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# Differences of Integers with Different Signs

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Have you ever watched a football game? Take a look at this dilemma.

After the football game Friday night, Sarah could hardly wait to write and tell her pen pal Emily all about it. It had been one of the most exciting games that Sarah had ever been to. The middle school team was evenly matched with a rival team from a neighboring school. The game had been very close. In fact it had come down to the last few minutes of play.

Sarah wrote this to her pen pal, “At the end of the fourth quarter, we were twenty yards away from a touchdown. The score was 14 to 14. We needed this touchdown to win the game. The running back took the football and began running. He made it 15 yards.”

“Then, on the next play, the defenders charged at our players. We had a loss of ten yards on that play. Next, our players earned a penalty of 15 yards, but the coach challenged the call and the referee took away a loss of ten yards. Then we ran for a gain of 5 yards. On the next play, the quarterback threw the ball for a touchdown and we won the game!”

Sarah reread her letter. All of the yards gained and lost seemed a bit confusing.

“I think I can write this clearer if I use integers,” Sarah thought to herself. “Then I can see how far the quarterback threw the ball for the touchdown.”

Writing about the football game will involve sums and differences of integers. That is what this Concept is all about. Follow along through this Concept and at the end Sarah will show you how she explains the yards lost and gained through an integer problem.

### Guidance

In the last Concept, you learned how to find the differences of integers that had the same sign. You learned how to find the differences of two positive integers and of two negative integers. Now we are going to apply what you learned in the last section when finding the differences of integers that have different signs.

-6 – 4 = ____

Just like the last Concept, there are two different ways to approach this problem. We can think of it in terms of losses and gains or we can change subtraction to addition and add the opposite.

Let’s start by thinking in terms of losses and gains.

This problem starts with a loss. There is a loss of six or a negative six.

-6

Next, we take away a gain of four. The subtraction is the taking away. We have a positive four, so we take away a gain of four. If you take away a gain it is the same as adding a loss.

-6 – 4 = -10

-6 – 4 = -6 + -4

The subtraction changed to addition. Positive four became its opposite, negative four.

-6 + -4 = 10

The answer is the same. It is still -10.

In this last problem we looked for the difference between a negative and a positive. What about finding the difference between a positive and a negative?

6 – -3 = ____

Once again, we can approach this problem in two ways. We can think in terms of losses and gains, and we can change the subtraction to addition and add the opposite.

Let’s start by thinking in terms of losses and gains.

We start with a gain because our first value is positive six.

6

Then we take away a loss. When you take away a loss of 3, it is the same as adding three.

6 – -3 = 9

6 – -3 = 6 + 3

The subtraction sign became an addition sign. The negative three became its opposite which is positive three.

6 + 3 = 9

Practice a few of these on your own. Choose a method and find the difference of each pair of integers.

-5 – 7 = ____

Solution: -12

2 – -8 = ____

Solution: 10

#### Example C

-13 – 5 = ____

Solution: -18

Remember the football game? Let's go back to the football game and integers.

After the football game Friday night, Sarah could hardly wait to write and tell her pen pal Emily all about it. It had been one of the most exciting games that Sarah had ever been to. The middle school team was evenly matched with a rival team from a neighboring school. The game had been very close. In fact it had come down to the last few minutes of play.

Sarah wrote this to her pen pal, “At the end of the fourth quarter, we were twenty yards away from a touchdown. The score was 14 to 14. We needed this touchdown to win the game. The running back took the football and began running. He made it 15 yards.”

“Then, on the next play, the defenders charged at our players. We had a loss of ten yards on that play. Next, our players earned a penalty of 15 yards , but the coach challenged the call and the referee took away a loss of ten yards . Then we ran for a gain of 5 yards . On the next play, the quarterback threw the ball for a touchdown and we won the game!”

Sarah reread her letter. All of the yards gained and lost seemed a bit confusing.

“I think I can write this clearer if I use integers,” Sarah thought to herself. “Then I can figure out how far the quarterback threw the ball for the touchdown.”

Let’s write out the integers that we are using in this problem.

“He made it 15 yards” = +15

“A loss of ten yards” = + -10

“A penalty of 15 yards” = + -15

“Referee took away a loss of ten yards” = – -10

“Then we ran for a gain of 5 yards” = +5

Now we can write a problem using sums and differences of the following integers.

15 + -10 + -15 – - 10 + 5

Let’s work from left to right adding integers.

15 + -10 = 5

5 + -15 = -10

-10 – -10 = 0 yards gained

0 + 5 = 5 yards gained.

Since the team originally needed 20 yards for a touchdown, after all of the gains and losses, they ended up with a gain of five.

20 – 5 = 15

The quarterback threw the ball 15 yards for the touchdown.

### Vocabulary

Here are the vocabulary words in this Concept.

Sum
the result of an addition problem.
Difference
the result of a subtraction problem.

### Guided Practice

Here is one for you to try on your own.

During the first quarter of Friday night’s game, Lawrence High School’s football team had a gain of 10 yards, then a loss of 20 yards then a gain of 5 yards, another gain of 3 yards and a loss of 2 yards before the coach called time out. If they started on the ten yard line, where were they when the coach called time out?

To work through this problem, we need to write an integer number sentence showing the losses and gains that the team had. Each loss is a negative number and each gain is a positive one. We know that they started on the ten yard line, so that is our first number.

10 + 10 – 20 + 5 + 3 – 2 = ____

Next, we add each integer in order.

$10 + 10 &= 20\\20 - 20 &= 0\\0 + 5 &= 5\\5 + 3 &= 8\\8 - 2 &= 6$

The team was on the six yard line when the coach called time out. At this point they had actually experienced a loss of four from their starting place on the ten yard line.

### Video Review

Here are videos for review.

### Practice

Directions: Find the differences of the following integer pairs.

1. -2 – 4 = ____
2. -8 – 9 = ____
3. -6 – 7 = ____
4. -11 – 12 = ____
5. -13 – 22 = ____
6. -89 – 11 = ____
7. 2 – 7 = ____
8. 4 – 9 = ____
9. 5 – 8 = ____
10. 13 – 20 = ____
11. 12 – 23 = ____
12. 25 – 30 = ____
13. 45 – 90 = ____
14. 34 – 67 = ____
15. -2 – -3 = ____
16. -8 – -3 = ____
17. -9 – -7 = ____
18. -5 – -10 = ____
19. -9 – -12 = ____
20. -10 – -10 = ____
21. -14 – -15 = ____
22. 5 – -8 = ____
23. 6 – -7 = ____
24. 10 – -9 = ____
25. 11 – -7 = ____
26. 18 – -9 = ____
27. 22 – -5 = ____
28. 34 – -3 = ____
29. 35 – -35 = ____
30. 45 – -10 = ____