers, the opposite is true. If we add a negative number, the result will be a number that is smaller. We know that numbers decrease as we move left on the number line, therefore adding a negative number results in a movement left.

Adding two Integers on a Number Line

- Start at the leftmost number of the addition problem
- Use the sign of the second number to determine the direction of the movement
- "-" : tells us to move left
- "+" : tells us to move right
- Use the absolute value of the second number to determine how many units to move

Example 2: Add (-4) + (-3) using a number line

- We start at -4 on the number line
- Our second number is -3, the sign is negative so we are moving left by 3 units to arrive at -7, our answer

Example 3: Add (-7) + 12 using a number line

- We start at -7 on the number line
- Our second number is 12, the sign is positive so we are moving right by 12 units to arrive at 5, our answer

### Adding Integers with the Same Sign

When we add integers without a number line, we break the process down into two different categories: adding integers with the same sign, and adding integers with different signs. When we add integers with the same sign, the process is really easy. First, we should know that adding two or more positive numbers, results in a positive sum.(+) + (+) = +

+4 + (+9) = +13

We should already know this addition fact from our studies with whole numbers. We see a similar result when adding two or more negative numbers. When we add two or more negative numbers, we obtain a negative result.

(-) + (-) = -

(-4) + (-9) = -13

In each case, we can add the absolute values and attaching the common sign to the answer. Let's look at a few examples.

Example 4: Add (-9) + (-11)

- Add the absolute values: 9 + 11 = 20
- Attach the common sign (-) to the answer, -20

Example 5: Add (-2) + (-5) + (-22)

- Add the absolute values: 2 + 5 + 22 = 29
- Attach the common sign (-) to the answer, -29

### Adding Integers with Different Signs

The harder scenario occurs when we add integers with different signs. When we add only two integers, we can think about the sign of the sum as the result of a tug of war between two values. The negative number moves us to the left on the number line, while the positive number moves us to the right. The sign of the larger absolute value will always prevail. Our sum will be negative and lie to the left of zero if the larger absolute value is negative. The opposite is true when our larger absolute value is positive. In that case, the sum is positive and will lie to the right of zero on the number line. Let’s think about 6 + (-11) using a number line.For this problem, we will start at 0 on the number line, and then move right by 6 units (+6), followed by a movement left of 11 units (-11): We can see from the above image that (-11) has the larger absolute value, and the sign of the sum ends up being negative. Once we know the sign, we can think about the number part. Notice how the two numbers work against each other. One is moving us to the right and the other to the left. This means the net movement right or left can be calculated by subtracting the larger absolute value minus the smaller. In this case, we can subtract 11 - 6 = 5. Then we can attach the known sign, which is "-". This leads us to an answer of -5.

6 + (-11) = -5

Adding two Integers with Different Signs

- Subtract the larger absolute value minus the smaller
- Attach the sign from the larger absolute value to the answer

- Subtract the larger absolute value minus the smaller
- 13 - 6 = 7
- Attach the sign from the larger absolute value to the answer
- |13| > |-6|, our sign will be (+)
- +7

Example 7: Add 34 + (-51)

- Subtract the larger absolute value minus the smaller
- 51 - 34 = 17
- Attach the sign from the larger absolute value to the answer
- |-51| > |34|, our sign will be (-)
- -17

### Adding more than two Integers

In many cases, we will be asked to add more than two integers. Remember that addition is commutative. This means the order is not important. When we see addition with more than two integers, we can make the process easier by reordering the addition. We will sum all numbers with like signs first, and then perform a final addition with different signs when needed.Example 8: Add 22 + (-13) + 19 + 7 + (-30)

- Reorder the addition
- 22 + 19 + 7 + (-13) + (-30)
- Perform the addition with like signs first
- 22 + 19 + 7 = 48
- -13 + (-30) = -43
- Perform the final addition with different signs
- 48 + (-43)
- 48 - 43 = 5
- |48| > |-43|, sign will be (+)
- +5

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