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# Sums of Integers on a Number Line

## Add positive and negative numbers using a number line.

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Sums of Integers on a Number Line

Cooper is having a great time getting to know his pen pal in New Zealand. He and his pen pal Riley are the same age and both love sports. The one thing that Cooper is having a hard time with is the whole idea of time zones.

“I get it if we are talking here, New York and California,” Cooper tells his friend Emma. “That’s a difference of three hours. It is a loss of three if I go to California and it is a gain of three if I travel from California back to here. It is 9 am right now, so it is 6 am in California.”

“Well if you get that, what’s so hard about New Zealand? You can use integers to figure out the time just like you did with New York and California,” Emma says smiling.

“What do you mean?”

“Well, you said it is nine am right now. Then there is a loss of four hours to travel to California. You can write an addition problem and figure it out.”

9 + -3 = 6

“See?” Emma says jotting the numbers on a piece of paper.

“Nope, not really,” Cooper says shaking his head. “I know that New Zealand is 16 hours ahead of us, but I don’t know what to do from there. Is that a loss or a gain?”

Cooper is definitely puzzled. Emma is correct though. Adding integers will definitely help Cooper figure out his time zone trouble. In this Concept, you will learn how Cooper can solve this problem by using a number line.

### Guidance

To review, an integer is a member of the set of whole numbers and their opposites. We can say that integers are both positive and negative whole numbers.

Besides writing, identifying, and using integers, we can add integers too. We can find the sum of more than one integer.

How can we find integer sums?

One of the best ways to find integer sums is to use a number line. We can add integers by looking at where they fall on the number line.

-5 + 7 = ____

Here we have a negative five plus a positive 7. You can think of this in terms of losses and gains. We start with a loss of five and then we have a gain of seven.

That is a great question! It can be a little confusing to try to figure out how to add a loss and a gain or a negative and a positive number. We can use a number line to help us clarify the sum of these two integers.

Next, we start with the first integer. It is a loss of 5 or we start at negative five.

Next we add a positive seven. Since we are adding a positive seven we move toward the positive side of the number line. We start at negative five and count up seven units.

-5 + 7 = 2

Let’s look at another one.

6 + -9 = _____

Our first integer is positive 6, so that is where we will begin.

We add negative nine next. Since we are adding a negative number, it is a loss, so we move toward the negative side of the number line. We are adding a negative nine, so we move to the left nine units.

6 + -9 = -3

We can also add two negative numbers.

-2 + -4 = ____

Since we are adding two negatives, we are adding a loss and another loss, so we have a greater loss. Negative plus negative is more negative. Let’s look at this on the number line.

Next, we add a negative four. Since we add a negative we move toward the negative or left side of the number line four units.

-2 + -4 = -6

You can use a number line for a reference anytime you would like. Many times this will help you until you develop more skill adding integers.

Practice finding each sum by adding integers on the number line.

-5 + 9 = ____

Solution: 4

-1 + -8 = ____

Solution: -9

#### Example C

5 + -7 = ____

Solution: -2

Now let's look at how to solve the problem about the time zones by using a number line. Draw a number line and use it to work through the problem as we go along. Here is the original problem.

Cooper is having a great time getting to know his pen pal in New Zealand. He and his pen pal Riley are the same age and both love sports. The one thing that Cooper is having a hard time with is the whole idea of time zones.

“I get it if we are talking here, New York and California,” Cooper tells his friend Emma. “That’s a difference of three hours. It is a loss of three if I go to California and it is a gain of three if I travel from California back to here. It is 9 am right now, so it is 6 am in California.”

“Well if you get that, what’s so hard about New Zealand? You can use integers to figure out the time just like you did with New York and California,” Emma says smiling.

“What do you mean?”

“Well, you said it is nine am right now. Then there is a loss of four hours to travel to California. You can write an addition problem and figure it out.”

9 + -3 = 6

“See?” Emma says jotting the numbers on a piece of paper.

“Nope, not really,” Cooper says shaking his head. “I know that New Zealand is 16 hours ahead of us, but I don’t know what to do from there. Is that a loss or a gain?”

Now that we know about positive and negative integers, let's learn the rest of the discussion.

“Alright, never mind about that,” Cooper says. “I’ll just add 16 to 8 am and that will give me the time in New Zealand.”

Cooper writes the following on his paper.

8 + 16 = 24

Emma smiles.

“You can’t do it that way silly. We don’t have 16 hour clocks. We have 12 hour clocks. You can use integers to solve this, but you will need to split up the 16 hours into 12 hours and 4 hours. If you start at 8 in the morning and then you gain 12 hours, you end up at 8 at night.”

8 am to 8 pm is 12 hours.

“Oh, I see, Cooper said. Now I have four hours more to add. So I start at 8 pm and add four hours.”

8 + 4 = 12 midnight

“When it is 8 am here, it is 12 midnight in New Zealand,” Cooper said.

What about if Cooper was in New Zealand at 2 pm in the afternoon and could travel instantly back to New York? What time would he arrive?

You can figure this out in the same way. There is still a 16 hour difference-except this time we are going back in time not forward in time.

-16 hours

2 pm to 2 am the day before = 12 hours

2 am + -4 hours = ____

We can count backwards for this one.

2, 1, 12, 11 pm

Cooper would arrive in New York at 11 pm the day before he left.

### Vocabulary

Here are the vocabulary words in this Concept.

Integer
the set of whole numbers and their opposites. Positive and negative whole numbers are integers.
Sum

### Guided Practice

Here is one for you to try on your own.

Draw a number line from -6 to 9 and find this sum.

$-5 + -1 + 7$ = _____

To find this sum, we are going to start at negative five.

Then we add negative 1 and end up at negative 6.

Next, we add positive seven to negative 6.

Our answer is $1$ .

### Video Review

Here is a video for review.

### Practice

Directions: Add the following integers that have the same sign by using a number line. You may figure out a pattern. If so, try finding the sums without a number line.

1. 6 + 7 = _____

2. -9 + -7 = _____

3. -3 + -4 = _____

4. 5 + 12 = _____

5. -12 + -23 = _____

6. 27 + 11 = _____

7. -34 + -13 = _____

8. 25 + 16 = _____

9. -9 + -29 = _____

10. -16 + -12 = _____

Directions: Add the following integers using a number line. Notice that they have different signs.

11. -9 + 3 = _____

12. -7 + 5 = _____

13. 1 + -12 = _____

14. 3 + -8 = _____

15. -19 + 11 = _____

16. 7 + -12 = _____

17. 23 + -10 = _____

18. -4 + 16 = _____

19. 15 + -18 = _____

20. -15 + 9 = _____

### Vocabulary Language: English

Integer

Integer

The integers consist of all natural numbers, their opposites, and zero. Integers are numbers in the list ..., -3, -2, -1, 0, 1, 2, 3...