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# Differences of Integers Using a Number Line

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

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Differences of Integers Using a Number Line

Have you ever watched a game show?

While on the airplane, Cameron enjoyed watching a game show. During the game show, contestants could earn points and could face penalties too.

One contestant had eight penalties, but then two of the penalties were taken away.

Do you know the final count of penalties for this contestant?

You will need to find differences of integers to figure this out.

This Concept is about finding differences of integers on a number line. You will learn how to accomplish this task during this Concept.

### Guidance

We can subtract integers by using a strategy. Using a strategy will allow us to find the difference between two integers. Remember that the word “difference” is a key word that means the answer in a subtraction problem.

Do you know what an integer is?

An integer is the set of whole numbers and their opposites. Essentially, we can think of integers as positive and negative whole numbers.

Previously we worked with number lines to add integers. Well, a similar strategy can be used to subtract integers as well. Let’s look at how to do this.

We will explore for subtracting integers involves using a number line, this is the first strategy. When using a number line to model subtraction, imagine a person standing at 0, facing the positive numbers on the line. The person will then move forward or backward to show the first quantity in the problem.

To model subtracting a number from that quantity, imagine that person turning and facing the negative numbers. Imagining the person turn around distinguishes the operation of subtraction from the operation of addition. (In Lesson 4.2, the person never turned around and faced the opposite direction because you were adding.)

To subtract a positive number, the person moves forward. To subtract a negative number, the person moves backward.

This first problem shows how we can use a number line to model the subtraction of two positive integers.

Use a number line to find the difference of 43\begin{align*}4-3\end{align*}.

You are subtracting two positive integers.

So, to model 43\begin{align*}4-3\end{align*}, imagine the person moving 4 units to the right of zero. This shows the quantity, 4.

Next, imagine the person turning around.

Since you are subtracting a positive integer, 3, the person moves forward. After turning around, imagine that person moving 3 units forward, or to the left.

The person ends up at 1. So, 43=1\begin{align*}4-3=1\end{align*}.

Now, let's imagine subtracting a negative integer from a positive integer.

Use a number line to find the difference of 4(3)\begin{align*}4-(-3)\end{align*}.

To model 4(3)\begin{align*}4-(-3)\end{align*}, imagine the person moving 4 units forward and to the right of zero. Then imagine the person turning around. Since a negative integer, -3, is being subtracted, imagine the person moving 3 units backward. That means the person will be moving to the left.

The person ends up at 7. So, 4(3)=7\begin{align*}4-(-3)=7\end{align*}.

Now, let's imagine subtracting a negative integer from a negative integer.

Use a number line to find the difference of 4(3)\begin{align*}-4-(-3)\end{align*}.

To model 4(3)\begin{align*}-4-(-3)\end{align*}, imagine the person moving 4 units backward and to the left. Then imagine the person turning around. Since a negative integer, -3, is being subtracted, imagine the person moving 3 units backward and to the right.

The person ends up at -1. So, 4(3)=1\begin{align*}-4-(-3)=-1\end{align*}.

#### Example A

52\begin{align*}-5 - 2\end{align*}

Solution: 7\begin{align*}-7\end{align*}

#### Example B

7(2)\begin{align*}7 - (-2)\end{align*}

Solution: 9\begin{align*}9\end{align*}

#### Example C

9(5)\begin{align*}-9 - (-5)\end{align*}

Solution: 4\begin{align*}-4\end{align*}

Here is the original problem once again.

While on the airplane, Cameron enjoyed watching a game show. During the game show, contestants could earn points and could face penalties too.

One contestant had eight penalties, but then two of the penalties were taken away.

Do you know the final count of penalties for this contestant?

You will need to find differences of integers to figure this out.

First, let's write a number sentence to show this situation.

8(2)\begin{align*}-8 - (-2)\end{align*}

The contestant earned eight penalties and then two were taken away.

6\begin{align*}-6\end{align*}

The contestant only earned six penalties.

### Vocabulary

Difference
the answer in a subtraction problem.
Integer
the set of whole numbers and their opposites.

### Guided Practice

Here is one for you to try on your own.

Use a number line to find the difference of 43\begin{align*}-4-3\end{align*}.

To model 43\begin{align*}-4-3\end{align*}, imagine the person moving 4 units backward and to the left. The person moves backward because the initial quantity, -4, is a negative integer.

Next, imagine the person turning around. Since a positive integer, 3, is being subtracted, imagine the person moving 3 units forward and to the left.

The person ends up at -7. So, 43=7\begin{align*}-4-3=-7\end{align*}.

### Practice

Directions: Subtract the following integers using a number line.

1. 86\begin{align*}8-6\end{align*}

2. 53\begin{align*}-5-3\end{align*}

3. 3(6)\begin{align*}-3-(-6)\end{align*}

4. 2(5)\begin{align*}2-(-5)\end{align*}

5. 63\begin{align*}-6-3\end{align*}

6. 8(3)\begin{align*}8-(-3)\end{align*}

7. 7(7)\begin{align*}-7-(-7)\end{align*}

8. 54\begin{align*}-5-4\end{align*}

9. 1(8)\begin{align*}1-(-8)\end{align*}

10. 47\begin{align*}-4-7\end{align*}

Directions: Subtract these integers without a number line.

11. 2437\begin{align*}-24-37\end{align*}

12. 34(7)\begin{align*}-34-(-7)\end{align*}

13. 44(37)\begin{align*}-44-(-37)\end{align*}

14. 82(7)\begin{align*}-82-(-7)\end{align*}

15. 6497\begin{align*}-64-97\end{align*}

### 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...