11.3: Subtracting Integers
Introduction
The Football Game
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 lesson is all about. Follow along through this lesson and at the end Sarah will show you how she explains the yards lost and gained through an integer problem.
What You Will Learn
By the end of this lesson, you will be able to demonstrate the following skills.
- Find differences of integers on a number line.
- Subtract integers with the same sign.
- Subtract integers with different signs.
- Solve real-world problems involving sums and differences of integers.
Teaching Time
I. Find Differences of Integers on a Number Line
In our last lesson, you learned how to add integers using a number line. This lesson focuses on finding the differences of integers. The word difference is a word that you have seen before. It is a key word that means subtract. When you see the word difference, you know that you will be subtracting values.
One of the best ways to learn about finding the differences of integers is to use a number line. We can subtract values using a number line. Let’s look at an example.
Example
-6 – 2 = ____
Here we have the value of negative six and we are subtracting two from it. Let’s use a number line to figure this out.
We start at the first value, which is negative six.
Next, we subtract two from this value. If we subtract two, we move further into the negatives. Think of it as another loss. We start at negative six and move two units to the left into the negatives.
-6 – 2 = -8
We start with a loss and we have more loss, so our answer is a greater loss.
The answer is negative eight.
Let’s look at another one.
Example
2 – 9 = ____
Here we have two minus nine. We start at positive two.
We have a loss of nine. We subtract nine from the two where we started.
The answer is negative 7.
2 – 9 = -7
The answer is negative seven.
Let’s look at a word problem and use a number line to solve it.
Example
Jamie earned ten dollars cutting grass. He owes his brother twelve dollars. Does Jamie still owe his brother money if he gives him the whole ten dollars he made? How much does he owe?
Let’s start by writing a number sentence to represent this problem.
Jamie earned $10.00. He owes his brother $12.00.
10 - 12 = ____
Let’s use a number line to figure out if Jamie still owes his brother money.
We start at positive ten and then subtract 12. This means that we move twelve units toward the negative side of the number line.
10 – 12 = -2
The answer is yes-he still owes more money. Jamie owes his brother $2.00.
Practice a few of these on your own. Use a number line to work through the problem.
- 4 – 10 = ____
- -9 – 3 = ____
- -12 – 4 = ____
Take a few minutes to check your work with a partner.
II. Subtract Integers with the Same Sign
In the last section, we practiced subtracting integers using a number line. You won’t always have time to draw a number line though, so this section will teach you how to subtract integers that have the same sign without the visual aid. Let’s begin.
First, let’s look at subtracting two positive numbers.
Example
9 – 4 = ____
In this problem, if we use the language of losses and gains, we could say that we have a gain of nine and a loss of four. Because our loss is not greater than the gain, our answer is positive.
This is a key point. If the loss is greater than the gain, then our answer would be negative. In this example, the loss of four is not greater than the gain of nine, so our answer remains positive.
9 – 4 = 5
The answer is positive 5.
Here is an example where we are still finding the difference between two positive numbers, but the loss is greater than the gain.
Example
3 – 8 = _____
In this example we start with a positive three or a gain of three. Then we have a loss of eight. The loss is greater than the gain that we started with.
3 – 8 = -5
Our answer is negative. It is a negative five.
Yes. Actually there is an easier way to think about subtracting any two integers. You can always think in terms of losses and gains, but if that is difficult, we can think of subtraction as being the opposite of addition-that is the key to making things simpler. Here is the hint.
What does this look like? How can we rewrite a subtraction problem as an addition problem?
Example
3 – 8 = ____
Here is the same example as before. We can change subtraction to addition by adding the opposite.
3 – 8 = 3 + -8 = ____
The subtraction became addition.
Positive three plus a negative 8 is still a negative 5.
Our answer did not change even though our method of solving it did. The answer is still -5.
In the last two examples, we were subtracting integers with the same sign, but both times the sign was positive.
How can we find the difference of two negative numbers?
Example
-6 – -3 = ____
We can find this difference in two ways. The first way is to think in terms of losses and gains. The second is to change subtraction to addition by adding the opposite. Let’s start with losses and gains.
If we think of this problem in terms of losses and gains, we start with a loss of 6.
-6
Next, we don’t add another loss, but we take away a loss. If you take away a loss, that is the same thing as a gain. So we have a gain of 3.
-6 combined with a gain of 3 = -3
Our answer is -3.
Now, let’s solve the problem by changing subtraction to addition by adding the opposite.
-6 – -3 = -6 + 3
We changed the subtraction to addition and added the opposite. The opposite of the given value of negative three is positive three. Now we can solve the addition problem.
-6 + 3 = -3
Notice that the answer is the same no matter which way you approach it. The answer is still -3.
Practice what you have learned by finding the differences of the following integer pairs.
- 5 – 10 = ____
- 14 – 7 = ____
- -4 – -8 = ____
Take a few minutes to check your work with a neighbor. How did you solve each problem? Explain your thinking to a friend. Is your work accurate?
III. Subtract Integers with Different Signs
In the last section, 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.
Let’s look at an example.
Example
-6 – 4 = ____
Just like the last section, 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
The answer is -10.
Now let’s change the subtraction to addition and add the opposite.
-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 example we looked for the difference between a negative and a positive. What about finding the difference between a positive and a negative?
Example
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
The answer is 9.
Now let’s change the subtraction to addition and add the opposite.
6 – -3 = 6 + 3
The subtraction sign became an addition sign. The negative three became its opposite which is positive three.
6 + 3 = 9
The answer is 9.
Practice a few of these on your own. Choose whichever method you would like and find the difference of each pair of integers.
- -5 – 7 = ____
- 2 – -8 = ____
- -13 – 5 = ____
Take a few minutes to check your answers with a partner. Is your work accurate? How did you choose to find each difference?
IV. Solve Real-World Problems Involving Sums and Differences of Integers
Now that you have learned how to find the sums and differences of integers, you can apply these skills to some real-life examples. In real-life, there may be both sums and differences in the same problem. To figure these out, you will need to think in terms of losses and gains.
Example
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.
\begin{align*}10 + 10 &= 20\\ 20 - 20 &= 0\\ 0 + 5 &= 5\\ 5 + 3 &= 8\\ 8 - 2 &= 6\end{align*}
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.
Sports are only one place where we see integers in real-life situations. But the strategies that you have learned in this lesson will help you no matter what the situation. You can figure out the sums and differences of integers.
Real Life Example Completed
The Football Game
Now that you have learned all about integers, you can apply what you have learned to Sarah’s letter to Emily. In the last example, you even saw how to use integers to write about yards lost and gained. Let’s apply that to this problem.
First, reread the problem and underline any important information.
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 that are found in this lesson.
- Sum
- the result of an addition problem.
- Difference
- the result of a subtraction problem.
Technology Integration
James Sousa, Subtracting Integers: The Basics
James Sousa, Subtracting Integers
James Sousa, Example of Subtracting Integers
Time to Practice
Directions: Find the differences of the following integer pairs.
- -2 – 4 = ____
- -8 – 9 = ____
- -6 – 7 = ____
- -11 – 12 = ____
- -13 – 22 = ____
- -89 – 11 = ____
- 2 – 7 = ____
- 4 – 9 = ____
- 5 – 8 = ____
- 13 – 20 = ____
- 12 – 23 = ____
- 25 – 30 = ____
- 45 – 90 = ____
- 34 – 67 = ____
- -2 – -3 = ____
- -8 – -3 = ____
- -9 – -7 = ____
- -5 – -10 = ____
- -9 – -12 = ____
- -10 – -10 = ____
- -14 – -15 = ____
- 5 – -8 = ____
- 6 – -7 = ____
- 10 – -9 = ____
- 11 – -7 = ____
- 18 – -9 = ____
- 22 – -5 = ____
- 34 – -3 = ____
- 35 – -35 = ____
- 45 – -10 = ____