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# Sums of Integers on a Number Line

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Sums of Integers on a Number Line

Have you ever had to count suitcases at an airport? Look at what happened to Cameron's family.

When Cameron's family arrived at the airport, they all lined up with their suitcases.

Cameron had one suitcase.

His sister had one suitcase.

His Mom has three suitcases.

His Dad had three suitcases, but Dad decided to carry two of his bags onto the plan.

What was the total number of suitcases checked?

Can you represent this situation using integers?

This Concept is about finding sums of integers using a number line. This problem is easy to figure out when you use the information in this Concept.

### Guidance

In this Concept, we will extend our understanding of integers by exploring different strategies for adding them.

What is an integer?

An integer is the set of whole numbers and their opposites. Said another way, integers are positive and negative whole numbers.

Previously we examined situations where integers were represented on number lines. Imagine a person standing at the zero mark and facing the positive numbers. If that person moved 4 units forward (in a positive direction), that person would end up at the point representing 4. That is because 4 is 4 units to the right of zero on a number line.

How do we add integers?

There are strategies to help us do this. The first strategy we will explore for adding integers involves using a number line. To model addition of integers on a number line, imagine a person standing at zero, facing the positive numbers. To represent a positive integer, the person moves forward. To represent a negative integer, the person moves backward.

Let’s look at a situation that shows how we can use a number line to model the addition of two positive integers.

Use a number line to find the sum of $4+6$ .

You are adding two positive numbers. The positive numbers are to the right of zero on a number line.

So, to model $4+6$ , imagine the person moving 4 units forward and then 6 more units forward. In other words, the person will move 4 units to the right of zero, and then moving 6 more units to the right.

All in all, the person moved 10 units to the right of zero and ended up at the tic mark representing 10. So, $4+6=10$ .

Now, let's examine how to find the sum of two negative integers on a number line.

Use a number line to find the sum of $-4+(-6)$ .

Imagine the person starting at zero on the number line. You are adding two negative numbers. The negative numbers are to the left of zero on a number line.

So, to model $-4+(-6)$ , imagine the person moving 4 units backward (in a negative direction), and then moving 6 more units backward.

All in all, the person moved 10 units to the left of zero and ended up at the tic mark representing -10. So, $-4+(-6)=-10$ .

Finally, let's explore how to use a number line to find the sum of two integers, each with a different sign. This may seem a little tricky, but if you think it through step by step you can come up with the correct sum .

Use a number line to find the sum of $4+(-6)$ .

Imagine starting at zero on the number line. You are adding a positive number, 4, to a negative number, -6.

To model $4+(-6)$ , imagine the person moving 4 units forward and to the right of zero. To model adding -6 to that integer, imagine the person moving 6 units backward and to the left.

The person moved 4 units to the right of zero and then 6 more units to the left from that point until the person reached the tic mark representing -2. So $4+(-6)=-2$ .

Use a number line to find the sum of $-4+6$ .

Imagine starting at zero on the number line. You are adding a negative number, -4, to a positive number, 6.

So, to model $-4+6$ , first represent the -4 by moving the person 4 units backward and to the left of zero. To model adding 6 to that integer, imagine the person then moving 6 units forward and to the right.

The person moved 4 units to the left of zero and then 6 more units to the right until reaching the tic mark representing 2. So, $-4+6=2$ .

Good! Noticing patterns will help you as you add integers. Now let’s practice with a few examples.

Add the following integers by using a number line.

#### Example A

$-5 + -8$

Solution: $-13$

#### Example B

$-9 + 3$

Solution: $-6$

#### Example C

$-9 + 12$

Solution: $3$

Now back to Cameron's family at the airport.

Have you ever had to count suitcases at an airport? Look at what happened to Cameron's family.

When Cameron's family arrived at the airport, they all lined up with their suitcases.

Cameron had one suitcase.

His sister had one suitcase.

His Mom has three suitcases.

His Dad had three suitcases, but Dad decided to carry two of his bags onto the plan.

What was the total number of suitcases checked?

Can you represent this situation using integers?

To represent the situation using integers, let's write it all out as a number sentence to solve.

$1 + 1 + 3 + 3 + -2$

Now we add them up.

$6$ suitcases were checked on the plane.

This is our answer.

### Vocabulary

Integer
the set of whole numbers and their opposites.
Sum
the answer in an addition problem.

### Guided Practice

Here is one for you to try on your own.

Use a number line to figure out the following sum.

$-8 + 3 + -2$

Answer

To figure this out, we move our values on the number line. If the value added is positive, then we move to the right. If it is negative, then we move to the left.

We start at $-8$ .

Then we move to the right three units.

Then we move back to the left two units.

Our answer is $-7$ .

### Practice

Directions: Add the integers using a number line as your guide.

1. $2+5$

2. $(-4)+4$

3. $(-3)+(-3)$

4. $8+(-6)$

5. $-8+(-6)$

6. $-2+(-6)$

7. $8+5+(-2)$

8. $5+(-6)+2$

9. $2+(-2)+7$

10. $-2+(-6)+5$

11. $-8+3+(-4)$

12. $-6+6+(-5)$

13. $-8+(-8)+3$

14. $-7+(-6)+3$

15. $-9+(-6)+11$

### 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...
Sum

Sum

The sum is the result after two or more amounts have been added together.

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